So you want to be an A-level Mathematician?

Mathematics is a wonderful, terrible, elegant, messy, exciting and tedious subject.  Students considering an A-level (or two) in this subject may rightly be daunted by the work-load, and intimidated by the huge stack of new ideas they are expected to assimilate and regurgitate two years later, but should take care not to get stuck in the rut of ‘what I need to be able to do for the exam’.

Many courses come with a recommended reading list – a mesmerizing heap of relevant literature which, if not exactly essential to your course, will broaden your horizons and develop you further than the course textbooks alone would do.  When a student starts A-level Maths or Further Maths, their ‘suggested purchases’ list consists of a £50 calculator and… um.  Yeah, that ought to do.  Inspiring, no?

The list that follows is deliberately not a recommended reading list.  There are plenty of more qualified reviewers of mathematical literature out there (Cambridge University have something of a reputation for that sort of thing, so check out their recommended reading list if that’s what you’re after.)  My list is a collection of people / websites / channels that I suggest my students follow.  They lean more towards the interesting, recreational and non-examinable, but are all the better for that, since there will be plenty of the must-learn mathematics going on in the classroom.  My motivation for these recommendations is to insure against students losing their love of the subject.  Mathematics in year 12 or 13 (and even more so Further Mathematics) is a very demanding course, and while it is full of interesting and exciting ideas, it is still possible to be overwhelmed and intimidated by it all, and end up losing track of what made you want study it in the first place.  Call it the reverse Tom Sawyer effect, if you will (he turned a chore into a sought-after activity, and if we’re not careful we can turn the privilege of maths education into a chore).  So if you’re a student of maths, or a teacher looking to inspire your A-level & would-be A-level mathematicians, these are a few of the wonderful internet people who inspire me:

Numberphile: Primarily a YouTube channel which specializes in short explanations of interesting mathematics by a whole range of enthusiastic mathematicians including some of my personal favourites, James Grime and Matt Parker.  The mathematics can be quite a high level, but is always presented in an accessible fashion.  There is also such a wide range of subject matter, from number theory to compass construction, picking up on maths in the news, in popular games, or discussing new mathematical proofs in interesting and engaging ways.  You will also get a sense of the development of mathematics through history with anecdotes and insights into famous mathematicians.  Below is one of my favourites – Matt Parker demonstrates how computers perform calculations by building his own out of dominoes:

Vi Hart: Another YouTube channel to follow, Vi gives a wonderfully creative take on the mathematics that students will be encountering as they progress through their A-level course.  The video style alone is a work of art, with hours of scribbling condensed into a few minutes, with a concise and superbly scripted narration to go with it.  She combines music with interesting mathematics in a unique presentation that is a joy to watch.  Her video ‘Reel‘ is probably the most beautiful description of the way we feel about imaginary numbers you will ever encounter.  Here’s another favourite of mine, where she leads you unsuspecting from counting to adding to multiplying and suddenly you find yourself understanding logarithms:

XKCD: ‘A webcomic of romance, sarcasm, math and language’, these comics are worth a look for some light – if geeky – entertainment, but the really clever stuff comes in the What If section, where some really fun questions are answered from a scientific point of view.  The author used to work at NASA, so he really knows his stuff, and the way he presents solutions, along with his intuitive approach to tackling each problem, is superb.  Plus, who doesn’t want to know what would happen if a baseball travelled near the speed of light, or how many BB guns you would need to fire at a freight train to get it to stop?  Sprinkled with amusing illustrations, these are a great way to investigate daft problems.  One of my favourites has to be Train Loop: this question was in response to a Norwegian ad where a train performs a loop-the-loop.  The original ad is here:

Veritasium: This YouTube channel mainly investigates physical phenomena, and yields surprising insights by means of identifying misconceptions.  Essentially, the host will ask a bunch of people a question, identify the common misconceptions that we have about something, and then explain (or get an expert or two to explain, if it’s really tricky) what’s actually going on, such as this one about how we perceive temperature.  Some videos are designed to have more of an interactive feel, where an experiment is carried out, and the audience (us) are given the chance to predict the outcome.  One that will come in handy for getting to grips with the mass vs weight confusion is Misconceptions about falling objects.  But a personal favourite of mine hits a very common area of confusion – is there really no gravity on the International Space Station?

Vsauce: Another Youtube channel that’s not strictly mathematical, but this guy crams so many interesting phenomena and ideas into every video that there’s bound to be a few things to make you think.  The mathematics is less explicitly involved, but it’s definitely there.  One of my favourites is a really clever description (plus visual simulation) of what would happen if the Earth were flat:

Brilliant: After getting a little distracted by all the cool science bits, let’s get back to a bit of maths.  I’ve only recently discovered this website, but I’ve been really impressed.  It’s essentially a bank of multiple choice maths questions, ranging from pre-GCSE to post-A-level, separated into categories and difficulty levels.  Because of the format, it lends itself really well to smart phones, so download the app and fill those spare minutes brushing up on anything from geometry to calculus.  Also includes sections on some of the more mathsy bits of science.

WolframAlpha: An incredibly useful tool for all sorts of calculations.  It works rather like Google’s built-in calculator (which, by the way, can draw graphs for you and all sorts), but is connected to Wolfram’s powerful ‘knowledge engine’, allowing you to integrate functions directly (not just get numerical approximations), work out your binomial expansions, solve equations and much much more.

GeoGebra: This free program has revolutionized the way I teach certain concepts.  It is easy to learn and intuitive to use, and has sufficient built-in capacity to develop all sorts of interesting demonstrations.  Students can download it for free, use an online version or download an app for their tablet.  They can use it as a graphical calculator, drawing graphs (including implicit functions and parametric curves), performing differentiation and integration, investigating shapes, learning matrix transformations, understanding projectile motion, and all sorts of cool stuff.  I’ve made my own short introduction to the program, designed for students to quickly get to grips with the basics.  By searching the uploaded files of other educators on GeoGebraTube, you can build an intuitive understanding of all sorts of hard-to-grasp concepts, as well as have your eyes opened to all sorts of new and interesting mathematics.

Finally, a quick mention of my own website, TheChalkface.net.  It doesn’t quite belong with the links above, but I do upload my teaching and learning materials, and the A-level Maths section is particularly good (in my opinion) for revision resources.  It has links to AQA exam papers (downloadable in bulk instead of one by one), hundreds of worked exam solutions, my Not-Formula Book revision booklets and their further condensed Essentials revision cards, as well as a host of worksheets and activities designed to fit within the A-level syllabus.

The links I’ve listed above are just the beginning – loads of people are doing fantastic work out there inspiring students (and teachers!)  If there are any sites you think should have been on my list, please put them in the comments below – thank you!


How the French Revolution made King Henry’s nose obsolete: The metric system

Should students who struggle with numeracy be learning to solve quadratics?  Is rote-learning trigonometry genuinely the best use of our time with every student regardless of ability or aspiration?  I’m not sure.  I am a firm believer in the utility of my subject, and also in the value of learning things that may never be used again.  Moreover, learning for the sake of learning, and curiosity for its own sake is what most often yields the amazing advances that justify our investment in the subject.

I have my own answers to the question “What’s the point of maths?”  In fact, if you’re interested, I even made a leaflet for the more skeptical of my students:


However, this post is about one of the (many) topics that is unquestionably applicable to everyday life.  Since the dawn of civilization, mankind has wanted to keep track of things; to count our herds, weigh our flour, measure our land.  And more recently, compare our smartphone specs, remodel our kitchen or track our puppy’s BMI (is that a thing?)

There is definitely value, I think, in glancing back over the rich tapestry of imperial units.  Too often students take things at face value without question, and it’s worth getting them to ask why we have centimetres: Where did they come from?  Why are they that size?  This is more easily done with the imperial system.  Oudated and outrageous in its complexity, its very confusion speaks of its widespread and disparate usage.  From farmers to tailors, and from sea-farers to road-builders, the applications come through in the units chosen: feet and hands, shackles and cables, rods and chains, furlongs and barley-corns.  The unit chosen is entirely arbitrary – the only value in a given unit is the value people place in it (just like the confidence we have in paper money).  So long as everyone used yards, yards were the logical choice.  Of course, depending on whether you were at sea or on land, ploughing a field or hemming a skirt, there were dozens of popular alternatives to choose from.  I started out the first year 7 lesson of the year with this wonderful video by Matt Parker:

As well as being something of a history lesson, this video – as was its intention, of course – highlights the greatest flaw in the imperial system: it’s a bugger to work with.  The system developed organically, over hundreds – if not thousands – of years, so it resembles a sprawling 18th century London – built up over time by separate groups with little cohesive narrative.  Until the upheaval of the French Revolution gave us the push we needed to make a drastic change, and along came the meticulously planned Milton Keynes of measurement systems.  Taking the most universal thing they could think of as the base unit, they split the circumference of our entire planet into quarters, then broke those quarters into 10 easy pieces.  Each of those was broken down into a million (more manageable) lengths (conveniently roughly equivalent to a yard, thus thumbing their noses at its inventor, Henry I, who is credited with defining it as the distance from his outstretched thumb to the tip of his nose).  Once the metre was established (even the word was audacious: ‘metre’, of course, meaning ‘measure’), the rest followed in a logical base-10 system: millimetres, kilometres, etc.  Of course, their original measurements were a little off (the Earth happens to be 40,075km around, not exactly 40,000km), but, like we said, the choice of unit is not as important as the value we attach to it.

And school children of today don’t know they’re born, with the luxury of only ever having to multiply or divide by 10, 100 or 1000.  The system was designed with ease-of-use in mind.  Having said that, old habits die hard, and it takes a generation or more before people are more comfortable with the new units than with the old.  This worksheet introduces the metric system while highlighting the cultural scenarios when the imperial system is alive and kicking in the UK:


And this one is something of a follow-on, giving details of imperial to metric conversions for students to use throughout:

unitconversionWhile our curriculum may be mostly tied up with unit conversion, the skill I am most keen to develop in my students is an intuitive sense of the units they are manipulating.  They should know that their iphone weighs 150g and that there’s a 10m drop from the classroom window (those two facts are not necessarily related…).  In my opinion, fostering a familiarity with units of measurement is one of the best ways a parent can equip their child to confidently use maths in the so-called ‘real world’.  Parents don’t need to love algebra or get all excited about circular motion, but we should all be able to provide opportunities for our children to measure, to estimate, and to budget.  This next one is simply an opportunity for students to find examples of what is measured in which units.  Just how heavy is that car ferry?  How far away is France?  What is the width of a pencil line?


At this stage I try my hardest not to get too hung up on the distinction between weight and mass (see ill-concealed rant / A-level info-sheet).  Students need to develop a ‘feel’ for this stuff first and foremost.  The following match-up activity I enjoy using.  Students (individually or as competing teams) have to first choose the most logical thing to measure (do we want temperature or distance for a high-jumper?), then pick a number that fits the given units.  Truth be told, some are hard to guess, but there’s a lot of mileage (sorry, km-age) in asking students which answers they’re really confident with – they may surprise you.


And this worksheet encourages rough estimation.  Given the number, can you choose the appropriate units?  Is a tin of beans more likely to be 415 grams, kilograms or tonnes?


Some quirk of human nature makes my students 12 times more eager to answer questions when I dole out plastic counters for their efforts.  Usually given out with no explanation or promise of reward, and collected back at the end with no totting up or declaration of a winner.  Competition alone is enough.  This next activity is, therefore, in the form of a quiz.  Put them in teams, and maybe even save it for the end of term so they view it differently to a normal lesson.  This has plenty of scope for curiosity and extra discussion (so read up on it before-hand, or better yet make your own) – students have to guess the big numbers that go with these big things.


Once your students get on to compound measure, test their confidence converting compound units of speed with Speed Conversion, and of course check out my earlier post on density (which packs in a surprising amount, for its size…)

A-level Mathematics Resources

Going in circles: Making sense of A-level Maths: Mechanics 2

If you feel like your Mechanics 2 lessons are going in circles, or you don’t think you’ve got the potential to get work done, this post may be helpful to you.  Remember, with great power comes great energy transfer per second.

Before I get too carried away, let me take a moment and pick up a sofa:

Sofa Moments

Moments are a great way to start the course.  Straight away you’re diving into a concept they thought they had sussed (equilibrium) and saying “yeah, but if I put the two equal and opposite forces here and here, the book spins round – does that look like equilibrium to you?”  We have to extend our definition to cope with rigid bodies rather than particles, but it gives scope for solving so many more problems.  The sofa one I like in particular because it explains the “I’ve got the heavy end” phenomena in a way that M1 – modelling everything as particles – would miss completely.  The most useful thing is that, like resolving forces in any direction you choose, you can take moments about any point and generate more information to solve a problem.  Next time you’re building a cabin out of railway sleepers, take a moment and write a worksheet.  Some of you may not have got round to building a cabin yet, so feel free to share a moment with me:

Sleeper Moments

(This idyllic cabin beside the river where I grew up in rural Leicestershire is actually available for holidays #shamelessplug)

Centres of mass naturally follows on from moments, and even if the class won’t be studying volumes of revolution any time soon, it can’t hurt to throw a bit of quick calculus their way here, even if it’s just the hand-wavy kind.  For instance, the centre of mass of a triangular lamina lies on each median line (from each corner to the midpoint of the opposite line).  Imagine cutting the triangle into many small rectangles.  The centre of mass of each is on the median, therefore so will their combined centre of mass.


I brought in a small cardboard box with a rock duct-taped to one end of the inside, and we threw it around a bit to get a feel for non-uniform objects.  Remember, the centre of mass of any object can be found experimentally by suspending it from a point, balancing it on a point or even spinning it as you throw it (what this last one lacks in accuracy it makes up for in student engagement).

Energy is the first really big challenge of the course, and it should not be underestimated how confusing students can find energy questions.  My booklet, All About Energy, makes for a nice introduction.  Taking inspiration from BetterExplained’s ADEPT method of introducing a concept, I’ve filled the booklet with analogies, pictures and some real facts and figures (let’s face it – how many of us can find the elastic potential energy in a bungee rope but don’t know how much energy is in a bowl of Weetabix?)  Best served as a full-colour, A5 booklet, it doesn’t go into great detail with the nuts and bolts, but makes a good accompaniment to your introduction:

All About Energy

Getting to grips with the distinctions between force, energy and power is crucial.  But it’s great to be able to apply what we know about energy immediately to problems which it would be possible to solve with SUVAT equations, but by no means straight-forward:

SUVAT vs Energy

My next example is usually a roller-coaster drawn on the board, with loop-the-loops, strange twists and turns, and definitely not constant acceleration.  By knowing the height at any point (and neglecting – for now – resistance forces) the speed can be calculated easily.  This will be handy later, when applying energy to circular motion.  Add in friction (keep it constant for now), and the only extra information you need is the length of the track.

Energy is definitely one of those topics where I’ll need all colours of whiteboard pen available.  I describe kinetic, gravitational potential and elastic potential in terms of bank accounts where we can save energy, or withdraw it.  Any work done by, or against, external forces are like money being paid into the system from outside (or paid out from the system).  We have no control over this energy, just like we don’t control money that isn’t ours.  The difference between work done against, say, friction, and that done by weight is that we can cash in the energy we’ve ‘lost’ to work done against gravity by moving downwards to release it again.  Just like money isn’t lost when I put it in an ISA in the same way that it is when I spend it in a shop.

As an improvement on SUVAT, I like to use the Felix Baumgartner example again, but this time with energy:

Space Jump Energy

The bungee jump is, of course, a favourite application of elastic potential energy, but we shouldn’t ignore the humble nerf gun (easier to demonstrate in the classroom, too):

Nerf Energy

Once you get to bungee jumping, though, make sure you understand exactly what’s going on at each moment of the jump.  Find a slow-motion bungee jump on youtube, or do what I did this year, and set “watch GoldenEye” as homework the lesson before:

And follow it up with a carefully researched worksheet:

Dam Jumpers

As well as solutions, this worksheet includes descriptions at the end of the different phases of motion: before the rope tightens, while the rope is stretching but before maximum speed, while decelerating to rest, and the bounce back.  It includes values for the energy at each point, as well as details of the forces involved.  For a long time I tried to make a program that would show me exactly what was going on at any given moment during a jump, but – like many energy problems – it becomes a mean differential equation because kinetic is in terms of velocity while gravitational and elastic potential are in terms of displacement.  However, I had a bit of a breakthrough and have made this little GeoGebra beauty in terms of displacement:

Bungee Energy Calculator

I like how it makes that middle section clearer: the brief time while the rope is being stretched but the force isn’t great enough to slow you down yet, only reduce your acceleration.  There’s also an Excel version which will give you more precise figures, but doesn’t illustrate the concept quite so well, I think.

If you enjoyed Top Gear’s Porsche versus Beetle race, you may like to throw this activity in there at the end.  It starts with SUVAT, then improves on the model using energy, assuming a constant air resistance, and then goes further by taking into account variable air resistance (don’t worry – I did the tricky maths already, so there’s a nice-ish formula for students to use).  Although the last bit isn’t necessary, it makes a good extension, as this is what more and more precise mathematical modelling looks like in reality.

Beetle Drop

PS – make sure you watch the race.  You know, to check your solutions:

Circular motion is the next big thing.  As always, start with the definitions and spend enough time messing around with them that the formulae are not only familiar but have practically been derived by students.  I like to start by asking which is going around faster; the Earth around the Sun, or the Moon around the Earth?  Students will, quite correctly, come up with two different answers, depending on their interpretation, and that’s the motivation for a definition of angular speed.

Going Round

Sticking with constant velocity for the moment (but that doesn’t have to be ‘horizontal circular motion’, remember), we can actually calculate the height of a geostationary orbit.  That’s pretty cool, right?

Geostationary Orbit

Don’t be too keen to dive into ‘vertical’ circular motion.  There’s so much fun to be had with the other kind:

Round the BendUp the Wall

(and, to help you out with investigating the Wall of Death further, a useful little Banked Curve Calculator in Excel.)

Finally, we’re at vertical circles.  If your department doesn’t have a loop-the-loop track and a little hot-wheels car, you’ll have to build one.  This is all about getting to grips with the possible scenarios.  A car performing a loop-the-loop might make it all the way round without a hitch, or not quite halfway up, and roll back down, or more than halfway, and fall inwards.  But what happens if the car is fixed to the track, like a roller-coaster?  Or if it’s on the outside of a circular track, like driving over a bridge?  While getting my head around all of this, I made a summary sheet that is as much use to me as it is to the students:

Vertical Circles

Use my GeoGebra illustration to show what happens to the forces as you go around the circle (but bear in mind the details in the previous document, as this demo is limited).

The only other thing to get your head around is that pretty much every single question will involve forces and energy.  Resolve towards the centre, because you know that will be equal to the centripetal force which we have a formula for.  And use conservation of energy, combined with the height, to find the speed.  The link here is the speed.

And today’s homework is to watch Fifth Gear make loop-the-loop work in reality:

And, of course, answer questions on it:

Loop the Loop

The final chapters of M2 take us beyond SUVAT at last.  The progression began back at GCSE (or earlier) with the constant speed equation, was developed further into the constant acceleration equations in M1, and now, in quick succession, using calculus to describe motion when acceleration is given as a function of time, then using differential equations to describe motion when acceleration is a function of displacement.  This opens the doors wide on all sorts of problems, and even though many of them can only be solved using much more advanced techniques, being able to generate the formulae is half the battle in the days of WolframAlpha which will solve your differential equations for you.

The nicest application of these topics, though, brings us right back to circular motion.  It is useful for a teacher to be able to derive the formulae for centripetal acceleration for constant angular velocity.  It’s worth being aware of the derivation for variable angular velocity also, but probably not worth the half-hour it would take to go through it in class!

Deriving Circular Motion

As with M1, I’ve made a Checklist; those key skills which are potential limiting factors for success in M2, but do not, on their own, constitute a full understanding of the module.

The M2 Ideas Test, on the other hand, is often a considerable challenge even for A-grade students, and will really test how comprehensive and thorough a student’s understanding truly is.

And, of course, the M2 Not-Formula Book and the one-page M2 Essentials revision card, both of which summarise key information from the course which will not be provided in the formula booklet, and are highly recommended for independent revision and reinforcement of ideas.

I’d love to hear your thoughts on any of these resources, and of course any others you have found particularly useful.

A-level Mathematics Resources

“Model the topics as particles connected by light, inextensible strings”: Making sense of A-level Maths: Mechanics 1

All through school I loved algebra for the power it gave me to describe patterns and solve problems.  And I loved physics for the insight it gave me into how and why things work the way they do.

A-level Maths & Mechanics was where they both came together for me.  I can describe archery using algebra, use calculus to explain a car race, even drag trigonometry out of triangles and use it to play conkers.

I’ve been teaching the course for over 5 years now, and the more in-depth my understanding becomes the more I enjoy it.  I love opening the eyes of students to the application of maths they enjoy to real world phenomena that intrigue them.  For a few lessons a week we get that science-y buzz of “now I’ve explained the theory, let me demonstrate it by making this thing spin around / fall off the table / go flying out of the window.”

Below are a few ideas and resources I’ve put together over the years for Mechanics 1.  Watch this space for a follow-up post on Mechanics 2.  This relates directly to our exam board of AQA, but content is broadly similar across the exam boards.

The Language of Maths: The Language of Maths

I like to kick off the year with this little translation activity.  It’s a bit of fun, but it brings out some of the specialist vocabulary used throughout the course, and prompts discussion of assumptions and the mathematical modelling process in general.  Chapter 1 is, on its own, not really examinable, but sets the scene for all of applied mathematics, so in my opinion is well worth a lesson.  I refer back to the mathematical modelling cycle frequently in reference to verifying results, an appropriate level of precision for answers, considering the suitability of assumptions, and so on.

The SUVAT equations (or ‘kinematics equations’, or ‘constant acceleration equaions’) were presented to me as a complete package but I would have loved to see where they came from.  If you’re not quite sure, check out this quick guide – they all come from our basic definitions of speed and acceleration:

Deriving the SUVAT equations

Or use this Equations of Motion investigation with a class to have them generate them for themselves.

Another really handy tool (mostly for teachers, when making up sensible questions, or quickly checking answers) is my SUVAT calculator in Excel.  Comes equipped with unit conversion too, to save you the job:

SUVAT Calculator

On the subject of unit conversions, one of the most useful is the ratio of metres per second to miles per hour.  Why not set this as an exercise for the class?  From 5 miles is roughly 8 km it’s easy to see that 1 m/s is roughly 2 mph.

If students are learning to drive, see if they can derive the stopping distances from the Highway Code.  With a bit of prompting, knowing the distance for one speed should be enough to calculate the rest. A bit of reverse-engineering reveals the standard deceleration rate to be around 6.5 m/s/s.  This worksheet looks at how much more deadly a crash at 80mph would be than a crash at 70mph:

Stopping Distances

The most interesting SUVAT questions involve freefall.  Why did Felix Baumgartner jump from nearly 40,000 metres high?  It wasn’t to give him time to accelerate to the speed of sound (his target); it was to get him far enough beyond our dense atmosphere to reduce air resistance for long enough for him to accelerate freely under the influence of gravity alone.  You can watch his jump here: http://youtu.be/FHtvDA0W34I , but I think I prefer this version:

Which brings us to forces.  When I first taught this I underestimated the confusion that weight and mass can cause, since we use the words interchangeably (often even in the maths classroom) until A-level.  This little fact sheet highlights the differences (and also explains why maths uses g=9.8 while physics still prefer 9.81!):

Weight and Mass

Or for a more in-depth investigation into the force of gravity, see the gravity investigation below, which asks, among other things, why astronauts on the space station, who should experience gravity only fractionally lower than ours, appear to be weightless:

Gravity Investigation

I describe friction as a ‘reactive’ or ‘passive’ force.  Much like the normal (contact) force, it will resist potential motion, or slow it down, but would never, on its own, cause motion.  This idea can be extended in M2 for the contact force when we have circular motion (we must have a centripetal force, and only if there’s no other force providing it, the normal reaction steps up).  I’ve made a simple dynamic illustration of a particle in equilibrium on a slope (and, if you hadn’t spotted it yet, did you know that the minimum coefficient of friction for a particle on an inclined plane is just tan of the angle to the horizontal?)  Friction is the best excuse to tip desks up and watch books slide off.  At 45 degrees?  Coefficient of friction must be 1. It’s worth asking conceptual questions about the friction formula, too.  Why are formula 1 cars designed to have a massive down-draft pushing them into the road?  What forces would change as a result?

Statics is important, and resolving forces for a mechanics student is like taking a blood sample for a medic, or adding VAT for an accountant.  It’s not the hardest thing you’ll ever do, but you’ll need it so often you’d better make sure you not only understand how it works but be able to do it in seconds without breaking your train of thought.   Students will need to get past drawing right-angled triangles for every force so that they can focus not just on a method that works but the best method for this particular question.  Sometimes that’s a vector triangle and cosine rule,  sometimes it’s resolving horizontally and vertically and sometimes it’s picking a better pair of directions to resolve.

Newton’s laws help us to describe how a bicycle can outpace a car – briefly – when the lights turn green, or how a child can tow a 10 tonne barge.  One example I like is dropping something too heavy to hold; instead of just dropping it, let’s say you try to slow its fall.  You can’t completely counteract the force of gravity, but you can reduce the overall downwards force, so that the object accelerates downwards less quickly and you may avoid a broken washing machine.

Hold On

Projectile motion is a great application of 2-D SUVAT equations, but I often like to solve the problems in two separate sections – horizontal motion on the left, with its nice constant speed equation, and vertical motion on the right, using SUVAT under gravity.  The only connection between horizontal and vertical motion is the time t.  This trebuchet worksheet encourages students to deal with more than just known speeds and angles.  And you’ll know you’ve properly grasped projectiles when you can explain why doubling the speed quadruples the range.  This next worksheet brings in a non-zero initial height (inspiration drawn from my time on board ship, where heaving lines are thrown from the deck to the quayside):

Projectiles Worksheet

For an easier, concept-based introduction to parabolic motion (perhaps as a homework prior to learning the maths behind it), try FlightPath:

FlightPath actifity

Finally, I’m going to recommend my M1 Checklist (a list of all the basic skills that, if lacking, will cause problems), the M1 Ideas Test (great for ironing out misconceptions and going beyond ‘can solve problems’ to ‘understands the concepts’), and my M1 Not-Formula Book and the one-page M1 Essentials revision card, both of which summarise key information from the course which will not be provided in the formula booklet, and are highly recommended for independent revision and reinforcement of ideas.

More Not Formula Books and Essentials Cards are available from my website:

How we learn, as shared by @magicalmaths
A-level Teaching

Why teaching A-level is the best preparation for teaching A-level

I thought I understood A-level Maths until I started teaching it.  As @magicalmaths tweeted the other day, we learn some of what we see or read and rather more of what we discuss or experience for ourselves, but none of it compares to how well we learn by teaching others.  If you’re looking for the ideal training course to equip you to teach A-level Maths, you need to go back to the A-level Maths classroom, but this time stand at the front.

How we learn, as shared by @magicalmaths

I was very proud of my own A-level results, but have become increasingly convinced that there is a yawning gulf between being able to get the best grade at A-level and being able to teach others to do so.

It’s now more than 5 years since I started teaching and the main difference between me now and NQT me (other than his gloriously unkempt facial hair and his lack of a wife (correlation/causation?)) is the fact that now I’m starting to realise just how much I don’t know.  If I had to quantify it (and I am a maths teacher, so I do), I’d say my level of understanding of A-level mathematics has more than doubled since I started teaching it.  And by far the most effective tool for increasing my understanding has been actually having to teach it.  Tough luck on the first class or two (don’t worry – unless you’re really clueless they’ll cope just fine), but by the time you’re teaching a topic for the second or third time you’ll know it inside out, back to front or under any other geometric transformation you can imagine (180 degree rotation is a useful one for correcting student errors from the front).  When teaching a topic, knowing what you know and what you don’t is crucial.  You’ll probably be somewhere along this scale with any given topic:

1. Grasp the key idea.

2. Answer simple questions.

3. Answer more complex questions.

4. Interpret/analyse the general form of solutions.

5. Map out a path to the solution without having to solve the problem.

6. Make up your own questions (including all relevant (and no unnecessary) information.)

7. Identify the potential pitfalls for a problem.

8. Grasp the key idea that extends/generalises this one.

For example, when learning about projectile motion in Mechanics 1:

1. Understand how horizontal and vertical motion can be treated separately.

2. Calculate horizontal range from a given initial speed and angle.

3. Calculate the possible launch angles for a given speed, horizontal range and initial vertical height.

4. Explain qualitatively why doubling the speed quadruples the horizontal range, and why 45 degrees gives the maximum range.

5. Predict the quadratic equation from vertical motion, the linear one from horizontal motion and how the key variable (time) connects the two.

6. Accurateley estimate the speed of a bullet / arrow / cricket ball and form a question with enough unknowns (but not too many).

7. Warn students to look out for negative displacement vertically, remind them that the initial velocity is not a force and therefore horizontal speed doesn’t change.

8. Describe the modifications necessary to our model should we choose to take into account air resistance.

Trebuchet Projectiles

And solving equations of the form a sin(x) + b cos (x) = c for Core 4:

1. Recognise how the sin(A+B) or cos(A+B) formulae can be rearranged to get a single function.

2. Solve questions where you’re told whether it’s sin or cos, and whether it’s + or -.

3. Rearrange and simplify trig equations in a variety of forms to get a single trig function in the equation.

4. Describe how the Rcos(x+a) format relates to the original functions through graphical transformations.

5. Consider the signs of the sin and cos components to determine whether you need Rcos(x+a), Rcos(x-a), Rsin(x+a) or Rsin(x-a), and what happens if you choose the ‘wrong’ one.

6. Show how interference of two waves simply generates another wave, each wave being a transformed trig function.

7. Remind students that, depending on their choice of function, the primary solution for the angle may not be the correct one, and how to find the right one.

8. Consider how graphs of functions may be modified in other ways than adding.  Does multiplying two trig functions give a trig function?  Are there similar patterns for tan(x)?

In the last few weeks I’ve had some moments where I’ve glanced at a student’s work and leapt straight in at point 7 identifying their misconception with almost uncanny accuracy in seconds.  But I’ve also jumped to point 6 in class without realising my understanding was still at a 3, wasting 45 minutes of lesson time getting things wrong on the board.

If you’ve only just begun to teach A-level, don’t be put off by how well another teacher may seem to understand a topic.  It’ll come; you just have to be prepared to get stuck in and teach it for a bit first!  I have an ever-increasing list of topics I want to ‘really get my head around’, but in the meantime, knowing my limitations enables me to plan properly and get support from more experienced teachers when I know I need it.

Don't take my word for it: try it yourself!
GCSE Mathematics Resources

A touch of density: 8 things I taught and 1 amazing thing I learned while teaching density

You can always tell which topics I enjoy teaching by the number of resources I create for it.  Density is definitely one of my favourites to teach, to any age group, possibly because it links in so nicely with ‘real world science’.  I can include apple bobbing in lessons, melt snowballs on the radiator, and discuss why a 12,000-tonne iron ship doesn’t sink.

1. Set a few things straight: “Which is heavier – gold or paper?” *compares wedding ring to pile of books* “Oh, so you need to compare the same amount of each?”  The classic ‘pound of gold or pound of feathers’ normally makes an appearance here.  The key thing to get across is that the more closely packed a particular material is, the heavier each cubic centimetre will be.  Then grams per cubic centimetre will make as much sense as miles per hour.  I have some 1cm dice handy for anything to do with volume: “If this were water, it would weigh 1g.  If it were gold, it would weigh about 20g.”

2. Get a feel for density: Start off with some common materials, and think in terms of ‘sink/float’.  A nice way to begin is my Density Ordering cards.  Did you realise that glass is more dense than granite?  Or that balsa wood is less dense even than freshly fallen snow?  This normally drags me off on a bit of a science tangent (why is ice less dense than water?  And what is up with platinum??)

3. Do a practical experiment: drop things in water, or weigh and measure some objects.  If there’s no snow around, show a video:

4. Tell a story: The classic Archimedes Eureka story may, it turns out, be a bit of an oversimplification due to the tiny tolerances involved (but don’t worry: Wikipedia gives a suggestion of some even cooler maths involving buoyancy that is probably closer to the truth, and can form extension work), but the story is useful nonetheless, and could easily be turned into a Eureka activity.

Worth its weight in gold?

5. Talk about gold: For some reason we love to think about expensive things.  The value of gold per gram, coupled with its high density, make it very valuable in small amounts.  A pencil made of gold would cost you upwards of £300.  Have them measure an eraser and work out how much it would be worth in solid gold.  A 6-inch diameter sphere is worth £1 million.  And if you want to go really big, consider All the Gold in the World.

All the Gold

6.  Conduct devastating thought experiments: Not content with melting all the gold in the world into a big lump (which, by the way, would be a cube barely 20m to a side), let’s Melt the Eiffel Tower.  This one is crazy: due to the highly efficient wrought-iron lattice structure, it’s incredibly light for its size.  And since it also has quite a wide base, if you melted it down to form a cuboid with the same square footprint the height would be a mere 2 inches.  7300 tonnes may seem a lot, but if you scaled it down perfectly to fit in a classroom it would weigh around 3kg (the same as a large laptop).  The metalwork this model requires would be ludicrously thin (think 3 layers of tinfoil).

Melting the Eiffel Tower

7.  And what’s the atmosphere like at this point in the lesson?  Denser than you thought?  At 1.225 kg per cubic metre it may well be.  Remember a cubic metre of water weighs a tonne, though, for comparison.  Students may be surprised to calculate that the air in their classroom weighs as much as them.  Bring it back to the Eiffel Tower and compare the weight of the iron (7300 tonnes) to the weight of the air within (modelled as a pyramid I get the air to weigh 2000 tonnes).  Not the same, as some websites would have you believe (and blithely tell your class for years before doing the maths yourself…) but pretty close!

8.  Shoe-horn density into other topics: When you’re doing volume of cylinders, have students work out the weight of a dug-out canoe.  Combine volume of a pyramid with the volume, and density, of The Great Pyramid (answers on a PowerPoint).

Dug-out Volume

And finally – enough about what I’ve taught, how about what I learnt?  While estimating the weights of mobile phones by holding one in each hand my class kept thinking the lighter one was heavier.  Turns out, our brains were tricking us into comparing density rather than weight!  An iphone was smaller than a Samsung, but denser.  Still weighed about the same, but I’d have sworn it felt heavier.  Try it yourself – your brain is automatically adjusting for volume!  Hold them in bags so you can’t tell the size, and suddenly you can compare weights accurately again!

Don't take my word for it: try it yourself!

The Samsung Galaxy Ace weighs almost exactly the same (in fact, slightly more), but because its density is 25% lower, it feels lighter!