Chapter 1 The Art of Problem Solving 2008

Chapter 1: The Art of Problem Solving 1. 1 1. 2 1. 3 1. 4 Solving Problems by Inductive Reasoning An Application of Inductive Reasoning: Number Patterns Strategies for Problem Solving Calculating, Estimating, and Reading Graphs 2 © 2008 Pearson Addison-Wesley. All rights reserved

Chapter 1 Section 1 -3 Strategies for Problem Solving © 2008 Pearson Addison-Wesley. All

Strategies for Problem Solving • • A General Problem-Solving Method Using a Table or Chart Working Backward Using Trial and Error Guessing and Checking Considering a Similar Simpler Problem Drawing a Sketch Using Common Sense © 2008 Pearson Addison-Wesley. All rights reserved 4

A General Problem-Solving Method 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. 5 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Using a Table or Chart 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? 6 © 2008 Pearson Addison-Wesley. All rights reserved

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 Produced Number of Pairs at the End 7 © 2008 Pearson Addison-Wesley. All rights reserved

Example (solution continued) Step 3 Carry out the plan. Month Number of Pairs at Start 1 st 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 8 © 2008 Pearson Addison-Wesley. All rights reserved

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. 9 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Working Backward Start with an unknown number. Triple it and then subtract 5. Now, take the new number and double it but then subtract 47. If you take this latest total and quadruple it you have 60. What was the original unknown number? 10 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution Step 1 Understand the problem. We are looking for a number that goes through a series of changes to turn into 60. Step 2 Devise a plan. Work backwards to undo the changes. Step 3 Carry out the plan. The final amount was 60. Divide by 4 to undo quadruple = 15. Add 47 to get 62, then divide by 2 = 31. Add 5 to get 36 and divide by 3 = 12. 11 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution The original unknown number was 12. Step 4 Look back and check.

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? 13 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution He lived in the nineteenth century, which means during the 1800 s. Find a perfect square that is between 1800 and 1900. 42 2 = 1764 43 2 = 1849 44 2 = 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. 14 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Guessing and Checking Find a positive natural number that satisfies the equation below.

Example: Solution Try this by guess and check: Solution x = 16 satisfies the

Example: Considering a Simpler Problem What is the ones (or units) digit in 3200?

Example: Solution Step 1 Understand the problem. We are looking for the last digit if 3200 is multiplied out. Step 2 Devise a plan. Look for a pattern with multiplication by 3 s. Step 3 Carry out the plan. 31 = 3, 32 = 9, 33 = 27, 34 = 81 35 = 243, 36 = 729, 37 = 2187, 38 = 6561, … Notice that if the power is divisible by 4 then the units digit is a 1. 18 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution The units digit in 3200 is 1 because the power, 200, is divisible by 4. Step 4 Look back and check. We can try a few more powers of 3 to make sure that the pattern continues and also check the multiplication. 19 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Drawing a Sketch An array of nine dots is arranged in a 3 x 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. 20 © 2008 Pearson Addison-Wesley. All rights reserved

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 $0. 30. One is not a quarter. What are the two coins? Solution This involves a “catch. ” The two coins are a quarter and a nickel. Note that one of the coins is not a quarter, it is a nickel. 22 © 2008 Pearson Addison-Wesley. All rights reserved