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https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.1%3A_Relationships_between_Quantities	6.1.1: Relationships between Quantities Lesson Let's try to solve some new kinds of problems. Exercise $\PageIndex{1}$: Pricing Theater Popcorn A movie theater sells popcorn in bags of different sizes. The table shows the volume of popcorn and the price of the bag. Complete one column of the table with prices where popcorn is priced at a constant rate. That is, the amount of popcorn is proportional to the price of the bag. Then complete the other column with realistic example prices where the amount of popcorn and price of the bag are not in proportion. \| volume of popcorn (ounces) \| price of bag, proportional ($) \| price of bag, not proportional ($) \| \|---\|---\|---\| \| $10$ \| $6$ \| $6$ \| \| $20$ \| \|\| \| $35$ \| \|\| \| $48$ \| Exercise $\PageIndex{2}$: Entrance Fees A state park charges an entrance fee based on the number of people in a vehicle. A car containing 2 people is charged $14, a car containing 4 people is charged $20, and a van containing 8 people is charged $32. - How much do you think a bus containing 30 people would be charged? - If a bus is charged $122, how many people do you think it contains? - What rule do you think the state park uses to decide the entrance fee for a vehicle? Exercise $\PageIndex{3}$: Making Toast A toaster has 4 slots for bread. Once the toaster is warmed up, it takes 35 seconds to make 4 slices of toast, 70 seconds to make 8 slices, and 105 seconds to make 12 slices. - How long do you think it will take to make 20 slices? - If someone makes as many slices of toast as possible in 4 minutes and 40 seconds, how many slices do think they can make? Are you ready for more? What is the smallest number that has a remainder of 1, 2, and 3 when divided by 2, 3, and 4, respectively? Are there more numbers that have this property? Summary In much of our previous work that involved relationships between two quantities, we were often able to describe amounts as being so much more than another, or so many times as much as another. We wrote equations like $x+3=8$ and $4x=20$ and solved for unknown amounts. In this unit, we will see situations where relationships between amounts involve more operations. For example, a pizza store might charge the amounts shown in the table for delivering pies. \| number of pies \| total cost in dollars \| \|---\|---\| \| $1$ \| $13$ \| \| $2$ \| $23$ \| \| $3$ \| $33$ \| \| $5$ \| $53$ \| We can see that each additional pie adds $10 to the total cost, and that each total includes a $3 additional cost, maybe representing a delivery fee. In this situation, 8 pies will cost $8\cdot 10+3$ and a total cost of $63 means 6 pies were ordered. In this unit, we will see many situations like this one, and will learn how to use diagrams and equations to answer questions about unknown amounts. Practice Exercise $\PageIndex{4}$ Lin and Tyler are drawing circles. Tyler's circle has twice the diameter of Lin’s circle. Tyler thinks that his circle will have twice the area of Lin’s circle as well. Do you agree with Tyler? (From Unit 3.2.2) Exercise $\PageIndex{5}$ Jada and Priya are trying to solve the equation $\frac{2}{3}+x=4$. - Jada says, “I think we should multiply each side by $\frac{3}{2}$ because that is the reciprocal of $\frac{2}{3}$.” - Priya says, “I think we should add $-\frac{2}{3}$ to each side because that is the opposite of $\frac{2}{3}$.” - Which person’s strategy should they use? Why? - Write an equation that can be solved using the other person’s strategy. (From Unit 5.5.1) Exercise $\PageIndex{6}$ What are the missing operations? - $48 ? (-8) = (-6)$ - $(-40)?8=(-5)$ - $12?(-2)=14$ - $18?(-12)=6$ - $18?(-20)=-2$ - $22?(-0.5)=-11$ (From Unit 5.4.1) Exercise $\PageIndex{7}$ In football, the team that has the ball has four chances to gain at least ten yards. If they don't gain at least ten yards, the other team gets the ball. Positive numbers represent a gain and negative numbers represent a loss. Select all of the sequences of four plays that result in the team getting to keep the ball. - $8, -3, 4, 21$ - $30, -7, -8, -12$ - $2, 16, -5, -3$ - $5, -2, 20, -1$ - $20, -3, -13, 2$ (From Unit 5.4.2) Exercise $\PageIndex{8}$ A sandwich store charges $20 to have 3 turkey subs delivered and $26 to have 4 delivered. - Is the relationship between number of turkey subs delivered and amount charged proportional? Explain how you know. - How much does the store charge for 1 additional turkey sub? - Describe a rule for determining how much the store charges based on the number of turkey subs delivered. Exercise $\PageIndex{9}$ Which question cannot be answered by the solution to the equation $3x=27$? - Elena read three times as many pages as Noah. She read 27 pages. How many pages did Noah read? - Lin has 27 stickers. She gives 3 stickers to each of her friends. With how many friends did Lin share her stickers? - Diego paid $27 to have 3 pizzas delivered and $35 to have 4 pizzas delivered. What is the price of one pizza? - The coach splits a team of 27 students into 3 groups to practice skills. How many students are in each group?	libretexts	2025-03-17T19:52:14.795536	2020-04-29T02:40:13	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.1%3A_Relationships_between_Quantities", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.1.1: Relationships between Quantities", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.2%3A_Reasoning_about_Contexts_with_Tape_Diagrams	6.1.2: Reasoning about Contexts with Tape Diagrams Lesson Let's use tape diagrams to make sense of different kinds of stories. Exercise $\PageIndex{1}$: Notice and Wonder: Remembering Tape Diagrams - What do you notice? What do you wonder? - What are some possible values for $a$, $b$, and $c$ in the first diagram? For $x$, $y$, and $z$ in the second diagram? How did you decide on those values? Exercise $\PageIndex{2}$: Every Picture Tells a Story Here are three stories with a diagram that represents it. With your group, decide who will go first. That person explains why the diagram represents the story. Work together to find any unknown amounts in the story. Then, switch roles for the second diagram and switch again for the third. 1. Mai made 50 flyers for five volunteers in her club to hang up around school. She gave 5 flyers to the first volunteer, 18 flyers to the second volunteer, and divided the remaining flyers equally among the three remaining volunteers. 2. To thank her five volunteers, Mai gave each of them the same number of stickers. Then she gave them each two more stickers. Altogether, she gave them a total of 30 stickers. 3. Mai distributed another group of flyers equally among the five volunteers. Then she remembered that she needed some flyers to give to teachers, so she took 2 flyers from each volunteer. Then, the volunteers had a total of 40 flyers to hang up. Exercise $\PageIndex{3}$: Every story Needs a Picture Here are three more stories. Draw a tape diagram to represent each story. Then describe how you would find any unknown amounts in the stories. - Noah and his sister are making gift bags for a birthday party. Noah puts 3 pencil erasers in each bag. His sister puts $x$ stickers in each bag. After filling 4 bags, they have used a total of 44 items. - Noah’s family also wants to blow up a total of 60 balloons for the party. Yesterday they blew up 24 balloons. Today they want to split the remaining balloons equally between four family members. - Noah’s family bought some fruit bars to put in the gift bags. They bought one box each of four flavors: apple, strawberry, blueberry, and peach. The boxes all had the same number of bars. Noah wanted to taste the flavors and ate one bar from each box. There were 28 bars left for the gift bags. Are you ready for more? Design a tiling that uses a repeating pattern consisting of 2 kinds of shapes (e.g., 1 hexagon with 3 triangles forming a triangle). How many times did you repeat the pattern in your picture? How many individual shapes did you use? Summary Tape diagrams are useful for representing how quantities are related and can help us answer questions about a situation. Suppose a school receives 46 copies of a popular book. The library takes 26 copies and the remainder are split evenly among 4 teachers. How many books does each teacher receive? This situation involves 4 equal parts and one other part. We can represent the situation with a rectangle labeled 26 (books given to the library) along with 4 equal-sized parts (books split among 4 teachers). We label the total, 46, to show how many the rectangle represents in all. We use a letter to show the unknown amount, which represents the number of books each teacher receives. Using the same letter, $x$, means that the same number is represented four times. Some situations have parts that are all equal, but each part has been increased from an original amount: A company manufactures a special type of sensor, and packs them in boxes of 4 for shipment. Then a new design increases the weight of each sensor by 9 grams. The new package of 4 sensors weighs 76 grams. How much did each sensor weigh originally? We can describe this situation with a rectangle representing a total of 76 split into 4 equal parts. Each part shows that the new weight, $x+9$, is 9 more than the original weight, $x$. Practice Exercise $\PageIndex{4}$ The table shows the number of apples and the total weight of the apples. \| number of apples \| weight of apples (grams) \| \|---\|---\| \| $2$ \| $511$ \| \| $5$ \| $1200$ \| \| $8$ \| $2016$ \| Estimate the weight of 6 apples. (From Unit 3.1.1) Exercise $\PageIndex{5}$ Select all stories that the tape diagram can represent. - There are 87 children and 39 adults at a show. The seating in the theater is split into 4 equal sections. - There are 87 first graders in after-care. After 39 students are picked up, the teacher put the remaining students into 4 groups for an activity. - Lin buys a pack of 87 pencils. She gives 39 to her teacher and shared the remaining pencils between herself and 3 friends. - Andre buys 4 packs of paper clips with 39 paper clips in each. Then he gives 87 paper clips to his teacher. - Diego’s family spends $87 on 4 tickets to the fair and a $39 dinner. Exercise $\PageIndex{6}$ Andre wants to save $40 to buy a gift for his dad. Andre’s neighbor will pay him weekly to mow the lawn, but Andre always gives a $2 donation to the food bank in weeks when he earns money. Andre calculates that it will take him 5 weeks to earn the money for his dad’s gift. He draws a tape diagram to represent the situation. - Explain how the parts of the tape diagram represent the story. - How much does Andre’s neighbor pay him each week to mow the lawn? Exercise $\PageIndex{7}$ Without evaluating each expression, determine which value is the greatest. Explain how you know. - $7\frac{5}{6}-9\frac{3}{4}$ - $(-7\frac{5}{6})+(-9\frac{3}{4})$ - $(-7\frac{5}{6})\cdot 9\frac{3}{4}$ - $(-7\frac{5}{6})\div (-9\frac{3}{4})$ (From Unit 5.4.1) Exercise $\PageIndex{8}$ Solve each equation. - $(8.5)\cdot (-3)=a$ - $(-7)+b=(-11)$ - $c-(-3)=15$ - $d\cdot (-4)=32$ (From Unit 5.5.1)	libretexts	2025-03-17T19:52:14.866818	2020-04-29T02:39:39	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.2%3A_Reasoning_about_Contexts_with_Tape_Diagrams", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.1.2: Reasoning about Contexts with Tape Diagrams", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.3%3A_Reasoning_about_Equations_with_Tape_Diagrams	6.1.3: Reasoning about Equations with Tape Diagrams Lesson Let's see how equations can describe tape diagrams. Exercise $\PageIndex{1}$: Find Equivalent Expressions Select all the expressions that are equivalent to $7(2-3n)$. Explain how you know each expression you select is equivalent. - $9-10n$ - $14-3n$ - $14-21n$ - $(2-3n)\cdot 7$ - $7\cdot 2\cdot (-3n)$ Exercise $\PageIndex{2}$: Matching Equations to Tape Diagrams - Match each equation to one of the tape diagrams. Be prepared to explain how the equation matches the diagram. - Sort the equations into categories of your choosing. Explain the criteria for each category. - $2x+5=19$ - $2+5x=19$ - $2(x+5)=19$ - $5(x+2)=19$ - $19=5+2x$ - $(x+5)\cdot 2=19$ - $19=(x+2)\cdot 5$ - $19\div 2=x+5$ - $19-2=5x$ Exercise $\PageIndex{3}$: Drawing Tape Diagrams to Represent Equations - Draw a tape diagram to match each equation. - $114=3x+18$ - $114=3(y+18)$ - Use any method to find values for $x$ and $y$ that make the equations true. Are you ready for more? To make a Koch snowflake: - Start with an equilateral triangle that has side lengths of 1. This is step 1. - Replace the middle third of each line segment with a small equilateral triangle with the middle third of the segment forming the base. This is step 2. - Do the same to each of the line segments. This is step 3. - Keep repeating this process. - What is the perimeter after step 2? Step 3? - What happens to the perimeter, or the length of line traced along the outside of the figure, as the process continues? Summary We have seen how tape diagrams represent relationships between quantities. Because of the meaning and properties of addition and multiplication, more than one equation can often be used to represent a single tape diagram. Let’s take a look at two tape diagrams. We can describe this diagram with several different equations. Here are some of them: - $26+4x=46$, because the parts add up to the whole. - $4x+26=46$, because addition is commutative. - $46=4x+26$, because if two quantities are equal, it doesn’t matter how we arrange them around the equal sign. - $4x=46-26$, because one part (the part made up of $4x$’s) is the difference between the whole and the other part. For this diagram: - $4(x+9)=76$, because multiplication means having multiple groups of the same size. - $(x+9) \cdot 4=76$, because multiplication is commutative. - $76\div 4=x+9$, because division tells us the size of each equal part. Glossary Entries Definition: Equivalent Expressions Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression. For example, $3x+4x$ is equivalent to $5x+2x$. No matter what value we use for $x$, these expressions are always equal. When $x$ is 3, both expressions equal 21. When $x$ is 10, both expressions equal 70. Practice Exercise $\PageIndex{4}$ Solve each equation mentally. - $2x=10$ - $-3x=21$ - $\frac{1}{3}x=6$ - $-\frac{1}{2}x=-7$ (From Unit 5.5.1) Exercise $\PageIndex{5}$ Complete the magic squares so that the sum of each row, each column, and each diagonal in a grid are all equal. (From Unit 5.2.2) Exercise $\PageIndex{6}$ Draw a tape diagram to match each equation. - $5(x+1)=20$ - $5x+1=20$ Exercise $\PageIndex{7}$ Select all the equations that match the tape diagram. - $35=8+x+x+x+x+x+x$ - $35=8+6x$ - $6+8x=35$ - $6x+8=35$ - $6x+8x=35x$ - $35-8=6x$ Exercise $\PageIndex{8}$ Each car is traveling at a constant speed. Find the number of miles each car travels in 1 hour at the given rate. - $135$ miles in $3$ hours - $22$ miles in $\frac{1}{2}$ hour - $7.5$ miles in $\frac{1}{4}$ hour - $\frac{100}{3}$ miles in $\frac{2}{3}$ hour - $97\frac{1}{2}$ miles in $\frac{3}{2}$ hour (From Unit 4.1.2)	libretexts	2025-03-17T19:52:14.940699	2020-04-29T02:39:06	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.3%3A_Reasoning_about_Equations_with_Tape_Diagrams", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.1.3: Reasoning about Equations with Tape Diagrams", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.4%3A_Reasoning_about_Equations_and_Tape_Diagrams_(Part_1)	6.1.4: Reasoning about Equations and Tape Diagrams (Part 1) Lesson Let's see how tape diagrams can help us answer questions about unknown amounts in stories. Exercise $\PageIndex{1}$: Algebra Talk: Seeing Structure Find a solution to each equation without writing anything down. $x+1=5$ $2(x+1)=10$ $3(x+1)=15$ $500=100(x+1)$ Exercise $\PageIndex{2}$: Situations and Diagrams Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable. - Diego has 7 packs of markers. Each pack has $x$ markers in it. After Lin gives him 9 more markers, he has a total of 30 markers. - Elena is cutting a 30-foot piece of ribbon for a craft project. She cuts off 7 feet, and then cuts the remaining piece into 9 equal lengths of $x$ feet each. - A construction manager weighs a bundle of 9 identical bricks and a 7-pound concrete block. The bundle weighs 30 pounds. - A skating rink charges a group rate of $9 plus a fee to rent each pair of skates. A family rents 7 pairs of skates and pays a total of $30. - Andre bakes 9 pans of brownies. He donates 7 pans to the school bake sale and keeps the rest to divide equally among his class of 30 students. Exercise $\PageIndex{3}$: Situations, Diagrams, and Equations Each situation in the previous activity is represented by one of the equations. - $7x+9=30$ - $30=9x+7$ - $30x+7=9$ - Match each situation to an equation. - Find the solution to each equation. Use your diagrams to help you reason. - What does each solution tell you about its situation? Are you ready for more? While in New York City, is it a better deal for a group of friends to take a taxi or the subway to get from the Empire State Building to the Metropolitan Museum of Art? Explain your reasoning. Summary Many situations can be represented by equations. Writing an equation to represent a situation can help us express how quantities in the situation are related to each other, and can help us reason about unknown quantities whose value we want to know. Here are three situations: - An architect is drafting plans for a new supermarket. There will be a space 144 inches long for rows of nested shopping carts. The first cart is 34 inches long and each nested cart adds another 10 inches. The architect want to know how many shopping carts will fit in each row. - A bakery buys a large bag of sugar that has 34 cups. They use 10 cups to make some cookies. Then they use the rest of the bag to make 144 giant muffins. Their customers want to know how much sugar is in each muffin. - Kiran is trying to save $144 to buy a new guitar. He has $34 and is going to save $10 a week from money he earns mowing lawns. He wants to know how many weeks it will take him to have enough money to buy the guitar. We see the same three numbers in the situations: 10, 34, and 144. How could we represent each situation with an equation? In the first situation, there is one shopping cart with length 34 and then an unknown number of carts with length 10. Similarly, Kiran has 34 dollars saved and then will save 10 each week for an unknown number of weeks. Both situations have one part of 34 and then equal parts of size 10 that all add together to 144. Their equation is $34+10x=144$. Since it takes 11 groups of 10 to get from 34 to 144, the value of $x$ in these two situations is $(144-34)\div 10$ or 11. There will be 11 shopping carts in each row, and it will take Kiran 11 weeks to raise the money for the guitar. In the bakery situation, there is one part of 10 and then 144 equal parts of unknown size that all add together to 34. The equation is $10+144x=34$. Since 24 is needed to get from 10 to 34, the value of $x$ is $(34-10)\div 144$ or $\frac{1}{6}$. There is $\frac{1}{6}$ cup of sugar in each giant muffin. Glossary Entries Definition: Equivalent Expressions Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression. For example, $3x+4x$ is equivalent to $5x+2x$. No matter what value we use for $x$, these expressions are always equal. When $x$ is 3, both expressions equal 21. When $x$ is 10, both expressions equal 70. Practice Exercise $\PageIndex{4}$ Draw a square with side length 7 cm. - Predict the perimeter and the length of the diagonal of the square. - Measure the perimeter and the length of the diagonal of the square. - Describe how close the predictions and measurements are. (From Unit 3.1.1) Exercise $\PageIndex{5}$ Find the products. - $(100)\cdot (-0.09)$ - $(-7)\cdot (-1.1)$ - $(-7.3)\cdot (5)$ - $(-0.2)\cdot (-0.3)$ (From Unit 5.3.2) Exercise $\PageIndex{6}$ Here are three stories: - A family buys 6 tickets to a show. They also pay a $3 parking fee. They spend $27 to see the show. - Diego has 27 ounces of juice. He pours equal amounts for each of his 3 friends and has 6 ounces left for himself. - Jada works for 6 hours preparing for the art fair. She spends 3 hours on a sculpture and then paints 27 picture frames. Here are three equations. - $3x+6=27$ - $6x+3=27$ - $27x+3=6$ - Decide which equation represents each story. What does $x$ represent in each equation? - Find the solution to each equation. Explain or show your reasoning. - What does each solution tell you about its situation? Exercise $\PageIndex{7}$ Here is a diagram and its corresponding equation. Find the solution to the equation and explain your reasoning. Exercise $\PageIndex{8}$ 1. Plot these points on the coordinate plane: $A=(3,2),\: B=(7.5,2),\: C=(7.5, -2.5),\: D=(3,-2)$ 2. What is the vertical difference between $D$ and $A$? 3. Write an expression that represents the vertical distance between $B$ and $C$? (From Unit 5.2.6)	libretexts	2025-03-17T19:52:15.010039	2020-04-29T02:38:37	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.4%3A_Reasoning_about_Equations_and_Tape_Diagrams_(Part_1)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.1.4: Reasoning about Equations and Tape Diagrams (Part 1)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.5%3A_Reasoning_about_Equations_and_Tape_Diagrams_(Part_2)	6.1.5: Reasoning about Equations and Tape Diagrams (Part 2) Lesson Let's use tape diagrams to help answer questions about situations where the equation has parentheses. Exercise $\PageIndex{1}$: Algebra Talk: Seeing Structure Solve each equation mentally. $x-1=5$ $2(x-1)=10$ $3(x-1)=15$ $500=100(x-1)$ Exercise $\PageIndex{2}$: More Situations and Diagrams Draw a tape diagram to represent each situation. For some of the situations, you need to decide what to represent with a variable. - Each of 5 gift bags contains $x$ pencils. Tyler adds 3 more pencils to each bag. Altogether, the gift bags contain 20 pencils. - Noah drew an equilateral triangle with sides of length 5 inches. He wants to increase the length of each side by $x$ inches so the triangle is still equilateral and has a perimeter of 20 inches. - An art class charges each student $3 to attend plus a fee for supplies. Today, $20 was collected for the 5 students attending the class. - Elena ran 20 miles this week, which was three times as far as Clare ran this week. Clare ran 5 more miles this week than she did last week. Exercise $\PageIndex{3}$: More Situations, Diagrams, and Equations Each situation in the previous activity is represented by one of the equations. - $(x+3)\cdot 5=20$ - $3(x+5)=20$ - Match each situation to an equation. - Find the solution to each equation. Use your diagrams to help you reason. - What does each solution tell you about its situation? Are you ready for more? Han, his sister, his dad, and his grandmother step onto a crowded bus with only 3 open seats for a 42-minute ride. They decide Han’s grandmother should sit for the entire ride. Han, his sister, and his dad take turns sitting in the remaining two seats, and Han’s dad sits 1.5 times as long as both Han and his sister. How many minutes did each one spend sitting? Summary Equations with parentheses can represent a variety of situations. - Lin volunteers at a hospital and is preparing toy baskets for children who are patients. She adds 2 items to each basket, after which the supervisor’s list shows that 140 toys have been packed into a group of 10 baskets. Lin wants to know how many toys were in each basket before she added the items. - A large store has the same number of workers on each of 2 teams to handle different shifts. They decide to add 10 workers to each team, bringing the total number of workers to 140. An executive at the company that runs this chain of stores wants to know how many employees were in each team before the increase. Each bag in the first story has an unknown number of toys, $x$, that is increased by 2. Then ten groups of $x+2$ give a total of 140 toys. An equation representing this situation is $10(x+2)=140$. Since 10 times a number is 140, that number is 14, which is the total number of items in each bag. Before Lin added the 2 items there were $14-2$ or 12 toys in each bag. The executive in the second story knows that the size of each team of employees has been increased by 10. There are now 2 teams of $y+10$ each. An equation representing this situation is $2(y+10)=140$. Since 2 times an amount is 140, that amount is 70, which is the new size of each team. The value of $y$ is $70-10$ or 60. There were 60 employees on each team before the increase. Glossary Entries Definition: Equivalent Expressions Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression. For example, $3x+4x$ is equivalent to $5x+2x$. No matter what value we use for $x$, these expressions are always equal. When $x$ is 3, both expressions equal 21. When $x$ is 10, both expressions equal 70. Practice Exercise $\PageIndex{4}$ Here are some prices customers paid for different items at a farmer’s market. Find the cost for 1 pound of each item. - $5 for 4 pounds of apples - $3.50 for $\frac{1}{2}$ pound of cheese - $8.25 for $1\frac{1}{2}$ pounds of coffee beans - $6.75 for $\frac{3}{4}$ pounds of fudge - $5.50 for a $6\frac{1}{4}$ pound pumpkin (From Unit 4.1.2) Exercise $\PageIndex{5}$ Find the products. - $\frac{2}{3}\cdot\left( \frac{-4}{5}\right)$ - $\left(\frac{-5}{7}\right)\cdot\left(\frac{7}{5}\right)$ - $\left(\frac{-2}{39}\right)\cdot 39$ - $\left(\frac{2}{5}\right)\cdot\left(\frac{-3}{4}\right)$ (From Unit 5.3.2) Exercise $\PageIndex{6}$ Here are two stories: - A family buys 6 tickets to a show. They also each spend $3 on a snack. They spend $24 on the show. - Diego has 24 ounces of juice. He pours equal amounts for each of his 3 friends, and then adds 6 more ounces for each. Here are two equations: - $3(x+6)=24$ - $6(x+3)=24$ - Which equation represents which story? - What does $x$ represent in each equation? - Find the solution to each equation. Explain or show your reasoning. - What does each solution tell you about its situation? Exercise $\PageIndex{7}$ Here is a diagram and its corresponding equation. Find the solution to the equation and explain your reasoning. Exercise $\PageIndex{8}$ Below is a set of data about temperatures. The range of a set of data is the distance between the lowest and highest value in the set. What is the range of these temperatures? $9^{\circ}\text{C},\: -3^{\circ}\text{C},\: 22^{\circ}\text{C},\: -5^{\circ}\text{C},\: 11^{\circ}\text{C},\: 15^{\circ}\text{C}$ (From Unit 5.2.6) Exercise $\PageIndex{9}$ A store is having a 25% off sale on all shirts. Show two different ways to calculate the sale price for a shirt that normally costs $24. (From Unit 4.3.2)	libretexts	2025-03-17T19:52:15.081917	2020-04-29T02:37:53	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.5%3A_Reasoning_about_Equations_and_Tape_Diagrams_(Part_2)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.1.5: Reasoning about Equations and Tape Diagrams (Part 2)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.6%3A_Distinguishing_between_Two_Types_of_Situations	6.1.6: Distinguishing between Two Types of Situations Lesson Let's think about equations with and without parentheses and the kinds of situations they describe. Exercise $\PageIndex{1}$: Which One Doesn't Belong: Seeing Structure Which equation doesn't belong? $4(x+3)=9$ $4\cdot x+12=9$ $4+3x=9$ $9=12+4x$ Exercise $\PageIndex{2}$: Card Sort: Categories of Equations Your teacher will give you a set of cards that show equations. Sort the cards into 2 categories of your choosing. Be prepared to explain the meaning of your categories. Then, sort the cards into 2 categories in a different way. Be prepared to explain the meaning of your new categories. Exercise $\PageIndex{3}$: Even More Situations, Diagrams, and Equations Story 1: Lin had 90 flyers to hang up around the school. She gave 12 flyers to each of three volunteers. Then she took the remaining flyers and divided them up equally between the three volunteers. Story 2: Lin had 90 flyers to hang up around the school. After giving the same number of flyers to each of three volunteers, she had 12 left to hang up by herself. - Which diagram goes with which story? Be prepared to explain your reasoning. - In each diagram, what part of the story does the variable represent? - Write an equation corresponding to each story. If you get stuck, use the diagram. - Find the value of the variable in the story. Are you ready for more? A tutor is starting a business. In the first year, they start with 5 clients and charge $10 per week for an hour of tutoring with each client. For each year following, they double the number of clients and the number of hours each week. Each new client will be charged 150% of the charges of the clients from the previous year. - Organize the weekly earnings for each year in a table. - Assuming a full-time week is 40 hours per week, how many years will it take to reach full time and how many new clients will be taken on that year? - After reaching full time, what is the tutor’s annual salary if they take 2 weeks of vacation? - Is there another business model you’d recommend for the tutor? Explain your reasoning. Summary In this unit, we encounter two main types of situations that can be represented with an equation. Here is an example of each type: - After adding 8 students to each of 6 same-sized teams, there were 72 students altogether. - After adding an 8-pound box of tennis rackets to a crate with 6 identical boxes of ping pong paddles, the crate weighed 72 pounds. The first situation has all equal parts, since additions are made to each team. An equation that represents this situation is $6(x+8)=72$, where $x$ represents the original number of students on each team. Eight students were added to each group, there are 6 groups, and there are a total of 72 students. In the second situation, there are 6 equal parts added to one other part. An equation that represents this situation is $6x+8=72$, where $x$ represents the weight of a box of ping pong paddles, there are 6 boxes of ping pong paddles, there is an additional box that weighs 8 pounds, and the crate weighs 72 pounds altogether. In the first situation, there were 6 equal groups, and 8 students added to each group. $6(x+8)=72$. In the second situation, there were 6 equal groups, but 8 more pounds in addition to that. $6x+8=72$. Glossary Entries Definition: Equivalent Expressions Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression. For example, $3x+4x$ is equivalent to $5x+2x$. No matter what value we use for $x$, these expressions are always equal. When $x$ is 3, both expressions equal 21. When $x$ is 10, both expressions equal 70. Practice Exercise $\PageIndex{4}$ A school ordered 3 large boxes of board markers. After giving 15 markers to each of 3 teachers, there were 90 markers left. The diagram represents the situation. How many markers were originally in each box? (From Unit 6.1.2) Exercise $\PageIndex{5}$ The diagram can be represented by the equation $25=2+6x$. Explain where you can see the 6 in the diagram. (From Unit 6.1.3) Exercise $\PageIndex{6}$ Match each equation to a story. (Two of the stories match the same equation.) - $3(x+5)=17$ - $3x+5=17$ - $5(x+3)=17$ - $5x+3=17$ - Jada's teacher fills a travel bag with 5 copies of a textbook. The weight of the bag and books is 17 pounds. The empty travel bag weighs 3 pounds. How much does each book weigh? - A piece of scenery for the school play is in the shape of a 5-foot-long rectangle. The designer decides to increase the length. There will be 3 identical rectangles with a total length of 17 feet. By how much did the designer increase the length of each rectangle? - Elena spends $17 and buys a $3 book and a bookmark for each of her 5 cousins. How much does each bookmark cost? - Noah packs up bags at the food pantry to deliver to families. He packs 5 bags that weigh a total of 17 pounds. Each bag contains 3 pounds of groceries and a packet of papers with health-related information. How much does each packet of papers weigh? - Andre has 3 times as many pencils as Noah and 5 pens. He has 17 pens and pencils all together. How many pencils does Noah have? Exercise $\PageIndex{7}$ Elena walked 20 minutes more than Lin. Jada walked twice as long as Elena. Jada walked for 90 minutes. The equation $2(x+20)=90$ describes this situation. Match each expression with the statement in the story with the expression it represents. - $x$ - $x+20$ - $2(x+20)$ - $90$ - The number of minutes that Jada walked - The number of minutes that Elena walked - The number of minutes that Lin walked	libretexts	2025-03-17T19:52:15.151470	2020-04-29T02:37:16	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.01%3A_New_Page/6.1.6%3A_Distinguishing_between_Two_Types_of_Situations", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.1.6: Distinguishing between Two Types of Situations", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page	6.2: Solving Equations of the Form px+q=r and p(x+q)=r and Problems That Lead to Those Equations Last updated Save as PDF Page ID 35013 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:15.223187	2020-01-25T01:41:38	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.2: Solving Equations of the Form px+q=r and p(x+q)=r and Problems That Lead to Those Equations", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.1%3A_Reasoning_about_Solving_Equations_(Part_1)	6.2.1: Reasoning about Solving Equations (Part 1) Lesson Let's see how a balanced hanger is like an equation and moving its weight is like solving the equation. Exercise $\PageIndex{1}$: Hanger Diagrams In the two diagrams, all the triangles weigh the same and all the squares weigh the same. For each diagram, come up with . . . - One thing that must be true - One thing that could be true - One thing that cannot possibly be true Exercise $\PageIndex{2}$: Hanger and Equation Matching On each balanced hanger, figures with the same letter have the same weight. - Match each hanger to an equation. Complete the equation by writing $x$, $y$, $z$, or $w$ in the empty box. - $2 ?+3=5$ - $3 ?+2=3$ - $6=2?+3$ - $7=3?+1$ - Find the solution to each equation. Use the hanger to explain what the solution means. Exercise $\PageIndex{3}$: Use Hangers to Understand Equation Solving Here are some balanced hangers where each piece is labeled with its weight. For each diagram: - Write an equation. - Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram. - Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation. Summary In this lesson, we worked with two ways to show that two amounts are equal: a balanced hanger and an equation. We can use a balanced hanger to think about steps to finding an unknown amount in an associated equation. The hanger shows a total weight of 7 units on one side that is balanced with 3 equal, unknown weights and a 1-unit weight on the other. An equation that represents the relationship is $7=3x+1$. We can remove a weight of 1 unit from each side and the hanger will stay balanced. This is the same as subtracting 1 from each side of the equation. An equation for the new balanced hanger is $6=3x$. So the hanger will balance with $\frac{1}{3}$ of the weight on each side: $\frac{1}{3}\cdot 6=\frac{1}{3}\cdot 3x$. The two sides of the hanger balance with these weights: 6 1-unit weights on one side and 3 weights of unknown size on the other side. Here is a concise way to write the steps above: $\begin{array}{lr}{7=3x+1}&{}\\{6=3x}&{\text{after subtracting 1 from each side}}\\{2=x}&{\text{after multiplying each side by }\frac{1}{3}}\end{array}$ Practice Exercise $\PageIndex{4}$ There is a proportional relationship between the volume of a sample of helium in liters and the mass of that sample in grams. If the mass of a sample is 5 grams, its volume is 28 liters. (5, 28) is shown on the graph below. - What is the constant of proportionality in this relationship? - In this situation, what is the meaning of the number you found in part a? - Add at least three more points to the graph above, and label with their coordinates. - Write an equation that shows the relationship between the mass of a sample of helium and its volume. Use $m$ for mass and $v$ for volume. (From Unit 2.4.2) Exercise $\PageIndex{5}$ Explain how the parts of the balanced hanger compare to the parts of the equation. $7=2x+3$ Exercise $\PageIndex{6}$ For the hanger below: - Write an equation to represent the hanger. - Draw more hangers to show each step you would take to find $x$. Explain your reasoning. - Write an equation to describe each hanger you drew. Describe how each equation matches its hanger.	libretexts	2025-03-17T19:52:15.290025	2020-04-29T02:43:37	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.1%3A_Reasoning_about_Solving_Equations_(Part_1)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.2.1: Reasoning about Solving Equations (Part 1)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.2%3A_Reasoning_about_Solving_Equations_(Part_2)	6.2.2: Reasoning about Solving Equations (Part 2) Lesson Let's use hangers to understand two different ways of solving equations with parentheses. Exercise $\PageIndex{1}$: Equivalent to $2(x+3)$ Select all the expressions equivalent to $2(x+3)$. - $2\cdot (x+3)$ - $(x+3)2$ - $2\cdot x+2\cdot 3$ - $2\cdot x+3$ - $(2\cdot x)+3$ - $(2+x)3$ Exercise $\PageIndex{2}$: Either Or 1. Explain why either of these equations could represent this hanger: $14=2(x+3)$ or $14=2x+6$ 2. Find the weight of one circle. Be prepared to explain your reasoning. Exercise $\PageIndex{3}$: Use Hangers to Understand Equation Solving, Again Here are some balanced hangers. Each piece is labeled with its weight. - Assign one of these equations to each hanger: \[\begin{array}{lll}{2(x+5)=16}&{\qquad}&{3(y+200)=3,000}\\{20.8=4(z+1.1)}&{\qquad}&{\frac{20}{3}=2(w+\frac{2}{3})}\end{array}\nonumber\] - Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram. - Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation. Summary The balanced hanger shows 3 equal, unknown weights and 3 2-unit weights on the left and an 18-unit weight on the right. There are 3 unknown weights plus 6 units of weight on the left. We could represent this balanced hanger with an equation and solve the equation the same way we did before. $\begin{aligned}3x+6&=18 \\ 3x&=12 \\ x&=4\end{aligned}$ Since there are 3 groups of $x+2$ on the left, we could represent this hanger with a different equation: $3(x+2)=18$. The two sides of the hanger balance with these weights: 3 groups of $x+2$ on one side, and 18, or 3 groups of 6, on the other side. The two sides of the hanger will balance with $\frac{1}{3}$ of the weight on each side: $\frac{1}{3}\cdot 3(x+2)=\frac{1}{3}\cdot 18$. We can remove 2 units of weight from each side, and the hanger will stay balanced. This is the same as subtracting 2 from each side of the equation. An equation for the new balanced hanger is $x=4$. This gives the solution to the original equation. Here is a concise way to write the steps above: $\begin{array}{lr}{3(x+2)=18}&{}\\{x+2=6}&{\text{after multiplying each side by }\frac{1}{3}}\\{x=4}&{\text{after subtracting }2\text{ from each side}}\end{array}$ Practice Exercise $\PageIndex{4}$ Here is a hanger: - Write an equation to represent the hanger. - Solve the equation by reasoning about the equation or the hanger. Explain your reasoning. Exercise $\PageIndex{5}$ Explain how each part of the equation $9=3(x+2)$ is represented in the hanger. - $x$ - $9$ - $3$ - $x+2$ - $3(x+2)$ - the equal sign Exercise $\PageIndex{6}$ Select the word from the following list that best describes each situation. - You deposit money in a savings account, and every year the amount of money in the account increases by 2.5%. - For every car sold, a car salesman is paid 6% of the car’s price. - Someone who eats at a restaurant pays an extra 20% of the food price. This extra money is kept by the person who served the food. - An antique furniture store pays $200 for a chair, adds 50% of that amount, and sells the chair for $300. - The normal price of a mattress is $600, but it is on sale for 10% off. - For any item you purchase in Texas, you pay an additional 6.25% of the item's price to the state government. - Tax - Commission - Discount - Markup - Tip or gratuity - Interest (From Unit 4.3.2) Exercise $\PageIndex{7}$ Clare drew this diagram to match the equation $2x+16=50$, but she got the wrong solution as a result of using this diagram. - What value of $x$ can be found using the diagram? - Show how to fix Clare's diagram to correctly match the equation. - Use th new diagram to find a correct value for $x$. - Explain how the mistake Clare made when she drew her diagram. (From Unit 6.1.3)	libretexts	2025-03-17T19:52:15.360746	2020-04-29T02:43:08	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.2%3A_Reasoning_about_Solving_Equations_(Part_2)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.2.2: Reasoning about Solving Equations (Part 2)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.3%3A_Dealing_with_Negative_Numbers	6.2.3: Dealing with Negative Numbers Lesson Let's show that doing the same to each side works for negative numbers too. Exercise $\PageIndex{1}$: Which One Doesn't Belong: Rational Number Arithmetic Which equation doesn't belong? $\begin{array}{llll}{15=-5\cdot -3}&{\qquad}&{\qquad}&{4--2=6}\\{2+-5=-3}&{\qquad}&{\qquad}&{-3\cdot -4=-12}\end{array}$ Exercise $\PageIndex{2}$: Old and New Ways to Solve Solve each equation. Be prepared to explain your reasoning. - $x+6=4$ - $x--4=-6$ - $2(x-1)=-200$ - $2x+-3=-23$ Exercise $\PageIndex{3}$: Keeping It True Here are some equations that all have the same solution. $\begin{aligned} x&=-6\\ x-3&=-9 \\ -9&=x-3 \\ 900&=-100(x-3) \\ 900&=(x-3)\cdot (-100) \\ 900&=-100x+300\end{aligned}$ - Explain how you know that each equation has the same solution as the previous equation. Pause for discussion before moving to the next question. - Keep your work secret from your partner. Start with the equation $-5=x$. Do the same thing to each side at least three times to create an equation that has the same solution as the starting equation. Write the equation you ended up with on a slip of paper, and trade equations with your partner. - See if you can figure out what steps they used to transform $-5=x$ into their equation. When you think you know, check with them to see if you are right. Summary When we want to find a solution to an equation, sometimes we just think about what value in place of the variable would make the equation true. Sometimes we perform the same operation on each side (for example, subtract the same amount from each side). The balanced hangers helped us to understand that doing the same to each side of an equation keeps the equation true. Since negative numbers are just numbers, then doing the same thing to each side of an equation works for negative numbers as well. Here are some examples of equations that have negative numbers and steps you could take to solve them. Example: $\begin{aligned} 2(x-5)&=-6 \\ \frac{1}{2}\cdot 2(x-5)&=\frac{1}{2}\cdot (-6)\qquad\text{multiply each side by }\frac{1}{2} \\ x-5&=-3 \\ x-5+5&=-3 +5 \qquad\text{ add }5\text{ to each side} \\ x&=2\end{aligned}$ Example: $\begin{aligned} -2x+-5&=6 \\ -2x+-5--5&=6--5 \qquad\text{ subtract }-5\text{ from each side} \\ -2x&=11 \\ -2x\div -2 &=11\div 2\qquad\text{ divide each side by }-2 \\ x&=-\frac{11}{2}\end{aligned}$ Doing the same thing to each side maintains equality even if it is not helpful to solving for the unknown amount. For example, we could take the equation $-3x+7=-8$ and add $-2$ to each side: $\begin{aligned} -3x+7&=-8 \\ -3x+7+-2&=-8+-2\qquad\text{ add }-2\text{ to each side} \\ -3x+5 &=-10\end{aligned}$ If $-2x+7=-8$ is true then $-3x+5=-10$ is also true, but we are no closer to a solution than we were before adding -2. We can use moves that maintain equality to make new equations that all have the same solution. Helpful combinations of moves will eventually lead to an equation like $x=5$, which gives the solution to the original equation (and every equation we wrote in the process of solving). Practice Exercise $\PageIndex{4}$ Solve each equation. - $4x=-28$ - $x--6=-2$ - $-x+4=-9$ - $-3x+7=1$ - $25x+-11=-86$ Exercise $\PageIndex{5}$ Here is an equation $2x+9=-15$. Write three different equations that have the same solution as $2x+9=-15$. Show or explain how you found them. Exercise $\PageIndex{6}$ Select all the equations that match the diagram. - $x+5=18$ - $18\div 3=x+5$ - $3(x+5)=18$ - $x+5=\frac{1}{3}\cdot 18$ - $3x+5=18$ (From Unit 6.1.3) Exercise $\PageIndex{7}$ There are 88 seats in a theater. The seating in the theater is split into 4 identical sections. Each section has 14 red seats and some blue seats. - Draw a tape diagram to represent the situation. - What unknown amounts can be found by by using the diagram or reasoning about the situation? (From Unit 6.1.2) Exercise $\PageIndex{8}$ Match each story to an equation. - A stack of nested paper cups is 8 inches tall. The first cup is 4 inches tall and each of the rest of the cups in the stack adds $\frac{1}{4}$ inch to the height of the stack. - A baker uses 4 cups of flour. She uses $\frac{1}{4}$ cup to flour the counters and the rest to make 8 identical muffins. - Elena has an 8-foot piece of ribbon. She cuts off a piece that is $\frac{1}{4}$ of a foot long and cuts the remainder into four pieces of equal length. - $\frac{1}{4}+4x=8$ - $4+\frac{1}{4}x=8$ - $8x+\frac{1}{4}=4$ (From Unit 6.1.4)	libretexts	2025-03-17T19:52:15.427776	2020-04-29T02:42:40	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.3%3A_Dealing_with_Negative_Numbers", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.2.3: Dealing with Negative Numbers", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.4%3A_Different_Options_for_Solving_One_Equation	6.2.4: Different Options for Solving One Equation Lesson Let's think about which way is easier when we solve equations with parentheses. Exercise $\PageIndex{1}$: Algebra Talk: Solve Each Equation $100(x-3)=1,000$ $500(x-3)=5,000$ $0.03(x-3)=0.3$ $0.72(x+2)=7.2$ Exercise $\PageIndex{2}$: Analyzing Solution Methods Three students each attempted to solve the equation $2(x-9)=10$, but got different solutions. Here are their methods. Do you agree with any of their methods, and why? Noah’s method: $\begin{aligned} 2(x-9)&=10 \\ 2(x-9)+9&=10+9\qquad\text{ add }9\text{ to each side} \\ 2x&=19 \\ 2x\div 2&=19\div 2\qquad\text{ divide each side by }2 \\ x&=\frac{19}{2}\end{aligned}$ Elena's method: $\begin{aligned} 2(x-9)&=10 \\ 2x-18&=10 \qquad\text{ apply the distributive property} \\ 2x-18-18&=10-18\qquad\text{ subtract }18\text{ from each side} \\ 2x&=-8 \\ 2x\div 2&=-8\div 2 \qquad\text{ divide each side by }2 \\ x&=-4\end{aligned}$ Andre's method: $\begin{aligned} 2(x-9)&=10 \\ 2x-18&=10 \qquad \text{ apply the distributive property} \\ 2x-18+18&=10+18 \qquad\text{ add }18\text{ to each side} \\ 2x&=28 \\ 2x\div 2&=28\div 2\qquad\text{ divide each side by }2 \\ x&=14\end{aligned}$ Exercise $\PageIndex{3}$: Solution Pathways For each equation, try to solve the equation using each method (dividing each side first, or applying the distributive property first). Some equations are easier to solve by one method than the other. When that is the case, stop doing the harder method and write down the reason you stopped. - $2,000(x-0.03)=6,000$ - $2(x+1.25)=3.5$ - $\frac{1}{4}(4+x)=\frac{4}{3}$ - $-10(x-1.7)=-3$ - $5.4 = 0.3(x+8)$ Summary Equations can be solved in many ways. In this lesson, we focused on equations with a specific structure, and two specific ways to solve them. Suppose we are trying to solve the equation $\frac{4}{5}(x+27)=16$. Two useful approaches are: - divide each side by $\frac{4}{5}$ - apply the distributive property In order to decide which approach is better, we can look at the numbers and think about which would be easier to compute. We notice that $\frac{4}{5}\cdot 27$ will be hard, because 27 isn't divisible by 5. But $16\div\frac{4}{5}$ gives us $16\cdot\frac{5}{4}$, and 16 is divisible by 4. Dividing each side by $\frac{4}{5}$ gives: $\begin{aligned} \frac{4}{5}(x+27)&=16 \\ \frac{5}{4}\cdot\frac{4}{5}(x+27)&=16\cdot\frac{5}{4} \\ x+27&=20 \\ x&=-7\end{aligned}$ Sometimes the calculations are simpler if we first use the distributive property. Let's look at the equation $100(x+0.06)=21$. If we first divide each side by 100, we get $\frac{21}{100}$ or 0.21 on the right side of the equation. But if we use the distributive property first, we get an equation that only contains whole numbers. $\begin{aligned} 100(x+0.06)&=21 \\ 100x+6&=21 \\ 100x&=15 \\ x&=\frac{15}{100}\end{aligned}$ Practice Exercise $\PageIndex{4}$ Andre wants to buy a backpack. The normal price of the backpack is $40. He notices that a store that sells the backpack is having a 30% off sale. What is the sale price of the backpack? (From Unit 4.3.2) Exercise $\PageIndex{5}$ On the first math exam, 16 students received an A grade. On the second math exam, 12 students received an A grade. What percentage decrease is that? (From Unit 4.3.3) Exercise $\PageIndex{6}$ Solve each equation. - $2(x-3)=14$ - $-5(x-1)=40$ - $12(x+10)=24$ - $\frac{1}{6}(x+6)=11$ - $\frac{5}{7}(x-9)=25$ Exercise $\PageIndex{7}$ Select all expressions that represent a correct solution to the equation $6(x+4)=20$. - $(20-4)\div 6$ - $\frac{1}{6}(20-4)$ - $20-6-4$ - $20\div 6-4$ - $\frac{1}{6}(20-24)$ - $(20-24)\div 6$ Exercise $\PageIndex{8}$ Lin and Noah are solving the equation $7(x+2)=91$. Lin starts by using the distributive property. Noah starts by dividing each side by 7. - Show what Lin's and Noah's full solution methods might look like. - What is the same and what is different about their methods?	libretexts	2025-03-17T19:52:15.495708	2020-04-29T02:42:13	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.4%3A_Different_Options_for_Solving_One_Equation", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.2.4: Different Options for Solving One Equation", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.5%3A_Using_Equations_to_Solve_Problems	6.2.5: Using Equations to Solve Problems Lesson Let's use tape diagrams, equations, and reasoning to solve problems. Exercise $\PageIndex{1}$: Remember Tape Diagrams - Write a story that could be represented by this tape diagram. - Write an equation that could be represented by this tape diagram. Exercise $\PageIndex{2}$: At the Fair 1. Tyler is making invitations to the fair. He has already made some of the invitations, and he wants to finish the rest of them within a week. He is trying to spread out the remaining work, to make the same number of invitations each day. Tyler draws a diagram to represent the situation. - Explain how each part of the situation is represented in Tyler’s diagram: How many total invitations Tyler is trying to make. How many invitations he has made already. How many days he has to finish the invitations. - How many invitations should Tyler make each day to finish his goal within a week? Explain or show your reasoning. - Use Tyler’s diagram to write an equation that represents the situation. Explain how each part of the situation is represented in your equation. - Show how to solve your equation. 2. Noah and his sister are making prize bags for a game at the fair. Noah is putting 7 pencil erasers in each bag. His sister is putting in some number of stickers. After filling 3 of the bags, they have used a total of 57 items. - Explain how the diagram represents the situation. - Noah writes the equation $3(x+7)=57$ to represent the situation. Do you agree with him? Explain your reasoning. - How many stickers is Noah's sister putting in each prize bag? Explain or show your reasoning. 3. A family of 6 is going to the fair. They have a coupon for $1.50 off each ticket. If they pay $46.50 for all their tickets, how much does a ticket cost without the coupon? Explain or show your reasoning. If you get stuck, consider drawing a diagram or writing an equation. Exercise $\PageIndex{3}$: Running Around Priya, Han, and Elena, are members of the running club at school. - Priya was busy studying this week and ran 7 fewer miles than last week. She ran 9 times as far as Elena ran this week. Elena only had time to run 4 miles this week. - How many miles did Priya run last week? - Elena wrote the equation $\frac{1}{9}(x-7)=4$ to describe the situation. She solved the equation by multiplying each side by 9 and then adding 7 to each side. How does her solution compare to the way you found Priya's miles? - One day last week, 6 teachers joined $\frac{5}{7}$ of the members of the running club in an after-school run. Priya counted a total of 31 people running that day. How many members does the running club have? - Priya and Han plan a fundraiser for the running club. They begin with a balance of -80 because of expenses. In the first hour of the fundraiser they collect equal donations from 9 family members, which brings their balance to -44. How much did each parent give? - The running club uses the money they raised to pay for a trip to a canyon. At one point during a run in the canyon, the students are at an elevation of 128 feet. After descending at a rate of 50 feet per minute, they reach an elevation of -472 feet. How long did the descent take? Are you ready for more? A musician performed at three local fairs. At the first he doubled his money and spent $30. At the second he tripled his money and spent $54. At the third, he quadrupled his money and spent $72. In the end he had $48 left. How much did he have before performing at the fairs? Summary Many problems can be solved by writing and solving an equation. Here is an example: Clare ran 4 miles on Monday. Then for the next six days, she ran the same distance each day. She ran a total of 22 miles during the week. How many miles did she run on each of the 6 days? One way to solve the problem is to represent the situation with an equation, $4+6x=22$, where $x$ represents the distance, in miles, she ran on each of the 6 days. Solving the equation gives the solution to this problem. $\begin{aligned} 4+6x&=22 \\ 6x&=18 \\ x&=3\end{aligned}$ Practice Exercise $\PageIndex{4}$ Find the value of each variable. - $a\cdot 3=-30$ - $-9\cdot b=45$ - $-89\cdot 12=c$ - $d\cdot 88=-88,000$ (From Unit 5.3.2) Exercise $\PageIndex{5}$ Match each equation to its solution and to the story it describes. Equations: - $5x-7=3$ - $7=3(5+x)$ - $3x+5=-7$ - $\frac{1}{3}(x+7)=5$ Solutions: - $-4$ - $\frac{-8}{3}$ - $2$ - $8$ Stories: - The temperature is $-7$. Since midnight the temperature tripled and then rose 5 degrees. What was temperature at midnight? - Jada has 7 pink roses and some white roses. She gives all of them away: 5 roses to each of her 3 favorite teachers. How many white roses did she give away? - A musical instrument company reduced the time it takes for a worker to build a guitar. Before the reduction it took 5 hours. Now in 7 hours they can build 3 guitars. By how much did they reduce the time it takes to build each guitar? - A club puts its members into 5 groups for an activity. After 7 students have to leave early, there are only 3 students left to finish the activity. How many students were in each group? Exercise $\PageIndex{6}$ The baby giraffe weighed 132 pounds at birth. He gained weight at a steady rate for the first 7 months until his weight reached 538 pounds. How much did he gain each month? Exercise $\PageIndex{7}$ Six teams are out on the field playing soccer. The teams all have the same number of players. The head coach asks for 2 players from each team to come help him move some equipment. Now there are 78 players on the field. Write and solve an equation whose solution is the number of players on each team. Exercise $\PageIndex{8}$ A small town had a population of 960 people last year. The population grew to 1200 people this year. By what percentage did the population grow? (From Unit 4.2.2) Exercise $\PageIndex{9}$ The gas tank of a truck holds 30 gallons. The gas tank of a passenger car holds 50% less. How many gallons does it hold? (From Unit 4.2.2)	libretexts	2025-03-17T19:52:15.641679	2020-04-29T02:41:41	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.5%3A_Using_Equations_to_Solve_Problems", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.2.5: Using Equations to Solve Problems", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.6%3A_Solving_Problems_about_Percent_Increase_or_Decrease	6.2.6: Solving Problems about Percent Increase or Decrease Lesson Let's use tape diagrams, equations, and reasoning to solve problems with negatives and percents. Exercise $\PageIndex{1}$: 20% Off An item costs $x$ dollars and then a 20% discount is applied. Select all the expressions that could represent the price of the item after the discount. - $\frac{20}{100}x$ - $x-\frac{20}{100}x$ - $(1-0.20)x$ - $\frac{100-20}{100}x$ - $0.80x$ - $(100-20)x$ Exercise $\PageIndex{2}$: Walking More Each Day 1. Mai started a new exercise program. On the second day, she walked 5 minutes more than on the first day. On the third day, she increased her walking time from day 2 by 20% and walked for 42 minutes. Mai drew a diagram to show her progress. Explain how the diagram represents the situation. 2. Noah said the equation $1.20(d+5)=42$ also represents the situation. Do you agree with Noah? Explain your reasoning. 3. Find the number of minutes Mai walked on the first day. Did you use the diagram, the equation, or another strategy? Explain or show your reasoning. 4. Mai has been walking indoors because of cold temperatures. On Day 4 at noon, Mai hears a report that the temperature is only 9 degrees Fahrenheit. She remembers the morning news reporting that the temperature had doubled since midnight and was expected to rise 15 degrees by noon. Mai is pretty sure she can draw a diagram to represent this situation but isn't sure if the equation is $9=15+2t$ or $2(t+15)=9$. What would you tell Mai about the diagram and the equation and how they might be useful to find the temperature, $t$, at midnight? Exercise $\PageIndex{3}$: A Sale on Shoes - A store is having a sale where all shoes are discounted by 20%. Diego has a coupon for $3 off of the regular price for one pair of shoes. The store first applies the coupon and then takes 20% off of the reduced price. If Diego pays $18.40 for a pair of shoes, what was their original price before the sale and without the coupon? - Before the sale, the store had 100 pairs of flip flops in stock. After selling some, they notice that $\frac{3}{5}$ of the flip flops they have left are blue. If the store has 39 pairs of blue flip flops, how many pairs of flip flops (any color) have they sold? - When the store had sold $\frac{2}{9}$ of the boots that were on display, they brought out another 34 pairs from the stock room. If that gave them 174 pairs of boots out, how many pairs were on display originally? - On the morning of the sale, the store donated 50 pairs of shoes to a homeless shelter. Then they sold 64% of their remaining inventory during the sale. If the store had 288 pairs after the donation and the sale, how many pairs of shoes did they have at the start? Are you ready for more? A coffee shop offers a special: 33% extra free or 33% off the regular price. Which offer is a better deal? Explain your reasoning. Summary We can solve problems where there is a percent increase or decrease by using what we know about equations. For example, a camping store increases the price of a tent by 25%. A customer then uses a $10 coupon for the tent and pays $152.50. We can draw a diagram that shows first the 25% increase and then the $10 coupon. The price after the 25% increase is $p+.25p$ or $1.25p$. An equation that represents the situation could be $1.25p-10=152.50$. To find the original price before the increase and discount, we can add 10 to each side and divide each side by 1.25, resulting in $p=130$. The original price of the tent was $130. Practice Exercise $\PageIndex{4}$ A backpack normally costs $25 but it is on sale for $21. What percentage is the discount? (From Unit 4.3.3) Exercise $\PageIndex{5}$ Find each product - $\frac{2}{5}\cdot (-10)$ - $-8\cdot\left(\frac{-3}{2}\right)$ - $\frac{10}{6}\cdot 0.6$ - $\left(\frac{-100}{37}\right)\cdot\left(-0.37\right)$ (From Unit 5.3.2) Exercise $\PageIndex{6}$ Select all expressions that show $x$ increased by 35%. - $1.35x$ - $\frac{35}{100}x$ - $x+\frac{35}{100}x$ - $(1+0.35)x$ - $\frac{100+35}{100}x$ - $(100+35)x$ Exercise $\PageIndex{7}$ Complete each sentence with the word discount , deposit , or withdrawal . - Clare took $20 out of her bank account. She made a _____. - Kiran used a coupon when he bought a pair of shoes. He got a _____. - Priya put $20 into her bank account. She made a _____. - Lin paid less than usual for a pack of gum because it was on sale. She got a _____. (From Unit 4.3.2) Exercise $\PageIndex{8}$ Here are two stories: - The initial freshman class at a college is 10% smaller than last year’s class. But then during the first week of classes, 20 more students enroll. There are then 830 students in the freshman class. - A store reduces the price of a computer by $20. Then during a 10% off sale, a customer pays $830. Here are two equations: - $0.9x+20=830$ - $0.9(x-20)=830$ - Decide which equation represents each story. - Explain why one equation has parentheses and the other doesn’t. - Solve each equation, and explain what the solution means in the situation.	libretexts	2025-03-17T19:52:15.710740	2020-04-29T02:41:09	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.02%3A_New_Page/6.2.6%3A_Solving_Problems_about_Percent_Increase_or_Decrease", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.2.6: Solving Problems about Percent Increase or Decrease", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page	Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset + - Margin Size Reset + - Font Type Enable Dyslexic Font Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 6: Expressions, Equations, and Inequalities Pre-Algebra I (Illustrative Mathematics - 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Grade 7) 6: Expressions, Equations, and Inequalities 6.3: Inequalities Expand/collapse global location 6.3: Inequalities Last updated Save as PDF Page ID 35014 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:15.784197	2020-01-25T01:41:39	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.3: Inequalities", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.1%3A_Reintroducing_Inequalities	6.3.1: Reintroducing Inequalities Lesson Let's work with inequalities. Exercise $\PageIndex{1}$: Greater Than One The number line shows values of $x$ that make the inequality $x>1$ true. - Select all the values of $x$ from this list that make the inequality $x>1$ true. - $3$ - $-3$ - $1$ - $700$ - $1.05$ - Name two more values of $x$ that are solutions to the inequality. Exercise $\PageIndex{2}$: The Roller Coaster A sign next to a roller coaster at an amusement park says, “You must be at least 60 inches tall to ride.” Noah is happy to know that he is tall enough to ride. - Noah is $x$ inches tall. Which of the following can be true: $x>60$, $x=60$, or $x<60$? Explain how you know. - Noah’s friend is 2 inches shorter than Noah. Can you tell if Noah’s friend is tall enough to go on the ride? Explain or show your reasoning. - List one possible height for Noah that means that his friend is tall enough to go on the ride, and another that means that his friend is too short for the ride. - On the number line below, show all the possible heights that Noah’s friend could be. 5. Noah's friend is $y$ inches tall. Use $y$ and any of the symbols $<$, $=$, $>$ to express this height. Exercise $\PageIndex{3}$: Is the Inequality True or False? The table shows four inequalities and four possible values for $x$. Decide whether each value makes each inequality true, and complete the table with “true” or “false.” Discuss your thinking with your partner. If you disagree, work to reach an agreement. \| $x$ \| $0$ \| $100$ \| $-100$ \| $25$ \| \|---\|---\|---\|---\|---\| \| $x\leq 25$ \| \|\|\|\| \| $100<4x$ \| \|\|\|\| \| $-3x>-75$ \| \|\|\|\| \| $100\geq 35-x$ \| Are you ready for more? Find an example of in inequality used in the real world and describe it using a number line. Summary We use inequalities to describe a range of numbers. In many places, you are allowed to get a driver’s license when you are at least 16 years old. When checking if someone is old enough to get a license, we want to know if their age is at least 16. If $h$ is the age of a person, then we can check if they are allowed to get a driver’s license by checking if their age makes the inequality $h>16$ (they are older than 16) or the equation $h=16$ (they are 16) true. The symbol $\geq$, pronounced “greater than or equal to,” combines these two cases and we can just check if $h\geq 16$ (their age is greater than or equal to 16). The inequality $h\geq 16$ can be represented on a number line: Practice Exercise $\PageIndex{4}$ For each inequality, find two values for $x$ that make the inequality true and two values that make it false. - $x+3>70$ - $x+3<70$ - $-5x<2$ - $5x<2$ Exercise $\PageIndex{5}$ Here is an inequality: $-3x>18$. - List some values for $x$ that would make this inequality true. - How are the solutions to the inequality $-3x\geq 18$ different from the solutions to $-3x>18$? Explain your reasoning. Exercise $\PageIndex{6}$ Here are the prices for cheese pizza at a certain pizzeria: \| pizza store \| price in dollars \| \|---\|---\| \| small \| $11.160$ \| \| medium \| \| \| large \| $16.25$ \| - You had a coupon that made the price of a large pizza $13.00. For what percent off was the coupon? - Your friend purchased a medium pizza for $10.31 with a 30% off coupon. What is the price of a medium pizza without a coupon? - Your friend has a 15% off coupon and $10. What is the largest pizza that your friend can afford, and how much money will be left over after the purchase? (From Unit 4.3.3) Exercise $\PageIndex{7}$ Select all the stories that can be represented by the diagram. - Andre studies 7 hours this week for end-of-year exams. He spends 1 hour on English and an equal number of hours each on math, science, and history. - Lin spends $3 on 7 markers and a $1 pen. - Diego spends $1 on 7 stickers and 3 marbles. - Noah shares 7 grapes with 3 friends. He eats 1 and gives each friend the same number of grapes. - Elena spends $7 on 3 notebooks and a $1 pen. (From Unit 6.1.4)	libretexts	2025-03-17T19:52:15.858974	2020-04-29T02:46:19	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.1%3A_Reintroducing_Inequalities", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.3.1: Reintroducing Inequalities", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.2%3A_Finding_Solutions_to_Inequalities_in_Context	6.3.2: Finding Solutions to Inequalities in Context Lesson Let's solve more complicated inequalities. Exercise $\PageIndex{1}$: Solutions to Equations and Solutions to Inequalities - Solve $-x=10$ - Find 2 solutions to $-x>10$ - Solve $2x=-20$ - Find 2 solutions to $2x>-20$ Exercise $\PageIndex{2}$: Earning Money For Soccer Stuff - Andre has a summer job selling magazine subscriptions. He earns $25 per week plus $3 for every subscription he sells. Andre hopes to make at least enough money this week to buy a new pair of soccer cleats. - Let $n$ represent the number of magazine subscriptions Andre sells this week. Write an expression for the amount of money he makes this week. - The least expensive pair of cleats Andre wants costs $68. Write and solve an equation to find out how many magazine subscriptions Andre needs to sell to buy the cleats. - If Andre sold 16 magazine subscriptions this week, would he reach his goal? Explain your reasoning. - What are some other numbers of magazine subscriptions Andre could have sold and still reached his goal? - Write an inequality expressing that Andre wants to make at least $68. - Write an inequality to describe the number of subscriptions Andre must sell to reach his goal. - Diego has budgeted $35 from his summer job earnings to buy shorts and socks for soccer. He needs 5 pairs of socks and a pair of shorts. The socks cost different amounts in different stores. The shorts he wants cost $19.95. - Let $x$ represent the price of one pair of socks. Write an expression for the total cost of the socks and shorts. - Write and solve an equation that says that Diego spent exactly $35 on the socks and shorts. - List some other possible prices for the socks that would still allow Diego to stay within his budget. - Write an inequality to represent the amount Diego can spend on a single pair of socks. Exercise $\PageIndex{3}$: Granola Bars and Savings - Kiran has $100 saved in a bank account. (The account doesn’t earn interest.) He asked Clare to help him figure out how much he could take out each month if he needs to have at least $25 in the account a year from now. - Clare wrote the inequality $-12x+100\geq 25$, where $x$ represents the amount Kiran takes out each month. What does $-12x$ represent? - Find some values of $x$ that would work for Kiran. - We could express all the values that would work using either $x\leq\underline{\quad}$ or $x\geq\underline{\quad}$. Which one should we use? - Write the answer to Kiran’s question using mathematical notation. - A teacher wants to buy 9 boxes of granola bars for a school trip. Each box usually costs $7, but many grocery stores are having a sale on granola bars this week. Different stores are selling boxes of granola bars at different discounts. - If $x$ represents the dollar amount of the discount, then the amount the teacher will pay can be expressed as $9(7-x)$. In this expression, what does the quantity $7-x$ represent? - The teacher has $36 to spend on the granola bars. The equation $9(7-x)=36$ represents a situation where she spends all $36. Solve this equation. - What does the solution mean in this situation? - The teacher does not have to spend all $36. Write an inequality relating 36 and $9(7-x)$ representing this situation. - The solution to this inequality must either look like $x\geq 3$ or $x\leq 3$. Which do you think it is? Explain your reasoning. Are you ready for more? Jada and Diego baked a large batch of cookies. - They selected $\frac{1}{4}$ of the cookies to give to their teachers. - Next, they threw away one burnt cookie. - They delivered $\frac{2}{5}$ of the remaining cookies to a local nursing home. - Next, they gave 3 cookies to some neighborhood kids. - They wrapped up $\frac{2}{3}$ of the remaining cookies to save for their friends. After all this, they had 15 cookies left. How many cookies did they bake? Summary Suppose Elena has $5 and sells pens for $1.50 each. Her goal is to save $20. We could solve the equation $1.5x+5=20$ to find the number of pens, $x$, that Elena needs to sell in order to save exactly $20. Adding -5 to both sides of the equation gives us $1.5x=15$, and then dividing both sides by $1.5$ gives the solution $x=10$ pens. What if Elena wants to have some money left over? The inequality $1.5x+5>20$ tells us that the amount of money Elena makes needs to be greater than $20. The solution to the previous equation will help us understand what the solutions to the inequality will be. We know that if she sells 10 pens, she will make $20. Since each pen gives her more money, she needs to sell more than 10 pens to make more than $20. So the solution to the inequality is $x>10$. Glossary Entries Definition: Solution to an Inequality A solution to an inequality is a number that can be used in place of the variable to make the inequality true. For example, 5 is a solution to the inequality $c<10$, because it is true that $5<10$. Some other solutions to this inequality are 9.9, 0, and -4. Practice Exercise $\PageIndex{4}$ The solution to $5-3x>35$ is either $x>-10$ or $-10>x$. Which solution is correct? Explain how you know. Exercise $\PageIndex{5}$ The school band director determined from past experience that if they charge $t$ dollars for a ticket to the concert, they can expect attendance of $1000-50t$. The director used this model to figure out that the ticket price needs to be $8 or greater in order for at least 600 to attend. Do you agree with this claim? Why or why not? Exercise $\PageIndex{6}$ Which inequality is true when the value of $x$ is -3? - $-x-6<-3.5$ - $-x-6>3.5$ - $-x-6>-3.5$ - $x-6>-3.5$ (From Unit 6.3.1) Exercise $\PageIndex{7}$ Draw the solution set for each of the following inequalities. 1. $x\leq 5$ 2. $x<\frac{5}{2}$ (From Unit 6.3.1) Exercise $\PageIndex{8}$ Write three different equations that match the tape diagram. (From Unit 6.1.3) Exercise $\PageIndex{9}$ A baker wants to reduce the amount of sugar in his cake recipes. He decides to reduce the amount used in 1 cake by $\frac{1}{2}$ cup. He then uses $4\frac{1}{2}$ cups of sugar to bake 6 cakes. - Describe how the tape diagram represents the story. - How much sugar was originally in each cake recipe? (From Unit 6.1.2) Exercise $\PageIndex{10}$ One year ago, Clare was 4 feet 6 inches tall. Now Clare is 4 feet 10 inches tall. By what percentage did Clare’s height increase in the last year? (From Unit 4.3.3)	libretexts	2025-03-17T19:52:15.933234	2020-04-29T02:45:48	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.2%3A_Finding_Solutions_to_Inequalities_in_Context", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.3.2: Finding Solutions to Inequalities in Context", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.3%3A_Efficiently_Solving_Inequalities	6.3.3: Efficiently Solving Inequalities Lesson Let's solve more complicated inequalities. Exercise $\PageIndex{1}$: Lots of Negatives Here is an inequality: $-x\geq -4$. - Predict what you think the solutions on the number line will look like. - Select all the values that are solutions to $-x\geq -4$: - $3$ - $-3$ - $4$ - $-4$ - $4.001$ - $-4.001$ - Graph the solutions to the inequality on the number line: Exercise $\PageIndex{2}$: Inequalities with Tables 1. Let's investigate the inequality $x-3>-2$. \| $x$ \| $-4$ \| $-3$ \| $-2$ \| $-1$ \| $0$ \| $1$ \| $2$ \| $3$ \| $4$ \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| $x-3$ \| $-7$ \| $-5$ \| $-1$ \| $1$ \| - Complete the table. - For which values of $x$ is it true that $x-3=-2$? - For which values of $x$ is it true that $x-3>-2$? - Graph the solutions to $x-3>-2$ on the number line: 2. Here is an inequality: $2x<6$. - Predict which values of $x$ will make the inequality $2x<6$ true. - Complete the table. Does it match your prediction? $x$ $-4$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$ $2x$ Table $\PageIndex{2}$ - Graph the solutions to $2x<6$ on the number line: 3. Here is an inequality: $-2x<6$. - Predict which values of $x$ will make the inequality $-2x<6$ true. - Complete the table. Does it match your prediction? $x$ $-4$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$ $-2x$ Table $\PageIndex{4}$ - Graph the solutions to $-2x<6$ on the number line: d. How are the solutions to $2x<6$ different from the solutions to $-2x<6$? Exercise $\PageIndex{3}$: Which Side are the Solutions? 1. Let's investigate $-4x+5\geq 25$. - Solve $-4x+5=25$. - Is $-4x+5\geq 25$ true when $x$ is 0? What about when $x$ is $7$? What about when $x$ is $-7$? - Graph the solutions to $-4x+5\geq 25$ on the number line. 2. Let's investigate $\frac{4}{3}x+3<\frac{23}{3}$. - Solve $\frac{4}{3}x+3=\frac{23}{3}$. - Is $\frac{4}{3}x+3<\frac{23}{3}$ true when $x$ is $0$? - Graph the solutions to $\frac{4}{3}x+3<\frac{23}{3}$ on the number line. 3. Solve the inequality $3(x+4)>17.4$ and graph the solutions on the number line. 4. Solve the inequality $-3\left(x-\frac{4}{3}\right)\leq 6$ and graph the solutions on the number line. Are you ready for more? Write at least three different inequalities whose solution is $x>-10$. Find one with $x$ on the left side that uses a $<$. Summary Here is an inequality: $3(10-2x)<18$. The solution to this inequality is all the values you could use in place of $x$ to make the inequality true. In order to solve this, we can first solve the related equation $3(10-2x)=18$ to get the solution $x=2$. That means 2 is the boundary between values of that make the inequality true and values that make the inequality false. To solve the inequality, we can check numbers greater than 2 and less than 2 and see which ones make the inequality true. Let’s check a number that is greater than 2: $x=5$. Replacing $x$ with 5 in the inequality, we get $3(10-2\cdot 5)<18$ or just $0<18$. This is true, so $x=5$ is a solution. This means that all values greater than 2 make the inequality true. We can write the solutions as $x>2$ and also represent the solutions on a number line: Notice that 2 itself is not a solution because it's the value of $x$ that makes $3(10-2x)$ equal to 18, and so it does not make $3(10-2x)<18$ true. For confirmation that we found the correct solution, we can also test a value that is less than 2. If we test $x=0$, we get $3(10-2\cdot 0)<18$ or just $30<18$. This is false, so $x=0$ and all values of $x$ that are less than 2 are not solutions. Glossary Entries Definition: Solution to an Inequality A solution to an inequality is a number that can be used in place of the variable to make the inequality true. For example, 5 is a solution to the inequality $c<10$, because it is true that $5<10$. Some other solutions to this inequality are 9.9, 0, and -4. Practice Exercise $\PageIndex{4}$ - Consider the inequality $-1\leq \frac{x}{2}$. - Predict which values of $x$ will make the inequality true. - Complete the table to check your prediction. $x$ $-4$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$ $\frac{x}{2}$ Table $\PageIndex{5}$ - Consider the inequality $1\leq\frac{-x}{2}$. - Predict which values of $x$ will make it true. - Complete the table to check your prediction. $x$ $-4$ $-3$ $-2$ $-1$ $0$ $1$ $2$ $3$ $4$ $-\frac{x}{2}$ Table $\PageIndex{6}$ Exercise $\PageIndex{5}$ Diego is solving the inequality $100-3x\geq -50$. He solves the equation $100-3x=-50$ and gets $x=50$. What is the solution to the inequality? - $x<50$ - $x\leq 50$ - $x>50$ - $x\geq 50$ Exercise $\PageIndex{6}$ Solve the inequality $-5(x-1)>-40$, and graph the solution on a number line. Exercise $\PageIndex{7}$ Select all values of $x$ that make the inequality $-x+6\geq 10$ true. - $-3.9$ - $4$ - $-4.01$ - $-4$ - $4.01$ - $3.9$ - $0$ - $-7$ (From Unit 6.3.1) Exercise $\PageIndex{8}$ Draw the solution set for each of the following inequalities. 1. $x>7$ 2. $x\geq -4.2$ (From Unit 6.3.1) Exercise $\PageIndex{9}$ The price of a pair of earrings is $22 but Priya buys them on sale for $13.20. - By how much was the price discounted? - What was the percentage of the discount? (From Unit 4.3.3)	libretexts	2025-03-17T19:52:16.025545	2020-04-29T02:45:15	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.3%3A_Efficiently_Solving_Inequalities", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.3.3: Efficiently Solving Inequalities", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.4%3A_Interpreting_Inequalities	6.3.4: Interpreting Inequalities Lesson Let's write inequalities. Exercise $\PageIndex{1}$: Solve Some Inequalities! For each inequality, find the value or values of $x$ that make it true. - $8x+21\leq 56$ - $56<7(7-x)$ Exercise $\PageIndex{2}$: Club Activities Matching Choose the inequality that best matches each given situation. Explain your reasoning. - The Garden Club is planting fruit trees in their school’s garden. There is one large tree that needs 5 pounds of fertilizer. The rest are newly planted trees that need $\frac{1}{2}$ pound fertilizer each. - $25x+5\leq\frac{1}{2}$ - $\frac{1}{2}x+5\leq 25$ - $\frac{1}{2}x+25\leq 5$ - $5x+\frac{1}{2}\leq 25$ - The Chemistry Club is experimenting with different mixtures of water with a certain chemical (sodium polyacrylate) to make fake snow. To make each mixture, the students start with some amount of water, and then add $\frac{1}{7}$ of that amount of the chemical, and then 9 more grams of the chemical. The chemical is expensive, so there can’t be more than a certain number of grams of the chemical in any one mixture.- $\frac{1}{7}x+9\leq 26.25$ - $9x+\frac{1}{7}\leq 26.25$ - $26.25x+9\leq\frac{1}{7}$ - $\frac{1}{7}x+26.25\leq 9$ - The Hiking Club is on a hike down a cliff. They begin at an elevation of 12 feet and descend at the rate of 3 feet per minute. - $37x-3\geq 12$ - $3x-37\geq 12$ - $12-3x\geq -37$ - $12x-37\geq -3$ - The Science Club is researching boiling points. They learn that at high altitudes, water boils at lower temperatures. At sea level, water boils at $212^{\circ}\text{F}$. With each increase of 500 feet in elevation, the boiling point of water is lowered by about $1^{\circ}\text{F}$. - $212-\frac{1}{500}e<195$ - $\frac{1}{500}e-195<212$ - $195-212e<\frac{1}{500}$ - $212-195e<\frac{1}{500}$ Exercise $\PageIndex{3}$: Club Activities Display Your teacher will assign your group one of the situations from the last task. Create a visual display about your situation. In your display: - Explain what the variable and each part of the inequality represent - Write a question that can be answered by the solution to the inequality - Show how you solved the inequality - Explain what the solution means in terms of the situation Are you ready for more? $\left\{ 3,4,5,6\right\}$ is a set of four consecutive integers whose sum is 18. - How many sets of three consecutive integers are there whose sum is between 51 and 60? Can you be sure you’ve found them all? Explain or show your reasoning. - How many sets of four consecutive integers are there whose sum is between 59 and 82? Can you be sure you’ve found them all? Explain or show your reasoning. Summary We can represent and solve many real-world problems with inequalities. Writing the inequalities is very similar to writing equations to represent a situation. The expressions that make up the inequalities are the same as the ones we have seen in earlier lessons for equations. For inequalities, we also have to think about how expressions compare to each other, which one is bigger, and which one is smaller. Can they also be equal? For example, a school fundraiser has a minimum target of $500. Faculty have donated $100 and there are 12 student clubs that are participating with different activities. How much money should each club raise to meet the fundraising goal? If $n$ is the amount of money that each club raises, then the solution to $100+12n=500$ is the minimum amount each club has to raise to meet the goal. It is more realistic, though, to use the inequality $100+12n\geq 500$ since the more money we raise, the more successful the fundraiser will be. There are many solutions because there are many different amounts of money the clubs could raise that would get us above our minimum goal of $500. Glossary Entries Definition: Solution to an Inequality A solution to an inequality is a number that can be used in place of the variable to make the inequality true. For example, 5 is a solution to the inequality $c<10$, because it is true that $5<10$. Some other solutions to this inequality are 9.9, 0, and -4. Practice Exercise $\PageIndex{4}$ Priya looks at the inequality $12-x>5$ and says “I subtract a number from 12 and want a result that is bigger than 5. That means that the solutions should be values of that are smaller than something.” Do you agree with Priya? Explain your reasoning and include solutions to the inequality in your explanation. Exercise $\PageIndex{5}$ When a store had sold $\frac{2}{5}$ of the shirts that were on display, they brought out another 30 from the stockroom. The store likes to keep at least 150 shirts on display. The manager wrote the inequality $\frac{3}{5}x+30\geq 150$ to describe the situation. - Explain what $\frac{3}{5}$ means in the inequality. - Solve the inequality. - Explain what the solution means in the situation. Exercise $\PageIndex{6}$ You know $x$ is a number less than 4. Select all the inequalities that must be true. - $x<2$ - $x+6<10$ - $5x<20$ - $x-2>2$ - $x<8$ (From Unit 6.3.1) Exercise $\PageIndex{7}$ Here is an unbalanced hanger. - If you knew each circle weighed 6 grams, what would that tell you about the weight of each triangle? Explain your reasoning. - If you knew each triangle weighed 3 grams, what would that tell you about the weight of each circle? Explain your reasoning. (From Unit 6.3.1) Exercise $\PageIndex{8}$ Match each sentence with the inequality that could represent the situation. - Han got $2 from Clare, but still has less than $20. - Mai spent $2 and has less than $20. - If Tyler had twice the amount of money he has, he would have less than $20. - If Priya had half the money she has, she would have less than $20. - $x-2<20$ - $2x<20$ - $x+2<20$ - $\frac{1}{2}x<20$ (From Unit 6.3.1) Exercise $\PageIndex{9}$ At a skateboard shop: - The price tag on a shirt says $12.58. Sales tax is 7.5% of the price. How much will you pay for the shirt? - The store buys a helmet for $19.00 and sells it for $31.50. What percentage was the markup? - The shop pays workers $14.25 per hour plus 5.5% commission. If someone works 18 hours and sells $250 worth of merchandise, what is the total amount of their paycheck for this pay period? Explain or show your reasoning. (From Unit 4.3.3)	libretexts	2025-03-17T19:52:16.099918	2020-04-29T02:44:48	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.4%3A_Interpreting_Inequalities", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.3.4: Interpreting Inequalities", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.5%3A_Modeling_with_Inequalities	6.3.5: Modeling with Inequalities Lesson Let's look at solutions to inequalities. Exercise $\PageIndex{1}$: Possible Values The stage manager of the school musical is trying to figure out how many sandwiches he can order with the $83 he collected from the cast and crew. Sandwiches cost $5.99 each, so he lets $x$ represent the number of sandwiches he will order and writes $5.99x\leq 83$. He solves this to 2 decimal places, getting $x\leq 13.86$. Which of these are valid statements about this situation? (Select all that apply.) - He can call the sandwich shop and order exactly 13.86 sandwiches. - He can round up and order 14 sandwiches. - He can order 12 sandwiches. - He can order 9.5 sandwiches. - He can order 2 sandwiches. - He can order -4 sandwiches. Exercise $\PageIndex{2}$: Elevator A mover is loading an elevator with many identical 48-pound boxes. The mover weighs 185 pounds. The elevator can carry at most 2000 pounds. - Write an inequality that says that the mover will not overload the elevator on a particular ride. Check your inequality with your partner. - Solve your inequality and explain what the solution means. - Graph the solution to your inequality on a number line. - If the mover asked, “How many boxes can I load on this elevator at a time?” what would you tell them? Exercise $\PageIndex{3}$: Info Gap: Giving Advice Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner. If your teacher gives you the problem card : - Silently read your card and think about what information you need to be able to answer the question. - Ask your partner for the specific information that you need. - Explain how you are using the information to solve the problem. Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently. - Read the data card and discuss your reasoning. If your teacher gives you the data card : - Silently read your card. - Ask your partner “What specific information do you need?” and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions. - Read the problem card and solve the problem independently. - Share the data card and discuss your reasoning. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner. Are you ready for more? In a day care group, nine babies are five months old and 12 babies are seven months old. How many full months from now will the average age of the 21 babies first surpass 20 months old? Summary We can represent and solve many real-world problems with inequalities. Whenever we write an inequality, it is important to decide what quantity we are representing with a variable. After we make that decision, we can connect the quantities in the situation to write an expression, and finally, the whole inequality. As we are solving the inequality or equation to answer a question, it is important to keep the meaning of each quantity in mind. This helps us to decide if the final answer makes sense in the context of the situation. For example: Han has 50 centimeters of wire and wants to make a square picture frame with a loop to hang it that uses 3 centimeters for the loop. This situation can be represented by $3+4s=50$, where $s$ is the length of each side (if we want to use all the wire). We can also use $3+4s\leq 50$ if we want to allow for solutions that don’t use all the wire. In this case, any positive number that is less or equal to 11.75 cm is a solution to the inequality. Each solution represents a possible side length for the picture frame since Han can bend the wire at any point. In other situations, the variable may represent a quantity that increases by whole numbers, such as with numbers of magazines, loads of laundry, or students. In those cases, only whole-number solutions make sense. Glossary Entries Definition: Solution to an Inequality A solution to an inequality is a number that can be used in place of the variable to make the inequality true. For example, 5 is a solution to the inequality $c<10$, because it is true that $5<10$. Some other solutions to this inequality are 9.9, 0, and -4. Practice Exercise $\PageIndex{4}$ 28 students travel on a field trip. They bring a van that can seat 12 students. Elena and Kiran’s teacher asks other adults to drive cars that seat 3 children each to transport the rest of the students. Elena wonders if she should use the inequality $12+3n>28$ or $12+3n\geq 28$ to figure out how many cars are needed. Kiran doesn’t think it matters in this case. Do you agree with Kiran? Explain your reasoning. Exercise $\PageIndex{5}$ - In the cafeteria, there is one large 10-seat table and many smaller 4-seat tables. There are enough tables to fit 200 students. Write an inequality whose solution is the possible number of 4-seat tables in the cafeteria. - 5 barrels catch rainwater in the schoolyard. Four barrels are the same size, and the fifth barrel holds 10 liters of water. Combined, the 5 barrels can hold at least 200 liters of water. Write an inequality whose solution is the possible size of each of the 4 barrels. - How are these two problems similar? How are they different? Exercise $\PageIndex{6}$ Solve each equation. - $5(n-4)=-60$ - $-3t+-8=25$ - $7p-8=-22$ - $\frac{2}{5}(j+40)=-4$ - $4(w+1)=-6$ (From Unit 6.2.3) Exercise $\PageIndex{7}$ Select all the inequalities that have the same graph as $x<4$. - $x<2$ - $x+6<10$ - $5x<20$ - $x-2>2$ - $x<8$ (From Unit 6.3.1) Exercise $\PageIndex{8}$ A 200 pound person weighs 33 pounds on the Moon. - How much did the person’s weight decrease? - By what percentage did the person’s weight decrease? (From Unit 4.3.3)	libretexts	2025-03-17T19:52:16.172425	2020-04-29T02:44:22	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.03%3A_New_Page/6.3.5%3A_Modeling_with_Inequalities", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.3.5: Modeling with Inequalities", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page	6.4: Writing Equivalent Expressions Last updated Save as PDF Page ID 35015 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:16.244771	2020-01-25T01:41:40	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.4: Writing Equivalent Expressions", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.1%3A_Subtraction_in_Equivalent_Expressions	6.4.1: Subtraction in Equivalent Expressions Lesson Let's find ways to work with subtraction in expressions. Exercise $\PageIndex{1}$: Number Talk: Additive Inverses Find each sum or difference mentally. $-30+-10$ $-10+-30$ $-30-10$ $10--30$ Exercise $\PageIndex{2}$: A Helpful Observation Lin and Kiran are trying to calculate $7\frac{3}{4}+3\frac{5}{6}-1\frac{3}{4}$. Here is their conversation: Lin: “I plan to first add $7\frac{3}{4}$ and $3\frac{5}{6}$, so I will have to start by finding equivalent fractions with a common denominator.” Kiran: “It would be a lot easier if we could start by working with the $1\frac{3}{4}$ and $7\frac{3}{4}$. Can we rewrite it like $7\frac{3}{4}+1\frac{3}{4}-3\frac{5}{6}$?” Lin: “You can’t switch the order of numbers in a subtraction problem like you can with addition; $2-3$ is not equal to $3-2$.” Kiran: “That’s true, but do you remember what we learned about rewriting subtraction expressions using addition? $2-3$ is equal to $2+(-3)$.” - Write an expression that is equivalent to that uses addition instead of subtraction. - If you wrote the terms of your new expression in a different order, would it still be equivalent? Explain your reasoning. Exercise $\PageIndex{3}$: Organizing Work 1. Write two expressions for the area of the big rectangle. 2. Use the distributive property to write an expression that is equivalent to $\frac{1}{2}(8y+-x+-12)$. The boxes can help you organize your work. 3. Use the distributive property to write an expression that is equivalent to $\frac{1}{2}(8y-x-12)$. Are you ready for more? Here is a calendar for April 2017. Let's choose a date: the 10th. Look at the numbers above, below, and to either side of the 10th: 3, 17, 9, 11. - Average these four numbers. What do you notice? - Choose a different date that is in a location where it has a date above, below, and to either side. Average these four numbers. What do you notice? - Explain why the same thing will happen for any date in a location where it has a date above, below, and to either side. Summary Working with subtraction and signed numbers can sometimes get tricky. We can apply what we know about the relationship between addition and subtraction—that subtracting a number gives the same result as adding its opposite—to our work with expressions. Then, we can make use of the properties of addition that allow us to add and group in any order. This can make calculations simpler. For example: $\frac{5}{8}-\frac{2}{3}-\frac{1}{8}$ $\frac{5}{8}+-\frac{2}{3}+-\frac{1}{8}$ $\frac{5}{8}+-\frac{1}{8}+-\frac{2}{3}$ $\frac{4}{8}+-\frac{2}{3}$ We can also organize the work of multiplying signed numbers in expressions. The product $\frac{3}{2}(6y-2x-8)$ can be found by drawing a rectangle with the first factor, $\frac{3}{2}$, on one side, and the three terms inside the parentheses on the other side: Multiply $\frac{3}{2}$ by each term across the top and perform the multiplications: Reassemble the parts to get the expanded version of the original expression: $\frac{3}{2}(6y-2x-8)=9y-3x-12$ Glossary Entries Definition: Term A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression $5x+18$ has two terms. The first term is $5x$ and the second term is 18. Practice Exercise $\PageIndex{4}$ For each expression, write an equivalent expression that uses only addition. - $20-9+8-7$ - $4x-7y-5x+6$ - $-3x-8y-4-\frac{8}{7}z$ Exercise $\PageIndex{5}$ Use the distributive property to write an expression that is equivalent to each expression. If you get stuck, consider drawing boxes to help organize your work. - $9(4x-3y-\frac{2}{3})$ - $-2(-6x+3y-1)$ - $\frac{1}{5}(20y-4x-13)$ - $8(-x-\frac{1}{2})$ - $-8(-x-\frac{3}{4}y+\frac{7}{2})$ Exercise $\PageIndex{6}$ Kiran wrote the expression $x-10$ for this number puzzle: “Pick a number, add -2, and multiply by 5.” Lin thinks Kiran made a mistake. - How can she convince Kiran he made a mistake? - What would be a correct expression for this number puzzle? Exercise $\PageIndex{7}$ The output from a coal power plant is shown in the table: \| energy in megawatts \| number of days \| \|---\|---\| \| $1,200$ \| $2.4$ \| \| $1,800$ \| $3.6$ \| \| $4,000$ \| $8$ \| \| $10,000$ \| $20$ \| Similarly, the output from a solar power plant is shown in the table: \| energy in megawatts \| number of days \| \|---\|---\| \| $100$ \| $1$ \| \| $650$ \| $4$ \| \| $1,200$ \| $7$ \| \| $1,750$ \| $10$ \| Based on the tables, is the energy output in proportion to the number of days for either plant? If so, write an equation showing the relationship. If not, explain your reasoning. (From Unit 2.3.1)	libretexts	2025-03-17T19:52:16.319581	2020-04-29T02:49:02	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.1%3A_Subtraction_in_Equivalent_Expressions", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.4.1: Subtraction in Equivalent Expressions", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.2%3A_Expanding_and_Factoring	6.4.2: Expanding and Factoring Lesson Let's use the distributive property to write expressions in different ways. Exercise $\PageIndex{1}$: Number Talk: Parentheses Find the value of each expression mentally. $2+3\cdot 4$ $(2+3)(4)$ $2-3\cdot 4$ $2-(3+4)$ Exercise $\PageIndex{2}$: Factoring and Expanding with Negative Numbers In each row, write the equivalent expression. If you get stuck, use a diagram to organize your work. The first row is provided as an example. Diagrams are provided for the first three rows. \| factored \| expanded \| \|---\|---\| \| $-3(5-2y)$ \| $-15+6y$ \| \| $5(a-6)$ \| \| \| $6a-2b$ \| \| \| $-4(2w-5z)$ \| \| \| $-2(2x-3y)$ \| \| \| $20x-10y+15z$ \| \| \| $k(4-17)$ \| \| \| $10a-13a$ \| \| \| $-2x(3y-z)$ \| \| \| $ab-bc-3bd$ \| \| \| $-x(3y-z+4w)$ \| Are you ready for more? Expand to create an equivalent expression that uses the fewest number of terms: $((((x+1)\frac{1}{2})+1)\frac{1}{2})+1$. If we wrote a new expression following the same pattern so that there were 20 sets of parentheses, how could it be expanded into an equivalent expression that uses the fewest number of terms? Summary We can use properties of operations in different ways to rewrite expressions and create equivalent expressions. We have already seen that we can use the distributive property to expand an expression, for example $3(x+5)=3x+15$. We can also use the distributive property in the other direction and factor an expression, for example $8x+12=4(2x+3)$. We can organize the work of using distributive property to rewrite the expression $12x-8$. In this case we know the product and need to find the factors. The terms of the product go inside: We look at the expressions and think about a factor they have in common. $12x$ and $-8$ each have a factor of 4. We place the common factor on one side of the large rectangle: Now we think: "4 times what is 12$x$?" "4 times what is -8?" and write the other factors on the other side of the rectangle: So, $12x-8$ is equivalent to $4(3x-2)$. Glossary Entries Definition: Expand To expand an expression, we use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression. For example, we can expand the expression $5(4x+7)$ to get the equivalent expression $20x+25$. Definition: Factor (an expression) To factor an expression, we use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression. For example, we can factor the expression $20x+35$ to get the equivalent expression $5(4x+7)$. Definition: Term A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression $5x+18$ has two terms. The first term is $5x$ and the second term is 18. Practice Exercise $\PageIndex{3}$ - Expand to write an equivalent expression: $\frac{-1}{4}(-8x+12y)$ - Factor to write an equivalent expression: $36a-16$ Exercise $\PageIndex{4}$ Lin missed math class on the day they worked on expanding and factoring. Kiran is helping Lin catch up. - Lin understands that expanding is using the distributive property, but she doesn’t understand what factoring is or why it works. How can Kiran explain factoring to Lin? - Lin asks Kiran how the diagrams with boxes help with factoring. What should Kiran tell Lin about the boxes? - Lin asks Kiran to help her factor the expression $-4xy-12xz+20xw$. How can Kiran use this example to Lin understand factoring? Exercise $\PageIndex{5}$ Complete the equation with numbers that makes the expression on the right side of the equal sign equivalent to the expression on the left side. $75a+25b=\underline{\quad}(\underline{\quad}a+b)$ Exercise $\PageIndex{6}$ Elena makes her favorite shade of purple paint by mixing 3 cups of blue paint, $1\frac{1}{2}$ cups of red paint, and $\frac{1}{2}$ of a cup of white paint. Elena has $\frac{2}{3}$ of a cup of white paint. - Assuming she has enough red paint and blue paint, how much purple paint can Elena make? - How much blue paint and red paint will Elena need to use with the $\frac{2}{3}$ of a cup of white paint? (From Unit 4.1.3) Exercise $\PageIndex{7}$ Solve each equation. - $\frac{-1}{8}d-4=\frac{-3}{8}$ - $\frac{-1}{4}m+5=16$ - $10b+-45=-43$ - $-8(y-1.25)=4$ - $3.2(s+10)=32$ (From Unit 6.2.3) Exercise $\PageIndex{8}$ Select all the inequalities that have the same solutions as $-4x<20$. - $-x<5$ - $4x>-20$ - $$4x$ - $$x$ - $x>5$ - $x>-5$ (From Unit 6.3.1)	libretexts	2025-03-17T19:52:16.394569	2020-04-29T02:48:32	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.2%3A_Expanding_and_Factoring", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.4.2: Expanding and Factoring", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.3%3A_Combining_Like_Terms_(Part_1)	6.4.3: Combining Like Terms (Part 1) Lesson Let's see how we can tell that expressions are equivalent. Exercise $\PageIndex{1}$: Why is it True? Explain why each statement is true. - $5+2+3=5+(2+3)$ - $9a$ is equivalent to $11a-2a$. - $7a+4-2a$ is equivalent to $7a+-2a+4$. - $8a-(8a-8)$ is equivalent to $8$. Exercise $\PageIndex{2}$: A's and B's Diego and Jada are both trying to write an expression with fewer terms that is equivalent to $7a+5b-3a+4b$ - Jada thinks $10a+1b$ is equivalent to the original expression. - Diego thinks $4a+9b$ is equivalent to the original expression. - We can show expressions are equivalent by writing out all the variables. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it. $7a+5b-3a+4b$ $(a+a+a+a+a+a+a) + (b+b+b+b+b) - (a+a+a)+(b+b+b+b)$ $(a+a+a+a) + (a+a+a)+(b+b+b+b+b)-(a+a+a)+(b+b+b+b)$ $(a+a+a+a)+(b+b+b+b+b)+(a+a+a)-(a+a+a)+(b+b+b+b)$ $(a+a+a+a)+(b+b+b+b+b)+(b+b+b+b)$ $(a+a+a+a)+(b+b+b+b+b+b+b+b+b)$ $4a+9b$ - Here is another way we can rewrite the expressions. Explain why the expression on each row (after the first row) is equivalent to the expression on the row before it. $7a+5b-3a+4b$ $7a+5b+(-3a)+4b$ $ya+(-3a)+5b+4b$ $(7+-3)a+(5+4)b$ $4a+9b$ Are you ready for more? Follow the instructions for a number puzzle: - Take the number formed by the first 3 digits of your phone number and multiply it by 40 - Add 1 to the result - Multiply by 500 - Add the number formed by the last 4 digits of your phone number, and then add it again - Subtract 500 - Multiply by $\frac{1}{2}$ - What is the final number? - How does this number puzzle work? - Can you invent a new number puzzle that gives a surprising result? Exercise $\PageIndex{3}$: Making Sides Equal Replace each ? with an expression that will make the left side of the equation equivalent to the right side. Set A - $6x+?=10x$ - $6x+?=2x$ - $6x+?=-10x$ - $6x+?=0$ - $6x+?=10$ Check your results with your partner and resolve any disagreements. Next move on to Set B. Set B - $6x-?=2x$ - $6x-?=10x$ - $6x-?=x$ - $6x-?=6$ - $6x-?=4x-10$ Summary There are many ways to write equivalent expressions that may look very different from each other. We have several tools to find out if two expressions are equivalent. - Two expressions are definitely not equivalent if they have different values when we substitute the same number for the variable. For example, $2(-3+x)+8$ and $2x+5$ are not equivalent because when $x$ is 1, the first expression equals 4 and the second expression equals 7. - If two expressions are equal for many different values we substitute for the variable, then the expressions may be equivalent, but we don't know for sure. It is impossible to compare the two expressions for all values. To know for sure, we use properties of operations. For example, $2(-3+x)+8$ is equivalent to $2x+2$ because: $\begin{array}{ccl}{2(-3+x)+8}&{\qquad}&{\qquad} \\ {-6+2x+8}&{\qquad}&{\text{by the distributive property}} \\ {2x+-6+8}&{\qquad}&{\text{by the commutative property}} \\ {2x+(-6+8)}&{\qquad}&{\text{by the associative property}} \\ {2x+2}&{\qquad}&{\qquad}\end{array}$ Glossary Entries Definition: Expand To expand an expression, we use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression. For example, we can expand the expression $5(4x+7)$ to get the equivalent expression $20x+25$. Definition: Factor (an expression) To factor an expression, we use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression. For example, we can factor the expression $20x+35$ to get the equivalent expression $5(4x+7)$. Definition: Term A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression $5x+18$ has two terms. The first term is $5x$ and the second term is 18. Practice Exercise $\PageIndex{4}$ Andre says that $10x+6$ and $5x+11$ are equivalent because they both equal 16 when $x$ is 1. Do you agree with Andre? Explain your reasoning. Exercise $\PageIndex{5}$ Select all expressions that can be subtracted from $9x$ to result in the expression $3x+5$. - $-5+6x$ - $5-6x$ - $6x+5$ - $6x-5$ - $-6x+5$ Exercise $\PageIndex{6}$ Select all the statements that are true for any value of $x$. - $7x+(2x+7)=9x+7$ - $7x+(2x-1)=9x+1$ - $\frac{1}{2}x+(3-\frac{1}{2}x)=3$ - $5x-(8-6x)=-x-8$ - $0.4x-(0.2x+8)=0.2x-8$ - $6x-(2x-4)=4x+4$ Exercise $\PageIndex{7}$ For each situation, would you describe it with $x<25$, $x>25$, $x\leq 25$, or $x\geq 25$? - The library is having a party for any student who read at least 25 books over the summer. Priya read $x$ books and was invited to the party. - Kiran read $x$ books over the summer but was not invited to the party. 4. (From Unit 6.3.1) Exercise $\PageIndex{8}$ Consider the problem: A water bucket is being filled with water from a water faucet at a constant rate. When will the bucket be full? What information would you need to be able to solve the problem? (From Unit 2.3.3)	libretexts	2025-03-17T19:52:16.468918	2020-04-29T02:48:05	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.3%3A_Combining_Like_Terms_(Part_1)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.4.3: Combining Like Terms (Part 1)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.4%3A_Combining_Like_Terms_(Part_2)	6.4.4: Combining Like Terms (Part 2) Lesson Let's see how to use properties correctly to write equivalent expressions. Exercise $\PageIndex{1}$: True or False? Select all the statements that are true. Be prepared to explain your reasoning. - $4-2(3+7)=4-2\cdot 3-2\cdot 7$ - $4-2(3+7)=4-2\cdot 3+-2\cdot 7$ - $4-2(3+7)=4-2\cdot 3+2\cdot 7$ - $4-2(3+7)=4-(2\cdot 3+2\cdot 7)$ Exercise $\PageIndex{2}$: Seeing it Differently Some students are trying to write an expression with fewer terms that is equivalent to $8-3(4-9x)$ Noah says, “I worked the problem from left to right and ended up with $20-45x$." $8-3(4-9x)$ $5(4-9x)$ $20-45x$ Lin says, "I started inside the parentheses and ended up with $23x$." $8-3(4-9x)$ $8-3(-5x)$ $8+15x$ $23x$ Jada says, “I used the distributive property and ended up with $27x-4$.” $8-3(4-9x)$ $8-(12-27x)$ $8-12-(-27x)$ $27x-4$ Andre says, “I also used the distributive property, but I ended up with $-4-27x$." $8-3(4-9x)$ $8-12-27x$ $-4-27x$ - Do you agree with any of them? Explain your reasoning. - For each strategy that you disagree with, find and describe the errors. Are you ready for more? - Jada’s neighbor said, “My age is the difference between twice my age in 4 years and twice my age 4 years ago.” How old is Jada’s neighbor? - Another neighbor said, “My age is the difference between twice my age in 5 years and and twice my age 5 years ago.” How old is this neighbor? - A third neighbor had the same claim for 17 years from now and 17 years ago, and a fourth for 21 years. Determine those neighbors’ ages. Exercise $\PageIndex{3}$: Grouping Differently Diego was taking a math quiz. There was a question on the quiz that had the expression $8x-9-12x+5$. Diego’s teacher told the class there was a typo and the expression was supposed to have one set of parentheses in it. - Where could you put parentheses in $8x-9-12x+5$ to make a new expression that is still equivalent to the original expression? How do you know that your new expression is equivalent? - Where could you put parentheses in $8x-9-12x+5$ to make a new expression that is not equivalent to the original expression? List as many different answers as you can. Summary Combining like terms allows us to write expressions more simply with fewer terms. But it can sometimes be tricky with long expressions, parentheses, and negatives. It is helpful to think about some common errors that we can be aware of and try to avoid: - $6x-x$ is not equivalent to 6. While it might be tempting to think that subtracting $x$ makes the $x$ disappear, the expression is really saying take $1x$ away from $6x$'s, and the distributive property tells us that $6x-x$ is equivalent to $(6-1)x$. - $7-2x$ is not equivalent to $5x$. The expression $7-2x$ tells us to double an unknown amount and subtract it from 7. This is not always the same as taking 5 copies of the unknown. - $7-4(x+2)$ is not equivalent to $3(x+2)$. The expression tells us to subtract 4 copies of an amount from 7, not to take $(7-4)$ copies of the amount. If we think about the meaning and properties of operations when we take steps to rewrite expressions, we can be sure we are getting equivalent expressions and are not changing their value in the process. Glossary Entries Definition: Expand To expand an expression, we use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression. For example, we can expand the expression $5(4x+7)$ to get the equivalent expression $20x+35$. Definition: Factor (an expression) To factor an expression, we use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression. For example, we can factor the expression $20x+35$ to get the equivalent expression $5(4x+7)$. Definition: Term A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression $5x+18$ has two terms. The first term is $5x$ and the second term is 18. Practice Exercise $\PageIndex{4}$ - Noah says that $9x-2x+4x$ is equivalent to $3x$, because the subtraction sign tells us to subtract everything that comes after $9x$. - Elena says that $9x-2x+4x$ is equivalent to $11x$, because the subtraction only applies to $2x$. Do you agree with either of them? Explain your reasoning. Exercise $\PageIndex{5}$ Identify the error in generating an expression equivalent to $4+2x-\frac{1}{2}(10-4x)$. Then correct the error. $4+2x+\frac{-1}{2}(10+-4x)$ $4+2x+-5+2x$ $4+2x-5+2x$ $-1$ Exercise $\PageIndex{6}$ Select all expressions that are equivalent to $5x-15-20x+10$. - $5x-(15+20x)+10$ - $5x+-15+-20x+10$ - $5(x-3-4x+2)$ - $-5(-x+3+4x+-2)$ - $-15x-5$ - $-5(3x+1)$ - $-15(x-\frac{1}{3})$ Exercise $\PageIndex{7}$ The school marching band has a budget of up to $750 to cover 15 new uniforms and competition fees that total $300. How much can they spend for one uniform? - Write an inequality to represent this situation. - Solve the inequality and describe what it means in the situation. (From Unit 6.3.2) Exercise $\PageIndex{8}$ Solve the inequality that represents each story. Then interpret what the solution means in the story. - For every $9 that Elena earns, she gives $x$ dollars to charity. This happens 7 times this month. Elena wants to be sure she keeps at least $42 from this month’s earnings. $7(9-x)\geq 42$ - Lin buys a candle that is 9 inches tall and burns down $x$ inches per minute. She wants to let the candle burn for 7 minutes until it is less than 6 inches tall. $9-7x<6$ (From Unit 6.3.4) Exercise $\PageIndex{9}$ A certain shade of blue paint is made by mixing $1\frac{1}{2}$ quarts of blue paint with 5 quarts of white paint. If you need a total of 16.25 gallons of this shade of blue paint, how much of each color should you mix? (From Unit 4.1.3)	libretexts	2025-03-17T19:52:16.543466	2020-04-29T02:47:38	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.4%3A_Combining_Like_Terms_(Part_2)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.4.4: Combining Like Terms (Part 2)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.5%3A_Combining_Like_Terms_(Part_3)	6.4.5: Combining Like Terms (Part 3) Lesson Let's see how we can combine terms in an expression to write it with less terms. Exercise $\PageIndex{1}$: Are They Equal? Select all expressions that are equal to $8-12-(6+4)$. - $8-6-12+4$ - $8-12-6-4$ - $8-12+(6+4)$ - $8-12-6+4$ - $8-4-12-6$ Exercise $\PageIndex{2}$: X's and Y's Match each expression in column A with an equivalent expression from column B. Be prepared to explain your reasoning. A - $(9x+5y)+(3x+7y)$ - $(9x+5y)-(3x+7y)$ - $(9x+5y)-(3x-7y)$ - $9x-7y+3x+5y$ - $9x-7y+3x-5y$ - $9x7y-3x-5y$ B - $12(x+y)$ - $12(x-y)$ - $6(x-2y)$ - $9x+5y+3x-7y$ - $9x+5y-3x+7y$ - $9x-3x+5y-7y$ Exercise $\PageIndex{3}$: Seeing Structure and Factoring Write each expression with fewer terms. Show or explain your reasoning. - $3\cdot 15+4\cdot 15-5\cdot 15$ - $3x+4x-5x$ - $3(x-2)+4(x-2)-5(x-2)$ - $3(\frac{5}{2}x+6\frac{1}{2})+4(\frac{5}{2}x+5\frac{1}{2})-5(\frac{5}{2}x+6\frac{1}{2})$ Summary Combining like terms is a useful strategy that we will see again and again in our future work with mathematical expressions. It is helpful to review the things we have learned about this important concept. - Combining like terms is an application of the distributive property. For example: $\begin{array}{c}{2x+9x}\\{(2+9)\cdot x} \\ {11x}\end{array}$ - It often also involves the commutative and associative properties to change the order or grouping of addition. For example: $\begin{array}{c}{2a+3b+4a+5b}\\{2a+4a+3b+5b}\\{(2a+4a)+(3b+5b)}\\{6a+8b}\end{array}$ - We can't change order or grouping when subtracting; so in order to apply the commutative or associative properties to expressions with subtraction, we need to rewrite subtraction as addition. For example: $\begin{array}{c}{2a-3b-4a-5b}\\{2a+-3b+-4a+-5b}\\{2a+-4a+-3b+-5b}\\{-2a+-8b}\\{-2a-8b}\end{array}$ - Since combining like terms uses properties of operations, it results in expressions that are equivalent. - The like terms that are combined do not have to be a single number or variable; they may be longer expressions as well. Terms can be combined in any sum where there is a common factor in all the terms. For example, each term in the expression $5(x+3)-0.5(x+3)+2(x+3)$ has a factor of $(x+3)$. We can rewrite the expression with fewer terms by using the distributive property: $\begin{array}{c}{5(x+3)-0.5(x+3)+2(x+3)}\\{(5-0.5+2)(x+3)}\\{6.5(x+3)}\end{array}$ Glossary Entries Definition: Expand To expand an expression, we use the distributive property to rewrite a product as a sum. The new expression is equivalent to the original expression. For example, we can expand the expression $5(4x+7)$ to get the equivalent expression $20x+35$. Definition: Factor (an expression) To factor an expression, we use the distributive property to rewrite a sum as a product. The new expression is equivalent to the original expression. For example, we can factor the expression $20x+35$ to get the equivalent expression $5(4x+7)$. Definition: Term A term is a part of an expression. It can be a single number, a variable, or a number and a variable that are multiplied together. For example, the expression $5x+18$ has two terms. The first term is $5x$ and the second term is 18. Practice Exercise $\PageIndex{4}$ Jada says, “I can tell that $\frac{-2}{3}(x+5)+4(x=5)-\frac{10}{3}(x+5)$ equals 0 just by looking at it.” Is Jada correct? Explain how you know. Exercise $\PageIndex{5}$ In each row, decide whether the expression in column A is equivalent to the expression in column B. If they are not equivalent, show how to change one expression to make them equivalent. A - $3x-2x+0.5x$ - $3(x+4)-2(x+4)$ - $6(x+4)-2(x+5)$ - $3(x+4)-2(x+4)+0.5(x+4)$ - $20(\frac{2}{5}x+\frac{3}{4}y-\frac{1}{2})$ B - $1.5x$ - $x+3$ - $2(2x+7)$ - $2(2x+7)$ - $1.5$ - $\frac{1}{2}(16x+30y-20)$ Exercise $\PageIndex{6}$ For each situation, write an expression for the new balance using as few terms as possible. - A checking account has a balance of -$126.89. A customer makes two deposits, one $3\frac{1}{2}$ times the other, and then withdraws $25. - A checking account has a balance of $350. A customer makes two withdrawals, one $50 more than the other. Then he makes a deposit of $75. (From Unit 6.4.3) Exercise $\PageIndex{7}$ Tyler is using the distributive property on the expression $9-4(5x-6)$. Here is his work: $9-4(5x-6)$ $9+(-4)(5x+-6)$ $9+-20x+-6$ $3-20x$ Mai thinks Tyler’s answer is incorrect. She says, “If expressions are equivalent then they are equal for any value of the variable. Why don’t you try to substitute the same value for $x$ in all the equations and see where they are not equal?” - Find the step where Tyler made an error. - Explain what he did wrong. - Correct Tyler's work. (From Unit 6.4.4) Exercise $\PageIndex{8}$ - If $(11+x)$ is positive, but $(4+x)$ is negative, what is one number that $x$ could be? - If $(-3+y)$ is positive, but $(-9+y)$ is negative, what is one number that $y$ could be? - If $(-5+z)$ is positive, but $(-6+z)$ is negative, what is one number that $z$ could be? (From Unit 6.3.1)	libretexts	2025-03-17T19:52:16.618717	2020-04-29T02:47:11	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.04%3A_New_Page/6.4.5%3A_Combining_Like_Terms_(Part_3)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.4.5: Combining Like Terms (Part 3)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.05%3A_New_Page	6.5: Let's Put it to Work Last updated Save as PDF Page ID 35016 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:16.689371	2020-01-25T01:41:40	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.05%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.5: Let's Put it to Work", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.05%3A_New_Page/6.5.1%3A_Applications_of_Expressions	6.5.1: Applications of Expressions Lesson Let's use expressions to solve problems. Exercise $\PageIndex{1}$: Algebra Talk: Equivalent to $0.75t-21$ Decide whether each expression is equivalent to $0.75t-21$. Be prepared to explain how you know. $\frac{3}{4}t-21$ $\frac{3}{4}(t-21)$ $0.75(t-28)$ $t-0.25t-21$ Exercise $\PageIndex{2}$: Two Ways to Calculate Usually when you want to calculate something, there is more than one way to do it. For one or more of these situations, show how the two different ways of calculating are equivalent to each other. - Estimating the temperature in Fahrenheit when you know the temperature in Celsius - Double the temperature in Celsius, then add 30. - Add 15 to the temperature in Celsius, then double the result. - Calculating a 15% tip on a restaurant bill - Take 10% of the bill amount, take 5% of the bill amount, and add those two values together. - Multiply the bill amount by 3, divide the result by 2, and then take $\frac{1}{10}$ of that result. - Changing a distance in miles to a distance in kilometers - Take the number of miles, double it, then decrease the result by 20%. - Divide the number of miles by 5, then multiply the result by 8. Exercise $\PageIndex{3}$: Which Way? You have two coupons to the same store: one for 20% off and one for $30 off. The cashier will let you use them both, and will let you decide in which order to use them. - Mai says that it doesn’t matter in which order you use them. You will get the same discount either way. - Jada says that you should apply the 20% off coupon first, and then the $30 off coupon. - Han says that you should apply the $30 off coupon first, and then the 20% off coupon. - Kiran says that it depends on how much you are spending. Do you agree with any of them? Explain your reasoning.	libretexts	2025-03-17T19:52:16.748912	2020-04-29T02:49:40	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/06%3A_Untitled_Chapter_6/6.05%3A_New_Page/6.5.1%3A_Applications_of_Expressions", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "6.5.1: Applications of Expressions", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7	7: Angles, Triangles, and Prisms Last updated Save as PDF Page ID 35018 Illustrative Mathematics OpenUp Resources 7.1: Angle Relationships 7.1.1: Relationships of Angles 7.1.2: Adjacent Angles 7.1.3: Nonadjacent Angles 7.1.4: Solving for Unknown Angles 7.1.5: Using Equations to Solve for Unknown Angles 7.2: Drawing Polygons with Given Conditions 7.2.1: Building Polygons (Part 1) 7.2.2: Building Polygons (Part 2) 7.2.3: Triangles with 3 Common Measures 7.2.4: Drawing Triangles (Part 1) 7.2.5: Drawing Triangles (Part 2) 7.3: Solid Geometry 7.3.1: Slicing Solids 7.3.2: Volume of Right Prisms 7.3.3: Decomposing Bases for Area 7.3.4: Surface Area of Right Prisms 7.3.5: Distinguishing Volume and Surface Area 7.3.6: Applying Volume and Surface Area 7.4: Let's Put It to work 7.4.1: Building Prisms	libretexts	2025-03-17T19:52:16.833725	2020-01-25T01:41:42	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7: Angles, Triangles, and Prisms", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page	7.1: Angle Relationships Last updated Save as PDF Page ID 35019 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:16.979633	2020-01-25T01:41:43	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.1: Angle Relationships", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.1%3A_Relationships_of_Angles	7.1.1: Relationships of Angles Lesson Let's examine some special angles. Exercise $\PageIndex{1}$: Visualizing Angles Use the applet to answer the questions. - Which angle is bigger, $a$ or $b$? - Identify an obtuse angle in the diagram. Exercise $\PageIndex{2}$: Pattern Block Angles 1. Look at the different pattern blocks inside the applet. Each block contains either 1 or 2 angles with different degree measures. Which blocks have only 1 unique angle? Which have 2? 2. If you place three copies of the hexagon together so that one vertex from each hexagon touches the same point, as shown, they fit together without any gaps or overlaps. Use this to figure out the degree measure of the angle inside the hexagon pattern block. 3. Figure out the degree measure of all of the other angles inside the pattern blocks. (Hint: turn on the grid to help align the pieces.) Be prepared to explain your reasoning. Are you ready for more? We saw that it is possible to fit three copies of a regular hexagon snugly around a point. Each interior angle of a regular pentagon measures $108^{\circ}$. Is it possible to fit copies of a regular pentagon snugly around a point? If yes, how many copies does it take? If not, why not? Exercise $\PageIndex{3}$: More Pattern Block Angles 1. Use pattern blocks to determine the measure of each of these angles. 2. If an angle has a measure of $180^{\circ}$ then the two legs form a straight line. An angle that forms a straight line is called a straight angle. Find as many different combinations of pattern blocks as you can that make a straight angle. Use the applet if you choose. (Hint: turn on the grid to help align the pieces.) Exercise $\PageIndex{4}$: Measuring Like This or That Tyler and Priya were both measuring angle $TUS$. Priya thinks the angle measures 40 degrees. Tyler thinks the angle measures 140 degrees. Do you agree with either of them? Explain your reasoning. Summary When two lines intersect and form four equal angles, we call each one a right angle . A right angle measures $90^{\circ}$. You can think of a right angle as a quarter turn in one direction or the other. An angle in which the two sides form a straight line is called a straight angle . A straight angle measures $180^{\circ}$. A straight angle can be made by putting right angles together. You can think of a straight angle as a half turn, so that you are facing in the opposite direction after you are done. If you put two straight angles together, you get an angle that is $360^{\circ}$. You can think of this angle as turning all the way around so that you are facing the same direction as when you started the turn. When two angles share a side and a vertex, and they don't overlap, we call them adjacent angles . Glossary Entries Definition: Adjacent Angles Adjacent angles share a side and a vertex. In this diagram, angle $ABC$ is adjacent to angle $DBC$. Definition: Right Angle A right angle is half of a straight angle. It measures 90 degrees. Definition: Straight Angle A straight angle is an angle that forms a straight line. It measures 180 degrees. Practice Exercise $\PageIndex{5}$ Here are questions about two types of angles. - Draw a right angle. How do you know it's a right angle? What is its measure in degrees? - Draw a straight angle. How do you know it’s a straight angle? What is its measure in degrees? Exercise $\PageIndex{6}$ An equilateral triangle’s angles each have a measure of 60 degrees. - Can you put copies of an equilateral triangle together to form a straight angle? Explain or show your reasoning. - Can you put copies of an equilateral triangle together to form a right angle? Explain or show your reasoning. Exercise $\PageIndex{7}$ Here is a square and some regular octagons. In this pattern, all of the angles inside the octagons have the same measure. The shape in the center is a square. Find the measure of one of the angles inside one of the octagons. Exercise $\PageIndex{8}$ The height of the water in a tank decreases by 3.5 cm each day. When the tank is full, the water is 10 m deep. The water tank needs to be refilled when the water height drops below 4 m. - Write a question that could be answered by solving the equation $10-0.035d=4$. - Is 100 a solution of $10-0.035d>4$? Write a question that solving this problem could answer. (From Unit 6.3.5) Exercise $\PageIndex{9}$ Use the distributive property to write an expression that is equivalent to each given expression. - $-3(2x-4)$ - $0.1(-90+50a)$ - $-7(-x-9)$ - $\frac{4}{5}(10y+-x+-15)$ (From Unit 6.4.1) Exercise $\PageIndex{10}$ Lin’s puppy is gaining weight at a rate of 0.125 pounds per day. Describe the weight gain in days per pound. (From Unit 2.1.3)	libretexts	2025-03-17T19:52:17.051863	2020-05-04T00:18:49	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.1%3A_Relationships_of_Angles", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.1.1: Relationships of Angles", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.2%3A_Adjacent_Angles	7.1.2: Adjacent Angles Lesson Let's look at some special pairs of angles. Exercise $\PageIndex{1}$: Estimating Angle Measures Estimate the degree measure of each indicated angle. Exercise $\PageIndex{2}$: Cutting Rectangles Your teacher will give you two small, rectangular papers. 1. On one of the papers, draw a small half-circle in the middle of one side. 2. Cut a straight line, starting from the center of the half-circle, all the way across the paper to make 2 separate pieces. (Your cut does not need to be perpendicular to the side of the paper.) 3. On each of these two pieces, measure the angle that is marked by part of a circle. Label the angle measure on the piece. 4. What do you notice about these angle measures? 5. Clare measured 70 degrees on one of her pieces. Predict the angle measure of her other piece. 6. On the other rectangular paper, draw a small quarter-circle in one of the corners. 7. Repeat the previous steps to cut, measure, and label the two angles marked by part of a circle. 8. What do you notice about these angle measures? 9. Priya measured 53 degrees on one of her pieces. Predict the angle measure of her other piece. Exercise $\PageIndex{3}$: Is It a Complement or Supplement? 1. Use the protractor in the picture to find the measure of angles $BCA$ and $BCD$. 2. Explain how to find the measure of angle $ACD$ without repositioning the protractor. 3. Use the protractor in the picture to find the measure of angles $LOK$ and $LOM$. 4. Explain how to find the measure of angle $KOM$ without repositioning the protractor. 5. Angle $BAC$ s a right angle. Find the measure of angle $CAD$. 6. Point $O$ is on line $RS$. Find the measure of angle $SOP$. Are you ready for more? Clare started with a rectangular piece of paper. She folded up one corner, and then folded up the other corner, as shown in the photos. - Try this yourself with any rectangular paper. Fold the left corner up at any angle, and then fold the right corner up so that the edges of the paper meet. - Clare thought that the angle at the bottom looked like a 90 degree angle. Does yours also look like it is 90 degrees? - Can you explain why the bottom angle always has to be 90 degrees? Hint: the third photo shows Clare’s paper, unfolded. The crease marks have dashed lines, and the line where the two paper edges met have a solid line. Mark these on your own paper as well. Summary If two angle measures add up to $90^{\circ}$, then we say the angles are complementary . Here are three examples of pairs of complementary angles. If two angle measures add up to $180^{\circ}$, then we say the angles are supplementary . Here are three examples of pairs of supplementary angles. Glossary Entries Definition: Adjacent Angles Adjacent angles share a side and a vertex. In this diagram, angle $ABC$ is adjacent to angle $DBC$. Definition: Complementary Complementary angles have measures that add up to 90 degrees. For example, a $15^{\circ}$ angle and a $75^{\circ}$ angle are complementary. Definition: Right Angle A right angle is half of a straight angle. It measures 90 degrees. Definition: Straight Angle A straight angle is an angle that forms a straight line. It measures 180 degrees. Definition: Supplementary Supplementary angles have measures that add up to 180 degrees. For example, a $15^{\circ}$ angle and a $165^{\circ}$ angle are supplementary. Practice Exercise $\PageIndex{4}$ Angles $A$ and $C$ are supplementary. Find the measure of angle $C$. Exercise $\PageIndex{5}$ - List two pairs of angles in square $CDFG$ that are complementary. - Name three angles that sum to $180^{\circ}$. Exercise $\PageIndex{6}$ Complete the equation with a number that makes the expression on the right side of the equal sign equivalent to the expression on the left side. $5x-2.5+6x-3=\underline{\quad}(2x-1)$ (From Unit 6.4.5) Exercise $\PageIndex{7}$ Match each table with the equation that represents the same proportional relationship. A: \| $x$ \| $y$ \| \|---\|---\| \| $2$ \| $8$ \| \| $3$ \| $12$ \| \| $4$ \| $16$ \| \| $5$ \| $20$ \| - $y=1.5x$ - $y=1.25x$ - $y=4x$ B: \| $x$ \| $y$ \| \|---\|---\| \| $3$ \| $4.5$ \| \| $6$ \| $9$ \| \| $7$ \| $10.5$ \| \| $10$ \| $15$ \| C: \| $x$ \| $y$ \| \|---\|---\| \| $2$ \| $\frac{5}{2}$ \| \| $4$ \| $5$ \| \| $6$ \| $\frac{15}{2}$ \| \| $12$ \| $15$ \| (From Unit 2.2.1)	libretexts	2025-03-17T19:52:17.133056	2020-05-04T00:18:24	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.2%3A_Adjacent_Angles", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.1.2: Adjacent Angles", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.3%3A_Nonadjacent_Angles	7.1.3: Nonadjacent Angles Lesson Let's look at angles that are not right next to one another. Exercise $\PageIndex{1}$: Finding Related Statements Given $a$ and $b$ are numbers, and $a+b=180$, which statements also must be true? $a=180-b\qquad a-180=b\qquad 360=2a+2b\qquad a=90\text{ and }b=90$ Exercise $\PageIndex{2}$: Polygon Angles Use any useful tools in the geometry toolkit to identify any pairs of angles in these figures that are complementary or supplementary. Exercise $\PageIndex{3}$: Vertical Angles Use a straightedge to draw two intersecting lines. Use a protractor to measure all four angles whose vertex is located at the intersection. Compare your drawing and measurements to the people in your group. Make a conjecture about the relationships between angle measures at an intersection. Exercise $\PageIndex{4}$: Row Game: Angles Find the measure of the angles in one column. Your partner will work on the other column. Check in with your partner after you finish each row. Your answers in each row should be the same. If your answers aren’t the same, work together to find the error and correct it. column A $P$ is on line $m$. Find the value of $a$. Find the value of $a$. column B Find the value of $b$. In right triangle $LMN$, angles $L$ and $M$ are complementary. Find the measure of angle $L$. column A Angle $C$ and angle $E$ are supplementary. Find the measure of angle $E$. Find the value of $c$. Two angles are complementary. One angle measures 37 degrees. Find the measure of the other angle. column B $X$ is on line $WY$. Find the value of $b$. $B$ is on line $FW$. Find the measure of angle $CBW$. Two angles are supplementary. One angle measures 127 degrees. Find the measure of the other angle. Summary When two lines cross, they form two pairs of vertical angles . Vertical angles are across the intersection point from each other. Vertical angles always have equal measure. We can see this because they are always supplementary with the same angle. For example: This is always true! $a+b=180$ so $a=180-b$. $c+b=180$ so $c=180-b$. That means $a=c$. Glossary Entries Definition: Adjacent Angles Adjacent angles share a side and a vertex. In this diagram, angle $ABC$ is adjacent to angle $DBC$. Definition: Complementary Complementary angles have measures that add up to 90 degrees. For example, a $15^{\circ}$ angle and a $75^{\circ}$ angle are complementary. Definition: Right Angle A right angle is half of a straight angle. It measures 90 degrees. Definition: Straight Angle A straight angle is an angle that forms a straight line. It measures 180 degrees. Definition: Supplementary Supplementary angles have measures that add up to 180 degrees. For example, a $15^{\circ}$ angle and a $165^{\circ}$ angle are supplementary. Definition: Vertical Angles Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal. For example, angles $AEC$ and $DEB$ are vertical angles. If angle $AEC$ measures $120^{\circ}$, then angle $DEB$ must also measure $120^{\circ}$. Angles $AED$ and $BEC$ are another pair of vertical angles. Practice Exercise $\PageIndex{5}$ Two lines intersect. Find the value of $b$ and $c$. Exercise $\PageIndex{6}$ In this figure, angles $R$ and $S$ are complementary. Find the measure of angle $S$. Exercise $\PageIndex{7}$ If two angles are both vertical and supplementary, can we determine the angles? Is it possible to be both vertical and complementary? If so, can you determine the angles? Explain how you know. Exercise $\PageIndex{8}$ Match each expression in the first list with an equivalent expression from the second list. - $5(x+1)-2x+11$ - $2x+2+x+5$ - $\frac{-3}{8}x-9+\frac{5}{8}x+1$ - $2.06x-5.53+4.98-9.02$ - $99x+44$ - $\frac{1}{4}x-8$ - $\frac{1}{2}(6x+14)$ - $11(9x+4)$ - $3x+16$ - $2.06x+(-5.53)+4.98+(-9.02)$ (From Unit 6.4.5) Exercise $\PageIndex{9}$ Factor each expression. - $15a-13a$ - $-6x-18y$ - $36abc+54ad$ (From Unit 6.4.2) Exercise $\PageIndex{10}$ The directors of a dance show expect many students to participate but don’t yet know how many students will come. The directors need 7 students to work on the technical crew. The rest of the students work on dance routines in groups of 9. For the show to work, they need at least 6 full groups working on dance routines. - Write and solve an inequality to represent this situation, and graph the solution on a number line. - Write a sentence to the directors about the number of students they need. (From Unit 6.3.5) Exercise $\PageIndex{11}$ A small dog gets fed $\frac{3}{4}$ cup of dog food twice a day. Using $d$ for the number of days and $f$ for the amount of food in cups, write an equation relating the variables. Use the equation to find how many days a large bag of dog food will last if it contains 210 cups of food. (From Unit 2.2.2)	libretexts	2025-03-17T19:52:17.215655	2020-05-04T00:18:01	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.3%3A_Nonadjacent_Angles", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.1.3: Nonadjacent Angles", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.4%3A_Solving_for_Unknown_Angles	7.1.4: Solving for Unknown Angles Lesson Let's figure out some missing angles. Exercise $\PageIndex{1}$: True or False: Length Relationships Here are some line segments. Decide if each of these equations is true or false. Be prepared to explain your reasoning. $CD+BC=BD$ $AB+BD=CD+AD$ $AC-AB=AB$ $BD-CD=AC-AB$ Exercise $\PageIndex{2}$: Info Gap: ANgle Finding Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner. If your teacher gives you the problem card : - Silently read your card and think about what information you need to be able to answer the question. - Ask your partner for the specific information that you need. - Explain how you are using the information to solve the problem. Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently. - Read the data card and discuss your reasoning. If your teacher gives you the data card : - Silently read your card. - Ask your partner “What specific information do you need?” and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions. - Read the problem card and solve the problem independently. - Share the data card and discuss your reasoning. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner. Exercise $\PageIndex{3}$: What's the Match? Match each figure to an equation that represents what is seen in the figure. For each match, explain how you know they are a match. - $g+h=180$ - $g=h$ - $2h+g=90$ - $g+h+48=180$ - $g+h+35=180$ Are you ready for more? - What is the angle between the hour and minute hands of a clock at 3:00? - You might think that the angle between the hour and minute hands at 2:20 is 60 degrees, but it is not! The hour hand has moved beyond the 2. Calculate the angle between the clock hands at 2:20. - Find a time where the hour and minute hand are 40 degrees apart. (Assume that the time has a whole number of minutes.) Is there just one answer? Summary We can write equations that represent relationships between angles. - The first pair of angles are supplementary, so $x+42=180$. - The second pair of angles are vertical angles, so $y=28$. - Assuming the third pair of angles form a right angle, they are complementary, so $z+64=90$. Glossary Entries Definition: Adjacent Angles Adjacent angles share a side and a vertex. In this diagram, angle $ABC$ is adjacent to angle $DBC$. Definition: Complementary Complementary angles have measures that add up to 90 degrees. For example, a $15^{\circ}$ angle and a $75^{\circ}$ angle are complementary. Definition: Right Angle A right angle is half of a straight angle. It measures 90 degrees. Definition: Straight Angle A straight angle is an angle that forms a straight line. It measures 180 degrees. Definition: Supplementary Supplementary angles have measures that add up to 180 degrees. For example, a $15^{\circ}$ angle and a $165^{\circ}$ angle are supplementary. Definition: Vertical Angles Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal. For example, angles $AEC$ and $DEB$ are vertical angles. If angle $AEC$ measures $120^{\circ}$, then angle $DEB$ must also measure $120^{\circ}$. Angles $AED$ and $BEC$ are another pair of vertical angles. Practice Exercise $\PageIndex{4}$ $M$ is a point on line segment $KL$. $NM$ is a line segment. Select all the equations that represent the relationship between the measures of the angles in the figure. - $a=b$ - $a+b=90$ - $b=90-a$ - $a+b=180$ - $180-a=b$ - $180=b-a$ Exercise $\PageIndex{5}$ Which equation represents the relationship between the angles in the figure? - $88+b=90$ - $88+b=180$ - $2b+88=90$ - $2b+88=180$ Exercise $\PageIndex{6}$ Segments $AB$, $EF$, and $CD$ intersect at point $C$, and angle $ACD$ is a right angle. Find the value of $g$. Exercise $\PageIndex{7}$ Select all the expressions that are the result of decreasing $x$ by 80%. - $\frac{20}{100}x$ - $x-\frac{80}{100}x$ - $\frac{100-20}{100}x$ - $0.80x$ - $(1-0.8)x$ (From Unit 6.2.6) Exercise $\PageIndex{8}$ Andre is solving the equation $4(x+\frac{3}{2})=7$. He says, “I can subtract $\frac{3}{2}$ from each side to get $4x=\frac{11}{2}$ and then divide by 4 to get $x=\frac{11}{8}$.” Kiran says, “I think you made a mistake.” - How can Kiran know for sure that Andre’s solution is incorrect? - Describe Andre’s error and explain how to correct his work. (From Unit 6.2.2) Exercise $\PageIndex{9}$ Solve each equation. $\begin{array}{lllll}{\frac{1}{7}a+\frac{3}{4}=\frac{9}{8}}&{\qquad}&{\frac{2}{3}+\frac{1}{5}b=\frac{5}{6}}&{\qquad}&{\frac{3}{2}=\frac{4}{3}c+\frac{2}{3}}\\{0.3d+7.9=9.1}&{\qquad}&{11.03=8.78+0.02e}&{\qquad}&{\qquad}\end{array}$ (From Unit 6.2.1) Exercise $\PageIndex{10}$ A train travels at a constant speed for a long distance. Write the two constants of proportionality for the relationship between distance traveled and elapsed time. Explain what each of them means. \| time elapsed (hr) \| distance (mi) \| \|---\|---\| \| $1.2$ \| $54$ \| \| $3$ \| $135$ \| \| $4$ \| $180$ \| (From Unit 2.2.2)	libretexts	2025-03-17T19:52:17.301003	2020-05-04T00:17:36	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.4%3A_Solving_for_Unknown_Angles", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.1.4: Solving for Unknown Angles", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.5%3A_Using_Equations_to_Solve_for_Unknown_Angles	7.1.5: Using Equations to Solve for Unknown Angles Lesson Let's figure out missing angles using equations. Exercise $\PageIndex{1}$: Is this Enough? Tyler thinks that this figure has enough information to figure out the values of $a$ and $b$. Do you agree? Explain your reasoning. Exercise $\PageIndex{2}$: What Does It Look Like? Elena and Diego each wrote equations to represent these diagrams. For each diagram, decide which equation you agree with, and solve it. You can assume that angles that look like right angles are indeed right angles. 1. Elena: $x=35$ Diego: $x+35=180$ 2. Elena: $35+w+41=180$ Diego: $w+35=180$ 3. Elena: $w+35=90$ Diego: $2w+35=90$ 4. Elena: $2w+35=90$ Diego: $w+35=90$ 5. Elena: $w+148=180$ Diego: $x+90=148$ Exercise $\PageIndex{3}$: Calculate the Measure Find the unknown angle measures. Show your thinking. Organize it so it can be followed by others. Lines $l$ and $m$ are perpendicular. Are you ready for more? The diagram contains three squares. Three additional segments have been drawn that connect corners of the squares. We want to find the exact value of $a+b+c$. - Use a protractor to measure the three angles. Use your measurements to conjecture about the value of $a+b+c$. - Find the exact value of $a+b+c$ by reasoning about the diagram. Summary To find an unknown angle measure, sometimes it is helpful to write and solve an equation that represents the situation. For example, suppose we want to know the value of $x$ in this diagram. Using what we know about vertical angles, we can write the equation $3x+90=144$ to represent this situation. Then we can solve the equation. $\begin{aligned} 3x+90&=144 \\ 3x+90-90&=144-90 \\ 3x&=54 \\ 3x\cdot\frac{1}{3}&=54\cdot\frac{1}{3} \\ x&=18\end{aligned}$ Glossary Entries Definition: Adjacent Angles Adjacent angles share a side and a vertex. In this diagram, angle $ABC$ is adjacent to angle $DBC$. Definition: Complementary Complementary angles have measures that add up to 90 degrees. For example, a $15^{\circ}$ angle and a $75^{\circ}$ angle are complementary. Definition: Right Angle A right angle is half of a straight angle. It measures 90 degrees. Definition: Straight Angle A straight angle is an angle that forms a straight line. It measures 180 degrees. Definition: Supplementary Supplementary angles have measures that add up to 180 degrees. For example, a $15^{\circ}$ angle and a $165^{\circ}$ angle are supplementary. Definition: Vertical Angles Vertical angles are opposite angles that share the same vertex. They are formed by a pair of intersecting lines. Their angle measures are equal. For example, angles $AEC$ and $DEB$ are vertical angles. If angle $AEC$ measures $120^{\circ}$, then angle $DEB$ must also measure $120^{\circ}$. Angles $AED$ and $BEC$ are another pair of vertical angles. Practice Exercise $\PageIndex{4}$ Segments $AB$, $DC$, and $EC$ intersect at point $C$. Angle $DCE$ measures $148^{\circ}$. Find the value of $x$. Exercise $\PageIndex{5}$ Line $l$ is perpendicular to line $m$. Find the value of $x$ and $w$. Exercise $\PageIndex{6}$ If you knew that two angles were complementary and were given the measure of one of those angles, would you be able to find the measure of the other angle? Explain your reasoning. Exercise $\PageIndex{7}$ For each inequality, decide whether the solution is represented by $x<4.5$ or $x>4.5$. - $-24>-6(x-0.5)$ - $-8x+6>-30$ - $-2(x+3.2)<-15.4$ (From Unit 6.3.3) Exercise $\PageIndex{8}$ A runner ran $\frac{2}{3}$ of a 5 kilometer race in 21 minutes. They ran the entire race at a constant speed. - How long did it take to run the entire race? - How many minutes did it take to run 1 kilometer? (From Unit 4.1.2) Exercise $\PageIndex{9}$ Jada, Elena, and Lin walked a total of 37 miles last week. Jada walked 4 more miles than Elena, and Lin walked 2 more miles than Jada. The diagram represents this situation: Find the number of miles that they each walked. Explain or show your reasoning. (From Unit 6.2.6) Exercise $\PageIndex{10}$ Select all the expressions that are equivalent to $-36x+54y-90$. - $-9(4x-6y-10)$ - $-18(2x-3y+5)$ - $-6(6x+9y-15)$ - $18(-2x+3y-5)$ - $-2(18x-27y+45)$ - $2(-18x+54y-90)$ (From Unit 6.4.2)	libretexts	2025-03-17T19:52:17.378987	2020-05-04T00:17:11	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.01%3A_New_Page/7.1.5%3A_Using_Equations_to_Solve_for_Unknown_Angles", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.1.5: Using Equations to Solve for Unknown Angles", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page	7.2: Drawing Polygons with Given Conditions Last updated Save as PDF Page ID 35020 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:17.454240	2020-01-25T01:41:44	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.2: Drawing Polygons with Given Conditions", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.1%3A_Building_Polygons_(Part_1)	7.2.1: Building Polygons (Part 1) Lesson Let's build shapes. Exercise $\PageIndex{1}$: True or False: Signed Numbers Decide whether each equation is true or false. Be prepared to explain your reasoning. $4\cdot (-6)=(-6)+(-6)+(-6)+(-6)$ $-8\cdot 4=(-8\cdot 3)+4$ $6\cdot (-7)=7\cdot (-7)+7$ $-10-6=-10-(-6)$ Exercise $\PageIndex{2}$: What Can You build? - Use the segments in the applet to build several polygons, including at least one triangle and one quadrilateral. - After you finish building several polygons, select one triangle and one quadrilateral that you have made. - Measure all the angles in the two shapes you selected. Note: select points in order counterclockwise, like a protractor. - Using these measurements along with the side lengths as marked, draw your triangle and quadrilateral as accurately as possible on separate paper. Exercise $\PageIndex{3}$: Building Diego's and Jada's Shapes - Diego built a quadrilateral using side lengths of 4 in, 5 in, 6 in, and 9 in. - Build such a shape. - Is your shape an identical copy of Diego’s shape? Explain your reasoning. - Jada built a triangle using side lengths of 4 in, 5 in, and 8 in. - Build such a shape. - Is your shape an identical copy of Jada’s shape? Explain your reasoning. Exercise $\PageIndex{4}$: Building Han's Shape Han built a polygon using side lengths of 3 in, 4 in, and 9 in. - Build such a shape. - What do you notice? Summary Sometimes we are given a polygon and asked to find the lengths of the sides. What options do you have if you need to build a polygon with some side lengths? Sometimes, we can make lots of different figures. For example, if you have side lengths 5, 7, 11, and 14, here are some of the many, many quadrilaterals we can make with those side lengths: Sometimes, it is not possible to make a figure with certain side lengths. For example, 18, 1, 1, 1 (try it!). We will continue to investigate the figures that can be made with given measures. Practice Exercise $\PageIndex{5}$ A rectangle has side lengths of 6 units and 3 units. Could you make a quadrilateral that is not identical using the same four side lengths? If so, describe it. Exercise $\PageIndex{6}$ Come up with an example of three side lengths that can not possibly make a triangle, and explain how you know. Exercise $\PageIndex{7}$ Find $x$, $y$, and $z$. (From Unit 7.1.3) Exercise $\PageIndex{8}$ How many right angles need to be put together to make: - 360 degrees? - 180 degrees? - 270 degrees? - A straight angle? (From Unit 7.1.1) Exercise $\PageIndex{9}$ Solve each equation. $\begin{array}{lll}{\frac{1}{7}(x+\frac{3}{4})=\frac{1}{8}}&{\qquad}&{\frac{9}{2}=\frac{3}{4}(z+\frac{2}{3})}\\{1.5=0.6(w+0.4)}&{\qquad}&{0.08(7.97+v)=0.832}\end{array}$ (From Unit 6.2.2) Exercise $\PageIndex{10}$ - You can buy 4 bottles of water from a vending machine for $7. At this rate, how many bottles of water can you buy for $28? If you get stuck, consider creating a table. - It costs $20 to buy 5 sandwiches from a vending machine. At this rate, what is the cost for 8 sandwiches? If you get stuck, consider creating a table. (From Unit 4.1.3)	libretexts	2025-03-17T19:52:17.521881	2020-05-04T00:22:05	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.1%3A_Building_Polygons_(Part_1)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.2.1: Building Polygons (Part 1)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.2%3A_Building_Polygons_(Part_2)	7.2.2: Building Polygons (Part 2) Lesson Let's build more triangles. Exercise $\PageIndex{1}$: Where is Lin? At a park, the slide is 5 meters east of the swings. Lin is standing 3 meters away from the slide. - Draw a diagram of the situation including a place where Lin could be. - How far away from the swings is Lin in your diagram? - Where are some other places Lin could be? Exercise $\PageIndex{2}$: How Long is the Third Side? Use the applet to answer the questions. - Build as many different triangles as you can that have one side length of 5 inches and one of 4 inches. Record the side lengths of each triangle you build. - Are there any other lengths that could be used for the third side of the triangle but aren't values of the sliders? - Are there any lengths that are values of the sliders but could not be used as the third side of the triangle? Are you ready for more? Assuming you had access to strips of any length, and you used the 9-inch and 5-inch strips as the first two sides, complete the sentences: - The third side can't be _____ inches or longer. - The third side can't be _____ inches or shorter. Exercise $\PageIndex{3}$: Swinging the Sides Around We'll explore a method for drawing a triangle that has three specific side lengths. Use the applet to answer the questions. - Follow these instructions to mark the possible endpoints of one side: - For now, ignore segment $AC$, the 3-inch side length on the left side Figure $\PageIndex{1}$ - Let segment $BD$ be the 3-unit side length on the right side. Right-click on point $D$, check Trace On. Rotate the point, drawing all the places where a 3-inch side could end. - For now, ignore segment $AC$, the 3-inch side length on the left side - What shape have you drawn while moving $BD$ around? Why? Which tool in your geometry toolkit can do something similar? - Use your drawing to create two unique triangles, each with a base of length 4 inches and a side of length 3 inches. Use a different color to draw each triangle. - Repeat the previous instructions, letting segment $AC$ be the 3-unit side length. - Using a third color, draw a point where the two traces intersect. Using this third color, draw a triangle with side lengths of 4 inches, 3 inches, and 3 inches. Summary If we want to build a polygon with two given side lengths that share a vertex, we can think of them as being connected by a hinge that can be opened or closed: All of the possible positions of the endpoint of the moving side form a circle: You may have noticed that sometimes it is not possible to build a polygon given a set of lengths. For example, if we have one really, really long segment and a bunch of short segments, we may not be able to connect them all up. Here's what happens if you try to make a triangle with side lengths 21, 4, and 2: The short sides don't seem like they can meet up because they are too far away from each other. If we draw circles of radius 4 and 2 on the endpoints of the side of length 21 to represent positions for the shorter sides, we can see that there are no places for the short sides that would allow them to meet up and form a triangle. In general, the longest side length must be less than the sum of the other two side lengths. If not, we can’t make a triangle! If we can make a triangle with three given side lengths, it turns out that the measures of the corresponding angles will always be the same. For example, if two triangles have side lengths 3, 4, and 5, they will have the same corresponding angle measures. Practice Exercise $\PageIndex{4}$ In the diagram, the length of segment $AB$ is 10 units and the radius of the circle centered at $A$ is 4 units. Use this to create two unique triangles, each with a side of length 10 and a side of length 4. Label the sides that have length 10 and 4. Exercise $\PageIndex{5}$ Select all the sets of three side lengths that will make a triangle. - $3, 4, 8$ - $7, 6, 12$ - $5, 11, 13$ - $4, 6, 12$ - $4, 6, 10$ Exercise $\PageIndex{6}$ Based on signal strength, a person knows their lost phone is exactly 47 feet from the nearest cell tower. The person is currently standing 23 feet from the same cell tower. What is the closest the phone could be to the person? What is the furthest their phone could be from them? Exercise $\PageIndex{7}$ Each row contains the degree measures of two complementary angles. Complete the table. \| measure of an angle \| measure of its complement \| \|---\|---\| \| $80^{\circ}$ \| \| \| $25^{\circ}$ \| \| \| $54^{\circ}$ \| \| \| $x$ \| (From Unit 7.1.2) Exercise $\PageIndex{8}$ Here are two patterns made using identical rhombuses. Without using a protractor, determine the value of $a$ and $b$. Explain or show your reasoning. (From Unit 7.1.1) Exercise $\PageIndex{9}$ Mai’s family is traveling in a car at a constant speed of 65 miles per hour. - At that speed, how long will it take them to travel 200 miles? - How far do they travel in 25 minutes? (From Unit 4.1.3)	libretexts	2025-03-17T19:52:17.593594	2020-05-04T00:21:35	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.2%3A_Building_Polygons_(Part_2)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.2.2: Building Polygons (Part 2)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.3%3A_Triangles_with_3_Common_Measures	7.2.3: Triangles with 3 Common Measures Lesson Let's contrast triangles. Exercise $\PageIndex{1}$: 3 Sides; 3 Angles Examine each set of triangles. What do you notice? What is the same about the triangles in the set? What is different? Set 1: Set 2: Exercise $\PageIndex{2}$: 2 Sides and 1 Angle Examine this set of triangles. - What is the same about the triangles in the set? What is different? - How many different triangles are there? Explain or show your reasoning. Exercise $\PageIndex{3}$: 2 Angles and 1 Side Examine this set of triangles. - What is the same about the triangles in the set? What is different? - How many different triangles are there? Explain or show your reasoning. Summary Both of these quadrilaterals have a right angle and side lengths 4 and 5. However, in one case, the right angle is between the two given side lengths; in the other, it is not. If we create two triangles with three equal measures, but these measures are not next to each other in the same order, that usually means the triangles are different. Here is an example: Practice Exercise $\PageIndex{4}$ Are these two triangles identical? Explain how you know. Exercise $\PageIndex{5}$ Are these triangles identical? Explain your reasoning. Exercise $\PageIndex{6}$ Tyler claims that if two triangles each have a side length of 11 units and a side length of 8 units, and also an angle measuring $100^{\circ}$, they must be identical to each other. Do you agree? Explain your reasoning. Exercise $\PageIndex{7}$ The markings on the number line are equally spaced. Label the other markings on the number line. (From Unit 5.3.1) Exercise $\PageIndex{8}$ A passenger on a ship dropped his camera into the ocean. If it is descending at a rate of -4.2 meters per second, how long until it hits the bottom of the ocean, which is at -1,875 meters? (From Unit 5.3.2) Exercise $\PageIndex{9}$ Apples cost $1.99 per pound. - How much do $3\frac{1}{4}$ pounds of apples cost? - How much do $x$ pounds of apples cost? - Clare spent $5.17 on apples. How many pounds of apples did Clare buy? (From Unit 4.1.3) Exercise $\PageIndex{10}$ Diego has a glue stick with a diameter of 0.7 inches. He sets it down 3.5 inches away from the edge of the table, but it rolls onto the floor. How many rotations did the glue stick make before it fell off of the table? (From Unit 3.1.5)	libretexts	2025-03-17T19:52:17.659345	2020-05-04T00:21:07	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.3%3A_Triangles_with_3_Common_Measures", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.2.3: Triangles with 3 Common Measures", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.4%3A_Drawing_Triangles_(Part_1)	7.2.4: Drawing Triangles (Part 1) Lesson Let's see how many different triangles we can draw with certain measurements. Exercise $\PageIndex{1}$: Which One Doesn't Belong: Triangles Which one doesn't belong? Exercise $\PageIndex{2}$: Does Your Triangle Match Theirs? Three students have each drawn a triangle. For each description: - Drag the vertices to create a triangle with the given measurements. - Make note of the different side lengths and angle measures in your triangle. - Decide whether the triangle you made must be an identical copy of the triangle that the student drew. Explain your reasoning. Jada’s triangle has one angle measuring 75°. Andre’s triangle has one angle measuring 75° and one angle measuring 45°. Lin’s triangle has one angle measuring 75°, one angle measuring 45°, and one side measuring 5 cm. Exercise $\PageIndex{3}$: How Many Can You Draw? - Draw as many different triangles as you can with each of these sets of measurements: - Two angles measure $60^{\circ}$, and one side measures 4 cm. - Two angles measure $90^{\circ}$, and one side measures 4 cm. - One angle measures $60^{\circ}$, one angle measures $90^{\circ}$, and one side measures 4 cm. - Which sets of measurements determine one unique triangle? Explain or show your reasoning. Are you ready for more? In the diagram, 9 toothpicks are used to make three equilateral triangles. Figure out a way to move only 3 of the toothpicks so that the diagram has exactly 5 equilateral triangles. Summary Sometimes, we are given two different angle measures and a side length, and it is impossible to draw a triangle. For example, there is no triangle with side length 2 and angle measures $120^{\circ}$ and $100^{\circ}$: Sometimes, we are given two different angle measures and a side length between them, and we can draw a unique triangle. For example, if we draw a triangle with a side length of 4 between angles $90^{\circ}$ and $60^{\circ}$, there is only one way they can meet up and complete to a triangle: Any triangle drawn with these three conditions will be identical to the one above, with the same side lengths and same angle measures. Practice Exercise $\PageIndex{4}$ Use a protractor to try to draw each triangle. Which of these three triangles is impossible to draw? - A triangle where one angle measures $20^{\circ}$ and another angle measures $45^{\circ}$ - A triangle where one angle measures $120^{\circ}$ and another angle measures $50^{\circ}$ - A triangle where one angle measures $90^{\circ}$ and another angle measures $100^{\circ}$ Exercise $\PageIndex{5}$ A triangle has an angle measuring $90^{\circ}$, an angle measuring $20^{\circ}$, and a side that is 6 units long. The 6-unit side is in between the $90^{\circ}$ and $20^{\circ}$ angles. - Sketch this triangle and label your sketch with the given measures. - How many unique triangles can you draw like this? Exercise $\PageIndex{6}$ - Find a value for $x$ that makes $-x$ less than $2x$. - Find a value for $x$ that makes $-x$ greater than $2x$. (From Unit 5.4.1) Exercise $\PageIndex{7}$ One of the particles in atoms is called an electron. It has a charge of -1. Another particle in atoms is a proton. It has charge of +1. The overall charge of an atom is the sum of the charges of the electrons and the protons. Here is a list of common elements. \| charge from electrons \| charge from protons \| overall charge \| \| \|---\|---\|---\|---\| \| carbon \| $-6$ \| $+6$ \| $0$ \| \| aluminum \| $-10$ \| $+13$ \| \| \| phosphide \| $-18$ \| $+15$ \| \| \| iodide \| $-54$ \| $+53$ \| \| \| tin \| $-50$ \| $+50$ \| Find the overall charge for the rest of the atoms on the list. (From Unit 5.2.2) Exercise $\PageIndex{8}$ A factory produces 3 bottles of sparkling water for every 7 bottles of plain water. If those are the only two products they produce, what percentage of their production is sparkling water? What percentage is plain? (From Unit 4.1.3)	libretexts	2025-03-17T19:52:17.730819	2020-05-04T00:20:30	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.4%3A_Drawing_Triangles_(Part_1)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.2.4: Drawing Triangles (Part 1)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.5%3A_Drawing_Triangles_(Part_2)	7.2.5: Drawing Triangles (Part 2) Lesson Let's draw some more triangles. Exercise $\PageIndex{1}$: Using a Compass to Estimate Length - Draw a $40^{\circ}$ angle. - Use a compass to make sure both sides of your angle have a length of 5 centimeters. - If you connect the ends of the sides you drew to make a triangle, is the third side longer or shorter than 5 centimeters? How can you use a compass to explain your answer? Exercise $\PageIndex{2}$: Revisiting How Many Can You Draw? Use the applet to draw triangles. - Draw as many different triangles as you can with each of these sets of measurements: - One angle measures $40^{\circ}$, one side measures 4 cm and one side measures 5 cm. - Two sides measure 6 cm and one angle measures $100^{\circ}$. - Did either of these sets of measurements determine one unique triangle? How do you know? Exercise $\PageIndex{3}$: Three Angles Use the applet to draw triangles. Sides can overlap. - Draw as many different triangles as you can with each of these sets of measurements: - One angle measures $50^{\circ}$, one measures $60^{\circ}$, and one measures $70^{\circ}$. - One angle measures $50^{\circ}$, one measures $60^{\circ}$, and one measures $100^{\circ}$. - Did either of these sets of measurements determine one unique triangle? How do you know? Are you ready for more? Using only the point, segment, and compass tools provided, create an equilateral triangle. You are only successful if the triangle remains equilateral while dragging its vertices around. GeoGebra Applet DsB2VFYv Summary A triangle has six measures: three side lengths and three angle measures. If we are given three measures, then sometimes, there is no triangle that can be made. For example, there is no triangle with side lengths 1, 2, 5, and there is no triangle with all three angles measuring $150^{\circ}$. Sometimes, only one triangle can be made. By this we mean that any triangle we make will be the same, having the same six measures. For example, if a triangle can be made with three given side lengths, then the corresponding angles will have the same measures. Another example is shown here: an angle measuring $45^{\circ}$ between two side lengths of 6 and 8 units. With this information, one unique triangle can be made. Sometimes, two or more different triangles can be made with three given measures. For example, here are two different triangles that can be made with an angle measuring $45^{\circ}$ and side lengths 6 and 8. Notice the angle is not between the given sides. Three pieces of information about a triangle’s side lengths and angle measures may determine no triangles, one unique triangle, or more than one triangle. It depends on the information. Practice Exercise $\PageIndex{4}$ Three pieces of information about a triangle’s side lengths and angle measures may determine no triangles, one unique triangle, or more than one triangle. It depends on the information. Exercise $\PageIndex{5}$ A triangle has one side that is 5 units long and an adjacent angle that measures $25^{\circ}$. The two other angles in the triangle measure $90^{\circ}$ and $65^{\circ}$. Complete the two diagrams to create two different triangles with these measurements. Exercise $\PageIndex{6}$ Is it possible to make a triangle that has angles measuring 90 degrees, 30 degrees, and 100 degrees? If so, draw an example. If not, explain your reasoning. Exercise $\PageIndex{7}$ Segments $CD$, $AB$, and $FG$ intersect at point $E$. Angle $FEC$ is a right angle. Identify any pairs of angles that are complementary. (From Unit 7.1.2) Exercise $\PageIndex{8}$ Match each equation to a step that will help solve the equation for $x$. - $3x=-4$ - $-4.5=x-3$ - $3=\frac{-x}{3}$ - $\frac{1}{3}=-3x$ - $x-\frac{1}{3}=0.4$ - $3+x=8$ - $\frac{x}{3}=15$ - $7=\frac{1}{3}+x$ - Add $\frac{1}{3}$ to each side. - Add $\frac{-1}{3}$ to each side. - Add $3$ to each side. - Add $-3$ to each side. - Multiply each side by $3$. - Multiply each side by $-3$. - Multiply each side by $\frac{1}{3}$. - Multiply each side by $\frac{-1}{3}$. (From Unit 5.5.1) Exercise $\PageIndex{9}$ - If you deposit $300 in an account with a 6% interest rate, how much will be in your account after 1 year? - If you leave this money in the account, how much will be in your account after 2 years? (From Unit 4.2.3)	libretexts	2025-03-17T19:52:17.799654	2020-05-04T00:20:04	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.02%3A_New_Page/7.2.5%3A_Drawing_Triangles_(Part_2)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.2.5: Drawing Triangles (Part 2)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page	Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset + - Margin Size Reset + - Font Type Enable Dyslexic Font Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 7: Angles, Triangles, and Prisms Pre-Algebra I (Illustrative Mathematics - 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Grade 7) 7: Angles, Triangles, and Prisms 7.3: Solid Geometry Expand/collapse global location 7.3: Solid Geometry Last updated Save as PDF Page ID 35021 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:17.873179	2020-01-25T01:41:45	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.3: Solid Geometry", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.1%3A_Slicing_Solids	7.3.1: Slicing Solids Lesson Let's see what shapes you get when you slice a three-dimensional object. Exercise $\PageIndex{1}$: Prisms, Pyramids, and Polyhedra Describe each shape as precisely as you can. Click on the applet and drag the mouse to show the object turning in 3D. Exercise $\PageIndex{2}$: What's the Cross Section Here are a rectangular prism and a pyramid with the same base and same height. Drag the large red point up and down to move the plane through the solids. - If we slice each solid parallel to its base halfway up, what shape cross sections would we get? What is the same about the cross sections? What is different? - If we slice each solid parallel to its base near the top, what shape cross sections would we get? What is the same about the cross sections? What is different? Are you ready for more? Describe the cross sections that would result from slicing each solid perpendicular to its base. Exercise $\PageIndex{3}$: Card Sort: Cross Sections Your teacher will give you a set of cards. Sort the images into groups that make sense to you. Be prepared to explain your reasoning. Exercise $\PageIndex{4}$: Drawing Cross Sections Use the applet to draw each cross section and describe it in words. Start each one by dragging the gray bar on the left side across the screen until you see the whole 3D image on the left and the controls on the right. - Here is an applet with a rectangular prism, 4 units by 2 units by 3 units. - A plane cuts the prism parallel to the bottom and top faces. - The plane moves up and cuts the prism at a different height. - A vertical plane cuts the prism diagonally. - A square pyramid has a base that is 4 units by 4 units. Its height is also 4 units. - A plane cuts the pyramid parallel to the base. - A vertical plane cuts the prism. - A cube has an edge of length 4. - A plane cuts off the corner of the cube. - The plane moves farther from the corner and makes a cut through the middle of the cube. Summary When we slice a three-dimensional object, we expose new faces that are two dimensional. The two-dimensional face is a cross section . Many different cross sections are possible when slicing the same three-dimensional object. Here are two peppers. One is sliced horizontally, and the other is sliced vertically, producing different cross sections. The imprints of the slices represent the two-dimensional faces created by each slice. It takes practice imagining what the cross section of a three-dimensional object will be for different slices. It helps to experiment and see for yourself what happens! Glossary Entries Definition: Base (of a prism or pyramid) The word base can also refer to a face of a polyhedron. A prism has two identical bases that are parallel. A pyramid has one base. A prism or pyramid is named for the shape of its base. Definition: Cross Section A cross section is the new face you see when you slice through a three-dimensional figure. For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section. Definition: Prism A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms. Here are some drawings of prisms. Definition: Pyramid A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex. Here are some drawings of pyramids. Practice Exercise $\PageIndex{5}$ A cube is cut into two pieces by a single slice that passes through points $A$, $B$, and $C$. What shape is the cross section? Exercise $\PageIndex{6}$ Describe how to slice the three-dimensional figure to result in each cross section. Three-dimensional figure: Cross sections: Exercise $\PageIndex{7}$ Here are two three-dimensional figures. Describe a way to slice one of the figures so that the cross section is a rectangle. Exercise $\PageIndex{8}$ Each row contains the degree measures of two supplementary angles. Complete the table. \| measure of an angle \| measure of its supplement \| \|---\|---\| \| $80^{\circ}$ \| \| \| $25^{\circ}$ \| \| \| $119^{\circ}$ \| \| \| $x$ \| (From Unit 7.1.2) Exercise $\PageIndex{9}$ Two months ago, the price, in dollars, of a cell phone was $c$. - Last month, the price of the phone increased by 10%. Write an expression for the price of the phone last month. - This month, the price of the phone decreased by 10%. Write an expression for the price of the phone this month. - Is the price of the phone this month the same as it was two months ago? Explain your reasoning. (From Unit 4.2.3)	libretexts	2025-03-17T19:52:17.948405	2020-05-04T00:25:36	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.1%3A_Slicing_Solids", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.3.1: Slicing Solids", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.2%3A_Volume_of_Right_Prisms	7.3.2: Volume of Right Prisms Lesson Let's look at volumes of prisms. Exercise $\PageIndex{1}$: Three Prisms with the Same Volume Rectangles A, B, and C represent bases of three prisms. - If each prism has the same height, which one will have the greatest volume , and which will have the least? Explain your reasoning. - If each prism has the same volume, which one will have the tallest height, and which will have the shortest? Explain your reasoning. Exercise $\PageIndex{2}$: Finding Volume with Cubes This applet has 64 snap cubes, all sitting in the same spot on the screen, like a hidden stack of blocks. You will always know where the stack is because it sits on a gray square. You can keep dragging blocks out of the pile by their red points until you have enough to build what you want. Click on the red points to change from left/right movement to up/down movement. There is also a shape on the grid. It marks the footprint of the shapes you will be building. - Using the face of a snap cube as your area unit, what is the area of the shape? Explain or show your reasoning. - Use snap cubes to build the shape from the paper. Add another layer of cubes on top of the shape you have built. Describe this three-dimensional object. - What is the volume of your object? Explain your reasoning. - Right now, your object has a height of 2. What would the volume be - if it had a height of 5? - if it had a height of 8.5? Exercise $\PageIndex{3}$: Can You Find the Volume? The applet has a set of three-dimensional figures. - For each figure, determine whether the shape is a prism. - For each prism: - Find the area of the base of the prism. - Find the height of the prism. - Calculate the volume of the prism. \| Is it a prism? \| area of prism base (cm$^{2}$) \| height (cm) \| volume (cm$^{3}$) \| \|---\|---\|---\|---\| - Begin by grabbing the gray bar on the left and dragging it to the right until you see the slider. - Choose a figure using the slider. - Rotate the view using the Rotate 3D Graphics tool marked by two intersecting, curved arrows. - Note that each polyhedron has only one label per unique face. Where no measurements are shown, the faces are identical copies. - Use the distance tool, marked with the "cm," to click on any segment and find the height or length. - Troubleshooting tip: the cursor must be on the 3D Graphics window for the full toolbar to appear. Are you ready for more? Imagine a large, solid cube made out of 64 white snap cubes. Someone spray paints all 6 faces of the large cube blue. After the paint dries, they disassemble the large cube into a pile of 64 snap cubes. - How many of those 64 snap cubes have exactly 2 faces that are blue? - What are the other possible numbers of blue faces the cubes can have? How many of each are there? - Try this problem again with some larger-sized cubes that use more than 64 snap cubes to build. What patterns do you notice? Exercise $\PageIndex{4}$: What's the Prism's Height? There are 4 different prisms that all have the same volume. Here is what the base of each prism looks like. - Order the prisms from shortest to tallest. Explain your reasoning. - If the volume of each prism is 60 units 3 , what would be the height of each prism? - For a volume other than 60 units 3 , what could be the height of each prism? - Discuss your thinking with your partner. If you disagree, work to reach an agreement. Summary Any cross section of a prism that is parallel to the base will be identical to the base. This means we can slice prisms up to help find their volume. For example, if we have a rectangular prism that is 3 units tall and has a base that is 4 units by 5 units, we can think of this as 3 layers, where each layer has $4\cdot 5$ cubic units. That means the volume of the original rectangular prism is $3(4\cdot 5)$ cubic units. This works with any prism! If we have a prism with height 3 cm that has a base of area 20 cm 2 , then the volume is $3\cdot 20$ cm 3 regardless of the shape of the base. In general, the volume of a prism with height $h$ and area $B$ is $V=B\cdot h$ For example, these two prisms both have a volume of 100 cm 3 . Glossary Entries Definition: Base (of a prism or pyramid) The word base can also refer to a face of a polyhedron. A prism has two identical bases that are parallel. A pyramid has one base. A prism or pyramid is named for the shape of its base. Definition: Cross Section A cross section is the new face you see when you slice through a three-dimensional figure. For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section. Definition: Prism A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms. Here are some drawings of prisms. Definition: Pyramid A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex. Here are some drawings of pyramids. Definition: Volume Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps. For example, the volume of this rectangular prism is 60 units 3 , because it is composed of 3 layers that are each 20 units 3 . Practice Exercise $\PageIndex{5}$ - Select all the prisms. - For each prism, shade one of its bases. Exercise $\PageIndex{6}$ The volume of both of these trapezoidal prisms is 24 cubic units. Their heights are 6 and 8 units, as labeled. What is the area of a trapezoidal base of each prism? Exercise $\PageIndex{7}$ Two angles are complementary. One has a measure of 19 degrees. What is the measure of the other? (From Unit 7.1.2) Exercise $\PageIndex{8}$ Two angles are supplementary. One has a measure that is twice as large as the other. Find the two angle measures. (From Unit 7.1.2) Exercise $\PageIndex{9}$ Match each expression in the first list with an equivalent expression from the second list. - $7(x+2)-x+3$ - $6x+3+4x+5$ - $\frac{-2}{5}x-7+\frac{3}{5}x-3$ - $8x-5+4-9$ - $24x+36$ - $\frac{1}{5}x-10$ - $6x+17$ - $2(5x+4)$ - $12(2x+3)$ - $8x+(-5)+4+(-9)$ (From Unit 6.4.5) Exercise $\PageIndex{10}$ Clare paid 50% more for her notebook than Priya paid for hers. Priya paid $x$ for her notebook and Clare paid $y$ dollars for hers. Write an equation that represents the relationship between $y$ and $x$. (From Unit 4.2.3)	libretexts	2025-03-17T19:52:18.032103	2020-05-04T00:25:09	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.2%3A_Volume_of_Right_Prisms", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.3.2: Volume of Right Prisms", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.3%3A_Decomposing_Bases_for_Area	7.3.3: Decomposing Bases for Area Lesson Let's look at how some people use volume. Exercise $\PageIndex{1}$: Are These Prisms? 1. Which of these solids are prisms? Explain how you know. 2. For each of the prisms, what does the base look like? - Shade one base in the picture. - Draw a cross section of the prism parallel to the base. Exercise $\PageIndex{2}$: A Box of Chocolates A box of chocolates is a prism with a base in the shape of a heart and a height of 2 inches. Here are the measurements of the base. To calculate the volume of the box, three different students have each drawn line segments showing how they plan on finding the area of the heart-shaped base. - For each student’s plan, describe the shapes the student must find the area of and the operations they must use to calculate the total area. - Although all three methods could work, one of them requires measurements that are not provided. Which one is it? - Between you and your partner, decide which of you will use which of the remaining two methods. - Using the quadrilaterals and triangles drawn in your selected plan, find the area of the base. - Trade with a partner and check each other’s work. If you disagree, work to reach an agreement. - Return their work. Calculate the volume of the box of chocolates. Are you ready for more? The box has 30 pieces of chocolate in it, each with a volume of 1 in 3 . If all the chocolates melt into a solid layer across the bottom of the box, what will be the height of the layer? Exercise $\PageIndex{3}$: Another Prism A house-shaped prism is created by attaching a triangular prism on top of a rectangular prism. - Draw the base of this prism and label its dimensions. - What is the area of the base? Explain or show your reasoning. - What is the volume of the prism? Summary To find the area of any polygon, you can decompose it into rectangles and triangles. There are always many ways to decompose a polygon. Sometimes it is easier to enclose a polygon in a rectangle and subtract the area of the extra pieces. To find the volume of a prism with a polygon for a base, you find the area of the base, $B$, and multiply by the height, $h$. $V=Bh$ Glossary Entries Definition: Base (of a prism or pyramid) The word base can also refer to a face of a polyhedron. A prism has two identical bases that are parallel. A pyramid has one base. A prism or pyramid is named for the shape of its base. Definition: Cross Section A cross section is the new face you see when you slice through a three-dimensional figure. For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section. Definition: Prism A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms. Here are some drawings of prisms. Definition: Pyramid A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex. Here are some drawings of pyramids. Definition: Volume Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps. For example, the volume of this rectangular prism is 60 units 3 , because it is composed of 3 layers that are each 20 units 3 . Practice Exercise $\PageIndex{4}$ You find a crystal in the shape of a prism. Find the volume of the crystal. The point $B$ is directly underneath point $E$, and the following lengths are known: - From $A$ to $B$: 2 mm - From $B$ to $C$: 3 mm - From $A$ to $F$: 6 mm - From $B$ to $E$: 10 mm - From $C$ to $D$: 7 mm - From $A$ to $G$: 4 mm Exercise $\PageIndex{5}$ A rectangular prism with dimensions 5 inches by 13 inches by 10 inches was cut to leave a piece as shown in the image. What is the volume of this piece? What is the volume of the other piece not pictured? Exercise $\PageIndex{6}$ A triangle has one side that is 7 cm long and another side that is 3 cm long. - Sketch this triangle and label your sketch with the given measures. (If you are stuck, try using a compass or cutting some straws to these two lengths.) - Draw one more triangle with these measures that is not identical to your first triangle. - Explain how you can tell they are not identical. (From Unit 7.2.4) Exercise $\PageIndex{7}$ Select all equations that represent a relationship between angles in the figure. - $90-30=b$ - $30+b=a+c$ - $a+c+30+b=180$ - $a=30$ - $a=c=30$ - $90+a+c=180$ (From Unit 7.1.4) Exercise $\PageIndex{8}$ A mixture of punch contains 1 quart of lemonade, 2 cups of grape juice, 4 tablespoons of honey, and $\frac{1}{2}$ gallon of sparkling water. Find the percentage of the punch mixture that comes from each ingredient. Round your answers to the nearest tenth of a percent. (Hint: 1 cup = 16 tablespoons) (From Unit 4.2.4)	libretexts	2025-03-17T19:52:18.107594	2020-05-04T00:24:38	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.3%3A_Decomposing_Bases_for_Area", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.3.3: Decomposing Bases for Area", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.4%3A_Surface_Area_of_Right_Prisms	7.3.4: Surface Area of Right Prisms Lesson Let's look at the surface area of prisms. Exercise $\PageIndex{1}$: Multifaceted Your teacher will show you a prism. - What are some things you could measure about the object? - What units would you use for these measurements? Exercise $\PageIndex{2}$: So Many Faces Here is a picture of your teacher's prism: Three students are trying to calculate the surface area of this prism. - Noah says, “This is going to be a lot of work. We have to find the areas of 14 different faces and add them up.” - Elena says, “It’s not so bad. All 12 rectangles are identical copies, so we can find the area for one of them, multiply that by 12 and then add on the areas of the 2 bases.” - Andre says, “Wait, I see another way! Imagine unfolding the prism into a net. We can use 1 large rectangle instead of 12 smaller ones.” - Do you agree with any of them? Explain your reasoning. - How big is the “1 large rectangle” Andre is talking about? Explain or show your reasoning. If you get stuck, consider drawing a net for the prism. - Will Noah’s method always work for finding the surface area of any prism? Elena’s method? Andre’s method? Be prepared to explain your reasoning. - Which method do you prefer? Why? Exercise $\PageIndex{3}$: Revisiting a Pentagonal Prism - Between you and your partner, choose who will use each of these two methods to find the surface area of the prism. - Adding the areas of all the faces - Using the perimeter of the base. - Use your chosen method to calculate the surface area of the prism. Show your thinking. Organize it so it can be followed by others. 3. Trade papers with your partner, and check their work. Discuss your thinking. If you disagree, work to reach an agreement. Are you ready for more? In a deck of cards, each card measures 6 cm by 9 cm. - When stacked, the deck is 2 cm tall, as shown in the first photo. Find the volume of this deck of cards. - Then the cards are fanned out, as shown in the second picture. The distance from the rightmost point on the bottom card to the rightmost point on the top card is now 7 cm instead of 2 cm. Find the volume of the new stack. Summary To find the surface area of a three-dimensional figure whose faces are made up of polygons, we can find the area of each face, and add them up! Sometimes there are ways to simplify our work. For example, all of the faces of a cube with side length $s$ are the same. We can find the area of one face, and multiply by 6. Since the area of one face of a cube is $s^{2}$, the surface area of a cube is $6s^{2}$. We can use this technique to make it faster to find the surface area of any figure that has faces that are the same. For prisms, there is another way. We can treat the prism as having three parts: two identical bases, and one long rectangle that has been taped along the edges of the bases. The rectangle has the same height as the prism, and its width is the perimeter of the base. To find the surface area, add the area of this rectangle to the areas of the two bases. Glossary Entries Definition: Base (of a prism or pyramid) The word base can also refer to a face of a polyhedron. A prism has two identical bases that are parallel. A pyramid has one base. A prism or pyramid is named for the shape of its base. Definition: Cross Section A cross section is the new face you see when you slice through a three-dimensional figure. For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section. Definition: Prism A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms. Here are some drawings of prisms. Definition: Pyramid A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex. Here are some drawings of pyramids. Definition: Surface Area The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps. For example, if the faces of a cube each have an area of 9 cm 2 , then the surface area of the cube is $6\cdot 9$, or 54 cm 2 . Definition: Volume Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps. For example, the volume of this rectangular prism is 60 units 3 , because it is composed of 3 layers that are each 20 units 3 . Practice Exercise $\PageIndex{4}$ Edge lengths are given in units. Find the surface area of each prism in square units. Exercise $\PageIndex{5}$ Priya says, “No matter which way you slice this rectangular prism, the cross section will be a rectangle.” Mai says, “I’m not so sure.” Describe a slice that Mai might be thinking of. (From Unit 7.3.1) Exercise $\PageIndex{6}$ $B$ is the intersection of line $AC$ and line $ED$. Find the measure of each of the angles. - Angle $ABF$ - Angle $ABD$ - Angle $EBC$ - Angle $FBC$ - Angle $DBG$ (From Unit 7.1.5) Exercise $\PageIndex{7}$ Write each expression with fewer terms. - $12m-4m$ - $12m-5k+m$ - $9m+k-(3m-2k)$ (From Unit 6.4.3) Exercise $\PageIndex{8}$ - Find 44% of 625 using the facts that 40% of 625 is 250 and 4% of 625 is 25. - What is 4.4% of 625? - What is 0.44% of 625? (From Unit 4.2.4)	libretexts	2025-03-17T19:52:18.273357	2020-05-04T00:24:09	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.4%3A_Surface_Area_of_Right_Prisms", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.3.4: Surface Area of Right Prisms", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.5%3A_Distinguishing_Volume_and_Surface_Area	7.3.5: Distinguishing Volume and Surface Area Lesson Let's work with surface area and volume in context. Exercise $\PageIndex{1}$: THe Science Fair Mai’s science teacher told her that when there is more ice touching the water in a glass, the ice melts faster. She wants to test this statement so she designs her science fair project to determine if crushed ice or ice cubes will melt faster in a drink. She begins with two cups of warm water. In one cup, she puts a cube of ice. In a second cup, she puts crushed ice with the same volume as the cube. What is your hypothesis? Will the ice cube or crushed ice melt faster, or will they melt at the same rate? Explain your reasoning. Exercise $\PageIndex{2}$: Revisiting the Box of Chocolates The other day, you calculated the volume of this heart-shaped box of chocolates. The depth of the box is 2 inches. How much cardboard is needed to create the box? Exercise $\PageIndex{3}$: Card Sort: Surface Area or Volume Your teacher will give you cards with different figures and questions on them. - Sort the cards into two groups based on whether it would make more sense to think about the surface area or the volume of the figure when answering the question. Pause here so your teacher can review your work. - Your teacher will assign you a card to examine more closely. What additional information would you need to be able to answer the question on your card? - Estimate reasonable measurements for the figure on your card. - Use your estimated measurements to calculate the answer to the question. Are you ready for more? A cake is shaped like a square prism. The top is 20 centimeters on each side, and the cake is 10 centimeters tall. It has frosting on the sides and on the top, and a single candle on the top at the exact center of the square. You have a knife and a 20-centimeter ruler. - Find a way to cut the cake into 4 fair portions, so that all 4 portions have the same amount of cake and frosting. - Find another way to cut the cake into 4 fair portions. - Find a way to cut the cake into 5 fair portions. Exercise $\PageIndex{4}$: A Wheelbarrow of Concrete A wheelbarrow is being used to carry wet concrete. Here are its dimensions. - What volume of concrete would it take to fill the tray? - After dumping the wet concrete, you notice that a thin film is left on the inside of the tray. What is the area of the concrete coating the tray? (Remember, there is no top.) Summary Sometimes we need to find the volume of a prism, and sometimes we need to find the surface area. Here are some examples of quantities related to volume: - How much water a container can hold - How much material it took to build a solid object Volume is measured in cubic units, like in 3 or m 3 . Here are some examples of quantities related to surface area: - How much fabric is needed to cover a surface - How much of an object needs to be painted Surface area is measured in square units, like in 2 or m 2 . Glossary Entries Definition: Base (of a prism or pyramid) The word base can also refer to a face of a polyhedron. A prism has two identical bases that are parallel. A pyramid has one base. A prism or pyramid is named for the shape of its base. Definition: Cross Section A cross section is the new face you see when you slice through a three-dimensional figure. For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section. Definition: Prism A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms. Here are some drawings of prisms. Definition: Pyramid A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex. Here are some drawings of pyramids. Definition: Surface Area The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps. For example, if the faces of a cube each have an area of 9 cm 2 , then the surface area of the cube is $6\cdot 9$, or 54 cm 2 . Definition: Volume Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps. For example, the volume of this rectangular prism is 60 units 3 , because it is composed of 3 layers that are each 20 units 3 . Practice Exercise $\PageIndex{5}$ Here is the base of a prism. - If the height of the prism is 5 cm, what is its surface area? What is its volume? - If the height of the prism is 10 cm, what is its surface area? What is its volume? - When the height doubled, what was the percent increase for the surface area? For the volume? Exercise $\PageIndex{6}$ Select all the situations where knowing the volume of an object would be more useful than knowing its surface area. - Determining the amount of paint needed to paint a barn. - Determining the monetary value of a piece of gold jewelry. - Filling an aquarium with buckets of water. - Deciding how much wrapping paper a gift will need. - Packing a box with watermelons for shipping. - Charging a company for ad space on your race car. - Measuring the amount of gasoline left in the tank of a tractor. Exercise $\PageIndex{7}$ Han draws a triangle with a $50^{\circ}$ angle, a $40^{\circ}$ angle, and a side of length 4 cm as shown. Can you draw a different triangle with the same conditions? (From Unit 7.2.4) Exercise $\PageIndex{8}$ Angle $H$ is half as large as angle $J$. Angle $J$ is one fourth as large as angle $K$. Angle $K$ has measure 240 degrees. What is the measure of angle $H$? (From Unit 7.1.3) Exercise $\PageIndex{9}$ The Colorado state flag consists of three horizontal stripes of equal height. The side lengths of the flag are in the ratio $2:3$. The diameter of the gold-colored disk is equal to the height of the center stripe. What percentage of the flag is gold? (From Unit 4.2.4)	libretexts	2025-03-17T19:52:18.349914	2020-05-04T00:23:38	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.5%3A_Distinguishing_Volume_and_Surface_Area", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.3.5: Distinguishing Volume and Surface Area", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.6%3A_Applying_Volume_and_Surface_Area	7.3.6: Applying Volume and Surface Area Lesson Let's explore things that are proportional to volume or surface area. Exercise $\PageIndex{1}$: You Decide For each situation, decide if it requires Noah to calculate surface area or volume. Explain your reasoning. - Noah is planning to paint the bird house he built. He is unsure if he has enough paint. - Noah is planning to use a box with a trapezoid base to hold modeling clay. He is unsure if the clay will all fit in the box. Exercise $\PageIndex{2}$: Foam Play Structure At a daycare, Kiran sees children climbing on this foam play structure. Kiran is thinking about building a structure like this for his younger cousins to play on. - The entire structure is made out of soft foam so the children don’t hurt themselves. How much foam would Kiran need to build this play structure? - The entire structure is covered with vinyl so it is easy to wipe clean. How much vinyl would Kiran need to build this play structure? - The foam costs 0.8¢ per in 3 . Here is a table that lists the costs for different amounts of vinyl. What is the total cost for all the foam and vinyl needed to build this play structure? \| vinyl (in 2 ) \| cost ($) \| \|---\|---\| \| $75$ \| $0.45$ \| \| $125$ \| $0.75$ \| Are you ready for more? When he examines the play structure more closely, Kiran realizes it is really two separate pieces that are next to each other. - How does this affect the amount of foam in the play structure? - How does this affect the amount of vinyl covering the play structure? Exercise $\PageIndex{3}$: Filling the Sandbox The daycare has two sandboxes that are both prisms with regular hexagons as their bases. The smaller sandbox has a base area of 1,146 in 2 and is filled 10 inches deep with sand. - It took 14 bags of sand to fill the small sandbox to this depth. What volume of sand comes in one bag? (Round to the nearest whole cubic inch.) - The daycare manager wants to add 3 more inches to the depth of the sand in the small sandbox. How many bags of sand will they need to buy? - The daycare manager also wants to add 3 more inches to the depth of the sand in the large sandbox. The base of the large sandbox is a scaled copy of the base of the small sandbox, with a scale factor of 1.5. How many bags of sand will they need to buy for the large sandbox? - A lawn and garden store is selling 6 bags of sand for $19.50. How much will they spend to buy all the new sand for both sandboxes? Summary Suppose we wanted to make a concrete bench like the one shown in this picture. If we know that the finished bench has a volume of 10 ft 3 and a surface area of 44 ft 2 we can use this information to solve problems about the bench. For example, - How much does the bench weigh? - How long does it take to wipe the whole bench clean? - How much will the materials cost to build the bench and to paint it? To figure out how much the bench weighs, we can use its volume, 10 ft 3 . Concrete weighs about 150 pounds per cubic foot, so this bench weighs about 1,500 pounds, because $10\cdot 150=1,500$. To figure out how long it takes to wipe the bench clean, we can use its surface area, 44 ft 2 . If it takes a person about 2 seconds per square foot to wipe a surface clean, then it would take about 88 seconds to clean this bench, because $44\cdot 2=88$. It may take a little less than 88 seconds, since the surfaces where the bench is touching the ground do not need to be wiped. Would you use the volume or the surface area of the bench to calculate the cost of the concrete needed to build this bench? And for the cost of the paint? Glossary Entries Definition: Base (of a prism or pyramid) The word base can also refer to a face of a polyhedron. A prism has two identical bases that are parallel. A pyramid has one base. A prism or pyramid is named for the shape of its base. Definition: Cross Section A cross section is the new face you see when you slice through a three-dimensional figure. For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section. Definition: Prism A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms. Here are some drawings of prisms. Definition: Pyramid A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex. Here are some drawings of pyramids. Definition: Surface Area The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps. For example, if the faces of a cube each have an area of 9 cm 2 , then the surface area of the cube is $6\cdot 9$, or 54 cm 2 . Definition: Volume Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps. For example, the volume of this rectangular prism is 60 units 3 , because it is composed of 3 layers that are each 20 units 3 . Practice Exercise $\PageIndex{4}$ A landscape architect is designing a pool that has this top view: - How much water will be needed to fill this pool 4 feet deep? - Before filling up the pool, it gets lined with a plastic liner. How much liner is needed for this pool? - Here are the prices for different amounts of plastic liner. How much will all the plastic liner for the pool cost? \| plastic liner (ft 2 ) \| cost ($) \| \|---\|---\| \| $25$ \| $3.75$ \| \| $50$ \| $7.50$ \| \| $75$ \| $11.25$ \| Exercise $\PageIndex{5}$ Shade in a base of the trapezoidal prism. (The base is not the same as the bottom.) - Find the area of the base you shaded. - Find the volume of this trapezoidal prism. (From Unit 7.3.3) Exercise $\PageIndex{6}$ For each diagram, decide if $y$ is an increase or a decrease of $x$. Then determine the percentage that $x$ increased or decreased to result in $y$. (From Unit 4.2.4) Exercise $\PageIndex{7}$ Noah is visiting his aunt in Texas. He wants to buy a belt buckle whose price is $25. He knows that the sales tax in Texas is 6.25%. - How much will the tax be on the belt buckle? - How much will Noah spend for the belt buckle including the tax? - Write an equation that represents the total cost, $c$, of an item whose price is $p$. (From Unit 4.3.1)	libretexts	2025-03-17T19:52:18.429928	2020-05-04T00:22:59	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.03%3A_New_Page/7.3.6%3A_Applying_Volume_and_Surface_Area", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.3.6: Applying Volume and Surface Area", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.04%3A_New_Page	7.4: Let's Put It to work Last updated Save as PDF Page ID 35022 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:18.500472	2020-01-25T01:41:46	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.04%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.4: Let's Put It to work", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.04%3A_New_Page/7.4.1%3A_Building_Prisms	7.4.1: Building Prisms Lesson Let's build a triangular prism from scratch. Exercise $\PageIndex{1}$: Nets Here are some nets for various prisms. - What would each net look like when folded? - What do you notice about the nets? Exercise $\PageIndex{2}$: Making the Base The base of a triangular prism has one side that is 7 cm long, one side that is 5.5 cm long, and one angle that measures $45^{\circ}$. - Draw as many different triangles as you can with these given measurements. - Select one of the triangles you have drawn. Measure and calculate to approximate its area. Explain or show your reasoning. Exercise $\PageIndex{3}$: Making the Prism Your teacher will give you an incomplete net. Follow these instructions to complete the net and assemble the triangular prism: - Draw an identical copy of the triangle you selected in the previous activity along the top of the rectangle, with one vertex on point $A$. - Draw another copy of your triangle, flipped upside down, along the bottom of the rectangle, with one vertex on point $C$. - Determine how long the rectangle needs to be to wrap all the way around your triangular bases. Pause here so your teacher can review your work. - Cut out and assemble your net. After you finish assembling your triangular prism, answer these questions. Explain or show your reasoning. - What is the volume of your prism? - What is the surface area of your prism? - Stand your prism up so it is sitting on its triangular base. - If you were to cut your prism in half horizontally, what shape would the cross section be? - If you were to cut your prism in half vertically, what shape would the cross section be? Exercise $\PageIndex{4}$: Combining Prisms - Compare your prism with your partner’s prism. What is the same? What is different? - Find a way you can put your prism and your partner’s prism together to make one new, larger prism. Describe your new prism. - Draw the base of your new prism and label the lengths of the sides. - As you answer these questions about your new prism, look for ways you can use your calculations from the previous activity to help you. Explain or show your reasoning. - What is the area of its base? - What is its height? - What is its volume? - What is its surface area? Are you ready for more? How many identical copies of your prism would it take you to put together a new larger prism in which every dimension was twice as long?	libretexts	2025-03-17T19:52:18.561740	2020-05-04T00:26:50	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/07%3A_Untitled_Chapter_7/7.04%3A_New_Page/7.4.1%3A_Building_Prisms", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "7.4.1: Building Prisms", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8	8: Probability and Sampling Last updated Save as PDF Page ID 35025 Illustrative Mathematics OpenUp Resources 8.1: Probabilities of Single Step Events 8.1.1: Mystery Bags 8.1.2: Chance Experiments 8.1.3: What are Probabilities? 8.1.4: Estimating Probabilities Through Repeated Experiments 8.1.5: More Estimating Probabilities 8.1.6: Estimating Probabilities Using Simulation 8.2: Probabilites of Multi-step Events 8.2.1: Simulating Multi-step Experiments 8.2.2: Keeping Track of All Possible Outcomes 8.2.3: Multi-step Experiments 8.2.4: Designing Simulations 8.3: Sampling 8.3.1: Comparing Groups 8.3.2: Larger Populations 8.3.3: What Makes a Good Sample? 8.3.4: Sampling in a Fair Way 8.4: Using Samples 8.4.1: Estimating Population Measures of Center 8.4.2: Estimating Population Proportions 8.4.3: More about Sampling Variability 8.4.4: Comparing Populations Using Samples 8.4.5: Comparing Populations With Friends 8.5: Let's Put it to Work 8.5.1: Memory Test	libretexts	2025-03-17T19:52:18.650459	2020-01-25T01:41:48	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8: Probability and Sampling", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page	8.1: Probabilities of Single Step Events Last updated Save as PDF Page ID 35026 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:18.721872	2020-01-25T01:41:49	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.1: Probabilities of Single Step Events", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.1%3A_Mystery_Bags	8.1.1: Mystery Bags Lesson Let's make predictions based on what we know. Exercise $\PageIndex{1}$: Going Fishing Andre and his dad have been fishing for 2 hours. In that time, they have caught 9 bluegills and 1 yellow perch. The next time Andre gets a bite, what kind of fish do you think it will be? Explain your reasoning. Exercise $\PageIndex{2}$: Playing the Block Game Your teacher will give your group a bag of colored blocks. - Follow these instructions to play one round of the game: - Everyone in the group records the color written on the bag in the first column of the table. - Without looking in the bag, one person takes out one of the blocks and shows it to the group. - If they get a block that is the same color as the bag, they earn: - 1 point during round 1 - 2 points during round 2 - 3 points during round 3 - Next, they put the block back into the bag, shake the bag to mix up the blocks, and pass the bag to the next person in the group. - Repeat these steps until everyone in your group has had 4 turns. - At the end of the round, record each person's score in the table. What color bag? person 1's score person 2's score person 3's score person 4's score round 1 round 2 round 3 Table $\PageIndex{1}$ - Pause here so your teacher can give you a new bag of blocks for the next round. - Repeat the previous steps to play rounds 2 and 3 of the game. - After you finish playing all 3 rounds, calculate the total score for each person in your group. Are you ready for more? Tyler’s class played the block game using purple, orange, and yellow bags of blocks. - During round 1, Tyler’s group picked 4 purple blocks and 12 blocks of other colors. - During round 2, Tyler’s group picked 11 orange blocks and 5 blocks of other colors. - During round 3, Tyler forgot to record how many yellow blocks his group picked. For a final round, Tyler’s group can pick one block from any of the three bags. Tyler’s group decides that picking from the orange bag would give them the best chance of winning, and that picking from the purple bag would give them the worst chance of winning. What results from the yellow bag could have lead Tyler’s group to this conclusion? Explain your reasoning. Summary One of the main ways that humans learn is by repeating experiments and observing the results. Babies learn that dropping their cup makes it hit the floor with a loud noise by repeating this action over and over. Scientists learn about nature by observing the results of repeated experiments again and again. With enough data about the results of experiments, we can begin to predict what may happen if the experiment is repeated in the future. For example, a baseball player who has gotten a hit 33 out of 100 times at bat might be expected to get a hit about 33% of his times at bat in the future as well. In some cases, we can predict the chances of things happening based on our knowledge of the situation. For example, a coin should land heads up about 50% of the time due to the symmetry of the coin. In other cases, there are too many unknowns to predict the chances of things happening. For example, the chances of rain tomorrow are based on similar weather conditions we have observed in the past. In these situations, we can experiment, using past results to estimate chances. Practice Exercise $\PageIndex{3}$ Lin is interested in how many of her classmates watch her favorite TV show, so she starts asking around at lunch. She gets the following responses: $\text{yes yes yes no no no no no no no yes no no no}$ If she asks one more person randomly in the cafeteria, do you think they will say “yes” or “no”? Explain your reasoning. Exercise $\PageIndex{4}$ An engineer tests the strength of a new material by seeing how much weight it can hold before breaking. Previous tests have held these weights in pounds: $1,200\quad 1,400\quad 1,300\quad 1,500\quad 950\qquad 1,600\qquad 1,100$ Do you think that this material will be able to hold more than 1,000 pounds in the next test? Explain your reasoning. Exercise $\PageIndex{5}$ A company tests two new products to make sure they last for more than a year. - Product 1 had 950 out of 1,000 test items last for more than a year. - Product 2 had 150 out of 200 last for more than a year. If you had to choose one of these two products to use for more than a year, which one is more likely to last? Explain your reasoning. Exercise $\PageIndex{6}$ Put these numbers in order from least to greatest. $\frac{1}{2}\qquad\frac{1}{3}\qquad\frac{2}{5}\qquad 0.6\qquad 0.3$ Exercise $\PageIndex{7}$ A small staircase is made so that the horizontal piece of each step is 10 inches long and 25 inches wide. Each step is 5 inches above the previous one. What is the surface area of this staircase? (From Unit 7.3.5)	libretexts	2025-03-17T19:52:18.791985	2020-05-11T07:53:34	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.1%3A_Mystery_Bags", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.1.1: Mystery Bags", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.2%3A_Chance_Experiments	8.1.2: Chance Experiments Lesson Let's investigate chance. Exercise $\PageIndex{1}$: Which is More Likely? Which is more likely to happen? - When reaching into a dark closet and pulling out one shoe from a pile of 20 pairs of shoes, you pull out a left shoe. - When listening to a playlist—which has 5 songs on it—in shuffle mode, the first song on the playlist plays first. Exercise $\PageIndex{2}$: How Likely Is it? 1. Label each event with one of these options: impossible, unlikely, equally likely as not, likely, certain - You will win grand prize in a raffle if you purchased 2 out of the 100 tickets. - You will wait less than 10 minutes before ordering at a fast food restaurant. - You will get an even number when you roll a standard number cube. - A four-year-old child is over 6 feet tall. - No one in your class will be late to class next week. - The next baby born at a hospital will be a boy. - It will snow at our school on July 1. - The Sun will set today before 11:00 p.m. - Spinning this spinner will result in green. - Spinning this spinner will result in red. 2. Discuss your answers to the previous question with your partner. If you disagree, work to reach an agreement. 3. Invent another situation for each label, for a total of 5 more events. Exercise $\PageIndex{3}$: Take a Chance This applet displays a random number from 1 to 6, like a number cube. With a partner, you will play a game of chance. - In the first round, one of you will score on an even roll and one of you will score on an odd roll. You decide that first. - In the second round, the winner of round 1 will score on numbers $1-4$, and the other player will score on numbers $5-6$. - Each round is 10 rolls. Be sure to turn on "History" after your first roll and wait for it to update before rolling again. - When each player had three numbers, dud one of them usually win? - When one player had four numbers, did you expect them to win? Explain your reasoning. Are you ready for more? On a game show, there are 3 closed doors. One door has a prize behind it. The contestant chooses one of the doors. The host of the game show, who knows where the prize is located, opens one of the other doors which does not have the prize. The contestant can choose to stay with their first choice or switch to the remaining closed door. - Do you think it matters if the contestant switches doors or stays? - Practice playing the game with your partner and record your results. Whoever is the host starts each round by secretly deciding which door has the prize. - Play 20 rounds where the contestant always stays with their first choice. - Play 20 more rounds where the contestant always switches doors. - Did the results from playing the game change your answer to the first question? Explain. Exercise $\PageIndex{4}$: Card Sort: Likelihood - Your teacher will give you some cards that describe events. Order the events from least likely to most likely. - After ordering the first set of cards, pause here so your teacher can review your work. Then, your teacher will give you a second set of cards. - Add the new set of cards to the first set so that all of the cards are ordered from least likely to most likely. Summary A chance experiment is something that happens where the outcome is unknown. For example, if we flip a coin, we don’t know if the result will be a head or a tail. An outcome of a chance experiment is something that can happen when you do a chance experiment. For example, when you flip a coin, one possible outcome is that you will get a head. An event is a set of one or more outcomes. We can describe events using these phrases: - Impossible - Unlikely - Equally likely as not - Likely - Certain For example, if you flip a coin: - It is impossible that the coin will turn into a bottle of ketchup. - It is unlikely the coin will land on its edge. - It is equally likely as not that you will get a tail. - It is likely that you will get a head or a tail. - It is certain that the coin will land somewhere. The probability of an event is a measure of the likelihood that an event will occur. We will learn more about probabilities in the lessons to come. Glossary Entries Definition: Chance Experiment A chance experiment is something you can do over and over again, and you don’t know what will happen each time. For example, each time you spin the spinner, it could land on red, yellow, blue, or green. Definition: Event An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes. Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.” Definition: Outcome An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails. Practice Exercise $\PageIndex{5}$ The likelihood that Han makes a free throw in basketball is 60%. The likelihood that he makes a 3-point shot is 0.345. Which event is more likely, Han making a free throw or making a 3-point shot? Explain your reasoning. Exercise $\PageIndex{6}$ Different events have the following likelihoods. Sort them from least to greatest: $60$% $8$ out of $10$ $0.37$ $20$% $\frac{5}{6}$ Exercise $\PageIndex{7}$ There are 25 prime numbers between 1 and 100. There are 46 prime numbers between 1 and 200. Which situation is more likely? Explain your reasoning. - A computer produces a random number between 1 and 100 that is prime. - A computer produces a random number between 1 and 200 that is prime. Exercise $\PageIndex{8}$ It takes $4\frac{3}{8}$ cups of cheese, $\frac{7}{8}$ cups of olives, and $2\frac{5}{8}$ cups of sausage to make a signature pizza. How much of each ingredient is needed to make 10 pizzas? Explain or show your reasoning. (From Unit 4.1.2) Exercise $\PageIndex{9}$ Here is a diagram of a birdhouse Elena is planning to build. (It is a simplified diagram, since in reality, the sides will have a thickness.) About how many square inches of wood does she need to build this birdhouse? (From Unit 7.3.6) Exercise $\PageIndex{10}$ Select all the situations where knowing the surface area of an object would be more useful than knowing its volume. - Placing an order for tiles to replace the roof of a house. - Estimating how long it will take to clean the windows of a greenhouse. - Deciding whether leftover soup will fit in a container. - Estimating how long it will take to fill a swimming pool with a garden hose. - Calculating how much paper is needed to manufacture candy bar wrappers. - Buying fabric to sew a couch cover. - Deciding whether one muffin pan is enough to bake a muffin recipe. (From Unit 7.3.5)	libretexts	2025-03-17T19:52:18.870421	2020-05-11T07:53:12	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.2%3A_Chance_Experiments", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.1.2: Chance Experiments", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.3%3A_What_are_Probabilities	8.1.3: What are Probabilities? Lesson Let's find out what's possible. Exercise $\PageIndex{1}$: Which Game Would You CHoose? Which game would you choose to play? Explain your reasoning. Game 1: You flip a coin and win if it lands showing heads. Game 2: You roll a standard number cube and win if it lands showing a number that is divisible by 3. Exercise $\PageIndex{2}$: What's Possible? - For each situation, list the sample space and tell how many outcomes there are. - Han rolls a standard number cube once. - Clare spins this spinner once. - Kiran selects a letter at random from the word “MATH.” - Mai selects a letter at random from the alphabet. - Noah picks a card at random from a stack that has cards numbered 5 through 20. - Next, compare the likelihood of these outcomes. Be prepared to explain your reasoning. - Is Clare more likely to have the spinner stop on the red or blue section? - Is Kiran or Mai more likely to get the letter T? - Is Han or Noah more likely to get a number that is greater than 5? - Suppose you have a spinner that is evenly divided showing all the days of the week. You also have a bag of papers that list the months of the year. Are you more likely to spin the current day of the week or pull out the paper with the current month? Are you ready for more? Are there any outcomes for two people in this activity that have the same likelihood? Explain or show your reasoning. Exercise $\PageIndex{3}$: What's in the Bag? Your teacher will give your group a bag of paper slips with something printed on them. Repeat these steps until everyone in your group has had a turn. - As a group, guess what is printed on the papers in the bag and record your guess in the table. - Without looking in the bag, one person takes out one of the papers and shows it to the group. - Everyone in the group records what is printed on the paper. - The person who took out the paper puts it back into the bag, shakes the bag to mix up the papers, and passes the bag to the next person in the group. \| Guess the sample space. \| What is printed on the paper? \| \| \|---\|---\|---\| \| person 1 \| \|\| \| person 2 \| \|\| \| person 3 \| \|\| \| person 4 \| - How was guessing the sample space the fourth time different from the first? - What could you do to get a better guess of the sample space? - Look at all the papers in the bag. Were any of your guesses correct? - Are all of the possible outcomes equally likely? Explain. - Use the sample space to determine the probability that a fifth person would get the same outcome as person 1. Summary The probability of an event is a measure of the likelihood that the event will occur. Probabilities are expressed using numbers from 0 to 1. - If the probability is 0, that means the event is impossible. For example, when you flip a coin, the probability that it will turn into a bottle of ketchup is 0. The closer the probability of some event is to 0, the less likely it is. - If the probability is 1, that means the event is certain. For example, when you flip a coin, the probability that it will land somewhere is 1. The closer the probability of some event is to 1, the more likely it is. If we list all of the possible outcomes for a chance experiment, we get the sample space for that experiment. For example, the sample space for rolling a standard number cube includes six outcomes: 1, 2, 3, 4, 5, and 6. The probability that the number cube will land showing the number 4 is $\frac{1}{6}$. In general, if all outcomes in an experiment are equally likely and there are $n$ possible outcomes, then the probability of a single outcome is $\frac{1}{n}$. Sometimes we have a set of possible outcomes and we want one of them to be selected at random . That means that we want to select an outcome in a way that each of the outcomes is equally likely . For example, if two people both want to read the same book, we could flip a coin to see who gets to read the book first. Glossary Entries Definition: Chance Experiment A chance experiment is something you can do over and over again, and you don’t know what will happen each time. For example, each time you spin the spinner, it could land on red, yellow, blue, or green. Definition: Event An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes. Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.” Definition: Outcome An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails. Definition: Probability The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen. For example, the probability of selecting a moon block at random from this bag is $\frac{4}{5}$. Definition: Random Outcomes of a chance experiment are random if they are all equally likely to happen. Definition: Sample Space The sample space is the list of every possible outcome for a chance experiment. For example, the sample space for tossing two coins is: \| heads-heads \| tails-heads \| \| heads-tails \| tails-tails \| Practice Exercise $\PageIndex{4}$ List the sample space for each chance experiment. - Flipping a coin - Selecting a random season of the year - Selecting a random day of the week Exercise $\PageIndex{5}$ A computer randomly selects a letter from the alphabet. - How many different outcomes are in the sample space? - What is the probability the computer produces the first letter of your first name? Exercise $\PageIndex{6}$ What is the probability of selecting a random month of the year and getting a month that starts with the letter “J?” If you get stuck, consider listing the sample space. Exercise $\PageIndex{7}$ $E$ represents an object’s weight on Earth and $M$ represents that same object’s weight on the Moon. The equation $M=\frac{1}{6}E$ represents the relationship between these quantities. - What does the $\frac{1}{6}$ represent in this situation? - Give an example of what a person might weigh on Earth and on the Moon. (From Unit 2.2.1) Exercise $\PageIndex{8}$ Here is a diagram of the base of a bird feeder which is in the shape of a pentagonal prism. Each small square on the grid is 1 square inch. The distance between the two bases is 8 inches. What will be the volume of the completed bird feeder? (From Unit 7.3.3) Exercise $\PageIndex{9}$ Find the surface area of the triangular prism. (From Unit 7.3.4)	libretexts	2025-03-17T19:52:18.951003	2020-05-11T07:52:43	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.3%3A_What_are_Probabilities", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.1.3: What are Probabilities?", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.4%3A_Estimating_Probabilities_Through_Repeated_Experiments	8.1.4: Estimating Probabilities Through Repeated Experiments Lesson Let's do some experimenting. Exercise $\PageIndex{1}$: Decimals on the Number Line - Locate and label these numbers on the number line. - $0.5$ - $0.75$ - $0.33$ - $0.67$ - $0.25$ - Choose one of the numbers from the previous question. Describe a game in which that number represents your probability of winning. Exercise $\PageIndex{2}$: In the Long Run Mai plays a game in which she only wins if she rolls a 1 or a 2 with a standard number cube. - List the outcomes in the sample space for rolling the number cube. - What is the probability Mai will win the game? Explain your reasoning. - If Mai is given the option to flip a coin and win if it comes up heads, is that a better option for her to win? - Begin by dragging the gray bar below the toolbar down the screen until you see the table in the top window and the graph in the bottom window. This applet displays a random number from 1 to 6, like a number cube. Mai won with the numbers 1 and 2, but you can choose any two numbers from 1 to 6. Record them in the boxes in the center of the applet. Click the Roll button for 10 rolls and answer the questions. - What appears to be happening with the points on the graph? - - After 10 rolls, what fraction of the total rolls were a win? - How close is this fraction to the probability that Mai will win? - Roll the number cube 10 more times. Record your results in the table and on the graph from earlier. - - After 20 rolls, what fraction of the total rolls were a win? - How close is this fraction to the probability that Mai will win? Exercise $\PageIndex{3}$: Due For A Win - For each situation, do you think the result is surprising or not? Is it possible? Be prepared to explain your reasoning. - You flip the coin once, and it lands heads up. - You flip the coin twice, and it lands heads up both times. - You flip the coin 100 times, and it lands heads up all 100 times. - If you flip the coin 100 times, how many times would you expect the coin to land heads up? Explain your reasoning. - If you flip the coin 100 times, what are some other results that would not be surprising? - You’ve flipped the coin 3 times, and it has come up heads once. The cumulative fraction of heads is currently $\frac{1}{3}$. If you flip the coin one more time, will it land heads up to make the cumulative fraction $\frac{2}{4}$? Summary A probabilityfor an event represents the proportion of the time we expect that event to occur in the long run. For example, the probability of a coin landing heads up after a flip is $\frac{1}{2}$, which means that if we flip a coin many times, we expect that it will land heads up about half of the time. Even though the probability tells us what we should expect if we flip a coin many times, that doesn't mean we are more likely to get heads if we just got three tails in a row. The chances of getting heads are the same every time we flip the coin, no matter what the outcome was for past flips. Glossary Entries Definition: Chance Experiment A chance experiment is something you can do over and over again, and you don’t know what will happen each time. For example, each time you spin the spinner, it could land on red, yellow, blue, or green. Definition: Event An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes. Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.” Definition: Outcome An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails. Definition: Probability The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen. For example, the probability of selecting a moon block at random from this bag is $\frac{4}{5}$. Definition: Random Outcomes of a chance experiment are random if they are all equally likely to happen. Definition: Sample Space The sample space is the list of every possible outcome for a chance experiment. For example, the sample space for tossing two coins is: \| heads-heads \| tails-heads \| \| heads-tails \| tails-tails \| Practice Exercise $\PageIndex{4}$ A carnival game has 160 rubber ducks floating in a pool. The person playing the game takes out one duck and looks at it. - If there’s a red mark on the bottom of the duck, the person wins a small prize. - If there’s a blue mark on the bottom of the duck, the person wins a large prize. - Many ducks do not have a mark. After 50 people have played the game, only 3 of them have won a small prize, and none of them have won a large prize. Estimate the number of the 160 ducks that you think have red marks on the bottom. Then estimate the number of ducks you think have blue marks. Explain your reasoning. Exercise $\PageIndex{5}$ Lin wants to know if flipping a quarter really does have a probability of $\frac{1}{2}$ of landing heads up, so she flips a quarter 10 times. It lands heads up 3 times and tails up 7 times. Has she proven that the probability is not $\frac{1}{2}$? Explain your reasoning. Exercise $\PageIndex{6}$ A spinner has four equal sections, with one letter from the word “MATH” in each section. - You spin the spinner 20 times. About how many times do you expect it will land on A? - You spin the spinner 80 times. About how many times do you expect it will land on something other than A? Explain your reasoning. Exercise $\PageIndex{7}$ A spinner is spun 40 times for a game. Here is a graph showing the fraction of games that are wins under some conditions. Estimate the probability of a spin winning this game based on the graph. Exercise $\PageIndex{8}$ Which event is more likely: rolling a standard number cube and getting an even number, or flipping a coin and having it land heads up? (From Unit 8.1.2) Exercise $\PageIndex{9}$ Noah will select a letter at random from the word “FLUTE.” Lin will select a letter at random from the word “CLARINET.” Which person is more likely to pick the letter “E?” Explain your reasoning. (From Unit 8.1.3)	libretexts	2025-03-17T19:52:19.027395	2020-05-11T07:52:13	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.4%3A_Estimating_Probabilities_Through_Repeated_Experiments", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.1.4: Estimating Probabilities Through Repeated Experiments", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.5%3A_More_Estimating_Probabilities	8.1.5: More Estimating Probabilities Lesson Let's estimate some probabilities. Exercise $\PageIndex{1}$: Is it Likely? - If the weather forecast calls for a 20% chance of light rain tomorrow, would you say that it is likely to rain tomorrow? - If the probability of a tornado today is $\frac{1}{10}$, would you say that there will likely be a tornado today? - If the probability of snow this week is 0.85, would you say that it is likely to snow this week? Exercise $\PageIndex{2}$: Making My Head Spin Work with your group to decide which person will use each spinner. Make sure each person selects a different spinner. Answer the first set of questions on your own. Spinner A Spinner B Spinner C Spinner D - Spin your spinner 10 times and record your outcomes. - Did you get all of the different possible outcomes in your 10 spins? - What fraction of your 10 spins landed on 3? Work with your group to answer the next set of questions. - Share your outcomes with your group and record their outcomes. - Outcomes for spinner A: - Outcomes for spinner B: - Outcomes for spinner C: - Outcomes for spinner D: - Do any of the spinners have the same sample space? If so, do they have the same probabilities for each number to result? - For each spinner, what is the probability that it ends on the number 3? Explain or show your reasoning. - For each spinner, what is the probability that it lands on something other than the number 3? Explain or show your reasoning. - Noah put spinner D on top of his closed binder and spun it 10 times. It never landed on the number 1. How might you explain why this happened? - Han put spinner C on the floor and spun it 10 times. It never landed on the number 3, so he says that the probability of getting a 3 is 0. How might you explain why this happened? Are you ready for more? Design a spinner that has a $\frac{2}{3}$ probability of landing on the number 3. Explain how you could precisely draw this spinner. Exercise $\PageIndex{3}$: How much Green? Your teacher will give you a bag of blocks that are different colors. Do not look into the bag or take out more than 1 block at a time. Repeat these steps until everyone in your group has had 4 turns. - Take one block out of the bag and record whether or not it is green. - Put the block back into the bag, and shake the bag to mix up the blocks. - Pass the bag to the next person in the group. - What do you think is the probability of taking out a green block from this bag? Explain or show your reasoning. - How could you get a better estimate without opening the bag? Summary Suppose a bag contains 5 blocks. If we select a block at random from the bag, then the probability of getting any one of the blocks is $\frac{1}{5}$. Now suppose a bag contains 5 blocks. Some of the blocks have a star, and some have a moon. If we select a block from the bag, then we will either get a star block or a moon block. The probability of getting a star block depends on how many there are in the bag. In this example, the probability of selecting a star block at random from the first bag is $\frac{1}{5}$, because it contains only 1 star block. (The probability of getting a moon block is $\frac{4}{5}$.) The probability of selecting a star block at random from the second bag is $\frac{3}{5}$, because it contains 3 star blocks. (The probability of getting a moon block from this bag is $\frac{2}{5}$.) This shows that two experiments can have the same sample space, but different probabilities for each outcome. Glossary Entries Definition: Chance Experiment A chance experiment is something you can do over and over again, and you don’t know what will happen each time. For example, each time you spin the spinner, it could land on red, yellow, blue, or green. Definition: Event An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes. Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.” Definition: Outcome An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails. Definition: Probability The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen. For example, the probability of selecting a moon block at random from this bag is $\frac{4}{5}$. Definition: Random Outcomes of a chance experiment are random if they are all equally likely to happen. Definition: Sample Space The sample space is the list of every possible outcome for a chance experiment. For example, the sample space for tossing two coins is: \| heads-heads \| tails-heads \| \| heads-tails \| tails-tails \| Practice Exercise $\PageIndex{4}$ What is the same about these two experiments? What is different? - Selecting a letter at random from the word “ALABAMA” - Selecting a letter at random from the word “LAMB” Exercise $\PageIndex{5}$ Andre picks a block out of a bag 60 times and notes that 43 of them were green. - What should Andre estimate for the probability of picking out a green block from this bag? - Mai looks in the bag and sees that there are 6 blocks in the bag. Should Andre change his estimate based on this information? If so, what should the new estimate be? If not, explain your reasoning. Exercise $\PageIndex{6}$ Han has a number cube that he suspects is not so standard. - Han rolls the cube 100 times, and it lands on a six 40 times. - Kiran rolls the cube 50 times, and it lands on a six 21 times. - Lin rolls the cube 30 times, and it lands on a six 11 times. Based on these results, is there evidence to help prove that this cube is not a standard number cube? Explain your reasoning. Exercise $\PageIndex{7}$ A textbook has 428 pages numbered in order starting with 1. You flip to a random page in the book in a way that it is equally likely to stop at any of the pages. - What is the sample space for this experiment? - What is the probability that you turn to page 45? - What is the probability that you turn to an even numbered page? - If you repeat this experiment 50 times, about how many times do you expect you will turn to an even numbered page? (From Unit 8.1.3) Exercise $\PageIndex{8}$ A rectangular prism is cut along a diagonal on each face to create two triangular prisms. The distance between $A$ and $B$ is 5 inches. What is the surface area of the original rectangular prism? What is the total surface area of the two triangular prisms together? (From Unit 7.3.5)	libretexts	2025-03-17T19:52:19.104925	2020-05-11T07:51:39	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.5%3A_More_Estimating_Probabilities", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.1.5: More Estimating Probabilities", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.6%3A_Estimating_Probabilities_Using_Simulation	8.1.6: Estimating Probabilities Using Simulation Lesson Let's simulate real-world situations. Exercise $\PageIndex{1}$: Which One Doesn't Belong: Spinners Which spinner doesn't belong? Exercise $\PageIndex{2}$: Deigo's Walk Your teacher will give your group the supplies for one of the three different simulations. Follow these instructions to simulate 15 days of Diego’s walk. The first 3 days have been done for you. - Simulate one day: - If your group gets a bag of papers, reach into the bag, and select one paper without looking inside. - If your group gets a spinner, spin the spinner, and see where it stops. - If your group gets two number cubes, roll both cubes, and add the numbers that land face up. A sum of 2–8 means Diego has to wait. - Record in the table whether or not Diego had to wait more than 1 minute. - Calculate the total number of days and the cumulative fraction of days that Diego has had to wait so far. - On the graph, plot the number of days and the fraction that Diego has had to wait. Connect each point by a line. - If your group has the bag of papers, put the paper back into the bag, and shake the bag to mix up the papers. - Pass the supplies to the next person in the group. \| day \| Does Diego have to wait more than 1 minute? \| total number of days Diego had to wait \| fraction of days Diego had to wait \| \|---\|---\|---\|---\| \| 1 \| no \| 0 \| $\frac{0}{1}=0.00$ \| \| 2 \| yes \| 1 \| $\frac{1}{2}=0.50$ \| \| 3 \| yes \| 2 \| $\frac{2}{3}\approx 0.67$ \| \| 4 \| \|\|\| \| 5 \| \|\|\| \| 6 \| \|\|\| \| 7 \| \|\|\| \| 8 \| \|\|\| \| 9 \| \|\|\| \| 10 \| \|\|\| \| 11 \| \|\|\| \| 12 \| \|\|\| \| 13 \| \|\|\| \| 14 \| \|\|\| \| 15 \| - Based on the data you have collected, do you think the fraction of days Diego has to wait after the 16th day will be closer to 0.9 or 0.7? Explain or show your reasoning. - Continue the simulation for 10 more days. Record your results in this table and on the graph from earlier. day Does Diego have to wait more than 1 minute? total number of days Diego had to wait fraction of days Diego had to wait 16 17 18 19 20 21 22 23 24 25 Table $\PageIndex{2}$ - What do you notice about the graph? - Based on the graph, estimate the probability that Diego will have to wait more than 1 minute to cross the crosswalk. Are you ready for more? Let's look at why the values tend to not change much after doing the simulation many times. - After doing the simulation 4 times, a group finds that Diego had to wait 3 times. What is an estimate for the probability Diego has to wait based on these results? - If this group does the simulation 1 more time, what are the two possible outcomes for the fifth simulation? - For each possibility, estimate the probability Diego has to wait. - What are the differences between the possible estimates after 5 simulations and the estimate after 4 simulations? - After doing the simulation 20 times, this group finds that Diego had to wait 15 times. What is an estimate for the probability Diego has to wait based on these results? - If this group does the simulation 1 more time, what are the two possible outcomes for the twenty-first simulation? - For each possibility, estimate the probability Diego has to wait. - What are the differences between the possible estimates after 21 simulations and the estimate after 20 simulations? - Use these results to explain why a single result after many simulations does not affect the estimate as much as a single result after only a few simulations. Exercise $\PageIndex{3}$: Designing Experiments For each situation, describe a chance experiment that would fairly represent it. - Six people are going out to lunch together. One of them will be selected at random to choose which restaurant to go to. Who gets to choose? - After a robot stands up, it is equally likely to step forward with its left foot or its right foot. Which foot will it use for its first step? - In a computer game, there are three tunnels. Each time the level loads, the computer randomly selects one of the tunnels to lead to the castle. Which tunnel is it? - Your school is taking 4 buses of students on a field trip. Will you be assigned to the same bus that your math teacher is riding on? Summary Sometimes it is easier to estimate a probability by doing a simulation . A simulation is an experiment that approximates a situation in the real world. Simulations are useful when it is hard or time-consuming to gather enough information to estimate the probability of some event. For example, imagine Andre has to transfer from one bus to another on the way to his music lesson. Most of the time he makes the transfer just fine, but sometimes the first bus is late and he misses the second bus. We could set up a simulation with slips of paper in a bag. Each paper is marked with a time when the first bus arrives at the transfer point. We select slips at random from the bag. After many trials, we calculate the fraction of the times that he missed the bus to estimate the probability that he will miss the bus on a given day. Glossary Entries Definition: Chance Experiment A chance experiment is something you can do over and over again, and you don’t know what will happen each time. For example, each time you spin the spinner, it could land on red, yellow, blue, or green. Definition: Event An event is a set of one or more outcomes in a chance experiment. For example, if we roll a number cube, there are six possible outcomes. Examples of events are “rolling a number less than 3,” “rolling an even number,” or “rolling a 5.” Definition: Outcome An outcome of a chance experiment is one of the things that can happen when you do the experiment. For example, the possible outcomes of tossing a coin are heads and tails. Definition: Probability The probability of an event is a number that tells how likely it is to happen. A probability of 1 means the event will always happen. A probability of 0 means the event will never happen. For example, the probability of selecting a moon block at random from this bag is $\frac{4}{5}$. Definition: Random Outcomes of a chance experiment are random if they are all equally likely to happen. Definition: Sample Space The sample space is the list of every possible outcome for a chance experiment. For example, the sample space for tossing two coins is: \| heads-heads \| tails-heads \| \| heads-tails \| tails-tails \| Definition: Simulation A simulation is an experiment that is used to estimate the probability of a real-world event. For example, suppose the weather forecast says there is a 25% chance of rain. We can simulate this situation with a spinner with four equal sections. If the spinner stops on red, it represents rain. If the spinner stops on any other color, it represents no rain. Practice Exercise $\PageIndex{4}$ The weather forecast says there is a 75% chance it will rain later today. - Draw a spinner you could use to simulate this probability. - Describe another way you could simulate this probability. Exercise $\PageIndex{5}$ An experiment will produce one of ten different outcomes with equal probability for each. Why would using a standard number cube to simulate the experiment be a bad choice? Exercise $\PageIndex{6}$ An ice cream shop offers 40 different flavors. To simulate the most commonly chosen flavor, you could write the name of each flavor on a piece of paper and put it in a bag. Draw from the bag 100 times, and see which flavor is chosen the most. This simulation is not a good way to figure out the most-commonly chosen flavor. Explain why. Exercise $\PageIndex{7}$ Each set of three numbers represents the lengths, in units, of the sides of a triangle. Which set can not be used to make a triangle? - $7, 6, 14$ - $4, 4, 4$ - $6, 6, 2$ - $7, 8,13$ (From Unit 7.2.2) Exercise $\PageIndex{8}$ There is a proportional relationship between a volume measured in cups and the same volume measured in tablespoons. 48 tablespoons is equivalent to 3 cups, as shown in the graph. - Plot and label some more points that represent the relationship. - Use a straightedge to draw a line that represents this proportional relationship. - For which value $y$ is ($1,y$) on the line you just drew? - What is the constant of proportionality for this relationship? - Write an equation representing this relationship. Use $c$ for cups and $t$ for tablespoons. (From Unit 2.5.1)	libretexts	2025-03-17T19:52:19.200526	2020-05-11T07:50:29	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.01%3A_New_Page/8.1.6%3A_Estimating_Probabilities_Using_Simulation", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.1.6: Estimating Probabilities Using Simulation", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page	8.2: Probabilites of Multi-step Events Last updated Save as PDF Page ID 35027 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:19.272228	2020-01-25T01:41:50	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.2: Probabilites of Multi-step Events", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.1%3A_Simulating_Multi-step_Experiments	8.2.1: Simulating Multi-step Experiments Lesson Let's simulate more complicated events. Exercise $\PageIndex{1}$: Notice and Wonder: Ski Business What do you notice? What do you wonder? Exercise $\PageIndex{2}$: Alpine Zoom Alpine Zoom is a ski business. To make money over spring break, they need it to snow at least 4 out of the 10 days. The weather forecast says there is a $\frac{1}{3}$ chance it will snow each day during the break. Use the applet to simulate the weather for 10 days of break to see if Alpine Zoom will make money. - Describe a chance experiment that you could use to simulate whether it will snow on the first day of break. - How could this chance experiment be used to determine whether Alpine Zoom will make money? - In each trial, spin the spinner 10 times, and then record the number of 1’s that appeared in the row. - The applet reports if the Alpine Zoom will make money or not in the last column. - Click Next to get the spin button back to start the next simulation. - Based on your simulations, estimate the probability that Alpine Zoom will make money. Exercise $\PageIndex{3}$: Kiran's Game Kiran invents a game that uses a board with alternating black and white squares. A playing piece starts on a white square and must advance 4 squares to the other side of the board within 5 turns to win the game. For each turn, the player draws a block from a bag containing 2 black blocks and 2 white blocks. If the block color matches the color of the next square on the board, the playing piece moves onto it. If it does not match, the playing piece stays on its current square. - Take turns playing the game until each person in your group has played the game twice. - Use the results from all the games your group played to estimate the probability of winning Kiran’s game. - Do you think your estimate of the probability of winning is a good estimate? How could it be improved? Are you ready for more? How would each of these changes, on its own, affect the probability of winning the game? - Change the rules so that the playing piece must move 7 spaces within 8 moves. - Change the board so that all the spaces are black. - Change the blocks in the bag to 3 black blocks and 1 white block. Exercise $\PageIndex{4}$: Simulation Nation Match each situation to a simulation . Situations: - In a small lake, 25% of the fish are female. You capture a fish, record whether it is male or female, and toss the fish back into the lake. If you repeat this process 5 times, what is the probability that at least 3 of the 5 fish are female? - Elena makes about 80% of her free throws. Based on her past successes with free throws, what is the probability that she will make exactly 4 out of 5 free throws in her next basketball game? - On a game show, a contestant must pick one of three doors. In the first round, the winning door has a vacation. In the second round, the winning door has a car. What is the probability of winning a vacation and a car? - Your choir is singing in 4 concerts. You and one of your classmates both learned the solo. Before each concert, there is an equal chance the choir director will select you or the other student to sing the solo. What is the probability that you will be selected to sing the solo in exactly 3 of the 4 concerts? Simulations: - Toss a standard number cube 2 times and record the outcomes. Repeat this process many times and find the proportion of the simulations in which a 1 or 2 appeared both times to estimate the probability. - Make a spinner with four equal sections labeled 1, 2, 3, and 4. Spin the spinner 5 times and record the outcomes. Repeat this process many times and find the proportion of the simulations in which a 4 appears 3 or more times to estimate the probability. - Toss a fair coin 4 times and record the outcomes. Repeat this process many times, and find the proportion of the simulations in which exactly 3 heads appear to estimate the probability. - Place 8 blue chips and 2 red chips in a bag. Shake the bag, select a chip, record its color, and then return the chip to the bag. Repeat the process 4 more times to obtain a simulated outcome. Then repeat this process many times and find the proportion of the simulations in which exactly 4 blues are selected to estimate the probability. Summary The more complex a situation is, the harder it can be to estimate the probability of a particular event happening. Well-designed simulations are a way to estimate a probability in a complex situation, especially when it would be difficult or impossible to determine the probability from reasoning alone. To design a good simulation, we need to know something about the situation. For example, if we want to estimate the probability that it will rain every day for the next three days, we could look up the weather forecast for the next three days. Here is a table showing a weather forecast: \| today (Tuesday) \| Wednesday \| Thursday \| Friday \| \| \|---\|---\|---\|---\|---\| \| probability of rain \| $0.2$ \| $0.4$ \| $0.5$ \| $0.9$ \| We can set up a simulation to estimate the probability of rain each day with three bags. - In the first bag, we put 4 slips of paper that say “rain” and 6 that say “no rain.” - In the second bag, we put 5 slips of paper that say “rain” and 5 that say “no rain.” - In the third bag, we put 9 slips of paper that say “rain” and 1 that says “no rain.” Then we can select one slip of paper from each bag and record whether or not there was rain on all three days. If we repeat this experiment many times, we can estimate the probability that there will be rain on all three days by dividing the number of times all three slips said “rain” by the total number of times we performed the simulation. Practice Exercise $\PageIndex{5}$ Priya’s cat is pregnant with a litter of 5 kittens. Each kitten has a 30% chance of being chocolate brown. Priya wants to know the probability that at least two of the kittens will be chocolate brown. To simulate this, Priya put 3 white cubes and 7 green cubes in a bag. For each trial, Priya pulled out and returned a cube 5 times. Priya conducted 12 trials. Here is a table with the results. \| trial number \| outcome \| \|---\|---\| \| 1 \| ggggg \| \| 2 \| gggwg \| \| 3 \| wgwgw \| \| 4 \| gwggg \| \| 5 \| gggwg \| \| 6 \| wwggg \| \| 7 \| gwggg \| \| 8 \| ggwgw \| \| 9 \| wwwgg \| \| 10 \| ggggw \| \| 11 \| wggwg \| \| 12 \| gggwg \| - How many successful trials were there? Describe how you determined if a trial was a success. - Based on this simulation, estimate the probability that exactly two kittens will be chocolate brown. - Based on this simulation, estimate the probability that at least two kittens will be chocolate brown. - Write and answer another question Priya could answer using this simulation. - How could Priya increase the accuracy of the simulation? Exercise $\PageIndex{6}$ A team has a 75% chance to win each of the 3 games they will play this week. Clare simulates the week of games by putting 4 pieces of paper in a bag, 3 labeled “win” and 1 labeled “lose.” She draws a paper, writes down the result, then replaces the paper and repeats the process two more times. Clare gets the result: win, win, lose. What can Clare do to estimate the probability the team will win at least 2 games? Exercise $\PageIndex{7}$ - List the sample space for selecting a letter a random from the word “PINEAPPLE.” - A letter is randomly selected from the word “PINEAPPLE.” Which is more likely, selecting “E” or selecting “P?” Explain your reasoning. (From Unit 8.1.5) Exercise $\PageIndex{8}$ On a graph of side length of a square vs. its perimeter, a few points are plotted. 1. Add at least two more ordered pairs to the graph. 2. Is there a proportional relationship between the perimeter and side length? Explain how you know. (From Unit 2.4.2)	libretexts	2025-03-17T19:52:19.352171	2020-05-11T07:57:02	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.1%3A_Simulating_Multi-step_Experiments", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.2.1: Simulating Multi-step Experiments", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.2%3A_Keeping_Track_of_All_Possible_Outcomes	8.2.2: Keeping Track of All Possible Outcomes Lesson Lets explore sample spaces for experiments with multiple parts. Exercise $\PageIndex{1}$: How Many Different Meals? How many different meals are possible if each meal includes one main course, one side dish, and one drink? \| main courses \| side dishes \| drinks \| \|---\|---\|---\| \| grilled chicken \| salad \| milk \| \| turkey sandwich \| applesauce \| juice \| \| pasta salad \| $\underline{\quad}$ \| water \| Exercise $\PageIndex{2}$: Lists, Tables, and Trees Consider the experiment: Flip a coin, and then roll a number cube. Elena, Kiran, and Priya each use a different method for finding the sample space of this experiment. - Elena carefully writes a list of all the options: Heads 1, Heads 2, Heads 3, Heads 4, Heads 5, Heads 6, Tails 1, Tails 2, Tails 3, Tails 4, Tails 5, Tails 6. - Kiran makes a table: 1 2 3 4 5 6 H H1 H2 H3 H4 H5 H6 T T1 T2 T3 T4 T5 T6 Table $\PageIndex{2}$ - Priya draws a tree with branches in which each pathway represents a different outcome: - Compare the three methods. What is the same about each method? What is different? Be prepared to explain why each method produces all the different outcomes without repeating any. - Which method do you prefer for this situation? Pause here so your teacher can review your work. - Find the sample space for each of these experiments using any method. Make sure you list every possible outcome without repeating any. - Flip a dime, then flip a nickel, and then flip a penny. Record whether each lands heads or tails up. - Han’s closet has: a blue shirt, a gray shirt, a white shirt, blue pants, khaki pants, and black pants. He must select one shirt and one pair of pants to wear for the day. - Spin a color, and then spin a number. d. Spin the hour hand on an analog clock, and then choose a.m. or p.m. Exercise $\PageIndex{3}$: How Many Sandwiches? 1. A submarine sandwich shop makes sandwiches with one kind of bread, one protein, one choice of cheese, and two vegetables. How many different sandwiches are possible? Explain your reasoning. You do not need to write out the sample space. - Breads: Italian, white, wheat - Proteins: Tuna, ham, turkey, beans - Cheese: Provolone, Swiss, American, none - Vegetables: Lettuce, tomatoes, peppers, onions, pickles 2. Andre knows he wants a sandwich that has ham, lettuce, and tomatoes on it. He doesn’t care about the type of bread or cheese. How many of the different sandwiches would make Andre happy? 3. If a sandwich is made by randomly choosing each of the options, what is the probability it will be a sandwich that Andre would be happy with? Are you ready for more? Describe a situation that involves three parts and has a total of 24 outcomes in the sample space. Summary Sometimes we need a systematic way to count the number of outcomes that are possible in a given situation. For example, suppose there are 3 people (A, B, and C) who want to run for the president of a club and 4 different people (1, 2, 3, and 4) who want to run for vice president of the club. We can use a tree , a table , or an ordered list to count how many different combinations are possible for a president to be paired with a vice president. With a tree, we can start with a branch for each of the people who want to be president. Then for each possible president, we add a branch for each possible vice president, for a total of $3\cdot 4=12$ possible pairs. We can also start by counting vice presidents first and then adding a branch for each possible president, for a total of $3\cdot 4=12$ possible pairs. A table can show the same result: \| 1 \| 2 \| 3 \| 4 \| \| \|---\|---\|---\|---\|---\| \| A \| $A, 1$ \| $A, 2$ \| $A, 3$ \| $A, 4$ \| \| B \| $B, 1$ \| $B, 2$ \| $B, 3$ \| $B, 4$ \| \| C \| $C, 1$ \| $C, 2$ \| $C, 3$ \| $C, 4$ \| So does this ordered list: A1, A2, A3, A4, B1, B2, B3, B4, C1, C2, C3, C4 Practice Exercise $\PageIndex{4}$ Noah is planning his birthday party. Here is a tree showing all of the possible themes, locations, and days of the week that Noah is considering. - How many themes is Noah considering? - How many locations is Noah considering? - How many days of the week is Noah considering? - One possibility that Noah is considering is a party with a space theme at the skating rink on Sunday. Write two other possible parties Noah is considering. - How many different possible outcomes are in the sample space? Exercise $\PageIndex{5}$ For each event, write the sample space and tell how many outcomes there are. - Lin selects one type of lettuce and one dressing to make a salad. Lettuce types: iceberg, romaine Dressings: ranch, Italian, French - Diego chooses rock, paper, or scissors, and Jada chooses rock, paper, or scissors. - Spin these 3 spinners. Exercise $\PageIndex{6}$ A simulation is done to represent kicking 5 field goals in a single game with a 72% probability of making each one. A 1 represents making the kick and a 0 represents missing the kick. \| trial \| result \| \|---\|---\| \| 1 \| 10101 \| \| 2 \| 11010 \| \| 3 \| 00011 \| \| 4 \| 11111 \| \| 5 \| 10011 \| (From Unit 8.2.1) Exercise $\PageIndex{7}$ There is a bag of 50 marbles. - Andre takes out a marble, records its color, and puts it back in. In 4 trials, he gets a green marble 1 time. - Jada takes out a marble, records its color, and puts it back in. In 12 trials, she gets a green marble 5 times. - Noah takes out a marble, records its color, and puts it back in. In 9 trials, he gets a green marble 3 times. Estimate the probability of getting a green marble from this bag. Explain your reasoning. (From Unit 8.1.4)	libretexts	2025-03-17T19:52:19.508289	2020-05-11T07:56:29	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.2%3A_Keeping_Track_of_All_Possible_Outcomes", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.2.2: Keeping Track of All Possible Outcomes", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.3%3A_Multi-step_Experiments	8.2.3: Multi-step Experiments Lesson Let's look at probabilities of experiments that have multiple steps. Exercise $\PageIndex{1}$: True or False? Is each equation true or false? Explain your reasoning. $8=(8+8+8+8)\div 3$ $(10+10+10+10+10)\div 5=10$ $6+4+6+4+6+4)\div 6=5$ Exercise $\PageIndex{2}$: Spinning a Color and Number The other day, you wrote the sample space for spinning each of these spinners once. What is the probability of getting: - Green and 3? - Blue and any odd number? - Any color other than red and any number other than 2? Exercise $\PageIndex{3}$: Cubes and Coins The other day you looked at a list, a table, and a tree that showed the sample space for rolling a number cube and flipping a coin. - Your teacher will assign you one of these three structures to use to answer these questions. Be prepared to explain your reasoning. - What is the probability of getting tails and a 6? - What is the probability of getting heads and an odd number? Pause here so your teacher can review your work. - Suppose you roll two number cubes. What is the probability of getting: - Both cubes showing the same number? - Exactly one cube showing an even number? - At least one cube showing an even number? - Two values that have a sum of 8? - Two values that have a sum of 13? - Jada flips three quarters. What is the probability that all three will land showing the same side? Exercise $\PageIndex{4}$: Pick a Card Imagine there are 5 cards. They are colored red, yellow, green, white, and black. You mix up the cards and select one of them without looking. Then, without putting that card back, you mix up the remaining cards and select another one. - Write the sample space and tell how many possible outcomes there are. - What structure did you use to write all of the outcomes (list, table, tree, something else)? Explain why you chose that structure. - What is the probability that: - You get a white card and a red card (in either order)? - You get a black card (either time)? - You do not get a black card (either time)? - You get a blue card? - You get 2 cards of the same color? - You get 2 cards of different colors? Are you ready for more? In a game using five cards numbered 1, 2, 3, 4, and 5, you take two cards and add the values together. If the sum is 8, you win. Would you rather pick a card and put it back before picking the second card, or keep the card in your hand while you pick the second card? Explain your reasoning. Summary Suppose we have two bags. One contains 1 star block and 4 moon blocks. The other contains 3 star blocks and 1 moon block. If we select one block at random from each, what is the probability that we will get two star blocks or two moon blocks? To answer this question, we can draw a tree diagram to see all of the possible outcomes. There are $5\cdot 4=20$ possible outcomes. Of these, 3 of them are both stars, and 4 are both moons. So the probability of getting 2 star blocks or 2 moon blocks is $\frac{7}{20}$. In general, if all outcomes in an experiment are equally likely, then the probability of an event is the fraction of outcomes in the sample space for which the event occurs. Practice Exercise $\PageIndex{5}$ A vending machine has 5 colors (white, red, green, blue, and yellow) of gumballs and an equal chance of dispensing each. A second machine has 4 different animal-shaped rubber bands (lion, elephant, horse, and alligator) and an equal chance of dispensing each. If you buy one item from each machine, what is the probability of getting a yellow gumball and a lion band? Exercise $\PageIndex{6}$ The numbers 1 through 10 are put in one bag. The numbers 5 through 14 are put in another bag. When you pick one number from each bag, what is the probability you get the same number? Exercise $\PageIndex{7}$ When rolling 3 standard number cubes, the probability of getting all three numbers to match is $\frac{6}{216}$. What is the probability that the three numbers do not all match? Explain your reasoning. Exercise $\PageIndex{8}$ For each event, write the sample space and tell how many outcomes there are. - Roll a standard number cube. Then flip a quarter. - Select a month. Then select 2020 or 2025. (From Unit 8.2.2) Exercise $\PageIndex{9}$ On a graph of the area of a square vs. its perimeter, a few points are plotted. - Add some more ordered pairs to the graph. - Is there a proportional relationship between the area and perimeter of a square? Explain how you know. (From Unit 2.4.2)	libretexts	2025-03-17T19:52:19.577809	2020-05-11T07:55:52	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.3%3A_Multi-step_Experiments", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.2.3: Multi-step Experiments", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.4%3A_Designing_Simulations	8.2.4: Designing Simulations Lesson Let's simulate some real-life scenarios. Exercise $\PageIndex{1}$: Number Talk: Division Find the value of each expression mentally. $(4.2+3)\div 2$ $(4.2+2.6+4)\div 3$ $(4.2+2.6+4+3.6)\div 4$ $(4.2+2.6+4+3.6+3.6)\div 5$ Exercise $\PageIndex{2}$: Breeding Mice A scientist is studying the genes that determine the color of a mouse’s fur. When two mice with brown fur breed, there is a 25% chance that each baby will have white fur. For the experiment to continue, the scientist needs at least 2 out of 5 baby mice to have white fur. To simulate this situation, you can flip two coins at the same time for each baby mouse. If you don't have coins, you can use this applet. - If both coins land heads up, it represents a mouse with white fur. - Any other result represents a mouse with brown fur. - Have each person in the group simulate a litter of 5 offspring and record their results. Next, determine whether at least 2 of the offspring have white fur. mouse 1 mouse 2 mouse 3 mouse 4 mouse 5 Do at least 2 have white fur? simulation 1 simulation 2 simulation 3 Table $\PageIndex{1}$ - Based on the results from everyone in your group, estimate the probability that the scientist’s experiment will be able to continue. - How could you improve your estimate? Are you ready for more? For a certain pair of mice, the genetics show that each offspring has a probability of $\frac{1}{16}$ that they will be albino. Describe a simulation you could use that would estimate the probability that at least 2 of the 5 offspring are albino. Exercise $\PageIndex{3}$: Designing Simulations Your teacher will give your group a paper describing a situation. - Design a simulation that you could use to estimate a probability. Show your thinking. Organize it so it can be followed by others. - Explain how you used the simulation to answer the questions posed in the situation. Summary Many real-world situations are difficult to repeat enough times to get an estimate for a probability. If we can find probabilities for parts of the situation, we may be able to simulate the situation using a process that is easier to repeat. For example, if we know that each egg of a fish in a science experiment has a 13% chance of having a mutation, how many eggs do we need to collect to make sure we have 10 mutated eggs? If getting these eggs is difficult or expensive, it might be helpful to have an idea about how many eggs we need before trying to collect them. We could simulate this situation by having a computer select random numbers between 1 and 100. If the number is between 1 and 13, it counts as a mutated egg. Any other number would represent a normal egg. This matches the 13% chance of each fish egg having a mutation. We could continue asking the computer for random numbers until we get 10 numbers that are between 1 and 13. How many times we asked the computer for a random number would give us an estimate of the number of fish eggs we would need to collect. To improve the estimate, this entire process should be repeated many times. Because computers can perform simulations quickly, we could simulate the situation 1,000 times or more. Practice Exercise $\PageIndex{4}$ A rare and delicate plant will only produce flowers from 10% of the seeds planted. To see if it is worth planting 5 seeds to see any flowers, the situation is going to be simulated. Which of these options is the best simulation? For the others, explain why it is not a good simulation. - Another plant can be genetically modified to produce flowers 10% of the time. Plant 30 groups of 5 seeds each and wait 6 months for the plants to grow and count the fraction of groups that produce flowers. - Roll a standard number cube 5 times. Each time a 6 appears, it represents a plant producing flowers. Repeat this process 30 times and count the fraction of times at least one number 6 appears. - Have a computer produce 5 random digits (0 through 9). If a 9 appears in the list of digits, it represents a plant producing flowers. Repeat this process 300 times and count the fraction of times at least one number 9 appears. - Create a spinner with 10 equal sections and mark one of them “flowers.” Spin the spinner 5 times to represent the 5 seeds. Repeat this process 30 times and count the fraction of times that at least 1 “flower” was spun. Exercise $\PageIndex{5}$ Jada and Elena learned that 8% of students have asthma. They want to know the probability that in a team of 4 students, at least one of them has asthma. To simulate this, they put 25 slips of paper in a bag. Two of the slips say “asthma.” Next, they take four papers out of the bag and record whether at least one of them says “asthma.” They repeat this process 15 times. - Jada says they could improve the accuracy of their simulation by using 100 slips of paper and marking 8 of them. - Elena says they could improve the accuracy of their simulation by conducting 30 trials instead of 15. - Do you agree with either of them? Explain your reasoning. - Describe another method of simulation the same scenario. Exercise $\PageIndex{6}$ The figure on the left is a trapezoidal prism. The figure on the right represents its base. Find the volume of this prism. (From Unit 7.3.3) Exercise $\PageIndex{7}$ Match each expression in the first list with an equivalent expression from the second list. - $(8x+6y)+(2x+4y)$ - $(8x+6y)-(2x+4y)$ - $(8x+6y)-(2x-4y)$ - $8x-6y-2x+4y$ - $8x-6y+2x-4y$ - $8x-(-6y-2x+4y)$ - $10(x+y)$ - $10(x-y)$ - $6(x-\frac{1}{3}y)$ - $8x+6y+2x-4y$ - $8x+6y-2x+4y$ - $8x-2x+6y-4y$ (From Unit 6.4.5)	libretexts	2025-03-17T19:52:19.650541	2020-05-11T07:54:47	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.02%3A_New_Page/8.2.4%3A_Designing_Simulations", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.2.4: Designing Simulations", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page	Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset + - Margin Size Reset + - Font Type Enable Dyslexic Font Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 8: Probability and Sampling Pre-Algebra I (Illustrative Mathematics - Grade 7) { "8.3.1:_Comparing_Groups" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.3.2:_Larger_Populations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.3.3:_What_Makes_a_Good_Sample" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.3.4:_Sampling_in_a_Fair_Way" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()" } { "8.01:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.02:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.03:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.04:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.05:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()" } { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "01:_Scale_Drawings" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "02:_Introducing_Proportional_Relationships" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "03:_Untitled_Chapter_3" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "04:_Untitled_Chapter_4" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "05:_Untitled_Chapter_5" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "06:_Untitled_Chapter_6" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "07:_Untitled_Chapter_7" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "08:_Untitled_Chapter_8" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "09:_Untitled_Chapter_9" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()" } Sat, 25 Jan 2020 01:41:51 GMT 8.3: Sampling 35028 35028 admin { } Anonymous Anonymous 2 false false [ "article:topic", "license:ccby", "authorname:im" ] [ "article:topic", "license:ccby", "authorname:im" ] https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FPreAlgebra%2FPre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)%2F08%253A_Untitled_Chapter_8%2F8.03%253A_New_Page Search site Search Search Go back to previous article Username Password Sign in Sign in Sign in Forgot password Expand/collapse global hierarchy Home Bookshelves Pre-Algebra Pre-Algebra I (Illustrative Mathematics - Grade 7) 8: Probability and Sampling 8.3: Sampling Expand/collapse global location 8.3: Sampling Last updated Save as PDF Page ID 35028 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:19.723361	2020-01-25T01:41:51	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.3: Sampling", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.1%3A_Comparing_Groups	8.3.1: Comparing Groups Lesson Let's compare two groups. Exercise $\PageIndex{1}$: Notice and Wonder: Comparing Heights What do you notice? What do you wonder? Exercise $\PageIndex{2}$: More Team Heights - How much taller is the volleyball team than the gymnastics team? - Gymnastics team’s heights (in inches) : 56, 59, 60, 62, 62, 63, 63, 63, 64, 64, 68, 69 - Volleyball team’s heights (in inches): 72, 75, 76, 76, 78, 79, 79, 80, 80, 81, 81, 81 - Make dot plots to compare the heights of the tennis and badminton teams. - Tennis team’s heights (in inches): 66, 67, 69, 70, 71, 73, 73, 74, 75, 75, 76 - Badminton team’s heights (in inches): 62, 62, 65, 66, 68, 71, 73 What do you notice about your dot plots? - Elena says the members of the tennis team were taller than the badminton team. Lin disagrees. Do you agree with either of them? Explain or show your reasoning. Exercise $\PageIndex{3}$: Family Heights Compare the heights of these two families. Explain or show your reasoning. - The heights (in inches) of Noah’s family members: 28, 39, 41, 52, 63, 66, 71 - The heights (in inches) of Jada’s family members: 49, 60, 68, 70, 71, 73, 77 Are you ready for more? If Jada's family adopts newborn twins who are each 18 inches tall, does this change your thinking? Explain your reasoning. Exercise $\PageIndex{4}$: Track Length Here are three dot plots that represent the lengths, in minutes, of songs on different albums. - One of these data sets has a mean of 5.57 minutes and another has a mean of 3.91 minutes. - Which dot plot shows each of these data sets? - Calculate the mean for the data set on the other dot plot. - One of these data sets has a mean absolute deviation of 0.30 and another has a MAD of 0.44. - Which dot plot shows each of these data sets? - Calculate the MAD for the other data set. - Do you think the three groups are very different or not? Be prepared to explain your reasoning. - A fourth album has a mean length of 8 minutes with a mean absolute deviation of 1.2. Is this data set very different from each of the others? Summary Comparing two individuals is fairly straightforward. The question "Which dog is taller?" can be answered by measuring the heights of two dogs and comparing them directly. Comparing two groups can be more challenging. What does it mean for the basketball team to generally be taller than the soccer team? To compare two groups, we use the distribution of values for the two groups. Most importantly, a measure of center (usually mean or median ) and its associated measure of variability (usually mean absolute deviation or interquartile range) can help determine the differences between groups. For example, if the average height of pugs in a dog show is 11 inches, and the average height of the beagles in the dog show is 15 inches, it seems that the beagles are generally taller. On the other hand, if the MAD is 3 inches, it would not be unreasonable to find a beagle that is 11 inches tall or a pug that is 14 inches tall. Therefore the heights of the two dog breeds may not be very different from one another. Glossary Entries Definition: Mean The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11. To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. $7+9+12+13+14=55$ and $55\div 5=11$. Definition: Mean Absolute Deviation (MAD) The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11. To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are. $4+2+1+2+3=12$ and $12\div 5=2.4$ Definition: Median The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order. For the data set 7, 9, 12, 13, 14, the median is 12. For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. $6+8=14$ and $14\div 2=7$. Practice Exercise $\PageIndex{5}$ Compare the weights of the backpacks for the students in these three classes. Exercise $\PageIndex{6}$ A bookstore has marked down the price for all the books in a certain series by 15%. - How much is the discount on a book that normally costs $18.00? - After the discount, how much would the book cost? (From Unit 4.3.2) Exercise $\PageIndex{7}$ Match each expression in the first list with an equivalent expression from the second list. - $6(x+2y)-2(y-2x)$ - $2.5(2x+4y)-5(4y-x)$ - $4(5x-3y)-10x+6y$ - $5.5(x+y)-2(x+y)+6.5(x+y)$ - $7.9(5x+3y)-4.2(5x+3y)-1.7(5x+3y)$ - $10(x-y)$ - $10(x+y)$ - $10x+6y$ - $10x-6y$ (From Unit 6.4.5) Exercise $\PageIndex{8}$ Angles $C$ and $D$ are complementary. The ratio of the measure of Angle $C$ to the measure of Angle $D$ is $2:3$. Find the measure of each angle. Explain or show your reasoning. (From Unit 7.1.2)	libretexts	2025-03-17T19:52:19.795887	2020-05-11T07:59:27	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.1%3A_Comparing_Groups", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.3.1: Comparing Groups", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.2%3A_Larger_Populations	8.3.2: Larger Populations Lesson Let's compare larger groups. Exercise $\PageIndex{1}$: First Name versus Last Name Consider the question: In general, do the students at this school have more letters in their first name or last name? How many more letters? - What are some ways you might get some data to answer the question? - The other day, we compared the heights of people on different teams and the lengths of songs on different albums. What makes this question about first and last names harder to answer than those questions? Exercise $\PageIndex{2}$: John Jacobjingleheimerschmidt Continue to consider the question from the warm-up: In general, do the students at this school have more letters in their first name or last name? How many more letters? - How many letters are in your first name? In your last name? - Do the number of letters in your own first and last names give you enough information to make conclusions about students' names in your entire school? Explain your reasoning. - Your teacher will provide you with data from the class. Record the mean number of letters as well as the mean absolute deviation for each data set. - The first names of the students in your class. - The last names of the students in your class. - Which mean is larger? By how much? What does this difference tell you about the situation? - Do the mean numbers of letters in the first and last names for everyone in your class give you enough information to make conclusions about students’ names in your entire school? Explain your reasoning. Exercise $\PageIndex{3}$: Siblings and Pets Consider the question: Do people who are the only child have more pets ? - Earlier, we used information about the people in your class to answer a question about the entire school. Would surveying only the people in your class give you enough information to answer this new question? Explain your reasoning. - If you had to have an answer to this question by the end of class today, how would you gather data to answer the question? - If you could come back tomorrow with your answer to this question, how would you gather data to answer the question? - If someone else in the class came back tomorrow with an answer that was different than yours, what would that mean? How would you determine which answer was better? Exercise $\PageIndex{4}$: Sampling the Population For each question, identify the population and a possible sample . - What is the mean number of pages for novels that were on the best seller list in the 1990s? - What fraction of new cars sold between August 2010 and October 2016 were built in the United States? - What is the median income for teachers in North America? - What is the average lifespan of Tasmanian devils? Are you ready for more? Political parties often use samples to poll people about important issues. One common method is to call people and ask their opinions. In most places, though, they are not allowed to call cell phones. Explain how this restriction might lead to inaccurate samples of the population. Summary A population is a set of people or things that we want to study. Here are some examples of populations: - All people in the world - All seventh graders at a school - All apples grown in the U.S. A sample is a subset of a population. Here are some examples of samples from the listed populations: - The leaders of each country - The seventh graders who are in band - The apples in the school cafeteria When we want to know more about a population but it is not feasible to collect data from everyone in the population, we often collect data from a sample. In the lessons that follow, we will learn more about how to pick a sample that can help answer questions about the entire population. Glossary Entries Definition: Mean The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11. To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. $7+9+12+13+14=55$ and $55\div 5=11$. Definition: Mean Absolute Deviation (MAD) The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11. To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are. $4+2+1+2+3=12$ and $12\div 5=2.4$ Definition: Median The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order. For the data set 7, 9, 12, 13, 14, the median is 12. For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. $6+8=14$ and $14\div 2=7$. Definition: Population A population is a set of people or things that we want to study. For example, if we want to study the heights of people on different sports teams, the population would be all the people on the teams. Definition: Sample A sample is part of a population. For example, a population could be all the seventh grade students at one school. One sample of that population is all the seventh grade students who are in band. Practice Exercise $\PageIndex{5}$ Suppose you are interested in learning about how much time seventh grade students at your school spend outdoors on a typical school day. Select all the samples that are a part of the population you are interested in. - The 20 students in a seventh grade math class. - The first 20 students to arrive at school on a particular day. - The seventh grade students participating in a science fair put on by the four middle schools in a school district. - The 10 seventh graders on the school soccer team. - The students on the school debate team. Exercise $\PageIndex{6}$ For each sample given, list two possible populations they could belong to. - Sample: The prices for apples at two stores near your house. - Sample: The days of the week the students in your math class ordered food during the past week. - Sample: The daily high temperatures for the capital cities of all 50 U.S. states over the past year. Exercise $\PageIndex{7}$ If 6 coins are flipped, find the probability that there is at least 1 heads. (From Unit 8.2.3) Exercise $\PageIndex{8}$ A school's art club holds a bake sale on Fridays to raise money for art supplies. Here are the number of cookies they sold each week in the fall and in the spring: \| fall \| 20 \| 26 \| 25 \| 24 \| 29 \| 20 \| 19 \| 19 \| 24 \| 24 \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| spring \| 19 \| 27 \| 29 \| 21 \| 25 \| 22 \| 26 \| 21 \| 25 \| 25 \| - Find the mean number of cookies sold in the fall and in the spring. - The MAD for the fall data is 2.8 cookies. The MAD for the spring data is 2.6 cookies. Express the difference in means as a multiple of the larger MAD. - Based on this data, do you think that sales were generally higher in the spring than in the fall? (From Unit 8.3.1) Exercise $\PageIndex{9}$ A school is selling candles for a fundraiser. They keep 40% of the total sales as their commission, and they pay the rest to the candle company. \| price of candle \| number of candles sold \| \|---\|---\| \| small candle: $11 \| $68$ \| \| medium candle: $18 \| $45$ \| \| large candle: $25 \| $21$ \| How much money must the school pay to the candle company? (From Unit 4.3.2)	libretexts	2025-03-17T19:52:19.876357	2020-05-11T07:58:55	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.2%3A_Larger_Populations", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.3.2: Larger Populations", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.3%3A_What_Makes_a_Good_Sample	8.3.3: What Makes a Good Sample? Lesson Let's see what makes a good sample. Exercise $\PageIndex{1}$: Number Talk: Division by Powers of 10 Find the value of each quotient mentally. $34,000\div 10$ $340\div 100$ $34\div 10$ $3.4\div 100$ Exercise $\PageIndex{2}$: Selling Paintings Your teacher will assign you to work with either means or medians. - A young artist has sold 10 paintings. Calculate the measure of center you were assigned for each of these samples: - The first two paintings she sold were for $50 and $350. - At a gallery show, she sold three paintings for $250, $400, and $1,200. - Her oil paintings have sold for $410, $400, and $375. - Here are the selling prices for all 10 of her paintings: $$50\quad $200\quad $250\quad $275\quad $280\quad $350\quad $375\quad $400\quad $410\quad $1,200$ Calculate the measure of center you were assigned for all of the selling prices. - Compare your answers with your partner. Were the measures of center for any of the samples close to the same measure of center for the population? Exercise $\PageIndex{3}$: Sampling the Fish Market The price per pound of catfish at a fish market was recorded for 100 weeks. - Here are dot plots showing the population and three different samples from that population. What do you notice? What do you wonder? - If the goal is to have the sample represent the population, which of the samples would work best? Which wouldn't work so well? Explain your reasoning. To use this applet, drag the gray bar at the bottom up to see the sample dot plots. Are you ready for more? When doing a statistical study, it is important to keep the goal of the study in mind. Representative samples give us the best information about the distribution of the population as a whole, but sometimes a representative sample won’t work for the goal of a study! For example, suppose you want to study how discrimination affects people in your town. Surveying a representative sample of people in your town would give information about how the population generally feels, but might miss some smaller groups. Describe a way you might choose a sample of people to address this question. Exercise $\PageIndex{4}$: Auditing Sales An online shopping company tracks how many items they sell in different categories during each month for a year. Three different auditors each take samples from that data. Use the samples to draw dot plots of what the population data might look like for the furniture and electronics categories. Auditor 1’s sample Auditor 2's sample Auditor 3’s sample Population Auditor 1’s sample Auditor 2's sample Auditor 3's sample Population Summary A sample that is representative of a population has a distribution that closely resembles the distribution of the population in shape, center, and spread. For example, consider the distribution of plant heights, in cm, for a population of plants shown in this dot plot. The mean for this population is 4.9 cm, and the MAD is 2.6 cm. A representative sample of this population should have a larger peak on the left and a smaller one on the right, like this one. The mean for this sample is 4.9 cm, and the MAD is 2.3 cm. Here is the distribution for another sample from the same population. This sample has a mean of 5.7 cm and a MAD of 1.5 cm. These are both very different from the population, and the distribution has a very different shape, so it is not a representative sample. Glossary Entries Definition: Mean The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11. To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. $7+9+12+13+14=55$ and $55\div 5=11$. Definition: Mean Absolute Deviation (MAD) The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11. To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are. $4+2+1+2+3=12$ and $12\div 5=2.4$ Definition: Median The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order. For the data set 7, 9, 12, 13, 14, the median is 12. For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. $6+8=14$ and $14\div 2=7$. Definition: Population A population is a set of people or things that we want to study. For example, if we want to study the heights of people on different sports teams, the population would be all the people on the teams. Definition: Representative A sample is representative of a population if its distribution resembles the population's distribution in center, shape, and spread. For example, this dot plot represents a population. This dot plot shows a sample that is representative of the population. Definition: Sample A sample is part of a population. For example, a population could be all the seventh grade students at one school. One sample of that population is all the seventh grade students who are in band. Practice Exercise $\PageIndex{5}$ Suppose 45% of all the students at Andre’s school brought in a can of food to contribute to a canned food drive. Andre picks a representative sample of 25 students from the school and determines the sample’s percentage. He expects the percentage for this sample will be 45%. Do you agree? Explain your reasoning. Exercise $\PageIndex{6}$ This is a dot plot of the scores on a video game for a population of 50 teenagers. The three dot plots together are the scores of teenagers in three samples from this population. Which of the three samples is most representative of the population? Explain how you know. Exercise $\PageIndex{7}$ This is a dot plot of the number of text messages sent one day for a sample of the students at a local high school. The sample consisted of 30 students and was selected to be representative of the population. - What do the six values of 0 in the dot plot represent? - Since this sample is representative of the population, describe what you think a dot plot for the entire population might look like. Exercise $\PageIndex{8}$ A doctor suspects you might have a certain strain of flu and wants to test your blood for the presence of markers for this strain of virus. Why would it be good for the doctor to take a sample of your blood rather than use the population? (From Unit 8.3.2) Exercise $\PageIndex{9}$ How many different outcomes are in each sample space? Explain your reasoning. (You do not need to write out the actual options, just provide the number and your reasoning.) - A letter of the English alphabet is followed by a digit from 0 to 9. - A baseball team’s cap is selected from 3 different colors, 2 different clasps, and 4 different locations for the team logo. A decision is made to include or not to include reflective piping. - A locker combination like 7-23-11 uses three numbers, each from 1 to 40. Numbers can be used more than once, like 7-23-7. (From Unit 8.2.2)	libretexts	2025-03-17T19:52:19.954162	2020-05-11T07:58:24	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.3%3A_What_Makes_a_Good_Sample", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.3.3: What Makes a Good Sample?", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.4%3A_Sampling_in_a_Fair_Way	8.3.4: Sampling in a Fair Way Lesson Let's explore ways to get representative samples. Exercise $\PageIndex{1}$: Ages of Moviegoers A survey was taken at a movie theater to estimate the average age of moviegoers. Here is a dot plot showing the ages of the first 20 people surveyed. - What questions do you have about the data from survey? - What assumptions would you make based on these results? Exercise $\PageIndex{2}$: Comparing Methods for Selecting Samples Take turns with your partner reading each option aloud. For each situation, discuss: - Would the different methods for selecting a sample lead to different conclusions about the population? - What are the benefits of each method? - What might each method overlook? - Which of the methods listed would be the most likely to produce samples that are representative of the population being studied? - Can you think of a better way to select a sample for this situation? - Lin is running in an election to be president of the seventh grade. She wants to predict her chances of winning. She has the following ideas for surveying a sample of the students who will be voting: - Ask everyone on her basketball team who they are voting for. - Ask every third girl waiting in the lunch line who they are voting for. - Ask the first 15 students to arrive at school one morning who they are voting for. - A nutritionist wants to collect data on how much caffeine the average American drinks per day. She has the following ideas for how she could obtain a sample: - Ask the first 20 adults who arrive at a grocery store after 10:00 a.m. about the average amount of caffeine they consume each day. - Every 30 minutes, ask the first adult who comes into a coffee shop about the average amount of caffeine they consume each day. Exercise $\PageIndex{3}$: That's the First Straw Your teacher will have some students draw straws from a bag. - As each straw is taken out and measured, record its length (in inches) in the table. straw 1 straw 2 straw 3 straw 4 straw5 sample 1 sample 2 Table $\PageIndex{1}$ - Estimate the mean length of all the straws in the bag based on: - the mean of the first sample. - the mean of the second sample. - Were your two estimates the same? Did the mean length of all the straws in the bag change in between selecting the two samples? Explain your reasoning. - The actual mean length of all of the straws in the bag is about 2.37 inches. How do your estimates compare to this mean length? - If you repeated the same process again but you selected a larger sample (such as 10 or 20 straws, instead of just 5), would your estimate be more accurate? Explain your reasoning. Exercise $\PageIndex{4}$: That's the Last Straw There were a total of 35 straws in the bag. Suppose we put the straws in order from shortest to longest and then assigned each straw a number from 1 to 35. For each of these methods, decide whether it would be fair way to select a sample of 5 straws. Explain your reasoning. - Select the straws numbered 1 through 5. - Write the numbers 1 through 35 on pieces of paper that are all the same size. Put the papers into a bag. Without looking, select five papers from the bag. Use the straws with those numbers for your sample. - Using the same bag as the previous question, select one paper from the bag. Use the number on that paper to select the first straw for your sample. Then use the next 4 numbers in order to complete your sample. (For example, if you select number 17, then you also use straws 18, 19, 20, and 21 for your sample.) - Create a spinner with 35 sections that are all the same size, and number them 1 through 35. Spin the spinner 5 times and use the straws with those numbers for your sample. Are you ready for more? Computers accept inputs, follow instructions, and produce outputs, so they cannot produce truly random numbers. If you knew the input, you could predict the output by following the same instructions the computer is following. When truly random numbers are needed, scientists measure natural phenomena such as radioactive decay or temperature variations. Before such measurements were possible, statisticians used random number tables, like this: Use this table to select a sample of 5 straws. Pick a starting point at random in the table. If the number is between 01 and 35, include that number straw in your sample. If the number has already been selected, or is not between 01 and 35, ignore it, and move on to the next number. Summary A sample is selected at random from a population if it has an equal chance of being selected as every other sample of the same size. For example, if there are 25 students in a class, then we can write each of the students' names on a slip of paper and select 5 papers from a bag to get a sample of 5 students selected at random from the class. Other methods of selecting a sample from a population are likely to be biased . This means that it is less likely that the sample will be representative of the population as a whole. For example, if we select the first 5 students who walk in the door, that will not give us a random sample because students who typically come late are not likely to be selected. A sample that is selected at random may not always be a representative sample, but it is more likely to be representative than using other methods. It is not always possible to select a sample at random. For example, if we want to know the average length of wild salmon, it is not possible to identify each one individually, select a few at random from the list, and then capture and measure those exact fish. When a sample cannot be selected at random, it is important to try to reduce bias as much as possible when selecting the sample. Glossary Entries Definition: Mean The mean is one way to measure the center of a data set. We can think of it as a balance point. For example, for the data set 7, 9, 12, 13, 14, the mean is 11. To find the mean, add up all the numbers in the data set. Then, divide by how many numbers there are. $7+9+12+13+14=55$ and $55\div 5=11$. Definition: Mean Absolute Deviation (MAD) The mean absolute deviation is one way to measure how spread out a data set is. Sometimes we call this the MAD. For example, for the data set 7, 9, 12, 13, 14, the MAD is 2.4. This tells us that these travel times are typically 2.4 minutes away from the mean, which is 11. To find the MAD, add up the distance between each data point and the mean. Then, divide by how many numbers there are. $4+2+1+2+3=12$ and $12\div 5=2.4$ Definition: Median The median is one way to measure the center of a data set. It is the middle number when the data set is listed in order. For the data set 7, 9, 12, 13, 14, the median is 12. For the data set 3, 5, 6, 8, 11, 12, there are two numbers in the middle. The median is the average of these two numbers. $6+8=14$ and $14\div 2=7$. Definition: Population A population is a set of people or things that we want to study. For example, if we want to study the heights of people on different sports teams, the population would be all the people on the teams. Definition: Representative A sample is representative of a population if its distribution resembles the population's distribution in center, shape, and spread. For example, this dot plot represents a population. This dot plot shows a sample that is representative of the population. Definition: Sample A sample is part of a population. For example, a population could be all the seventh grade students at one school. One sample of that population is all the seventh grade students who are in band. Practice Exercise $\PageIndex{5}$ The meat department manager at a grocery store is worried some of the packages of ground beef labeled as having one pound of meat may be under-filled. He decides to take a sample of 5 packages from a shipment containing 100 packages of ground beef. The packages were numbered as they were put in the box, so each one has a different number between 1 and 100. Describe how the manager can select a fair sample of 5 packages. Exercise $\PageIndex{6}$ Select all the reasons why random samples are preferred over other methods of getting a sample. - If you select a random sample, you can determine how many people you want in the sample. - A random sample is always the easiest way to select a sample from a population. - A random sample is likely to give you a sample that is representative of the population. - A random sample is a fair way to select a sample, because each person in the population has an equal chance of being selected. - If you use a random sample, the sample mean will always be the same as the population mean. Exercise $\PageIndex{7}$ Jada is using a computer’s random number generator to produce 6 random whole numbers between 1 and 100 so she can use a random sample. The computer produces the numbers: 1, 2, 3, 4, 5, and 6. Should she use these numbers or have the computer generate a new set of random numbers? Explain your reasoning. Exercise $\PageIndex{8}$ A group of 100 people is divided into 5 groups with 20 people in each. One person’s name is chosen, and everyone in their group wins a prize. Noah simulates this situation by writing 100 different names on papers and putting them in a bag, then drawing one out. Kiran suggests there is a way to do it with fewer paper slips. Explain a method that would simulate this situation with fewer than 100 slips of paper. (From Unit 8.1.6) Exercise $\PageIndex{9}$ Data collected from a survey of American teenagers aged 13 to 17 was used to estimate that 29% of teens believe in ghosts. This estimate was based on data from 510 American teenagers. What is the population that people carrying out the survey were interested in? - All people in the United States. - The 510 teens that were surveyed. - All American teens who are between the ages of 13 and 17. - The 29% of the teens surveyed who said they believe in ghosts. (From Unit 8.3.2) Exercise $\PageIndex{10}$ A computer simulates flipping a coin 100 times, then counts the longest string of heads in a row. Based on these results, estimate the probability that there will be at least 15 heads in a row. \| trial \| most heads in a row \| \|---\|---\| \| $1$ \| $8$ \| \| $2$ \| $6$ \| \| $3$ \| $5$ \| \| $4$ \| $11$ \| \| $5$ \| $13$ \| (From Unit 8.2.1)	libretexts	2025-03-17T19:52:20.038586	2020-05-11T07:57:55	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.03%3A_New_Page/8.3.4%3A_Sampling_in_a_Fair_Way", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.3.4: Sampling in a Fair Way", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page	Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset + - Margin Size Reset + - Font Type Enable Dyslexic Font Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 8: Probability and Sampling Pre-Algebra I (Illustrative Mathematics - Grade 7) { "8.4.1:_Estimating_Population_Measures_of_Center" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.4.2:_Estimating_Population_Proportions" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.4.3:_More_about_Sampling_Variability" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.4.4:_Comparing_Populations_Using_Samples" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.4.5:_Comparing_Populations_With_Friends" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()" } { "8.01:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.02:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.03:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.04:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "8.05:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()" } { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "01:_Scale_Drawings" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "02:_Introducing_Proportional_Relationships" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "03:_Untitled_Chapter_3" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "04:_Untitled_Chapter_4" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "05:_Untitled_Chapter_5" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "06:_Untitled_Chapter_6" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "07:_Untitled_Chapter_7" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "08:_Untitled_Chapter_8" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "09:_Untitled_Chapter_9" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.<PageSubPageProperty>b__1]()" } Sat, 25 Jan 2020 01:41:52 GMT 8.4: Using Samples 35029 35029 admin { } Anonymous Anonymous 2 false false [ "article:topic", "license:ccby", "authorname:im" ] [ "article:topic", "license:ccby", "authorname:im" ] https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FPreAlgebra%2FPre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)%2F08%253A_Untitled_Chapter_8%2F8.04%253A_New_Page Search site Search Search Go back to previous article Username Password Sign in Sign in Sign in Forgot password Expand/collapse global hierarchy Home Bookshelves Pre-Algebra Pre-Algebra I (Illustrative Mathematics - Grade 7) 8: Probability and Sampling 8.4: Using Samples Expand/collapse global location 8.4: Using Samples Last updated Save as PDF Page ID 35029 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:20.110514	2020-01-25T01:41:52	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.4: Using Samples", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.1%3A_Estimating_Population_Measures_of_Center	8.4.1: Estimating Population Measures of Center Lesson Let's use samples to estimate measures of center for the population. Exercise $\PageIndex{1}$: Describing the Center Would you use the median or mean to describe the center of each data set? Explain your reasoning. Heights of 50 basketball players Ages of 30 people at a family dinner party Backpack weights of sixth-grade students How many books students read over summer break Exercise $\PageIndex{2}$: Three Different TV Shows Here are the ages (in years) of a random sample of 10 viewers for 3 different television shows. The shows are titled, “Science Experiments YOU Can Do,” “Learning to Read,” and “Trivia the Game Show.” \| sample 1 \| 6 \| 6 \| 5 \| 4 \| 8 \| 5 \| 7 \| 8 \| 6 \| 6 \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| sample 2 \| 15 \| 14 \| 12 \| 13 \| 12 \| 10 \| 12 \| 11 \| 10 \| 8 \| \| sample 3 \| 43 \| 60 \| 50 \| 36 \| 58 \| 50 \| 73 \| 59 \| 69 \| 51 \| - Calculate the mean for one of the samples. Make sure each person in your group works with a different sample. Record the answers for all three samples. - Which show do you think each sample represents? Explain your reasoning Exercise $\PageIndex{3}$: Who's Watching What? Here are three more samples of viewer ages collected for these same 3 television shows. \| sample 4 \| 57 \| 71 \| 5 \| 54 \| 52 \| 13 \| 59 \| 65 \| 10 \| 71 \| \|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\| \| sample 5 \| 15 \| 5 \| 4 \| 5 \| 4 \| 3 \| 25 \| 2 \| 8 \| 3 \| \| sample 6 \| 6 \| 11 \| 9 \| 56 \| 1 \| 3 \| 11 \| 10 \| 11 \| 2 \| - Calculate the mean for one of these samples. Record all three answers. - Which show do you think each of these samples represents? Explain your reasoning. - For each show, estimate the mean age for all the show's viewers. - Calculate the mean absolute deviation for one of the shows' samples. Make sure each person in your group works with a different sample. Record all three answers. Learning to Read Science Experiments YOU Can Do Trivia the Game Show Which sample? MAD Table $\PageIndex{3}$ - What do the different values for the MAD tell you about each group? - An advertiser has a commercial that appeals to 15- to 16-year-olds. Based on these samples, are any of these shows a good fit for this commercial? Explain or show your reasoning. Exercise $\PageIndex{4}$: Movie Reviews A movie rating website has many people rate a new movie on a scale of 0 to 100. Here is a dot plot showing a random sample of 20 of these reviews. - Would the mean or median be a better measure for the center of this data? Explain your reasoning. - Use the sample to estimate the measure of center that you chose for all the reviews. - For this sample, the mean absolute deviation is 19.6, and the interquartile range is 15. Which of these values is associated with the measure of center that you chose? - Movies must have an average rating of 75 or more from all the reviews on the website to be considered for an award. Do you think this movie will be considered for the award? Use the measure of center and measure of variability that you chose to justify your answer. Are you ready for more? Estimate typical temperatures in the United States today by looking up current temperatures in several places across the country. Use the data you collect to decide on the appropriate measure of center for the country, and calculate the related measure of variation for your sample. Summary Some populations have greater variability than others. For example, we would expect greater variability in the weights of dogs at a dog park than at a beagle meet-up. Dog park: Mean Weight: 12.8 kg MAD: 2.3 kg Beagle meet-up: Mean Weight: 10.1 kg MAD: 0.8 kg The lower MAD indicates there is less variability in the weights of the beagles. We would expect that the mean weight from a sample that is randomly selected from a group of beagles will provide a more accurate estimate of the mean weight of all the beagles than a sample of the same size from the dogs at the dog park. In general, a sample of a similar size from a population with less variability is more likely to have a mean that is close to the population mean. Glossary Entries Definition: Interquartile range (IQR) The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile. For example, the IQR of this data set is 20 because $50-30=20$. \| 22 \| 29 \| 30 \| 31 \| 32 \| 43 \| 44 \| 45 \| 50 \| 50 \| 59 \| \| Q1 \| Q2 \| Q3 \| Practice Exercise $\PageIndex{5}$ A random sample of 15 items were selected. For this data set, is the mean or median a better measure of center? Explain your reasoning. Exercise $\PageIndex{6}$ A video game developer wants to know how long it takes people to finish playing their new game. They surveyed a random sample of 13 players and asked how long it took them (in minutes). $1,235\quad 952\quad 457\quad 1,486\quad 1,759\quad 1,148\quad 548\quad 1,037\quad 1,864\quad 1,245\quad 976\quad 866\quad 1,431$ - Estimate the median time it will take all players to finish this game. - Find the interquartile range for this sample. Exercise $\PageIndex{7}$ Han and Priya want to know the mean height of the 30 students in their dance class. They each select a random sample of 5 students. - The mean height for Han’s sample is 59 inches. - The mean height for Priya’s sample is 61 inches. Does it surprise you that the two sample means are different? Are the population means different? Explain your reasoning. Exercise $\PageIndex{8}$ Clare and Priya each took a random sample of 25 students at their school. - Clare asked each student in her sample how much time they spend doing homework each night. The sample mean was 1.2 hours and the MAD was 0.6 hours. - Priya asked each student in her sample how much time they spend watching TV each night. The sample mean was 2 hours and the MAD was 1.3 hours. - At their school, do you think there is more variability in how much time students spend doing homework or watching TV? Explain your reasoning. - Clare estimates the students at her school spend an average of 1.2 hours each night doing homework. Priya estimates the students at her school spend an average of 2 hours each night watching TV. Which of these two estimates is likely to be closer to the actual mean value for all the students at their school? Explain your reasoning.	libretexts	2025-03-17T19:52:20.197472	2020-05-11T08:02:50	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.1%3A_Estimating_Population_Measures_of_Center", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.4.1: Estimating Population Measures of Center", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.2%3A_Estimating_Population_Proportions	8.4.2: Estimating Population Proportions Lesson Let's estimate population proportions using samples. Exercise $\PageIndex{1}$: Getting to School A teacher asked all the students in one class how many minutes it takes them to get to school. Here is a list of their responses: $20\quad 10\quad 15\quad 8\quad 5\quad 15\quad 10\quad 5\quad 20\quad 5\quad 15\quad 10\quad 3\quad 10\quad 18\quad 5\quad 25\quad 5\quad 5\quad 12\quad 10\quad 30\quad 5\quad 10\quad $ - What fraction of the students in this class say: - it takes them 5 minutes to get to school? - it takes them more than 10 minutes to get to school? - If the whole school has 720 students, can you use this data to estimate how many of them would say that it takes them more than 10 minutes to get to school? Be prepared to explain your reasoning. Exercise $\PageIndex{2}$: Reacting Times The track coach at a high school needs a student whose reaction time is less than 0.4 seconds to help out at track meets. All the twelfth graders in the school measured their reaction times. Your teacher will give you a bag of papers that list their results. - Work with your partner to select a random sample of 20 reaction times, and record them in the table. Table $\PageIndex{1}$ - What proportion of your sample is less than 0.4 seconds? - Estimate the proportion of all twelfth graders at this school who have a reaction time of less than 0.4 seconds. Explain your reasoning. - There are 120 twelfth graders at this school. Estimate how many of them have a reaction time of less than 0.4 seconds. - Suppose another group in your class comes up with a different estimate than yours for the previous question. - What is another estimate that would be reasonable ? - What is an estimate you would consider unreasonable ? Exercise $\PageIndex{3}$: A New Comic Book Hero Here are the results of a survey of 20 people who read The Adventures of Super Sam regarding what special ability they think the new hero should have. \| response \| what new ability? \| \|---\|---\| \| 1 \| fly \| \| 2 \| freeze \| \| 3 \| freeze \| \| 4 \| fly \| \| 5 \| fly \| \| 6 \| freeze \| \| 7 \| fly \| \| 8 \| super strength \| \| 9 \| freeze \| \| 10 \| fly \| \| 11 \| freeze \| \| 12 \| freeze \| \| 13 \| fly \| \| 14 \| invisibility \| \| 15 \| freeze \| \| 16 \| fly \| \| 17 \| freeze \| \| 18 \| fly \| \| 19 \| super strength \| \| 20 \| freeze \| - What proportion of this sample want the new hero to have the ability to fly? - If there are 2,024 dedicated readers of The Adventures of Super Sam , estimate the number of readers who want the new hero to fly. Two other comic books did a similar survey of their readers.- In a survey of people who read Beyond Human , 42 out of 60 people want a new hero to be able to fly. - In a survey of people who read Mysterious Planets , 14 out of 40 people want a new hero to be able to fly. - Do you think the proportion of all readers who want a new hero that can fly are nearly the same for the three different comic books? Explain your reasoning. - If you were in charge of these three comics, would you give the ability to fly to any of the new heroes? Explain your reasoning using the proportions you calculated. Exercise $\PageIndex{4}$: Flying to the Shelves The authors of The Adventures of Super Sam chose 50 different random samples of readers. Each sample was of size 20. They looked at the sample proportions who prefer the new hero to fly. - What is a good estimate of the proportion of all readers who want the new hero to be able to fly? - Are most of the sample proportions within 0.1 of your estimate for the population proportion? - If the authors of The Adventures of Super Sam give the new hero the ability to fly, will that please most of the readers? Explain your reasoning. The authors of the other comic book series created similar dot plots. 4. For each of these series, estimate the proportion of all readers who want the new hero to fly. - Beyond Human : - Mysterious Planets : 5. Should the authors of either of these series give their new hero the ability to fly? 6. Why might it be more difficult for the authors of Mysterious Planets to make the decision than the authors of the other series? Are you ready for more? Draw an example of a dot plot with at least 20 dots that represent the sample proportions for different random samples that would indicate that the population proportion is above 0.6, but there is a lot of uncertainty about that estimate. Summary Sometimes a data set consists of information that fits into specific categories. For example, we could survey students about whether they have a pet cat or dog. The categories for these data would be {neither, dog only, cat only, both}. Suppose we surveyed 10 students. Here is a table showing possible results: \| option \| number of responses \| \|---\|---\| \| neither dog nor cat \| $2$ \| \| dog only \| $4$ \| \| cat only \| $1$ \| \| both dog and cat \| $3$ \| In this sample, 3 of the students said they have both a dog and a cat. We can say that the proportion of these students who have a both a dog and a cat is $\frac{3}{10}$ or 0.3. If this sample is representative of all 720 students at the school, we can predict that about $\frac{3}{10}$ of 720, or about 216 students at the school have both a dog and a cat. In general, a proportion is a number from 0 to 1 that represents the fraction of the data that belongs to a given category. Glossary Entries Definition: Interquartile range (IQR) The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile. For example, the IQR of this data set is 20 because $50-30=20$. \| 22 \| 29 \| 30 \| 31 \| 32 \| 43 \| 44 \| 45 \| 50 \| 50 \| 59 \| \| Q1 \| Q2 \| Q3 \| Definition: Proportion A proportion of a data set is the fraction of the data in a given category. For example, a class has 18 students. There are 2 left-handed students and 16 right-handed students in the class. The proportion of students who are left-handed is $\frac{2}{20}$, or 0.1. Practice Exercise $\PageIndex{5}$ Tyler wonders what proportion of students at his school would dye their hair blue, if they were allowed to. He surveyed a random sample of 10 students at his school, and 2 of them said they would. Kiran didn’t think Tyler’s estimate was very accurate, so he surveyed a random sample of 100 students, and 17 of them said they would. - Based on Tyler's sample, estimate what proportion of the students would dye their hair blue. - Based on Kiran's sample, estimate what proportion of the students would dye their hair blue. - Whose estimate is more accurate? Explain how you know. Exercise $\PageIndex{6}$ Han surveys a random sample of students about their favorite pasta dish served by the cafeteria and makes a bar graph of the results. Estimate the proportion of the students who like lasagna as their favorite pasta dish. Exercise $\PageIndex{7}$ Elena wants to know what proportion of people have cats as pets. Describe a process she could use to estimate an answer to her question. Exercise $\PageIndex{8}$ The science teacher gives daily homework. For a random sample of days throughout the year, the median number of problems is 5 and the IQR is 2. The Spanish teacher also gives daily homework. For a random sample of days throughout the year, the median number of problems is 10 and the IQR is 1. If you estimate the median number of science homework problems to be 5 and the median number of Spanish problems to be 10, which is more likely to be accurate? Explain your reasoning. (From Unit 8.4.1) Exercise $\PageIndex{9}$ Diego wants to survey a sample of students at his school to learn about the percentage of students who are satisfied with the food in the cafeteria. He decides to go to the cafeteria on a Monday and ask the first 25 students who purchase a lunch at the cafeteria if they are satisfied with the food. Do you think this is a good way for Diego to select his sample? Explain your reasoning. (From Unit 8.3.4)	libretexts	2025-03-17T19:52:20.289174	2020-05-11T08:02:20	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.2%3A_Estimating_Population_Proportions", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.4.2: Estimating Population Proportions", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.3%3A_More_about_Sampling_Variability	8.4.3: More about Sampling Variability Lesson Let's compare samples from the same population. Exercise $\PageIndex{1}$: Average Reactions The other day, you worked with the reaction times of twelfth graders to see if they were fast enough to help out at the track meet. Look back at the sample you collected. - Calculate the mean reaction time for your sample. - Did you and your partner get the same sample mean? Explain why or why not. Exercise $\PageIndex{2}$: Reaction Population Your teacher will display a blank dot plot. - Plot your sample mean from the previous activity on your teacher's dot plot. - What do you notice about the distribution of the sample means from the class? - Where is the center? - Is there a lot of variability? - Is it approximately symmetric? - The population mean is 0.442 seconds. How does this value compare to the sample means from the class? Pause here so your teacher can display a dot plot of the population of reaction times. - What do you notice about the distribution of the population? - Where is the center? - Is there a lot of variability? - Is it approximately symmetric? - Compare the two displayed dot plots. - Based on the distribution of sample means from the class, do you think the mean of a random sample of 20 items is likely to be: - within 0.01 seconds of the actual population mean? - within 0.1 seconds of the actual population mean? Explain or show your reasoning. Exercise $\PageIndex{3}$: How Much Do You Trust the Answer? The other day you worked with 2 different samples of viewers from each of 3 different television shows. Each sample included 10 viewers. Here are the mean ages for 100 different samples of viewers from each show. - For each show, use the dot plot to estimate the population mean. - Trivia the Game Show - Science Experiments YOU Can Do - Learning to Read - For each show, are most of the sample means within 1 year of your estimated population mean? - Suppose you take a new random sample of 10 viewers for each of the 3 shows. Which show do you expect to have the new sample mean closest to the population mean? Explain or show your reasoning. Are you ready for more? Market research shows that advertisements for retirement plans appeal to people between the ages of 40 and 55. Younger people are usually not interested and older people often already have a plan. Is it a good idea to advertise retirement plans during any of these three shows? Explain your reasoning. Summary This dot plot shows the weights, in grams, of 18 cookies. The triangle indicates the mean weight, which is 11.6 grams. This dot plot shows the means of 20 samples of 5 cookies, selected at random. Again, the triangle shows the mean for the population of cookies. Notice that most of the sample means are fairly close to the mean of the entire population. This dot plot shows the means of 20 samples of 10 cookies, selected at random. Notice that the means for these samples are even closer to the mean for the entire population. In general, as the sample size gets bigger, the mean of a sample is more likely to be closer to the mean of the population. Glossary Entries Definition: Interquartile range (IQR) The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile. For example, the IQR of this data set is 20 because $50-30=20$. \| 22 \| 29 \| 30 \| 31 \| 32 \| 43 \| 44 \| 45 \| 50 \| 50 \| 59 \| \| Q1 \| Q2 \| Q3 \| Definition: Proportion A proportion of a data set is the fraction of the data in a given category. For example, a class has 18 students. There are 2 left-handed students and 16 right-handed students in the class. The proportion of students who are left-handed is $\frac{2}{20}$, or 0.1. Practice Exercise $\PageIndex{4}$ One thousand baseball fans were asked how far they would be willing to travel to watch a professional baseball game. From this population, 100 different samples of size 40 were selected. Here is a dot plot showing the mean of each sample. Based on the distribution of sample means, what do you think is a reasonable estimate for the mean of the population? Exercise $\PageIndex{5}$ Last night, everyone at the school music concert wrote their age on a slip of paper and placed it in a box. Today, each of the students in a math class selected a random sample of size 10 from the box of papers. Here is a dot plot showing their sample means, rounded to the nearest year. - Does the number of dots on the dot plot tell you how many people were at the concert or how many students are in the math class? - The mean age for the population was 35 years. If Elena picks a new sample of size 10 from this population, should she expect her sample mean to be within 1 year of the population mean? Explain your reasoning. - What could Elena do to select a random sample that is more likely to have a sample mean within 1 year of the population mean? Exercise $\PageIndex{6}$ A random sample of people were asked which hand they prefer to write with. “l” means they prefer to use their left hand, and “r” means they prefer to use their right hand. Here are the results: $l\quad r\quad r\quad r\quad r\quad r\quad r\quad r\quad r\quad r\quad l\quad r\quad r\quad r\quad r$ Based on this sample, estimate the proportion of the population that prefers to write with their left hand. (From Unit 8.4.2) Exercise $\PageIndex{7}$ Andre would like to estimate the mean number of books the students at his school read over the summer break. He has a list of the names of all the students at the school, but he doesn’t have time to ask every student how many books they read. What should Andre do to estimate the mean number of books? (From Unit 8.4.1) Exercise $\PageIndex{8}$ A hockey team has a 75% chance of winning against the opposing team in each game of a playoff series. To win the series, the team must be the first to win 4 games. - Design a simulation for this event. - What counts as a successful outcome in your simulation? - Estimate the probability using your simulation. (From Unit 8.2.4)	libretexts	2025-03-17T19:52:20.365751	2020-05-11T08:01:53	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.3%3A_More_about_Sampling_Variability", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.4.3: More about Sampling Variability", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.4%3A_Comparing_Populations_Using_Samples	8.4.4: Comparing Populations Using Samples Lesson Let's compare different populations using samples. Exercise $\PageIndex{1}$: Same Mean? Same MAD? Without calculating, tell whether each pair of data sets have the same mean and whether they have the same mean absolute deviation. - set A 1 3 3 5 6 8 10 14 set B 21 23 23 25 26 28 30 34 Table $\PageIndex{1}$ - set X 1 2 3 4 5 set Y 1 2 3 4 5 6 Table $\PageIndex{2}$ - set P 47 53 58 62 set Q 37 43 68 72 Table $\PageIndex{3}$ Exercise $\PageIndex{2}$: With a Heavy Load Consider the question: Do tenth-grade students' backpacks generally weigh more than seventh-grade students' backpacks? Here are dot plots showing the weights of backpacks for a random sample of students from these two grades: - Did any seventh-grade backpacks in this sample weigh more than a tenth-grade backpack? - The mean weight of this sample of seventh-grade backpacks is 6.3 pounds. Do you think the mean weight of backpacks for all seventh-grade students is exactly 6.3 pounds? - The mean weight of this sample of tenth-grade backpacks is 14.8 pounds. Do you think there is a meaningful difference between the weight of all seventh-grade and tenth-grade students' backpacks? Explain or show your reasoning. Exercise $\PageIndex{3}$: Do They Carry More? Here are 10 more random samples of seventh-grade students' backpack weights. \| sample number \| mean weight (pounds) \| \|---\|---\| \| 1 \| 5.8 \| \| 2 \| 9.2 \| \| 3 \| 5.5 \| \| 4 \| 7.3 \| \| 5 \| 7.2 \| \| 6 \| 6.6 \| \| 7 \| 5.2 \| \| 8 \| 5.3 \| \| 9 \| 6.3 \| \| 10 \| 6.4 \| - - Which sample has the highest mean weight? - Which sample has the lowest mean weight? - What is the difference between these two sample means? - All of the samples have a mean absolute deviation of about 2.8 pounds. Express the difference between the highest and lowest sample means as a multiple of the MAD. - Are these samples very different? Explain or show your reasoning. - Remember our sample of tenth-grade students' backpacks had a mean weight of 14.8 pounds. The MAD for this sample is 2.7 pounds. Your teacher will assign you one of the samples of seventh-grade students' backpacks to use. - What is the difference between the sample means for the the tenth-grade students' backpacks and the seventh-grade students' backpacks? - Express the difference between these two sample means as a multiple of the larger of the MADs. - Do you think there is a meaningful difference between the weights of all seventh-grade and tenth-grade students' backpacks? Explain or show your reasoning. Exercise $\PageIndex{4}$: Steel From Different Regions When anthropologists find steel artifacts, they can test the amount of carbon in the steel to learn about the people that made the artifacts. Here are some box plots showing the percentage of carbon in samples of steel that were found in two different regions: - Was there any steel found in region 1 that had: - more carbon than some of the steel found in region 2? - less carbon than some of the steel found in region 2? - Do you think there is a meaningful difference between all the steel artifacts found in regions 1 and 2? - Which sample has a distribution that is not approximately symmetric? - What is the difference between the sample medians for these two regions? sample median (%) IQR (%) region 1 $0.64$ $0.05$ region 2 $0.47$ $0.03$ Table $\PageIndex{5}$ - Express the difference between these two sample medians as a multiple of the larger interquartile range. - The anthropologists who conducted the study concluded that there was a meaningful difference between the steel from these regions. Do you agree? Explain or show your reasoning. Summary Sometimes we want to compare two different populations. For example, is there a meaningful difference between the weights of pugs and beagles? Here are histograms showing the weights for a sample of dogs from each of these breeds: The red triangles show the mean weight of each sample, 6.9 kg for the pugs and 10.1 kg for the beagles. The red lines show the weights that are within 1 MAD of the mean. We can think of these as “typical” weights for the breed. These typical weights do not overlap. In fact, the distance between the means is $10.1-6.9$ or 3.2 kg, over 6 times the larger MAD! So we can say there is a meaningful difference between the weights of pugs and beagles. Is there a meaningful difference between the weights of male pugs and female pugs? Here are box plots showing the weights for a sample of male and female pugs: We can see that the medians are different, but the weights between the first and third quartiles overlap. Based on these samples, we would say there is not a meaningful difference between the weights of male pugs and female pugs. In general, if the measures of center for two samples are at least two measures of variability apart, we say the difference in the measures of center is meaningful. Visually, this means the range of typical values does not overlap. If they are closer, then we don't consider the difference to be meaningful. Glossary Entries Definition: Interquartile range (IQR) The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile. For example, the IQR of this data set is 20 because $50-30=20$. \| 22 \| 29 \| 30 \| 31 \| 32 \| 43 \| 44 \| 45 \| 50 \| 50 \| 59 \| \| Q1 \| Q2 \| Q3 \| Definition: Proportion A proportion of a data set is the fraction of the data in a given category. For example, a class has 18 students. There are 2 left-handed students and 16 right-handed students in the class. The proportion of students who are left-handed is $\frac{2}{20}$, or 0.1. Practice Exercise $\PageIndex{5}$ Lin wants to know if students in elementary school generally spend more time playing outdoors than students in middle school. She selects a random sample of size 20 from each population of students and asks them how many hours they played outdoors last week. Suppose that the MAD for each of her samples is about 3 hours. Select all pairs of sample means for which Lin could conclude there is a meaningful difference between the two populations. - elementary school: 12 hours, middle school: 10 hours - elementary school: 14 hours, middle school: 9 hours - elementary school: 13 hours, middle school: 6 hours - elementary school: 13 hours, middle school: 10 hours - elementary school: 7 hours, middle school: 15 hours Exercise $\PageIndex{6}$ These two box plots show the distances of a standing jump, in inches, for a random sample of 10-year-olds and a random sample of 15-year-olds. Is there a meaningful difference in median distance for the two populations? Explain how you know. Exercise $\PageIndex{7}$ The median income for a sample of people from Chicago is about $60,000 and the median income for a sample of people from Kansas City is about $46,000, but researchers have determined there is not a meaningful difference in the medians. Explain why the researchers might be correct. Exercise $\PageIndex{8}$ A farmer grows 5,000 pumpkins each year. The pumpkins are priced according to their weight, so the farmer would like to estimate the mean weight of the pumpkins he grew this year. He randomly selects 8 pumpkins and weighs them. Here are the weights (in pounds) of these pumpkins: $2.9\quad 6.8\quad 7.3\quad 7.7\quad 8.9\quad 10.6\quad 12.3\quad 15.3$ 1. Estimate the mean weight of the pumpkins the farmer grew. This dot plot shows the mean weight of 100 samples of eight pumpkins, similar to the one above. 2. What appears to be the mean weight of the 5,000 pumpkins? 3. What does the dot plot of the sample means suggest about how accurate an estimate based on a single sample of 8 pumpkins might be? 4. What do you think the farmer might do to get a more accurate estimate of the population mean? (From Unit 8.4.3)	libretexts	2025-03-17T19:52:20.456481	2020-05-11T08:01:23	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.4%3A_Comparing_Populations_Using_Samples", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.4.4: Comparing Populations Using Samples", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.5%3A_Comparing_Populations_With_Friends	8.4.5: Comparing Populations With Friends Lesson Let's ask important questions to compare groups. Exercise $\PageIndex{1}$: Features of Graphic Representations Dot plots, histograms, and box plots are different ways to represent a data set graphically. Which of those displays would be the easiest to use to find each feature of the data? - the mean - the median - the mean absolute deviation - the interquartile range - the symmetry Exercise $\PageIndex{2}$: Info Gap: Comparing Populations Your teacher will give you either a problem card or a data card . Do not show or read your card to your partner. If your teacher gives you the problem card : - Silently read your card and think about what information you need to be able to answer the question. - Ask your partner for the specific information that you need. - Explain how you are using the information to solve the problem. Continue to ask questions until you have enough information to solve the problem. - Share the problem card and solve the problem independently. - Read the data card and discuss your reasoning. If your teacher gives you the data card : - Silently read your card. - Ask your partner “What specific information do you need?” and wait for them to ask for information. If your partner asks for information that is not on the card, do not do the calculations for them. Tell them you don’t have that information. - Before sharing the information, ask “ Why do you need that information? ” Listen to your partner’s reasoning and ask clarifying questions. - Read the problem card and solve the problem independently. - Share the data card and discuss your reasoning. Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner. Are you ready for more? Is there a meaningful difference between top sports performance in two different decades? Choose a variable from your favorite sport (for example, home runs in baseball, kills in volleyball, aces in tennis, saves in soccer, etc.) and compare the leaders for each year of two different decades. Is the performance in one decade meaningfully different from the other? Exercise $\PageIndex{3}$: Comparing to Known Characteristics - A college graduate is considering two different companies to apply to for a job. Acme Corp lists this sample of salaries on their website: $$45,000\quad $55,000\quad $140,000\quad $70,000\quad $60,000\quad $50,000$ What typical salary would Summit Systems need to have to be meaningfully different from Acme Corp? Explain your reasoning. - A factory manager is wondering whether they should upgrade their equipment. The manager keeps track of how many faulty products are created each day for a week. $6\quad 7\quad 8\quad 6\quad 7\quad 5\quad 7$ The new equipment guarantees an average of 4 or fewer faulty products per day. Is there a meaningful difference between the new and old equipment? Explain your reasoning. Summary When using samples to comparing two populations, there are a lot of factors to consider. - Are the samples representative of their populations? If the sample is biased, then it may not have the same center and variability as the population. - Which characteristic of the populations makes sense to compare—the mean, the median, or a proportion? - How variable is the data? If the data is very spread out, it can be more difficult to make conclusions with certainty. Knowing the correct questions to ask when trying to compare groups is important to correctly interpret the results. Glossary Entries Definition: Interquartile Range (IQR) The interquartile range is one way to measure how spread out a data set is. We sometimes call this the IQR. To find the interquartile range we subtract the first quartile from the third quartile. For example, the IQR of this data set is 20 because $50-30=20$. \| $22$ \| $29$ \| $30$ \| $31$ \| $32$ \| $43$ \| $44$ \| $45$ \| $50$ \| $50$ \| $59$ \| \| Q1 \| Q2 \| Q3 \| Definition: Proportion A proportion of a data set is the fraction of the data in a given category. For example, a class has 18 students. There are 2 left-handed students and 16 right-handed students in the class. The proportion of students who are left-handed is $\frac{2}{20}$, or 0.1. Practice Exercise $\PageIndex{4}$ An agent at an advertising agency asks a random sample of people how many episodes of a TV show they watch each day. The results are shown in the dot plot. The agency currently advertises on a different show, but wants to change to this one as long as the typical number of episodes is not meaningfully less. - What measure of center and measure of variation would the agent need to find for their current show to determine if there is a meaningful difference? Explain your reasoning. - What are the values for these same characteristics for the data in the dot plot? - What numbers for these characteristics would be meaningfully different if the measure of variability for the current show is similar? Explain your reasoning. Exercise $\PageIndex{5}$ Jada wants to know if there is a meaningful difference in the mean number of friends on social media for teens and adults. She looks at the friend count for the 10 most popular of her friends and the friend count for 10 of her parents’ friends. She then computes the mean and MAD of each sample and determines there is a meaningful difference. Jada’s dad later tells her he thinks she has not come to the right conclusion. Jada checks her calculations and everything is right. Do you agree with her dad? Explain your reasoning. Exercise $\PageIndex{6}$ The mean weight for a sample of a certain kind of ring made from platinum is 8.21 grams. The mean weight for a sample of a certain kind of ring made from gold is 8.61 grams. Is there a meaningful difference in the weights of the two types of rings? Explain your reasoning. Exercise $\PageIndex{7}$ The lengths in feet of a random sample of 20 male and 20 female humpback whales were measured and used to create the box plot. Estimate the median lengths of male and female humpback whales based on these samples. (From Unit 8.4.1)	libretexts	2025-03-17T19:52:20.531084	2020-05-11T08:00:48	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.04%3A_New_Page/8.4.5%3A_Comparing_Populations_With_Friends", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.4.5: Comparing Populations With Friends", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.05%3A_New_Page	8.5: Let's Put it to Work Last updated Save as PDF Page ID 35030 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:20.601725	2020-01-25T01:41:53	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.05%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.5: Let's Put it to Work", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.05%3A_New_Page/8.5.1%3A_Memory_Test	8.5.1: Memory Test Lesson Let's put it all together. Exercise $\PageIndex{1}$: Collecting a Sample You teacher will give you a paper that lists a data set with 100 numbers in it. Explain whether each method of obtaining a sample of size 20 would produce a random sample. Option 1: A spinner has 10 equal sections on it. Spin once to get the row number and again to get the column number for each member of your sample. Repeat this 20 times. Option 2: Since the data looks random already, use the first two rows. Option 3: Cut up the data and put them into a bag. Shake the bag to mix up the papers, and take out 20 values. Option 4: Close your eyes and point to one of the numbers to use as your first value in your sample. Then, keep moving one square from where your finger is to get a path of 20 values for your sample. Exercise $\PageIndex{2}$: Sample Probabilities Continue working with the data set your teacher gave you in the previous activity. The data marked with a star all came from students at Springfield Middle School. - When you select the first value for your random sample, what is the probability that it will be a value that came from a student at Springfield Middle School? - What proportion of your entire sample would you expect to be from Springfield Middle School? - If you take a random sample of size 10, how many scores would you expect to be from Springfield Middle School? - Select a random sample of size 10. - Did your random sample have the expected number of scores from Springfield Middle School? Exercise $\PageIndex{3}$: Estimating a Measure of Center for the Population - Decide which measure of center makes the most sense to use based on the distribution of your sample. Discuss your thinking with your partner. If you disagree, work to reach an agreement. - Estimate this measure of center for your population based on your sample. - Calculate the measure of variability for your sample that goes with the measure of center that you found. Exercise $\PageIndex{4}$: Comparing Populations Using only the values you computed in the previous two activities, compare your sample to your partner's. Is it reasonable to conclude that the measures of center for each of your populations are meaningfully different? Explain or show your reasoning.	libretexts	2025-03-17T19:52:20.735072	2020-05-11T08:03:40	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/08%3A_Untitled_Chapter_8/8.05%3A_New_Page/8.5.1%3A_Memory_Test", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "8.5.1: Memory Test", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9	9: Putting it All Together Last updated Save as PDF Page ID 35032 Illustrative Mathematics OpenUp Resources 9.1: Running a Restaurant 9.1.1: Planning Recipes 9.1.2: Costs of Running a Restaurant 9.1.3: More Costs of Running a Restaurant 9.1.4: Restaurant Floor Plan 9.2: Making Connections 9.2.1: How Crowded Is this Neighborhood? 9.2.2: Fermi Problems 9.2.3: More Expressions and Equations 9.2.4: Measurement Error (Part 1) 9.2.5: Measurement Error (Part 2) 9.3: Designing a Course 9.3.1: Measuring Long Distances Over Uneven Terrain 9.3.2: Building a Trundle Wheel 9.3.3: Using a Trundle Wheel to Measure Distances 9.3.4: Designing a 5K Course	libretexts	2025-03-17T19:52:20.816658	2020-01-25T01:41:54	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9: Putting it All Together", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page	9.1: Running a Restaurant Last updated Save as PDF Page ID 35033 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:20.888454	2020-01-25T01:41:55	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.1: Running a Restaurant", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.1%3A_Planning_Recipes	9.1.1: Planning Recipes Lesson Let's choose some recipes for a restaurant. Exercise $\PageIndex{1}$: A Recipe for Your Restaurant Imagine you could open a restaurant. - Select a recipe for a main dish you would like to serve at your restaurant. - Record the amount of each ingredient from your recipe in the first two columns of the table. You may not need to use every row. ingredient amount amount per serving calories per serving Table $\PageIndex{1}$ - How many servings does this recipe make? Determine the amount of each ingredient in one serving, and record it in the third column of the table. - Restaurants are asked to label how many calories are in each meal on their menu. - Use the information to calculate the amount of calories from each ingredient in your meal, and record it in the last column of the table. - Next, find the total calories in one serving of your meal. - If a person wants to eat 2,000 calories per day, what percentage of their daily calorie intake would one serving of your meal be? Grains \| common measure \| mass (g) \| calories \| fat (g) \| sodium (mg) \| \| \|---\|---\|---\|---\|---\|---\| \| bagels \| 1 bagel \| $99$ \| $261$ \| $1.31$ \| $418$ \| \| biscuit dough, refrigerated \| 1 biscuit \| $58$ \| $178$ \| $6.14$ \| $567$ \| \| biscuits, dry mix \| 1 cup \| $120$ \| $514$ \| $18.48$ \| $1531$ \| \| bread crumbs \| 1 oz \| $28.35$ \| $112$ \| $1.5$ \| $208$ \| \| bread stuffing, dry mix \| 1 oz \| $28.35$ \| $109$ \| $0.96$ \| $398$ \| \| cornbread, dry mix \| 1 oz \| $28.35$ \| $119$ \| $3.46$ \| $232$ \| \| cornmeal \| 1 cup \| $157$ \| $581$ \| $2.75$ \| $11$ \| \| couscous \| 1 cup \| $173$ \| $650$ \| $1.11$ \| $17$ \| \| croissants \| 1 oz \| $28.35$ \| $115$ \| $5.95$ \| $132$ \| \| croutons \| $\frac{1}{2}$ oz \| $14.2$ \| $66$ \| $2.6$ \| $155$ \| \| egg noodles \| 1 cup \| $38$ \| $146$ \| $1.69$ \| $8$ \| \| English muffins \| 1 oz \| $28.35$ \| $67$ \| $0.51$ \| $132$ \| \| focaccia bread \| 1 piece \| $57$ \| $142$ \| $4.5$ \| $320$ \| \| French bread \| 1 oz \| $28.35$ \| $77$ \| $0.69$ \| $171$ \| \| frybread \| 1 piece \| $152$ \| $502$ \| $18.57$ \| $500$ \| \| grits \| 1 tbsp \| $9.7$ \| $36$ \| $0.12$ \| $0$ \| \| hamburger or hotdog buns \| 1 roll \| $44$ \| $123$ \| $1.72$ \| $217$ \| \| millet \| 1 cup \| $200$ \| $756$ \| $8.44$ \| $10$ \| \| multigrain bread \| 1 oz \| $28.35$ \| $75$ \| $1.2$ \| $108$ \| \| naan bread \| 1 piece \| $90$ \| $262$ \| $5.09$ \| $418$ \| \| oats \| 1 cup \| $81$ \| $307$ \| $5.28$ \| $5$ \| \| pasta \| 1 cup \| $91$ \| $338$ \| $1.37$ \| $5$ \| \| phyllo dough \| 1 oz \| $28.35$ \| $85$ \| $1.7$ \| $137$ \| \| pie crust, refrigerated \| 1 crust \| $229$ \| $1019$ \| $58.3$ \| $937$ \| \| pita bread \| 1 oz \| $28.35$ \| $8$ \| $0.34$ \| $152$ \| \| potato chips \| 1 oz \| $28$ \| $149$ \| $9.51$ \| $148$ \| \| potato flour \| 1 cup \| $160$ \| $571$ \| $0.54$ \| $88$ \| \| pretzels \| 1 oz \| $28.35$ \| $109$ \| $0.83$ \| $352$ \| \| puff pastry, frozen \| 1 oz \| $28.35$ \| $156$ \| $10.8$ \| $71$ \| \| puffed rice cereal \| 1 cup \| $14$ \| $56$ \| $0.07$ \| $0$ \| \| quinoa \| 1 cup \| $170$ \| $626$ \| $10.32$ \| $8$ \| \| ramen noodles \| 1 package \| $81$ \| $356$ \| $14.25$ \| $1503$ \| \| rice flour \| 1 cup \| $158$ \| $578$ \| $2.24$ \| $0$ \| \| rice noodles \| 2 oz \| $57$ \| $207$ \| $0.32$ \| $104$ \| \| rice, brown \| 1 cup \| $185$ \| $679$ \| $5.92$ \| $9$ \| \| rice, white \| 1 cup \| $185$ \| $675$ \| $1.22$ \| $9$ \| \| rice, wild \| 1 cup \| $160$ \| $571$ \| $1.73$ \| $11$ \| \| rye flour \| 1 cup \| $128$ \| $416$ \| $2.84$ \| $3$ \| \| saltine crackers \| 5 crackers \| $14.9$ \| $62$ \| $1.29$ \| $140$ \| \| taco shells \| 1 shell \| $12.9$ \| $61$ \| $2.81$ \| $42$ \| \| tortilla chips \| 1 oz \| $28.35$ \| $134$ \| $5.86$ \| $93$ \| \| tortillas \| 1 tortilla \| $49$ \| $146$ \| $3.71$ \| $364$ \| \| tostada shells \| 1 piece \| $12.3$ \| $58$ \| $2.88$ \| $81$ \| \| wheat bread \| 1 slice \| $29$ \| $79$ \| $1.31$ \| $137$ \| \| wheat flour \| 1 cup \| $125$ \| $455$ \| $1.22$ \| $2$ \| \| wheat rolls \| 1 roll \| $28$ \| $76$ \| $1.76$ \| $147$ \| \| white bread \| 1 slice \| $29$ \| $77$ \| $0.97$ \| $142$ \| \| wonton wrappers \| 1 oz \| $28.35$ \| $82$ \| $0.43$ \| $162$ \| Fruits and Vegetables \| common measure \| mass (g) \| calories \| fat (g) \| sodium (mg) \| \| \|---\|---\|---\|---\|---\|---\| \| acorn squash \| 1 cup \| $140$ \| $56$ \| $0.14$ \| $4$ \| \| ancho peppers, dried \| 1 pepper \| $17$ \| $48$ \| $1.39$ \| $7$ \| \| apple juice \| 1 cup \| $248$ \| $114$ \| $0.32$ \| $10$ \| \| apple juice, frozen concentrate \| 6 fl oz \| $211$ \| $350$ \| $0.78$ \| $53$ \| \| apples \| 1 cup \| $110$ \| $53$ \| $0.14$ \| $0$ \| \| applesauce \| 1 cup \| $246$ \| $167$ \| $0.42$ \| $5$ \| \| apricots \| 1 cup \| $155$ \| $74$ \| $0.6$ \| $2$ \| \| apricots, canned \| 1 cup \| $219$ \| $182$ \| $0.24$ \| $9$ \| \| artichokes \| 1 artichoke \| $128$ \| $60$ \| $0.19$ \| $120$ \| \| asparagus \| 1 cup \| $134$ \| $27$ \| $0.16$ \| $3$ \| \| avocados \| 1 cup \| $150$ \| $240$ \| $22$ \| $10$ \| \| bamboo shoots, canned \| 1 cup \| $131$ \| $25$ \| $0.52$ \| $9$ \| \| bananas \| 1 cup \| $225$ \| $200$ \| $0.74$ \| $2$ \| \| beets \| 1 cup \| $136$ \| $58$ \| $0.23$ \| $106$ \| \| beets, canned \| 1 cup \| $246$ \| $74$ \| $0.22$ \| $352$ \| \| bell peppers, green \| 1 cup \| $149$ \| $30$ \| $0.25$ \| $4$ \| \| bell peppers, red \| 1 cup \| $149$ \| $46$ \| $0.45$ \| $6$ \| \| blackberries \| 1 cup \| $144$ \| $62$ \| $0.71$ \| $1$ \| \| blueberries \| 1 cup \| $148$ \| $84$ \| $0.49$ \| $1$ \| \| blueberries, frozen \| 1 cup \| $155$ \| $79$ \| $0.99$ \| $2$ \| \| brocolli \| 1 cup \| $91$ \| $31$ \| $0.34$ \| $30$ \| \| broccoli, frozen \| 10 oz \| $95$ \| $28$ \| $0.32$ \| $16$ \| \| brussels sprouts \| 1 cup \| $88$ \| $38$ \| $0.26$ \| $22$ \| \| brussels sprouts, frozen \| $\frac{1}{2}$ cup \| $95$ \| $39$ \| $0.39$ \| $10$ \| \| butternut squash \| 1 cup \| $140$ \| $63$ \| $0.14$ \| $6$ \| \| cabbage \| $\frac{1}{2}$ cup \| $35$ \| $8$ \| $0.06$ \| $6$ \| \| cantaloupe \| 1 cup \| $177$ \| $60$ \| $0.34$ \| $28$ \| \| carrots \| 1 cup \| $128$ \| $52$ \| $0.31$ \| $88$ \| \| cauliflower \| 1 cup \| $107$ \| $27$ \| $0.3$ \| $32$ \| \| celery \| 1 cup \| $101$ \| $16$ \| $0.17$ \| $81$ \| \| chayote \| 1 cup \| $132$ \| $25$ \| $0.17$ \| $3$ \| \| cherries \| 1 cup \| $138$ \| $87$ \| $0.28$ \| $0$ \| \| chives \| 1 tbsp \| $3$ \| $1$ \| $0.02$ \| $0$ \| \| clementines \| 1 fruit \| $74$ \| $35$ \| $0.11$ \| $1$ \| \| coconut \| 1 cup \| $80$ \| $283$ \| $26.79$ \| $16$ \| \| collards \| 1 cup \| $36$ \| $12$ \| $0.22$ \| $6$ \| \| corn \| 1 cup \| $145$ \| $125$ \| $1.96$ \| $22$ \| \| corn, frozen \| 1 cup \| $136$ \| $120$ \| $1.06$ \| $4$ \| \| cranberries \| 1 cup \| $110$ \| $51$ \| $0.14$ \| $2$ \| \| cranberries, dried \| $\frac{1}{4}$ cup \| $40$ \| $123$ \| $0.44$ \| $2$ \| \| cranberry sauce, canned \| 1 cup \| $277$ \| $440$ \| $0.42$ \| $14$ \| \| cucumber \| 1 cup \| $133$ \| $16$ \| $0.21$ \| $3$ \| \| currants \| 1 cup \| $112$ \| $63$ \| $0.22$ \| $1$ \| \| dates \| 1 date \| $24$ \| $66$ \| $0.04$ \| $0$ \| \| edamame, frozen \| 1 cup \| $118$ \| $129$ \| $5.58$ \| $7$ \| \| eggplant \| 1 cup \| $82$ \| $20$ \| $0.15$ \| $2$ \| \| feijoa \| 1 cup \| $243$ \| $148$ \| $1.02$ \| $7$ \| \| figs \| 1 large \| $64$ \| $47$ \| $0.19$ \| $1$ \| \| fruit cocktail, canned \| 1 cup \| $242$ \| $138$ \| $0.17$ \| $15$ \| \| grape juice \| 1 cup \| $253$ \| $152$ \| $0.33$ \| $13$ \| \| grapefruit \| 1 cup \| $230$ \| $74$ \| $0.23$ \| $0$ \| \| grapes \| 1 cup \| $151$ \| $104$ \| $0.24$ \| $3$ \| \| green beans \| 1 cup \| $100$ \| $31$ \| $0.22$ \| $6$ \| \| green beans, canned \| 1 cup \| $135$ \| $28$ \| $0.55$ \| $362$ \| \| green beans, frozen \| 1 cup \| $121$ \| $47$ \| $0.25$ \| $4$ \| \| guavas \| 1 cup \| $165$ \| $112$ \| $1.57$ \| $3$ \| \| hearts of palms, canned \| 1 cup \| $146$ \| $41$ \| $0.91$ \| $622$ \| \| hominy, canned \| 1 cup \| $165$ \| $119$ \| $1.45$ \| $569$ \| \| honeydew \| 1 cup \| $170$ \| $61$ \| $0.24$ \| $31$ \| \| jalapenos \| 1 cup \| $90$ \| $26$ \| $0.33$ \| $3$ \| \| jicama \| 1 cup \| $120$ \| $46$ \| $0.11$ \| $5$ \| \| kale \| 1 cup \| $16$ \| $8$ \| $0.15$ \| $6$ \| \| leeks \| 1 cup \| $89$ \| $54$ \| $0.27$ \| $18$ \| \| lemon juice \| 1 cup \| $244$ \| $54$ \| $0.59$ \| $2$ \| \| lemon peel \| 1 tbsp \| $6$ \| $3$ \| $0.02$ \| $0$ \| \| lemonade, frozen concentrate \| 1 fl oz \| $36.5$ \| $72$ \| $0.26$ \| $3$ \| \| lemons \| 1 cup \| $212$ \| $61$ \| $0.64$ \| $4$ \| \| lettuce, butterhead \| 1 cup \| $55$ \| $7$ \| $0.12$ \| $3$ \| \| lettuce, green leaf \| 1 cup \| $36$ \| $5$ \| $0.05$ \| $10$ \| \| lettuce, iceberg \| 1 cup \| $72$ \| $10$ \| $0.1$ \| $7$ \| \| lettuce, red leaf \| 1 cup \| $28$ \| $4$ \| $0.06$ \| $7$ \| \| lettuce, romaine \| 1 cup \| $47$ \| $8$ \| $0.14$ \| $4$ \| \| limes \| 1 lime \| $67$ \| $20$ \| $0.13$ \| $1$ \| \| Mandarin oranges, canned \| 1 cup \| $252$ \| $154$ \| $0.25$ \| $15$ \| \| mangoes \| 1 cup \| $165$ \| $99$ \| $0.63$ \| $2$ \| \| maraschino cherries \| 1 cherry \| $5$ \| $8$ \| $0.01$ \| $0$ \| \| mashed potatoes, dehydrated flakes \| 1 cup \| $60$ \| $212$ \| $0.25$ \| $46$ \| \| mushrooms \| 1 cup \| $70$ \| $15$ \| $0.24$ \| $4$ \| \| nopales \| 1 cup \| $86$ \| $14$ \| $0.08$ \| $18$ \| \| okra \| 1 cup \| $100$ \| $33$ \| $0.19$ \| $7$ \| \| olives, canned \| 1 tbsp \| $8.4$ \| $10$ \| $0.9$ \| $62$ \| \| onions \| 1 cup \| $160$ \| $64$ \| $0.16$ \| $6$ \| \| onions, frozen, chopped \| 10 oz \| $95$ \| $28$ \| $0.1$ \| $11$ \| \| orange juice \| 1 cup \| $249$ \| $122$ \| $0.3$ \| $5$ \| \| orange juice, frozen concentrate \| 1 cup \| $262$ \| $388$ \| $0.66$ \| $18$ \| \| orange peel \| 1 tbsp \| $6$ \| $6$ \| $0.01$ \| $0$ \| \| oranges \| 1 cup \| $180$ \| $85$ \| $0.22$ \| $0$ \| \| papayas \| 1 cup \| $145$ \| $62$ \| $0.38$ \| $12$ \| \| parsnips \| 1 cup \| $133$ \| $100$ \| $0.4$ \| $13$ \| \| peaches \| 1 cup \| $154$ \| $60$ \| $0.38$ \| $0$ \| \| peaches, canned \| 1 cup \| $251$ \| $136$ \| $0.08$ \| $13$ \| \| pears \| 1 cup \| $140$ \| $80$ \| $0.2$ \| $1$ \| \| pears, canned \| 1 cup \| $251$ \| $143$ \| $0.08$ \| $13$ \| \| peas \| 1 cup \| $145$ \| $117$ \| $0.58$ \| $7$ \| \| peas, frozen \| 1 cup \| $134$ \| $103$ \| $0.54$ \| $145$ \| \| pickles, dill \| 1 spear \| $35$ \| $4$ \| $0.1$ \| $283$ \| \| pickles, sweet \| 1 cup \| $160$ \| $146$ \| $0.66$ \| $731$ \| \| pineapple \| 1 cup \| $165$ \| $82$ \| $0.2$ \| $2$ \| \| pineapple, canned \| 1 cup \| $181$ \| $109$ \| $0.2$ \| $2$ \| \| plantains \| 1 cup \| $148$ \| $181$ \| $0.55$ \| $6$ \| \| plums \| 1 cup \| $165$ \| $76$ \| $0.46$ \| $0$ \| \| pomegranate juice \| 1 cup \| $1249$ \| $134$ \| $0.72$ \| $22$ \| \| pomegranates \| $\frac{1}{2}$ cup \| $87$ \| $72$ \| $1.02$ \| $3$ \| \| portabella mushrooms \| 1 cup \| $86$ \| $19$ \| $0.3$ \| $8$ \| \| potatoes, frozen hash browns \| $\frac{1}{2}$ cup \| $105$ \| $86$ \| $0.65$ \| $23$ \| \| potatoes, red \| $\frac{1}{2}$ cup \| $75$ \| $52$ \| $0.11$ \| $14$ \| \| potatoes, russet \| $\frac{1}{2}$ cup \| $75$ \| $59$ \| $0.06$ \| $4$ \| \| prickly pears \| 1 unit \| $149$ \| $61$ \| $0.76$ \| $7$ \| \| pumpkin \| 1 cup \| $116$ \| $30$ \| $0.12$ \| $1$ \| \| pumpkin, canned \| 1 cup \| $245$ \| $83$ \| $0.69$ \| $12$ \| \| radishes \| 1 cup \| $116$ \| $19$ \| $0.12$ \| $45$ \| \| raisins \| 1 cup \| $165$ \| $493$ \| $0.76$ \| $18$ \| \| raspberries \| 1 cup \| $123$ \| $64$ \| $0.8$ \| $1$ \| \| rhubarb \| 1 cup \| $122$ \| $26$ \| $0.24$ \| $5$ \| \| sauerkraut \| 1 cup \| $142$ \| $27$ \| $0.2$ \| $939$ \| \| serrano peppers \| 1 cup \| $105$ \| $34$ \| $0.46$ \| $10$ \| \| shallots \| 1 tbsp \| $10$ \| $7$ \| $0.01$ \| $1$ \| \| shiitake mushrooms \| 1 piece \| $19$ \| $6$ \| $0.09$ \| $2$ \| \| soybeans \| 1 cup \| $256$ \| $376$ \| $17.41$ \| $38$ \| \| spinach \| 1 cup \| $30$ \| $7$ \| $0.12$ \| $24$ \| \| spinach, frozen \| 1 cup \| $156$ \| $45$ \| $0.89$ \| $115$ \| \| strawberries \| 1 cup \| $152$ \| $49$ \| $0.46$ \| $2$ \| \| strawberries, frozen \| 1 cup \| $221$ \| $77$ \| $0.24$ \| $4$ \| \| summer squash \| 1 cup \| $113$ \| $18$ \| $0.2$ \| $2$ \| \| sweet onions \| 1 serving \| $148$ \| $47$ \| $0.12$ \| $12$ \| \| sweet potato \| 1 cup \| $133$ \| $114$ \| $0.07$ \| $73$ \| \| sweet potato, canned \| 1 cup \| $196$ \| $212$ \| $0.63$ \| $76$ \| \| Swiss chard \| 1 cup \| $36$ \| $7$ \| $0.07$ \| $77$ \| \| tangerines \| 1 cup \| $195$ \| $103$ \| $0.6$ \| $4$ \| \| taro \| 1 cup \| $104$ \| $116$ \| $0.21$ \| $11$ \| \| tomatillos \| 1 tomatillo \| $34$ \| $11$ \| $0.35$ \| $0$ \| \| tomato juice, canned \| 1 cup \| $243$ \| $41$ \| $0.7$ \| $615$ \| \| tomato sauce, canned \| 1 cup \| $245$ \| $59$ \| $0.74$ \| $1161$ \| \| tomato soup, canned, condensed \| 1 cup \| $148$ \| $98$ \| $0.65$ \| $558$ \| \| tomatoes \| 1 cup \| $149$ \| $27$ \| $0.3$ \| $7$ \| \| tomatoes, sun-dried \| 1 cup \| $54$ \| $139$ \| $1.6$ \| $58$ \| \| turnip greens \| 1 cup \| $55$ \| $18$ \| $0.16$ \| $22$ \| \| turnips \| 1 cup \| $130$ \| $36$ \| $0.13$ \| $87$ \| \| water chestnuts, canned \| $\frac{1}{2}$ cup \| $70$ \| $35$ \| $0.04$ \| $6$ \| \| watermelon \| 1 cup \| $154$ \| $46$ \| $0.23$ \| $2$ \| \| winter squash \| 1 cup \| $116$ \| $39$ \| $0.15$ \| $5$ \| \| yams \| 1 cup \| $150$ \| $177$ \| $0.26$ \| $14$ \| \| zucchini \| 1 cup \| $124$ \| $21$ \| $0.4$ \| $10$ \| Meat \| common measure \| mass (g) \| calories \| fat (g) \| sodium (mg) \| \| \|---\|---\|---\|---\|---\|---\| \| bacon \| 1 slice \| $26$ \| $106$ \| $10.21$ \| $122$ \| \| bratwurst \| 2.33 oz \| $66$ \| $196$ \| $17.38$ \| $560$ \| \| chicken leg \| 3 oz \| $85$ \| $182$ \| $13.56$ \| $71$ \| \| chicken thigh \| 1 piece \| $193$ \| $427$ \| $32.06$ \| $156$ \| \| chicken wing \| 1 piece \| $107$ \| $204$ \| $13.75$ \| $90$ \| \| chicken, light meat \| 1 lb \| $88$ \| $100$ \| $1.45$ \| $60$ \| \| chorizo \| 1 oz \| $28.35$ \| $129$ \| $10.85$ \| $350$ \| \| cod \| 3 oz \| $85$ \| $61$ \| $0.17$ \| $93$ \| \| crab \| 3 oz \| $85$ \| $73$ \| $0.82$ \| $251$ \| \| crayfish \| 3 oz \| $85$ \| $61$ \| $0.82$ \| $53$ \| \| elk \| 1 oz \| $28.35$ \| $31$ \| $0.41$ \| $16$ \| \| ground beef, 70% lean \| 4 oz \| $113$ \| $375$ \| $33.9$ \| $75$ \| \| ground beef, 90% lean \| 4 oz \| $113$ \| $199$ \| $11.3$ \| $75$ \| \| ground pork, 84% lean \| 4 oz \| $113$ \| $246$ \| $18.08$ \| $77$ \| \| ground pork, 96% lean \| 4 oz \| $113$ \| $137$ \| $4.52$ \| $76$ \| \| ground turkey, 85% lean \| 4 oz \| $85$ \| $153$ \| $10.66$ \| $46$ \| \| ground turkey, 93% lean \| 1 oz \| $28.35$ \| $43$ \| $2.36$ \| $20$ \| \| halibut \| 3 oz \| $85$ \| $77$ \| $1.13$ \| $58$ \| \| ham \| 1 oz \| $28.35$ \| $38$ \| $1.53$ \| $319$ \| \| hot dogs \| 1 piece \| $51$ \| $141$ \| $12.33$ \| $498$ \| \| kielbasa \| 3 oz \| $85$ \| $276$ \| $25.19$ \| $789$ \| \| lamb \| 1 oz \| $28.35$ \| $38$ \| $1.5$ \| $18$ \| \| lobster \| 1 lobster \| $150$ \| $116$ \| $1.12$ \| $634$ \| \| mahimahi \| 3 oz \| $85$ \| $72$ \| $0.6$ \| $75$ \| \| pepperoni \| 3 oz \| $85$ \| $428$ \| $39.34$ \| $1345$ \| \| pork ribs \| 1 rib \| $128$ \| $242$ \| $15.13$ \| $81$ \| \| pork sausage \| 1 piece \| $25$ \| $72$ \| $6.2$ \| $185$ \| \| pork tenderloin \| 3 oz \| $85$ \| $102$ \| $3$ \| $44$ \| \| salmon \| 1 fillet \| $108$ \| $373$ \| $12.34$ \| $55$ \| \| shrimp \| 3 oz \| $85$ \| $72$ \| $0.43$ \| $101$ \| \| tofu \| $\frac{1}{2}$ cup \| $126$ \| $98$ \| $5.25$ \| $15$ \| \| trout \| 1 fillet \| $79$ \| $111$ \| $4.88$ \| $40$ \| \| tuna, canned \| 1 oz \| $28.35$ \| $24$ \| $0.27$ \| $70$ \| \| turkey \| 1 serving \| $85$ \| $92$ \| $2.12$ \| $105$ \| Nuts, Beans, and Seeds \| common measure \| mass (g) \| calories \| fat (g) \| sodium (mg) \| \| \|---\|---\|---\|---\|---\|---\| \| almonds \| 1 cup \| $143$ \| $828$ \| $71.4$ \| $1$ \| \| baked beans, canned \| 1 cup \| $254$ \| $239$ \| $0.94$ \| $871$ \| \| black beans, canned \| 1 cup \| $240$ \| $218$ \| $0.7$ \| $331$ \| \| black beans, dry \| 1 cup \| $194$ \| $662$ \| $2.75$ \| $10$ \| \| balck-eyed peas, canned \| 1 cup \| $240$ \| $185$ \| $1.32$ \| $703$ \| \| black-eyed peas, dry \| 1 cup \| $167$ \| $561$ \| $2.1$ \| $27$ \| \| broadbeans (fava beans), canned \| 1 cup \| $256$ \| $182$ \| $0.56$ \| $1160$ \| \| broadbeans (fava beans), dry \| 1 cup \| $150$ \| $512$ \| $2.3$ \| $20$ \| \| cashews \| 1 oz \| $28.35$ \| $157$ \| $12.43$ \| $3$ \| \| chia seeds \| 1 oz \| $28.35$ \| $138$ \| $8.71$ \| $5$ \| \| chickpeas (garbanzo beans), canned \| 1 cup \| $240$ \| $211$ \| $4.68$ \| $667$ \| \| chickpeas (garbanzo beans), dry \| 1 cup \| $200$ \| $756$ \| $12.08$ \| $48$ \| \| flaxseed \| 1 tbsp \| $10.3$ \| $55$ \| $4.34$ \| $3$ \| \| great northern beans, canned \| 1 cup \| $262$ \| $299$ \| $1.02$ \| $969$ \| \| great northern beans, dry \| 1 cup \| $183$ \| $620$ \| $2.09$ \| $26$ \| \| kidney beans, canned \| 1 cup \| $256$ \| $215$ \| $1.54$ \| $758$ \| \| kidney beans, dry \| 1 cup \| $184$ \| $613$ \| $1.53$ \| $44$ \| \| lentils, dry \| 1 cup \| $192$ \| $676$ \| $2.04$ \| $12$ \| \| lima beans, canned \| 1 cup \| $241$ \| $190$ \| $0.41$ \| $810$ \| \| lima beans, frozen \| 1 cup \| $164$ \| $216$ \| $0.72$ \| $85$ \| \| macadamia nuts \| 1 cup \| $134$ \| $962$ \| $101.53$ \| $7$ \| \| navy beans, canned \| 1 cup \| $262$ \| $296$ \| $1.13$ \| $880$ \| \| navy beans, dry \| 1 cup \| $208$ \| $701$ \| $3.12$ \| $10$ \| \| peanut butter \| 2 tbsp \| $32$ \| $191$ \| $16.22$ \| $136$ \| \| peanuts \| 1 oz \| $28.35$ \| $166$ \| $14.08$ \| $116$ \| \| pecans \| 1 cup \| $109$ \| $753$ \| $78.45$ \| $0$ \| \| pine nuts \| 1 cup \| $135$ \| $909$ \| $92.3$ \| $3$ \| \| pinto beans, canned \| 1 cup \| $240$ \| $197$ \| $1.34$ \| $643$ \| \| pinto beans, dry \| 1 cup \| $193$ \| $670$ \| $2.37$ \| $23$ \| \| pistachios \| 1 cup \| $123$ \| $689$ \| $55.74$ \| $1$ \| \| pumpkin seeds \| 1 cup \| $129$ \| $721$ \| $63.27$ \| $9$ \| \| sesame seeds \| 1 cup \| $144$ \| $825$ \| $71.52$ \| $16$ \| \| sunflower seeds \| 1 cup \| $46$ \| $269$ \| $23.67$ \| $4$ \| Dairy \| common measure \| mass (g) \| calories \| fat (g) \| sodium (mg) \| \| \|---\|---\|---\|---\|---\|---\| \| almond milk \| 1 cup \| $262$ \| $39$ \| $2.88$ \| $186$ \| \| blue cheese \| 1 oz \| $28.35$ \| $100$ \| $8.15$ \| $325$ \| \| butter \| 1 pat \| $5$ \| $36$ \| $4.06$ \| $1$ \| \| cheddar cheese \| 1 cup \| $132$ \| $533$ \| $43.97$ \| $862$ \| \| coconut milk \| 1 cup \| $226$ \| $445$ \| $48.21$ \| $29$ \| \| cottage cheese \| 4 oz \| $113$ \| $111$ \| $4.86$ \| $411$ \| \| cream cheese \| 1 tbsp \| $14.5$ \| $51$ \| $4.99$ \| $46$ \| \| egg white \| 1 large \| $33$ \| $17$ \| $0.06$ \| $55$ \| \| egg yolk \| 1 large \| $17$ \| $55$ \| $4.51$ \| $8$ \| \| eggs \| 1 large \| $50$ \| $72$ \| $4.76$ \| $71$ \| \| evaporated milk \| 1 cup \| $252$ \| $270$ \| $5.04$ \| $252$ \| \| feta cheese \| 1 cup \| $150$ \| $396$ \| $31.92$ \| $1376$ \| \| heavy whipped cream \| 1 cup \| $120$ \| $408$ \| $43.3$ \| $32$ \| \| margarine \| 1 tbsp \| $14.2$ \| $101$ \| $11.38$ \| $4$ \| \| milk, 1% \| 1 cup \| $244$ \| $102$ \| $2.37$ \| $107$ \| \| milk, 2% \| 1 cup \| $244$ \| $122$ \| $4.83$ \| $115$ \| \| milk, skim \| 1 cup \| $245$ \| $83$ \| $0.2$ \| $103$ \| \| milk, whole \| 1 cup \| $244$ \| $149$ \| $7.93$ \| $105$ \| \| mozzarella cheese \| 1 cup \| $132$ \| $389$ \| $26.11$ \| $879$ \| \| Parmesan cheese \| 1 cup \| $100$ \| $420$ \| $27.84$ \| $1804$ \| \| powdered milk \| 1 cup \| $68$ \| $243$ \| $0.49$ \| $373$ \| \| queso fresco cheese \| 1 cup \| $122$ \| $365$ \| $29.06$ \| $916$ \| \| rice milk \| 1 cup \| $240$ \| $113$ \| $2.33$ \| $94$ \| \| sour cream, light \| 1 tbsp \| $12$ \| $16$ \| $1.27$ \| $10$ \| \| soymilk \| 1 cup \| $243$ \| $80$ \| $3.91$ \| $90$ \| \| Swiss cheese \| 1 cup \| $132$ \| $519$ \| $40.91$ \| $247$ \| \| yogurt \| 6 oz \| $170$ \| $107$ \| $2.64$ \| $119$ \| Other \| common measure \| mass (g) \| calories \| fat (g) \| sodium (mg) \| \| \|---\|---\|---\|---\|---\|---\| \| agave syrup \| 1 tsp \| $6.9$ \| $21$ \| $0.03$ \| $0$ \| \| allspice \| 1 tsp \| $1.9$ \| $5$ \| $0.17$ \| $1$ \| \| baking powder \| 1 tsp \| $4.6$ \| $2$ \| $0$ \| $488$ \| \| baking soda \| 1 tsp \| $4.6$ \| $0$ \| $0$ \| $1259$ \| \| balsamic vinegar \| 1 tbsp \| $16$ \| $14$ \| $0$ \| $4$ \| \| barbecue sauce \| 1 tbsp \| $17$ \| $29$ \| $0.11$ \| $175$ \| \| basil \| 1 tsp \| $0.7$ \| $2$ \| $0.03$ \| $1$ \| \| bay leaf \| 1 tsp \| $0.6$ \| $2$ \| $0.05$ \| $0$ \| \| black pepper \| 1 tsp \| $2.3$ \| $6$ \| $0.07$ \| $0$ \| \| brown sugar \| 1 tsp \| $3$ \| $11$ \| $0$ \| $1$ \| \| canola oil \| 1 tbsp \| $14$ \| $124$ \| $14$ \| $0$ \| \| chicken bouillon \| 1 cube \| $4.8$ \| $10$ \| $0.23$ \| $1152$ \| \| chicken broth \| 1 cup \| $249$ \| $15$ \| $0.52$ \| $924$ \| \| chili powder \| 1 tsp \| $2.7$ \| $8$ \| $0.39$ \| $77$ \| \| chocolate syrup \| 1 cup \| $304$ \| $1064$ \| $27.06$ \| $1052$ \| \| cilantro \| $\frac{1}{4}$ cup \| $4$ \| $1$ \| $0.02$ \| $2$ \| \| cinnamon \| 1 tsp \| $2.6$ \| $6$ \| $0.03$ \| $0$ \| \| cloves \| 1 tsp \| $2.1$ \| $6$ \| $0.27$ \| $6$ \| \| cocoa powder \| 1 cup \| $86$ \| $196$ \| $11.78$ \| $18$ \| \| coconut oil \| 1 tbsp \| $13.6$ \| $121$ \| $13.47$ \| $0$ \| \| coffee \| 1 fl oz \| $29.6$ \| $0$ \| $0.01$ \| $1$ \| \| corn syrup \| 1 cup \| $341$ \| $965$ \| $0.68$ \| $211$ \| \| cornstarch \| 1 cup \| $138$ \| $488$ \| $0.06$ \| $12$ \| \| cream of chicken soup \| $\frac{1}{2}$ cup \| $126$ \| $113$ \| $7.27$ \| $885$ \| \| cream of mushroom soup \| $\frac{1}{2}$ cup \| $126$ \| $100$ \| $6.68$ \| $871$ \| \| cream of tartar \| 1 tsp \| $3$ \| $8$ \| $0$ \| $2$ \| \| cumin \| 1 tsp \| $2.1$ \| $8$ \| $0.47$ \| $4$ \| \| curry powder \| 1 tsp \| $2$ \| $6$ \| $0.28$ \| $1$ \| \| dill \| 1 tsp \| $2.1$ \| $6$ \| $0.31$ \| $0$ \| \| dulce de leche \| 1 tbsp \| $19$ \| $60$ \| $1.4$ \| $25$ \| \| enchilada sauce \| $\frac{1}{4}$ cup \| $56$ \| $17$ \| $0.51$ \| $306$ \| \| garlic \| 1 clove \| $3$ \| $4$ \| $0.01$ \| $0.5$ \| \| garlic powder \| 1 tsp \| $3.1$ \| $10$ \| $0.02$ \| $2$ \| \| ginger \| 1 tsp \| $1.8$ \| $6$ \| $0.08$ \| $0$ \| \| gravy, canned \| 1 cup \| $233$ \| $1$ \| $5.5$ \| $1305$ \| \| honey \| 1 cup \| $339$ \| $25$ \| $0$ \| $14$ \| \| horseradish \| 1 tsp \| $5$ \| $35$ \| $0.03$ \| $21$ \| \| hot sauce \| 1 tsp \| $4.7$ \| $56$ \| $0.02$ \| $124$ \| \| hummus \| 1 tbsp \| $15$ \| $17$ \| $1.44$ \| $57$ \| \| Italian dressing \| 1 tbsp \| $14.7$ \| $35$ \| $3.1$ \| $146$ \| \| jams, jellies, and preserves \| 1 tbsp \| $20$ \| $56$ \| $0.01$ \| $6$ \| \| ketchup \| 1 tbsp \| $17$ \| $17$ \| $0.02$ \| $154$ \| \| lard \| 1 tbsp \| $12.8$ \| $115$ \| $12.8$ \| $0$ \| \| maple syrup \| 1 tbsp \| $20$ \| $52$ \| $0.01$ \| $2$ \| \| marshmallows, mini \| 1 cup \| $50$ \| $159$ \| $0.1$ \| $40$ \| \| mayonnaise \| 1 tbsp \| $15$ \| $103$ \| $11.67$ \| $73$ \| \| molasses \| 1 cup \| $337$ \| $977$ \| $0.34$ \| $125$ \| \| mustard \| 1 tsp \| $5$ \| $3$ \| $0.17$ \| $55$ \| \| mustard powder \| 1 tsp \| $2$ \| $10$ \| $0.72$ \| $0$ \| \| olive oil \| 1 tbsp \| $13.5$ \| $119$ \| $13.5$ \| $0$ \| \| onion powder \| 1 tsp \| $2.4$ \| $8$ \| $0.02$ \| $2$ \| \| onion soup mix \| 1 tbsp \| $7.5$ \| $22$ \| $0.03$ \| $602$ \| \| oregano \| 1 tsp \| $1$ \| $3$ \| $0.04$ \| $0$ \| \| paprika \| 1 tsp \| $2.3$ \| $6$ \| $0.3$ \| $2$ \| \| parsley \| 1 tsp \| $0.5$ \| $1$ \| $0.03$ \| $2$ \| \| pasta sauce \| $\frac{1}{2}$ cup \| $132$ \| $66$ \| $2.13$ \| $577$ \| \| pesto \| $\frac{1}{4}$ cup \| $63$ \| $263$ \| $23.69$ \| $380$ \| \| pizza sauce \| $\frac{1}{4}$ cup \| $63$ \| $34$ \| $0.72$ \| $219$ \| \| poultry seasoning \| 1 tsp \| $1.5$ \| $5$ \| $0.11$ \| $0$ \| \| powdered sugar \| 1 cup \| $120$ \| $467$ \| $0$ \| $2$ \| \| ranch dressing \| 1 tbsp \| $15$ \| $64$ \| $6.68$ \| $135$ \| \| rosemary \| 1 tsp \| $1.2$ \| $4$ \| $0.18$ \| $1$ \| \| sage \| 1 tsp \| $0.7$ \| $2$ \| $0.09$ \| $0$ \| \| salsa \| 2 tbsp \| $36$ \| $10$ \| $0.06$ \| $256$ \| \| salt \| 1 tsp \| $6$ \| $0$ \| $0$ \| $2325$ \| \| shortening \| 1 tbsp \| $12.8$ \| $113$ \| $12.8$ \| $0$ \| \| soy sauce \| 1 tbsp \| $16$ \| $8$ \| $0.09$ \| $879$ \| \| sugar \| 1 packet \| $2.8$ \| $11$ \| $0$ \| $0$ \| \| sweet and sour sauce \| 2 tbsp \| $35$ \| $52$ \| $0.01$ \| $130$ \| \| taco seasoning \| 2 tsp \| $5.7$ \| $18$ \| $0$ \| $411$ \| \| tartar sauce \| 2 tbsp \| $30$ \| $63$ \| $5.01$ \| $200$ \| \| tea, black \| 1 fl oz \| $29.6$ \| $0$ \| $0$ \| $1$ \| \| tea, green \| 1 cup \| $245$ \| $2$ \| $0$ \| $0$ \| \| teriyaki sauce \| 1 tbsp \| $18$ \| $16$ \| $0$ \| $690$ \| \| thyme \| 1 tsp \| $1$ \| $3$ \| $0.01$ \| $1$ \| \| vanilla extract \| 1 tsp \| $4.2$ \| $12$ \| $0$ \| $0$ \| \| vegetable broth \| 1 cup \| $221$ \| $11$ \| $0.15$ \| $654$ \| \| vinegar \| 1 tbsp \| $14.9$ \| $3$ \| $0$ \| $0$ \| \| wasabi \| 1 tbsp \| $20$ \| $58$ \| $2.18$ \| $678$ \| \| water \| 1 fl oz \| $29.6$ \| $0$ \| $0$ \| $1$ \| \| watercress \| 1 cup \| $34$ \| $4$ \| $0.03$ \| $14$ \| \| Worcestershire sauce \| 1 tbsp \| $17$ \| $13$ \| $0$ \| $167$ \| \| yeast \| 1 tsp \| $4$ \| $13$ \| $0.3$ \| $2$ \| The labels on packaged foods tell how much of different nutrients they contain. Here is what some different food labels say about their sodium content. - cheese crackers, 351 mg, 14% daily value - apple chips, 15 mg, <1% daily value - granola bar, 82 mg, 3% daily value Estimate the maximum recommended amount of sodium intake per day (100% daily value). Explain your reasoning. Exercise $\PageIndex{2}$: Health Claims For a meal to be considered: - “low calorie”—it must have 120 calories or less per 100 grams of food. - “low fat”—it must have 3 grams of fat or less per 100 grams of food. - “low sodium”—it must have 140 milligrams of sodium or less per 100 grams of food. g - Does the meal you chose in the previous activity meet the requirements to be considered: - low calorie? - low fat? - low sodium? Be prepared to explain your reasoning. - Select or invent another recipe you would like to serve at your restaurant that does meet the requirements to be considered either low calorie, low fat, or low sodium. Show that your recipe meets that requirement. Organize your thinking so it can be followed by others. \| ingredient \| amount per serving \| calories per serving \| fat per serving \| sodium per serving \| \|---\|---\|---\|---\|---\|	libretexts	2025-03-17T19:52:21.323995	2020-01-25T01:41:58	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.1%3A_Planning_Recipes", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.1.1: Planning Recipes", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.2%3A_Costs_of_Running_a_Restaurant	9.1.2: Costs of Running a Restaurant Lesson Let's explore how much the food will cost. Exercise $\PageIndex{1}$: Introducing Spreadsheets - Type each formula into the cells of a spreadsheet program and press enter. Record what the cell displays. Make sure to type each formula exactly as it is written here. A B C D 1 $=40-32$ $=1.5+3.6$ $=14/7$ $=0.56$ Table $\PageIndex{1}$ - - Predict what will happen if you type the formula =A1C1 into cell C2 of your spreadsheet. - Type in the formula, and press enter to check your prediction. - - Predict what will happen next if you delete the formula in cell A1 and replace it with the number 100. - Replace the formula with the number, and press enter to check your prediction. - - Predict what will happen if you copy cell C2 and paste it into cell D2 of your spreadsheet. - Copy and paste the formula to check your prediction. Exercise $\PageIndex{2}$: Cost per Serving - Set up a spreadsheet with these column labels in the first row. Table $\PageIndex{2}$A B C D 1 ingredient unit in recipe amount in recipe amount per serving 2 - Type the information about the ingredients in your recipe into the first 3 columns of the spreadsheet. - Type a formula into cell D2 to automatically calculate the amount per serving for your first ingredient. - Copy cell D2 and paste it into the cells beneath it to calculate the amount per serving for the rest of your ingredients. Pause here so your teacher can review your work. - Add these column labels to your spreadsheet. Table $\PageIndex{3}$E F G H purchase price purchase amount purchase unit cost per purchase unit - Research the cost of each ingredient in your meal, and record the information in columns E, F, and G. - Type a formula into cell H2 to automatically calculate the cost per purchase unit for your first ingredient. - Copy cell H2, and paste it into the cells beneath it to calculate the cost per purchase unit for the rest of your ingredients. - Add these column labels to your spreadsheet. Table $\PageIndex{4}$I J K conversion from purchase units to recipe units cost per recipe unit cost per serving - Complete column I with how many of your recipe unit are in 1 of your purchase unit for each ingredient. For example, if your recipe unit was cups and your purchase unit was gallons, then your conversion would be 16 because there are 16 cups in 1 gallon. - Type a formula into cell J2 to calculate the cost per recipe unit for your first ingredient. - Type a formula into cell K2 to calculate the cost per serving for your first ingredient. - Compare formulas with your partner. Discuss your thinking. If you disagree, work to reach an agreement. - Copy cells J2 and K2, and paste them into the cells beneath them to calculate the cost per recipe unit and cost per serving for the rest of your ingredients. - Type a formula into the first empty cell below your last ingredient in column K to calculate the total cost per serving for all of the ingredients in your recipe. Record the answer here.	libretexts	2025-03-17T19:52:21.396708	2020-01-25T01:41:59	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.2%3A_Costs_of_Running_a_Restaurant", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.1.2: Costs of Running a Restaurant", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.3%3A_More_Costs_of_Running_a_Restaurant	9.1.3: More Costs of Running a Restaurant Lesson Let's explore how much it costs to run a restaurant. Exercise $\PageIndex{1}$: Are We Making Money? - Restaurants have many more expenses than just the cost of the food. - Make a list of other items you would have to spend money on if you were running a restaurant. - Identify which expenses on your list depend on the number of meals ordered and which are independent of the number of meals ordered. - Identify which of the expenses that are independent of the number of meals ordered only have to be paid once and which are ongoing. - Estimate the monthly cost for each of the ongoing expenses on your list. Next, calculate the total of these monthly expenses. - Tell whether each restaurant is making a profit or losing money if they have to pay the amount you predicted in ongoing expenses per month. Organize your thinking so it can be followed by others. - Restaurant A sells 6,000 meals in one month, at an average price of $17 per meal and an average cost of $4.60 per meal. - Restaurant B sells 8,500 meals in one month, at an average price of $8 per meal and an average cost of $2.20 per meal. - Restaurant C sells 4,800 meals in one month, at an average price of $29 per meal and an average cost of $6.90 per meal. - - Predict how many meals your restaurant would sell in one month. - How much money would you need to charge for each meal to be able to cover all the ongoing costs of running a restaurant? - What percentage of the cost of the ingredients is the markup on your meal? Exercise $\PageIndex{2}$: Disposable or Reusable? A sample of full service restaurants and a sample of fast food restaurants were surveyed about the average number of customers they serve per day. - How does the average number of customers served per day at a full service restaurant generally compare to the number served at a fast food restaurant? Explain your reasoning. - About how many customers do you think your restaurant will serve per day? Explain your reasoning. - Here are prices for plates and forks: Table $\PageIndex{1}$plates forks disposable 165 paper plates for $12.50 600 plastic forks for $10 reusable 12 ceramic plates for $28.80 24 metal forks for $30 - Using your predicted number of customers per day from the previous question, write an equation for the total cost, $d$, of using disposable plates and forks for every customer for $n$ days. - Is $d$ proportional to $n$? Explain your reasoning. - Use your equation to predict the cost of using disposable plates and forks for 1 year. Explain any assumptions you make with this calculation. - - How much would it cost to buy enough reusable plates and forks for your predicted number of customers per day? - If it costs $10.75 a day to wash the reusable plates and forks, write an expression that represents the total cost, $r$, of buying and washing reusable plates and forks after $n$ days. - Is $r$ proportional to $n$? Explain your reasoning. - How many days can you use the reusable plates and forks for the same cost that you calculated for using disposable plates and forks for 1 year?	libretexts	2025-03-17T19:52:21.457678	2020-01-25T01:41:59	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.3%3A_More_Costs_of_Running_a_Restaurant", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.1.3: More Costs of Running a Restaurant", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.4%3A_Restaurant_Floor_Plan	9.1.4: Restaurant Floor Plan Lesson Let's design the floor plan for a restaurant. Exercise $\PageIndex{1}$: Dining Area - Restaurant owners say it is good for each customer to have about 300 in 2 of space at their table. How many customers would you seat at each table? - It is good to have about 15 ft 2 of floor space per customer in the dining area. - How many customers would you like to be able to seat at one time? - What size and shape dining area would be large enough to fit that many customers? - Select an appropriate scale, and create a scale drawing of the outline of your dining area. - Using the same scale, what size would each of the tables from the first question appear on your scale drawing? - To ensure fast service, it is good for all of the tables to be within 60 ft of the place where the servers bring the food out of the kitchen. Decide where the food pickup area will be, and draw it on your scale drawing. Next, show the limit of how far away tables can be positioned from this place. - It is good to have at least $1\frac{1}{2}$ ft between each table and at least $1\frac{1}{2}$ ft between the sides of tables where the customers will be sitting. On your scale drawing, show one way you could arrange tables in your dining area. Are you ready for more? The dining area usually takes up about 60% of the overall space of a restaurant because there also needs to be room for the kitchen, storage areas, office, and bathrooms. Given the size of your dining area, how much more space would you need for these other areas? Exercise $\PageIndex{2}$: Cold Storage Some restaurants have very large refrigerators or freezers that are like small rooms. The energy to keep these rooms cold can be expensive. - A standard walk-in refrigerator (rectangular, 10 feet wide, 10 feet long, and 7 feet tall) will cost about $150 per month to keep cold. - A standard walk-in freezer (rectangular, 8 feet wide, 10 feet long, and 7 feet tall) will cost about $372 per month to keep cold. Here is a scale drawing of a walk-in refrigerator and freezer. About how much would it cost to keep them both cold? Show your reasoning.	libretexts	2025-03-17T19:52:21.515911	2020-05-20T18:27:56	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.01%3A_New_Page/9.1.4%3A_Restaurant_Floor_Plan", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.1.4: Restaurant Floor Plan", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page	9.2: Making Connections Last updated Save as PDF Page ID 35034 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:21.586414	2020-01-25T01:41:56	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.2: Making Connections", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.1%3A_How_Crowded_Is_this_Neighborhood	9.2.1: How Crowded Is this Neighborhood? Lesson Let's see how proportional relationships apply to where people live. Exercise $\PageIndex{1}$: Dot Density The figure shows four squares. Each square encloses an array of dots. Squares A and B have side length 2 inches. Squares C and D have side length 1 inch. - Complete the table with information about each square. square area of the square in square inches number of dots number of dots per square inch A B C D Table $\PageIndex{1}$ - Compare each square to the others. What is the same and what is different? Exercise $\PageIndex{2}$: Dot Density with a Twist The figure shows two arrays, each enclosed by a square that is 2 inches wide. - Let $a$ be the area of the square and $d$ be the number of dots enclosed by the square. For each square, plot a point that represents its values of $a$ and $d$. Start by choosing an appropriate scale for each axis, using the sliders to adjust the intervals. - Draw lines from $(0,0)$ to each point. For each line, write an equation that represents the proportional relationship. - What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares? Exercise $\PageIndex{3}$: Housing Density Here are pictures of two different neighborhoods. This image depicts an area that is 0.3 kilometers long and 0.2 kilometers wide. This image depicts an area that is 0.4 kilometers long and 0.2 kilometers wide. For each neighborhood, find the number of houses per square kilometer. Exercise $\PageIndex{4}$: Population Density - New York City has a population of 8,406 thousand people and covers an area of 1,214 square kilometers. - Los Angeles has a population of 3,884 thousand people and covers an area of 1,302 square kilometers. - The points labeled $A$ and $B$ each correspond to one of the two cities. Which is which? Label them on the graph. - Write an equation for the line that passes through $(0,0)$ and $A$. What is the constant of proportionality? - Write an equation for the line that passes through $(0,0)$ and $B$. What is the constant of proportionality? - What do the constants of proportionality tell you about the crowdedness of these two cities? Are you ready for more? - Predict where these types of regions would be shown on the graph: - a suburban region where houses are far apart, with big yards - a neighborhood in an urban area with many high-rise apartment buildings - a rural state with lots of open land and not many people - Next, use this data to check your predictions: \| place \| description \| population \| area (km 2 ) \| \|---\|---\|---\|---\| \| Chalco \| a suburb of Omaha, Nebraska \| $10,994$ \| $7.5$ \| \| Anoka County \| a county in Minnesota, near Minneapolis/ St. Paul \| $339,534$ \| $1,155$ \| \| Guttenberg \| a city in New Jersey \| $11,176$ \| $0.49$ \| \| New York \| a state \| $19,746,227$ \| $141,300$ \| \| Rhode Island \| a state \| $1,055,173$ \| $3,140$ \| \| Alaska \| a state \| $736,732$ \| $1,717,856$ \| \| Tok \| a community in Alaska \| $1,258$ \| $342.7$ \|	libretexts	2025-03-17T19:52:21.657323	2020-05-20T18:31:07	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.1%3A_How_Crowded_Is_this_Neighborhood", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.2.1: How Crowded Is this Neighborhood?", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.2%3A_Fermi_Problems	9.2.2: Fermi Problems Lesson Let's estimate some quantities. Exercise $\PageIndex{1}$: How Old Are You? What is your exact age at this moment? Exercise $\PageIndex{2}$: A Heart Stoppingly Large Number How many times has your heart beat in your lifetime? Exercise $\PageIndex{3}$: All the Hairs on Your Head How many strands of hair do you have on your head?	libretexts	2025-03-17T19:52:21.712760	2020-05-20T18:30:32	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.2%3A_Fermi_Problems", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.2.2: Fermi Problems", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.3%3A_More_Expressions_and_Equations	9.2.3: More Expressions and Equations Lesson Let's solve harder problems by writing equivalent expressions. Exercise $\PageIndex{1}$: Tickets for the School Play Student tickets for the school play cost $2 less than adult tickets. - If $a$ represents the price of one adult ticket, write an expression for the price of a student ticket. - Write an expression that represents the amount of money they collected each night: - The first night, the school sold 60 adult tickets and 94 student tickets. - The second night, the school sold 83 adult tickets and 127 student tickets. - Write an expression that represents the total amount of money collected from ticket sales on both nights. - Over these two nights, they collected a total of $1,651 in ticket sales. - Write an equation that represents this situation. - What was the cost of each type of ticket? - Is your solution reasonable? Explain how you know. Exercise $\PageIndex{2}$: A Souvenir Strand The souvenir stand sells hats, postcards, and magnets. They have twice as many postcards as hats, and 100 more magnets than post cards. - Let $h$ represent the total number of hats. Write an expression in terms of $h$ for the total number of items they have to sell. - The owner of the stand pays $8 for each hat, $0.10 for each post card, and $0.50 for each magnet. Write an expression for the total cost of the items. - The souvenir stand sells the hats for $11.75 each, the postcards for $0.25 each, and the magnets for $3.50 each. Write an expression for the total amount of money they would take in if they sold all the items. - Profits are calculated by subtracting costs from income. Write an expression for the profits of the souvenir stand if they sell all the items they have. Use properties to write an equivalent expression with fewer terms. - The souvenir stand sells all these items and makes a total profit of $953.25. - Write an equation that represents this situation. - How many of each item does the souvenir stand sell? Explain or show your reasoning. Exercise $\PageIndex{3}$: Jada Crochets a Scarf Basic crochet stitches are called single, double, and triple. Jada measures her average stitch size and sees that a “double crochet” stitch is not really twice as long; it uses $\frac{1}{2}$ inch less than twice as much yarn as a single crochet stitch. Jada’s “triple crochet” stitch uses 1 inch less than three times as much yarn as a single crochet stitch. - Write an expression that represents the amount of yarn Jada needs to crochet a scarf that includes 800 single crochet stitches, 400 double crochet stitches, and 200 triple crochet stitches. - Write an equivalent expression with as few terms as possible. - If Jada uses 5540 inches of yarn for the entire scarf, what length of yarn does she use for a single crochet stitch?	libretexts	2025-03-17T19:52:21.771312	2020-05-20T18:30:06	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.3%3A_More_Expressions_and_Equations", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.2.3: More Expressions and Equations", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.4%3A_Measurement_Error_(Part_1)	9.2.4: Measurement Error (Part 1) Lesson Let's check how accurate our measurements are. Exercise $\PageIndex{1}$: How Long Are These Pencils? - Estimate the length of each pencil. - How accurate are your estimates? - For each estimate, what is the largest possible percent error? Exercise $\PageIndex{2}$: How Long Are These Floor Boards? A wood floor is made by laying multiple boards end to end. Each board is measured with a maximum percent error of 5%. What is the maximum percent error for the total length of the floor?	libretexts	2025-03-17T19:52:21.899499	2020-05-20T18:29:39	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.4%3A_Measurement_Error_(Part_1)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.2.4: Measurement Error (Part 1)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.5%3A_Measurement_Error_(Part_2)	9.2.5: Measurement Error (Part 2) Lesson Let's check how accurate our calculations are. Exercise $\PageIndex{1}$: Measurement Error for Area Imagine that you measure the length and width of a rectangle and you know the measurements are accurate within 5% of the actual measurements. If you use your measurements to find the area, what is the maximum percent error for the area of the rectangle? Exercise $\PageIndex{2}$: Measurement Error for Volume - The length, width, and height of a rectangular prism were measured to be 10 cm, 12 cm, and 25 cm. Assuming that these measurements are accurate to the nearest cm, what is the largest percent error possible for: - each of the dimensions? - the volume of the prism? - If the length, width, and height of a right rectangular prism have a maximum percent error of 1%, what is the largest percent error possible for the volume of the prism?	libretexts	2025-03-17T19:52:21.954544	2020-05-20T18:29:00	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.02%3A_New_Page/9.2.5%3A_Measurement_Error_(Part_2)", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.2.5: Measurement Error (Part 2)", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page	9.3: Designing a Course Last updated Save as PDF Page ID 35035 Illustrative Mathematics OpenUp Resources	libretexts	2025-03-17T19:52:22.026722	2020-01-25T01:41:57	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.3: Designing a Course", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.1%3A_Measuring_Long_Distances_Over_Uneven_Terrain	9.3.1: Measuring Long Distances Over Uneven Terrain Lesson Let's measure long distances over uneven terrain. Exercise $\PageIndex{1}$: How Far Is It? How do people measure distances in different situations? What tools do they use? Come up with at least three different methods and situations where those methods are used. Exercise $\PageIndex{2}$: Planning a 5K Course The school is considering holding a 5K fundraising walk on the school grounds. Your class is supposed to design the course for the walk. - What will you need to do to design the course for the walk? - Come up with a method to measure the course. Pause here so your teacher can review your plan. Exercise $\PageIndex{3}$: Comparing Methods Let’s see how close different measuring methods are to each other. Your teacher will show you a path to measure. - Use your method to measure the length of the path at least two times. - Decide what distance you will report to the class. - Compare your results with those of two other groups. Express the differences between the measurements in terms of percentages. - Discuss the advantages and disadvantages of each group’s method.	libretexts	2025-03-17T19:52:22.087847	2020-05-20T18:33:19	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.1%3A_Measuring_Long_Distances_Over_Uneven_Terrain", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.3.1: Measuring Long Distances Over Uneven Terrain", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.2%3A_Building_a_Trundle_Wheel	9.3.2: Building a Trundle Wheel Lesson Let's build a trundle wheel. Exercise $\PageIndex{1}$: What Is a Trundle Wheel? A tool that surveyors use to measure distances is called a trundle wheel. - How does a trundle wheel measure distance? - Why is this method of measuring distances better than the methods we used in the previous lesson? - How could we construct a simple trundle wheel? What materials would we need? Exercise $\PageIndex{2}$: Building a Trundle Wheel Your teacher will give you some supplies. Construct a trundle wheel and use it to measure the length of the classroom. Record: - the diameter of your trundle wheel - the number of clicks across classroom - the length of the classroom (Be prepared to explain your reasoning.)	libretexts	2025-03-17T19:52:22.144331	2020-05-20T18:32:48	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.2%3A_Building_a_Trundle_Wheel", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.3.2: Building a Trundle Wheel", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.3%3A_Using_a_Trundle_Wheel_to_Measure_Distances	9.3.3: Using a Trundle Wheel to Measure Distances Lesson Let's use our trundle wheels. Exercise $\PageIndex{1}$: Measuring Distances with the Trundle Wheel Earlier you made trundle wheels so that you can measure long distances. If you do not have a trundle wheel, you can use an applet to simulate measuring. Your teacher will tell you a path whose distance you will measure. - Measure the path with your trundle wheel three times and calculate the distance. Record your results in the table. diameter number of rotations computation distance trial 1 trial 2 trial 3 Table $\PageIndex{1}$ - Decide what distance you will report to the class. Be prepared to explain your reasoning. - Compare this distance with the distance you measured the other day for this same path. - Compare your results with the results of two other groups. Express the differences between the measurements in terms of percentages. To use this applet: - Choose a diameter for the wheel, and enter a number 1 -- 5 for the path you will measure. - Watch carefully to keep track of the number of rotations that the wheel makes before it stops. - Try three different diameters for the same path, and compare your results. - Answer the questions.	libretexts	2025-03-17T19:52:22.203839	2020-05-20T18:32:19	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.3%3A_Using_a_Trundle_Wheel_to_Measure_Distances", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.3.3: Using a Trundle Wheel to Measure Distances", "author": "Illustrative Mathematics" }
https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.4%3A_Designing_a_5K_Course	9.3.4: Designing a 5K Course Lesson Let's map out the 5K course. Exercise $\PageIndex{1}$: Make a Proposal Your teacher will give you a map of the school grounds. - On the map, draw in the path you measured earlier with your trundle wheel and label its length. - Invent another route for a walking course and draw it on your map. Estimate the length of the course you drew. - How many laps around your course must someone complete to walk 5 km? Exercise $\PageIndex{2}$: Measuring and Finalizing the Course - Measure your proposed race course with your trundle wheel at least two times. Decide what distance you will report to the class. - Revise your course, if needed. - Create a visual display that includes: - A map of your final course - The starting and ending locations - The number of laps needed to walk 5 km - Any other information you think would be helpful to the race organizers Are you ready for more? The map your teacher gave you didn't include a scale. Create one.	libretexts	2025-03-17T19:52:22.259535	2020-05-20T18:31:51	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://math.libretexts.org/Bookshelves/PreAlgebra/Pre-Algebra_I_(Illustrative_Mathematics_-_Grade_7)/09%3A_Untitled_Chapter_9/9.03%3A_New_Page/9.3.4%3A_Designing_a_5K_Course", "book_url": "https://commons.libretexts.org/book/math-34975", "title": "9.3.4: Designing a 5K Course", "author": "Illustrative Mathematics" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/02%3A_The_Self/2.05%3A_Art_and_Identity-_Introduction	2.5: Art and Identity- Introduction - - Last updated - Save as PDF One of the more important themes emerging from the last century has been the individual’s search for identity. For example, genealogical websites have proliferated and special television programs are devoted to the subject. Since it first aired on PBS in 2012, Henry Louis Gates Jr.’s Finding Your Roots has been a popular program. The British version, The Guardian , has been successful since 2006. Some anthropologists suggest that the deep-rooted interest in identity or ancestry is partly shaped by evolutionary forces dating back to early humans supporting each other in extended family groups. Anthropologist Dwight Read theorizes that the Neolithic people were the first to understand the concept of the family tree and the perception of self in a family unit and in society. 1 If connected through blood, people have the tendency to be more willing to care for each other; a common interest and support system is readily realized within a clan or a group. Early humans created two- and three-dimensional likenesses of themselves in their environment to help understand who they were in relation to the other members of their group. Con- temporary humans do the same; they make records of themselves with family members, most commonly in photographs and Selfies, and on Instagram. It is the same fundamental concept and placement in an environment that collectively identifies who we are in society, for example, in social gatherings, organizations, and religious settings. This means, above all, that we must place ourselves within the world in order to obtain identity. Children search for their identity at a very young age by observing and recognizing their parents and family members. Their markings with in a simple drawing of self and family similar to those of early humans help them to vindicateand confirm who they are and how they are perceived by their family group. Like children, artists sometimes explore their identity through self-portraits and symbolically in works of art that relate to ancestry or culture. Doing so allows them to take a look inside their core and see how they fit within their contemporary culture; this investigation of self plays an important role in how artists understand their environment and the world. Vincent van Gogh is known as a person who spent much of his time in solitude. He painted more than thirty self-portraits between the years 1886 and 1889, placing him among the most prolific self-portraitists of all time. Indeed, some of his most respected works are his self-portraits that trace his image throughout the last years of his life, the most crucial to his career. (Figures 8.1, 8.2, and 8.3) While Van Gogh used the study of his own image to help develop his skills as an artist, these self portraits also give us insights into the artist’s life and well being, how he fit in society, and his place among the groups with whom he associated. Like Van Gogh, Pablo Picasso painted a number of self-portraits. Throughout his career, Picasso painted various likenesses that reflected changes in himself, his style, his artistic development, as well as in his life style and beliefs—all of which may be viewed closely from the content of his paintings. (Figures 8.4 and 8.5) The first self-portrait, painted in 1901 while he was establishing himself as an artist in Paris, France, and still spending time in Barcelona, Spain, reflects the somber mode and tones of his Blue Period (1901-1904). The second, dated to 1906, at the very end of his Rose Period (1904-1906), Picasso depicts himself as the artist who by that time was moving in artistic circles, gaining respect, and acquiring patrons. Frida Kahlo (1907-1954, Mexico) used the iconography of her Mexican heritage to paint herself and the pain that had become an integral part of her life following a bus accident at the age of 18 in which she suffered numerous injuries. She identified as a group member of her country, with Mexican culture and ancestry, and as belonging to the female gender. Kahlo’s self-portraits are dramatic, bloody, brutal, and at times overtly political. ( Self-Portrait , Frida Kahlo: upload.wikimedia.org/Wikipedia/en/1/1e/ Frida_Kahlo_%28self_portrait%29.jpg) In seeking her roots, she voiced concern for her country as it struggled for an independent cultural identity. She spoke to her country and people through her art. Kahlo’s art was inspired by her public beliefs and personal sufferings; she wanted her art to speak from her consciousness. Although self-portraits of today may be slightly different from those of earlier decades, they still depict self-exploration and identity through society and groups that communicate who we are. Cai Guo-Qiang (b. 1958, China, lives USA) exploded small charges of gunpowder to create an image of himself. ( Self- Portrait: A Subjugated Soul , Cai Guo-Qiang: http:// www.caiguoqiang.com/sites/default/files/styles/medium/public/1989_SelfPortrait_0389_001ltr-web. jpg) Different from those by Van Gogh, Picasso, and Kahlo, Cai’s self-portrait does not have any likeness or resemblance to his personal features, but it too sends a message about our society and how Cai relates to it. For example, the artist associates the lack of identifying information, rendering him anonymous, with contemporary society, and the fired gunpowder with both chaos and transformation. Despite the distance in time that separates early and modern humans, the search for their place in society and who they are remains of fascination and a mystery to all humans regardless of their time in history. - Ghose, Tia (Oct. 26, 2012). Why we care about our ancestry, Live Science. http://www.livescience.com/24313-why-ancestry.html	libretexts	2025-03-17T19:52:22.625207	2021-06-15T23:26:21	{ "license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/02%3A_The_Self/2.05%3A_Art_and_Identity-_Introduction", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "2.5: Art and Identity- Introduction", "author": "Pamela Sachant, Peggy Blood, Jeffery LeMieux, & Rita Tekippe" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/03%3A_Morality/3.04%3A_Art_and_Ethics-_Introduction	3.4: Art and Ethics- Introduction - - Last updated - Save as PDF This chapter is concerned with the perception, susceptibility, and ethics of art. It will explore and analyze the moral responsibility of artists and their rights to represent and create without censorship. Morality and art are connected usually in art that provokes and disturbs. Such art stirs up the artist’s or viewer’s personal beliefs, values, and morals due to what is depicted. Works that seem to purposely pursue or strongly communicate a message may cause controversies to flair up: controversies over the rights of artistic freedom or over how society evaluates art. That judgment of works created by artists has to do with society’s value judgment in a given time in history. The relationship between the artist and society is intertwined and sometimes at odds as it relates to art and ethics. Neither has to be sacrificed for the other, however, and neither needs to bend to the other in order to create or convey the work’s message. Art is subjective: it will be received or interpreted by different people in various ways. What may be unethical to one may be ethical to another. Because art is subjective, it is vulnerable to ethical judgment. It is most vulnerable when society does not have a historical context or understanding of art in order to appreciate a work’s content or aesthetics. This lack does not make ethical judgment wrong or irrational; it shows that appreciation of art or styles changes over time and that new or different art or styles can come to be appreciated. The general negative taste of society usually changes with more exposure. Still, taste remains subjective. Ethics has been a major consideration of the public and those in religious or political power throughout history. For many artists today, the first and major consideration is not ethics, but the platform from which to create and deliver the message through formal qualities and the medium. Consideration of ethics may be established by the artist but without hindrance of free expression. It is expected that in a work of art an artist’s own beliefs, values, and ideology may contrast with societal values. It is the art that speaks and adds quality value to what is communicated. This is what makes the power of free artistic expression so important. The art is judged not by who created the work or the artist’s character, but based on the merits of the work itself. However, through this visual dialogue existing between artist and society, there must be some mutual understanding. Society needs to understand that freedom of expression in the arts encourages greatness while artists need to be mindful of and open to society’s disposition. When the public values art as being a positive spiritual and physical addition to society, and the artist creates with ethical intentions, there is a connection between viewer and creator. An artist’s depiction of a subject does not mean that the creator approves or disapproves of the subject being presented. The artist’s purpose is to express, regardless of how the subject matter may be interpreted. Nevertheless, this freedom in interpretation does not mean that neither the artist nor society holds responsibility for their actions. Art and ethics, in this respect, demands that artists use their intellectual faculties to create a true expressive representation or convey psychological meaning. This type of art demands a capability on the viewer’s part to be moved by many sentiments from the artist. It demands the power of art to penetrate outward appearances, and seize and capture hidden thoughts and in- terpretations of the momentary or permanent emotions of a situation. While artists are creating, capturing visual images, and interpreting for their viewers, they are also giving them an unerring measure of the artists’ own moral or ethical sensibilities. Ethical dilemmas are not uncommon in the art world and often arise from the perception or interpretation of the artwork’s content or message. Provocative themes of spirituality, sexuality, and politics can and may be interpreted in many ways and provoke debates as to their being unethical or without morality. For example, when Dada artist Marcel Duchamp (1887-1968, France) created Fountain in 1917 , it was censored and rejected by contemporary connoisseurs of the arts and the public. ( Fountain , Marcel Duchamp: http://www.sfmoma.org/explore/collection/ artwork/25853#ixzz3mwCWDOxZ) A men’s urinal turned on its side, Duchamp considered this work to be one of his Readymade , manufactured objects that were turned into or designated by him as art. Today, Fountain is one of Duchamp’s most famous works and is widely considered an icon of twentieth-century art. More recently, The Holy Virgin Mary by Chris Ofili (b. 1968, England) shocked viewers when it was included in the 1997-2000 Sensation exhibition in London, Berlin, and New York. ( The Holy Virgin Mary , Chris Ofili: www.khanacademy.org/humaniti...ly-virgin-mary) The image caused considerable outrage from some members of the public across the country, including then-mayor of New York City Rudolph Giuliani. With its collaged images of women’s buttocks, glitter-mixed paint, and applied balls of elephant dung, many considered the painting blasphemous. Ofili stated that was not hisintention; he wanted to acknowledge both the sacred and secular, even sensual, beauty of the Virgin Mary, and that the dung, in his parents’ native country of Nigeria, symbolized fertility and the power of the elephant. Nevertheless, and probably unaware of the artist’s meaning, people were outraged. Traditionally, aesthetics in art has been associated with beauty, enjoyment, and the viewer’s visual, intellectual, and emotional captivation. Scandalous art may not be beautiful, but it very well could be enjoyable and hold one captive. The viewer is taken in and is attracted to something that is neither routine nor ordinary. All are considered to be meaningful experiences that are distinctive to Fine Arts. Aesthetic judgment goes hand in hand with ethics. It is part of the decision-making process people use when they view a work of art and decide if it is “good” or “bad.” The process of aesthetic judgment is a conceptual model that describes how people decide on the quality of artworks created and, for them individually or societally, makes an ethical decision about a certain work of art. As we can see, art indubitably has had the power to shock and, as a source of social provoca- tion, art will continue to shock unsuspecting viewers. Audiences will continue to feel scandalized, disturbed, or offended by art that is socially, politically, and religiously challenging. Being consid- ered scandalous or radical, as already observed, does not take away from experiencing or appre- ciation of the art, nor do such responses speak to the artist’s ethics or morality. Art may, however, fail in some eyes to offer an aesthetic experience. Such a failure also depends on the complexrelationship between art and the viewer, living in a given moment of time.	libretexts	2025-03-17T19:52:23.042976	2021-06-15T23:26:33	{ "license": "Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/03%3A_Morality/3.04%3A_Art_and_Ethics-_Introduction", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "3.4: Art and Ethics- Introduction", "author": "Pamela Sachant, Peggy Blood, Jeffery LeMieux, & Rita Tekippe" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/04%3A_Happiness/4.01%3A_The_Pursuit_of_Happiness	4.1: The Pursuit of Happiness Learning Objectives - Define and discuss happiness, including its determinants - Describe the field of positive psychology and identify the kinds of problems it addresses - Explain the meaning of positive affect and discuss its importance in health outcomes - Describe the concept of flow and its relationship to happiness and fulfillment Although the study of stress and how it affects us physically and psychologically is fascinating, it is—admittedly—somewhat of a grim topic. Psychology is also interested in the study of a more upbeat and encouraging approach to human affairs—the quest for happiness. Happiness America’s founders declared that its citizens have an unalienable right to pursue happiness. But what is happiness? When asked to define the term, people emphasize different aspects of this elusive state. Indeed, happiness is somewhat ambiguous and can be defined from different perspectives (Martin, 2012). Some people, especially those who are highly committed to their religious faith, view happiness in ways that emphasize virtuosity, reverence, and enlightened spirituality. Others see happiness as primarily contentment—the inner peace and joy that come from deep satisfaction with one’s surroundings, relationships with others, accomplishments, and oneself. Still others view happiness mainly as pleasurable engagement with their personal environment—having a career and hobbies that are engaging, meaningful, rewarding, and exciting. These differences, of course, are merely differences in emphasis. Most people would probably agree that each of these views, in some respects, captures the essence of happiness. Elements of Happiness Some psychologists have suggested that happiness consists of three distinct elements: the pleasant life, the good life, and the meaningful life, as shown in figure $\PageIndex{1}$ (Seligman, 2002; Seligman, Steen, Park, & Peterson, 2005). The pleasant life is realized through the attainment of day-to-day pleasures that add fun, joy, and excitement to our lives. For example, evening walks along the beach and a fulfilling sex life can enhance our daily pleasure and contribute to the pleasant life. The good life is achieved through identifying our unique skills and abilities and engaging these talents to enrich our lives; those who achieve the good life often find themselves absorbed in their work or their recreational pursuits. The meaningful life involves a deep sense of fulfillment that comes from using our talents in the service of the greater good: in ways that benefit the lives of others or that make the world a better place. In general, the happiest people tend to be those who pursue the full life—they orient their pursuits toward all three elements (Seligman et al., 2005). For practical purposes, a precise definition of happiness might incorporate each of these elements: an enduring state of mind consisting of joy, contentment, and other positive emotions, plus the sense that one’s life has meaning and value (Lyubomirsky, 2001). The definition implies that happiness is a long-term state—what is often characterized as subjective well-being—rather than merely a transient positive mood we all experience from time to time. It is this enduring happiness that has captured the interests of psychologists and other social scientists. The study of happiness has grown dramatically in the last three decades (Diener, 2013). One of the most basic questions that happiness investigators routinely examine is this: How happy are people in general? The average person in the world tends to be relatively happy and tends to indicate experiencing more positive feelings than negative feelings (Diener, Ng, Harter, & Arora, 2010). When asked to evaluate their current lives on a scale ranging from $0$ to $10$ (with $0$ representing “worst possible life” and $10$ representing “best possible life”), people in more than $150$ countries surveyed from 2010–2012 reported an average score of $5.2$. People who live in North America, Australia, and New Zealand reported the highest average score at $7.1$, whereas those living Sub-Saharan Africa reported the lowest average score at $4.6$ (Helliwell, Layard, & Sachs, 2013). Worldwide, the five happiest countries are Denmark, Norway, Switzerland, the Netherlands, and Sweden; the United States is ranked 17th happiest (See fig. 14.5.2) (Helliwell et al., 2013). Several years ago, a Gallup survey of more than $1,000$ U.S. adults found that $52\%$ reported that they were “very happy.” In addition, more than $8$ in $10$ indicated that they were “very satisfied” with their lives (Carroll, 2007). However, a recent poll of $2,345$ U.S. adults surprisingly revealed that only one-third reported they are “very happy.” The poll also revealed that the happiness levels of certain groups, including minorities, recent college graduates, and the disabled, have trended downward in recent years (Gregoire, 2013). Although it is difficult to explain this apparent decline in happiness, it may be connected to the challenging economic conditions the United States has endured over the last several years. Of course, this presumption would imply that happiness is closely tied to one’s finances. But, is it? This question brings us to the next important issue: What factors influence happiness? Factors Connected to Happiness What really makes people happy? What factors contribute to sustained joy and contentment? Is it money, attractiveness, material possessions, a rewarding occupation, a satisfying relationship? Extensive research over the years has examined this question. One finding is that age is related to happiness: Life satisfaction usually increases the older people get, but there do not appear to be gender differences in happiness (Diener, Suh, Lucas, & Smith, 1999). Although it is important to point out that much of this work has been correlational, many of the key findings (some of which may surprise you) are summarized below. Family and other social relationships appear to be key factors correlated with happiness. Studies show that married people report being happier than those who are single, divorced, or widowed (Diener et al., 1999). Happy individuals also report that their marriages are fulfilling (Lyubomirsky, King, & Diener, 2005). In fact, some have suggested that satisfaction with marriage and family life is the strongest predictor of happiness (Myers, 2000). Happy people tend to have more friends, more high-quality social relationships, and stronger social support networks than less happy people (Lyubomirsky et al., 2005). Happy people also have a high frequency of contact with friends (Pinquart & Sörensen, 2000). Can money buy happiness? In general, extensive research suggests that the answer is yes, but with several caveats. While a nation’s per capita gross domestic product (GDP) is associated with happiness levels (Helliwell et al., 2013), changes in GDP (which is a less certain index of household income) bear little relationship to changes in happiness (Diener, Tay, & Oishi, 2013). On the whole, residents of affluent countries tend to be happier than residents of poor countries; within countries, wealthy individuals are happier than poor individuals, but the association is much weaker (Diener & Biswas-Diener, 2002). To the extent that it leads to increases in purchasing power, increases in income are associated with increases in happiness (Diener, Oishi, & Ryan, 2013). However, income within societies appears to correlate with happiness only up to a point. In a study of over $450,000$ U.S. residents surveyed by the Gallup Organization, Kahneman and Deaton (2010) found that well-being rises with annual income, but only up to $\$75,000$. The average increase in reported well-being for people with incomes greater than $\$75,000$ was null. As implausible as these findings might seem—after all, higher incomes would enable people to indulge in Hawaiian vacations, prime seats as sporting events, expensive automobiles, and expansive new homes—higher incomes may impair people’s ability to savor and enjoy the small pleasures of life (Kahneman, 2011). Indeed, researchers in one study found that participants exposed to a subliminal reminder of wealth spent less time savoring a chocolate candy bar and exhibited less enjoyment of this experience than did participants who were not reminded of wealth (Quoidbach, Dunn, Petrides, & Mikolajczak, 2010). What about education and employment? Happy people, compared to those who are less happy, are more likely to graduate from college and secure more meaningful and engaging jobs. Once they obtain a job, they are also more likely to succeed (Lyubomirsky et al., 2005). While education shows a positive (but weak) correlation with happiness, intelligence is not appreciably related to happiness (Diener et al., 1999). Does religiosity correlate with happiness? In general, the answer is yes (Hackney & Sanders, 2003). However, the relationship between religiosity and happiness depends on societal circumstances. Nations and states with more difficult living conditions (e.g., widespread hunger and low life expectancy) tend to be more highly religious than societies with more favorable living conditions. Among those who live in nations with difficult living conditions, religiosity is associated with greater well-being; in nations with more favorable living conditions, religious and nonreligious individuals report similar levels of well-being (Diener, Tay, & Myers, 2011). Clearly the living conditions of one’s nation can influence factors related to happiness. What about the influence of one’s culture? To the extent that people possess characteristics that are highly valued by their culture, they tend to be happier (Diener, 2012). For example, self-esteem is a stronger predictor of life satisfaction in individualistic cultures than in collectivistic cultures (Diener, Diener, & Diener, 1995), and extraverted people tend to be happier in extraverted cultures than in introverted cultures (Fulmer et al., 2010). So we’ve identified many factors that exhibit some correlation to happiness. What factors don’t show a correlation? Researchers have studied both parenthood and physical attractiveness as potential contributors to happiness, but no link has been identified. Although people tend to believe that parenthood is central to a meaningful and fulﬁlling life, aggregate findings from a range of countries indicate that people who do not have children are generally happier than those who do (Hansen, 2012). And although one’s perceived level of attractiveness seems to predict happiness, a person’s objective physical attractiveness is only weakly correlated with her happiness (Diener, Wolsic, & Fujita, 1995). Life Events and Happiness An important point should be considered regarding happiness. People are often poor at affective forecasting: predicting the intensity and duration of their future emotions (Wilson & Gilbert, 2003). In one study, nearly all newlywed spouses predicted their marital satisfaction would remain stable or improve over the following four years; despite this high level of initial optimism, their marital satisfaction actually declined during this period (Lavner, Karner, & Bradbury, 2013). In addition, we are often incorrect when estimating how our long-term happiness would change for the better or worse in response to certain life events. For example, it is easy for many of us to imagine how euphoric we would feel if we won the lottery, were asked on a date by an attractive celebrity, or were offered our dream job. It is also easy to understand how long-suffering fans of the Chicago Cubs baseball team, which has not won a World Series championship since 1908, think they would feel permanently elated if their team would finally win another World Series. Likewise, it easy to predict that we would feel permanently miserable if we suffered a crippling accident or if a romantic relationship ended. However, something similar to sensory adaptation often occurs when people experience emotional reactions to life events. In much the same way our senses adapt to changes in stimulation (e.g., our eyes adapting to bright light after walking out of the darkness of a movie theater into the bright afternoon sun), we eventually adapt to changing emotional circumstances in our lives (Brickman & Campbell, 1971; Helson, 1964). When an event that provokes positive or negative emotions occurs, at first we tend to experience its emotional impact at full intensity. We feel a burst of pleasure following such things as a marriage proposal, birth of a child, acceptance to law school, an inheritance, and the like; as you might imagine, lottery winners experience a surge of happiness after hitting the jackpot (Lutter, 2007). Likewise, we experience a surge of misery following widowhood, a divorce, or a layoff from work. In the long run, however, we eventually adjust to the emotional new normal; the emotional impact of the event tends to erode, and we eventually revert to our original baseline happiness levels. Thus, what was at first a thrilling lottery windfall or World Series championship eventually loses its luster and becomes the status quo (See figure $\PageIndex{3}$). Indeed, dramatic life events have much less long-lasting impact on happiness than might be expected (Brickman, Coats, & Janoff-Bulman, 1978). Recently, some have raised questions concerning the extent to which important life events can permanently alter people’s happiness set points (Diener, Lucas, & Scollon, 2006). Evidence from a number of investigations suggests that, in some circumstances, happiness levels do not revert to their original positions. For example, although people generally tend to adapt to marriage so that it no longer makes them happier or unhappier than before, they often do not fully adapt to unemployment or severe disabilities (Diener, 2012). Figure $\PageIndex{4}$, which is based on longitudinal data from a sample of over $3,000$ German respondents, shows life satisfaction scores several years before, during, and after various life events, and it illustrates how people adapt (or fail to adapt) to these events. German respondents did not get lasting emotional boosts from marriage; instead, they reported brief increases in happiness, followed by quick adaptation. In contrast, widows and those who had been laid off experienced sizeable decreases in happiness that appeared to result in long-term changes in life satisfaction (Diener et al., 2006). Further, longitudinal data from the same sample showed that happiness levels changed significantly over time for nearly a quarter of respondents, with 9% showing major changes (Fujita & Diener, 2005). Thus, long-term happiness levels can and do change for some people. Increasing Happiness Some recent findings about happiness provide an optimistic picture, suggesting that real changes in happiness are possible. For example, thoughtfully developed well-being interventions designed to augment people’s baseline levels of happiness may increase happiness in ways that are permanent and long-lasting, not just temporary. These changes in happiness may be targeted at individual, organizational, and societal levels (Diener et al., 2006). Researchers in one study found that a series of happiness interventions involving such exercises as writing down three good things that occurred each day led to increases in happiness that lasted over six months (Seligman et al., 2005). Measuring happiness and well-being at the societal level over time may assist policy makers in determining if people are generally happy or miserable, as well as when and why they might feel the way they do. Studies show that average national happiness scores (over time and across countries) relate strongly to six key variables: per capita gross domestic product (GDP, which reflects a nation’s economic standard of living), social support, freedom to make important life choices, healthy life expectancy, freedom from perceived corruption in government and business, and generosity (Helliwell et al., 2013). Investigating why people are happy or unhappy might help policymakers develop programs that increase happiness and well-being within a society (Diener et al., 2006). Resolutions about contemporary political and social issues that are frequent topics of debate—such as poverty, taxation, affordable health care and housing, clean air and water, and income inequality—might be best considered with people’s happiness in mind. Positive Psychology In 1998, Seligman (the same person who conducted the learned helplessness experiments mentioned earlier), who was then president of the American Psychological Association, urged psychologists to focus more on understanding how to build human strength and psychological well-being. In deliberately setting out to create a new direction and new orientation for psychology, Seligman helped establish a growing movement and field of research called positive psychology (Compton, 2005). In a very general sense, positive psychology can be thought of as the science of happiness; it is an area of study that seeks to identify and promote those qualities that lead to greater fulfillment in our lives. This field looks at people’s strengths and what helps individuals to lead happy, contented lives, and it moves away from focusing on people’s pathology, faults, and problems. According to Seligman and Csikszentmihalyi (2000), positive psychology, "at the subjective level is about valued subjective experiences: well-being, contentment, and satisfaction (in the past); hope and optimism (for the future); and… happiness (in the present). At the individual level, it is about positive individual traits: the capacity for love and vocation, courage, interpersonal skill, aesthetic sensibility, perseverance, forgiveness, originality, future mindedness, spirituality, high talent, and wisdom." (p. 5) Some of the topics studied by positive psychologists include altruism and empathy, creativity, forgiveness and compassion, the importance of positive emotions, enhancement of immune system functioning, savoring the fleeting moments of life, and strengthening virtues as a way to increase authentic happiness (Compton, 2005). Recent efforts in the field of positive psychology have focused on extending its principles toward peace and well-being at the level of the global community. In a war-torn world in which conflict, hatred, and distrust are common, such an extended “positive peace psychology” could have important implications for understanding how to overcome oppression and work toward global peace (Cohrs, Christie, White, & Das, 2013). DIG DEEPER: The Center for Investigating Healthy Minds On the campus of the University of Wisconsin–Madison, the Center for Investigating Healthy Minds at the Waisman Center conducts rigorous scientific research on healthy aspects of the mind, such as kindness, forgiveness, compassion, and mindfulness. Established in 2008 and led by renowned neuroscientist Dr. Richard J. Davidson, the Center examines a wide range of ideas, including such things as a kindness curriculum in schools, neural correlates of prosocial behavior, psychological effects of Tai Chi training, digital games to foster prosocial behavior in children, and the effectiveness of yoga and breathing exercises in reducing symptoms of post-traumatic stress disorder. According to its website, the Center was founded after Dr. Davidson was challenged by His Holiness, the 14th Dalai Lama, “to apply the rigors of science to study positive qualities of mind” (Center for Investigating Health Minds, 2013). The Center continues to conduct scientific research with the aim of developing mental health training approaches that help people to live happier, healthier lives). Positive Affect and Optimism Taking a cue from positive psychology, extensive research over the last $10-15$ years has examined the importance of positive psychological attributes in physical well-being. Qualities that help promote psychological well-being (e.g., having meaning and purpose in life, a sense of autonomy, positive emotions, and satisfaction with life) are linked with a range of favorable health outcomes (especially improved cardiovascular health) mainly through their relationships with biological functions and health behaviors (such as diet, physical activity, and sleep quality) (Boehm & Kubzansky, 2012). The quality that has received attention is positive affect , which refers to pleasurable engagement with the environment, such as happiness, joy, enthusiasm, alertness, and excitement (Watson, Clark, & Tellegen, 1988). The characteristics of positive affect, as with negative affect (discussed earlier), can be brief, long-lasting, or trait-like (Pressman & Cohen, 2005). Independent of age, gender, and income, positive affect is associated with greater social connectedness, emotional and practical support, adaptive coping efforts, and lower depression; it is also associated with longevity and favorable physiological functioning (Steptoe, O’Donnell, Marmot, & Wardle, 2008). Positive affect also serves as a protective factor against heart disease. In a $10$-year study of Nova Scotians, the rate of heart disease was $22\%$ lower for each one-point increase on the measure of positive affect, from $1$ (no positive affect expressed) to $5$ (extreme positive affect) (Davidson, Mostofsky, & Whang, 2010). In terms of our health, the expression, “don’t worry, be happy” is helpful advice indeed. There has also been much work suggesting that optimism —the general tendency to look on the bright side of things—is also a significant predictor of positive health outcomes. Although positive affect and optimism are related in some ways, they are not the same (Pressman & Cohen, 2005). Whereas positive affect is mostly concerned with positive feeling states, optimism has been regarded as a generalized tendency to expect that good things will happen (Chang, 2001). It has also been conceptualized as a tendency to view life’s stressors and difficulties as temporary and external to oneself (Peterson & Steen, 2002). Numerous studies over the years have consistently shown that optimism is linked to longevity, healthier behaviors, fewer postsurgical complications, better immune functioning among men with prostate cancer, and better treatment adherence (Rasmussen & Wallio, 2008). Further, optimistic people report fewer physical symptoms, less pain, better physical functioning, and are less likely to be rehospitalized following heart surgery (Rasmussen, Scheier, & Greenhouse, 2009). Flow Another factor that seems to be important in fostering a deep sense of well-being is the ability to derive flow from the things we do in life. Flow is described as a particular experience that is so engaging and engrossing that it becomes worth doing for its own sake (Csikszentmihalyi, 1997). It is usually related to creative endeavors and leisure activities, but it can also be experienced by workers who like their jobs or students who love studying (Csikszentmihalyi, 1999). Many of us instantly recognize the notion of flow. In fact, the term derived from respondents’ spontaneous use of the term when asked to describe how it felt when what they were doing was going well. When people experience flow, they become involved in an activity to the point where they feel they lose themselves in the activity. They effortlessly maintain their concentration and focus, they feel as though they have complete control of their actions, and time seems to pass more quickly than usual (Csikszentmihalyi, 1997). Flow is considered a pleasurable experience, and it typically occurs when people are engaged in challenging activities that require skills and knowledge they know they possess. For example, people would be more likely report flow experiences in relation to their work or hobbies than in relation to eating. When asked the question, “Do you ever get involved in something so deeply that nothing else seems to matter, and you lose track of time?” about $20\%$ of Americans and Europeans report having these flow-like experiences regularly (Csikszentmihalyi, 1997). Although wealth and material possessions are nice to have, the notion of flow suggests that neither are prerequisites for a happy and fulfilling life. Finding an activity that you are truly enthusiastic about, something so absorbing that doing it is reward itself (whether it be playing tennis, studying Arabic, writing children’s novels, or cooking lavish meals) is perhaps the real key. According to Csikszentmihalyi (1999), creating conditions that make flow experiences possible should be a top social and political priority. How might this goal be achieved? How might flow be promoted in school systems? In the workplace? What potential benefits might be accrued from such efforts? In an ideal world, scientific research endeavors should inform us on how to bring about a better world for all people. The field of positive psychology promises to be instrumental in helping us understand what truly builds hope, optimism, happiness, healthy relationships, flow, and genuine personal fulfillment. Summary Happiness is conceptualized as an enduring state of mind that consists of the capacity to experience pleasure in daily life, as well as the ability to engage one’s skills and talents to enrich one’s life and the lives of others. Although people around the world generally report that they are happy, there are differences in average happiness levels across nations. Although people have a tendency to overestimate the extent to which their happiness set points would change for the better or for the worse following certain life events, researchers have identified a number of factors that are consistently related to happiness. In recent years, positive psychology has emerged as an area of study seeking to identify and promote qualities that lead to greater happiness and fulfillment in our lives. These components include positive affect, optimism, and flow. Glossary - flow - state involving intense engagement in an activity; usually is experienced when participating in creative, work, and leisure endeavors - happiness - enduring state of mind consisting of joy, contentment, and other positive emotions; the sense that one’s life has meaning and value - optimism - tendency toward a positive outlook and positive expectations - positive affect - state or a trait that involves pleasurable engagement with the environment, the dimensions of which include happiness, joy, enthusiasm, alertness, and excitement - positive psychology - scientific area of study seeking to identify and promote those qualities that lead to happy, fulfilled, and contented lives	libretexts	2025-03-17T19:52:23.343777	2021-06-15T23:26:42	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/04%3A_Happiness/4.01%3A_The_Pursuit_of_Happiness", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "4.1: The Pursuit of Happiness", "author": "OpenStax" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will	8.1: Freedom of the Will Free will is the ability to choose between different possible courses of action. It is closely linked to the concepts of responsibility, praise, guilt, sin, and other judgments which apply only to actions that are freely chosen. It is also connected with the concepts of advice, persuasion, deliberation, and prohibition. Traditionally, only actions that are freely willed are seen as deserving credit or blame. There are numerous different concerns about threats to the possibility of free will, varying by how exactly it is conceived, which is a matter of some debate. Some conceive free will to be the capacity to make choices in which the outcome has not been determined by past events. Determinism suggests that only one course of events is possible, which is inconsistent with the existence of such free will. This problem has been identified in ancient Greek philosophy, and remains a major focus of philosophical debate. This view that conceives free will to be incompatible with determinism is called incompatibilism , and encompasses both metaphysical libertarianism, the claim that determinism is false and thus free will is at least possible, and hard determinism, the claim that determinism is true and thus free will is not possible. It also encompasses hard incompatibilism, which holds not only determinism but also its negation to be incompatible with free will, and thus free will to be impossible whatever the case may be regarding determinism. In contrast, compatibilists hold that free will is compatible with determinism. Some compatibilists even hold that determinism is necessary for free will, arguing that choice involves preference for one course of action over another, requiring a sense of how choices will turn out. Compatibilists thus consider the debate between libertarians and hard determinists over free will vs determinism a false dilemma. Different compatibilists offer very different definitions of what "free will" even means, and consequently find different types of constraints to be relevant to the issue. Classical compatibilists considered free will nothing more than freedom of action, considering one free of will simply if, had one counterfactually wanted to do otherwise, one could have done otherwise without physical impediment. Contemporary compatibilists instead identify free will as a psychological capacity, such as to direct one's behavior in a way responsive to reason. And there are still further different conceptions of free will, each with their own concerns, sharing only the common feature of not finding the possibility of determinism a threat to the possibility of free will. In Western philosophy The underlying questions are whether we have control over our actions, and if so, what sort of control, and to what extent. These questions predate the early Greek stoics (for example, Chrysippus), and some modern philosophers lament the lack of progress over all these millennia. On one hand, humans have a strong sense of freedom, which leads us to believe that we have free will. On the other hand, an intuitive feeling of free will could be mistaken. It is difficult to reconcile the intuitive evidence that conscious decisions are causally effective with the scientific view that the physical world can be explained to operate perfectly by physical law. The conflict between intuitively felt freedom and natural law arises when either causal closure or physical determinism (nomological determinism) is asserted. With causal closure, no physical event has a cause outside the physical domain, and with physical determinism, the future is determined entirely by preceding events (cause and effect). The puzzle of reconciling 'free will' with a deterministic universe is known as the problem of free will or sometimes referred to as the dilemma of determinism . This dilemma leads to a moral dilemma as well: How are we to assign responsibility for our actions if they are caused entirely by past events? Compatibilists maintain that mental reality is not of itself causally effective. Classical compatibilists have addressed the dilemma of free will by arguing that free will holds as long as we are not externally constrained or coerced. Modern compatibilists make a distinction between freedom of will and freedom of action , that is, separating freedom of choice from the freedom to enact it. Given that humans all experience a sense of free will, some modern compatibilists think it is necessary to accommodate this intuition. Compatibilists often associate freedom of will with the ability to make rational decisions. A different approach to the dilemma is that of incompatibilists, namely, that if the world is deterministic then, our feeling that we are free to choose an action is simply an illusion. Metaphysical libertarianism is the form of incompatibilism which posits that determinism is false and free will is possible (at least some people have free will). This view is associated with non-materialist constructions, including both traditional dualism, as well as models supporting more minimal criteria; such as the ability to consciously veto an action or competing desire. Yet even with physical indeterminism, arguments have been made against libertarianism in that it is difficult to assign Origination (responsibility for "free" indeterministic choices). Free will here is predominately treated with respect to physical determinism in the strict sense of nomological determinism, although other forms of determinism are also relevant to free will. For example, logical and theological determinism challenge metaphysical libertarianism with ideas of destiny and fate, and biological, cultural and psychological determinism feed the development of compatibilist models. Separate classes of compatibilism and incompatibilism may even be formed to represent these. Below are the classic arguments bearing upon the dilemma and its underpinnings. Incompatibilism Incompatibilism is the position that free will and determinism are logically incompatible, and that the major question regarding whether or not people have free will is thus whether or not their actions are determined. "Hard determinists", such as d'Holbach, are those incompatibilists who accept determinism and reject free will. In contrast, "metaphysical libertarians", such as Thomas Reid, Peter van Inwagen, and Robert Kane, are those incompatibilists who accept free will and deny determinism, holding the view that some form of indeterminism is true. Another view is that of hard incompatibilists, which state that free will is incompatible with both determinism and indeterminism. Traditional arguments for incompatibilism are based on an "intuition pump": if a person is like other mechanical things that are determined in their behavior such as a wind-up toy, a billiard ball, a puppet, or a robot, then people must not have free will. This argument has been rejected by compatibilists such as Daniel Dennett on the grounds that, even if humans have something in common with these things, it remains possible and plausible that we are different from such objects in important ways. Another argument for incompatibilism is that of the "causal chain". Incompatibilism is key to the idealist theory of free will. Most incompatibilists reject the idea that freedom of action consists simply in "voluntary" behavior. They insist, rather, that free will means that man must be the "ultimate" or "originating" cause of his actions. He must be causa sui , in the traditional phrase. Being responsible for one's choices is the first cause of those choices, where first cause means that there is no antecedent cause of that cause. The argument, then, is that if man has free will, then man is the ultimate cause of his actions. If determinism is true, then all of man's choices are caused by events and facts outside his control. So, if everything man does is caused by events and facts outside his control, then he cannot be the ultimate cause of his actions. Therefore, he cannot have free will. This argument has also been challenged by various compatibilist philosophers.	libretexts	2025-03-17T19:52:24.433205	2021-06-15T23:27:14	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "8.1: Freedom of the Will", "author": "Noah Levin" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.01%3A_The_Turing_Test	8.1.1: The Turing Test - - Last updated - Save as PDF “ Alan Turing ” 29 Alan Mathison Turing OBE FRS (/ˈtjʊərɪŋ/; 23 June 1912 – 7 June 1954) was an English computer scientist, mathematician, logician, cryptanalyst and theoretical biologist. He was highly influential in the development of theoretical computer science, providing a formalisation of the concepts of algorithm and computation with the Turing machine, which can be considered a model of a general purpose computer. Turing is widely considered to be the father of theoretical computer science and artificial intelligence. During the Second World War, Turing worked for the Government Code and Cypher School (GC&CS) at Bletchley Park, Britain's codebreaking centre. For a time he led Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for speeding the breaking of German ciphers, including improvements to the pre-war Polish bombe method, an electromechanical machine that could find settings for the Enigma machine. Turing played a pivotal role in cracking intercepted coded messages that enabled the Allies to defeat the Nazis in many crucial engagements, including the Battle of the Atlantic; it has been estimated that this work shortened the war in Europe by more than two years and saved over fourteen million lives. After the war, he worked at the National Physical Laboratory, where he designed the ACE, among the first designs for a stored-program computer. In 1948 Turing joined Max Newman's Computing Machine Laboratory at the Victoria University of Manchester, where he helped develop the Manchester computers and became interested in mathematical biology. He wrote a paper on the chemical basis of morphogenesis, and predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, first observed in the 1960s. Turing was prosecuted in 1952 for homosexual acts, when by the Labouchere Amendment, "gross indecency" was still criminal in the UK. He accepted chemical castration treatment, with DES, as an alternative to prison. Turing died in 1954, 16 days before his 42nd birthday, from cyanide poisoning. An inquest determined his death as suicide, but it has been noted that the known evidence is also consistent with accidental poisoning. In 2009, following an Internet campaign, British Prime Minister Gordon Brown made an official public apology on behalf of the British government for "the appalling way he was treated." Queen Elizabeth II granted him a posthumous pardon in 2013. “The Turing Test” Turing’s work “Computing Machinery and Intelligence” 30 was groundbreaking in his looking toward the future of how we can understand and deal with increasingly capable machines. The quotes from Turing below that are combined with my commentary are from this article. The first section of Turing’s article is labeled “The Imitation Game.” The recent biopic abut Turing (2014, Directed by Morten Tyldum) uses this as its title, making use both of Turing’s own most popular contribution to Philosophy and the fact that Turing had a lot of difficulties fitting socially, so that he had to “imitate” being a normal person. It’s worth a watch if you have the time. In this first section of the article, Turing outlines what his project is going to be, and he makes it very clear what he wants to argue for, and what he does not want to argue for, “I propose to consider the question, "Can machines think?" This should begin with definitions of the meaning of the terms "machine" and "think." The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous, If the meaning of the words "machine" and "think" are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to the question, "Can machines think?" is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words. The new form of the problem can be described in terms of a game which we call the 'imitation game." It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart front the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either "X is A and Y is B" or "X is B and Y is A."” The form that the test will take is rather easy to understand given today’s technologies, but Turing had to go to great lengths to basically just describe the judge communicating with the two through electronic form, like a chat room, instant messaging, or texting. Communicating this way ensures that only the content of what is said is considered when making a decision. He then says, “We now ask the question, "What will happen when a machine takes the part of A in this game?" Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, "Can machines think?"” So what he wants to do is rather simple: a judge (you, for example) chats with 2 beings through a computer. One is a computer and the other is a person. They both try to convince you that they are the person, and if you can’t decide which is which or guess wrong – and this happens repeatedly to both yourself and others – then Turing believes we ought to conclude that the computer is intelligent like we are since it has properly imitated us (hence calling it the “imitation game”). Turing then goes on to explain the advantages of approaching machine intelligence this way as opposed to trying to define thinking and showing that a machine can do it, “The new problem has the advantage of drawing a fairly sharp line between the physical and the intellectual capacities of a man. No engineer or chemist claims to be able to produce a material which is indistinguishable from the human skin. It is possible that at some time this might be done, but even supposing this invention available we should feel there was little point in trying to make a "thinking machine" more human by dressing it up in such artificial flesh. The form in which we have set the problem reflects this fact in the condition which prevents the interrogator from seeing or touching the other competitors, or hearing -their voices… We do not wish to penalise the machine for its inability to shine in beauty competitions, nor to penalise a man for losing in a race against an aeroplane. The conditions of our game make these disabilities irrelevant. The "witnesses" can brag, if they consider it advisable, as much as they please about their charms, strength or heroism, but the interrogator cannot demand practical demonstrations…. It might be urged that when playing the "imitation game" the best strategy for the machine may possibly be something other than imitation of the behaviour of a man. This may be, but I think it is unlikely that there is any great effect of this kind. In any case there is no intention to investigate here the theory of the game, and it will be assumed that the best strategy is to try to provide answers that would naturally be given by a man.” Why approach artificial intelligence in this way? It’s rather simple: how do we know other people are intelligent? Do we define “thinking” and then say “I know that you are thinking and thus can be intelligent”? Or do we just assume others are intelligent if we interact with them and they show they are intelligent? Turing just wants to extend this courtesy that we extend to other people to machines as well. After presenting his reasons for believing the test to be a good one, Turing goes on to deal with many counter arguments to his own. This is a solid philosophical move and is his attempt to deal with all of the most reasonable and common objections to what he has proposed. He dealt with a number of technological issues which we can now take for granted: there are few who doubt that artificial intelligence will happen some time, and now it appears to just be a matter of when and how. Computers advance in power every year, so some real form of artificial intelligence beyond Siri-like helpers is on the distant horizon. He does deal with other, more timeless objections as well, and those follow below. The Mathematical Objection Despite the advances in technology, it is still possible that computers will never be capable of computing everything necessary to mimic or have human intelligence. He summarizes these objections as follows, “The result in question refers to a type of machine which is essentially a digital computer with an infinite capacity. It states that there are certain things that such a machine cannot do. If it is rigged up to give answers to questions as in the imitation game, there will be some questions to which it will either give a wrong answer, or fail to give an answer at all however much time is allowed for a reply. There may, of course, be many such questions, and questions which cannot be answered by one machine may be satisfactorily answered by another. We are of course supposing for the present that the questions are of the kind to which an answer "Yes" or "No" is appropriate, rather than questions such as "What do you think of Picasso?" The questions that we know the machines must fail on are of this type…This is the mathematical result: it is argued that it proves a disability of machines to which the human intellect is not subject.” His response to this is rather simple: So what? How do you know that people aren’t limited? And even if a computer has limitations, does it really matter? His response is below and it finishes with his most important point for the purposes of the usefulness of the Imitation Game, “The short answer to this argument is that although it is established that there are limitations to the Powers If any particular machine, it has only been stated, without any sort of proof, that no such limitations apply to the human intellect. But I do not think this view can be dismissed quite so lightly. Whenever one of these machines is asked the appropriate critical question, and gives a definite answer, we know that this answer must be wrong, and this gives us a certain feeling of superiority. Is this feeling illusory? It is no doubt quite genuine, but I do not think too much importance should be attached to it. We too often give wrong answers to questions ourselves to be justified in being very pleased at such evidence of fallibility on the part of the machines. Further, our superiority can only be felt on such an occasion in relation to the one machine over which we have scored our petty triumph. There would be no question of triumphing simultaneously over all machines. In short, then, there might be men cleverer than any given machine, but then again there might be other machines cleverer again, and so on. Those who hold to the mathematical argument would, I think, mostly he willing to accept the imitation game as a basis for discussion…” So, even if there is something to this argument, the Imitation Game itself can still function as a test for intelligence. The Argument from Consciousness This argument is one of the more interesting and strongest ones that still stands today against the possibility of a true Artificial Intelligence akin to our own, human intelligence. Turing explains this argument as follows, “This argument is very, well expressed in Professor Jefferson's Lister Oration for 1949, from which I quote. "Not until a machine can write a sonnet or compose a concerto because of thoughts and emotions felt, and not by the chance fall of symbols, could we agree that machine equals brain-that is, not only write it but know that it had written it. No mechanism could feel (and not merely artificially signal, an easy contrivance) pleasure at its successes, grief when its valves fuse, be warmed by flattery, be made miserable by its mistakes, be charmed by sex, be angry or depressed when it cannot get what it wants."” This objection calls into the question validity of the Imitation Game test and essentially says, “Even if it passes the test, it’s not intelligent because it lacks consciousness or the ability to do something new and emotional.” Turing responds by asking whether or not we really know that others are thinking (or even ourselves). He has a point here since what constitutes “consciousness” and “thinking” is still mostly a mystery. We know how brains work during those processes, but we can’t exactly define them yet, hence why Turing likes his test as the method of determining intelligence. Turing goes on to say that the real force of the argument is that it calls into question a machines ability to understand anything like a human does. John Searle’s “Chinese Room” argument that follows below is a stronger presentation of this argument, and Turing gives his own response to these objections by saying that a machine could certainly behave as if it has understanding, and whether or not that understanding would be “genuine” is a separate issue; the test of the Imitation Game for intelligence can still work, especially since it’s how we know that other people understand things. Arguments from Various Disabilities Turing summarizes this objection as follows, “These arguments take the form, "I grant you that you can make machines do all the things you have mentioned but you will never be able to make one to do X." Numerous features X are suggested in this connexion I offer a selection: Be kind, resourceful, beautiful, friendly, have initiative, have a sense of humour, tell right from wrong, make mistakes, fall in love, enjoy strawberries and cream, make some one fall in love with it, learn from experience, use words properly, be the subject of its own thought, have as much diversity of behaviour as a man, do something really new.” Turing’s response is surprisingly simple: So what if they can’t do these things since they could still be intelligent without having to do these things, and more importantly, couldn’t we just make the machine do these things? Can’t we make a computer make mistakes, learn, behave in a certain way, have taste buds, self-program, etc.? And if it’s not funny, is not human then? While I would like to call those without a sense of humor inhuman, I won’t. They’re just boring, and computers can certainly be boring. Finally, Turing drives home his main point yet again: none of this calls into question the appropriateness of his test in determining when a machine has achieved intelligence. Lady Lovelace's Objection This is perhaps one of the more cited objections to computer intelligence, and predates Turing by over a hundred years. He portrays it, and his response, as follows, “Our most detailed information of Babbage's Analytical Engine comes from a memoir by Lady Lovelace (1842). In it she states, "The Analytical Engine has no pretensions to originate anything. It can do whatever we know how to order it to perform" (her italics). This statement is quoted by Hartree (1949) who adds: "This does not imply that it may not be possible to construct electronic equipment which will 'think for itself,' or in which, in biological terms, one could set up a conditioned reflex, which would serve as a basis for 'learning.' Whether this is possible in principle or not is a stimulating and exciting question, suggested by some of these recent developments But it did not seem that the machines constructed or projected at the time had this property." I am in thorough agreement with Hartree over this. It will be noticed that he does not assert that the machines in question had not got the property, but rather that the evidence available to Lady Lovelace did not encourage her to believe that they had it. It is quite possible that the machines in question had in a sense got this property. For suppose that some discrete-state machine has the property. The Analytical Engine was a universal digital computer, so that, if its storage capacity and speed were adequate, it could by suitable programming be made to mimic the machine in question. Probably this argument did not occur to the Countess or to Babbage. In any case there was no obligation on them to claim all that could be claimed… A variant of Lady Lovelace's objection states that a machine can "never do anything really new." This may be parried for a moment with the saw, "There is nothing new under the sun." Who can be certain that "original work" that he has done was not simply the growth of the seed planted in him by teaching, or the effect of following well-known general principles.” Again, to come back to Turing’s main point, this doesn’t seem to create a genuine problem for his test. Whether or not a machine – or anyone – can do anything new is a separate question than whether something is intelligent. Argument from Continuity in the Nervous System This objection to Turing is a more technical one, and he explains it like this, “The nervous system is certainly not a discrete-state machine. A small error in the information about the size of a nervous impulse impinging on a neuron, may make a large difference to the size of the outgoing impulse. It may be argued that, this being so, one cannot expect to be able to mimic the behaviour of the nervous system with a discrete-state system.” This is an early version of the distinction between what is now known as “hard AI” versus “soft AI”. Hard AI is the type of intelligence that Turing has been talking about: a computer, as we understand them, runs a program that is intelligent. This objection he is discussing basically says that there is no way this can work since a computer is, in some sense, binary (just using 1’s and 0’s to do everything) and the human brain and mind work in an entirely different way. This problem is still being discussed, since we just don’t know whether or not our brains do function like a very complex personal computer or not. Because there is still something to this objection, there are those that are going for what is known as “soft AI” where, instead of creating a traditional program on a traditional computer, the goal is to create an artificial mechanical brain that completely replicates our own brain in every way. Rather than having a program be intelligent, the idea is that the resultant thing would be intelligent since it would be, piece for piece, an exact (but artificial) replica of the human brain. Many people think this is the only way to get a real AI and people are laying the foundation to try to create an “artificial” brain that can think for itself and mimic the human mind. Time will give us the answer to whether or not this works. “The Chinese Room” 31 The Chinese room argument holds that a program cannot give a computer a "mind", "understanding" or "consciousness", regardless of how intelligently or human-like the program may make the computer behave. The argument was first presented by philosopher John Searle in his paper, "Minds, Brains, and Programs", published in Behavioral and Brain Sciences in 1980. It has been widely discussed in the years since. The centerpiece of the argument is a thought experiment known as the Chinese room. The argument is directed against the philosophical positions of functionalism and computationalism, which hold that the mind may be viewed as an information-processing system operating on formal symbols. Specifically, the argument refutes a position Searle calls Strong AI: The appropriately programmed computer with the right inputs and outputs would thereby have a mind in exactly the same sense human beings have minds. Although it was originally presented in reaction to the statements of artificial intelligence (AI) researchers, it is not an argument against the goals of AI research, because it does not limit the amount of intelligence a machine can display. The argument applies only to digital computers running programs and does not apply to machines in general. The Chinese Room Thought Experiment Searle's thought experiment begins with this hypothetical premise: suppose that artificial intelligence research has succeeded in constructing a computer that behaves as if it understands Chinese. It takes Chinese characters as input and, by following the instructions of a computer program, produces other Chinese characters, which it presents as output. Suppose, says Searle, that this computer performs its task so convincingly that it comfortably passes the Turing test: it convinces a human Chinese speaker that the program is itself a live Chinese speaker. To all of the questions that the person asks, it makes appropriate responses, such that any Chinese speaker would be convinced that they are talking to another Chinese-speaking human being. The question Searle wants to answer is this: does the machine literally "understand" Chinese? Or is it merely simulating the ability to understand Chinese? Searle calls the first position "strong AI" and the latter "weak AI". Searle then supposes that he is in a closed room and has a book with an English version of the computer program, along with sufficient paper, pencils, erasers, and filing cabinets. Searle could receive Chinese characters through a slot in the door, process them according to the program's instructions, and produce Chinese characters as output. If the computer had passed the Turing test this way, it follows, says Searle, that he would do so as well, simply by running the program manually. Searle asserts that there is no essential difference between the roles of the computer and himself in the experiment. Each simply follows a program, step-by-step, producing a behavior which is then interpreted as demonstrating intelligent conversation. However, Searle would not be able to understand the conversation. ("I don't speak a word of Chinese," he points out.) Therefore, he argues, it follows that the computer would not be able to understand the conversation either. Searle argues that, without "understanding" (or "intentionality"), we cannot describe what the machine is doing as "thinking" and, since it does not think, it does not have a "mind" in anything like the normal sense of the word. Therefore, he concludes that "strong AI" is false. The Turing Test The Chinese room implements a version of the Turing test. Alan Turing introduced the test in 1950 to help answer the question "can machines think?" In the standard version, a human judge engages in a natural language conversation with a human and a machine designed to generate performance indistinguishable from that of a human being. All participants are separated from one another. If the judge cannot reliably tell the machine from the human, the machine is said to have passed the test. Turing then considered each possible objection to the proposal "machines can think", and found that there are simple, obvious answers if the question is de-mystified in this way. He did not, however, intend for the test to measure for the presence of "consciousness" or "understanding". He did not believe this was relevant to the issues that he was addressing. He wrote: I do not wish to give the impression that I think there is no mystery about consciousness. There is, for instance, something of a paradox connected with any attempt to localise it. But I do not think these mysteries necessarily need to be solved before we can answer the question with which we are concerned in this paper. To Searle, as a philosopher investigating in the nature of mind and consciousness, these are the relevant mysteries. The Chinese room is designed to show that the Turing test is insufficient to detect the presence of consciousness, even if the room can behave or function as a conscious mind would. “What it’s like to be a Bat” 32 The Philosopher Thomas Nagel devised a thought experiment which I believe can act as an interesting response to Searle’s Chinese Room argument. Searle does have a very strong and interesting point about understanding, but he doesn’t show that there is no understanding taking place – just that it won’t be the same type of understanding we (normal, conscious humans) believe that we have. Nagel asks us to try to understand what it would be like to be a bat, and concludes that we could not possibly begin to grasp how bats experience the world, but that doesn’t mean there’s not something happening, it’s just that it’s in a form we cannot comprehend. If you take what he says and apply it to the Chinese Room, then couldn’t a computer have its own type of non-human understanding that we can’t comprehend? I have included a summary of his arguments below, and it is very technical, but hopefully it makes sense. " What is it like to be a bat? " is a paper by American philosopher Thomas Nagel, first published in The Philosophical Review in October 1974, and later in Nagel's Mortal Questions (1979). In it, Nagel argues that materialist theories of mind omit the essential component of consciousness, namely that there is something that it is (or feels) like to be a particular, conscious thing. He argued that an organism had conscious mental states, "if and only if there is something that it is like to be that organism—something it is like for the organism." Daniel Dennett called Nagel's example "The most widely cited and influential thought experiment about consciousness." The thesis attempts to refute reductionism (the philosophical position that a complex system is nothing more than the sum of its parts). For example, a physicalist reductionist's approach to the mind–body problem holds that the mental process humans experience as consciousness can be fully described via physical processes in the brain and body. Nagel begins by arguing that the conscious experience is widespread, present in many animals (particularly mammals), and that for an organism to have a conscious experience it must be special, in the sense that its qualia or "subjective character of experience" are unique. Nagel stated, “An organism has conscious mental states if and only if there is something that it is like to be that organism - something that it is like for the organism to be itself.” The paper argues that the subjective nature of consciousness undermines any attempt to explain consciousness via objective, reductionist means. A subjective character of experience cannot be explained by a system of functional or intentional states. Consciousness cannot be explained without the subjective character of experience, and the subjective character of experience cannot be explained by a reductionist being; it is a mental phenomenon that cannot be reduced to materialism. Thus for consciousness to be explained from a reductionist stance, the idea of the subjective character of experience would have to be discarded, which is absurd. Neither can a physicalist view, because in such a world each phenomenal experience had by a conscious being would have to have a physical property attributed to it, which is impossible to prove due to the subjectivity of conscious experience. Nagel argues that each and every subjective experience is connected with a “single point of view,” making it unfeasible to consider any conscious experience as “objective”. Nagel uses the metaphor of bats to clarify the distinction between subjective and objective concepts. Bats are mammals, so they are assumed to have conscious experience. Nagel used bats for his argument because of their highly evolved and active use of a biological sensory apparatus that is significantly different from that of many other organisms. Bats use echolocation to navigate and perceive objects. This method of perception is similar to the human sense of vision. Both sonar and vision are regarded as perceptional experiences. While it is possible to imagine what it would be like to fly, navigate by sonar, hang upside down and eat bugs like a bat, that is not the same as a bat's perspective. Nagel claims that even if humans were able to metamorphose gradually into bats, their brains would not have been wired as a bat's from birth; therefore, they would only be able to experience the life and behaviors of a bat, rather than the mindset. Such is the difference between subjective and objective points of view. According to Nagel, “our own mental activity is the only unquestionable fact of our experience”, meaning that each individual only knows what it is like to be them (Subjectivism). Objectivity, requires an unbiased, non-subjective state of perception. For Nagel, the objective perspective is not feasible, because humans are limited to subjective experience. Nagel concludes with the contention that it would be wrong to assume that physicalism is incorrect, since that position is also imperfectly understood. Physicalism claims that states and events are physical, but those physical states and events are only imperfectly characterized. Nevertheless, he holds that physicalism cannot be understood without characterizing objective and subjective experience. That is a necessary precondition for understanding the mind-body problem.	libretexts	2025-03-17T19:52:24.512935	2021-06-15T23:27:17	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.01%3A_The_Turing_Test", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "8.1.1: The Turing Test", "author": "Noah Levin" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.02%3A_Determinism	8.1.2: Determinism - - Last updated - Save as PDF “Physical Determinism” 33 Physical determinism generally refers to the assertion of a deterministic physical universe (greater physical system). This holds that a complete description of the physical state of the world at any given time and a complete statement of the physical laws of nature together entail every truth as to what physical events happen after that time. In a scientific context, the term 'event' is somewhat technical, depending upon which theory one is considering, and basically is the occurrence of some 'state' peculiar to that theory, for example, a quantum state or a thermodynamic state. Physical determinism includes (but is not restricted to) nomological determinism, which holds that all future events are governed by the past or present according to all-encompassing deterministic laws. The concept of physical determinism has also been used to denote the predictability of a physical system, although this usage is uncommon. Physical determinism can also be viewed as an observed phenomenon of our experience, or a thesis only relevant to mathematical models of physics and other physical sciences. Physical determinism has also been used as a specific deterministic hypothesis about human behavior. Although somewhat unrelated to its standard context, physical determinism has also been used in social engineering theory. History The notion of physical determinism takes its classical form in the ideas of Laplace, who posited (in agreement with the physics of his time) that an omniscient observer (called sometimes Laplace's demon) knowing with infinite precision the positions and velocities of every particle in the universe could predict the future entirely. Although such an omniscient observer is a hypothetical construct, and infinite precision exceeds the capacities of human measurement, the illustration is presented as a statement of what in principle would be possible if physical determinism were true, and so reduction to practice is not an issue. Physical determinism is currently under heavy debate in modern science. For example, physical indeterminism has been proposed to accommodate various interpretations of quantum mechanics. Suggestions have also been made to reformulate the conception of determinism with respect to its application to physical law. Causal completeness Physical determinism is related to the question of causal completeness of physics, which is synonymous with the weaker form of causal closure. This is the idea that every real event has a scientific explanation, that science need not search for explanations beyond itself. If causal completeness does not apply to everything in the universe, then the door is open to events that are not subject to physical law. For example, a relatively common view of mental events is that they are an epiphenomenon produced as a by-product of neurological activity, and without causal impact. In this case, only a failure of deterministic physical law would allow room for their causal significance. Other formulations A more modern formulation of physical determinism skirts the issue of causal completeness. It is based upon connections between 'events' supplied by a theory: "a theory is deterministic if, and only if, given its state variables for some initial period, the theory logically determines a unique set of values for those variables for any other period." — Ernest Nagel, Alternative descriptions of physical state p. 292 This quote replaces the idea of 'cause-and-effect' with that of 'logical implication' according to one or another theory that connects events. In addition, an 'event' is related by the theory itself to formalized states described using the parameters defined by that theory. Thus, the details of interpretation are placed where they belong, fitted to the context in which the chosen theory applies. Using the definition of physical determinism above, the limitations of a theory to some particular domain of experience also limits the associated definition of 'physical determinism' to that same domain. “Determinism” 34 Determinism is the philosophical position that for every event there exist conditions that could cause no other event. "There are many determinisms, depending on what pre-conditions are considered to be determinative of an event or action." Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and considerations. Some forms of determinism can be empirically tested with ideas from physics and the philosophy of physics. The opposite of determinism is some kind of indeterminism (otherwise called nondeterminism). Determinism is often contrasted with free will. Determinism often is taken to mean causal determinism, which in physics is known as cause-and-effect. It is the concept that events within a given paradigm are bound by causality in such a way that any state (of an object or event) is completely determined by prior states. This meaning can be distinguished from other varieties of determinism mentioned below. Other debates often concern the scope of determined systems, with some maintaining that the entire universe is a single determinate system and others identifying other more limited determinate systems (or multiverse). Numerous historical debates involve many philosophical positions and varieties of determinism. They include debates concerning determinism and free will, technically denoted as compatibilistic (allowing the two to coexist) and incompatibilistic (denying their coexistence is a possibility). Determinism should not be confused with self-determination of human actions by reasons, motives, and desires. Determinism rarely requires that perfect prediction be practically possible. Varieties Below are some of the more common viewpoints meant by, or confused with "determinism". - Causal determinism is "the idea that every event is necessitated by antecedent events and conditions together with the laws of nature". However, causal determinism is a broad enough term to consider that "one's deliberations, choices, and actions will often be necessary links in the causal chain that brings something about. In other words, even though our deliberations, choices, and actions are themselves determined like everything else, it is still the case, according to causal determinism, that the occurrence or existence of yet other things depends upon our deliberating, choosing and acting in a certain way". Causal determinism proposes that there is an unbroken chain of prior occurrences stretching back to the origin of the universe. The relation between events may not be specified, nor the origin of that universe. Causal determinists believe that there is nothing in the universe that is uncaused or self-caused. Historical determinism (a sort of path dependence) can also be synonymous with causal determinism. Causal determinism has also been considered more generally as the idea that everything that happens or exists is caused by antecedent conditions. In the case of nomological determinism, these conditions are considered events also, implying that the future is determined completely by preceding events—a combination of prior states of the universe and the laws of nature. Yet they can also be considered metaphysical of origin (such as in the case of theological determinism). - Nomological determinism is the most common form of causal determinism. It is the notion that the past and the present dictate the future entirely and necessarily by rigid natural laws, that every occurrence results inevitably from prior events. Quantum mechanics and various interpretations thereof pose a serious challenge to this view. Nomological determinism is sometimes illustrated by the thought experiment of Laplace's demon. Nomological determinism is sometimes called 'scientific' determinism, although that is a misnomer. Physical determinism is generally used synonymously with nomological determinism (its opposite being physical indeterminism). - Necessitarianism is closely related to the causal determinism described above. It is a metaphysical principle that denies all mere possibility; there is exactly one way for the world to be. Leucippus claimed there were no uncaused events, and that everything occurs for a reason and by necessity. - Predeterminism is the idea that all events are determined in advance. The concept of predeterminism is often argued by invoking causal determinism, implying that there is an unbroken chain of prior occurrences stretching back to the origin of the universe. In the case of predeterminism, this chain of events has been pre-established, and human actions cannot interfere with the outcomes of this pre-established chain. Predeterminism can be used to mean such pre-established causal determinism, in which case it is categorised as a specific type of determinism. It can also be used interchangeably with causal determinism—in the context of its capacity to determine future events. Despite this, predeterminism is often considered as independent of causal determinism. The term predeterminism is also frequently used in the context of biology and hereditary, in which case it represents a form of biological determinism. - Fatalism is normally distinguished from "determinism" and it is a form of teleological determinism. Fatalism is the idea that everything is fated to happen, so that humans have no control over their future. Fate has arbitrary power, and need not follow any causal or otherwise deterministic laws. Types of Fatalism include hard theological determinism and the idea of predestination, where there is a God who determines all that humans will do. This may be accomplished either by knowing their actions in advance, via some form of omniscience or by decreeing their actions in advance. - Theological determinism is a form of determinism which states that all events that happen are pre-ordained, or predestined to happen, by a monotheistic deity, or that they are destined to occur given its omniscience. Two forms of theological determinism exist, here referenced as strong and weak theological determinism. The first one, strong theological determinism, is based on the concept of a creator deity dictating all events in history: "everything that happens has been predestined to happen by an omniscient, omnipotent divinity". The second form, weak theological determinism, is based on the concept of divine foreknowledge—"because God's omniscience is perfect, what God knows about the future will inevitably happen, which means, consequently, that the future is already fixed". There exist slight variations on the above categorisation. Some claim that theological determinism requires predestination of all events and outcomes by the divinity (i.e. they do not classify the weaker version as 'theological determinism' unless libertarian free will is assumed to be denied as a consequence), or that the weaker version does not constitute 'theological determinism' at all. With respect to free will, "theological determinism is the thesis that God exists and has infallible knowledge of all true propositions including propositions about our future actions", more minimal criteria designed to encapsulate all forms of theological determinism. Theological determinism can also be seen as a form of causal determinism, in which the antecedent conditions are the nature and will of God. - Logical determinism or Determinateness is the notion that all propositions, whether about the past, present, or future, are either true or false. Note that one can support Causal Determinism without necessarily supporting Logical Determinism and vice versa (depending on one's views on the nature of time, but also randomness). The problem of free will is especially salient now with Logical Determinism: how can choices be free, given that propositions about the future already have a truth value in the present (i.e. it is already determined as either true or false)? This is referred to as the problem of future contingents. - Often synonymous with Logical Determinism are the ideas behind Spatio-temporal Determinism or Eternalism : the view of special relativity. J. J. C. Smart, a proponent of this view, uses the term "tenselessness" to describe the simultaneous existence of past, present, and future. In physics, the "block universe" of Hermann Minkowski and Albert Einstein assumes that time is a fourth dimension (like the three spatial dimensions). In other words, all the other parts of time are real, like the city blocks up and down a street, although the order in which they appear depends on the driver (see Rietdijk–Putnam argument). - Adequate determinism is the idea that quantum indeterminacy can be ignored for most macroscopic events. This is because of quantum decoherence. Random quantum events "average out" in the limit of large numbers of particles (where the laws of quantum mechanics asymptotically approach the laws of classical mechanics). Stephen Hawking explains a similar idea: he says that the microscopic world of quantum mechanics is one of determined probabilities. That is, quantum effects rarely alter the predictions of classical mechanics, which are quite accurate (albeit still not perfectly certain) at larger scales. Something as large as an animal cell, then, would be "adequately determined" (even in light of quantum indeterminacy). - The Many-worlds interpretation accepts the linear casual sets of sequential events with adequate consistency yet also suggests constant forking of casual chains creating "multiple universes" to account for multiple outcomes from single events. Meaning the casual set of events leading to the present are all valid yet appear as a singular linear time stream within a much broader unseen conic probability field of other outcomes that "split off" from the locally observed timeline. Under this model causal sets are still "consistent" yet not exclusive to singular iterated outcomes. The interpretation side steps the exclusive retrospective casual chain problem of "could not have done otherwise" by suggesting "the other outcome does exist" in a set of parallel universe time streams that split off when the action occurred. This theory is sometimes described with the example of agent based choices but more involved models argue that recursive causal splitting occurs with all particle wave functions at play. This model is highly contested with multiple objections from the scientific community. Philosophical connections With nature/nurture controversy Although some of the above forms of determinism concern human behaviors and cognition, others frame themselves as an answer to the debate on nature and nurture. They will suggest that one factor will entirely determine behavior. As scientific understanding has grown, however, the strongest versions of these theories have been widely rejected as a single-cause fallacy. In other words, the modern deterministic theories attempt to explain how the interaction of both nature and nurture is entirely predictable. The concept of heritability has been helpful in making this distinction. Biological determinism, sometimes called genetic determinism, is the idea that each of human behaviors, beliefs, and desires are fixed by human genetic nature. Behaviorism involves the idea that all behavior can be traced to specific causes—either environmental or reflexive. John B. Watson and B. F. Skinner developed this nurture-focused determinism. Cultural determinism or social determinism is the nurture-focused theory that the culture in which we are raised determines who we are. Environmental determinism, also known as climatic or geographical determinism, proposes that the physical environment, rather than social conditions, determines culture. Supporters of environmental determinism often also support Behavioral determinism. Key proponents of this notion have included Ellen Churchill Semple, Ellsworth Huntington, Thomas Griffith Taylor and possibly Jared Diamond, although his status as an environmental determinist is debated. With particular factors Other 'deterministic' theories actually seek only to highlight the importance of a particular factor in predicting the future. These theories often use the factor as a sort of guide or constraint on the future. They need not suppose that complete knowledge of that one factor would allow us to make perfect predictions. Psychological determinism can mean that humans must act according to reason, but it can also be synonymous with some sort of Psychological egoism. The latter is the view that humans will always act according to their perceived best interest. Linguistic determinism claims that our language determines (at least limits) the things we can think and say and thus know. The Sapir–Whorf hypothesis argues that individuals experience the world based on the grammatical structures they habitually use. Economic determinism is the theory which attributes primacy to the economic structure over politics in the development of human history. It is associated with the dialectical materialism of Karl Marx. Technological determinism is a reductionist theory that presumes that a society's technology drives the development of its social structure and cultural values. With free will Philosophers have debated both the truth of determinism, and the truth of free will. Compatibilism refers to the view that free will is, in some sense, compatible with determinism. The three incompatibilist positions, on the other hand, deny this possibility. The hard incompatibilists hold that both determinism and free will do not exist, the libertarianists that determinism does not hold, and free will might exist, and the hard determinists that determinism does hold and free will does not exist. The standard argument against free will, according to philosopher J. J. C. Smart focuses on the implications of determinism for 'free will'. However, he suggests free will is denied whether determinism is true or not. On one hand, if determinism is true, all our actions are predicted and we are assumed not to be free; on the other hand, if determinism is false, our actions are presumed to be random and as such we do not seem free because we had no part in controlling what happened. In his book, The Moral Landscape, author and neuroscientist Sam Harris also argues against free will. He offers one thought experiment where a mad scientist represents determinism. In Harris' example, the mad scientist uses a machine to control all the desires, and thus all the behavior, of a particular human. Harris believes that it is no longer as tempting, in this case, to say the victim has "free will". Harris says nothing changes if the machine controls desires at random - the victim still seems to lack free will. Harris then argues that we are also the victims of such unpredictable desires (but due to the unconscious machinations of our brain, rather than those of a mad scientist). Based on this introspection, he writes "This discloses the real mystery of free will: if our experience is compatible with its utter absence, how can we say that we see any evidence for it in the first place?" adding that "Whether they are predictable or not, we do not cause our causes." That is, he believes there is compelling evidence of absence of free will. Some research (founded by the John Templeton Foundation) suggested that reducing a person's belief in free will is dangerous, making them less helpful and more aggressive. This could occur because the individual's sense of self-efficacy suffers. With the soul Some determinists argue that materialism does not present a complete understanding of the universe, because while it can describe determinate interactions among material things, it ignores the minds or souls of conscious beings. A number of positions can be delineated: - Immaterial souls are all that exist (Idealism). - Immaterial souls exist and exert a non-deterministic causal influence on bodies. (Traditional free-will, interactionist dualism). - Immaterial souls exist, but are part of deterministic framework. - Immaterial souls exist, but exert no causal influence, free or determined (epiphenomenalism, occasionalism) - Immaterial souls do not exist — there is no mind-body dichotomy, and there is a Materialistic explanation for intuitions to the contrary. With ethics and morality Another topic of debate is the implication that Determinism has on morality. Hard determinism (a belief in determinism, and not free will) is particularly criticized for seeming to make traditional moral judgments impossible. Some philosophers, however, find this an acceptable conclusion. Philosopher and incompatibilist Peter van Inwagen introduces this thesis as such: Argument that Free Will is Required for Moral Judgments - The moral judgment that you shouldn't have done X implies that you should have done something else instead - That you should have done something else instead implies that there was something else for you to do - That there was something else for you to do implies that you could have done something else - That you could have done something else implies that you have free will - If you don't have free will to have done other than X we cannot make the moral judgment that you shouldn't have done X. However, a compatibilist might have an issue with Inwagen's process because one can not change the past like his arguments center around. A compatibilist who centers around plans for the future might posit: - The moral judgment that you should not have done X implies that you can do something else instead - That you can do something else instead implies that there is something else for you to do - That there is something else for you to do implies that you can do something else - That you can do something else implies that you have free will for planning future recourse - If you have free will to do other than X we can make the moral judgment that you should do other than X, and punishing you as a responsible party for having done X that you know you should not have done can help you remember to not do X in the future.	libretexts	2025-03-17T19:52:24.596566	2021-06-15T23:27:17	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.02%3A_Determinism", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "8.1.2: Determinism", "author": "Noah Levin" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.03%3A_The_Illusion_of_Free_Will	8.1.3: The Illusion of Free Will - - Last updated - Save as PDF The Illusion of Free Will 35 80. Theologians repeatedly tell us, that man is free, while all their principles conspire to destroy his liberty. By endeavouring to justify the Divinity, they in reality accuse him of the blackest injustice. They suppose, that without grace, man is necessitated to do evil. They affirm, that God will punish him, because God has not given him grace to do good! Little reflection will suffice to convince us, that man is necessitated in all his actions, that his free will is a chimera, even in the system of theologians. Does it depend upon man to be born of such or such parents? Does it depend upon man to imbibe or not to imbibe the opinions of his parents or instructors? If I had been born of idolatrous or Mahometan parents, would it have depended upon me to become a Christian? Yet, divines gravely assure us, that a just God will damn without pity all those, to whom he has not given grace to know the Christian religion! Man's birth is wholly independent of his choice. He is not asked whether he is willing, or not, to come into the world. Nature does not consult him upon the country and parents she gives him. His acquired ideas, his opinions, his notions true or false, are necessary fruits of the education which he has received, and of which he has not been the director. His passions and desires are necessary consequences of the temperament given him by nature. During his whole life, his volitions and actions are determined by his connections, habits, occupations, pleasures, and conversations; by the thoughts, that are involuntarily presented to his mind; in a word, by a multitude of events and accidents, which it is out of his power to foresee or prevent. Incapable of looking into futurity, he knows not what he will do. From the instant of his birth to that of his death, he is never free. You will say, that he wills, deliberates, chooses, determines; and you will hence conclude, that his actions are free. It is true, that man wills, but he is not master of his will or his desires; he can desire and will only what he judges advantageous to himself; he can neither love pain, nor detest pleasure. It will be said, that he sometimes prefers pain to pleasure; but then he prefers a momentary pain with a view of procuring a greater and more durable pleasure. In this case, the prospect of a greater good necessarily determines him to forego a less considerable good. The lover does not give his mistress the features which captivate him; he is not then master of loving, or not loving the object of his tenderness; he is not master of his imagination or temperament. Whence it evidently follows, that man is not master of his volitions and desires. "But man," you will say, "can resist his desires; therefore he is free." Man resists his desires, when the motives, which divert him from an object, are stronger than those, which incline him towards it; but then his resistance is necessary. A man, whose fear of dishonour or punishment is greater than his love of money, necessarily resists the desire of stealing. "Are we not free, when we deliberate?" But, are we masters of knowing or not knowing, of being in doubt or certainty? Deliberation is a necessary effect of our uncertainty respecting the consequences of our actions. When we are sure, or think we are sure, of these consequences, we necessarily decide, and we then act necessarily according to our true or false judgment. Our judgments, true or false, are not free; they are necessarily determined by the ideas, we have received, or which our minds have formed. Man is not free in his choice; he is evidently necessitated to choose what he judges most useful and agreeable. Neither is he free, when he suspends his choice; he is forced to suspend it until he knows, or thinks he knows, the qualities of the objects presented to him, or, until he has weighed the consequences of his actions. "Man," you will say, "often decides in favour of actions, which he knows must be detrimental to himself; man sometimes kills himself; therefore he is free." I deny it. Is man master of reasoning well or ill? Do not his reason and wisdom depend upon the opinions he has formed, or upon the conformation of his machine? As neither one nor the other depends upon his will, they are no proof of liberty. "If I lay a wager, that I shall do, or not do a thing, am I not free? Does it not depend upon me to do it or not?" No, I answer; the desire of winning the wager will necessarily determine you to do, or not to do the thing in question. "But, supposing I consent to lose the wager?" Then the desire of proving to me, that you are free, will have become a stronger motive than the desire of winning the wager; and this motive will have necessarily determined you to do, or not to do, the thing in question. "But," you will say, "I feel free." This is an illusion, that may be compared to that of the fly in the fable, who, lighting upon the pole of a heavy carriage, applauded himself for directing its course. Man, who thinks himself free, is a fly, who imagines he has power to move the universe, while he is himself unknowingly carried along by it. The inward persuasion that we are free to do, or not to do a thing, is but a mere illusion. If we trace the true principle of our actions, we shall find, that they are always necessary consequences of our volitions and desires, which are never in our power. You think yourself free, because you do what you will; but are you free to will, or not to will; to desire, or not to desire? Are not your volitions and desires necessarily excited by objects or qualities totally independent of you? 81. "If the actions of men are necessary, if men are not free, by what right does society punish criminals? Is it not very unjust to chastise beings, who could not act otherwise than they have done?" If the wicked act necessarily according to the impulses of their evil nature, society, in punishing them, acts necessarily by the desire of self-preservation. Certain objects necessarily produce in us the sensation of pain; our nature then forces us against them, and avert them from us. A tiger, pressed by hunger, springs upon the man, whom he wishes to devour; but this man is not master of his fear, and necessarily seeks means to destroy the tiger. 82. "If every thing be necessary, the errors, opinions, and ideas of men are fatal; and, if so, how or why should we attempt to reform them?" The errors of men are necessary consequences of ignorance. Their ignorance, prejudice, and credulity are necessary consequences of their inexperience, negligence, and want of reflection, in the same manner as delirium or lethargy are necessary effects of certain diseases. Truth, experience, reflection, and reason, are remedies calculated to cure ignorance, fanaticism and follies. But, you will ask, why does not truth produce this effect upon many disordered minds? It is because some diseases resist all remedies; because it is impossible to cure obstinate patients, who refuse the remedies presented to them; because the interest of some men, and the folly of others, necessarily oppose the admission of truth. A cause produces its effect only when its action is not interrupted by stronger causes, which then weakens or render useless, the action of the former. It is impossible that the best arguments should be adopted by men, who are interested in error, prejudiced in its favour, and who decline all reflection; but truth must necessarily undeceive honest minds, who seek her sincerely. Truth is a cause; it necessarily produces its effects, when its impulse is not intercepted by causes, which suspend its effects. 83. "To deprive man of his free will," it is said, "makes him a mere machine, an automaton. Without liberty, he will no longer have either merit or virtue." What is merit in man? It is a manner of acting, which renders him estimable in the eyes of his fellow-beings. What is virtue? It is a disposition, which inclines us to do good to others. What can there be contemptible in machines, or automatons, capable of producing effects so desirable? Marcus Aurelius was useful to the vast Roman Empire. By what right would a machine despise a machine, whose springs facilitate its action? Good men are springs, which second society in its tendency to happiness; the wicked are ill-formed springs, which disturb the order, progress, and harmony of society. If, for its own utility, society cherishes and rewards the good, it also harasses and destroys the wicked, as useless or hurtful. 84. The world is a necessary agent. All the beings, that compose it, are united to each other, and cannot act otherwise than they do, so long as they are moved by the same causes, and endued with the same properties. When they lose properties, they will necessarily act in a different way. God himself, admitting his existence, cannot be considered a free agent. If there existed a God, his manner of acting would necessarily be determined by the properties inherent in his nature; nothing would be capable of arresting or altering his will. This being granted, neither our actions, prayers, nor sacrifices could suspend, or change his invariable conduct and immutable designs; whence we are forced to infer, that all religion would be useless. 85. Were not divines in perpetual contradiction with themselves, they would see, that, according to their hypothesis, man cannot be reputed free an instant. Do they not suppose man continually dependent on his God? Are we free, when we cannot exist and be preserved without God, and when we cease to exist at the pleasure of his supreme will? If God has made man out of nothing; if his preservation is a continued creation; if God cannot, an instant, lose sight of his creature; if whatever happens to him, is an effect of the divine will; if man can do nothing of himself; if all the events, which he experiences, are effects of the divine decrees; if he does no good without grace from on high, how can they maintain, that a man enjoys a moment's liberty? If God did not preserve him in the moment of sin, how could man sin? If God then preserves him, God forces him to exist, that he may sin.	libretexts	2025-03-17T19:52:24.658346	2021-06-15T23:27:18	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.03%3A_The_Illusion_of_Free_Will", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "8.1.3: The Illusion of Free Will", "author": "Noah Levin" }
https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.04%3A_Compatibilism	8.1.4: Compatibilism - - Last updated - Save as PDF Compatibilism 36 Frankfurt Cases - The Principle of Alternate Possibilities Alternative Possibilities are one of the key requirements for the freedom component of free will, critically needed for libertarian free will. Alternative Possibilities have been part of the problem of free will at least from the time of Thomas Hobbes, who denied anyone ever "could have done otherwise". In 1961, Harry Frankfurt famously defined what he called "The Principle of Alternate Possibilities" or PAP. "a person is morally responsible for what he has done only if he could have done otherwise.” Frankfurt developed sophisticated arguments (thought experiments) to disprove this principle using what is known as a Frankfurt controller, but might be called Frankfurt's Demon. The Frankfurt is a hypothetical agent who can control the minds of others, either a "nefarious neuroscientist or a demon inside one's mind that can intervene in our decisions. Considering the absurd nature of his counterfactual intervener, the recent philosophical literature is surprisingly full of articles with "Frankfurt-style cases" supporting Frankfurt, and logical counterexamples to his attack on the principle of alternate possibilities. This work is based on a logical fallacy Frankfurt's basic claim is as follows: "The principle of alternate possibilities is false. A person may well be morally responsible for what he has done even though he could not have done otherwise. The principle's plausibility is an illusion, which can be made to vanish by bringing the relevant moral phenomena into sharper focus." Libertarians like Robert Kane, David Widerker, and Carl Ginet have mounted attacks on Frankfurt-type examples, in defense of free will. The basic idea is that in an indeterministic world Frankfurt's demon cannot know in advance what an agent will do. As Widerker put it, there is no "prior sign" of the agent's de-liberate choice. This is the epistemic Kane-Widerker Objection to Frankfurt-style cases. In information theoretic and ontological terms, the information about the choice does not yet exist in the universe. So in order to block an agent's decision, the intervening demon would have to act in advance. That would eliminate the agent's control and destroy the presumed "responsibility" of the agent for the choice, despite no available alternative possibilities. This is the ontological Information Objection. According to Daniel Dennett's Default Responsibility Principle, the Frankfurt controller is now responsible, not the agent. Here is a discussion of the problem, from Kane's A Contemporary Introduction to Free Will , 2005, (p.87) 5. The Indeterminist World Objection While the "flicker of freedom" strategy will not suffice to refute Frankfurt, it does lead to a third objection that is more powerful. This third objection is one that has been developed by several philosophers, including myself, David Widerker, Carl Ginet, and Keith Wyma. 5 We might call it the Indeterministic World Objection. I discuss this objection in my book Free Will and Values. Following is a summary of this discussion: Suppose Jones's choice is undetermined up to the moment when it occurs, as many incompatibilists and libertarians require of a free choice. Then a Frankfurt controller, such as Black, would face a problem in attempting to control Jones's choice. For if it is undetermined up to the moment when he chooses whether Jones will choose A or B, then the controller Black cannot know before Jones actually chooses what Jones is going to do. Black may wait until Jones actually chooses in order to see what Jones is going to do. But then it will be too late for Black to intervene. Jones will be responsible for the choice in that case, since Black stayed out of it. But Jones will also have had alternative possibilities, since Jones's choice of A or B was undetermined and therefore it could have gone either way. Suppose, by contrast, Black wants to ensure that Jones will make the choice Black wants (choice A). Then Black cannot stay out of it until Jones chooses. He must instead act in advance to bring it about that Jones chooses A. In that case, Jones will indeed have no alternative possibilities, but neither will Jones be responsible for the outcome. Black will be responsible since Black will have intervened in order to bring it about that Jones would choose as Black wanted. In other words, if free choices are undetermined, as incompatibilists require, a Frankfurt controller like Black cannot control them without actually intervening and making the agent choose as the controller wants. If the controller stays out of it, the agent will be responsible but will also have had alternative possibilities because the choice was undetermined. If the controller does intervene, by contrast, the agent will not have alternative possibilities but will also not be responsible (the controller will be). So responsibility and alternative possibilities go together after all, and PAP would remain true—moral responsibility requires alternative possibilities—when free choices are not determined. 6 If this objection is correct, it would show that Frankfurt-type examples will not work in an indeterministic world in which some choices or actions are undetermined. In such a world, as David Widerker has put it, there will not always be a reliable prior sign telling the controller in advance what agents are going to do. 7 Only in a world in which all of our free actions arc determined can the controller always be certain in advance how the agent is going to act. This means that, if you are a compatibilist, who believes free will could exist in a determined world, you might be convinced by Frankfurt-type examples that moral responsibility does not require alternative possibilities. But if you are an incompatibilist or libertarian, who believes that some of our morally responsible acts must be undetermined you need not be convinced by Frankfurt-type examples that moral responsibility does not require alternative possibilities. 5. See Robert Kane, Free Will and Values (Albany, NY: SUNY Press 1985) p. 51; David Widerker "Libertarianism and Frankfurt's Attack or the Principle of Alternative Possibilities," Philosophical Review , 104 1995: 247-61; Carl Ginet "In Defense of the Principle of Alternative Possibilities: Why I Don't Find Frankfurt's Argument Convincing,' Philosophical Perspectives 10, 1996: 403-17; Keith Wyma, "Moral Responsibility and the Leeway for Action," American Philosophical Quarterly 34 (1997): 57-70. 6. Kane, 1985, p. 51. 7. Widerker, 1995, 248ff. Alternative Possibilities Are NOT Probabilities One of the major errors in thinking about alternative possibilities is to assume that they are the direct cause of action. This leads many philosophers to make the oversimplified assumption that if there are two possibilities, for example, that they are equally probable, or perhaps one has thirty percent chance of leading to action, the other seventy percent. Alternative possibilities are simply that - possibilities. They only lead to action following an act of determination by the will that the action is in accord with the agent's character and values. And the will is adequately determined. Most philosophers who use the standard argument against free will in their work assume that chance alternative possibilities will show up as random behavior. Here is an example from leading libertarian incompatibilist Peter van Inwagen. He imagines a God who can "replay" exactly the same circumstnces to demonstrate the random willings. Now let us suppose that God a thousand times caused the universe to revert to exactly the state it was in at t 1 (and let us suppose that we are somehow suitably placed, metaphysically speaking, to observe the whole sequence of "replays"). What would have happened? What should we expect to observe? Well, again, we can't say what would have happened, but we can say what would probably have happened: sometimes Alice would have lied and sometimes she would have told the truth. As the number of "replays" increases, we observers shall — almost certainly — observe the ratio of the outcome "truth" to the outcome "lie" settling down to, converging on, some value. We may, for example, observe that, after a fairly large number of replays, Alice lies in thirty percent of the replays and tells the truth in seventy percent of them—and that the figures 'thirty percent' and 'seventy percent' become more and more accurate as the number of replays increases. But let us imagine the simplest case: we observe that Alice tells the truth in about half the replays and lies in about half the replays. If, after one hundred replays, Alice has told the truth fifty-three times and has lied forty-eight times, we'd begin strongly to suspect that the figures after a thousand replays would look something like this: Alice has told the truth four hundred and ninety-three times and has lied five hundred and eight times. Let us suppose that these are indeed the figures after a thousand [1001] replays. Is it not true that as we watch the number of replays increase we shall become convinced that what will happen in the next replay is a matter of chance. ("Free Will Remains a Mystery," in Philosophical Perspectives , vol. 14, 2000, p.14) In our two-stage model of free will, if Alice is a generally honest person, her character will ensure that she rarely lies even if lying frequently "comes to mind" as one of her alternative possibilities.	libretexts	2025-03-17T19:52:24.724350	2021-06-15T23:27:19	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "url": "https://human.libretexts.org/Courses/American_River_College/HUM_301%3A_Introduction_to_Humanities_Textbook_(Collom)/08%3A_Freedom/8.01%3A_Freedom_of_the_Will/8.1.04%3A_Compatibilism", "book_url": "https://commons.libretexts.org/book/human-104688", "title": "8.1.4: Compatibilism", "author": "Noah Levin" }

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