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Extraneous Solutions – Part 1 of 3?
Disclaimer Within my small inner circle of math teachers, the mystery of extraneous solutions seems to be the issue of the year. I have so much to say on this topic (algebraic, logical, pedagogical,...
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Extraneous Solutions – Part 2 of 3
Solving an Equation as a Sequence of Equation Replacement Operations Part 1 was so long because I wanted to be extremely thorough and to present things to an audience that perhaps hadn’t thought much...
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The Product Rule
At some point in every calculus class, we must discover and prove the product rule for derivatives. How a calculus teacher chooses to do this probably says a lot about their pedagogy and educational...
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Derivatives of Trigonometric Functions
First, let’s present the standard approach. This is from the calculus textbook I teach out of. This was, as far as I was concerned, the only possible proof. The pedagogical flexibility lay entirely in...
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Geometric Proofs of Trigonometric Identities
Sparked by a conversation this past weekend about the usefulness of the half-angle identities, I constructed geometric proofs for and . Since I’ve never seen these anywhere before, I thought I’d share....
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