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What’s the fastest way of helping this student?

Thanks to Jonathan Newman for the submission. You should click on his name to go to his blog.

  • Forget about sine and cosine. Provide a few fraction division problems like 1/2 div by 2/3, then perhaps x/1 div by 1/x, and see if the student then realizes how the same operations apply here.

  • The error is that they cancel sinx and 1/sinx as if they were multiplied, but instead are divided. I think the confusion here is that converting from (co)secant to (co)sine involves reciprocals, and then they’re divided so you need to use reciprocals *again*.

    Mechanically, I think writing “double decker” fractions, a/b/c/d where the middle bar is longer, will help reinforce the distinction between multiplication and division. The student needs to be comfortable with this notation using plain numbers before trig identities. Alternatively, write sinx/1 * 1/cscx …, making the division flipping first before worrying about simplifying trig identities. The รท symbol isn’t used much in algebra precisely because of confusion like this. Division is a fraction. It “feels” different.

    Analytically, the student should look at the identity and see that it is similar to the Pythagorean identity. They should see that this will become that before diving into blind symbol manipulation.

  • It might help a bit to simply rewrite every step in full, rather than trying to keep track of things in the student’s head. That might not help with the cancelling in division, but at least it would help with the adding 1 and 1 to get 1.

    I also wonder if the forgetting it needs to be switched to multiplication in order to simplify isn’t something the student would immediately recognize as a silly mistake that they made. In which case it definitely would help to simply see simpler examples for a bit.