My new favourite vector resource - via Back to Back activities!

Walking around my department towards the middle of last week (which I try and do whenever I get the chance, which unfortunately is not as often as I would like) I spied a fantastic image that one of teachers was using as a part of a "no pens day"; having pupils sit back to back whilst one describes and the other draws this picture:

perhaps it was because I was due to teach it the following week, but my mind raced immediately to this picture, which I promptly designed at the end of last week

of course the topic being...Vectors!

I love vector mathematics - it is such a useful way of visualising so many key concepts in maths and science; I use them to conceptualise negatives, translations (I actually draw on the vector arrows), all sorts of things. What particularly struck me about this image is the way it ties vectors nicely with similar triangles and scale factors, For the top end pupils the discussion as to why B to E is 2a and why K to G is 2b and building up the whole picture from there, is a great discussion to come out of this picture, along with then all of the other vectors is as good a top end vector resource I have seen - eventually it is possible to generate this picture:

In terms of trying to define vectors in terms of other vectors, what a great activity! It won't stop there either - tomorrow I will be using the image to explore ideas like:

(a) Are the points KLJ on a straight line? What would the vector be? What about FDE?
(b) Do the points KHE divide the diagonal MC into 5 equal sections?
(c) If the line from N to D is extended so that it intersects the line segment between A and B, into what ratio does it divide the line segment AB?

Of course that is not the only way to use this image - over the half term I will likely create something that uses it for trigonometry, scale diagrams, Pythagoras' Theorem etc as well - all out of a simple image that an NQT was using for drawing.

My resources around this image can be here and here.

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.