CPM Homework Help : INT2 Problem 7-13

Write an equation that represents the number of tiles in Figure $x$ for the tile pattern below.

tile	tile	tile
	tile	tile
	tile	tile	tile

Figure 1

tile	tile	tile	tile
	tile	tile	tile
	tile	tile	tile
	tile	tile	tile	tile	tile

Figure 2

tile	tile	tile	tile	tile
	tile	tile	tile	tile
	tile	tile	tile	tile
	tile	tile	tile	tile
	tile	tile	tile	tile	tile	tile	tile

Figure 3

Hint:

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.

Added to the 3 tile pattern figures, 1 tile on top left, in each figure is circled in red. The centers, as follows, are each circled in green: Figure 1, 2 by 3 tiles, Figure 2, 3 by 4 tiles, Figure 3, 4 by 5 tiles. The bottom right tiles are circled in blue as follows: Figure 1, 1 tiles, Figure 2, 2 tiles, Figure 3, 3 tiles.

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 arm. Its length matches its corresponding figure number.

Partial answer:

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