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.

If students are having trouble labeling their angles, consider saying:

• “Tell me more about the relationship between and and where they are on the circle.”

• “How could you use the location of to help you draw and label the other angles related to ?”

During the discussion, display a unit circle with all 24 angles marked off, like the one in the Student Response. Begin by selecting previously identified students to share their strategies for marking off angles from the 4 that were given.

Next, ask questions to highlight the structure of the angles in the unit circle:

• “What do you notice about the angle measures in the different quadrants?” (The angles in Quadrants I and II are all less than , the angles in Quadrants III and IV are all greater than . All of the angles are apart.)
• “Which angle measures differ by radians? What do you notice about them?” (These are angles whose rays are opposite one another and that combine to make a line through the center, like and . This is because the length of the arc is units, half the circle, and radians is the angle measurement of a half circle.)
• "What exact coordinate values can you identify on the circle?" (, , , )
  Mark and label the points as students identify them.

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).

The goal of this discussion is for students to highlight important patterns in the coordinates of the 24 points around the unit circle. Some of these patterns include:

• For angles that differ by radians, the coordinates have the same magnitude but opposite signs. For example, a -radian angle has coordinates that are approximately , while a -radian angle has coordinates that are approximately .
• Complementary angles in the first quadrant have reversed - and -coordinates. For example, a -radian angle has coordinates that are approximately while a -radian angle has coordinates that are approximately .
• Supplementary angles have the same -coordinate, but the -coordinates have different signs. For example, a -radian angle has coordinates that are approximately and a -radian angle has coordinates that are approximately .
• The coordinates of the angles in the first quadrant are repeated, with different signs and a different order, in all four quadrants.