## Circles

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.

**Activity one**

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.

**The argument**

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?”

**The lesson**

Objective:

- Demonstrate the shapes can be reconfigured to calculate area easier.
- Gain an understanding of the formula a = πr
^{2}

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!”

True!

"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?”

“Um, yes”.

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).

You rock

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