Challenge Level

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

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

Can you make a rectangle to represent $x^2 + 7x + 12$?

Can you do it in more than one base?

Watch the video below 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 $x^2 + 7x + 12$ 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.*