Last year my students struggled so much with proving trig identities. You know, this kind of thing:
Part of it is that my students never did proofs in Geometry, so they have no former experience with the idea of “proofs” (they don’t get it in any other classes either ). Another part of it is that they flip out when they see a bunch of letters and numbers and have no idea where to begin (mostly a confidence issue). But another big part of it is that my Precalculus students have forgotten (1) how to manipulate fractions (any operations with them), (2) how to factor, and (3) how to distribute or multiply polynomials (I dislike using the word FOIL, but I find myself repeating it over and over). And if they remember how to, they have no conception of checking whether their work is accurate or not and more often than not make mistakes which throw off their entire equation.
SO I had the idea of starting with what they know (or should know), e.g. and moving slowly into progressively more and more complicated equations, moving from just numbers, to variables, then trig functions. I then begin another thread where the least common denominator is less than multiplying both denominators together (i.e. relatively prime), and work through “numbers –> variables –> trig functions”. I think this is what my education grad teacher meant when she kept repeating the word “scaffold”.
Well, here are the results of my efforts from last year, and when looking through my previous lesson plans, I re-discovered it and thought “hey, I actually had a decent idea last year!” so I thought I’d share that here. I don’t remember this helping a whole lot–mostly because I had the larger issue of students simply not doing the work, but I’ve already had students exclaim “I needed this, because I was always bad at it!” which makes me feel good. (Even if it’s not helping, they think it’s helping and that’s a step…)