Yesterday struggled with misconceptions around the Triangle Inequality Theorem. Today, Pythagorean Theorem. They’ve seen it before, but not this year–and all they really knew how to do coming in was to find the hypotenuse when it works out to a whole number.
I had three goals for the class.
1. Find the hypotenuse when it works out to a whole number
2. Find the hypotenuse when it is irrational, expressing it as both a radical and between two whole numbers
3. Identify that a triangle is a right triangle if it satisfies the Pythagorean Theorem
Talked a little bit about how cool Pythagoras was to start class. Took some notes on the Theorem, worked examples of all three types with the class, then had them practice on their own. They mostly all crushed the first two–they’re really good at working with squares and radicals from earlier this year. Number three was a breeze for about two-thirds of the class. The rest were all dutifully working away–and my first hint that something was wrong came when a bunch of my lower skilled kids were moving way faster than was reasonable for squaring a ton of numbers. I realized that, despite being able to find the hypotenuse of a right triangle, when asked to determine if a triangle was right, they went back to what we did yesterday–they were all checking each triangle with the Triangle Inequality Theorem. They were doing a great job of it. But this was pretty worrying for me. We had just spent half an hour working exclusively with the Pythagorean Theorem. If they reverted to something else they liked better after practicing the Pythagorean Theorem for most of a class, how could they apply their knowledge in a new context?
My solution in the past to objectives that students tend to confuse has been to spend a day exploring and practicing how to tell the difference and apply each concept. But do I really spend a day exploring the difference between the Triangle Inequality Theorem and the Pythagorean Theorem?