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Kitchen Math: The Pythagorean Theorem Is Lurking Behind Your Sink
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.
Published in Education
I was thinking something similar earlier in the week. My suspicion is that if you go back early enough, it was probably brute force to start with. For square roots, just multiply numbers together as they get closer and closer to the number you are looking for. Likely something similar with trig functions.
Geometry was traditionally the vehicle used to teach logic. Proofs have a structure that can transfer as an outline to constructing a persuasive argument, building a case, carefully establishing points as you go along.
As a homeschool dad 15 years ago I taught geometry from an 1960 vintage text we found in a used bookstore. It was fine. Homeschooling has grown so much that now there are a couple of books available that are marketed to homeschoolers.
Avoid the public school texts. They have been prettied up with so many meaningless illustrations and distracting text boxes that they are hard to read, and the modern texts emphasize geometric identity memorization to the detriment of instruction in logic.
MJBubba (View Comment):
Geometry was traditionally the vehicle used to teach logic…
As a homeschool dad 15 years ago I taught geometry from an 1960 vintage text we found in a used bookstore…
Avoid the public school texts. They have been prettied up with so many meaningless illustrations and distracting text boxes that they are hard to read, and the modern texts emphasize geometric identity memorization to the detriment of instruction in logic.
Here’s my problem: I did well in high school math, finishing Calculus and scoring high enough on the ACT to choose whatever college I wanted. However, the text used (Saxon) was heavy on the memorization of algorithms; at least, I was able to get by without understanding much of what I was doing. I was mostly self-taught in our small private school.
The way I’ve approached choosing math curriculum and teaching math to my kids is to err on the side of understanding. If my kids can reason their way through, I don’t feel like I’ve cheated them. If they do what I did and get good grades with a bunch of shortcuts, I think they’ll have missed the point of math, which is so much more beautiful than getting a high score on a test.
What that means for geometric proofs, a la Euclid, I do not know. I have two textbooks with two different ways of approaching the subject, and we’ll be doing both. I shall be a mathematician by the end of the year.
I’d have to disagree with you in some cases.
Sometimes a clear diagram can convey more than a page of formulas.
For example, using a contour map with north / south and east / west slopes helped me understand partial derivatives.
Further, the illustrations in newer books are often more clear, as they can use a wider range of colors. Late Eighties and Nineties texts are often superior to their predecessors in by virtue of better printing methods.
Try to avoid books made too recently, though, as they have been pushing more and more information onto CDs and websites. This is less helpful for the homeschooler.
That’s admirable.
I chose my college because it had the reputation of the hardest physics undergraduate program in the country at the time (as well as a general reputation for student burnout – I was a teenage masochist. In retrospect, I doubt I made the wisest choice, and my folly prompted a switch from physics to math.)
The physics students were probably the brightest at our university, and the curriculum gave them thorough understanding. But not instant understanding. It was common for physics students to “just go through the motions” during a course, not understanding why they were doing all they were doing until the next course. My insistence on proving to myself I understood what I was doing while I was doing it was actually rather counterproductive.
Understanding is the goal, and too much just going through the motions drives a sensitive child batty. But some going through the motions without knowing why is an effective step to understanding.
My geometry teacher lived on my street. He owned an HP-35. It was the first electronic calculator I ever saw. He told my parents that I always wanted to learn another way to do the problems. (That was so I could check myself.)
Having a lot of tools simplifies a lot of problems. It is only by knowing the theory that you can learn which tool to reach for.
And show your work.
I got some of the Euclidean Proofs too; but even when I was schooling, it was downplayed. I picked up “Mathematics for the Millions” by Hogben later, and it brought back a lot of that Euclidean math.
Matlab said so.
I got the idea (from watching over someone else’s shoulder) that Matlab wouldn’t be particularly difficult to learn. How accurate is that?
I’ve not used it, but I used a couple products in college that were straightforward.
I was wrong though – it was Mathmatica, not Matlab
http://www.smbc-comics.com/comic/2013-01-20