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MU MATH 100 - The Art of Problem Solving

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Slide 1Chapter 1: The Art of Problem SolvingSlide 3Strategies for Problem SolvingA General Problem-Solving MethodExample: Using a Table or ChartExample: SolutionExample (solution continued)Slide 9Example: Working BackwardSlide 11Slide 12Example: Using Trial and ErrorSlide 14Example: Guessing and CheckingSlide 16Example: Considering a Simpler ProblemSlide 18Slide 19Example: Drawing a SketchSlide 21Example: Using Common SenseChapter 1The Art of Problem Solving © 2008 Pearson Addison-Wesley.All rights reserved© 2008 Pearson Addison-Wesley. All rights reserved1-3-2Chapter 1: The Art of Problem Solving1.1 Solving Problems by Inductive Reasoning 1.2 An Application of Inductive Reasoning: Number Patterns 1.3 Strategies for Problem Solving 1.4 Calculating, Estimating, and Reading Graphs© 2008 Pearson Addison-Wesley. All rights reserved1-3-3Chapter 1Section 1-3Strategies for Problem Solving© 2008 Pearson Addison-Wesley. All rights reserved1-3-4Strategies 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 reserved1-3-5A General Problem-Solving MethodPolya’s Four-Step MethodStep 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 reserved1-3-6Example: Using a Table or ChartA 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 5th month?© 2008 Pearson Addison-Wesley. All rights reserved1-3-7Example: SolutionStep 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 StartNumber ProducedNumber of Pairs at the End© 2008 Pearson Addison-Wesley. All rights reserved1-3-8Example (solution continued)Step 3 Carry out the plan. Month Number of Pairs at StartNumber ProducedNumber of Pairs at the End1st 1 0 12nd 1 1 23rd 2 1 34th 3 2 55th 5 3 8© 2008 Pearson Addison-Wesley. All rights reserved1-3-9Example (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.© 2008 Pearson Addison-Wesley. All rights reserved1-3-10Example: Working BackwardStart 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 reserved1-3-11Example: SolutionStep 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 reserved1-3-12Example: SolutionSolutionThe 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 reserved1-3-13Example: Using Trial and ErrorThe 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?© 2008 Pearson Addison-Wesley. All rights reserved1-3-14Example: SolutionHe lived in the nineteenth century, which means during the 1800s. Find a perfect square that is between 1800 and 1900. 42 2 = 176443 2 = 184944 2 = 193643 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 reserved1-3-15Example: Guessing and CheckingFind a positive natural number that satisfies the equation below.248xx x- =© 2008 Pearson Addison-Wesley. All rights reserved1-3-16Example: SolutionTry this by guess and check:24If 4 : 4 4 48x = - =164(2) 48� - �29If 9 : 4 9 98x = - =814(3) 98� - �216If 16 : 4 16 168x = - =2564(4) 168� - =Solutionx = 16 satisfies the equation.© 2008 Pearson Addison-Wesley. All rights reserved1-3-17Example: Considering a Simpler ProblemWhat is the ones (or units) digit in 3200?© 2008 Pearson Addison-Wesley. All rights reserved1-3-18Example: SolutionStep 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 reserved1-3-19Example: SolutionSolutionThe 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 reserved1-3-20Example: Drawing a SketchAn 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.© 2008 Pearson Addison-Wesley. All rights reserved1-3-21Example: SolutionThrough trial and error with different attempts such asWe find an answer is© 2008 Pearson Addison-Wesley. All rights reserved1-3-22Example: Using Common SenseTwo currently minted United States coins together have a total value of $0.30. One is not a quarter. What are the two coins?SolutionThis involves a “catch.” The two


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