Maybe you hate the backsplash tiles behind your kitchen sink. I know I do. Not your tiles, I mean – mine. Their pattern is a boring grid; their color, a grubby off-white (if you’ve ever dug up grubs in your garden, you know what color I mean). A few of the tiles are stamped with chintzy brown sunflowers in a listless attempt at cheer. It probably doesn’t help that I hate our kitchen sink as well, a sink wedged in a countertop corner for added inconvenience.

But maybe you’re lucky. Maybe you have a convenient kitchen sink and exciting backsplash tile. If your backsplash tile is laid in a hopscotch pattern, you even have the Pythagorean theorem lurking right behind your sink. To the left is a backsplash much nicer than mine, tastefully tiled in a hopscotch pattern (well, tasteful aside from the peculiar choice of blue grout). To mathematicians, the hopscotch pattern is known as “Pythagorean tiling”, because of how beautifully it illustrates the Pythagorean theorem. Several proofs of the Pythagorean theorem exist, some of which are more intuitive than others. To me, the diagram superimposed on the hopscotch tiling produces one of the most intuitive proofs, especially for children. If you like, you can knock off reading right now and do the proof yourself: the diagram has all the labeling you need.

A right triangle is one containing a right angle (an angle measuring **90˚**, or **π/2** in radians). Its hypotenuse is the side of the triangle opposite the right angle. For the theorem, the length of the hypotenuse is called **c**, and the length of the triangle’s other two sides are called **a** and **b**.

The area of the square outlined in white is **c^2**. The area of the dark brown square inside it is **(a – b)^2**. The four right triangles surrounding the dark brown square each have area **1/2 ab**. Because the dark brown square and the four triangles surrounding it add up to make the square outlined in white,

**(a – b)^2 + 4 * 1/2 ab = c^2**. *Now expand (a – b)^2 and rewrite 4 * 1/2 as 2:*

**a^2 – 2ab + b^2 + 2ab = c^2**. *Notice 2ab and -2ab cancel:*

**a^2 + b^2 = c^2**. *Ta da!*

That’s the Pythagorean theorem.

Simple. Kids only need to know how to expand a binomial in order to do this computation, something we learned at age 12, but which we could have been taught sooner. So why isn’t proving the Pythagorean theorem more widely taught?

I’m not sure. The same teacher who taught us binomials taught us the Pythagorean theorem and showed us *a* proof of it – the proof she happened to remember. Her proof was an asymmetric dissection – I don’t remember which one: the asymmetry made it harder for a kid to believe it would generalize to any right triangle, and *very* hard to remember the initial setup. A symmetric setup for proving the Pythagorean theorem, on the other hand, makes it easier to visualize how changing the lengths of **a** and **b** nonetheless leaves the formula **a^2 + b^2 = c^2 **(where **c** is the resulting hypotenuse) unchanged, and the symmetry also makes the setup memorable. I believe I was lucky, though, to have a teacher who bothered to show us any proof *at all*.

If your kitchen-sink backsplash is like mine, square tiles all of the same size in the typical boring grid, you can still use them to diagram a proof of the Pythagorean theorem, by nesting a square askew inside another square. That’s how Pythagoras did it. Follow the link and you’ll see a diagram different from the one above, but with the same symmetry. I happen to like the proof-by-hopscotch-tile, though, if only because I’m annoyed at my kitchen sink.

*This post is the second in a series on kitchen math. The first one in the series is here.*

It’s funny this should come up. I’ve recently taken up an interest in re-learning some of my trigonometry. Glad to know I am not the only person with right triangles on the brain.

It’s always in the last place you look.

I know @percival has a proof of the Pythagorean theorem in his image cache.

If any of you have a favorite proof that is asymmetric, I’d be curious to hear why it’s your favorite.

As I mentioned earlier, I occasionally used trig when I was carpentering. I used the Pythagorean theorem

a lot. Six feet, eight feet and ten feet was the most common way to make sure you had a right angle.Very strictly speaking, don’t you need a lemma to prove the area of a right triangle with catheti

aandbis ½ab? This is trivially done by constructing the rectangle with sidesaandband observing that the right triangles that compose it obviously have half the area of the rectangle.My tile eschews such modern innovations as Greek geometry. At Fourmilab, it’s Jurassic all the way!

I can’t say it’s my favourite, because the symmetric proof in the original post is my favourite, but I have a certain fondness for Einstein’s boyhood proof because a) it was done by Einstein, and b) it is the first known instance where he employed a symmetry argument, which would be the basis of much of his work in adulthood. Here, the symmetry is more subtle because he used an invariant under scaling.

Euclid’s proof, like so much in the Elements, is an example of why I have always considered him the Original Hacker. He would slog his way from axioms to proof in the manner a programmer under a shipping deadline would write code, always getting there but never going back to refine things afterward. Most of Euclid’s proofs have since been re-done in a form which is not only shorter, but more elegant and easily understood.

If we’re doing everything from the bottom up, I should also include a demonstration that the expansion of

(a – b)^2really isa^2 – 2ab + b^2, like I said it was. The area of a right triangle is one of those things that can be understood, by a simple slice, as soon as a child is prepared to believe the area of anaxbrectangle isab. At least for me, knowing how to expand(a – b)^2came considerably after I understood the area of a right triangle!The various slice-and-dice arguments used to show the area of triangles – and, if you believe the circumference of a circle is 2π, the area of a circle, too – should be another kitchen math post. These are very simple ideas, and the kitchen conceit is hokey, but when I think back to the math I actually

learnedin elementary school, a lot of it was from stuff I read on my own that made the math similarly hokey – though I still maintain bananas are too bent to make slicing them a convincing demonstration of conic sections!Stop talkin’ about me

Lol. I thought about saying @rightangles

President Garfield and the Pythagorean Theorem

I got to Garfield’s setup and paused to sketch it for myself. The darn thing is half the diagram used in Pythagoras’ original proof!

We hardly needed the quadratic formula, either… funny, that…

One of the rules I learned in engineering school is that in every problem you encountered, the triangle was always a right triangle and either 3–4–5 or 30°–60°–90°. In the real world, not so much.

I loved geometry in high school. Proofs were my favorite thing. I can’t remember how I did them anymore, I just remember liking it.

You know me so well.

That is a fun kind of symmetry!

In intro geometry I struggled with proofs a bit, because I’d make up theorems. Not intentionally – I just couldn’t keep straight which geometric relationships the class had already proven, and which, although they were intuitive, we hadn’t. Aside from that frustration – mostly frustrating because remembering which was which

wasthe hardest part for me – I’ve always enjoyed proofs, and wish kids doing proofs for themselves was introduced into the curriculum sooner.It’s a little deflating to be told, no, kids your age just aren’t developmentally ready to understand

whyyou gotta use this or that formula. I’m sure some aren’t, but if you are, being told you aren’t and you wouldn’t understand only makes math scarier than it has to be.Kitchen calculation is actual a solid method for teaching the principle of arithmatic.

Give me M&Ms and ziploc bags, and I can do all grade school math.

Me too! But it might have had something to do with my hunk of a geometry teacher. Oh, sorry, off topic.

Uh oh, the Ricochet Comment Temporal Paradox is back!

I liked geometric proofs too. And later, I found writing programs to be so similar, I immediately took to it.

For the same reason? ;-P

Probably not. I don’t even remember the teacher.

A strong kitchen cleanser and Brillo will get rid of it.

You people have this all wrong! The correct statement is

“The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

I have it on good authority:

Ewwww!!

Math!!!!!!!Eek!!!!“Math is hard.”–Barbie.

Don’t look now, but it’s lurking… It is always lurking…

I’ll be saving this post for the fall: my homeschooled son is heading into geometry/trig, and it’s been a long, long time, folks.