6th Grade Math Textbook, Fundamentals

Shortest Length

in Inches Possible Sides Total

1 none 0

2 2-3-4 2-4-5 2-5-6 2-6-7 2-7-8 5

3 3-4-5 3-4-6 3-5-6 3-5-7 3-6-7 3-6-8 3-7-8 7

4 4-5-6 4-5-7 4-5-8 4-6-7 4-6-8 4-7-8 6

5 5-6-7 5-6-8 5-7-8 3

6 6-7-8 1

7 none 0

8 none 0

22

 &KDSWHU

Problem-Solving Strategy:

Account for all Possibilities

Objective To solve problems using the strategy Account for All Possibilities

Problem 1: Eight rods are on a table. Their lengths are 1, 2, 3, 4, 5, 6, 7, and 8 inches.

How many different triangles can be formed using exactly three rods at a time?

(You may make different triangles using the same rod again, but you may not use

the same rod more than once in the same triangle.)

10-13

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 in.

2 in.

Swing these around.

They will never meet.

So no triangle is possible.

6 in.

In all, 22 possible triangles can be formed.

Read to understand what is being asked.

List the facts and restate the question.

Facts: You are given eight rods of lengths

1, 2, 3, 4, 5, 6, 7, and 8 inches.

You form triangles using the rods as sides.

You may use the same rod in different

triangles but not within the same triangle.

Question:How many different triangles can

you make?

Select a strategy.

You could attempt to account for all such

possible triangles.

Apply the strategy.

The key to this problem is the Triangle Inequality Theorem,

which says that the sum of the lengths of any two sides of a

triangle must be greater than the length of the third side. So

to be able to form a triangle with three rods, the sum of the

two shortest lengths must be greater than the longest length.

For example, you cannot form a triangle from the 1-in., 2-in.,

and 6-in. rods because 1 2 is not greater than 6.

To find all the possible triangles, take each rod in turn, beginning with

the shortest, and write all possible groups of rods that can form triangles,

as shown in the following table.