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Mastering Problem Solving Techniques in Mathematics

Learn Polya's Four-Step Problem-Solving Procedure: Understand the problem, Devise a plan, Carry out the plan, Check the answer. Practice solving problems involving diagrams, trial and error, money, and calculations. Apply problem-solving skills to real-world scenarios in this interactive math learning session.

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Mastering Problem Solving Techniques in Mathematics

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  1. Math in Our World Section 1.3 Problem Solving

  2. Learning Objectives • State the four steps in the basic problem solving procedure. • Solve problems by using a diagram. • Solve problems by using trial and error. • Solve problems involving money. • Solve problems by using calculation.

  3. Polya’s Four-Step Problem-Solving Procedure • Step 1 Understand the problem. Read the problem slowly, jotting down the key ideas • Step 2 Devise a plan to solve the problem. Draw a diagram, find a formula, look for patterns • Step 3 Carry out the plan to solve the problem. Solve the problem, follow the numbers, create an equation • Step 4 Check the answer. Does your answer make sense? Did you solve for the requested unknown?

  4. EXAMPLE 1 Solving a Problem by Using a Diagram A gardener is asked to plant eight tomato plants that are 18 inches tall in a straight line with 2 feet between each plant. How much space is needed between the first plant and the last one? Be careful—what seems like an obvious solution is not always correct! You might be tempted to just multiply 8 by 2, but instead we will use Polya’s method.

  5. EXAMPLE 1 Solving a Problem by Using a Diagram SOLUTION • Step 1 Understand the problem. In this case, the key information given is that there will be eight plants in a line, with 2 feet between each. We are asked to find the total distance from the first to the last. • Step 2 Devise a plan to solve the problem. When a situation is described that you can draw a picture of, it’s often helpful to do so. • Step 3 Carry out the plan to solve the problem. The figure would look like this: Now we can use the picture to add up the distances: 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14 feet • Step 4 Check the answer. There are eight plants, but only seven spaces of 2 feet between them. So 7 x 2 = 14 feet is right.

  6. EXAMPLE 2 Solving a Problem Using Trial and Error Suppose that you have 10 coins consisting of quarters and dimes. If you have a total of $1.90, find the number of each type of coin.

  7. EXAMPLE 3 Solving a Problem Involving Salary So you’ve graduated from college and you’re ready for that first real job. In fact, you have two offers! One pays an hourly wage of $19.20 per hour, with a 40-hour work week. You work for 50 weeks and get 2 weeks’ paid vacation. The second offer is a salaried position, offering $41,000 per year. Which job will pay more?

  8. EXAMPLE 4 Solving a Problem by Using Calculation A 150-pound person walking briskly for 1 mile can burn about 100 calories. How many miles per day would the person have to walk to lose 1 pound in one week? It is necessary to burn 3,500 calories to lose 1 pound.

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