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

If students have difficulty reading the axes, consider asking:

• “Consider the temperature graph. What do you notice about the axes?”

• “Mark a point on the temperature graph. Tell me more about that point.”

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. 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 telling students that functions whose values repeat at regular intervals are called 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.

Students sort different graphs during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and to make connections (MP2, MP7).

Monitor for the different ways in which groups choose to categorize the graphs, but especially for categories that distinguish between types of functions. 

As students work, encourage them to refine their descriptions of graphs of functions using more precise language and mathematical terms (MP6).

Select groups to share their categories and how they sorted their graphs. 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.”