For our latest Creative Math Club, Sonja (who I’m happy to say is rested and recovered from her family’s bout with the flu!) prepared some activities having to do with the Pythagorean Theorem. We geared this one towards the older set of kids, those between the ages of 10-15. We had an amazing turnout for the day so ended up splitting off into groups arranged by age…the under 6 set of kids, the 7-9 set of kids, and then the 10-15+. There were kids and parents everywhere!
We had lots of inch squares for everyone to play around with and mosaic for the first 10 minutes or so, just to get in the mode of squares and numbers. Once the older kids had settled into the routine, Sonja wrote out a number line from 1-10 and then had the kids square each number:
Sonja was starting to teach them the Pythagorean Theorem, but in a backwards way. Or, rather, in an investigative way. So instead of giving them the formula a(squared) + b(squared)=c(squared) she led them in an investigation to discover that using squares. Once they had squared all the numbers between 1-10 (some kids knew the answers by doing it mentally, other kids made a square using the square pieces of paper…)
Sonja challenged them to find a sequence of 3 numbers where the square of that number plus the square of the next number equaled the next number squared.
I hope I’m explaining that clearly, lol. So, looking at the numbers 1-10, they were to find the sequential numbers where the first number squared plus the second number squared equaled the third number squared. Lots of number crunching ensued. They were using their mental math skills, or the square pieces, or calculators. They were also, because they are in fact creative kids, taking larger square sheets of paper and constructing paper houses out of them. Which is totally off topic, but I figure that any kind of investigation (on or off topic) is better than no investigation at all. And with creative kids, it’s the most random experiences that can lead to the best kind of learning. So, all forms of square ‘play’ was allowed, whether it was a direct contribution or not.
In any case, they eventually discovered that the numbers 3,4, and 5 are the only consecutive numbers where the squares of the first two numbers equals the square of the third number.
Sonja had prepared some new squares…not the square inch shapes, but 3 inch squares, 4 inch squares, and 5 inch squares. Each kid got one of each, and were asked to construct a triangle with them.
What they came up with was a right triangle:
At this point, we had a discussion about angles and triangles. The definition of ‘right angle’ was important, and then Sonja was able to tie the triangle they had made with squares, into the Pythagorean Theorem. Which is to say, anytime there is a right triangle, the Pythagorean Theorem of a(squared) + b(squared) = c(squared) applies. This also introduced the idea of the hypoteneuse being like the third number, or, the c part of the triangle.
By this time, the kids were starting to get antsy from sitting at a table, so we ended that part of it without doing much application of the formula as far as plugging in numbers and measuring–we can do that another math club. What we did was one last cool application of the idea by using a 12 foot long piece of string.
Sonja asked, if we are building a house, and we need all the corners to be square, or have right angles, so our house is a perfect square, how can we do it using only this rope and our knowlege of the Pythagorean Theorem? The kids gave different ideas, and then Sonja had them measure and mark the rope off into sections of 1 foot. They ended up with a rope marked off at every foot. She told them to think about what consecutive numbers made up the answer in the first part of the lesson, they replied “3,4 and 5” so she had someone grab hold at the 3 feet mark, then had someone grab hold 4 marks after that, and the last person grabbed hold 5 marks after that which happened to be at the 12 foot mark. She then had them take it down to the floor and make a triangle out of it. This is what they saw….a right triangle appeared!:
So. Freaking. Cool!
So, that was our brush with the Pythagorean Theorem. Personally, I remember learning the Theorem as just a string of numbers to memorize without any connection to why or how it works. I really enjoyed re-learning it this way!