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:
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|
And then, what is the smallest possible area?
Mimi answered this one - use biggest and smallest lengths on the same rectangle:
|Area = 110|
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