Introducing Loci - Cones and chalk outside!

I was very fortunate this week that the recent wet and nasty weather that had permeated the East Midlands following our lovely September weather cleared away on Thursday to leave a dry (if somewhat crisp) day as it allowed me to run one of my favourite lessons - outdoor loci.

Having grouped the pupils accordingly it was off out on to the hard standing outside the main school building with some tape measures, some small cones (borrowed from PE) and some bits of chalk to start building and drawing loci.

I set up the 7 standard stations for loci - a point, two points, a line, 2 lines at an angle, 2 parallel lines, a point and a line, and a corner (we actually used the corner of the building for this). Each group was assigned a picture and had to use the chalk and/or cones to build the given "locus" - i.e. either show all the points at a fixed distance (I told them 1 metre) or equidistant as appropriate.They had about 15 to 20 minutes or so to build their picture (an interesting mix of chalk and cones used in some places) and then we came back together and toured each group's diagram, with the group explaining what they had (with varying degrees of success) drawn. All of the groups could at least articulate what they were trying to create, even though in some cases the accuracy suffered through working on concrete, and so no learning was lost here.

Having studied construction techniques in a previous lesson, and so with pupils already familiar with terminology like "perpendicular" and "angle bisector" we were able to tease out where these pictures were appearing, as well as some interesting ones like the parabola from the point and the line (we had studied sequences, including quadratic sequences in the previous topic, and so was able to allude to the link even if I didn't want to explore it in any great depth). What was great though is then in today's lesson, when we were then constructing loci using the standard techniques, the lesson was much smoother because there was already an understanding of the sorts of pictures we were expecting to see. This meant the focus could be entirely on interpreting the instructions and then constructing the appropriate shape - without the usual having to work out what the shape looked like first.

So in summary, why not do something different the next time you teach loci? Not only will it help when it comes to the actual construction skills, it will also show the kids that different side of maths that is not just done with pens and paper.

Expanding three binomials - developing conceptual understanding

I have been doing some work with my NQTs recently on developing their conceptual understanding of maths and looking at how we can ensure that our approaches in one area allow proper generalisation and links to other areas. One of the things we were discussing was the idea of brackets expansion, partly motivated by the need to expand products of 3 brackets in the new GCSE. We started looking at why brackets expand the way they do, using images like this:

and drawing the links between the expansion and area. Logically of course this leads to the idea of volume for 3 brackets expansion, but this led me to a problem (one I was discussing a little on Twitter recently), how does this generalise the approach? Allow me to elaborate with an example

This cuboid could be broken down and so the concept of multiplying all of the parts together can be demonstrated, but as a method for completing the calculation this is inefficient to say the least. So how to generalise a method and retain the conceptual understanding? I was puzzling over this when it hit me - treat the cuboid as a prism and apply the prism volume formula volume = area of cross section x depth.

This may seem like a small thing, but what it allows is a generalisation of methods for multiplying two brackets. Whilst I don't think this could be applied to the FOIL method, certainly applying to the grid method would work:

Now this might seem to be an obvious approach to most people, but the real power of it to me was that I could link it back to the volume, with the first calculation giving the area of the side face, and then this area being multiplied by the third length. My NQTs and I can also apply a direct expansion method; i.e.:

x(x - 2) + 3(x - 2) = x² + x - 6       then

2x(x² + x - 6) + 5(x² + x - 6) = 2x³ + 7x² - 7x - 30

So why am I blogging this? Well I think it is important that pupils do gain that conceptual understanding that links two factor expansion to area, and then 3 factor expansion to volume, and I think the methods applied to expand products of 2 or 3 brackets should reflect this conceptual understanding. This is the approach I will be using with my Year 10 set 5 of 6 in the coming week, and I think it has a great chance of success.

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.