Lesson 4

The Unit Circle (Part 2)

4.1: Notice and Wonder: Angles Around the Unit Circle (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

What do you notice? What do you wonder?

A circle on a coordinate plane, center at the origin, radius 1.Points are scattered in irregular intervals around the circle.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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 \(\frac{\pi}{6}\).

4.2: Angles Everywhere (15 minutes)

Activity

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.

Launch

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.

Student Facing

Here is a circle of radius 1 with some radii drawn.

A circle with center at the origin of an x y plane.
  1. Draw and label angles, with the positive \(x\)-axis as the starting ray for each angle, measuring \(\frac{\pi}{12},\frac{\pi}{6},\frac{\pi}{4} \ldots, 2\pi\) in the counterclockwise direction. Four of these angles, one in each quadrant, have been drawn for you. There should be a total of 24 angles labeled when you are finished, including those that line up with the axes. Be prepared to share any strategies you used to make the angles.
  2. Label the points, where the rays meet the unit circle, for which you know the exact coordinate values.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

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.

Activity Synthesis

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.)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. At the appropriate time, give groups 2–3 minutes to plan what they will say when they share their strategies for marking off angles. Encourage students to consider what details are important to share, and to think about how they will explain their reasoning using mathematical language. 
Design Principle(s): Support sense-making; Maximize meta-awareness

4.3: Angle Coordinates Galore (15 minutes)

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 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)

Launch

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.”

Action and Expression: Internalize Executive Functions. Provide students with a three-column table to organize. Use the column headings: angle measure, horizontal coordinate, and vertical coordinate. The table will provide visual support for students to organize their information.
Supports accessibility for: Language; Organization

Student Facing

Your teacher will assign you a section of the unit circle.

  1. Find and label the coordinates of the points assigned to you where the angles intersect the circle.
  2. Compare and share your values with your group.
  3. What relationships or patterns do you notice in the coordinates? Be prepared to share what you notice with the class.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Other than \((1,0)\), \((0,1)\), \((\text-1,0)\), and \((0,\text-1)\), the coordinates we used in this activity involved approximations. The point \((0.8,0.6)\), however, lies exactly on our unit circle.

  1. Explain why this must be true.
  2. Find all other points on the unit circle that also lie exactly at the intersection of two grid lines.
  3. What are the approximate angle measures needed to intersect at \((0.8, 0.6)\) and each of these new points?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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.
Conversing: MLR2 Collect and Display. During the whole-class discussion, listen for and collect the language students use to discuss the relationships or patterns in the coordinates of the unit circle. Write the students’ words and phrases on a visual display. Be sure to emphasize words and phrases such as “reversed \(x\)- and \(y\)-coordinates,” “different signs,” or “complementary angles.” Remind students to borrow language from the display as needed. This will provide students with a resource to draw language from during small-group and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making 

Lesson Synthesis

Lesson Synthesis

Arrange students in groups of 4. Tell students to create a visual display of the unit circle that includes any angles that are multiples of \(\frac{\pi}{12}\) and the \((x,y)\) coordinates associated with each angle. These displays will be helpful throughout the rest of the unit as students refer back to them. The challenge making this display is identifying the angles in radians since most protractors are in degrees. Students have several options to approximate the radian measurements for their displays:

  • Fold a paper circle along its diagonals to identify \(\frac{\pi}{4}\) radians and its match in the other 3 quadrants.
  • Fold a sheet of paper in order to trisect the right angle of a corner, which can then be used to mark multiples of \(\frac{\pi}{6}\) radians in each quadrant.
  • Use the unit circle created in the activity “Angles Everywhere” as a guide and transfer them to the visual display.

4.4: Cool-down - What are the Circle Coordinates? (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Given any point in a quadrant on the unit circle and its associated angle, like \(R\) shown here, we can make some statements about other points that must also be on the unit circle.

A circle with center at the origin of an x y plane.

For example, if the coordinates of \(R\) are \((\text-0.87,0.5)\) and \(a\) is \(\frac{5\pi}{6}\) radians, then there is a point \(S\) in quadrant 1 with coordinates \((0.87,0.5)\). Since \(R\) is \(\frac{\pi}{6}\) radians from a half circle, the angle associated with point \(S\) must be \(\frac{\pi}{6}\) radians. Similarly, there is a point \(T\) at \((\text-0.87,\text-0.5)\) with an angle \(\frac{\pi}{6}\) radians greater than a half circle. This means point \(T\) is at angle \(\frac{7\pi}{6}\) radians, since \(\pi+\frac{\pi}{6}=\frac{7\pi}{6}\).

What is the matching point to \(R\) in quadrant 4? (A point at \((0.87,\text-0.5)\) and angle \(\frac{11\pi}{6}\) radians.)

In future lessons, we’ll learn about how to find the coordinates of point \(R\) ourselves using its angle \(a\) and what we know about right triangles.