## Wednesday, 23 April 2014

### Five Rectangles

In Year 4 we've been looking at a puzzle, taken from Gary Antonick's 'Numberplay' in the New York Times:
Create a set of five rectangles that have sides of length 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 units.
The really quick and easy way to explore this is with Cuisenaire rods:

After we'd found five rectangles with those side lengths, we recorded them using the Cuisenaire Environment:

Here are more pictures:

Then we started looking at the area the rectangles covered.

We got total areas of
120, 121, 123, 125, 130, 154, 161 and 184.

Next question: what is the biggest possible area?

Alicia answered this - use the biggest side lengths on the same rectangles:

 Area = 190
Can you see why this is the maximum?

And then, what is the smallest possible area?

Mimi answered this one - use biggest and smallest lengths on the same rectangle:
 Area = 110
Can you see why this is the minimum?

This is a great investigation - manageable, easy to understand, and susceptible to taking off in many directions. We didn't try to see how many possible ways of making the rectangles there are - that was a step too far.

But just seeing some of the ways was worthwhile. It works well because the rods and the numbers 1 to 10 are easy to grasp. And the maths it takes us into is worth it - length, area, 2 X 3 = 3 X 2, multiplication facts, addition of five numbers... And then a bit of more abstract thinking - what would the maximum and minimum be, and why.

It may be the first time a class has tackled this particular puzzle. We can certainly recommend it to other classes!

We rounded this investigation off with a small puzzle:
Make these rectangles, and then see if you can make a square by putting them together:
1x6 4x7 5x8 3x9 2x10
3x6 4x7 2x8 1x9 5x10
1x2 4x5 3x8 7x9 6x10
1x2 4x6 3x7 8x9 5x10

#### 1 comment:

1. Beautiful... You can also extend this to the concept of perimeter.... Ask them what is the total of all the side lengths ..and they would probably feel the need for the formula (short cut) to find the sum of first n numbers.... Secondly, tell them to make rectangles of same areas but different perimeters etc.. ..Thanks for sharing..