IM UNIT 2 Equations and Inequalities Teacher Guide

Student Learning Goals

• Let’s graph equations in two variables.

5.1 Which One Doesn't Belong: Hours and
Dollars

Warm Up: 5 minutes
This warm-up prompts students to carefully analyze and compare features of graphs of
linear equations. In making comparisons, students have a reason to use language
precisely (MP6). The activity also enables the teacher to hear the terminology students
know and learn how they talk about characteristics of graphs.

The work here prepares students to reason about solutions to equations by graphing,
which is the focus of this lesson.

Building On

• 8.EE.B
• 8.F.B.5

Instructional Routines

• Which One Doesn’t Belong?

Launch
Arrange students in groups of 2–4. Display the graphs for all to see.

Give students 1 minute of quiet think time and then time to share their thinking with their
small group. In their small groups, ask each student to share their reasoning as to why a
particular graph does not belong, and together find at least one reason each item doesn't
belong.

Student Task Statement
Which one doesn’t belong?

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ABCD

Student Response
Sample response:

• A is the only graph that intersects the vertical axis at 0 (or that starts at the origin).
• B is the only graph where the horizontal axis represents dollars and the vertical axis

represents hours.

• C is the only one that shows plotted points (which lie along a line) instead of a

continuous line.

• D is the only one with a negative slope (or which slants downward from left to right).

Activity Synthesis
Ask each group to share one reason why a particular item does not belong. Record and
display the responses for all to see. After each response, ask the class if they agree or
disagree. Since there is no single correct answer to the question of which one does not
belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use,
such as " -intercept" or "negative slope." Also, press students on unsubstantiated claims.
(For example, if a student claims that graph A is the only one with a slope greater than 1,
ask them to explain or show how they know.)

5.2 Snacks in Bulk

10 minutes
Previously, students saw that an equation in two variables can have many solutions
because there are many pairs of values that satisfy the equation. This activity illustrates
that idea graphically. Students see that the coordinates of all points on the graphs are
pairs of values that make the equation true, which means that they are all solutions to the
equation.

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They also see that, because the given equation models the quantities and constraints in a
situation, not all points on the graph are meaningful. For example, only positive or
values on the graph (that is, only points in the first quadrant of the coordinate plane) have
meanings in this context, because almonds and figs cannot have negative values for their
weight.

During the activity, look for students who perform numerical computations straightaway
and those who first write a variable equation and then use it to answer the first two
questions.

Addressing

• HSA-CED.A.2
• HSA-CED.A.3
• HSA-REI.D.10

Instructional Routines

• MLR6: Three Reads
• 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 share their responses with a partner.

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Support for English Language Learners

Reading, Listening, Conversing: MLR6 Three Reads. Use this routine to support reading
comprehension of this word problem. Use the first read to orient students to the
situation. Ask students to describe what the situation is about without using numbers
(Clare purchased some snacks: salted almonds and dried figs). Use the second read to
identify quantities and relationships. Ask students what can be counted or measured
without focusing on the values. Listen for, and amplify, the important quantities that
vary in relation to each other in this situation: the cost per pound of each snack food,
the amount of each snack food purchased, and the total amount of money spent
before tax. After the third read, ask students to brainstorm possible strategies to
determine the amount of one snack food purchased if the amount of the other snack
food is known. This helps students connect the language in the word problem and the
reasoning needed to solve the problem.
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 say that the points not on the line are impossible given that Clare
spent $75. Encourage these students to think about what those points would mean if we
didn’t know how much money Clare spent.

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Student Task Statement
To get snacks for a class trip, Clare went to the “bulk”
section of the grocery store, where she could buy any
quantity of a product and the prices are usually good.

Clare purchased some salted almonds at $6 a pound
and some dried figs at $9 per pound. She spent
$75 before tax.

1. If she bought 2 pounds of almonds, how many pounds of figs did she buy?

2. If she bought 1 pound of figs, how many pounds of almonds did she buy?

3. Write an equation that describes the relationship between pounds of figs and
pounds of almonds that Clare bought, and the dollar amount that she paid. Be
sure to specify what the variables represent.

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4. Here is a graph that represents the quantities in this situation.

a. Choose any point on the line, state its coordinates, and explain what it
tells us.

b. Choose any point that is not on the line, state its coordinates, and explain
what it tells us.

Student Response
1. 7 pounds

2. 11 pounds

3. Sample response:

4. Sample response: : Clare bought 2 pounds of almonds and 7 pounds of figs for a
a. Point A or
total of $75.

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b. Point D or : Clare bought 1 pound of almonds and 1 pound of figs and the

total was not $75.

Activity Synthesis

Display the graph for all to see. Invite students to share their equation for the situation
and their interpretations of the points on and off the graph. Make sure students
understand that a point on the graph of an equation in two variables is a solution to the
equation. Discuss questions such as:

• "What does the point mean in this situation?" (Clare purchased 10 pounds of

almonds and 3 pounds of figs.)

• "Is that a possible combination of pounds of figs and almonds? Why or why not?" (No.

It doesn't lie on the graph. Also, if Clare bought 10 pounds of almonds and 3 pounds

of figs, it would cost her $87, not $75.)

• "From the graph, it looks like might be a solution, but it is hard to know for

sure. Is there a way to verify?" (Substitute the values into the equation and see if they

make the equation true.)

• "Suppose we extend the two ends of the graph beyond the first quadrant. Would a

point on those parts of the line—say, —be a solution to the equation

? Why or why not?" (It would still be a solution to the equation, but it

wouldn't make sense in this context. The weight of almonds or figs cannot be

negative.)

5.3 Graph It!

20 minutes (there is a digital version of this activity)
In the previous activity, students analyzed and interpreted points on a graph relative to an
equation and a situation. In this activity, they write a linear equation to model a
situation, use graphing technology to graph the equation, and then use the graph to solve
problems. Each given situation involves an initial value and a constant rate of change.

Before students begin the activity, introduce them to the graphing technology available in
the classroom. Offer a quick tutorial on how to graph equations, adjust the graphing
window, and plot points. This tutorial could happen independently of the activity as long as
it precedes the activity.

Addressing

• HSA-CED.A.2

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• HSA-CED.A.3
• HSA-REI.D.10

Instructional Routines

• Graph It

Launch
Give all students access to graphing technology. Tell students that in this course they will
frequently use technology to create a graph that represents an equation and use the
graph to solve problems.

Demonstrate how to use the technology available in your classroom to create and view
graphs of equations. Explain how to enter equations, adjust the graphing window, and plot
a point. If using Desmos, please see the digital version of this activity for suggested
instructions.

Arrange students in groups of 2–4. Assign one situation to each group. Ask students to
answer the first few questions, including writing an equation, and then graph the equation
and answer the last question.

Support for Students with Disabilities

Representation: Access for Perception. Provide students with a physical copy of written
directions for using graphing technology and read them aloud. Include step-by-step
directions for how to enter equations, adjust the graphing window, and plot a point.
Supports accessibility for: Language; Memory

Student Task Statement
1. A student has a savings account with $475 in it. She deposits $125 of her
paycheck into the account every week. Her goal is to save $7,000 for college.

a. How much will be in the account after 3 weeks?

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b. How long will it take before she has $1,350?

c. Write an equation that represents the relationship between the dollar
amount in her account and the number of weeks of saving.

d. Graph your equation using graphing technology. Mark the points on the
graph that represent the amount after 3 weeks and the week she has
$1,350. Write down the coordinates.

e. How long will it take her to reach her goal?

2. A 450-gallon tank full of water is draining at a rate of 20 gallons per minute.
a. How many gallons will be in the tank after 7 minutes?

b. How long will it take for the tank to have 200 gallons?

c. Write an equation that represents the relationship between the gallons
of water in the tank and minutes the tank has been draining.

d. Graph your equation using graphing technology. Mark the points on the
graph that represent the gallons after 7 minutes and the time when the
tank has 200 gallons. Write down the coordinates.

e. How long will it take until the tank is empty?

Student Response
1. a. $850
b. 7 weeks

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c. Sample response: , where is the dollar amount in the account

and is the number of weeks of savings

d.

e. 53 weeks

2. a. 310 gallons

b. 12.5 minutes, or 12 minutes 30 seconds

c. Sample equation: , where is the amount in the tank and is time
in minutes

d.
e. 22.5 minutes, or 22 minutes 30 seconds

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Are You Ready for More?
1. Write an equation that represents the relationship between the gallons of
water in the tank and hours the tank has been draining.

2. Write an equation that represents the relationship between the gallons of
water in the tank and seconds the tank has been draining.

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3. Graph each of your new equations. In what way are all of the graphs the
same? In what way are they all different?

4. How would these graphs change if we used quarts of water instead of gallons?
What would stay the same?

Student Response
1. where is time in hours
2. where is time in seconds

3.
Sample response: All three graphs are lines with negative slopes that are different.
They all have the same vertical intercept. Their horizontal intercepts are all different,
but represent the same point in time.

4. The graphs would still be lines with negative slopes that are different. They would
have the same horizontal intercepts. The vertical intercept would be different, but
represent the same amount of water.

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Activity Synthesis

Select a group who analyzed the first situation and one group who analyzed the second
situation to share their responses. Display their graphs for all to see.

Focus the discussion on two things: the meanings of the points on the graph, and how the
graph could be used to answer questions about the quantities in each situation. Discuss
questions such as:

• “How did you find the answers to the first two questions?” (By calculation, for

example, computing , or finding , and then dividing by 125.)

• “How did you find the answer to the last question?” (By calculation, for

instance, finding , and then dividing by 125. Or, alternatively, by using the

graph.)

Highlight how the graph of the equations could be used to answer the questions. If not

already mentioned by students, discuss how the graph of can be used to

find the answers to all the questions about the student's savings account, and the graph

of can help us with the questions about draining the water tank.

Keep the graphs of the two equations displayed for the lesson synthesis.

Support for English Language Learners

Representing, Conversing: MLR7 Compare and Connect. As students share their responses
with the class, call attention to the different ways the quantities are represented
graphically and within the context of each situation. Take a close look at both graphs
to distinguish what the points represent in each situation. Wherever possible, amplify
student words and actions that describe the connections between a specific feature
of one mathematical representation and a specific feature of another representation.
Design Principle(s): Maximize meta-awareness; Support sense-making

Lesson Synthesis

To help students sum up the key ideas of the lesson, display dynamic graphs of the two
equations from the "Graph It!" activity. Also, display these questions for all to see:

• In the first situation, will the student have $5,000 after saving for 20 weeks?

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• In the second situation, after how many minutes should we stop the draining if we

want to leave 150 gallons of water in the tank?

Discuss with students:

• "How can we use the graph of to answer the first question?" (See if

the point is on or below the graph. It is not, so the answer is no.)

• "How can we use the graph of to answer the second question?" (Find

the point on the graph where the -value is 150, and see what the -value is.)

• "On the graph for the first situation, what does the point mean?" (After 15

weeks, there is $3,000 in the bank account.)

• "Is that ordered pair a solution to the equation ? How can we tell?"

(No. The point is above the graph, not on the graph. After 15 weeks, there

is less than $3,000 in the account.)

• "Is a solution to the equation ? Why or why not?" (No. The

point is not on the graph. A negative -value also has no meaning in this situation.

The point means that after 25 minutes, there are -50 gallons of water in the

tank, which doesn't make sense.)

• In general, how can a graph helps us find solutions to two-variable equations?" (Any

point on the graph of the equation is a solution to that equation.)

5.4 A Spoonful of Sugar

Cool Down: 5 minutes
Addressing

• HSA-CED.A.2

• HSA-CED.A.3

• HSA-REI.D.10

Student Task Statement

A ceramic sugar bowl weighs 340 grams when empty. It is then filled with
sugar. One tablespoon of sugar weighs 12.5 grams.

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1. Write an equation to represent the relationship between the total weight of
the bowl in grams, , and the tablespoons of sugar, .

2. When the sugar bowl is full, it weighs 740 grams. How many tablespoons of
sugar can the bowl hold? Show your reasoning.

3. The graph represents the relationship between the number of teaspoons of
sugar in the bowl and the total weight of the bowl.

Which point on the graph could represent your answer to the previous
question?
4. About how many tablespoons of sugar are in the bowl when the total weight is
600 grams?

Student Response
1. , or
2. 32 tablespoons
3. Point D, as its coordinates could be

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4. About 21 tablespoons

Student Lesson Summary

Like an equation, a graph can give us information about the relationship between
quantities and the constraints on them.

Suppose we are buying beans and rice to feed a large gathering of people, and we
plan to spend $120 on the two ingredients. Beans cost $2 a pound and rice costs
$0.50 a pound.

If represents pounds of beans and pounds of rice, the equation
can represent the constraints in this situation.

The graph of shows a straight line.

Each point on the line is a pair of - and -values that make the equation true and is
thus a solution. It is also a pair of values that satisfy the constraints in the situation.

• The point is on the line. If we buy 10 pounds of beans and 200

pounds of rice, the cost will be , which equals 120.

• The points and are also on the line. If we buy only beans—60

pounds of them—and no rice, we will spend $120. If we buy 45 pounds of

beans and 60 pounds of rice, we will also spend $120.

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What about points that are not on the line? They are not solutions because they
don't satisfy the constraints, but they still have meaning in the situation.

• The point is not on the line. Buying 20 pounds of beans and 80

pounds of rice costs or 80, which does not equal 120. This

combination costs less than what we intend to spend.

• The point means that we buy 70 pounds of beans and 180 pounds of
or 230, which is over our budget of 120.
rice. It will cost

Lesson 5 Practice Problems .

1. Problem 1

Statement

Select all the points that are on the graph of the equation
A.
B.
C.
D.
E.
F.

Solution

["A", "B", "D"]

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2. Problem 2

Statement .

Here is a graph of the equation

Select all coordinate pairs that represent a solution
to the equation.

A.
B.
C.
D.
E.
F.

Solution

["A", "D"]

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3. Problem 3

Statement

A theater is selling tickets to a play. Adult tickets cost $8 each and children’s
tickets cost $5 each. They collect $275 after selling adult tickets and
children’s tickets.

What does the point mean in
this situation?

Solution

It means that 30 adult tickets and 7 children’s tickets were sold.

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4. Problem 4
Statement

Technology required. Priya starts with $50 in her bank account. She then
deposits $20 each week for 12 weeks.

a. Write an equation that represents the relationship between the dollar
amount in her bank account and the number of weeks of saving.

b. Graph your equation using graphing technology. Mark the point on the
graph that represents the amount after 3 weeks.

c. How many weeks does it take her to have $250 in her bank account?
Mark this point on the graph.

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Solution

a. Sample response: where is the dollar amount in bank account

and is the number of weeks.

b. See first graph.

c. 10 weeks. See second graph.

5. Problem 5

Statement

During the month of August, the mean of the daily rainfall in one city was 0.04
inches with a standard deviation of 0.15 inches. In another city, the mean of
the daily rainfall was 0.01 inches with a standard deviation of 0.05 inches.

Han says that both cities had a similar pattern of precipitation in the month of
August. Do you agree with Han? Explain your reasoning.

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Solution

Sample response. Yes, Han is correct. Both cities had very little rain on a daily basis.
The large standard deviation means that the rain came all at once or possibly just on
a few days. Although one city had more rain than the other, this pattern of most days
being dry and a few days being wet is the same in both cities.

(From Unit 1, Lesson 13.)

6. Problem 6

Statement

In a video game, players form teams and work together to earn as many
points as possible for their team. Each team can have between 2 and 4
players. Each player can score up to 20 points in each round of the game. Han
and three of his friends decided to form a team and play a round.
Write an expression, an equation, or an inequality for each quantity described
here. If you use a variable, specify what it represents.

a. the allowable number of players on a team

b. the number of points Han's team earns in one round if every player
earns a perfect score

c. the number of points Han's team earns in one round if no players earn a
perfect score

d. the number of players in a game with six teams of different sizes:
two teams have 4 players each and the rest have 3 players each

e. the possible number of players in a game with eight teams

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Solution

Sample response:

a. and (or ), where is the number of players on a team

b.

c. , where is the score of the team

d.

e. , where is the total number of players in a game

(From Unit 2, Lesson 1.)

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7. Problem 7
Statement

A student on the cross-country team runs 30 minutes a day as a part of her
training.
Write an equation to describe the relationship between the distance she runs
in miles, , and her running speed, in miles per hour, when she runs:

a. at a constant speed of 4 miles per hour for the entire 30 minutes

b. at a constant speed of 5 miles per hour the first 20 minutes, and then at
4 miles per hour the last 10 minutes

c. at a constant speed of 6 miles per hour the first 15 minutes, and then at
5.5 miles per hour for the remaining 15 minutes

d. at a constant speed of miles per hour the first 6 minutes, and then at
6.5 miles per hour for the remaining 24 minutes

e. at a constant speed of 5.4 miles per hour for minutes, and then at
miles per hour for minutes

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Solution

Equations that are equivalent to these are also acceptable.
a.
b.
c.
d.
e.

(From Unit 2, Lesson 2.)

8. Problem 8
Statement

In the 21st century, people measure length in feet and meters. At various
points in history, people measured length in hands, cubits, and paces. There
are 9 hands in 2 cubits. There are 5 cubits in 3 paces.

a. Write an equation to express the relationship between hands, , and
cubits, .

b. Write an equation to express the relationship between hands, , and
paces, .

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Solution number of dollar
months amount
Sample responses: 5
a. 6 1,200
b. 7 1,300
8 1,400
(From Unit 2, Lesson 3.) 1,500

9. Problem 9 28

Statement

The table shows the amount of money,
, in a savings account after months.

Select all the equations that represent
the relationship between the amount
of money, , and the number of
months, .

A.
B.
C.
D.
E.
F.
G.

Solution

["C", "E", "G"]

(From Unit 2, Lesson 3.)

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Lesson 6: Equivalent Equations

Goals
• Comprehend that “equivalent equations” are equations that have exactly the same

solutions, and that multiple equivalent equations can represent the same
relationship.

• Determine and explain (orally and in writing) whether two equations are equivalent.

• Identify operations that can be performed on an equation to create equivalent

equations.

Learning Targets
• I can tell whether two expressions are equivalent and explain why or why not.

• I know and can identify the moves that can be made to transform an equation into an

equivalent one.

• I understand what it means for two equations to be equivalent, and how equivalent

equations can be used to describe the same situation in different ways.

Lesson Narrative

In middle school, students learned that two expressions are equivalent if they have the
same value for all values of the variables in the expressions. They wrote equivalent
expressions by applying properties of operations, combining like terms, or rewriting parts
of an expression.

In this lesson, students learn that equivalent equations are equations with the exact same
solutions. Students see that the moves that generate equivalent expressions (for example,
applying the distributive property or combining like terms) can also create equivalent
equations. Additionally, an equivalent equation can be created by adding the same
number to both sides or multiplying both sides by the same non-zero number. Students
have seen moves like this before, when solving one-variable equations in middle school.
What is new here is an awareness that each of the equations created as a part of the
solving process is equivalent to the original equation.

Students also regard equivalent equations as synonymous statements about a
relationship. They use context to interpret the solution to equivalent equations, and to

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think about why it makes sense that equivalent equations have the same solution. In doing
so, students reason abstractly and quantitatively (MP2).

The emphasis of this lesson is on equations in one variable. Students will have many
opportunities to study equivalent equations in two variables in future lessons.

Alignments

Building On

• 6.EE.A.4: Identify when two expressions are equivalent (i.e., when the two

expressions name the same number regardless of which value is substituted into

them). For example, the expressions and are equivalent because they

name the same number regardless of which number stands for.

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-REI.A.1: Explain each step in solving a simple equation as following from the

equality of numbers asserted at the previous step, starting from the assumption that
the original equation has a solution. Construct a viable argument to justify a solution
method.

• HSA-SSE.A.1: Interpret expressions that represent a quantity in terms of its context.

Building Towards

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

reasoning.

Instructional Routines

• MLR8: Discussion Supports

Required Materials
Four-function calculators

Student Learning Goals

• Let's investigate what makes two equations equivalent.

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

6.1 Two Expressions

Warm Up: 5 minutes
The purpose of this warm-up is to help students recall what it means for two expressions
to be equivalent. The given expressions are in forms that are unfamiliar to students but
are not difficult to evaluate for integer values of the variable. This is by design—to pique
students' curiosity while keeping the mathematics accessible.

Building On

• 6.EE.A.4

Building Towards

• HSA-REI.A

Launch

Assign the first expression to one half of the class and the second expression to the other
half. Give students a couple of minutes to evaluate their assigned expression at as many
values of as listed in the first question.

Anticipated Misconceptions

When evaluating their expression, some students may perform the operations in an

incorrect order. For example, when finding the value of in the second expression,

they may find and then multiply by 2. Ask them whether the subtraction or

multiplication should be performed first. Remind them about the order of operations as

needed.

Student Task Statement
Your teacher will assign you one of these expressions:

Evaluate your expression when is:
1. 5
2. 7
3. 13

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4. -1

Student Response
For both expressions, the values are:

1. 8

2. 20

3. 80

4. -4

Activity Synthesis

Ask a few students from each group for their results. Then, ask students what they wonder
about the results. Students are likely curious if the values of the two expressions will be
the same for other values of . If they noticed that all the given values of are odd
numbers, they might wonder if even values of would give the same result. If time
permits, consider allowing students to try evaluating the expressions using a value of their
choice.

Discuss questions such as:

• "Were you surprised that these expressions have the same result for different values

of ?"

• "If (or when) you tried using other values of , what did you find?"

• "Do you think that the two expressions will have the same value no matter what

value of is used? How do you know?"

Tell students that it would be impossible to check every value of to see if the expressions
would give the same value. There are, however, ways to show that these expressions must
have the same value for any value of . We call expressions that are equal no matter what
value we use for the variable equivalent expressions.

Remind students that in middle school they had seen simpler equivalent expressions. For

example, they know that is equivalent to by the distributive property

(without trying different values of ).

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Explain to students that they'll learn more about how to identify or write equivalent
expressions and about equivalent equations in this unit.

6.2 Much Ado about Ages

15 minutes
In this activity, students examine simple relationships that can each be expressed with
many equations. Writing multiple equations for the same relationship and doing so in
context prepares students to later consider more rigorously what makes equations
equivalent.

As students write equations for the first relationship, identify students who use all
numbers and those who use both numbers and variables.

For the second question, students may explain Tyler's claim concretely (using values of
and ) or more abstractly (using the structure of the equations). For instance, they may
reason that:

• When the middle child is 12, the youngest child is 7, and that substituting and

to gives or , which is 10. At all other ages of the

two children, the expression always has a value of 10.

• is twice of . If the difference between and is 5, then twice

the difference between and must be twice of 5, which is 10, so the two equations

are still describing the same relationship.

Identify students who reason in these ways so they can share later.

Addressing

• HSA-CED.A.2

Building Towards

• HSA-REI.A

Launch

Give students a couple of minutes to answer the first question and ask them to pause for a
discussion before moving on.

Invite students to share their equations. Record and display the equations for all to
see. Along the way, consider organizing them into categories (numerical equations,
equations in one variable, and equations in two variables).

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 5

Solicit from students some thoughts on the following questions. (It is not necessary to
resolve the questions at the moment.)

• "Consider the equations you wrote for each situation. Are they all equivalent? Why or

why not?"

• "What do you think it means for two equations to be equivalent?"

Some students might say that each equation accurately represents the same relationship
between the ages, so the equations must be equivalent. Those who wrote variable
equations might say that the equations are equivalent because the same value for the
variable makes each equation true.

After students have had a chance to consider these questions, prompt them to complete
the remaining questions.

Anticipated Misconceptions

Some students may say that doesn’t describe the relationship between the

ages because it doesn’t involve the number 5. Encourage students to think of some values

for and that do make the equation true. After they find some pairs of and , ask

them what they notice about the pairs.

Student Task Statement
1. Write as many equations as possible that could represent the relationship
between the ages of the two children in each family described. Be prepared to
explain what each part of your equation represents.

a. In Family A, the youngest child is 7 years younger than the oldest, who is
18.

b. In Family B, the middle child is 5 years older than the youngest child.

2. Tyler thinks that the relationship between the ages of the children in Family B

can be described with , where is the age of the middle child

and the age of the youngest. Explain why Tyler is right.

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

3. Are any of these equations equivalent to one another? If so, which ones?
Explain your reasoning.

Student Response
1. a. Sample responses:

▪ , where 11 is age of the youngest child.
▪
▪
▪ , where is the age of the youngest child.
▪
▪

b. Sample responses: is the age of the middle child and the age of the

▪ , where

youngest.

▪

▪

2. See Activity Narrative for sample responses.

3. Yes, they are all equivalent. Sample reasoning:
◦ They are all true when
. They are also all not true when is not 3.

◦ We can perform the same operation to both sides of one equation and get

another equation.

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 7

Are You Ready for More?
Here is a puzzle:

Which expressions could be equal to ?

Student Response
and

Activity Synthesis

Select previously identified students to share their explanation of why Tyler's claim is true,
in the order shown in the Activity Narrative. If no students mention the last approach,
bring it up.

Next, emphasize two main points:

• Equivalent equations have exactly the same solutions, or exactly the same values

that make each of the equations true. All of the equations in the last question are

equivalent because they have 3 as the solution and they all have no other solutions.

• Suppose we start with a true equation, where the two sides are equal. If we perform

the same operation to both sides of the equation and get a new equation where the

two sides are also equal, we can say that the two equations are equivalent. For

instance: by 6 gives . If and 15 are

◦ Subtracting both sides of

equal, then the expressions or numbers we get by subtracting 6 from each one

are also equal. We can conclude that and are equivalent.

◦ Dividing both sides of by 3 gives . If is equal to 15,

the result of dividing by 3 is equal to dividing 15 by 3. We can conclude

that and are equivalent.

Ask students:

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 8

• "How can we show that is equivalent to ?" (Multiplying both sides of

by 9 gives .)

• "How can we show that is equivalent to ?" (Subtracting 2 from both

sides of and then multiplying both sides of the resulting equation, , by
gives .)

If no students notice that we have made these moves when solving equations, bring it to
their attention. Highlight that solving an equation essentially involves writing a series of
equivalent equations that eventually isolates the variable on one side.

6.3 What's Acceptable?

15 minutes
This activity further develops the idea of equivalent equations and does so in the context
of a situation. Students pay attention to the moves that create equations with the same
solution and those that don't. They also make sense of both the moves and the
solutions in terms of the given situation. The work here gives students opportunities to
reason concretely and abtractly (MP2). In an upcoming lesson, they will think about why
these moves lead to equations with the same solution.

Addressing

• HSA-REI.A.1

• HSA-SSE.A.1

Instructional Routines

• MLR8: Discussion Supports

Launch

Arrange students in groups of 2 and provide access to calculators. Ask students to read the

opening paragraph and display the equation for all to see. Ask

students to explain how the equation represents Noah's purchase.

Then, give students a couple of minutes to discuss the first question with their partner and
ask them to pause for a whole-class discussion. Once students understand that the
solution to the equation is the original price of a pair of jeans and that substituting 60 for
makes the equation true, move on to the rest of the activity.

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

Tell students that they will see a series of equations that are related to the original
equation and that can be interpreted in terms of the situation. For each equation, their job
is to determine either what move was made to the original equation, or what the equation
means in context. Next, they should check if the equation has the same solution as the
original. (To help students understand the task and if needed, consider working on
Equation A as a class.)

If time is limited, ask one partner in each group to take Equations A, C, E and the other
partner to take B, D, F.

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. Some students may benefit from a graphic organizer that displays
and the original equation before each of the related equations. This will help focus
students’ attention on the moves that may have been made.
Supports accessibility for: Organization; Attention

Student Task Statement
Noah is buying a pair of jeans and using a coupon for 10% off. The total price is
$56.70, which includes $2.70 in sales tax. Noah's purchase can be modeled by the
equation:

1. Discuss with a partner:

a. What does the solution to the equation mean in this situation?

b. How can you verify that 70 is not a solution but 60 is the solution?

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

2. Here are some equations that are related to . Each

equation is a result of performing one or more moves on that original

equation. Each can also be interpreted in terms of Noah’s purchase.

For each equation, determine either what move was made or how the
equation could be interpreted. (Some examples are given here.) Then, check if
60 is the solution of the equation.

Equation A [The price is expressed in cents instead of dollars.]

◦ What was done?
◦ Interpretation?
◦ Same solution?

Equation B [Subtract 2.70 from both sides of the equation.]

◦ What was done?
◦ Interpretation?
◦ Same solution?

Equation C [10% off means paying 90% of the original price.
90% of the original price plus sales tax is $56.70.]
◦ What was done?
◦ Interpretation?

◦ Same solution?

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 11

3. Here are some other equations. For each equation, determine what move was
made or how the equation could be interpreted. Then, check if 60 is the
solution to the equation.

Equation D [The price after using the coupon for 10% off and
before sales tax is $56.70.]
◦ What was done?
◦ Interpretation?

◦ Same solution?

Equation E [Subtract 2.70 from the left and add 2.70 to the
right.]
◦ What was done?

◦ Interpretation?
◦ Same solution?

Equation F [The price of 2 pairs of jeans, after using the coupon
for 10% off and paying sales tax, is $56.70.]
◦ What was done?
◦ Interpretation?

◦ Same solution?

4. Which of the six equations are equivalent to the original equation? Be
prepared to explain how you know.

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 12

Student Response

1. Sample response:
a. The solution is the value of that makes the equation a true statement. It tells
us the original price of the jeans.

b. Evaluating the expression on the left when gives 65.70, not 56.70.

Evaluating the expression for gives 56.70, so , which is a

true statement.

2. Sample response: . Move: Multiply each side by 100. Same

◦ Equation A:

solution.

◦ Equation B: . Interpretation: The price after using the coupon for

10% off but before adding sales tax is $54. Same solution.

◦ Equation C: . Move: Combining like terms in the expression

on the left. Same solution.

3. Sample response: . Move: Subtract 2.70 from the left side. Not the

◦ Equation D:

same solution.

◦ Equation E: . Interpretation: The price after using the coupon

for 10% off and before sales tax is $59.40. Not the same solution.

◦ Equation F: . Move: Multiply the left side by 2. Not the

same solution.

4. Equations A, B, and C. They have the same solution as the original equation.

Activity Synthesis

Display the original equation and Equations A–F for all to see. Invite students to share
what was done to the original equation to get each of those equations and whether they
have the same solution. Along the way, compile a list of moves that lead to equations with
the same solution and those that lead to different solutions.

Ask students to observe the list and see what kinds of moves produce the Equations A–C
and D–F. Record the moves that create equations with the same solutions, such as:

• adding or multiplying both sides of the equal sign by the same number
• applying properties of operations (commutative, associative, or distributive)

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 13

• combining like terms

Also record the moves that create equations with different solutions, such as:

• adding or multiplying a different number to the two sides, or performing an

operation to only one side

• adding different expressions to the two sides

• performing different operations on each side

(The lists don't need to be comprehensive, as students will examine these moves more
closely later.)

Next, give students a couple of examples of how the equations and their solutions could
be interpreted in context. For example:

• Equation A shows the cost calculation in cents, so the original price of a pair of jeans

is unchanged.

• Equation D shows the discounted price, excluding tax, to be $56.70, but in the

original equation, the price $56.70 included tax. This means that, in Equation D, the
original price of one pair of jeans is different than in the first equation.

Prompt students to interpret 1–2 other equations and to explain why the solution in each
equation (or the price of one pair of jeans) is equal or unequal to that in the initial
equation. Here are possible interpretations:

• Equation B shows the discounted price before sales tax to be $54, which is $2.70 less

than $56.70. This is also the case in the initial equation, so the original price of the
jeans is still the same.

• Equation C shows the discounted price as 90% of the original price, which is the same

as 10% less than the original price (100%). The original price is unchanged.

• Equation E shows the discounted price, before tax, to be $59.40. This is more than

the pre-tax price in the initial equation, so the original price of the jeans here must be
higher.

• Equation F shows the price of 2 pairs of jeans, including the discount and tax, to

be $56.70. This means the price of a pair of jeans must be much less than in the
initial equation.

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 14

Support for English Language Learners

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion.
For each observation that is shared, ask students to restate what they heard using
precise mathematical language. Consider providing students time to restate what
they hear to a partner before selecting one or two students to share with the class.
Ask the original speaker if their peer was accurately able to restate their thinking. Call
students' attention to any words or phrases that helped clarify the original statement.
This provides more students with an opportunity to produce language as they
interpret the reasoning of others.
Design Principle(s): Support sense-making

Support for Students with Disabilities

Representation: Develop Language and Symbols. Create a display of important terms and
vocabulary. Keep this display visible throughout the remainder of the unit. Invite
students to suggest language or diagrams to include that will support their
understanding of combining like terms, and the commutative, associative, and
distributive properties.
Supports accessibility for: Conceptual processing; Language

Lesson Synthesis

To help students connect the various ideas in this lesson and articulate their
understanding, ask them questions such as:

• "How would you explain "equivalent equations" to a classmate who is absent

today?"

• "The equation represents purchasing 5 tubs of yogurt for $6. In this equation,

what does the solution represent?" (the cost of one tub of yogurt)

• "Which of these equations are equivalent to the equation (about the yogurt)?

How do you know?"

◦

◦

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 15

◦

◦

(The first and the third equations are equivalent to the original equation
because they have the same solution. There is an acceptable move that was
done to the original equation to get to these equivalent equations: multiplying
each side of the equal sign by 3, and adding 4 to each side of the equation.)

• "For the equations that you think are equivalent, what do they mean in the context of

the yogurt purchase?" (The equation can be interpreted as: buying 3 times

as many tubs of yogurt costs 3 times as much. The third equation can be interpreted

as: the total cost of 5 tubs of yogurt and something else that costs $4 is $10.)

6.4 Box of Beans and Rice

Cool Down: 5 minutes
Addressing

• HSA-REI.A.1

Launch
Give students continued access to calculators.

Student Task Statement

A cardboard box, which weighs 0.6 pound when empty, is filled with 15 bags of

beans and a 4-pound bag of rice. The total weight of the box and the

contents inside it is 25.6 pounds. One way to represent this situation is with the

equation .

1. In this situation, what does the solution to the equation represent?

2. Select all equations that are also equivalent to .

◦ Equation A:
◦ Equation B:

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 16

◦ Equation C:
◦ Equation D:
◦ Equation E:

Student Response
1. The weight, in pounds, of one bag of beans
2. B, C, and E

Student Lesson Summary

Suppose we bought two packs of markers and a $0.50 glue stick for $6.10. If is the

dollar cost of one pack of markers, the equation represents this

purchase. The solution to this equation is 2.80.

Now suppose a friend bought six of the same packs of markers and three $0.50

glue sticks, and paid $18.30. The equation represents this

purchase. The solution to this equation is also 2.80.

We can say that and are equivalent equations

because they have exactly the same solution. Besides 2.80, no other values of

make either equation true. Only the price of $2.80 per pack of markers satisfies the

constraint in each purchase.

How do we write equivalent equations like these?

There are certain moves we can perform!

In this example, the second equation, , is a result of multiplying

each side of the first equation by 3. Buying 3 times as many markers and glue sticks

means paying 3 times as much money. The unit price of the markers hasn't

changed.

Here are some other equations that are equivalent to , along with
the moves that led to these equations.

• Add 3.50 to each side of the original equation.

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 17

• Subtract 0.50 from each side of the original equation.
• Multiply each side of the original equation by .
• Apply the distributive property to rewrite the left side.

In each case:

• The move is acceptable because it doesn't change the equality of the two sides

of the equation. If has the same value as 6.10, then multiplying

by and multiplying 6.10 by keep the two sides equal.

• Only makes the equation true. Any value of that makes an equation

false also makes the other equivalent equations false. (Try it!)

These moves—applying the distributive property, adding the same amount to both
sides, dividing each side by the same number, and so on—might be familiar
because we have performed them when solving equations. Solving an equation
essentially involves writing a series of equivalent equations that eventually isolates
the variable on one side.

Not all moves that we make on an equation would create equivalent equations,
however!

For example, if we subtract 0.50 from the left side but add 0.50 to the right side, the

result is . The solution to this equation is 3.30, not 2.80. This means that

is not equivalent to .

Glossary
• equivalent equations

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 18

Lesson 6 Practice Problems ?

1. Problem 1

Statement

Which equation is equivalent to the equation
A.
B.
C.
D.

Solution

B

2. Problem 2

Statement

Select all the equations that have the same solution as the

equation .

A.

B.

C.

D.

E.

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 19

Solution

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

3. Problem 3

Statement

Jada has a coin jar containing nickels and dimes worth a total of $3.65. The

equation is one way to represent this situation.

Which equation is equivalent to the equation ?

A.

B.

C.

D.

Solution

C

4. Problem 4

Statement

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 20

Select all the equations that have the same solution as .
A.
B.
C.
D.
E.
F.

Solution

["B", "D", "E"]

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 21

5. Problem 5
Statement

The number of hours spent in an airplane on a single flight is recorded on a
dot plot. The mean is 5 hours and the standard deviation is approximately
5.82 hours. The median is 4 hours and the IQR is 3 hours. The value 26 hours
is an outlier that should not have been included in the data.

When the outlier is removed from the data set:
a. What is the mean?

b. What is the standard deviation?

c. What is the median?

d. What is the IQR?

Solution

a. 3.5 hours
b. 1.92 hours
c. 3.5 hours
d. 3 hours

(From Unit 1, Lesson 14.)

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 22

6. Problem 6

Statement

A basketball coach purchases bananas for the players on his team. The table
shows total price in dollars, , of bananas.
Which equation could represent the total price in dollars for bananas?

number of bananas total price in dollars
7 4.13
8 4.72
9 5.31
10 5.90

A.
B.
C.
D.

Solution

A

(From Unit 2, Lesson 3.)

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 23

7. Problem 7

Statement

Kiran is collecting dimes and quarters in a jar. He has collected $10.00 so far

and has dimes and quarters. The relationship between the numbers of

dimes and quarters, 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.

Solution

["A", "C", "E"]

(From Unit 2, Lesson 4.)

Algebra1 Unit 2 Lesson 6 CC BY 2019 by Illustrative Mathematics 24