# Tool for the Field: Polya’s Problem-Solving Method

## Purpose

The purpose of this tool for the field is to help paraprofessionals become more familiar with, and practice using, Polya’s four-step problem-solving method.

## Procedures

1. Read the section below entitled “Background Information,” and familiarize yourself with the chart of Polya’s four-step problem-solving method.
2. Read the example below about Mrs. Byer’s class, and then look over the example of how Polya’s method was used to solve the problem.
3. Choose one of the following problems and try to solve it yourself using Polya’s method:
1. Every person at a party of 12 people said hello to each of the other people at the party exactly once. How many “hellos” were said at the party?
2. A new burger restaurant offers two kinds of buns, three kinds of meats, and two types of condiments. How many different burger combinations are possible that have one type of bun, one type of meat, and one condiment type?
3. A family has five children. How many different gender combinations are possible, assuming that order matters? (For example, having four boys and then a girl is distinct from having a girl and then four boys.)
4. Hillary and Marco are both nurses at the city hospital. Hillary has every fifth day off, and Marco has off every Saturday (and only Saturdays). If both Hillary and Marco had today off, how many days will it be until the next day when they both have off?
4. Reflect on your experience. In which types of situations do you think students would find Polya’s method helpful? Are there types of problems for which students would find the method more cumbersome than it is helpful? Can you think of any students who would particularly benefit from a structured problem-solving approach such as Polya’s?

## Background Information

Nearly 100 years ago, a man named George Polya designed a four-step method to solve all kinds of problems: Understand the problem, make a plan, execute the plan, and look back and reflect. Because the method is simple and generalizes well, it has become a classic method for solving problems. In fact, the method is applicable to all areas of our lives where we encounter problems—not just math. Although the method appears to be a straightforward method where you start at Step 1, and then go through Steps 2, 3, and 4, the reality is that you will often need to go back and forth through the four steps until you have solved and reflected on a problem.

## Polya’s Problem-Solving Chart: An Example

A version of Polya’s problem-solving chart can be found below, complete with descriptions of each step and an illustration of how the method can be used systematically to solve the following problem:

There are 22 students in Mrs. Byer’s third grade class. Every student is required to either play the recorder or sing in the choir, although students have the option of doing both. Eight of Mrs. Byer’s students chose to play the recorder, and 20 students sing in the choir. How many of Mrs. Byer’s students both play the recorder and sing in the choir?

 Step Number Description of Step Example 1. Understand the problem. Figure out what is being asked. What is known? What is not known? What type of answer is required? Is the problem similar to other problems you’ve seen? Are there any important terms for which you should look up definitions? There are 22 total students. There are three groups of students: Students who only play recorder, students who only sing in choir, and students who do both. Initially, we do not know how many students are in any of these groups, but we know the total of the three groups adds up to 22. We also know that a total of 8 students play the recorder, and a total of 20 students sing in the choir. We must find the number of students who do both. 2. Make a plan. Come up with some strategies for solving the problem. Common strategies include making a list, drawing a picture, eliminating possibilities, using a formula, guessing and checking, and solving a simpler, related problem. We could list out the 22 students and then assign to each either recorder, choir, or both until we got the right totals. We could draw a Venn Diagram that separates out the three types of groups. We could try solving a similar problem with a class of fewer students. 3. Execute the plan. Use the strategy chosen in Step 2 to solve the problem. If you encounter difficulties using the strategy, you may want to use resources such as the textbook to help. If the strategy itself appears not to be working, return to Step 2 and select a different strategy. Let’s try solving a similar problem with a class of 6 students, 5 of whom play recorder and 3 of whom are in the choir. In this case, we know that there is only one student who doesn’t play recorder, and so this student must sing in the choir. That means the other two choir singers must play the recorder, so there are 2 students who do both. Now, let’s try that same method with the original problem. Since only 8 of the 22 students play recorder, the other 14 must sing in the choir and not play recorder. But there are 20 students in the choir, so 6 of these choir students also play the recorder. So the answer is 6. 4. Look back and reflect. Part of Step 4 is to find a way to check your answer, preferably using a different method than what you used to solve the problem. Another part of Step 4 is to evaluate the method you used to solve the problem. Was it effective? Are there ways you could have made it more effective? Are there other types of problems with which you might be able to use this type of solution method? Let’s check our answer with a Venn Diagram, which was one of the other strategies we considered in Step 2. We first fill in each region based on the results we found in Step 3. Now we check to see if the numbers match the original problem. Notice that 2 + 6 + 14 = 22 total students, 2 + 6 = 8 students playing the recorder, and 6 + 14 = 20 students in choir. So our answer checks out! Looking back on our answer, we now see that our process of subtracting from the total can be used in any similar situation, as long as all students must be in at least one of the two groups. In the future, we wouldn’t even have to use the simpler related problem since we’ve found a more general pattern!