Lesson 1

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.”

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.

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. 

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. 

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.

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.

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:

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!

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.

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.

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