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The mathematician George Polya captured the problem solving principles and strategies he used in his discipline in the book How to Solve It: A New Aspect of Mathematical Method(Princeton University Press, 1957). The book includes a summary of Polya’s problem solving heuristic as well as advice on the teaching of problem solving.

• Strategies Problem Solvers Use
• Strategies Problem Solving Problems
• Strategies For Problem-solving In Counselling
• Strategies Problem Solving Engel
• Problem Solving Skills

Related Pages
Solving Word Problems Using Block Models
Heuristic Approach to Problem-Solving
Algebra Lessons

Problem Solving Strategies

The strategies used in solving word problems:

1. What do you know?
2. What do you need to know?
3. Draw a diagram/picture

• K stands for “What do I know,” F stands for “What do I need to Find,” C stands for “Come up with a plan” which includes 2 parts: the operation (+, -, x, and /) and the problem-solving strategy, and lastly, the S stands for “solve the problem” (which includes all the work that is involved in solving.
• A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic.

Solution Strategies
Label Variables
Verbal Model or Logical Reasoning
Algebraic Model - Translate Verbal Model to Algebraic Model
Solve and Check.

Solving Word Problems

Step 1: Identify (What is being asked?)
Step 2: Strategize
Step 3: Write the equation(s)
Step 4: Answer the question
Step 5: Check

• Show Video Lesson

Problem Solving Strategy: Guess And Check

Using the guess and check problem solving strategy to help solve math word problems.

Example:
Jamie spent $40 for an outfit. She paid for the items using $10, $5 and $1 bills. If she gave the clerk 10 bills in all, how many of each bill did she use?

Problem Solving : Make A Table And Look For A Pattern

1. Identify - What is the question?
2. Plan - What strategy will I use to solve the problem?
3. Solve - Carry out your plan.
4. Verify - Does my answer make sense?

Example:
Marcus ran a lemonade stand for 5 days. On the first day, he made $5. Every day after that he made $2 more than the previous day. How much money did Marcus made in all after 5 days?

• Show Video Lesson

Find A Pattern Model (Intermediate)

In this lesson, we will look at some intermediate examples of Find a Pattern method of problem-solving strategy.

Example:
The figure shows a series of rectangles where each rectangle is bounded by 10 dots.
a) How many dots are required for 7 rectangles?
b) If the figure has 73 dots, how many rectangles would there be?

Solution:

a) The number of dots required for 7 rectangles is 52.

b) If the figure has 73 dots, there would be 10 rectangles.

Strategies Problem Solvers Use

Example:
Each triangle in the figure below has 3 dots. Study the pattern and find the number of dots for 7 layers of triangles.

Solution:

The number of dots for 7 layers of triangles is 36.

Strategies Problem Solving Problems

Example:
The table below shows numbers placed into groups I, II, III, IV, V and VI. In which groups would the following numbers belong?
a) 25
b) 46
c) 269

Solution:
The pattern is: The remainder when the number is divided by 6 determines the group.
a) 25 ÷ 6 = 4 remainder 1 (Group I)
b) 46 ÷ 6 = 7 remainder 4 (Group IV)
c) 269 ÷ 6 = 44 remainder 5 (Group V)

Example:
The following figures were formed using matchsticks.

a) Based on the above series of figures, complete the table below.

Strategies For Problem-solving In Counselling

b) How many triangles are there if the figure in the series has 9 squares?

c) How many matchsticks would be used in the figure in the series with 11 squares?

Solution:
a)

b) The pattern is +2 for each additional square.
18 + 2 = 20
If the figure in the series has 9 squares, there would be 20 triangles.

c) The pattern is + 7 for each additional square
61 + (3 x 7) = 82
If the figure in the series has 11 squares, there would be 82 matchsticks.

Example:
Seven ex-schoolmates had a gathering. Each one of them shook hands with all others once. How many handshakes were there?

Solution:

Total = 6 + 5 + 4 + 3 + 2 + 1 = 21 handshakes.

The following video shows more examples of using problem solving strategies and models.
Question 1: Approximate your average speed given some information
Question 2: The table shows the number of seats in each of the first four rows in an auditorium. The remaining ten rows follow the same pattern. Find the number of seats in the last row.
Question 3: You are hanging three pictures in the wall of your home that is 16 feet wide. The width of your pictures are 2, 3 and 4 feet. You want space between your pictures to be the same and the space to the left and right to be 6 inches more than between the pictures. How would you place the pictures?

The following are some other examples of problem solving strategies.

• Explore it/Act it/Try it (EAT)

  Explore it/Act it/Try it (EAT) Method (Basic)
  Explore it/Act it/Try it (EAT) Method (Intermediate)
  Explore it/Act it/Try it (EAT) Method (Advanced)

• Finding A Pattern

  Finding A Pattern (Basic)
  Finding A Pattern (Intermediate)
  Finding A Pattern (Advanced)

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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Guest post by Kady Dupre

Here’s a word problem for you:

Miss Friday’s class does a daily word problem. Ten of her students are great at word problems involving addition, and only 7 seem to understand subtraction word problems. Five of her students are bored with the easy problems. Thirteen students are still struggling with basic math facts and 3 have trouble reading the word problems at all. How many of her students are engaged and learning?

Here’s a better question: “How do you grow confident and effective problem solvers?”

Why Students Struggle with Math Word Problems

Students struggle with math word problems for many reasons, but three of the biggest I’ve encountered include:

Issue #1: Student Confidence
For many students, just looking at a word problem leads to anxiety. No one can think clearly with a sense of dread or fear of failure looming!

Issue #2: Flexible Thinking
Many kids are taught to solve word problems methodically, with a prescriptive step-by-step plan using key words that don’t always work. Plans are great, but not when students use them as a crutch rather than a tool. Today’s standardized tests and real-world applications require creative thinking and flexibility with strategies.

Issue #3: Differentiation
Teachers want students to excel quickly and often push too fast, too soon. In the case of word problems, you have to go slow to go fast. Just like in Guided Reading, you’ll want to give lots of practice with “just-right” problems and provide guided practice with problems just-above the students’ level.

3 Problem Solving Strategies

The solution is to conquer math word problems with engaging classroom strategies that counteract the above issues!

1. Teach a Problem-Solving Routine

Kids (and adults) are notoriously impulsive problem solvers. Many students see a word problem and want to immediately snatch out those numbers and “do something” with them. When I was in elementary school, this was actually a pretty reliable strategy! But today, kids are asked to solve much more complex problems, often with tricky wording or intentional distractors.

Grow flexible thinkers and build confidence by teaching a routine. A problem solving routine simply encourages students to slow down and think before and after solving. I’ve seen lots of effective routines but my favorites always include a “before, during, and after” mindset.

To make the problem solving routine meaningful and effective:

• Use it often (daily if possible)
• Incorporate “Turn & Teach” (Students orally explain their thinking and process to a partner.)
• Allow for “Strategy Share” after solving (Selected students explain their method and thinking.)

Strategies Problem Solving Engel

2. Differentiate Word Problems

No, this doesn’t mean to write a different word problem for every student! This can be as simple as adjusting the numbers in a problem or removing distractors for struggling students. Scaffolding word problems will grow confidence and improve problem solving skills by gradually increasing the level difficulty as the child is ready. This is especially effective when you are trying to teach students different structures of word problems to go with a certain operation.

For example, comparison subtraction problems are very challenging for some students. By starting with a simple version, you allow students to focus on the problem itself, rather than becoming intimidated or frustrated.

I’ve had great success in using scaffolded problems with my guided math groups. After solving the easier problem, students realize that it’s not that tricky and are ready to take on the tougher ones!

3. Compare Problems Side-by-Side

Problem Solving Skills

To develop flexible thinking, nothing is more powerful than analyzing and comparing word problems. Start by using problems that have similar stories and numbers, but different problem structures. Encourage conversation, use visual representations, and have students explain the difference in structure and operation. Here’s an example showing student work on two similar problems about monkeys. Click here to download a blank copy of these problems. My freebie includes several variations to help you differentiate.

Use these three strategies to get kids thinking and talking about their problem solving strategies while building that “oh-so-important” confidence, and you CAN conquer math word problems!

Kady Dupre has worked as a classroom teacher, instructional coach, and intervention teacher in elementary grades. She loves creating learning resources for students and teachers. She authors Teacher Trap, a blog aimed at sharing her challenges, successes, and insights as a teacher.