Polya’s Four Steps in Problem Solving

Step 1) Understand the Problem Read problem and then write down what you know and

determine what you are trying to find out

Step 2) Devise a Plan Tackle the problem using one of the following strategies:

Inductive reasoning to find a pattern, make a list or table, use trial and error, make a sketch or diagram, relate to a similar problem you have solved before, use given information to eliminate possibilities, etc.

Polya’s Four Steps in Problem Solving

Step 3) Carry out the plan and solve the problem

Solve the problem using your strategy

Step 4) Look back and check your answer

Answer should satisfy the conditions of the problem

Answer should make sense in context to the problem

Problem SolvingQuestions to ask yourself before solving:

Is there enough information to solve the problem?

Is the information relevant to the problem’s solution?

Are some facts not necessary to arrive at a solution?

What is missing? Which necessary piece of information is missing and

prevents you from solving the following problem?

“The bill for your meal totaled $20.36, including the tax. How much change should you receive from your cashier?”

“If a steak sells for $8.15, what is the cost per pound?”

What is unnecessary? One more piece of information is given than is

necessary for solving the problem. Identify it and then solve the problem.

“A manufacturer packages its apple juice in bottles and boxes. A 128-ounce bottle costs $5.39, and a 9-pack of 6.75 ounce boxes costs $3.15. If you need 20 ounces of apple juice, which packaging option is the better value?”

Step 1) Understand the Problem

Given

Bottle – 128 ounces: $5.39

Box – Nine 6.75 ounces: $3.15

Needed: 20 ounces

Step 2) Device a Plan

The one with the better value is the one that has the lower price per ounce. We want to find and compare the cost per ounce for the bottle and boxes. The cost per ounce is the unit price and is calculated by dividing the price of the juice by the number of ounces.

The fact that we need 20 ounces of juice is the unnecessary piece of information.

Step 3) Carry out the plan and solve the problem

Bottle

Box

What is the better option?

Note:

Step 4) Look back and check your answer

The product of each unit price and the number of ounces per box/bottle should give you back the price for the original box/bottle package.

Note: Use unrounded result for unit price

Bottle Box

The unit prices satisfy the problem’s conditions

Apply the Four-Step Procedure

“By paying $350 cash up front and the balance at $45 per month, how long will it take to pay for a computer costing $980?”

Note: Balance is the amount still to be paid

“By paying $350 cash up front and the balance at $45 per month, how long will it take to pay for a computer costing $980?”

Balance after initial Payment: $980 - $350 = $630

Monthly Payment: $45/month

Solution: Divide $630 balance by $45, the payment per month

Solve by Making a List “Suppose you are an engineer programming the

automatic gate for a 30-cent toll. The gate should accept exact change only. It should not accept pennies. How many coin combinations must you program the gate to accept?”

Solve by Making a ListList all combinations that add up to 30 cents:

5 combinations

Solve by Using a Diagram

“Your wardrobe is limited to two pairs of jeans (one blue and one black) and three T-shirts (one red, one yellow, and one blue) to choose from. How many different outfits can you form?”

Solve by Using a Diagram

Jeans T-Shirt Possibilities

BlueRedYellowBlue

Blue-RedBlue-YellowBlue-Blue

BlackRedYellowBlue

Black-RedBlack-YellowBlack-Blue

6 outfits

Solving with More Than One Solution

“A sales director lives in city A and is required to fly to regional offices in cities B, C, D, and E. Other than starting and ending the trip in city A, there are no restrictions as to the order in which the other four cities are visited. Give the sales director an order for visiting each city once and then returning back home to city A that costs less than $1460.”

One-way airfares:

Magic Square A square array of numbers arranged so that the

numbers in all rows, columns, and the two diagonals have the same sum.

Magic Square

#47a (Page 40) #48a (Page 40)

Practice ProblemsPage 37

What is Missing? #3, 4

Identify Unnecessary Info: #5, 7

Polya’s Method: #9, 11, 13, 25, 28, 29, 31, 35, 37, 40, 41, 44, 47, 48, 50