Question

Dear Mitch,

I'm in elementary school in London and in our class right now we're studying ancient civilizations. I'm really lucky because I go to an international school, and so the people in my class are from all different parts of the world. Some kids are even from right here in London! Anyway, we just started studying ancient civilizations and our teacher (he's a really cool guy) is going to base a lot of our math lessons on whatever ancient stuff we're going to do. I'm pretty sure we're going to study pyramids, at least a little, but I'm not sure about the other stuff we're going to do. Anyway, our teacher really likes it sometimes when somebody from the class does their own lesson for the rest of the class, and so I was wondering if you knew of any lessons I could try?

From,
Janie

Answer

Dear Janie,

Your class definitely does sound cool, and so does your teacher (and so do you!).

There's a lesson about pyramids that students and their families really seem to enjoy a lot, so here it is.

First, a quick review of the basics:

1. Pyramids are 3-dimensional, which means that they are not flat like drawings in a book and have form to them so they could actually hold real things inside. This, as you might already know, is exactly why they were built in the first place, to hold things.

2. Even though a lot of people think of pyramids as '3-dimensional triangles' or 'solid triangles', they actually come in different shapes, What I mean is that although when you stand back and look at a pyramid, the first thing most people notice is the triangular look because any side of a pyramid a person is seeing has 3 edges and is a triangle. However, as a three-dimensional (solid) shape the pyramid can have three flat sides or four flat sides or 20 flat sides (but you probably won't find a whole lot that really have 20 sides). Four flat sides is the most common shape for a pyramid, not three,. The way to make the whole subject less confusing is if you are careful to use different words for different things. So, it's best not to use the word 'side' for any of the surfaces the pyramid has, (or the flat parts or the "planes"), just as it will help if we don't use the word 'side' for any of the three edges of those triangles. Let's call the sides of triangles 'edges', and let's call the sides of pyramids 'flat sides'.

3. Confusing? It won't be once you make a few models out of paper or cardboard. (It's easy to do). and I'm going to tell you how to make one because you'll need to do this before you do the math lesson for your class: (And don't worry because it's easy and fun!) It get's easy after you try it at home or at school on your own, or with someone else helping a little. Then it's fun for the whole class to try in your lesson.

STEP ONE: Get a sheet of the plain paper that copying machines and printers use, like this white rectangle (which has been framed so you won't miss it!):

STEP TWO: Carefully fold it in half the long way. So when you open it you end up with a fold line that goes from the middle of one short side to the middle of the other short side, top to bottom.

Like this:

STEP THREE: Keeping it unfolded on your desk (or table or floor), make a little mark, like a dot, like this:

STEP FOUR: Now using a ruler or the side of a book or anything else with straight edge, make a solid line connecting the bottom left corner to the top mark; Repeat for the bottom right corner. Now you have one triangle, an isosceles triangle, which just means two sides are the same size and the third side is a different size, like this:

For a quick trick that helps you really remember the names of all three kinds of triangles, and remember them forever, go back to the page that lists the QUESTIONS & ANSWERS and double-click on the one called "Grouping and Triangles", but don't do that until you've finished this, so you don't get stuck or lost in the middle before you learn the lesson you want to teach!

Okay, so now we have one of the right shape triangles for our pyramid. We need 4 of these pyramids to do the lesson. AND (now here's the only tricky part, you're really going to need to make all four of the sides out of something stiffer than a sheet of paper. You'll want to use something at least as sturdy as card stock, or poster board, and if you can get an adult to help with the cutting, regular brown cardboard from a box makes them turn out awesome!

NEXT: After you trace your paper triangle onto the poster board (or whatever you use), cut it out (with help!). Now, you may have found tracing a paper triangle to be difficult, with your pencil sliding all over the place, so use your new, stiffer triangle as your new tracer for the next three triangles. When you have four triangles, set them down on a flat surface like this:

NEXT: Tape them all together leaving a tiny bit of space, as in the picture. That space should disappear when you pick it up and fold it into a closed square pyramid.

Finally, tape the last side together. When you turn what you've made the right way you should have a 3-d pyramid with an open bottom

TAKE A BREAK!

VOLUME & ESTIMATION:

Estimation is one of the most important areas of math and is even something that people practice all the time in real life whether they know it or not, and whether they want to or not! For example: About how much time will your homework take you tonight so you can figure out what time to begin so you are not up too late? OR: About how much room should you leave in your stomach if you know that tonight a special dessert is coming? That's all estimation: It's just guessing using your experience and ability to reason.

The word VOLUME means a few different things, and a lot of people think of it as how loud something is, like when they say, "Turn up the volume, because I love this song!"

But in math, it most often means how much space something can hold, or how much space something takes up. If you push a small ball into a bowl of water, the surface of the water will only rise a little, but if you push a big ball into the same bowl, it might overflow - because the bigger ball has more volume and takes up more room down there.

Volume is three dimensional, so it requires three dimensions in the calculating (figuring). But for estimation you just sort of guess your way closer to the answer, and with practice you get better and better at it.

Here's the interesting part: Most people (even adults!) find it tricky to estimate volume. It's a matter of practice and anyone can improve!

NEXT: have the class repeat the whole 4-triangle procedure, this time using a different sized triangle - either bigger or smaller, it does not matter. And when it's complete, you will, of course have two different sized pyramids (the one you made and the one the class makes).

Have students volunteer their guesses how much more the larger one will hold than the smaller one. Two times as much? Three times?? Write down the predictions on the board, and leave them there until the whole lesson is completed.

Finally, using sand or dry rice or cereal or bottle caps, have one group fill the small one while the other group holds the larger one with the opening facing up. Have the group with the smaller one empty it into the larger one. Have them repeat this move until the larger is full and level at its opening.

Compare how many times this had to be done with the predictions of how many times it would have to be done. That is estimation with volume. Congratulations, you made it to the end of this very long description. Good luck with your lesson!

Hope this helps,

Mitch