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.

This Warm-up prompts students to make sense of a problem before solving it, by familiarizing themselves with a context and the mathematics that might be involved (MP1).

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

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 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.

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).