# Lesson 12

Radian Sense

## 12.1: Which One Doesn’t Belong: Angle Measures (5 minutes)

### Warm-up

This warm-up prompts students to compare four angle measurements. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

### Launch

Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning why a particular item does not belong, and together find at least one reason each item doesn’t belong.

### Student Facing

Which one doesn’t belong?

### Student Response

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### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as *arc length* and *radian*. Also, press students on unsubstantiated claims.

## 12.2: Degrees Versus Radians (10 minutes)

### Activity

Students use a double number line to convert between degree and radian measures. A double number line diagram includes a pair of parallel number lines marked in equal increments and numbered. The tick marks on the lines are aligned. A pair of aligned numbers on the diagram represents a ratio that is equivalent to that of every other pair of aligned numbers on the diagram. These diagrams are helpful when comparing quantities that are proportional.

### Launch

Explain to students that a double number line is a diagram that can help compare proportional quantities. Tell them that today they will use a double number line to compare degree and radian measurements.

Let students work for 2 minutes. Then, pull the class together to make sure all students have labeled 360 degrees with \(2\pi\) radians and 180 degrees with \(\pi\) radians, and that they understand how to proceed from there.

*Representation: Internalize Comprehension.*Represent the same information through different modalities by using images. Ask students what 0 degrees and 360 degrees look like, then sketch images of circles with no shading and the entire circle shaded above the matching angles. Invite students to look at the image in the warm-up with \(\frac{3\pi}{2}\) radians shaded, and locate where that angle fits on the double number line.

*Supports accessibility for: Conceptual processing; Visual-spatial processing*

### Student Facing

This double number line shows degree measurements on one line and radians on another.

- Fill in the radian measures on the bottom line for 0\(^\circ\), 90\(^\circ\), 180\(^\circ\), 270\(^\circ\), and 360\(^\circ\).
- Express each radian measurement in degrees.
- \(\frac{\pi}{3}\) radians
- \(\frac{5\pi}{4}\) radians

- Express each degree measurement in radians.
- 30\(^\circ\)
- 120\(^\circ\)

### Student Response

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### Student Facing

#### Are you ready for more?

Your boat is heading due south when you hear that you must head north-east to return home. You turn the boat, traveling in a counterclockwise circular path until you’re facing north-east.

- Sketch the path of the boat.
- If the circle you traced had a radius of 40 feet, what distance did you travel?

### Student Response

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### Activity Synthesis

The goal is to discuss how to use proportional reasoning to convert between radians and degrees. Here are some questions for discussion:

- “How many degrees is \(\frac{\pi}{12}\) radians equivalent to? How do you know?” (It is equivalent to 15 degrees. Sample response: \(\frac{\pi}{12}\) is half of \(\frac{\pi}{6}\), which is 30 degrees. Half of 30 is 15.)
- “How many radians is 1 degree equivalent to?” (It is equivalent to \(\frac{\pi}{180}\) radians. Sample response: \(2\pi\) is equivalent to 360 degrees, so divide \(2\pi\) by 360 to find the radian equivalent.)
- “How could you find the degree angle measure equivalent to 1 radian?” (Sample response: We could set up a proportion that looks like this: \(\frac{\pi}{180}=\frac{1}{x}\). If we rewrite this proportion, we find that \(x\approx 57\) degrees.)

*Speaking: MLR8 Discussion Supports.*Use this routine to support whole-class discussion. After each student shares their responses, provide the class with the following sentence frames to help them respond: "I agree because . . .” or "I disagree because . . . .” If necessary, revoice student ideas to demonstrate mathematical language use by restating a statement as a question in order to clarify, apply appropriate language, and involve more students. For example, a statement such as, “It’s equal to 15,” can be restated as a question such as, “Is \(\frac{\pi}{12}\) radians equivalent to 15 degrees?”

*Design Principle(s): Support sense-making*

## 12.3: Pie Coloring Contest (20 minutes)

### Activity

Students build their sense for the sizes of angles measured in radians by shading sectors with given central angle measures.

### Launch

Arrange students in groups of 2. Distribute 1 set of pre-cut slips to each group.

### Student Facing

Your teacher will give you a set of cards with angle measures on them. Place the cards upside down in a pile. Choose 1 student to go first. This student should draw a card, then on either circle shade a sector of the circle whose central angle is the measure on the card that was drawn.

Take turns repeating these steps. If you are shading in a circle that already has a shaded sector, choose a spot next to the already-shaded sectors—don’t leave any gaps. You might have to draw additional lines to break the sectors into smaller pieces.

Continue until an angle is drawn that won’t fit in any of the sectors that are still blank.

When you’re finished, answer these questions about each circle:

- What is the central angle measure for the remaining unshaded sector?
- What is the central angle measure for the block of shaded sectors?

### Student Response

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### Anticipated Misconceptions

If students struggle to figure out the size of the angles on the cards they draw, ask them what radian measure 180 degrees is equivalent to (\(\pi\) radians). Then, ask them to find the measures of the central angles of each sector on the 2 circles (\(\frac{\pi}{4}\) and \(\frac{\pi}{6}\)).

### Activity Synthesis

The goal is to gather strategies for understanding the sizes of angles measured in radians. Here are some questions for discussion:

- “For groups that drew a card with \(\frac{\pi}{8}\) radians, how did you figure out what size sector to shade?” (Sample response: Each sector in the first circle had a central angle measuring \(\frac{\pi}{4}\) radians, so \(\frac{\pi}{8}\) radians is half that.)
- “What is the sum of the angle measures for the shaded and unshaded sector blocks, and why?” (The angle measures add to \(2\pi\), because together they make up the whole circle.)
- “How could you shade a sector with central angle measure \(\frac{\pi}{24}\) radians in each of the circles?” (This would be one-sixth of a sector in the first circle, and one-quarter of a sector in the second circle.)

## Lesson Synthesis

### Lesson Synthesis

For each pair of angle measurements, ask students to decide which is larger. Students may use any method of their choosing to arrive at their answers. Some may use their double number lines. Others may refer to the circles in the pie coloring activity.

- \(\pi\) radians or 200 degrees
- \(\frac{3\pi}{2}\) radians or 200 degrees
- 1 radian or 90 degrees
- \(\frac{\pi}{3}\) radians or 45 degrees

Student responses:

- 200 degrees
- \(\frac{3\pi}{2}\) radians
- 90 degrees
- \(\frac{\pi}{3}\) radians

## 12.4: Cool-down - Order Up (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

We can divide circles into congruent sectors to get a sense for the size of an angle measured in radians.

Suppose we want to draw an angle that measures \(\frac{2\pi}{3}\) radians. We know that \(\pi\) radians is equivalent to 180 degrees. If we divide a sector with a central angle of \(\pi\) radians into thirds, we can shade in 2 of them to create an angle measuring \(\frac{2\pi}{3}\) radians.

Another way to understand the size of an angle measured in radians is to create a double number line with degrees on one line and radians on the other. On the double number line in the image, the degree measurements are aligned with their equivalent radian measures. For example, \(\pi\) radians is equivalent to 180\(^\circ\).

Suppose we need to know the size of an angle that measures \(\frac{3\pi}{4}\) radians. The left half of the double number line represents \(\pi\) radians. Divide the left half of the top and bottom number lines into fourths, then count out 3 of them on the radians line to land on \(\frac{3\pi}{4}\). On the top line, each interval we drew represents 45 degrees because \(180\div 4 = 45\). If we count 3 of those intervals, we find that \(\frac{3\pi}{4}\) radians is equivalent to 135 degrees because \(45\boldcdot 3=135\).