##
* *Reflection: Developing a Conceptual Understanding
Modeling Proportional Relationships - Section 1: Warm Up

In this warm-up activity, where the cost of an adult ticket is inversely proportional to the number of adult tickets sold, students are using tables and graphs to discover a new function shape. Once they have their graphs completed and determine that it is a nonlinear relationship, I can build upon their conceptual understanding by asking “What are the intercepts of your graph?” They should be able to see on their graph that they have not plotted any intercepts yet, but does that necessarily mean that they will not exist? Some hypothetical questions help the students come to a greater internalization of this type of relationship. “How much do I need to charge if 2000 adults come? How about if 4000 come? 8000?” “Will I ever have so many adults coming that I can just charge nothing?” “What if I could hypothetically sell half a ticket: how much would I need to charge for a ticket in order for that ‘half-ticket’ to cover the costs?” “What if I sold 0 adult tickets… is there any price that would allow those 0 tickets to cover my costs?” This kind of discussion can be richer now, when the scenario is concrete. Later, when we generalize to rational functions in their symbolic form, we can recall this discussion and clarify our understanding of asymptotes.

*Questioning Techniques to Build Understanding*

*Developing a Conceptual Understanding: Questioning Techniques to Build Understanding*

# Modeling Proportional Relationships

Lesson 1 of 8

## Objective: SWBAT mathematically represent direct and inverse proportional relationships between variables.

## Big Idea: A relationship between two quantities can often be described with a mathematical function. Which type of function is most appropriate is dictated by the scenario.

*90 minutes*

#### Warm Up

*20 min*

In this warm up, students take 10 minutes to practice making sense of the relationship between two variables. Warm-up Party Planning is a short activity in which students consider different combinations of ticket prices and number of paying guests that will cover the costs of a party. Warm-up Party Planning describes the scenario and asks students to create a graph to depict the relationship. I include this warm up before the relationship card sort in order to remind students that calculating and plotting ordered pairs can be a useful first step in accurately describing a relationship between two variables.

#### Resources

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#### Paired Brainstorming

*20 min*

Students work in pairs to illustrate their Relationship Cards and begin to discuss how to sort them. I print the cards on heavy paper and ask students to illustrate them in order to facilitate sorting. Depending on time constraints, I may ask students to do this for homework the night before.

The goal of the activity is to help students see that relationships between quantitative variables can often be expressed mathematically and that different types of relationships are represented by different functions. In the current unit, we are studying rational functions, so many of the cards contain inversely proportional relationships. I want my students to see that this type of relationship can be modeled with a rational function.

#### Resources

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Pairs of students will join together to form groups of 4 to plan and create a poster explaining their classification system. Through their brainstorming efforts, students should recognize that

- some relationships are "positive" in the sense that as one of the variables goes up, the other one goes up too.
- some relationships are "negative" in the sense that as one of the variables goes up, the other one goes down.
- some relationships are linear, in the sense that when one of the variables doubles, the other variables either doubles (positive relationship) or halves (negative relationship)
- other relationships are non-linear and there are several different models that may apply, including exponential, quadratic, and inversely proportional.

Students will create the categories into which the cards will be sorted and there could be a wide variety of answers, but the categories above are typical of what students come up with.

When the categories have been determined, students make a poster that classifies the relationships mathematically. They will use the Problem Solving Demo Rubric to assess their progress on the poster. Over the next few days, we will begin class with students presenting these posters to their peers.

#### Resources

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- LESSON 1: Modeling Proportional Relationships
- LESSON 2: Simplifying Rational Expressions
- LESSON 3: Operations with Rational Expressions
- LESSON 4: Solving Rational Equations
- LESSON 5: Quiz and Intro to Inverse Functions
- LESSON 6: Inverse Functions
- LESSON 7: Review Stations
- LESSON 8: Unit Assessment: Rational and Inverse Functions