Unit 6 Plan - Grade 6

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Complete the Diagrams (1 problem)

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

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How Do You Know a Solution Is a Solution? (1 problem)

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

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Weight of the Circle (1 problem)

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

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Solve It! (1 problem)

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

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More Storytime (1 problem)

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

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Fundraising for the Animal Shelter (1 problem)

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

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Growth (1 problem)

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

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Decisions about Equivalence (1 problem)

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

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Complete the Equation (1 problem)

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)
$5\boldcdot3-5\boldcdot2=\boxed{5} \boldcdot (3-2)$

Lesson 10

The Distributive Property, Part 2

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Which Expressions Represent Area? (1 problem)

Select all the expressions that represent the large rectangle's total area.

A rectangle with a height of 5 partition into two smaller rectangles.

$3\,(5+b)$
$5\,(b+3)$
$5b +15$
$15+5b$
$3 \boldcdot 5 + 3b$

Show Solution

$5(b+3)$
$5b +15$
$15+5b$

Lesson 11

The Distributive Property, Part 3

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Writing Equivalent Expressions (1 problem)

Use the distributive property to write an expression that is equivalent to each expression. If you get stuck, consider drawing a diagram.

$(3r - 1)\, 8$
$p\,(6+2t+5y)$
$12+4x$

Show Solution

$24r-8$ (or equivalent)
$6p+2pt+ 10py$ (or equivalent)
Sample responses:
- $4(3+x)$
- $2(6+2x)$
- $\frac{1}{2} (24+8x)$

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

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More 3's (1 problem)

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 13

Expressions with Exponents

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Grain Calculation (1 problem)

Recall the story about the inventor of chess who asked for grains of rice to be doubled every day, starting with 2 grains on the first day.

Andre and Elena knew that after 28 days, the inventor would have $2^{28}$ grains of rice, but they wanted to find out how many grains that actually is.

Andre wrote:
$\displaystyle 2^{28} = 2 \boldcdot 28 = 56$

Elena said, “No, exponents mean repeated multiplication. It should be $28 \boldcdot 28$ , which works out to be 784.”

Do you agree with either of them? Explain your reasoning.

Show Solution

I disagree with both Andre and Elena. Sample reasoning: Andre thinks exponents are just a different way of writing multiplication of two numbers. Elena calculates $28^2$ rather than $2^{28}$ . To find the value of $2^{28}$ , we have to multiply 2 by itself 28 times.

Lesson 14

Evaluating Expressions with Exponents

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Calculating Volumes (1 problem)

Jada and Noah want to find the combined volume of two gift boxes. One is shaped like a cube and the other is shaped like a rectangular prism that is not a cube. The prism has a volume of 20 cubic inches. The cube has edge lengths of 10 inches.

Jada says the total volume is 27,000 cubic inches. Noah says it is 1,020 cubic inches. Here is how each of them reasoned:

Jada's method:

$20 + 10^3$
$30^3$
$27,000$

Noah's method:

$20 + 10^3$
$20 + 1,000$
$1,020$

Do you agree with either of them? Explain your reasoning.

Show Solution

I agree with Noah. Sample reasoning: The cube has a volume of 1,000 cubic inches, and the additional 20 cubic inches from the prism makes the total volume 1,020 cubic inches. The exponent calculation comes before addition.

Lesson 15

Equivalent Exponential Expressions

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Expressions with Exponents (1 problem)

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

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Kitchen Cleaner (1 problem)

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

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Interpret the Point (1 problem)

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: $\frac13(7\frac12)=t$ , $t=2\frac12$

Lesson 18

More Relationships

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Table, Equation, and Graph (1 problem)

The table shows the relationship between the perimeter of a square and its side length.

perimeter (cm)	8	10	12	14	23
side length (cm)	2	$2 \frac12$	3	$3 \frac12$	$5 \frac34$

Write an equation that shows how the side length of the square, $s$ , is related to the perimeter of the square, $P$ .
Plot the ordered pairs from the table on the graph to show the relationship.
What does the point $(12,3)$ represent in this situation?

Show Solution

$\frac{1}{4}P=s$ (or equivalent)
Graph with points plotted at $(8, 2)$ , $(10, 2.5)$ , $(12, 3)$ , $(14, 3.5)$ , and $(23, 5.75)$ .
Sample response: When the perimeter of square is 12 cm, its side length is 3 cm.

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!

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No cool-down

Unit 6 Expressions And Equations — Unit Plan