Area and Perimeter Exploration

So we were looking at area a while ago. This led us into a conversation about the properties of 2D shapes. And then we spent some time with perimeter.

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?

Same Area, Different Perimeter

One boy's personal exploration into Infinity

We'd been doing all that investigation into 2D shapes, looking at area and perimeter. We had  found shapes that had the same perimeter but different area. You may have read about it on a previous post.

This provoked one boy to inquire in the opposite direction.

"What if I keep the area the same, but change the length of the sides?" he mused.

Not content with that, he continued, "...and what if I use triangles instead of rectangles?"

Here is the page of diagrams that he drew to explore this idea:

He started with a right angled triangle with sides 8cm and 4cm = an area of 32 square cm (Sorry - I don't think I can do superscript for index notation on this text editor.)

Then he doubled the long side and halved the short side: 16cm x 2cm = 32 square cm

Realising that he would go off the page the next time he doubled the dimensions of the long side, he decided to use a scale of 1:2 for the next diagram:  32cm x 1cm = 32 square cm

His Stunning Conclusion

"You know what?" he asked. "I think there is an infinite number of triangles with an area of 32 square cms. I could keep on doubling and halving forever."