In this activity, students are given an equation, and then they generate a corresponding table and graph. Then they respond to some questions in which they interpret these representations in terms of the situation. The purpose of this exercise is to reinforce meaningful connections between different representations of the same situation.

When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5). However, we don’t recommend technology if the focus turns to pressing buttons to get answers, without understanding. Students also may be better equipped to attempt the analysis questions if they create the table and graph by hand. Consider these points before deciding whether it would be productive to allow use of graphing technology.

The purpose of this discussion is to make connections between the verbal description, equation, table, and graph.

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to what the 40 and -4 mean in the situation and where the numbers can be seen in the graph. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.

If time allows, display these prompts for feedback:

• “ makes sense, but what do you mean when you say ?”
• “Can you describe that another way?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.

After Stronger and Clearer Each Time, invite students to share their responses. Highlight the connections between the equation, the graph, and the quantities in the situation. In particular, show the point alongside the equation , and demonstrate on the graph how the graph shows that the rate of change in this situation is -4 errors per additional hour of sleep.

Here are some questions for discussion:

• “How can we use an equation to express the number of errors after 3.5 hours of sleep?” ( )
• “Where on the graph is the number of errors made by someone who got no sleep?” (The vertical intercept)
• “Will the graph continue to decrease indefinitely?” (No. The number of errors is limited to the number of questions on the task. Also, humans can’t sleep indefinitely.)

The goal of this discussion is to connect the work from the linear examples to the new example involving exponential decay. Invite students to share their responses and their reasoning. Record their responses to show connections between each graph and the corresponding equation. Here are some questions for discussion:

• “How did you figure out the slope of the linear graph?” (I used the coordinates and subtracted a -value from the next -value. For example, and .)
• “How did you figure out the growth factor of an exponential graph?” (I used the coordinates and divided a -value by the previous -value. For example, and .)