The goal of this Warm-up is to motivate the need for a notation that can be used to communicate about functions.

Students analyze three graphs from an earlier lesson, interpret various points on the graphs, and use their analyses to answer questions about the situations. This work requires students to make careful connections between points on the graphs, pairs of input and output values, and verbal descriptions of the functions. Students find that unless each feature and the function being referred to is clearly articulated, which could be tedious to do, what they wish to communicate about the function may be ambiguous or unclear.

When answering the last two questions, students are likely to find the prompts lacking in specificity and to probe: “for which day?” Suggest that they answer based on their interpretation of the questions.

Then, look for students who assume that the questions refer to one particular function and those who assume they refer to all three functions (and consequently answer them for each function). Ask them to share their interpretations during the whole-class discussion.

Invite students to share their response to the first set of questions.

To help illustrate that it could be tedious to refer to a specific part of a function fully and precisely, ask each question completely for each of the three days. (For instance, “How far away was the dog from the post 60 seconds after the owner left on Day 1? How far away was the dog from the post 60 seconds after the owner left on Day 2?”) If students offer a numerical value (for instance, “1.5 feet”) without stating what question it answers or to what quantity it corresponds to, ask them to clarify.

Next, select previously identified students to share their responses to the last two questions. Regardless of whether students chose to answer them for a particular day or for all three days, point out that the answers depend on the day. When the day (or the function) is not specified, it is unclear what information is sought.

Explain that sometimes we need to be pretty specific when talking about functions. But being specific could require many words and become burdensome. Tell students that they will learn about a way to describe functions clearly and succinctly.

In this activity, students learn that function notation can be used as a handy shorthand for communicating about functions and specific parts or features of a function. They interpret statements that are written in this notation and use the notation to refer to points on a graph or to represent simple verbal statements about a function.

This activity reinforces students’ understanding about what makes a relationship between two variables a function—namely, that it gives a unique output for each input. It also prompts students to use function notation to express a functional relationship that does not involve numerical values for its input and output.