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<urn:uuid:bbf04a05-4eeb-4cbb-9c21-97e3cd5b1a6e>
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Graphing Logarithmic Functions Graphing Logarithmic Functions
Graphing Logarithmic Functions
⇐ Use this menu to view and help create subtitles for this video in many different languages. You'll probably want to hide YouTube's captions if using these subtitles.
- We are asked to graph y is equal to log base 5 of x
- and just to remind us what this is saying,
- this is saying that y is equal to the power that I have to raise 5 to, to get to x.
- Or if i would write this logarithmic equation as an exponential equation
- 5 is my base
- y is the exponent that i have to raise my base to,
- and then x is what i get when I raise 5 to the yth power.
- So anther way of writing this equation would be
- 5 to the yth power is going to be equal to x.
- These are the same thing.
- Here we have y as a function of x, here we have x as a function of y.
- But they're really saying the exact same thing:
- "Raise 5 to the yth power to get x".
- But when you put this as a logarithm, you are saying:
- "To what power do I have to raise 5 to, to get x?
- Well, I have to raise it to y."
- Here, what do I get when I raise 5 to the yth power? I get x.
- Now that that is out of the way,
- let's make ourselves a little table
- that we can use to plot some points,
- and we can connect the dots to see what this curve looks like.
- So let me pick some Xs and some Ys.
- And in general we want to pick some numbers
- that give us some nice, round answers.
- Some nice, fairly simple numbers for us to deal with
- so we don't have to get a calculator.
- And so in general, you want to pick x values,
- you want to pick x values where the power that you have to raise 5 to to get that x value
- is a pretty straightforward power.
- Or another way to think about it is, you could just think about the different y values
- that you want to raise 5 to the power of
- and then you can get you x values.
- So we could actually think about this one to come up with our actual x values.
- But we want to be clear that when we express it like this,
- the independent variable is x and the dependent variable is y.
- We might just look at this one to pick some nice x's that give us nice, clean answers for y.
- So what happens, here I'm actually going to fill out the y first
- Just so that we get nice, clean x's.
- So let's say that we are going to raise 5 to the --I'm going to pick some new colours--
- to the negative 2 power, -- and let me do some other colours --
- negative 1, zero, 1, and I'll do one more, and then 2.
- So once again, this is a little nontraditional
- where I'm filling in the dependent variable first,
- but the way that we have written it over here, ......
- it's easy to to find out what the independent variable must be for this logarithmic function.
- So, what x gives me the y of negative 2?
- What does x have to be for y to be equal to -2?
- Well, 5 to the negative 2 power is going to be equal to x,
- so 5 to the negative 2, is 1 over 25, so we get 1/25.
- So another way, if we go back to this earlier one,
- if we say log, base 5, of 1/25.
- What power do I have to raise 5 to to get 1/25?
- Well, I have to raise it to the negative 2 power.
- Or you could say 5 to the negative 2 is equal to 1/25.
- These are saying the exact same thing.
- Now, let's do another one.
- What happens when I raise 5 to the negative 1 power?
- Well, I get one fifth. For this original one over there,
- we are just saying that log base 5 of 1/5, you want to be careful
- this is saying: "what power do I have to raise 5 to, in order to get 1/5?"
- Well, I have to raise it to the negative 1 power.
- Here, what happens when I take 5 tot the 0 power? I get 1.
- And so this relationship is saying the same thing as log, base 5, of 1,
- what power do I have to raise 5 to to get 1?
- Wel, I've just got to raise it to the 0 power.
- Let's... what happens when I raise 5 tot the first power?
- Well, I get 5.
- So if you go over here, that is just saying, what power do I have to raise 5 to to get 5?
- Well, I have to just raise it to the 1st power.
- And then finally, if I take 5 squared, I get 25.
- So if you look at it from the logarithm point of view, you say
- what power do I have to raise 5 to to get to 25?
- Well, I have to raise it to the second power.
- So, I kind of took the inverse of the logarithmic function. I wrote it as an exponential function.
- I switched the dependent and independent variables.
- So I could pick, or derive, nice clean x's that would give me nice, clean y's.
- Now, with that out of the way, but I do want you to remind,
- I could have just picked random numbers over here,
- but then I probably would have gotten less clean numbers over here, so I would have had to use a calculator.
- The only reason why I did it this way, is so I get nice clean results that I can plot by hand
- So let's actually graph it.
- So the y's go between -2 and 2,
- the x's go from 1/25 all the way to 25
- So let's graph it.
- So that is my y-axis, and this is my x-axis.
- So I'll drw it like that, that is my x-axis and then the y's
- start at zero and then you get to positive 1, positive 2,
- and then you have -1, -2 and then on the x-axis it is all positive
- and I'll let you think about whether the domain here is, well we can think about it,
- Is a logarithmic function defined for an x that is not positive?
- Is there any power that I could raise 5 to so that I would get 0?
- No. You could raise 5 to an infinitely negative power to get a very very small number
- that approaches 0, but you can never get-
- there is no power that you can raise 5 to to get 0.
- So x cannot be 0. There is no power that you can raise 5 to
- to get a negative number. So x can also not be a negative number.
- So the domain of the function right over here-
- and this is relevant becausewe want to think about what we are graphing -
- The domain here is x has to be greater than 0.
- Let me write that down.
- The domain here is that x has to be greater than 0.
- So we are only going to be able to graph this function in the positive x-axis.
- So with that out of the way, x gets as large as 25,
- so let me put those points here, so that's 5, 10, 15, 20
- and 25.
- And then let's plot these.
- ... and is blue and x is 1.25 and y is -2.
- When x is 1/25, is it going to be really close to there, then y is negative 2.
- So that's going to be right over there.
- Not quite at the y-axis, 1/25 .. of the y-axis, but pretty close.
- So that right over there is 1/25 comma -2 right over there.
- Then when x is 1/5, which is slihtly further to the right,
- 1/5 with y = -2. So right over there.
- This is 1/5, -1. And then when x is 1, y is 0.
- So 1 might be there, so this is the point (1,0).
- And then when x is 5, y is 1.
- I;ve covered it over here, y is 1.
- So that's the point (5,1).
- And then finally when xis 25, y is 2.
- So this is (25,2). And then I can graph the function.
- And I'll do it in the colour pink.
- So as x get's super super super small, y goes to negative infinity.
- So what power hve you have to rais e5 to to get point .0001
- That has to be a very negative power.
- So we get very negative as we approach 0,
- and then it kind of moves up like that.
- And then starts to curve to the right like that.
- And this thing right over here is going to keep going down at an ever steeper rate
- and it's never going to quite touch the y-axis.
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| 4.375 | 5 | 4.071927 | 4.482309 |
<urn:uuid:7a877c85-f966-4c94-821f-a2ad28e9de4a>
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This worksheet is X-actly what your child needs to practice writing X! First, kids trace lines to practice the fine motor skills they need to form the letter X. Then, they trace the letter several times for practice. Finally, they trace the letter X in the word "Xylophone."
Check out the rest of the alphabet here.
| 4.1875 | 5 | 4.259243 | 4.482248 |
<urn:uuid:fc7d382e-f4c7-4a4e-b192-69f9086c695f>
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Bang The Drum
Rationale: This lesson will help children identify /b/, the phoneme represented by B. Students will learn to recognize /b/ in spoken words by learning a meaningful representation (banging the drum) and the letter symbol B, practice--finding /b/ in words, and apply phoneme awareness with /b/ in phonetic cue reading by distinguishing rhyming words beginning letters.
chart with "Bob bought blue birds"
drawing paper and crayons
Book Bouncing on the Bed( Orchard Book 1999)
word cards with –bold, flat, meet, beat, teeth, brush, bang, sang
assessment worksheet (url below)
1. Say, "Our written language is a secret code. The tricky part is learning what stand the mouth moves, when we say words. Today we are going to work spotting the mouth move /b/. We spell /b/ with letter B. /b/ sounds like what a drum makes when you bang it.
2. Let's pretend to bang the drum,/b//b/b/.Making the sound of the drum. Notice where your lips come together, (touch one another). When we say /b/, we blow air between our lips.
3. Let me show you how to find /b/ in the word babble. I am going to stretch babble out in supper slow motion and listen for my drum sound. b--aaaa--bbbb--bbbb--llleeeee. There it was I felt my lips come together and blow air. I can hear the drum sound in babble.
4. Let's try a tongue tickler (on chart). "Bob bought blue birds." Everybody say it three times together. Now say it again, and this time, stretch the /b/ at the beginning of the words. "Bbbbbob bbbought bbblue bbbbirds." Try it again, and this time break the word: /B/ ob /b/ ought /b/ lue /b/irds.
5. Have students take out primary paper and pencil. We use letter B to spell /b/. Capital B looks like two bubbles. Let us write the lowercase b. Start at the rooftop, make a straight line down and bbbb bounce back up and around. I want to see everybody's b. After I put a smiley face on it, I want you to make nine more just like it.
6. Call on student to answer and tell how if they hear /b/ in birthday or party? bird or dog? fall or build? store or bank? Say: let us see if you can spot the mouth move /b/ in some words. Bang your drum if you hear /b/ in: The bird flew below the bathtub to the barn.
7. Say: Let's read the book, Bouncing On The Bed. Book talk: A child describes the bouncing, sliding, wiggling, running, splashing, reading, snuggling, and more that fills the day from sun up to bedtime. Every time you hear the /b/ sound, bang your drum. Ask children if they can think of other words they know that begin with /b/. Then have each student write the name of his or her picture with invented spelling. Display their work.
8. Show BIG and model how to decide if it is big or pig. The B tells me to bang my drum, /b/, so this word is bbb--iii--ggg. You try some. BOLD: bold or flat? BEAT: meet or beat? Brush: teeth or brush? BANG: bang or sang?
9. For assessment, distribute the worksheet. Students are to complete the partial spelling and color the pictures that begin with B. Call on students individually to read the phonetic cue words from step #8.
Click here for Awakenings index
| 4.15625 | 5 | 4.290126 | 4.482125 |
<urn:uuid:ce3ee540-a480-46b2-a12f-17e62efd02ab>
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Rationale: To become a successful reader, children must learn to read fluently. To do this, children must read a lot! Children need to perform repeated readings of the same text in order to gain the characteristics of a successful reader. If the children read the same story over and over, they become more confident in the story, they know the text better and they can read with more fluency. They should gain the ability to read fast, smooth, and with expression. When children become fluent readers, they increase their comprehension which is the ultimate goal of reading.
- Chalk board
-stopwatches – one for every two students
-Cat’s Trip by: Sharon Fear – one for every two students
Your Partner’s Name:
1. Explain Why: “Today we are going to talk about being a fluent reader. Does anyone know what that means? Well, there are all sorts of things that make a good reader, a fluent reader.”
2. Review: “To review, we will talk a little bit about what makes a good reader. Some of the things that make a good reader are reading fast, smoothly and with expression. I believe you all know what reading fast means. What about reading smoothly? It means that you can read through without getting stumped on a word or things like that. All of the words flow. What about reading with expression? That means you read with the kind of emotions that the characters are having. If the characters are mad, you read like you were mad, things like that. We will also discuss some strategies that can make us a good reader. Can anyone think of a way that they figure out what a word is if they don't know it?" I may have to lead the students, but what I am looking for are things like sounding out a word or reading the rest of a sentence to figure out what the word may be. It would also be wonderful if the students thought about covering up parts of the words and starting with the vowel in the middle, or something like that.
3. Explain How: “Today we are going to practice rereading the same story and some sentences so that we can learn to develop fluency.”
4. Model: Write a practice sentence on the board: (My cat has never been on a trip.) Read the sentence very slow to the children. For example, “Mmyy caat haaas neeever beeenn oonnn aaa tttriiip. Sound them out slowly again and practice the silent cover-up method. Read the sentence again smoothly and using expression. To read this sentence better you could enunciate never. “Which way did you like it better, slow or fast? Why did you like it better?” (Hopefully the children will say it sounds better fast because you can understand it better).
5. Simple Practice: Write another sentence on the board. (My cat wants to go to the beach.) This time, divide the class into partners. Have them practice reading it to each other several times. “Make sure that each time you read it you are reading it more smoothly and with more expression than the time before. Read it three times and then have your partner read it three times. Did you see a difference in the way you read the sentence the first time and the way you read it the third time?”
6. “Does anyone have a pet cat? ... Did your pet cat ever go on a trip with you? The book, Cat’s Trip is about a cat that goes on a trip. Do you think the cat had fun on the trip? To find out we have to read the book.” Whole Texts: First read Cat’s Trip to the students so that they know what the story is about and they are prepared to read the story on their own. Then keep the students in their pairs and give each pair a copy of Cat’s Trip. Then give each student a copy of the timesheet and each pair of students a stopwatch. Have the students read the story three times to each other. This may take a while, but allow the students the time they need. At the end ask them “Did your times improve as you read? Did you begin to read more smoothly as well?”
7. Assessment: I will use the timesheets that the children completed as assessment. In this way I will be able to see that the children completed the activity and hopefully see that they improved along the way. For those who did not improve very much I will work with more often to improve on their fluency.
Fear, Shannon. Cat’s Trip. Modern Curriculum Press, 1996.
Quick – Follow that Bear
by: Leslie McGill
as a Mouse by: Gina Thomas
Hurry on Your Trop
Cat by: Rebecca Smith
| 4.4375 | 5 | 4.008857 | 4.482119 |
<urn:uuid:71b51812-14b7-4709-b71d-beef287dcc28>
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Welcome to 1st grade!
About the Teacher
Routines and Procedures
Watch our caterpillars grow into butterflies!
Fun Websites for Reading and Phonics!
Available Conference Times
Helping your child read at home
Tell your child to look at the picture. You may tell the child the word is something that can be seen in the picture, if that is the case.
Tell the child to look for chunks in the word, such as
in mat or
Ask the child to get his/her mouth ready to say the word by shaping the mouth for the beginning letter.
Ask the child if the word looks like another word she/he knows. Does
? for example.
Ask the child to go on and read to the end of the sentence. Often by reading the other words in context, the child can figure out the unknown word.
If the child says the wrong word while reading, ask questions like: Does it make sense? Does it sound right? Does it look right?
Helping your child with addition/subtraction
Use tangible objects to represent numbers. Coins or blocks are great options.
Try "Touch Math" or number lines to help with counting up(adding) or counting down(subtracting).
Flash cards and math games are fun to use. Make a family math night.
Last Modified: Tuesday, Aug. 23, 2011
© 2013 TeacherWeb, Inc.
Content on this site is the responsibility of the Subscriber. Additional information is available in the
TeacherWeb Terms & Conditions
| 4.53125 | 5 | 3.914823 | 4.482024 |
<urn:uuid:21918d7b-4ab5-4306-934f-4bb9a6b1e2c9>
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Complete the following test so you can be sure you understand the material. Your answers are private, and test results are not scored.
Be careful adding formatting and styles. You can't make changes afterwards.
The best way to create a heading in a document is to:
Apply a larger font size to it than the body text.
Add bold formatting by clicking the Bold button on the Mini toolbar.
Apply a heading style.
You learned in the practice how to create a numbered list as you type. You type 1, add your text, and press ENTER.
You want to add emphasis to a few words of text that you have typed. The first step is to:
Click Bold on the Mini toolbar.
Select the text you want to format.
Click Bold in the Font group on the Home tab.
You can change the color or fonts in a Quick Style set.
In the practice you created a numbered list as you typed. To end the list, you pressed:
TAB key once.
| 4.0625 | 5 | 4.381991 | 4.481497 |
<urn:uuid:a56d07d4-addf-40d3-a394-0d13a4853459>
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Learning letters need not be a painful process where your child is drilled on letter sounds. This is a game that not only teaches your child the consonant sounds, but also helps them learn how to recognize sounds as they naturally occur in words.
- Sturdy paper or cardboard-one piece is enough for four letters
- Objects or pictures of objects beginning with each consonant sound ( be careful NOT to choose a word with a blend. For example, book instead of bread). You can glue the pictures on cardboard or laminate them for durability and to make it easier for your child to pick up.
How to Make:
- Fold each paper in half.
- Now fold the same paper in half again, so that you have four sections.
- Cut the paper into four sections, and print four letters on each section.
- Laminate or cover with clear contact paper for durability.
- Glue the pictures on small pieces of cardboard about half the size of each letter. On the back of each picture write the sound that it starts with.
How to Play:
- Choose two letters that look and sound different. Place them in front of your child.
- Point to one and say, “This is “S”. This is “B .” Make sure to tell your child the sound the letter makes, NOT the letter name. Many children get confused between letter names and letter sounds, and so it is better to teach the letter names at a later stage. You should also teach the hard consonant sounds first: c for cat, not c(s) for circle.
- Ask your child, “Show me the “S.” Show me the “B.” If your child gets confused, simply tell them the correct answer.
If your child has trouble remembering the letter sounds, don’t spend time drilling him on it over and over again. Children (and adults too) learn and remember better when there is a space between learning periods.
Come back to the letters a few hours later; even if it takes your child a few days to remember each set, she will still be finished in only a few weeks.
Once your child knows at least two letters, you can introduce the object pictures.
- Choose two letters. Place each one level with each other, with a bit of a distance between them.
- Take the pictures of objects that go with those sounds, and mix them up in a pile to the left of the letters.
- Choose one, asking your child to tell you what it is. Model sounding out the words, stretching out the sound of the first letter. Then place that picture under the corresponding letter.
- Do one more, and then let your child try it out. Show her how to check her work when she is done, by flipping over the picture.
TIP: Stay with your child the first few times she plays this game; she will need help stretching out the first sound.
| 4.3125 | 5 | 4.131899 | 4.481466 |
<urn:uuid:16e0ac5c-74c4-4178-bc10-c17f33be93e5>
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Photograph by Marli Bryant Miller
Written by Martin Richard
This huge rock is in the Willamette Valley in Oregon. It is different from all the other rocks around it.
How did it get there? Where did it come from?
Here is one clue: it has parallel scratch marks on it! Those scratch marks are found on rocks that have been dragged by glaciers. (We wrote about that right here.)
So we are very sure that this rock was scratched when it was dragged by a glacier.
What else do we know about this rock?
Scientists have taken pieces of this rock and analyzed the minerals. The minerals are the same as rocks from northwest Montana. That’s over 500 miles away!
So we have good evidence that this rock came from northwest Montana and that a glacier dragged it around.
Were there glaciers in northwest Montana?
Yes, at the end of the last ice age, northwestern Montana was partially covered by glaciers. But this rock is in the Willamette Valley of Oregon! The glacier was not 500 miles long.
But there is a way that a piece of a glacier could have gotten to Oregon. It could have floated there, as an iceberg!
Floated on what?
There is no ocean between Montana and Oregon. There is a river though. Well, not one river, but a system of rivers, which flow from Montana all the way to the ocean. Those rivers flow into the Columbia River, and this rock is near the Willamette River which flows into the Columbia.
Could an iceberg from a Montana glacier have made it all the way to Oregon?
There is a lot off evidence that there was a huge lake in northwestern Montana at the end of the last ice age. A huge glacier from Canada made an ice dam on the Clark Fork River. The lake that formed behind the dam is called Glacial Lake Missoula. It was more than a thousand feet deep at the dam, and it was huge. Glacial Lake Missoula covered almost 3,000 square miles, and held more water than Lake Erie and Lake Ontario combined!
(Actually, the glacier dammed the river several times. But the first lake that formed seems to have been the deepest and the biggest, and we are going to talk about that one.)
When that first ice dam broke, all that water behind it burst out in a huge wall. In some places, the wall of water was more than 500 feet high!! There is evidence that the water moved through the narrow valleys at almost 60 miles an hour!
We can tell how deep the water was, because it stripped off the soil of the valley walls, stripped it right down to the rock. Multiply the height of the valley by the width and multiply that by the speed of the flow and you get the volume of flow per hour: 9 cubic MILES of water per hour.
Let’s try to imagine 9 cubic miles of water. The next time you go outside, look to the east, where the sun comes up. Look at something you think is two miles away, and imagine a line going there. Now look to your right, to the south. Look at something in that direction that is two miles away. Imagine that line. Those two lines make a square two miles on a side. It has an area of 2 x 2 = 4 square miles.
Now look straight up and try to imagine a line two miles high. You are now standing at the corner of an imaginary cube, 2 miles on each side. Your cube has a volume of 2 x 2 x 2 = 8 cubic miles.
Now fill that cube with water.
Make the sides just a bit longer, to just under two and a tenth miles (2.08, actually), and you have 9 cubic miles of water.
That’s how much water flowed at the peak of the flood from Glacial Lake Missoula. In an hour!
How does that compare to other rivers?
The Amazon has the largest flow of any river in the world: about one hundredth of a cubic mile per hour. So the flow of the Glacial Lake Missoula Flood was 900 times bigger!
Try to imagine 900 Amazons in one flood!
A cube of water one yard on each side weighs just about 1,700 pounds, just under a ton. Imagine the weight of YOUR cube of water, two MILES on each side. Imagine the force and the power of that water as it raged through the canyons at 60 miles an hour, with giant whirlpools breaking and stripping the rock itself. Imagine the roar of the water, the crashing of great boulders slamming into each other and into the valley walls.
That would make a great movie!
But let’s get back to our rock.
The rock has parallel scratch marks. That tells us it was once in a glacier. The minerals of the rock tell us it came from northwest Montana. Other evidence tells us there was once a huge lake in behind an ice dam in northern Montana. That lake had glaciers on its shores, which means there were icebergs on the lake.
When the ice dam broke, it unleashed a gigantic, humongous, terrifying, titanic, stupendous, colossal, awesome monster of a flood, which ripped apart and re-arranged the land as it raged to the sea, carrying icebergs which carried rocks and carried this rock 500 miles.
What a trip this rock has taken, what a tale this rock could tell!
Oh wait. It IS telling us.
All we have to do is pay attention, collect the facts, and imagine the story that explains the facts. When we do that, this rock tells us a great story: of the humungous floods from Glacial Lake Missoula.
You want to learn more about the story of Glacial Lake Missoula and the Humungous Flood? You could start here, and explore the links.
| 4.34375 | 5 | 4.099321 | 4.481024 |
<urn:uuid:a68cbcd4-5f89-4988-a8a2-b200b015650a>
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Letter Sound Accuracy, Egg Carton Game
|Activity Type: Build Accuracy|
|Activity Form: Game|
|Group Size: Small Group|
|Length: 10 minutes|
|Goal: Given printed letters, the student can identify them and say the sound of each ( a -> /a/ )|
|Items: All letter sounds learned so far|
What to do
- Put a letter sticker (or write a letter) in the bottom of each egg cup in the carton. The letters should be a mix of all letters learned so far. Then put a counter (or a dime) in the carton and close it. (Note that you can also play this game with letter combinations or irregular words, or a mix of all of these.)
- Would you like to play the egg carton game? Here's how we play. I shake the carton and then open the lid and say the sound of the letter that the counter landed on. My turn first.
- Open the lid, look underneath the counter, pause, say the sound, and hold the carton up so students can see the letter. You get one point for each correct sound. First player to earn 10 points wins. So that's one point to me.
- Now I close the lid again and pass it to my neighbor. Pass the carton to the student on your left. Make sure that, when they say the letter sound, they show the letter to the other students.
- If they are correct say: Well done. One point to you. Remember your score. Okay, now pass the carton to your neighbor.
- If they are incorrect or don't know the sound, say: Can anyone help? What's the sound? Have the student repeat the sound and then pass the carton to his or her neighbor. If a student has trouble with multiple letters, make a note in an Activity Log.
- Continue playing and keeping score. You can either pass when it is your turn or keep playing. Try to get some letters wrong and ask the students for help.
View this video to see an example of how the egg carton game is played.
Photos of the egg carton game.
| 4.1875 | 5 | 4.253328 | 4.480276 |
<urn:uuid:de5addab-948f-4925-8d18-34ee7963629b>
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There are different types of math transformation, one of which is the type y = f(bx). This type of math transformation is a horizontal compression when b is greater than one. We can graph this math transformation by using tables to transform the original elementary function. Other important transformations include vertical shifts, horizontal shifts, and reflections.
Let's investigate another transformation. I want to know what's the transformation y=f of bx do? So let's start with the function f of x =4-x I'm going to graph that function and then I want to graph f of 4x. So let's make this our parent function it's actually pretty easy to come up with values for it and you know what the shape is going to be, it's a radical function. So I'll make this u and root 4-u, let's pick values to plug in that'll give us nice perfect squares inside the radical. So for example u=0 is a good choice because it gives you 4-0, 4 and 4 is 2.
If I want to get 1 inside here I'd pick u=3, so I'll pick 3 I get 1 inside and the square root of that is 1 and if I want to get 0 I'll pick u=4. So 4-4 is 0, 0 is 0, so these are the three points I'll use to graph it. And let me graph it right now 0 2, 3 1, 4 0 and you can see that if it's a radical function. This is it's end point so it's going to open to the left and it'll look something like this, now what does the transformation do? Well let me make a substitution, I want to graph f of 4x so let me make the substitution, first of all this is y equals the square root of 4-4x right? f of 4x is replacing the x by 4x, so I'll substitute u for 4x. Now if u equals 4x that means x equals one quarter u. That's why I take these u values and I multiply them by a quarter. And I get 0 times a quarter is 0, 3 times a quarter is three quarters and four times a quarters 1. These are my x values, and then here I'll have root 4-4x and this would be exactly the same as this because 4x is u, so 4-u exactly these values 2, 1, 0.
Alright so I'm going to plot 0, 2 three quarters 1 and 1, 0. So here is 0, 2, three quarters, 1 is here and 1, 0 is here. Now 1, 0 is the transformation of 4, 0 the old end point. So the old end point which is way out here has moved in to here, and this is what my new graph looks like. This is a horizontal compression the graph has been squeezed in to the y axis and it's a compression by a factor 1 quarter right 1 to 4, so just remember when you see the transformation f of 4x the number 4 it's greater than 1 you might expect this to be a horizontal stretch but it's actually a horizontal compression. So when you describe the transformation y=f of bx, if the b value is bigger than 1 you get a horizontal compression of the original graph by a factor of one over b.
Just like we saw here this is compression by a factor of one quarter. And those are our two graphs.
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|Saturday 25th May 2013|
CBSE Maths eBooks
Chapter 17: Co-ordinate Geometry
We are already familiar with plotting a point on a plane graph paper. For this we take two perpendicular lines XoX’ and YoY’ intersecting at O. XOX’ is called x-axis or abscissa and YoY’ is called y-axis or ordinate.
Point in a plane
Let us take a point P in a plane. Let XOX’ and YOY’ be pendicualr to each other at O. are drawn. If OM = x and ON = y then x-coordinate of P is x and y-coordinate of P is y. Here we write x-coordinate first. Hence (x, y) and (y, x) are different point whenever .
The two lines XOX’ and YOY’ divides the plane into four parts called quadrants. XOY, YOX’, X’OY’ and Y’OX are respectively the first second, third and and fourth quadrants. We take the direction from O to X and O to Y as positive and the direction from O to X’ and O to Y’ as negative.
Distance between two points
Let P (x1, y1) and Q (x2, y2) be the two points. We have to find PQ.
Let P (x, y) divided a line AB such that AP : PB = m1 : m2.
Let coordinates of A are (x1, y1) and B are (x2, y2).
© 2003-2012, CBSE Guess.com
POWERED BY DREAMZSOP
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A fun activity to teach about the effects of salt on ice is just a baking sheet away!
What you'll need:
- rimmed baking sheet/cookie sheet
- Matchbox cars or something of the like
- salt (table salt is fine!)
- sandwich bag
- Fill a baking sheet up with a thin layer of water. Place in freezer and let it freeze completely.
- Put ice cube in a bag.
- I started by filling an ice tray with her and freezing it. This way she could see that ice is just frozen water.
- Give a bag with 1 cube of ice. Let them roll it around. Instruct them to rub it, blow on it, etc. Ask questions like, "What is happening to the ice? Why do you think it's melting?" Make sure you explain after that it was the heat from the air and their hands that helps it melt.
- Pull out tray of frozen water. Let them roll and skid cars across it. Point out how slippery it is, asking what would happen if there was ice on our roads outside? If there isn't heat to melt the ice, our cars would slip.
- I then made comments like, "I wonder if we can put something on the ice to help it melt. What do you think?"
- If desired, try other items in the house like cornstarch, flour, cocoa powder. Let them pour salt on the ice. Make observations together about what happens when the salt is put on top of the ice.
- Let them roll the cars. You should find some traction, as well as the ice melting. Explain that we also put salt on the roads to help melt the ice and keep cars from slipping. Point it out next time you see it!
This is a fun activity to do, especially if your kiddos are into cars.
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Fuzz Gets a Big Buzz!
By: Jenna Gore
Rationale: To enable children to be successful readers, it is imperative that they learn to identify letters and the sounds that those letters make. The letter-sound correspondence is the foundation of reading and therefore requires explicit instruction. This is why I have chosen to teach the short vowel u and its sound /u/. In my lesson we will go over the u = /u/, spell words with this correspondent, and identify it in print and spoken language.
Letterbox set for each student Dry Erase board
Letterbox set for the teacher (This is a piece of 8 x 10 colored card stock for each box. This provides a letterbox that is big enough for the teacher to model for the class.)
Copy of Fuzz and the Buzz copy for my student
Letter blocks for students and letters for the teacher Letters needed: p, u, f, f, a, l, t, o, b,
g, z, z, s, h, n, m, c, r, k,
Poster of Tongue Twister “Ugly duckling was upset because he was unusual”
1.)“Today we are going to learn a new sound for one of our alphabet letters. That letter is u. The sound that we are learning is the sound that u makes when it is alone in a word and that sound is /u/. To make it easy to remember our new sound, we are going to learn a fun movement we can do with our bodies. Have any of you ever had a time when you couldn’t remember something, so you put your hand to your head and say uuu, to help you remember! Well, I know I have and whenever that happens I always put my hand on my head and say /u/. Demonstrate by putting hands on your head and saying /u/. Now let’s see if you can do it. Remember to make the /u/ sound when you put your hands on your head. I like that attitude! Learning is fun and it’s important that we have fun while we learn.”
2.) “Now let’s learn a fun tongue twister with our /u/ sound in it. Put up the poster for all the children to see. Our tongue twister says “Ugly duckling was upset because he was unusual!” Can anyone tell me what unusual means? That’s right it means it different. Now I want you to read the tongue twister with me. Thank you for reading with me! This time when we read our tongue twister I want us to stretch out the /u/ sound and when we do that let’s do our hand on our head movement. Everyone ready? “Uuugly duuuuckling was uuuupset because he was uuunusual”! Wow you are great at this.
3.)Now have the student tell whether they hear the /u/ sound in different words. “I’m going to give you a choice between to different words and I want you to tell me which word has the /u/ sound. Does bat or tug have the /u/ sound? Good. job or fun? Jump or dance?
4.)“Take out your letters and letterboxes. Make sure you spread out your letters on your desk so that you can see all your letters. (Use the big model taped on the board so that you can see the boxes and the letters.) Now, we are going to spell words that have the /u/ sound in them. Remember to put only one sound in each box. Watch me as I spell the word bug. /B-b-b-u-u--g-g-g/ I am saying the word out loud so I can hear all the sounds. It’s ok to do that, it helps to say the word out loud so we can hear all of the sounds. The first sound I hear in bug is /b/. So, I am going to put the b in my first box. What is the second sound I hear? (Do the hand on head move) Correct! I hear the /u/ sound. So, I put the u in the next box. The last sound I hear is /g/, and so I put the g in the last box. Let’s read our word. B-u-g. Good job! We have just spelled the word bug! Let’s give you a try now and see if you can spell these words that have /u/ in them. Use your letters and letterboxes to spell these words: (2) phonemes: up (3) puff, hug, buzz, mud (4) blush, stun, (5) strum. Tell the student how many boxes they will need to use for each set of words. We’re ready to practice reading and spelling our words with the /u/ sound. Do your best to spell them just the way I showed you. When you are finished, raise your hand and I will come check your work!
5.)After your student has finished spelling the words and reading them, pass out Fuzz and the Buzz to each pair of students. “Today we are going to read a story called Fuzz and the Buzz. This story is about Fuzz the cub. One day Fuzz decides to go out for some nuts. To get the nuts Fuzz has to shake the tree that the nuts are in, but when he shakes the tree he makes some bees, that also lived in the tree, very mad. What will happen to Fuzz? Will he be able to get away?! You'll have to read to find out! So everyone start reading with your partner. Make sure you both read too. I am going to come around if you have any questions.” Have your student read on their own. When they are finished reading, ask the students what words they read that had the /u/ sound in them. Write the words on the board.
Pass out the picture worksheet and go over the names of the different pictures to avoid confusion. Have the students’ circle the pictures that have the /u/ sound and let them color after they have completed the worksheet. For other assessments, the teacher could have each child come up to the teacher's desk individually and read Fuzz and the Buzz and assess the reading using a running record.
Icky Sticky Fingers! by Meg Betbeze
& Lesniak, T (1999). The Letterbox
Lesson: A hands-on approach for teaching decoding. The
(1990). Phonics Reader Short Vowel, Fuzz
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10 ways to help your child with phonics
Easy tips for revving up this essential reading skill.
By Bruce Johnson
1. Continue with the read-aloud. Include alphabet books. Some alphabet books tell a story or share information. Look for these.
2. Sing the alphabet often. Be careful that “l-m-n-o” doesn’t come too quickly and sound like one letter. It is a group of letters with four distinct sounds; slow down at this part, maybe even pause after the letter n, and then continue with saying the alphabet slowly and clearly.
3. Point to alphabet letters and say their names. Mix the letters and say their names. 4. Work on names. Teach your child to spell his or her name. Write the name on a piece of paper. Ask your child to trace over it and then copy. Warning: this may become a bit tricky with names that do not follow conventional sound–symbol relationships. Point out the irregularities.
5. Make letters in fun ways with paint, play clay, sticks, sugar, or sand.
6. Look for letters wherever you go. Examples: signs, cereal boxes, book covers.
7. Look at letters, say the letter name, say the letter sound, then say a word that begins with that sound.
8. Make flash cards. Play letter games such as Memory or Go Fish with letters or sounds, and when you find a match say a word that begins with that matched sound. Play Tic-Tac-Toe using letters other than X and O.
9. Start with simple words, like bat. Write the word on a piece of paper, point to the first letter, and ask for the sound. Continue with each subsequent letter.
10. Go on a letter hunt. Write a letter on top of a sheet of paper, like b. Look for all of the words of objects around the house that begin with that letter or sound. Draw pictures or write words.
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SAMPLE ACTIVITIES, Part 2
Send a letter to the parents with directions and suggestions for how they can use the summer fun projects. Be enthusiastic and encouraging as you remind parents that they are, indeed, their child’s first and most important teacher!
Do a job around the house.
Write your name on a piece of paper – first, middle, and last. What are your initials?
How many toes does your family have? How many thumbs? How many noses?
Read a book together. Retell the story in your own words.
Tell someone your whole address. What city do you live in? What state? What’s the name of your country?
Have someone call out these letters, and see if you can write them:
W, J, B, K, N, S, T.
How many eggs are in a dozen? How many days in a week? How many months in the year?
Act out the opposites of these words: hot, up, tall, fast, loud, front, sad, over, top. Check out the PowerPoint in the Downloads section. It goes with the "Opposite Song" on the CD, Keep On Singing with Dr. Jean.
Draw a person and put as many body parts as you can.
Can you say your phone number? Can you write it?
Help your parents cook something for dinner.
Sing the alphabet song. Can you sing it backwards?
How many nursery rhymes can you say? Can your parents teach you a new one?
How many months are there in a year? Can you name them? When is your birthday?
Count backwards from 10 to 0. Can you write the numbers?
Ask your parents to teach you a game they liked when they were your age.
What does a plant need to live? What do animals need to live?
Find an interesting picture in a magazine and make up a story about it.
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The picture above shows the beginning of a song. After the 4 sharps that represent the key you see a symbol that is called the time signature or meter of the song. In this case it is the common which means that there are 4 quarter notes in a measure. A measure is the part of the staff between two vertical lines.
Any of these means there are 4 beats in each measure. Count 1, 2, 3, 4.
Both 3/4 and 6/8 have 3 beats in a measure. Count 1, 2, 3. This is a waltz.
2 beats in a measure. Count 1, 2. This is a march.
Other meters are rare and you can define how long they are if you know that the upper value represents the number of notes that fit in a measure and the lower value the type of these notes. So 12/4 means 12 quarter notes in a measure. That is a waltz with a lot of notes in each measure.
Each measure should be filled to it's full lenght with notes and rests. An exception is the first measure. As you can see in the picture at the top of this page, the first measure only contains a note 1 beat long. There are no rest symbols added before it.
To play the notes at the right time you should count as the meter implies. In a 4/4 meter you must count in each measure:
one - two - three - four
like you are marching. You can play a quarter note at each count (beat). If there are 1/8 notes then count
one -and - two - and - three - and - four -and
in the same length of time. If there are 1/16 notes then count:
one - po - ta - to - two - po - ta -to - three - po - ta - to - four - po - ta - to
also in the same time as you would count 1-2-3-4. So you will have to say potato very fast.
In the picture above you see how you must count two get the timing of the notes right. In the last measure you see how three notes are bundled in a triplet. That means three notes use the same time as two. I would add dal-da to the count to get the timing right. Usually the short flagged notes are beamed together into bundles that represent one count each. Each horizontal beam represents one flag like pictured below.
A bow between notes means that these notes are played as one. So in this case you play E-F-G-A. The first E is played only once and lasts 5 beats. You count 1 (play E), 2, 3, 4, 1, 2 (play F), 3 (play G), 4 (play A).
Back to the note reading homepage
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To help each child understand that we are wise when we obey the teachings of Jesus Christ.
Prayerfully study Matthew 7:24–27.
Prepare two signs, using folded pieces of paper as shown:
Prepare to sing or say the words to
“The Wise Man and the Foolish Man” (Children’s Songbook, p. 281) and “Choose the Right Way” (Children’s Songbook, p. 160). The words to “Choose the Right Way” are included at the back of the manual.
Write some wise actions and some foolish actions on small pieces of paper. Use the actions below and add others that are appropriate for your class. Place the pieces of paper in a small container such as a box, bowl, or bag.Say your prayers.Obey your parents.Be reverent at Primary.Be kind to others.Share.Tell the truth.Quarrel.Say a bad word.Be selfish.Tell lies.Be unkind.
Make the necessary preparations for any enrichment activities you want to use.
Suggested Lesson Development
Invite a child to give the opening prayer.
Follow up with the children if you encouraged them to do something during the week.
We Can Make Wise Decisions
Display the gloves, hat, and shoes (adapt this activity as needed if you are using other articles of clothing). Put the gloves on top of your head.
Is this how we should wear gloves? Why?
What should we use gloves for?
Point out that it would be foolish to put gloves on your head because then they could not protect your hands.
Invite a child to demonstrate the wise way to wear gloves.
Put the hat on one of your feet.
Is this how we should wear a hat? Why?
What should we use hats for?
Point out that it would be foolish to put a hat on your foot because then it could not protect your head.
Invite another child to demonstrate the wise way to wear a hat.
Put the shoes on your hands.
Is this how we should wear shoes? Why?
What should we use shoes for?
Point out that it would be foolish to put shoes on your hands because then they could not protect your feet.
Invite another child to demonstrate the wise way to wear shoes.
Point out that few people would wear gloves, hats, or shoes on the wrong parts of their bodies. But choices about what to wear can also be wise or foolish. Briefly discuss with the children when it would be wise or foolish to wear different articles of clothing (for example, it would be wise to wear gloves when you are going outside in cold weather, but it would be foolish to wear shoes when you are taking a bath).
Explain that we make many choices each day. Foolish choices can harm us and make us unhappy. Wise choices help keep us safe and happy.
Jesus Christ Wants Us to Be Wise and Obey His Teachings
Explain that Jesus Christ often used stories called parables to teach people. He told one parable about a man who made a foolish decision and a man who made a wise decision. Tell the children that you are going to read this parable from the Bible. Read aloud Matthew 7:24–27.
How do you think the man who built his house on the sand felt when the storm knocked his house down?
How do you think the man who built his house on the rock felt when the storm did not hurt his house?
Explain that Jesus compared us to the men in the parable. If we follow Jesus’ teachings, we are like the wise man who built his house on the rock. We will be happy. If we do not follow Jesus’ teachings, we are like the foolish man who built his house on the sand. We will be unhappy.
Help the class sing or say the words to “The Wise Man and the Foolish Man,” using the actions indicated below:
We Are Wise When We Choose the Right
Story and discussion
Explain that everyone makes choices every day. We can make wise choices or we can make foolish ones. Tell in your own words the following story about B. H. (Harry) Roberts, who grew up to become a General Authority. Have the children decide whether Harry’s choices in the story are wise or foolish.
Harry was born in England a long time ago. His family was very poor, so Harry did not have a chance to go to school. He wanted very much to learn to read and write. He thought that if he could have a chance to learn he would not only read books but write them too.
When Harry was ten years old, he came to America and crossed the plains with his sister Polly and other pioneers.
Harry had many adventures. One day he heard that his group would be crossing a large river the next day. This sounded exciting, so Harry crept out of camp early the next morning and set out to see the river. This was against the camp rules.
Do you think Harry’s decision to sneak out of camp was wise or foolish?
The river was farther away than Harry thought, and he was tired when he finally got there. He lay down and fell sound asleep among some willows.
When Harry woke up, all the wagons had crossed the river. He ran to the riverbank and shouted to get someone’s attention. He was told to swim across, so he took off his coat and shoes and jumped into the river. He almost made it across the river, but he became too tired, and the captain of the company had to bring him the rest of the way on his horse. Harry was happy to be safe, but there was no way to get his coat and shoes back. Every night he wished he still had his coat, and every day he wished he still had his shoes.
Harry had to walk nine hundred miles barefoot. This made his feet very sore. Prickly cactus grew near the trail, and Harry was so hungry that he often gathered it for food. The sharp spines stuck in his sore feet. Polly pulled the spines out while they both cried. Harry cried because his feet hurt, and Polly cried because she felt sorry for Harry.
Why was Harry’s choice to sneak out of camp a foolish choice?
How did Harry feel about his choice afterward?
After Harry arrived in Salt Lake City, he finally had an opportunity to go to school. When he was eleven years old, a teacher took him into her school and taught him the alphabet. The only book Harry had was the Bible, and he read it over and over. He studied hard in school and became a very good student.
What choices did Harry make? (He chose to learn to read, and he chose to study the scriptures.)
Were these choices wise or foolish?
When Harry grew up, he became a wise and important man in the Church. He loved to read and enjoyed reading the scriptures. He also wrote many books about the Church. Many people have read his books and learned more about the Church. (See Church News, 19 July 1980, pp. 8–9; and Truman G. Madsen, Defender of the Faith: The B. H. Roberts Story [Salt Lake City: Bookcraft, 1980], pp. 19–21, 37–40, 56–57.)
Display the CTR chart and have the children repeat the words with you: “I will choose the right.” Explain to the children that when we choose the right we are making wise decisions.
Display the signs labeled “wise” and “foolish” on the floor or table. Have the children say with you the word on each sign. Point out that the “wise” sign has a happy face on it because being wise makes us happy. Point out that the “foolish” sign has a sad face on it because being foolish makes us unhappy.
Show the container with the small pieces of paper and invite the children to take turns choosing a paper from the container. As each child chooses a paper, read or have the child read the action written on it. Have the child decide whether the action is wise or foolish and place the paper by the appropriate sign.
Encourage the children to do the actions they identified as wise choices.
What kind of choices make us happy?
In order to be wise and happy, whose teachings should we follow?
Why is it foolish to disobey Jesus Christ and do wrong?
Testify to the children that we will be happy when we make wise choices. Express your appreciation for Jesus Christ and the things he taught that help us be happy.
Encourage the children to make a special effort this week to think about their decisions and try to make wise choices.
Sing or say the words to “Choose the Right Way” with the children.
Invite a child to give the closing prayer. Ask the child to pray that each class member will be wise and choose the right.
Choose from the following activities those that will work best for the children in your class. You can use them in the lesson itself or as a review or summary. For additional guidance, see “Class Time” in “Helps for the Teacher.”
Show picture 2-7, The Wise Men, and have the children tell you about the picture. Remind the children that the Wise Men came looking for Jesus after he was born (see Matthew 2:1–12). Explain that these men were wise not only because they had great knowledge but also because they sought and worshiped Jesus Christ. They made wise choices to follow Jesus.
You may want to have the children role-play the story of the Wise Men seeking Jesus.
Ask each child to tell you about a wise choice he or she has made recently (give suggestions if necessary). Write each child’s response on a separate piece of paper and have the children draw pictures of themselves on their papers. Remind the children to put smiles on the faces of their drawings because making wise choices helps us be happy.
Demonstrate the following object lesson (practice this demonstration at home before trying it in class).
Eight or ten children’s play blocks to build two simple identical houses, as shown:
Two deep pans the same size.
A large flat rock to put in one of the pans.
A mound of sand to put in the other pan.
Water in a watering can or pitcher.
Pour the sand into a mound in one pan, and flatten the mound. Put the flat rock in the other pan.
Show the blocks and explain that you will use them to build two houses. One house will be built on the sand and the other on the rock.
You may wish to ask the children to help build the house on the rock. The house should be similar to the house you build on the sand.
After the houses are built, have the children imagine that a storm comes and beats upon the houses. It rains hard and the winds blow.
What do you think a storm would do to these two houses?
After the children have given their ideas, do not comment. Carefully sprinkle or pour water on the house built on sand until the sand slides away and the house falls down. Put the same amount of water on the house built on the rock, and let the children observe that the house does not fall.
Help the children determine that building a house on sand was foolish, while building a house on the rock was wise.
Read aloud the first part of Helaman 5:12 (through foundation). Point out that this scripture compares Jesus Christ to a rock. Remind the children that following the teachings of Jesus is like building a house on a rock. It is wise.
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|The first thing to notice is that an egg is a very
symmetric object. If we look at the egg
from lots of different directions, then the view from many
of the directions is the same. Suppose, for example, that
you glue the the
egg to the top of a table, with the middle of the fat end
stuck to the table and the the egg pointing straight up, as
in the picture.
If you now walk around the table, you notice that the egg looks exactly the same no matter from which direction you look at it. (Move your mouse on top of the picture, to see an animation of this motion.)
You get the same effect if you stick a long needle right
through the egg, from the middle of the thick end through to
the middle of the thin end, as in the picture at the top of
the page. If you hold the needle straight up in front of
your eyes and slowly rotate it, the view you see does not
change at all as you rotate the egg around.
The needle is called an axis of rotational symmetry for the egg.
The symmetry of the egg is very helpful in describing its
shape. Here's how: we can imagine slicing the egg in half
from top to bottom by taking a sharp knife and running it
along the axis of symmetry. If we separate the two halves,
each now has a flat surface.
The surface we see is called the cross-section of the egg. We can reconstruct the shape of the whole three-dimensional egg simply by rotating the two-dimensional cross-section around the axis of symmetry. In mathematical language, the surface of the egg (the egg shell) is thus a surface of revolution.
All we have to do now is to describe the shape of this cross-section! The shape is called an oval (the word `oval' comes from the Latin word for egg, and means `having the shape of an egg'). You can see that it looks like an ellipse which has been slightly squished at one end. Nevertheless, an ellipse is pretty close to the right shape. So for our first attempt at an accurate description of the shape of an egg, let's explore ellipses.
There is a well-known method for drawing ellipses which uses pins and string. It can be used to derive equations for an ellipse.
As pretty as the ellipses are, you can easily see that they are not really egg shaped! Both ends of an ellipse look the same, but the cross section of an egg is sharper at one end than at the other. So how can we do better? Here are two types of curves which look more like the cross-section of a real egg.
Actually, there are many other possibilities. If you want to find out more about curves, you might want to take a look at `A Book of Curves' by Edward H. Lockwood (Cambridge University Press, 1961), or, if you can't drag yourself away from cyberspace, there is a nice site about plane curves in Scotland.
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Lesson 18 Section 4
RATIO AND PROPORTION 2
Example 1. Joan earns $1600 a month, and pays $400 in rent. Express that fact in the language of ratio.
Answer. "A quarter of Joan's salary goes for rent."
That sentence, or one like it, expresses the ratio of $400 to $1600, of the part that goes for rent to the whole. We are not concerned with the numbers themselves, but only their ratio.
Example 2. In Jim's class there are 30 pupils, while in Jane's there are only 10. Express that fact in the language of ratio.
Answer. "In Jim's class there are three times as many as in Jane's."
This expresses the ratio of 30 pupils to 10.
Example 3. In a class of 24 students, there were 16 B's. Express that fact in the language of ratio.
Answer. "Two thirds of the class got B."
This expresses the ratio of the part that got B to the whole number of students: 16 out of 24. Their common divisor is 8. 8 goes into 16 two times and into 24 three times. 16 is two thirds of 24.
Example 10. This month's bill is $75, while last month's was only $30. Express that fact in the language of ratio.
Answer. "This month's bill is two and a half times last month's."
75 is equal to two times 30 (60) with a remainder of 15, which is half of 30.
75 = 60 + 15.
75 is two and a half times 30.
The student will hear this language, will read it, and should be able to understand it and speak it.
At this point, please "turn" the page and do some Problems.
Continue on to the next Lesson.
Please make a donation to keep TheMathPage online.
Copyright © 2012 Lawrence Spector
Questions or comments?
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Parallel and perpendicular lines - Higher
We have seen graphs in the form y = mx + c before.
When a graph is written in the form y = mx + c, m represents the gradient and c represents the y intercept.
Take the example of the cost of a taxi ride at £1 + £3 per mile. In this case the gradient is the cost per mile and the intercept the £1 standing charge.
This can be written as y = 3x + 1
This line has a gradient of 3 and a y intercept of 1.
We can use this information to plot the graph, without drawing a table.
The line cuts the y-axis at 1 so the y intercept is 1.
The line has a gradient of 3 because every time it goes along one square it moves up 3 squares (every mile costs £3).
Plot the graph of y = -x + 2 without using a table
m = -1 and c = 2, so the line cuts the y axis at 2 and has a gradient of -1.
The graph should look like this:
Remember, if the gradient is negative, the graph will slope up to the left.
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A good way for the children to learn about THEIR currency or any currency in English and to also review sentences.
Photocopied money. Lots of it, in all denominations.
A pre-typed sheet of paper indicating what each number between 3 and 18 is worth. Example: 3=1,000, 4=5,000, 5=10,000, 6 and 15 = miss a turn, 7 and 14 = bankrupt and so on.
Write a sentence on the board, hangman style. _ _ _ _ _ _ _ _ _ _ _ _ _ _. (Today is Tuesday).
The first student rolls the three dice. They roll a 10. Look on the sheet. 10 = 5,000. They then say a letter. If they 'T', then you write on the board. T_ _ _ _ _ _ T_ _ _ _ _ _. The student will then receive 10,000 from you as they have two t's at 5,000 each.
If someone says a letter that has already been said, then they forfeit their turn. If someone knows the whole sentence, they must put up their hand and say "Answer!". If they are right, give them 10,000 and let them start the next round once you have put a new sentence on the board.
To start with, review sentences you have been doing in the class and later you can tailor the sentences to fit the class and the students. "Peter was late today." Jo didn't come because she is sick." And so on.
If they roll a 7 or a 14, then you take all their money back.
Make them count the money in English. When you finish, they have to tell you how much they have. Reward the winner with sweets, candy or even a small amount of cash.
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Tick Tock /t/
Letter recognition and phonemic awareness are very important in learning to read. During this lesson we will review letter recognition for the letter /t/. I will also introduce the sound of the letter /t/. This will tie in phonemic awareness along with reviewing the letter recognition. The students will learn not only the sound of /t/ but also the mouth gesture and how to correctly write an upper- and lower-case /t/. My goal is for the students to be able to recognize the sound of t as well as the letter when combined with other letters in the form of a word and also for the students to write both upper- and lower-case letter /t/s correctly.
Poster with upper- and lower-case Tt
Flash cards with pictures that may contain the letter t
Yes and no signs for children
Book: Tiny Turtles by Wendy McLean
Worksheet with words for children to circle the letter t
Tongue twister poster
Dry erase boards and markers
1. We will begin the lesson by reviewing what the letter t looks like. I will show the poster of the correct way to write upper- and lower-case t. I will tell them: For capital T, go down and cross at the top, For lowercase t is just a teenager, not as tall as his daddy, but not short either; cross at the fence.
2. I will have the students write both the upper- and lower-case ts on their paper. I will go around and check off the students who do them correctly then I will have all the children write the letters on their dry erase board and hold it up.
3. Next I will ask the students what sound t makes. Yes! T makes the sound tttt as in cat and top.
4. I will ask the students what does your mouth do when saying the sound tttt? Great! Your tongue touches the back of your top teeth. Can anyone think of a hand gesture that we can do when we hear the /t/ sound? That will work but lets try moving our finger like a clock: tick tock, tick tock! Great!
5. Ok now lets try a tongue twister! Let䴜s say it together: Tommy tricked Tim and took his train off the track. That was great! Now lets say it slowly and stretch out the /t/ sound: TTTommy tttricked TTTim and tttook his tttrain off the tttrack. Great!
6. I am going to hold up a picture. I will say what the picture is‰¥Ï if you hear the /t/ sound in the name of the picture I want you to hold up the t on your sign. If you do not hear the t I want you to hold up the frowny face on the sign. Ready: dog, cat, train, mat, apple, trick, tear, paper, great, computer, student
7. Now I am going to read a few sentences. Every time you hear the /t/ sound I want you to do the tick tock with your finger. Remember the /t/ sound can be at the beginning, middle, or end of a word. Ok get your fingers ready: Ted the turtle was trained to walk the tight rope by Trevor the turkey. Together Ted and Trevor taught others to walk to tight rope. Ted and Trevor were terrific in their trade. Great job! Your fingers kept busy in those sentences didn䴜t they.
8. For assessment: I am passing out papers with words on them. Every time you see the letter t I want you to circle it. I am so proud of you all for finding all the t䴜s. Now I am going to read the words. If you hear the /t/ sound in the word I want you to highlight the word. Ready: time, must, only, fast, crate, them, tape, turtle, attitude, apple, create, tiger, I am so proud of you; You are finally t-masters.
9. Here is a coloring sheet that has the upper and lower case letter t䴜s on it. You can color it using your favorite colors while I read the book Tiny Turtles!
Timmy the Turtle Goes Tick Tock
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What did you say, EEEHHH?
morning boys and girls, today we are going to learn about the letter e
and the sound it makes. Does anyone now what the
letter e looks like? If so raise your
hand and I will call on you to come write it for us to see on the
board." Very good that is what the letter e
Can anyone tell me what sound the letter e makes? No
one knows? Well the letter e says
/e/. I am sure that you
remember the other vowel that we learned last week.
Can someone raise their hand and tell me what letter we learned."
says:"we learned about the vowel a. "Very good, we
did learn about the letter a. Does anyone know
why we need to know what the
letter e looks like and what sound it makes? No
response from students. We need to know because the
letter e is in a lot of words that
we read and write. If we know
what it looks like and what sound it makes then it will help us to be
able to read and write much easier
and faster. Today,
we are going to learn how to spell and read words that have an e in it and the
sound that it makes."
will show the children a special gesture for remembering e = /e/. "Has
anyone every try to talk to someone and they could not
understand you? I have to. Have you had a difficult time hearing
someone else? Do you remember what they said to you?
Sometimes people say
EEEEHH." Show students picture of the man putting
his hand to his ear and saying EEEEHH. "Look at the
picture if you
forget what to do. I want everyone to cup their
hand and put it right behind their ear. Now say
EEEEHHH. This is what we are going to
say to remember the sound that e makes."
Practice doing it a few times with class. Now I am
going to speak very softly so that you can
not hear me. I want everyone to
practice their hand gesture and say EEEHHH. Very
good participation class. I am going to read a couple
of words to you and I want to see if everyone can
hear our /e/ sound in spoken words. If you hear the
/e/ sound in the word do our
special hand gesture.
3. Practice finding /e/ in spoken words. "I need everyone to listen very closely to the /e/ sound. Don't forget to do the hand gesture. Do you hear /e/ in eggs or grape? hedge or bush? ten or nine? fed or food? Great job!"
Get out sentence strip with tongue twister on it. First,
model the tongue twister and hand gesture. "Now
I want everyone to practice saying the
tongue twister with me. When we say it stretch out
the /e/ and do the hand gesture. Is everyone ready?
Eddie and the Eskimo enter the elevator on the elephant." Do
it a few more times. "Did everyone hear the /e/
sound? I know I did."
out the letters and the letterboxes to every student. Tell students
that each box represents a sound and when you hear the sound
put the letter in its box. Tel
them that you only turn over as many boxes over as you hear sounds.
Model the lesson with the word get.
First I am going to sound out the word. Then
once I have done that I am going to place the first letter that I hear
in the first box. I heard g
so I well put it in the first box while saying /g/. The
second letter I heard was an e so I will put the o in the
second letter box while
saying /e/ and t while saying /t/
in the last box to make the word get. "Today, we are
going to do a letterbox lesson with the letter e =
/e/. Everyone turn all of
your lower case letter face up so that you can see them because we are
only going to be working with lower case.
Remember that each of your boxes represents a sound in a word." Start
the letterbox lesson. We will begin with only two
phonemes and work our way up to 5 phonemes. I
will say each word one at a time giving them enough time to think about
the sounds and the letters
since that is what I am covering in today's lesson.
The words are egg, pet, ten, fed, pet,bed, help, nest, crept,and slept. Walk around the
room and observe, assisting any students who need help. If a student
misspells a word, pronounce the word as it appears and the student to
try to fix the word to make it the correct word. After
checking each student's work, model the correct spelling for each word
(just like you did get) in your large
letterboxes to the entire class.
the students have spelled all of the words successfully I will have
them read the words as I spell them out for them. I will write
each word on the board and ask the class to read it aloud. I will
not use the letterboxes for this part of the lesson. "Since
we are done spelling all of our
words we are at this point going to read them. I am
going to spell them for you on the board. After I
spell them I want you to read
them." I will make sure each child can read the words I
might even ask them individually. If I see a
student who is having trouble I will go over the
word with the class by using the vowel first, then the first letters,
and then the last letters.
will Hand out copies of Red Gets Fed to each student.
Then I will give a brief book talk to peak the
student's interest in the book. Today
we are going to read a book called Red Gets Fed. Red
is Meg's dog. One morning while Meg is sleeping Red goes into her room
and tries to wake her up. He is
very hungry. Meg gets up and feds Red. However, Red
is still hungry after eating. Meg goes back to bed
tries to go wake up dad. Will dad wake up and
fed Red? To find out if Red gets fed again, you'll have to read the book."
will have the students read Red Gets Fed. While they are
reading it I will walk around the
room to observe. "Now, everyone I
want you to read Red Gets Fed while I listen to you read."
9. Have each student write a message while other students are being called to the teacher's desk. "I am going to pass out the primary paper and you will need to get out a pencil to write your message. Does anyone have a suggestion on what to write about? One student's says since it is almost Halloween we should write about what we are going to be. Does everyone agree? Okay, then everyone write about what you will be for Halloween."
One at a time I will call each child up to the teacher's desk. I will ask each child a few words that have the /e/ sound and words that do not. They should be able to tell me which words do have the /e/ sound. The students should be assessed on the understanding that e = /e/. I will say I want you to tell me all the words that have the /e/ sound. The teacher should grade each child according to their ability to identify the correct words that make the /e/sound. I could also walk around the room and informally assess the students while doing the letterbox lesson or do a running record on Red Gets Fed and note their miscues to see if I need to go over the lesson again.
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Busy as a Bee
Rationale: In order to become fluent readers, children must learn to break the alphabetic code. After they learn individual phonemes, they are ready to learn digraphs. This lesson will help children learn to recognize ee=/E/ by spelling and reading words containing ee. This lesson will give children the opportunity to practice reading and spelling words with a double ee in them.
Materials: Elkonin letterboxes and
a set of lower case alphabet letters for each child, list of ee=/E/
words for teacher, list of necessary letters on the chalkboard-
Letterbox words: bee, see, feet, deep,
need, sweep, bleed, sleep.
Necessary letters: b,d,s,f,t,p,n,w,l,e,e, (e's taped together)
The Foot Book by Dr. Seuss, Published in 1968.
1. Introduce the lesson by explaining that when two vowels are side by side in a word, they make a single mouth move. Although this is a rule the children should understand that this is only 45% reliable. "Today we are going to learn the sound that two e's make when they are side by side, so that we can recognize that sound in words. We will practice by spelling and reading words with ee in them."
2. When two e's are together, they say /E/. Repeat after me: ee says /E/. Do you hear /E/ in meet or night? Say or feed? When you go down a slide, you say "WEEEE!" ee makes that /E/ sound.
3. I'm going to read you a few sentences. When you hear a word with ee in the middle, clap your hands twice. If you don't hear a word with ee, just stay silent. What does ee say? /E/!
a. I see my friend.
b. Are you going to school?
c. Jim is a baseball player.
d. I sleep under the sheets.
4. Now we are going
to use our letterboxes to spell words with ee. Since the two e's
make only one sound when put together, they will both go in the same
bee with me. You spell it out loud as I spell it on the board.
(Teacher draws a letterbox with two squares on the board. The first box
contains b and the second square contains ee.) Now take
out your letterboxes and only the letters I have written on the board.
5. Fold your letterbox so that you have two letterboxes showing for two sounds. Now spell bee. Now try see. (Give them time to finish before writing it on the board.) Now fold your box where 3 squares are showing. (Have students spell feet, red, deep, and need.) Now change to 4 squares. (Have students spell sweep, bleed, and sleep.)
6. Put away your letters and letterboxes. Now read the words as I point to them. (Students read word list teacher has written on the board.
"7. 'Now we are going to read The Foot Book. Before we start, what does ee say? /E/! Great!" Each child reads a page and the teacher finishes the book. Children will be assessed by teacher taking miscue notes as each child reads.
References: Murray, B.A., and Lesniak, T. 1999. The letterbox lesson: A hands-on approach for teaching decoding. The Reading Teacher, 52, 644-650.
The Reading Genie: Rachel Williams, Busy as A Bee- http://www.auburn.edu/rdggenie/insights/millsbr.html
Click Here to go back to the
Click Here to go back to the Journeys Web Site
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Quantitative Aptitude Questions On Time and Distance
Time and distance
We have the relation between speed, time and distance as follows:
Speed = distance / time.
So the distance covered in unit time is called speed.
This forms the basis for Time and Distance. It can be re-written as Distance = Speed X Time or
Time = Distance / Speed.
Units of Speed:
The units of speed are kmph (km per hour) or m / s.
1 kmph = 5 / 18 m / s
1 m / s = 18 / 5 kmph
When the travel comprises of various speeds then the concept of average speed is to be applied.
Average Speed = Total distance covered / Total time of travel
Note: In the total time above, the time of rest is not considered.
If a car travels along four sides of a square at 100 kmph, 200 kmph, 300 kmph and 400 kmph find its average speed.
Average Speed = Total distance / Total time.
Let each side of square be x km. Then the total distance = 4x km.
The total time is sum of individual times taken to cover each side.
To cover x km at 100 kmph, time = x / 100.
For the second side time = x / 200.
Using this we can write average speed = 4x / (x/100 + x/200 + x/300 + x/400) = 192 kmph.
A man if travels at 5/6 th of his actual speed takes 10 min more to travel a distance. Find his usual time.
Let s be the actual speed and t be the actual time of the man.
Now the speed is (5/6)s and time is (t+10) min. But the distance remains the same.
So distance 1 = distance 2 => s X t = (5/6)s X (t+10) => t = 50 min.
If a person walks at 30 kmph he is 10 min late to his office. If he travels at 40 kmph then he reaches to his office 5 min early. Find the distance to his office.
Let the distance to his office be d. The difference between the two timings is given as 15 min = 1 / 4 hr.
Now if d km are covered at 30 kmph then time = d/30. Similarly second time = d/40.
So, d/30 – d/40 = 1 / 4 => d = 30 km.
When two objects move with speeds s1 and s2
- In opposite directions their combined speed = s1 + s2
- In same direction their combined speed = s1 ~ s2.
Two people start moving from the same point at the same time at 30 kmph and 40 kmph in opposite directions. Find the distance between them after 3 hrs.
Speed = 30 + 40 = 70 kmph (since in opposite directions)
Time = 3 hrs
So distance = speed X time = 70 X 3 = 210 km.
A starts from X to Y at 6 am at 40 kmph and at the same time B starts from Y to X at 50 kmph. When will they meet if X and Y are 360 km apart?
Distance = 360 km
Speed = 40 + 50 = 90 kmph.
Time = distance / speed = 360 / 90 = 4hrs from 6 am => 10 am.
A starts from X to Y at 6 am at a speed of 50 kmph. After two hours B starts from Y to X at 60 kmph. When will they meet if X and Y are 430 km apart?
By the time B started A traveled for 2 hrs => 2 X 50 = 100 km.
So at 8 am, distance = 430 – 100 = 330 km
Speed = 50 + 60 = 110 kmph.
Time = distance / speed = 330 / 110 = 3 hrs from 8 am => 11 am.
When a train crosses a negligible length object (man / pole / tree) the distance that it has to travel is its own length.
When a train has to cross a lengthy object (train / bridge / platform) the distance it has to travel is the sum of its length and the length of the object.
If a train traveling at 40 kmph crosses another train of length 100m traveling at 14 kmph in opposite direction in 30 s find the length of the train.
Let length of train be d.
Distance to be covered = d + 100.
Speed = 40 + 14 = 54 kmph = 54 X 5 / 18 = 15 m / s
Time = 30 s.
Distance = speed X time => d+100 = 15 X 30 => d = 350 m.
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Watch It Go
1st Grade Oral Language Resources
Children will:• Learn about the concept of how things move.
• Access prior knowledge and build background about how things move.
• Explore and apply the concept of how things move and the different ways of getting from one place to another.
Children will:• Demonstrate an understanding of the concept of how things move.
• Orally use words that describe how things move.
• Extend oral vocabulary by speaking about things that become airborne.
• Use key concept words [wind, wheel, kite; airborne, ascend].
Explain• Use the slideshow to review the key concept words.
• Explain that children are going to learn about how things move:
• What is wind.
• What does wind move.
• What things can become airborne.
• What things can ascend.
• What is a wheel.
• What things have wheels.
Model• After the host introduces the slideshow, point to the photo on screen. Ask children: What do you see in the photo? (sailboat). How is the sailboat moving across the water? (wind).
• Ask children: Why is wind not a dependable way of moving things? (sometimes it's too windy, or there's not enough wind, or the wind isn't strong enough to move really heavy and big objects, hard to control).
• Say: The wind is the air that moves all around us. Wind can move many things such as leaves, a kite, and sometimes if it's really windy, it moves us as well. Scientists are trying to move things using wind power but wind is tricky since it isn't predictable. Sometimes the wind is too strong such as in a hurricane where it can destroy homes. Sometimes the wind is light, such as a gentle breeze. When have you enjoyed the wind? When have you not liked the wind? (answers will vary).
Guided Practice• Guide children through the next four slides, showing them the different ways that things can move. Always have the children describe how the things are moving.
Apply• Have children play the games that follow. Have them discuss with their partner the different topics that appear during the Talk About It feature.
• After the first game, ask children to talk about why the things that have wheels have more than one wheel. After the second game, have them discuss the different things that they can actually use everyday to get around the city.
Close• Ask children: How do you move around?
• Say: We move around with our legs or in wheelchairs. We can run, skip, bike, or walk to where we have to go or we can get into a car, taxi, bus, plane, carriage, boat, or train. All these things are moved by wind or with wheels. However, a plane is the only one which can become airborne and ascend. Think of other ways that you can travel by air without a plane or helicopter.
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This sheet contains instructions to help guide students through the experiment.
Go through this experiment with students to help them visualize what might change on the earth’s surface when the tectonic plates move.
- Divide students into groups of two partners.
- Have every group fill a pie pan with 1 inch of water.
- Pass out a Styrofoam® cup to each child.
- Have each student tear the cup into about 12 pieces to represent the major tectonic plates underlying the earth’s surface, and float them on the water, in turns. They have just modeled the lithosphere—the place deep below the surface of the earth where the tectonic plates are located. In the real lithosphere, the tectonic plates are floating on magma. Here they are floating on water.
- Students should gently experiment with their Styrofoam® tectonic plates. First, they should pull them apart. Ask them, “What do you see in the space where the Styrofoam® pieces once touched?” (Water. In real life, this is magma.) What might it create?
- Now, students should gently bump two plates together. Ask them, “What might happen on the surface from a bump like this below? Could it push magma into a mountain range? Cause an earthquake?” (Yes.)
- Now, students should push one plate under the other (water squirts a bit). Tell students, “Remember, water is magma in our model. So what might you get here on earth from magma shooting up—a volcano?”
- Now model the Haiti earthquake that struck in January 2010. It was caused by the motions of two plates grinding past each other in opposite directions. In the case of Haiti, the Caribbean plate moved east past the North American plate. It’s called a strike-slip fault.
- Ask students to experiment for five minutes in different ways with all 12 plates. Ask them to think of what would happen on the surface.
- Now, ask students to turn to the nearest person who isn’t their partner and show him/her one plate interaction. The partners should interpret it—that is tell what happens on the surface when tectonic plates behave the way you have shown them. Then students should trade places.
This teacher sheet is a part of the Shape It Up
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Rationale: Children must have phonemic
awareness in order to read and spell words; that is, they must have the
ability to identify phonemes or vocal gestures from words which are constructed,
when they are found in their natural context ö spoken words.
Materials: Tog the Dog (book); chalk or marker board; primary paper; pencils; pictures of dog, cat, frog, turtle, fox, snake, mop, broom, pot, plate; sheet with pseudo words like mod, lop, roz, tog, etc. written on it.
1. Introduce the lesson by telling the child or children, "Today we are going to talk about a new sound; the /o/ sound. We have to open our mouths really wide to make this sound, and our mouths will make the shape of an "o," which is the letter that makes the /o/ sound.
2. Show the children the grapheme "o" by writing it up on the board, and then show them the phoneme /o/. Ask them, "Have you ever been to the doctor for a check-up, and the doctor asks you to open your mouth really wide so he can look down your throat? He probably asked you to say "/ooooooooooo/!" Can you say that with me? Open your mouth wide and make the doctor sound! "/ooooooooooo/!"
3. "Let's say a little tongue twister to help us remember the /o/ sound. I'll say it first, and then you say it with me. "Ollie the Octopus ate a hot dog." "Can you say that with me now?" "Ollie the Octopus ate a hot dog." "Now, let's say it one more time, but let's say it like this: OOOOOOOOllie the OOOOOOOOOOctopus ate a hooooooooooot doooooooooooog." "This way, we can stretch out the /o/ sound. Say it with me now! OOOOOOOOOOOOOllie the OOOOOOOOOOctopus ate a hooooooooooot dooooooooooooog." "Good job!"
4. "Now, I'm going to show you some pictures, and I want you to tell me what each picture is." (Show pictures of a dog, cat, frog, turtle, fox, snake, mop, broom, pot, plate.) "Good! Now, I'm going to show you the pictures again, but this time, Iâll show you two pictures at a time, and, each pair will have one picture that makes the /o/ sound, and one that doesn't. Raise your hand when you see the picture that makes the /o/ sound. (Show dog and cat, frog and turtle, fox and snake, and pot and plate.)
5. "Let's practice writing the lowercase letter "o," because it is the letter that makes the /o/ sound. I'm going to write it on the board, and you write it on your paper. We'll start by placing our pencil a little below the fence, curve up and around to the ground, like when we make letter "c," and keep going up and around to where we started to close it up. Now we've made the letter "o!" See if you can practice that a few times on your paper."
6. "Now, I want to see if we can spell some words by listening to each word and deciding where we hear the /o/ sound so we'll know where to put the letter "o," and then we'll try to figure out where the other letters go. I'll write our letters and words on the board, and you see if you can write them on your paper. If I say the word "hop," where do we hear the /o/ sound? H-O-P. Is the /o/ sound the first sound we hear? Is it the between the first and last sounds? Is it the last sound? Good, that's right, it's the second sound we hear. Well, now we need a beginning and an ending sound. H-O-P. What was that first sound we heard? Do you know what letter makes the /h/ sound? You're right, it's the "h." Now, we have H-O, and we need our /p/ sound. What letter makes our /p/ sound? It is the "p!" Great! Now we've spelled "hop." (Continue to do this with other words like dog, got, etc.)
7. "I'm going to read a book to you called Tog the Dog, and I want you to listen for the /o/ sound. (Read book). Now, I'm going to read it again, and whenever you hear a word that has the /o/ sound in it, raise your hand." (Read the book again so the child or children can raise their hand when they hear the /o/ sound.
8. For assessment, have some pseudo words written out that contain the /o/ sound. "You've done a really good job with recognizing the /o/ sound. The last thing we're going to do is look at this sheet where I've printed some words that aren't real words, but they have the /o/ sound in them. I want to see if you can read them or sound them out, even though they aren't real. (Have words like mod, lop, roz, tog, etc. written out for the child or children to decode.)
Reference: The Reading Genie (www.auburn.edu/rdggenie)
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Rationale: In order to learn to read and spell words, children need to understand digraphs so they can match letters to their phonemes. In this lesson, children will learn to recognize the ou = /ow/ in both spoken and written words They will learn to do this by practicing reading and spelling words containing ou = /ow/. I will use a letterbox lesson to help with instruction of this correspondence.
Materials: Primary paper, pencil. The Napping House by Audrey Wood. Elkonin letter boxes. letter: s.h,o.u.t.l,d,m.n,f,a,and b, flashcards of the letterbox words, and a chart with the following tongue twister on it: I found out about the loud shout.
Procedures: 1. Introduce the lesson by asking the
children if they have ever gotten hurt and said "ow!" I will then tell
them that when the two letters o and u get together, they make that sound.
I will write these letters on the hoard. Today we are going to work on
words that have the ow sound in them and are spelt with an on. 2. I want
all of you to repeat this tongue twister after me. "I found out about the
loud shout" Good! Did you hear the ow sound in those words. Now I want
you to hold out the ow sound when we say the tongue twister. I fou------nd
ou-------t abou------t the lou-------d shou--------t. Excellent job! 3.
Next. I will have the children take out their letterboxes and the letters
listed ahove. Now we are going to spell the words that I've just said.
Remember the ow sound is made when the o and u get together. I will model
one example on the hoard for the students and then we will go through words
one at a time as the students spell the words in their letterboxes. Because
the o and u together make one sound, they should put them in one box.
Words to spell: out = ou/t loud = l/ou/d
shout = sh/ou/t found = f/ou/n/d
mound = m/ou/n/d
5. We will then practice this sound by reading the hook The Napping House. I will read The book to the students and they will say ow! when they hear the ou sound. The students will then read the book themselves and write down the words that have the ou correspondence in them. 6. For review, I will hold up flash cards of the words we have learned with the ou sound and the children will be called upon one at a time to say what is on the card. 7. For assessment. I will pass out a worksheet in which the students will have to circle the words that have the ou = /ow/ correspondence in them. They will fill it out and turn it in.
References: Murray, Bruce and Lesniak T.(1999) The
Letterbox Lesson: A Hands-On
Approach to teaching decoding.
Wood. Audrey. The Napping House
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2nd Grade Oral Language Resources
Children will:• Learn about the concept of our moon.
• Access prior knowledge and build background about what the moon looks like.
• Explore and apply the concept of why the moon looks different on different nights.
Children will:• Demonstrate an understanding of the moon.
• Orally use words that describe what the moon looks like.
• Extend oral vocabulary by talking about how the moon moves in the sky.
• Use key concept words [orbit, rotate, full, craters, satellite, phase].
Explain• Use the slideshow to review the key concept words.
• Explain that children are going to learn about:
• What orbit means.
• How the moon and sun orbit.
• Why the moon looks different on different nights.
• What the moon looks like up close.
• What astronauts learned from visiting the moon.
Model• After the host introduces the slideshow, point to the photo on screen. Ask children: What do you see in this picture? (water, trees, the moon). What does the moon look like in this picture? (small, round, gray with dark spots, glowing).
• Proceed to the next slide. Ask children: Does the moon orbit the Earth or the sun? (the Earth).
• Say: We can see the moon with only our eyes at night. What are some other ways we can learn more about the moon? (look through a telescope, visit in a space shuttle, satellite photos).
Guided Practice• Guide children through the next three slides, showing the different views of the moon. Always have the children talk about what they can learn about the moon from the photo.
Apply• Have children play the games that follow. Have them discuss with their partner the different topics that appear during the Talk About It feature.
• After the first game, ask children to talk about how the moon and the sun are different and help them learn more about the moon facts they're interested in. After the second game, have them discuss their favorite phase of the moon.
Close• Ask children: What is a new way we could learn about the moon? Explain.
• Summarize for children that our moon looks different when we look at it in different ways. It also looks different on different nights. Encourage children to think about what they could learn from a trip to the moon.
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|Target Language:||Present Simple|
|Resources Needed:||One copy of the worksheet for each pair of students. Each worksheet should consist of:
...Two ID cards that contain a person's:
...And two blank ID cards.
Write your name, age, marital status, job, country, address and phone number on the board.
Elicit the questions from the students, such as: 'Where do I live?', 'What is my phone number?
Get the students to do the same activity in pairs.
Write a famous person’s information on the board.
Elicit the changes in the form of the questions (e.g. What’s your name? - What’s his/her name?)
Tell the students they will ask for two people’s information. Then divide students into pairs.
Give one student Worksheet A - which has four ID cards on it. Two of the ID cards are totally filled out. Two of the ID cards just have people's names. Give the other student Worksheet B - this will also have four ID cards, including the details which Student A needs to fill in the blanks on their worksheet and vice versa. Give them a minute to look at the information and ask questions about information they don’t understand.
Students take turns to ask questions to fill in the missing information using the person’s name for the first question (to make it clear). E.g. 'What is xxx's name'. Monitor, help and correct where necessary.
Check and conclude with class.
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Rambling Rabbits Make /a/'s
Rationale: To become successful in reading, individuals must acknowledge that each letter has a particular corresponding sound. This lesson will assist children in identifying ���a,��� the phoneme represented by /a/. The students will learn to recognize /a/ in spoken words by relating the letter a, practice finding /a/ in words, writing the letter ���a,��� and identifying the sound in spoken words through a read-aloud book.
Materials: Primary Paper
Chart with ���Abby ate Aunt Ally���s apples.���
Pre-made cards with: Do you hear /a/ in cat or dog?
Do you hear /a/ in mouse or rat?
Do you hear /a/ in hat or bow?
Do you hear /a/ in rabbit or bunny?
Do you hear /a/ in him or man?
Do you hear /a/ in fat or skinny?
Do you hear /a/ in cry or laugh?
Do you hear /a/ in white or black?
Do you hear /a/ in bathtub or shower?
Do you hear /a/ in crab or fish?
Assessment worksheets identifying pictures with /a/. (URL���s below)
1. Say: Our written language is a hassle at times. The hard part is figuring out which letter makes each sound and the movement that your mouth makes for each letter. Today we are going to work on spotting the mouth move /a/. We will spell /a/ with ���a.��� a looks like a bunny, and /a/ sounds like a rabbit rambling on.
2. Let���s pretend to talk excessively /a/, /a/, /a/, /a/ [making mouth movement with hands like one is talking and hopping like a rabbit.] Notice what position your mouth is in when you say /a/. When we say /a/, you open your mouth wide and your tongue stays down. You try it out! It is like you are running out of breath when you have been rambling on too much.
3. Let me show you how to find /a/ in the word rabbit. I���m going to stretch rabbit out in super slow motion and listen for my rambling stream of sound. R-a-a-a-bit. Slower! R-r-r-a-a-a-a-b-i-t. You got it! I felt my mouth open and stay open after the /a/ sound and then I said the rest of the word, closing my mouth as doing so. I can feel my mouth make that ramble /a/ in rabbit.
4. Let���s say a tongue twister (on chart). ���Abby ate Aunt Ally���s apples.��� Everybody say it 3 times together. Now say it again, and this time, stretch /a/ out at the beginning of each word. ���Aaabby aaate Aaaunt Aaally���s aaapples.��� Try it again and this time break off the word: ���/a/bby /a/te /a/unt /a/lly���s /a/pples.���
5. [Have students take out primary paper and pencil] we use ���a��� to spell /a/. Capital A looks like a rabbit���s cage. Let���s write the lowercase letter ���a.��� Start on the fence line, and make a loop, lift your hand and put it back on the fence line touching the top of your loop and make a line from there all the way down to the sidewalk staying on your loop���s side. I want to see everyone���s ���a.��� After I put a smile on it for just fine repeat that nine more times.
6. Call on students to answer and tell how they knew: Do you hear /a/ in cat or dog? Mouse or rat? Hat or bow? Cry or laugh? Crab or fish? Say: Let���s see if you can spot the /a/ sound in some of these words. Clap if you hear /a/: cat, dog, cast, last, first, math, sit, sat, crab, fish.
7. Say: ���In this book we will see what will happen when a rabbit rambles on. Clap your hands every time you hear the /a/ sound.
8. Show CAT and model how to decide if it is cat or dog: The /a/ tells me to ramble on, /a/ so this word is c-a-a-a-t, cat. You try some MAT: mat or met? Fast: fist or fast? RABBIT: rabbit or ribbit?
9. For assessment, distribute the worksheets. Students are to complete the papers by 1) circling the correct word and writing it and 2) line up to correct /a/ picture.
Bruce Murray. Brush Your Teeth With F.
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Writing Poems by "Talking Back"
One of the best ways to learn how to construct a simple poem is to take someone else's poem and use it as your structure. This is not cheating or copying, because you write your own version with your own ideas. For example your students could write their own poems based on William Carlos Williams' poem The Red Wheelbarrow, but instead write about a topic of their choice. They use Willams' simple one or two word lines as a starting point.
For a whole book on this topic, try Talking Back to Poems by Daniel Alderson. It has lots of examples and poems to use.
A haiku is a three-line poem with a total of 17 syllables (5/7/5). There are websites such as this that explain how to write a haiku and give a number of examples.
Along with haikus, your students can then go on to write cinquains and tankas. These are all simple short poems with some rules to make it more fun!
First Lines/Last Lines
Give your students several first and last lines
to choose from, then ask them to write a poem using the lines they have picked. If you like, you can use lines from published poems, or make up your own, or use mine:
This was not the day
Beneath the bridge
The photos on the piano
In the small town, on the widest street
In the mirror
You must not call me
and the sound went on.
you ended with nothing.
painted on the wall.
like the black night.
too many flies in the soup.
only you will know the answer.
in the box.
There are many ways to use repetition in a poem:
1. Choose one word (e.g. nose, which can also be knows or nos) and use it at least ten times. It's good to use a word that sounds the same but is spelled differently and has different meanings.
2. Repeat the same word or words at the beginning of each line. E.g. Someday I will, Can you hear, You will be, When I am - these are simple ones.
3. Write a line that can act as a refrain, then use it three or four times throughout the poem.
Write a poem in which you compare two opposite things (such as rocks and water, or smile and frown), or a poem about two completely different things (such as car and cloud, or tennis ball and snail). It's often better to use concrete objects or actions rather than emotions or abstract concepts such as honesty or courage.
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Verbs: Return To Your Writing
Read aloud the paragraph you wrote at the beginning of this chapter. Where are the verbs? Look for the words that indicate time. Which are verbs and which aren't? Use Chart 1 in Part IV to help you find the auxiliaries in your paragraph. Wherever you find one, check to see if a main verb follows. Where you're not sure of a verb, turn the sentence into a negative statement and use the not or n't as a flag that waves in the middle of your verb. Mark all your verbs. Have you used any compound verbs? Trade marked paragraphs with a classmate, and check each other's work. Wherever you disagree, explain your reasons. Raise questions and get a tutor or teacher to answer them.
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Copyright © University of Cambridge. All rights reserved.
This challenge is a bit different to my usual ones. I used it many years ago when I wanted some quite young children to do lots of adding in a more fun way.
|You start with four numbers at the corners of a square. Then you add up the numbers at the two ends of each side and put the total in the middle of that side. So in my example 3 + 5 = 8, 5 + 4 = 9, 3 + 1 = 4, 1 + 4 = 5.|
|These four new answers give us the corners of a new square. The corners are 8, 9, 5 and 4.|
|These four new numbers are added up and the answers put in the centre of the edges of this new square. And so on and so on.|
The diagram gets more and more complicated, growing as shown below:-
There is not much more to say, apart from have a go yourself. Use any starting numbers at the corners. Can you estimate what the size of the last four numbers will be?
What would happen if you used different shapes, for example pentagons or hexagons?
What would happen if you used subtraction, always taking the smaller from the bigger?
What would happen if you multiplied? Divided? What ...??
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How would you find 0.5% of 64?
0.5% means 'when the number is 100 your answer will be 0.5'.
If other words for every 100 we have, we have 0.5, therefore we must find how many 100s are in 64, because that will tell us how many 0.5s there are.
So we divide by 100, that gives us 0.64, this is how many 100s there are in 64, and so we know there is that many 0.5s.
So if our answer is 0.64 0.5s then we just multiply them 0.64*0.5 = 0.32.
0.32 is your answer.
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|(pi = = 3.141592...)
Note: "ab" means
"a" multiplied by "b". "a2" means
"a squared", which is the same as "a" times "a".
"b3" means "b cubed", which is the same
as "b" times "b" times
Be careful!! Units count.
Use the same units for all measurements. Examples
cube = a 3
rectangular prism = a b c
irregular prism = b h
cylinder = b h = pi r 2 h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 pi r 2 h
sphere = (4/3) pi r 3
ellipsoid = (4/3) pi r1 r2 r3
Volume is measured in "cubic" units. The volume
of a figure is the number of cubes required to fill it completely, like
blocks in a box.
Volume of a cube = side times side times side. Since
each side of a square is the same, it can simply be the length of one
If a square has one side of 4 inches, the volume would
be 4 inches times 4 inches times 4 inches, or 64 cubic inches. (Cubic
inches can also be written in3.)
Be sure to use the same units for all measurements.
You cannot multiply feet times inches times yards, it doesn't make
a perfectly cubed measurement.
The volume of a rectangular prism is the length on
the side times the width times the height. If the width is 4 inches, the
length is 1 foot and the height is 3 feet, what is the volume?
NOT CORRECT .... 4 times 1 times 3 = 12
CORRECT.... 4 inches is the same as 1/3 feet.
Volume is 1/3 feet times 1 foot times 3 feet = 1 cubic foot (or 1 cu.
ft., or 1 ft3).
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The North Wind and The Sun
The North Wind boasted of great strength. The Sun argued that there was great power in gentleness.
"We shall have a contest," said the Sun.
Far below, a man traveled a winding road. He was wearing a warm winter coat.
"As a test of strength," said the Sun, "Let us see which of us can take the coat off of that man."
"It will be quite simple for me to force him to remove his coat," bragged the Wind.
The Wind blew so hard, the birds clung to the trees. The world was filled with dust and leaves. But the harder the wind blew down the road, the tighter the shivering man clung to his coat.
Then, the Sun came out from behind a cloud. Sun warmed the air and the frosty ground. The man on the road unbuttoned his coat.
The sun grew slowly brighter and brighter.
Soon the man felt so hot, he took off his coat and sat down in a shady spot.
"How did you do that?" said the Wind.
"It was easy," said the Sun, "I lit the day. Through gentleness I got my way."
Storytelling in the Classroom | Lesson Plans & Activities
Copyright © 2000 Story Arts
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Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
Decide which charts and graphs represent the number of goals two football teams scored in fifteen matches.
Have a look at this table of how children travel to school. How
does it compare with children in your class?
Kaia is sure that her father has worn a particular tie twice a week
in at least five of the last ten weeks, but her father disagrees.
Who do you think is right?
How have the numbers been placed in this Carroll diagram? Which
labels would you put on each row and column?
Use the interactivities to complete these Venn diagrams.
What statements can you make about the car that passes the school
gates at 11am on Monday? How will you come up with statements and
test your ideas?
Class 5 were looking at the first letter of each of their names. They created different charts to show this information. Can you work out which member of the class was away on that day?
This activity is based on data in the book 'If the World Were a Village'. How will you represent your chosen data for maximum effect?
This problem explores the range of events in a sports day and which ones are the most popular and attract the most entries.
95% of people in Britain should live within 10 miles of the route of the Olympic Torch tour. Is this true?
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Radical expressions are square roots of monomials, binomials, or polynomials. One way of simplifying radical expressions is to break down the expression into perfect squares multiplying each other. A perfect square, such as 4, 9, 16 or 25, has a whole number square root. Simplifying radical expressions becomes especially important in Geometry when solving formulas and in using the Pythagorean Theorem.
When you guys start working with radical expressions, it's important that you know what a radical expression is. Radical expression just means it's an expression that has a square root sign. I call it a "square rootie" sometimes. I don't know why. It just comes out of my mouth. It's not an official term. So the other thing is that the thing that's under the radical sign, or under the "square rootie" is called a radicand. And you'll see that more when we start doing some problems.
So when you're asked to simplify radical expressions, we have a really important property and here's what it is: If you have the square root of the product AB that's equal to the product of their individual square roots. It's equal to the square root of A times the square root of B. Just be really careful. This is only true as long as A and B are both positive and not 0.
Let me show you an example. If I had the square root of 10, that's not something that you guys have probably worked with very much. The square root of 10 is equal to the square root of two times the square root of 5. If you don't trust me, grab yourself a calculator and check it out.
The square root of 10 is the decimal of 3.16. The square root of two is the decimal 1.41. And I'm claiming that 3.16 is equal to 1.41 times whatever the square root of five is; 2.23. It's going to be a tiny bit off because I'm rounding, but you guys get the idea. This decimal times that decimal gives me that answer.
And sometimes it doesn't make a whole lot of sense if you don't have a calculator handy, but it can be really useful in problems like this, like the square root of 18. I don't know what the square root of 18 is, but I do know that 18 is the equal to the product of nine times two.
So the simplified form of the square root of 18 would be, let's see, square root of nine is three, times the square root of two. This would be my answer in simplified form. Simplified form means there are no perfect factors in the radicand, or no square numbers under the "square rootie", if that makes more sense to you.
So when you're approaching these problems, it's really important that you're good at recognizing perfect square factors. I've gone through and listed all of the perfect squares for the numbers one through 15, like one times itself is one, two times two, three times three, four times four.
Pretty much, you just have to memorize these. Get really comfortable with all the squares of the numbers one through 15, so that when you're doing these problems, you can recognize these factors. These are important numbers. One last thing, I want to leave you with before you start your homework problems is to watch out for this property. The negative square root of 144 is not the same thing as the square root of 144. That's really important.
The negative square root of 144 would be 12. It's like I square rooted 144 and then negativize it, as opposed to this: The square root of 144, if you try it on your calculator, you'll see it says "Error". This is no real solution, no real number. There's no real number that when you multiply it by itself you get the answer 144. So watch out for that. Those are two really important distinctions.
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The See-Saw: How to Blend
Materials: Chalkboard with chalk and letter cards for a, c, t, n, b, p, h, l, s, and m
Procedure: 1. Today we are going to find out how we take sounds and make them into words. Here’s an example. Tell me what I’m saying…cccccaaaaaaattttt. I said cat. It took me a long time to say that word like that. Imagine if all words were that long. It would take a really long time just to say one sentence. We are going to look at a lot of words but we are going to focus on words that have the letter a=/a/ in them.
2. Today we are going to blend words. Blending words is really easy to do if you understand how to do it. First I will draw a see-saw on the board and explain how it moves because of the weight differences on either side. When I play on the see-saw, I can go up and down. “C” wanted to play on the see-saw and so did “t”. (I will demonstrate on the chalkboard). Let’s just say for fun that “t” is a lot bigger than ”c” so “c” asks “a” to come and sit with him. Now, on one side of the see-saw “c” and “a” are together and they say /ca/. The other side says /t/. If the “c” and “a” slide to the other end of the see-saw, they say cat. When the “t” wants to play somewhere else, n, b, and p can take his spot saying can, cab and cap. I can then switch other letters in and out to suit the situation. (I will also use the words nap, hat, last, and slam.
3. Now I want you to figure out some words on your own. I will pronounce some words in a funny way. The sounds will be said one at a time. I want you to guess what I am saying. Here is an example; n-a-p. That’s right, I said nap. (If incorrect, I will say, sorry, the word is nap).
4. I will pass out copies of the book A Cat Nap to the children in the room. I will then ask them to find a quiet place in the classroom to read the book out loud to themselves. I will walk around the classroom and assess what they have learned. I will then help the one’s who are still having trouble with blending.
Reference: Murray, Bruce ed. (1998). Lessons For Learning to Read. (p.26). “See-Saw: How to Blend” by: Tiffany Hellwagner
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"Baaaaa", said the Sheep
RATIONALE: A strong sense of phonemic awareness is essential in order for children to learn how to read and spell words. To help further phonetic cue reading, they must be taught significant letter-sound correspondences necessary for decoding words. Of all the phonemes, short vowels are probably the hardest to identify. In the lesson I will be focusing on the vowel correspondence a=/a/. Students will learn how to recognize the /a/ sound in spoken words, practice spelling /a/ words in the letterbox, and spot the /a/ sound in words on their own.
Materials: Poster with "Ann and Adam ate apples and ran from ants" on it, letterbox for each child and the letters a, b, d, f, g, l, m, n, r, s, and t, individual copies of Fat Cat by Nora Gaydos, primary paper and pencil
1. Introduce the lesson by telling the students we will be learning about the letter a, and the sound that it makes. Explain to the class the importance of learning the sounds, particularly vowels. Also explain that all words in books are made up of letters, which make sounds. The letter a is an important and fun letter and sound. Today we will learn methods on how to read and spell words with the /a/ sound.
2. Ask the students: have you ever heard the noise that sheep make? What do they sound like? That's right!!! Baaaaa Notice the a sound /a/. You can also hear this sound in words like cat, nap, and sad. You hear the /a/ sound in the middle of all those words. Be listening today for other words that may have that /a/ sound.
3. Now I have a fun tongue twister. Display the poster. Read the twister to the class, "Ann and Adam ate apples then ran from the ants". Now lets all read it together. Now we will read it again, but this time it will be a little different. We are going to stretch out the /a/ sound in each word, Ann would be said Aaaaaan. Lets all say it together. Great. Can anyone tell me a word they heard with the /a/ sound in it? How about words without the /a/ sound? Good Job!
4. Ask the students to take out primary paper and pencil. Most of you already know how to write the letter a. We are going to practice incase someone has forgotten. On the board write an a, then go through the steps: for lowercase a, you start under the fence then you go around and touch the sidewalk, around and straight down. Can anyone tell me what sound this make? On three lets all say it together! Please write five a's on your paper to represent the /a/ sound. The teacher can also be modeling while students are writing. Walk around and view everyone's a's. Wonderful work everyone!
5. Now it is time for our letterboxes! Handout letterboxes and the appropriate letters. Now we will use what we have learned today, about the letter a=/a/, to spell words. Let's start with the word cat. This word has three sounds so we will open three boxes, /c/ /a/ /t/. The first sound I hear, in cat, is c-c-c-at. I hear the /c/ sound, so I will place the letter c in the first box. (Modeling throughout as students follow along with their letterboxes) The next sound I hear is our /a/ sound, from baaaaaa. I then place an a in the second box. Now I have /c/ /a/, ca. To finish cat I need the last letter, /t/. Now we have the word cat. Do you hear /a/ in: mat or sit, bed or bad, stamp or truck? I will have the following words written on the board: fat, man, sad, and bad. I will ask the students to continue with the words on the board. When finished I will ask each student to explain to their neighbor how they spelled the word fat. Next the group will try four phoneme words: flag, glad, and cram. While the students are working on this I will go around to students and ask them to read a word out loud to me and explain.
6. I will then hand out a book to each student. I will ask the students to reread the book that I read aloud the previous day. This book has a lot of words with the /a/ sound. This book will be easier because we learned today that a=/a/. Please read the book quietly. If you need to space yourself out so you can concentrate find a quite spot in the room. I will come around and listen to you while you are reading quietly. When you are finished reading come back to your desk, I would like for you to write down some of those /a/ words from the book. Can someone tell me a /a/ word from the book? Write the words on the board, so the all the students can see them.
7. For assessment I will give the student
picture pages and ask them to circle the pictures that have the /a/ sound
in the word. Then go back and write the word underneath the picture.
(Web page entitled The Baby is Crying by Meagan Strider)
(Web page entitled "Aaaa!!" Who Woke Up the Baby? By Allison Felton)
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Let's Get Icky and Sticky
Rationale: This lesson will help beginning readers to learn to spell and read words. They will learn to recognize i=/i/ in written and spoken words. They will learn a meaningful representation and practice spelling and reading words with i=/i/ using a letterbox lesson. Also, they will read a new book.
· Letterboxes: A set of 2, 3, 4, and 5 for each student and teacher
· Letterbox letters for each student and teacher: i (2), s, t, f, a, n, b (2), k, l (2), e, d, r, F, h, p, m, c
· Overhead projector
· Picture of "icky sticky glue" with hand motion description
· Poster with tongue twister: "Nikki is an icky sticky piggy living in an igloo made of itchy, icky, sticky mud. "
· Primary Paper (2 sheets, one for teacher and one for student) Pencils (2 wooden)
· Book Liz Is Six
· Worksheet with pictures for assessment (pictures of two choices, which picture do you hear i=/i/? (pig or horse? Spill or drip? Cook or grill? Stink or smell?)
1. First, I will show the students the letter i on the overhead projector.
· I will use the upper and lower case I from my set of letterbox tiles.
· "Can you tell me what letter this is?" "That's correct, it is the letter I." "Who can tell me what sound it makes?" "Great job!"
· Now I will place the picture of someone touching icky sticky glue on the overhead.
· "The i makes the sound /i/ like you have icky sticky glue all over your hands."
· I will then stretch out the i sound to sound like''I have glue all over my hands,'' just like the picture shows.
· "Now I want everyone to try and get the icky sticky glue off their hands!"
2. Next, I will show the tongue twister on the overhead projector. "I am going to read this silly sentence to you and then I want you to read it after me."
· I will read the sentence stretching out the i to sound like icky sticky glue .
· "Now it's your turn to repeat after me: "Nikki is an icky sticky piggy living in an igloo made of itchy, icky, sticky mud. "
3. Now, I want you to pay really close attention because I am going to ask you some questions. "I am going to read two words to you and I want you to be listening for the icky sticky i. After I read the words, I want you to raise your hand and tell me what word you heard icky sticky in, and then show me the hand motion that represents it."
· Word list:
o Bed or sit
o Fix or kite
o Lip or nose
o Trick or Treat
4. Hand out letterbox tiles and have students turn them over to the lowercase side. Now I want everyone watch me as I model how to use our letterboxes.
· For this word, I am going to need three letterboxes. That means there are three sounds in my word. This also means that our mouths are only going to move three times when we say this word. The word is…fix.
· The f says /f/ so we need to put the letter f in our first letterbox.
· The second sound is i so we need to put the letter i in the second letterbox.
· The last sound is /x/ so we need to put the letter x in the last letterbox.
· Now it is your turn. The students will begin by reading each word and then spelling it.
o Words: (2) is, it (3) fat, fit, sit, tin, lip, kit (4) bled, bred, fled, sled, spill, grip (5) drift, twist, split, blest, slept.
o The student will use their letterboxes and letter tiles to spell the words. I will walk around the room and monitor the students and help them if needed.
5. I will now have students read words off the overhead projector. I will show a list of words that they spelled in step 3. If a child cannot read a word, I will use body-coda blending to facilitate reading. I will start with the vowel i and then add the letters that correspond with the phoneme from left to right.
6. Next, I will introduce the decodable book: Liz Is Six. We are going to read Liz Is Six.
· This story is about a girl who is your age and gets a brand new mitt for her birthday. They decide to play a baseball game and pig gets a great big hit! Let's read to see what happens in the rest of the game with the pig and Liz.
· The student will then read the book aloud to me, using his yellow pointer to help him read each word.
7. Finally, we are going to write a message to each other about what our favorite animal is. I remind him how to write an /i/ and have him write a couple of words for him to practice. He may use invented spelling when writing.
As I work with my student, I will note miscues that I hear as he is reading. I will be able to check his phonemic awareness by anecdotal notes that will collaborate throughout the semester to check reading progress. To end the lesson, I will present my student with a worksheet with pictures on it, some containing the /i/ sound in them. The goal will be to circle the picture that contains the /i/ sound. Under the picture, he will write the word of the picture. He will receive a sticker if he completes the entire lesson.
Shumock, Emily. "Icky Piggy."
Battles, Ellen. "The Old Creaky Door."
Murray, B.A. & Lesniak, T (1999). The Letterbox Lesson: A hands-on approach for teaching decoding. The Reading Teacher, 52, 644-650. Cushman, S (1990). Liz Is Six. Carson, CA: Educational Insights. Picture of Icky Sticky /i/.
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Archemedes' principle states that an object immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object. The weight of a fluid is equal to it it's density times it's volume times gravity.
Spring scale, 500 ml beaker, 500 g mass, styrofoam cup.
1. Fill the 500 ml beaker to the 300 ml level.
2. Measure the weight of the 500 g mass using the spring scale and record.
3. Lower the 500 g mass into the water until it is submerged, but not touching the bottom.
4. Observer and record the new water level, and observe and record the new weight of the 500 g mass.
5. Remove the 500 g mass from the water and place in the styrofoam cup. Place the cup and 500 g mass in the beaker. Record the new water level.
1. Calculate the buoyant force of the water on the mass.
2. Calculate the volume of water displaced by the 500 g mass.
3. Calculate the weight of the displaced water. How does it compare to the buoyant force? Explain.
4. Compare the water level of the mass in the cup to the water level of the mass submerged without the cup. Which was higher? Explain.
5. Explain how a metal object could float in water.
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Rationale: Children need to be able to recognize phonemes and see that letters in the alphabet represents these sounds. Before learning to read one must be able to read and blend phonemes. My goal is to help students locate s=/s/ in the beginning, middle, and end of words. By the end of the lesson, my students should be able to recognize /s/ in spoken words, written words, and use it in writing.
Materials: note card (for tongue twister), picture page (hand drawn with square, circle, snake, sun, moon, and six), primary paper/pencil, S-card
1. Introduce- “Boys and girls, we will discover what sounds Sam the silly snake says when slithering through the sand. Can you tell me what sound a snake makes? Very good!” Make the sound a few times and have them feel the way their mouth moves (teeth together and tongue hissing and the back of their teeth).
2. Let’s say a tongue twister together: “Sam said he was sorry he put salt in Sally’s sandwich.” Have the students repeat several times. “Now let’s pretend we are snakes and stretch out each /s/ sound.” (Repeat twister putting emphasis on S’s). Review: “What sound does s make?”
3. Give them a picture page with various words some containing the letter s. Have each student draw and S on the ones containing the s phoneme (square, circle, snake, sun, moon, six). Now listen for the /s/ sound in words I call out. If you hear /s/ then hold up your S-card (sit, stand, reach, star, finish, start, run, swim, win, loose). Great job you used the /s/ phoneme perfectly.
4. Write the letter S on primary paper, Model starting at just below the roof making a little c the sit on the fence. Then go down the sidewalk and make a curve in the opposite direction. Now have the student practice a line of S’s. Wonderful!
5. I will read them a paragraph and have the students hold up the S-card every time they hear a word with the letter s.
“ Six students went to school on Sunday morning. The class was empty. Silly kids forgot it was a day off, so they went swimming. The sun was out and Sally got a sunburn. So she went home.”
6. As an assessment have each student think of three words containing the letter S and share them with the class. Repeat the words as a group after the individual reads them.
Reference: Reading Genie website:www.auburn.edu/rdggenie
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By: Cynthia Kinsaul
Rational: The phoeme /p/ can sometimes be difficult for children to learn. This lesson will help them develop this discriminatory skill. The students will learn to recognize /p/ in spoken words and objects provided. By doing this the children will increase their ability to discriminate the p=/p/ sound.
Materials: (1) cooking pot, (1) picture of a pink pig, (1) ping pong ball, (1) cotton ball, (1) shoe string, (1) crayon, (1) brush, (1) rubbermaid box, picture pages, construction paper, scissors, glue, If You Give a Pig a Pancake by Harper Collins.
1. Explain to the children that written language consists of connecting letters to sounds. In today's lesson we will be looking for the p=/p/ sound in words.
2. Ask the students what the p=/p/ sound makes. Then model the /p/ sound by saying "watch my mouth and listen as I make the p=/p/ sound. "Let's all make the p=/p/ sound with our mouths." "Can anyone tell me a word where we make the sound /p/? Here are some examples, show the children the items brought to class that begin with the /p/ sound. Pot, picture of a pink pig, ping-pong ball, etc.
3. "Let's try a tongue twister". Pink pigs play ping-pong in a pen. Have the tongue twister written on the board with the letter p in every word written in a different letter than the other words so the children can see the p=/p/ sound. "Everyone say the tongue twister together on my signal". "We are going to say it three times". Then say it one more time and really emphasize the /p/ sound in the words.
4. Give each student a cotton ball to place on their desk. Tell them to put their mouths close to the cotton, but not touching it, and make the /p/ sound. Each time they make the /p/ sound the cotton will move. This will help them remember how to make the /p/ sound.
5. After we have practiced saying the p=/p/ sound tell the children we are going to play a game to identify which items in the box have the /p/ sound and which one's don't. Have children take turns coming up and pulling an object out of the box and along with the rest of the class decide if the object begins with the /p/ sound or if it does not. Objects in the box include a ping-pong ball, a picture of a pink pig, a pot, a shoestring, a crayon, and a brush. The children should leave the objects that have the/p/ sound sitting out on the table for the class to observe through out the lesson. The objects without the /p/ sound should be placed back in the box.
6. Read the book If You Give A Pig a Pancake by Laura Numeroff. The story is about a little pink pig who eats a pancake and what all she has to do after she has eaten the pancake. For a response from the students I will ask them to clap every time they hear the p=/p/ sound.
7. For the assessment step, give each child a piece of construction paper and a pair of scissors. Make sure each child has a picture page with items that begin with the letter p. Have each child go through their page and cut out pictures of anything they find that has the /p/ sound. Then have the children glue the pictures to their piece of construction paper and write under the picture what it is. As a review have each child stand up and share with the class the pictures they found.
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The Megalodon was the biggest shark ever, but it is extinct. The word Megalodon means giant tooth. The Megalodon is an ancient shark that could have been 40 feet long, and some people say it could have been 50-100 feet long. It is at least two to three times the size of the great white shark. One tooth was the size of a personís hand. It ate mostly whales. It also ate 2% of its body weight a day. Its jaw opened 6 feet wide and 7 feet high. It did not chew. It ate big chunks and swallowed them whole. It lived roughly 1.5 to 25 million years ago. It is now extinct, yet its exact time of extinction is debated.
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Imagine a pyramid which is built in square layers of small cubes.
If we number the cubes from the top, starting with 1, can you
picture which cubes are directly below this first cube?
Can you make a cycle of pairs that add to make a square number
using all the numbers in the box below, once and once only?
Can you use this information to work out Charlie's house number?
Add the sum of the squares of four numbers between 10 and 20 to the
sum of the squares of three numbers less than 6 to make the square
of another, larger, number.
In 1871 a mathematician called Augustus De Morgan died. De Morgan
made a puzzling statement about his age. Can you discover which
year De Morgan was born in?
Mrs Morgan, the class's teacher, pinned numbers onto the backs of
three children. Use the information to find out what the three
Can you make square numbers by adding two prime numbers together?
Find another number that is one short of a square number and when
you double it and add 1, the result is also a square number.
These squares have been made from Cuisenaire rods. Can you describe
the pattern? What would the next square look like?
Think of a number, square it and subtract your starting number. Is
the number you’re left with odd or even? How do the images
help to explain this?
Cut differently-sized square corners from a square piece of paper
to make boxes without lids. Do they all have the same volume?
Each light in this interactivity turns on according to a rule. What
happens when you enter different numbers? Can you find the smallest
number that lights up all four lights?
The discs for this game are kept in a flat square box with a square
hole for each disc. Use the information to find out how many discs
of each colour there are in the box.
Use the interactivities to complete these Venn diagrams.
Complete the magic square using the numbers 1 to 25 once each. Each
row, column and diagonal adds up to 65.
Can you arrange the numbers 1 to 17 in a row so that each adjacent
pair adds up to a square number?
Does a graph of the triangular numbers cross a graph of the six
times table? If so, where? Will a graph of the square numbers cross
the times table too?
One block is needed to make an up-and-down staircase, with one step
up and one step down. How many blocks would be needed to build an
up-and-down staircase with 5 steps up and 5 steps down?
A challenge that requires you to apply your knowledge of the
properties of numbers. Can you fill all the squares on the board?
What is the value of the digit A in the sum below: [3(230 + A)]^2 =
A square patio was tiled with square tiles all the same size. Some
of the tiles were removed from the middle of the patio in order to
make a square flower bed, but the number of the remaining tiles. . . .
A woman was born in a year that was a square number, lived a square
number of years and died in a year that was also a square number.
When was she born?
How many four digit square numbers are composed of even numerals?
What four digit square numbers can be reversed and become the
square of another number?
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The purpose of this activity is to learn to use numbers, an important step toward learning how to add and subtract.From the Virtual Pre-K: Ready For Math toolkit
Cut out 20 playing-card sized rectangles from the empty cereal box. Make two sets of cards from 1 to 10. You can make dots to represent each number (one dot for 1, two dots for 2), and add the written numbers as your child starts recognizing numerals. You can also make cards with the written number on one side and the number of dots on the other.
Number Card Game Ideas:
Which is More?: Help your child count out 10 cards for each of you. Each player puts down one card, then together read the two numbers and decide which is more. The player with the larger number wins both cards. Continue playing until one player has all the cards.
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Transcript: Let's get going. Yesterday we learned about flux and we have seen the first few examples of how to set up and compute integrals for a flux of a vector field for a surface. Remember the flux of a vector field F through the surface S is defined by taking the double integral on the surface of F dot n dS where n is the unit normal to the surface and dS is the area element on the surface. As we have seen, for various surfaces, we have various formulas telling us what the normal vector is and what the area element becomes. For example, on spheres we typically integrate with respect to phi and theta for latitude and longitude angles.
On a horizontal plane, we would just end up degrading dx, dy and so on. At the end of lecture we saw a formula. A lot of you asked me how we got it. Well, we didn't get it yet. We are going to try to explain where it comes from and why it works. The case we want to look at is if S is the graph of a function, it is given by z equals some function in terms of x and y. Our surface is out here. Z is a function of x and y. And x and y will range over some domain in the x, y plane, namely the region that is the shadow of the surface on the x, y plane.
I said that we will have a formula for n dS which will end up being plus/minus minus f sub x, minus f sub y, one dxdy, so that we will set up and evaluate the integral in terms of x and y. Every time we see z we will replace it by f of xy, whatever the formula for f might be. Actually, if we look at a very easy case where this is just a horizontal plane, z equals constant, the function is just a constant, well, the partial derivatives become just zero. You get <0, 0, 1> dx, dy.
That is what you would expect for a horizontal plane just from common sense. This is more interesting, of course, if a function is more interesting. How do we get that? Where does this come from? We need to figure out, for a small piece of our surface, what will be n delta S. Let's say that we take a small rectangle in here corresponding to sides delta x and delta y and we look at the piece of surface that is above that. Well, the question we have now is what is the area of this little piece of surface and what is its normal vector? Observe this little piece up here. If it is small enough, it will look like a parallelogram.
I mean it might be slightly curvy, but roughly it looks like a parallelogram in space. And so we have seen how to find the area of a parallelogram in space using cross-product. If we can figure out what are the vectors for this side and that side then taking that cross-product and taking the magnitude of the cross-product will give us the area. Moreover, the cross-product also gives us the normal direction. In fact, the cross-product gives us two in one.
It gives us the normal direction and the area element. And that is why I said that we will have an easy formula for n dS while n and dS taken separately are more complicated because you would have to actually take the length of a direction of this guy. Let's carry out this problem. Let's say I am going to look at a small piece of the x, y plane. Here I have delta x, here I have delta y, and I am starting at some point (x, y). Now, above that I will have a parallelogram on my surface. This point here, the point where I start, I know what it is. It is just (x, y). And, well, z is f(x, y).
Now what I want to find, actually, is what are these two vectors, let's call them U and V, that correspond to moving a bit in the x direction or in the y direction? And then U cross V will be, well, in terms of the magnitude of this guy will just be the little piece of surface area, delta S. And, in terms of direction, it will be normal to the surface. Actually, I will get just delta S times my normal vector. Well, up to sign because, depending on whether I do U V or V U, I might get the normal vector in the direction I want or in the opposite direction. But we will take care of that later. Let's find U and V. And, in case you have trouble with that small picture, I have a better one here.
Let's keep it just in case this one gets really too cluttered. It really represents the same thing. Let's try to figure out these vectors U and V. Vector U starts at the point x, y, f of x, y and it goes to -- Whereas, its head, well, I will have moved x by delta x. So, x plus delta x and y doesn't change. And, of course, the z coordinate has to change. It becomes f of x plus delta x and y. Now, how does f change if I change x a little bit? Well, we have seen that it is given by the partial derivative f sub x. This is approximately equal to f of x, y plus delta x times f sub x at the given point x, y.
I am not going to add it because the notation is already long enough. That means my vector U, well, approximately because I am using this linear approximation,
I have my two sides. Now I can take that cross-product. Well, maybe I will first factor something out. See, I can rewrite this as one, zero, f sub x times delta x. And this one I will rewrite as zero, one, f sub y delta y. And so now the cross-product, n hat delta S up to sign is going to be U cross V. We will have to do the cross-product, and we will have a delta x, delta y coming out. I am just saving myself the trouble of writing a lot of delta x's and delta y's, but if you prefer you can just do directly this cross-product.
Let's compute this cross-product. Well, the i component is zero minus f sub x. The y component is going to be, well, f sub y minus zero but with the minus sign in front of everything, so negative f sub y. And the z component will be just one times delta x delta y. Does that make sense? Yes. Very good. And so now we shrink this rectangle, we shrink delta x and delta y to zero, that is how we get this formula for n dS equals negative fx, negative fy, one, dxdy. Well, plus/minus because it is up to us to choose whether we want to take the normal vector point up or down.
See, if you take this convention then the z component of n dS is positive. That corresponds to normal vector pointing up. If you take the opposite signs then the z component will be negative. That means your normal vector points down. This one is with n pointing up. I mean when I say up, of course it is still perpendicular to the surface. If the surface really has a big slope then it is not really going to go all that much up, but more up than down. OK. That is how we get the formula. Any questions? No. OK. That is a really useful formula.
You don't really need to remember all the details of how we got it, but please remember that formula. Let's do an example, actually. Let's say we want to find the flux of the vector field z times k, so it is a vertical vector field, through the portion of the paraboloid z equals x^2 y^2 that lives above the unit disk. What does that mean? z = x^2 y^2. We have seen it many times. It is this parabola and is pointing up. Above the unit disk means I don't care about this infinite surface. I will actually stop when I hit a radius of one away from the z-axis. And so now I have my vector field which is going to point overall up because, well, it is z times k. The more z is positive, the more your vector field goes up.
Of course, if z were negative then it would point down, but it will live above. Actually, a quick opinion poll. What do you think the flux should be? Should it be positive, zero, negative or we don't know? I see some I don't know, I see some negative and I see some positive. Of course, I didn't tell you which way I am orienting my paraboloid. So far both answers are correct. The only one that is probably not correct is zero because, no matter which way you choose to orient it you should get something. It is not looking like it will be zero. Let's say that I am going to do it with the normal pointing upwards.
Second chance. I see some people changing back and forth from one and two. Let's draw a picture. Which one is pointing upwards? Well, let's look at the bottom point. The normal vector pointing up, here we know what it means. It is this guy. If you continue to follow your normal vector, see, they are actually pointing up and into the paraboloid. And I claim that the answer should be positive because the vector field is crossing our paraboliod going upwards, going from the outside out and below to the inside and upside. So, in the direction that we are counting positively.
We will see how it turns out when we do the calculation. We have to compute the integral for flux. Double integral over a surface of F dot n dS is going to be -- What are we going to do? Well, F we said is <0, 0, z>. What is n dS. Well, let's use our brand new formula. It says negative f sub x, negative f sub y, one, dxdy. What does little f in here? It is x^2 y^2. When we are using this formula, we need to know what little x stands for.
It is whatever the formula is for z as a function of x and y. We take x^2 y^2 and we take the partial derivatives with minus signs. We get negative 2x, negative 2y and one, dxdy. Well, of course here it didn't really matter because we are going to dot them with zero. Actually, even if we had made a mistake we somehow wouldn't have had to pay the price. But still. We will end up with double integral on S of z dxdy. Now, what do we do with that? Well, we have too many things. We have to get rid of z. Let's use z equals x^2 y^2 once more.
That becomes double integral of x^2 y^2 dxdy. And here, see, we are using the fact that we are only looking at things that are on the surface. It is not like in a triple integral. You could never do that because z, x and y are independent. Here they are related by the equation of a surface. If I sound like I am ranting, but I know from experience this is where one of the most sticky and tricky points is.
OK. How will we actually integrate that? Well, now that we have just x and y, we should figure out what is the range for x and y. Well, the range for x and y is going to be the shadow of our region. It is going to be this unit disk. I can just do that for now. And this is finally where I have left the world of surface integrals to go back to a usual double integral. And now I have to set it up. Well, I can do it this way with dxdy, but it looks like there is a smarter thing to do. I am going to use polar coordinates. In fact, I am going to say this is double integral of r^2 times r dr d theta. I am on the unit disk so r goes zero to one, theta goes zero to 2pi. And, if you do the calculation, you will find that this is going to be pi over two.
Any questions about the example. Yes? How did I get this negative 2x and negative 2y? I want to use my formula for n dS. My surface is given by the graph of a function. It is the graph of a function x^2 y^2. I will use this formula that is up here. I will take the function x^2 y^2 and I will take its partial derivatives. If I take the partial of f, so x^2 y^2 with respect to x, I get 2x, so I put negative 2x. And then the same thing, negative 2y, one, dxdy. Yes?
Which k hat? Oh, you mean the vector field. It is a different part of the story. Whenever you do a surface integral for flux you have two parts of the story. One is the vector field whose flux you are taking. The other one is the surface for which you will be taking flux. The vector field only comes as this f in the notation, and everything else, the bounds in the double integral and the n dS, all come from the surface that we are looking at. Basically, in all of this calculation, this is coming from f equals zk. Everything else comes from the information paraboloid z = x^2 y^2 above the unit disk.
In particular, if we wanted to now find the flux of any other vector field for the same paraboloid, well, all we would have to do is just replace this guy by whatever the new vector field is. We have learned how to set up flux integrals for this paraboloid. Not that you should remember this one by heart. I mean there are many paraboloids in life and other surfaces, too. It is better to remember the general method. Any other questions? No. OK. Let's see more ways of taking flux integrals. But, just to reassure you, at this point we have seen the most important ones. 90% of the problems that we will be looking at we can do with what we have seen so far in less time and this formula.
Let's look a little bit at a more general situation. Let's say that my surface is so complicated that I cannot actually express z as a function of x and y, but let's say that I know how to parametize it. I have a parametric equation for my surface. That means I can express x, y and z in terms of any two parameter variables that might be relevant for me. If you want, this one here is a special case where you can parameterize things in terms of x and y as your two variables. How would you do it in the fully general case? In a way, that will answer your question that, I think one of you, I forgot, asked yesterday how would I do it in general? Is there a formula like M dx plus N dy? Well, that is going to be the general formula. And you will see that it is a little bit too complicated, so the really useful ones are actually the special ones.
Let's say that we are given a parametric description -- -- of a surface S. That means we can describe S by formulas saying x is some function of two parameter variables. I am going to call them u and v. I hope you don't mind. You can call them t1 and t2. You can call them whatever you want. One of the basic properties of a surface is because I have only two independent directions to move on. I should be able to express x, y and z in terms of two variables. Now, let's say that I know how to do that. Or, maybe I should instead think of it in terms of a position vector if it helps you. That is just a vector with components
It works like a parametric curve but with two parameters. Now, how would we actually set up a flux integral on such a surface. Well, because we are locating ourselves in terms of u and v, we will end up with an integral du dv. We need to figure out how to express n dS in terms of du and dv. N dS should be something du dv. How do we do that? Well, we can use the same method that we have actually used over here. Because, if you think for a second, here we used, of course, a rectangle in the x, y plane and we lifted it to a parallelogram and so on. But more generally you can think what happens if I change u by delta u keeping v constant or the other way around?
You will get some sort of mesh grid on your surface and you will look at a little parallelogram that is an elementary piece of that mesh and figure out what is its area and normal vector. Well, that will again be given by the cross-product of the two sides. Let's think a little bit about what happens when I move a little bit on my surface. I am taking this grid on my surface given by the u and v directions.
And, if I take a piece of that corresponding to small changes delta u and delta v, what is going to be going on here? Well, I have to deal with two vectors, one corresponding to changing u, the other one corresponding to changing v. If I change u, how does my point change? Well, it is given by the derivative of this with respect to u. This vector here I will call, so the sides are given by, let me say, partial r over partial u times delta u. If you prefer, maybe I should write it as partial x over partial u times delta u. Well, it is just too boring to write.
And so on. It means if I change u a little bit, keeping v constant, then how x changes is, given by partial x over partial u times delta u, same thing with y, same thing with z, and I am just using vector notation to do it this way. That is the analog of when I said delta r for line integrals along a curve, vector delta r is the velocity vector dr dt times delta t. Now, if I look at the other side --
Let me start again. I ran out of space. One side is partial r over partial u times delta u. And the other one would be partial r over partial v times delta v. Because that is how the position of your point changes if you just change u or v and not the other one. To find the surface element together with a normal vector, I would just take the cross-product between these guys. If you prefer, that is the cross-product of partial r over partial u with partial r over partial v, delta u delta v. And so n dS is this cross-product times du dv up to sign.
It depends on which choice I make for my normal vector, of course. That, of course, is a slightly confusing equation to think of. A good exercise, if you want to really understand what is going on, try this in two good examples to look at. One good example to look at is the previous one. What is it? It is when u and v are just x and y. The parametric equations are just x equals x, y equals y and z is f of x, y. You should end up with the same formula that we had over there. And you should see why because both of them are given by a cross-product. The other case you can look at just to convince yourselves even further. We don't need to do that because we have seen the formula before, but in the case of a sphere we have seen the formula for n and for dS separately.
We know what n dS are in terms of d phi, d theta. Well, you could parametize a sphere in terms of phi and theta. Namely, the formulas would be x equals a sine phi cosine theta, y equals a sign phi sine theta, z equals a cosine phi. The formulas for circle coordinates setting Ro equals a . That is a parametric equation for the sphere. And then, if you try to use this formula here, you should end up with the same things we have already seen for n dS, just with a lot more pain to actually get there because cross-product is going to be a bit complicated. But we are seeing all of these formulas all fitting together. Somehow it is always the same question. We just have different angles of attack on this general problem. Questions?
No. OK. Let's look at yet another last way of finding n dS. And then I promise we will switch to something else because I can feel that you are getting a bit overwhelmed for all these formulas for n dS. What happens very often is we don't actually know how to parametize our surface. Maybe we don't know how to solve for z as a function of x and y, but our surface is given by some equation. And so what that means is actually maybe what we know is not really these kinds of formulas, but maybe we know a normal vector.
And I am going to call this one capital N because I don't even need it to be a unit vector. You will see. It can be a normal vector of any length you want to the surfaces. Why would we ever know a normal vector? Well, for example, if our surface is a plane, a slanted plane given by some equation, ax by cz = d. Well, you know the normal vector. It is . Of course, you could solve for z and then go back to that case, which is why I said that one is very useful. But you can also just stay with a normal vector. Why else would you know a normal vector? Well, let's say that you know an equation that is of a form g of x, y, z equals zero. Well, then you know that the gradient of g is perpendicular to the level surface. Let me just give you two examples.
If you have a plane, ax by cz = d, then the normal vector would just be . If you have a surface S given by an equation, g(x, y, z) = 0, then you can take a normal vector to be the gradient of g. We have seen that the gradient is perpendicular to the level surface. Now, of course, we don't necessarily have to follow what is going to come. Because, if we could solve for z, then we might be better off doing what we did over there. But let's say that we want to do it this. What can we do? Well, I am going to give you another way to think geometrically about n dS.
Let's start by thinking about the slanted plane. Let's say that my surface is just a slanted plane. My normal vector would be maybe somewhere here. And let's say that I am going to try -- I need to get some handle on how to set up my integrals, so maybe I am going to express things in terms of x and y. I have my coordinates, and I will try to use x and y. Then I would like to relate delta S or dS to the area in the x y plane. That means I want maybe to look at the projection of this guy onto a horizontal plane.
Let's squish it horizontally. Then you have here another area. The guy on the slanted plane, let's call that delta S. And let's call this guy down here delta A. And delta A would become ultimately maybe delta x, delta y or something like that. The question is how do we find the conversion rate between these two areas? I mean they are not the same. Visually, I hope it is clear to you that if my plane is actually horizontal then, of course, they are the same. But the more slanted it becomes the more delta A becomes smaller than delta S. If you buy land and it is on the side of a cliff, well, whether you look at it on a map or whether you look at it on the actual cliff, the area is going to be very different.
I am not sure if that is a wise thing to do if you want to build a house there, but I bet you can get really cheap land. Anyway, delta S versus delta A depends on how slanted things are. And let's try to make that more precise by looking at the angel that our plane makes with the horizontal direction. Let's call this angle alpha, the angle that our plane makes with the horizontal direction. See, it is all coming together. The first unit about cross-products, normal vectors and so on is actually useful now.
I claim that the surface element is related to the area in the plane by delta A equals delta S times the cosine of alpha. Why is that? Well, let's look at this small rectangle with one horizontal side and one slanted side. When you project this side does not change, but this side gets shortened by a factor of cosine alpha. Whatever this length was, this length here is that one times cosine alpha.
That is why the area gets shrunk by cosine alpha. In one direction nothing happens. In the other direction you squish by cosine alpha. What that means is that, well, we will have to deal with this. And, of course, the one we will care about actually is delta S expressed in terms of delta A. But what are we going to do with this cosine? It is not very convenient to have a cosine left in here. Remember, the angle between two planes is the same thing as the angle between the normal vectors. If you want to see this angle alpha elsewhere, what you can do is you can just take the vertical direction. Let's take k. Then here we have our angle alpha again.
In particular, cosine of alpha, I can get, well, we know how to find the angle between two vectors. If we have our normal vector N, we will do N dot k, and we will divide by length N, length k. Well, length k is one. That is one easy guy. That is how we find the angle. Now I am going to say, well, delta S is going to be one over cosine alpha delta A. And I can rewrite that as length of N divided by N dot k times delta A.
Now, let's multiply that by the unit normal vector. Because what we are about is not so much dS but actually n dS. N delta S will be, I am just going to multiply by N. Well, let's think for a second. What happens if I take a unit normal N and I multiply it by the length of my other normal big N? Well, I get big N again. This is a normal vector of the same length as N, well, up to sign. The only thing I don't know is whether this guy will be going in the same direction as big N or in the opposite direction. Say that, for example, my capital N has, I don't know, length three for example. Then the normal unit vector might be this guy, in which case indeed three times little n will be big n.
Or it might be this one in which case three times little n will be negative big N. But up to sign it is N. And then I will have N over N dot k delta A. And so the final formula, the one that we care about in case you don't really like my explanations of how we get there, is that N dS is plus or minus N over N dot k dx dy. That one is actually kind of useful so let's box it. Now, just in case you are wondering, of course, if you didn't want to project to x, y, you would have maybe preferred to project to say the plane of a blackboard, y, z, well, you can do the same thing. To express n dS in terms of dy dz you do the same argument.
Simply, the only thing that changes, instead of using the vertical vector k, you use the normal vector i. So you would be doing N over N dot i dy dz. The same thing. So just keep an open mind that this also works with other variables. Anyway, that is how you can basically project the vectors of this area element onto the x, y plane in a way. Let's look at the special case just to see how this fits with stuff we have seen before.
Let's do a special example where our surface is given by the equation z minus f of x, y equals zero. That is a strange way to write the equation. z equals f of x, y. That we saw before. But now it looks like some function of x, y, z equals zero. Let's try to use this new method. Let's call this guy g(x, y, z). Well, now let's look at the normal vector. The normal vector would be the gradient of g, you see. What is the gradient of this function? The gradient of g --
Well, partial g, partial x, that is just negative partial f, partial x. The y component, partial g, partial y is going to be negative f sub y, and g sub z is just one. Now, if you take N over N dot k dx dy, well, it looks like it is going to be negative f sub x, negative f sub y, one divided by -- Well, what is N dot k? If you dot that with k you will get just one, so I am not going to write it, dx dy. See, that is again our favorite formula. This one is actually more general because you don't need to solve for z, but if you cannot solve for z then it is the same as before.
I think that is enough formulas for n dS. After spending a lot of time telling you how to compute surface integrals, now I am going to try to tell you how to avoid computing them. And that is called the divergence theorem. And we will see the proof and everything and applications on Tuesday, but I want to at least the theorem and see how it works in one example. It is also known as the Gauss-Green theorem or just the Gauss theorem, depending in who you talk to.
The Green here is the same Green as in Green's theorem, because somehow that is a space version of Green's theorem. What does it say? It is 3D analog of Green for flux. What it says is if S is a closed surface -- Remember, it is the same as with Green's theorem, we need to have something that is completely enclosed. You have a surface and there is somehow no gaps in it. There is no boundary to it. It is really completely enclosing a region in space that I will call D.
And I need to choose my orientation. The orientation that will work for this theorem is choosing the normal vector to point outwards. N needs to be outwards. That is one part of the puzzle. The other part is a vector field. I need to have a vector field that is defined and differentiable -- -- everywhere in D, so same instructions as usual. Then I don't have actually to compute the flux integral. Double integral of f dot n dS of a closed surface S.
I am going to put a circle just to remind you it is has got to be a closed surface. It is just a notation to remind us closed surface. I can replace that by the triple integral of a region inside of divergence of F dV. Now, I need to tell you what the divergence of a 3D vector field is. Well, you will see that it is not much harder than in the 2D case. What you do is just -- Say that your vector field has components P, Q and R. Then you will take P sub x Q sub y R sub z. That is the definition. It is pretty easy to remember. You take the x component partial respect to S plus partial respect to y over y component plus partial respect to z of the z component.
For example, last time we saw that the flux of the vector field zk through a sphere of radius a was four-thirds pi a cubed by computing the surface integral. Well, if we do it more efficiently now by Green's theorem, we are going to use Green's theorem for this sphere because we are doing the whole sphere. It is fine. It is a closed surface. We couldn't do it for, say, the hemisphere or something like that. Well, for a hemisphere we would need to add maybe the flat face of a bottom or something like that.
Green's theorem says that our flux integral can actually be replaced by the triple integral over the solid bowl of radius a of the divergence of zk dV. But now what is the divergence of this field? Well, you have zero, zero, z so you get zero plus zero plus one. It looks like it will be one. If you do the triple integral of 1dV, you will get just the volume -- -- of the region inside, which is four-thirds by a cubed. And so it was no accident. In fact, before that we looked at also xi yj zk and we found three times the volume.
That is because the divergence of that field was actually three. Very quickly, let me just say what this means physically. Physically, see, this guy on the left is the total amount of stuff that goes out of the region per unit time. I want to figure out how much stuff comes out of there. What does the divergence mean? The divergence means it measures how much the flow is expanding things. It measures how much, I said that probably when we were trying to understand 2D divergence.
It measures the amount of sources or sinks that you have inside your fluid. Now it becomes commonsense. If you take a region of space, the total amount of water that flows out of it is the total amount of sources that you have in there minus the sinks. I mean, in spite of this commonsense explanation, we are going to see how to prove this. And we will see how it works and what it says.
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Beginning Reading Lesson
In order for children to become fluent
readers, they must first begin to understand that letters are mapped
graphemes, which are the vocal gestures they hear in words. These
gestures are called phonemes which are mapped onto graphemes.
can be a single letter or 2 letters that together still make one
When a combination of letters makes one single sound we call this a
digraph. The goal of this lesson is to help students understand
digraphs are made up of more than one letter but only has one vocal
phoneme. The digraph taught in this lesson is /sh/. Students will be
recognize audibly and visually the phoneme and grapheme /sh/ in text by
of the letterboxes as well as learn to spell and read /sh/
primary paper, chart with “Shirley shuffled while
shopping for fish, shoes, and shells”, Elkonin boxes for each student,
(d, a, s, h, i, p, t, o, c, h, e) for each student. Sheep on a Ship
Nancy E. Shaw, one for each student.
- “When you are at the library and people start
talking too loud, what does the librarian say? Shhh….that’s
right. Shhh is the special sound we are going to talk about today. We hear and use this a lot when we want people
to lower their voices. Did you know that sh is a
sound? Many times when we hear a sound it is written with one
letter, but sometimes there are special sounds that are written with
two letters. Sh is an example of a sound that is written
with two letters and those two letters are s and h.
When we see s and h next to each other in a word that
tells us that the sound they make is /sh/.
(Teacher will use the board while talking about this.)
- “Now, class let’s practice our new sound.
I want you to pretend that everyone is talking really loud and we need
to say /sh/ together.
Shhh…great job! Now let’s say our sound, but put your index
finger over your lips when you say the sound. Shhh…very
good! (Teacher models) Throughout the rest of our lesson
whenever you hear the sound /sh/
I want you to put your index finger over you lips as a signal that you
hear the sound.”
- “Now I have a tongue twister for everyone to
read together. Ready? Here we go.
shark shopped for shirts, shoes, and shorts by the sea shore.”
- “Now this time when we say our tongue twister I
want you to stretch out the /sh/ sound in the words, just like
this wassshhhhh. Ready? Ssshhhelly the ssshhark ssshhhhopped for
ssshhhirts, ssshhhhoes, and ssshhhorts by the sea ssshhhore. Great
- “Class, please take out your letterboxes that I
gave you. Please pay attention while I demonstrate how to spell
the word wish.” Draw letterboxes on the board for modeling.
I will slowly stretch out the word and remind the students that each
box represents a sound and sh is one sound that goes in one
box. I will model slowly stretching out the word
wwwwiiiisssshhhhh. I will put w in box one, i in box two, and /sh/ in box three,
because the sh makes one sound so it goes in one box.
“Ok, class now it’s your turn to spell out some words. Open up
three letterboxes and spell the word dash, fish, ship, and shut.
Next, open up four boxes and spell flash, and brush. Great
job! Now, that you have spelled these words let’s see if you can
read the words outside of the letterboxes.” I will walk around
the room and observe the students while we are doing the letterbox
lesson. (assessment) Write the words on the board and have
the class say the words together. “Wonderful job class.”
- “I am going to write some words on my small
white board and when I turn it around. I will say the word that I
have written and you will have five seconds to think about and I want
you to say /sh/ if you see the /sh/ in the
word.” Ask the students if the /sh/ is at the
beginning, middle, or end of the word? “Great job!”
- “Now with a partner you are going to read Sheep
on a Ship. You will take turns reading and when come to word
with /sh/ in it I want you to write it down on the paper I will
give you. (The papers with the words will be turned in for
assessment.) “With the words from our book are going to make a
poster with a ship in the sea and we will place all these words on a
Sheep Ship. Great job class!!”
Ssshhopping for Ssshhells. http://www.auburn.edu/rdggenie/innov/kendrickbr.html
Katrina. Sound of the Week Lesson
for /sh/. http://www.letteroftheweek.com/sound_sh.html
Shaw, Nancy E. Sheep on a ship. Houghton
Reprint edition 1992
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Find the number of which 79.5% is 101.Often, solving these kinds of problems is taught with the idea that you "translate" certain words in the problem into certain symbols, and thus build an equation.
Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:
Y × 79.5% = 101.
0.795 Y = 101
Y = 101/0.795 = 127.044025157
I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?
So while this idea is great and works, it is also necessary for students to understand the concept of percent well.
In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:
"79.5% of that number is 101"... so 0.795 Y = 101.
Ideas for using MENTAL math for calculating percent problems
I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.
- Find 10% of some example numbers (by dividing by 10).
- Find 1% of some example numbers (by dividing by 100).
- Find 20%, 30%, 40% etc. of these numbers.
FIRST find 10% of the number, then multiply by 2, 3, 4, etc.
For example, find 20% of 18. Find 40% of $44. Find 80% of 120.
I know you can teach the student to go 0.2 × 18, 0.4 × 0.44, and 0.8 × 120 - however when using mental math, the above method seems to me to be more natural.
- Find 3%, 4%, 6% etc. of these numbers.
FIRST find 1% of the number, then multiply.
- Find 15% of some numbers.
First find 10%, halve that to find 5%, and add the two results.
- Calculate some simple discounts. If an item is discounted 20%, 15%, etc., then find the new price.
- "40% of a number is 56. What is the number?" - types of problems.
You can do this mentally, too: First FIND 10% and then multiply that result by 10, to find 100% of the number (which is the number itself).
If 40% is 56, then 10% is 14. So 100% of the number is 140. This result is reasonable, because 40% of this number was 56, so the actual number (140) needs to be more than double that.
- "34% of a number is 129. What is the number?" (Now you need a calculator.)
You don't need to write an equation. You could also first find 1% of this number, and then find 100% of the number.
If 34% of a number is 129, then 1% of that number is 129/34. Find that, and multiply the result by 100.
I recently got this sort of homework question sent to me:
I have a problem and I don't know how to solve it so here is the problem: At a popular clothing store clothes are on sale when they have hung on the rack too long. When an item is first put on sale, the store marks the prices down 30% off. If some shoes are regular-priced at $50.00, how much will they cost after the discount?
You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!
(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)
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I'm Thinking of a Sound
Playing with the sounds of words will help your child listen to sounds and words, and understand the importance of language.
What you need:
Names of objects
What to do:
1. Say to your child, "I'm thinking of the sound sssss, as in sat. Can you tell me a word that begins with sssss?" Your child names one word. "Can you tell me another word that begins with sssss?"
2. Repeat this game using different sounds.
More on: Activities for Preschoolers
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Choose any three by three square of dates on a calendar page.
Circle any number on the top row, put a line through the other
numbers that are in the same row and column as your circled number.
Repeat this for a number of your choice from the second row. You
should now have just one number left on the bottom row, circle it.
Find the total for the three numbers circled. Compare this total
with the number in the centre of the square. What do you find? Can
you explain why this happens?
Can you go through this maze so that the numbers you pass add to
Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?
I was in Bewdley, in Worcestershire recently. It's a very old
town and well known for a pretty bridge over the river in the
centre of town. But it's also known for another reason, one that
most people don't know about. Mr Edward Benbow, from Bewdley, holds
the palindromic record! "The what?" you may be asking.
A palindrome is a word, sentence or verse that reads the same
forward or backwards, that is to say from right to left and left to
right. EVE and ANNA are both short palindromic names, then there's
HANNAH and BOB. You probably have some palindromic people in your
family! Not sure? Well how about MUM or DAD, and you might even
have a SIS!
Was it a car or a cat I saw? The letters need a little
adjusting, but that is a palindromic sentence, or rather
Mr Benbow has put together 22,500 words to make a "palindromic
composition"! They don't make a great deal of sense but that's a
lot of words to read backwards.
Numbers can also be palindromes. For instance 121, this can be
read backwards or
forwards. Palindromic numbers are very easy to create from other
numbers with the aid of addition.
1. Write down any number that is more than one digit. (e.g.
2. Write down the number reversed beneath the first number.
3. Add the two numbers together. (121)
4. And 121 is indeed a palindrome.
Try a simple one first, such as 18.
Sometimes you need to use the first addition answer and repeat
process of reversing and adding.
You will nearly always arrive at a palindrome answer within six
steps. Try one of these numbers 68 or 79.
If you choose a number greater than 89 arriving at the
palindrome answer takes more steps but it still works.
But don't try 196! In fact, avoid it like the plague.... A
computer has already gone through several thousand stages and still
hasn't come up with a palindrome answer!
I wonder if you can think how many 2 digit palindromes there
are? How about finding all of the 3, 4, 5 or 6 digit ones? How
about to a million digits! Some people have already beaten you to
that, you'll be glad to know. You can find the results of their
work on the web here
Don't get palindromes mixed up with inversions . Inversions describe
numbers that read the same upside down as the right way up. Look at
Can you think which year in the last century read the same when
inverted? How about a year in the century before last? When will
the next one occur?
So how and where did the idea of Palindromes come from? Well we
know a great deal more about word palindromes. The word itself is
from the Greek palindromus , which means to run back again. The
palindromes that were made up by organising groups of words not
letters. A Greek poet, Sotades , who
lived in Egypt in about 276BC during the reign of Ptolemy II, wrote
a palindrome about the king which wasn't appreciated. Poor old
Sotades was sealed in a box and cast out to sea for his efforts.
Palindromes are found written in Egyptian hieroglyphics and in
Latin texts. Despite quite a search, I have not been able to
uncover anything significant about the history of the number
palindrome. There is a journal written especially for people who
are passionate about palindromes and there are many, many
palindrome web sites. Some provide information on the findings of
who investigate how they can be formed and the patterns found when
they create palindromes. Other sites provide palindrome puzzles and
problems for you to solve. Maybe you can become the new Harvey
Duner who, from July 18th 2001, became a record breaker with a
palindrome prime of 39,027 digits. What can we say Harvey, except
it could have been 39,093 digits!
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To become fluent readers, children need to learn how to read faster, smoother, and more expressively. Students will be able to work on their reading fluency through repeated and dyad reading. By rereading text, students will learn to read more words per minute. By working with partners, students may learn new decoding skills and will get more practice reading. The more students read, the more their reading skills will improve.
stopwatch (one per two people),
one race track chart per child (The chart will have a race track drawn on it. The goal is to win the race, which means to make it around the track. There will be three different levels: start, middle of the track, and finish),
class set of a children's book about racing Eat My Dust! Henry’s Ford First Race, by Monica Kulling,
worksheet with three or four simple sentences to read aloud to practice speed (ex. She has made a mess.)
1. Introduce the lesson by explaining the concept of cross-checking. Give an example of a sentence read the wrong way such as; I put my hat on my foot." Did that sentence make any sense? No, it should be I put my shoe on my foot. One thing to remember when you read is to make sure that the sentence makes sense when you read it aloud. Tell them that reading fluently results in reading that is more enjoyable and that today we are going to work on becoming fluent readers.
2. Today we are going to work on reading words as fast as we can. The point of the activity is not to skip any words or read them incorrectly. We want to read correctly as fast as we can. Model reading a sentence slowly decoding every word. Then read the same sentence faster to show the difference and the goal for the lesson.
3. Now, I want each of you to get a partner. I am going to hand out a worksheet with some sentences on them. I want you to practice reading the sentences out loud to your partner. Start slow to make sure you read all of the words correctly. Then try to say the sentences faster and smoother. Take turns and make sure each of you gets practice.
4. Read the racecar book aloud using the shared reading concept. Make sure the students follow along in their copy of the book.
5. Explain to the students how fast fluent readers read and how they should all practice reading at a quicker pace. Now that we have read the book, I want each of you to pick out two pages that are your favorite. I am going to pass out another worksheet with a race track on it along with a stopwatch. The goal of this activity is to see how fast you can read the pages you have chosen in order to make it around the track on the chart. Now boys and girls, you will each take turns reading to your partner. While one person reads, the other will keep the time on the watch. Then the next time you read, if your time has improved, you can move the car to the start position and so on. The car only moves if you increase your speed. I want you to do this activity until you win the race or make it around the track. Once this is done they should then prepare to read the book for the third time. This time they should read the book to a peer. The peer should use a checklist like the one attached to this page to help assess their reading. They should take turns reading to one another. After this is done they should discuss the book. Tell what they liked and what they did not like. They should write a few things they talked about down and turn it in along with the checklists.
6. Once you’ve won the race on the chart, I want each of you to read the book silently to yourself until everyone is finished.
Observe each group of students by walking around to be able to hear the fluency develop with the repeated readings. Look at the charts to see if they are improving their times. Allow students more time to practice reading silently.
At the end of the day just before they are about to go home I will allow the children to check out a book from the classroom with me and carry it home with them to read by themselves or with a parent. Hopefully this will be a way to get the parents involved with their child’s learning as well as motivate the child to learn. This book does not have to be on the child’s level. It can be any book that they would like to read. These books will be different than the books we use in our speed lessons. They will come from our classroom library.
Lloyd. Teaching Decoding in Holistic Classrooms.
Click here to return to Explorations
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Make Up a Cereal
What you need:
Large self-stick notes (or paper, scissors, and tape)
What to do:
1. Ask your child to point to words and numbers on the box and guess what they say.
2. Read aloud some of the words and numbers. Talk about the different kinds of information they convey. For example, words tell the name of the cereal, what's in it, and how it tastes. Numbers tell the price and how much the box weighs.
3. Make up a new cereal together. What is its name? Print the new name on a self-stick note and stick it on top of the old name. How does the new cereal taste? Stick on words like "yummy." How much does it cost? Stick on a new price. Let your child draw a picture of the new cereal to stick on, too.
4. Ask your child to "read" back the new print on the box.
Why it works:
Your child becomes aware of the differences among pictures, words, and numbers.
More on: Diagnosing Lds
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On this page, we hope to clear up any problems you might have with
coordinate geometry (circles, ellipses, midpoints, etc.). If
you're in to graphs, this is your section!
Systems of equations
Quiz on Coordinate Geometry
When dealing with lines and points, it is very important to be able to find out how long a line segment is or to find a midpoint. However, since the midpoint and distance formulas are covered in most geometry courses, you can click here to better your understanding of the midpoint and distance formulas.
Circles, when graphed on the coordinate plane, have an equation of x2 + y2 = r2 where r is the radius (standard form) when the center of the circle is the origin. When the center of the circle is (h, k) and the radius is of length r, the equation of a circle (standard form) is (x - h)2 + (y - k)2 = r2. Example:
1. Problem: Find the center and radius of (x - 2)2 + (y + 3)2 = 16. Then graph the circle. Solution: Rewrite the equation in standard form. (x - 2)2 + [y - (-3)]2 = 42 The center is (2, -3) and the radius is 4. The graph is easy to draw, especially if you use a compass. The figure below is the graph of the solution.
Ellipses, or ovals, when centered at the origin, have an equation (standard form) of (x2/a2) + (y2/b2) = 1. When the center of the ellipse is at (h, k), the equation (in standard form) is as follows:
(x - h)2 (y - k)2 -------- + -------- = 1 a2 b2Example:
1. Problem: Graph x2 + 16y2 = 16. Solution: Multiply both sides by 1/16 to put the equation in standard form. x2 y2 -- + -- = 1 16 1 a = 4 and b = 1. The vertices are at (±4, 0) and (0, ±1). (The points are on the axes because the equation tells us the center is at the origin, so the vertices have to be on the axes.) Connect the vertices to form an oval, and you are done! The figure below is the graph of the ellipse.
The equation of a hyperbola (in standard form) centered at the origin is as follows:
x2 y2 -- - -- = 1 a2 b2
1. Problem: Graph 9x2 - 16y2 = 144. Solution: First, multiply each side of the equation by 1/144 to put it in standard form. x2 y2 -- - -- = 1 16 9 We now know that a = 4 and b = 3. The vertices are at (±4, 0). (Since we know the center is at the origin, we know the vertices are on the x axis.) The easiest way to graph a hyperbola is to draw a rectangle using the vertices and b, which is on the y-axis. Draw the asymptotes through opposite corners of the rectangle. Then draw the hyperbola. The figure below is the graph of 9x2 - 16y2 = 144.
The easiest way to solve systems of equations that include circles, ellipses, or hyperbolas, is graphically. Because of the shapes (circles, ellipses, etc.), there can be more than one solution. Example:
1. Problem: Solve the following system of equations: x2 + y2 = 25 3x - 4y = 0 Solution: Graph both equations on the same coordinate plane. The points of intersection have to satisfy both equations, so be sure to check the solutions. Both intersections do check. The figure below shows the solution.
Take the Quiz on coordinate geometry. (Very useful to review or to see if you've really got this topic down.) Do it!
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This is a fun-filled, engaging activity to practice number recognition by 5s through 120.
This game includes 4 cards each of the numbers 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, and 120 plus special cards Skip, Draw 2, Go Again, and WILD.
An easy to read Directions page is included for students.
This number game is for 2-6 players. The teacher simply prints, laminates, and cuts the cards apart.
Directions for Play:
1. Students sit in a circle. 5-6 cards are distributed to each player.
2. The extra cards are placed in a stack in the middle of the group.
3. The top card in the stack is turned over.
4. The first player must match the number or the character. If he can match, he places the card on top of the first card. If he cannot match, he draws a card from the stack or plays a special card.
5. Play then moves to the second player.
6. The first player to run out of cards wins.
Options for cards:
Sort the cards by characters and use cards to practice skip counting by 5s.
Play other number games such as Top-it or Battle/War.
Common Core State Standards
1.NBT.1 Read and write numbers to 120.
2.NBT.3 Read and write numbers to 1000.
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Lesson 17 Music Theory - Notes on String
I've purposely held off on this topic, I can't even begin to tell you just how important it truely is. Music theory can get quite confusing, if you go on to learn more, take it one part at a time. What I am going to talk about here is basic theory and it's vital to learning the guitar.
Notes in Music
Whole Notes: A-B-C-D-E-F-G
In "Western" music there are simply seven whole notes, however, within these whole notes we have sharp and flat notes.
Sharps vs. Flats
The scale shown above is one of the most important things to memorize, and memorizing it is very simple. If you look closely at figure 1, you will see that the interval between B and C has no sharp or flat, and the interval between E and F has no sharp or flat - these intervals are known as natural half steps. All of the other notes have a sharp or flat.
What is a sharp note and what is a flat note?
The "distance" between two notes is a whole step, the "distance" between a whole note and sharp/flat note is a half step. Therefore, playing an A note then a B note is a whole step, playing an A note then a A# note is a half step. Simply put, a sharp note is a half step above a whole note, and a flat note is a half step below a whole note.
What is the difference between a sharp and flat note?
Believe it or not they are the same! For example, C sharp is the same note as D flat, this concept applies to all of the sharp/flat combinations. When ascending a scale you refer to the notes as sharps, when descending you refer to them as flats.
Notes on the Guitar
Remember the talk about "distance" being a whole step or half step. When you apply this concept to the
guitar, each fret represents a half step. However, keep in mind B-C and E-F are deemed as natural half steps.
Let's take a look at an example, in Part 4 of Lesson 3 guitar finger exercises we covered the notes E-F-G on the top E-String:
E String: |--0--1--3--| Note: |--E--F--G--|
What note is fret 2?
If you said F sharp or G flat then you are correct! Now lets break this down, remember in figure 1 we said there is no half step between E and F? Well look at the example above, indeed we play the open string E and the very first fret is F, in other words, there is no fret seperating the two notes. However, since F to G has a sharp/flat note fret 2 represents that F-sharp/G-flat note.
The key to this lesson is that you start to memorize where the notes lie and that you know how to figure out what notes you are playing. So let's keep moving up the string and refer to Figure 1 to tell us what note we are playing:
E String: |--5--7--8--| Note: |--?--?--?--|
Since we know fret 3 is a G note, and we know from Figure 1 that G does have a sharp, then we know that fret four is a G#/Ab note. Therefore, fret 5 is an A note!
Using that same logic, fret 6 is A#/Bb and fret 7 is a B note! Now onto fret 8, in Figure 1 we said B-C has no half step, therefore fret 8 is a C note!
Finish it Off!
E String: |--10--12--| Note: |--D---E---|You are almost there! We know fret 9 is C#/Db so fret 10 is a D note! Now were about to start all over again! On all strings the 12th fret is the same note as the open string. Therefore, fret 12 on the sixth string is an E note, so what is fret 13? If you said an F note, then you got it! It repeats in the same formation starting at the 12th fret.
Your assignment is to do this on the remaining strings. Just go through it one note at a time, say the note out loud as you play it to help memorize it. However, take your time, it's a lot to remember! Trust me on this, knowing the notes on the fifth and sixth strings will be invaluable to you as you start to memorize your barre chords.
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Frogs can hear using big round ears on the sides of their head called a tympanum. Tympanum means drum. The size and distance between the ears depends on the wavelength and frequency of a male frogs call. On some frogs, the ear is very hard to see!
Ever wonder how frogs that can get so LOUD manage not to hurt their own ears? Some frogs make so much noise that they can be heard for miles! How do they keep from blowing out their own eardrums?
Well, actually, frogs have special ears that are connected to their lungs. When they hear noises, not only does the eardrum vibrate, but the lung does too! Scientists think that this special pressure system is what keeps frogs from hurting themselves with their noisy calls!
Back to Strange but True Facts.
Back to FROGLAND.
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Smile and Say
Rationale: To become a successful reader a child must be able to recognize phonemes in spoken words as well as their corresponding graphemes in written words. Children need to know their long vowels and /E/ is one importance because of its frequency in the English language. This lesson will help children to master the /E/ sound through gestures, tongue twisters, writing practice, and independent work.
Primary paper and pencil
Poster with “Eagles eat electric eels easily.”
Poster with lower case e written on it (lines drawn like primary paper)
Marker for poster
Dry erase board/chalkboard and marker/chalk
Yellow circles cut from construction paper
Worksheet with pictures of objects that contain /E/ (green, bee, eel etc.) and objects that do not (crab, brick, elephant, etc.)
2) Ask the students: Have you ever had your picture taken? What are the two things that the photographer tells you to do? He tells you to smile and say CHEEEEEEESE!!!! Well that is our mouth movement for the day. Let’s pretend that we having our picture taken. We need to smile really big and say cheeeeeeese!!!! Lets all do it together!
3) Let’s try a tongue twister (on poster): “Eagles eat electric eels easily.” “Now let’s see if we can say it together 3 times. Good job everyone! Now say it again, but this time lets stretch out the /E/.” EEEEEagles eeeeeeeeat eeeeeeelectric eeeeeeeels eeeeeeeeaslily. “Lets do it together one more time, but as we say it lets break off the /E/.” /E/ agles /E/ at /E/ lectric /E/ els /E/ asily.
4) (Have students take out primary paper and pencil) The sound /E/ is represented by two lower case e’s. “Let’s practice writing one lower case e.To do this we get in the center of the space right below the fence, go toward the door (right), up to touch the fence, then around and up. I would like to see everyone’s e. After I put a check mark on your paper, I want you to write nine more e’s.” “Now let’s put two lower case e’s together to represent our /E/ sound. After I put a check mark on your paper, I want you to write nine ee’s.
5) Let me show you how to find the /e/ in the word leaf. I’m going to stretch out the word leaf in super slow motion and listen to hear the doctor sound. lll. llleeeaaa. There it is! I do hear our smile and say cheese noise in leaf.
6) ) Call on students to answer and ask them how they knew. Do you hear the /e/ sound in green or purple? tea or juice ? bee or grasshopper ? flower or tree? Ask children to raise their hands if they can think of a work with the /E/ sound in it. Write their responses on the board.
7) Next pass out yellow circles made from construction paper and crayons. Instruct the students to draw a “smile and say cheese face” on their circle. When the students have completed this task, give them instructions. Tell them that you are going to read a book full of words that have our “smile and say cheese” sound in them and to help show that we hear our sound; we are going to hold up our faces that we just made.
8) Introduce the book Lee and the Team. Give a brief booktalk to get the student’s engaged. Lee is the leader of his baseball team. One day the team is late for a big game. Lee needs to get his team to the game. Will he be able to get them there on time?
9) Read Lee and the Team to the students. After you finish reading, ask the students to give examples of words form the book that had the /E/ sound in them.
Pass out the worksheet that
has pictures of things that with the /E/ sound and others that do not
the /e/ sound. Instruct students to circlepictures that contain
the /E/ sound and make an X through
the pictures that do not. If time permits, children may color in
Lynch. Reading Genie Website. Stick Out Your Tongue….and say ah!
to Odysseys index
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Programmed Math/Order of operations
Order of Operations in math is dealing with the order in which you work out an equation, which can solve many problems.
2 x 3 + 5 = x
In this example, what is x? We could either multiply 2 by 3, then add 5 to it OR we could add 3 and 5 together and multiply that by two. But how do we know?
Using BODMAS we can. BODMAS stands for Brackets Over Division Multiplication Addition Subtraction. This is the order we go in.
So using BODMAS we will multiply first and then add. So it is really:
(2x3) + 5 = x x = 11
Order of operation is also inportant in boolean algebra (see: Boolean Algebra). When a problem needs to be solved, it can be turned into BODMAS form to solve it. For example:
1 AND 1 OR 0 NAND 1. How would this work? If we convert it, it is:
So we would go for multiplication first, 1x1 = 1.
so it is now:
We do multiplication again: 0x1 = 0, but because this is the inverse, we make it 1.
So it is now: 1+1 = 2 (or in this case 1, because it is OR).
And our problem gets solved.Last modified on 21 September 2007, at 22:30
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Age Range: 5 to 7
1) Prepare trays of red, blue and yellow paint.
2) The children can then choose two colours and put a hand in each.
3) They then print each hand and rub their hands together to mix the paint.
4) Next, they can print with the new colour.
This activity can be used in a number of situations:
- When the children are making a painting
- When you want to teach children about how colours can be made by mixing others. A worksheet to help with this activity can be found here.
In the top row of boxes, children should make a mark (a blob, a finger print, a line etc.) with the colour indicated at the top of each column.
In the middle row of boxes, children should mix the pairs of colours indicated at the top of each column, and make a mark in the boxes below, using the colour that is made.
The bottom row of boxes are for children to experiment with colours. They can try to make new colours and make marks in the boxes (showing which colours were used to make the new colour in the boxes at the top of the column).
This worksheet could be saved for future reference when the children are painting.
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About one-tenth of Earth’s dry land and one-eighth of its oceans are covered with ice. This ice is made of snow that collects and becomes compacted (pressed down). Most ice occurs in thick ICE SHEETS that cap the land in the polar regions. In the past, during long cold eras called Ice Ages, ice covered much more of the Earth’s surface than it does today. Scientists estimate that there have been over 15 Ice Ages in the last 2 million years.
Ninety per cent of the world’s ice is found in Antarctica. The ice cap here is 4,200 m (13,000 ft) deep in places. Over thousands of years, a thick ice sheet builds up over land when more snow falls during the winter months than melts each summer. The enormous weight of the ice pushes much of this vast, high landmass down below sea level.
Icebergs are not formed from salty sea ice, but from land ice that calves (breaks off) from ice sheets or glaciers on the coast. Only 12 per cent of the iceberg’s mass appears above the sea surface. The rest is hidden below. A fringe of sea ice also edges the Antarctic landmass, expanding in winter and melting in summer.
Glaciers are slow-moving rivers of ice that begin high on mountains. Fallen snow pressed down by new snow forms a dense ice called firn. When enough ice builds up, gravity and the glacier’s own weight set it sliding downhill at a rate of 1–2 m (3–6 1/2 ft) per day.
Moving ice is a powerful erosive force. As glaciers slip downhill they carve deep, U-shaped valleys, sharp peaks, and steep ridges. The gouging power of the ice is increased by rocks and boulders carried along at the front, sides, and beneath the glacier. When the glacier reaches the warmer lowlands, it melts.
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Sam's Home || Chameleon Home || Chameleon Graphing Java Applet || Ask Dr. Math
Here is a picture of a plane. Two lines are drawn inside the plane. Each of these lines is an axis. (Together they are called axes.) The axes are like landmarks that we can use to find different places in the plane.
We can label the axes to make them easier to tell apart. The axis that goes from side to side is the x-axis, and the axis that goes straight up and down is the y-axis.
Let's zoom in on one corner of the plane. (This corner is called the first quadrant.)
We have marked some of the points on each axis to make them easier to find. The point where the two axes cross has a special name: it is called the origin.
The gray lines will help us find points. When you make your own graphs, you can use the lines on your graph paper to help you.
Please send questions, comments, and suggestions
to Ursula Whitcher
Home || The Math Library || Quick Reference || Search || Help
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These two group activities use mathematical reasoning - one is
numerical, one geometric.
An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
I was on a train the other day and I was looking out of the
window as we went through a station and I saw lots and lots of
carriages waiting in another line. They were going to be linked
together to make a new train going on a long journey - so I thought
It was then that my train went round a curve in the track and I
looked out of the window and saw the front of my train and quickly
turned my head and saw the back of the train.
After a few more miles I had seen that my train had ten
carriages to make up the whole train.
I thought back about the carriages I had seen at the station and
wondered about making them into several trains.
I thought my train was rather special having ten carriages so I
want to put this challenge to you all.
That's all really.
Here are some to start you off.
You could draw these.
You could use cubes/ beads/ boxes or whatever to stand for the
carriages. So it might also look like:-
and if you used squares or cubes it might look like:-
So now it's your turn. See what different train arrangements you
Remember the four "Rules" above.
It would be good to think if you have been using some kind of
special way of getting new ones. Maybe you've found a pattern that
helps you get lots of answers?
Please write to us and tell us about your ways of doing
You simply have to ask "I WONDER WHAT WOULD HAPPEN IF ...?"
"I wonder what would happen if I had to only make
"I wonder what would happen if I had to make only
"I wonder what would happen if I had to have all the trains
and so on
and so on
and so on.
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A narrative is a story. When you write a narrative, you are writing about an interesting event and what it means. A narrative can be short or long. The narrative that you are about to write will be about two pages long.
ELA Standard 2.1
Before you write a narrative,
Come up with a story that you want to share. It can be a story that you made up or it can be based on another story you know. The main thing is to tell the story in a way that is all your own.
Jot down ideas! Be sure to answer the following questions:
Use a Story Map like the one below to plan your story. Be sure to include the: Title, Setting, Characters, Problem, and Action at the Beginning, Middle, and End of your story. Create extra circles if you need to.
Be sure to add details to the setting and characters.
Now that you've come up with a plan for your story, it's time to write a first draft! Use your story map to help you remember all the details you want to include. If you can, try to write the whole story all at once. Don't worry about any spelling or grammar mistakes. Since this is a first draft, you don't even need to make sure that your handwriting is neat!
Read your narrative over. You might even read it aloud to a friend or teacher. You and your friend or teacher should decide:
Rewrite your narrative so that it fully answers these questions.
You should also edit your work by checking your spelling and punctuation. Look up any words that you are unsure of in the dictionary.
Now copy out the final version of your narrative in your best handwriting or print it out on a computer. Read through your final draft twice. The first time, check that the story is exactly the way you want it. (If it isn't, go back and make more changes.) If you like your final draft, check that there are no spelling or punctuation errors. Be sure to put your name, the date and title on the first page. Now, you can share your story with your friends!
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When trying to find the angle in a right triangle, we can use the inverse trigonometric functions, or arc-trig functions.
Do not confuse this with the reciprocal functions.
An inverse sine function, (also called arcsine) runs the sine function in reverse.
If SIN θ = x, then ArcSIN x = θ
Here is an example where we are given two sides of a triangle. The hypotenuse is 100, and the opposite side is 86.6 We want to find an unknown angle θ.
Since SIN θ = o/h = 86.6/100 = 0.866 then we take the ArcSIN of 0.866 to get 60°
Here is a harder vector problem that you will see in the homework. Let us say you travel for 2000 meters east. Then you walk 1410 meters Northeast at 45°. You want to find the total displacement.
First you must break the 1410 into horizontal and vertical components. Since it is a 45° triangle, we don’t even need trigonometry. The components are each 1000 m.
Then we use the Pythagorean theorem to find the total displacement.
However, displacement has a magnitude and a direction. How do we find the angle θ? We use the ArcTAN function.
The side opposite θ is 1000, and the adjacent side is 3000. So we take the ArcTAN of (1/3) to get 18°
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Perfect square trinomials
Lesson 18, Level 2
Problem 8. Without multiplying out
a) explain why (1 − x)² = (x − 1)².
Because (1 − x) is the negative of (x − 1). And (−a)² = a² for any quantity a.
b) explain why (1 − x)³ = −(x − 1)³.
(−a)³ = −a³ for any quantity a.
The following problems show how we can go from what we know to what we do not know.
Problem 9. Use your knowledge of (a + b)² to multiply out (a + b)3.
Hint: (a + b)3 = (a + b)(a + b)².
To see the answer, pass your mouse over the colored area.
Problem 10. Multiply out (x + 2)3.
Problem 11. Multiply out (x − 1)3.
Problem 12. The square of a trinomial. Use your knowledge of
[Hint: Treat as a binomial with as the first term.]
Show that it will equal the sum of the squares of each term, plus twice the product of all combinations of the terms.
Problem 13. Can you generalize the result of the previous problem? Can you immediately write down the square of (a + b + c + d)?
Completing the square
x² + 8x + _?_ = (x + _?_)²
When the coefficient of x² is 1, as in this case, then to make the quadratic on the left a perfect square trinomial, we must add a square number. What square number must we add?
We must add the square of half of coefficient of x. The trinomial will then be the square of (x + half-that-coefficient).
x² + 8x + 16 = (x + 4)²
We add the square of half the coefficient of x -- half of 8 is 4 -- because when we multiply (x + 4)², the coefficient of x will be twice that number.
Example 7. Complete the square: x² − 7x + ? = (x − ?)²
And since the middle term of the trinomial has a minus sign, then the binomial also must have a minus sign.
Problem 15. Complete the square. The trinomial is then the square of what binomial?
a) x² + 4x + ? x² + 4x + 4 = (x + 2)²
b) x² − 2x + ? x² − 2x + 1 = (x − 1)²
c) x² + 6x + ? x² + 6x + 9 = (x + 3)²
d) x² − 10x + ? x² − 10x + 25 = (x − 5)²
e) x² + 20x + ? x² + 20x + 100 = (x + 10)²
In Lesson 37 we will see how to solve a quadratic equation by completing the square.
Here is a square whose side is a + b.
It is composed of a square whose side is a, a square whose side is b, and two rectangles ab. That is,
(a + b)² = a² + 2ab + b².
Now, here is how to complete a rectangle to make it a square
The rectangle on the left is composed of a square whose side is x, and a rectangle 8x. We will now add a square to it and make it a perfect square. How? By taking half of the rectangle and placing it on top of the square. The figure to complete the square is the square of half the side of the rectangle
Please make a donation to keep TheMathPage online.
Copyright © 2012 Lawrence Spector
Questions or comments?
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Representing Relationships as Expressions, Equations,
Without realizing it, your students simplified expressions and solved equations when they were asked to add and subtract given numbers. They probably didn't know that 5 + 3 is an expression and 7 + 2 = 9 is an equation. When working with inequalities, it is important for students to understand these terms. This will allow them to clearly understand the similarities and differences between equations and inequalities.
By third grade, students have had prior work comparing and ordering whole numbers. Some students may have already been introduced to some inequality symbols, such as > (greater than) or < (less than). It is extremely important that children fully understand the meaning of these and other inequality symbols such as (not equal to), (greater than or equal to), and (less than or equal to). Therefore, getting children to use the correct mathematical language as well as learning the appropriate symbols to use is essential.
Materials: Overhead projector or front board
Prerequisite Skills and Concepts: Students should have a working knowledge of simplifying expressions using addition and subtraction.
- Write 5 + 3 and 5 + 3 = 8 on the board or overhead. Have a student real aloud the expression 5 + 3 and the equation 5 + 3 = 8.
- Ask: What is the difference between 5 + 3 and 5 + 3 = 8?
Many students will quickly respond by stating that one has an answer and the other does not.
- Ask: Besides the numerical answer, what else is different between 5 + 3 and 5 + 3 = 8?
The goal is for students to realize that 5 + 3 = 8 has an equal sign.
- Say: 5 + 3 is called an expression. An expression consists of a number or a combination of numbers joined by operation symbols. An expression does not have an equal sign.
Erase 5 + 3. Point to 5 + 3 = 8.
- Ask: What is on the left side of the equal sign? What is one the right side?
Elicit from students that an expression is on both sides of 5 + 3 = 8. Try to have students describe the numbers as well as the operation. The left side has 5 + 3 while the right side only has 8.
- Say: 5 + 3 = 8 is called an equation. The left side of an equation equals the right side.
- Ask: Does the left side of the equation 5 + 3 = 8 equal the right side?
The goal is for students to notice that the expression on the left side of the
equation equals the expression on the right side of the equation.
- Write 3 + 4 ____ 10 on the board.
- Ask: What is the value of the expression on the left side of the blank? What is the value of the expression on the right side?
The goal is for students to realize that the expression on the left side is less than the value of the expression on the right.
- Ask: Does it make sense if I write the words "equal to" in the blank of 3 + 4 ____ 10? Why not?
We want students to make connections between the mathematical symbols and language used.
- Ask: What words would make sense in the blank?
You are trying to get students to use the words "less than" in the blank. Write < in the blank.
- Write 7 + 6 ____ 5 on the board or overhead.
- Ask: What words would make sense in the blank for 7 + 6 ____ 5?
You are trying to get the students to use the words "greater than" in the blank. Write > in the blank.
- Continue with additional examples using different inequalities. Try using two numbers on each side as well. When introducing the concept, use the words "less than" and "greater" than before using the symbols.
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Investigate this balance which is marked in halves. If you had a
weight on the left-hand 7, where could you hang two weights on the
right to make it balance?
Investigate what happens when you add house numbers along a street
in different ways.
48 is called an abundant number because it is less than the sum of
its factors (without itself). Can you find some more abundant
If the answer's 2010, what could the question be?
In this section from a calendar, put a square box around the 1st,
2nd, 8th and 9th. Add all the pairs of numbers. What do you notice
about the answers?
What happens when you add the digits of a number then multiply the
result by 2 and you keep doing this? You could try for different
numbers and different rules.
Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?
Well now, what would happen if we lost all the nines in our number
system? Have a go at writing the numbers out in this way and have a
look at the multiplications table.
Start with four numbers at the corners of a square and put the
total of two corners in the middle of that side. Keep going... Can
you estimate what the size of the last four numbers will be?
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
Which times on a digital clock have a line of symmetry? Which look
the same upside-down? You might like to try this investigation and
Find out why these matrices are magic. Can you work out how they were made? Can you make your own Magic Matrix?
Place the numbers 1 to 10 in the circles so that each number is the
difference between the two numbers just below it.
Peter, Melanie, Amil and Jack received a total of 38 chocolate
eggs. Use the information to work out how many eggs each person
Max and Mandy put their number lines together to make a graph. How
far had each of them moved along and up from 0 to get the counter
to the place marked?
How could you put eight beanbags in the hoops so that there are
four in the blue hoop, five in the red and six in the yellow? Can
you find all the ways of doing this?
Winifred Wytsh bought a box each of jelly babies, milk jelly bears,
yellow jelly bees and jelly belly beans. In how many different ways
could she make a jolly jelly feast with 32 legs?
You have 5 darts and your target score is 44. How many different
ways could you score 44?
The value of the circle changes in each of the following problems.
Can you discover its value in each problem?
Use the information to work out how many gifts there are in each
Can you design a new shape for the twenty-eight squares and arrange
the numbers in a logical way? What patterns do you notice?
Cassandra, David and Lachlan are brothers and sisters. They range
in age between 1 year and 14 years. Can you figure out their exact
ages from the clues?
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2
litres. Find a way to pour 9 litres of drink from one jug to
another until you are left with exactly 3 litres in three of the
There are 44 people coming to a dinner party. There are 15 square
tables that seat 4 people. Find a way to seat the 44 people using
all 15 tables, with no empty places.
There are 78 prisoners in a square cell block of twelve cells. The
clever prison warder arranged them so there were 25 along each wall
of the prison block. How did he do it?
Complete these two jigsaws then put one on top of the other. What
happens when you add the 'touching' numbers? What happens when you
change the position of the jigsaws?
You have two egg timers. One takes 4 minutes exactly to empty and
the other takes 7 minutes. What times in whole minutes can you
measure and how?
Annie cut this numbered cake into 3 pieces with 3 cuts so that the
numbers on each piece added to the same total. Where were the cuts
and what fraction of the whole cake was each piece?
Using the statements, can you work out how many of each type of
rabbit there are in these pens?
Cherri, Saxon, Mel and Paul are friends. They are all different
ages. Can you find out the age of each friend using the
These sixteen children are standing in four lines of four, one
behind the other. They are each holding a card with a number on it.
Can you work out the missing numbers?
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99
How many ways can you do it?
This magic square has operations written in it, to make it into a
maze. Start wherever you like, go through every cell and go out a
total of 15!
In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?
Arrange eight of the numbers between 1 and 9 in the Polo Square
below so that each side adds to the same total.
Can you arrange 5 different digits (from 0 - 9) in the cross in the
There are three buckets each of which holds a maximum of 5 litres.
Use the clues to work out how much liquid there is in each bucket.
Zumf makes spectacles for the residents of the planet Zargon, who
have either 3 eyes or 4 eyes. How many lenses will Zumf need to
make all the different orders for 9 families?
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
On the planet Vuv there are two sorts of creatures. The Zios have 3 legs and the Zepts have 7 legs. The great planetary explorer Nico counted 52 legs. How many Zios and how many Zepts were there?
This problem is based on a code using two different prime numbers
less than 10. You'll need to multiply them together and shift the
alphabet forwards by the result. Can you decipher the code?
The clockmaker's wife cut up his birthday cake to look like a clock
face. Can you work out who received each piece?
Add the sum of the squares of four numbers between 10 and 20 to the
sum of the squares of three numbers less than 6 to make the square
of another, larger, number.
Find at least one way to put in some operation signs (+ - x ÷)
to make these digits come to 100.
Find out what a Deca Tree is and then work out how many leaves
there will be after the woodcutter has cut off a trunk, a branch, a
twig and a leaf.
A group of children are using measuring cylinders but they lose the
labels. Can you help relabel them?
Tell your friends that you have a strange calculator that turns
numbers backwards. What secret number do you have to enter to make
141 414 turn around?
Can you make square numbers by adding two prime numbers together?
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
A game for 2 players. Practises subtraction or other maths
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Play In The Park
The park is a great place for children to run, jump, play
Going to the park gives your child the chance to exercise and
meet other children. Spending time with other children is very
important as they learn behavior which will help when they start
school. Your child will also be able to speak and practice language
skills. Here are some things you can do to help your child learn
while having fun:
- Sing songs and do action rhymes. If other children are
present, invite them to join and make it a game.
- Talk about the kind of day it is – is it sunny, cool, bright,
cloudy, windy? Are there clouds? What color is the sky?
Is there a rainbow?
- Ask questions about what you see – do you see birds,
dogs, squirrels, butterflies, flowers, grass?
- Repeat some of the sounds you hear – if a dog is barking,
allow your child to repeat the sound. Chirp like a bird.
You can even pretend there are other animals and make
- If you have a ball, play a game and talk about the rules.
- Have a picnic. You can plan ahead what to bring and let
your child help. Say the names of the food you brought.
- Sit on a blanket and read a book together.
- On the way home, talk about your day together, and how your child is feeling. Was it fun, boring, or just ok? Was it nice to meet and play with other children? Is your child happy, tired or hungry? Is he glad to go home and have a bath?
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need to be
aware that spoken words have phonemes which are the sounds of the
letters. This will help them in reading and writing. They
to understand the relationship between these sounds and the letters
represent them. Letter recognition is one of the two best
beginning reading success (Adams 36).
lesson students will learn to identify d = /d/ in written and
words and will gain experience with this letter through guided practice.
Primary paper and pencil
Chart with the tongue twister: “Daniel the dog digs a ditch deep in the
Bag (paper grocery store bag will be sufficient)
Picture cards of the following words: duck, dog, cat, dime, pig,
dress, candle, ball, bear, diamond, fish, sword, clock, radio, pot
for each child in the classroom
to be put
into the bag]
“Dawdle Duckling” by Toni Buzzeo - Published by: Dial
Books for Young Readers
Picture page with: bag, sand, phone, towel,
bread, dollar, calendar, strawberry, desk, and dog
Each of the words we say and write are made up of the twenty-six
in the alphabet. Each letter making its own sound. Our
in different ways to form each sound. Today we are going to be
the mouth movement for /d/. Let’s all practice moving our mouths
we say “ddddd” Very good! Sometimes the /d/ will be
hidden in words, but pay close attention and I know you will be able to
you ever heard a car or boat motor that is trying to start, but just
can’t? The motor seems to be saying “ddddduh.”
Think about what your mouth does when you make that noise. The
tip of your
tongue barely touches the roof of your mouth, right behind your top
then your mouth opens a little bit and your tongue just POPS
me say it: (model) "ddddduh". Now it’s your turn!
Let’s all try and start our motors together, “dddduh.”
let’s try a fun tongue twister!” [on chart] Read through the
tongue twister once before explaining the activity to the
“As we read through the tongue twister this time I want you to listen
the /d/ sound, every time you hear the /d/ sound I want you to pretend
your motors. Dddaniel the dddog dddigs a ddditch dddeep in the
sanddd.” Repeat reading through the tongue twisters several
times until all of the students are starting their motors at the
Also, have students break the /d/ apart from the word as they read it
once or twice. “/D/ aniel the /d/ og /d/ igs a /d/ itch
/d/eep in the san /d/.” Great job!
paper and pencil] “Now that we know what the letter d sounds
like we are going to learn and practice how to write the letter d on
paper. I want everyone to watch as I show you how to write the
Then we will all practice. Start by reviewing how to make the
which will be used to make the little d. To make our
start a little below the fence, come up and touch the fence then around
touch the sidewalk and then come up a little above the sidewalk.
little c, then little d. I want you all to
the letter d on your paper. I am going to walk around and look at
wonderful letters you are making!” Teacher will walk around the
room offering extra guidance to struggling students. After each
mastered the letter d, proceed to the next part of the lesson.
[Children sitting in
their seats at desks your tables] “Now we are going to play a
game. I have cards with different pictures on them in my
bag. I am
going to come around and each of you will draw a card. Read the card
yourself. Be careful not to tell us what your word is yet.
card has the /d/ sound in its name raise your hand, if your card does
the /d/ sound in its name put your head down on your desk. When
has drawn a card and determined if it has the /d/ sound in it we will
the room and share our cards with each other.
be altered to be a more active game: [Teacher would bring
children to the
floor in a semi-circle around the teacher] “Now we are going to play a
game. I have cards with different pictures on them. When it
turn, I want you to draw a card. Read the card silently to
yourself. Be careful not to tell us what your word is
When everyone has drawn a card, I will count to three. On the
three I want you to go to this side of the room [point to left] if your
has the /d/ sound in it and this side of the room [point to right] if
does not have the /d/ sound. We will then read them aloud as a
class.” Cards will be set out like memory where they are upside
down so that the student cannot pick a word on purpose. Once the
all get to a side of the room have them go through and read their cards
aloud. Make sure that the class agrees with each student’s
and have them move accordingly.
For a little more review, go through a simple list of a few pairs of
the students to determine which word the /d/ sound is found in.
“Now I am going to give you a few pairs of words and I want you to tell
me which of the two words you hear the /d/ sound in. For example,
hear the /d/ sound in dog or cat? ddd-ooo- ggg. dog. ccc-aaa-ttt.
cat. [Wait for student response] Very good! The /d/ sound
dog! Now here are your words, listen carefully:
6. Read “Dawdle
Duckling” by Toni Buzzeo. Have students raise their hands as
they hear the /d/ sound while you read the story aloud.
Give students the picture page handout with various pictures. The
pictures will be of words that do have the d = /d/
well as, pictures that do not. Have students write the letter d
each picture that has the /d/ sound in it. Writing the letter
the teacher to check for mastery of the written letter, and writing the
under only pictures that have the /d/ in them will allow the teacher to
for mastery of the d = /d/ correspondence.
to Read: Thinking and Learning About Print. 1990.
It’s D! http://www.auburn.edu/rdggenie/discov/adamsel.html.
Decoding; Why and How. Upper
Saddle River, NJ.
(2005). pg. 60-82.
here to return to Constructions.
Click here to return to Reading
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Throughout mathematics, we find function notation. Function notation is a way to write functions that is easy to read and understand. Functions have dependent and independent variables, and when we use function notation the independent variable is commonly x, and the dependent variable is F(x). In order to write a relation or equation using function notation, we first determine whether the relation is a function.
One thing you'll find in your math classes is that there's lots of different ways to write the same thing. And although sometimes it can be confusing, it's actually pretty cool, because it helps you keep track of what variable or what letter is affecting which one.
Like, for example, most of the time X is the independent variable. And Y is the dependent variable. What that means is that your Y value depends on whatever X was. X can usually change on its own. And whatever X does affects what Y is. That's why we call Y the dependent variable.
So one way this is written is it's kind of different from what you've seen before. What you've seen before is probably something like this. Y equals 3X plus 2. A new way to write that is using what's called function notation. Instead of Y, I'm going to say F of X. What that means is the function that's using X is equal to 3X plus 2. It's the same thing written in two different ways.
I think it's kind of neat when we get into doing some more problems with numbers is that I can stick a number in there for X and I can show what number that is here as well. You can kind of tell what X number you're plugging in, the independent variable, to get your Y value output variable.
One last thing to keep in mind when you're working with functions and function notation is the ideas of domain and range. Domain is the set of all X values or the independent variables. Range is the set of Y values or the dependent variables. Which means that your domain changes on its own. It's independent. Your range is dependent on that domain stuff.
It's a lot to keep track in your head. As you can tell my head is kind of struggling with it, too. But you can see as we get going with some problems you can keep it all straight and do a good job with notation function.
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Sine and cosine are periodic functions, which means that sine and cosine graphs repeat themselves in patterns. You can graph sine and cosine functions by understanding their period and amplitude. Sine and cosine graphs are related to the graph of the tangent function, though the graphs look very different.
I want to talk about graphing the sine and cosine functions. But first, I need to go over a property that the sine and cosine functions have and that these three functions have. So I have the question, what do these functions have in common? I have three, very differently shaped functions, but they all have something in common: They repeat themselves periodically. This triangular shape repeats itself in the graph of y=t(x). This alternating interval pattern repeats itself in y=r(x). And this wave pattern repeats in y=s(x). How would we describe this property mathematically?
Well the property is called periodicity and these functions are called periodic functions. The definition's a little bit tricky, but let's see if we can walk through it and understand what it means. It says, if there is a number P, such that f(x)+P=f(x) for all of x in the domain of f, then f is a periodic function. What does this mean, f(x+P)=f(x)? It means that if I find the right value of P, I can always add that value to the x, to the input and get the same output.
Let's take a look at the functions. Suppose I start with an x value of 2. What could I add to 2 and get exactly the same output that I have here, which is 0? I could add two and that will give me 0. If I add two to 2, I get x equals 0, which has an output of 0. And if I add two to 0, I get 2, and that has an output of 0. If I add two again, I get 4, and that has an output of 0.
So have I found the P value that I need? The answer is no, because that P value won't work for all inputs. Let me give you an example, 3. If I add two to 3, I get 1. And the output of 1 is 1, it's not 1, so I have different outputs. Again, if I start at 1 and I add 2, I get to 1. The output at one is +1, and the output of 1 is 1, different outputs.
So I have to find another P value, one that works for all x's. And it turns out that the value is this difference, 4. I can get from one maximum to another by adding 4: 3+4=1. So four is the number that's going to work. And I would say that t(x)+4=t(x).
Let's take a look at y=r(x). Now here, if I started a nice number like 2 and I add 2, I do get the same output. Here the output is 0, and here the output's 0. And if I add two again, the output is still 0. But now, I'm a little skeptical; I want to try this out for other inputs. So let me try it out for something like 1.5. And for 1.5, the output is 1. If I add two to that I get 0.5, and there the output is 1. And if I add two to that I get 2.5, and there the output is 1.
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This is a lesson for a Weather ESL Quiz that will cover the water cycle, and include some storms that produce precipitation.
The water cycle is a continuous process all over the world and is made of three main parts:
Some forms of precipitation are:
- Rain - Big drops of water
- Hail - Balls of frozen water
- Snow - Flakes of frozen water
- Sleet - Partially melted snow
Part of the Weather ESL Quiz will cover severe storms;
Storms form when a warm front and a cold front meet. A cold front is the edge of a mass of cold air, and a warm front is the edge of a mass of warm air. Depending on the season and temperatures, the precipitation that falls may be rain, snow, sleet, or hail. Sometimes, you can see several of these during the same storm, especially in extreme storms.
These are the terms that are key to understanding weather and that will typically be covered in a Weather ESL Quiz.
MULTIPLE CHOICE QUESTIONS:
1. In what part of the water cycle does water vapor change into a liquid?
2. What kind of a storm has a funnel shape?
3. What kind of storm has a lot of snow?
B. cold front
4. When water falls from a cloud to the ground, it is called:
C. warm front
5. What is a ball of frozen water?
TRUE or FALSE QUESTIONS:
6. A cold front is the edge of a mass of cold air. _____
7. Evaporation is a liquid changing into a gas. _____
8. Sleet is big drops of water. _____
9. Hurricanes are formed over cold areas of snow. _____
10. Storms occur when a warm front meets a cold front. _____
Here are the answers for the Weather ESL Quiz.
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Teaching Fraction Addition
A response to the question:
Might you have any ideas as to what I can present fraction addition using some form of technology or physical manipulation?
My goal is to demonstrate my ingenuity in the use of "hands on" objects to teach the concepts of adding numerical fractions.-------------
Have you ever had students make fractions strips, and then use them to compare fractional amounts? In case you haven't here is a quick set of directions. (You could also buy these materials, but if a child makes them, that is part of the learning.)
Cut a set of strips that are all the same size. You can do that by using lined paper and cutting them so that the lines are running vertically through the strips ( up and down). If you make each strip 24 lines long, that will make some of the folding easier, too.
Now, label the first strip "one whole". It will be worth one. Take another strip, fold it into two equal pieces, and label each one "one half" (you can also use the fraction 1/2, to help your students link the word and the symbol). Take another strip, and fold it into four equal pieces (to be labeled fourths). There is more than one way to do these folds. To make it easier to compare them later, making the folds vertically, so the segments are all in line, is best. It is a good idea to let your students see how many different ways they can make these folds to create "equal" pieces, though.
Continue to make all the fraction pieces from halves to twelfths. Some will be more difficult to make than others. Working through the activity will help your students see the relationships between the fractions though. Resist the temptation to do it for them so the materials look neat...
When you are done, you can use these strips to compare fractions to see how many of one strip make the same sized piece as another strip. You can also compare to see which pieces are larger, or smaller than other pieces, for example, 12 is smaller than 5/8 and 8/12.
Finally, you can use them to add and subtract fractions. Suppose you want to add three fourths and one half. You find the one half strip, and the fourths strip. Lay them side by side to notice that one half is the same as two fourths. Now lay them one after the other, like a long train. You have the half, which is the same as two fourths, and the three fourths. That is five fourths all together. It is longer that the one whole strip, so it must be worth more than one. you can compare to find out how much longer than one it is.
If the fraction were 4 sixths and 3 fourths, you could still add them. Put them like a train, and note the length. Find a strip that compares with both the sixths and the fourths ( twelfths work) Change the names of the two fractions into twelfth, by comparing. Then figure out by substituting how many twelfths you have...
-Gail, for the T2T service
Join a discussion of this topic in T2T.
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EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.
Find a great variety of ways of asking questions which make 8.
If you had any number of ordinary dice, what are the possible ways
of making their totals 6? What would the product of the dice be
Suppose that there were 100 people and 100 rats. Supposing that all the people and rats have the usual number of legs, there will be 600 legs in the town belonging to People and Rats.
But now, what if you were only told that there were 600 legs belonging to people and rats but you did not know how many people/rats there were?
The first part of this month's challenge is to investigate how many people/rats there could be if the number of legs was 600. To start you off, it is not too hard to see that you could have 150 people and 75 rats; you could have had 250 people and 25 rats. See what other numbers you can come up with. Remember that you have to have 600 legs altogether and rats will have 4 legs and people will
have 2 legs.
I just chose 600 because that lets you have 100 people and 100 rats. You could now extend this idea by having 120 people and 120 rats (just as an example) using 720 legs and see what numbers you can come up with. Again, to start you off, you could have 270 people and 45 rats; or 320 people with 20 rats. Now you carry on.
You could choose any number you like for the total number of legs; try some out of your own. I had a hidden rule that whatever number I chose it meant that we could have the same number of people and rats. (600 legs 100 people and 100 rats; 720 legs 120 people and 120 rats.)
So you could choose the number of legs for yourself using that same rule. (Allowing you to have the same number of people and rats.)
It's probably time to have a look at all the results that you have got and see what things you notice. You can then tell me about them.
When you look at the examples that I did for you, you might notice:-
In the next example:-
This seems as if it could be worth looking at more deeply. I guess there are other things which will "pop up'' to explore.
Then there is the chance to put the usual question "I wonder what would happen if ...?''
Good luck, please send solutions in and any further ideas which came from asking "I wonder what would happen if ...?''
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Smooth, Speedy Reading
Rationale: In order for children to become expert readers and to enjoy reading, they must become fluent readers. To increase fluency in reading, students' focus should be on reading faster, smoother, and with more feeling rather than on accuracy. Children should learn to recognize words effortlessly and also be able to decode instantly. This allows children to comprehend easier and enjoy their reading. This lesson will develop fluency through reading and rereading.
Class set of Book "Julius" by Syd Hoff with marks after every ten words;
2 Sentence Strips per group:
“The tall boy went to the park.
The birds sang in the trees.”
Stopwatch for every group
Pencils to mark errors
“What is fluency? "Fluency is the ability to read smoother, faster, and with more expression" I will give the children an example of what it sounds like when someone is not a fluent reader. I will read the sentence "The tall boy went to the park." I will read if very slow and choppy. I will ask the children "Do you think this was an example of fluent reading?" Great!
“Now, how do we figure out a word that we have trouble reading? Right, we use cover-ups. Which part of the word do we look at first? Yes, the vowel. Then what do we add? That’s right, the beginning sound.”
“Okay, now I'm going to figure out this word as an example of our vowel-first cover-ups” (write the word park on the chalkboard). “First, I'm going to cover everything other than the vowel up. Okay, this vowel says /a/. Now I'm going to look at the beginning p. P says /p/. So far I have /p/ /a/, and now we’re adding r. So I have, /p/ /a/ /r/, /par/. Now the end, it says k. K says /k/. So /par/ /k/, park. So don't forget to use the vowel-first cover-up method when you need help figuring out a word.
“It's important to become faster at our reading because the faster we can read the more we can understand what we're reading and we'll like reading more. Now we're going to work on reading faster. Have you ever noticed that the first time you read something it sounds broken up and slow, a little bit like a robot? Demonstrate reading the sentence, " The tall boy went to the park," very slowly and haltingly. The- tall- boy- went- to – the – park. Then say, if you read that sentence again it sounds better. Demonstrate reading the same sentence a bit faster. The tall boy went to the park. Say, the more you practice, the faster you get, and you can add feeling and different voices. Read the sentence one more time, this time with more expression. The birds sang in the trees. I will also explain that it's easier every time because you learn the words in the sentence and become familiar with them.
"Great! Now everyone get the books. I want one each person to read the first two pages of "Julius" by Syd Hoff, to their partner. "When you finish with the two pages, give it to your partner to read aloud to you.”
Have each student take out paper and pencil to record how many words per minute he/she reads during each timed read. Have each student do four one-minute reads.
Now, we’re going to get with a partner. (Pass out a stop-watch to each pair of students.) “Here are some stopwatches. I want you to take turns reading the pages. While your partner reads you will time them. Here is how you will do it. As soon as your partner begins to read push the start button. When you partner reads the last word they will say finished. When you hear them say finished press the stop button and write down their time." "Before we begin are there any questions?" "Ok! Great! You can begin!"
Walk around the room as students begin, making sure everyone understands the process. Continue monitoring students as they engage in the fluency activity.
Assessment: "Great! I am so proud of you! Now, I will call you up one at a time to read me your pages. If you are not up here with me you should be practicing to be a more fluent reader. I want you to continue reading "Julius" at your seat. Later, we will be timing ourselves reading the entire book and use our time sheets to mark our progress, so you want to be practiced!
Encourage students to choose a book from the classroom library to read several times at home, and then show their family how well they can read. Mention that the next day you'll be eager to have a few volunteers read their book to the class.
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Direct and Indirect Speech Exercise II
Turn the following sentences into indirect speech.
1. ‘What do you want?’ she asked him.
2. ‘Are you coming with us?’ he asked me.
3. He asked, ‘When do you intend to make the payment?’
4. ‘Do you come from China?’ said the prince to the girl.
5. The poor man exclaimed, ‘Will none of you help me?’
6. ‘Which way should I go?’ asked the little girl.
7. Alladin said to the magician, ‘What have I done to deserve so severe a punishment?’
8. ‘Don’t you know the way home?’ I said to her.
9. ‘Do you write a good hand?’ the teacher said to the student.
10. ‘Have you anything to say on behalf of the accused?’ said the judge finally.
11. ‘Have you anything to tell me, little bird?’ asked Ulysses.
12. ‘Who are you, sir, and what do you want?’ they asked.
13. The king was impressed with the magician and asked, ‘What can I do for you?’
14. She asked, ‘What is it that makes you stronger and braver than other men?’
15. ‘Can you solve this problem?’ he asked me.
1. She asked him what he wanted.
2. He asked me if I was coming/going with them.
3. He enquired when I/he/she intended to make the payment.
4. The prince asked the girl if she came from China.
5. The poor man exclaimed whether none of them would help him.
6. The little girl asked which way she should go.
7. Alladin asked the magician what he had done to deserve so severe a punishment.
8. I asked her whether she did not know the way home.
9. The teacher asked the student if he/she wrote a good hand.
10. The judge finally asked whether he/she had anything to say on behalf of the accused.
11. Ulysses asked the little bird whether it had anything to tell him.
12. They asked who he was and what he wanted.
13. The king was impressed with the magician and asked what he could do for him.
14. She asked him what was it that made him stronger and braver than other men.
15. He asked me if I could solve that problem.
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Fun With Grammar
- Grades: PreK–K, 1–2, 3–5
Get students excited about grammar with these fun games and activities to reinforce the basics.
With this silly sentences activity, we practiced what we have learned about nouns and verbs. We gave each student two cards. On one card they wrote a noun and on the other card, they wrote a verb. After they had written the noun and the verb, we collected all of the cards. We placed all of the nouns together in a basket and all of the verbs together in a basket. Each child then drew one card from each basket and wrote and illustrated a silly sentence using the words on the cards. Then, students shared their silly sentences with their classmates.
Way Cool Wallets
These wallets came in handy when we needed to review plural nouns with our classes. To make the wallet, we folded an 8.5" X 11" sheet of paper in half lengthwise; then we creased the folded paper into thirds. Next, we stapled the paper along each crease and outside edge to create the wallet. We guided students to label the compartments “-s,” “-es,” and “-ies.” We then gave each student nine paper cards and a list of singular nouns to write on the cards. They had to copy the singular form of the noun on the front of the card and then write the plural form on the back. After they finished writing the plural nouns, they then placed them in the appropriate compartments in their wallet.
Let’s Have a Lotto Fun
We used this game to review parts of speech. We gave each student a blank lotto board and had them fill in the squares with four nouns, four verbs, four adjectives, and four pronouns. We prepared the same number of cards with the words "noun," "verb," "pronoun," and "adjective" and placed them in a container.
To play the game, we drew a part of speech card from the container and read it aloud. Students used paper markers to cover the correct squares. The first student to cover four words in any direction called out “Lotto!” In order to win the game, students had to read aloud each word and state its part of speech for verification.
Make a Mobile
We had our students make banana-split mobiles to review the parts of speech. We traced out the shapes for each child. Students were given a white, brown, and pink scoop of ice cream along with a banana and a bowl. After students cut out their shapes, they wrote an adjective on the white scoop, a noun on the brown scoop, and a verb on the pink scoop. Finally, students used all three parts of speech to write a sentence on the banana cutout. These mobiles added some fun decoration to our classrooms.
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Mmm Mmm Good!
Rationale: Children need to learn to recognize the sound /m/ in words and learn to match the phoneme /m/ with the letter m. The consonant m is one of the easier phonemes to recognize.
Materials: Primary paper and pencil; chart with "Molly made a mess with her milk"; one rubber band for the teacher; class set of cards containing pictures of objects beginning with the letter m; sheet of paper for each child with a large letter M printed on it; pretend paper money; glue; assessment worksheet with ten boxes on it.
1. Start by introducing the letter m. Tell the children that we will be learning the letter m and the mouth move that represents the letter m so that they will be able to recognize m in words.
2. Ask students: Did you ever have some kind of food that tasted really good and say "Mmm?" That is the mouth move we are going to look for in words. When you say /m/, you press your lips together. I will show you how to spot /m/ in words. I will stretch out the word to see if I press my lips together just like when my ice cream is "Mmm, Mmm, good." Stretch out the word map to see if you hear the /m/. I'll try. Mmm-aa-pp. Yes, you are right. The /m/ is at the beginning of the word.
3. Let's try a tongue twister. Molly made a mess with her milk. Now, let's say it together. Good. Now watch me stretch out the /m/ sound. (Teacher uses a rubberband when stretching out the /m/.) Mmmmolly-mmmade-a-mmmess… Now, you pretend like you have a rubber band and stretch out the M sound. (Children will move their hands pretending to stretch a rubber band.) Try it again and this time let's break the /m/ off of each word. /M/olly /m/ade a /m/ess with her /m/ilk.
4. Now, let's try writing the letter M. (Students take out primary paper and pencil.) We use the letter M to spell /m/. Start at the ground. Climb the mountain all the way to the sky. Slide down the mountain all the way back down to the ground. Do not pick up your pencil. Now there's another mountain. Climb up the mountain to the sky and back down to the ground again. After I check your paper you can make a whole line of M's.
5. Have the children tell whether they hear /m/ in the words that the picture cards represent. Have several pictures of obvious objects. Some should start with the letter m and others should not so that the children can try to recognize the /m/ sound. Tape the pictures on the chalkboard. Allow the children to take turns coming up to the board. If the object starts with the /m/ sound they can circle it with the chalk. If it does not, they can take the picture off of the board and place it aside. When all of the answers are correct only the objects starting with the letter m will be remaining on the board. You can then go over the words again, stressing the /m/ sound in the words. Some pictures you can use are: monkey, mouse, map, mother, mirror, macaroni, milk, bird, dog, apple, car, nail, nose.
6. To help the children remember the /m/ letter-sound connection, give each child a sheet of paper that has a large letter m written on it. Give each child some play money (coins or dollars) and allow them to glue the money onto the letter m. This will remind them that the letter m makes the /m/ sound like in /m/oney.
7. Sing a song to the tune of "Old McDonald Had A Farm" but instead
of saying e-i-e-i-o use the /m/ phoneme. For example,
Old McDonald had a monkey, me-my-me-my-mo
Old McDonald had a mouse, me-my-me-my-mo.
8. Read the book Miss Spider's ABC's by David Kirk and have the children clap one time when they hear the /m/ sound.
Use a worksheet that has ten blank boxes on it. Show the children the pictures from the earlier picture card game one at a time. Make sure the children know what word the picture is representing. If the word has the /m/ sound the children put a check in the box. If the word does not have the /m/ sound they put an X in the box.
Eldredge, Lloyd L. Teaching Decoding in Holistic Classrooms. (ch. 5) Merrill; Upper Saddle River, New Jersey, 1995.
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Our goal is to make a trebuchet capable of throwing a ball as big as a inch and a half across 50 feet. We will only be able to use wood or fiber board but not any type of metal. Each piece will be milled out and assemble with glue or any other type of adhesive. Each team member will have an assignment to turn in at the end of each week. We picked a trebuchet because it is a simple and efficient machine that we can make in little amount of time. The project will last 5 weeks
For our project we need to write a research paper about a trebuchet that we are building in class, a trebuchet is a war machine used to throw large objects at castles walls. By dropping heavy weight on the lever or the sling, the lever will rise with such spedd pulling the sling which the object was in heavy object .
The trebuchet was the dominant siege weapon in Europe from 850AD to 1350AD, larger versions were able to throw large stones, cows or even people who were shunned for treason. The trebuchet works by gathering the potential energy of the weight, and demonstrates There are multiple parts you have to consider in the design which can be improved or changed to change the range and throw-weight.
The trebuchet reached Europe during the in 500 A.D and was used by the French. During this time , the plan of the trebuchet was revised so instead of having men pulling it down they had levers to make it easier and quicker to pull the arm down.
The Trebuchet is an accurate machine and was widely used. The Medieval Trebuchet consisted of a lever and a sling. A force was applied to the shorter end of the arm , the weight is on the other longer end of the arm. The trebuchet arm could measure up to 60ft in length. Heavy weights or a box filled with sand or stones were fixed to the short end of the Medieval trebuchet arm. A heavy stone, or other large objects, was placed in a pouch that was attached by two ropes to the other, long end of the arm.
When the arm was released, the force created by the weight propelled the long end upward and caused the object to be thrown in the air towards the selected target .The Trebuchet was capable of hurling heavy stones weighing about 200 pounds with a distance of up to about 300 yards after maximum distance was achieved.
Step 1: Materials
• 19/32 in. x 4 ft. x 8 ft. Rtd Sheathing Syp (plywood) - $21.97 Home Depot
• Elmer's Carpenter's Max 16 fl. oz. Wood Glue - $6.78 Home Depot
• Crystal Sand 50 lb. Pool Filter Sand - $5.48 Home Depot
• Charlotte Pipe 3/4 in. x 2 ft. PVC - $1.24 Home Depot
• Charlotte Pipe 1 in. x 2 ft. PVC - $1.67 Home Depot
• Painter's Touch 12 oz. Flat Black General Purpose Spray - $3.87 Home Depot
• Painter's Touch 12 oz. Gloss Apple Red General Purpose Spray Paint - $3.87 Home Depot
• Red Heart Super Saver Yarn - $2.99 Jo-Ann Fabric and Craft Stores
• Great Value Heavy Duty Foil 12"x50' - $2. 52 Walmart
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There are three tables in a room with blocks of chocolate on each.
Where would be the best place for each child in the class to sit if
they came in one at a time?
The discs for this game are kept in a flat square box with a square
hole for each disc. Use the information to find out how many discs
of each colour there are in the box.
Use the lines on this figure to show how the square can be divided
into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Here is a picnic that Chris and Michael are going to share equally.
Can you tell us what each of them will have?
Watch the video to see how to fold a square of paper to create a
flower. What fraction of the piece of paper is the small triangle?
Can you find different ways of showing the same fraction? Try this matching game and see.
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Newton's Second Law
In the presence of a NET FORCE, an object experiences an ACCELERATION
directly proportional to the NET FORCE
inversely proportional to the MASS of the object.
-- F is the NET force
-- m is the mass which that net force acts on.
We often turn this around and write it as
F is the NET force acting on an object
m is the mass of the object which the force F acts upon.
What are the UNITS of force in
F = m a
A force of ONE unit
will give an object of 1.0 kg mass
an acceleration of 1.0 m/s/s ;
this force is known as
ONE NEWTON (1.0 N) .
1 N = ( 1 kg ) ( 1 m/s/s )
F = m a
Force will be measured in newtons
A force of 1 N will give
a mass of 1 kg
an accelertion of 1 m/s/s.
12 N = ( 3 kg ) ( 4 m/s/s ) A force of 12 N could give
a mass of 3 kg
an accelertion of 4 m/s/s
12 N = ( 2 kg ) ( 6 m/s/s ) A force of 12 N could give
a mass of 2 kg
an accelertion of 6 m/s/s .
We have already seen that all objects fall with the same acceleration, 9.8 m/s/s (which we approximate as nearly 10 m/s/s).
We call this free fall.
When such objects fall, the only force acting on them is their weight, the force of gravity.
The only force on a body in freefall is the force of gravity. We call this its weight.
Since it accelerates at 9.8 m/s/s,
that weight must be
w = (mass ) x (9.8 m/s/s),
w = m g
The weight of an object is the force of gravity on that object.
Weight, since it is a force, will be measured in units of newtons (N).
Mass will be measured in kilograms (kg).
If an object is in equilibrium -- at rest -- then, the net force on the object must be zero.
That is, the sum of all the forces on an object is zero when the object is in equilibrium.
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Parts of Speech
Verbs: All the Right Moves
Verbs are words that name an action or describe a state of being. Verbs are seriously important, because there's no way to have a sentence without them.
While we're on the topic, every sentence must have two parts: a subject and a predicate.
- A subject tells who or what the sentence is about. The subject is a noun or a pronoun.
- A predicate tells what the subject is or does. The verb is found in the predicate.
There are four basic types of verbs: action verbs, linking verbs, helping verbs, verb phrases.
You Could Look It Up
Verbs are words that name an action or describe a state of being.
The action of an action verb can be a visible action (such as gamble, walk, kvetch) or a mental action (such as think, learn, cogitate).
Quoth the Maven
To determine if a verb is transitive, ask yourself, “Who?” or “What?” after the verb. If you can find an answer in the sentence, the verb is transitive.
Action Verbs: Jumping Jack Flash
Action verbs tell what the subject does. For example: jump, kiss, laugh.
- The mobsters broke Irving's kneecaps.
- Some people worry about the smallest things.
An action verb can be transitive or intransitive. Transitive verbs need a direct object.
- The boss dropped the ball.
- The workers picked it up.
Intransitive verbs do not need a direct object.
- Who called?
- Icicles dripped from his voice.
Chain Gang: Linking Verbs
Linking verbs join the subject and the predicate. Linking verbs do not show action. Instead, they help the words at the end of the sentence name and describe the subject. Here are the most common linking verbs: be, feel, grow, seem, smell, remain, appear, sound, stay, look, taste, turn, become.
Although small in size as well as number, linking verbs are used a great deal. Here are two typical examples:
Quoth the Maven
To determine whether a verb is being used as a linking verb or an action verb, use am, are, or is for the verb. If the sentence makes sense with the substitution, the original verb is a linking verb.
- The manager was happy about the job change.
- He is a fool.
Many linking verbs can also be used as action verbs. For example:
- Linking: The kids looked sad.
- Action: I looked for the dog in the pouring rain.
Mother's Little Helper: Helping Verbs
Helping verbs are added to another verb to make the meaning clearer. Helping verbs include any form of to be. Here are some examples: do, does, did, have, has, had, shall, should, will, would, can, could, may, might, must.
Verb phrases are made of one main verb and one or more helping verbs.
- They will run before dawn.
- They do have a serious problem.
Identify each of the verbs in the following sentences. Remember to look for action verbs, linking verbs, and helping verbs.
- A group of chess enthusiasts had checked into a hotel.
- They were standing in the lobby discussing their recent tournament victories.
- After about an hour, the manager came out of the office and asked them to disperse.
- “But why?” they asked as they moved off.
- “Because,” he said, “I can't stand chess nuts boasting in an open foyer.”
- had checked
- were standing, discussing
- came, asked, disperse
- asked, moved
- said, can't stand, boasting
One more time, with gusto! Underline the verbs in each of these sentences.
- I can please only one person per day. Today is not your day.
- I love deadlines. I especially like the whooshing sound they make as they fly by.
- Tell me what you need, and I'll tell you how to get along without it.
- Accept that some days you are the pigeon and some days the statue.
- Everybody is somebody else's weirdo.
- I don't have an attitude problem; you have a perception problem.
- Last night I lay in bed looking up at the stars in the sky, and I thought to myself, where the heck is the ceiling?
- My reality check bounced.
- On the keyboard of life, always keep one finger on the escape key.
- Never argue with an idiot. They drag you down to their level, then beat you with experience.
- can please, is
- love, like, make, fly
- tell, tell, get
- accept, are
- don't have, have
- lay, looking, thought, is
- argue, drag, beat
Excerpted from The Complete Idiot's Guide to Grammar and Style © 2003 by Laurie E. Rozakis, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
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Program Arcade GamesWith Python And Pygame
Now that you can create loops, it is time to move on to learning how to create graphics. This chapter covers:
- How the computer handles x, y coordinates. It isn't like the coordinate system you learned in math class.
- How to specify colors. With millions of colors to choose from, telling the computer what color to use isn't as easy as just saying “red.”
- How to open a blank window for drawing. Every artist needs a canvas.
- How to draw lines, rectangles, ellipses, and arcs.
The Cartesian coordinate system, shown in Figure 5.1 (Wikimedia Commons), is the system most people are used to when plotting graphics. This is the system taught in school. The computer uses a similar, but somewhat different, coordinate system. Understanding why it is different requires a quick bit of computer history.
During the early '80s, most computer systems were text-based and did not support graphics. Figure 5.2 (Wikimedia Commons) shows an early spreadsheet program run on an Apple ][ computer that was popular in the '80s. When positioning text on the screen, programmers started at the top calling it line 1. The screen continued down for 24 lines and across for 40 characters.
Even with plain text, it was possible to make rudimentary graphics by just using characters on the keyboard. See this kitten shown in Figure 5.3 and look carefully at how it is drawn. When making this art, characters were still positioned starting with line 1 at the top.
Later the character set was expanded to include boxes and other primitive drawing shapes. Characters could be drawn in different colors. As shown in Figure 5.4 the graphics got more advanced. Search the web for “ASCII art” and many more examples can be found.
Once computers moved to being able to control individual pixels for graphics, the text-based coordinate system stuck.
The $x$ coordinates work the same as the Cartesian coordinates system. But the $y$ coordinates are reversed. Rather than the zero $y$ coordinate at the bottom of the graph like in Cartesian graphics, the zero $y$ coordinate is at the top of the screen with the computer. As the $y$ values go up, the computer coordinate position moved down the screen, just like lines of text rather than standard Cartesian graphics. See Figure 5.5.
Also, note the screen covers the lower right quadrant, where the Cartesian coordinate system usually focuses on the upper right quadrant. It is possible to draw items at negative coordinates, but they will be drawn off-screen. This can be useful when part of a shape is off screen. The computer figures out what is off-screen and the programmer does not need to worry too much about it.
To make graphics easier to work with, we'll use the Pygame. Pygame is a library of code other people have written, and makes it simple to:
- Draw graphic shapes
- Display bitmapped images
- Interact with keyboard, mouse, and gamepad
- Play sound
- Detect when objects collide
The first code a Pygame program needs to do is load and initialize the Pygame library. Every program that uses Pygame should start with these lines:
# Import a library of functions called 'pygame' import pygame # Initialize the game engine pygame.init()
If you haven't installed Pygame yet, directions for installing Pygame are available in the before you begin section. If Pygame is not installed on your computer, you will get an error when trying to run import pygame.
Important: The import pygame looks for a library file named pygame. If a programmer creates a new program named pygame.py, the computer will import that file instead! This will prevent any pygame programs from working until that pygame.py file is deleted.
Next, we need to add variables that define our program's colors. Colors are defined in a list of three colors: red, green, and blue. Have you ever heard of an RGB monitor? This is where the term comes. Red-Green-Blue. With older monitors, you could sit really close to the monitor and make out the individual RGB colors. At least before your mom told you not to sit so close to the TV. This is hard to do with today's high resolution monitors.
Each element of the RGB triad is a number ranging from 0 to 255. Zero means there is none of the color, and 255 tells the monitor to display as much of the color as possible. The colors combine in an additive way, so if all three colors are specified, the color on the monitor appears white. (This is different than how ink and paint work.)
Lists in Python are surrounded by either square brackets or parentheses. (Chapter 7 covers lists in detail and the difference between the two types.) Individual numbers in the list are separated by commas. Below is an example that creates variables and sets them equal to lists of three numbers. These lists will be used later to specify colors.
# Define some colors black = ( 0, 0, 0) white = ( 255, 255, 255) green = ( 0, 255, 0) red = ( 255, 0, 0)
Using the interactive shell in IDLE, try defining these variables and printing them
If the five colors above aren't the colors you are looking for, you can define your
own. To pick a color, find an on-line “color picker” like the one shown in
Figure 5.6. One such color picker is at:
Extra: Some color pickers specify colors in hexadecimal. You can enter hexadecimal numbers if you start them with 0x. For example:
white = (0xFF, 0xFF, 0xFF)
Eventually the program will need to use the value of $\pi$ when drawing arcs, so this is a good time in our program to define a variable that contains the value of $\pi$. (It is also possible to import this from the math library as math.pi.)
pi = 3.141592653
So far, the programs we have created only printed text out to the screen. Those programs did not open any windows like most modern programs do. The code to open a window is not complex. Below is the required code, which creates a window sized to a width of 700 pixels, and a height of 500:
# Set the width and height of the screen size = (700,500) screen = pygame.display.set_mode(size)
Why set_mode? Why not open_window? The reason is that this command can actually do a lot more than open a window. It can also create games that run in a full-screen mode. This removes the start menu, title bars, and gives the game control of everything on the screen. Because this mode is slightly more complex to use, and most people prefer windowed games anyway, we'll skip a detailed discussion on full-screen games. But if you want to find out more about full-screen games, check out the documentation on pygame's display command.
Also, why size=(700,500) and not size=700,500? The same reason why we put parentheses around the color definitions. Python can't normally store two numbers (a height and width) into one variable. The only way it can is if the numbers are stored as a list. Lists need either parentheses or square brackets. (Technically, parenthesis surrounding a set of numbers is more accurately called a tuple or an immutable list. Lists surrounded by square brackets are just called lists. An experienced Python developer would cringe at calling a list of numbers surrounded by parentheses a list rather than a tuple.) Lists are covered in detail in Chapter 7.
To set the title of the window (which shown in the title bar) use the following line of code:
pygame.display.set_caption("Professor Craven's Cool Game")
With just the code written so far, the program would create a window and immediately hang. The user can't interact with the window, even to close it. All of this needs to be programmed. Code needs to be added so that the program waits in a loop until the user clicks “exit.”
This is the most complex part of the program, and a complete understanding of it isn't needed yet. But it is necessary to have an idea of what it does, so spend some time studying it and asking questions.
#Loop until the user clicks the close button. done = False # Used to manage how fast the screen updates clock = pygame.time.Clock() # -------- Main Program Loop ----------- while done == False: # ALL EVENT PROCESSING SHOULD GO BELOW THIS COMMENT for event in pygame.event.get(): # User did something if event.type == pygame.QUIT: # If user clicked close done = True # Flag that we are done so we exit this loop # ALL EVENT PROCESSING SHOULD GO ABOVE THIS COMMENT # ALL GAME LOGIC SHOULD GO BELOW THIS COMMENT # ALL GAME LOGIC SHOULD GO ABOVE THIS COMMENT # ALL CODE TO DRAW SHOULD GO BELOW THIS COMMENT # ALL CODE TO DRAW SHOULD GO ABOVE THIS COMMENT # Limit to 20 frames per second clock.tick(20)
Eventually we will add code to handle the keyboard and mouse clicks. That code will go between the comments for event processing. Code for determining when bullets are fired and how objects move will go between the comments for game logic. We'll talk about that in later chapters. Code to draw will go in between the appropriate draw-code comments.
Alert! One of the most frustrating problems programmers have is to mess up the event processing loop. This “event processing” code handles all the keystrokes, mouse button clicks, and several other types of events. For example your loop might look like:
for event in pygame.event.get(): if event.type == pygame.QUIT: print("User asked to quit.") if event.type == pygame.KEYDOWN: print("User pressed a key.") if event.type == pygame.KEYUP: print("User let go of a key.") if event.type == pygame.MOUSEBUTTONDOWN: print("User pressed a mouse button")
The events (like pressing keys) all go together in a list. The program uses a for loop to loop through each event. Using a chain of if statements the code figures out what type of event occured, and the code to handle that event goes in the if statement.
All the if statements should go together, in one for loop. A common mistake when doing copy and pasting of code is to not merge loops from two programs, but to have two event loops.
# Here is one event loop for event in pygame.event.get(): if event.type == pygame.QUIT: print("User asked to quit.") if event.type == pygame.KEYDOWN: print("User pressed a key.") if event.type == pygame.KEYUP: print("User let go of a key.") # Here the programmer has copied another event loop # into the program. This is BAD. The events were already # processed. for event in pygame.event.get(): if event.type == pygame.QUIT: print("User asked to quit.") if event.type == pygame.MOUSEBUTTONDOWN: print("User pressed a mouse button")
The for loop on line 2 grabbed all of the user events. The for loop on line 13 won't grab any events because they were already processed in the prior loop.
Another typical problem is to start drawing, and then try to finish the event loop:
for event in pygame.event.get(): if event.type == pygame.QUIT: print("User asked to quit.") if event.type == pygame.KEYDOWN: print("User pressed a key.") pygame.rect.draw(screen,green,[50,50,100,100]) # This is code that processes events. But it is not in the # 'for' loop that processes events. It will not act reliably. if event.type == pygame.KEYUP: print("User let go of a key.") if event.type == pygame.MOUSEBUTTONDOWN: print("User pressed a mouse button")
This will cause the program to ignore some keyboard and mouse commands. Why? The for loop processes all the events in a list. So if there are two keys that are hit, the for loop will process both. In the example above, the if statements are not in the for loop. If there are multiple events, the if statements will only run for the last event, rather than all events.
The basic logic and order for each frame of the game:
- While not done:
- For each event (keypress, mouse click, etc.):
- Use a chain of if statements to run code to handle each event.
- Run calculations to determine where objects move, what happens when objects collide, etc.
- Clear the screen
- Draw everything
- For each event (keypress, mouse click, etc.):
It makes the program easier to read and understand if these steps aren't mixed togther. Don't do some calculations, some drawing, some more calculations, some more drawing. Also, see how this is similar to the calculator done in chapter one. Get user input, run calculations, and output the answer. That same pattern applies here.
The code for drawing the image to the screen happens inside the while loop. With the clock tick set at 10, the contents of the window will be drawn 10 times per second. If it happens too fast the computer is sluggish because all of its time is spent updating the screen. If it isn't in the loop at all, the screen won't redraw properly. If the drawing is outside the loop, the screen may initially show the graphics, but the graphics won't reappear if the window is minimized, or if another window is placed in front.
Right now, clicking the “close” button of a window while running this Pygame program in IDLE will still cause the program to crash. This is a hassle because it requires a lot of clicking to close a crashed program.
The problem is, even though the loop has exited, the program hasn't told the computer to close the window. By calling the command below, the program will close any open windows and exit as desired.
The following code clears whatever might be in the window with a white background. Remember that the variable white was defined earlier as a list of 3 RGB values.
# Clear the screen and set the screen background screen.fill(white)
This should be done before any drawing command is issued. Clearing the screen after the program draws graphics results in the user only seeing a blank screen.
When a window is first created it has a black background. It is still important to clear the screen because there are several things that could occur to keep this window from starting out cleared. A program should not assume it has a blank canvas to draw on.
Very important! You must flip the display after you draw. The computer will not display the graphics as you draw them because it would cause the screen to flicker. This waits to display the screen until the program has finished drawing. The command below “flips” the graphics to the screen.
Failure to include this command will mean the program just shows a blank screen. Any drawing code after this flip will not display.
# Go ahead and update the screen with what we've drawn. pygame.display.flip()
Let's bring everything we've talked about into one full program. This code can be used as a base template for a Pygame program. It opens up a blank window and waits for the user to press the close button.
# Sample Python/Pygame Programs # Simpson College Computer Science # http://programarcadegames.com/ # http://simpson.edu/computer-science/ # Explanation video: http://youtu.be/vRB_983kUMc import pygame # Define some colors black = ( 0, 0, 0) white = ( 255, 255, 255) green = ( 0, 255, 0) red = ( 255, 0, 0) pygame.init() # Set the width and height of the screen [width,height] size = [700,500] screen = pygame.display.set_mode(size) pygame.display.set_caption("My Game") #Loop until the user clicks the close button. done = False # Used to manage how fast the screen updates clock = pygame.time.Clock() # -------- Main Program Loop ----------- while done == False: # ALL EVENT PROCESSING SHOULD GO BELOW THIS COMMENT for event in pygame.event.get(): # User did something if event.type == pygame.QUIT: # If user clicked close done = True # Flag that we are done so we exit this loop # ALL EVENT PROCESSING SHOULD GO ABOVE THIS COMMENT # ALL GAME LOGIC SHOULD GO BELOW THIS COMMENT # ALL GAME LOGIC SHOULD GO ABOVE THIS COMMENT # ALL CODE TO DRAW SHOULD GO BELOW THIS COMMENT # First, clear the screen to white. Don't put other drawing commands # above this, or they will be erased with this command. screen.fill(white) # ALL CODE TO DRAW SHOULD GO ABOVE THIS COMMENT # Go ahead and update the screen with what we've drawn. pygame.display.flip() # Limit to 20 frames per second clock.tick(20) # Close the window and quit. # If you forget this line, the program will 'hang' # on exit if running from IDLE. pygame.quit()
Here is a list of things that you can draw:
A program can draw things like rectangles, polygons, circles, ellipses, arcs, and lines. We will also cover how to display text with graphics. Bitmapped graphics such as images are covered in Chapter 12. If you decide to look at that pygame reference, you might see a function definition like this:
pygame.draw.rect(Surface, color, Rect, width=0): return Rect
A frequent cause of confusion is the part of the line that says width=0. What this means is that if you do not supply a width, it will default to zero. Thus this function call:
pygame.draw.rect(screen, red, [55,500,10,5])
Is the same as this function call:
pygame.draw.rect(screen, red, [55,500,10,5], 0)
The : return Rect is telling you that the function returns a rectangle, the same one that was passed in. You can just ignore this part.
What will not work, is attempting to copy the line and put width=0 in the quotes.
# This fails and the error the computer gives you is # really hard to understand. pygame.draw.rect(screen, red, [55,500,10,5], width=0)
The code example below shows how to draw a line on the screen. It will draw on the screen a green line from (0,0) to (100,100) that is 5 pixels wide. Remember that green is a variable that was defined earlier as a list of three RGB values.
# Draw on the screen a green line from (0,0) to (100,100) # that is 5 pixels wide. pygame.draw.line(screen,green,[0,0],[100,100],5)
Use the base template from the prior example and add the code to draw lines. Read the comments to figure out exactly where to put the code. Try drawing lines with different thicknesses, colors, and locations. Draw several lines.
Programs can repeat things over and over. The next code example draws a line over and over using a loop. Programs can use this technique to do multiple lines, and even draw an entire car.
Putting a line drawing command inside a loop will cause multiple lines being drawn to the screen. But here's the catch. If each line has the same starting and ending coordinates, then each line will draw on top of the other line. It will look like only one line was drawn.
To get around this, it is necessary to offset the coordinates each time through the loop. So the first time through the loop the variable y_offset is zero. The line in the code below is drawn from (0,10) to (100,110). The next time through the loop y_offset increased by 10. This causes the next line to be drawn to have new coordinates of (0,20) and (100,120). This continues each time through the loop shifting the coordinates of each line down by 10 pixels.
# Draw on the screen several green lines from (0,10) to (100,110) # 5 pixels wide using a while loop y_offset = 0 while y_offset < 100: pygame.draw.line(screen,red,[0,10+y_offset],[100,110+y_offset],5) y_offset = y_offset+10
This same code could be done even more easily with a for loop:
# Draw on the screen several green lines from (0,10) to (100,110) # 5 pixels wide using a for loop for y_offset in range(0,100,10): pygame.draw.line(screen,red,[0,10+y_offset],[100,110+y_offset],5)
Run this code and try using different changes to the offset. Try creating an offset with different values. Experiment with different values until exactly how this works is obvious.
For example, here is a loop that uses sine and cosine to create a more complex set of offsets and produces the image shown in Figure 5.7.
for i in range(200): radians_x = i/20 radians_y = i/6 x=int( 75 * math.sin(radians_x)) + 200 y=int( 75 * math.cos(radians_y)) + 200 pygame.draw.line(screen,black,[x,y],[x+5,y], 5)
Multiple elements can be drawn in one for loop, such as this code which draws the multiple X's shown in Figure 5.8.
for x_offset in range(30,300,30): pygame.draw.line(screen,black,[x_offset,100],[x_offset-10,90], 2 ) pygame.draw.line(screen,black,[x_offset,90],[x_offset-10,100], 2 )
When drawing a rectangle, the computer needs coordinates for the upper left rectangle corner (the origin), and a height and width.
Figure 5.9 shows a rectangle (and an ellipse, which will be explained later) with the origin at (20,20), a width of 250 and a height of 100. When specifying a rectangle the computer needs a list of these four numbers in the order of (x, y, width, height).
The next code example draws this rectangle. The first two numbers in the list define the upper left corner at (20,20). The next two numbers specify first the width of 250 pixels, and then the height of 100 pixels.
The 2 at the end specifies a line width of 2 pixels. The larger the number, the thicker the line around the rectangle. If this number is 0, then there will not be a border around the rectangle. Instead it will be filled in with the color specified.
# Draw a rectangle pygame.draw.rect(screen,black,[20,20,250,100],2)
An ellipse is drawn just like a rectangle. The boundaries of a rectangle are specified, and the computer draws an ellipses inside those boundaries.
The most common mistake in working with an ellipse is to think that the starting point specifies the center of the ellipse. In reality, nothing is drawn at the starting point. It is the upper left of a rectangle that contains the ellipse.
Looking back at Figure 5.9 one can see an ellipse 250 pixels wide and 100 pixels tall. The upper left corner of the 250x100 rectangle that contains it is at (20,20). Note that nothing is actually drawn at (20,20). With both drawn on top of each other it is easier to see how the ellipse is specified.
# Draw an ellipse, using a rectangle as the outside boundaries pygame.draw.ellipse(screen,black,[20,20,250,100],2)
What if a program only needs to draw part of an ellipse? That can be done with the arc command. This command is similar to the ellipse command, but it includes start and end angles for the arc to be drawn. The angles are in radians.
The code example below draws four arcs showing four difference quadrants of the circle. Each quadrant is drawn in a different color to make the arcs sections easier to see. The result of this code is shown in Figure 5.10.
# Draw an arc as part of an ellipse. Use radians to determine what # angle to draw. pygame.draw.arc(screen,green,[100,100,250,200], pi/2, pi, 2) pygame.draw.arc(screen,black,[100,100,250,200], 0, pi/2, 2) pygame.draw.arc(screen,red, [100,100,250,200],3*pi/2, 2*pi, 2) pygame.draw.arc(screen,blue, [100,100,250,200], pi, 3*pi/2, 2)
The next line of code draws a polygon. The triangle shape is defined with three points at (100,100) (0,200) and (200,200). It is possible to list as many points as desired. Note how the points are listed. Each point is a list of two numbers, and the points themselves are nested in another list that holds all the points. This code draws what can be seen in Figure 5.11.
# This draws a triangle using the polygon command pygame.draw.polygon(screen,black,[[100,100],[0,200],[200,200]],5)
Text is slightly more complex. There are three things that need to be done. First, the program creates a variable that holds information about the font to be used, such as what typeface and how big.
Second, the program creates an image of the text. One way to think of it is that the program carves out a “stamp” with the required letters that is ready to be dipped in ink and stamped on the paper.
The third thing that is done is the program tells where this image of the text should be stamped (or “blit'ed”) to the screen.
Here's an example:
# Select the font to use. Default font, 25 pt size. font = pygame.font.Font(None, 25) # Render the text. "True" means anti-aliased text. # Black is the color. The variable black was defined # above as a list of [0,0,0] # Note: This line creates an image of the letters, # but does not put it on the screen yet. text = font.render("My text",True,black) # Put the image of the text on the screen at 250x250 screen.blit(text, [250,250])
Want to print the score to the screen? That is a bit more complex. This does not work:
text = font.render("Score: ",score,True,black)
Why? A program can't just add extra items to font.render like the print statement. Only one string can be sent to the command, therefore the actual value of score needs to be appended to the “Score: ” string. But this doesn't work either:
text = font.render("Score: "+score,True,black)
If score is an integer variable, the computer doesn't know how to add it to a string. You, the programmer, must convert the score to a string. Then add the strings together like this:
text = font.render("Score: "+str(score),True,black)
Now you know how to print the score. If you want to print a
timer, that requires print formatting, discussed in a chapter later on.
Check in the example code for
section on-line for the timer.py example:
This is a full listing of the program discussed in this chapter. This program,
along with other programs, may be downloaded from:
# Sample Python/Pygame Programs # Simpson College Computer Science # http://programarcadegames.com/ # http://simpson.edu/computer-science/ # Import a library of functions called 'pygame' import pygame # Initialize the game engine pygame.init() # Define the colors we will use in RGB format black = [ 0, 0, 0] white = [255,255,255] blue = [ 0, 0,255] green = [ 0,255, 0] red = [255, 0, 0] pi = 3.141592653 # Set the height and width of the screen size = [400,500] screen = pygame.display.set_mode(size) pygame.display.set_caption("Professor Craven's Cool Game") #Loop until the user clicks the close button. done = False clock = pygame.time.Clock() while done == False: # This limits the while loop to a max of 10 times per second. # Leave this out and we will use all CPU we can. clock.tick(10) for event in pygame.event.get(): # User did something if event.type == pygame.QUIT: # If user clicked close done=True # Flag that we are done so we exit this loop # All drawing code happens after the for loop and but # inside the main while done==False loop. # Clear the screen and set the screen background screen.fill(white) # Draw on the screen a green line from (0,0) to (100,100) # 5 pixels wide. pygame.draw.line(screen,green,[0,0],[100,100],5) # Draw on the screen several green lines from (0,10) to (100,110) # 5 pixels wide using a loop for y_offset in range(0,100,10): pygame.draw.line(screen,red,[0,10+y_offset],[100,110+y_offset],5) # Draw a rectangle pygame.draw.rect(screen,black,[20,20,250,100],2) # Draw an ellipse, using a rectangle as the outside boundaries pygame.draw.ellipse(screen,black,[20,20,250,100],2) # Draw an arc as part of an ellipse. # Use radians to determine what angle to draw. pygame.draw.arc(screen,black,[20,220,250,200], 0, pi/2, 2) pygame.draw.arc(screen,green,[20,220,250,200], pi/2, pi, 2) pygame.draw.arc(screen,blue, [20,220,250,200], pi,3*pi/2, 2) pygame.draw.arc(screen,red, [20,220,250,200],3*pi/2, 2*pi, 2) # This draws a triangle using the polygon command pygame.draw.polygon(screen,black,[[100,100],[0,200],[200,200]],5) # Select the font to use. Default font, 25 pt size. font = pygame.font.Font(None, 25) # Render the text. "True" means anti-aliased text. # Black is the color. This creates an image of the # letters, but does not put it on the screen text = font.render("My text",True,black) # Put the image of the text on the screen at 250x250 screen.blit(text, [250,250]) # Go ahead and update the screen with what we've drawn. # This MUST happen after all the other drawing commands. pygame.display.flip() # Be IDLE friendly pygame.quit()
Click here for a multiple-choice review quiz.
After answering the review questions below, try writing a computer program that creates an image of your own design. For details, see the Create-a-Picture lab.
- Before a program can use any functions like pygame.display.set_mode(), what two things must happen?
- What does the pygame.display.set_mode() function do?
- What is pygame.time.Clock used for?
- What does this for event in pygame.event.get() loop do?
- For this line of code:
- What does screen do?
- What does [0,0] do? What does [100,100] do?
- What does 5 do?
- Explain how the computer coordinate system differs from the standard Cartesian coordinate system.
- Explain how white = ( 255, 255, 255) represents a color.
- When drawing a rectangle, what happens if the specified line width is zero?
- Sketch the ellipse drawn in the code below, and label the origin coordinate,
the length, and the width:
- Describe, in general, what are the three steps needed when printing text to the screen using graphics?
- What are the coordinates of the polygon that the code below draws?
- What does pygame.display.flip() do?
- What does pygame.quit() do?
Complete Lab 3 “Create-a-Picture” to create your own picture, and show you understand how to use loops and graphics.
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In the next 2 lessons there is an explanation of the constructions [di̲], [mïth], and [wïnïth], meaning "how", "which?" and "where?"
Someone comes and tells you of a conversation he has just had with a 3rd person who perhaps is at your door, and
you go to find out for yourself because it isn't clear, or he hasn't explained enough to suit you. So you begin by saying:
What is it you were saying?
||Ki̲i̲m! Ɣä̲n ta̲a̲ kɛ riɛk. Cä̲ gatdä̲ nöŋ duɛ̲l wal mïn wa̲a̲r ru̲n kä̲ cɛ maath, kä̲ tä̲ä̲mɛ ci̲ duŋdɛ wä̲ di̲t puɔ̲nydɛ.
Doctor! I have trouble. I brought my child to the clinc this morning and he drank, but now his (condition) went big (has increased) in his body.
||Kä̲ gatdu̲ ɛ gat ïmïth? Ci̲ gaat ti̲ nyïn ŋuan bɛ̲n walɛ.
But your child is which child? Many children came today.
||Cɔalɛ ï̲ di̲, ɛn gat?
How is he called, the child?
||Cɔalɛ i̲ di̲, ɛn gat?
He is called Deŋ. Deŋ Köör.
||Kä̲ du̲ŋdɛɛŋu̲? Du̲ŋdɛ kiɛ̩l kiɤ ɤ ŋu̲?
And what is his (condition)? Is his (condition) a cough or what?
||Ɣɔ̲ɔ̲n, du̲ŋdɛ kiɛ̩l.
Yes, his (situation) is a cough.
||Kä̲ jɛn a ni̲ tä̲ä̲mɛ, ɛn gat puɔ̲nydɛ?
And where is he now, the child himself?
||Jɛn a wanɔ.
He is here.
||Jɛn a thaar jiaath ɛmɔ.
He is under that tree.
||Ɣɔ̲ɔ̲n, kä̲ wër, no̲o̲ni̲ jɛ wanɛmɛ.
Yes, and go, bring him here.
||Bä̲ jɛ wä̲ nööŋ.
will go bring him.
- This consonant is a voiced th- and is spoken in the position of the tongue for the English word "the".
- Practice drills.
- [di̲] is an interrogative word. It is not a pronoun. It is commonly translated to mean: "how?" "how's that?" (meaning
you didn't hear), "what do you mean?", "how many?", or "how much?". The meaning depends on the context and/or
the grammatical environment.
- In an elliptical sentence [i̲ di̲?] which means "how's that?", "what were you saying?". The full sentence
would be [Ɛ jï̲n ï̲ di̲? ] and finds its meaning in whatever the [ï̲] refers to. As for example, what do you think
about it? What do you say about it? How do you feel about it? In other words, the setting defines
- In accompaniment with the numbers particle [da̲ŋ] to mean "how much?" [da̲ŋ] may be omitted.
- [Ti̲ti̲ kɛ yio̲w da̲ŋ di̲?] -- These are with how much money? i.e. How much do these cost?
- [Go̲o̲ri̲ pɛ̲k da̲ŋ di̲? (or) Go̲o̲ri̲ pɛ̲k di̲?] -- You want the limit how much? i.e. How much do you want?
- With the noun [guä̲th] -- places to mean "how many times?". e.g. [Ci̲ bɛ̲n kɛ guä̲th di̲?] -- You came with how many times?
i.e. How many times did you come?
- With the Nuerized Arabic word [thaa] -- watch, to mean "what time is it?"
- [Ɛ thaa di̲ tä̲ä̲mɛ?] -- What time is it now?
- [Ci̲ bɛ̲n kɛ tha di̲?] -- What time did you come?
- With the clausal particel [i̲] (implying "you tell me") following all aspects.
- [Bi̲i̲ i̲ di̲?] -- How do you come?
- [Ci̲ bɛ̲n ï̲ di̲?] -- How did you come?
- [Bi̲ bɛ̲n ï̲ di̲?] -- How will you come?
- In the same type of construction as above with the verb [tɛ̲] -- exist/be, to mean "how is it?" "how is
so and so?" etc. It is used when wanting to inquire as to someone's health or the state of something.
- [Ti̲i̲ ï̲ di̲?] -- How are you?
- [Tëë ï̲ di̲?] -- How is it, etc?
- [Puɔ̲nydu̲ tëë ï̲ di̲?] -- How is your body?
- Notice in the dialogue that the verb [ɛ] is often omitted in the sentences initiated with [du̲ŋdɛ]. This is because the
verb [ɛ] is slurred together with the [-dɛ], or else is simply dropped.
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An American Hero: Martin Luther King Jr.
This lesson plan introduces young students to the contributions of Dr. King and has them think about how to make the world a better place.
- Grades: PreK–K
Step 1: Read Happy Birthday, Martin Luther King Jr. by Jean Marzollo.
Step 2: Discuss the story and focus on how Martin Luther King made the world a better place.
Step 3: Do a Think-Pair-Share with the students. Ask the students to think about how they make the world a better place.
Step 4: Pair up the students.
Step 5: Have the students share with their partner how they make the world a better place.
Step 6: Have the students share in a whole group and write down their responses on chart paper.
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Discussing gas density is slightly more complex than discussing solid/liquid density. Since gas volume is VERY responsive to temperature and pressure, these two factors must be included in EVERY gas density discussion.
By the way, solid and liquid volumes are responsive to temperature and pressure, but the response is so little that it can usually be ignored in introductory classes.
So, for gases, we speak of "standard gas density." This is the density of the gas (expressed in grams per liter) at STP. If you discuss gas density at any other set of conditions, you drop the word standard and specify the pressure and temperature. Also, when you say "standard gas density," you do not need to add "at STP." STP is part of the definition of the term. It does no harm to say "standard gas density at STP," it's just a bit redundant.
You can calculate the standard gas density fairly easily. Just take the mass of one mole of the gas and divide by the molar volume.
For nitrogen, we would have:
28.014 g mol¯1 / 22.414 L mol¯1 = 1.250 g/L
For water, we have:
18.015 g mol¯1 / 22.414 L mol¯1 = 0.8037 g/L
You could see this: "ideal standard gas density" or some other variation that uses ideal in addition to standard. The behavior of "real" gases diverges from predictions based on ideal conditions. Small gases like H2 at high temperatures approach ideal behavior almost exactly while larger gas molecules (NH3) at low temperatures diverge the greatest amount. These "real" gas differences are small enough to ignore right now, but in later classes they will become important.
The official IUPAC unit for gas density is kg/m3 (not g/L). However, it turns out that one kg/m3 equals one g/L. Here is a brief video explaining the conversion.
One place teachers like to bring gas density into play is when you calculate a molar mass of a gas using PV = nRT.
Example #1: The density of a gas is measured at 1.853 g / L at 745.5 mmHg and 23.8 °C. What is its molar mass?
Solution #1: Convert mmHg to atm and °C to K. Use 1.000 L. Plug into PV = nRT and solve for n. Then divide 1.853 g by n, the number of moles, for your answer.
(745.5/760) (1.000 L) = (n) (0.08206) (296.8 K)
The (745.5/760) term converts mmHg to atm. Notice I left the units off everything.
n = 0.0402753 mol (I'll keep a couple guard digits.)
1.853 g / 0.0402753 mol = 46.01 g/mol
Solution #2: This solution exploits a rearrangement of the ideal gas law. Here it is:
molar mass = [(1.853 g / 1.000 L) (0.08206) (296.8)] / (745.5/760)
Example #2: What is the molar mass of a gas which has a density of 0.00249 g/mL at 20.0 °C and 744.0 mm Hg?
Solution: Convert mmHg to atm (744.0/760.0) and °C to K (20.0 + 273.15). Use 0.001 L (which is 1 mL converted to liters). Plug into PV = nRT and solve for n (the value of which is calculated to be 4.069 x 10¯5 mol).
Then, divide 0.00249 g by the moles just calculated for an answer of 61.2 g/mol.
Please note that I used 273.15 (rather than 273) for the Celsius to Kelvin conversion. Some teachers require the use of 273.15. The ChemTeam always felt that 273 was good enough, but some teachers disagree. Yours may be one of those people (Are teachers people?).
Example #3: Anhydrous aluminum chloride sublimes at high temperatures. What density will the vapor have at 225 degrees Celsius and 0.939 atm of pressure?
1) Use PV = nRT to determine moles of gas present in vapor:
(0.939 atm) (1.00 L) = (n) (0.08206) (498 K)
n = 0.0229776 mol
I assumed 1.00 L because gas densities are measured in g/L.
2) Get grams of AlCl3 in the calculated moles:
0.0229776 mol x 133.34 g/mol = 3.06 g (to three sig figs)
3) Get density:
3.06 g / 1.00 L = 3.06 g/L
I could have assumed any volume I wanted back in the PV = nRT calc. However, I would have then divided the grams by that volume in this last step and wound up with 3.06.
Example #4: Air is a mixture of 21% oxygen gas and 79% nitrogen gas (neglect minor components and water vapor). What is the density of air at 30.0 °C and 1.00 atm?
Comment: For both solutions, we need the "molecular weight" of air:
MW(air) = (%O2 x MWO2) + (%N2 x MWN2)
(0.21 x 32) + (0.79 x 28) = 29 g/mol
1) Use PV = nRT and assume 1.00 L:
(1.00 atm) (1.00 L) = (n) (0.08206) (303 K)
n = 0.0402185 mol (of air at 303 K)
2) Calculate grams of air:
0.0402185 mol times 29 g/mol = 1.17 g
3) Determine density:
1.17 g / 1.00 L = 1.17 g/L
1) Density of air at STP
29 g/mol divided by 22.4 L/ mol = 1.29383 g/L (I'll keep some guard digits
2) As air is heated it gets less dense, so apply a temperature correction which makes the density smaller:
d (g/L) = 1.29383 g/L x (273 K / 303 K) = 1.16 g/L
This is actually a disguised Charles' Law correction:V1 / T1 = V2 / T2
(1.00 L / 273 K) = (x / 303 K)
x = 1.10989 L
1.29383 g / 1.10989 L = 1.16 g/L (to three sig figs)
If the problem had been written with a change in pressure from standard, we would have used the Combined Gas Law rather than Charles' Law.
Example #5: Two equal-volume balloons contain the same number of atoms. One contains helium and one contains argon. Comment on the relative densities of the balloons.
Here is a wrong answer: To determine density, you have to divide mass by volume. They are equal volume containers, and they contain the same number of atoms, then the densities of the balloons are equal."
Response to wrong answer: The answer above would be right, if the two sets of atoms had identical weights. However, an atom of argon weighs more than an atom of helium.
Therefore, at a condition of equal numbers of atoms, the argon balloon would be the denser one.
Note the implied Avogadro's Hypothesis in this question: equal volumes of gas contain equal numbers of atoms. Therefore, they MUST be at equal pressure and equal temperature.
Return to KMT & Gas Laws Menu
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The purpose of this activity is to help your child compare different items and put them in order, estimate (guess) their sizes, and then measure to find out. These are all important steps toward understanding measurement.From the Virtual Pre-K: Ready For Math toolkit
Safety note: Small items can be choking hazards for young children. Be sure this activity is closely supervised by you.
On recycled paper, trace the foot or shoe of each family member. If you have a small family, invite friends to join in. With your child’s help, write each person’s name inside his/her foot outline. Cut out and compare the paper feet – first two at a time and then the whole group. Who has the biggest foot? Smallest? Help your child put the paper feet in order from smallest to biggest.
Use small household items, such as pennies or paperclips, to measure each foot. First guess how many of the item each foot will be, and then find out (e.g., baby’s foot is five pennies long and Dad’s foot is 15 pennies long). Help your child write the number on each paper foot. When all the feet have been measured ask your child, “Whose foot is longest?” and “Who has the shortest foot?”
Next, compare the paper feet with household items by asking questions such as, “Is your foot longer or shorter than the telephone?” “Is Daddy’s foot wider or narrower than this straw?” “Is baby’s foot bigger or smaller than this cup?”
You can also use the paper feet to measure household objects, "How many of Daddy's feet is the rug? How many of baby's feet is the rug? Why?"
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On this page, we hope to clear up problems that you might
have with finding the equation of a line.
Many times, you'll have the graph of an equation shown to you and you'll need to find the equation. This seems a very daunting task, but it's actually quite easy!
For example, take a horizontal line such as y = 2. Every point on that graph is 2 units above the x axis. All horizontal lines have equations that are written in the same format, such as y = -4.5. Because they're all written the same way, we can come up with a general formula for horizontal lines. It is the following equation (where k represents any real number): y = k. Under the same assumptions, the general formula for vertical lines can be written as follows (where k represents any real number): x = k.
Linear equations such as y = .009x + 34 also have a general equation that can represent any linear equation. It is written as follows: y = mx + b.
The things to remember about the above formula, which is called the slope-intercept formula, are outlined below.
1. Since you know a line with an equation in that form cannot be horizontal or vertical, all you need to find are m and b to find the equation.
2. b is called the intercept. It is the point when the line crosses the y axis.
3. m is called the slope. The slope has both a sign (either + or -) and a value (the number behind the sign). For example, the equation y = -3x + 4 has a negative sign and a value of three. Therefore, the slope is -3. When looking at a graph, you can always tell if the slope is negative or positive by the direction it points. When looking for the sign of a slope, look at the left side of the graph. Then, look at the right side of the graph. If the right side is lower than the left side, the line has a negative slope, if the right side is higher, the line has a positive slope.
To find the value of the slope, you compute the rise over the run To do that, pick two points on the line at random and then draw a line through each of those points that run parallel to the coordinate axes. Count the number of units between the point on the line and where the two addition lines you drew intersect. The number of units on the horizontal line is the run and the number of units on the vertical line is the rise. Dividing the rise by the run gives you the value of the slope.
1. Find the equation of the line graphed in the accompanying figure:
Solution: The desired equation is in the slope-intercept form. You need to find m and b.
Back to Graphing on the Coordinate Plane.
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Date: 04/16/97 at 14:24:23 From: JARED ROSS Subject: Algebra If two consecutive numbers are less than one hundred, what is the larger number?
Date: 06/25/97 at 17:10:03 From: Doctor Sydney Subject: Re: Algebra Dear Jared, Hello! Thanks for writing. In reading your question, we came up with a couple of different ways to interpret it, so we aren't quite sure what you mean. One way we might intepret the question is the following: 1. Suppose we have two consecutive numbers, each of which is less than 100. What is the larger number? Or, you might mean: 2. Suppose that we have two consecutive numbers such that their sum is less than 100. Then what is the larger number? This second question is more difficult than the first, so we'll assume that that is what you mean. However, if you were really asking the first question and still need help with it, write back, and we can help out. So, let's look at the second interpretation. Well, there are many pairs of numbers that when added up, equal less than 100. Think of it this way: If the two numbers must add up to be less than 100 (and we are only dealing with integers, which are numbers of the type ...,-3, -2, -1, 0, 1, 2, 3,...), then we can express what we want as: n + (n + 1) < 100 This says that n, plus the number that comes right after n, must be equal to a number less than 100. But we can make this even simpler! The equation can also be written as: 2n + 1 < 100 2n < 99 n < 49.5 So, for any pair of consecutive numbers (n, n+1) such that n < 49.5, the sum of the consecutive numbers is less than 100. For instance, the consecutive numbers (49,50) add up to less than 100. In fact 49 and 50 are the largest consecutive numbers that have this property. See if you can figure out why. Can you find other examples of consecutive numbers that work? I hope this helps you. Please write back if we answered the wrong question or if you need more help. -Doctors Matthew and Sydney, The Math Forum Check out our web site! http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994-2013 The Math Forum
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Reading Cards for Young Learners - Card 6
These cards are practice materials for young people who are learning English. They are set at elementary or pre-intermediate level.
Waylink Young Learner's Reading Card
Sarah is a good student. She goes to school every day. She goes to Kings Heath School. She is never late. Sarah works hard at school. She has many friends at school. She walks to school with Emma. They walk to school every morning. They leave home at 7.30am. They get to school at 8 o'clock. They get home at half past four.
Answer the questions below:
Write short sentences.
- Who is a good student?
Sarah is a good student.
- How often does she go to school?
She goes to school every day.
- Is she ever late?
No, she is never late.
- Why does she do well at school?
She works hard at school.
- Why do you think she likes going to school?
She has many friends at school.
- Who does she go to school with?
She walks to school with Emma.
- What time does Sarah leave home?
She leaves home at 8 o'clock.
- How long does it take her to get to school?
It takes her 30 minutes/half an hour to get to school.
- What time does she leave school?
She leaves school at 4 o'clock.
- How many hours does she stay at school?
She stays at school for eight hours.
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The five main parts of a sentence are:
In general, the subject refers to the part of the sentence which tells whom or what the sentence is addressing. The subject is going to be either a noun or a noun phrase.
For example, "Kelly walked down the street." Kelly is the subject, because she is the actor, or subject, in the sentence.
There are a few different types of subjects. The underlined word is the subject.
Let us return to our example "Kelly walked down the street." In this sentence, "walked" is the predicate because it is the verb that tells us what Kelly is doing. A sentence can have just a subject and a predicate. For example, you could just say "Kelly walked" and you have a complete sentence.
Here are the types of predicates.
A clause is usually some sort of additional information to the sentence. We could say "They like ice cream." However, we could also say "They like ice cream on hot days." "They like ice cream" can stand by itself, but "on hot days" adds something extra to the sentence. Therefore, "on hot days" is a clause.
There are two different types of clauses:
A phrase is sort of like a dependent clause. It is a group of words that cannot stand alone as a sentence, but it can be used to add something to a sentence. There are a few different types of phrases:
As you can see from above, there are many different types of ways to add additional information to a sentence. All of these examples are known under the general category of modifiers.
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You are racing a child that you are twice as fast as and you want to give him a head start in a race from one side of a field to the other. How much of a head start should you allow so that you will come close to finishing the race even? Draw a picture and label the length of the field as L and the head start distance as H. What is the length that the child needs to run? If your speed is S what is the child's speed? You want to finish at nearly equal times; the time that it takes you would be L/S (the distancerun divided by the speed run), what would it be for the child? Set those equal and solve for H.
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When young children begin to read they love to go to the library and choose their own books. Yet, parents often wonder how a child can pick out a book on their own that is appropriate for their reading level.
Here is a simple rule that we use in our school library to determine if a book is the right level for a child. We call it the "Five Finger Rule" and it’s a great way to test a book for an appropriate reading level.
When your child finds a book of interest, apply these simple standards:
- Have your child fold down all five fingers on one hand.
- Open the book to the middle, and ask your child to read a page out loud.
- Your child puts up one finger every time a word is missed
- If all five fingers are up, the book is too hard. If only one or two fingers are up, the book is too easy. Three or four fingers up mean that the book is probably just right!
This simple rule is effective to insure that young readers are happy readers!
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Introduction to Pythagorean Triples
The Pythagorean Theorem tells us that a right triangle with legs a and b and hypotenuse c will always have the relationship a2+b2 = c2. (Do you know how to prove that?) When all three sides are whole numbers, we have a Pythagorean triple. The most famous of these is 32+42 = 52, often referred to in this context as (3,4,5). The 3-4-5 triangle was used in Egypt to help make perpendicular sides for their magnificent buildings. A loop of rope with 12 equally spaced knots (3+4+5 = 12) was pulled taut at knots 0, 3 and 7 to make a precise right angle.
If the three sides don't all have a factor in common, then they make a primitive Pythagorean triple (PPT). 62+82 = 102 is not a PPT because all three sides have a factor of 2.
Starting to Explore
When I'm making up problems for students, I often want another Pythagorean triple, so I have one more sitting in my brain, 52+122 = 132. Are there others? Are there infinitely many PPT's? How would we find more?
One approach to exploring these involves thinking about parity. (Parity refers to whether a number is odd or even.) I see that both of the PPT's above, (3,4,5) and (5,12,13), are odd+even = odd. Can we have odd+odd = even? What about even+even = even? If we have odd+even = odd, does the odd leg have to be the shorter one?
Do other questions occur to you?
Recommendation: stop reading and start playing as soon as you have a thought about how you might proceed.
I knew there were more PPT's, but couldn't remember any others. I wanted to find a few more, so I could see any obvious patterns. So I made a list of the first 25 perfect squares and looked for pairs that would add to equal another perfect square, or subtract to equal another one. I found (8,15,17) and (7,24,25). Well, that's one question answered: the odd leg does not have to be the shortest side. I see that all 4 hypotenuses are odd. So I want to address the question of whether odd+odd = even is possible.
Question1: Is odd+odd = even possible?
Here's how I start: Suppose we have two odd legs. We can let a = 2n+1 and b=2m+1. Then a2+b2 = ... Can we come to a contradiction? [See the hint at the end for a bit more direction.]
Question2: If a is an odd number, can I find a PPT for it?
I notice that all 3 triples in which the odd leg is the short one include consecutive numbers for the other leg and the hypotenuse. Hmm. If a is the short leg, then b+c = a2 in all 3 of those cases. Is that important? What if I write a2+b2 = c2 as c2-b2 = a2? Oh! A square minus a square can factor. So I'd have (c-b)(c+b) = a2. What does that get me? [I found a way to get a PPT for every odd number. Can you?]
Question3: Must the even side be a multiple of 4?
I wanted more triples, in case it would help me see more patterns. So I set up a spreadsheet with column a holding 1 through 100, row 1 holding 1 through 254 (first time I've ever used all available rows!). Column b and row 2 had the squares of these numbers. The rest of the spreadsheet showed a 0 if the square root of the sum of these squares was not a whole number, and otherwise showed the number. [Here's the formula in cell c3: =IF(INT(SQRT($B3+C$2))=SQRT($B3+C$2),SQRT($B3+C$2),0)]
For most multiples of 4, I found a PPT. And I didn't find any for the other even numbers. So I wanted to know whether the even side had to be a multiple of 4. If the even side is a, then b and c are odd, and c2-b2, with b=2m+1 and c=2n+1, can be explored.
Questions 4 and 5: Multiples of 3, 4 and 5
This reminded me that I had read (whose blog was that on?) that in every PPT, 3 will be a factor of one side, 4 will be a factor of one side, and 5 will be a factor of one side. (As in (5,12,13), one side may contain more than one of these factors.) I realized I'd already proved it for 4. I started trying to prove it for 3.
I mentioned parity earlier. Even numbers can be expressed as a=2m, and odd numbers can be expressed as b=2n+1. Similarly, if we want to think about whether side c will be a multiple of 3, we can look at three cases: c=3m, c=3m+1, or c=3m+2. Using this, I started with the question of whether c would be a multiple of 3. (It wasn't in any of the triples I'd found.) If it is, then neither a nor b can be. (Why?) Once I solved that problem, I wanted to prove that one of the legs would be a multiple of 3. Suppose b is not a multiple of 3 and consider c2-b2. There will be 4 cases, each of b and c can either be 3x+1 or 3x+2. What does this make a?
I tried to think about 5 in this way but got nowhere. I'm writing this blog post in hopes that explaining my thinking will help me get further on some of my dead ends.
I have a few other questions I haven't answered:
- Given a multiple of 4, how can I come up with a PPT?
- Can the same number show up in 3 different PPT's?
Hint: c2 is a multiple of 4. (Why?) What about a2+b2?
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Ask your students what color they think light is. A silly question? Sunlight or light from a lamp may look "white" but it is really made up of different colors. Red, blue, and green are the primary colors of light. That means you can make any color from combinations of these three. We see an object in a certain color because that color is reflected by the object, while other colors are absorbed.
3 big flashlights
pieces of red, blue, and green cellophane paper
30 cm white card
white toy or ornament
1. Tape one piece of cellophane over the bulb end of each flashlight.
2. In a dark room, turn on the flashlights.
3. Shine the red and green flashlights on to the white card. What happens? (You make the color yellow.) Try mixing the blue and green lights. (You get cyan.) Now mix the blue and red lights. (You get magenta.)
4. Put the white toy in the center of the card and shine the three flashlights on it. Have the students circle around the toy to see it in different colors.
5. Ask them what color you will get if you shine all three lights on the white card at the same time. (You get white. White light is made up of red, blue, and green. These are called the primary colors. To get black, you would have mixed colored paints, not lights.)
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Percent and Probability
A percent is a ratio of a number to 100. A percent can be expressed using the percent symbol %.
Example: 10 percent or 10% are both the same, and stand for the ratio 10:100.
A percent is equivalent to a fraction with denominator 100.
Example: 5% of something = 5/100 of that thing.
Example: 2 1/2% is equal to what fraction?
2 1/2% = (2 1/2)/100 = 5/200 = 1/40
Example: 52% most nearly equals which one of 1/2, 1/4, 2, 8, or 1/5?
Answer: 52% = 52/100. This is very close to 50/100, or 1/2.
Example: 13/25 is what %?
We want to convert 13/25 to a fraction with 100 in the denominator: 13/25 = (13 × 4)/(25 × 4) = 52/100, so 13/25 = 52%.
Alternatively, we could say: Let 13/25 be n%, and let us find n. Then 13/25 = n/100, so cross multiplying, 13 × 100 = 25 × n, so 25n = 13 × 100 = 1300. Then 25n ÷ 25 = 1300 ÷ 25, so n = 1300 ÷ 25 = 52. So 13/25 = n% = 52%.
Example: 8/200 is what %?
Method 1: 8/200 = (4 × 2)/(100 × 2), so 8/200 = 4/100 = 4%.
Method 2: Let 8/200 be n%. Then 8/200 = n/100, so 200 × n = 800, and 200n ÷ 200 = 800 ÷ 200 = 4, so n% = 4%.
Example: Write 80% as a fraction in lowest terms.
80% = 80/100, which is equal to 4/5 in lowest terms.
Percent and hundredths are basically equivalent. This makes conversion between percent and decimals very easy.
To convert from a decimal to a percent, just move the decimal 2 places to the right. For example, 0.15 = 15 hundredths = 15%.
0.0006 = 0.06%
Converting from percent to decimal form is similar, only you move the decimal point 2 places to the left. You must also be sure, before doing this, that the percentage itself is expressed in decimal form, without fractions.
Express 3% in decimal form. Moving the decimal 2 to the left (and adding in 0's to the left of the 3 as place holders,) we get 0.03.
Express 97 1/4% in decimal form. First we write 97 1/4 in decimal form: 97.25. Then we move the decimal 2 places to the left to get 0.9725, so 97 1/4% = 0.9725. This makes sense, since 97 1/4% is nearly 100%, and 0.9725 is nearly 1.
When estimating percents, it is helpful to remember the fractional equivalent of some simple percents.
100% = 1
(100% of any number equals that number.)
50% = 1/2 = 0.5
(50% of any number equals half of that number.)
25% = 1/4 = 0.25
(25% of any number equals one-fourth of that number.)
10% = 1/10 = 0.1
(10% of any number equals one-tenth of that number.)
1% = 1/100 = 0.01
(1% of any number equals one-hundredth of that number.)
Because it is very easy to switch between a decimal and a percent, estimating a percent is as easy as estimating a fraction as a decimal, and converting to a percent by multiplying by 100.
Estimate 19 as a percent of 80.
As a fraction, 19/80 20/80 = 1/4 =0.25 = 25%. The step used to estimate the percent occurred when we estimated 19/80 as 20/80.
The exact percent is actually 23.75%, so the estimate of 25% is only 1.25% off. (About 1 part in 100.)
Estimate 7 as a percent of 960.
As a fraction, 7/960 7/100 = 0.007 = 0.7%. The step used to estimate the percent occurred when we estimated 7/960 as 7/1000.
The exact percent, to the nearest thousandth of a percent, is actually 0.729%.
To estimate the percent of a number, we may convert the percent to a fraction, if useful, to estimate the percent.
Estimate 13% of 72.
Twice 13% is 26%, which is very close to 25%, and 25%=1/4. We may multiply both sides by 1/2 to get an estimate for 13%: 13% 12.5% = 1/2 × 25% = 1/2 × 1/4 = 1/8. Using our estimate of 1/8 for 13%, 1/8 × 72 = 9, so we get an estimate of 9 for 13% of 72.
If we had calculated this exactly, 13% of 72 equals 9.36. It may look like we did a lot more work to get the estimate of 9 that just multiplying 72 by 0.13, but with practice, keeping in mind some simple percents and the fractions they are equal to will enable you to estimate some number combinations very quickly.
Estimate 9.6% of 51.
Method 1: We could estimate 9.6% of 50. It would be easy to estimate 9.6% of 100, which is just 9.6. Since 50 is half of 100, we can just take half of 9.6, which is 4.8. The actual value of 9.6% of 51 is 4.896, so an estimate of 4.8 is pretty good.
Method 2: We could estimate 10% of 51, which is just 5.1. This is not as close an estimate as method 1, but is still a good estimate of the actual answer of 4.896.
Interest is a fee paid to borrow money. It is usually charged as a percent of the total amount borrowed. The percent charged is called the interest rate. The amount of money borrowed is called the principal. There are two types of interest, simple interest and compound interest.
Example: A bank charges 7% interest on a $1000 loan. It will cost the borrower 7% of $1000, which is $70, for each year the money is borrowed. Note that when the loan is up, the borrower must pay back the original $1000.
Simple interest is interest figured on the principal only, for the duration of the loan. Figure the interest on the loan for one year, and multiply this amount by the number of years the money is borrowed for.
Example: A bank charges 8% simple interest on a $600 loan, which is to be paid back in two years. It will cost the borrower 8% of $600, which is $48, for each year the money is borrowed. Since it is borrowed for two years, the total charge for borrowing the money will be $96. After the two years the borrower will still have to pay back the original $600.
Compound interest is interest figured on the principal and any interest owed from previous years. The interest charged the first year is just the interest rate times the amount of the loan. The interest charged the second year is the interest rate, times the sum of the loan and the interest from the first year. The interest charged the third year is the interest rate, times the sum of the loan and the first two years' interest amounts. Continue figuring the interest in this way for any additional years of the loan.
Example: A bank charges 8% compound interest on a $600 loan, which is to be paid back in two years. It will cost the borrower 8% of $600 the first year, which is $48. The second year, it will cost 8% of $600 + $48 = $648, which is $51.84. The total amount of interest owed after the two years is $48 + $51.84 = $99.84. Note that this is more than the $96 that would be owed if the bank was charging simple interest.
Example: A bank charges 4% compound interest on a $1000 loan, which is to be paid back in three years. It will cost the borrower 4% of $1000 the first year, which is $40. The second year, it will cost 4% of $1000 + $40 = $1040, which is $41.60. The third year, it will cost 4% of $1040 + $41.60 = $1081.60, which is $43.26 (with rounding). The total amount of interest owed after the three years is $40 + $41.60 + 43.26 = $124.86.
Percent increase and decrease of a value measure how that value changes, as a percentage of its original value.
Example: A collectors' comic book is worth $120 in 1994, and in 1995 its value is $132. The change is $132 - $120 = $12, an increase in price of $12; since $12 is 10% of $120, we say its value increased by 10% from 1994 to 1995.
Example: A bakery makes a chocolate cake that has 8 grams of fat per slice. A new change in the recipe lowers the fat to 6 grams of fat per slice. The change is 8g - 6g = 2g, a decrease of 2 grams; since 2 grams is 25% of 8, we say that the new cake recipe has 25% less fat, or a 25% decrease in fat.
Example: Amy is training for the 1500 meter run. When she started training she could run 1500 meters in 5 minutes and 50 seconds. After a year of practice her time decreased by 8%. How fast can she run the race now? Her old time was 5 × 60 + 50 = 350 seconds, and 8% of 350 is 28, so she can run the race in 350 - 28 = 322 seconds (5 minutes and 22 seconds).
Example: A fishing magazine sells 110000 copies each month. The company's president wants to increase the sales by 6%. How many extra magazines would they have to sell to reach this goal? This problem is easy, since it only asks for the change in sales: 6% of 110000 equals 6600 more magazines.
A discount is a decrease in price, so percent discount is the percent decrease in price.
Example: Chocolate bars normally cost 80 cents each, but are on sale for 40 cents each, which is 50% of 80, so the chocolate is on sale at a 50% discount.
Example: A compact disc that sells for $12 is on sale at a 20% discount. How much does the disc cost on sale? The amount of the discount is 20% of $12, which is $2.40, so the sale price is $12.00 - $2.40 = $9.60.
Example: Movie tickets sell for $8.00 each, but if you buy 4 or more you get $1.00 off each ticket. What percent discount is this? We figure $1 as a percentage of $8: $1.00/$8.00 × 100% = 12.5%, so this is a 12.5% discount.
An event is an experiment or collection of experiments.
The following are examples of events.
1) A coin toss.
2) Rolling a die.
3) Rolling 5 dice.
4) Drawing a card from a deck of cards.
5) Drawing 3 cards from a deck.
6) Drawing a marble from a bag of different colored marbles.
7) Spinning a spinner in a board game.
8) Tossing a coin and rolling a die.
Possible outcomes of an event are the results which may occur from any event. (Remember, they may not occur.)
The following are possible outcomes of events.
1) A coin toss has two possible outcomes. The outcomes are "heads" and "tails".
2) Rolling a regular six-sided die has six possible outcomes. You may get a side with 1, 2, 3, 4, 5, or 6 dots.
3) Drawing a card from a regular deck of 52 playing cards has 52 possible outcomes. Each of the 52 playing cards is different, so there are 52 possible outcomes for drawing a card.
4) How many different outcomes are there for the color of marble that may be drawn from a bag containing 3 red, 4 green, and 5 blue marbles? This event has 3 possible outcomes. You may get a red marble, a green marble, or a blue marble. Even if the marbles are different sizes, the outcome we are considering is the color of the marble that is drawn.
5) How many different outcomes are there for the colors of two marbles that may be drawn from a bag containing 3 red, 4 green, and 5 blue marble? This event has 6 possible outcomes: you may get two reds, two greens, two blues, a red and blue, a red and green, or a blue and green.
6) Rolling two regular dice, one of them red and one of them blue, has 36 possible outcomes. The outcomes are listed in the table below.
|Blue1||Blue1, Red1||Blue1, Red2||Blue1, Red3||Blue1, Red4||Blue1, Red5||Blue1, Red6|
|Blue2||Blue2, Red1||Blue2, Red2||Blue2, Red3||Blue2, Red4||Blue2, Red5||Blue2, Red6|
|Blue3||Blue3, Red1||Blue3, Red2||Blue3, Red3||Blue3, Red4||Blue3, Red5||Blue3, Red6|
|Blue4||Blue4, Red1||Blue4, Red2||Blue4, Red3||Blue4, Red4||Blue4, Red5||Blue4, Red6|
|Blue5||Blue5, Red1||Blue5, Red2||Blue5, Red3||Blue5, Red4||Blue5, Red5||Blue5, Red6|
|Blue6||Blue6, Red1||Blue6, Red2||Blue6, Red3||Blue6, Red4||Blue6, Red5||Blue6, Red6|
Note that the event tells us how to think of the outcomes. Even though there are 12 different marbles in example 4, the event tells us to count only the color of the die, so there are three outcomes. In example 6, the two dice are different, and there are 36 possible outcomes. Suppose we don't care about the color of the dice in example 6. Then we would only see 21 different outcomes: 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-2, 2-3, 2-4, 2-5, 2-6, 3-3, 3-4, 3-5, 3-6, 4-4, 4-5, 4-6, 5-5, 5-6, and 6-6. (We think of a 1 and a 2, a 1-2, as being the same as a 2 and a 1.)
The probability of an outcome for a particular event is a number telling us how likely a particular outcome is to occur. This number is the ratio of the number of ways the outcome may occur to the number of total possible outcomes for the event. Probability is usually expressed as a fraction or decimal. Since the number of ways a certain outcome may occur is always smaller or equal to the total number of outcomes, the probability of an event is some number from 0 through 1.
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball numbered 1? There are 2 ways to draw a 1, since there are two balls numbered 1. The total possible number of outcomes is 10, since there are 10 balls.
The probability of drawing a 1 is the ratio 2/10 = 1/5.
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball with a number greater than 4? There are 3 ways this may happen, since 3 of the balls are numbered greater than 4. The total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number greater than 4 is the ratio 3/10. Since this ratio is larger than the one in the previous example, we say that this event has a greater chance of occurring than drawing a 1.
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball with a number greater than 6? Since none of the balls are numbered greater than 6, this can occur in 0 ways. The total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number greater than 6 is the ratio 0/10 = 0.
Suppose there are 10 balls in a bucket numbered as follows: 1, 1, 2, 3, 4, 4, 4, 5, 6, and 6. A single ball is randomly chosen from the bucket. What is the probability of drawing a ball with a number less than 7? Since all of the balls are numbered greater than 7, this can occur in 10 ways. The total possible number of outcomes is 10, since there are 10 balls. The probability of drawing a number less than 7 is the ratio 10/10 = 1.
Note in the last two examples that a probability of 0 meant that the event would not occur, and a probability of 1 meant the event definitely would occur.
Suppose a card is drawn at random from a regular deck of 52 cards. What is the probability that the card is an ace? There are 4 different ways that the card can be an ace, since 4 of the 52 cards are aces. There are 52 different total outcomes, one for each card in the deck. The probability of drawing an ace is the ratio 4/52 = 1/13.
Suppose a regular die is rolled. What is the probability of getting a 3 or a 6? There are a total of 6 possible outcomes. Rolling a 3 or a 6 are two of them, so the probability is the ratio of 2/6 = 1/3.
| 4.375 | 5 | 3.989601 | 4.454867 |
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