Back to: Module: Helping with Instruction

## Background Material

What’s called, in a perhaps scary way, “the Pythagorean Theorem” is a simple fact. Maybe you already know what the fact is. And most of us heard about it in school. We explain it carefully, next. One warning: you might think this is a difficult assignment. It’s supposed to be a little bit difficult because the point is for you to be a little bit challenged and then use what you learn by being challenged to help a child learn something very unfamiliar–with the help of infographics. See how it goes!

Triangles, of course, have three sides. And in a “right triangle” two of those sides meet at a 90-degree angle.

Here’s a “right” angle:

The sides that make the right angle can be any length, and they don’t have to be the same length.

And so, here are a few “right triangles”:

Notice that in a right triangle, the two sides that form the right angle are joined by another side, called the “hypotenuse.” You’ve probably heard all this before, a long time ago.

Now here’s the tricky part. The three sides of a a right triangle, like all triangles can have different lengths. To allow for that difference, we call them c (for the hypotenuse) and *a* and *b* for the two sides that form the right angle.

We are now ready to state the fact, using those three letters:

Don’t freak out with the “2s” by the letters: we will explain next.

First, and maybe you know this already, the formula is read aloud like this: “a-squared plus b-squared equals c-squared.” To square a number is to multiply it by itself, that’s all. So what the formula says is that to find the square of the length of the hypotenuse, you need to add together the squares of the lengths of the other two sides.

What a mouthful! But wait. Why do we call a number that’s multiplied by itself a “square.” Sounds weird.

We call is a square because the area of a square is one side of the square multiplied by the other side of the square. But of course, the two sides of a square (its length and width) are *the same*. Bingo. A number multiplied by itself (gives you the area of a square with sides of that length).

So c^{2} looks like this as a square:

And c times c = c^{2} = the area of the square.

So we can apply that idea to illustrate the relationships of the lengths of the sides of a right triangle, as in the following infographic (which is a classic infographic, perhaps 2500 years old):

What’s the point? Read on. We’re getting there.

First, get oriented.

- In this triangle the hypotenuse is at the bottom.
- Ignore the different lines and colors and focus on the triangle and the squares.

OK. Now, the infographic shows the relationship expressed by a^{2} + b^{2} = c^{2} in a visual display *instead of* a formula. But it’s the same thing.

Most people go “Blech!” when they see a formula. But the infographic shows what the formula means in a way that many people find more acceptable than a “boring” formula/ You can see what the relationships are. Visually.

In words: the combined area of the two small squares is the same as the area of the large square.

This idea is incredibly useful throughout the world. It’s embedded in the tables on a carpenter’s square (used to cut rafters); it’s a basis of trigonometry and therefore of navigation (planes and ships; GPS devices automate all the calculations in part, by using this simple relationship).

The thing to note and to try to explain to a young child (your assignment) is that the formula and the figure express a relationship between three things. This relationship applies everywhere, and it never varies. The lengths of the sides can vary infinitely. You can draw right triangles of and size, and the relationship holds—*always. *It’s kind of amazing, really, this fact that was discovered perhaps 2500 years or so back. And it was unknown until then! Thank Pythagoras and probably some of his teachers.

By the way, the relationship a^{2} + b^{2} = c^{2} is useful. If you know the length of two sides of the triangle, you can find the length of the hypotenuse. In fact, if you know *any* the length of any two sides, you can find the length of the third. This use of the relationship is a key piece of why GPS systems work.

## Assignment

Explain the relationship of the lengths of the three sides of a right angle to a small child. By “small” we mean any a child of any age 3-10. You should choose a child whom you think doesn’t know the fact already—because the fun part of the activity is the conversation you will have trying to explain something unknown. Use anything in this assignment sheet. You can redraw the infographics to make them simpler, for instance. You can practice first on an adult who doesn’t remember what the Pythagorean Theorem is. So learn the ideas here, and have fun with a child of your choosing!

What is a “theorem?” Good question. A theorem is a step-by-step proof. There are many proofs that the a^{2}+ b^{2} = c^{2} relationship really is true. An American president even created one that hadn’t been seen before.

## Debriefing

If you are working alone, you can find someone who likes math to talk over what happened when you talked to the child. This discussion will be more fun, of course, if several paraprofessional have the experience (perhaps in a workshop) and talk about what happened.

## Sources

figure 1: “Right angle”. Licensed under CC BY-SA 3.0 via Wikimedia Commons – https://commons.wikimedia.org/wiki/File:Right_angle.svg#/media/File:Right_angle.svg

figure 2: “Rtriangle”. Licensed under CC BY-SA 3.0 via Wikimedia Commons –

https://commons.wikimedia.org/wiki/File:Rtriangle.svg#/media/File:Rtriangle.svg

figure 3: “Teorema” by Sérgio Schmiegelow utilizando o software inkscape, original uploader was Sergioschmiegelow at pt.wikipedia – Transferred from pt.wikipedia to Commons.. Licensed under Public Domain via Wikimedia Commons –

https://commons.wikimedia.org/wiki/File:Teorema.png#/media/File:Teorema.png

figure 4: http://www.mathwarehouse.com/geometry/triangles/right-triangle.php

figure 5: Pierce, Rod. (19 Oct 2014). “Definition of Square”. Math Is Fun. Retrieved 1 Jul 2015 from

http://www.mathsisfun.com/definitions/square.html

figure 6: from Wikipedia, at “right triangle” (image is in the public domain)