Polya's 4-Step Process
George Polya was a mathematician in the 1940s. He devised a systematic process for solving problems that is now referred to by his name: the Polya 4-Step Problem-Solving Process.
In this lesson, we will discuss each step of the Polya process while working through the solution to a problem. At the end of the lesson, you will have the opportunity to try more examples before taking your quiz.
Understanding the Problem
So, to start, let's think about a party. Sally was having a party. She invited 20 women and 15 men. She made 1 dozen blue cupcakes and 3 dozen red cupcakes. At the end of the party there were only 5 cupcakes left. How many cupcakes were eaten?
The first step of Polya's Process is to Understand the Problem. Some ways to tell if you really understand what is being asked is to:
- State the problem in your own words.
- Pinpoint exactly what is being asked.
- Identify the unknowns.
- Figure out what the problem tells you is important.
- Identify any information that is irrelevant to the problem.
In our example, we can understand the problem by realizing that we don't need the information about the gender of the guests or the color of the cupcakes - that is irrelevant. All we really need to know is that we are being asked, 'How many cupcakes are left of the total that were made?' So, we understand the problem.
Devise a Plan
Now that we understand the problem, we have to Devise a Plan to solve the problem. We could:
- Look for a pattern.
- Review similar problems.
- Make a table, diagram or chart.
- Write an equation.
- Use guessing and checking.
- Work backwards.
- Identify a sub-goal.
In our example, we need a sub-goal of figuring out the actual total number of cupcakes made before we can determine how many were left over.
We could write an equation to show what is unknown and how to find the solution:
(1 dozen + 3 dozen) - 5 = number eaten
Carry Out the Plan
The third step in the process is the next logical step: Carry Out the Plan. When you carry out the plan, you should keep a record of your steps as you implement your strategy from step 2.
Our plan involved the sub-goal of finding out how many cupcakes were made total. After that, we needed to know how many were eaten if only 5 remained after the party. To find out, we wrote an equation that would resolve the sub-goal while working toward the main goal.
So, (1 dozen + 3 dozen) - 5 = number eaten. Obviously, we would need the prior knowledge that 1 dozen equals 12.
1 x 12 = 12, and 3 x 12 = 36, so what we really have is (12 + 36) - 5 = number eaten.
12 + 36 = 48 and 48 - 5 = 43
That means that the number of cupcakes eaten is 43.
Look Back
The final step in the process is very important, but many students skip it, feeling like they have an answer so they can move on now. The final step is to Look Back, which really means to check your work.
- Does the answer make sense?
Sometimes you can add wrong or multiply when you should have divided, then your answer comes out clearly wrong if you just stop and think about it. In our problem, we wanted to know how many cupcakes were eaten out of a total of 48. We got the answer 43. 43 is less than 48, so this answer does make sense. (It would not have made sense if we got an answer greater than 48 - how could they eat more than were made?)
- Check your result.
Checking your result could mean solving the problem in another way to make sure you come out with the same answer. Basically, in mathematical terms, we are saying that 48 - 5 = 43. If we were to draw out a diagram of the 1 dozen blue cupcakes and 3 dozen red ones, then separate out the 5 that did not get eaten, we would see that we do, indeed, have 43 represented as the eaten cupcakes. Our answer checks out!
And that is all there is to Polya's 4-Step Process to Problem Solving:
- Understand
- Plan
- Carry out
- Check
Example
So how about you try? Try using Polya's 4-Step Process to solve this riddle: There are 10 people at a party. Each person must say hello to each other person exactly once. How many times is the word 'Hello' said?
Step 1 - Understand the problem
Okay, so we have 10 people saying hello, but they don't have to say hello to themselves, only to the 9 other people. I need to know how many times the word 'hello' is said. Got it.
Step 2 - Devise a plan
A diagram might be a great to show me what is happening here. If I draw the diagram as a circle with 10 points (representing each of the 10 people), I can visualize each saying hello.
Step 3 - Carry out the plan
Drawing the diagram of one person saying hello, we see that each person will have to say hello 9 times, thus there will be 10 people each saying hello 9 times. 10 x 9 = 90 hellos said.
Finally, Step 4 - Check your work
90 hellos might not make sense if there are 10 people; you might think the answer should have been 100. Well, to check our work with a problem like this, we could set up a different diagram. If we put the people in a straight line and then count them saying hello to each other one at a time, we will again see that the final tally is 90 hellos. 90 must be the correct answer. Remembering that they do not have to say hello to themselves may help you see why the answer can't be 100.
Good work.
Lesson Summary
In this lesson, we reviewed Polya's 4-Step Process for Problem Solving, which is simply a systematic process used to reach a solution to a problem.
- Understand the Problem
Restating the problem and identifying necessary information is a key to this step. - Devise a Plan
Use equations, diagrams, tables or any other tool needed to create a plan for solving the problem. - Carry Out Your Plan
Just do it! - Look Back
This means to review your work to double check your answer.
If you use these four steps when you approach any problem, be it math or otherwise, you will find your path to the solution much more direct and easy. Good luck!
Learning Outcomes
Following this lesson, you should have the ability to:
- Describe the steps in Polya's 4-Step Process for Problem Solving
- Explain the importance of having a plan to solve problems
- Apply Polya's process to problems