Unit 6 Expressions Equations And Inequalities — Unit Plan

Title

Assessment

Lesson 1

Relationships between Quantities

Movie Theater Popcorn, Revisited

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

Red and Yellow Apples

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

Three of These Equations Belong Together

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)

Finding Solutions

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)

More Finding Solutions

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

After-School Tutoring

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 9

Dealing with Negative Numbers

Solve Two More Equations

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

Solve Two Equations

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 $y=\text{-}3$ .

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

What Is Different?

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

Colder and Colder

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

Testing for Solutions

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

Party Decorations

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

Playlist Timing

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

Equivalent to $4-x$

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.

Show Solution

B, C
$\text-10x-15$ (or equivalent)

Lesson 21

Combining Like Terms (Part 2)

Subtracting Linear Expressions

Write an equivalent expression with fewer terms. Explain or show your reasoning.

$(16x+5)-4(3+2x)$

Show Solution

$8x-7$ (or equivalent). Sample reasoning: Using the distributive property gets $16x+5-12 -8x$ and then combining like terms gets $8x-7$ .

Section D Check

Section D Checkpoint

Problem 1

For each expression, write an equivalent expression with fewer terms.

$7x + 10y - 2x + 8y$
$(7x + 10y) - (2x + 8y)$

Show Solution

$5x + 18y$ (or equivalent)
$5x + 2y$ (or equivalent)

Problem 2

Expand the expression $\text-5(4f - 3g)$ .
Factor the expression $\text-12a + 30b - 18$ .

Show Solution

$\text-20f + 15g$ (or equivalent)
$6(\text-2a + 5b - 3)$ (or equivalent)

Lesson 22

Applications of Expressions

No cool-down

Unit 6 Assessment

End-of-Unit Assessment

Problem 1

Lin uses a $50 gift card to buy a game on her phone for $9.99. She also uses the gift card to buy upgrades for her characters in the game. Each upgrade costs $1.29.

Which of these inequalities describes this situation, where $n$ is the number of upgrades Lin can buy?

$9.99 + 1.29n \ge 50$

$9.99 + 1.29n \le 50$

$9.99 - 1.29n \ge 50$

$9.99 - 1.29n \le 50$

Show Solution

$9.99 + 1.29n \le 50$

Problem 2

Which number line shows all the values of $x$ that make the inequality $\text-3x + 1 < 7$ true?

A number line labeled A with the numbers negative 5 through 5 indicated. There is a closed circle at negative 2 and an arrow is drawn from the closed circle extending to the left.

A number line labeled B with the numbers negative 5 through 5 indicated. There is an open circle at negative 2 and an arrow is drawn from the open circle extending to the left.

A number line labeled C with the numbers negative 5 through 5 indicated. There is a closed circle at negative 2 and an arrow is drawn from the closed circle extending to the right.

A number line labeled D with the numbers negative 5 through 5 indicated. There is an open circle at negative 2 and an arrow is drawn from the open circle extending to the right.

Show Solution

Problem 3

Select all expressions that are equivalent to $6x + 1 - (3x - 1)$ .

$6x + 1 - 3x - 1$

$6x + \text-3x + 1 + 1$

$3x+2$

$6x - 3x +1 -1$

$6x + 1 +\text - 3x -\text - 1$

Show Solution

B, C, E

Problem 4

At midnight, the temperature in a city was 5 degrees Celsius. The temperature was dropping at a steady rate of 2 degrees Celsius per hour.

Write an inequality that represents $t$ , the number of hours past midnight, when the temperature was colder than -4 degrees Celsius. Explain or show your reasoning.
On the number line, show all the values of $t$ that make your inequality true.

Show Solution

$5 - 2t < \text-4$ or $t > \frac 9 2$ (or equivalent). Sample reasoning:
- $5-2t$ shows the temperature starting at 5 degrees Celsius and decreasing by 2 degrees every hour after midnight, $t$ . Because we want this quantity to be less than -4 degrees Celsius, write $5-2t < \text-4$ .
- The temperature will reach -4 degrees Celsius after $\frac 9 2$ hours. Since the temperature needs to be colder, $t$ must be greater than $\frac 9 2$ .
The graph shows an open circle at $4\frac12$ with shading to the right.

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$5 - 2t < \text-4$ . The temperature starts at 5 degrees, then it goes down 2 degrees every hour. The temperature has to be less than -4 degrees.
The graph shows an open circle at $4\frac12$ with shading to the right.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: Bases graph in part b on an incorrect inequality in part a.
Sample errors: Gives no explanation for a correct inequality; writes an inequality with only one mistake, like $5+2t<\text-4$ ; does not use “open circle” notation for graph; uses incorrect direction of inequality in parts a, b, or both.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Writes an inequality in part a that is not close to correct; omits graph in part b or it is simply incorrect.

Problem 5

Expand to write an equivalent expression:

$\text{-} \frac{1}{4}(\text-8x+12y)$
Factor to write an equivalent expression:

$36a-16$

Show Solution

$2x-3y$ (or equivalent)
$4(9a-4)$ or $2(18a-8)$ (or equivalent)

Problem 6

Tyler is rewriting the expression $6 - 2x + 5 + 4x$ with fewer terms. Here is his work:

$\displaystyle 6 - 2x + 5 + 4x$

$\displaystyle (6-2)x + (5+4)x$

$\displaystyle 4x+9x$

$\displaystyle 13x$

Tyler’s work is incorrect. Circle the step where he made a mistake. Explain or show why this expression is not equivalent to $6 - 2x + 5 + 4x$ .
Write an expression equivalent to $6 - 2x + 5 + 4x$ that only has two terms.

Show Solution

Students circle $(6-2)x + (5+4)x$ . Sample reasoning:
- $(6-2)x$ is not equivalent to $6-2x$ . I can show this because when $x$ is 0, one expression equals 0 but the other equals 6. The same is true for $(5-4)x$ and $5+4x$ .
- Tyler did not use the distributive property correctly. $(6-2)x = 6x-2x$ , not $6-2x$ . The same is true for $5−4x$ and $5+4x$ .
Sample response:

$\displaystyle 6 - 2x + 5 + 4x$

$\displaystyle 6 + \text-2x + 5 + 4x$

$\displaystyle \text-2x + 4x + 6 + 5$

$\displaystyle (\text-2+4)x+11$

$\displaystyle 2x+11$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$6-2x$ does not equal $(6-2)x$ .
$2x+11$

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Identifies Tyler’s error correctly, but the explanation is either incorrect or vague; Identifies the problem step by substituting a value like $x=0$ at each step, but no algebra mistake is identified; has an algebra mistake in part b with otherwise correct work shown.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Fails to identify the error in Tyler’s reasoning; gives incorrect answer to part b with no work shown.

Problem 7

A teacher only uses his car to drive to and from work each day, so the car only uses 0.6 gallon of gas each day. The car holds 14 gallons of gas. A warning light comes on when the remaining gas is 1.5 gallons or less.

If $d$ represents the number of days of driving, what does $14−0.6d$ represent?
Write and solve an equation to determine the number of days the teacher can drive the car without the warning light coming on.
Write and solve an inequality that represents this situation. Explain clearly what the solution to the inequality means in the context of this situation.

Show Solution

Sample responses:

$14 - 0.6d$ is the amount of gas in the tank, in gallons, after $d$ days of driving.
$14 - 0.6d = 1.5$ . Subtract 14 from each side to get $\text-0.6d = \text-12.5$ . Divide each side by $\text-0.6$ to get $d \approx 20.83$ . The car can drive for about 20 days, and the warning light will come on near the end of the 21st day.
The inequality $14 - 0.6d > 1.5$ represents the times when the warning light is off. The solution to this inequality is $d < 20.83$ , so the warning light is off for all times until near the end of the 21st day.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: Says the teacher can drive for 21 days if it is specified that the light will come on during the 21st day.
Sample:

The expression is how much gas is left. $t$ is the number of days.
$14 - 0.6d = 1.5$ , $\text-0.6d = \text-12.5$ , $d \approx 20.83$ . He can drive for 20 days.
$14 - 0.6d > 1.5$ . $d<20.83$ because 20.83 is when the light comes on and before that the light would be off. This means that Diego’s father can drive for 20 full days before the warning light comes on.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Has any arithmetic errors with work shown; gives poor explanations (especially in part c) with otherwise correct work; asserts that since $d \approx 20.83$ is a solution to the equation, the teacher can drive for 21 full days; fails to justify the direction of the inequality in the solution to part c (where justification can involve algebra, testing points, or referring to the context).

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Misinterprets of the situation, leading to incorrect answers for part a; writes incorrect equation in part b; reverses the inequality sign in either the original inequality or the solution to part c; omits real-world interpretation in parts b and c.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Shows little progress on most problem parts; has three or more error types under Tier 3 response.