Functions and Function Notation (Lesson 1.1)
Unit 1 Day 1
CED Topic(s): 1.1
Understand that a function describes the relationship between an independent variable and a dependent variable where each input value is mapped to exactly one output value. Functions can be expressed with an equation, table, graph, or verbal description.
Describe the set of inputs of a function (the domain) and the set of outputs (the range).
Use and interpret function notation.
Quick Lesson Plan
Welcome to the first EFFL (Experience First, Formalize Later) lesson of AP Precalculus! If you are new to this model, be sure to check out the following resources first!
What is Experience First, Formalize Later?
Making the Most of Your EFFL Lesson Debrief
Why Should We EFFL in AP (Pre) Calculus?
If your students are new to the EFFL model, be sure to take extra time today to explain what they can expect during the Activity portion of the lesson. You may wish to include things like:
You’re going to be working in groups of 4 (or 3) on the front page of this handout.
There is nothing that you shouldn’t be able to do on here, but I am going to ask you to think deeply and rely on your groupmates to help come up with strategies for solving these problems. If you get stuck on something, that’s OKAY. Challenging your brain is a good thing. Keep working at it and don’t give up.
The “Reader” in your group is the person who…(make up some attribute or criteria). That person will read the question out loud, then you will discuss it as a group, and only then will you pick up your pencil and write down your thinking. So the protocol is read, discuss, write.
I expect that your group will be working on this TOGETHER. I shouldn’t see one of you on question 4 while another is on question 1.
In this activity, students explore the relationship between a person’s age and their maximum heart rate (HRmax). By interpreting and using different representations of this relationship (table, graph, and equation) students reason about how a function’s independent variable (age) predicts or controls a function’s dependent variable (HRmax). In question 2, students identify and predict output values using a graph and knowledge of linear functions. In question 2c, students write an equation that models this relationship. We do not expect students to use function notation when writing their equation. This will be added on in the debrief. In questions 2d and 2e students think about a reasonable domain and range based on the context.
In question 3, students are given a function in function notation and must evaluate the function and interpret their answer. While students have much familiarity with function notation from previous grades, interpreting input-output pairs in context is more challenging for students and solidifies important function concepts such as covariation.
As students are working, hand out whiteboard markers so that groups can write up their responses on the board. You can ask specific students to write up specific questions based on what you noticed while monitoring students, or you can have each group send up one person and choose which question they want to write up.
What happens to a person’s maximum heart rate as they get older? Why do you think this is?
What’s an estimate you know is too high and too low for the 35 year old’s heart rate?
What does the graph show you about the type of relationship between age and maximum heart rate? How can you use this to write an equation for this relationship?
How did you determine what is “reasonable”?
What is the same and what is different between the function we looked at earlier and this function specifically for women?
When groups are done working through the front page, reconvene the class for the whole-class debrief. Have students switch out their pencils for red pens. Point out that as a class you’re going to add some margin notes to the activity in red. These can be considered their class “notes.”
The debrief should focus on the idea of the independent variable determining the dependent variable. Function notation is layered on in the margins and as labels on the table and graph. Be sure to point out how 2a and 2b differ in that 2a asks them to evaluate a function whereas 2b asks them to solve a function.
For questions 2d and 2e remind students that domain and range can be thought of analytically (what input values have a defined output as the domain and the corresponding outputs as the range) and contextually (what input values make sense based on the context and the corresponding outputs as the range).
When discussing the function notation in the QuickNotes be sure to emphasize that f(a) represents a y-value, it is the output value corresponding to an input value of a. Many students can easily identify “b” as the output but still think f(a) is just a naming convention for the entire function, instead of an actual output.