6th Grade Math Textbook, Fundamentals

 &KDSWHU

Problem-Solving Strategy:

Reason Logically

Objective To solve problems using the strategy Reason Logically.

Problem 1: Is it possible to cover a typical 8-by-8
checkerboard, such as the one pictured below, with

tiles of this shape?

7-15

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

Read to understand what is being asked.

List the facts and restate the question.

Facts: A checkerboard is an 8-by-8 arrangement of small squares.

Each tile is made up of 5 small checkerboard squares in a

fixed pattern.

Question:Can the checkerboard be covered exactly with tiles shaped

like? If so, how?

Select a strategy.

Try using the strategy Reason Logically.

Apply the strategy.

Suppose it is possible to cover a checkerboard with such tiles.

This would mean that 64 little squares can be covered exactly

by tiles made up of 5 little squares each.

However, that would mean that 64 is a multiple of 5

(or, equivalently, that 64 is evenly divisible by 5).

You know that 64 is nota multiple of 5 (because 64  5 12 R4).

So it is notpossible to cover the checkerboard with the tiles.

Check to make sure your answer makes sense.

Twelve tiles would have 12 • 5, or 60, small squares.

This is not enough to cover the board.

Thirteen tiles have 13 • 5, or 65, small squares.

This is too many to cover the board exactly.

Therefore, since it is necessary to use a whole number of tiles, it is not

possible to cover the board exactly with tiles of the given shape.