Blog: John Jackson , Secondary Development Lead | Posted: 5/01/16

In the beginning was the number ten.

OK so that’s not quite how the Old Testament goes, but our commitment to the number ten certainly *is* of biblical proportions.

Not only is the entire Western decimal number system organised in tens – ones are grouped in tens, tens are grouped in hundreds, hundreds are grouped in thousands etc – the language we use is an exact description of this system (i.e. we know ‘thirteen’ means one ten and three ones – even though in many other languages it’s said as “one ten three” which is much easier!).

In primary school, years are spent teaching children about place value in base ten (the value of a digit depends on its place in the number, and each place has a value of ten times the place to its right). Children learn to compare the size of numbers and then progress to add, subtract, multiply and divide – all within the base ten system.

For higher-attaining secondary students, place value and multiplication are often regarded as done and dusted issues. However, our mastery approach emphasises challenging students through *depth* of subject. This can mean taking a so-called ‘easy’ topic and digging further and further into the concept to extract new challenges, new ways of thinking and problem-solving. After all, one of the many allures of mathematics is how it enables us to push boundaries and experiment with ideas.

For those students requiring an extra level of challenge in place value and multiplication (and to challenge the misconception that these are basic topics!), try asking them to answer the following conundrum:

*13 × 4 = 100. When is this true?*

To answer this, let’s reflect on place value a little more.

We use place value to help us count and group numbers and, as mentioned, we find counting in base ten very easy. We’ve done it all our lives, we have ten fingers (probably why we chose this system!) and the language we use and knowledge we have supports this system.

But what if we set aside the base ten system and introduce an entirely new system – base six for example.

Why do this? Well for starters, it really is a challenge. When we change the way we group (and therefore calculate) numbers, a lot of our support structure disappears. We no longer have the stored knowledge or the right language and – excuse the hyperbole – our mathematical belief system begins to shake.

Base six would mean we only have symbols 0, 1, 2, 3, 4, 5 at our disposal. The nature of base six means we now group every six ones into one six, every six sixes into one thirty-six (and even this is misleading as we can’t write ‘36’ in base six – because ‘36’ itself is reflective of base ten) and so on.

So in this scenario, six now plays the role of ten as we previously knew it. Counting would go something like this: 1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, 21… and here 15 means one six and five ones (i.e. what we are used to calling ‘eleven’).

Looks confusing, right? It can be tricky at first, because we’re changing an organisational system that many of us believe to be intrinsic to mathematics. The realisation that place value in base ten is, in fact, nothing more than a man-made ‘filing’ system can be surprisingly difficult to accept!

And this is where language intensifies the challenge. We can’t accurately call 15_{6} ‘fifteen’ because this specifically means one ten and five ones – whereas in base six 15_{6 } we mean one *six* and five ones.

As you can see, this makes calculations in different bases are a lot harder! When I do long multiplication in a different base, I find myself having to say it out loud to myself, reminiscent of how I first learnt it as a young child.

This brings us back to our conundrum: *13 × 4 = 100. When is this true?*

Think about the calculation in base six. The key thing to keep in mind is the language. Remember ‘13’ is not ‘thirteen’ and ‘100’ is not ‘one hundred’ as we know them.

In base six:

- ‘13’ now represents one six and three ones (i.e. what we’d normally call 9)
- ‘4’ is still ‘four’ (because it means the same in base ten or base six)
- ‘100’ now represents one group of thirty-six (and no sixes and no ones)

So 13_{6} × 4_{6} = 100_{6} is equivalent to saying 9_{10} × 4_{10} = 36_{10}.

So maybe multiplication isn’t always quite so easy after all!

The point of this is to illustrate how any topic can be deepened to enable students with different levels of attainment to be continually stretched and supported (as well as engaged and excited!) within the same conceptual idea. After all, place value in different bases is not only relevant from a conceptual standpoint. Computer data, for example, is represented using binary or base two, a number system that uses 0s and 1s.

Whilst those struggling with a topic may spend longer on the basics, the quicker-grasping students can continue to enhance their conceptual understanding by investigating some of the beauty of mathematics – what stays the same in different bases? What changes? This also helps build up mathematical flexibility and problem solving skills, which will again enable them to grasp complex ideas more rapidly and deeply in the future.

Have you tried teaching your students about calculations in different bases? Tweet us @MathsMastery.

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