Unit 6 Plan - Grade 7

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Movie Theater Popcorn, Revisited (1 problem)

A movie theater sells popcorn in bags of different sizes. The table shows the volume of popcorn and the price of the bag.

volume of popcorn (ounces)	price of bag ($)
10	6
20	8
35	11
48	13.6

If the theater wanted to offer a 60-ounce bag of popcorn, what would be a good price? Explain your reasoning.

Show Solution

Sample responses:

$16, because there is a pattern of $4 plus $0.20 per ounce
$15, because there should be a discount for buying a larger bag of popcorn

Lesson 2

Reasoning about Contexts with Tape Diagrams

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Red and Yellow Apples (1 problem)

Here is a story: Lin bought 4 bags of apples. Each bag had the same number of apples. After eating 1 apple from each bag, she had 28 apples left.

Which diagram best represents the story? Explain why the diagram represents it.
A

B

C
Describe how you would find the unknown amount in the story.

Show Solution

C. Sample reasoning: When she ate 1 apple from each bag, there were $x-1$ apples left in each bag.
Each of the 4 pieces of the diagram represents 7 apples, because $28 \div 4 = 7.$ If $x-1=7$ , then $x$ is 8.

Lesson 3

Reasoning about Equations with Tape Diagrams

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Three of These Equations Belong Together (1 problem)

Here is a diagram.

Tape diagram, one part marked 6, three parts marked x, total 30.

Which equation matches the diagram?
1. $6+3x=30$
2. $6x+3=30$
3. $3x=30+6$
4. $30=3x-6$
Draw a diagram that matches the equation $3(x+6)=30$ .

Show Solution

$6+3x=30$
Sample response:

Lesson 4

Reasoning about Equations and Tape Diagrams (Part 1)

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Finding Solutions (1 problem)

Here is a diagram and its corresponding equation. Find the solution to the equation and explain your reasoning.

Tape diagram, 4 small parts each labeled x, 1 large part labeled 17, total 23.

$4x+17=23$

Show Solution

$x=1\frac12$ . Sample explanation: The diagram and equation show that 4 groups plus 17 more equals a total of 23. If we take aways the 17 more, we have 4 groups that equal a total of 6, and $\frac64=1\frac12.$

Lesson 5

Reasoning about Equations and Tape Diagrams (Part 2)

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More Finding Solutions (1 problem)

Here is a diagram and its corresponding equation. Find the solution to the equation and explain your reasoning.

Tape diagram, 4 parts labeled x + 7, total 38.

$4(x+7)=38$

Show Solution

$x=2\frac12$ . Sample reasoning: The tape diagram has 4 equal pieces, each of which represents $\frac{38}{4}$ (or $9\frac12$ ). $x+7=9\frac12,$ so $x$ must be $2\frac12$ .

Lesson 6

Distinguishing between Two Types of Situations

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After-School Tutoring (1 problem)

Write an equation for each story. Then find the number of problems originally assigned by each teacher. If you get stuck, try drawing a diagram to represent the story.

Five students came for after-school tutoring. Lin’s teacher assigned each of them the same number of problems to complete. Then he assigned each student 2 more problems. In all, 30 problems were assigned.
Five students came for after-school tutoring. Priya’s teacher assigned each of them the same number of problems to complete. Then she assigned 2 more problems to one of the students. In all, 27 problems were assigned.

Show Solution

$5(x+2)=30$ (or equivalent), solution: $x=4$ ; The teacher originally assigned 4 problems to each student.
$5x+2=27$ (or equivalent), solution: $x=5$ ; The teacher originally assigned 5 problems to each student.

Section A Check

Section A Checkpoint

Problem 1

$5(x+4) = 80$

<p>Tape diagram, 5 parts, x + 4, x + 4, x + 4, x + 4, x + 4, total 80.</p>

Explain how the equation represents the diagram.

Show Solution

Sample response: They both have 5 equal parts of an unknown amount that has been increased by 4, and both have a total of 80.

Problem 2

The seventh-grade teachers plan to order 36 new workbooks. When they place the order, they learn that shipping will cost a total of $17. The final cost for the workbooks is now $468.

Write an equation to represent the situation. If you get stuck, try drawing a diagram.
What does your variable represent?

Show Solution

$36x+17 = 468$ (or equivalent)
Sample response: $x$ represents the cost of each workbook.

Lesson 7

Reasoning about Solving Equations (Part 1)

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Solve the Equation (1 problem)

Solve the equation. If you get stuck, use the diagram.

$\displaystyle 5x+\frac14=\frac{61}{4}$

Balanced hanger. Left side, 5 circles labeled x, square labeled 1 fourth. Right side, rectangle labeled 61 fourths.

Show Solution

$x=3$

Lesson 8

Reasoning about Solving Equations (Part 2)

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Solve Another Equation (1 problem)

Solve the equation $3(x+4.5)=36$ . If you get stuck, use the diagram.

Balanced hanger diagram, left side, circle x, square 4 point 5, circle x, square 4 point 5, circle x, square 4 point 5, right side, rectangle 36.

Show Solution

7.5. Sample reasoning:

Divide each side by 3 leaving $x+4.5=12$ , then subtract 4.5 from each side.
The distributive property gives $3x+13.5=36$ . Subtract 13.5 from each side leaving $3x=22.5$ . Divide each side by 3.

Lesson 9

Dealing with Negative Numbers

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Solve Two More Equations (1 problem)

Solve each equation. Show your work, or explain your reasoning.

$\text-3x-5=16$
$\text-4(y-2)=12$

Show Solution

$x=\text-7$ . Sample reasoning: After adding 5 to both sides, we get $\text{-}3x=21$ . After dividing both sides by -3, we get $x=\text{-}7$ .
$y=\text-1$ . Sample reasoning: After dividing both sides by -4, we get $y-2=\text{-}3$ . After adding 2 to both sides, we get $y=\text{-}1$ .

Lesson 10

Different Options for Solving One Equation

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Solve Two Equations (1 problem)

Solve each equation. Explain or show your reasoning.

$8.88=4.44(x-7)$

$5\left(y+\frac25\right)=\text-13$

Show Solution

$x=9$ . Sample reasoning: After dividing both sides by 4.44, the equation is $2=x-7$ . After adding 7 to both sides, the equation is $x=9$ .
$y=\text-3$ . Sample reasoning: After distributing the 5, the equation is $5y+2=\text{-}13$ . After subtracting 2 from each side, it is $5y=\text{-}15$ . After dividing both sides by 5, it is $y=\text{-}3$ .

Lesson 11

Using Equations to Solve Problems

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The Basketball Game (1 problem)

Diego scored 9 points less than Andre in the basketball game. Noah scored twice as many points as Diego. If Noah scored 10 points, how many points did Andre score? Explain or show your reasoning.

Show Solution

14 points. Sample reasoning:

Equation: $2(x-9)=10$ , where $x$ is the number of points scored by Andre. $x-9=5$ , $x=14$ .
Reasoning: Diego scored half as many points as Noah, so he scored 5 points. Andre scored 9 points more than Diego, or 14 points.
Diagram: One possibility is two boxes each with $x-9$ showing a total of 10. Each box represents 5 points, so $x$ is 14.

Lesson 12

Solving Problems about Percent Increase or Decrease

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Timing the Relay Race (1 problem)

The track team is trying to reduce their time for a relay race. First, they reduce their time by 2.1 minutes. Then they are able to reduce that time by $\frac{1}{10}$ . If their final time is 3.96 minutes, what was their beginning time? Show or explain your reasoning.

Show Solution

6.5 minutes. Sample reasoning:

With equation: $0.9(x-2.1) = 3.96$ , $x-2.1=4.4$ , $x=6.5$ .
Reasoning with or without a diagram: 9 out of 10 parts represent 3.96 minutes, so the $\frac{1}{10}$ reduction was $3.96\div9$ or 0.44 minutes. That makes the time before the 2.1 minute reduction $3.96+0.44$ or 4.4 minutes. The original time was $4.4+2.1$ , or 6.5 minutes.

Section B Check

Section B Checkpoint

Problem 1

Solve each equation. Explain or show your reasoning.

$2(a+3.6)=44$
$7p-8=-22$
$\text{-}4(x+\frac{3}{2})=16$

Show Solution

$a=18.4$ . Sample reasoning: Students show dividing each side by 2 and subtracting 3.6 from each side.
$p=\text{-}2$ . Sample reasoning. Students show adding 8 to both sides and dividing both sides by 7.
$x=\text{-}\frac{11}{2}$ . Sample reasoning: Students show dividing each side by -4 and subtracting $\frac32$ from each side.

Problem 2

Andre ran 3.1 miles each day last week. This week he plans to increase the number of miles he runs each day so that he runs a total of 35 miles by the end of the week. He plans to run the same distance each day. What distance will Andre add to his run each day this week?

Show Solution

1.9 miles. Sample responses:

$7(3.1+x)=35$ where $x$ is the increase in daily miles. $7(3.1+x)=35$, $3.1 + x = 5$, $x=1.9$
Students draw a tape diagram that shows 7 parts each labeled $3.1+x$ and a total of 35.

Lesson 13

Reintroducing Inequalities

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What Is Different? (1 problem)

List some values for $x$ that would make the inequality $\text-2x > 10$ true.
What is different about the values of $x$ that make $\text-2x \geq 10$ true, compared to $\text-2x > 10$ ?

Show Solution

Sample responses: -6, -7, -100, -5.001 (any number less than -5)
Sample response: When $x$ is -5, the inequality $\text-2x \geq 10$ is true, but the inequality $\text-2x > 10$ is false.

Lesson 14

Finding Solutions to Inequalities in Context

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Colder and Colder (1 problem)

It is currently 10 degrees outside. The temperature is dropping 4 degrees every hour.

Explain what the equation $10 - 4h=\text-2$ represents.
What value of $h$ makes the equation true?
Explain what the inequality $10 -4h < \text-2$ represents.
Does the solution to this inequality look like $h < \text{\_\_}$ or $h > \text{\_\_}$ ? Explain your reasoning.

Show Solution

Sample response: when the temperature is exactly -2 degrees
$h=3$
Sample response: When the temperature is colder than -2 degrees
$h > \text{\_\_}$ . Sample reasoning: The solution is $h > 3$ . Since the temperature is dropping, it will be colder than -2 degrees after 3 hours.

Lesson 15

Efficiently Solving Inequalities

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Testing for Solutions (1 problem)

For each inequality, decide whether the solution is represented by $x < 2.5$ or $x > 2.5$ .

$\text-4x + 5 > \text-5$

$\text-25>\text-5(x+2.5)$

Show Solution

$x<2.5$
$x>2.5$

Lesson 16

Interpreting Inequalities

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Party Decorations (1 problem)

Andre is making paper cranes to decorate for a party. He plans to make one large paper crane for a centerpiece and several smaller paper cranes to put around the table. It takes Andre 10 minutes to make the centerpiece and 3 minutes to make each small crane. He will only have 30 minutes to make the paper cranes once he gets home.

Andre wrote the inequality $3x + 10 \leq 30$ to plan his time. Describe how this inequality represents the situation.
Solve Andre’s inequality, and explain what the solution means.

Show Solution

Sample response: The variable $x$ represents the number of small paper cranes Andre can make. $3x$ is the amount of time it takes to make $x$ small cranes. 10 is the number of minutes it takes to make the centerpiece. 30 is Andre’s time limit in minutes.
$x \leq 6\frac23$ . Sample response: Andre can make up to 6 small cranes.

Lesson 17

Modeling with Inequalities

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Playlist Timing (1 problem)

Elena is trying to create a playlist that lasts no more than 2 hours (120 minutes). She has already added songs that total 15 minutes. She reads that the average song length on her music streaming service is 3.5 minutes. Elena writes the inequality $3.5x + 15 \geq 120$ and solves it to find the solution $x \geq 30$ .

Explain how you know Elena made a mistake based on her solution.
Fix Elena’s inequality and explain what each part of the inequality means.

Show Solution

Sample response: $x \geq 30$ means Elena can add more than 30 songs on the playlist. This doesn’t make sense because there should be a maximum limit on songs rather than a minimum limit.
The correct inequality is $3.5x+15 \leq 120$ . The number 3.5 represents the average length of each song. The variable $x$ represents the number of songs that Elena adds. The 15 represents the 15 minutes of songs that are already on the playlist. The $\leq 120$ represents that the total number of minutes has to be less than or equal to 120.

Section C Check

Section C Checkpoint

Problem 1

Here is a situation: A farmer has 120 cubic yards of sawdust. She uses 7 cubic yards of sawdust each week as bedding for her animals. When will the farmer have less than 50 cubic yards of sawdust left?

Write an inequality that represents the situation. Make sure to explain what your variable represents.
Solve the inequality. Describe what the solution tells us about the situation.
Graph the solution to the inequality on the number line.

Show Solution

$\text{-}7x + 120 < 50$ (or equivalent) where $x$ is the number of weeks from now
$x > 10$ (or equivalent). Sample response: The farmer will have less than 50 cubic yards of sawdust anytime after 10 weeks from now.
Sample response:

Lesson 18

Subtraction in Equivalent Expressions

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Equivalent to $4-x$ (1 problem)

Select all the expressions that are equivalent to $4-x$ .
1. $x-4$
2. $4 + \text- x$
3. $\text- x + 4$
4. $\text- 4+x$
5. $4+x$
Use the distributive property to write an expression that is equivalent to $5(\text- 2x - 3)$ . If you get stuck, use the boxes to help organize your work.