5.2: George Pólya's Strategy

George Pólya was a great champion in the field of teaching effective problem-solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.

In 1945, Pólya published the short book How to Solve It, which gave a four-step method for solving mathematical problems. It consists of four main steps:

George Pólya, a Hungarian mathematician, is renowned for his substantial contributions to problem-solving in mathematics. Born in 1887, Pólya had a prolific career that spanned several decades and included significant work in probability, combinatorics, and mathematical physics. However, he is perhaps best known for his book "How to Solve It," published in 1945, which has become a classic in mathematical education. Pólya's work laid the foundation for modern mathematical problem-solving techniques and has influenced generations of educators and students.

Pólya's problem-solving strategy is a systematic approach designed to tackle mathematical problems effectively. His method consists of four principal steps: understanding the problem, devising a plan, carrying out the plan, and looking back. This structured approach encourages critical thinking and fosters a deeper comprehension of the problem-solving process. Polya believed that by following these steps, students could develop their analytical skills and become more proficient in tackling a wide range of mathematical challenges.

Polya's Problem-Solving Strategy

1. Understand the Problem:
   • Read the problem carefully: Ensure you understand all the terms and the problem’s requirements.
   • Identify what is given and what needs to be found: Distinguish between the known and unknown variables.
   • Restate the problem in your own words: This helps clarify the problem and ensures you have grasped the main idea.
   • Draw a diagram if necessary: Visual aids can make complex problems more manageable and understandable.
2. Devise a Plan:
   • Think of possible strategies: Consider various approaches, such as working backwards, looking for a pattern, or breaking the problem into smaller parts.
   • Choose the most promising strategy: Select the method that seems most likely to lead to a solution.
   • Relate the problem to similar problems: Use insights from previously solved problems that are similar in nature.
3. Carry Out the Plan:
   • Implement the chosen strategy: Apply the steps of your plan systematically.
   • Be thorough and patient: Carefully execute each step without rushing.
   • Keep track of your work: Document each part of the process to make it easier to review and correct if necessary.
4. Look Back:
   • Review the solution: Check the results to ensure they make sense and the problem is fully solved.
   • Reflect on the process: Consider what worked well and what didn’t, which helps improve future problem-solving skills.
   • Generalize the solution: Think about how the strategy could be applied to other problems.

George Pólya's problem-solving strategy remains a cornerstone of mathematical education. By providing a clear, step-by-step method for approaching problems, Polya empowers students to develop their analytical skills and confidence in their mathematical abilities. His emphasis on understanding, planning, executing, and reflecting offers a comprehensive framework that is as applicable today as it was when he first introduced it. Polya’s legacy endures in classrooms around the world, where his strategies continue to inspire and guide students in their mathematical journeys.

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Example \(\PageIndex{1}\)

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Solution

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Table \(\PageIndex{1}\)

Chickens	Cows	Number of chicken feet	Number of cow feet	Total number of feet
20	5	40	20	60
21	4	42	16	58
Notice we are going in the wrong direction! The total number of feet is decreasing!
19	6	38	24	62
Better! The total number of feet are increasing!
15	10	30	40	70
12	13	24	52	76

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found a solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Check in question 1:

Figure \(\PageIndex{1}\)

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31.

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet. How many of each animal does he have? Make sure you use Polya’s 4 problem-solving steps.

Problem Solving Strategy 2

Example \(\PageIndex{2}\)

Problem:

A bakery sells 24 cupcakes in a day. If each cupcake costs $2, how much money does the bakery make from selling cupcakes in a day?

Solution

Step 1: Understand the Problem:

• We need to find out how much money the bakery makes from selling cupcakes in a day.
• Given information: The bakery sells 24 cupcakes in a day, and each cupcake costs $2.

Here's how a teacher might approach this step:

1. Read the Problem Aloud: The teacher reads the problem aloud to the class, ensuring that all students are listening and paying attention.
2. Clarify the Goal: The teacher asks questions to clarify the goal of the problem. For example, "What are we trying to find out in this problem?"
3. Identify the Given Information: The teacher helps students identify the information given in the problem. For example, "What do we know about the number of cupcakes sold and the price per cupcake?"
4. Discuss the Context: The teacher discusses the context of the problem to make it more relatable for students. For example, "Imagine you own a bakery and you sell cupcakes. How much money would you make if each cupcake costs $2 and you sell 24 cupcakes?"
5. Encourage Student Questions: The teacher encourages students to ask questions if they are unsure about any part of the problem.
6. Summarize the Problem: The teacher summarizes the problem in simple terms, reiterating the goal and the given information. For example, "So, we want to find out how much money the bakery makes if they sell 24 cupcakes at $2 each."

By approaching the problem in this way, the teacher helps students understand the context and purpose of the problem, making it easier for them to devise a plan for solving it.

Step 2: Devise a Plan:

• We can multiply the number of cupcakes sold by the price per cupcake to find the total amount of money made.

Here's how a teacher might approach this step:

1. Brainstorm Strategies: The teacher leads a class discussion to brainstorm different strategies that could be used to solve the problem. For example, "How can we figure out the total amount of money the bakery makes from selling cupcakes?"
2. Discuss Strategies: The teacher discusses the pros and cons of each strategy, helping students understand when each strategy might be most effective. For example, "We could multiply the number of cupcakes by the price per cupcake, or we could add $2 for each cupcake sold. Which strategy do you think would be easier?"
3. Choose a Strategy: The teacher helps students choose a strategy that they think will be most effective for solving the problem. For example, "Let's use multiplication to find the total amount of money made by the bakery."
4. Plan the Solution: The teacher guides students in planning how they will use their chosen strategy to solve the problem. For example, "We will multiply 24 cupcakes by $2 to find the total amount of money made."
5. Encourage Student Input: The teacher encourages students to share their ideas and thoughts about the chosen strategy, fostering a collaborative problem-solving environment.

By approaching the problem in this way, the teacher helps students think critically about the problem and choose a strategy that will lead to a successful solution.

Step 3: Carry Out the Plan:

• Number of cupcakes sold = 24
• Price per cupcake = $2
• Total money made = Number of cupcakes sold × Price per cupcake = 24 × $2 = $48

Here's how a teacher might approach this step:

1. Demonstrate the Solution: The teacher demonstrates how to use the chosen strategy to solve the problem. For example, the teacher might write out the multiplication problem 24 × $2 on the board and solve it step by step.
2. Model the Process: The teacher models the process of solving the problem, explaining each step as they go. For example, "First, we multiply 24 by 2 to find out how much money the bakery makes from selling one cupcake. Then, we multiply that amount by 24 to find the total amount."
3. Provide Guided Practice: The teacher provides guided practice opportunities for students to solve similar problems with support. For example, the teacher might give students a similar problem to solve in pairs or small groups, offering guidance and feedback as needed.
4. Encourage Independent Practice: Once students have a good understanding of how to use the chosen strategy, the teacher encourages them to solve similar problems independently.
5. Review the Solution: After students have solved the problem, the teacher reviews the solution with the class, ensuring that everyone understands how the problem was solved and why.

By approaching the problem in this way, the teacher helps students develop confidence in their problem-solving abilities and reinforces the importance of using a systematic approach to problem solving.

Step 4: Look Back and Review the Solution:

• Does $48 make sense as the answer? Yes, because if each cupcake costs $2 and the bakery sells 24 cupcakes, the total amount should be 24 × $2 = $48.

Here's how a teacher might approach this step:

1. Check the Answer The teacher asks students to check their answer to ensure that it makes sense in the context of the problem. For example, "Does $48 seem like a reasonable answer for the total amount of money made by selling cupcakes?"
2. Review the Steps: The teacher reviews the steps that were taken to solve the problem, asking students to explain each step in their own words. For example, "Can you explain why we multiplied 24 by $2 to find the total amount of money made?"
3. Discuss Alternative Strategies: The teacher discusses alternative strategies that could have been used to solve the problem, encouraging students to think about different approaches. For example, "Could we have solved this problem by adding $2 twenty-four times instead of multiplying?"
4. Verify the Solution: The teacher helps students verify that their solution is correct by checking their calculations and reasoning. For example, the teacher might ask students to recalculate the total amount of money made to double-check their answer.
5. Reflect on the Process: Finally, the teacher encourages students to reflect on their problem-solving process, asking questions such as, "What did you learn from solving this problem? Is there anything you would do differently next time?"

By approaching the problem in this way, the teacher helps students develop a deeper understanding of the problem-solving process and reinforces the importance of checking their work and reflecting on their strategies.

In this problem, we applied Van de Walle's problem-solving strategy by understanding the problem, devising a plan, carrying out the plan, and reviewing the solution to find out how much money the bakery makes from selling cupcakes in a day.

Check in question 3:

Find the 320th term of 7, 10, 13, 16 …

Problem Solving Strategy 3 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Example \(\PageIndex{3}\)

Karen is thinking of a number. If you double it and subtract 7, you obtain 11. What is Karen’s number?

Solution

1. We start with 11 and work backward.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backward. (5 points)

Problem Solving Strategy 4 (Looking for a Pattern)

Definition: Sequence

A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example \(\PageIndex{4}\)

1, 4, 7, 10, 13… Find the next 2 numbers.

Solution

The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example \(\PageIndex{5}\)

1, 4, 9, 16 … Find the next 2 numbers.

Solution

It looks like each number is a perfect square. \(1^2=1\), \(2^2=4\)

\(3^2=9\)

\(4^2=16\)

So the next numbers would be

\(5^2=25\)

\(6^2=36\)

Example \(\PageIndex{6}\)

10, 7, 4, 1, -2… Find the next 2 numbers.

Solution

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be

-2 -3 = -5

-5 – 3 = -8

Example \(\PageIndex{7}\)

1, 2, 4, 8 …Ffind the next two numbers.

Solution

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

1 x 2 = 2

2 x 2 = 4

4 x 2 =8

So each number is being multiplied by 2.

8 x 2 = 16

16 x 2 = 32

Problem Solving Strategy 5 (Make a List or a Table)

Example \(\PageIndex{8}\)

Can perfect squares end in a 2 or a 3?

Solution

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8. But note that this is just an observation. To answer this question, one would need a mathematically rigorous proof.

Example \(\PageIndex{9}\)

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

q = quarter = 25 cents
d = dime = 10 cents

Solution

Table \(\PageIndex{2}\)

Quarters	Dimes	Calculation	Total Cents
0	3	\(0q + 3d = 0 \times 25 + 3 \times 10\)	30 cents
1	2	\(1q + 2d = 1 \times 25 + 2 \times 10\)	45 cents
2	1	\(2q + 1d = 2 \times 25 + 1 \times 10\)	60 cents
3	0	\(3q + 0d = 3 \times 25 + 0 \times 10\)	75 cents

Check in question 5:

How many ways can you make 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 6 (Process of Elimination)

This strategy can be used when there is only one possible solution.

Example \(\PageIndex{10}\)

I’m thinking of a number.

• The number is odd.
• It is more than 1 but less than 100.
• It is greater than 20.
• It is less than 5 times 7.
• The sum of the digits is 7.
• It is evenly divisible by 5.

Solution

• We know it is an odd number between 1 and 100.
• It is greater than 20 but less than 35
  • 21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.
• The sum of the digits is 7
  • 21 (2+1=3) No
  • 23 (2+3 = 5) No
  • 25 (2 + 5= 7) Yes

Using the same process we see there are no other numbers that meet this criteria. Also, we notice 25 is divisible by 5. By using the strategy of elimination, we have found our answer.

Reference

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