If students do not yet use accurate terminology—for example, if they mix up the hour and minute hands of the clock, consider saying:

• “Tell me more about what you mean by .”

• “If image B shows 1:30, what does each hand of the clock represent? What time does image A show?”

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 purpose of this discussion is for students to share their observations about the height of a ladybug on the clock hands over time and to introduce students to the idea of period and periodic functions.

Direct students' attention to the reference created using Collect and Display. Ask students to share their descriptions of how the height of the ladybug changes over time. Invite students to borrow language from the display as needed, and update the reference to include additional phrases as they respond. (We knew that the height would be the same every minute, so the pattern would repeat. The height matched going down and then going up, so it was symmetric.)

Tell students that the type of motion represented by the height of the ladybug on the end of the clock hand over time can be described as periodic. An important feature of periodic motion is the period, which is the length of the interval in which the motion repeats. In this situation, we say that the second hand has a period of 60 seconds. Movement around and around a circle is one type of periodic motion, and we’ll explore more types throughout the unit.

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.

If students have trouble getting started, consider asking:

• “Can you explain how you calculated the -coordinate of the point for the clock at the start of this activity.”

• “What could you draw on the first circle to help you visualize the length of the radius?”

The purpose of this discussion is for students to share how they calculated the unknown values. Highlight students who drew in right triangles as a strategy. If time allows, pair partners into groups of 4 to first share strategies with each other before selecting students to share their responses, including any visuals made, with the class.

For the last question, an important takeaway for students is that without more information, point could be in one of two places on the circle since there are two quadrants where the -value is positive. This repeating feature of coordinates on a circle is one that students will work with more in the future and connects to the periodic nature of trigonometric functions.