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

A General Problem-Solving Method Polya’s Four-Step Method Step 1 Understand the problem. Read and 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. © 2008 Pearson Addison-Wesley. All rights reserved

Strategies for Devising a Plan Use a Table or Chart Look for a Pattern Solve a Similar Simpler Problem Draw a Sketch Use Inductive Reasoning Write an equation and solve it Use a formula Working Backward Using Trial and Error Guess and Check Using Common Sense Look for a “catch” if an answer seems too obvious © 2008 Pearson Addison-Wesley. All rights reserved

Example: Working Backward or Use Equation 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? © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution (working backwards) 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. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution (working backwards) The original unknown number was 12. Step 4 Look back and check. We can take 12 and run through the computations to get 60. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution (using an equation) Plan: Set up an equation. Carry out the plan: Let x = unknown number Triple the number and subtract 5: 3x – 5 Double the new number and subtract 47: 2(3x – 5) – 47 Quadruple this number and set it equal to 60: 4[2(3x - 5) - 47] = 60 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Solution (using an equation) Carry out the plan: Solve the equation: 4[2(3x - 5) - 47] = 60 x = 12 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Common Sense I’m thinking of a number between 4 and 10. If I square the number and find the difference of the digits I will end up with the cube root of the number. What is my number? Test Question © 2008 Pearson Addison-Wesley. All rights reserved

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? Test Question © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Solution He lived in the nineteenth century, which means during the 1800s. 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. © 2008 Pearson Addison-Wesley. All rights reserved

Example: Considering a Simpler Problem / Pattern What is the ones (or units) digit in 3200? Test Problem © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved 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 3s. 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. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Solution 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. © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Use a table You have brought two unmarked buckets to a stream. The buckets hold 7 gallons and 3 gallons of water, respectively. How can you obtain exactly 5 gallons of water to take home? Solution Lg 7 4 1 5 Sm 3 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Look for a pattern Find the missing digit x 3 2 4 8 7 1 X 5 6 9 © 2008 Pearson Addison-Wesley. All rights reserved

Example: Look for a pattern Solution 3 2 4 8 3 x 8 = 24 7 1 7 x 3 = 21 X 5 8 x 5 = 40 6 9 4 x 9 = 36 so X must be 0 © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Magic Square Use each number 5 through 13 once to complete the magic square 8 6 9 11 5 Test Problem © 2008 Pearson Addison-Wesley. All rights reserved

© 2008 Pearson Addison-Wesley. All rights reserved Example: Magic Square Solution 8 13 6 7 9 11 12 5 10 © 2008 Pearson Addison-Wesley. All rights reserved