Can you find out what is special about the dimensions of rectangles you can make with squares, sticks and units?

For this problem, you will either need multilink cubes or these sets of number base sheets to cut out: Base Three    Base Four   Base Five   Base Six.

In the video, Charlie and Becky show how you can make rectangles using sets of squares, sticks and units.

Can you make a rectangle to represent x²+7x+12?
Can you do it in more than one base?

Watch the video to see how Charlie and Becky tackled this question:

Take 1 square and 12 units in your chosen base. Put them together with some sticks to make rectangles that will work in all bases.

Charlie and Becky made x²+7x+12 into a rectangle with length x+4 and width x+3.

How many different rectangles can you make?
What do you notice about the dimensions of your rectangles?

Imagine you had 1 square, lots of sticks and 100 units. What can you say about the dimensions of the rectangles it is possible to make?

Now, take 1 square and 12 sticks in your chosen base. Put them together with some units to make rectangles that will work in all bases.

How many different rectangles can you make?
What do you notice about the dimensions of your rectangles?

Imagine you had 1 square, 100 sticks and lots of units. What can you say about the dimensions of the rectangles it is possible to make?

If you had 1 square, p sticks and q units, what can you say about the dimensions of the rectangles it is possible to make?

Extension

Think about the rectangles it’s possible to make if you use two, three, four… squares, some sticks and some units.

You may also be interested in the other problems in our Getting started, getting stuck Feature.

Many thanks to Kenneth Ruthven and Paul Andrews whose ideas inspired this problem.

Age 14 to 16