CPM Homework Help : CCA Problem 9-40

Find a rule that represents the number of tiles in Figure $x$ for the tile pattern below.

tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$

Figure 1

tile $\space$	tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$	tile $\space$	tile $\space$

Figure 2

tile $\space$	tile $\space$	tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$	tile $\space$
	tile $\space$	tile $\space$	tile $\space$	tile $\space$	tile $\space$	tile $\space$	tile $\space$

Figure 3

Hint:

The single unit tile is circled in red. The middle 2 by 3 rectangle is circled in green. The one unit tile on the bottom right is circled in blue.
Figure 1

The single unit tile is circled in red. The middle 3 by 4 rectangle is circled in green. The 2 unit tiles on the bottom right are circled in blue.
Figure 2

The single unit tile is circled in red. The middle 4 by 5 rectangle is circled in green. The 3 unit tiles on the bottom right are circled in blue.
Figure 3

The red circles point out the left arms, the green circles point out the rectangles, and the blue circles point out the right arms.

More Help:

Notice the following:
The length of the rectangle in Figure 1 is $3$ units $× 2$ units; the rectangle in Figure 2 is $4$ units $× 3$ units; and the rectangle in Figure 3 is $5$ units $× 4$ units.

Step 1:

First, look at the left arm. It remains constant in each of these figures, with a value of $1$ .

Step 2:

Let $x=$ Figure number.
There is a rule for the size of the rectangles: $(x+2)(x+1)$ , where $(x+2)$ represents the length and $(x+1)$ represents the width.

Step 3:

Now, consider the right arms. Its length matches its corresponding figure number.

Partial Answer:

$y=$ left arm $+$ rectangle $+$ right arm
$y=1+(x+2)(x+1)+x$
Now simplify.