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.
- Make a circle of radius r centimeters.
- Make a square of side-length r centimeters.
- How many of these radius-squares can you fit inside the circle? Let's experiment to find out.
- 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.
- The area is about 3.14 radius-squares or...
- 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.
- Create a circle of radius r. Cut the circle into three segments and glue them together, so that they kind of form a parallelogram.
- Repeat, but cut the circle into 4 pieces.
- Repeat, but cut the circle into 8 pieces.
- Repeat, but cut the circle into 16 pieces.
- 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.
- 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).
- So the area of the rectangle will be base x height or...
A= πr•r or A= πr².
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.