6th Grade Math Textbook, Fundamentals

A

B

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Problem 2: Suppose an ant is at point Aof the figure
at the right and can walk only on the lines in the directions
indicated by the arrows. In how many different ways can
this ant walk from point Ato point B?

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Lesson 6-11 for exercise sets.

Read to understand what is being asked.

List the facts and restate the question.

Facts: An ant travels from Ato B. It can travel only on the line segments

and in the directions indicated by the arrows when walking between

the “nodes” (points where the line segments are joined).

Question:In how many different ways can this be done?

Select a strategy.

By solving the simpler problem of finding how many ways the ant can

get to the closer nodes, you might see how to find the number of ways

it can travel to reach the farther nodes.

Apply the strategy.

In the figure below, each node is labeled with a letter.

The circled numbers indicate how many paths lead to each node from A.

Node Dcan be reached in one of two ways:

from node Cor directly from node A. Node F

can be reached in three ways: directly from

node Aor from either of the two ways

through node D.

Continuing in this manner, you see that Ecan

be reached only from D(2 ways), but Gcan be

reached from F(3 ways) or from E(2 ways),

for a total of 5 ways. Node Hcan only be

reached from node G(5 ways).

Finally, node Bcan be reached 12 different

ways from any one of the following routes:

from node E(2 ways), node G(5 ways), or

node H(5 ways).

Check to make sure your answer makes sense.

There are 2 paths to E: ACDE, ADE.

There are 5 paths to G: ACDEG, ADEG, ACDFG, ADFG, AFG.

There are 5 paths to H: ACDEGH, ADEGH, ACDFGH, ADFGH, AFGH.

Finally, the 12 paths to Bare precisely these 12 paths, with a Battached to their ends.

A

C B

H

1

DE

FG

2

3

2

A 55

C B

H

1

DE

FG

2

3

2 12

A 55

C B

H

1

DE

FG

2

3