In Circles

Today we looked at circles. I thought that starting with the most favorite circle they know might be a good idea, so we started our math on the trampoline.

Since a circle is defined as a set of all points in a plane that are the same distance from a given point, I thought the best place to start would be to find the center of this circle.

We took yarn and measured from one point on the trampoline, straight across to the other side, and then we curved the yarn back on itself, folding it in half, to find the center point. We marked this with a big dot. We understood that this wasn't an exactly precise measurement, but it gave us an idea of where the center point was.

Then, using the diagram at the top of this post, we made sections on our trampoline circle. We discussed each term, drew the lines and then labeled them. They remembered some of the terms from when we had read this book a few weeks ago.

 Even if they didn't quite understand some of the different terms at first, as we kept talking about them, asking questions and using them, they caught on.

"Just as when we folded the yarn in half and found the center we can draw a line from the centre to any point along the circle. This is called the radius.

Is there just one radius?

No, there are many radii. Picture it like the spokes of a bicycle wheel.

A radius is also a line segment – it has two endpoints.

You could label the endpoints with letters, but let's not this time so we won't get confused with too much written on the trampoline.

Does the length of the radius change at all? No – they are all the same length.
Now draw a line from one point of the circle to another point on the circle. This is called a chord.

What is the longest chord we can make? Through the center.What is it called? Diameter.

Not only is it the longest chord, what else does it do? It divides the circle into two equal parts called a semi-circle.

You can also see by the way we folded the yarn in half to make the diameter turn into the radius that the diameter of a circle is twice the length of it’s radius.

Next we covered perimeter and circumference.

We talked about how the path around any geometric shape is called a perimeter.

Then I had them walk around the edge of the trampoline and told them the name for the perimeter of a circle is the circumference. It is the distance around the circle. 

How about a tangent? Yes, it is a straight line or plane that touches a curve at a point but does not intersect it at that point.

And a secant? A straight line that intersects a curve at two or more points. Is it the same as a chord? Yes, a chord is the bit of a secant that lies within the circle.

I showed them how to show that the line goes on; by putting arrows on the ends.

Now, for some fun.

Simon says, "Lay on the radius."

Simon says, "Put your elbow on the diameter."

Simon says, "Hop on one foot to the secant."

Simon says, "Run around a half circle."

Simon says, "Make a tangent with your body."

Simon says, "Put your foot on a chord and one hand on the center."

Now that they got some of the wiggles out of them, we went inside and made circles on construction paper and cut them out. The little boys traced around canning lids while Sam used a compass.

Speaking of using a compass, I have found this lovely book, Circles by Mindel and Harry Sitomer, to be excellent at teaching kids how to use a compass. Not only does it show you how, but it also gives you fun exercises using the compass to make art. It is out of print, but you may be able to find it at your library, or a used copy through Amazon.

 Back to  our circles, I made a chart on the white board listing circumference, diameter, radius and ratio. Along the top I listed their names.

We used some more yarn to measure the circumference of each of the circles. We listed these measurements on the chart. We then measured the diameter of each of the circles and had them figure out and then confirm by measuring each of the circle's radius. I told them how to write the fraction of c/d for each circle and wrote this on the ratio row. Then Sam determined roughly what the fraction equaled and they quickly saw the pattern that the circumference was always a little more than three times longer than the diameter. It was then I gave them the name for this ratio, Pi and showed them its symbol. I told them that the exact ratio, had our measurements been precise would always be 3.146.

I read them of the story of Archimedes from Mathematicians are People Too.

We then took our circles and found the center by folding them first one way and then again the other way (a 90 degree angle) and putting a dot where both the folds met. I then had them cut out angled sections.

We had fun calling each of them Pacman. Then we measured them with a protractor and I wrote down what their measurements were. If they were way off, I helped them measure them correctly.

We then added the measurements of the two pieces together and they were thrilled to find out they always equaled 360 degrees. We then went on to cutting them into three or four pieces and finding out the degrees of those pieces, adding them and always getting 360 degrees. By this time they were getting pretty good at guessing the degrees of the angles.

And what better thing is there to practice measuring angles than this?
By the way, did you know that 3/14 is National Pi Day?
You could have a little fun with it then.

sources and inspiration: 

Other lessons like this are at Jimmie's Collage, where she has her daughter make her own protractor and angle maker, at A Pilgrim's Heart and at Child's Play, where I got the idea for using the trampoline to teach math.

It's Hip to be Square!

"You might think I'm crazy,
but I don't even care
I can tell what's going on
It's hip to be square"

Squares can be so much fun to play around with.

First, I made a huge pile of multi-colored and multi-patterned squares out of scrapbooking paper so that the boys would enjoy playing with them. I let them just have some freeplay fun with them. When it seemed that they were ready to move on, I asked them "How many squares does it take to make the smallest square?"

This was such a simple question that the boys were confused at first and then someone piped up, "One square is a square, isn't it?"

"Yes! Put one square on your work space -any color and pattern you like. And what is the next smallest number that makes a square?”

The kids looked around, played a bit with the papers, and then set four of them together.

“Four!" They shouted almost at once.

"Okay. Wrap enough of another type square around your first square, so that they all equal a four square-square." They did so.

"What is next?" This time the answer came a bit quicker even though it was more squares to put together because they were getting the pattern.

"Nine!" was practically shouted.

“Numbers that can form a square when you put them together in a grid are called Square Numbers. See how many square numbers you can form.”

Once they had been at this a while, I got out a small white board and asked them, "Let's look at your squares in a little bit of a different way. Okay, we started with one, and then we added how many more squares to get the next square?" Again, a pause while the took in what I was asking them. "You mean how many of the next color squares?"
"Yes."
"Three?" Still a little hesitancy.
"Yes. Okay. 1+3=4 because four is the total amount of color A and color B, right? Okay, what is next then? We will start with four since that is how many we have totally now. What did we add to that?"
"We added five of the next color."
"Yes and 4+5=9, right? So, let's look at the pattern in the numbers that is emerging..." There was some silence as they pondered what I meant.
"You mean that you take the answer from the first problem as the starting point for the next problem?"
"Yes, that is true. And what about the number you are adding to it?"
They thought a moment, a little frustration showing as they struggled to understand what I meant?"
I circled the numbers to make the pattern more clear.
"They are always just two more."
"Yes, and are they odd numbers or even?"
"Odd."
"Can you predict the next number then?"
Of course they could.

"Now get out that number of the next color/pattern. Then add them to the square."

"So, what kind of number do you get from adding two odd numbers together, odd or even?"

Easily answered.

"Like I said, these are square numbers."
I showed them how these with the superscript.

"You just count the number of squares that make up each side of the square. So, two on each side make four totally, or two squared is four."

"Okay, now we can look at it from the other angle. This is called a Square Root symbol. It tells us that like the roots of a tree the Square Root is the root of the number. Look at the square with four squares in it. How many squares are on the side?"

"Two"

"That’s the square root, or what make up the square."

We looked at the square roots of all the squares we had before us.

Then we got out calculators and played with the square root button, first confiming the ones we had already looked at, and then going on to play with other numbers."

Then we made mosaics with our lovely square patterns.
And their reward for all their hard work?

We found some rewarding squares!

I found a similar lesson at A Child's Play, where she uses mosaic tiles.

And another at A Pilgrim's Heart.
And another at Educating Risa with counters.

If you have a similar lesson, or know of one, let me know and I will add it to the list.

The Store

"...make the child practically familiar with the process of exchange; either by using money and getting twelve pence for a shilling, or by some game in which a red counter represents ten white ones. Begin teaching each of the four elementary operations, by giving a few easy examples in relation to the coins or counters to which he is accustomed..."

Home Arithmetic by Mary Everest Boole. Parent's Review, Volume 4, 1893/94, pgs. 649-654

I bought a set of "Garage Sale Labels" from Walmart a few weeks ago when I was buying school supplies with this math game in mind. My youngest two boys saw them and have been waiting anxiously to see what the game was. When I asked them what they wanted to do first on the first day of school, without hesitation they said, "The math game with the money stickers." First I review money denominations with my 6-year old. I got him to sort them into piles and then to tell me what their names were and how much the different coins were worth. The only one he was a little hesitant on was the nickle.
We then matched up the coins with Math-U-See blocks to make sure he understood their worth. We also looked at their worth on an abacus.

I had printed out some blank addition cards, but I would have done better with these because they have the hundreds place. We talked about the difference between dollars and cents and place value. I wrote them above the correct positions. We also looked at the symbols for them and I added them to the right positions as well. This was all review, but I wanted to make sure we had all the piece in place before we played store.
In advance I had gotten out some of my school supplies and put on the money stickers. I tried to make them somewhat realistic. I used to hate it when in workbooks they had wildly unrealistic prices for things. This only hurts their concept of money and the value of things in the real world.
Now it was time to play store. I try to use real money when I can so that they can get the feel of it and a sense of how it really looks. We did use James' play bills for the dollar amounts as he was looking forward to using them and I didn't have that many spare real dollar bills. We took turns being customers and store clerks. We added up what we
were buying, and we paid the store clerk, who confirmed our totals. We had to decide the best denominations with which to pay and the store clerk sometimes had to give change.
With everything I did, I narrated what I was doing aloud. After awhile, I stepped back and just let them play on their own.
They played quite a long while, not because it was math time to them, but because they wanted to.
It was a game and it was fun.

Multiplication with Crossed Lines

I have found it very helpful for students who are just learning multiplication facts to have methods at their disposal for determining the facts they need. One of these methods is using crossed lines. For example, if I student needs to figure out what 3 x 7 is, they can make three lines going down on a piece of paper and then cross those lines with seven lines going across. They are to count the places in which these lines intersect. If they make dots at each of the intersections, then they are less likely to get misplaced or skip one. By doing this, it reinforces the concept of what 3 x 7 means, and is not just rote memorization.

You can even play games with these crossed line matrices. Once they have made up a few of these, you can take them and cover up where they cross with a piece of paper, only leaving the lines sticking out at the top and left hand side. Then they have to visualize the dot matrix that is now under the paper. This is like flashcards, and yet you are encouraging the student to visualize the meaning behind these multiplication facts.

source: Mathematics; a Way of Thinking, Bob Baratta-Lorton

Math Adventures: Cooking & Math

There are so many math concepts that you can cover while cooking with your children.

Today I made bread with James and used only a half cup and a half teaspoon measure.

Not only does it review the concept that one-half and one-half equals a whole,

but he had to convert 4 1/2 cups of flour to 1/2 cups. If I had been baking with Quentin, I would have gotten out a cup measure for him to see that it held two half-cup measures of flour.

When we needed to add 2 Tablespoons of yeast, we reviewed that 3 teaspoons equals 1

Tablespoon, so he had to determine how many half-teaspoons that would make, which is a two step process; converting first to teaspoons and then to half-teaspoons. Because the application is tangible, he happily and easily tackled the adding and subtracting of fractions with like denominators.

We completed these activities several times, as we added new ingredients.

Later when it was time to preheat the oven, we looked at the temperature dial and talked about relative temperatures. He then set the dial to preheat the oven to 350 degrees Fahrenheit.

For older students, I could decide to increase or decrease a recipe to practice multiplying or dividing fractions. I could also have them use ratios and proportions.