Deriving centripetal acceleration

When I was an A-level physics student (many, many years ago, when the world was young LOL) I found the derivation of the centripetal acceleration formula really hard to understand. What follows is a method that I have developed over the years that seems to work well. The PowerPoint is included at the end.

Step 1: consider an object moving on a circular path

Let’s consider an object moving in circular path of radius r at a constant angular speed of ω (omega) radians per second.

The object is moving anticlockwise on the diagram and we show it at two instants which are time t seconds apart. This means that the object has moved an angular distance of ωt radians.

Step 2: consider the linear velocities of the object at these times

The linear velocity is the speed in metres per second and acts at a tangent to the circle, making a right angle with the radius of the circle. We have called the first velocity v1 and the second velocity at the later time v2.

Since the object is moving at a constant angular speed ω and is a fixed radius r from the centre of the circle, the magnitudes of both velocities will be constant and will be given by v = ωr.

Although the magnitude of the linear velocity has not changed, its direction most certainly has. Since acceleration is defined as the change in velocity divided by time, this means that the object has undergone acceleration since velocity is a vector quantity and a change in direction counts as a change, even without a change in magnitude.

Step 3a: Draw a vector diagram of the velocities

We have simply extracted v1 and v2 from the original diagram and placed them nose-to-tail. We have kept their magnitude and direction unchanged during this process.

Step 3b: close the vector diagram to find the resultant

The dark blue arrow is the result of adding v1 and v2. It is not a useful operation in this case because we are interested in the change in velocity not the sum of the velocities, so we will stop there and go back to the drawing board.

Step 3c switch the direction of velocity v1

Since we are interested in the change in velocity, let’s flip the direction of v1 so that it going in the opposite direction. Since it is opposite to v1, we can now call this -v1.

It is preferable to flip v1 rather than v2 since for a change in velocity we typically subtract the initial velocity from the final velocity; that is to say, change in velocity = v2 – v1.

Step 3d: Put the vectors v2 and (-v1) nose-to-tail

Step 3e: close the vector diagram to find the result of adding v2 and (-v1)

The purple arrow shows the result of adding v2 + (-v1); in other words, the purple arrow shows the change in velocity between v1 and v2 due to the change in direction (notwithstanding the fact that the magnitude of both velocities is unchanged).

It is also worth mentioning that that the direction of the purple (v2v1) arrow is in the opposite direction to the radius of the circle: in other words, the change in velocity is directed towards the centre of the circle.

Step 4: Find the angle between v2 and (-v1)

The angle between v2 and (-v1) will be ωt radians.

Step 5: Use the small angle approximation to represent v2-v1 as the arc of a circle

If we assume that ωt is a small angle, then the line representing v2-v1 can be replaced by the arc c of a circle of radius v (where v is the magnitude of the vectors v1 and v2 and v=ωr).

We can then use the familiar relationship that the angle θ (in radians) subtended at the centre of a circle θ = arc length / radius. This lets express the arc length c in terms of ω, t and r.

And finally, we can use the acceleration = change in velocity / time relationship to derive the formula for centripetal acceleration we a = ω2r.


Well, that’s how I would do it. If you would like to use this method or adapt it for your students, then the PowerPoint is attached.

Please Like or leave a comment if you find this useful 🙂

Understanded of the pupils

It is a thing plainly repugnant . . . to Minister the Sacraments in a Tongue not understanded of the People.

Gilbert, Bishop of Sarum. An exposition of the Thirty-nine articles of the Church of England (1700)

How can we help our students understand physics better? Or, in more poetic language, how can we make physics a thing that is more ‘understanded of the pupils’?

Redish and Kuo (2015: 573) suggest that the Resources Framework being developed by a number of physics education researchers can be immensely helpful.

In summary, the Resources Framework models a student’s reasoning as based on the activation of a subset of cognitive resources. These ‘thinking resources’ can be classified broadly as:

  • Embodied cognition: these are simple, irreducible cognitive resources sometimes referred to as ‘phenomenological primitives’ or p-prims such as ‘if-resistance-increases-then-the-output-decreases‘ and ‘two-opposing-effects-can-result-in-a-state-of-dynamic-balance‘. They are typically straightforward and ‘obvious’ generalisations of our lived, everyday experience as we move through the physical world. Embodied cognition is perhaps summarised as our ‘sense of mechanism’.
  • Encyclopedic (ancillary) knowledge: this is a complex cognitive resource made of a large number of highly interconnected elements: for example, the concept of ‘banana’ is linked dynamically with the concept of ‘fruit’, ‘yellow’, ‘curved’ and ‘banana-flavoured’ (Redish and Gupta 2009: 7). Encyclopedic knowledge can be thought of as the product of both informal and formal learning.
  • Contextualisation: meaning is constructed dynamically from contextual and other clues. For example, the phrase ‘the child is safe‘ cues the meaning of ‘safe‘ = ‘free from the risk of harm‘ whereas ‘the park is safe‘ cues an alternative meaning of ‘safe‘ = ‘unlikely to cause harm‘. However, a contextual clue such as the knowledge that a developer had wanted to but failed to purchase the park would make the statement ‘the park is safe‘ activate the ‘free from harm‘ meaning for ‘safe‘. Contextualisation is the process by which cognitive resources are selected and activated to engage with the issue.

Using the Resources Framework for teaching

I have previously used aspects of the Resources Framework in my teaching and have found it thought provoking and helpful to my practice. However, the ideas are novel and complex — at least to me — so I have been trying to think of a way of conveniently organising them.

What follows in my ‘first draft’ . . . comments and suggestions are welcome!

The RGB Model of the Resources Framework

The RGB Model of the Resources Framework

The red circle (the longest wavelength of visible light) represents Embodied Cognition: the foundation of all understanding. As Kuo and Redish (2015: 569) put it:

The idea is that (a) our close sensorimotor interactions with the external world strongly influence the structure and development of higher cognitive facilities, and (b) the cognitive routines involved in performing basic physical actions are involved in even in higher-order abstract reasoning.

The green circle (shorter wavelength than red, of course) represents the finer-grained and highly-interconnected Encyclopedic Knowledge cognitive structures.

At any given moment, only part of the [Encyclopedic Knowledge] network is active, depending on the present context and the history of that particular network

Redish and Kuo (2015: 571)

The blue circle (shortest wavelength) represents the subset of cognitive resources that are (or should be) activated for productive understanding of the context under consideration.

A human mind contains a vast amount of knowledge about many things but has limited ability to access that knowledge at any given time. As cognitive semanticists point out, context matters significantly in how stimuli are interpreted and this is as true in a physics class as in everyday life.

Redish and Kuo (2015: 577)

Suboptimal Understanding Zone 1

A common preconception held by students is that the summer months are warmer because the Earth is closer to the Sun during this time of year.

The combination of cognitive resources that lead students to this conclusion could be summarised as follows:

  • Encyclopedic knowledge: the Earth’s orbit is elliptical
  • Embodied cognition: The closer to a heat source you are the warmer it is.

Both of these cognitive resources, considered individually, are true. It is their inappropriate selection and combination that leads to the incorrect or ‘Suboptimal Understanding Zone 1’.

To address this, the RF(RGB) suggests a two pronged approach to refine the contextualisation process.

Firstly, we should address the incorrect selection of encyclopedic knowledge. The Earth’s orbit is elliptical but the changing Earth-Sun distance cannot explain the seasons because (1) the point of closest approach is around Jan 4th (perihelion) which is winter in the northern hemisphere; (2) seasons in the northern and southern hemispheres do not match; and (3) the Earth orbit is very nearly circular with an eccentricity e of 0.0167 where a perfect circle has e = 0.

Secondly, the closer-is-warmer p-prim is not the best embodied cognition resource to activate. Rather, we should seek to activate the spread-out-is-less-intense ‘sense of mechanism’ as far as we are able to (for example by using this suggestion from the IoP).

Suboptimal Understanding Zone 2

Another common preconception held by students is all waves have similar properties to the ‘breaking’ waves on a beach and this means that the water moves with the wave.

The structure of this preconception could be broken down into:

  • Embodied cognition: if I stand close to the water on a beach, then the waves move forward to wash over my feet.
  • Encyclopaedic knowledge: the waves observed on a beach are water waves

Considered in isolation, both of these cognitive resources are unproblematic: they accurately describes our everyday, lived experience. It is the contextualisation process that leads us to apply the resources inappropriately and places us squarely in Suboptimal Understanding Zone 2.

The RF(RGB) Model suggests that we can address this issue in two ways.

Firstly, we could seek to activate a more useful embodied cognition resource by re-contextualising. For example, we could ask students to imagine themselves floating in deep water far from the shore: do the waves carry them in any particular direction or simply move them up or down as they pass by?

Secondly, we could seek to augment their encyclopaedic knowledge: yes, the waves on a beach are water waves but they are not typical water waves. The slope of the beach slows down the bottom part of the wave so the top part moves faster and ‘topples over’ — in other words, the water waves ‘break’ leading to what appears to be a rhythmic back-and-forth flow of the waves rather than a wave train of crests and troughs arriving a constant wave speed. (This analysis is over a short period of time where the effect of any tidal effects is negligible.)

Both processes try to ‘tug’ student understanding into the central, optimal zone.

Suboptimal Understanding Zone 3

Redish and Kuo (2015: 585) recount trying to help a student understand the varying brightness of bulbs in the circuit shown.

4 bulbs in a circuit: Bulbs A, B and D are in series with the cell but bulb C is in parallel across bulb B.
All bulbs are identical. Bulbs A and D are bright; bulbs B and C are dim.

The student said that they had spent nearly an hour trying to set up and solve the Kirchoff’s Law loop equations to address this problem but had been unsuccessful in accounting for the varying brightnesses.

Redish suggested to the student that they try an analysis ‘without the equations’ and just look at the problems in simpler physical terms using just the concept of electric current. Since current is conserved it must split up to pass through bulbs B and C. Since the brightness is dependent on the current, the smaller currents in B and C compared with A and D accounts for their reduced brightness.

When he was introduced to [this] approach to using the basic principles, he lit up and was able to solve the problem quickly and easily, saying, ‘‘Why weren’t we shown this way to do it?’’ He would still need to bring his conceptual understanding into line with the mathematical reasoning needed to set up more complex problems, but the conceptual base made sense to him as a starting point in a way that the algorithmic math did not.

Analysing this issue using the RF(RGB) it is plausible to suppose that the student was trapped in Suboptimal Understanding Zone 3. They had correctly selected the Kirchoff’s Law resources from their encyclopedic knowledge base, but lacked a ‘sense of mechanism’ to correctly apply them.

What Redish did was suggest using an embodied cognition resource (the idea of a ‘material flow’) to analyse the problem more productively. As Redish notes, this wouldn’t necessarily be helpful for more advanced and complex problems, but is probably pedagogically indispensable for developing a secure understanding of Kirchoff’s Laws in the first place.

Conclusion

The RGB Model is not a necessary part of the Resources Framework and is simply my own contrivance for applying the RF in the context of physics education at the high school level. However, I do think the RF(RGB) has the potential to be useful for both physics and science teachers.

Hopefully, it will help us to make all of our subject content more ‘understanded of the pupils’.


References

Redish, E. F., & Gupta, A. (2009). Making meaning with math in physics: A semantic analysis. GIREP-EPEC & PHEC 2009, 244.

Redish, E. F., & Kuo, E. (2015). Language of physics, language of math: Disciplinary culture and dynamic epistemology. Science & Education24(5), 561-590.

2021 Retrospective

A huge thank you to everyone who has viewed, read or commented on one of my posts in 2021: whether you agreed or disagreed with my point of view, you are the people that make the work of writing this blog so enjoyable and rewarding.

The top 3 most-read posts of 2021 were:

1. What to do if your school has a batshit crazy marking policy

Extract from blog post "What to do if your school has a batshit crazy marking policy"

This particular post was written back in 2019 and it’s sobering to realise that it is still relevant enough that it was featured by @TeacherTapp in November 2021. As edu-blog writers will know, this unlooked for honour generates thousands of views — thanks, @TeacherTapp!

Note to schools: please, if you haven’t done so already, please please please sort out your marking policy and make sure it is workable and fit for purpose. It would seem that, even now, teachers are being made to mark for the sake of marking, rather than for any tangible educational benefit.

2. FIFA for the GCSE Physics Calculation Win!

Extract from "FIFA for the GCSE Physics Calculation Win" blog.

The next post is one I am very proud of, even though FIFA is just a silly mnemonic to help students follow the “substitute-first-and-then-rearrange” method favoured by AQA mark schemes. Yes, FIFA did start life as a “mark-grubbing” dodge; however, somewhat to my own surprise, I found that the vast majority of students (LPAs included), can rearrange successfully if they substitute the numbers in first. Many other teachers have found the same thing as well — search #FIFAcalc on Twitter for some illustrative tweets from FIFAcalc’s biggest fans.

However, it is clear that the formula triangle method still has many adherents. I think this is unfortunate because: (a) they only work for a limited subset of formulas with the format y=mx; (b) they are a cognitive dead end that actively block students from accessing higher level STEM courses; and (c) as Ed Southall argues effectively, they are a form of procedural teaching rather than conceptual teaching.

3. Why does kinetic energy = 1/2mv^2?

Extract from blog explaining how to derive the kinetic energy equation

This post is a surprise “sleeper” hit also dating from 2019. It outlines an accessible method for deriving the kinetic energy formula. From getting a respectable but niche 200 views per year in 2019 and 2020, in 2021 it shot up to over 3K views. What is very encouraging for me is that most of these views come from internet searches by individuals from a wide range of backgrounds and not just my fellow denizens of the online edu-Bubble!

Here’s to 2022, folks!

The Coulomb Train Revisited (Part 2)

In this post, we will look at understanding potential difference (or voltage) using the Coulomb Train Model.

This is part 2 of a continuing series. You can read part 1 here.

The Coulomb Train Model (CTM) is a helpful model for both explaining and predicting the behaviour of real electric circuits which I think is suitable for use with KS3 and KS4 students (that’s 11-16 year olds for non-UK educators).

To summarise what has been discussed so far:


Modelling potential difference using the CTM

Potential difference is the ‘push’ needed to make electric charge move around a closed circuit. On the CTM, we can represent the ‘push’ as a gain in the energy of the coulomb. (This is consistent with the actual definition of the volt V = E/Q, where one volt is a change in energy of one joule per coulomb.)

How can we observe this gain in energy? Simple, we use a voltmeter.

Kudos to https://www.circuit-diagram.org/editor/ for the lovely circuit diagrams

On the CTM, this would look like this:

What the voltmeter does is compare the energy contained by two coulombs: one at A and the other at B. The coulombs at B, having passed through the 1.5 V cell, each have 1.5 joules of energy more than than the coulombs at A. This means that the voltmeter in this position reads 1.5 volts. We would say that the potential difference across the cell is 1.5 V. (Try and avoid talking about the potential difference ‘through’ or ‘of’ any part of the circuit.)


More potential difference measurements using the CTM

Let’s move the voltmeter to a different position.

On the CTM, this would look like this:

Let’s make the very reasonable assumption that the connecting wires have zero resistance. This would mean that the coulombs at C have 1.5 joules of energy and that the coulombs at D have 1.5 joules of energy. They have not lost any energy since they have not passed through any part of the circuit that actually has a resistance. The voltmeter would therefore read 0 volts since it cannot detect any energy difference.

Now let’s move the voltmeter one last time.

On the CTM, this would look like this:

Notice that the coulombs at F have 1.5 fewer joules than the coulombs at E. The coulombs transfer 1.5 joules of energy to the bulb because the bulb has a resistance.

Any part of the circuit that has non-zero resistance will ‘rob’ coulombs of their energy. On this very simple model, we assume that only the bulb has a resistance and so only the bulb will ‘push back’ against the movement of the coulombs and cost them energy.

Also on this simple model, the potential difference across the bulb is identical to the potential difference across the cell — but this is not always the case. For example, if the wires had a small but non-negligible resistance and if the cell had an internal resistance, but these would only come into play at A-level.

The bulb is shown as ‘flashing’ on the CTM to provide a visual cue to help students mentally model the transfer of energy from the coulombs to the bulb. In reality, instead of just one coulomb transferring a largish ‘chunk’ of energy, there would be approximately 1.25 billion billion electrons continuously transferring a tiny fraction of this energy over the course of one second (assuming a d.c. current of 0.2 amps) so we wouldn’t see the bulb ‘flash’ in reality.


How do the coulombs ‘know’ how much energy to drop off?

This section is probably more of interest to specialist physics teachers, but all are welcome.

One frequent criticism of donation models like the CTM is how do the coulombs ‘know’ to drop off all their energy at the bulb?

The response to this, of course, is that they don’t. This criticism is an artefact of an (arguably) over-simplified model whereby we assume that only the bulb has resistance. The energy carried by the coulombs according to this model could be shown as a sketch graph, and let’s be honest it does look a little dodgy…

But, more accurately, of course, the energy loss is a process rather than an event. And the connecting wires actually have a small resistance. This leads to this graph:

Realistically speaking, the coulombs don’t lose all their energy passing through the bulb: they merely lose most of their energy here due to the process of passing through a high resistance part of the circuit.

In part 3 of this series, we’ll look at how resistance can be modelled using the CTM.

You can read part 3 here.

Physics Six Mark Calculation Question? Give it the old FIFA-One-Two!

Batman gives a Physics-Six-Marker the ol’ FIFA-One-Two,

Many students struggle with Physics calculation questions at KS3 and KS4. Since 40% of the marks on GCSE Physics papers are for maths, this is a real worry for their teachers.

The FIFA system (if that’s not too grandiose a description) provides a minimal and flexible framework that helps students to successfully attempt calculation questions.

Since adopting the system, we encounter far fewer blanks on test and exam scripts where students simply skip over a calculation question. A typical student can gain 10-20 marks.

The FIFA system is outlined here but essentially consists of:

  • Formula: students write the formula or equation
  • Insert values: students insert the known data from the question.
  • Fine-tune: rearrange, convert units, simplify etc.
  • Answer: students state the final answer.

The “Fine-tune” stage is not — repeat, not — synonymous with re-arranging and is designed to be “creatively ambiguous” and allow space to “do what needs to be done” and can include unit conversion (e.g. kilowatts to watts), algebraic rearrangement and simplification.

The FIFA-One-Two

Uniquely for Physics, instead of the dreaded “Six Marker” extended writing question, we have the even-more-dreaded “Six Marker” long calculation question. (Actually, they can be awarded anywhere between 4 to 6 marks, but we’ll keep calling them “Six Markers” for convenience.)

The “FIFA-one-two” strategy can help students gain marks in these questions.

Let’s look how it could be applied to a typical “Six mark” long calculation question. We prepare the ground like this:

FIFA-one-two: the set up. (Note that since the expected unit of the final answer is given, this is actually a five marker not a six marker; however, the system works equally well in both cases.)

Since the question mentions the power output of the kettle first, let’s begin by writing down the energy transferred equation.

Next we insert the values. It’s quite helpful to write in any “non standard” units such as kilowatts, minutes etc as a reminder that these need to be converted in the Fine-tune phase.

And so we arrive at the final answer for this first section:

Next we write down the specific heat capacity equation:

And going through the second FIFA operation:

Conclusion

I think every “Six Marker” extended calculation question can be approached in a productive way using the FIFA-One-Two approach.

This means that, even if students can’t reach the final answer, they will pick up some method marks along the way.

I hope you give the FIFA-One-Two method a go with your students.

You can read more about using the FIFA system here: ‘Using the FIFA system for really challenging GCSE physics calculations‘.

Talk from Chat Physics 2021 https://chatphysics.org/chat-physics-live-2021

Update: Ed Southall makes a very persuasive case against formula triangle in this 2016 article.

FIFA for the GCSE Physics calculation win

Student: Did you know FIFA is also the name of a video game, Sir?

Me: Really?

Student: Yeah. It’s part of a series. I just got FIFA 20. It’s one of my favourite games ever.

Me: Goodness me. I had no idea. I just chose the letters ‘FIFA’ completely and utterly at random!

The FIFA method is an AQA mark scheme-friendly* way of approaching GCSE Physics calculation questions. (It is also useful for some Y12 Physics students.)

I mentioned it in a previous blog and @PedagogueSci was kind enough to give it a boost here, so I thought I’d explain the method in a separate blog post. (Update: you can also watch my talk at ChatPhysics Live 2021 here.)

The FIFA method:

  1. Avoids the use of formula triangles
  2. Minimises the cognitive load on students when approaching calculations.

Why we shouldn’t use formula triangles

Formula triangles are bad news. They are a cognitive dead end.

Screenshot 2019-10-27 at 15.34.54

During a university admissions interview for veterinary medicine, I asked a prospective student to explain how they would make up a solution for infusion into a dog. Part of the answer required them to work out the volume required for a given amount and concentration. The candidate started off by drawing a triangle, then hesitated, eventually giving up in despair. […]

They are a trick that hides the maths: students don’t apply the skills they have previously learned. This means students don’t realise how important maths is for science.

I’m also concerned that if students can’t rearrange simple equations like the one above, they really can’t manage when equations become more complex.

— Jenny Koenig, Why Are Formula Triangles Bad? [Emphases added]

[Update: this 2016 article from Ed Southall also makes a very persuasive case against formula triangle.]

I believe the use of formula triangle also increases (rather than decreases) the cognitive load on students when carrying out calculations. For example, if the concentration c is 0.5 mol dm-3 and the number of moles n required is 0.01 mol, then in order to calculate the volume V they need to:

  • recall the relevant equation and what each symbol means and hold it in working memory
  • recall the layout of symbols within the formula triangle and either (a) write it down or (b) hold it in working memory
  • recall that n and c are known values and that V is the unknown value and hold this information in working memory when applying the formula triangle to the problem

The FIFA method in use (part 1)

The FIFA acronym stands for:

  • FORMULA
  • INSERT VALUES
  • FINE TUNE (this often, but not always, equates to rearranging the formula)
  • ANSWER

Lets look at applying it for a typical higher level GCSE Physics calculation question

Screenshot 2019-10-27 at 16.04.29.png

We add the FIFA rubric:

Screenshot 2019-10-27 at 16.13.00.png

Students have to recall the relevant equation as it is not given on the Data and Formula Sheet. They write it down. This is an important step as once it is written down they no longer have to hold it in their working memory.

Screenshot 2019-10-27 at 16.18.15.png

Note that this is less cognitively demanding on the student’s working memory as they only have to recall the formula on its own; they do not have to recall the formula triangle associated with it.

Students find it encouraging that on many mark schemes, the selection of the correct equation may gain a mark, even if no further steps are taken.

Next, we insert the values. I find it useful to provide a framework for this such as:

Screenshot 2019-10-27 at 16.27.41.png

We can ask general questions such as: “What data are in the question?” or more focused questions such as “Yes or no: are we told what the kinetic energy store is?” and follow up questions such as “What is the kinetic energy? What units do we use for that?” and so on.

Screenshot 2019-10-27 at 16.35.54.png

Note that since we are considering each item of data individually and in a sequence determined by the written formula, this is much less cognitively demanding in terms of what needs to be held in the student’s working memory than the formula triangle method.

Note also that on many mark schemes, a mark is available for the correct substitution of values. Even if they were not able to proceed any further, they would still gain 2/5 marks. For many students, the notion of incremental gain in calculation questions needs to be pushed really hard otherwise they will not attempt these “scary” calculation questions.

Next we are going to “fine tune” what we have written down in order to calculate the final answer. In this instance, the “fine tuning” process equates to a simple algebraic rearrangement. However, it is useful to leave room for some “creative ambiguity” here as we can also use the “fine tuning” process to resolve difficulties with units. Tempting though it may seem, DON’T change FIFA to FIRA.

We fine tune in three distinct steps (see addendum):

Screenshot 2019-10-29 at 12.17.55.png

Finally, we input the values on a calculator to give a final answer. Note that since AQA have declined to provide a unit on the final answer line, a mark is available for writing “kg” in the relevant space — a fact which students find surprising but strangely encouraging.

Screenshot 2019-10-29 at 12.16.46.png

The key idea here is to be as positive and encouraging as possible. Even if all they can do is recall the formula and remember that mass is measured in kg, there is an incremental gain. A mark or two here is always better than zero marks.

The FIFA method in use (part 2)

In this example, we are using the creative ambiguity inherent in the term “fine tune” rather than “rearrange” to resolve a possible difficulty with unit conversion.

Screenshot 2019-10-27 at 17.20.42.png

In this example, we resolve another potential difficulty with unit conversion during the our creatively ambiguous “fine tune” stage:

Screenshot 2019-10-27 at 17.33.05.png

The emphasis, as always, is to resolve issues sequentially and individually in order to minimise cognitive overload.

The FIFA method and low demand Foundation tier calculation questions

I teach the FIFA method to all students, but it’s essential to show how the method can be adapted for low demand Foundation tier questions. (Note: improving student performance on these questions is probably a more significant and quicker and easier win than working on their “extended answer” skills).

For the treatment below, the assumption is that students have already been taught the FIFA method in a number of contexts and that we are teaching them how to apply it to the calculation questions on the foundation tier paper, perhaps as part of an examination skills session.

For the majority of low demand questions, the required formula will be supplied so students will not need to recall it. What they will need, however, is support in inserting the values correctly. Providing a framework as shown below can be very helpful:

Screenshot 2019-10-27 at 17.47.24.png

Also, clearly indicating where the data came from is useful.

Screenshot 2019-10-27 at 17.55.45.png

The fine tune stage is not needed, so we can move straight to the answer.

Screenshot 2019-10-27 at 18.01.07.png

Note also that the FIFA method can be applied to all calculation questions, not just the ones that could be answered using formula triangle methods, as in part (c) of the question above.

Screenshot 2019-10-27 at 18.06.16.png

And finally…

I believe that using FIFA helps to make our thinking and methods in Physics calculations more explicit and clearer for students.

My hope is that science teachers reading this will give it a go.

You can read about using the FIFA system for more challenging questions by clicking on these links: ‘Physics six mark calculation? Give it the old FIFA-one-two!‘ and ‘Using the FIFA system for really challenging GCSE calculations‘.

PS If you have enjoyed this, you might also enjoy Dual Coding SUVAT Problems and also Magnification using the Singapore Bar Model.

*Disclaimer: AQA has not endorsed the FIFA method. I describe it as “AQA mark scheme-friendly” using my professional own judgment and interpretation of published AQA mark schemes.

Addendum

I am embarrassed to admit that this was the original version published. Somehow I missed the more straightforward way of “fine tuning” by squaring the 30 and multiplying by 0.5 and somehow moved straight to the cross multiplication — D’oh!

My thanks to @BenyohaiPhysics and @AdamWteach for pointing it out to me.

Screenshot 2019-10-27 at 16.58.23.png

The Twelve Physics Pracs of Gove (Part Two)

A true-devoted pilgrim is not weary
To measure kingdoms with his feeble steps

–William Shakespeare, The Two Gentlemen of Verona

 

A picture [of reality]  . . .  is laid against reality like a measure  . . .   Only the end-points of the graduating lines actually touch the object that is to be measured  . . .   These correlations are, as it were, the feelers of the picture’s elements, with which the picture touches reality.

–Ludwig Wittgenstein, Tractatus-Logico-Philosophicus 2.141-2.1515

 

What they say of disc jockeys is also true of teachers: that someone, somewhere will remember some of your words forever; or, at least, for the duration of their lifetime. The downside is, of course, that you never know which of your words are going to be remembered. The wittily-crafted, near-Wildean aphorism pregnant with socratic wisdom — probably not. The unintentionally hilarious malapropism that makes you sound like a complete plonker — almost certainly.

To this day, I still remember Dr Prys’ sharp and appropriate response to a flippant comment (possibly from the callow 6th form me) about whether the scientific constants listed in the data book were truly trustworthy: “Look,” he said, “people have dedicated their whole lives to measuring just one of these numbers to one extra decimal place!” True devoted pilgrims indeed, mapping out the Universe step by tiny step, measurement by measurement.

I have written before on what I consider to be the huge importance of practical work in Physics education. Without hands-on experience of the hard work involved in the process of precise measurement, I do not believe that students can fully appreciate the magnificent achievement of the scientific enterprise: in essence, measurement is how scientific theories “touch” reality.

I am encouraged that parts of this view seem to be shared by the writers of the Subject Content guidance. (All hail our Govean apparatchik overlords!)

Of course, this has to be balanced with the acknowledgement that (as I understand it at least) teacher-assessed practical work will no longer count towards a student’s final exam grade. Many are concerned that this is actually a downgrading of the importance of practicals in Science and thus a backward step.

Sadly, they may turn out to be right: “We have to have this equipment for the practical/controlled assessment!” will no longer be a password for unlocking extra funding from recalcitrant SLTs (and from the exam budget too — double win!)

And, undoubtedly, some “teach-to-the-test” schools will quietly mothball their lab equipment (except for the showy stuff — like the telescope that no-one knows how to use — that they bring out for prospective pupil tours).

That would be sad, and although the DfE have, to be fair, nailed their pro-practical colours to the mast, we all know that the dreaded Law of Unintended Consequences may have the last laugh.

I would say it all depends on how the new A levels are actually put together. I will be attending some “launch events” in the near future. I will blog on whether I think we can expect an Apollo 11 or an Apollo 13 at that time.

In the meantime, I will be setting practicals galore as usual, as I’m old-fashioned enough to think that they give a lovely baroque feel to a scheme of work…

Look at me, I design coastlines, I got an award for Norway. Where’s the sense in that? None that I’ve been able to make out. I’ve been doing fiords all my life, for a fleeting moment they become fashionable and I get a major award. In this replacement Earth we’re building they’ve given me Africa to do, and of course, I’m doing it will all fjords again, because I happen to like them. And I’m old fashioned enough to think that they give a lovely baroque feel to a continent. And they tell me it’s not equatorial enough…
–Slartibartfast, from The Hitch-Hikers Guide to the Galaxy by Douglas Adams

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The Twelve Physics Pracs of Gove (Part One)

It’s not often that a DfE publication makes me feel like Kent Brockman, the newsreader from The Simpsons.

I’d like to remind them that as a trusted TV personality, I can be helpful in rounding up others to toil in their underground sugar caves.
Kent Brockman: “I, for one, welcome our new insect overlords.”

This feeling stems from reading the “Use of apparatus and techniques – physics” section from the DfE’s April 2014 Subject Content for AS and A level Biology, Chemistry, Physics and Psychology publication (p.23).

I had the rather novel feeling that it’s actually a sound list: and I, for one, welcome this intervention from our Govean-apparatchik overlords.

Why do I welcome this? Well, I feel that all too often we lose sight of the fact that, at its heart, Physics is, and must remain, a practical subject, the foundation of so much of the modern world.

Miroslav Holub’s poem “A Brief Reflection on Accuracy” paints a haunting and disturbing picture of what could be described as an entirely postmodernist, deconstructed and relativist (rather than relativistic) universe:

A certain soldier

    had to fire a cannon at six o’clock sharp every evening.

    Being a soldier he did so. When his accuracy was

    investigated he explained:

I go by

    the absolutely accurate chronometer in the window

    of the clockmaker down in the city.

   [ . . . ]

Oh, said the clockmaker,

    this is one of the most accurate instruments ever. Just imagine,

    for many years now a cannon has been fired at six o’clock sharp.

    And every day I look at this chronometer

    and always it shows exactly six.

[ . . . ]

So much for accuracy.
And fish move in the water, and from the skies
comes a rushing of wings while

Chronometers tick and cannons boom.

Without the grounding supplied by the art and science of measurement, I believe that we would all inhabit a castle-in-the-air universe as outlined above by Holub (whose experiences as an immunological research scientist are said to have influenced much of his poetry).


Is Holub’s nightmarish scenario even a remote possibility? Would we ever be in a world where “chronometers tick and cannons boom” but no-one actually checks the actual time by, say, looking out of the window to see if it’s daylight or not?

As with most nightmares, it’s probably closer than you think: “The sleep of reason brings forth monsters” as Goya suggested, and the steps that produce the monsters are often small, seemingly-harmless compromises of apparently little consequence.

One of my Y13 students, who has been attending a number of interviews for Physics courses, reports that some university departments have told him that “We spend a lot of the first year teaching students how to write formal laboratory reports as we find many of them have not learned how to do this during their A level courses.

Whaaa-aat? I nearly fell off my lab stool when Sam* told me this. In my opinion, that is unconscionable. “Oh, yeah,” Sam went on, “some of the students there said things like ‘Oh, our A level course content makes it unsuitable for practical teaching’.”


Opinions like that, if they genuinely reflect the views of the schoolteachers involved, are steps on the road to bringing forth monsters. Of course, it may not seem like a big deal to either the students or the teachers who are probably following what they see as a reasonable path of little resistance. But it is a big deal, it really is.

“And what did you say, Sam?” I asked.

“I said that we do a formal write up with a full analysis of experimental uncertainties every lesson.”

“Do we, Sam? Every lesson? Really?”
“Yeah, well,” said Sam with a smile, “I lied about that, didn’t I?”

“Exaggerated, Sam. I think you mean exaggerated.”

“Whatever you say, sir,” said Sam.

More on the 12 pracs of Gove in a later post..

* not his real name

The Physicist’s Eulogy

“You want a physicist to speak at your funeral. You want the physicist to talk to your grieving family about the Principle of Conservation of Energy, so that they will understand that your energy has not died. You want the physicist to remind your sobbing mother about the First Law of Thermodynamics: that no energy gets created in the universe, and none is destroyed. You want your mother to know that all your energy, every vibration, every joule of heat, every wave of every particle that was her beloved child remains with her in this universe. You want the physicist to tell your weeping father that amid the energies of the cosmos, you gave as good as you got.

“And at one point you’d hope that the physicist would step down from the pulpit and walk to your broken-hearted spouse there in the pew and tell him that all the photons that ever bounced off your face, all the particles whose paths were interrupted by your smile, by the touch of your hair — those hundreds of trillions of particles — have raced off like children, their ways forever changed by you. And as your spouse rocks in the arms of a loving family, may the physicist let him know that the photons that bounced from you and that were gathered in the particle detectors that are his eyes, that those photons have created within his brain constellations of electromagnetically charged neurons whose energy will go on forever.

“And the physicist will remind the congregation of how much of all our energy is given off as heat. There may be a few fanning themselves with their programs as she says it. And she will tell them that the warmth that flowed through you in life is still here, still part of all that we are, even as we who mourn continue the heat of our own lives.

“And you’ll want the physicist to explain to those who loved you that they need not have faith; indeed, they should not have faith. Let them know that they can measure, that scientists have measured precisely the conservation of energy and found it accurate, verifiable and consistent across space and time. You can hope that your family will examine the evidence and satisfy themselves that the science is sound and that they will be comforted to know that your energy is still around.

“Because, according to both the First and Second Laws of Thermodynamics, not one bit of you is gone: you’re just less orderly.”

Original author unknown. Quoted by ‘WelshmanEC2’ in The Guardian http://www.theguardian.com/commentisfree/2014/feb/14/below-the-line-people-angry-science-astronomy-enthusiast [accessed 14/2/14].  NB: some minor stylistic amendments made in the version presented here.

Update: the original author is Aaron Freeman who performed it on NPR Radio in 2005. Original transcript here. Audio of performance (with added slideshow) here.

Through Other Eyes

To see a World in a Grain of Sand
And a Heaven in a Wild Flower,
Hold Infinity in the palm of your hand
And Eternity in an hour.

–William Blake, Auguries of Innocence

Yet another whiny email from a Year 12 student. He requests a special selection of past paper questions on a particular topic. My answer? “Go to the flipping website that I have so laboriously set up for your benefit which has resources galore of that particular ilk and more, as well as digital bells and whistles, you clod!”

I did express the above sentiments somewhat more diplomatically in the email. And, to be honest, I was glad to get even that whiny missive: I feel we might be on the verge of that tipping point where the Year 12s stop being passive GCSE Spongebobs and become a little more independent, a little more grown up, a little more like proper 6th form students. Maybe. Just maybe. It might be a sign. I loved it when I heard him say to the other students in the class that “there’s a lot of good stuff on the website.”

Now I know that the student concerned had seen the website previously. He had even complimented me on it. But he obviously hadn’t seen it properly. And, strangely enough, it started me thinking about how we do not always see the world as others see it.

To my mind, one the finest descriptions and “thought experiments” on this topic comes from a short story by the incomparable R. A. Lafferty:

“It may be that I am the only one who sees the sky black at night and the stars white,” he said to himself, “and everyone else sees the sky white and the stars shining black. And I say the sky is black, and they say the sky is black; but when they say black they mean white.”
— R. A. Lafferty, Through Other Eyes, “Nine Hundred Grandmothers and other stories”

Do we genuinely ever see the world as others see it? The truth is — ultimately at least — we don’t rightly know.

Charles Cogsworth, the scientist in R. A. Lafferty’s short story, invents a machine called the Cerebral Scanner which literally allows its user to see out through other people’s eyes, and to truly see the world as others see it.

Charles makes the mistake of using the Scanner to look out through the eyes of his girlfriend, Valery. He is horrified: “she hears sounds that I thought nobody could ever hear. Do you know what worms sound like inside the earth? They’re devilish, and she would writhe and eat dirt with them.”

Valery also uses the Cerebral Scanner to look out through the eyes of Charles, and is equally disturbed. She confronts the hapless Charles:

“You can look at a hill and your heart doesn’t even skip a beat. You don’t even tingle when you walk over a field.”

“You see grass like clumps of snakes.”

“That’s better than not even seeing it alive.”

“You see rocks like big spiders.”

“That’s better than just seeing them like rocks. I love snakes and spiders. You can watch a bird fly by and not even hear the stuff gurgling in its stomach. How can you be so dead? And I always liked you so much. But I didn’t know you were dead like that.”

“How can one love snakes and spiders?”

“How can one not love anything? It’s even hard not to love you, even if you don’t have any blood in you. By the way, what gave you the idea that blood was that dumb colour? Don’t you even know that blood is red?

“ I see it red.”

“You don’t see it red. You just call it red. That silly colour isn’t red. What I call red is red.”

And he knew that she was right.

–R. A. Lafferty, Through Other Eyes

The phrase that has stayed with me over all the years since I first read this story as a callow youth is Valery’s description of what is, to her, Charles’ unforgivable deadness to the wonders of the world: “You can watch a bird fly by and not even hear the stuff gurgling in its stomach.

That is the experience of Physics that I want to communicate to my students. I want them to look at the universe and hear the stuff gurgling in its stomach. I want them to be able to experience their understanding, not just on an intellectual level, but also on a visceral level. This, to my mind, is what makes studying Physics fun.

Do I always succeed? Absolutely not. Do I sometimes succeed? Maybe, sometimes.

Do I have fun in classroom? A significant part of the time, yes. This is why I wanted to become a teacher. This is why I have stayed a teacher. And what about the other rubbish that is constantly being foisted on us?

Well, just for now, I think I’ll let it all go hang. I’ll worry about that on Monday.