Sunday, February 24, 2013

Preparing for the area of a circle, two conceptual area models...

So, I'm getting ready to tackle the area of a circle next week with my students. So far, there are two conceptual area models I have come across:

How Many Radius-Squares Can Fit inside a Circle?
Okay, so this is the model that is used by the singapore math folks. I used it last year and it was pretty successful.
  1. Make a circle of radius r centimeters.
  2. Make a square of side-length r centimeters.
  3. How many of these radius-squares can you fit inside the circle? Let's experiment to find out.
  4. Use scissors to cut up area-squares to fit them inside the circle. Three radius-squares will completely fill-up the first three quadrants of the circle. But all those left-over bits of the radius-squares can be cut-up and will fit into the last quadrant and there will be leftover space. So approximately three radius-squares plus a little more will fill the circle. That little bit more is about 0.14 radius-squares.
  5. The area is about 3.14 radius-squares or...
  6. A= π×r×r or A= π×r².
A Sliced-Up Circle Approximates a Rectangle
And this model is one used by Waldorf and most other math curricula.
  1. Create a circle of radius rCut the circle into three segments and glue them together, so that they kind of form a parallelogram.
  2. Repeat, but cut the circle into 4 pieces.
  3. Repeat, but cut the circle into 8 pieces.
  4. Repeat, but cut the circle into 16 pieces.
  5. Now, at this point, students can see that the shape is REALLY close to a parallelogram. If we were to continue repeating this proces, the parallelogram would get steeper and would become a rectangle. 
  6. The height of that rectangle will eventually be the radius, r. The base of that rectangle will be the circumference divided by two (or πr).
  7. So the area of the rectangle will be base x height or...
    A= πr•r or A= πr
    ².
I think that for 6th graders, the more direct way of doing it is the Singapore math method and I'll probably use it. But the other one might be a great extension for my higher-skills kids.

No comments:

Post a Comment