Writing Equations for Patterns with Squares

7.3.2 Writing Equations for Patterns with Squares

7.3.2 • Writing Equations for Patterns with Squares

Activity

Three steps of a growing pattern with squares. In Step 1, there is a square in the center, with four more squares at each vertex of the center square. The overall image of step 1 looks like a big X consisting of small squares. Step 2 replaces the center square with four squares arranged in two rows of two squares. There are still 4 squares at the corners. Step 3 places nine squares arranged in three rows of three squares at the center of the figure. There are still 4 squares at the corners.

1. If the pattern continues, what will we see in Step 5 and Step 18?

a. Sketch or describe the figure in each of these steps.

Compare your answer:

Step 5 is a collection of 25 small squares, in a 5-by-5 arrangement, surrounded by 4 other small squares at the corners. Step 18 is a collection of $18^{2}$ small squares, in an 18 by 18 arrangement, surrounded by 4 other small squares at the corners.

b. How many small squares are in each of these steps? Explain how you know.

Compare your answer:

Step 5 - 29, because it is $5^{2} + 4$ or $(5 \cdot 5) + 4$ . Step 18: 328, because it is $18^{2} + 4$ or $(18 \cdot 18) + 4$ .

2. Write an equation to represent the relationship between the step number $n$ and the number of squares $y$ . Be prepared to explain how each part of your equation relates to the pattern. (If you get stuck, try making a table.)

Compare your answer:

$y = n^{2} + 4$ or equivalent. The $n^{2}$ term represents the number of small squares in the central $n$ -by- $n$ large square. The 4 represents the 4 small squares that are added to the large square at its 4 corners.

3. Sketch the first three steps of a pattern that can be represented by the equation $y = n^{2} - 1$ .

Compare your answer:

An $n$ -by- $n$ array of squares with one square missing at the top right. The first step (for $n = 1$ ) will have no squares.

A pattern with squares. Step 1 has zero squares. Step 2 has a column of 2 squares and then a column with 1 square. Step 3 has 2 columns with three squares each and then a column of two squares.

Are you ready for more?

Extending Your Thinking

For the original step pattern in the statement, $y = n^{2} + 4$ , write an equation to represent the relationship between the step number, $n$ , and the perimeter, $P$ .

Compare your answer:

$P = 4 n + 16$

For the step pattern you created in Part 3 of the activity, $y = n^{2} - 1$ , write an equation to represent the relationship between the step number, $n$ , and the perimeter, $P$ .

Compare your answer:

If the pattern was an $n$ -by- $n$ array of squares with a square missing from the corner, the perimeter could be given by $P = 0$ when $n = 1$ , and by $P = 4 n$ when $n > 1$ .

Are these linear functions?

Compare your answer:

Yes, in both cases, these are linear functions.

Self Check

Three steps of a growing pattern. Step 1: Total of two squares, stacked one atop the other. Step 2: Total of five squares, two squares on the bottom row and the second row and one square on the top row. Step 3: Total of ten squares, three squares on the bottom row, row 2, and row 3, and one square on the top row.

Which equation represents the relationship between the step number $n$ and the number of squares $y$ ?

$y = n(n+1)$
$y=n^2+1$
$y = 2n+1$
$y = n + 1$

Additional Resources

Patterns Represented by Quadratics

See Steps 1 – 3 of a pattern of squares below. Write an equation representing the relationship between the step number $n$ and the number of squares $y$ .

A pattern of squares. Step 1 has a column of three squares. Step 2 has two columns with six squares in each column. Step 3 has three columns with 9 squares in each column.

Make a table for the number of columns and rows of each step and the number of squares:

Step	# of columns	# of rows	# of squares
1	1	3	3
2	2	6	12
3	3	9	27
$n$	$n$	$3 n$	$3 n^{2}$

First, count how many columns there are in each step (the length):

For each, there are $n$ columns.

Next, count how many rows there are in each step (the height):

For each, there are $3 n$ rows.

The area of this figure is then $l \times h$ or $n \times 3 n$ , which becomes $y = 3 n^{2}$ .

Try it

Patterns Represented by Quadratics

See Steps 1 – 3 of a pattern of squares below. Write an equation representing the relationship between the step number $n$ and the number of squares $y$ .

Here is how to represent this pattern as a quadratic:

Make a table for the number of columns and rows of each step and the number of squares:

Step	# of columns	# of rows	# of squares
1	0	0	0
2	2	2	3
3	3	3	8
$n$	$n$	$n$	$n^{2} - 1$

For this pattern, notice that the area would be the number of columns times the number of rows, but then one block is subtracted.

Note that for Step 1, there was a $1 \times 1$ block there, but then it was subtracted with the equation given.

The equation, then, is $y = n^{2} - 1$ .