IM UNIT 2 Equations and Inequalities Teacher Guide

Lesson 1: Planning a Pizza Party

Goals
• Comprehend the term “constraint” to mean a limitation on the possible or

reasonable values a quantity could have.

• Use variables and the symbols =, , and to represent simple constraints in a

situation.

• Write expressions with numbers and letters to represent the quantities in a situation.

Learning Targets
• I can explain the meaning of the term “constraints.”

• I can tell which quantities in a situation can vary and which ones cannot.

• I can use letters and numbers to write expressions representing the quantities in a

situation.

Lesson Narrative

This opening lesson invites students to experiment with expressions and equations to
model a situation. Students think about relevant quantities, whether they might be fixed or
variable, and how they might relate to one another. They make assumptions and
estimates, and use numbers and letters to represent the quantities and relationships. The
lesson also draws attention to the idea of constraints and how to represent them.

There is not one correct set of expressions or equations governing the potential quantities
involved in the pizza party. The focus is on the modeling process itself—identifying
relevant quantities, making assumptions, creating a model, and evaluating the model
(MP4). Discussions are built in to foster an environment of collaboration and active
thinking and listening. Encourage students to share their ideas and questions at these
times.

In subsequent lessons, students will continue to write and interpret expressions,
equations, and inequalities that represent situations and constraints.

Making internet-enabled devices available gives students an opportunity to choose
appropriate tools strategically (MP5).

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Alignments
Building On

• 7.EE.A: Use properties of operations to generate equivalent expressions.

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.

• HSN-Q.A.2: Define appropriate quantities for the purpose of descriptive modeling.

Instructional Routines

• Aspects of Mathematical Modeling
• MLR7: Compare and Connect

Student Learning Goals

• Let’s write expressions to estimate the cost of a pizza party.

1.1 A Main Dish and Some Side Dishes

Warm Up: 5 minutes
This warm-up elicits the idea that an equation can contain only letters, with each letter
representing a value. It also reminds students that an equation is a statement that two
expressions are equal, and that different expressions could be used to represent a
quantity. Later in this lesson and throughout the unit, students will create, interpret, and
reason about equations with letters representing quantities.

Building On

• 7.EE.A

Addressing

• HSA-CED.A.2

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Launch
Arrange students in groups of 2. For the last question, ask each partner to come up with a
new equation.

Student Task Statement
Here are some letters and what they represent. All costs are in dollars.

• represents the cost of a main dish.
• represents the number of side dishes.
• represents the cost of a side dish.
• represents the total cost of a meal.

1. Discuss with a partner: What does each equation mean in this situation?
a.
b.
c.
d.

2. Write a new equation that could be true in this situation.

Student Response
1. Sample response:
a. A main dish costs $7.50.
b. A main dish costs $4.50 more than a side dish.
c. Ordering side dishes at dollars each costs $6.00.
d. The total cost of the meal is the cost of a main dish plus the cost of side dishes
at dollars.
2. Sample responses:

◦

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◦
◦

Activity Synthesis

Invite students to share their interpretations of the given equations and the new equations
they wrote. Then, discuss with students:

• "What is an equation? What does it tell us?" (An equation is a statement that an

expression has the same value as another expression.)

• "Can equations contain only numbers?" (Yes.) "Only letters?" (Yes.) "A mix of numbers

and letters?" (Yes.)

• "The last question asked you to write an equation that could be true. Could this

equation be true: ? How do you know?" (It could be. If is 7.50 and is

12.50, then the equation is true.)

• "When might the equation be false?" (If is 7.50 and is anything but 12.50, or if the

value of is not 5 more than , then it is false.)

• "One equation tells us that a main dish is $7.50. Another equation tells us that it is

equal to the expression . Could both be true? Are they both appropriate for

expressing the cost of a main dish?" (Yes. The first one tells us the price in dollars.

The second tells us how the price compares to a side dish.)

1.2 How Much Will It Cost?

20 minutes
This activity prompts students to create expressions to represent the quantities and
relationships in a situation and engages them in mathematical modeling.

Students plan a pizza party and present a cost estimate. To do so, they need to consider
relevant variables, make assumptions and estimates, perform calculations, and adjust
their thinking along the way (MP4). Some students may choose to perform research and
revise their models as they gather new information. There are many possible solutions to
the task.

As students discuss their ideas, monitor for those who:

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• find and use actual data or exact values (for example, count the number of students

in the class, research the cost of a large pizza at a nearby shop, or quickly survey the
class for topping preferences)

• estimate quantities based on prior knowledge (for example, the cost of a large pizza

in a recent purchase, or the number of slices they and their friends generally
consume at lunch time)

• make assumptions about behaviors, preferences, or quantities (for instance, assume

that a certain percentage of the class prefers a certain topping)

Addressing

• HSA-CED.A.2

• HSA-CED.A.3

• HSN-Q.A.2

Instructional Routines

• Aspects of Mathematical Modeling

Launch
Ask students if they have ever been in charge of planning a party. Solicit a few ideas of
what party planners need to consider. Ask students to imagine being in charge of a class
pizza party. Explain that their job is to present a plan and a cost estimate for the party.

Arrange students in groups of 4. Provide access to calculators and, if feasible and desired,
access to the internet for researching pizza prices. Students can also make estimates
based on prior experience, refer to printed ads, or use their personal device to look up
pricing information.

Limit the time spent on the first question to 7–8 minutes and pause the class before
students move on to subsequent questions. Give groups of students 1–2 minutes to share
their proposals with another group. Then, select a few groups who used contrasting
strategies (such as those outlined in the Narrative) to briefly share their plans with the
class. Record or display their plans for all to see.

Next, ask students to complete the remaining questions. If needed, give an example of an
expression that can be written to represent a cost calculation.

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Anticipated Misconceptions
If students do not understand what is meant by “quantities that might change,” ask them if
it is more likely that the cost of a pepperoni pizza increases on the day of the party, or that
some students are absent that day. In a model that incorporates both of these quantities,
they may wish to use a number for the cost of each type of pizza and a letter for the
number of students present that day.

Student Task Statement
Imagine your class is having a pizza party.

Work with your group to plan what to order and to
estimate what the party would cost.

1. Record your group’s plan and cost estimate. What would it take to convince
the class to go with your group’s plan? Be prepared to explain your reasoning.

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2. Write down one or more expressions that show how your group’s cost
estimate was calculated.

3. a. In your expression(s), are there quantities that might change on the day
of the party? Which ones?

b. Rewrite your expression(s), replacing the quantities that might change
with letters. Be sure to specify what the letters represent.

Student Response
Sample response:

1. There are 28 people in the class. or 56 slices are needed.

◦ If each person gets 2 slices of pizza,

◦ One pizza has 8 slices, so 7 pizzas are needed.

◦ A large cheese pizza is $11 and a large pepperoni pizza is $13.

◦ Many students prefer cheese, so let’s order 4 cheese pizzas and 3 pepperoni

pizzas. .

◦ Tax is about 8%, and 8% of $83 is $6.64, so the total cost would be about $90.

2. Number of pizzas: . Cost of pizzas, excluding tax: . Total cost
including tax: .

3. a. Some students might be absent on the day of the party, so the number of
students might change. The pizza shop might have a special deal on pizzas, so
the cost per pizza might change.

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b. Number of pizzas: , where is the number of students. Cost of food

and drinks: , where is the cost per cheese pizza and is the cost per

pepperoni pizza.

Are You Ready for More?

Find a pizza place near you and ask about the diameter and cost of at least two
sizes of pizza. Compare the cost per square inch of the sizes.

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Student Response
Answers vary.

Activity Synthesis
Invite groups who did not previously share their plans to share the expressions they wrote
and explain what the expressions represent. After each group shares, ask if others
calculated the costs the same way but wrote different expressions.

As students present their expressions, record the quantities that they mention and display
them for all to see. Some examples:

• the number of students in the class
• the number of pizza slices per person
• the cost of delivery
• the price per topping

Briefly discuss the quantities that students anticipate would change (and therefore would
replace with letters).

Explain to students that the expressions they have written are examples of mathematical
models. They are mathematical representations of a situation in life that can be used to
make sense of problems and solve them. We will look more closely at how expressions
could represent the quantities in a situation like party planning, which involves certain
conditions or requirements.

Support for English Language Learners

Representing, Conversing: MLR7 Compare and Connect. As students share their plans with
the class, call students’ attention to the different ways the quantities are represented
in expressions, and to the different expressions that represent the same quantity.
Take a close look at the contrasting strategies used within the context of the situation.
Wherever possible, amplify student words and actions that students use to revise
their models by estimating quantities and making assumptions about the situation.
Design Principle(s): Maximize meta-awareness; Support sense-making

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1.3 What are the Constraints?

10 minutes
Writing and solving equations often revolves around the idea of representing and
satisfying constraints. This activity introduces the term “constraints” and begins to develop
the idea that expressions and equations can help us describe constraints on quantities. It
prompts students to recognize that quantities are sometimes constrained in terms of
the values they could take or in terms of how they relate to another quantity.

Addressing

• HSA-CED.A.3

Instructional Routines

• MLR7: Compare and Connect

Launch

Tell students that they will now look at some constraints of the pizza party. Explain that a
constraint is something that limits what is possible or what is reasonable in a situation.
For example, one constraint a teacher has to work with is the amount of time in a class
period or the number of school days in a year (both of these might be a fixed number).
Another constraint might be the number of students in a class (which may vary by class,
but is usually no more than a certain number).

Consider keeping students in groups of 4. Give students 2 minutes of quiet work time and
then time to briefly share their responses with their group. Follow with a whole-class
discussion.

Support for Students with Disabilities

Representation: Develop Language and Symbols. Display or provide charts with symbols

and meanings. Invite students to name additional examples of other variables that

might be considered constraints to encourage critical thinking and application to the

expressions created in this activity. Examples of additional constraints might include

to represent how long the party could last, or to say that each student will

get less than 3 beverages.

Supports accessibility for: Conceptual processing; Memory

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Anticipated Misconceptions

If students have trouble thinking of constraints for a chosen variable, ask about extreme
values. For instance, ask: "Do you think a large pizza might cost $100? How about $3?"

Some students may struggle with translating the written descriptions of the constraints

into inequalities. For example, “the greatest number of students in the class would be 30”

might be mistakenly written as . Ask these students to explain the meaning of the

“>” symbol. Ask: "If 29 students come to class, can we write ?"

Student Task Statement
A constraint is something that limits what is possible or reasonable in a situation.

For example, one constraint in a pizza party might be the number of slices of pizza

each person could have, . We can write to say that each person gets fewer

than 4 slices.

1. Look at the expressions you wrote when planning the pizza party earlier.
a. Choose an expression that uses one or more letters.

b. For each letter, determine what values would be reasonable. (For
instance, could the value be a non-whole number? A number greater
than 50? A negative number? Exactly 2?)

2. Write equations or inequalities that represent some constraints in your pizza
party plan. If a quantity must be an exact value, use the symbol. If it must be
greater or less than a certain value to be reasonable, use the or symbol.

Student Response

1. Sample response: the price of
a. , where represents the price of a large cheese pizza and
a large pepperoni pizza.

b. and can be whole or non-whole numbers. They represent prices of pizza, so
they can’t be negative. A large cheese pizza could be as low as $7.50 with a

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coupon or when there is a special promotion and is usually less than $12. A
large one-topping pizza is usually more than $11 but no more than $20.

2. Sample response: and

◦ or
◦ and

Activity Synthesis

Invite students to share the equations or inequalities that represent the constraints in
their party plan. Emphasize the following points:

• Sometimes a constraint is an exact value. For example: Ordering pizza online involves

a fee of $2.50, or the price of a large cheese pizza is $9.

• Other times, a constraint involves a boundary or a limit. For example: An order must

be at least $25 in value to qualify for free delivery, or the party must cost no more
than $90.

Lesson Synthesis

Some key takeaways from this lesson are the ideas that real-life situations often involve
constraints, that we can use expressions, equations, and inequalities to represent these
constraints.

To help students see these points, discuss questions such as:

• "In planning a pizza party, what were some ways we gathered information to

estimate the cost?" (counting the number of students, researching actual prices)

• "What were some assumptions we made?" (pizza preferences, number of slices per

person, possible prices of pizza)

• "Suppose we had gathered information differently, for instance, by asking every

student the exact pizza toppings and number of slices. Would that have been a
reasonable approach? How would that have changed the cost estimate? " (It would be
inefficient to take exact orders, cost much more, and likely mean a lot of leftovers.)

• "Suppose we had made a different set of assumptions, for instance, assuming that

everyone loved pepperoni and would like only 1 slice. How would that have changed
the cost estimate?" (If the assumption was a slice of pepperoni pizza per person, it
would probably cost less, but many students might not be able to enjoy the pizza or
might not have enough.)

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• "In planning the party, we saw some examples of constraints. Can you think of some

constraints in other situations? What might be some constraints in, say, planning a
field trip, or in organizing a community service event?"

Tell students that expressions, equations, and inequalities are mathematical models. A
model is a mathematical representation of a real-life situation. When people create
models, they rely on the information they have, but they also make assumptions and
decisions which affect the models. If the information or assumptions change, the model
would also change.

1.4 Ice Cream Party

Cool Down: 5 minutes
Addressing

• HSA-CED.A.2
• HSA-CED.A.3
• HSN-Q.A.2

Student Task Statement
As a reward for achieving their goals, all students in the ninth grade are invited to
an ice cream party.

1. Write an expression that could represent an estimated cost for the party. Use
at least one letter. State what each part of the expression represents.

2. Choose a letter in your expression. Describe the values that would be
reasonable for the quantity that the letter represents.

Student Response

1. Sample responses: . There are students in the ninth grade. A packaged ice

◦

cream cone is about $1.50, and tax is about 8%.

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◦ . There are 125 students in the ninth grade. Suppose a tub of

ice cream costs $5 and is enough for 6 students. A pack of 24 cones costs about
$3 and 6 packs are needed. A pack of 150 paper napkins costs dollars.

2. Sample responses:

◦ A whole number between 90 and 100 is a reasonable value for the number of

students in ninth grade.

◦ A number less than $4 but more than $1 is reasonable for the cost of a pack of

paper napkins.

Student Lesson Summary

Expressions, equations, and inequalities are mathematical models. They
are mathematical representations used to describe quantities and their
relationships in a real-life situation. Often, what we want to describe are
constraints. A constraint is something that limits what is possible or what is
reasonable in a situation.

For example, when planning a birthday party, we might be dealing with these
quantities and constraints:

quantities constraints

• the number of guests • 20 people maximum
• the cost of food and drinks • $5.50 per person
• the cost of birthday cake • $40 for a large cake
• the cost of entertainment • $15 for music and $27 for games
• the total cost • no more than $180 total cost

We can use both numbers and letters to represent the quantities. For example, we
can write 42 to represent the cost of entertainment, but we might use the letter to
represent the number of people at the party and the letter for the total cost in
dollars.

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We can also write expressions using these numbers and letters. For instance, the

expression is a concise way to express the overall cost of food if it costs $5.50

per guest and there are guests.

Sometimes a constraint is an exact value. For instance, the cost of music is $15.
Other times, a constraint is a boundary or a limit. For instance, the total cost must
be no more than $180. Symbols such as <, >, and = can help us express these
constraints.

quantities constraints

• the number of guests •
• the cost of food and drinks •
• the cost of birthday cake •
• the cost of entertainment •
• the total cost •

Equations can show the relationship between different quantities and constraints.
For example, the total cost of the party is the sum of the costs of food, cake,
entertainment. We can represent this relationship with:

Deciding how to use numbers and letters to represent quantities, relationships, and
constraints is an important part of mathematical modeling. Making
assumptions—about the cost of food per person, for example—is also important in
modeling.

A model such as can be an efficient way to make estimates or

predictions. When a quantity or a constraint changes, or when we want to know

something else, we can adjust the model and perform a simple calculation, instead

of repeating a series of calculations.

Glossary
• constraint

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• model

Lesson 1 Practice Problems

1. Problem 1

Statement

The videography team entered a contest and won a monetary prize of $1,350.
Which expression represents how much each person would get if there were
people on the team?

A.
B.
C.
D.

Solution

A

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2. Problem 2

Statement

To support a local senior citizens center, a student club sent a flyer home to
the students in the school. The flyer said, "Please bring in money to support
the senior citizens center. Paper money and coins accepted!" Their goal is to
raise dollars.

Match each quantity to an expression, an equation, or an inequality that
describes it.

A. the dollar amount the club would 1.
have if they reached half of their 2.
goal 3.
4.
B. the dollar amount the club would 5.
have if every student at the
school donated 50 cents to the
cause

C. the dollar amount the club could
donate if they made $50 more
than their goal

D. the dollar amount the club would
still need to raise to reach its goal
after every student at the school
donated 50 cents

E. the dollar amount the club would
have if half of the students at the
school each gave 50 cents

Solution

3. A: 2
4. B: 4
5. C: 1

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6. D: 5

7. E: 3

8. Problem 3

Statement

Each of the 10 students in the baking club made 2 chocolate cakes for a
fundraiser. They all used the same recipe, using cups of flour in total.

Write an expression that represents the amount of flour required for one
cake.

Solution or

Sample response:

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9. Problem 4

Statement

A student club started a fundraising effort to support animal rescue
organizations. The club sent an information flyer home to the students in
the school. It says, "We welcome donations of any amount, including any
change you could spare!" Their goal is to raise dollars, and to donate to a cat
shelter and a dog shelter.

Match each quantity to an expression, an equation, or an inequality that
describes it.

A. The dollar amount the club would 1.
have if they reached one-fourth 2.
of their goal.

B. The dollar amount the club would 3.
have if every student at the 4.
school donated a quarter to the 5.
cause.

C. The dollar amount the club could
donate to the cat shelter if they
reached their goal and gave a
quarter of the total donation to a
dog shelter.

D. The dollar amount the club would
still need to raise to reach its goal
after every student at the school
donated a quarter.

E. The dollar amount the club would
have if three-fourths of the
students at the school each gave
50 cents.

Solution

10. A: 2

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11. B: 5
12. C: 4
13. D: 3
14. E: 1

15. Problem 5
Statement

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A softball team is ordering pizza to eat after their tournament. They plan to
order cheese pizzas that cost $6 each and four-topping pizzas that cost $10
each. They order cheese pizzas and four-topping pizzas.

Which expression represents the total cost of all of the pizzas they order?

A.

B.

C.

D.

Solution

C

16. Problem 6

Statement

The value of coins in the pockets of several students is recorded. What is the
mean of the values: 10, 20, 35, 35, 35, 40, 45, 45, 50, 60

A. 10 cents
B. 35 cents
C. 37.5 cents
D. 50 cents

Solution

C

(From Unit 1, Lesson 9.)

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17. Problem 7
Statement

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The dot plot displays the number of hits a baseball team made in several
games. The distribution is skewed to the left.

If the game with 3 hits is considered to be recorded in error, it might be
removed from the data set. If that happens:

a. What happens to the mean of the data set?

b. What happens to the median of the data set?

Solution

a. The mean increases from 8.9 hits to approximately 9.21 hits.
b. The median remains 9 hits.

(From Unit 1, Lesson 10.)

18. Problem 8
Statement

A set of data has MAD 0 and one of the data values is 14. What can you say
about the data values?

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Solution

Sample response: There is no variability, so all the values are the same. Since one of
the values is 14, all the values must also be 14.

(From Unit 1, Lesson 11.)

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Lesson 2: Writing Equations to Model
Relationships (Part 1)

Goals
• Given a description of a situation or an equation, identify quantities that vary and

quantities that don’t.

• Understand that letters can be used to represent both quantities that vary and those

that are constant.

• Write equations with numbers and variables to describe relationships and

constraints.

Learning Targets
• I can tell which quantities in a situation can vary and which ones cannot.

• I can use letters and numbers to write equations representing the relationships in a

situation.

Lesson Narrative

This is the first of two lessons where students write equations to model various situations.
The work here progresses in two ways—in terms of the complexity of the relationships
and in terms of the amount of scaffolding built into the prompts.

Students begin by revisiting ways to calculate a given percentage of a given number, in
preparation for computations they'll need to do in the lesson. Then, they look at a
geometric context where three quantities can be related by addition and subtraction.
Next, they look at a couple of contexts on spending, earning, and sales tax, which involve
multiplication, multiplication and addition, and increasing a number by a percentage.

In each case, students begin by creating models where the values of the quantities are
known (or mostly known), and move toward models where the quantities are unknown or
can change. The repeated reasoning allows students to practice looking for and expressing
regularity (MP8). As they interpret verbal descriptions and write equations, students
develop their understanding of equations as a way to represent constraints and practice
reasoning quantitatively and abstractly (MP2).

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Alignments
Building On

• 6.RP.A.3.c: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity

means 30/100 times the quantity); solve problems involving finding the whole, given
a part and the percent.

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.

Instructional Routines

• Math Talk
• MLR1: Stronger and Clearer Each Time
• MLR2: Collect and Display
• MLR8: Discussion Supports
• Think Pair Share

Student Learning Goals

• Let's look at how equations can help us describe relationships and constraints.

2.1 Math Talk: Percent of 200

Warm Up: 5 minutes
This Math Talk invites students to use what they know about fractions, decimals, and the
meaning of percent to mentally solve problems. The strategies elicited here will be helpful
later in the lesson when students calculate prices that involve a percent increase and write
an equation to generalize the calculation.

Finding different percents of the same value (200) is also an opportunity to reason
repeatedly and look for and make use of structure (MP7, MP8).

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Students are likely to approach the problems in different ways. They may:

• Convert each percentage into a fraction and multiply the fraction by 200. For

example, they may think of 25% as , 12% as or , and 8% as or .

• Convert each percentage into a decimal and multiply it by 200.

• Notice that 1% of 200 is 2, and that any percentage of 200 can be found by

multiplying the percentage by 2. For example, 25% of 200 is , and % of 200 is

or .

Building On

• 6.RP.A.3.c

Instructional Routines

• Math Talk

• MLR8: Discussion Supports

Launch

Display one problem at a time. Give students quiet think time for each problem and ask
them to give a signal when they have an answer and a strategy. Keep all problems
displayed throughout the talk. Follow with a whole-class discussion.

Support for Students with Disabilities

Representation: Internalize Comprehension. To support working memory, provide
students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Task Statement
Evaluate mentally.
25% of 200
12% of 200
8% of 200

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% of 200

Student Response

• 50
• 24
• 16

• or

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their
responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ’s reasoning in a different way?”

• “Did anyone have the same strategy but would explain it differently?”

• “Did anyone solve the problem in a different way?”

• “Does anyone want to add on to ’s strategy?”

• “Do you agree or disagree? Why?"

If one of the strategies shown in the Narrative is not mentioned, consider sharing it with
students.

Support for English Language Learners

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when
they explain their strategy. For example, “First, I _____ because….” or “I noticed _____ so
I….” Some students may benefit from the opportunity to rehearse what they will say
with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

2.2 A Platonic Relationship

10 minutes

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This activity gives students an opportunity to examine multiple quantities and
relationships in a geometric context, and to use letters to represent quantities.

As students analyze the number of faces, vertices, and edges in several Platonic solids and
try to identify relationships, they practice looking for structure (MP7). Some of the
relationships could be represented by inequalities. One particular relationship can be
represented by equations, setting the stage for upcoming work on equivalent equations.

If work time is coming to an end and no students are able to find an equation that relates
the parts of the Platonic solids, suggest that students try adding the vertices and faces in
each row.

Addressing

• HSA-CED.A.2
• HSA-CED.A.3

Instructional Routines

• MLR2: Collect and Display

Launch
Ask students to keep their books or devices closed.

Display the images of the three Platonic solids. If physical polyhedra are available, consider
displaying them as well. Ask students: "In what ways are the three figures alike? In what
ways are they different?"

Students may say that the figures are alike in that:

• They are all three-dimensional figures.
• Each Platonic solid has only one kind of polygon for its faces (triangle for the

tetrahedron, square for the cube, and pentagon for the dodecahedron).

• The faces of each figure seem to be regular polygons (or polygons whose sides are

equal in length).

• They all have an even number of faces, vertices, and edges.

They may say that the figures are different in that:

• Each figure has a different polygon for its faces (a triangle for the tetrahedron, a

square for the cube, and a pentagon for the dodecahedron).

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

• They all have a different number of faces, vertices, and edges.

If students refer to edges and vertices as “lines” and “points," ask if they remember the
“math names” for these things. Review the terms "vertices," "edges," and "faces" as
needed.

Tell students that they will now investigate the relationships between the faces, vertices,
and edges in each polyhedron.

Support for Students with Disabilities

Representation: Develop Language and Symbols. Create a display of important terms and
vocabulary. During the launch, invite students to use the diagrams and the
information provided in the table about the tetrahedron and dodecahedron to come
up with brief definitions for: vertices, edges, and faces. Invite students to suggest
language or diagrams to include that will support their memory and understanding of
these terms.
Supports accessibility for: Conceptual processing; Language

Anticipated Misconceptions
Some students may get the terms "vertex," "faces," and "edges" confused. As students
work on the activity, check to make sure that they understand what should be counted.

Some students may see the relationship between vertices, edges, and faces, but be unsure
of how to express that relationship using an equation. If students can say in words
something like, “you always get two more,” ask them to try writing an equation that might
be correct. Then suggest that they test the equation for one of the solids. If it doesn't work,
ask them to make changes to the equation until it works.

Student Task Statement
These three figures are called Platonic solids.

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

Tetrahedron Cube Dodecahedron

The table shows the number of vertices, edges, and faces for the tetrahedron and
dodecahedron.

faces vertices edges

tetrahedron 4 4 6

cube

dodecahedron 12 20 30

1. Complete the missing values for the cube. Then, make at least two
observations about the number of faces, edges, and vertices in a Platonic
solid.

2. There are some interesting relationships between the number of faces ( ),

edges ( ), and vertices ( ) in all Platonic solids. For example, the number of

edges is always greater than the number of faces, or . Another example:

The number of edges is always less than the sum of the number of faces and

the number of vertices, or .

There is a relationship than can be expressed with an equation. Can you find
it? If so, write an equation to represent it.

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

Student Response
1. The cube has 6 faces, 8 vertices, and 12 edges. Sample observations:

◦ The number of edges is greater than either the number of faces or the number

of vertices.

◦ All the numbers are even numbers.
◦ The number of faces is less than or equal to the number of vertices.
◦ The number of edges is 2 more than the sum of the other two numbers.

2. (or equivalent)

Are You Ready for More?
There are two more Platonic solids: an octahedron which has 8 faces that are all
triangles and an icosahedron which has 20 faces that are all triangles.

1. How many edges would each of these solids have? (Keep in mind that each
edge is used in two faces.)

2. Use your discoveries from the activity to determine how many vertices each of
these solids would have.

3. For all 5 Platonic solids, determine how many faces meet at each vertex.

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

Student Response
1. The octahedron has 12 edges. The icosahedron has 30 edges.

2. The octahedron has 6 vertices. The icosahedron has 12 vertices.

3. The tetrahedron, cube, and dodecahedron have 3 faces meeting at each vertex. The
octahedron has 4 faces meeting at each vertex. The icosahedron has 5 faces meeting
at each vertex.

Activity Synthesis

Invite students to share their observations about the quantities and relationships in the
table. Some of the hypotheses students make about the relationships might not be true
for all Platonic solids. For now, it is sufficient that they are supported by the values in the
table.

Next, elicit the relationship between the quantities that could be represented by

. Record and display all correct equations for all to see. If students produce

only one correct equation, introduce a variant such as or

. Ask students whether these equations all represent the same

relationship and how they know. Students can simply show that each equation captures

the pattern in the table. It is not necessary for them to articulate why the equations are

equivalent, as they will have many opportunities to do so in upcoming lessons.

Support for English Language Learners

Conversing: MLR2 Collect and Display. During the launch, listen for and collect language
students use to describe how the three Platonic solids are alike and different. Record
informal student language alongside the mathematical terms (vertices, edges, faces)
on a visual display of the three solids and update it throughout the remainder of the
lesson. Remind students to borrow language from the display as needed. This will
provide students with a resource to draw language from during small-group and
whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

2.3 Blueberries and Earnings

10 minutes
In this activity, students write equations to represent quantities and relationships in two
situations. In each situation, students express the same relationship multiple times:

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

initially using numbers and variables and later using only variables. The progression helps
students see that quantities can be known or unknown, and can stay the same or vary,
but both kinds of quantities can be expressed with numbers or letters.

Addressing

• HSA-CED.A.2

Instructional Routines

• MLR1: Stronger and Clearer Each Time

Launch

Support for English Language Learners

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to
help students improve their written responses to the final question. Give students
time to meet with 1–2 partners to share and get feedback on their responses. Display
feedback prompts that will help students strengthen their ideas and clarify their
language. For example, “Can you describe the quantities?”, “What operation was
used?”, and “Can you try to explain this using a different example?” Invite students to
go back and revise their written explanation based on the feedback from peers. This
will help students understand situations in which quantities are related through
communicating their reasoning with a partner.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

Support for Students with Disabilities

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived
challenge. Invite students to write equations for 3–4 of the situations they select.
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
Students may translate “Mai earned dollars, which is 45 more dollars than Noah did” as

, not paying attention to where the plus sign should go. As with other problems

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

throughout this unit, encourage students to try using numbers in their equation to see if
the equation really says what they want it to say.

Student Task Statement
1. Write an equation to represent each situation.
a. Blueberries are $4.99 a pound. Diego buys pounds of blueberries and
pays $14.95.

b. Blueberries are $4.99 a pound. Jada buys pounds of blueberries and
pays dollars.

c. Blueberries are dollars a pound. Lin buys pounds of blueberries and
pays dollars.

d. Noah earned dollars over the summer. Mai earned $275, which is $45
more than Noah did.

e. Noah earned dollars over the summer. Mai earned dollars, which is
45 dollars more than Noah did.

f. Noah earned dollars over the summer. Mai earned dollars, which is
dollars more than Noah did.

2. How are the equations you wrote for the blueberry purchases like the
equations you wrote for Mai and Noah’s summer earnings? How are they
different?

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

Student Response
1. a.

b.

c.

d. or (or equivalent)

e. or (or equivalent)

f. or (or equivalent)

2. Sample response:

◦ The two sets of equations are alike in that in each set:
▪ Each equation involves three quantities.
▪ The first equation has two known quantities, the second has one known

quantity, and the last one has no known quantities.

◦ They are different in that:
▪ The three quantities in each set are different. In the first set, they are unit

price, pounds of blueberries, and total cost. In the second set, they are
Noah’s earnings, Mai’s earnings, and the difference between the two.

▪ In the first set, the relationship involves multiplication. In the second, it

involves addition (or subtraction).

Activity Synthesis
Focus the discussion on students' observations about how the two sets of equations are
alike. Then, ask how the equations within each set are different. If students mention that
some quantities are known or are fixed and others are not, ask them to specify which ones
are which.

Highlight the idea that sometimes we know how quantities are related, but the value of
each quantity may be unknown or may change. We often use letters to represent those
unknown or changing quantities.

There might be times, however, when we use letters to represent quantities that are
known or are constant. Doing so may help us focus on the relationship rather than the
numbers. Tell students we will look at examples of such situations in upcoming activities.

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

2.4 Car Prices

10 minutes
This activity gives students another opportunity to represent a relationship with numbers
and letters, to reason repeatedly, and to see more clearly that equations are helpful for
generalizing a relationship.

From the given descriptions, students are aware that there are four relevant quantities in
the car purchase situations. In each situation, the value of at least one quantity is not
given, creating a need for students to name it, either using a word or a phrase (for
instance, "total price" or "original price") or to use a variable (for example, or ). Some
students might choose to use a symbol. Monitor for the different ways students represent
these quantities.

Addressing

• HSA-CED.A.2

Instructional Routines

• Think Pair Share

Launch
Ask students if they have had to pay sales tax when making a purchase and, if so, to briefly
explain how sales tax works.

Explain to students that a car purchase also involves a sales tax. Car buyers pay not only
the price of a car, but also a tax that is a certain percentage of the car price. Car
dealerships also often charge their customers various fees.

Tell students that they will now write equations to describe the relationship between the
price of the car, the tax, a fee, and the total price. Emphasize that it is not necessary to
evaluate any expressions or perform any computations.

Arrange students in groups of 2. Give them a few minutes of quiet work time and then a
minute to discuss their responses with a partner. Follow with a whole-class discussion.

Anticipated Misconceptions
Some students may be taken aback by the prompt to write an expression relating four
quantities. If they have trouble getting started, suggest that they simply calculate the cost
of buying a $9,500 car, taking care to show their work.

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

One part of the first question gives the total price of purchase rather than the original
price of the car. If students use the given value as an original price, ask them to double
check the given information.
In the last question, students may struggle to represent algebraically. Some students
may multiply the price by , others may write . Urge students to look at their work in
the first question. Ask them: "By what number did you multiply the car price? What
operation turns the number 6 into 0.06?"

Student Task Statement
The tax on the sale of a car in Michigan is 6%. At a dealership in Ann Arbor, a car
purchase also involves $120 in miscellaneous charges.

1. There are several quantities in this situation: the original car price, sales tax,
miscellaneous charges, and total price. Write an equation to describe the
relationship between all the quantities when:
a. The original car price is $9,500.

b. The original car price is $14,699.

c. The total price is $22,480.

d. The original price is .

2. How would each equation you wrote change if the tax on car sales is % and
the miscellaneous charges are dollars?

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

Student Response , or

1. Sample response:
a.

b. , or

c. , or

d. , or

2. Equations equivalent to these are also acceptable:
a.

b.

c.

d.

Activity Synthesis

Select students whose equations are equivalent but in different forms to share their
responses. Record and display them for all to see. Then, draw students' attention to the
first and last equation in each question.

From the first question, those equations might be: and
. Ask students:

• "In the first equation, what quantities do we know?" (the original price of the car, the

tax rate, and the miscellaneous charges) "When might it be useful to write an

equation like this?" (when we know all relevant quantities except for one)

• "In the other equation, what quantities do we know?" (the tax rate and the

miscellaneous charges) "When might it be useful to write an equation like this?"
(when we want to have a kind of formula for finding the total price for a car of any
price, assuming the tax rate and miscellaneous charges are fixed at 6% and $120
respectively)

For the second question, the equations might be: and
. Ask students:

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

• "In the first equation, what quantities do we know?" (the original price of the car)

"When might it be useful to write an equation like this?" (when we want to know the
total cost of a $9,500 car but don't know the tax rate or other fees)

• "In the other equation, what quantities do we know?" (none) "Why might it be

helpful to write an equation like this?" (It helps us see the relationship of all the
quantities that are relevant in the situation. It is flexible in that it can be used to find
the total cost for any car price, tax rate, and miscellaneous charges.)

Emphasize that we might choose to use letters to represent quantities that vary or those
that are constant, depending on what we want to understand or know.

Lesson Synthesis

To help students synthesize their work in the lesson, consider asking them to write a
response to one or both of the following prompts:

• We could use numbers or letters to represent the quantities in a situation. When

might it make sense to use only numbers? When might it make sense to use letters?

• You've heard the phrases "a quantity that varies" and "a quantity that stays constant"

in this lesson. Desribe what they mean in your own words. If possible, give an
example of a situation that has a quantity that varies and a quantity that stays
constant.

2.5 Shirt Colors

Cool Down: 5 minutes
Addressing

• HSA-CED.A.2
• HSA-CED.A.3

Student Task Statement
A school choir needs to make T-shirts for its 75 members and has set aside some
money in their budget to pay for them. The members of the choir decided to order
from a printing company that charges $3 per shirt, plus a $50 fee for each color to
be printed on the shirts.

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

1. Write an equation that represents the relationship between the number of
T-shirts ordered, the number of colors on the shirts, and the total cost of the
order. If you use a variable, specify what it represents.

2. In this situation, which quantities do you think can vary? Which might be fixed?

Student Response
1. (or equivalent). represents the number of colors on the shirts.
represents the total cost in dollars.

2. Sample response: The cost of per shirt and the fee-per-color are fixed (they are set by
the printing company). The number of colors on the shirts can vary, so can the total
cost, depending on the number of colors being printed.

Student Lesson Summary

Suppose your class is planning a trip to a museum. The cost of admission is $7 per
person and the cost of renting a bus for the day is $180.

• If 24 students and 3 teachers are going, we know the cost will be:

or .

• If 30 students and 4 teachers are going, the cost will be: .

Notice that the numbers of students and teachers can vary. This means the cost of
admission and the total cost of the trip can also vary, because they depend on how
many people are going.

Letters are helpful for representing quantities that vary. If represents the number
of students who are going, represents the number of teachers, and represents
the total cost, we can model the quantities and constraints by writing:

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

Some quantities may be fixed. In this example, the bus rental costs $180 regardless
of how many students and teachers are going (assuming only one bus is needed).
Letters can also be used to represent quantities that are constant. We might do this
when we don’t know what the value is, or when we want to understand the
relationship between quantities (rather than the specific values).
For instance, if the bus rental is dollars, we can express the total cost of the trip as

. No matter how many teachers or students are going on the trip,
dollars need to be added to the cost of admission.

Lesson 2 Practice Problems

1. Problem 1

Statement

Large cheese pizzas cost $5 each and large one-topping pizzas cost $6 each.
Write an equation that represents the total cost, , of large cheese pizzas
and large one-topping pizzas.

Solution

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

2. Problem 2

Statement

Jada plans to serve milk and healthy cookies for a book club meeting. She is
preparing 12 ounces of milk and 4 cookies per person. Including herself, there
are 15 people in the club. A package of cookies contains 24 cookies and costs
$4.50.

A 1-gallon jug of milk contains 128 ounces and costs $3. Let represent
number of people in the club, represent the ounces of milk, represent the
number of cookies, and represent Jada'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", "E"]

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

3. Problem 3

Statement

A student on the track team runs 45 minutes each day as a part of her
training. She begins her workout by running at a constant rate of 8 miles per
hour for minutes, then slows to a constant rate of 7.5 miles per hour for
minutes.

Which equation describes the relationship between the distance she runs in
miles, , and her running speed, in miles per hour?

A.

B.

C.

D.

Solution

C

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

4. Problem 4
Statement

Elena bikes 20 minutes each day for exercise.
Write an equation to describe the relationship between her distance in miles,

, and her biking speed, in miles per hour, when she bikes:
a. at a constant speed of 13 miles per hour for the entire 20 minutes

b. at a constant speed of 15 miles per hour for the first 5 minutes, then at
12 miles per hour for the last 15 minutes

c. at a constant speed of miles per hour for the first 5 minutes, then at
miles per hour for the last 15 minutes

Solution

a.
b.
c.

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

5. Problem 5
Statement

The dot plot displays the number of marshmallows added to hot cocoa by
several kids. What is the MAD of the data represented in the dot plot?

A. 0.6 marshmallows
B. 3 marshmallows
C. 4 marshmallows
D. 5 marshmallows

Solution

A

(From Unit 1, Lesson 11.)

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

6. Problem 6

Statement

Here is a data set: 5 10 10 10 15 100

a. After studying the data, the reasearcher realized that the value 100 was
meant to be recorded as 15. What happens to the mean and standard
deviation of the data set when the 100 is changed to a 15?

b. For the original data set, with the 100, would the median or the mean be
a better choice of measure for the center? Explain your reasoning.

Solution

a. The mean changes from 25 to about 10.83. The standard deviation changes
from about 33.7 to about 3.44.

b. The median would be a better choice, because the value of 100 skews the
median but not the mean.

(From Unit 1, Lesson 12.)

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

7. Problem 7

Statement

A coach for a little league baseball team is ordering trophies for the team.
Players on the team are allowed to choose between 2 types of trophies. The
gold baseball trophies cost $5.99 each and the uniform baseball trophies cost
$6.49 each. The team orders gold baseball trophies and uniform baseball
trophies.

Write an expression that could represent the total cost of all of the trophies.

Solution

Sample response:

(From Unit 2, Lesson 1.)

8. Problem 8

Statement

The robotics team needs to purchase $350 of new equipment. Each of the
students on the team plans to fundraise and contribute equally to the
purchase.

Which expression represents the amount that each student needs to
fundraise?

A.

B.

C.

D.

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

Solution

C

(From Unit 2, Lesson 1.)

9. Problem 9
Statement

In a trivia contest, players form teams and work together to earn as many
points as possible for their team. Each team can have between 3 and 5
players. Each player can score up to 10 points in each round of the game.
Elena and four of her 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 number of points that Elena’s team earns in one round

b. the number of points Elena’s team earns in one round if every player
scores between 6 and 8 points

c. the number of points Elena’s team earns if each player misses one point

d. the number of players in a game if there are 5 teams of 4 players each

e. the number of players in a game if there are at least 3 teams

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

Solution ), where is the score of the team

1. (and possibly

2. and

3.

4.

5. , where is the number of players on a team

(From Unit 2, Lesson 1.)

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