Lesson 12

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.

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.

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.

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.

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.

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.)

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

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

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

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.)