Easy Multiples

In 2018 I decided to write a series of short posts about approaches or methods that teachers might not have seen before. When I share these posts, I am well aware that there will be many people who already know the thing I'm blogging about, but I figured that it's still worth sharing even if it's only new to a handful of people. My first post in this series was about using vectors for enlargements and my second post was about factorising by inspection. I then got really busy writing my book, and didn't add to this blog post series for four years! Oops. So today I'm relaunching the series with a very simple little 'trick' (not a trick at all, just maths).

My Year 6 daughter has recently learnt long division. To be clear on what I'm referring to, long division looks like this:

Whereas 'short division' looks like this (this is sometimes colloquially referred to as a 'bus stop method'):

The only difference between the two methods is that in short division we work out the remainders in our head and jot them down in the dividend, but in long division we work out the remainders on paper in a more structured format. If your divisor is greater than twelve (for example if you're dividing by 28) then it might be tricky to work out remainders in your head, so that's typically when the long division format might be preferred. But they're essentially the same method, just with a slightly different structure for processing the calculations.

It was funny to see my daughter learning long division as it's something that I literally never teach in secondary school. I was pleased with myself for remembering how it works. For many students it exists in Year 6 alone, never to be seen again. A typical Key Stage 2 SATs question might look like this:

But something like this is highly unlikely to come up at GCSE. Students do sometimes have to do divisions by hand in their non-calculator GCSE exam (an example is shown below, from the Foundation tier), but I think most students would choose to use short division.

Some people argue that the long division algorithm is used again when students learn algebraic division in Year 12. This may have been the case ten years ago, but I think that most(?) A level teachers now prefer more intuitive methods of polynomial division, like the factor method shown below for example. 

So for the most part, long division resides solely in Year 6. And my daughter, who is in the 'middle' group for maths, was coping fine with it, but she told me that she finds it tricky to write out the multiples at the start. For example when she's dividing by 28, she's been told to begin by writing out some multiples of 28. She finds this time-consuming, a bit tricky, and rather dull.

But don't worry, because there's a really simple way to write out the multiples of 28. My colleague Sian showed me this - she picked it up a few years ago from her daughter's Year 6 teacher. I showed my daughter, who loved it - she was then able to master long division as she'd found a way round the tricky bit.

To quickly and easily write out the multiples of 28, just write the multiples of 20 and the multiples of 8 and add them together:

As long as the child knows their standard times tables fairly well, listing the two sets of multiples is straightforward. And the addition is pretty straightforward too, as they are always adding to a multiple of ten.

Here's another example: multiples of 17.

This may already be really widely used by Year 6 teachers. But in case anyone hadn't thought about this super simple way of listing multiples, I thought it worth sharing here. As I've always said, even if it just helps one person then it's worth taking the time to write about it.

Algebraic Division

During public exam season I'm afraid I can't resist looking at The Student Room - not to read all the infuriating student posts where they obsess about lost marks and grade boundaries, but simply to see each exam paper as soon as possible. I have no patience. Arsey, presumably an anonymous teacher, posts model answers very soon after the exams have taken place (in previous years not until the next day, but now there are different papers for different time zones the model answers are posted straight after the UK sitting). When I was reading Arsey's recent C2 solutions, I noticed his method for polynomial division:

I've seen him do this before. It always occurs to me that this is far more straightforward than the long division method I teach my students. Why do I teach long division?  I suppose it's because it features in the C2 textbook so I've always assumed it's the 'best' method. Perhaps next year I'll try an alternative. In this post I look at four methods for polynomial division. 

1. Long Division

This method often features in A level textbooks. It just involves following a series of steps (divide, multiply, subtract, bring down, repeat) - an algorithm learnt by drill rather than through understanding. The steps are familiar to those who learnt long division at primary school. Those who were taught alternative division methods (eg chunking) are at a slight disadvantage but do catch up quickly. Practice makes perfect. This isn't an elegant method - it's totally procedural and isn't particularly nice to teach, and students have to know special rules for special cases (eg including a 0x term) - but it does the job just fine. It works well when there's a remainder.

2. Grid/Box Method
I've just tried this method for the first time and I can't believe how easy it is - and so much quicker than long division! All you have to do is set up a multiplication grid - start by filling in the bits you know and then the rest follows by logic. This video from Bon Crowder explains the method very clearly. James Tanton calls this the Galley Method - his Curriculum Essay about how it works includes exercises and interesting questions. The picture below from Esther (@MrsMathematica) shows a sensible layout which makes dealing with remainders very easy.

3. Inspection
This is like the grid method but set out differently. All you have to do is write your polynomial as the product of a linear function and an unknown quadratic (or cubic, quartic etc, depending on the question) then use logic and algebra to work out the numbers by equating the coefficients. It's quick and fairly straightforward. It's also easier to follow what's going on than in the confusing algorithm of long division.

In a 1940s textbook I spotted an alternative layout for inspection that I love. In the example below we want to divide by x - 1 so we write (x - 1) three times and then just fill in the rest. Again, it's quick and logical, and easy to keep track of each term.

4. Synthetic Division

I don't like this method so I don't really want to mention it here, but for completeness I suppose I should. I've found lots of (often negative) reference to it on American websites but I've never seen it used in England. I'm told that it's commonly used in Scotland (thanks to @mrallanmaths, @kenniejp23 and Paul Smith for commenting). The reason I dislike it is it appears to be one of those 'remember the steps but have no clue what's actually happening' methods.

Source: acedemic.utep.edu

I followed the rules above and it did work, plus it was pretty quick, but it was a bit like doing a magic trick.

The animation below shows the equivalence of long division and synthetic division. It looks to me that synthetic division is just a confusing method made even more abstract. There are many defenders of synthetic division though - they say that it's an acceptable method providing students are taught the underlying concept before they start applying the super-efficient algorithm.

Source: purplemath.com

So that's it - four methods for dividing polynomials. This PowerPoint from the Further Mathematics Support Programme summarises the first three methods.

Which do you prefer?