Lesson 4

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). 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.

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

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. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

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 \(x\)-axis going counterclockwise. After a brief quiet think time, invite students to suggest the measurements of 3–4 other angles, recording 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 \(\frac{\pi}{6}\).

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

The unit circle in this activity highlights 24 different angles in increments of \(\frac{\pi}{12}\). 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 1 would only display coordinates and angle measurements for \(\frac{\pi}{6}\) radians, \(\frac{\pi}{4}\) radians, and \(\frac{\pi}{3}\) radians between 0 radians and \(\frac{\pi}{2}\) radians. By using increments of \(\frac{\pi}{12}\), students can count by \(\frac{\pi}{12}\), \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), \(\frac{\pi}{2}\), or \(\pi\) 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 to share during the whole-class discussion:

• 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 \(\frac{\pi}{12}\) 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 \(\frac{\pi}{4}\) or \(\frac{3\pi}{4}\).
• Fold the paper, for example, over the line \(y = x\) to use \(\frac{\pi}{12}\) radians to find \(\frac{5\pi}{12}\) 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.

Display the image from the activity for all to see. Tell students that their task is to identify and label radian angle measurements on a circle centered at \((0,0)\). Each angle is measured from the positive \(x\)-axis and then rotating counterclockwise. Use two pencils to demonstrate how to think about the first given angle of \(\frac{\pi}{3}\) radians by starting with both pencils lined up at \((0,0)\) on the \(x\)-axis, and then rotating the top pencil to line up with the angle while keeping one end at the origin.

Arrange students in groups of 2–3. Give 2–5 minutes of quiet work time, then tell students to check their work and share their strategies for identifying all the angles with their group before continuing the rest of the activity.

For students confused about the units in the coordinate grid and why some are darker, explain that the gridlines are in increments of 0.1 unit. The dashed lines at -1 and 1 have been included for clarity.

Display a unit circle with all 24 angles marked off, like the one in the Student Response, for all to see during the discussion. 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 such as:

• “What do you notice about the angle measures in the different quadrants?” (The angles in quadrants 1 and 2 are all less than \(\pi\), the angles in quadrants 3 and 4 are all greater than \(\pi\).)
• “Which angle measures differ by \(\pi\) 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 \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\). This is because the length of the arc is \(\pi\) units, half the circle, since \(\pi\) radians is the angle measurement of a half circle.)

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 different methods 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 \(x\)- and \(y\)-values for a \(\frac{\pi}{6}\) radians angle are the same as those of a \(\frac{\pi}{3}\) radians angle, but in the opposite order)

Arrange students in groups of 4. Assign one person in each group the angles in the right half of the unit circle, one person the left half, one person the upper half, and one person the lower half. Encourage students to either make a table of values, with columns for angle, horizontal coordinate, and vertical coordinate, or to add coordinates onto the unit circle from the previously completed activity “Angles Everywhere.”

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 which differ by \(\pi\) radians, the coordinates have the same magnitude but opposite signs. For example, a \(\frac{\pi}{12}\) radians angle has coordinates approximately \((0.97,0.26)\), while a \(\frac{13\pi}{12}\) radians angle has coordinates approximately \((\text-0.97,\text-0.26)\).
• Complementary angles in the first quadrant have reversed \(x\)- and \(y\)-coordinates. For example, a \(\frac{\pi}{6}\) radians angle has coordinates approximately \((0.87,0.50)\) while a \(\frac{\pi}{3}\) radians angle has coordinates approximately \((0.50,0.87)\).
• Supplementary angles have the same \(y\)-coordinate but the \(x\)-coordinates have different signs. For example, a \(\frac{\pi}{6}\) radians angle has coordinates approximately \((0.87,0.5)\) and a \(\frac{5\pi}{6}\) radians angle has coordinates approximately \((\text-0.87, 0.5)\).
• The coordinates of the angles in the first quadrant are repeated, with different signs and a different order, in all four quadrants.