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!