Things that are NOT TRUE

A colleague at a previous school produced this poster (it goes on to 29 points: click the picture to download the full version)

If you’ll allow me a brief aside, credit for this poster goes to Robin Bhattacharyya. Robin was the most knowledgeable person I have ever met and at the time, one of my heroes. Originally a University Challenge champion in 1995, he went back for more in 2011 Christmas special, in which he led his college to victory over a more recent hero of mine, Daisy Christodoulou. (strictly, it was Trinity vs Warwick but from memory, it may as well have been Bhattacharyya vs Christodoulou!)

Anyway, back to the maths… A recent tweet from Stuart Price reminded me of this poster.

Stuart was focussing in on one specific type of misunderstanding which features heavily in Robin’s poster: the over- or under-generalising of the distributive property.  Trying to head off these problems early, I’ve been making a specific effort to teach year 8 about distributivity this week. It has gone better than in the past, so I thought I’d share what (I think) I’ve learned.

In the past, the phrase “multiplication is distributive over subtraction” sounds clunky and just saying it, let alone understanding it, has been trouble enough for my pupils. I’ve replaced it with “multiplication distributes over subtraction“. This small change has had a surprisingly large effect. I think this is partly because it relates more closely to the usual English usage of the verb ‘to distribute’; indeed, this also allows for visualisation: the multiplication is literally being distributed over the subtraction.

Secondly, Having checked whether several operations distribute over each other, I asked my pupils to generalise which operations distribute over others. In this section, Colombe introduced a metaphor which helped her peers to remember the general rule: Multiplication and Division are the government: they distribute (resources) to the citizens, Addition and Subtraction but not to themselves. This also ties in very nicely with Order of Operations.

I also used a three-act-math approach in these lessons, introducing the topic with a classic moment from countdown (if you’re a maths teacher, you must have seen this!)

This was my hook: how on earth did he do this? Did you notice that he didn’t actually know the intermediate numbers? This gained interest, and later on in the lessons I tried to set some questions in which pupils applied the distributive property to simplify some expressions, leading up to (318 x 75 – 50) / 25, but it didn’t go well. If I had used the strategy that John Mason suggested on Craig Barton’s podcast, of imagining what I was going to say and how the lesson would proceed, I feel like I may have anticipated this. It wasn’t clear that the equivalent of 318 x 3 – 2 was the ‘final answer’ in the simplification problems I had created. In my updated lesson plan, I have removed these questions, leaving just the explanation of the countdown genius as a teacher-led section of the lesson.

Another issue that came up as part of these lessons was a good question from my pupils: If it doesn’t distribute, what does happen? My answer at the time was fluffy… something like “it depends on the situation, usually you just apply the operation to one of the numbers”. I feel like I need to work on this, perhaps I need to add another sequence of lessons to my combining operations thread.

Overall though, I feel that this is the first time I’ve been even vaguely successful when teaching the distributive property, mostly because I managed to distill it, with the help of a pupil’s metaphor, to this:

One final reason why I think it went better than in the past: I was more committed. Previously, I would always question the value of teaching such lessons, because I suspected that the pupil’s future teachers wouldn’t make reference to the distributive property. Now I know that I will be teaching these pupils for several years and as HoD, I can better integrate this topic into our scheme of work, it is worth the investment.

Hyperspace Hopping

The Resource

This is a worksheet which introduces some of the ideas contained within the 3-d geometry section of A-level maths: vector equations of lines and planes and testing whether or not points lie on these planes.  The latter questions also give students an idea of linear dependence, although this understanding isn’t strictly necessary for the A-level course.

It’s in the context of getting from one planet (coordinate) to another using the buttons (direction vectors) in a spacecraft.

I’ve used this as an initial introduction to vectors, as was the norm at my previous school (from where I stole the idea – I think it comes almost directly from Robert Wilne). More recently I’ve used it as an introduction to 3-d vectors, after previously teaching 2-d vectors and this has been more effective as the pupils have a better toolkit of terminology to help them solve the problems, but the context seems to make the leap from lines to planes more intuitive.

And it’s called hyperspace hopping, so it’s fun! Enjoy.

Equations as Balances

The Resource

I think this is a fairly standard introduction to equations, attempting to embed the idea that the rule is that you perform the same operation to both sides in order to keep it balanced.

This works particularly well if you have some balance scales in your department with which you can demonstrate an example of finding an unknown mass by removing objects from both sides etc.

At Highgate, we used to follow it up by taking the model further, to cover other operations / situations which don’t really fit the traditional balance model. This worksheet does that, but I’m not sure I’d particularly advice it unless your whole department is working together to get the pupils thinking about equations as balances in this way.

A Circle Theorem

The Resource

This worksheet encourages pupils to practice the deductive reasoning required to solve problems involving angles in circles.  At the same time, they develop the theorem that ‘the angle at the centre is twice the angle at the circumference’, either by spotting the pattern, or by generalising in question 5.  I would usually go through the general case the class afterwards.

The back of the worksheet contains lots of hints and tips for pupils who are finding it difficult, allowing those who can to create the arguments for themselves.

One weakness of this worksheet is that it requires pupils to be familiar with the AOB (three points) method of describing angles.  I should really make a version with a simpler notation for pupils who are less familiar with that.

Although this general case proves it for all angles, it’s important to follow this up with a demonstration (I like geogebra, but I’m sure there are alternatives) to show that this rule works even upon moving around the points on the circle, in particular demonstrating the alternative cases such as this…

circle6

Trigonometry

The Resource

This is an approach to introducing trigonometry I took from my previous department at Highgate.  It focuses on just the tangent scale factor in this initial intro – we would then return to introduce sine and cosine a month or so later, following a spiral curriculum.

I would start with an introduction on the aim of the lesson: we want to find a link between the sides of a triangle and the angles within it. This could make use of the 3-act math format alongside a real life problem such as finding the height of a mountain: personally I often do this with one of my favourite tracks playing the background: Everest by Public Service Broadcasting.

The worksheet guides pupils to discover that if you divide opposite by adjacent for similar right angled triangles, you always get the same number.  This gives us the link we are looking for. It can be followed up (or proceeded) with a note on why you would expect triangles with the same internal angles to have the same scale factor.

Then for homework, pupils can be each given their own angle to work out this special number for.  At the start of next lesson, I would tabulate the pupil’s results and encourage the class to look for patterns: any incorrect/wildly inaccurate values for tan will stand out and can be quickly corrected.

I’d then demonstrate how we can use this table (for now avoiding the calculator tan button) to estimate sides / angles.  Using the pupils’ own numbers gives them a sense of ownership over the method, which adds a lot of value to the approach.

Suggestions for alternative approaches welcome as ever.

Pythagoras

The resource

A cutting and sticking exercise which leads pupils towards the discovery of pythagoras’ theorem. Beyond introducing the aim of the lesson, this requires very little whole class input from the teacher.

I’ve used this with higher-attaining pupils and it’s worked very well. I’m not sure whether more structure in the questions would be needed for others – I don’t have too many ideas how to break it down further: each individual question is relatively simple so at least the first two numerical examples should be achievable by most.

At Highgate, we then spent a lesson or two in which pupils found missing sides in a triangle by drawing both diagrams out each time – quite a substantial use of time, but it meant that they were very comfortable with the ‘proof’ when we came to generalise it.

I now tend to dive straight into the algebraic generalisation (either individually, in small groups or as a class, depending on how the class are finding it) before proceeding with more traditional applications of Pythagoras.

A follow up to show the pupils which demonstrates the idea in a subtlety different way was tweeted by @solvemymaths. This would make a nice follow up to show pupils.

Suggested improvements please!

Differentiation from First Principles

The Resource: https://goo.gl/NRq3fc

This is my worksheet which helps pupils to discover the rules for differenting polynomials.

I’d use this after an introductory discussion of a problem in which the gradient of a curve is sought. Often I’ve asked “How would you work out Usain Bolt’s speed 20m into a race?” This usually leads to a discussion about how speed guns work.. I inevitably sketch a distance time graph and we talk about how to find the gradient at a point. Tangents are often suggested and I may have to push students towards the idea of chords and limits, perhaps unsurprisingly, given how major an idea this was in the history of maths.

Writing this, I’ve started wondering if all this build up could be introduced through discovery, but I think the idea is just not intuitive enough.

So then the worksheet guides the students through an investigation of some standard curves before encouraging generalisation. In my experience, it works well with a range of A-level students, and allows scope for those who work more quickly through it to generalise and prove more thoroughly.

As always, I’d love any suggestions for improvements.