12 = ? Expanding Numbers

multiplication drills

There is a way to practise maths that looks like doing it backwards, but actually is a far more effective and creative way to practice recall of arithmetic facts.

Expanding Numbers

Start with a number such as 12 and write it in the centre of the sheet. Your goal is to see how many calculations you can find that equal it. You can also find information about special numbers nearby that number.

This is a much more creative and open approach to practise. You do not need to spend a long time on it but try to do it regularly. Even better, do this with friends/family and see who comes up with the most unusual calculation.

Expanding using Prime Colours

Prime Colours lends itself to this style of practice. Each card gets you to think about the calculations that result in the number. The card on the right is the number 20.

Depending on the Prime Colours’ activities that you’ve done, you could find any of the following calculations.

4 x 5 | 2 x 10 | 10 + 10 | 12 + 8 | 2 + 3 + 4 + 5 + 6

Getting to know numbers in this way is what real fluency is about. You can work collaboratively on a sheet and of course upgrade a sheet that is too easy by finding something else to find.

If you can take a few cards with you, in your pocket say, you can do this practice any time you have a spare minute and before long, you will have build up valuable connections between numbers.

Download the worksheet

Modelling Multiplication

multiplication drills

Concepts and misconceptions are at the heart of maths education.  Our goal is to give out children an understanding of concepts that they trust and can use.  We may have to do this through knowledge and skills because (except in The Matrix) we cannot download whole concepts into people’s heads.  We can run into difficulties if we assume that certain concepts are in place based only on the assessment of a few key facts and skills.


Let’s start with a simple one.  Lots of us either learn or realise that if we multiply whole numbers by 10, we can just add a zero to the end of the number to get the answer.  Most educators realise that this is a very limited trick but it allows children to develop ways to respond to questions before they can deal with the concept of place value.

The problem is that the technique suggests that you can just append numbers here and there.  It is a way to get a correct answer but not a mathematical operation that works in any other situation.

This type of problem is commonly seen in algebra when people talk about moving numbers or variables to the other side.  It’s not so bad to do this once you have a strong concept of algebra but before that, it tends to develop real misconceptions about how maths works.


If all you learn about multiplication is times tables and perhaps the column method, it won’t be too surprising if your concept of multiplication doesn’t go any further than repeated addition.  Repeated addition may be a form of multiplication but there is a lot more to it.  To have a robust internal conception of multiplication, you would want experiences of other ways that multiplication is used.  Natural Math have a great poster that shows 12 different models of multiplication

It is really important for developing understanding of more advanced topics, that your internal conception is able to accept all of these models.  In a way, you want your ideas to be open to more than the first technique that you learnt.  That way you form connections with other areas of maths and will be able to solve problems more creatively.


Here are a few things that will really help you broaden the concept of multiplication for your children or students.

1) Try the Multiplication Explorers course by Natural Math

2) Use grids, tables and webs to model multiplication as well as lists

3) Practise with Prime Colours.  These numbers are coded according to the fundamental theorem of arithmetic and develop more robust thinking about multiplication by making the process visible and physical.

Prime Colours’ Chatterbox

There are lots of ways to practise. There aren’t many ways to practice that work if you don’t keep it up. But sometimes the routine can get to you. Here’s a way to change things up just enough in the beginning to keep a mindful practice going.

To download or for instructions go to Prime Colours chatterbox page

Prime Colours’ Chatterbox

If you’re at the West Norwood Feast today, come by to get a free DIY chatterbox.