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

7.3.3 • Quadratic Sequences

Activity

1.

Sketch the next step in the pattern. Explain what you sketched.

Compare your answer:

Step 4 is a 4-by-6 rectangle.

2.

Kiran says that the pattern is growing linearly because as the step number goes up by 1, the number of rows and the number of columns also increase by 1. Do you agree? Be prepared to show your reasoning.

Compare your answer:

Kiran is correct that the number of rows and the number of columns each grow by 1, but the total number of squares does not increase by the same amount from step to step. So, the total number of squares does not grow linearly.

3.

To represent the number of squares after $n$ steps, Diego and Jada wrote different equations. Diego wrote the equation $f (n) = n (n + 2)$ . Jada wrote the equation $f (n) = n^{2} + 2 n$ . Is either Diego or Jada correct? Be prepared to show your reasoning.

Compare your answer:

Diego’s expression $n (n + 2)$ is correct because there are $n$ columns and $n + 2$ rows of small squares in Step $n$ . Jada’s expression is also correct because the $n (n + 2)$ array of small squares can be divided into an $n$ -by- $n$ square (having $n^{2}$ small squares) and an $n$ -by-2 rectangle (having $2 n$ small squares).

Video: Quadratic Sequences

Watch the following video to learn more about quadratic sequences.

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Self Check

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

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

Additional Resources

More Complex Quadratic Patterns

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 $y$ .

First, try to find the pattern if the missing squares were actually there.

Make a table:

Step	# of columns	# of rows	# of squares
1	2	2	2
2	2	2	7
3	4	4	14
$n$	$n + 1$	$n + 1$	$(n + 1)^{2} - 2$

There would be $(n + 1) (n + 1)$ squares, but there are 2 missing from each step, so the equation would be:

$y = (n + 1)^{2} - 2$

Try it

More Complex Quadratic Patterns

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 the quadratic sequence with an equation:

Notice there are no missing squares. Complete the table for each step:

Step	# of columns	# of rows	# of squares
1	4	4	16
2	5	5	25
3	6	6	36
$n$	$n + 3$	$n + 3$	$(n + 3)^{2}$

The number of columns is always $n + 3$ , and the number of rows is $n + 3$ .

The number of squares will be represented by $y = (n + 3)^{2}$ .

7.3.3 Quadratic Sequences

Activity

Video: Quadratic Sequences

Additional Resources

More Complex Quadratic Patterns

More Complex Quadratic Patterns