Arc Length and Sector Area
Lesson 13 of 17
Objective: Students will be able to practice finding arc length and sector areas for different radius lengths and central angle measures, laying groundwork for a definition of radian measure in the next class.
Opener: Arc Length
Today's opener (see arc length and sector area intro slide 2) begins with two straight-forward arc length exercises. Apart from the small role it played on the Defining Pi Project, we have not yet focused specifically on arc length in this class. We're jumping right in anyway, because I want to emphasize the common sense behind this idea. I look forward to seeing how kids will respond when so non-chalantly confronted with a "new" topic.
The first problem is slightly easier than the second, because 45 degrees is 1/8 of the circle - a unit fraction like those on the Defining Pi Project. The second problem involves 3/8 of the circle, which we didn't see on the project. Again, I'll be interested to see what kids come up with. The third problem is a very informal preview of sector area, which we'll get to later in the lesson. For the scaffolding it lays, and for how easy it becomes for the teacher to glance over the shoulder of a student and see what they understand, it's very important to encourage kids to draw a diagram on this third problem.
After at most 5 minutes, I'll start asking volunteers to put their work on the board. I really don't want to make a big deal out of this. I'm not rushing to formulas, because there's no need - we're just talking about fractions of the area and circumference of a circle, and I want to make feel obvious to students that nothing could feel more natural.
The Arc Length Chart
I show students the new learning target (arc length and sector area slide 3), which is the third and final one for our circles unit. The learning target includes a reference to radian measure, which I explain is the focus of tomorrow's class. Today we're laying groundwork for tomorrow's discussion.
In order to help us lay that groundwork, the main focus of today's lesson is a table (arc length chart) in which students will fill in the lengths of arcs intercepted by angles in different-sized circles.
Teacher's Note: This is a powerful exercise, and in order for you to get the most out of teaching it, I recommend trying it on your own first. It involves the repetition of a drill, but there's something bigger going on here.
I try not to say too much when I distribute this handout to the class, but I make a brief note that they should not think of this chart as "90 different problems." Rather, the 90 cells in this table are going to reveal some patterns, and I encourage kids to look for these patterns, because doing so will save them time. We set out to work, and I wait for students to start noticing whatever they may. When I hear exciting observations, I quietly write them on the board (MP2, MP7).
Take a look at the Common Core Standard G-C.B.5, which calls for us to "use similarity" to "define radian measure." It sounds a lot like how we defined the trig ratios at the start of the semester. Today, I'll only mention the word similarity to students, but I won't offer many explanations until tomorrow. Instead, I'll ask them if they can find it (Slide 5) by thinking about why numbers repeat in their chart.
(SPOILER ALERT: Along these lines, think of the "scale factor" that can be used to "dilate" one row or column into another. There's the similarity.)
After I see that most students are done with the chart, which could take anywhere from 20-40 minutes, I post the first turn-and-talk question on Slide_6. I post one question at a time, and depending on how much time we have left, I'll give between 1 and 3 minutes for students to discuss each of these. Again, we're constructing knowledge here. My role is really to stay out of the way and let arguments and agreements and conjectures happen. Only if students insist (and they usually do) will I quickly go over the (rather unsightly) answer to the first question. For the second one, I leave it alone, because there will be plenty of time to get to that this semester. I like to tell kids that you could argue "yes" or "no," depending on how you supported your argument. We will certainly come back to this quesiton soon!
Finally, we have a brief discussion of a few example problems regarding sector area. Again, I'm trying to make the point throughout the lesson that there is some common sense here. If we were able to think of slices of the circle in the Defining Pi Project, the big leap here is that now we're talking about multiple slices - no longer unit fractions. So when students see that 210 degree angle on the first problem (Slide 7), I ask first if anyone saw an angle in the Defining Pi Project that divides nicely into 210. Hopefully, they'll recognize the number 30, but 15 or 10 will serve our purposes too. I ask them how many 30, 15 or 10 degree slices you would need to make 210 degrees. If they can find the area of one 30-degree slice, it's just a little multiplication to get to 210 degrees.
For an Exit_Slip, one question on arc length and one on sector area are all I need to see what kids can grasp so far. The only purpose here is that I want to know what kinds of scaffolds I'll have to provide. I emphasize to students that I want them to sketch a diagram for each problem, because that will help me understand their thinking.
This new topic requires some old-fashioned skill drill homework. I use Kuta Software to make such worksheets. If you have the software, you can adjust the settings so central angles are given in degrees only. I don’t grade it – rather, I’ll just give a brief quiz tomorrow to see how they’re doing.