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Students Inventing Theorems

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Students Inventing Theorems

I was working through a gometry proof with one of my students. It was a proof that involved segment AB and a point M which turned out to be the midpoint of the segment. In the course of writing the proof, at one point the student said, "The next statement is that AB = 2AM."

"Really?" I asked. "How do you know that?"

"The definition of midpoint," he replied confidently.

"Okay," I replied, "let's go take a look at the midpoint definition and see if that's what it says. The definition read something like this: "A point M is the midpoint of segment AB if and only if M is between A and B, and AM = MB."

"So," I said, "the definition of midpoint doesn't actually say that AB = 2AM."

"Well, no, but it's still true."

"If it's true," I said, "prove it. If you can prove it, We can add it to our arsenal of theorems. I'll even name it after you. Jake's Midpoint Theorem."

So we worked through the proof. It looked something like this:

Prove: If M is the midpoint of segment AB then AB = 2AM.

1. M is the midpoint of AB
2. M is between A and B
Definition of Midpoint
3. AM = MB
Definition of midpoint
4. AM + MB = AB
Segment Addition Postulate (2)
5. AM + AM = AB
Substitution (3 and 4)
6. 2AM = AB
7. AB = 2AM
Symmetric Property of Equality

As promised, we named this "Jake's Midpoint Theorem," and I shared it with the rest of the class. They thought it was "cool" - but I'm not sure whether they really saw the point of it until the next day, when we were working on a completely different problem, and someone said excitedly, "We could use Jake's Midpoint Theorem here!" Sure enough - Jake's theorem fit in nicely in the problem. After we finished the proof, I said, "Now let's go back to the point where we used Jake's theorem, and pretend that Jake had never made that theorem. Let's see what would have happened."

So we went back to that point, and did the rest of the proof a different way, without relying on Jake's theorem. When we were finished, I said, "Now how many steps did Jake's theorem save us in this proof?" Suddenly, the rest of the students understood that Jake's theorem wasn't just "cool" - it was also helpful; we'd saved three steps in our proof by using it!

This happens almost every year that I teach geometry. Sometimes one of the more creative students will come up with three or four theorems of their own over the course of the year. Sometimes they turn out to be useful, and sometimes not so much. Either way, I always encourage the class to get creative. "If you can invent a theorem, and prove it, then you can use it from then on through the rest of the course."

Lesson by Mr. Twitchell

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