I’m not kidding. I seriously dreamt recently about a variation on the Spiral of Theodorus , woke up and thought, hey, that might be a way for students to create some math— specifically, the well-known theorem attributed to Pythagoras.
I’m not trying to dis Pythagorus here; I’m just trying to figure out how to create opportunities for students to arrive at the theorem WITHOUT just showing them “how” and having them practice finding missing lengths umpteen times, with some “real world” applications thrown in involving a leaning ladder or a shadow and a tree. Maybe that’s why my brain came up with something while I was sleeping.
For some reason, people tend to like the PT, maybe because it feels magical, maybe because it’s an easy equation to memorize and recall and sound smart (unless you are The Scarecrow).
It certainly isn’t due to dutifully copying how-to examples from the board (or a book or a video or website), memorizing the steps without ever knowing/wondering/asking what it all means. (I’m describing my school-aged self here, but I know you know/teach people who are just like this.)
There are thankfully ample visual and symbolic proofs available that help students make some sense of the theorem (usually after they have been introduced to it), so that may be part of the attraction. Its cool factor seems to be sufficient enough to make it memorable. (Oh, that’s what it all means? Cool.)
Just imagine the way studens would feel if they were empowered to discover it themselves!
Mind you, I don’t have research-based data or definitive lesson plans that guarantee deriving the PT. I just have a fleeting dream about a seed of an idea, and wonder if there is something worthwhile here. I know it needs considerable fleshing out and trial runs with real life teachers and students. Hint hint.
This is not part of the dream, but I would begin by having students use graph paper to make as many squares with whole number areas as they possibly can. (This idea is not original to me. If and when I find the source, I will include it.) Challenge them to make all areas from 1-10 (or even higher), label the side lengths, and justify the areas. (Dammit, I can’t figure out areas of 3, 6, and 7!!) This is a genuinely engaging activity, and depending on learning goals, there are lots of connections to similar figures, parallel and perpendicular slopes, and similfying radicals. For me, the essential learning this activity offers is a concrete, visual understanding of the relationship between the area of a square and the length of its sides, between “square” numbers and their “roots”, even the not-so-perfect ones.
In my dream, I made a right isosceles triangle with legs of 1 unit, then used the hypotenuse as the leg of the next right triangle. This is where the Spiral of Pat differs from that of Theodorus– each right triangle in my spiral is isosceles. His goal was (according to Wikipedia) to “prove that all the of the square roots of non-square integers from 3 to 17 are irrational.” My goal is for students to look for patterns and make and test conjectures. All the while honing their skills and fortifying their understanding of squares and roots.
When I tried my spiral out during daylight hours, I saw potential. Do you?
What do you think students will notice and wonder about? What conjectures do you think students will make? Do you think they will arrive at the “right” one? Why or why not? Would you also have students test this relationship with other polygons? What about cubic units? What would you do with this?
PS. If you have a way of helping me find areas of 3, 6, and 7 WITHOUT actually telling/showing me, I’d appreciate getting un-stuck.