# Lesson 1

Relationships between Quantities

## 1.1: Pricing Theater Popcorn (10 minutes)

### Warm-up

A context is described, and students generate two sets of values. The purpose of this warm-up is to remind students of some characteristics that make a relationship proportional or not proportional, so that later in the lesson, they are better equipped to recognize that a relationship is not proportional and explain why.

The numbers were deliberately chosen to encourage different ways of viewing a proportional relationship. For 20 ounces and 35 ounces, students might move from row to row and think in terms of scale factors. This approach is less straightforward for 48 ounces, and some students may shift to thinking in terms of unit rates.

There are many possible rationales for choosing numbers so that size is not proportional to price. As long as the numbers are different from those in the “proportional” column, the relationship between size and price is guaranteed to be not proportional. Look for students who have a reasonable way to explain why their set of numbers is not proportional, like “the unit price is different for each size,” or “each size costs a different amount per ounce.”

### Launch

Ask students to remember the last time they went to the movies. What do they know about the popcorn for sale? What sizes does it come in? About how much does it cost? Tell students that in this activity, they will come up with prices for different sizes of popcorn—one set of prices in which the price is in proportion to the size, and another set of prices in which the price is not in proportion to the size, but is still reasonable. Ask students to be ready to explain the reasons they chose the numbers they did.

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

### Student Facing

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 example 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 |

### Student Response

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### Activity Synthesis

Invite a student to share their prices for the proportional relationship and how they decided on those numbers. Ask if any students thought of it in a different way.

Then, invite a student to share their prices for the relationship that is not proportional and record these for all to see. Ask students to explain ways you can tell that the relationship is not proportional.

## 1.2: Entrance Fees (10 minutes)

### Activity

This context was used in an earlier unit about proportional relationships 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.

### Launch

Tell students that unlike in the previous activity where they could make up any numbers, this activity has a relationship where there *is* a pattern, and part of the work is to figure out the pattern. 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.

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.*Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to highlight the number of people with its corresponding entrance fee, making annotations of how the fee was calculated.

*Supports accessibility for: Visual-spatial processing*

*Reading, Representing: MLR6 Three Reads.*Use this routine to help students understand the question and to represent the relationships between quantities. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (the park charges an entrance fee that includes the number of people in the car). After the second read, ask students what can be counted or measured in this situation. Listen for, and amplify, the quantities that vary in relation to each other: number of people in a vehicle; entrance fee amount, in dollars. After the third read, ask students to organize the information (using a list, table, or diagram) and brainstorm ideas for how much the park charges for each person in the car.

*Design Principle(s): Support sense-making*

### Student Facing

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

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### Anticipated Misconceptions

Students may misunderstand that the first two questions require noticing and extending a pattern, and (because of the warm-up) think that any reasonable number is acceptable. Encourage them to organize the given information and think about what rule the park might use to determine the entrance fee based on the number of people in the vehicle.

Students may come up with “rules” that aren’t supported by the context or the given information. For example, they may notice that each additional person costs $3, but then reason that 30 people must cost $90. Whatever their rule, ask them to check that whether it works for all of the information given. For example, since 2 people cost $14, we can tell that “$3 per person” is not the rule.

### Activity Synthesis

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?” Possible responses:

- 2 people to 4 people is double, but 14 to 20 is not double.
- \(14 \div 2=7\), but \(20 \div 4 = 5\). If the entrance fee were in proportion to the number of people, each quotient would be equal.
- You can’t describe the situation with an equation like \(px=q\).

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

- 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
- \(8 + 3 \boldcdot \text{ people}\)

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: Making Toast (10 minutes)

### Optional activity

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 \(y=px+q\) (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.

### 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.

*Action and Expression: Internalize Executive Functions.*Provide students with a two column table for processing and organizing 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: Language; Organization*

### Student Facing

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 think they can make?

### Student Response

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### Student Facing

#### Are you ready for more?

What is the smallest number that has a remainder of 1, 2, and 3 when divided by 2, 3, and 4, respectively? Are there more numbers that have this property?

### 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.

*Reading, Speaking: MLR7 Compare and Connect.*During the whole-class discussion, invite students to look for what is the same and what is different between the various approaches to solving the problem. Display and discuss differences in the tables and diagrams. Invite students to make connections by looking for the same quantity (e.g., 20 slices) in each representation. These exchanges strengthen students’ mathematical language use and reasoning based on ratios.

*Design Principle(s): Maximize meta-awareness*

## Lesson Synthesis

### Lesson Synthesis

The goal of this lesson is to recognize that there are situations in the world that are more complicated than what we have studied until this point, and to let students know this unit is about developing tools to solve some more sophisticated problems. Questions for discussion:

- “Describe some rules we encountered in this lesson for how one quantity was related to another quantity.”
- “What made these situations more complicated than relationships we have seen in the past?”
- “What were some tools or strategies we used that were particularly helpful?”

## 1.4: Cool-down - Movie Theater Popcorn, Revisited (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

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 \(x+3=8\) and \(4x=20\) 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 pies.

number of pies | total cost in dollars |
---|---|

1 | 13 |

2 | 23 |

3 | 33 |

5 | 53 |

We can see that each additional pie adds \$10 to the total cost, and that each total includes a \$3 additional cost, maybe representing a delivery fee. In this situation, 8 pies will cost \(8\boldcdot 10 + 3\) 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.