Chapter 1: Arithmetic & Prealgebra Section 1.5: Problem Solving Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

The Problem Solving Process In his book, How to Solve It, George Polya outlined a step-by-step process for problem solving. Understand the Problem Devise a Plan Carry Out the Plan Look Back A fifth step should also be considered. Answer the Question Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

The Problem Solving Process Understand the Problem Do we understand all the words? Can we restate the problem in our own words? Do we know what is given? Do we know what the goal is? Is there enough information? Is there extraneous information? Is the problem similar to another problem we have solved? Draw a picture of the problem statement, if applicable. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

The Problem Solving Process Devise a Plan - Using One of the Following Strategies: Guess and Test Look for a Pattern Use a Variable Make a List Solve a Simpler Problem Draw a Picture Work Backwards Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

The Problem Solving Process Carry Out the Plan Implement the strategy or strategies we have chosen until the problem is solved or a new course of action is suggested. Give ourselves a reasonable amount of time in which to solve the problem. Don't be afraid of starting over. Often, a fresh start and a new strategy will lead to success. Look Back Is our solution correct? Does our answer satisfy the statement of the problem? Can we see an easier solution? Can we use this solution to solve other problems? Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

The Problem Solving Process Answer the Question In written or verbal responses, we should always answer the asked question in a complete sentence. Answering the question in a complete sentence forces you to examine the logistics of the situation. If you solve a problem and get an answer of 13.5, you may think you are done. If, however, the question was, “How many cats are in the barn?” forcing yourself to write, “There are 13.5 cats in the barn.” may get you to realize 13.5 may not be correct. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Strategy: Guess and Test Clues: The Guess and Test strategy may be appropriate when: There are a limited number of possible answers. We want to gain a better understanding of the problem. We have a good idea of what the answer is. We can systematically try possible answers. There is no other obvious strategy to try. Example: Fill in the blanks using some combination of the symbols +, −, ×, and ÷ to make a true statement. 7 __ 7 __ (7 __ 7) = 13 Possible answer: 7+7 − 7 ÷7 =13 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Strategy: Look for a Pattern Clues: The Look for a Pattern strategy may be appropriate when: A list of data is given. Listing special cases helps you deal with complex problems. We are asked to make a prediction or generalization. Example: Determine the pattern and then generate the next two iterations for the following. 37 ✕ 3 = 111 37 ✕ 6 = 222 37 ✕ 9 = 333 37 ✕ 12 = 444 Answer: The pattern is 37 ✕ 3n = nnn. 37 ✕ 15 = 555 37 ✕ 18 = 666 Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Strategy: Use a Variable (Algebra) Clues: The Use a Variable strategy may be appropriate when: A phrase similar to "for any number" is present or implied. A problem suggests an equation. A proof or a general solution is required. There is a large number of cases. A proof is asked in a problem involving numbers. We are trying to develop a general formula. Example: A backyard fair charged $1.00 admission for adults and $0.50 for children. The fair made $25 and sold 38 tickets. How many adult tickets were sold? Answer: If x represents the number of adult tickets, then 38 – x is the number of children tickets. Then, $1(x) + $0.50(38 – x) = $25 $1x + $19 - $0.50x = $25 $0.50x + $19 = $25 $0.50x = $6 x = $6/$0.50 = 12 12 adult tickets were sold. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates Strategy: Make a List Clues: The Make a List strategy may be appropriate when: The information can easily be organized and presented. Data can be easily generated. Asked "in how many ways" something can be done.  Example: A person has 10 coins consisting of dimes and quarters. If the person has a total of $1.90, find the number of quarters. Answer: Making a list… 10 quarters and no dimes is $2.50 9 quarters and 1 dime is $2.35 8 quarters and 2 dimes is $2.20 7 quarters and 3 dimes is $2.05 6 quarters and 4 dimes is $1.90 … The person has 6 quarters. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Strategy: Solve a Simpler Problem Clues: The Solve a Simpler Problem strategy may be appropriate when: The problem involves complicated computations. The problem involves very large or small numbers. We are asked to find the sum of a series of numbers. A direct solution is too complex. Example: A fruit stand is selling 8 apples for $1.25. At this rate, how much will 24 apples cost? Answer: Think of them in bags of 8 apples. Three bags will give you 24 apples and will cost $3.75. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Strategy: Draw a Picture Clues: The Draw a Picture strategy may be appropriate when: A physical situation is involved. Geometric figures or measurements are involved. A visual representation of the problem is possible. Example: A farmer is building a pen for his hogs. He will use the side of his barn, which is 30 feet long, as one side of the pen, and wants the width to be 12 feet. How many feet of fencing is needed? Answer: Drawing the picture… 54 ft of fencing is needed. BARN 12 ft 12 ft 30 ft Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates

Strategy: Work Backwards Clues: The Work Backwards strategy may be appropriate when: We know the answer and need to find the process. Example: Cindy was given her allowance on Monday. On Tuesday she spent $1.50 on candy. On Wednesday, Cindy found $1 on the ground. If Cindy now has $2, how much was her allowance? Answer: Working backwards, since she now has $2, that means she had $1 before the money she found. Then, since she had $1 left after spending $1.50, she started with $2.50. Applied Mathematics, 2nd Ed, Copyright 2018, Matovina & Yates