Sixth Sense
If you’re anything like me, you’re looking forward to a few weeks away from school and probably don’t want to think too much about September just yet. However, at this time of year – and better now than at midnight on 31 August! – it is worth reflecting on what has gone well and where there is room for improvement in one’s teaching. If you’re a subject lead then this reflection might well feed into a redrafting of your schemes of work: at KCLMS, as at most schools, these are very much working documents which are constantly being tweaked and (we hope!) improved.
Take the beginning of teaching AS Mechanics as an example. If, like us, you teach this to Year 12, this is probably the first time students will encounter “Applied Mathematics” and there is certainly a conceptual jump from “mathematicians learn things which are true” (Pythagoras’ Theorem, how to solve quadratic equations, the uniqueness of prime factorisation) to “mathematicians use models to make predictions that will be compared to observations made in the real world” (stock prices, rates of decay, punctuality of trains). Even if you leave Mechanics until Year 13, I’d argue that much of the following still applies.
In order to reach the point where students can tackle confidently the typical disastermoviescenario exam question in which “a train and N trucks are driving up a hill when the coupling breaks”, there is substantial conceptual understanding and procedural fluency that they need to develop: the forces model, Newton’s Laws, components, the meaning of the assumptions regarding lightness and inextensibility, the constant acceleration formulae, etc. Each of these ideas, I suggest, needs approaching separately and pulling apart in some detail before they will be wellprepared to attempt this sort of question.
At KCLMS, we start by focusing on the principles of Mechanics in one dimension, and we leave any discussion of components until the spring term in Year 12. This gives the Core teacher time to review, revise, and improve students’ trigonometric fluency before they need these skills in Mechanics lessons. You can go a long way in one dimension (pun fully intended!): even the Further Mathematicians study 1D impulse, momentum, moments and work done before they start to use components. I know that this isn’t the standard approach – think of all the textbooks that start with “Chapter 1: Vectors” – but it’s how I was encouraged to teach this material early in my career by my mentor who, in turn, picked it up from the Head of Maths at Westminster School. I’ve become ever more convinced by the sense of it each time I’ve taught the AS applied material.
Let’s look in some detail at the first few lessons on our scheme: I’m hoping to convince you that a bit of showmanship and a high level of pedantry at the start of the course will enable students to solve difficult problems with confidence later on.
Lesson 1
Objective: to identify the number of separate objects in a system, and to describe (without formal labelling) the forces acting on each object.
Teaching: I will start this lesson by dropping a textbook, and asking “what just happened?” When the students say “it fell”, I put it on a table and we discuss why it no longer falls. At this stage, I discourage the use (often misuse!) of formal language that they have picked up elsewhere: rather than using “tension”, “reaction”, etc., I will ask students to describe everything that each object “feels”, in plain English for now. Most students talk about weight, and they have to talk about this to realise that this is something that the object experiences in and of itself: the table doesn’t “feel” the weight of the book, but it does “feel” the book pushing down on it.
Example: Two boxes sit on the ground, one on top of the other. Draw each object separately and add the forces that each object “feels” on your diagram.
Solution on board [click image to enlarge]:
(the picture of the ground is quite uncluttered, but be prepared for some good questions from students here – does the ground have weight?; shouldn’t there be something below the ground pushing up?; shouldn’t we draw another picture of whatever this is? Ultimately I argue that we have to stop “zooming out” somewhere and the three objects shown above are the only three mentioned in the question!).
Lesson 2
Objective: to draw force diagrams, now with more formal labels, and to introduce the concept of a resultant force.
Teaching: There aren’t new ideas here: as mentioned above, the students almost certainly used words like “tension”, “reaction”, “friction” in lesson 1 and I almost certainly said “I don’t know what you mean by that”, so in this lesson we’re going to define those words and then begin to use them appropriately. In the latter part of this lesson, there’s time for a discussion of the resultant force (the single force that has the same dynamic effect as the original set of forces) – this is where we will start to see some numbers.
Example: A book sits on a rough tabletop. The book is pushed horizontally by a force of 50N. Draw a diagram showing all of the forces acting on the book, and find the size of the frictional force if the resultant force acting horizontally is 30N.
Solution on board [click image to enlarge]:
Lesson 3
Objective: to identify pairs of forces that Newton’s 3rd Law says can be considered as being equal.
Teaching: I will tell my students that Newton was a genius (though surprisingly obsessed with alchemy and hocuspocus as well!), but stress that he didn’t prove that forces exist … for the very good reason that they don’t. The key point I want to get across to my students is that forces are a model, and that we’re trying to find a model that produces consistent explanations for what we have seen, and/or we’re trying to use mathematics to make predictions that can then be compared with what we do see, in the real world. If our model is not good enough (and we will talk about what that means), we have to come up with something better. The impressive thing about Newton’s model is that it has been “good enough” for over 300 years. Students may say that they’ve learned about his 3rd Law in Physics, and they’ll chirp “action and reaction are equal and opposite”. I will fauxwince and discuss with them the much better formulation that “if the forces which two objects cause to act on each other are modelled as equal and opposite, then the predictions that this model makes are not inconsistent with experimental observations”.
Example: A person carrying a briefcase stands in a lift, which is held up by a cable. Draw force diagrams for each of these four objects, and then identify the pairs of forces that Newton’s 3rd Law says can be modelled as equal and opposite.
Solution on board [click image to enlarge]:
Lesson 4
Objective: to state Newton’s 2nd Law in full, and to apply this in simple situations (including to calculate weight from mass).
Teaching: Starting from Newton’s 1st Law, which in essence says that “if there’s no resultant force acting on an object, then the object won’t change its velocity”, I will want my students to see that if there is a resultant force acting on an object, then that object will accelerate. I will explain to them that Newton’s 2nd Law gives the detail: “we can model the resultant force acting on an object in a certain direction as being equal to the mass of the object multiplied by the acceleration in the same direction”. I won’t let my students shorten this to “F = ma”: concision very rapidly leads to confusion.
Example: A car accelerates along a horizontal road. The car’s engine generates a driving force of 12000N. Find the frictional force between the tyres and the road if the 1500kg car accelerates at 6ms^{2}.
Solution on board [click image to enlarge]:
Lesson 5
Objective: to use Newton’s 2nd Law to solve more complex onedimensional problems
Teaching: What’s important here is making it clear that Newton’s 2nd Law acts on AN object, in A direction, so I will expect my students to specify these explicitly.
Example: Revisiting the man in the lift with the briefcase that we saw in an earlier lesson, find the reaction force between the man and the lift if his mass is 90kg, the mass of the briefcase is 4kg, and the lift accelerates downwards at 2ms^{2}.
Solution on board [click image to enlarge]:
I agree that we haven’t yet done any questions as hard as those that come up on M1 papers, but these lessons should have laid strong foundations of the new ideas and the crucial concepts. We’re now in a position to start looking at connected particles: three or four lessons on problems involving strings, pulleys, cars and caravans should ensure that these ideas now bed in.
Only several weeks later, once we’ve covered the constant acceleration formulae and used projectiles as a way into components, will we be ready to come back to our train and its dangerously connected trucks chugging uphill. With force diagrams and Newton’s 2nd Law covered already, and trigonometry revised by the Core teacher, we won’t be teaching deep concepts and relying on secure technical skills in one go: even with the most confident students, this is a good thing!
Image credit
Page header by takomabibelot (adapted), some rights reserved
