Lesson 10

In this activity, students continue to explore the relationship between arc length, radius, and central angle. They recognize that a particular central angle defines arc lengths of different sizes in circles with different radii. This idea leads in to a subsequent activity in which students compare ratios of arc length to radius for angles and circles of different sizes.

Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.

The goal of the discussion is to start to build the idea that the length of an arc and the radius of the circle give us information about the central angle that defines the arc. Ask students these questions:

• “There are actually 2 circles that make up each progress bar, an inner circle and an outer circle. Which one should we focus on when measuring the progress?” (It doesn’t matter; either contains the same information.)
• “If the length of the arc on Tyler’s circle is longer, why does that not necessarily mean he has completed more tasks?” (The completed proportion depends not only on the arc length but on the size of a circle.)
• “For the second question, what information do you think would be helpful to accurately compare the progress of the two students?” (Students may discuss proportions and fractions. If the phrase central angle does not come up, ask students what information this measurement would give us.)
• “What measurement can we use to compare the ‘sizes’ of the circle?” (Radius gives relative size.)

In this activity, students revisit the fact that when a figure is dilated, corresponding angles do not change measure, all lengths and distances change by the scale factor, and areas change by the square of the scale factor. They are also reminded that all circles are similar, so any one circle is a dilation of any other circle. Students note that the ratio of a circle’s circumference to its diameter is a constant, and that this is the origin of the circumference formula.

These concepts will be useful in subsequent activities in which students determine that arc length to radius ratios are constant for given central angles.

If students aren’t sure how to find the ratio of the circles’ circumferences and diameters, remind them that circumference can be calculated with the formula \(C=2\pi r\). Ask if this formula can be written in terms of diameter instead of radius.

The goal is to discuss ratios that arise from the similarity of all circles. Ask students:

• “Suppose we draw a tiny circle with a 90 degree sector. Is it a dilation of the original circle? Why or why not?” (Yes. All circles are similar, so they are all dilations of each other.)
• “Suppose we dilated the circle not by a factor of 3, but by a factor of 1,000,000. What would happen to the radius, arc length, and central angle, and why?” (The radius and arc length would increase by a factor of 1,000,000, because distances are multiplied by the scale factor under dilation. Angles don’t change under dilation, so the central angle would still measure 90 degrees.)
• “What would be the ratio of our enormous circle’s circumference to its diameter?” (It would still be \(\pi\).)
• “How does this constant ratio help us when we’re calculating circle circumferences?” (Because the circumference divided by the diameter always equals \(\pi\), we can multiply the diameter by \(\pi\) to find a circle’s circumference.)
• “In what other context did we talk about ratios that didn’t change, no matter the size of the object?” (The trigonometric ratios were like this.)

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.

In a subsequent activity, students will prove that the ratio of arc length to radius is indeed constant for a particular central angle regardless of the size of the circle, and they’ll define this ratio as the radian measure of the angle.

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 \(\frac14\) the radius for cards B and H).

Arrange students in groups of 2. Distribute pre-cut slips. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the cards, they should work with their partner to come up with categories.

Give students 7–8 minutes of work time, then follow with a whole-class discussion. Be sure to leave enough time for students to calculate arc length to radius ratios in the synthesis.

If no groups appear to be creating 2 categories of congruent angles, ask students what they can figure out about the circles on the cards that show arc lengths and radius measurements.

Select previously identified groups to share their categories and how they sorted their cards. If possible, choose a group that created a group of 45 degree angles (cards B, D, F, and H) and another group of 270 degree angles (cards A, C, E, and G). Attend to the language that students use to describe their categories, giving them opportunities to describe the categories more precisely. Highlight the use of terms like central angle and ratio.

Then, if students have not already done so, ask them to calculate the 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 1 problem to work. Invite them 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 arc length to radius ratios equivalent to \(\frac{\pi}{4}\) or \(0.25\pi\).
• Cards C and G each represent 270 degree angles and have arc length to radius ratios equivalent to \(\frac{3\pi}{2}\) or \(1.5\pi\).
• 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 \(30\pi\) units and a radius of 20 units. Without actually doing a calculation, what central angle measure do you think might go with that, and why?” (The ratio of arc length to radius is equivalent to \(\frac{3\pi}{2}\), so it is probably a 270 degree angle.)