IM UNIT 2 Equations and Inequalities Teacher Guide

Solution

Sample response: Andre is not correct. 320 people need to get on the buses.
which is less than 320.

(From Unit 2, Lesson 4.)

7. Problem 7

Statement

Elena says that equations A and B are not equivalent.

◦ Equation A:
◦ Equation B:

Write a convincing explanation as to why this is true.

Solution gives ,

Sample response: Subtracting 13 from both sides of
not .

(From Unit 2, Lesson 7.)

Algebra1 Unit 2 Lesson 8 CC BY 2019 by Illustrative Mathematics 25

8. Problem 8

Statement

To grow properly, each tomato plant needs 1.5 square feet of soil and each
broccoli plant needs 2.25 square feet of soil. The graph shows the different
combinations of broccoli and tomato plants in an 18 square foot plot of soil.

Match each point to the statement that describes it.

A. Point A 1. The soil is fully used when
B. Point B 6 tomato plants and 4 broccoli
C. Point C plants are planted.
D. Point D
2. Only broccoli was planted, but
the plot is fully used and all plants
can grow properly.

3. After 3 tomato plants and 2
broccoli plants were planted,
there is still extra space in the
plot.

4. With 4 tomato plants and 6
broccoli plants planted, the plot is
overcrowded.

Solution

9. A: 2

Algebra1 Unit 2 Lesson 8 CC BY 2019 by Illustrative Mathematics 26

10. B: 3
11. C: 1
12. D: 4

(From Unit 2, Lesson 5.)

13. Problem 9

Statement

Algebra1 Unit 2 Lesson 8 CC BY 2019 by Illustrative Mathematics 27

Select all the equations that are equivalent to the equation .
A.
B.
C.
D.
E.

Solution

["A", "B", "E"]

(From Unit 2, Lesson 6.)

Algebra1 Unit 2 Lesson 8 CC BY 2019 by Illustrative Mathematics 28

14. Problem 10

Statement

Han is solving an equation. He took steps that are acceptable but ended up
with equations that are clearly not true.

What can Han conclude as a result of these acceptable steps?

A. There’s no value of that can make the equation
true.

B. Any value of can make the equation true.

C. is a solution to the equation .

D. is a solution to the equation .

Solution

A

(From Unit 2, Lesson 7.)

Algebra1 Unit 2 Lesson 8 CC BY 2019 by Illustrative Mathematics 29

Lesson 9: Which Variable to Solve for? (Part 2)

Goals
• Practice writing equations in two or more variables and solving for a particular

variable.

• Solve for a variable by performing acceptable operations, including when the values

of other quantities in a multi-variable equation are not known.

Learning Targets
• I can write an equation to describe a situation that involves multiple quantities whose

values are not known, and then solve the equation for a particular variable.

• I know how solving for a variable can be used to quickly calculate the values of that

variable.

Lesson Narrative

Previously, students have repeatedly solved an equation for one of the variables after the
values of the other variables are specified. They then generalized that process and learned
that it is possible, and sometimes preferable, to first solve for the variable, before
substituting known values or performing calculations.

In this lesson, they practice solving for the variable first and obtaining an expression that
defines it in terms of the others. They see that doing so allows them to solve problems
more efficiently.

Spreadsheets and other computer programs can help illustrate the benefits of isolating a
variable. Students see that once a variable of interest is isolated and expressed in terms of
all the other variables, a computer program can use that expression (even if it seems
complicated) and speedily calculate its value when the other variables take on different
values. The interactive model is a powerful way to test different assumptions in a situation
(MP4).

Alignments
Addressing

• HSA-CED.A.3: Represent constraints by equations or inequalities, and by systems of

equations and/or inequalities, and interpret solutions as viable or nonviable options
in a modeling context. For example, represent inequalities describing nutritional and
cost constraints on combinations of different foods.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 1

• HSA-CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same

reasoning as in solving equations. For example, rearrange Ohm's law to

highlight resistance .

• HSA-REI.B.3: Solve linear equations and inequalities in one variable, including

equations with coefficients represented by letters.

Instructional Routines

• Aspects of Mathematical Modeling

• MLR6: Three Reads

Required Materials
Four-function calculators

Spreadsheet technology

Required Preparation

Acquire devices that can run GeoGebra (recommended) or other spreadsheet technology.
It is ideal if each student has their own device. (A GeoGebra Spreadsheet is available under
Math Tools.) Devices are required for the digital version of the activity "Cargo Shipping."

Student Learning Goals

• Let’s solve an equation for one of the variables.

9.1 Faces, Vertices, and Edges

Warm Up: 5 minutes
In this warm-up, students are given a simple equation in three variables and are prompted
to rearrange it to pin down a particular variable. In each question, only the value of one
variable is given, so students need to manipulate the equation even when some quantities
are unknown. The work here prepares students to rearrange other variable equations
later in the lesson.

As students work, look for those who substitute the given value before rearranging and
those who first isolate the variable of interest before substituting. Invite them to share
their approaches during class discussion.

Addressing

• HSA-CED.A.4

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 2

Anticipated Misconceptions

Some students may be unclear what it means to write an equation that "makes it easier to

find the number of vertices (or faces)." Remind them of the work in an earlier activity. In

the post-parade clean-up activity, for instance, they wrote the equation to quickly

find the length of the road section each volunteer would clean up, , if there were

volunteers. They wrote to quickly find the number of volunteers, , if each volunteer

were to clean up miles.

Student Task Statement , which relates the number

In an earlier lesson, you saw the equation
of vertices, faces, and edges in a Platonic solid.

1. Write an equation that makes it easier to find the number of vertices in each
of the Platonic solids described:

a. An octahedron (shown here), which has 8
faces.

b. An icosahedron, which has 30 edges.

2. A Buckminsterfullerene (also called a “Buckyball”) is a polyhedron with 60
vertices. It is not a Platonic solid, but the numbers of faces, edges, and vertices
are related the same way as those in a Platonic solid.

Write an equation that makes it easier to find the number of faces a Buckyball
has if we know how many edges it has.

Student Response

1. a. (or equivalent)

b. (or equivalent)

2. (or equivalent)

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 3

Activity Synthesis
Select previously identified students to share their responses and strategies. Record and
display for all to see the steps they take to rearrange the equations. Emphasize how each
step constitutes an acceptable move and how it keeps the equation true.

Make sure students see that we can either substitute known values into the given
equation before rearranging it, or we can rearrange the equation first before substituting
known values. In the examples here, it doesn't matter which way it is done. Ultimately, we
were solving for in the first question and for in the second question.

Explain that there will be times when one strategy might be more helpful than the other,
as students will see in subsequent activities.

9.2 Cargo Shipping

20 minutes (there is a digital version of this activity)
This activity encourages students to write an equation in two variables to represent a
constraint and then solve for each of the variables. One motivation for rearranging the
equation is to find an expression that, when entered into a calculator or computer, can
then be used to quickly find the value of one quantity given the value of the other.

As they use spreadsheet technology to create mathematical models, test them, and solve
problems, students engage in aspects of mathematical modeling (MP4).

Addressing

• HSA-CED.A.3
• HSA-CED.A.4
• HSA-REI.B.3

Instructional Routines

• Aspects of Mathematical Modeling

Launch
Arrange students in groups of 3–4 and provide access to devices with spreadsheet or
graphing technology.

Give students a few minutes of quiet time to think about the first question and then time
to discuss their responses with their group. Ask students to pause for a class discussion,
and ensure everyone is using a correct equation before proceeding.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 4

Once students have responses for the second question, invite them to share how they
found the number of cars that can be shipped if the cargo already has some number of
trucks. Next, ask for the expressions they wrote to find the number of cars that can fit if
there are trucks. Record the expressions for all to see.

Tell students that we can test the expressions by using a calculator or computer. Consider
demonstrating how to use technology to calculate the number of cars given the number of
trucks. For examples using a table, sliders, and a spreadsheet, see the digital version of
this activity. You may need to prepare alternate instructions if using different technology in
the classroom.

Tell students that their job in the last question is to find the number of trucks when the
number of cars is known. Encourage them to test their equations using available
technology.

Support for Students with Disabilities

Representation: Access for Perception. Provide students with a physical copy of written
directions for using the spreadsheet or graphing technology available and read them
aloud. Include step-by-step directions for how to enter equations, find the value of
one variable by entering the value of another, and test equations.
Supports accessibility for: Language; Memory

Student Task Statement
An automobile manufacturer is preparing a shipment of
cars and trucks on a cargo ship that can carry 21,600
tons.

The cars weigh 3.6 tons each and the trucks weigh 7.5
tons each.

1. Write an equation that represents the weight constraint of a shipment. Let
be the number of cars and be the number of trucks.

2. For one shipment, trucks are loaded first and cars are loaded afterwards.
(Even though trucks are bulkier than cars, a shipment can consist of all trucks
as long as it is within the weight limit.)

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Find the number of cars that can be shipped if the cargo already has:
a. 480 trucks
b. 1,500 trucks
c. 2,736 trucks
d. trucks

3. For a different shipment, cars are loaded first, and then trucks are loaded
afterwards.
a. Write an equation you could enter into a calculator or a spreadsheet tool
to find the number of trucks that can be shipped if the number of cars is
known.

b. Use your equation and a calculator or a computer to find the number of
trucks that can be shipped if the cargo already has 1,000 cars. What if the
cargo already has 4,250 cars?

Student Response
1.
2. a. 5,000
b. 2,875
c. 300

d.

3. Sample response:

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 6

a. , where is the number of cars.

b. 2,400 and 840

Are You Ready for More?
For yet another shipment, the manufacturer is also shipping motorcycles, which
weigh 0.3 ton each.

1. Write an equation that you could enter into a calculator or a spreadsheet tool
to find the number of motorcycles that can be shipped, , if the number of
cars and trucks are known.

2. Use your equation to find the number of motorcycles that can be shipped if
the cargo already contains 1,200 trucks and 3,000 cars.

Student Response
Sample response:

1.

2. 6,000

Activity Synthesis

When finding the number of cars, , given trucks, students may have arrived at the

expression by generalizing the calculation they performed when the

number of trucks was a numerical value. Likewise, in the last question they may have

arrived at by using some numerical values for and generalizing the

process.

While this strategy is expected and perfectly reasonable, make sure students also see that
we can arrive at the same expression for and for by rearranging the equation

.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 7

Highlight that we can solve for when we know the number of trucks and want to
compute the number of cars, and solve for when we know the number of cars and want
to find the number of trucks.

9.3 Streets and Staffing

10 minutes
This activity gives students another opportunity to write and rearrange equations in two
variables and to do so in context.

Unlike in previous activities, students no longer start by computing one quantity given
numerical values of the other quantity and then generalizing the process. Instead, they are
prompted to write one equation, solve for each variable, and interpret the solution.
Students also articulate why one model might be more helpful than the other under a
certain circumstance. Along the way, students reason quantitatively and abstractly (MP2)
and engage in aspects of modeling (MP4).

Making spreadsheet technology available gives students an opportunity to choose
appropriate tools strategically (MP5).

Addressing

• HSA-CED.A.3
• HSA-CED.A.4
• HSA-REI.B.3

Instructional Routines

• Aspects of Mathematical Modeling
• MLR6: Three Reads

Launch
Familiarize students with the idea of organizational budgets, if needed. Explain that every
governmental agency has a budget (not unlike a budget for a pizza party) and needs to
decide how to allocate a limited amount of funds. Ask students to think of examples of

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 8

expenses that a city's Department of Streets might have (road repairs, new street
construction, additional workers, new equipment, and so on). Given a set budget,
spending more on one thing means having less to spend on something else, so the agency
would need to weigh their options and prioritize their needs.

Support for English Language Learners

Reading, Listening, Conversing: MLR6 Three Reads. Ask students to keep their books or
devices closed and display only the image and the task statement without revealing
the questions that follow. Use the first read to orient students to the situation. Ask
students to describe what the situation is about without using numbers (a city
department has money to resurface roads this year). Use the second read to identify
quantities and relationships. Ask students “What can be counted or measured?” Listen
for, and amplify, the important quantities that vary in relation to each other in this
situation: the total amount of money available, the cost to resurface 1 mile of a 2-lane
road, the average worker’s salary, and the number of miles of road to resurface. After
the third read, reveal the first question and ask students to brainstorm possible
strategies to write an equation. This routine helps students interpret the language
within a given situation needed to create an equation.
Design Principle(s): Support sense-making

Support for Students with Disabilities

Representation: Internalize Comprehension. Provide appropriate reading
accommodations and supports to ensure student access to written directions, word
problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing

Anticipated Misconceptions
Some students may have trouble sorting out the given information because of
unfamiliarity with the context or with certain terms. Ask them to explain the setup of the
problem as best they understand it, and then point out any information that might be
missing.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 9

Student Task Statement
The Department of Streets of a city has a budget of
$1,962,800 for resurfacing roads and hiring additional
workers this year.
The cost of resurfacing a mile of 2-lane road is
estimated at $84,000. The average starting salary of a
worker in the department is $36,000 a year.

1. Write an equation that represents the relationship between the miles of 2-lane
roads the department could resurface, , and the number of new workers it
could hire, , if it spends the entire budget.

2. Take the equation you wrote in the first question and:
a. Solve for . Explain what the solution represents in this situation.

b. Solve for . Explain what the solution represents in this situation.

3. The city is planning to hire 6 new workers and to use its entire budget.
a. Which equation should be used to find out how many miles of 2-lane
roads it could resurface? Explain your reasoning.

b. Find the number of miles of 2-lane roads the city could resurface if it
hires 6 new workers.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 10

Student Response
1.

2. a. . It represents the number of new workers the

department could hire if it resurfaces miles of roads and spends its entire
budget.

b. . It represents the miles of roads the department
could resurface if it spends the entire budget and hires new workers.

3. a. The equation where is already isolated should be used. We can simply
substitute 6 for in the expression that is equal to to find the miles of roads
that could be resurfaced.

b. about 20.77 or 20.8 miles

Activity Synthesis

Verify that the equations students wrote correctly isolate each variable. If students wrote

expressions in different forms for a variable (for instance, and

for ), discuss how the expressions are equivalent.

If time permits, consider asking students to use their equations to answer these questions:

• "How many people could be hired if the department is resurfacing 16 miles of

roads?" (17 people) "3 miles?" (47 people)

• "How many miles of roads could be resurfaced if the department is hiring

2 new workers?" (22.5 miles) "If no new workers are hired?" (About 23.37 miles)

Make sure students understand that solving for makes it possible to quickly find the
number of people that the department could hire given some miles of road to be
resurfaced (and while sticking to the budget). Similarly, solving for makes it possible to
quickly find the miles that could be resurfaced given any number of new hires.

Lesson Synthesis

Decribe and display the following situation to students.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 11

Suppose you are organizing a party and have a budget of dollars for the appetizers. You

plan to order vegetarian spring rolls at $0.75 each and shrimp rolls at $0.95 each. The

equation represents this constraint.

Ask one half of the class to solve for and the other half to solve for .

Then, ask students: "When might it be most handy to use each of these equations in your
party planning?"

• (When we want to find out how much different combinations or

rolls would cost.)

• (When we know the budget and want to find out how many vegetarian

rolls we could get if we order different numbers of shrimp rolls.)

• (When we know the budget and want to find out how many shrimp

rolls we could get if we order different numbers of vegetarian rolls.)

9.4 Carnival Tickets

Cool Down: 5 minutes
Addressing

• HSA-CED.A.4

• HSA-REI.B.3

Student Task Statement

A school is holding a carnival and hopes to raise $500. Child tickets cost $3 and

adult tickets cost $5. If the school sells child tickets and adult tickets, then the

equation expresses the fact that the school raised exactly $500 from

ticket sales.

1. Solve the equation for .

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 12

2. Explain when it might be helpful to rewrite the equation this way.

Student Response
1.

2. Sample response: Solving for helps us find out the number of child tickets to sell if
we know the number of adults attending the carnival and we want to make exactly
$500 from ticket sales.

Student Lesson Summary

Solving for a variable is an efficient way to find out the values that meet the
constraints in a situation. Here is an example.
An elevator has a capacity of 3,000 pounds and is being loaded with boxes of two
sizes—small and large. A small box weighs 60 pounds and a large box weighs 150
pounds.
Let be the number of small boxes and the number of large boxes. To
represent the combination of small and large boxes that fill the elevator to capacity,
we can write:

If there are 10 large boxes already, how many small boxes can we load onto the
elevator so that it fills it to capacity? What if there are 16 large boxes?
In each case, we can substitute 10 or 16 for and perform acceptable moves to
solve the equation. Or, we can first solve for :

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 13

This equation allows us to easily find the number of small boxes that can be loaded,
, by substituting any number of large boxes for .

Now suppose we first load the elevator with small boxes, say, 30 or 42, and want to
know how many large boxes can be added for the elevator to reach its capacity.
We can substitute 30 or 42 for in the original equation and solve it. Or, we can first
solve for :

Now, for any value of , we can quickly find by evaluating the expression on the
right side of the equal sign.
Solving for a variable—before substituting any known values—can make it easier to
test different values of one variable and see how they affect the other variable. It
can save us the trouble of doing the same calculation over and over.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 14

Lesson 9 Practice Problems

1. Problem 1

Statement

A car has a 16-gallon fuel tank. When driven on a highway, it has a gas mileage
of 30 miles per gallon. The gas mileage (also called "fuel efficiency") tells us the
number of miles the car can travel for a particular amount of fuel (one gallon
of gasoline, in this case). After filling the gas tank, the driver got on a highway
and drove for a while.

a. How many miles has the car traveled if it has the following amounts of
gas left in the tank?
i. 15 gallons
ii. 10 gallons
iii. 2.5 gallons

b. Write an equation that represents the relationship between the distance
the car has traveled in miles, , and the amount of gas left in the tank in
gallons, .

c. How many gallons are left in the tank when the car has traveled the
following distances on the highway?
i. 90 miles
ii. 246 miles

d. Write an equation that makes it easier to find the the amount of gas left
in the tank, , if we know the car has traveled miles.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 15

Solution

a. i. 30 miles
ii. 180 miles
iii. 405 miles

b.
c. i. 13 gallons

ii. 7.8 gallons

d.

2. Problem 2

Statement where is the

The area of a rectangle is represented by the formula
length and is the width. The length of the rectangle is 5.

Write an equation that makes it easy to find the width of the rectangle if we
know the area and the length.

Solution or

Sample responses:

3. Problem 3

Statement

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 16

Noah is helping to collect the entry fees at his school's sports game. Student
entry costs $2.75 each and adult entry costs $5.25 each. At the end of the
game, Diego collected $281.25.
Select all equations that could represent the relationship between the number
of students, , the number of adults, , and the dollar amount received at the
game.

A.
B.

C.
D.
E.

Solution

["A", "B"]

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 17

4. Problem 4

Statement

is an equation to calculate the volume of a cylinder, , where
represents the radius of the cylinder and represents its height.
Which equation allows us to easily find the height of the cylinder because it is
solved for ?

A.
B.
C.

D.

Solution

["C"]

5. Problem 5

Statement

The data represents the number of hours 10 students slept on Sunday night.

6677788
889
Are there any outliers? Explain your reasoning.

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 18

Solution

There are no outliers. The quartiles are Q1: 7 hours and Q3: 8 hours, so the
interquartile range is 1 hour. An outlier would need to be larger than 9.5 hours or
less than 5.5 hours.

(From Unit 1, Lesson 14.)

6. Problem 6

Statement

The table shows the volume of water in

cubic meters, , in a tank after water time after volume of water
has been pumped out for a certain pumping begins (cubic meters)
number of minutes.

0 30

Which equation could represent the 5 27.5
volume of water in cubic meters after 10 20
minutes of water being pumped out? 15 7.5

A.
B.
C.
D.

Solution

D

(From Unit 2, Lesson 4.)

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 19

7. Problem 7

Statement

A catering company is setting up for a wedding. They expect 150 people to
attend. They can provide small tables that seat 6 people and large tables that
seat 10 people.

a. Find a combination of small and large tables that seats exactly 150
people.

b. Let represent the number of small tables and represent the number
of large tables. Write an equation to represent the relationship between
and .

c. Explain what the point means in this situation.

d. Is the point a solution to the equation you wrote? Explain your
reasoning.

Solution

a. Sample response: 10 small tables and 9 large tables

b.

c. It represents the number of seats available if there are 20 small tables and 5
large tables.

d. No. Sample reasoning: With 20 small tables and 5 large tables, there will be
seating for 170 people, so some seats won’t be used.

(From Unit 2, Lesson 5.)

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 20

8. Problem 8
Statement

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 21

Which equation has the same solution as ?
A.
B.
C.
D.

Solution

D

(From Unit 2, Lesson 6.)

9. Problem 9

Statement

Noah is solving an equation and one of his moves is unacceptable. Here are
the moves he made.

Which answer best explains why the “divide each side by step” is
unacceptable?

A. When you divide both sides of by you get .

B. When you divide both sides of by it could lead us to think that

there is no solution while in fact the solution is .

C. When you divide both sides of by you get .

D. When you divide both sides of by it could lead us to think that

there is no solution while in fact the solution is .

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 22

Solution

B

(From Unit 2, Lesson 7.)

10. Problem 10

Statement

Lin says that a solution to the equation must also be a solution to

the equation .

Write a convincing explanation about why this is true.

Solution

Sample response: The two equations are equivalent. Adding to both sides of

gives . Adding the same expression to both sides of an

equation keeps the two sides equal, so the value of that is a solution to the first

equation is still a solution to the second equation.

(From Unit 2, Lesson 7.)

Algebra1 Unit 2 Lesson 9 CC BY 2019 by Illustrative Mathematics 23

Lesson 10: Connecting Equations to Graphs
(Part 1)

Goals are reflected on its graph and
• Analyze how the numbers in an equation

are related to the rate of change in the relationship.

• Graph linear equations of the form and interpret points on the graph in

context.

• Understand that different forms of a linear equation can give different insights about

the relationship it represents and about the graph.

Learning Targets , the
• I can describe the connections between an equation of the form

features of its graph, and the rate of change in the situation.

• I can graph a linear equation of the form .

• I understand that rewriting the equation for a line in different forms can make it

easier to find certain kinds of information about the relationship and about the

graph.

Lesson Narrative

Previously, students have written and interpreted equations that model quantitative
relationships and constraints. They have also rearranged and solved equations, isolated
one of the variables, and explained why the steps taken to rewrite equations are
legitimate.

In this lesson, students consider how parts of two-variable linear equations—the
parameters and variables—relate to features of the graphs of those equations. They also
think about how different forms of two-variable equations affect the information we could
gain about the relationships between the quantities and about the graphs. Throughout the
lessons, students practice reasoning quantitatively and abstractly (MP2) as they interpret
equations and graphs in context.

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 1

Alignments

Building On

• 8.EE.B.6: Use similar triangles to explain why the slope is the same between any

two distinct points on a non-vertical line in the coordinate plane; derive the equation

for a line through the origin and the equation for a line

intercepting the vertical axis at .

• HSA-SSE.A.1.a: Interpret parts of an expression, such as terms, factors, and

coefficients.

Addressing

• HSA-CED.A.3: Represent constraints by equations or inequalities, and by systems of

equations and/or inequalities, and interpret solutions as viable or nonviable options
in a modeling context. For example, represent inequalities describing nutritional and
cost constraints on combinations of different foods.

• HSA-CED.A.4: Rearrange formulas to highlight a quantity of interest, using the same

reasoning as in solving equations. For example, rearrange Ohm's law to

highlight resistance .

• HSA-REI.D.10: Understand that the graph of an equation in two variables is the set of

all its solutions plotted in the coordinate plane, often forming a curve (which could be

a line).

Instructional Routines

• Aspects of Mathematical Modeling
• MLR2: Collect and Display
• MLR5: Co-Craft Questions
• Think Pair Share

Student Learning Goals

• Let’s investigate what graphs can tell us about the equations and relationships

they represent.

10.1 Games and Rides

Warm Up: 5 minutes

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 2

Throughout this lesson, students will use a context that involves two variables—the
number of games and the number of rides at an amusement park—and a budgetary
constraint. This warm-up prompts students to interpret and make sense of some
equations in context, familiarizing them with the quantities and relationships (MP2). Later
in the lesson, students will dig deeper into what the parameters and graphs of the
equations reveal.

Building On

• HSA-SSE.A.1.a

Instructional Routines

• Think Pair Share

Launch
Arrange students in groups of 2. Give students a couple of minutes of quiet work time and
then another minute to share their response with their partner. Follow with a whole-class
discussion.

Student Task Statement
Jada has $20 to spend on games and rides at a carnival. Games cost $1 each and
rides are $2 each.

1. Which equation represents the relationship between the number of games, ,
and the number of rides, , that Jada could do if she spends all her money?

A: B: C:

2. Explain what each of the other two equations could mean in this situation.

Student Response
1. Equation C

2. Sample responses:

◦ A: Jada spends $20 on games and rides, which are $1 each.

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 3

◦ A: The combined number of games and rides that Jada enjoys is 20.

◦ B: Jada spends $20 on games and rides, where games are $2 each and rides are

$1 each.

Activity Synthesis
Invite students to share their interpretations of the equations.

Most students are likely to associate the 20 in the equation with the $20 that Jada has, but

some students may interpret it to mean the combined number of games and rides Jada

enjoys. (This is especially natural to do for .) If this interpretation comes up,

acknowledge that it is valid.

10.2 Graphing Games and Rides

20 minutes
This activity is the first of several that draw students' attention to the structure of linear
equations in two variables, how it relates to the graphs of the equations, and what it tells
us about the situations.

Students start by interpreting linear equations in standard form, , and using

them to answer questions and create graphs. They see that this form offers useful insights

about the quantities and constraints being represented. They also notice that graphing

equations in this form is fairly straighforward. We can use any two points to graph a line,

but the two intercepts of the graph (where one quantity has a value of 0) can be quickly

found using an equation in standard form.

Students then analyze the graphs to gain other insights. They determine the rate of
change in each relationship and find the slope and vertical intercept of each graph. Next,
they rearrange the equations to isolate . They make new connections here—the
rearranged equations are now in slope-intercept form, which shows the slope of the graph
and its vertical intercept. These values also tell us about the rate of change and the value
of one quantity when the other quantity is 0.

Building On

• 8.EE.B.6

Addressing

• HSA-CED.A.4

• HSA-REI.D.10

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 4

Instructional Routines

• MLR2: Collect and Display

Launch
Tell students that they will now interpret some other equations about games and rides.
They will also use graphs to help make sense of what combinations of games and rides are
possible given certain prices and budget constraints.

Read the opening paragraph in the task statement and display the three equations for all
to see. Give students a minute of quiet time to think about what each equation means in
the situation and then discuss their interpretations. Make sure students share these
interpretations:

• Equation 1: Games and rides cost $1 each and the student is spending $20 on them.
• Equation 2: Games cost $2.50 each and rides cost $1 each. The student is spending

$15 on them.

• Equation 3: Games cost $1 each and rides cost $4 each. The student is spending $28

on them.

Arrange students in groups of 3–4. Assign one equation to each group (or ask each group
to choose an equation). Ask them to answer the questions for that equation.

Give students 7–8 minutes of quiet work time, and then a few minutes to discuss their
responses with their group and resolve any disagreements. Ask groups that finish early to
answer the questions for a second equation of their choice. Follow with a whole-class
discussion.

Support for English Language Learners

Conversing: MLR2 Collect and Display. During the launch, listen for and collect language
students use to describe the meaning of the three equations. Record a written
interpretation next to each of the three equations on a visual display. Use arrows or
annotations to highlight connections between specific language of the interpretations
and the parts of the equations. This will provide students with a resource to draw
language from during small-group and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 5

Anticipated Misconceptions
Some students may not know how to interpret the phrase “for every additional game that
a student plays.” Suggest to students that they compare how many rides they could take if
they played 3 games, to the number of rides they could take if they played 4 games. What
about if they played 5 games? Ask them to notice how the number of rides changes when
one more game is played.

Student Task Statement
Here are the three equations. Each represents the relationship between the
number of games, , the number of rides, , and the dollar amount a student is
spending on games and rides at a different amusement park.
Equation 1:
Equation 2:
Equation 3:

Your teacher will assign to you (or ask you to choose) 1–2 equations. For each
assigned (or chosen) equation, answer the questions.

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 6

First equation:

1. What’s the number of rides the student could get on if they don’t play any
games? On the coordinate plane, mark the point that represents this situation
and label the point with its coordinates.

2. What’s the number of games the student could play if they don’t get on any
rides? On the coordinate plane, mark the point that represents this situation
and label the point with its coordinates.

3. Draw a line to connect the two points you’ve drawn.

4. Complete the sentences: “If the student played no games, they can get

on rides. For every additional game that the student plays, , the

possible number of rides, , (increases or decreases)

by .”

5. What is the slope of your graph? Where does the graph intersect the vertical
axis?

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 7

6. Rearrange the equation to solve for .

7. What connections, if any, do you notice between your new equation and the
graph?

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 8

Second equation:

1. What’s the number of rides the student could get on if they don’t play any
games? On the coordinate plane, mark the point that represents this situation
and label the point with its coordinates.

2. What’s the number of games the student could play if they don’t get on any
rides? On the coordinate plane, mark the point that represents this situation
and label the point with its coordinates.

3. Draw a line to connect the two points you’ve drawn.

4. Complete the sentences: “If the student played no games, they can get

on rides. For every additional game that a student plays, , the

possible number of rides, , (increases or

decreases) by .”

5. What is the slope of your graph? Where does the graph intersect the vertical
axis?

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 9

6. Rearrange the equation to solve for .

7. What connections, if any, do you notice between your new equation and the
graph?

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 10

Student Response
Equation 1:

1. 20 rides
2. 20 games

3.
4. 20, decreases by 1
5. Slope: -1. Vertical intercept:
6. or
7. Sample response: The new equation shows where the graph intersects the -axis

(at 20) and the slope of the graph (-1).
Equation 2: 2.50x + y = 15$

1. 15 rides
2. 6 games

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 11

3.
4. 15, decreases by 2.5
5. Slope: -2.5. Vertical intercept:
6. or
7. Sample response: The new equation shows where the graph intersects the -axis (at

15) and the slope of the graph (-2.5).
Equation 3:

1. 7 rides
2. 28 games

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 12

3.
4. 7, decreases by

5. Slope: . Vertical intercept:

6. or

7. Sample response: The new equation shows where the graph intersects the -axis
(at 7) and the slope of the graph ( ).

Activity Synthesis

Select students to briefly share the graphs and responses. Keep the original equations, the
rearranged equations, and their graphs displayed for all to see during discussion.

To help students see the connections between linear equations in standard form and their
graphs, ask students:

• “How did you find the number of possible rides when the student plays no games?”

(Subsitute 0 for and solve for .)

• “How did you find the number of possible games when the student gets on no rides?”

(Substitute 0 for and solve for .)

• “Where on the graph do we see those two situations (all games and no rides, or all

rides and no games)?” (On the vertical and horizontal axes or the - and -intercepts.)

• “The three equations are all given in the same form: . What information

can you get from an equation in this form? What do the , , and represent in each

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 13

equation?” ( is the price per game, is the price per ride, and is the amount of
money the student spends on games and rides.)

To help students see that an equivalent equation in slope-intercept form reveals other
insights about the situation and the graph, discuss:

• “If we rearrange the first equation and solve for , we get the equation . Is

the graph of this equation different from that of the original equation?” (No, the

equations are equivalent, so they have the same graph.)

• “You were asked to complete some sentences about what would happen if the

student played more games. How did the graph help you complete the sentences?”
(The graph shows how many rides the student can get on if they played no games.
The line slants downward, which means that the more games are played, the fewer
rides are possible. The graph shows how much the -value (number of rides) drops
when the -value (number of games) goes up by 1.)

• “Would you have been able to see the trade-offs between games and rides by looking

at the original equations in standard form?” (No, not easily.)

• “Do the rearranged equations still describe the same relationships between games

and rides?” (Yes. They are equivalent to the original.)

• “What new insights does this form of equation give us?” (Isolating gives an equation

in the form of , which reveals the slope of the graph and where it

intersects the -axis. The slope tells us how the number of rides changes if the

student plays additional games. The -intercept tells us the possible number of rides

when no games are played.)

Highlight that each form of equation gives us some insights about the relationship
between the quantities. Solving for gives us the slope and -intercept, which are handy
for creating or visualizing a graph. Even without a graph, the slope and -intercept can tell
us about the relationship between the quantities.

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 14

Support for Students with Disabilities

Representation: Internalize Comprehension. Demonstrate, and encourage students to use
color coding and annotations to highlight connections between representations in a
problem. For example, use the same color to illustrate where the slope appears in
each equation and corresponding graph. Continue to use colors consistently as
students discuss “What do the , , and represent in each equation?”
Supports accessibility for: Visual-spatial processing

10.3 Nickels and Dimes

10 minutes
This activity serves two practice goals: writing and graphing linear equations of the form

to represent a constraint, and interpreting points on a graph in terms of the
situation it represents. In this case, only whole-number values are meaningful for both
variables (number of dimes and number of nickels). Students need to consider
whether decimal solutions are reasonable in the situation.

Graphing the equation involves some decisions. The axes of the blank coordinate plane
are not labeled, so students need to decide which quantity goes on which axis (and to
recognize that the decision affects what each point on the graph represents). Students
could also choose to draw a continuous graph (a line) or a discrete graph (points at
whole-number values of one variable or both variables).

As students work, notice the graphing decisions students make. Identify students
who draw a discrete graph so they could share their rationale during class discussion.

Students engage in quantitative and abstract reasoning (MP2) as they think about the
solutions and graph of an equation in context. They practice aspects of modeling (MP4) as
they write an equation for a constraint, decide on representations for the model,
and reflect on whether the mathematical results make sense in the given situation.

Addressing

• HSA-CED.A.3
• HSA-REI.D.10

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 15

Instructional Routines

• Aspects of Mathematical Modeling
• MLR5: Co-Craft Questions

Launch
Consider keeping students in groups of 3–4.

Support for English Language Learners

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to provide opportunities
for students to analyze how different mathematical forms and symbols can represent
different situations. Display only the problem statement without revealing the
questions that follow. Invite students to write down possible mathematical questions
that could be asked about the situation. Invite students to compare their questions
before revealing the remainder of the question. Listen for and amplify any questions
that address quantities of each type of coin.
Design Principle(s): Maximize meta-awareness; Support sense-making

Support for Students with Disabilities

Representation: Internalize Comprehension. Activate or supply background knowledge
about generalizing a process to create an equation for a given situation. Some
students may benefit by first calculating how many nickels Andre would have if there
were 0, 1, 5, or 10 dimes in the jar, and then how many dimes if there were 1, 5, or 10
nickels in the jar. Invite students to use what they notice about the processes they
used to create an equation.
Supports accessibility for: Visual-spatial processing; Conceptual processing

Anticipated Misconceptions

Some students who wish to change their equation from standard form to slope-intercept

form may get stuck because they are not sure whether to solve for or . Either choice is

acceptable, but this is a good opportunity for students to think through the implications of

their choice. Ask students: “In , which variable goes on the horizontal axis?

Which goes on the vertical?”

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 16

Other students might wish to graph using the equation in standard form without first
rewriting it into another form. Ask if they could identify two points on the graph.
Alternatively, ask them to think about how many nickels there would be if there were 0
dimes, 1 dime, 2 dimes, and so on, and plot some points accordingly.

Student Task Statement
Andre’s coin jar contains 85 cents. There are no quarters or pennies in the jar, so
the jar has all nickels, all dimes, or some of each.

1. Write an equation that relates the
number of nickels, , the number of
dimes, , and the amount of money,
in cents, in the coin jar.

2. Graph your equation on the
coordinate plane. Be sure to label
the axes.

3. How many nickels are in the jar if
there are no dimes?

4. How many dimes are in the jar if
there are no nickels?

Student Response
1. , where is the number of nickels and is the number of dimes.
2. See sample graphs in the Activity Synthesis.
3. 17 nickels

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 17

4. It’s not possible for the coin jar to have no nickels if the amount of money in it is 85
cents.

Are You Ready for More?
What are all the different ways the coin jar could have 85 cents if it could also
contain quarters?

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 18

Student Response
Listed as nickels, dimes, quarters: (17,0,0), (15,1,0), (13,2,0), (12,0,1), (11,3,0), (10,1,1), (9,4,0),
(8,2,1), (7,0,2), (7,5,0), (6,3,1), (5,1,2),(5,6,0), (4,4,1), (3,2,2), (2,0,3), (3,7,0), (2,5,1), (1,3,2),
(0,1,3), (1,8,0), (0,6,1)

Activity Synthesis
Select previously identified students to share their graphs. For each graph, ask if anyone
else also drew it the same way. If no one drew discrete graphs and no one mentioned that
fractional values of or have no meaning or are not possible in the situation, ask
students about it.

Display the following graphs (or comparable graphs by students) for all to see.

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 19

Make sure students understand that all of these graphs are acceptable representations of
the relationship between the quantities. A graph showing only points with whole-number
coordinate values represents the solutions to the equation accurately but may be time
consuming to draw. A line may be a quicker way to see the possible solutions and can be
used for problem solving as long as we are aware that only points with whole-number
values make sense.

For example, when reasoning about the last question, students who used a continuous
graph might see that the jar would contain 8.5 dimes if it has no nickels. It is important
that they recognize that this is impossible. The same reflection about the context is also
necessary if students answered the question by solving the equation for when is 0.

If time permits, discuss these questions to reinforce the connections to earlier work on
equivalent equations and their graphs:

• "Suppose you were to express the relationship between the same quantities but in

dollars instead of in cents. What would the equation look like?" ( )

• "What would the graph of this equation look like? Try graphing it on the same

coordinate plane." (It'd be the same line as the graph for .)

• "Why would the graph of this equation be identical to the other one?" (The two

equations are equivalent. Dividing the first equation—representing the relationship
in cents—by 100 gives the second equation—representing the relationship in dollars.
The same combinations of nickels and dimes make both equations true.)

Lesson Synthesis

To help students consolidate their work in this lesson, discuss questions such as:

• "We saw equations in different forms representing the same constraint. For

example, and both represent the games and rides that a

student could do with a fixed budget. What information about the situation and

about the graph can we gain from the standard form, ?" (In this

example, the standard form allows us to see the cost per ride, the cost per game, and

the budget.)

• "What information does the slope-intercept form give us?" (It gives us the slope and

-intercept of the graph. The slope tells us what is given up in terms of rides for each
additional game played. The -intercept tells us how many rides are possible when
no games are played.)

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 20

• "What might be an efficient way to graph an equation of the form ?"

(Substituting 0 for or for in the equation. Doing so gives us and ( , which

are the horizontal and vertical intercepts of the graph. We could choose two other

points, as well, but using 0 eliminates one of the variables, simplifying the calculation.

Alternatively, we could isolate and rearrange the equation into slope-intercept

form, which shows us the -intercept and the slope.)

10.4 Kiran at the Carnival

Cool Down: 5 minutes
Addressing

• HSA-CED.A.3

• HSA-REI.D.10

Student Task Statement
Kiran is spending $12 on games and rides at another carnival, where a game costs
$0.25 and a ride costs $1.

1. Write an equation to represent the relationship between the dollar amount
Kiran is spending and the number of games, , and the number of rides, , he
could do.

2. Which graph represents the relationship between the quantities in this
situation? Explain how you know.

ABC

Student Response
1.

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 21

2. Graph C (for ). Sample explanations:

◦ If Kiran plays 0 games, he could get on 12 rides. If he gets on 0 rides, he could

play 48 games. Both the points and are on the line in Graph C.

◦ Rearranging the equation into slope-intercept form gives , so the

graph has a slope of and intersects the -axis at 12, which matches Graph C.

Student Lesson Summary

Linear equations can be written in different forms. Some forms allow us to better
see the relationship between quantities or to predict the graph of the equation.

Suppose an athlete wishes to burn 700 calories a day by running and swimming. He
burns 17.5 calories per minute of running and 12.5 calories per minute of freestyle
swimming.

Let represents the number of minutes of running and the number of minutes of
swimming. To represent the combination of running and swimming that would
allow him to burn 700 calories, we can write:

We can reason that the more minutes he runs, the fewer minutes he has to swim to
meet his goal. In other words, as increases, decreases. If we graph the equation,
the line will slant down from left to right.

Algebra1 Unit 2 Lesson 10 CC BY 2019 by Illustrative Mathematics 22