We drew a circle and measured the diameter and radius. The diameter was 3 and radius was 1.5. The formula was C=2ΠR so we multiplied 2 x 3.14 x 1.5 and it equaled 9.42. Then we divided 9.42/3 (diameter) and it equaled 3.14. Which is C/D and was the info that was given to us. So we proved that circumference is C=2ΠR. -A student proof.
The proof I was expecting was something like “Since Π is defined as C/D, then C=ΠD, and since D=2R, then C=2ΠR.” What’s the feedback that will cause them to be concerned about thinking in circles? The best (only) idea I’ve come up with is: “If you had multiplied 2x7x1.5 to get 21, then divided C/D = 21/3 = 7, you would have shown that Π=7.” But I have ’till Jan 4th to find something better. Any suggestions?
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First, it’s interesting to point out that they measured the diameter, then cut in half to find the radius. Then for the 2\pi r formula, they doubled it, getting the diameter back. Simply ask them if that seems strange.
Second, they are measuring, so perhaps they are looking at confirmation of the formula? Ask them. Trouble is, they say 3.14 = C/D, when they haven’t measured C. Ask them to consider or explain that, maybe.
I mean, it seems to me, this is not a finished product. It needs student editing and thought.
The question seems rather awkward to me, since pi is equally well defined as C/D or C/(2R). Asking someone to prove something trivial is usually not a good idea, since it is very difficult to tell what the teacher is looking for. It is hard to avoid circular reasoning, when the desired proof is just to state the definitions of pi and radius.
Asking them to prove that pi is >2 sqrt(2) (using just geometric arguments, not an already computed value of pi) would be a more productive introduction to proofs.
Ask, is it still true with a different radius or diameter?
My initial thought was that they were not clear about the difference in meanings between knowns and unknowns, variables and constants (I’ve been reading Torigoe), and that they don’t realize they are begging the question. My second thought is that they see algebra as a bunch of mathematical sleight of hand — a matching game where you lay out your formulas and your numbers, stir, and wait for the formulas to eject something, which must be the answer. They will not see Paul’s question as strange; they will see it as “algebra.”
My third thought that they are attempting a proof by contradiction, which is a pretty sophisticated move. Have you worked with them on the difference between a valid and invalid inference? It’s going to be awfully difficult for them to put aside their knowledge that pi is supposed to be 3.14, even if it’s not given as one of the premises for this proof.
I don’t disagree with other commenters — you might need to get away from this question, because it might not be clear enough to them why they are doing this. Another strategy: ask them to solve it symbolically (which should prevent them from using the numeric value of pi), and/or ask them to translate their thoughts (above) into symbols. Then ask if they can do it in fewer steps.
Thanks everyone:
@Paul and Ian: I will ask them about measuring C, and about changing the radius. That should get them scratching their heads about exactly what they said.
@GSWP and Mylène: I didn’t give quite enough background. The givens were π=C/D and D=2R. But combining these algebraically to get C=2πR is a brand new activity for many of my students. Two students saw it as obvious, 46 didn’t understand what they were supposed to do–and didn’t express anything like “Is that all you meant?” once shown. So they make up numbers, or they draw something and measure it, so they can substitute. I’ve gotten thinking about this from your post and comment, respectively, on Torigoe; I have a post in the making on that. [Update: it's made.]
Happy New Year to all ….
[...] the same thing. But that’s the rub: to them, it’s not the same thing!. Torigoe (Thanks, Mylène) has got me thinking about what might be going [...]