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).

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 students to check 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.

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:

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!

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.

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