## 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.*

## A lovely circle problem – two ways

So, I was working with some year 12s on a few problems around circles out of the new Pearson A Level textbook. (Incidentally, it’s this book, and I think it’s probably the best textbook I’ve come across. I would certainly recommend it.)

This question appears in a mixed exercise on circles:

It’s a lovely question. Before reading on, have a go at it – or at least have a think about what approach you’d take – as I’m going to discuss a couple of methods and I’d be interested to know how everyone else approached it.

Method 1:

I looked at this problem and saw right angled triangles with the hypotenuse root 52. I knew the gradient of the radii must be -2/3 as each radius met a tangent of radius gradient 3/2. From there it followed logically that the ratio of vertical side : horizontal side is 2 : 3.

Using this I could call the vertical side 2k and the horizontal side 3k. Pythagoras’s Theorem then gives 13k^2 = 52, which leads to k^2 is 4 and then k is 2 (or -2).

So the magnitude of the vertical side is 4 and of the horizontal side is 6.

From here it follows nicely that p is (-3,1) and q is (9, -7).

Finally there was just the case working out the equation given a gradient and a point.

L1: y – 1 = (3/2)(x +3)

2y – 2 = 3x + 9

3x – 2y + 11 = 0

L2: y + 7 = (3/2)(x -9)

2y + 14 = 3x – 27

3x – 2y – 41 = 0

I thought this was a lovely solution, but it seemed like a rather small amount of work for an 8 mark question. This made me wonder what the marks would be for, and then it occurred to me that perhaps this wasn’t the method the question writer had planned. Perhaps they had anticipated a more algebraic approach.

Method 2:

I had the equation of a circle: (x – 3)^2 + (y + 3)^2 = 52. I also knew that each tangent had the equation y = (3/2)x + c. It follows that if I solve these simultaneously I will end up with a quadratic that has coefficients and constants in terms of c. As the lines are tangents, I need the solution to be equal roots, so by setting the discriminant equal to zero I should get a quadratic in c which will solve to give me my 2 y intercepts. Here are the photos of my workings.

As you can see, this leads to the same answer, but took a lot more work.

*I’d love to know how you, or your students, would tackle this problem.*

## A little circle problem

I’ve just seen this post from Colin Beveridge (@icecolbeveridge) answering this question:

Naturally I had a go at it before reading Colin’s solution. When I read his I found a lovely concise solution that we slightly different to mine, so I thought I’d share mine.

I started by just drawing a right angled triangle from the centre of the circle like so.

I seem to have cut off the denominator of 6 on the angle there. I know that the hypotenuse is 12-r (where r is radius) and the side opposite the known angle is r.

This means I can use the sine ratio of pi/6 to get

r/(12-r) = 1/2

Which leads to:

2r = 12 – r

Then

3r = 12

r = 4

Which is the same as Colin got.

I’ve seen questions like this on A level papers before and I know they often throw students, so I make sure I explore lots of geometry based problems and puzzles to combat this. I’d be interested to know which way you would approach this. Colin and I used a very similar approach, just differing in the point at which we introduced the 12. Which way did you do it?

## Proving Products

Just now one of the great maths based pages I follow shared this:

So naturally I figured I would have a go. I thoughts just get stuck in with the algebra and see what happens, normally a good approach to these things.

My first thought was that if I use 2n – 3, 2n -1, 2n + 1 and 2n +3 then tgere would be less to simplify later. I know that (2n + 1)(2n – 1) = 4n^2 – 1 and (2n – 3)(2n + 3) = 4n^2 – 9 so I multiples these together.

(4n^2 – 1)(4n^2 -9) = 16n^4 – 40n^2 + 9

I thought the best next move would be to complete the square:

(4n^2 – 5)^2 – 16

This shows me that the product of 4 consecutive odd numbers is always 16 less than a perfect square and as such that the product of 4 consecutive odd numbers plus 16 is always a square.

(4n^2 – 5)^2 – 16 + 16 = (4n^2 – 5)^2

A nice little proof to try next time you teach it to your year 11s.

## A lovely old problem

Recently Ed Southall shared this problem from 1976:

I’m not entirely sure if it is from an A level or and O level paper. It covers topics that currently sit on the A level, but I think calculus was on the O level at some point. *Edit: it’s O level* I saw the question and couldn’t help but have a try at it.

First, I drew the diagram – of course:

I have the coordinates of P, and hence N so I needed to work out the coordinates of Q. To do this I differentiated to get the gradient of a tangent and followed to get the gradient of a tangent at P.

Next I found the equation, and hence the X intercept.

And then, because I’m am idiot, I decided to work out the Y coordinate I already knew and had used!

The word in brackets is duh…..

Now I had all three point.

It was a simple division to find the tangent ratio of the angle.

The next 2 parts were trivial:

And then I misread the question and assumed I’d been asked to find the shaded region (actually part d).

Because I decided calculators were probably not widely available in 1976 I did it without one:

I thought it was quite a lot of complicated simplifying, but then I saw part c and the nice answer it gives:

Which makes the simplifying in part d simpler:

*I thought this was a lovely question and I found it enjoyable to do. It tests a number of skills together and although it is scaffolded it still requires a little bit of thinking. I hope to see some nice big questions like this on the new specification.*

*Edit: The front cover of the paper:*

## Mathematical Style

Yesterday I read this post from Tom Bennison (@DrBennison). The post was written to start a conversation for a twitter chat that I unfortunately couldn’t make. It did, however, make me think.

He was questioning whether mathematical elegance and style should be assessed at A level. Suggesting that solutions with more elegance should be awarded more marks.

Bizarrely, the example he used was almost exactly the same as a discussion if had with a year 13 class not long before I read his post. His example was finding the turning point of a quadratic. He looked at two methods – completing the square and differentiation – and suggested that as CTS is more elegant that should be worth more.

I agree immensely that CTS is a preferable method with far more elegance, but I don’t think the marks should be different depending on the method you choose. I feel that we should be encouraging mathematical thought, trying to create young mathematicians who can apply themselves to a problem and find their own way through. I feel if we start assigning marks for elegance and style them we would be moving towards the “guess what’s in my head” style of assessment that I feel we need to be moving away from. The way to do well would be to spot from a question what the examiner wants, rather than to apply the mathematical tools at ones disposal and find a solution.

Back to that Y13 lesson I mentioned, we were looking back over some C3 functions work and one of the questions involved finding the range of a quadratic function – so obviously it was necessary to find the minimum. A discussion ensued as to how to do this with students coming up with 3 valid methods. The two mentioned above, both of which I find quite elegant, although I do much prefer CTS. The third method was suggested by one student who said “it’s -b/a – you just do -b/a” I knew what he meant – he was saying that this was the x value where the minimum occurred and that you put that in to find y, but he didn’t really understand what it was or why. He’d come across the method online and has learned it as a trick. When I showed him it came from completing the square and looking at it as a graph transformation, I saw the light bulb come on.

It is an interesting discussion. Some methods are far more elegan, and some are just algorithmic tricks. I think that the lack of understanding with these tricks will lead to marks being lost. So perhaps this will self regulate.

*I’d love to hear your views in this, which way would you tackle finding the minimum of a quadratic? And do you think we should assign marks to elegance and style?*

## Passivity in the maths classroom

Today I managed to find a few minutes to browse the latest issue of Maths Teaching, the ATM journal. One article that caught my eye was the “from the archive” section, where Danny Brown (@dannytybrown) introduced an article that was first published in 1957. The article was written by Ruben Schramm and is entitled “The student’s passive attitude towards mathematics and his activities.”

The article discusses mathematics teaching, particularly the nature of students who often, for whatever reason try to find an algorithmic method to follow to solve a problem, looking to recognise the problem and answer it in a similar way to how they have answered questions before. This is a problem that was obviously prevalent in the 1950s, as evidenced in the paper, but it is still prevalent now, and I feel the nature of our exam system must at least hold a portion of the blame. The questions on exams tend to be very similar and students will learn methods to answer them whether the teachers like it or not. This is one issue I hope will be dampened a little with the upcoming changes to the exams.

Schaum suggests that this passivity in maths, this tendency to look for algorithms, is in part down to how students see mathematics. He suggests that when they see teachers solve problems on the board by delivering a slick, scripted solution they can get a feeling that it is via “witchcraft” and see the whole process of uncoordinated steps, rather than a series of interconnected mathematical ideas. The latter would encourage the students to drive the mathematics from their internal ideas, and this would lead to them being more able to apply their knowledge in new contexts. If we can develop this at all levels then I feel we really would be educating mathematicians – ie giving students the skills to be able to apply their knowledge in new contexts, rather than teaching them to follow a recipe to answer a question.

Schaum goes on to discuss authority, the infallible authority that students see in their teachers and in the mathematical theorems and formulae. It is suggested that students see these theorems as infallible, and as such they reach out for them in their memories and try to apply them to problems. This can mean that the problem they are applying them too is only vaguely similar to the problem the theorem or method is actually there to solve. Schaum calls these “analogy mistakes”, and suggests that it is down to how comfortable with the content students feel that mean they revert to them. I feel that this is true in part, but that also the pressure of exams can lead students to confuse things in their head if they have opted to learn algorithms rather than looking to develop a deeper understanding.

I’ve had a couple of examples of these “analogy mistakes” in lessons and exams recently. A year 12 student came to an afterschool elective as she was trying to solve some coordinate geometry problems involving tangents. She had gotten herself really confused because in her notes she had written tangent gradient is perpendicular (when discussing circles) but she didn’t think it should be perpendicular because a tangent at a point should have the same gradient as the curve. I spend a little time discussing where her misconception had come from (her notes should have said “perpendicular to the radius”) and discussed how she could remember this more easily if she has thought about the graphs and sketched them.

Another example was in a recent exam one of my students had answered part of a question on alternative from incorrectly, she had done the alternative form bit well and the answer was 25 Sin(x + a), but it then asked her for the maximum she had written -25. When I questioned her about this after it seems she had fallen victim to an “analogy mistake”, she had remembered that “maximum is positive” when discussing second derivatives and in the pressure of the exam this memory had taken over, rather than the rational thought process that should have flagged up that the maximum or the function would be 25, which is definitely bigger than -25.

In his preface Danny Brown suggested that one way to counteract this would be by questioning and discussion, if we remove the authority from the discussion and don’t validate the answers by issuing statements saying they are correct or incorrect, but rather open them as conjecture to the class who then can discuss this, then we can allow students to develop their own mathematical ideas. Lampert (2001) also discussed this idea and suggests that as teachers we need to be striking the right balance between allowing students to discuss and conjecture and ensuring they understand what is important and aren’t making mistakes. This is something I strive for in my own classroom, and something I am currently working on trying to improve.

*This post was cross posted to Betterqs here.*

## An Interesting Conics Question

My AS further maths class and I have finished the scheme for learning for the year, leaving oodles of time for review, recap and plenty of practice. Currently we are revisiting topics that they either didn’t score too well on the last mock in or that they have requested we look at again due to lacking confidence.

Today we were looking at conic sections, by request of the students, and this past paper question caught my eye:

We had recapped the topic quickly together then the students attempted so past paper questions. This one was one they rook to well, all competing the first 2 sections quickly and without trouble. Likewise, part c was mostly uneventful – bar one or two silly substitution errors and someone missing a difference of two squares factorisation.

Then they got to part d.

This caused them a little bit of worry so we worked together on it as a class, after we’d made sure everyone had a, b and c right.

Here’s what the board looked like when we’d done:

I asked the class what we should do to start, one suggested drawing a diagram *all this nagging about always drawing a diagram, especially if your stuck is paying off! *We drew it, but it didn’t help much:

Then one said, “if they’re parallel then one gradient is -1 over the other” – I refrained from scolding and calmly said “indeed, those gradients will be *negative reciprocals *of each other”. I think asked what the gradients were and quickly had “y1 – y2 over x1 minus x2” thrown at me.

So we worked out the gradient of the line join in n to the origin:

Then the gradient of PQ:

One then suggested we put the equal to each other, but he was corrected by another student before I could react. So we set one equal to the negative reciprocal of the other and solved:

A lovely question, with a lovely neat answer and a load of fun algebra on the way. I enjoyed watching the students tackle it and was glad I didn’t need to put too much input in myself.

We then discussed the answer and the class expressed surprise that 1 was the final answer and that a complicated algebraic journey could end so simply. It was a nice discussion and a nice way to start the day, and the week.

*This post was cross-posted to One Good Thing and Betterqs*

## Activity Networks and Critical Paths

When I first came to teach Activity Networks and Critical Paths I couldn’t really remember being taught them. I knew I had studied critical path analysis, but my memories of them were no less vague than that. In order to teach them I refreshed my memory using an edexcel textbook, as it was the board we used, and I got to grips with the “on arc” method.

This year I’ve been teaching a class as part of the core maths early adopters programme and we have selected the AQA qualification. For our optional content we chose the critical path option (option 2b). When browsing the mark scheme for the specimen paper I saw this:

Apparently the “on node” method. When I first looked at it it seemed strange. I had a good think about it and once I’d got my head past the fact that the “late time” was the latest an activity could finish, rather than the latest an activity could start – as in the “on arc” method, I think I actually prefer it.

The thing my students find hardest every year is drawing an activity network, especially where dummies are involved, and it seemed to me that this “on node” method would be much easier in this respect, and that dummies wouldn’t need to be considered. I did worry that actually identifying critical activities, and hence the critical path, might be a little more difficult, but I still didn’t think it would be too taxing.

So when I came to teach my year 12 core maths class activity Networks I went with this “on node” method. I’ve never known a class get the hang of drawing activity networks so quickly, and as this was a core maths class, rather than an a level class, their GCSE grades are much lower – mainly Cs compared to mainly As.

They also didn’t have a problem with the early and late times or identifying critical activities. I much prefer this method.

It got me thinking about the validity of each method and whether either would be allowable in an exam. I feel that both methods are valid and that we should be teaching how to solve these problems using maths, so each method should be allowable, but I’m not sure whether the exam boards agree.

Close inspection of the specimen mark scheme for AQA core maths paper 2b certainly implies that either method is allowable:

However a look at the Edexcel markscheme doesn’t:

If you look at the question paper and answer book there is in fact the heavy implication that only one method will suffice, as there is an explicit mention of dummies and the network is already set up for this method to be filled in:

*Do you have a strong preference to either method? Do you think exam boards should be prescribing which method to use? Do you have any further insights into which method the other exam boards favour or prescribe? If so, I’d love to hear about it.*

## Terminal Exams

Earlier this week I came across this post from 2014 in which I was thinking about the then upcoming switch from modular A Levels to linear A Levels. A move from short exams occurring throughout the course to 3 long ones at the end.

I was excited by the prospect, mainly because the earlier stuff is easier once you have learned the harder stuff, and on top of that their is more time to teach if you dont have to go into revision for exam mode every few months.

We’ve been through it a few times now, so I thought it might be a good time to revisit the subject.

What’s happened?We switched from 6 short modular exams to 3 terminal exams, and the link between AS and A Level was severed so marks gained in Y12 no longer count toward your final grade. This to me made a lot of sense and I didn’t want to enter any for AS exams. The first year we didn’t enter them, but since the decision has been made at a whole trust level to enter, so that those who leave to go on to apprentices etc still get a qualification.

That first year we were the only subject in our school not to enter, and we gave them internal exams at the same time. We didn’t have anything to base grade boundaries on and it didn’t tell us much about the students abilities. The next year we struggled to get through all the year 12 content in time and didn’t have as much exam prep time as we would have liked before sitting AS exams. This year we re did the SFL and were done teaching new content by mid march, which was handy in the end.

I’m in two minds about AS exams, they don’t count for anything once they finish year 13 and it adds additional pressure. But it does give them additional impetus to revise and ensure the Y12 content is thoroughly stuck in their minds. When all the subjects but us sat them we found that the students revised a lot more for their other subjects which makes me think that it should at least be a school wide decision.

What about the other exams?Losing the repeated exam windows has definitely helped with the scheduling if learning. Its given me the flexibility to teach in the order I feel is lost sensible, rather than teaching stuff in a certain order as that happens to be the module it was put in. The fact we dont need to go into exam mode and revision mode as often is also of benefit allowing more time to focus on understanding the mathematics and the concepts that students need to understand in order to succeed. For certain students though, I feel they would benefit from that “oh crap” moment when they don’t revise for the first exam and fluff it. But I think thise students are less common than some suspect.

How do the exams compare?Obviously the spec has changed, so it’s not a direct comparison, but I feel that the linear approach allows a wider range of questions, different topics can be merged into different questions and there have been some really nice racing questions so far. My worries in 2014 about the length of the exams seem unfounded as I’ve not heard anyone complaining about the length, although have heard some of them say they needed more time.

What about the spec?I like the new spec. It has lots of fun maths in it and I feel it’s a good broad range if maths to know for moving forward into higher education. I’ve very much enjoyed teaching it. I miss some of the stuff that isn’t there, but all in all I think it’s a good spec.

In summaryI still think that the linear model with terminal assessments is preferable over a modular model. I’m torn as to whether I think AS exams should be sat or not and I’m a fan of the new spec on the whole.

What are your opinions? Do you agree or do you think modular was/is better? Why have you come to this conclusion? I’d love to hear all views from both sides, please let me know in the comments, via email or via social media.

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