6th Grade Math Textbook, Fundamentals

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Problem Solving: Review of Strategies

Objective To solve problems using a variety of strategies

Problem: At the right is a part of a design made up of
concentric squares (that is, squares with the same center).
The entire design is made up of 50 squares. What is the
sum of the perimeters of all 50 squares?

Read to understand what is being asked.

List the facts and restate the question.

Facts: A design is made up of concentric squares

with side lengths of 1 foot, 3 feet, 5 feet, 7 feet,

9 feet, and so on.

There is a total of 50 squares in the design.

Question:What is the sum of the perimeters of all the

squares in the design?

Select a strategy.

There are many ways to approach this problem.

Here are two possibilities:

• You can use the strategy Find a Pattern.


• You can use the strategy Organize the Data
  to analyze the situation.

Apply the strategy.

Method 1:Find a Pattern

Starting from the smallest square and moving outward,

you find that the side lengths are the sequence of the first

50 odd numbers: 1, 3, 5, 7,..., 99. Because each square

has four sides, if you add the numbers representing each

side length in this sequence and then multiply the sum

by 4, the result will be the total of the perimeters.

So you need to compute the sum 1  3  5  7 ... 99.

Find the sum of the first term, the sum of the first two terms,

the sum of the first three terms, and so on.

See if you can discover a pattern.

1  1

1  3  4

1  3  5  9

1  3  5  7  16

1  3  5  7  9  25

4-16

Problem-Solving Strategies

1.Guess and Test

2.Organize Data

3.Find a Pattern

4.Make a Drawing

5.Solve a Simpler Problem

6.Reason Logically

7.Adopt a Different Point of View

8.Account for All Possibilities

9.Work Backward

10.Consider Extreme Cases

1 ft

3 ft

5 ft

7 ft