The purpose of this Warm-up is to elicit the idea that graphs can be discrete or continuous, which will be useful when students interpret exponential functions and make sense of whether a graph should be discrete or continuous in the associated Algebra 1 lesson. While students may notice and wonder many things about these graphs, the fact that one is a line and one is a discrete set of points are the important discussion points.

When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the graphs. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the situation being represented does not come up during the conversation, ask students to discuss this idea. What are some situations that might be appropriate to represent with just points, versus a connected line? (Students will see many examples in this lesson, so it’s not necessary to dwell on this question for long.)

The purpose of this activity is to elicit the idea that graphs can be discrete or continuous, depending on the context, which will be useful when students interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson. When students pay close attention to the appropriate domain and range, both restricted by the context, they are engaging in work that is important in modeling with mathematics (MP4). Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Given a description of a context, students determine the restrictions on the domain and range imposed by the context. Considering how a context restricts the domain or range of a function modeling it is an important part of mathematical modeling (MP4). This prepares students to interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson.