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Functions and Function Notation (Lesson 1.1)

Unit 1 - Day 1

​Learning Objectives​
  • Understand that functions represent situations where one quantity determines another and each input has exactly one output.

  • Evaluate functions in function notation to find outputs for provided inputs and solve equations to find an input that yields a given output

  • Analyze functions in multiple representations

  • Interpret statements that use function notation in a given context

Quick Lesson Plan
Activity: How Does the Turtle Cross the Sand



Lesson Handout

Answer Key

Experience First

Before you teach the lesson, make your own class code for the Turtle Crossing activity on Desmos. Students will be following along with this activity in the experience portion of the lesson. We invite students to play with the animation on the first slide to begin exploring how their distance vs. time graph affects the path of the turtle. We want to ignite students’ curiosity before offering the more formalized idea of functions and their properties. 

Question 3 asks students to informally consider the difference between evaluating a function (finding an output for a given input) and solving a function (finding the input that yields a given input). Note that part c) is not possible because 12 is outside of the range of the function.

Question 4 gets at the idea that each input can have only one output, but students should have contextual language to describe why this is the case. Ask students if they can come up with a situation where each input has more than one output.

The last question plants the seed for one-to-one functions, an idea that students need to grasp in order to understand later lessons on inverse functions (Lesson 1.7).

Formalize Later

Make sure that students get opportunities to analyze functions given in multiple representations. We want students to build deep connections between algebraic representations of a function (given by equations) and graphical representations. Furthermore students should be able to determine if a relation is a function from a table, set of ordered pairs, graph, or verbal context. 

Use a turn-and-talk to have students explain why the vertical line test works. Many students have heard of this test and can apply its basic principles, but struggle to connect it to the definition of a function, namely that if a vertical line passes through a graph more than once, that there must be two points “stacked” on top of each other, which implies that a single input has more than one output.

It is critical that students understand how to evaluate a function at a variety of inputs. Evaluating f(x+h) is crucial for later work in AP Calculus on difference quotients.

Interpreting statements like in question 3 of the Check Your Understanding is a great way to assess student understanding of function notation, specifically where the inputs and outputs are located. It is valuable to use statements with different variable names that makes students think critically about if the variable represents an input or an output (see question 3c).

Although this lesson surfaces the ideas of domain and range, tomorrow’s lesson will delve into these topics at a much deeper level.

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