Module V: Problem Solving

The PRAXIS II focuses on mathematical understanding the middle school teachers must have. This includes the ability to reason, justify, and apply concepts and procedures to a variety of problem solving situations. Problems are an integral component of the mathematics curriculum. As such, the mathematics curriculum should help students develop a wide range of problem solving strategies. In the book, How to Solve It, George Poyla outlined the four major steps in problem solving. These steps and common strategies are outlined below.

Step 1: Understand the Problem 
 2: Devise a Plan
Step 3: Carry out the Plan 
Step 4: Look Back

Step 1:




Understand the problem

Read the problem once to get a sense of the problem

Read the problem a second time to start getting information to solve the problem

  • Have you seen a similar problem before? If so, how is this problem similar? How is it different?
  • What facts or key information do you have?
  • What do you know that is not stated in the problem?
  • Can you simplify the problem?

Step 2:

Choose a strategy / Devise a Plan

  1. Draw a diagram/picture
  2. Make a systematic list
  3. Eliminate possibilities
  4. Look for a pattern
  5. Guess-check-reflect
  6. Identify sub-problems
  7. Analyze the units
  8. Solve an easier related problem
  9. Create a physical representation
  10. Work backwards
  11. Draw Venn diagrams
  12. Convert to algebra
  13. Use a formula

Step 3:

Solve it / Carry out the Plan

Show your work as you solve the problem. It justifies/proves your answer.

Step 4:




Look Back

Reread the problem a third time.

  • Did you answer the question that was asked?
  • Is your answer in the correct units?
  • Does your answer seem reasonable?
  • Does it make sense?

Write your answer in a complete sentence.

When we encounter a problem solving situation, the most difficult step in Polya’s procedure is devising a plan. Learning and practicing the specific strategies outlined in the table will help you develop into a good mathematical problem solver. In the section below, you will investigate these strategies to solve a series of mathematical tasks. Most of the Learning Objects you have encountered so far in the four content modules are premised in a problem solving task.



Draw a Picture

Allan, Betsy, Clyde and Deloris are going to the movie, The Great Adventures of a Mathematician. If Betsy and Deloris want to next to each other, in how many ways can they be seated in a row of four chairs?


CDs and DVDs are on sale. Amanda buys 4 CDs and 1 DVD for $38. Carlos buys 1 CD and 4 DVDs for $47. There is only one price for all CDs in the store and only one price for all DVDs. If a DVD costs more than CD, what is the price for each one?

Use a Variable

Mr. Bailey has the same number of nickels as quarters. The value of the quarters is $1.80 more than the value of the nickels. What is the total value of her nickels and quarters?

Look for a Pattern

There are seven teams playing in the Milwaukee Little League tournament. Each team is scheduled to play every other team exactly once. How many games are scheduled for the tournament?

Make a List

A deli shop advertises over 40 different sandwiches for lunch. To make a sandwich, a customer chooses a type of bread, one type of mean, and one type of cheese. If there are three types of breads (wheat, rye, and French roll), four types of meats (ham, turkey, roast beef, and chicken), and two types of cheese (cheddar and swiss), is the deli shop’s advertisement correct?

Venn Diagram

In Lincoln Middle School, there are 60 students in seventh grade. If 25 of these students take art only, 18 take music only, and 9 do not take either art or music, how many take both art and music?

Work Backwards

Jill received her allowance on Monday. On Tuesday she put ½ of her money in her piggy bank. On Wednesday, she spent 1/3 of the remaining allowance for a birthday present for her mother. On Thursday, she bought lunch for $4.00 with her remaining allowance. On Friday, she was left with $2.00. How much was her allowance on Monday?

To learn more, visit the materials linked below.

Individual Learning Objects