devising a plan in problem solving

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Polya's Problem Solving

George Polya was a famous Hungarian mathematician who developed a framework for problem-solving in mathematics in 1957. His problem-solving approach is still used widely today and can be applied to any problem-solving discipline (i.e. chemistry, statistics, computer science). Below you will find a description of each step along with strategies to help you accomplish each step. Having a specific strategy like this one may help to reduce anxiety around math tests.

Understand the Problem  

Understanding the problem is a crucial first step as this will help you identify what the question is asking and what you need to calculate. Strategies to help include: 

• Identify (i.e. highlight or circle) the unknowns in the problem or question.
• Draw or visualize a picture that can help you understand the problem. 

Devise a Plan  

Devising a plan is a process in which you find the connection between the data/information you are given and the unknown. However, you may not have been given enough data/information to find a connection immediately, so this process may involve calculating/finding additional variables before the final unknown can be solved. Strategies to help you devise a plan include:  

• List the unknowns and knowns. 
• Identify if a theorem would help you calculate the unknown (i.e. a2 + b2 = c2). 
• Decide what variables you need to know the value of to solve for the unknown. 
• Select which variable you will solve for first.

Carry Out the Plan  

This step involves calculating the steps identified in the “Devise a Plan” stage. Strategies to help you carry out the plan include:  

• Focus on solving one part of the problem at a time.
• Clearly write out each step. 
• Double check each variable or step as you solve.
• Repeat this process until you solve for the final unknown. 

Look Back 

This step involves reviewing your answer and steps to confirm that your final calculation is correct. Strategies to help you review your work include:  

• Recalculate each step to see if you get the same answer.
• Check if your final calculation has the appropriate units (i.e. m/s, N/m2). 
• Repeat steps to correct any errors found.

The problem-solving process in a mathematics classroom

• January 2016

• University of Central Florida

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Intermediate Algebra Tutorial 8

• Use Polya's four step process to solve word problems involving numbers, percents, rectangles, supplementary angles, complementary angles, consecutive integers, and breaking even. 

Whether you like it or not, whether you are going to be a mother, father, teacher, computer programmer, scientist, researcher, business owner, coach, mathematician, manager, doctor, lawyer, banker (the list can go on and on),  problem solving is everywhere.  Some people think that you either can do it or you can't.  Contrary to that belief, it can be a learned trade.  Even the best athletes and musicians had some coaching along the way and lots of practice.  That's what it also takes to be good at problem solving.

George Polya , known as the father of modern problem solving, did extensive studies and wrote numerous mathematical papers and three books about problem solving.  I'm going to show you his method of problem solving to help step you through these problems.

If you follow these steps, it will help you become more successful in the world of problem solving.

Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:

Step 1: Understand the problem.  

Step 2:   Devise a plan (translate).  

Step 3:   Carry out the plan (solve).  

Step 4:   Look back (check and interpret).  

Just read and translate it left to right to set up your equation

Since we are looking for a number, we will let 

x = a number

*Get all the x terms on one side

*Inv. of sub. 2 is add 2  

FINAL ANSWER:  The number is 6.

We are looking for two numbers, and since we can write the one number in terms of another number, we will let

x = another number 

ne number is 3 less than another number:

x - 3 = one number

*Inv. of sub 3 is add 3

*Inv. of mult. 2 is div. 2  

FINAL ANSWER:  One number is 90. Another number is 87.

When you are wanting to find the percentage of some number, remember that ‘of ’ represents multiplication - so you would multiply the percent (in decimal form) times the number you are taking the percent of.

We are looking for a number that is 45% of 125,  we will let

x = the value we are looking for

FINAL ANSWER:  The number is 56.25.

We are looking for how many students passed the last math test,  we will let

x = number of students 

FINAL ANSWER: 21 students passed the last math test.

We are looking for the price of the tv before they added the tax,  we will let

x = price of the tv before tax was added. 

*Inv of mult. 1.0825 is div. by 1.0825

FINAL ANSWER: The original price is $500.

Perimeter of a Rectangle = 2(length) + 2(width)

We are looking for the length and width of the rectangle.  Since length can be written in terms of width, we will let

length is 1 inch more than 3 times the width:

1 + 3 w = length

*Inv. of add. 2 is sub. 2

*Inv. of mult. by 8 is div. by 8  

FINAL ANSWER: Width is 3 inches. Length is 10 inches.

Complimentary angles sum up to be 90 degrees.

We are already given in the figure that

x = one angle

5 x = other angle

*Inv. of mult. by 6 is div. by 6

FINAL ANSWER: The two angles are 30 degrees and 150 degrees.

If we let x represent the first integer, how would we represent the second consecutive integer in terms of x ?  Well if we look at 5, 6, and 7 - note that 6 is one more than 5, the first integer. 

In general, we could represent the second consecutive integer by x + 1 .  And what about the third consecutive integer. 

Well, note how 7 is 2 more than 5.  In general, we could represent the third consecutive integer as x + 2.

Consecutive EVEN integers are even integers that follow one another in order.     

If we let x represent the first EVEN integer, how would we represent the second consecutive even integer in terms of x ?   Note that 6 is two more than 4, the first even integer. 

In general, we could represent the second consecutive EVEN integer by x + 2 . 

And what about the third consecutive even integer?  Well, note how 8 is 4 more than 4.  In general, we could represent the third consecutive EVEN integer as x + 4.

Consecutive ODD integers are odd integers that follow one another in order.     

If we let x represent the first ODD integer, how would we represent the second consecutive odd integer in terms of x ?   Note that 7 is two more than 5, the first odd integer. 

In general, we could represent the second consecutive ODD integer by x + 2.

And what about the third consecutive odd integer?  Well, note how 9 is 4 more than 5.  In general, we could represent the third consecutive ODD integer as x + 4.  

Note that a common misconception is that because we want an odd number that we should not be adding a 2 which is an even number.  Keep in mind that x is representing an ODD number and that the next odd number is 2 away, just like 7 is 2 away form 5, so we need to add 2 to the first odd number to get to the second consecutive odd number.

We are looking for 3 consecutive integers, we will let

x = 1st consecutive integer

x + 1 = 2nd consecutive integer

x + 2  = 3rd consecutive integer

*Inv. of mult. by 3 is div. by 3  

FINAL ANSWER: The three consecutive integers are 85, 86, and 87.

We are looking for 3 EVEN consecutive integers, we will let

x = 1st consecutive even integer

x + 2 = 2nd consecutive even integer

x + 4  = 3rd  consecutive even integer

*Inv. of add. 10 is sub. 10  

FINAL ANSWER: The ages of the three sisters are 4, 6, and 8.

In the revenue equation, R is the amount of money the manufacturer makes on a product.

If a manufacturer wants to know how many items must be sold to break even, that can be found by setting the cost equal to the revenue.

We are looking for the number of cd’s needed to be sold to break even, we will let

*Inv. of mult. by 10 is div. by 10

FINAL ANSWER: 5 cd’s.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem .  At the link you will find the answer as well as any steps that went into finding that answer.

  Practice Problems 1a - 1g: Solve the word problem.

(answer/discussion to 1e)

http://www.purplemath.com/modules/translat.htm This webpage gives you the basics of problem solving and helps you with translating English into math.

http://www.purplemath.com/modules/numbprob.htm This webpage helps you with numeric and consecutive integer problems.

http://www.purplemath.com/modules/percntof.htm This webpage helps you with percent problems.

http://www.math.com/school/subject2/lessons/S2U1L3DP.html This website helps you with the basics of writing equations.

http://www.purplemath.com/modules/ageprobs.htm This webpage goes through examples of age problems,  which are like the  numeric problems found on this page.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

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