The class and I have worked hard on gaining a conceptual understanding of 'area'. We have made great gains toward understanding and using the Pythagoras theorem, and now we are moving into finding the area of circles. The attached video is a brief overview of the general approach. It's raw. And by 'raw' I mean budget. Read the rest first, then watch the video.
I handed out graph paper and gave the class this scenario:
The company you work for has built a new workshop. Because you are so good at your job the boss has let you go out and choose your own work area. He tells you to go into the empty workshop and tape out your work area. But… you are only allowed 24 metres of tape. How would you do it?
The learners said things like “I’d do it up against a wall”. That’s great thinking, but the wall will count in the 24 meters. “So it’s 24 meters around the outside?” – “Yes and the word for that is… perimeter.”
So with a perimeter of 24m what is the largest area you could make?
Most of the learners got engaged and began to draw rectangles on the graph paper. Others didn't quite get the idea so we drew some rectangles on the board and began to compare the area. A major moment of learning was when learners realised that the area can change even when the perimeter remains the same.
We finally came to the conclusion that a square has the best perimeter to area ratio.
At this point an argument started in the class. A learner said “See, a square is always the best”. The other learner looked at me and said “Damon, can you beat a square?”
“Yes, with a circle”.
“How would that work?”
“Good question” I drew a circle on the grid on the whiteboard. "Seeing as we always measure area in squares, how do you think we deal with shapes like circles?”
- Demonstrate the shapes can be reconfigured to calculate area easier.
- Gain an understanding of the formula a = πr2
I bought paper plates and handed them out to the class. We folded the plate in half and drew a line down the middle. Again in half, then again, then again. We then cut out the eighths.
Now arrange the eighths into a rectangular shape. A learner Pipes up “that’s not a rectangle!”
"Cut each piece in half and do it again. Now it starts to look like a rectangle. Cut each piece in half one more time". Finally it resembles a rectangle.
“If we kept cutting these pieces in half would it continue to approach a rectangle?”
They agree that it would.
“And how do you find the area of a rectangle?”
Ahhh, now they start to see it and someone says “length times height”.
“Is the height the same as the radius?”
I ended up talking through the end and explained that the sides were half the circumference and the height was the radius. A few students seemed to ‘get’ it. Unfortunately I was running out of time and learner concentration and made a fairly lame link between this approach and the formula. I’ll hit this hard again on Monday. So much for discovery learning!
So, in summary, a great idea, a reasonable execution, and a fairly lowly lesson (story of my life).