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George Polya’s Four Steps in Problem Solving

Aug 02, 2014

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By: Taylor Schultz MATH 3911. George Polya’s Four Steps in Problem Solving. George Polya. George Polya was a teacher and mathematician. Lived from 1887-1985 Published a book in 1945: How To Solve It , explaining that people could learn to become better problem solvers.

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By: Taylor Schultz MATH 3911 George Polya’s Four Steps in Problem Solving

George Polya • George Polya was a teacher and mathematician. • Lived from 1887-1985 • Published a book in 1945: How To Solve It, explaining that people could learn to become better problem solvers.

Polya’s Four Steps • 1. Understand the problem. • 2. Devise a plan. • 3. Carry out the plan. • 4. Look back.

The Problem To Solve • Find the square root of 1,444 without using a calculator. • √1,444

Understanding the Problem • When first looking at a problem, you must first read the problem carefully and see if you understand it. • Ask yourself, what do you know, and what do you want to figure out? • We know that: A number b is a square root of a number a if b2 = a. • In order to find a square root of a, you need a # that, when squared, equals a. • We want to figure out: What number squared would equal 1,444. • (b2=1,444)

Devising a Plan • For this second step, you need to develop a strategy for using what you know. • Consider how the problem relates to concepts you know or other problems you have solved. • You can solve this problem by using a guess-and-check (trial and error) approach, or by using an algebraic square root method.

Devising a Plan Continued.. • So, how do we find the square root? • IT’S EASY! • Just ask what times itself is the number in the root symbol? • Examples: • √9 is 3 because 3 times 3 is 9 ( 3×3=9) • √16 is 4 because 4 times 4 is 16 ( 4×4=16) • √49 is 7 because 7 times 7 is 49 (7×7=49)

Devising a Plan: Guessing and Checking • This strategy requires you to start by making a guess and then checking how far off your answer is. • Then, you revise your guess and try again! • So, we want to know what the b is in b2 =1,444. • Plan: Find what b is to equal 1,444. (b×b=1,444)

Carrying Out the Plan • This is the step where you carry out the steps of your plan. • We have came up with the guessing and checking method, so let’s put it to use!

Carrying Out the Plan: Guessing and Checking Process • You could start by multiplying any of the two same numbers together. • Let’s try: 20×20, which equals 400. • This answer is obviously way lower than 1,444, so I’ll revise my guess and try again. • This time I’ll try: 30×30, which equals 900. • This answer is still too low, but I am getting closer. • This time I’ll try 34×34, which equals 1,156. • I am still not quite there, but I am getting closer. • I have now started to narrow down my guesses, so this time I’ll try 38×38, which equals 1,444! • Through guessing and checking, I have now figured out that b=38 (382=1,444)

Looking Back • Finally, in this last step you look back reviewing and checking your results. • Have you answered the original question? • Yes, we have answered that the √1,444=38. • Is there a way to check your answer to see if it is reasonable? • Yes, by multiplying 38×38 to equal 1,444. • Also, if you have a calculator, you can plug in the √1,444 giving you 38. • You can use this knowledge to solve related problems in the future.

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Polya’s Four Step Problem Solving Process

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• Content: Polya’s Problem-Solving Method

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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.

• 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.

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?           

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?

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.)

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?

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:

Scenario 

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?

Helping Students Do Math

• Introductory Scenario and Pre-Test
• Content: Does Anyone Know What Math Is?
• Introductory Scenario
• Content: The Fennema-Sherman Attitude Scales
• Content: Past Experience with Math
• Content: Learning About Math
• Content: What is it like to teach math?
• Content: Using a Frayer Model
• Content: Helping a Child Learn from a Textbook
• Content: Using Online Math Resources
• Content: Helping a Student Learn to use a Calculator
• Links for More Information
• Content: Better Questions
• Content: Practice Asking Good Questions
• Content: Applying Poly’s Method to a Life Decision
• Content: Learning Progression Activities
• Content: Connecting Concepts and Procedures
• Content: Resources
• Activity: The Old Guy’s No-Math Test
• Take Notes and Post-Test
• Open All · Close All

Contact: [email protected] 

The Ohio Partnership for Excellence in Paraprofessional Preparation is primarily supported through a grant with the Ohio Department of Education and Workforce, Office for Exceptional Children. Opinions expressed herein do not necessarily reflect those of the Ohio Department of Education or Offices within it, and you should not assume endorsement by the Ohio Department of Education and Workforce.

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Math Lesson 1 Polyas Problem Solving Strategy

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2.3.1: George Polya's Four Step Problem Solving Process

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Step 1: Understand the Problem

• Do you understand all the words?
• Can you restate the problem in your own words?
• Do you know what is given?
• Do you know what the goal is?
• Is there enough information?
• Is there extraneous information?
• Is this problem similar to another problem you have solved?

Step 2: Devise a Plan: Below are some strategies one might use to solve a problem. Can one (or more) of the following strategies be used? (A strategy is defined as an artful means to an end.)