In this activity students are presented with a series of questions similar to “A table is 60 inches wide. Do you know its width in feet?” For some of the questions, the answer is yes (because we can convert from inches to feet by dividing by 12). In other cases, the answer is no (for example, “A person is 14 years old. Do you know their height?”). The purpose is to develop students’ understanding of the structure of a function as something that has one and only one output for each allowable input. In cases where the answer is yes, students draw an input-output diagram with the rule in the box. In cases where the answer is no, they give examples of an input with two or more outputs. In the Activity Synthesis, the word “function” is introduced to students for the first time.

The goal of this discussion is for students to understand that functions are rules that have one distinct output for each input.

Direct students’ attention to the reference created using Collect and Display. Ask previously selected students to share the rules they identified for questions they answered yes to. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. For example, the display may already have a rule for finding the area of the square from its perimeter as “Find the side length, then square it,” and next to it can be added “Divide by four, then square it.” For the final two problems where the input is not a specific value, if no student used a letter to stand in for the input, invite them to consider how to do so now, and record their thinking on the display. For example, next to “ of a number” could go  “.”

Tell students that each time they answered one of the questions with a yes, the sentence defined a function, and that one way to represent a function is by writing a rule to define the relationship between the input and the output. Functions are special types of rules where each input has only one possible output. Functions are useful since once we know an input, we can find the single output that goes with it. Contrast this with something like a rolling a number cube where the input “roll” has many possible outputs. For the questions students responded to with no, these are not functions because there is no single output for each input.

To highlight how rules that are not functions do not determine outputs in a unique way, end the discussion by asking:

• “Was the problem in the Warm-up where you had to square numbers an example of a function?” (Yes, each input has only one output, even though some inputs have the same output.)
  “Is the reverse—that is, knowing what number was squared to get a specific number—a function?” (No, if a number squared is 16, we don't know if the number was 4 or -4.)

In this activity students revisit the questions in the previous activity and start using the language of functions to describe the way one quantity depends on another. For the “yes” questions, students write a statement like “[The output] depends on [the input]” or “[The output] is a function of [the input].” For the “no” questions, they write a statement like “[The output] does not depend on [the input].” Students will use this more precise language throughout the rest of the unit and course when describing functions (MP6).

Depending on the time available and students’ needs, only a subset of the questions, such as just the odds, may be assigned.

The goal of this discussion is for students to use the language like “[The output] depends on [the input]” and “[The output] is a function of [the input]” while recognizing that a “function” means each input gives exactly one output.

Begin the discussion by asking students if any of them had a different response from their partner that they were not able to reach agreement on. If any groups say yes, ask both partners to share their responses. Next, select groups to briefly share their responses for the other questions, and address any questions. For example, students may have a correct answer but be unsure since they used different wording than the person who shared their answer verbally with the class.

If time permits, give groups 1–2 minutes to invent a new question like the ones in the task that is not a function. Select 2–3 groups to share their question, and ask a different group to explain why it is not a function using language like “[The input] does not determine [the output] because . . . .”

The activity calls back to a previous lesson in which students filled out tables of values from input-output diagrams. Here, students determine if a rule is describing the same function but with different words, giving them an opportunity to look for and make use of the structure of a function (MP7).

Students are given three different input-output diagrams and need to determine which rules could describe the same function. A key point in this activity is that context plays an important role. For example, if the first rule is limited to positive inputs and the second rule is about sides of squares (which also has only positive inputs), then the two input-output rules describe the same function.

The goal of this discussion is for students to explain how two different rules can describe the same function and that two functions are the same if and only if all of their input-output pairs are the same.

Consider asking some of the following questions:

• “Do the latter two input-output rules describe the same function since they both take an input of 10 to an output of 100?” (No, every input-output pair needs to match in order for the two rules to describe the same function, not just a few input-output pairs.)
• “Do any of the input-output rules describe the same function?” (Yes, if the first is restricted to positive inputs and the second is about areas of squares, then they share the same input-output pairs and describe the same function.)