The purpose of this Warm-up is to prepare students for the following activities in which they will mark and label angles and coordinates on the unit circle. While students may notice and wonder many things about the image, the relationships between the angles are the important discussion points.

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. 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 identifying the specific angle measurements for some of the other indicated angles does not come up during the conversation, ask students to consider what the measurements would be. Remind students that on a unit circle, angles are measured from the starting ray of the positive -axis, going counterclockwise. After a brief quiet think time, invite students to suggest the measurements of three to four other angles. Record these measurements on the image for all to see. It is not necessary for students to simplify the radian measurements at this time, and not doing so helps to highlight some of the structure of the unit circle when counting by .

The purpose of this activity is for students to explore the structure of radian angles in the unit circle. Students may notice patterns, including rotational or reflective symmetry, while drawing line segments at repeated intervals.

The unit circle in this activity highlights 24 different angles in increments of . Often, a unit circle is depicted with only 3 angles between the axes, for a total of 18 angles (and their associated points, which is the focus of the next activity). For example, Quadrant I would display coordinates and angle measurements for only radian, radian, and radians between 0 radians and radians. By using increments of , students can count by , , , , , or  as they move around the unit circle, and they have more opportunity to make sense of the structure of the unit circle. Additionally, when students graph cosine and sine in a future lesson, using 24 points creates a smoother curve than using only 18 points.

Many strategies are available for drawing the angles efficiently. Monitor for students who use strategies like the ones listed here:

• Continue each ray from the origin to the unit circle in the opposite direction, making another one of the 24 rays.
• Use tracing paper to mark a -radian angle, and use it around the circle.
• Use the edge of an index card or another right angle to “rotate” an angle in one quadrant into an adjacent quadrant.
• Fold the paper to identify the angles halfway between the given right angles, such as  or .
• Fold the paper, for example, over the line to use radian to find radians.

An optional blackline master of the unit circle has been included. Consider using this to make it easier for students to use folding to identify other angles and as something students can keep and refer back to throughout the unit. Provide access to tracing paper, index cards, and straightedges throughout the activity.

The purpose of this activity is to investigate the repeated structure in the coordinates of points on the unit circle (MP8). While not explicitly prompted to look for patterns until after labeling points on the unit circle, students will likely notice the same numbers occurring as they work, sometimes in a different order and sometimes with different signs. They may also be motivated to use structure to simplify their work taking so many measurements.

Watch for the different methods that students use for estimating or calculating the coordinates of the different points, including:

• Using the grid and estimating (or using a ruler).
• Using trigonometry and a calculator.
• Observing and using structure in the coordinates (for example, the - and -values for a -radian angle are the same as those of a -radian angle, but in the opposite order).