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

Unit 5: Rational and Inverse Functions
Lesson 1 of 8

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.

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Subject(s):
Math, direct proportion, Algebra, inverse functions, inverse proportionality, function, domain, range, arithmetic with rational expressions, rational function
90 minutes

Colleen Werner

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