Students sort different diagrams and measurements during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7). In this task, students examine relationships between arc lengths, radii, and central angles. They observe that the ratio between arc length and radius appears to be constant for a given central angle, and may be a proxy for angle measurement.

Monitor for groups who recognize that central angles can be calculated for the measurements given on Cards B, C, G, and H, and for those who consider ratios between the arc lengths and the radii. It’s appropriate for students’ thinking to be informal at this point (for example, they may notice that the integer portion of the arc length is  the radius for Cards B and H).

If students are unsure how to match the cards with arc length and radius measurements to a diagram, ask them what additional information would be helpful to know. When a student suggests the circumference or the size of the circle, remind students that they know how to calculate the circumference. 

Once all groups have completed the Card Sort, discuss the following:

• “Which cards were tricky to sort? Explain why.”
• “Did you need to make adjustments in your sorting? What might have caused an error? What adjustments were made?”

Then, if students have not already done so, ask them to calculate the precise central angle and the ratio of arc length to radius for each of Cards B, C, G, and H. If time is short, consider dividing the problems amongst the class so each student has only one problem to complete. Invite students to share observations about their results. Be sure the following points come up in the discussion:

• Cards B and H each represent 45-degree angles and have ratios of arc length to radius that are equivalent to , or .
• Cards C and G each represent 270-degree angles and have ratios of arc length to radius that are equivalent to , or .
• The ratio appears to be the same for a particular central angle.

Ask students, “Suppose I gave you another card that showed an arc length of units and a radius of 20 units. What central angle measure do you think might go with that, and why?” (The ratio of arc length to radius is equivalent to , so it is probably a 270-degree angle.)

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 a subsequent 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.

If students are stuck trying to show that , ask if they can rewrite by substituting in other expressions for and .

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 two 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 such that and .” (The value of is .)
• “How does this example relate to what you proved in the activity?” (We proved that the ratio is equal to the value . In this example, we showed both are equal to .)