Chapter 1 The Art of Problem Solving Copyright

Section 1 -3 Strategies for Problem Solving Copyright © 2016, 2012, and 2008 Pearson

Strategies for Problem Solving - Objectives: • Know George Polya’s four-step method of problem solving. • Be able to apply various strategies for solving problems. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 3

A General Problem-Solving Method p. 19 Polya’s Four-Step Method Step 1 Understand the problem. Read analyze carefully. What are you to find? Step 2 Devise a plan. Step 3 Carry out the plan. Be persistent. * Step 4 Look back and check. Make sure that your answer is reasonable and that you’ve answered the question. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 4

Strategies for Problem Solving p. 20 Make a table or a chart. Look for a pattern. Solve a similar, simpler problem. Draw a sketch. Use inductive reasoning. Write an equation and solve it. If a formula applies, use it. Work backward. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 5

Strategies for Problem Solving Guess and check. Use trial and error. Use common sense. Look for a “catch” if an answer seems too obvious or impossible. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 6

Example: Using a Table or Chart p. 20 A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring but each month thereafter produced one new pair of rabbits. If each new pair produced reproduces in the same manner, how many pairs of rabbits will there be at the end of the 5 th month? Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 7

Example: Solution Step 1 Understand the problem. How many pairs of rabbits will there be at the end of five months? The first month, each pair produces no new rabbits, but each month thereafter each pair produces a new pair. Step 2 Devise a plan. Construct a table to help with the pattern. Month Number of Pairs at Start Number of Pairs Produced at the End Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 8

Example (solution continued) Step 3 Carry out the plan. Month 1 st Number of Pairs at Start 1 Number Produced 0 Number of Pairs at the End 1 2 nd 1 1 2 3 rd 2 1 3 4 th 3 2 5 5 th 5 3 8 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 9

Example (solution continued) Solution: There will be 8 pairs of rabbits. Step 4 Look back and check. This can be checked by going back and making sure that it has been interpreted correctly. Double-check the arithmetic. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 10

Example: Working Backward p. 21 Ronnie goes to the racetrack with his buddies on a weekly basis. One week he tripled his money, but then lost $12. He took his money back the next week, doubled it, but then lost $40. The following week he tried again, taking his money back with him. He quadrupled it, and then played well enough to take that much home, a total of $224. How much did he start with the first week? Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 11

Example: Solution Because his final amount was $224 and this represents four times the amount he started with on the third week, we divide $224 by 4 to find that he started the third week with $56. Before he lost $40 the second week, he had this $56 plus the $40 he lost, giving him $96. The $96 represented double what he started with, so he started with $96 divided by 2, or $48, the second week. Repeating this process once more for the first week, before his $12 loss he had $48 + $12 = $60. This represents triple what he started with, so we divide $60 by 3 to find that he started with $20. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 12

Example: Solution To check, 1 st week: (3 × $20) – $12 = $60 – $12 = $48 2 nd week: (2 × $48) – $40 = $96 – $40 = $56 3 rd week: (4 × $56) = $224 (His final amount) Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 13

Example: Using Trial and Error The mathematician Augustus De Morgan lived in the nineteenth century. He made the following statement: “I was x years old in the year x 2. ” In what year was he born? (Don’t forget the question. ) Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 14

Example: Solution He lived in the nineteenth century, which means during the 1800 s. Find a perfect square that is between 1800 and 1900. 422 = 1764 432 = 1849 442 = 1936 43 is the only natural number that works. De Morgan was 43 in 1849. Subtract 43 from 1849 to get that he was born in 1806. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 15

Example: Considering a Simpler Problem p. 23 -34 The digit farthest to the right in a counting number is called the ones or units digit, because it tells how many ones are contained in the number when grouping by tens is considered. What is the ones (or units) digit in 24000? Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 16

Example: Solution Recall that 24000 means that 2 is used as a factor 4000 times. To answer the question, we examine some smaller powers of 2 and then look for a pattern. We start with the exponent 1 and look at the first twelve powers of 2. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 17

Example: Solution Notice that in any one of the four rows above, the ones digit is the same all the way across the row. The final row, which contains the exponents 4, 8, and 12, has the ones digit 6. Each of these exponents is divisible by 4 and, because 4000 is divisible by 4, we can use inductive reasoning to predict that the units digit in 24000 is 6. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 18

Example: Drawing a Sketch An array of nine dots is arranged in a 3 × 3 square as shown below. Join the dots with exactly four straight lines segments. You are not allowed to pick up your pencil from the paper and may not trace over a segment that has already been drawn. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 19

Example: Solution Through trial and error with different attempts such as: We find an

Example: Using Common Sense Two currently minted United States coins together have a total value of $1. 05. One is not a dollar. What are the two coins? * Pay attention to wording… Brainteaser type problems often rely more on common sense. Solution Our initial reaction might be, “The only way to have two such coins with a total of $1. 05 is to have a nickel and a dollar, but the problem says that one of them is not a dollar. ” This statement is indeed true. What we must realize here is that the one that is not a dollar is the nickel, and the other coin is a dollar! So the two coins are a dollar and a nickel. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 21