Experience First:

To prepare for today’s lesson, print and cut out the 8 scenarios in the Lesson 3.4 Scenarios file. Each group will need their own set of cards. We recommend printing these on cardstock and/or laminating them so you can use them for years to come. 

 

Today’s lesson follows a slightly different format in that the majority of the Activity will consist of students determining a function type to go with each scenario. It is important to note that students are not writing the equation that models the scenario, simply selecting what type of function (constant, linear, quadratic, cubic, rational, etc.) best describes the relationship between the variables in the scenario. Expect that this will be challenging for the students and prepare to give them plenty of time to discuss in their groups. 

 

For the second part of the activity, groups will look at two scenarios in greater detail and identify some of the assumptions made by the model and some of its limitations. Function models are helpful because they help us make sense of a scenario but they also often include some kind of simplification, for example the models we will look at in this course consider the effect of only one independent variable, when there are most likely several others at play. Models also often assume that change happens according to some predictable pattern, when in reality, external circumstances often disrupt our expectations. One of the goals of today’s lesson is for students to consider these ambiguities and learn to question the assumptions behind a model. Using monitoring questions will be critical to getting students to consider some of these nuances.

 

Here are a few possible pairings of scenarios that will allow students to work with a variety of function types and data representations.

A and D

B and H

F and G

E and C

H and D

E and H

A and B

A and G

C and D

F and B

F and C

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Monitoring Questions:

• How has the world record for the 5K changed over time? Is there a predictable pattern?

• What other variables might be influencing this data?

• Do you think this trend will continue?

• Is the pattern for these years the same as for later years?

• What does the weight of the bottle depend on? 

• What will happen to the time required if there are more volunteers? 

• What happens to the time needed to complete the mailing when the number of volunteers doubles?

• Why are there many dots at the same input value?

• Do you think the trend in salaries will continue for people with 30 or 40 years of experience? Why or why not?

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Formalize Later:

In math class we tend to look at scenarios that are somewhat contrived in order that it can be modeled by a function we’re trying to study. With the focus on modeling in this course, the goal is slightly different. We give real world data and ask what function type best models the scenario. Students won’t find perfectly constant third differences in the restaurant data (Scenario E) but they will find roughly constant third differences, which clue them in that a cubic function might be a good fit. Students also know, however, that cubic functions increase or decrease without bound, which is unlikely to happen in a real world scenario. Selecting a function model also requires articulating the assumptions made by the model, acknowledging its limitations, and determining for which values the function is actually valid. 

 

By the end of this lesson, students should be able to use a variety of strategies to determine a function type that can model a scenario. Contextual clues include inversely proportional relationships, geometric relationships such as area and volume, and scenarios that have different rules/structures for different values of the domain (think pricing structures). Numerical clues include constant first, second, third, …, nth differences. Graphical clues include shape and symmetry.