Constructing a Function Model (Lesson 3.5)
Unit 3 Day 8
CED Topic(s): 1.14
Learning Targets

Construct a function model based on the constraints of a mathematical or contextual scenario.

Construct a function model using transformations from a parent function.

Use rational functions to model quantities that are inversely proportional.

Apply a function model to answer questions about a data set or contextual scenario.
Quick Lesson Plan
Additional Materials:
Lesson 3.5 Data: PDF
Experience First:
Today we unpack one of the scenarios presented in yesterday’s lesson. Students will construct a quadratic model for the average difference in life expectancy between females and males. After introducing the context of the lesson, hand out one copy of the data for each group, found under the additional materials. Students will use the graph to determine the function type and look for key features, but the data is necessary so students can get specific values of the average difference in life expectancy for the years between 1940 and 2014.
On the first page of the activity, students use an algebraic approach to finding the parameters of the quadratic function. Understanding that the value of c represents the yintercept (in this case the average difference in life expectancy in the year 1940), students are now left to solve a system of linear equations for the values of a and b. This is a nice opportunity for students to review elimination or substitution, but remember this is not the main goal of the lesson. If students choose to use technology to solve for a and b, we think this is completely fine. In the debrief, point out that sometimes parameters can be found by hand and sometimes technology is necessary. You might even have two groups who used different approaches both share their answers. Be careful that you do not focus on the values for a and b as much as the method for finding a and b. It is likely that every group will have slightly different values because their chosen data points will be different. This is a key takeaway in mathematical modeling. There is not necessarily one correct model (unless the model is deterministic).
On the second page, students use a transformations approach to find a function model for the same data. This works especially well for quadratic functions where students can identify the maximum or minimum value. As you are monitoring students, look for different strategies students might use to solve for the vertical stretch.
One of the key takeaways from the lesson is that models are helpful in predicting other outputs. Make a big deal about how their model was wrong for 1983 and get them to see that this is not due to a calculation error but the modeling process itself. This is new thinking for students! Students are quick to think that “errors” signal that they made a mistake somewhere. You could even have one group member check in with other groups only to see that every group got it “wrong”!
Monitoring Questions:

What other data point did you choose? Why did you choose that one? Does it matter which one we choose?

Is the “a” value going to be positive or negative? How do you know?

Why do your two function models give differing predictions for 1983?

Which function model did you think was easier to find?

How do the points you choose affect the values for which the model works well?

Do you think it’s good to pick values close to each other or further away from each other?

Your prediction for 1983 was wrong! Maybe there’s a mistake in your work somewhere…

Are there any years for which your model will have no error?
Formalize Later:
Constructing function models is something we will come back to time and time again in this course. This is not a skill students master in a day (or even an entire course). The goal is for students to understand the thought process that is behind selecting and constructing a function model. The first three units of this course have been all about exploring how two quantities change and culminate in determining how to capture that change with a function.
Note that there are 3 different types of “models” students will be asked to construct in this course.

Deterministic model: the question gives all the information needed and there is one correct answer. (Examples: The number of bacteria starts at 7 and triples every week. Write an equation for a function that has shifted three units to the left from the parent function.)

Matching criteria: the question asks students to generate a function that satisfies certain criteria and there are multiple right answers (Ex: Write a function that has a horizontal asymptote at y=2)

Bivariate data sets: the question asks students to identify the function that best fits the data based on how the quantities change with respect to each other. (Ex: Data is given in a scatterplot).
Question 1 of the Check Your Understanding falls into the second category (matching criteria), question 2 falls into the first category (deterministic), and question 3 falls into the third category (bivariate data set), as does the Activity in this lesson.