## Exploring the link between addition and multiplication

Today’s Cuisenaire rod session was quite interesting. After aying and looking at some stuff that was similar to previous posts my daughter came up with this sequence:

(Again, please forgive the ordering the table is quite small).

She decided that she wanted to add how much each sequence was worth:

She started with tallys as she had used before, then asked if there was a quicker way. I got her to think about what was going on and she decided she could use multiplying :

After she did the one with 3s we had a discussion about the = symbol and what it meant and why it was wrong to use it the way she had initially.

When she did the 5 one she said “is that the wrong way round”, which led to a nice discussion on the commutative law.

After she had done a few she realised she could miss out a step:

When she did the one with the 8 I said she didn’t need to +1 on a different line and explained why, but she said she wanted to keep doing it to show it was separate. We then discussed the order of operations.

I think this task is an excellent way of seeing why multiplication would take precedence over addition when we come to looking at the order of operations.

I think this task and discussion were a good way to embed the link between repeated addition and multiplication, and to lay foundations for algebraic reasoning when it comes to collecting like terms. I can see that for older students it would also be a great way to show and think about the position to term relationship in a sequence.

This is the 9th post in a series about the use of manipulatives in teaching mathematics. The others can be viewed from here.

## Part whole division – why?

I’ve been thinking a lot about division recently. I wrote this here a short while ago about dividing by fractions, then I was sent a document by Andrew Harris (2001) entitled “Multiplication and Division”, which I was asked to read as part of a series if CPD sessions from the local maths hub, then a number of different people have asked me questions about division recently too. I think probably for most this is due to helping their own kids with maths and meeting methods and structures that they aren’t familiar with, as they weren’t taught when they were at school themselves.

The main thing from friends that keeps popping up is using part whole models for division. And funnily enough it is one of the structures I was considering after reading the Harris document and looking at the distributive law and what higher level topics this underpins in later maths.

*So what is it?*

Using the part whole method for division is where you split a number into 2 or more parts before dividing then add your answers back at the end. For instance, if you want to divide 486 by 6 you can split it into 480 and 6. The benefit of choosing these numbers is that 48 is in the six times table. So you can see that 48÷6=8 so 480÷6=80, then you have 6÷6=1, add them together and you get 486÷6=81.

This structure, or method, is a very common mental strategy used by lots of people when dividing numbers in their head. Lots of those people will never have heard the term “part whole model” and will not have seen it laid out in a pictorial manner as students today will, but they will use that structure nonetheless. I myself was using it as a mental strategy a long time before I’d heard anyone refer to a part whole model or seen the visual representations.

What we are doing when we do this is using the distributive law of multiplication and division to break our problem into chunks that are easier to manage.

One of the questions I was asked was “is there a rule to how you split it up?” The person who asked me was wondering if you always split it up into hundreds, tens, ones etc or if you could do any. I explained that it didn’t matter, and that actually the divisor would normally be important in deciding. For instance if you were dividing 423 by 3 it wouldn’t make much sense to use 400 as this isnt divisible by 3. It would be more sensible to choose 300 (÷3=100), 120 (÷3=40) and 3(÷3=1).

*But why not use short or long division?*

This is a question I’ve seen a lot of times from a lot of people. They see the part whole method as a long and clumsy way to solve problems that they can solve easily using one of the two standard algorithms. I can see the point in asking, the algorithms are far more efficient as written methods. But that’s not why this model is taught. No one expects students to get to their GCSE and start drawing part whole models to solve division problems. The visual representations are their to help build an understanding of what is going on, an understanding of the relationship between numbers and mathematical operations. In this case it’s to build an understanding of how the distributive law works and to give a good mental strategy for division. It even helps understand how the long and short division algorithms work, as they are both based on splitting the dividend up into parts. There must come a point when these structures and representations are removed and students move to the abstract, but that doesn’t devalue their importance to that learning journey.

*What else is the model used for?*

The idea of a part whole diagram is introduced way earlier than this. Students get used to partitioning numbers into part whole models while working on addition and subtraction. It helps then see at that level that they have a relationship, that they are the inverse of each other. So when students come to meet this model for division it’s a small step on what they were already doing.

These are similar to some of the earliest part whole models my daughter did when she started school. They were being used to show place value, and also to show how addition and subtraction work and interact. For both these tasks this model is an excellent visual representation to help students understand the concepts.

Part whole models can also be represented as bar models. Here the one on the left can again be used to show either place value or addition/subtraction. The one on the right is an early algebraic model, and if we are told that x+2=9 we can use this representation to show why x must equal 7. This representation is more effective if students are familiar with it from their earlier mathematics.

Building on this we can show the distributive law when it comes to multiplication:

And show how that links to division:

As we go further into maths this idea of part whole division comes up again and again. One place that springs to mind is when calculus is first introduced at A level. One of the first things that we teach is how do differentiate and integrate polynomials with different powers of x. And a favourite style of question from examiners is this:

Or its derivative equivalent.

The easiest way to do this, when it comes to integration or differentiation, is to rewrite the fraction as separate terms:

What we have done here is used the part whole model to divide the expression on the numerator by x^2. We could draw that in our part whole model:

I wouldn’t advise that, its unnecessary, but having a secure knowledge of that model and how it works due to the distributive law is key to understanding how and why we can simplify this fraction in that way.

*I’ve thought a lot about division recently, and I’m sure I will continue to do so, so if you agree,disagree or have anything else to add please get in touch either in the comments or via social media as I’d love to hear your views.*

## Vedic Multiplication

On Friday, timehop told me it was a year since I wrote this post on multiplication methods. I’d forgotten about the post, and tweeted a link to it. A number of people commented and tweeted about it and a nice discussion ensued. Part of the discussion moved to Vedic Multiplication.

I know a few Vedic Multiplication methods, and believe there to be many more. Most of the ones I know link to the most common algorithms but there is this curious one used for numbers close to one hundred.

Start with a multiplication problem:

First, take bother numbers away from 100:

Multiply those numbers and make them your last two digits:

Take one of the differences from 100 away from the other number (it should be the same):

That becomes your first two digits:

It’s an interesting little trick. I don’t see it as something that there is any reason to teach, and I don’t think it promotes understanding at all, but it’s interesting nonetheless. I think it may have a use in lessons, as an interesting introduction to algebraic proof.

**How would you prove it?**

First, consider the product ab, and apply the same steps:

The last two digits are (100-a)(100-b). The first two are a-(100-b) which equals a+b-100. Or b-(100-a) which also equals a+b-100. To make these the first two digits of a four digit number we need to multiply the expression by 100.

This gives:

**(100-a)(100-b)+100(a+b-100)**

Which expands to:

**10000-100a-100b+ab+100a+100b-10000**

Which cancels to:

**ab** As required.

A nice, accessible, algebraic proof that proves this works. It works for all numbers, not just those close to 100, but if your product (100-a)(100-b) > 99 (ie more than 2 digits) you need to carry the digits.

## Bizarre multiplication

A while ago I wrote this post about multiplication. In it I explored a few different ways of multiplying and what I felt worked best and why. I covered the main ones, and I realise there are many different methods.

However, today, while marking my year 8 books I discovered a method I had never seen before:

I looked at it and thought, “bizarre”. I asked the girl in question who taught her it and she said she’d didn’t know, but that it made sense to her but grid and column methods don’t.

I like the method, it reminds me of expanding a pair of binomial brackets, but it the numbers were 19 + 4 etc. It’s fantastic for 2 digit by 2 digital or 2 digit by 3 digit and I think it shows the distributive property quite well. There are major limitations though. It looks really neat in these two examples, but the questions that were 3 digit by 3 digit were a little spaghetti like and the ones with 4 digits were totally illegible.

*Have you seen this method before? Do you like it? I’d love to know!*

## Multiplication Methods

Last night I saw an intriguing tweet from @mr_chadwick . Mr Chadwick is a primary teacher and he was worried that his daughter who is in year 8 was still multiplying using the grid method. This caused a fascinating conversation regarding the different methods of completing multiplication tasks and the pros and cons of each and it got me thinking quite a lot about the subject.

There are many ways to complete multiplication problems, the main ones being Grid, Column (aka Standard or Long), and Chinese Grid (or Spagetti Method or Lattice Method). (Also, I have recently been shown this ancient method by an A Level pupil who grew up in the Democratic Republic of Congo).

I was wondering if any one was particularly faster than the other, so I tried some out:

Each method took me a little over a minute to compute the product of 2 three digit numbers. (Except the grid method which took about 15 seconds longer). From experience I know that people tend to prefer the method they were shown first, and I don’t see a problem using any of them as long as there is an understanding of the concepts and that the person in question is proficient at using the method they have chosen.

I did think my prefered method for teaching someone who didn’t know any methods would be the Lattice Method, as I see a much larger potential for silly errors in the other methods than i do for this, but last nights discussion has got me thinking a little differently. The discussion moved onto applications in algebra. I know a lot of people prefer to use the grid method to expand double brackets, I personally prefer crab claw method, but i teach both and allow my students to decide, and some much preefer the grid. It also works quite well for larger polynomials, as shown here (in a video which rather confusingly calls it the lattice method!). The grid method can also be applied to matrices, as I have written about previously.

I’m still unsure as to which i prefer. The Lattice method gives a far lower chance of making silly errors, and I think it is the best one fro ensuring the decimal point ends up in the correct place when multiplying decimals, but the grid does have the benefits of being applied to much higher levels! I’d welcome your views on the subject.

I feel that i should also include another multiplication method which I discovered a few years ago, I was introduced to it as Japanese multiplication, but I’ve recently heard of it referrred to Gorilla Multiplication. I think it may have its origins in india and if you want to know more then Alex Bellos has written about it in his book Alex’s Adventures in Numberland (Another on my christmas list… and a book who’s american title is the most amazing title for a maths book I’ve ever heard: “Here’s Looking at Euclid”!)

## Another Multiplication Technique

I’ve written a few posts over the years on different multiplication techniques (see this and this), there are many and each has its own appeals and pitfalls.

Today I discovered another. I was looking over Q D1 exam paper and came across this flowchart:

The questions were all fairly reasonable and one of my students was completing the question to see if he had for it right. Afterwards I asked if he knew. What the algorithm was doing, he wasn’t sure at first but when another student explained it was finding the product of x and y he realised.

Then he asked, “

but why does it work?”I looked at the algorithm and initially it didn’t jump out at me. I tried the algorithm with 64 and 8.

I could see it worked through mocking factors of 2 from the left to the right but this time there was no odd numbers, so I picked some other numbers:

And that’s when it all made sense. Basically, what’s happening is you are moving factors of 2 from x to y thus keeping the product equal. When x is odd, you remove “one” of y from your multiplication and put it in column t. Your product is actually xy + t all the way down, it’s just that until you take any out your value for t is 0. T is a running total of all you have taken from your product.

The above becomes:

40 × 20= 20 × 40= 10 × 80= 5 × 160= 4 × 160 + 160= 2 × 320 + 160= 1 × 640 + 160= 0 × 640 + 160 + 640= 0 × 640 + 800= 800I tried it out again to be sure:

This is an nice little multiplication method that works, I’m not sure it’s very practical, buy interesting nontheless.

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