Lesson 2 Growing Dots Develop Understanding

Ready

Use the given values in the table to determine a pattern and complete the table.

1.

Term	$1 st$	$2 nd$	$3 rd$	$4 th$	$5 th$	$6 th$	$7 th$	$8 th$
Value	$3$	$6$	$12$	$24$

2.

Term	$1 st$	$2 nd$	$3 rd$	$4 th$	$5 th$	$6 th$	$7 th$	$8 th$
Value	$76$	$60$	$44$	$28$

3.

Term	$1 st$	$2 nd$	$3 rd$	$4 th$	$5 th$	$6 th$	$7 th$	$8 th$
Value	$320$	$160$	$80$	$40$

4.

Term	$1 st$	$2 nd$	$3 rd$	$4 th$	$5 th$	$6 th$	$7 th$	$8 th$
Value	$- 15$	$- 8$	$- 1$	$6$

Some equations have two variables. You may recall seeing an equation written in slope-intercept form: $y = 5 x + 2$ . When given an $x$ -value, the equation can be used to determine the associated $y$ -value. A solution to an equation is the value that makes the equation a true statement. Therefore, a solution to an equation with two variables must include both the $x$ -value and the $y$ -value. Often the answer is written as an ordered pair. The $x$ -value is always first. Example: $(x, y)$ .

Determine the $y$ -value of each ordered pair based on the given $x$ -value.

5.

$y = 7 x - 13$

$(6, \underset{―}{} \underset{―}{})$ , $(- 3, \underset{―}{} \underset{―}{})$ , $(1, \underset{―}{} \underset{―}{})$

6.

$y = - 3 x + 8$

$(- 8, \underset{―}{} \underset{―}{})$ , $(5, \underset{―}{} \underset{―}{})$ , $(0, \underset{―}{} \underset{―}{})$

7.

$y = 5 x - 2$

$(0, \underset{―}{} \underset{―}{})$ , $(3, \underset{―}{} \underset{―}{})$ , $(- 9, \underset{―}{} \underset{―}{})$

8.

$y = - x + 4$

$(- 4, \underset{―}{} \underset{―}{})$ , $(1, \underset{―}{} \underset{―}{})$ , $(7, \underset{―}{} \underset{―}{})$

Set

In the pictures shown, each square represents $1$ tile.

9.

Draw Step 4 and Step 5.

10.

Dan explained that the middle tower is always the same number as the step number. He also pointed out that the two arms on each side of the tower contain one less block than the step number.

Create an equation that fits Dan’s way of seeing the relationship.

11.

Sarah counted the number of tiles at each step and made a table. She explained that the number of tiles in each figure was always $3$ times the step number minus $2$ .

step number	$1$	$2$	$3$	$4$	$5$	$6$
number of tiles	$1$	$4$	$7$	$10$	$13$	$16$

Create an equation that fits Sarah’s way of seeing the relationship.

12.

Nancy focused on the number of blocks in the base compared to the number of blocks above the base. She said the number of base blocks were the odd numbers starting at $1$ . She also noticed the number of tiles above the base followed the pattern $0$ , $1$ , $2$ , $3$ , $4$ . She organized her work in the table.

Step number	# in base + # on top
$1$	$1 + 0$
$2$	$3 + 1$
$3$	$5 + 2$
$4$	$7 + 3$
$5$	$9 + 4$

Create an equation that fits Nancy’s way of seeing the relationship.

Based on the function notation provided, evaluate the function to determine the output.

Example:

Instead of using $x$ and $y$ in an equation, like $y = 5 x + 1$ , mathematicians often write $f (n) = 5 n + 1$ because it can give more information. With this notation, the direction to find $f (2)$ means to replace the value of $n$ with $2$ and perform the operations to find $f (n)$ . The point $(n, f (n))$ is in the same location on the graph as $(x, y)$ , where $n$ describes the location along the $x$ -axis, and $f (n)$ is the height of the graph.

Given that $f (n) = 8 n - 3$ and $g (n) = 3 n - 10$ , evaluate the following functions with the indicated values.

13.

$f (1) =$

14.

$g (0) =$

15.

$f (5) =$

16.

$g (- 4) =$

Go

Write each expression using an exponent.

17.

$7 \times 7 \times 7 \times 7$

18.

$3 \times 3 \times 3 \times 3 \times 3 \times 3$

19.

$\frac{1}{5} \times \frac{1}{5}$

For problems 20–22: Write each expression in expanded form (a). Then calculate the value of the expression (b).

20.

$7^{1}$

a.

expanded form:

b.

value of the expression:

21.

$5^{4}$

a.

expanded form:

b.

value of the expression:

22.

$7 {(2)}^{3}$

a.

expanded form:

b.

value of the expression:

Evaluate each expression.

23.

$2^{x}$ for $x = 9$

24.

$6 n^{2}$ for $n = 3$

25.

$4 s^{3}$ for $s = 5$