Lesson: Area of a Circle (or How I Got Students Hungry for the Formula)

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):

MCA Warm-up (The MCA’s are the big comprehensive state test in MN. We start each class by doing a couple of review questions similar to what they will see on the test. After 6 warm-ups, they take a quiz online to further mimic the testing experience and hold them accountable to remembering the material.)

Go over homework

  • Discuss how you know proportional.
  • Make proportional conclusions:
    • Constant of proportionality
    • Proportional equation: C = 3.14d

Notes

  • add pi and Circumference formula to notes
  • have them draw a circle with diameter, calculate circumference
  • Area = ?? still need a way to calculate area of a circle…

Launch Inv 3.3:

  • Hand out circle drawn on cm grid paper.
  • Talk about diameter and radius of circle then ask, “If I asked you to find the area of the circle, what would I be asking you to do?”
  • Work with partner to estimate the area of the circle using whatever strategy they thought would give them the most accurate estimate.
  • Entered their data on a Google Form when done.

Explore

  • As a class:
    • Look at Form data, talk about estimates and strategies
      • Area of 72 (area of rectangle)
      • Area of 144 (area of large square circle inscribed in) These two estimates give up good lower and upper bound
      • Show student strategies for estimates in between 72 and 144
      • If someone used formula, ask them why it works? are they sure that makes sense?
    • Say, “We have these estimates that seem like they’re probably somewhat accurate, but there must be a more exact method where we don’t have to count squares. I don’t know how to find area of a circle, so I’m going to see if I can turn my circle into a shape that I do know how to find the area of.” Here’s where it got fun!
      • Method 1 (this is what I did first and second hour):
        • have them cut out circle
        • fold circle in half – ask them for circumference of half of the circle: πr , outline this distance with a highlighter and label
        • then fold circle in half again, talk about length of straight edges, r
        • fold in half again, talk about length of straight edges again, still r
        • unfold and cut out along fold lines, should have 8 slices of pizza
        • Start arranging on desk to form rectangle, talk about dimensions of rectangle, area = πr•r
      • 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
    • Add area formula to notes
      • Find area of practice circle in notes
      • (let students eat pizza upon showing you their completed area)

Practice – labsheet

  • in groups
  • circumference and area practice WS

Summary

Resources:

Circle notes:

Screen Shot 2016-02-07 at 6.33.56 PM.png

Circle they estimated area of (wish I had saved some student work here. They did some really great reasoning. I had a couple of students who were only a hundredth off of the actual area):

Screen Shot 2016-02-07 at 6.35.45 PM.png

The great pizza discussion:

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What I liked:

  • The hook: Having students estimate area first got them hooked. They wanted to make sense of formula so that they could see how close their estimate was to the actual area
  • Composing and decomposing: Christopher Danielson tweets about this a lot and it’s the basis of all math talks, but seeing how visibly wowed students were by decomposing the circle into a rectangle was so fun to see. Before the lesson, I thought the circle as rectangle might be a hard sell since visually it’s more of a trapezoid, but students didn’t get hung up on this at all. They were the ones that told me it was rectangle.
  • Ordering a pizza: my students have never been more engaged than they were when I pulled that pizza out.
  • More students seemed to follow this logic than the pi groups of logic in previous years

What I would want to continue to improve:

  • I did a lot of the leading, especially in the classes where ordering a pizza wasn’t possible. Is there a way to get students to do this or do they need to be lead here?
  • More circumference practice before adding area of the mix. We talked a lot about the proportionality of circumference and diameter, but students needed practice just finding some circumferences before throwing in area.
  • Still only had about half of the class following the area discussion and where the dimensions came from. This may be due to the fact that I went to straight to the abstract, I didn’t use numbers when talking about length and wdith of the rectangle. If I had students do a couple of specific examples first, would that help all students generalize better and come to their own conclusion about dimensions and area?
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Better Questions

The best compliment I have ever received came from a student at the end of my first year of teaching. (Yes, my first year, when I was naive, clueless and a general mess of a human being and teacher.) I had a student named Emma. She was one of those rare students who enjoyed thinking deeply about mathematics. She was never satisfied with a procedure. She wanted to know why and how. She often stayed after class just to talk about a math problem we had done in class. At the end of our year together, she approached me in her quiet, always thoughtful manner and said, “Ms. Lee, I really liked having you as a teacher because you asked me really good questions. You asked me questions that really made me think about math.” Cue, Ms. Lee, standing still, mouth agape, speechless. She will never understand the compliment she paid me, the late nights I stayed up that first year just thinking about questions–what would I ask them, how would I ask them? If they came up with this strategy or got stuck here, what would I ask them to move them forward or connect back?

All of this is to say, I take questioning seriously. I think it’s vitally important to getting kids to reason, critique, and problem solve. Never say anything a kid can say. Ask them a question. Make them say it instead. I have two question moments to share from this school year:

  1. The question I’m always trying to get better at asking:

What does it mean to be proportional?

Proportionality if the central concept of 7th grade math in Minnesota. Every year I teach proportional reasoning strategies-unit rates, scale factors, scaling ratios, setting ratios equal to each other, etc-but every year I feel like students never quite get a good clear grasp of what it means for two quantities to be proportional to each other. This year, I tried to be more deliberate in my attempt to tie proportionality to fairness. I came up with a unit question, “How can proportional relationships help us make decisions about how resources should be distributed?” This helped me in planning the questions/problems I asked students to work on and helped ground the complexities of proportionality in something 7th graders are always keen to argue about, fairness. One problem that finally got them to reason, think and question fairness while using their proportional reasoning strategies came when I asked this warm-up question:Screen Shot 2016-01-30 at 12.09.24 PM.pngSome students were hell-bent on 20/20. Other students had a gut-feeling that this wasn’t fair because there were more girls. Students who found unit rates (questions per gender) were finally able to shed some viable proof on why the 20/20 plan wasn’t fair. They were using math to construct arguments and their arguments were grounded in the concept of proportionality. My only regret, this question came at the end of the unit. I need to give them more questions like this throughout the unit that make them wrestle with fairness by using their proportional reasoning strategies.

2. The question I’m still searching for:

My friend tells me he was born on Thanksgiving Day, November 28. Instantly my noticing/wondering wheels start spinning. I wanted to know how often since then his birthday has fallen on Thanksgiving.  At first, I thought maybe every 6 or 7 years, depending on leap year. The actual answer? So incredibly more fascinating than I could have ever hoped for. I’m left with more noticings and wonderings than when I started. In fact, I have so many questions now that I don’t know what exactly I would ask students.

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What do I ask students?

What if he was born Nov. 27, 1990? Would his birthday repeat in the same pattern? How does the timing of Leap Year effect the pattern? What are all of the possible dates that Thanksgiving can fall on? Do they all repeat in the same pattern?

Any suggestions here would be greatly appreciated!

#MTBoS

My Favorite

I have so many “favorites” when it comes to why I teach and what happens in my classroom everyday. Here just a couple:

  • My favorite way to stay organized: Notability. I hate paper. I lose paper within minutes of first being handed it. Notability helps keep all of my papers organized in a place where I will be able to find those papers weeks and even years later. I use Notability to store and organize notes that I take at workshops, articles I find, answer keys, and student worksheets. My students are 1:1 with iPads, so I also have them using Notability to do all their note-taking. They groan at first, but then realize how incredibly organized it keeps them since they don’t have to keep track of papers.
  • My favorite part during a math lesson: The few times I get students to challenge and question each other. It’s the space between the right answer and the wrong answer. The space where a student either knows something doesn’t make sense and asks a question or he/she knows it does make sense but wants to know why.
  • My favorite part about teaching junior high students: When you finally figure out what makes a kid tick. That kid that’s been challenging for me all year. He/she is unorganized, disruptive, and unreceptive to any help or feedback until that moment when I figure out how exactly he/she wants me to relate to him/her.

#MTBoS