Problem Solving (Homework and Tests)
- The higher the math class, the more types of problems: in earlier classes, problems often required just one step to find a solution. Increasingly, you will tackle problems which require several steps to solve them. Break these problems down into smaller pieces and solve each piece - divide and conquer!
- Problem types:
- Problems testing memorization (”drill”),
- Problems testing skills (”drill”),
- Problems requiring application of skills to familiar situations (”template” problems),
- Problems requiring application of skills to unfamiliar situations (you develop a strategy for a new problem type),
- Problems requiring that you extend the skills or theory you know before applying them to an unfamiliar situation
- When you work problems on homework, write out complete solutions, as if you were taking a test. Don’t just scratch out a few lines and check the answer in the back of the book. If your answer is not right, rework the problem; don’t just do some mental gymnastics to convince yourself that you could get the correct answer. If you can’t get the answer, get help.
- The practice you get doing homework and reviewing will make test problems easier to tackle.
Tips on Problem Solving
- Apply Pólya’s four-step process:
- The first and most important step in solving a problem is to understand the problem, that is, identify exactly which quantity the problem is asking you to find or solve for (make sure you read the whole problem).
- Next you need to devise a plan, that is, identify which skills and techniques you have learned can be applied to solve the problem at hand.
- Carry out the plan.
- Look back: Does the answer you found seem reasonable? Also review the problem and method of solution so that you will be able to more easily recognize and solve a similar problem.
- Some problem-solving strategies: use one or more variables, complete a table, consider a special case, look for a pattern, guess and test, draw a picture or diagram, make a list, solve a simpler related problem, use reasoning, work backward, solve an equation, look for a formula, use coordinates.
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