Lesson 8

The purpose of this warm-up is to elicit the idea that some graphs have repeated behavior, which will be useful when students graph periodic motion in a later activity. While students may notice and wonder many things about the image, the repeating pattern of the outputs of the graph is the important discussion point.

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

If the repetitive behavior of the graph does not come up during the conversation, ask students to discuss this idea. If none of the students notice or wonder anything about repeating, here are some possible prompts for discussion:

• “How would you describe the graph to someone who cannot see it?”
• “Are there any parts of this graph that happen more than once?”
• “Does anyone notice anything that repeats?”

In this activity, students examine relationships shown in graphs. They consider different types of events that cause repeated outputs (MP2). They examine one situation which is periodic (the distance of a car from the start line as it goes around a racetrack) and two situations which are not periodic, but do rise and fall at regular intervals. The long-term goal in this unit is to develop mathematical tools which can help model these complex phenomena.

Monitor for students who connect the repetitive nature of these graphs to the context of the clock hands or Ferris wheel from earlier lessons.

Some students may be unsure how to read the axes for all but the first relationship, as they look different than what students have seen so far. Encourage these students to focus on a specific coordinate on a graph to help them make sense of the axis labels as a way to get started.

Begin the discussion by selecting previously identified students to share their observations about connections between the repeating nature of these graphs and the clock hands or Ferris wheel contexts from earlier lessons. If no groups make this connection, invite students to do so now—for example, by comparing the period of the race car to the period of the minute hand on a clock or by contrasting how the height of a point on a Ferris wheel always moves between the same high and low while the city high and low temperatures change from day to day.

Display the graphs for all students to see. Here are some questions for discussion:

• “What similarities do you see between the graphs representing these situations?” (There are highs and lows in all the graphs, and these highs and lows come at pretty regular intervals.)
• “Do these graphs all represent functions? How can you tell?” (Yes, in these situations for each time, there is a single value.)
• “What do you think it means for a function to be periodic?” (The output values follow a repeating pattern.)
• “What other words could you use to describe this type of behavior?” (repeating, cyclic)

Conclude the discussion by reminding students that previously, we named functions whose values repeat at regular intervals periodic functions. The graph of the race track is an example of what a periodic function can look like. The other two situations shown, temperature and population over time, are not quite periodic functions since the output values are not the same over and over again, but they have a definite periodic pattern to them.

The purpose of this activity is for students to compare different function types with a focus on periodic and non-periodic functions. The card sort allows students to compare a variety of graphs, helping students to construct their understanding of what the graphs of periodic functions can look like in preparation for future lessons that focus on the graphs of cosine and sine.

Monitor for different ways groups choose to categorize the graphs, but especially for categories that distinguish between function types.

Arrange students in groups of 2. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the graphs, they should work with their partner to come up with categories. Select 2–3 groups to share the categories they chose, then instruct students to choose new categories and sort their cards again. 

Select groups to share their categories and how they sorted their equations. You can choose as many different types of categories as time allows, but ensure that one set of categories distinguishes between function types (for example, linear, quadratic, exponential, periodic). Attend to the language that students use to describe their categories and graphs, giving them opportunities to describe their graphs more precisely. Highlight the use of terms like linear, exponential, quadratic, and periodic.