Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

Activity

A pattern of squares. Step 1 has three squares in one row. Step 2 has four squares in one row and then two squares centered in the row above it. Step 3 has five squares on the bottom row and then two rows with three squares each stacked above.

Here is a pattern of squares.

1. Is the number of small squares growing linearly? Explain how you know.

Compare your answer:

No, there are 3 squares in the first step, 6 squares in the second step, and 11 squares in the third step. So the number of additional squares is 3 going from the first to the second step and 5 going from the second to the third step.

2. Answer questions a–e to complete the table.

Step	Number of small squares
1	3
2	6
3	11
4	a. _____
5	b. _____
10	c. _____
12	d. _____
$n$	e. _____

a. In Step 4, what is the number of small squares?

Step 4 has 18 small squares.

b. In Step 5, what is the number of small squares?

Step 5 has 27 small squares.

c. In Step 10, what is the number of small squares?

Step 10 has 102 small squares.

d. In Step 12, what is the number of small squares?

Step 12 has 146 small squares.

e. In Step $n$ , what is the number of small squares?

Compare your answer:

$n^{2} + 2$

3. Is the number of small squares growing exponentially? Explain how you know.

Compare your answer:

No, from Step 1 to Step 2, the number of squares doubles. From Step 2 to Step 3, the number of squares increases by a factor of $\frac{11}{6}$ . From Step 3 to Step 4, the number of squares increases by a factor of $\frac{18}{11}$ . These growth factors are all different; in contrast, for an exponential function, the values increase by equal growth factors over equal intervals.

The nth expression, $n^{2} + 2$ , is an example of a quadratic expression. A quadratic expression can be written using a squared term. It can be written in other ways as well, but the expression should be able to be written with the highest exponent on the variable as 2.

A quadratic relationship is not like a linear relationship because as one quantity increases by a certain amount, the second quantity doesn’t increase by the same amount.
A quadratic relationship is not like an exponential relationship because as one quantity increases by a certain amount, the second quantity doesn’t change by the same factor. (In this case, if the number of squares grew exponentially, this would mean that, from each step to the next, the number of squares is multiplied by the same factor.)

Are you ready for more?

Extending Your Thinking

1.

Han wrote $n (n + 2) - 2 (n - 1)$ for the number of small squares in the design at Step $n$ in the pattern at the beginning of the activity.

Explain why Han is correct.

Compare your answer:

At Step $n$ , there is an $n$ by $n + 2$ rectangle containing $n (n + 2)$ small squares with two columns of $n - 1$ squares removed from either side.

2.

On your own, draw and label what Step 3 would look like when Han was writing his expression.

Compare your answer:

I drew a rectangle with three small squares down and five small squares across. Then I removed the two small squares from the first column and the last column.

A set of squares has been drawn with the bottom row consisting of 5 squares. Then, there are two rows with three squares each stacked above. On the outside edges of rows two and three, where the missing squares are located, dashed lines have been drawn in to indicate their location. The left side of the image is labeled n and the bottom of the image is labeled n plus 2. With the dashed lines added to the original image, the full figure resembles a rectangle.

Video: Quadratic Expressions

Watch the following video to learn more about quadratic expressions.

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Self Check

Which of the following is an example of a quadratic expression?

$3(n+2)$
$2^n+3$
$n^2+3$
$2n+4$

Additional Resources

Writing Expression from Patterns

Find the number of dots in Step 5, and write an expression to find $n$ number of dots.

A pattern of dots. Step 0 has 2 rows with 2 dots each. Step 1 has 3 rows of dots. In rows 1 and 2, there are 2 dots each and then in row 3 there are 3 dots. Step 2 has 2 rows with 2 dots each and then 2 more rows with 3 dots each for a total of 4 rows. Step 3 has 2 rows with 2 dots each and then 3 rows with 3 dots each for a total of 5 rows.

Step 1 - Make a table to help you find the pattern.

Step number	Number of dots
0	4
1	7
2	10
3	13

Step 2 - Notice that the start has 2 rows and 2 columns, or 4 dots. Each step adds a row of 2 dots, plus 1 more dot. So, each step adds 3 dots.

Step 3 - Continuing this linear pattern, Step 5 would have 19 dots.

Step 4 - An expression could be $3 n + 4$ . This is a linear expression.

Try it

Try It: Writing Expression from Patterns

Find the number of dots in Step 5, and write an expression to find $n$ number of dots.

A pattern of dots. In Step 0, there is 1 row with 1 dot. In Step 1, there is 1 row with 3 dots. In Step 2 there are 2 rows. In the top row there are 5 dots and in the bottom row there are 4 dots. The dots in the bottom row are arranged so there are two on the left and two on the right. A space is open in the middle of the row. Step 3 has 3 rows of dots. In the top row there are 7 dots. In the bottom two rows, there are 6 dots each that are arranged with a space in the middle of the row.

Here is how to write an expression from a pattern:

Step	Number of dots
0	1
1	3
2	9
3	19

Each step has that number of dots squared, then doubled, and then one is added.

So, Step 5 would have 51 dots.

The expression would be $2 n^{2} + 1$ , where $n$ is the step number.

This is a quadratic expression.

7.2.3 Quadratic Expressions

Activity

Are you ready for more?

Extending Your Thinking

Video: Quadratic Expressions

Additional Resources

Writing Expression from Patterns

Try It: Writing Expression from Patterns