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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:

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