This Warm-up prompts students to compare four clock faces. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another, such as how students describe circular motion or the directionality of the clock hands. The context of a clockface is used throughout this unit, so this Warm-up is an opportunity for students to start building familiarity with the context in preparation to solve problems (MP1).

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure that the reasons given are correct.

During the discussion, prompt students to explain the meaning of any terminology they use, such as "angle," "arc," or "arc length," and to clarify their reasoning as needed. Consider asking:

• “How do you know ?”
• “What do you mean by ?”
• “Can you say that in another way?

The goal of this activity is to get students thinking about an input-output relationship whose outputs repeat at regular intervals. This leads to defining “period.” Clocks provide a familiar repeating context for students to reason about. Several relationships between time, the angles created by clock hands, and the height of the end of a clock hand will be explored in this unit.

Monitor for students who use clear language to describe the repetition in the height of the ladybug. Students will refine their language regarding periodic functions throughout the unit, so here their language can be less formal and more context-based. For example, in reflecting on the differences and similarities between the motion of the second hand and the motion of the minute hand, students can use the context to reason about period and amplitude informally without using those terms.

The goal of this activity is for students to make connections between points on a circle, the coordinates of those points if the circle is centered at the origin, and how right triangles and the Pythagorean Theorem can help determine unknown information. 

The idea that we can choose to overlay the coordinate plane to reason about the location of a point on a circle is part of reasoning abstractly and quantitatively (MP2). By adding the familiar structure, we can use a greater number of mathematical tools to determine information about a situation.

Monitor for students who have clear explanations about why point could be in two different places on the circle, particularly students who have created a sketch to help them think about the situation.