Pressure - a rich new vein for compound measures and proportionality

First off - apologies for the lack of post in the last week. A combination of mounds of marking, open evening, and of course preparing for the session at #mathsconf5, has left me a little short on time! Nonetheless I am slowly getting back into the swing, and thought I would talk about my teaching of compound measures this week.

Typically, teaching compound measures at GCSE has meant teaching kids how to calculate speed, distance and time and then looking at density as mass over volume. Throw in a bit of time as a decimal hour and having to find volume from given lengths and that was that. However the new GCSE has provided a new rich source of teaching for compound measures that can highlight much more strongly the proportional and inversely proportional relationships - the calculation of pressure.

Having downloaded a solid exam question worksheet resource from mrbuckton4maths on TES to give the pupils an opportunity for consolidation, I felt the need for a final top end question to challenge the best and brightest in the class - and came up with this:

The box below exerts a pressure of 2.5 Pascals when in the orientation pictured. Calculate the pressure when the box is turned onto the shaded side.

………………………… Pascals

[2]

 I am sure I have seen a similar question in one of the SAMs (I am sure I got the inspiration from somewhere), however what I particularly like about this question is that despite 1 Pascal being the pressure exerted by a force of 1 N when spread over 1 metre squared, there is no need to convert the cm into metres to solve this problem. The inverse proportionality between area and pressure is all that is needed, as the start and end units are both Pascals. Simply multiplying 2.5 by 8 (=20) and then dividing 20 by 6 (= 3⅔) is enough to solve the problem correctly (hence only 2 marks), without considering the units of measure themselves. Although some won't like this approach I do really like the exploitation of the inverse proportion as an abstract process. What I also like is that the area of this block in contact with the surface changes depending on its orientation, unlike the volume which is fixed for each shape (in this case 24 cm2) which means that questions like the one above can be asked for pressure where they cannot be asked for density.

Now pressure is not a quantity that lots of maths teachers will have an in-depth knowledge of, but take my advice and talk to your science department colleagues about getting some questions about pressure into your lessons.

Shape and ratio - a nice place to mix topics.

It has become quite clear that one of the key aspects of the new GCSE will be pupils having to draw from different areas of maths to solve problems. As well as the standard "form the equation" from shape or angle properties we will be looking at mixes across the algebra, number, shape, ratio and data strands. I designed what I think is a nice problem with a mix of shape and ratio, an area I think will be a rich source of mixing for examiners given the renewed focus on proportional thinking in the new qualification.

I like this because it uses ratio in two different ways, as well as including area of a trapezium, which seems suitably challenging for KS3 or borderline GCSE pupils. So feel free to use (the resource and markscheme is linked here) and share any other interesting mixes of these topics, or others.

How many different... - a simple approach to depth and breadth

A long overdue change in maths teaching is taking place at the minute - a change from teaching techniques that work, with an appreciation of why the techniques work sometimes lacking, into teaching for a much greater understanding of the underlying concepts and a greater focus on comprehending why techniques work in the way they do to give the result they give. The term du jour is 'mastery', and with goes hand in hand the ideas of depth and breadth. No longer do we seek to provide our most able or 'high-fliers' with access to work that takes them beyond the maths they are studying; instead we seek to provide them with the opportunity to gain a deeper, more fuller understanding of the maths they are studying and experience a breadth of situations where that mathematics may apply. For many teachers this can be quite a challenge, particularly as many of us are the product of the previous approach, myself included, which means that any deeper understanding of this mathematics that we have come to, we have had to find ourselves. I mean no disrespect at all to my own teachers, most of whom I remember fondly, it was simply that the focus of the time was different. So now, with the need to stimulate a depth and breadth of understanding in pupils that we never achieved ourselves at the same point, the question remains how do we go about it. Well one way I have been using is to ask questions that immediately prompt pupils to explore similarities or differences, rather than just stop at recall. I am going to briefly outline two examples I have used recently, one with Year 7 and one with Year 10.

Like many schools we have a big focus on assuring basic numeracy. My Year 7 bottom set have a numeracy starter at the beginning of every lesson, and a starter I am using this week revolves around this image:

Now rather than simply ask what time is shown on the clock, or ask for times away from this time, my starter is simply "How many different ways can you find of writing the time on shown on this clock?". I have even put a hint up (as part of structuring the support for my bottom set) that reminds them to think about ways they might not write or say. The reason I feel that this has the potential to provide depth is that ultimately I have no idea what pupils will say. I have come up with about half a dozen responses, but the pupils could come up with more than I have, or simply different ones to me. Provided I am brave enough not to just reject wrong answers out of hand, but to explore answers and see how pupils have arrived at them, I am likely to be deepening both the pupils understanding of telling the time (a real problem for some) as well as my understanding of the pupils.

Again like many schools, we have our 'borderline' pupils working toward entry onto a reduced Higher tier (reduced in that we are realistic about the amount of the paper the pupils will focus their attention towards). Of course in the new GCSE a crucial part of this is the ability to reason proportionally, and a lot of our early work with these pupils is exploring different proportions and different ways of representing proportional relationships. I did this using Cuisenaire rods where pupils found the relationship shown in this image.

Most pupils were able to describe something like "5 purple bars = 2 yellows", but then I challenged them as to how many different ways of writing this relationship they could think of. There weren't many, but we did come up collectively with writing the ratio, the idea that there were 2.5 purple bars for every yellow etc. I also showed them a graphical representation and algebraically defining the relationship. The nice thing again though was the links we were then able to make in their previously studied maths.

There are a number of other opportunities to ask the question "How many different..." in maths lessons, a few others I will be using/have used:

• ways of calculating the perimeter/area?
• ways of writing this probability?
• ways of displaying this data?
• ways of changing the answer by putting in brackets?
• ways of drawing the factor tree.

I am sure other people will come up with others, but the point here is that if you need a relatively straightforward way of providing a bit of depth or breadth, try thinking about whether you can ask how many different ways of doing or seeing something your pupils can come up with.

Adding and Subtracting Fractions with bar models - worth sticking with.

Over my years in the profession I have seen lots of different ways of adding fractions; multiplying the bottoms immediately, Battenburg, going straight for the lowest common multiple, the list goes on. Recently as I have been deepening my understanding of fractions and moving to a point in my pedagogy where I really want pupils to really understand their maths I have eschewed these methods as giving pupils a 'how' without giving them the necessary 'why' - as Skemp wrote they would provide pupils with an instrumental understanding rather than a relational one. So going back to the fundamental representation of fractions I began working with pupils to understand how the visualisation of fractions leads to the addition and subtraction of fractions. The approach works a little like this:

Consider these two fractions:

The bar model approach to adding fractions would be thinking about how to combine these two pictures into a single picture. If we try and do this directly, this is the picture that results.

Of course the problem here is that we cannot evaluate the fraction, because there are parts that are different sizes (my kids were quite happy that this wasn't 5/12). Once we have demonstrated that the major stumbling block is that the pieces in the combined picture are different sizes, the natural conclusion is that we need to adapt the original pictures so that they have the same size pieces.

The crucial next step, which took me a little time to get right, was generating the understanding that starting from a picture with 8 pieces, we can only create pictures with a multiple of 8 pieces, i.e. 16 pieces, 24 pieces etc (like so)

drawing similar pictures with the sixths makes the point that you can only create twelths, eighteenths, etc.

The reason this is so important is because it allows us to justify why the lowest common denominator is useful in adding and subtracting fractions - if we need the diagrams to have the same size (and hence number of) pieces, and we can only change each diagram to have a multiple of the number of pieces it starts with, then this implies that the number of pieces each diagram should be broken into is the lowest common multiple of the two. I must admit that this took me a little while to get through with pupils, mainly because I didn't explain that crucial step properly until the third lesson when I really identified what was holding the pupils up. Once I had explained this properly, and pupils caught the concept quite quickly, everyone was ready to accept that the pictures should be changed into twenty-fourths.

At that point everything fell into place quite nicely; once we changed both pictures into twenty-fourths we were able to show that the addition of the two fractions was just the combining of the two diagrams into a single diagram like so:

and that the result of subtracting the two fractions was the same as looking at the difference between the two shaded areas (I physically removed the two shaded areas that were the same size):

The thing that really struck me was that although it took 3 lessons to get through (again partly because I had to re-think my way past a wall part way through), once those 3 lessons were over a significant number of pupils understood the adding and subtracting of fractions.One girl even said "I need the diagrams" when she was still consolidating her learning. Importantly, from the vibe and atmosphere I was getting in the classroom it felt that this was a deeper understanding than just the monotonous work of a group of pupil shaped robots answering question after question requiring the addition and subtraction of fractions using a method that they honestly have no idea of why it works. Instead it felt like the pupils had begun to really appreciate how the representation of the fractions led to the need to find lowest common denominators (or at least common denominators), and how this led to the idea of multiplying both numerator and denominator by the same value (if you are slicing each piece into three smaller pieces, as in the case of the eighths above, then you are slicing all the shaded pieces into 3 smaller ones at the same time).

So can I take this opportunity to urge everyone who has made it this far down the blog to try and use the visual models to really give pupils an understanding of how fraction calculations work (I am using a similar approach for multiplication next week); it may take longer, it may seem like pupils are struggling (they probably are) and it may seem like you are wasting time when you could just teach them 'a method'; but trust me if my experience is anything to go by, if you can nurse them through those struggles their appreciation and understanding of why fractions add and subtract in the way that they do will be so much greater.

Team Challenge, inspired by UKMT

This week my new GCSE class will get their first taster of one of my favourite activities, the UKMT inspired team challenge. I find this sort of activity really does get pupils thinking and discussing (and sometimes even arguing) about the maths they are doing, so I thought I would take the time to share how it works.

The idea is inspired by the UKMT Team challenge round known as 'Shuttle' (formerly 'Mini-Relay' or 'Head to Head') in as far as it has 4 questions and pupils are scored 3 points if they get the question right first time, or 1 point if they get it right eventually. Typically I don't have each team split into pairs like the UKMT do, nor do I use the answer to the last question in the next one; instead the team are only given question 1 to start with, once they answer it they bring it to me at the front of the room. If they answer correctly they get their points as above and the next question, if not they get sent back with their previous question. Of course like any good challenge, the questions get harder as you go (at least IMO!)

The challenge I am doing on Friday with my new Year 10 set 5 (of 6, so higher tier pupils, but needing plenty of support) is about ratio and proportion, so following the link here will provide you with the four questions I am using (good example of more involved ratio questions on their own, even if you don't fancy using the team challenge) as well as a generic score recording sheet to use with any shuttle challenge (note you will need to create an account on TES if you don't have one in order to download).

Of course the beauty of this idea is that it allows you to get kids doing really hard questions (can't make them too easy or they will be finished in no time!) with a smile on their faces (most of the time anyway). Watching the way they try and convince each other is great fun; particularly if you have a member of support staff (like I do) there to act as the scorer so you can get involved with the pupils where necessary. Next time you are stuck for a lesson idea why not give it a go? It means you only have to write 4 questions instead of 10!

The Migrant crisis and Maths

I am sure many of us have been following the growing migrant crisis affecting, in particular, the Mediterranean region. The number of deaths during travel, coupled with the logistical problems of settling the migrants once they have left their country of origin are two of the biggest migration issues faced by Europe for a long time. A colleague of mine recently sent me a link to a BBC news article about the issue containing some truly thought-provoking statistics; and of course I couldn't read the article without that little maths teacher area of my brain firing with uses for the graphs and charts shown. Here are some of the statistical representations that the article used to report on the issue:

The mathematical possibilities here are quite striking. There are some great representations of proportion and percentage, probability, circles etc in addition to the obvious bar charts and pie charts problems that can be posed.

So taken was I that I immediately came home this evening and created 3 worksheets/questions that can used with these stimuli; one on bar charts, on on pie charts and one on proportion. They are just a flavour of the sort of questions that can be asked about these stimuli, but are useful enough for themselves. The worksheets can be found here and I would love to see other people develop questions from these or other representations around this very important issue.