Lesson 11
A New Way to Measure Angles
11.1: A One-Unit Radius (5 minutes)
Warm-up
In this activity, students find arc lengths for common angle measurements in a circle with a radius of 1 unit. This will be helpful when arc lengths in unit circles are used as radian angle measurements in an upcoming activity.
Student Facing
A circle has radius 1 unit. Find the length of the arc defined by each of these central angles. Give your answers in terms of \(\pi\).
- 180 degrees
- 45 degrees
- 270 degrees
- 225 degrees
- 360 degrees
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
Ask students to share their strategies, especially for the 225 degree angle. Some students may create the fraction \(\frac{225}{360}\) and multiply this fraction by \(2\pi\), writing it in equivalent form with smaller numbers either before or after multiplying. Others may observe that 225 is the sum of 180 and 45, so they can add \(\pi\) and \(\frac{\pi}{4}\) to get the answer.
11.2: A Constant Ratio (15 minutes)
Activity
In this activity, students examine and complete a narrative proving that the length of the arc intercepted by a central angle is proportional to the radius of the circle. This will lead directly to the definition of radian measure in the next activity. Analyzing ratios that are invariant under dilation and giving them names is analogous to defining the trigonometric ratios of similar right triangles in a previous unit.
Launch
Design Principle(s): Support sense-making
Supports accessibility for: Conceptual processing
Student Facing
Diego and Lin are looking at 2 circles.
Diego says, “It seems like for a given central angle, the arc length is proportional to the radius. That is, the ratio \(\frac{\ell}{r}\) has the same value as the ratio \(\frac{L}{R}\) because they have the same central angle measure. Can we prove that this is true?”
Lin says, “The big circle is a dilation of the small circle. If \(k\) is the scale factor, then \(R=kr\).”
Diego says, “The arc length in the small circle is \(\ell=\frac{\theta}{360}\boldcdot 2\pi r\). In the large circle, it’s \(L=\frac{\theta}{360}\boldcdot 2\pi R\). We can rewrite that as \(L=\frac{\theta}{360}\boldcdot 2\pi rk\). So \(L=k\ell\).”
Lin says, “Okay, from here I can show that \(\frac{\ell}{r}\) and \(\frac{L}{R}\) are equivalent.”
- How does Lin know that the big circle is a dilation of the small circle?
- How does Lin know that \(R=kr\)?
- Why could Diego write \(\ell=\frac{\theta}{360}\boldcdot 2\pi r\)?
- When Diego says that \(L=k\ell\), what does that mean in words?
- Why could Diego say that \(L=k\ell\)?
- How can Lin show that \(\frac{\ell}{r}=\frac{L}{R}\)?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students are stuck trying to show that \(\frac{\ell}{r}=\frac{L}{R}\), ask if they can rewrite \(\frac{L}{R}\) by substituting in other expressions for \(L\) and \(R\).
Activity Synthesis
The goal is to ensure that students recognize that for a given angle, the length of the arc is proportional to the radius. Ask students, “What does it mean for 2 quantities to be proportional?” (It means that there is some constant multiplier between them.)
Then, display this image for all to see.
Ask:
- “What is the multiplier, or constant of proportionality, between the radius and arc length for these 90 degree central angles? That is, we’re looking for a value \(w\) such that \(\ell=wr\) and \(L=wR\).” (The value of \(w\) is \(\frac{\pi}{2}\).)
- “How does this example relate to what you proved in the activity?” (We proved that the ratio \(\frac{\ell}{r}\) is equal to the value \(\frac{L}{R}\). In this example, we showed both are equal to \(\frac{\pi}{2}\).)
11.3: Defining Radians (15 minutes)
Activity
In this activity, students learn the definition of the radian measure of an angle. They observe relationships between the radius of a circle and the length of an arc on that circle, and they determine that 180 degrees is equivalent to \(\pi\) radians.
Monitor for a variety of strategies for the last problem, including calculations based on the definition of radian measure and proportional reasoning using the equivalence of 180 degrees and \(\pi\) radians.
Launch
Ask students to list some different units for measuring lengths (miles, kilometers, light years) and volumes (gallons, liters, cubic feet). Tell students that angles also have different units of measurement, and that they will learn about a new angle measurement unit in this activity.
Remind students that they proved that for central angles in circles, arc length and radius are proportional. That is, for a given angle, the ratio of the arc length defined by the angle to the circle’s radius is a constant value. This is such a useful property that it is used as a way to measure angles.
Give students a few minutes to work, then pull the class back together to ensure all students have calculated the measure of the angle in the first part as 1 radian.
Supports accessibility for: Language; Organization
Student Facing
Suppose we have a circle that has a central angle. The radian measure of the angle is the ratio of the length of the arc defined by the angle to the circle’s radius. That is, \(\theta = \frac{\text{arc length}}{\text{radius}}\).
- The image shows a circle with radius 1 unit.
- Cut a piece of string that is the length of the radius of this circle.
- Use the string to mark an arc on the circle that is the same length as the radius.
- Draw the central angle defined by the arc.
- Use the definition of radian to calculate the radian measure of the central angle you drew.
- Draw a 180 degree central angle (a diameter) in the circle. Use your 1-unit piece of string to measure the approximate length of the arc defined by this angle.
- Calculate the radian measure of the 180 degree angle. Give your answer both in terms of \(\pi\) and as a decimal rounded to the nearest hundredth.
- Calculate the radian measure of a 360 degree angle.
Student Response
For access, consult one of our IM Certified Partners.
Student Facing
Are you ready for more?
Research where the “360” in 360 degrees comes from. Why did people choose to define a degree as \(\frac{1}{360}\) of the circumference of a circle?
Student Response
For access, consult one of our IM Certified Partners.
Anticipated Misconceptions
If students are confused by the fact that their radian measures of angles are the same as the arc lengths they’ve calculated, remind them that the radius of the circle is 1 unit, so it makes sense that when we calculate the ratio of arc length to radius for angles in these circles, the answer is the arc length divided by 1—that is, it’s simply the arc length.
Activity Synthesis
The goal is to discuss the idea that by relating arc length to a circle’s radius, the radian measure of an angle shows how many radii make up the arc defined by a central angle.
Ask students to share their strategies for the last problem. If possible, select one student who calculated that the circumference of the circle in the activity is \(2\pi\) units, and another who reasoned that 360 degrees is 2 times 180 degrees, so the radian measure of a 360 degree angle must be twice that of a 180 degree angle. Then ask students:
- “Does the word radian seem to relate to a particular part of a circle?” (The word radian sounds like radius.)
- “How is a radian connected to the radius?” (The radian measure of an angle is the ratio of the arc length to the radius, so it tells how many radii fit along the arc defined by the angle.)
- “How did your answers to the second and third questions connect?” (The string measurement showed that the arc length was about 3 units in length. The radian measure of the angle ended up being \(\pi\) or 3.14 units, which matches our approximate string measurements.)
- “What if the circle had a radius of 2 units instead of 1? How would the radian measures change?” (They wouldn’t change. The arc length to radius ratio is constant no matter the size of the circle.)
- “We defined the radian measure of an angle as \(\theta = \frac{\text{arc length}}{\text{radius}}\). How does this relate to the idea that the arc length and radius are proportional?” (We can think of the angle measure as being the constant of proportionality.)
Design Principle(s): Support sense-making
Lesson Synthesis
Lesson Synthesis
Students may be wondering why we don’t just stick with degree measurements for angles. Tell students that we often choose units depending on the situation. For example, we’re much more likely to see a water bottle labeled “1 liter” than “61 cubic inches” even though both are equivalent volume measurements. Similarly, degrees and radians come in handy for different things. Degrees are familiar, but they are also somewhat arbitrary. Radians, on the other hand, are connected to fundamental properties of a circle, so they are useful in many mathematical situations.
Display this applet for all to see.
Move the slider slowly all the way to the right so students can see the radii distances traveling along the circumference of the circle. Then, move the slider so that the angle measures 1 radian. Ask students:
- “How does this picture describe what an angle with measure 1 radian looks like?” (Sample response: The arc length is 1 unit and so is the radius. The ratio of the 2 measurements is 1. Therefore, that angle measures 1 radian.)
- “What would a 3 radian angle look like? How do you know?” (Since 180 degrees is equivalent to \(\pi\) or about 3.14 radians, a 3 radian angle would be a little less than 180 degrees.)
- “How many radians would a 90 degree angle measure? How do you know?” (Since 180 degrees is equivalent to \(\pi\) radians, 90 degrees must be equivalent to \(\frac{\pi}{2}\) radians.)
- “How does this answer fit in with this applet image?” (Based on the size of a 1 radian angle, 90 degrees would be about 1.5 radians. The value \(\frac{\pi}{2}\) is approximately equal to 1.57, so this makes sense.)
Students may wonder what the unit symbol is for radian measure. For example, the symbol \(^\circ\) denotes degrees. Tell students that because radian measure is a ratio, it is a unitless measure. If students have done dimensional analysis in other classes, point out that when calculating a radian measure such as \(\theta = \frac{6\pi\text{ cm}}{12\text{ cm}}\), the units in the numerator and denominator “cancel.”
Finally, ask students to add this definition to their reference charts as you add it to the class reference chart:
For any angle, imagine drawing a circle with the angle’s vertex at its center. Then, the radian measure of the angle is the ratio of the length of the arc defined by the angle to the circle’s radius. That is, \(\theta = \frac{\text{arc length}}{\text{radius}}\). (Definition)
11.4: Cool-down - Find a Radian Measure (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
Degrees are one way to measure the size of an angle. Radians are another way to measure angles. Assume an angle’s vertex is the center of a circle. The radian measure of the angle is the ratio between the length of the arc defined by the angle and the radius of the circle. We can write this as \(\theta=\frac{\text{arc length}}{\text{radius}}\). This ratio is constant for a given angle, no matter the size of the circle.
Consider a 180 degree central angle in a circle with radius 3 units. The arc length defined by the angle is \(3\pi\) units. The radian measure of the angle is the ratio of the arc length to the radius, which is \(\pi\) radians because \(\frac{3\pi}{3}=\pi\).
Another way to think of the radian measure of the angle is that it measures the number of radii that would make up the length of the arc defined by the angle. For example, if we draw an arc that is the same length as the radius, both the arc length and the radius are 1 unit. The radian measure of the central angle that defines this arc is the quotient of those values, or 1 radian.