Student 1: “Wait. How are you interfering between the angle measurements and the side ratios?”
Student 2: “That’s what the trig ratios do!”

So very satisfying to get them to this point.  It first involves clarifying that there is a big difference between side ratios and the angle measurements (the sides don’t add up to 180, for one commonly busted belief), and then reestablishing that they are related.  Not just reestablishing: making it clear what a remarkable relationship this is.  When lightening does strike, it’s good to be there.


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