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.

Archimedes Fun

It can be so much fun to study a person in history by looking at what things he has done. Even people like mathematicians, which are not always thought of as fun. Take Archimedes, for example. The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. A new crown had been made for King Hiero II, and Archimedes was asked to determine whether it was of solid gold, or whether silver had been added by a dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density. While taking a bath, he noticed that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the volume of the crown. The submerged crown would displace an amount of water equal to its own volume. By dividing the weight of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" meaning in Greek "I have found it!"

With this story in mind, we set up a simple demonstration to show this concept. Don't worry no bathtub involved. We took a bowl and set it in a baking dish. We then filled the bowl to the brim with water.

We then took two objects of almost identical weight. We determined that the rubber ball symbolized the lump of gold, and placed it in the bowl of water. It displaced a certain amount of water. We then placed our candle, which represented the crown, and we saw that it displaced some more water, showing that although its weight was similar, its density was not.

Update: I have found out that both 4 quarters and 5 Hershey's Chocolate Kisses weigh 1 ounce, so these can be used. You can use a postal scale if you have one, to prove this as part of your experiment. Carry out the experiment the same as above and you should see that the candies have more mass and displace more water.

  Archimedes also was the first to study balance or the center of gravity of shapes. I had out a few triangles with different angles and asked him to draw lines from the corners to the middle of the side opposite it. The lines intercepted in the middle. Then, giving the boys the triangles with the lines down, I asked the boys to balance the triangles on the tips of their fingers...then we looked under the triangles to see where their fingers were.

The results were the same with all the triangles...
much to their delight.

Resources:

• This last activity AIMS education foundation and I got it through Living Math.
• A fun book to read is Archimedes and the Door of Science.
• Another fun and simple experiment involving bouyancy is from Almost Unschoolers.

Pythagoras

We all remember learning A squared plus B squared equals C squared and for many of us it was just one more meaningless fact to memorize. We never really saw it in action and we didn't learn who first made this discovery. I don't want my kids to just memorize to forget. I want them to really understand and know math facts in some sort of context. During our study of Ancient Greece, I wanted us to learn about Pythagoras. We can contribute many math discoveries to him, but we don't know a whole lot more about him as a person, so the book What's Your Angle, Pythagoras? A Math Adventure by Julie Ellis and Phyllis Hornung is a purely fictionalized story of the young Pythagoras. It is, however, a wonderful introduction to looking at number patterns and being able to apply them to real problems. It covers topics like right triangles and their properties as well as his famous theorem.

And, yes, I know that the sketch that Sam made has an extra row on his 5 squared section...he might still be working on attention to detail, but he certainly knows what the theorem means and can picture what it means in his head. I am satisfied.

Sam copied a page of the book in his math journal because it was so clear and visually appealing. As a group, we played with right triangles and Math-U-See blocks. We also played with knotted ropes and right triangles, also pictured in the book. 

Tangrams: The Seven Magic Shapes

Three Pigs, One Wolf and the Seven Magic Shapes by Grace Maccarone is a wonderful introduction for younger students to tangrams. The Tangram is an ancient Chinese puzzle in which one is to form a specific shape given only in outline or silhouette using all seven pieces, which may not overlap. In this book, the seven shapes make different characters or objects that further the plot. The tangrams in this book are not a single silhouette; the pieces are set slightly apart so that the young readers can manipulate the tans (the individual shapes that make up the set of tangrams) and copy the pictures. After copying several different pictures from the book, Quentin felt confidant enough to make his own pictures. These can be traced onto paper and given to other people to solve. It is a lot of fun for young children to see others working on a puzzle they have created.

resources:

• Another resource about tangrams can be found at Love2Learn2day.