Lesson 3

The Unit Circle (Part 1)

3.1: Finding Coordinates of Points on the Unit Circle (10 minutes)

Warm-up

The goal of this warm-up is to begin introducing students to the unit circle and some of its features, such as the symmetry inherent in the \(x\)- and \(y\)-coordinates of its points. In the following activity, students will have a more formal introduction to the unit circle and see how points on it can be defined using only their angle of rotation. Later, students will combine the ideas of angle and \((x,y)\) coordinates as they define cosine and sine for radian values beyond quadrant 1. This work also builds toward the Pythagorean Identity.

Monitor for these approaches to the problems:

  • drawing vertical or horizontal lines (with the given \(x\)- or \(y\)-coordinate values) and estimating where they intersect the circle using the grid
  • using the Pythagorean Theorem to calculate the other part of the coordinate

Launch

Tell students to close their books or devices (or to keep them closed). Remind students that in the previous lesson, we learned to use right triangles to explore the \((x,y)\) coordinates of points in quadrant 1. In particular, we looked at points 1 unit away from the origin because those points can be scaled as needed to fit different situations (like different sizes of clocks).

Arrange students in groups of 2. Ask students to imagine or sketch what the group of points 1 unit away from the origin looks like. Give students brief quiet work time and then time to share their work with a partner. Invite groups to share what they think this situation looks like along with their reasoning and then display the image from the task for all to see.

Tell students that a circle centered at the origin with radius 1 is called the unit circle and that it has several features we’re going to explore over the next few lessons.

Student Facing

A circle with center at the origin of an x y plane.
  1. The \(x\)-coordinate of a point on the unit circle is \(\frac{3}{5}\). What does this tell you about where the point might lie on the unit circle? Find any possible \(y\)-coordinates of the point and plot them on the unit circle.
  2. The \(y\)-coordinate of a point on the unit circle is \(\text-0.4\). What does this tell you about where the point might lie on the unit circle? Find any possible \(x\)-coordinates of the point and plot them on the unit circle.

Student Response

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

Anticipated Misconceptions

If students are not sure where to begin, invite them to label specific values on the axis, such as 0.5, to help them make sense of the image. Encourage students to use their work from earlier lessons where they drew in the right triangles and labeled side lengths to identify coordinates of points on the circle.

Activity Synthesis

Begin the discussion by selecting students to explain why there are two possible values of the \(x\)-coordinate for each given \(y\) value and two possible \(y\)-coordinate values for each given \(x\) value. This can be explained using strictly the coordinate plane or by using right triangles. Specifically, students may

  • draw the appropriate vertical or horizontal line and see where it meets the unit circle to estimate the other coordinate
  • draw the two possible right triangles inside the unit circle and use the Pythagorean Theorem to algebraically determine the other part of the coordinate

For students who take the second approach, highlight that, for the first problem, the coordinates of one of the points on the unit circle are \(\left(\frac{3}{5}, \text-\frac{4}{5}\right)\). The Pythagorean Theorem applies to the side lengths which are \(\frac{3}{5}\) and \(\frac{4}{5}\) and it says \(\left(\frac{3}{5}\right)^2 + \left(\frac{4}{5}\right)^2 = 1\). Draw in a right triangle with labeled side lengths to help students visualize how the triangle is oriented inside the circle. If possible, use patty paper for the triangle and then move the patty paper to show how the coordinates \(\left(\frac{3}{5}, \text-\frac{4}{5}\right)\) must also be on the circle since \(\left(\frac{3}{5}\right)^2+ \left(\text-\frac{4}{5}\right)^2 = 1\).

This works not just for the point \(\left(\frac{3}{5}, \text-\frac{4}{5}\right)\), but for any point on the unit circle, because the sign of either coordinate becomes positive when that coordinate is squared.

3.2: Which Point? (10 minutes)

Activity

The purpose of this activity is to continue to build students’ understanding of the unit circle by focusing on a different way to define a point on the circle than a coordinate pair by using just an angle. This activity is meant to build on the previous activity and help students establish language for talking about points on the unit circle which they will use throughout the unit (MP6).

Launch

Display the image for all to see. Tell students to each pick a point and be prepared to identify its location using words. Challenge them to use as few words as possible.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Invite students to pick a point with a partner and discuss how to describe its location. Display sentence frames to support students when they explain their strategy. For example, “It looks like. . .”, “I noticed . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

All points are 1 unit from the origin.

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

Choose one of the points. Be prepared to describe its location using only words.

Student Response

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

Activity Synthesis

Select 3–5 students to identify their selected point using only words for the rest of the class. After each explanation and brief quiet think time, invite other students to silently guess the point. Next, ask the original student to circle or identify their point in some physical way on the displayed image so the rest of the class can check their guess. Finish each round by taking an informal show of hands to find how successful the explanation was at determining the point.

Tell students that for any point on a unit circle, we can define it using just one feature: an angle. Specifically, an angle that starts at the positive \(x\)-axis and rotates counterclockwise. On the displayed image, add a segment from the origin to a point in quadrant 2, labeling the point as \(P\). Mark the angle created, as seen in this image, as \(\theta\).

Points on a unit circle.

Often, a point on the unit circle is described simply as “a point \(P\) at angle \(\theta\)” since there is only one point on the circle that could meet that description. If time allows, tell students that if we had picked a point in quadrant 3 or 4, it would have a greater angle of rotation while a point in quadrant 1 would have a smaller angle of rotation.

Students will continue to work with the unit circle and points on the circle closely for the next several lessons, so they do not need to be fluent with the vocabulary of unit circles at this time. The remainder of this lesson will focus on \(\theta\) and how to think about the angle of rotation for a point on the unit circle using radians.

3.3: Measuring Circles (15 minutes)

Optional activity

This activity is optional.

The goal of this activity is to measure circles using the radius as a unit of measure. In grade 7, students measure the circumference and diameter of different-sized circles and observe that the pairs of measurements appear to be proportional. In this task, students measure the circumference of different-sized circles using the radius of their circle as a unit of measurement. This allows students to observe that the number does not appear to depend on the size of the circle and recall that this number is \(2\pi\) since \(\pi\) is the constant of proportionality relating the circumference and diameter of a circle.

Monitor for students who have clear explanations for the exact number of radii needed to fit around the circle, such as by reasoning about an equation for the circumference of a circle.

Launch

Arrange students in groups of 4. Provide each student in a group with a different circle to measure.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with their partners, display sentence frames to support conversation such as: “First, I _____ because . . .”, “It looks like. . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills

Student Facing

  1. Your teacher will give you a circular object.
    1. About how many radii does it take to go halfway around the circle?
    2. About how many radii does it take to go all the way around the circle?
    3. Compare your answers to the previous two questions with your partners.
  2. What is the exact number of radii that fit around the circumference of the circle? Explain how you know.
  3. Why doesn’t the number of radii that fit around the circumference of a circle depend on the radius of the circle? Explain how you know.

Student Response

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

Activity Synthesis

The purpose of this discussion is to ensure that all students understand why the number of radii that fit around the circumference of a circle does not depend on the size of the radius of the circle. Begin the discussion by inviting students to share how many radii it took them to go all the way around a circle, recording responses for all to see.

Invite previously identified students to share their reasoning about the exact number of radii that fit around the circumference of a circle. If not mentioned, make sure students recall that the circumference of a circle is proportional to the diameter, \(d\), and that the constant of proportionality is \(\pi\), which we can see in the equation \(C = 2\pi r\) or \(C=\pi d\).

Next, select students to share their explanation for why the number of radii that fit around the circumference of a circle doesn’t depend on the radius of the circle. Record student explanations and any diagrams used for all to see.

Conclude the discussion by displaying the applet for all to see.

Use the slider to first show the arc of the circle equal to 1 radius length. This recalls the definition of a radian from geometry: the angle made by wrapping a length of one radius around an arc of a circle of radius \(r\). Continue with the slider to show how the entire circumference is a bit more than 6 radius-lengths since \(2\pi\) is about 6.2.

Alternatively, display this image of a circle where an angle of 1 radian is marked:

Graph of unit circle centered at origin 

Use a piece of string, or other flexible material such as ribbon, to show how the arc of the circle intersected by the angle has length equal to the radius.

Conversing, Representing: MLR8 Discussion Supports. Use this routine to amplify mathematical uses of language to explain why the number of radii that fit around the circumference of a circle does not depend on the size of the radius of the circle. For each response that is shared, ask students to restate what they heard using precise mathematical language. If time allows, ask students to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement such as proportional, scaled, and constant of proportionality. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

3.4: Around a Bike Wheel (15 minutes)

Activity

The purpose of this activity is to help students recall radian measurement of angles, focusing on some key understandings, such as the relationship between arc length, 1 radian, and the radius of a circle.

As in the previous activities of this lesson, the circle is a unit circle, but now students are asked to think in feet, which can feel more like a “real” measurement to students than “1 unit,” and connects to how odometers on cars and bikes work by measuring the number of revolutions of the wheel and relating this to the distance traveled. Another way to say this is that the odometer keeps track of the angle the wheel has rotated: when this angle is measured in radians and the radius of the wheel is 1 foot, as it is here, the angle of rotation in radians is the distance traveled in feet.

Launch

Arrange students in groups of 2. Tell students that they are now going to consider the relationship between the angle of rotation for a point on a unit circle and the arc length made by a point rotating through the angle.

Ask students to read the opening statement and complete the first 2 questions individually. After quiet work time, ask students to compare their responses to their partner’s. Follow with a whole-class discussion about the location of point \(Q\). Once students are in agreement on the location of \(Q\) (possibly by reasoning about the 12 equal parts that make up the circle and the circumference of the circle), remind students that one way to measure angles is using radians. The radian measure of an angle is the ratio of arc length to radius. Ask, “What angle, in radians, does \(P\) rotate through to get to \(Q\)?” (Since the arc length from \(P\) to \(Q\) is 1 foot and the radius is 1 foot, the angle is 1 radian.) Display this definition of radian for all to see throughout the activity.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students consider the context of this problem and to increase awareness of unit circle language. Without revealing the questions that follow, display the image of the bicycle wheel and the statement, “A bicycle wheel has a 1 foot radius. The wheel rolls to the left (counterclockwise).” Ask students to write down a list of possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving the location of point \(P\) after a certain number of rolls. 
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

A bicycle wheel has a 1 foot radius. The wheel rolls to the left (counterclockwise).

A circle, subdivided by 12 congruent central angles. Point P is on the circle on the  right side of the horizontal diagonal. Radius is 1 foot.
  1. What is the circumference of this wheel?
  2. Mark the point \(Q\) where \(P\) will be after the wheel has rolled 1 foot to the left. Be prepared to explain your reasoning.
  3. Mark the point \(R\) where \(P\) will be after the wheel has rolled 3 feet to the left. What angle, in radians, does \(P\) rotate through to get to \(R\)? Explain your reasoning.
  4. Where will point \(P\) be after the bike has traveled \(\pi\) feet to the left? What about \(10 \pi\) feet? \(100 \pi\) feet? Mark these points on the circle. Explain your reasoning.
  5. After traveling some distance to the left, the point \(P\) is at the lowest location in its rotation. How far might the bike have traveled? Explain your reasoning.

Student Response

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

Student Facing

Are you ready for more?

Picture the bicycle with a bright light at point \(P\) and moving now from left to right. As the bike passes in front of you going left to right, what shape do you think the light would trace in the air?

Student Response

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

Launch

Arrange students in groups of 2. Tell students that they are now going to consider the relationship between the angle of rotation for a point on a unit circle and the arc length made by a point rotating through the angle.

Ask students to read the opening statement and complete the first 2 questions individually. After quiet work time, ask students to compare their responses to their partner’s. Follow with a whole-class discussion about the location of point \(Q\). Once students are in agreement on the location of \(Q\) (possibly by reasoning about the 12 equal parts that make up the circle and the circumference of the circle), remind students that one way to measure angles is using radians. The radian measure of an angle is the ratio of arc length to radius. Ask, “What angle, in radians, does \(P\) rotate through to get to \(Q\)?” (Since the arc length from \(P\) to \(Q\) is 1 foot and the radius is 1 foot, the angle is 1 radian.) Display this definition of radian for all to see throughout the activity.

Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students consider the context of this problem and to increase awareness of unit circle language. Without revealing the questions that follow, display the image of the bicycle wheel and the statement, “A bicycle wheel has a 1 foot radius. The wheel rolls to the left (counterclockwise).” Ask students to write down a list of possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving the location of point \(P\) after a certain number of rolls.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

A bicycle wheel has a 1 foot radius. The wheel rolls to the left (counterclockwise).

A circle, subdivided by 12 congruent central angles. Point P is on the circle on the  right side of the horizontal diagonal. Radius is 1 foot.
  1. What is the circumference of this wheel?
  2. Mark the point \(Q\) where \(P\) will be after the wheel has rolled 1 foot to the left. Be prepared to explain your reasoning.
  3. Mark the point \(R\) where \(P\) will be after the wheel has rolled 3 feet to the left. What angle, in radians, does \(P\) rotate through to get to \(R\)? Explain your reasoning.
  4. Where will point \(P\) be after the bike has traveled \(\pi\) feet to the left? What about \(10 \pi\) feet? \(100 \pi\) feet? Mark these points on the circle. Explain your reasoning.
  5. After traveling some distance to the left, the point \(P\) is at the lowest location in its rotation. How far might the bike have traveled? Explain your reasoning.

Student Response

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

Student Facing

Are you ready for more?

Picture the bicycle with a bright light at point \(P\) and moving now from left to right. As the bike passes in front of you going left to right, what shape do you think the light would trace in the air?

Student Response

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

Anticipated Misconceptions

Students may want to use approximations instead of leaving answers in terms of \(\pi\) when working with radians. There is nothing incorrect with doing so, but encourage these students to use exact values as much as possible instead of approximations. Remind them that a full rotation is \(2\pi\) radians, and we can think of all other angles in a unit circle as some fraction of that measurement. For the purposes of this unit, most angles given are an integer multiple of \(\frac{\pi}{12}\) radians.

Activity Synthesis

The purpose of this discussion is for students to share their reasoning about the points they plotted on the wheel with their classmates. Pair groups and tell students to take turns sharing their reasoning for how they plotted the new points determined by the different counterclockwise rotations of \(P\). After 2–3 minutes of discussion time, select students to share the angle they identified that brings \(P\) to its lowest location in its rotation. If not mentioned by students, highlight how rotating \(\frac34\) of the way around the circle means the radian measurement is \(\frac{3\pi}{2}\) radians (the ratio of arc length to radius is \(\frac{3\pi}{2} \div 1 = \frac{3\pi}{2}\) since the arc length is \(\frac34 \boldcdot 2\pi = \frac{3\pi}{2}\)).

If time allows, display the applet for all to see, setting the diameter to 2 (the radius to 1).

In the video, the wheel is rotating in a clockwise fashion so the angle measures are negative. There is no need at this time to stress that clockwise rotation is negative, so allow students to answer using positive values. Negative angles are a focus of a future lesson, and following lessons will stress that angles measured going counterclockwise are positive. Here are some questions for discussion:

  • “How far does the bike travel in one half revolution of the wheel?” (\(\pi\) feet)
  • “What angle is that angle of revolution?” (\(\text-\pi\) radians)
  • “How far does the bike travel in one full revolution of the wheel?” (\(2\pi\) feet)
  • “What angle of revolution is that?” (\(\text-2\pi\) radians)
  • “What is the relationship between the distance the bike travels and the angle measure in radians?” (They are the same, though the signs are different here because the wheel is moving in a clockwise direction.)

Students will continue to develop their fluency with radian measurement in the following lessons.

Lesson Synthesis

Lesson Synthesis

The goal of this synthesis is to help students build their confidence working with radian angle measurements and consider what the sign of the \(x\)- or \(y\)-coordinates tell us about the angle of a point on the unit circle.

Display a blank unit circle for all to see with a point \(P\) labeled at \((1,0)\) and mark off the radian measurements for 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) radians. Ask students to imagine the point slowly rotating counterclockwise around the unit circle. Here are some questions for discussion:

  • “How far does the point have to rotate before the \(x\)-values of \(P\) start to repeat in the same order?” (1 complete rotation of \(2\pi\) radians.)
  • “How many radians can \(P\) rotate and have a negative \(y\)-value?” (Any between \(\pi\) and \(2\pi\).)
  • “How many radians can \(P\) rotate and have a negative \(x\)-value?” (Any between \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\).)
  • “If point \(P\) rotates through \(\frac{5\pi}{4}\) radians, where is it? How do you know?” (Point \(P\) would be halfway through quadrant 3. \(\frac{5\pi}{4}\) radians is \(\frac{\pi}{4}\) radians larger than \(\pi\) and \(\frac{\pi}{4}\) radians smaller than \(\frac{3\pi}{2}\), which is the same as \(\frac{6\pi}{4}\).)

3.5: Cool-down - Radian measure (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

One way to define a circle centered at \((0,0)\) is by the equation \(x^2+y^2=r^2\), where \(r\) is the radius of the circle. A unit circle has \(r=1\), so the equation for this unit circle with center \(O\) must be \(x^2+y^2=1\). Points on the unit circle have several interesting properties, such as having matching points on opposite sides of the axes due to symmetry. Another feature of points on a unit circle is that they can be defined solely by an angle of rotation which is measured in radians.

Radians are a natural tool to use to measure the distance traveled on a circle. Let’s say that the wheels on a bike have a radius of 1 foot. When the bike starts to move to the left, rotating the wheel counterclockwise, let’s think about what happens to point \(P\).

A circle, subdivided by 12 congruent central angles. Point P is on the circle on the  right side of the horizontal diagonal. Radius is 1 foot.

The point \(P\) will return to its starting location when the wheel has rotated through an angle of \(2\pi\) radians. During this rotation, the bike will move a length equal to the circumference of the wheel, which is \(2\pi\) feet. In general, the angle of rotation of the wheel with radius 1 foot, in radians, is the same as the number of feet this bike has traveled. So what do we do when a wheel doesn’t have a radius of 1 unit? Since all circles are similar, we can use the same type of thinking but scaled up or down to match the size of the wheel, which is something we’ll do in future lessons.

Thinking about the wheel as a unit circle, as shown in this image, the arc length of the circle from \(P\) to \(Q\) has length equal to 1 unit, the radius of the unit circle. Because of this, the angle \(POQ\) is said to measure one radian. If we continue to measure off radian lengths around the circle, it takes a little more than 6 to measure the entire circumference.

Unit circle inscribed in a coordinate plane. Point 1 comma 0 is labeled P, Q is a point on the circle in the first quadrant. The minor arc P Q is colored for emphasis.

This makes sense because the ratio of the circumference to the diameter for a circle is \(\pi\), and so the circumference is \(2\pi\) times the radius, or about 6.3 radii.

Let’s think about some other angles on the unit circle. Here, angle \(POR\) measures \(\pi\) radians because its arc is \(\frac{1}{2}\) of a full circle (counterclockwise) or \(\frac{1}{2}\) of \(2\pi\). Angle \(POU\) is three quarters of a full circle (counterclockwise), so that’s \(\frac{3\pi}{2}\) radians.

Unit circle inscribed in a coordinate plane. Point P at 1 comma 0, Point R at negative 1 comma 0, point U at 0 comma negative 1. Point Q on the circle in the first quadrant.