In my Pre-calc class, I lectured. I thought I could pull them in at various steps along the way, but class felt a bit boring to me today. I wondered if two days in a row of facing front put them back into that passive student mode. As part of their homework I told them to email me with one thing they like about this class, one thing they don't, one thing they wish were different, and one question. 10-15 students wrote to me, and I started a list of pros and cons. I want to get their feedback so I know what the quieter students are thinking.
In Calc II, I had asked them (as part of their homework) to either figure out a proof of the Pythagorean Theorem or look it up. Only 4 of them had done that (about what I expected), but 3 of the 4 gave this...
The 4th said she had a 'proof using words', but it was also just a description. I was amazed. They seem to have no conception of the meaning of proof. One student was making a very accurate diagram of a 3-4-5 triangle with the squares attached, and when I called it an example, he said he could change the sides to a, b, and c.
I asked the class, "How do we know this is true?" And they said we could measure it. I asked, "What if the third side is 4.9 inches, instead of 5 inches? Or 4.95?" They had trouble seeing why measurement wasn't enough. I'm going to learn so much from this class.
I showed them a visual proof and an algebraic proof, both starting with this tilted square inside a square. In the visual proof, you swing two of the triangles around, so the 4 triangles make two rectangular areas. What's left is one square area with side length a, and another with side length b. So elegant.
The algebraic proof describes the total area two ways:
(a+b)2= 4*(1/2*a*b)+c2
A few simple algebra steps will do it.
It felt important to discuss the meaning of proof. I have no idea if it stuck, and I'd like to come back to this during this course.
Edit on 9-5: Unknown pointed me to another great Vi Hart video. She does pretty much the same proof I showed above, but using paper that she folds (and rips). It's a great demo.
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The 4th said she had a 'proof using words', but it was also just a description. I was amazed. They seem to have no conception of the meaning of proof. One student was making a very accurate diagram of a 3-4-5 triangle with the squares attached, and when I called it an example, he said he could change the sides to a, b, and c.
I asked the class, "How do we know this is true?" And they said we could measure it. I asked, "What if the third side is 4.9 inches, instead of 5 inches? Or 4.95?" They had trouble seeing why measurement wasn't enough. I'm going to learn so much from this class.
I showed them a visual proof and an algebraic proof, both starting with this tilted square inside a square. In the visual proof, you swing two of the triangles around, so the 4 triangles make two rectangular areas. What's left is one square area with side length a, and another with side length b. So elegant.
The algebraic proof describes the total area two ways:
(a+b)2= 4*(1/2*a*b)+c2
A few simple algebra steps will do it.
It felt important to discuss the meaning of proof. I have no idea if it stuck, and I'd like to come back to this during this course.
Edit on 9-5: Unknown pointed me to another great Vi Hart video. She does pretty much the same proof I showed above, but using paper that she folds (and rips). It's a great demo.
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