Five Rectangles

Here's a maths challenge, 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.

We tried it 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?

We finished 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

This one bent the rules a little...!