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?