PROBLEM SOLVING Polya’s 4 Steps to Problem Solving 1. 2. 3. 4. Understand the problem Devise a plan. Strategies to use? Carry out the plan Look back! Does your response answer the question? Does your response seem correct? Guess and Test • Clues suggesting Guess and Test strategy – – – – Limited number of possible answers to check Have a good idea what the answer is Can systematically try all possibilities No other obvious strategy to try Use a Variable Variable: A letter used in place of numbers. • Clues suggesting Use a Variable strategy – – – – Problem suggests an equation A proof or a general solution is required An unknown quantity is related to know ones Problem says things like “for any #”, “consecutive”, “odd”, “even”. Draw a Picture • Clues suggesting Draw a Picture strategy – – – – Problem involves a physical situation Involves geometric figures or measurements To gain a better understanding of the problem Visual representation of the problem is possible Is the sum of five consecutive numbers a multiple of 5? ____ _____ _____ _____ _____ _x__ _x+1_ _x+2_ _x+3_ _x+4_ = 5x + 10 5x + 10 = 5 ( x + 2) A rectangular field has a perimeter of 220 meters and an area of 2,856 square meters. What are its dimensions? 220 m = l + w + l + w = 2 l + 2 w l l + w = 110 50 x 60 = 3000 40 x 70 = 2800 42 x 68 = 2856 w w l Ans: 42 m by 68 m It takes 64 cubes to fill a cubical box that has no top. How many cubes are NOT touching a side or the bottom? 16 16 It takes 64 cubes to fill a cubical box that has no top. How many cubes are NOT touching a side or the bottom? 16 16 It takes 64 cubes to fill a cubical box that has no top. How many cubes are NOT touching a side or the bottom? 8 16 16 8 It takes 64 cubes to fill a cubical box that has no top. How many cubes are NOT touching a side or the bottom? 8 16 + 16 + 8 + 8 + 4 = 52 16 16 8 4 64 – 52 = 12 After the first two terms in the sequence, each number is the sum of the preceding two terms. Find the missing numbers. 4 ___ ___ ___ ___ 67 Tom and Barb have only nickels and dimes. Barb has 75 cents and Tom has 90 cents. Both have the same number of coins and Tom has the same number of dimes as Barb has nickels. How many nickels does Barb have? The surface of Big Lake is 31 feet above the surface of Long Lake. Long Lake is half as deep as Big Lake and the bottom of Long Lake is 8 feet below the bottom of Big Lake. How deep is each lake? Will the sum of any n consecutive integers have a factor of n?