6th Grade Math Textbook, Fundamentals

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

 &KDSWHU

Problem-Solving Strategy:

Solve a Simpler Problem

Objective To solve problems using the strategy Solve a Simpler Problem

Problem 1:There are six steps at the entrance to Shirley’s school.
She likes to walk up the steps in different ways, but she always goes
up either one or two steps at a time. In how many different ways
can Shirley go up the steps in this manner?

6-11

floor

1

2

3

5

8

13

1st step

2nd step

3rd step

4th step

5th step

6th step

Read to understand what is being asked.

List the facts and restate the question.

Facts: Shirley takes only one or two steps

at a time.

Question:In how many different ways can Shirley

go up the steps?

Select a strategy.

You can use the strategy Solve a Simpler Problem

and find the solution for Steps 1, 2, and 3.

Apply the strategy.

For the first step, there is only one way Shirley can climb it.

For the second step, there are two possibilities: She could get to the second

step from the floor, or she could get to the second step from the first step.

Consider the third step. She could get to the third step from the first step

or from the second step. Because there is 1 way to get to the first step and

because there are 2 ways to get to the second step, there are 1 2, or 3,

ways to get to the third step.

Continue this line of thinking for the subsequent steps. Shirley can

arrive at any step only from twosteps lower or from onestep lower.

So the number of ways to get to a step is the sum of the number

of ways to get to the two steps below.

This leads to the diagram at the right.

Each circled number shows the number

of different ways Shirley can get to

that step. In each case, this number

is the sum of the numbers on the two

steps below.

So Shirley can go up the six steps in 13 different ways.

Check to make sure your answer makes sense.

Here are the 3 ways Shirley could use to get to Step 3: 1,1,1; 1,2; 2,1.

Here are the 5 ways Shirley could use to get to Step 4: 1,1,1,1; 1,1,2; 1,2,1; 2,1,1; 2,2.

These ways to arrive at Steps 3 and 4 form the 8 ways to arrive at Step 5.

The 13 ways to arrive at Step 6 are the 5 ways to get to Step 4 plus the 8 ways

to get to Step 5.