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"The student and I worked through problems concerning deducing angle and side measures of special right triangles, 30-60-90, and 45-45-90. As her class as not yet begun to use trigonometry, we solved for them using specific equations given by her teacher. The equations required us to identify the shortest and longest base sides of the 30-60-90 triangles, and then work with the equations to find the answer. Her biggest problem was with the logic of deducing which side must be longer based on the angle that was opposite of it. However, we shortly figured out the trick to that, and had it solved.
We then moved on to finding the area on the inside of any common polygon, using the equation Area= (1/2)(apothem)(perimeter). The problems gave us very limited information, like just the apothem, or just the radius of the shape, and we had to find everything else from there. She worked through with the conceptual problem of carving up these polygons into right triangles, identifying the radii and apothems of the shapes, and multiplying square roots. After walking through two examples with her step by step, she started to gain confidence, and after about 5 additional examples, she was confidently solving the problems with very little help. We had a light bulb moment where I could tell that she was finally seeing the smaller triangles that we had carved out in the context of the area they were taking up in the greater shape as a whole, and how that problem solving fit into the specific steps of solving that particular shape.
The most effective strategy for her was making sure that we labeled everything before making any calculations, so for example, if we were given the right angle and the 60 degree angle, we would go ahead and write in the third angle's measure, just to keep the big picture in mind.
The session went very well as a whole. The student has a test on this subject this week, and she will practice with the remaining book problem examples, of which there are many, and be able to reference the notes of the work that she has already done. The biggest win of the session was her attitude shift, which was uncertain at the start, to gaining some confidence in her own abilities and being able to work through the problems independently at the end of the session. "