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A movie theater sells popcorn in bags of different sizes. The table shows the volume of popcorn and the price of the bag.

Complete one column of the table with prices where popcorn is priced at a constant rate. That is, the amount of popcorn is proportional to the price of the bag. Then complete the other column with realistic prices where the amount of popcorn and price of the bag are not in proportion.

volume of popcorn (ounces)	price of bag, proportional (\$)	price of bag, not proportional (\$)
10	6	6
20
35
48

1.2

Activity

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR6: Three Reads

Materials

None

Activity Narrative

In this activity, students use any strategy to extend a non-proportional pattern. This context was used in an earlier unit as an example of a relationship that is not proportional. However, a different rule for determining the entrance fee is used here.

Watch for students who organize the given information in a table or another visual representation, and for unique, correct approaches to the first two questions.

As students analyze several pairs of values in the relationship and then encapsulate the relationship with a rule, they look for and express regularity in repeated reasoning (MP8).

The referenced unit where students compared proportional and non-proportional relationships is in the previous course.

This is the first time Math Language Routine 6: Three Reads is suggested in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation. During the second read, students identify quantities. During the third read, the final prompt is revealed and students brainstorm possible starting points for answering the question. The intended question is withheld until the third read so students can make sense of the whole context before rushing down a solution path. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation with a partner.

Launch

Tell students that unlike the previous activity where they could choose any numbers, this activity has a relationship where there is a pattern, and part of the work is to figure out what that pattern is. This activity has to do with an entrance fee to a park, where the fee is based on the number of people in the vehicle.

MLR6 Three Reads. Keep books or devices closed. Display only the problem stem, without revealing the questions. Say, “We are going to read this problem 3 times.”

After the 1st read: “Tell your partner what this situation is about.”
After the 2nd read: “List the quantities. What can be counted or measured?”
For the 3rd read: Reveal and read the first question. Ask, “What are some ways we might get started on this?”

Advances: Reading, Representing

Representation: Access for Perception. Provide visual representations of anything that is read aloud.
Supports accessibility for: Language, Memory

Activity

None

A state park charges an entrance fee based on the number of people in a vehicle. A car containing 2 people is charged \$14, a car containing 4 people is charged \$20, and a van containing 8 people is charged \$32.

How much do you think a bus containing 30 people would be charged?
If a bus is charged \$122, how many people do you think it contains?
What rule do you think the state park uses to decide the entrance fee for a vehicle?

Student Response

Activity Synthesis

The purpose of this discussion is to elicit different ways of describing the rule for determining the entrance fee based on the number of people, and to notice that the relationship is not proportional.

Invite a student who organized the given information in a table to share. If no students did this, display this table for all to see:

number of people	entrance fee in dollars
2	14
4	20
8	32
30
	122

Ask: “What are some ways that you can tell that this relationship is not proportional?” Sample responses:

2 people to 4 people is double, but 14 to 20 is not double.
, but . If the entrance fee was in proportion to the number of people, each quotient would be equal.
You can’t describe the situation with an equation like .

Invite students who had different strategies for answering the first two questions to share their responses. Include as many unique strategies as time allows. For each strategy, ask students to state their rule that the park uses to decide the entrance fee. Record all unique, correct rules, so all students can see different ways of expressing the same idea. For example, the rule might be expressed in the following ways:

8 dollars for the vehicle plus 3 dollars per person
3 dollars for every person and an additional \$8
3 times the number of people plus 8

Note: We have the entire rest of the unit to systematically develop relationships like these. There is no need to formalize or generalize anything yet!

1.3

Activity

Optional

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

MLR7: Compare and Connect

Materials

None

Activity Narrative

In this activity, students are presented with a different relationship that is not proportional and also doesn’t fit a pattern that can be characterized by an equation in the form (like the previous activity could be). This optional activity is a good opportunity for students to interpret another context and describe a relationship, but it can be safely skipped if the previous activity takes too much time.

Students must make sense of the problem and persevere in problem solving (MP1) in this activity because there are many viable ways to represent the relationship and solve the problems, none of which are demonstrated first.

Launch

Keep students in the same groups. Give 2 minutes of quiet work time and then invite students to share their sentences with their partner, followed by whole-class discussion.

Representation: Internalize Comprehension. Provide a blank two-column table for students to process and organize information. Invite students to share their column labels (for example, number of slices and number of seconds) and how they organized the given information.
Supports accessibility for: Organization, Attention

Activity

None

A toaster has 4 slots for bread. Once the toaster is warmed up, it takes 35 seconds to make 4 slices of toast, 70 seconds to make 8 slices, and 105 seconds to make 12 slices.

How long do you think it will take to make 20 slices?
If someone makes as many slices of toast as possible in 4 minutes and 40 seconds, how many slices do you think they can make?

Student Response

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Activity Synthesis

Invite students to share their responses and their reasoning. Select as many unique approaches as time allows.

MLR7 Compare and Connect. Lead a discussion comparing, contrasting, and connecting the different representations. Ask,

“How are the tables or diagrams the same? How are they different?”
“How does the quantity 20 slices show up in each representation?”
“Are there any benefits or drawbacks to one representation compared to another?”

Advances: Representing, Conversing

Student Lesson Summary

In much of our previous work that involved relationships between two quantities, we were often able to describe amounts as being so much more than another, or so many times as much as another. We wrote equations like and and solved for unknown amounts.

In this unit, we will see situations where relationships between amounts involve more operations. For example, a pizza store might charge the amounts shown in the table for delivering pizzas.

number of pizzas	total cost in dollars
1	13
2	23
3	33
5	53

We can see that each additional pizza adds \$10 to the total cost, and that each total includes a \$3 additional cost, maybe representing a delivery fee. In this situation, 8 pizzas will cost and a total cost of \$63 means 6 pies were ordered.

In this unit, we will see many situations like this one, and will learn how to use diagrams and equations to answer questions about unknown amounts in the situation.