22 CHAPTER 1 Critical Thinking Skills Problem-Solving Procedures Following is a general procedure for problem solving as given by George Polya. Note that Example 1 demonstrates many of these guidelines. Large Boxes and Small Boxes Number of Tacos Cost 6 large and 0 small (6 12) (0 8) 72 × + × = (6 $14) (0 $10) $84 × + × = 5 large and 2 small (5 12) (2 8) 76 × + × = (5 $14) (2 $10) $90 × + × = 4 large and 3 small (4 12) (3 8) 72 × + × = (4 $14) (3 $10) $86 × + × = 3 large and 5 small (3 12) (5 8) 76 × + × = (3 $14) (5 $10) $92 × + × = 2 large and 6 small (2 12) (6 8) 72 × + × = (2 $14) (6 $10) $88 × + × = 1 large and 7 small × + × = (1 12) (7 8) 68 × + × = (1 $14) (7 $10) $84 0 large and 9 small × + × = (0 12) (9 8) 72 (0 $14) (9 $10) $90 × + × = Profile in Mathematics George Polya George Polya (1887–1985) was educated in Europe and taught at Stanford University. In his book How to Solve It , Polya outlines four steps in problem solving. We will use Polya’s four steps as guidelines for problem solving. GUIDELINES FOR PROBLEM SOLVING 1. Understand the problem. • Read the problem carefully at least twice. In the first reading, get a general overview of the problem. In the second reading, determine (a) exactly what you are being asked and (b) what information the problem provides. • Try to make a sketch to illustrate the problem. Label the information given. • Make a list of the given facts that are pertinent to the problem. • Determine if the information you are given is sufficient to solve the problem. 2. Devise a plan to solve the problem. • Have you seen the problem or a similar problem before? Are the procedures you used to solve the similar problem applicable to the new problem? • Can you express the problem in terms of an algebraic equation? (We explain how to write algebraic equations in Chapter 6.) • Look for patterns or relationships in the problem that may help in solving it. • Can you express the problem more simply? • Can you substitute smaller or simpler numbers to make the problem more understandable? • Will listing the information in a table help in solving the problem? • Can you make an educated guess at the solution? Sometimes if you know an approximate solution, you can work backward and eventually determine the correct procedure to solve the problem. 3. Carry out the plan. • Use the plan you devised in Step 2 to solve the problem. 4. Check the results. • Ask yourself, “Does the answer make sense?” and “Is the answer reasonable?” If the answer is not reasonable, recheck your method for solving the problem and your calculations. • Can you check the solution using the original statement? • Is there an alternative method to arrive at the same conclusion? • Can the results of this problem be used to solve other problems? PROCEDURE From the second column, we see that the maximum number of tacos that can be purchased for $93 or less is 76 tacos. This result can be obtained in two different ways: either 5 large boxes and 2 small boxes or 3 large boxes and 5 small boxes. b) From the table, we see that the most economical way to purchase 76 tacos is to purchase 5 large boxes and 2 small boxes. The cost is $90. 7 Now try Exercise 11 Associated Press
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