February 2, 2013

Proof: algebraic vs geometric

Proof seems to be the dividing line in our school between honors, regular and fundamentals courses.  In geometry the fundamentals course does no formal proof (though they are required to conjecture and justify solutions), regular level does some proof and the honors group does quite a bit of formal proof.  In PreCalculus I'm trying to have the honors class prove every rule before they use it, in contrast to the other classes.  In the regular level the teacher occasionally presents a proof but students never derive anything independently (which is not to say they never justify their work or understand where a rule comes from, but they're not going through the process of proving every formula they use).

To begin identities we proved sin^2+cos^2=1 geometrically (using triangle trig).  I then asked students to prove algebraically or geometrically: 1+cot^2=csc^2 and tan^2+1=sec^2.  Some students missed the part that once we had proved an identity, we knew it was true and could use it.  They were confused how we could start with sin^2+cos^2=1 when we wanted to prove 1+cot^2=csc^2 and instead started over with the triangles just like we had proven the first equation.  So much of proof in geometry class is random- we don't use what we ask them to prove.  But in reality, the great thing about proving something is that we now know it is true!

At NCTM I learned how to prove sin(a+b) and cos(a+b) geometrically using paper folding, which was a novel approach.  I used the setup outlined in these powerpoint slides.  Again, we used the new equations to prove some other ones algebraically: sin(a-b), cos(a-b), sin(2a), cos(2b)

After the second set of proofs I asked if students preferred algebraic or geometric proofs:

F Block:
4 algebraic
8 geometric
1 no preference

G Block:
8 algebraic
2 geometric
3 no preference

Conclusion: good thing we do some of each! (and, my response rate wasn't very good in F Block)

The next set of rules (law of sines, law of cosines, Area=1/2absin(C)) we did both geometrically and algebraically.  For these, each proof starts with a diagram which is used to set up the first equation(s), then students manipulate terms algebraically to find the rule.  I am enjoying the way algebra and geometry really complement each other in pre-calculus.  The separation of concepts into the seemingly neat categories of algebra and geometry for 3 years presents a rather false notion of mathematics as being partitioned.  In truth, the different methods support each other and when a mathematician hits a dead end using one representation, they switch to another, which often provides a hidden insight.  I've convinced students (some more than others) to be flexible in their approach- going from the unit circle, to a graph, to a triangle, to identities as they work to solve trig problems.

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