Now had come the moment I dread - when area and perimeter collide in the minds of the students and they come away dazed and confused. (How many times have you seen that? Hopefully we had clarified each of the concepts clearly enough.)

Anyway, the time had come to pose a favourite question of mine:

In fact, I wrote a post about this almost 2 years ago.

Here's what we came up with this time:

This group chose to use a hexagon for their example. The perimeter of the yellow hexagon was 15cm. Good stuff.

They found that 2 trapeziums (trapezia?) had the same area as the hexagon but a different perimeter. Well done.

Another group chose to use the Base-10 blocks. They made a 10x10 square (a=100; p=40) and then reorganised the blocks to make a 5x20 rectangle (a=100; p=50).

So just to be annoying, I asked them what would happen if they put all the blocks end-to-end? Same area but what would be the new perimeter? And then if you cut that rectangle in half long ways....

So, everybody happy. Yes - you can have the same area but with different perimeters.

Next question:

Previously I have done this as two separate activities but today we had time and we were on a roll so we kept going.

And here is what the kids came up with:

*Happy with this one - nicely set out and very clearly explained.*

*An interesting solution - not using regular shapes.*

*Solves the problem but not a lot of explanation.*

*Another way to do this.*

The post I did 2 years ago that showed this activity was with Year 4 students. This time I am working with Year 5. Not quite as bright and colourful as the younger kids, not quite as diverse in their solutions and somewhat more structured in their presentation of solutions.

Wonder whose expectations have changed - theirs as students or mine as a teacher?

Bruce, it's neat to hear about a course of study in which students encounter area and perimeter separately and then encounter this important question about their relationship.

ReplyDeleteI find this problem, when used with students from 8 years old and up (our grade 3), elicits a lot of wishful thinking that given an area perimeter can be known through some magic formula...

http://mathforum.org/workshops/delcocff2009/packet2703.pdf

The solution comes from recognizing that the smaller rectangles give us a clue about the dimensions of this rectangle and not just its area... teasing out why the small rectangles matter gives me lots of information about students' thoughts about area and perimeter.