One recent problem I had to solve that I think exhibits some good problem solving technique was on my last Precalc II exam. It was a simple multiple choice problem that had to do with solving the triangle (finding the length of all of the side and the measure of all of the angles). This sounds easy but I blanked out on the law of cosines equation that was supposed to be used to solve this problem. I felt foolish after the exam but I did not let it stop me from finding the correct answer as opposed to completely guessing. Instead, I looked at the as it was and I ignored the fact that the intention was that students would use the law of cosines to solve it. I looked at it with the goal of solving the triangle with the information given. I do not remember the exact values given in the problem but I remember that the length of two of the sides of the triangle were given and used that fact that the angle corresponding to the longer side would have to be larger which allowed me to eliminate 2 of the four choices. This made it a 50/50 chance as opposed to ¼. Then I looked at the fact that the angle given was larger than the two in left over solutions and therefore the missing side must be larger than the other two and then the choices for the right answer were brought down to one without even using the law of cosines. I feel my approach to that problem attacked the idea of functional fixedness and mental sets ( I was able to ignore the fact that the law of cosines was the equation--algorithm--that would yield the right answer 100% of the time and I was still able to work my way to the right answer). In this case a mental set wasn't that bad to have either because I was able to use the law of sines that I used to solve the previous problems on the exam to check my answer to the problem I just discussed.
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