## Tuesday, January 16, 2018

### Lesson 8-9: The Area of a Circle (Day 89)

Lesson 8-9 of the U of Chicago text is called "The Area of a Circle." We all know the famous formula that appears in this lesson.

Two years ago, my Pi Day activity was more geared towards the area. Therefore, I'm posting that Pi Day worksheet today for Lesson 8-9.

Meanwhile, many chapters in the second half of the book are longer than those in the first half -- and this causes a problem in setting up the chapter review and chapter test. Thursday is Day 91, which is when Lesson 9-1 will be taught, and tomorrow is the Chapter 8 Test. This means that today needs to be the Chapter 8 Review as well as Lesson 8-9. Get used to this, since there are several more long chapters coming up in the text.

This is what I wrote two years ago about Lesson 8-9:

I visited several other teacher blogs for ideas on lessons. One of these blogs has a lesson that's perfect for Pi Day:

https://theinfinitelee.wordpress.com/2016/02/08/lesson-area-of-a-circle-or-how-i-got-students-hungry-for-the-formula/

Laura Lee is a middle school math teacher from Minnesota. Here is how she teaches her seventh graders about pi:

I teach CMP. I love discovering pi in Investigation 3.1 of Filling and Wrapping. It’s hands-on and engaging. Students love discovering that pi is a real thing not just a random number and excuse to eat pie on 3/14. If the lesson goes really well, I’ll even get a kid to ask, “Isn’t this a proportional relationship? Isn’t pi a constant of proportionality?” To which I want to use every happy emoji ever created in large, bold, capitalized letters with lots of exclamation points!!!
But then Investigation 3.2 comes along, where students discover the formula for area of a circle as pi groups of radius squares. I’ve never been able to get students to make sense of this lesson. There seem to be lots of understandings (what a radius square is, how to calculate how many times one number goes into another, understanding multiplication as groups of…) needed in order to get valid data that would allow students to conclude pi groups fit inside the circle. It just gets messy and doesn’t seem to make any sense to students, so this year I decided to take a different approach.
I decided to go with composing and decomposing. I would show them how to decompose the circle into a rectangle that is πr by r. Here’s how the lesson went (spoiler alert- a Domino’s pizza makes an appearance):
Notice that last year, I posted a lesson that actually covered area before circumference. Lee's lesson restores the order from the U of Chicago text, with circumference (Lesson 8-8) before area (8-9).

Let's just skip to the part where, as Lee writes, a pizza makes an appearance:

Method 2 (I had been talking about pizza all morning with my first 2 classes, so last hour I decided I should just order a pizza and use that to derive the area formula)
• Order a pizza (Domino’s large cheese worked great!)
• Reveal pizza to class, watch them go insane!
• Have students gather around your front table
• Slice pizza into 16 slices,
• talk about circumference of 8 of the slices or half of the pizza: πr, record this on the pizza box
• then start arranging pizza slices into a rectangle, listen to student “Ahas!” and “No ways!” when they see it is clearly starting to form a rectangle
• Talk about dimensions of rectangle and then the area

The U of Chicago text does something similar in its Lesson 8-9. The difference, of course, is that the text doesn't use an actual pizza.

Lee writes that for her, the key is proportionality. This fits perfectly with the Common Core:

CCSS.MATH.CONTENT.HSG.C.A.1
Prove that all circles are similar.

Then again, notice that Common Core seems to expect a proof here. How does Common Core expect students to prove the similarity of all circles without Calculus?

Unfortunately, none of our sources actually prove that all circles are similar. What I'm expecting is something like this -- to prove that two circles are similar, we prove that there exists a dilation mapping one to the other. For simplicity, let's assume the circles are concentric, and the radii of the two circles are r and s. So we let D be the dilation of scale factor s/r whose center is -- where else -- the common center O of the two circles. If R is a point on the circle of radius r, then OR = r, and so its image R' must be a point whose distance from O is r * s/r = s, and so it must lie on the other circle of radius s. Likewise, if R' is a point on the circle of radius s, its preimage must be a point whose distance from O is s / (s/r) = r, and so it must like on the circle of radius r. Therefore the image of the circle of radius r is exactly the circle of radius s.

Of course, this only works if the circles are concentric. If the circles aren't concentric, then it's probably easiest just to compose the dilation with an isometry -- here a translation is easiest -- mapping the center of one circle to that of the other. Therefore there exists a similarity transformation mapping any circle to any other circle. Therefore all circles are similar. QED

To get from the area of the unit disk (pi) to the area of any disk (pi * r^2), we are basically using the Fundamental Theorem of Similarity from Section 12-6 of the U of Chicago. This time, though, we are using part (b) of that theorem:

Fundamental Theorem of Similarity:
If G ~ G' and k is the ratio of similitude [the scale factor -- dw], then
(b) Area(G') = k^2 * Area(G) or Area(G') / Area(G) = k^2.

We skipped this formula back when we covered Lesson 12-6 because at the time, we hadn't learned about area yet. [2018 update: Actually, this year we reverse the order -- so far we've covered area but not similarity at all.] Although Wu attempts to prove a special case of the Fundamental Theorem of Similarity using triangles, it's much easier to do it using squares, as the U of Chicago does. If G can be divided into A unit squares, then G' can be divided into A squares each of length k. And the area of a square of length k is clearly k^2, so the area of G' must be Ak^2. For the circle problem k is the radius r, and A is the area of the unit circle or pi, so the area of a circle is pi * r^2. We can do this right on the same worksheet -- there's already a circle drawn of radius 10 times the length of a square, so instead of the length of each square being 1/10, let it be 1 instead. Then the area of the circle of radius 10 is equal to the number of shaded boxes, or 314, since the old unit square has been divided into 100 unit squares.

Now here are some Pi Day videos:

OK, I didn't need to reblog the Pi Day video. But I posted a pi video for Lesson 8-8, and so I might as well post a video for Lesson 8-9.

And this is what I wrote two years ago about Review for Chapter 8 Test:

As I mentioned earlier, the Chapter 8 Test is on Wednesday, which means that the review for the Chapter 8 Test must be today.

In earlier posts, I mentioned the problems that occur when a teacher blindly assigns a worksheet that doesn't correspond to what the student just learned in the text. Since I'm posting a review worksheet today, we should ask ourselves whether the students really learned the material that is to be assessed in this worksheet.

For example, most students learn about area at some point in their geometry texts, but only the U of Chicago text includes tessellations in the area chapter. Yet the very first question on this area test is about -- tessellations. So a teacher who assigns this worksheet to the class will then have the students confused on the very first question!

Let's review the purpose of this blog and the reason why I post worksheets here. The purpose of this blog is to inform teachers about the transformations (isometries, similarity transformations) and other ideas that are unique to Common Core method of teaching geometry. The worksheets don't make up a complete course, but instead are intended to be used with a non-Common Core text -- the one that teachers already use in the classroom, in order to supplement the non-Common Core text with Common Core ideas. Another intent is for those teachers who do have Common Core texts, but are unfamiliar with Common Core, to understand what Common Core Geometry is all about. My worksheets are based mainly on the U of Chicago text because both this old text and the Common Core Standards were influenced by NCTM, National Council of Teachers of Mathematics.

So this means that a teacher interested in Common Core Geometry may read this blog, see this worksheet, decide to assign it to the class, and then have all the students complain after seeing the first question because their own text doesn't mention tessellations at all.

I decided to include the tessellation question because it appear in the U of Chicago text. But as of now, it's uncertain that tessellations even appear on the PARCC or SBAC exams. So it would be OK, and preferable, for a teacher to cross out the question or even change it. There's a quadrilateral, a kite, that's already given in the question, so the question could be changed to, say, find the area of the kite, especially if the school's text highlights, instead of tessellations, the formula for the area of a kite.

I admit that it's tricky to accommodate all the various texts on a single worksheet. I included tessellations since this is a drawing assignment that is fun, and I'd try to include them if I were teaching a class of my own. But I also want to include questions that may be similar to those that may appear on the PARCC or SBAC exams.

[2018 update: The decision to include tessellations on this worksheet is even more controversial than it was two years ago, since this year we began with Lesson 8-4 and didn't even cover the earlier Lesson 8-2 on tessellations!]

For example, Questions 2 and 9 are exactly the type of "explain how the..." questions that many people say will appear on those Common Core exams. And so it was an easy decision for me to include those questions.

Another issue that came up is the Pythagorean Theorem. Such questions appear in the SPUR section of Chapter 8 in the U of Chicago, but we didn't cover Lesson 8-7. [2018 update: This year, of course, we covered the Pythagorean Theorem.] I dropped the questions that were purely on the Pythagorean Theorem, but I kept two trapezoid questions where the Pythagorean Theorem is needed to find the length of a side or the height. But these questions will confuse a student who has reached the area chapter, but not the Pythagorean Theorem chapter that may be several chapters away.

Then there is a question where students derive the area of a parallelogram from that of a trapezoid. I point out that in other texts -- especially those where trapezoid is defined inclusively -- this isn't how one derives the area of a parallelogram. In the U of Chicago, the chain of area derivations is:

rectangle --> triangle --> trapezoid --> parallelogram

But in other texts, it may be different, such as:

square --> rectangle --> parallelogram --> triangle --> trapezoid