## Circles Galore

This lesson is from MARS. Using graphics from the original source, I created this recording sheet — in pdf and in docx.

But instead of passing out the worksheet and starting right in like I had intended, I placed the stack in my desk drawer and decided to ask the kids to follow my instructions to create the diagrams. (I'm getting better at asking kids to do more, including following a set of directions.)

I gave each kid one sheet of unlined paper and a compass. These were my instructions to them, and I followed along also:

1. Put a dot in middle of paper.
2. Draw a large circle using the dot as its center.
3. Draw in the diameter.
4. On this diameter, mark half the radius. (I could ask them to bisect the radius, but to save time, we just estimated "half" using a ruler.)
5. Now draw a circle, centered at this half-radius mark, using a radius half that of the original circle, draw two adjacent circles along the diameter.
6. Shade in the outside of the two small circles — their drawings should now look like Stage 1 on worksheet.

We stopped here, and I asked Question 1) What fraction of the large circle is shaded? They quietly did their thing.

After seeing about half of the kids finished answering this question. I said, "I know not everyone is done with #1, which is perfectly okay, but I'm going on with my instructions because some people are done, and I need you to follow along. Then if you need to, you can go back to #1."

I repeated directions 4 and 5 above, and before I asked Question 2, Josh said, "What fraction is shaded now?" They went to work again.

I then repeated the instructions to get their drawings to Stage 3. However, our compass can't make a circle smaller than 0.5-inch radius, so we freehanded them in. Everyone knew next came Question 3) Now, what fraction is shaded?

And Josh immediately asked, "Are we doing this again, make even smaller circles?!" We drew in the next 16 shaded circles for Stage 4.

You guessed it. Question 4) What fraction of the large circle is shaded? I asked them to work alone on the questions, but that they would have a chance to discuss with other classmates during the last 15 minutes of class. They plugged away.

Then the talking began; and there was a lot of it, so I started recording their conversations. I was more interested in those who did not get 1/2 for Question 1. I happened upon Zach telling his two classmates that the answer was 2/pi:

Did you catch what Zach said? He reasoned that (16*pi) + (16*pi) = 2*pi*32 by saying "You can't get rid of stuff because we haven't done any mathematical operation to get rid of pi." I was so glad this came up! Sometimes I feel a wrong answer is just as important as the correct one.

(But I thought it was funny too how Slater reacted, "That's wrong." Then there was peer pressure, "He made me change my answer." And there was alleged copying. Then there was, "Taj showed me the way." I will miss this class.)

Daniel started the discussion with this group.

The "formula" that Daniel had trouble articulating was never written on his paper, it was all in his head. Then I noticed that he'd carried his calculations to stage 8! I was so pleasantly surprised by this that I was speechless like a complete dork in this next clip.

Class ended and everyone agreed that the answer to question 1 was 1/2.

I passed out the worksheet above for homework, asking them to try to figure out the answers for stages 2 through 4 if they hadn't already, then to give stage n a try if they had time. Daniel never saw the worksheet to know that I'd wanted him to keep thinking of the pattern, but he was already heading there on his own.

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Tomorrow we'll finish out this lesson, but I wanted to capture what we've had so far before my amnesia sets in.

Some thoughts:

1. When I search for lessons/activities for my kids to do, I always search for lessons "recommended" for kids at least one grade level above mine. Meaning, I look under 9th and 10th grade activities for my 8th graders. Why not? I could always work back from there.
2. I remember adding "stage n" to the worksheet just because there was some left over space at the bottom. This made Daniel happy.
3. I'm glad I had them draw the circles. This was a last minute decision that paid off — they made pretty sketches!
4. I learn that my kids are smarter than I am. (But let's not tell them this.) Daniel's "equation" was so elegant that he could do all of it in his head. I had to work hard to get to his answers!
5. Stage n, if we get it or not tomorrow, is icing on the cake. I feel pretty full.
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Speaking of stage n. What do you think? Is there a limit that we are approaching?

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• May 9, 2012 9:20 PM Peter Price wrote:
I'm officially out of my depth.

I thought it was tending towards 1/2, but the fractions didn't work out that way.

When I cheated and used a spreadsheet up to n=10, it got to 0.66699218750000, which looked a lot like 2/3. So I extended it to n=20, and got 0.66666698455811.

I've forgotten more math I once learned than I've remembered, so I don't have an elegant proof. But I'm convinced that the answer is that the area approaches 2/3. I suspect that your students will get there long before I do.

Great activity! I'm looking forward to part II.
1. May 10, 2012 9:16 PM fawnnguyen wrote:
Are you kidding, I use Excel all the time, not cheating at all, you're smart to use available tools! Yep, 2/3 is the limit. Funny that you used n=20. I'd have used n=5635353262598000000 in Wolfram just to be really sure!

I thought YOU were doing Part II. Thanks, Peter!!

• April 2, 2013 4:38 AM Paul Smith wrote:
That's a lovely problem, thank you, and a nice way to spend a morning in the school holidays. Finding the formula for the nth term is easy given a few examples and the On-Line Encyclopedia of Integer Sequences (oeis.org): it is (2^(n+1) + (-1)^n)/(3 * 2^n). And thus, yes, I am convinced that the limit is 2/3.
1. April 2, 2013 8:32 PM fawnnguyen wrote:
Well, it's very lovely of you to drop in and leave a comment, Paul. Thank you for looking up the equation and for reminding me about this problem from last year. I shall do it again next week!