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 center 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: 

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