Unit 6 Expressions And Equations — Unit Plan

Title

Assessment

Lesson 1

Tape Diagrams and Equations

Complete the Diagrams

Complete the first diagram so it represents $5 \boldcdot x = 15$ . Complete the second diagram so it represents $5 + y = 15$ .

Two blank tape diagrams. Each tape diagram totals 15.

Show Solution

Sample response:

Lesson 2

Truth and Equations

How Do You Know a Solution Is a Solution?

Explain how you know that 88 is a solution to the equation $\frac18 x = 11$ by completing the sentences:

The word “solution” means . . .

88 is a solution to $\frac18x = 11$ because . . .

Show Solution

Sample responses:

The word “solution” means a value for the variable that makes the equation true.

88 is a solution to $\frac18x = 11$ , because if $x$ is 88, the equation is $\frac18 \boldcdot 88=11$ , which is true.

Lesson 3

Staying in Balance

Weight of the Circle

Here is a balanced hanger diagram.

Balanced hanger. Left side, 4 identical circles, w. Right side, 1 rectangle, 25.

Write an equation that represents the diagram.
Find the weight of one circle. Explain or show your reasoning.
What is the solution to your equation?

Show Solution

$4w=25$
$\frac{25}{4}$ or $6\frac14$ units. Sample reasoning: The left side of the diagram has 4 circles, so I divided the right side into 4 equal pieces. Each of those pieces weighs $6\frac14$ units. This shows that each circle piece weighs $6\frac14$ units.
$w=\frac{25}{4}$ or $6\frac14$

Lesson 4

Practice Solving Equations

Solve It!

Solve each equation. Explain or show your reasoning.

$x+1\frac34=10$ .
$5.7x = 17.1$
$\frac{1}{10}x=\frac25$

Show Solution

Sample responses:

$x=6\frac14$ . Sample reasoning: $x+1\frac34-1\frac34=8-1\frac34$
$x=3$ . Sample reasoning: $5.7x\div5.7=17.1\div5.7$
$x=4$ . Sample reasoning: I divided both sides of the equation by $\frac{1}{10}$ .

Lesson 5

Represent Situations with Equations

More Storytime

For each situation:

Choose an equation that represents it.
Solve the equation.
Explain what the solution means in the situation.

Lin needs 10 cups of flour for a bread recipe. She only has $2\frac12$ cups. How much more flour does she need?
- $x+2\frac12=10$
- $x=10+2\frac12$
- $2\frac12x=10$
Each notebook costs 5.70. How many notebooks does Diego buy if he spends a total of 17.10?
- $x+5.7=17.1$
- $5.7x=17.1$
- $17.1x=5.7$

Show Solution

$x+2\frac12=10$ , $x=7\frac12$ . Lin needs $7\frac12$ cups of flour.
$5.7x=17.1$ , $x=3$ . Diego buys 3 notebooks.

Lesson 6

Percentages and Equations

Fundraising for the Animal Shelter

Noah raised $54 to support the animal shelter, which is 60% of his fundraising goal. What is Noah’s fundraising goal?

Write an equation with a variable to represent the situation.
Answer the question. Show or explain your reasoning.

Show Solution

$54=\frac{60}{100}x$ (or equivalent)
$90. Sample reasoning:
- Divide both sides of the equation by $\frac{60}{100}$ or $\frac35$ to get $x=54\boldcdot \frac53=90$ .
- I wrote $0.6x=54$ and thought about six-tenths of what number would be equal to 54. $6 \boldcdot 9 = 54$ , so 0.6 needs to be multiplied by 90 to get 54.

Section A Check

Section A Checkpoint

Problem 1

Answer each question, and explain or show your reasoning.

Is $\frac38$ a solution to $y+\frac12=\frac78$ ?
Is 15 a solution to $0.4x=20$ ?

Show Solution

Yes. Sample reasoning: Substituting $\frac38$ for $y$ gives a true equation, $\frac38+\frac12=\frac78$ .
No. Sample reasoning: Substituting 15 for $x$ gives the equation $(0.4) \boldcdot 15= 20$ , which is false.

Problem 2

Solve each equation, and explain or show your reasoning.

$a + 123 = 468$
$2.5b=20.5$

Show Solution

$a=345$ . Sample reasoning: Subtract $123$ from each side.
$b=8.2$ . Sample reasoning: Divide each side by $2.5$ .

Lesson 7

Write Expressions with Variables

Growth

A plant measured $x$ inches tall last week and 8 inches tall this week.

Circle the expression that represents the number of inches the plant grew this week. Explain how you know.
- $x-8$
- $8-x$
Each tree needs 1.2 liters of water. Write an expression that represents the amount of water needed for $n$ trees.

Show Solution

$8-x$ . Sample reasoning: Since the plant grew taller this week, 8 is greater than $x$ . The difference of 8 and $x$ is the amount that the plant grew.
$1.2n$ (or equivalent)

Lesson 8

Equal and Equivalent

Decisions about Equivalence

Decide if the expressions in each pair are equivalent. Explain or show how you know.

$x+x+x+x$ and $4x$
$5x$ and $x+5$

Show Solution

Equivalent. Sample reasoning: The diagrams representing these expressions would have the same length for any value of $x$ .
Not equivalent. Sample reasoning: if $x=1$ , then $5x=5$ and $x+5=6$ , so they do not have the same value.

Lesson 9

The Distributive Property, Part 1

Complete the Equation

Write a number or expression in each empty box to create true equations.

$7 \boldcdot (3+5)=\boxed{\phantom{\huge3333}}+\boxed{\phantom{\huge3333}}$
$5\boldcdot3-5\boldcdot2=\boxed{\phantom{\huge33}} \boldcdot (3-2)$

Show Solution

$7\boldcdot(3+5)=\boxed{21}+\boxed{35}$ or $7\boldcdot(3+5)=\boxed{7 \boldcdot 3}+ \boxed{7 \boldcdot 5}$ (or equivalent)
$5\boldcdot3-5\boldcdot2=\boxed{5} \boldcdot (3-2)$

Section B Check

Section B Checkpoint

Problem 1

Andre says that $2x + 5$ and $7x$ are equivalent expressions because they have the same value when $x$ is 1. Do you agree with Andre’s reasoning? Explain your reasoning. Use a diagram if it helps.

Show Solution

No, I do not agree. Sample reasoning: To be equivalent expressions, they need to have the same value for every value of the variable. When

x

is 0,

2x+5

is 5, but

7x

is 0. Since they do not have the same value when

x

is 0, they are not equivalent expressions.

Problem 2

Use the distributive property to write an expression that is equivalent to each expression.

$3(x+4)$
$a(3.5-2.7)$
$5y+10$

Show Solution

Sample responses:

$3x+12$
$0.8a$
$5(y+2)$

Lesson 12

Meaning of Exponents

More 3's

What is the value of the expression $3^5$ ?
Explain how to use that value to quickly find the value of $3^6$ .

Show Solution

243
Sample response: $3^6=3^5 \boldcdot 3=729$

Lesson 15

Equivalent Exponential Expressions

Expressions with Exponents

Find the value of each expression for the given value of $x$ .

$(\frac14)^x$ when $x$ is 3
$4+x^5$ when $x$ is 2
$4x^2$ when $x$ is 10
$(4x)^2$ when $x$ is 10

Show Solution

$\frac{1}{64}$
36
400
1,600

Section C Check

Section C Checkpoint

Problem 1

Decide whether each equation is true or false. Explain how you know.

$2^4=2 \boldcdot 4$
$3 \boldcdot 3^4 = 3^5$
$2 \boldcdot 3^2 = 6^2$

Show Solution

False. Sample reasoning: $16=8$ is not true.
True. Sample reasoning: Each side is $3 \boldcdot 3 \boldcdot 3 \boldcdot 3 \boldcdot 3$ .
False. Sample reasoning: $18=36$ is not true.

Problem 2

Find the value of

7+x^3

when

x

is 2.

Show Solution

15. Sample reasoning:

$7 + x^3 \\ 7 + 2^3 \\ 7 + 8 \\15$

Lesson 16

Two Related Quantities, Part 1

Kitchen Cleaner

To remove grease from kitchen surfaces, a recipe says to use 1 cup of baking soda for every $\frac{1}{2}$ cup of water.

cups of baking soda	cups of water
1	$\frac12$
2	1
3	$\frac32$

Which graph represents the relationship between cups of baking soda and cups of water? Explain how you know.

A
Five points graphed on a coordinate plane with the origin labeled O. The horizontal axis is labeled “cups of water and the numbers 0 through 5 are indicated. The vertical axis is labeled “cups of baking soda” and the numbers 0 through 5 are indicated. The data are as follows: 1 comma one half, 2 comma 1, 3 comma 1 and one half, 4 comma 2, and 5 comma 2 and one half.

B
Five points graphed on a coordinate plane with the origin labeled O. The horizontal axis is labeled “cups of water” and the numbers 0 through 5 are indicated. The vertical axis is labeled “cups of baking soda” and the numbers 0 through 5 are indicated. The data are as follows: one half comma 1, 1 comma 2, 1 and one half comma 3, and 2 comma 4.

C
Five points graphed on a coordinate plane with the origin labeled O. The horizontal axis is labeled “cups of water” and the numbers 0 through 5 are indicated. The vertical axis is labeled “cups of baking soda” and the numbers 0 through 5 are indicated. The data are as follows: 1 comma 1, 2 comma 2, 3 comma 3, and 4 comma 4.
Select all equations that can represent the relationship between $b$ , cups of baking soda, and $w$ , cups of water, in this situation.
1. $w = \frac{1}{2}b$
2. $b=\frac{1}{2}w$
3. $b = w$
4. $b = 2w$
5. $w = 2b$

Show Solution

Graph B. Sample reasoning:
- In all graphs, the first value of the coordinates represents the amount of water. The amount of baking soda is twice the amount of water, so the coordinates of the points should be $(\frac{1}{2}, 1)$ , $(1, 2)$ , $(\frac{3}{2}, 3)$ , and so on.
- I matched the coordinates of the points to the values in the table: $\frac{1}{2}$ cup of water goes with 1 cup of baking soda, 1 cup of water goes with 2 cups of baking soda, and so on.
- The ratio of cups of water to cups of baking soda is 2 to 1, so I looked at the coordinate points that show the same ratio.
A, D

Lesson 17

Two Related Quantities, Part 2

Interpret the Point

Noah built a robot that travels at a constant rate. The equation $\frac{1}{3}d=t$ and the graph both represent the relationship between the distance traveled in meters, $d$ , and the travel time in minutes, $t$ .

Five points plotted on a coordinate grid.

Which variable is independent variable?
What does the point $(12,4)$ represent in this situation?
What does the coefficient $\frac{1}{3}$ tell us about the situation?
What point on the graph would represent the time it takes the robot to travel $7\frac12$ meters?

Show Solution

Distance, $d$ , is the independent variable.
Sample responses:
- Noah’s robot can travel 12 meters in 4 minutes.
- It takes Noah’s robot 4 minutes to travel 12 meters.
Sample response: The $\frac{1}{3}$ tells us that it takes the robot $\frac{1}{3}$ minute to travel 1 meter.
$(7\frac12,2\frac12)$ . Sample reasoning: $\frac13(7\frac12)=t$ , $t=2\frac12$

Section D Check

Section D Checkpoint

Problem 1

Tyler bought 4 ounces of vegetable seeds for $10 from an online store that sells seeds in bulk.

weight of seeds (ounces)	cost (dollars)
4	10
10
	35
50

Complete the table to show the costs for different amounts of seeds.
Write an equation that shows the relationship between the weight of seeds in ounces, $w$ , and the cost in dollars, $c$ .

Show Solution

weight of seeds (ounces) cost (dollars)

4 10

10 25

14 35

50 125
$2.5w = c$ or $w = 0.4c$ (or equivalent)

Problem 2

Diego worked out a deal with his parents. For every hour that he reads a book, he earns $\frac14$ hour of screen time. Diego uses the equation $s=\frac14r$ to represent this relationship.

What does each variable in the equation represent?
Which is the independent variable? Which is the dependent variable? Explain how you know.

Show Solution

The variable $s$ represents the number of hours of screen time Diego earns and the variable $r$ represents the number of hours Diego reads.
The independent variable is $r$ and the dependent variable is $s$ . Sample reasoning: The number of hours of screen time Diego earns depends on the number of hours he reads.

Lesson 19

Tables, Equations, and Graphs, Oh My!

No cool-down

Unit 6 Assessment

End-of-Unit Assessment

Problem 1

Which expression is equal to $6^4$ ?

$4^6$

$6 \boldcdot 6 \boldcdot 6 \boldcdot 6$

Show Solution

$6 \boldcdot 6 \boldcdot 6 \boldcdot 6$

Problem 2

Select all the expressions that are equivalent to $3x^4$ .

$3(x \boldcdot x \boldcdot x \boldcdot x)$

$3+x^4$

$3x + 3x + 3x + 3x$

3(x^2 \boldcdot x^2)

$3x \boldcdot 3x \boldcdot 3x \boldcdot 3x$

Show Solution

A, D

Problem 3

Which expression is equivalent to $20c-8d$ ?

$2(10c+4d)$

$4(5c-8d)$

$4(5c-2d)$

$c(20-8d)$

Show Solution

$4(5c-2d)$

Problem 4

Here is an expression: $3 \boldcdot 2^t$

Find the value of the expression when $t$ is 1.
Find the value of the expression when $t$ is 4.

Show Solution

Problem 5

Write two expressions that are equivalent to $m + m + m + m$ .

The first expression should be a product of a coefficient and a variable.
The second expression should be a sum of two terms.

Show Solution

$4m$ (or equivalent)
Sample responses: $2m + 2m$ or $3m + m$

Problem 6

Jada makes sparkling juice by mixing 2 cups of sparkling water with every 3 cups of apple juice.

How much sparkling water does Jada need if she uses 15 cups of apple juice? Explain or show your reasoning.
How much apple juice does Jada need if she uses 6 cups of sparkling water?
Plot these pairs of measurements as points on the graph.

A blank coordinate plane with the origin labeled "O." The j-axis is labeled “cups of apple juice” and the numbers 0 through 22, in increments of 2, are indicated. Vertical gridlines are drawn at each integer from 1 to 22. The s-axis is labeled “cups of sparkling water” and the numbers 0 through 22, in increments of 2, are indicated. Horizontal gridlines are drawn at each integer from 1 to 22.
The variable $s$ represents the number of cups of sparkling water, and the variable $j$ represents the number of cups of apple juice. Write an equation that shows how $s$ and $j$ are related.

Show Solution

10 cups. Sample reasoning: For each cup of apple juice used, $\frac{2}{3}$ cup of water is needed. $15 \boldcdot \frac{2}{3} = 10$ .
9 cups
Graph shows points at $(9,6)$ and $(15,10)$ .
$j = \frac{3}{2} s$ (or equivalent)

Minimal Tier 1 response:

Work is complete and correct.
Sample:

10. There are 5 times as many cups of apple juice, so there needs to be 5 times as many cups of sparkling water.
9
Graph shows points at $(9,6)$ and $(15,10)$ .
$j=1.5s$

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: 1–2 incorrect answers but correct reasoning for the first question, or mostly correct answers but incomplete reasoning for the first question; the unit rate $\frac32$ or $\frac23$ being multiplied by the incorrect variable in the equation.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: More than 2 incorrect problem parts; use of a unit rate other than $\frac32$ or $\frac23$ (or their equivalents) in equation ; incorrect interpretation of the relationship—that it is additive (for instance, that there is always 1 more cup of apple juice than sparkling water), or that there is more sparkling water than apple juice.

Problem 7

This rectangle has a perimeter of 36 units.

Complete the table to show the length and width of at least 3 different rectangles that also have a perimeter of 36 units.

length ( $\ell$ ) width ( $w$ )

12 6
Describe the relationship between the values in the two columns of the table.
Write an equation to represent the relationship, where one variable is written in terms of the other. Identify the independent and dependent variables.
Plot the values in your table as points on the graph. Make sure to label the axes.

Show Solution

Sample response: See table.
Sample responses:
- If we know the length of the rectangle, we can find its width by subtracting the length from 18.
- The sum of the width and length is 18.
- If we know the width, we can find the length by subtracting the width from 18.
Sample responses:
- In the equation $w = 18 - \ell$ , the rectangle’s length is the independent variable and width is the dependent variable.
- In the equation $\ell = 18 - w$ , the rectangle’s width is the independent variable and length is the dependent variable.
Sample response: See graph. (Students may label the horizontal axis with "width" and vertical axis with "length" to match their equation.)

length ( $\ell$ )	width ( $w$ )
12	6
10	8
12	6
4	14

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

See table.
$w + \ell = 18$
$\ell = 18 - w$ . The independent variable is the width, and the dependent variable is the length.
See graph.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: the equation correctly represents the situation, but is not structured to express the dependent variable in terms of the independent variable.
Sample errors: Arithmetic error leads to an incorrect solution. Axes on the graph are not labeled.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: Values chosen for the length and width add to 36 instead of 18. Rectangles chosen have an area of 36 square units instead of a perimeter of 36 units.
Sample errors: Work involves a misinterpretation of the situation that affects all or most problem parts, but work does show understanding of using equations, tables, and graphs to represent a situation.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Work shows multiple Tier 3 errors and there are major omissions.