Lesson 3

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

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.

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.

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.

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).

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.

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 \(\theta\).

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.

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.

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

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:

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.

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.

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.

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.