IM UNIT 2 Equations and Inequalities Teacher Guide

Lesson 3: Writing Equations to Model
Relationships (Part 2)

Goals
• Identify and describe (orally and in writing) patterns in tables of values and in

calculations.

• Use observed patterns to generalize the relationships between quantities and to

write equations.

Learning Targets
• I can use words and equations to describe the patterns I see in a table of values or in

a set of calculations.

• When given a description of a situation, I can use representations like diagrams and

tables to help make sense of the situation and write equations for it.

Lesson Narrative

In this lesson, students continue to develop their ability to identify, describe, and model
relationships with mathematics.

Previously, students worked mostly with descriptions of familiar relationships and were
guided to reason repeatedly, which enabled them to see a general relationship between
two quantities. Here, students are given tables of values and asked to generalize the
relationship between pairs of quantities—by studying the values and looking for patterns
(MP8), and by interpreting them in context (MP2). Some of the relationships they
encounter here are novel or otherwise require perseverance and careful reasoning to pin
down (MP1).

Alignments
Addressing

• HSA-CED.A.2: Create equations in two or more variables to represent relationships

between quantities; graph equations on coordinate axes with labels and scales.

• 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.

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Building Towards

• HSA-CED.A.2: Create equations in two or more variables to represent relationships

between quantities; graph equations on coordinate axes with labels and scales.

• HSF-LE.A.2: Construct linear and exponential functions, including arithmetic and

geometric sequences, given a graph, a description of a relationship, or two
input-output pairs (include reading these from a table).

Instructional Routines

• Anticipate, Monitor, Select, Sequence, Connect
• MLR8: Discussion Supports
• Take Turns

Student Learning Goals

• Let's use patterns to help us write equations.

3.1 Finding a Relationship

Warm Up: 5 minutes
The activities in this lesson require students to observe tables of values, look for patterns,
and generalize their observations into equations. This warm-up prompts students to think
about how they could go about analyzing the values in the table and to articulate their
reasoning.

Building Towards

• HSA-CED.A.2
• HSF-LE.A.2

Launch
Arrange students in groups of 2. Display the table for all to see. Explain that the quantities
in each column are related.

Ask groups to try to find a relationship and pay attention to how they go about doing
so. Emphasize that the goal is not to successfully find a relationship. It is to notice
the strategies they use when attempting to figure out what the relationship might be.

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Student Task Statement
Here is a table of values. The two quantities, and , are related.

What are some strategies you could use to find a relationship
between and ? Brainstorm as many ways as possible.

10

38

5 24

7 48

Student Response
Sample responses:

• See if there is an operation that could be done to that would produce .
• Compare how the -values change and the -values change, and see if there is any

pattern.

• See if the numbers in one column follow a special pattern and if that pattern could be

connected to the numbers in the other column.

• Plot the - and -values on a coordinate plane and see if the graph looks like one of a

familiar relationship.

Activity Synthesis
Invite groups to share their strategies and record them for all to see. If not already
described by students, apply each strategy using the values in the table, or ask students to
give an example of how it could be applied.

Some students may notice that each time increases by 2, increases by 8 more than the
previous time. Others may notice that the -values are 1 less than the square numbers 1,
4, 9, 25, and 49, and that these numbers are the squares of the listed -values, and from
there concluded that the relationship is along the lines of: "square and subtract 1 to get

." Neither observations are essential, but consider asking if they see any special patterns
in either column that could help determine the relationship.

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If no one mentions plotting the pairs of values as a way to
understand the relationship between and , and if time
permits, consider diplaying a graph such as shown (or
displaying a blank coordinate plane and plotting the points
together).

Ask students to keep in mind the different strategies as they work on the activities in the
lesson.

3.2 Something about 400

15 minutes
Previously, students wrote equations to model relationships presented via verbal
descriptions. In this partner activity, students are presented with pairs of values that
represent quantities and take turns describing the relationship between the
quantities—first using words, and then using equations.

To do so, they need to interpret the values in context, look for structure or patterns, and
generalize them (MP7). As they do so, students also practice reasoning quantitatively and
abstractly (MP2). As students trade roles explaining their thinking and listening, they have
opportunities to explain their reasoning and critique the reasoning of others (MP3).

As students discuss their thinking, listen for the different ways they describe the same
relationship. For example, here are some ways students might describe the relationship in
the second table ("meters from home" and "meters from school"):

• The distance from home and the distance from school always add up to 400.
• The distance from school is always 400 minus the distance from home.
• As the distance from home, , increases by a number, the distance from school, ,

decreases by the same number. starts at 0 and starts at 400.

• The distance between home and school is 400 meters. The table seems to be telling

us about a person traveling from home to school and how their distance to home
and distance to school change along the way.

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Select students with different analyses and descriptions to share later.

Addressing

• HSA-CED.A.2

Instructional Routines

• Anticipate, Monitor, Select, Sequence, Connect
• MLR8: Discussion Supports
• Take Turns

Launch
Keep students in groups of 2. Give students a few minutes of quiet time to study the tables
of values in the first question. Then, ask partners to take turns describing the relationships
in the tables before moving on to the second question. As one person explains, the
partner's job is to listen and make sure they agree. If they don't agree, the partners discuss
until they reach an agreement.

Consider asking partners to also take turns matching the tables to the equations in the
second question.

Some students may not know what is meant by “amount deposited” in Table D. Clarify this
term to students if needed.

If time is limited, consider asking each group to analyze only two of the tables.

Support for English Language Learners

Conversing: MLR8 Discussion Supports. Arrange students in groups of 2. Students should
take turns finding a match and explaining their reasoning to their partner. Display the
following sentence frames for all to see: “_____ and _____ are equivalent because….”,
and “I noticed _____ , so I matched….” Encourage students to challenge each other
when they disagree. This will help students clarify their reasoning about equations
that represent the relationship.
Design Principle(s): Support sense-making; Maximize meta-awareness

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Support for Students with Disabilities

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived
challenge. Invite students to select 1–2 tables to describe in words, and to use the
tables they described as a starting point to match equations. Chunking this task into
more manageable parts may also benefit students who benefit from additional
processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Anticipated Misconceptions

The relationship in Table D may not be obvious to students. Encourage students who get
stuck to look at the equations in the last question and try to figure out which equation
desribes the relationship between the numbers in the table.

Student Task Statement
1. Describe in words how the two quantities in each table are related.

◦ Table A

number of laps, 01 2.5 6 9
meters run,
0 400 1,000 2,400 3,600

◦ Table B

meters from home, 0 75 128 319 396
meters from school,
400 325 272 81 4

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◦ Table C

electricity bills in dollars, 85 124 309 816
total expenses in dollars, 485 524 709 1,216

◦ Table D

monthly salary in dollars, 872 998 1,015 2,110
amount deposited in dollars, 472 598 615 1,710

2. Match each table to an equation that represents the relationship.

◦ Equation 1:
◦ Equation 2:
◦ Equation 3:
◦ Equation 4:

Student Response
1. Sample response:

◦ Table A: Every lap is 400 meters, so the distance run is 400 times the number of

laps.

◦ Table B: The distance between home and school is 400 meters. Every meter

traveled away from home is a meter closer to school.

◦ Table C: Total expenses can be found by adding $400 to electricity bills. (The

$400 could represent rent payment.)

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◦ Table D: The amount deposited is $400 less than monthly salary. (The employer

might deduct $400 per paycheck for taxes, health insurance, retirement savings,
and so on. Or, the employee might first pay $400 in rent or other expenses
before depositing the rest of the monthly salary.)

2. ◦ Table A: Equation 4
◦ Table B: Equation 3
◦ Table C: Equation 1
◦ Table D: Equation 2

Are You Ready for More?
Express every number between 1 and 20 at least one way using exactly four 4’s and
any operation or mathematical symbol. For example, 1 could be written as

.

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Student Response
Sample response:

Activity Synthesis

Invite previously identified students to share how they thought about one of
the relationships in the first question. Start with students who reasoned only in terms of
numerical operations, and move toward those who interpreted the quantities in context
(as shown in the Activity Narrative). If possible, record and display their descriptions for all
to see, and highlight the connections between the different responses.

Discuss with students whether or how their ways of thinking about each relationship
affected the work of matching the tables and equations. If not brought up in students'
comments, point out that some ways of describing a relationship could make it easier to
identify or write a corresponding equation. To really understand what's happening in the
situation, however, often requires carefully interpreting the operations that relate the two
quantities.

3.3 What are the Relationships?

15 minutes
In an earlier activity, students discerned the relationship between two quantities by
analyzing and looking for patterns in tables of values. They then described each
relationship in words and identified a corresponding equation. In that activity, all
relationships were linear.

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This activity offers new opportunities to identify and represent relationships between pairs
of quantities. In each of the first two situations, the relationship is an inverse variation.
(Students do not need to know the term or the concept to be able to reason about the
relationships.) The last situation involves a proportional relationship. Because the given
information involves three quantities (volume in gallons, in cups, and in fluid ounces),
students need to reason carefully about how two of them (volume in gallons and in fluid
ounces) are related.

Monitor for different ways students use to figure out and articulate how the quantities are
related. For instance, students may:

• Use broad and qualitative descriptions ("As the base length increases, the height

decreases").

• Give specific and quantitative descriptions ("The product of the base length and the

height is always 48").

• Use diagrams, tables, graphs, or equations to make sense of the relationships and to

illustrate them.

Identify students using varying strategies and ask them to share during discussion later.

Addressing

• HSA-CED.A.2
• HSA-CED.A.3

Instructional Routines

• Anticipate, Monitor, Select, Sequence, Connect

Launch
Tell students that they will now describe the relationship between two quantities in some
new situations.

If time is limited, ask students to focus on the first two questions.

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Support for Students with Disabilities

Action and Expression: Develop Expression and Communication. Provide options for
communicating understanding. Invite students to describe the relationships between
quantities in different ways, for example, using verbal (written or oral) descriptions,
tables, diagrams, or other representations.
Supports accessibility for: Language; Organization

Anticipated Misconceptions
If students assume that the relationships must be expressed as equations and they get
stuck, clarify that verbal descriptions, tables, or other representations are just as welcome.

When answering the first question, students may look only at the relationship between the
first few rows of the table and say that the -values are decreasing by 8 each time, not
noticing that this is not always the case. Encourage them to look farther down the table
and to also look at the relationships between the values in the columns.

Some students may have trouble getting started on the question about the volume of milk
or setting up a table. Ask students which units are given in the problem, or suggest the
headings “gallons,” “cups,” and “fluid ounces.” Then, ask them to use the given information
to complete a row in the table. This might involve trying a different unit to start with. (For
example, if they start with gallon and struggle to find the equivalent amount in cups and
fluid ounces, try starting with "4 cups" or "8 cups.") A more direct hint is to suggest finding
the number of fluid ounces in 8 cups.

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Student Task Statement
1. The table represents the relationship between the base length and the height
of some parallelograms. Both measurements are in inches.

base length (inches) height (inches)
1 48
2 24
3 16
4 12
68

What is the relationship between the base length and the height of these
parallelograms?

2. Visitors to a carnival are invited to guess the number of beans in a jar. The
person who guesses the correct number wins $300. If multiple people guess
correctly, the prize will be divided evenly among them.
What is the relationship between the number of people who guess correctly
and the amount of money each person will receive?

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3. A -gallon jug of milk can fill 8 cups, while 32 fluid ounces of milk can fill 4
cups.

What is the relationship between number of gallons and ounces? If you get
stuck, try creating a table.

Student Response
1. Sample responses:

◦ As the base length increases by 1, the height decreases, but not by a steady

amount.

◦ Multiplying the base length and the height gives 48. The area of each

parallelogram is 48 square inches.

◦ , or (or equivalent)

2. Sample responses:

◦ number of winners amount in dollars each winner receives

1 300

2 150

3 100

5 60

15 20

◦ The amount in dollars received by each winner is 300 divided by the number of

winners.

◦ , or (or equivalent), where is the dollar amount a winner

receives and is the number of winners.

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3. Sample responses:

◦ If 4 cups contain 32 fluid ounces, then 8 cups must contain 64 fluid ounces.

Because there are 16 cups in 1 gallon and 16 is twice of 8, there must be 128

fluid ounces in a gallon.

◦ gallons cups fluid ounces

8 64

1 16 128

◦ Multiplying the number of gallons by 128 gives the number of fluid ounces.

◦ , or (or equivalent), where is the number of fluid

ounces and is the number of gallons.

Activity Synthesis

Select previously identified students to share their responses and thinking. Sequence the
presentation in the order of precision, starting with the broader descriptions or
illustrations and ending with equations. (Remind students who use equations to specify
what the variables represent.) If students write equations in different forms to describe
the same relationship, record and display the equivalent equations for all to see.

If no students considered using tables to make sense of the pairs of quantities, ask how
these would help and show an example. A table, for instance, can be particularly helpful
for reasoning about the last two relationships.

Display the equations that can represent the three situations. Highlight that writing
equations is an efficient way to capture the constraints in a situation.

To help students connect the equations to prior work, ask students which quantities vary
and which remain constant in each equation. Point out that these equations are also
equations in two variables, but unlike the equations we saw in the previous activity, not all
of these represent linear relationships.

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Lesson Synthesis

In the lesson, students used a number of ways to reason about the relationship between
quantities and to write an equation to represent that relationship. Invite them to
summarize those reasoning strategies, which might include:

• creating a table to help us see how a quantity changes or how two quantities might

be related

• looking for a pattern in the table: noticing how the values in a table change from one

row to the next, or from one column to the next

• trying different numbers for one variable and observing how they affect the other

variables

Select a couple of tables and descriptions of situations from the lesson to elicit students'
reflections. Or, if time permits, consider using these two new situations:

• A student starts a new semester with $30 in their lunch account. Each lunch at school

costs $1.75. What's the relationship between the number of school lunches
purchased, , and the dollar amount in the account, ?

• A chef is pouring oil from a large jug into equal-size bottles. This table shows the

relationship between the number of bottles used and the volume of oil, in fluid
ounces, in each bottle.

number of bottles fluid ounces per bottle

3 24

4 18

6 12

89

3.4 Labeling Books

Cool Down: 5 minutes

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Addressing

• HSA-CED.A.2
• HSA-CED.A.3

Student Task Statement
Clare volunteers at a local library during the summer. Her work includes putting
labels on 750 books.

1. How many minutes will she need to finish labeling all books if If she takes no
breaks and labels:
a. 10 books a minute

b. 15 books a minute

2. Suppose Clare labels the books at a constant speed of books per minute.
Write an equation that represents the relationship between her labeling
speed and the number of minutes it would take her to finish labeling.

Student Response , where is the number of minutes and is the speed in
1. a. 75 minutes

b. 50 minutes

2. , or
books per minute.

Student Lesson Summary

Sometimes, the relationship between two quantities is easy to see. For instance, we

know that the perimeter of a square is always 4 times the side length of the square.

If represents the perimeter and the side length, then the relationship between

the two measurements (in the same unit) can be expressed as , or .

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Other times, the relationship between quantities might take a bit of work to figure
out—by doing calculations several times or by looking for a pattern. Here are two
examples.

• A plane departed from New Orleans and is heading to San Diego. The table

shows its distance from New Orleans, , and its distance from San Diego, , at
some points along the way.

miles from New Orleans miles from San Diego

100 1,500
300 1,300
500 1,100

1,020
900 700
1,450

What is the relationship between the two distances? Do you see any patterns
in how each quantity is changing? Can you find out what the missing values
are?
Notice that every time the distance from New Orleans increases by some
number of miles, the distance from San Diego decreases by the same number
of miles, and that the sum of the two values is always 1,600 miles.
The relationship can be expressed with any of these equations:

• A company decides to donate $50,000 to charity. It will select up to 20

charitable organizations, as nominated by its employees. Each selected
organization will receive an equal amount of donation.

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What is the relationship between the number of selected organizations, , and
the dollar amount each of them will receive, ?

◦ If 5 organizations are selected, each one receives $10,000.
◦ If 10 organizations are selected, each one receives $5,000.
◦ If 20 organizations are selected, each one receives $2,500.

Do you notice a pattern here? 10,000 is , 5,000 is , and 2,500 is
.

We can generalize that the amount each organization receives is 50,000

divided by the number of selected organizations, or .

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Lesson 3 Practice Problems

1. Problem 1

Statement number of loads total weight in pounds
of crushed stone
A landscaping company is
delivering crushed stone 00
to a construction site. The
table shows the total 1 2,000
weight in pounds, , of
loads of crushed stone. 2 4,000

Which equation could 3 6,000
represent the total
weight, in pounds, for
loads of crushed stone?

A.

B.
C.
D.

Solution

C

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2. Problem 2

Statement

Members of the band sold juice and popcorn at a college football game to
raise money for an upcoming trip. The band raised $2,000. The amount raised
is divided equally among the members of the band.
Which equation represents the amount, , each member receives?

A.

B.

C.
D.

Solution

B

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3. Problem 3
Statement

Tyler needs to complete this table for his consumer science class. He knows
that 1 tablespoon contains 3 teaspoons and that 1 cup contains 16
tablespoons.

number of teaspoons number of tablespoons number of cups
2

36 12
48 3

a. Complete the missing values in the table.
b. Write an equation that represents the number of teaspoons, , contained

in a cup, .

Solution

Sample response:

number of teaspoons number of tablespoons number of cups
a.

96 32 2
36 12 0.75
144 48 3

b.

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4. Problem 4
Statement

The volume of dry goods, like apples or peaches, can be measured usings
bushels, pecks, and quarts. A bushel contains 4 pecks, and a peck contains 8
quarts.
What is the relationship between number of bushels, , and the number of
quarts, ? If you get stuck, try creating a table.

Solution

Sample response:

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5. Problem 5
Statement

The data show the number of free throws attempted by a team in its first ten
games.

2 11 11 11 12 12 13
14 14 15
The median is 12 attempts and the mean is 11.5 attempts. After reviewing the
data, it is determined that 2 should not be included, since that was an
exhibition game rather than a regular game during the season.
a. What happens to the median if 2 attempts is removed from the data set?

b. What happens to the mean if 2 attempts is removed from the data set?

Solution

a. The median remains 12 attempts.

b. The mean increases to about 12.6 attempts.

(From Unit 1, Lesson 10.)

6. Problem 6

Statement

The standard deviation for a data set is 0. What can you conclude about the
data?

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Solution

Sample responses:

◦ The standard deviation measures the variation of the data. If the standard

deviation is 0, then there is no variation, so all the data is the same.

◦ All the data must be equal to the mean, because the sum of the squares of the

differences of the data from the mean is 0. So, all the data take the same value.

(From Unit 1, Lesson 12.)

7. Problem 7

Statement

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Elena has $225 in her bank account. She takes out $20 each week for weeks.
After weeks she has dollars left in her bank account.
Write an equation that represents the amount of money left in her bank
account after weeks.

Solution

Sample response:

(From Unit 2, Lesson 2.)

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8. Problem 8

Statement

Priya is hosting a poetry club meeting this week and plans to have fruit punch
and cheese for the meeting. She is preparing 8 ounces of fruit punch per
person and 2 ounces of cheese per person. Including herself, there are 12
people in the club.

A package of cheese contains 16 ounces and costs $3.99. A one-gallon jug of
fruit punch contains 128 ounces and costs $2.50. Let represent number of
people in the club, represent the ounces of fruit punch, represent the
ounces of cheese, and represent Priya's budget in dollars.

Select all of the equations that could represent the quantities and constraints
in this situation.

A.

B.

C.

D.

E.

Solution

["A", "C", "D"]

(From Unit 2, Lesson 2.)

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9. Problem 9
Statement

The density of an object can be found by taking its mass and dividing by its
volume.
Write an equation to represent the relationship between the three quantities
(density, mass, and volume) in each situation. Let the density, , be
measured in grams/cubic centimeters (or g/cm3).

a. The mass is 500 grams and the volume is 40 cubic centimeters.

b. The mass is 125 grams and the volume is cubic centimeters.

c. The volume is 1.4 cubic centimeters and the density is 80 grams per
cubic centimeter.

d. The mass is grams and the volume is cubic centimeters.

Solution , or equivalent)

a. (or

b.
c.

d.

(From Unit 2, Lesson 2.)

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Lesson 4: Equations and Their Solutions

Goals
• Explain (orally and in writing) the meaning of solutions to equations in one variable

and two variables.

• Find solutions to equations in one variable and in two variables by reasoning about

the relationships in context.

• Interpret solutions to equations in one variable and in two variables.

Learning Targets
• I can explain what it means for a value or pair of values to be a solution to an

equation.

• I can find solutions to equations by reasoning about a situation or by using algebra.

Lesson Narrative

In middle school, students learned that a solution to an equation is a value or values that
make the equation true. In this lesson, they revisit what they learned about solutions to
equations in one variable and two variables. They also continue to practice modeling
relationships with equations and to make sense of equations and their solutions in context
(MP2, MP4).

Students verify and find solutions to given equations by checking if the values satisfy the
equations and by reasoning. Some students may choose to solve equations algebraically
or by performing certain sequences of steps they learned in middle school, but students
are not expected to rely on algebraic methods to answer questions here. A little later in
the unit, students will take a close look at the moves for rewriting and solving equations.

In the next lesson, students will revisit the idea that coordinate pairs that are on a graph of
an equation in two variables are solutions to the equation.

Alignments
Addressing

• HSA-REI.A: Understand solving equations as a process of reasoning and explain the

reasoning.

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

equations with coefficients represented by letters.

Instructional Routines

• MLR3: Clarify, Critique, Correct

• Think Pair Share

Student Learning Goals

• Let’s recall what we know about solutions to equations.

4.1 What is a Solution?

Warm Up: 5 minutes
This warm-up prompts students to recall what they know about the solution to an
equation in one variable. Students interpret a given equation in the context of a situation,
explain why certain values are not solutions to the equation, and then find the value that is
the solution.

Addressing

• HSA-REI.A

• HSA-REI.B.3

Student Task Statement

A granola bite contains 27 calories. Most of the calories come from grams of
carbohydrates. The rest come from other ingredients. One gram of carbohydrate
contains 4 calories.

The equation represents the relationship between these quantities.

1. What could the 5 represent in this situation?

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2. Priya said that neither 8 nor 3 could be the solution to the equation. Explain
why she is correct.

3. Find the solution to the equation.

Student Response

1. Sample response: It could represent the calories from other ingredients in the
granola bite.

2. No, 8 and 3 could not be solutions. Sample explanation: is 37, not 27, and
is 17, not 27.

3.

Activity Synthesis

Focus the discussion on how students knew that 8 and 3 are not solutions to the equation
and how they found the solution. Highlight strategies that are based on reasoning about
what values make the equation true.

Ask students: "In general, what does a solution to an equation mean?" Make sure students
recall that the solution to an equation in one variable is a value for the variable that makes
the equation a true statement.

4.2 Weekend Earnings

15 minutes
In this activity, students write equations in one variable to represent the constraints in a
situation. They then reason about the solutions and interpret the solutions in context.

To solve the equation, some students may try different values of until they find one that
gives a true equation. Others may perform the same operations to each side of the

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equation to isolate . Identify students who use different strategies and ask them to share
later.

Addressing

• HSA-REI.A
• HSA-REI.B.3

Instructional Routines

• MLR3: Clarify, Critique, Correct
• Think Pair Share

Launch
Arrange students in groups of 2 and provide access to calculators. Give students a few
minutes of quiet work time, and then time to discuss their responses. Ask them to share
with their partner their explanations for why 4 and 7 are or are not solutions.

If students are unsure how to interpret “take-home earnings,” clarify that it means the
amount Jada takes home after paying job-related expenses (in this case, the bus fare).

Support for English Language Learners

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students interpret
the language of writing equations, and to increase awareness of language used to talk
about representing situations with equations. Display only the task statement that
describes the context, without revealing the questions that follow. Invite students to
discuss possible mathematical questions that could be asked about the situation.
Listen for and amplify any questions involving equations that connect the quantities
in this situation.
Design Principle(s): Maximize meta-awareness; Support sense-making

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

Support for Students with Disabilities

Action and Expression: Internalize Executive Functions. Chunk this task into more
manageable parts for students who benefit from support with organizational skills in
problem solving. Check in with students after the first 2–3 minutes of work time. Invite
1–2 students to share how they determined an equation that represents Jada’s
take-home earnings. Record their thinking on a display and keep the work visible as
students continue to work.
Supports accessibility for: Organization; Attention

Anticipated Misconceptions
If students struggle to write equations in the first question, ask them how they might find
out Jada's earnings if she works 1 hour, 2 hours, 5 hours, and so on. Then, ask them to
generalize the computation process for hours.

Student Task Statement
Jada has time on the weekends to earn some money. A local bookstore is looking
for someone to help sort books and will pay $12.20 an hour. To get to and from the
bookstore on a work day, however, Jada would have to spend $7.15 on bus fare.

1. Write an equation that represents Jada’s take-home earnings in dollars, , if
she works at the bookstore for hours in one day.

2. One day, Jada takes home $90.45 after working hours and after paying the
bus fare. Write an equation to represent this situation.

3. Is 4 a solution to the last equation you wrote? What about 7?

◦ If so, be prepared to explain how you know one or both of them are

solutions.

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 5

◦ If not, be prepared to explain why they are not solutions. Then, find the

solution.

4. In this situation, what does the solution to the equation tell us?

Student Response
1.

2. (or equivalent)

3. Neither 4 nor 7 is a solution. Sample reasoning: or

◦ Substituting 4 into the equation gives

, which is not a true statement.

◦ Substituting 7 into the equation gives , which is also false.

◦ The solution is 8. Substituting 8 into the equation gives , which is

true.

4. It tells us the number of hours that Jada worked that allowed her to take home
$90.45 after paying for her bus fare.

Are You Ready for More?
Jada has a second option to earn money—she could help some neighbors with
errands and computer work for $11 an hour. After reconsidering her schedule, Jada
realizes that she has about 9 hours available to work one day of the weekend.

Which option should she choose—sorting books at the bookstore or helping her
neighbors? Explain your reasoning.

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 6

Student Response

Students could argue either way, depending on the assumptions they make. Sample
responses:

• Jada should work at the bookstore because she would earn more there. Her pay

would be $109.80, and after subtracting $7.15 for the bus pass, she would still earn
$102.65. She would earn $99 from the other option.

• Jada should help her neighbors. Working 9 hours at the bookstore would mean a few

extra dollars than working 9 hours helping her neighbors, but it would also mean
losing some personal time because of the travel involved.

Activity Synthesis

Ask a student to share the equation that represents Jada earning $90.45. Make sure

students understand why describes that constraint.

Next, invite students to share how they knew if 4 and 7 are or are not solutions to the
equation. Highlight that substituting those values into the equation and evaluating them
lead to false equations.

Then, select students using different strategies to share how they found the solution.

Some students might notice that the solution must be greater than 7 (because when

, the expression has a value less than 90.45) and start by checking if

is a solution. If no students mention this, ask them about it.

Make sure students understand what the solution means in context. Emphasize that 8 is
the number of hours that meet all the constraints in the situation. Jada gets paid $12.20 an
hour, pays $7.15 in bus fare, and takes home $90.45. For all of these to be true, she must
have worked 8 hours.

4.3 Calories from Protein and Fat

15 minutes
In the previous activity, students recalled what it means for a number to be a solution to
an equation in one variable. In this activity, they review the meaning of a solution to an
equation in two variables.

Addressing

• HSA-REI.A
• HSA-REI.B.3

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 7

Launch
Give students continued access to calculators.

Student Task Statement

One gram of protein contains 4 calories. One gram of fat contains 9 calories. A
snack has 60 calories from grams of protein and grams of fat.

The equation represents the relationship between these quantities.

1. Determine if each pair of values could be the number of grams of protein and
fat in the snack. Be prepared to explain your reasoning.

a. 5 grams of protein and 2 grams of fat

b. 10.5 grams of protein and 2 grams of fat

c. 8 grams of protein and 4 grams of fat

2. If there are 6 grams of fat in the snack, how many grams of protein are there?
Show your reasoning.

3. In this situation, what does a solution to the equation tell us?
Give an example of a solution.

Student Response

1. No
a. Yes

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 8

b. No

2. 1.5 grams. Sample reasoning: , so must be 1.5 for the equation to be
true.

3. Sample response: It means a pair of grams of protein and fat in the snack that add up
to 60 calories. An example: 6 grams of protein and 4 grams of fat.

Activity Synthesis

The goal of the discussion is to make sure students understand that a solution to an
equation in two variables is any pair of values that, when substituted into the equation and
evaluated, make the equation true. Discuss questions such as:

• “In this situation, what does it mean when we say that and are not

solutions to the equation?” (They are not a combination of protein and fat that would

produce 60 calories. Substituting them for the variables in the equation leads to a

false equation of .)

• “How did you find out the grams of protein in the snack given that there are 6 grams

of fat?” (Substitute 6 for and solve the equation.)

• “Can you find another combination that is a solution?”

• “How many possible combinations of grams of protein and fat (or and ) would add

up to 60 calories?” (Many solutions)

As a segue to the next lesson, solicit some ideas on how we know that there are many
solutions to the equation. If no one mentions using a graph, bring it up and tell students
that they will explore the graphs of two-variable equations next.

Lesson Synthesis

To summarize the lesson, refer back to the activity about protein and fat. Remind students
that a gram of protein has 4 calories and a gram of fat has 9 calories. Discuss questions
such as:

• "What does the equation tell us about the calories in a snack?" (It has

110 calories from some grams of protein and some grams of fat.)

• "In this situation, what does it mean to solve the equation?" (To find the combination

of grams of protein and fat that produce 110 calories.)

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

• "Is the combination of 11 grams of protein and 5 grams of fat a solution to the

equation? Why or why not?" (No, they don't add up to 110 calories. Substituting 11 for

and 5 for into the equation doesn't lead to a true equation.)

• "Consider the equation . What does it tell us about the snack?"

(The snack has 5 grams of protein and a total of 110 calories.)

• "What does it mean to solve this equation?" (To find the grams of fat that, when

combined with 5 grams of calories, give a total of 110 calories. To find the value of

that would make the equation true.)

4.4 Box of T-shirts

Cool Down: 5 minutes
Addressing

• HSA-REI.A

Student Task Statement

An empty shipping box weighs 250 grams. The box is then filled with T-shirts. Each
T-shirt weighs 132.5 grams.

The equation represents the relationship between the

quantities in this situation, where is the weight, in grams, of the filled box and

the number of shirts in the box.

1. Name two possible solutions to the equation . What do the
solutions mean in this situation?

2. Consider the equation . In this situation, what does the
solution to this equation tell us?

Student Response
1. Sample response:

◦ and
◦ and

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

◦ Each solution tells us the number of T-shirts in the box and the corresponding

total weight in grams.

2. It tells us the number of T-shirts in the box that result in a total weight of 2,900
grams.

Student Lesson Summary

An equation that contains only one unknown quantity or one quantity that can vary
is called an equation in one variable.

For example, the equation represents the relationship between the

length, , and the width, , of a rectangle that has a perimeter of 72 units. If we

know that the length is 15 units, we can rewrite the equation as:

.

This is an equation in one variable, because is the only quantity that we don't
know. To solve this equation means to find a value of that makes the equation
true.

In this case, 21 is the solution because substituting 21 for in the equation results
in a true statement.

An equation that contains two unknown quantities or two quantities that vary is
called an equation in two variables. A solution to such an equation is a pair of
numbers that makes the equation true.

Suppose Tyler spends $45 on T-shirts and socks. A T-shirt costs $10 and a pair of
socks costs $2.50. If represents the number of T-shirts and represents the
number of pairs of socks that Tyler buys, we can can represent this situation with
the equation:

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 11

This is an equation in two variables. More than one pair of values for and make
the equation true.

and and and

In this situation, one constraint is that the combined cost of shirts and socks must
equal $45. Solutions to the equation are pairs of and values that satisfy this
constraint.

Combinations such as and or and are not solutions

because they don’t meet the constraint. When these pairs of values are substituted

into the equation, they result in statements that are false.

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 12

Lesson 4 Practice Problems

1. Problem 1

Statement

An artist is selling children's crafts. Necklaces cost $2.25 each, and
bracelets cost $1.50 per each.
Select all the combinations of necklaces and bracelets that the artist could sell
for exactly $12.00.

A. 5 necklaces and 1 bracelet
B. 2 necklaces and 5 bracelets
C. 3 necklaces and 3 bracelet
D. 4 necklaces and 2 bracelets
E. 3 necklaces and 5 bracelets
F. 6 necklaces and no bracelets
G. No necklaces and 8 bracelets

Solution

["B", "D", "G"]

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 13

2. Problem 2

Statement

Diego is collecting dimes and nickels in a jar. He has collected $22.25 so far.

The relationship between the numbers of dimes and nickels, and the amount

of money in dollars is represented by the equation .

Select all the values that could be solutions to the equation.

A.

B.

C.

D.

E.

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 14

Solution

["A", "D", "E"]

3. Problem 3

Statement

Volunteer drivers are needed to bring 80 students to the championship

baseball game. Drivers either have cars, which can seat 4 students, or vans,

which can seat 6 students. The equation describes the

relationship between the number of cars, , and number of vans, , that can

transport exactly 80 students.

Select all statements that are true about the situation.

A. If 12 cars go, then 2 vans are needed.

B. and are a pair of solutions to the equation.

C. If 6 cars go and 11 vans go, there will be extra space.

D. 10 cars and 8 vans isn’t enough to transport all the students.

E. If 20 cars go, no vans are needed.

F. 8 vans and 8 cars are numbers that meet the constraints in this situation.

Solution

["B", "C", "E", "F"]

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 15

4. Problem 4

Statement

The drama club is printing t-shirts for its members. The printing company
charges a certain amount for each shirt plus a setup fee of $40. There are
21 students in the drama club.

a. If there are 21 students in the club and the t-shirt order costs a total of
$187, how much does each t-shirt cost? Show your reasoning.

b. The equation represents the cost of printing the

shirts at a second printing company. Find the solution to the equation

and state what it represents in this situation.

Solution

a. Each t-shirt costs $7. Sample reasoning: Without the set-up fee, the order costs
$147. Dividing 147 by 21 gives 7.

b. The solution is 65. It represents the setup fee, in dollars, at the second printing
company.

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5. Problem 5

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 17

Statement

The box plot represents the distribution of the number of children in 30
different familes.

After further examination, the value of 12 is removed for having been
recorded in error. The box plot represents the distribution of the same data
set, but with the maximum, 12, removed.

The median is 2 children for both plots.
a. Explain why the median remains the same when 12 was removed from
the data set.

b. When 12 is removed from the data set, does mean remain the same?
Explain your reasoning.

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 18

Solution

a. The median remains the same because removing an extreme value from a data
set tends not to have much effect or no effect on the median. In this case, there
may be several families with 2 children.

b. Sample response: The mean changes because the mean uses all values in its
calculation, so removing one of the values usually affects the mean. Since 12 is
the greatest value in the set, the mean of the data set should decrease.

(From Unit 1, Lesson 10.)

6. Problem 6

Statement

The number of points Jada's basketball team scored in their games have a
mean of about 44 and a standard deviation of about 15.7 points.

Interpret the mean and standard deviation in the context of Jada's basketball
team.

Solution

Jada's team scored about 44 points on average but there was a lot of variability. The
standard deviation of 15.7 means that there was substantial variability in the scores.
This number could be increased by outliers, that is, games where Jada's team scored
very few points or a lot of points. Otherwise, we can predict that most of the time, the
team scored between 28 points and 62 points, which is a very large range.

(From Unit 1, Lesson 13.)

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 19

7. Problem 7

Statement

Kiran’s family is having people over to watch a football game. They plan to
serve sparkling water and pretzels. They are preparing 12 ounces of sparkling
water and 3 ounces of pretzels per person. Including Kiran’s family, there will
be 10 people at the gathering.

A bottle of sparkling water contains 22 ounces and costs $1.50. A package of
pretzels contains 16 ounces and costs $2.99. Let represent number of
people watching the football game, represent the ounces of sparkling water,

represent the ounces of pretzels, and represent Kiran’s budget in dollars.
Which equation best represents Kiran’s budget?

A.

B.

C.

D.

Solution

D

(From Unit 2, Lesson 2.)

8. Problem 8

Statement

Algebra1 Unit 2 Lesson 4 CC BY 2019 by Illustrative Mathematics 20

The speed of an object can be found by taking the distance it travels and
dividing it by the time it takes to travel that distance. An object travels 100 feet
in 2.5 seconds. Let the speed, , be measured in feet per second.

Write an equation to represent the relationship between the three quantities
(speed, distance, and time).

Solution or

Sample response:

(From Unit 2, Lesson 2.)

9. Problem 9

Statement

A donut shop made 12 dozen donuts to give to a school’s math club.
Which expression represents how many donuts each student would get if the
donuts were equally distributed and there were students in math club?

A.

B.

C.
D.

Solution

B

(From Unit 2, Lesson 1.)

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Lesson 5: Equations and Their Graphs

Goals
• Comprehend that the graph of a linear equation in two variables represents all pairs

of values that are solutions to the equation.

• Interpret points on a graph of a linear equation to answer questions about the

quantities in context.

• Use graphing technology to graph linear equations and identify solutions to the

equations.

Learning Targets
• I can use graphing technology to graph linear equations and identify solutions to the

equations.

• I understand how the coordinates of the points on the graph of a linear equation are

related to the equation.

• When given the graph of a linear equation, I can explain the meaning of the points on

the graph in terms of the situation it represents.

Lesson Narrative

So far in the unit, students have primarily used descriptions, expressions, and equations to
represent relationships and constraints. In this lesson, they revisit the idea that graphs can
be a useful way to represent relationships. Students are reminded that each point on a
graph is a solution to an equation the graph represents. They analyze points on and off a
graph and interpret them in context. In explaining correspondences between equations,
verbal descriptions, and graphs, students hone their skill at making sense of problems
(MP1).

In this lesson, students are also introduced to the use of graphing technology to graph
equations. This introduction could happen independently as long as it precedes the
second activity in the lesson.

Alignments
Building On

• 8.EE.B: Understand the connections between proportional relationships, lines, and

linear equations.

Algebra1 Unit 2 Lesson 5 CC BY 2019 by Illustrative Mathematics 1

• 8.F.B.5: Describe qualitatively the functional relationship between two quantities by

analyzing a graph (e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a function that has
been described verbally.

Addressing

• HSA-CED.A.2: Create equations in two or more variables to represent relationships

between quantities; graph equations on coordinate axes with labels and scales.

• 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-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

• Graph It

• MLR6: Three Reads

• Think Pair Share

• Which One Doesn’t Belong?

Required Materials

Graphing technology
Examples of graphing technology are: a handheld graphing calculator, a computer with a
graphing calculator application installed, and an internet-enabled device with access to a
site like desmos.com/calculator or geogebra.org/graphing. For students using the digital
materials, a separate graphing calculator tool isn't necessary; interactive applets are
embedded throughout, and a graphing calculator tool is accessible on the student digital
toolkit page.

Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is
ideal if each student has their own device. (If students typically access the digital version of
the materials, Desmos is always available under Math Tools.)

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