Students use a double number line to convert between measures in degrees and radians. 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. As students use these diagrams to compare quantities that are proportional, they are increasing their precision of language in describing the relationship between degrees and radians (MP6).

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 radians equivalent to? How do you know?” (It is equivalent to 15 degrees. is half of , which is 30 degrees. Half of 30 is 15.)
• “How many radians is 1 degree equivalent to?” (It is equivalent to  radians.  is equivalent to 360 degrees, so divide by 360 to find the radian equivalent.)
• “How could you find the measure in degrees that is equivalent to 1 radian?” (We could set up a proportion that looks like this:  . If we rewrite this proportion, we find that degrees.)

In this partner activity, students take turns building their sense for the sizes of angles measured in radians by shading sectors with given central angle measures. Students practice converting from degrees to radians when they pull a card with a degree measure and record the total shaded sector in radians. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3).

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. ( radians) Then ask them to find the measures of the central angles of each sector on the two circles. ( and )

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 radian, how did you figure out what size sector to shade?” (Each sector in the first circle had a central angle measuring radian, so radian 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  radians or 360°, because together they make up the whole circle.)
• “How could you shade a sector with central angle measure radian 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.)