## A Level maths - career based scenario problems

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I'm writing a set of resources for A Level Maths/Further Maths, which are intended to be 20-30 minute group tasks built around a workplace scenario.  I'm looking for ideas for interesting jobs which might theoretically encounter a problem which could be tackled with A Level maths.

So far I have...

Formula 1 strategist - problem involves deterioration of lap times due to tyre wear, pit stops and integration of a polynomial to determine the optimal strategy.

High rope access supervisor - rigging a tight rope, resolving forces and calculating tension.

Prototype engineer - designing a bearing which will run in one direction and run 'lumpilly' in the other direction based around the volume of revolution of a squircle.

Statistical modelling - giving Boris some advice about how many deaths he will expect based on the sum of a geometric series.

Lab technician - flow rate of a gas into a cylinder, leading to an integration problem to solve for a required volume.

I realise I have got 3 integration focussed tasks there.  I'm not looking for more of the same, so other areas of the course which could get you thinking.

Parametric equations (including calculus perhaps)
Equations of circles.
Binomial expansion.
Complex numbers
Vectors (up to vector equations of planes and lines, angles between planes and lines, intersection of planes)
Standard deviation
Histograms
Binomial probabilities
Normal Distribution
Venn diagrams/Tree diagrams
Condititonal Probability/Bayes Theorem
F=ma including friction on an inclined plane and connected particles
Moments
Work/Energy principle (IKE+IPE+work in=FKE+FPE + energy lost)
Bouncing balls in 1 or 2 dimensions

I could write a standard exam question on any of the above and make it notionally in the context of "Bob is a statistician, he wants to find the standard deviation of this data set".  What I want are more open ended tasks, which the students can make assumptions, discuss and decide on the best way to answer the question, but which will get them thinking about an interesting career while they do it.  We are an Engineering college, so anything in that direction would be great.

Many thanks for any useful suggestions.

It's hard to imagine it happening, but a public health type would need working knowledge of about about half of those should a global pandemic ever come along.

> Work/Energy principle (IKE+IPE+work in=FKE+FPE + energy lost)

Formula 1 engineer matching the size of a KERS unit to a car.  Also includes quadratics in terms of cable sizes, ohmic losses and melting things.  Moments feature as well.

Daft question, but what does that mean?

> Standard deviation

Quality control engineer at DMM setting tolerances for testing parts

friction, inclined plane, moments - hard braking whilst going downhill without going over bicycle handlebars. it applies to any vehicle.

I don't understand squaricle (I had to look that up) - clockwise/anticlockwise no difference to rotational symmetry so it does not have a lumpy and smooth direction.

Post edited at 19:53

Apologies for the negative tone and the fact that I'm not going to be of much/any help, but is it worth the effort?

http://mrbartonmaths.com/teachers/research/real.html

If you really want to produce these resources, my takeaway from the research is unless you have a really able group of students, you need to make the problems as straightforward as possible to compensate for the fact that they will find them less accessible than "normal" questions.

> > Work/Energy principle (IKE+IPE+work in=FKE+FPE + energy lost)

> Formula 1 engineer matching the size of a KERS unit to a car.  Also includes quadratics in terms of cable sizes, ohmic losses and melting things.  Moments feature as well.

I think this is going above my head to squeeze all that into a 20 minute A Level task, however I will go with KERS and see how much I can get out of that.

> Daft question, but what does that mean?

Y12 A Level stuff, taking a quadratic function that represents a model and understanding how the roots and turning points relate to the model.

> > Standard deviation

> Quality control engineer at DMM setting tolerances for testing parts

Cheers.

I'm getting them to rotate the Squircle around the y-axis to make sort of a cylinder with rounded corners then find the volume of it.

> I don't understand squaricle (I had to look that up) - clockwise/anticlockwise no difference to rotational symmetry so it does not have a lumpy and smooth direction.

I'll take a read - thanks Rob, if it's done right, then yes - I think it is worth the bother.  I'm not putting A Level problems in context, i'm giving them a problem, getting them to talk about careers, getting them inspired to think about what they might be doing in 3-5 years time.  It's not going to be easy, i'm going to push them to work together and do something - anything, to try and get towards a solution.  I'm ultimatly not bothered if the maths is right that they do, i'll give them a regular test to assess that.

> I don't understand squaricle (I had to look that up) - clockwise/anticlockwise no difference to rotational symmetry so it does not have a lumpy and smooth direction.

Sounds like a sprag clutch, though they completely jam in one direction rather than just being lumpy

Not sure how the math would work but how about something about how the petrol station / electric car charger question will pan out. Plenty of scope for assumptions there.

> getting them inspired to think about what they might be doing in 3-5 years time.  It's not going to be easy, i'm going to push them to work together and do something

IME this certainly has a positive effect on the motivation of the students you can inspire, although for me it was more at GCSE level and along the lines of if I want to be X Ineed to get a grade ? in Maths. You have reminded me of the days when their was a coursework element to the A level mechanics modules. One involved designing a water slide into a plunge pool. One student suggested reducing friction applying baby oil to the customers first, he got a place at Cambridge and you can guess where he is employed now.

Posh massage emporium

mathematical reasons for a shell not hitting the target?

windage, barrel wear, problems with datum used for indirect fire, effect of charge v elevation/trajectory

Post edited at 22:29

> Posh massage emporium

Inspiring students was one of the most rewarding parts of the job.

I recently had a problem to solve, namely how to shot blast a parabolic dish out of stone. The stone is sat on a turntable (of course it is, this is UKC) the blasting nozzle travels from the outside in towards the centre at a variable speed, cutting at a constant rate as it goes (that is a constant rate in a static block of stone). The problem was to find a formula to define the speed the cutting head had to travel as it approaches the centre. I did get there in the end, the maths was not above A-level and the results were fairly good.

Yaw miss alignment on wind turbines. Power lost as a result of wind not directly perpendicular to the rotor.

Using wind roses and wind speed distributions for resource yield assessment and increased yield performance as a result of corrected yaw missalignment.

Take a look at yield line analysis in structural engineering.  Work, energy,  geometry.

A standard introductory econometrics text book will have plenty of real world examples to get them thinking about standard deviation, normal distributions, conditional probabilities, etc. Stock and Watson is good.

Professional poker player? Kids would love that.

I only teach up to middle school but every time we do probability and mention gambling they are hooked

Does it work well in a single pass?  Impressive that you can CNC machine stone with a sandblaster.

It's only possible in a single pass (at least with the machine I used). The products made are garden / outdoor mounted decorative pieces, the client wanted to make them parabolic to reflect sounds to a point and he's planning to coat/paint the insides for lamps. It's not the Hubble telescope (Flintstones edition)!

Most of my regular use of maths is based around the costs of growing an animal to the point at which it is sold and speculation on the way that the price will move whilst I am doing so.

On conditional / Bayes you could do something topical, given that many people are testing themselves for covid by one test or another.

If we have a disease with prevalence 1%, say, and a test for it that is 95% likely to give a positive result if someone has the disease, and 97% likely to give a negative result if they don't, what is the probability that someone who tests positive actually has the disease.

They can work this out with a simple tree diagram.

The result surprises most people. You could ask them, is such a test useful? Is the probability of testing positive if you have the disease the same as the probability of having the disease, if you test positive? What does a p-value (if you do such things) tell you, and what do you really want to know?

Jacob Cohen did an accessible paper on this stuff years ago, and goes through the example I have given, on schizophrenia. The fabulous title is: The Earth is Round (p<0.05).

Post edited at 12:32

> every time we do probability and mention gambling they are hooked

Gambling does seem to be addictive...

> We are an Engineering college, so anything in that direction would be great.

A few things I have worked on in a long electronic engineering career, that were more system design, and used basic maths/physics.

Cell planning for mobile phones (power and coverage)

Redundancy of components with a given MTTF rate to meet a given availability target

GNSS position computation

Satellite orbits

Train stopping distances

Train lookahead sensor systems (required resolution, sense distance, real world problems)

Circuit board track separation calculations (measuring minimum separation between tracks)

Odometry using accelerometers (with drift...)

Measuring temperature with thermistors, and compensating for manufacturing variation (of all components in the system) to hit set trip points, plus the differences in measurement method (pot divider, RC network), including computational complexity...

Simple upsampling/downsampling filter using linear interoplation

I'm not a mathematician; my maths declined at university, and never recovered (from a very easy A at A level). But many of my colleagues are. And they produce amazing stuff with the likes of Mathematica, and then convert it into real world implementations. Things like simulating a GNSS satellite constellation, modelling all the perturbations, and generating a signal, as received by a terrestrial receiver, with millimetric accuracy. Or figuring out where mysteriously vanished planes might be found in an ocean...

Post edited at 13:09

Labs in hospitals need to decide from time to time whether to keep going with an assay process they have been using, or whether to switch to a new one which may be faster or cheaper. Before they can do this they need to assure themselves that the new test will be as effective as the current test.

For example, biomedical scientists in the histology lab are testing a new protocol for an immunological assay that detect antigens in tissue samples.Each of 25 samples are tested with the both new and the old protocols. Each protocol gives a binary result, but neither test is perfect, so it is to be expected that there will be some differences between the outcomes of the two tests, with one giving a 1 while the other gives 0, and vice-versa.

If, under a null hypothesis that the two tests are as effective as each other, then of those results that change, the change is as likely to be from 1 to 0 as from 0 to 1. ie if p is the probability of change from 0 to 1, then under the null, p=0.5.

If N tests flip, and x of them flip from 0 to 1 with probability p, then you can write an exact expression for the probability that this will happen, using the binomial distribution.

You can write a simplified version of this under the null, where p=0.5.

You can then work out the probability under the null of getting x or fewer flips, or N-x or more - the very definition of a p-value.

You can then decide whether to reject the null.

(Apologies to any BMSs out there, but I think it goes something like this.)

I work on Video content piracy prevention.   We add a very faint (imperceptible) watermark to TV programs as they are broadcast. the watermark is unique to each end-user.  We recover the watermark from pirated content using a Viterbi Algorithm. The watermark adds and the tv signal (being the noise) cancels over time.

A nice example of applying a Viterbi Algorithm to images here. MatLab. https://www.mathworks.com/matlabcentral/fileexchange/47359-viterbi-algorithm.

I can send you some sample content with one of your student's names burnt in as a watermark. and then they can compete to see who can find out who stole the content.

Optical fibre Comms - nice example of dB for attenuation, can also do bandwidth calculation.

How about dipping a toe into machine learning via a spot of linear regression, decision trees, and (possibly) neural networks?

Just to play the devil's advocate: My problem is that my students lack ABSTRACT understanding of maths, knowledge they can then apply to the concrete problem cropping up in their research which you will never present in your exams. And I am a biologist, just think of my poor physics  and engineering  colleagues!

Making maths concrete is the original sin, dumbing down to mere calculating!

Otherwise, using Covid testing to introduce Bayesian probabilities is the obvious idea.

CB

A rectangular four walled room has one entrance on three of its walls.

Each of the three entrances is at a slightly different level.

The x,y,z coordinates at the mid point of each entrance describe a plane.

To remove steps in the floor, for the benefit of wheelchair access, it is proposed to lay a floor screed to the levels of the plane described above.

Given the x,y coordinates for each corner of the room calculate the screed level z at all four corners...

This problem is similar to something I had to do for a railway station concourse.

> Making maths concrete is the original sin, dumbing down to mere calculating!

I disagree. My maths declined at university because it was taught by the mathematics department, and was abstract and bore no relationship to the needs of our electronics course. I could not see what it was for, and could not visualise the abstract concepts they were trying to teach. I struggle with intangible concepts I cannot visualise; without a good mental model, my brain simply refuses to remember stuff.

Up to A level, I could see the practical application of everything we were taught. I have continued to apply my limited maths to real world problems, such as those I listed above.

Post edited at 02:36

> Just to play the devil's advocate: My problem is that my students lack ABSTRACT understanding of maths, knowledge they can then apply to the concrete problem cropping up in their research which you will never present in your exams.

At ALevel 75% of the exam is abstract, here’s a maths problem, solve it. 25% is often pseudo applied - tide height follows this trig function, find the maximum tide height, that sort of thing.

Consequently most of what we do is abstract problems.

What I’m trying to do is show them that in a problem you might encounter in your job, the useful maths techniques might not be obvious, and there may be several valid approaches.

Well said.

About a quarter of a century ago, we had two outstanding maths lecturers from the maths department on the EEE courses, as I first took on some course management responsibilities. They were both exceptional mathematicians (one, now in his 80s, still co-publishes with Cambridge maths profs) and very talented at teaching the subject in a way that inspired young engineers through application, with stories from consultancy work they had done with engineering companies. When one retired and the other was moved, the service teaching quality plummeted, alongside student satisfaction. The new service lecturer was a pure maths research Prof, who had just moved from Russia.

Following some successful remedial support we were forced to put on within our team, we tried an experiment: designing a new module, integrating half the year 2 maths module with some content from control and communication modules, with a big increase in utilising the visualisation benefits of MATLAB (the original maths module content mainly focussed on Laplace and Fourier methods). Unfortunately, (despite the irony, the proposed new structure had more formally labelled maths syllabus content) the IEE told us if we made the change we would lose validation. Back to square one, we dumped the maths department servicing and employed an able applied mathematician with engineering experience who would use the same teaching style as the two earlier stars.

I'm now thinking that you are trying to achieve two different things.

> What I’m trying to do is show them that in a problem you might encounter in your job

If the aim of this is to inspire them/make then think of future courses it could well be very worthwhile

>the useful maths techniques might not be obvious, and there may be several valid approaches.

This seems like an attempt to address the "problem solving" issue we have had since the "new" GCSE. I will be interested if you have any success with middle ability students. I found the UKMT challenges and oxbridge extrance papers useful with the more able students, but hey tend very much towards the abstract side of things. I think I had some success (at GCSE not A level) producing resources where there were 5/6 problems that all looked very similar in appearance

but required different techniques to solve them.

> I'm now thinking that you are trying to achieve two different things.

Yes, that’s a fair summary. I don’t think they are mutually exclusive though. In a job they will encounter problems which don’t have an obvious method to reach a solution, while discussing this problem they can also discuss the career that it is linked to.

> producing resources where there were 5/6 problems that all looked very similar in appearance but required different techniques to solve them.

I’d be interested in having a look at these if you don’t mind sharing? andyblackett “at” googlemail dot com

I've been retired a few years but they were basically copied or adapted from here

https://ssddproblems.com/

If you have time, there are some interesting papers here on problem solving. I do find it ironic that I have time to read this stuff now I have retired.

http://mrbartonmaths.com/teachers/research/problem.html

I disagree in turn. For engineering, you may have a point, I simply do not know enough about engineering studies.

For the natural sciences, grasping abstract mathematical concepts is IMO essential, even if the specific concepts may never be applied again anywhere after the linear algebra class in the 2nd semester. The point is precisely to train and develop the ability to move on from any concrete cases at hand to the underlying, abstract principles*. This is the essence of science, and you have to start hammering this idea in from day one, ideally already starting in school!

CB

* and to be brutal, to weed out those students early who chose biology solely because they were incapable of abstract thought and dealing with the level of maths required by other subjects, hated maths already in school, but like petting rabbits and want to rescue whales.....

It's fair enough that a small minority are not able enough to continue but when a teacher turns off a whole class that's just bad teaching. All the science departments I've worked alongside have faced occasional similar problems to the prof who couldn't bring himself to teach my students and applying maths to any subject area increases interest, understanding and focus.

> applying maths to any subject area increases interest, understanding and focus.

I'm not qualified to comment on University departments, but in my experience most A level students enjoy pure maths far more than the applied modules. My original degree was Engineering and I really liked teaching mechanics, but only a small minority of students did compared to pure modules.

Sure. I was talking about students on a degree course with specific maths skills as an important part of the syllabus. The hard  reality is that many courses that cared about a good A level equivalent ability at entry have had to run levelling up classes to try and ensure everyone was actually at that standard for the parts that related to the course.

Post edited at 10:19

Vectors - working out the route/course and start time for a long channel crossing on a slow boat with varying tidal flows at different places to arrive at the destination at a certain time. Might not be up to A level standard perhaps as it's simply lots of iteration of a sub-A level concept? Could be complicated by making the boat a yacht with different speeds depending on heading and wind strength.

Post edited at 10:59

Many Project Euler problems are fiendishly difficult and nearly all require both coding and mathematical insight (neither alone will get you over the line), which makes them good fun to tackle and often, I would think, a good exercise for engineers, but, IIRC,  this one (PE #607) required not much more than basic trig, plus knowledge of what makes refraction happen:

Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and so the journey would take 10 days. However, their path is crossed by a long marsh which runs exactly South-West to North-East, and walking through the marsh will slow them down. The marsh is 50 leagues wide at all points, and the mid-point of AB is located in the middle of the marsh. A map of the region is shown in the diagram below.

The marsh consists of 5 distinct regions, each 10 leagues across, as shown by the shading in the map. The strip closest to point A is relatively light marsh, and can be crossed at a speed of 9 leagues per day. However, each strip becomes progressively harder to navigate, the speeds going down to 8, 7, 6 and finally 5 leagues per day for the final region of marsh, before it ends and the terrain becomes easier again, with the speed going back to 10 leagues per day.

If Frodo and Sam were to head directly East for point B, they would travel exactly 100 leagues, and the journey would take approximately 13.4738 days. However, this time can be shortened if they deviate from the direct path.

Find the shortest possible time required to travel from point A to B, and give your answer in days, rounded to 10 decimal places [That 10 dp is a Project Euler thing. They do that to stop you brute-forcing the answer.]

Texas Republican congressman Louie Gohmert has asked a senior US government official if changing the moon’s orbit around the Earth, or the Earth’s orbit around the sun, might be a solution for climate change

What forces would be required? Would the long term stabilty of solar system be affected - Use Henri Poincaré entry to Oscar II, King of Sweden's mathematical competition to lead into chaos.

Or the magnetised blood theory expounded by Sherri Tenpenny, a physician. Using the experimental evidence (not) provide by nurse practitioner student Joanna Overhol together with Maxwells equations, fluid mechanics to calculate what items could be hung from various body parts without piecings.  Then pull the speed of light out of them.

> What forces would be required? Would the long term stabilty of solar system be affected - Use Henri Poincaré entry to Oscar II, King of Sweden's mathematical competition to lead into chaos.

> Or the magnetised blood theory expounded by Sherri Tenpenny, a physician. Using the experimental evidence (not) provide by nurse practitioner student Joanna Overhol together with Maxwells equations, fluid mechanics to calculate what items could be hung from various body parts without piecings.  Then pull the speed of light out of them.

Is this the result of a basic AI system trying to answer my question? I have no idea what you are saying.

Circle theorems are useful to provide an estimate of minimum curve radius before a rail wheel falls off the track (inside track)/ flange strikes the track (outside track).

Conditional probability is heavily used in life extensions of components and in determination of machine lifetime (things like L10 bearing life, population data on component failures across large fleets) .

6 Sigma manufacturing principles and cost of tightening specs (see WC Rocks in 2012).

Tipping/ Friction comparison can be used to support calculation of an SPMT's stability carrying a tall load (further complicated by stability planes created by suspension which will collapse if heavy load breaks edge of plane).

Trig to determine likelihood of a crane's boom clashing with a building (that was a total pain, some good datasheets on the Mammoet/ Sarens websites).

Parametric equations are routinely used in CAD modelling, often take people by surprise when they don't set them up right.

If you want to go down that route, why not follow the Indiana approach to making maths easier and use legislature to let

pi := 3

by law.

CB

A real world problem I have come across while designing insulin injector pens (but would apply to lots of products):

You rotate a knob attached to a leadscrew which winds up a spring. How should the pitch of the leadscrew change along its length to keep the applied torque constant?

This is simple (not trivial) with a frictionless screw, but becomes moderately involved if you consider friction.

It's an inclined plane problem at heart, but can be tackled via an energy argument. Or you can even bring in parametric equations.

And a bonus design challenge: how do you even make a leadscrew with a variable pitch along its length? Or more specifically, what should the nut look like?

> If you want to go down that route, why not follow the Indiana approach to making maths easier and use legislature to let

>      pi := 3

> by law.

This made me laugh - my wife (ex Cambridge astrophysicist) insists that you may as well say that Pi =3, because it certainly isn't 3.141 and when calculating something like the size of a star millions of light years away it's close enough!

I am sure she knows this XKCD gem then:

https://xkcd.com/2205/

CB

She didn't, but she did enjoy it!

> The hard  reality is that many courses that cared about a good A level equivalent ability at entry have had to run levelling up classes to try and ensure everyone was actually at that standard for the parts that related to the course.

You sound like me moaning about having to design Year 7 catch up courses, although to be fair I never worked in a selective Secondary

The simple version of this that catches many sailors out is: you are on a yacht, about to sail due East across the North Sea. The trip will take ~24 hours. The tide flows North for six hours, then South for six hours etc. The wind is blowing from due East and should hold steady throughout your crossing. You can only sail at least 45 degrees to the apparent wind.

Which tack (so heading ~NE or ~SE) should you start on, and when/how frequently should you switch?

>  Or more specifically, what should the nut look like?

A pair of pins on opposite sides of the bolt? The pins run in the thread and don't care about the pitch because they're pins. If pins don't work then make them into pivoting sections of thread that can follow any pitch.

It's not really moaning as we turned a practical problem into a valuable solution. Coventry Poly/Uni set the ball rolling with a more comprehensive research look at the issues and actions. We took maths qualifications from many sources all of which were very variable and gave different skills even when done well.  Good teaching and support matters from lectures to individual help and fitting that into the bigger picture makes courses better.

In reply to cb294 & captain paranoia:

One of the key issues to me is learning style; be it mathematics or programming, some people are drawn in by the subject itself and thrive in the abstract; for others the abstract makes little-to-no sense until they apply it to a tangible, real world problem, at which point the theory underpinning the abstract goes "click".

Education systems should recognise this big distinction and not cut one group or the other off; at some point people can be sent their separate ways via degree programs, but before that a single curriculum runs the risk of failing both groups in equal measure.

>  and to be brutal, to weed out those students early who chose biology solely because they were incapable of abstract thought and dealing with the level of maths required by other subjects, hated maths already in school, but like petting rabbits and want to rescue whales.....

An issue that seems very accentuated in the UK compared to other places I have (more limited) experience of.  The conventional view of biology here is often that it is applied chemistry, which is applied physics, which is applied maths, with the maths safely abstracted away  I increasingly see the key parts of biology as applied information theory*; far less removed from the maths. As well as that, the ability to build and understand complex paper and mathematical models at different levels is so important, and needs the kind of analytic thinking fusing pure and applied maths that is not well taught in the UK, and that is what the OP is looking to build their teaching around - so, good one ablackett.

* - then again, perhaps everything in existence comes down to information theory.  If so, my saying so is perhaps equal parts profound and stating the bloody obvious.

Post edited at 00:09

I'm sat here not sleeping with in excess of 60 midge bites, so your post is just what I needed to take my mind off it...

> You rotate a knob attached to a leadscrew which winds up a spring. How should the pitch of the leadscrew change along its length to keep the applied torque constant?

> This is simple (not trivial) with a frictionless screw, but becomes moderately involved if you consider friction.

Surely this is not a solvable problem unless you further qualify it to "... to keep the applied torque constant for a given speed of rotation of the knob"?

> and a bonus design challenge: how do you even make a leadscrew with a variable pitch along its length?

Machine a metal rod, rotating it at a fixed speed against a point-like cutting tool which traverses the length of the rod with a speed set as a function of time. (Edit: or equivalently, speed set as a function of distance along the rod)

> Or more specifically, what should the nut look like?

You can't have a nut in the conventional sense of the engagement with the thread being more than 360°.  You need a "nut" that subtends a very small part of the circle; I guess 10° at most. It needs some flex or rotational DOF to allow it to adjust to the changing angle of the thread.  To ensure repeatable engagement for something relatively long, narrow and thin it's going to need the thread cutting much deeper than for a conventional nut, I'm guessing you had to machine in a recess at least 1.5 mm into the worm thread?

I suppose being a spring, you had the thread run through the spring with the top of the spring having multiple parts protruding radially inwards to engage with the thread.

Perhaps there's some way of having a full 360° inwards protrusion from the spring that engages with the variable pitch thread, but I can't figure it out.

Fascinating to hear about the level of design work that goes in to an epipen.  Having been shown how to use one I had no idea that it had such smarts in to to make it so intuitive to use.

Post edited at 00:27

Spot on!

> > You rotate a knob attached to a leadscrew which winds up a spring. How should the pitch of the leadscrew change along its length to keep the applied torque constant?

> Surely this is not a solvable problem unless you further qualify it to "... to keep the applied torque constant for a given speed of rotation of the knob"?

That would be true if there were a viscous drag term (which there probably is in reality), but neither spring rate nor friction is time/speed dependent.

> Machine a metal rod, rotating it at a fixed speed against a point-like cutting tool which traverses the length of the rod with a speed set as a function of time. (Edit: or equivalently, speed set as a function of distance along the rod)

In fact all these things are injection moulded, so the question becomes how do you get it out of the mould? That's not something a layperson could easily answer!

> You can't have a nut in the conventional sense of the engagement with the thread being more than 360°.  You need a "nut" that subtends a very small part of the circle; I guess 10° at most. It needs some flex or rotational DOF to allow it to adjust to the changing angle of the thread.  To ensure repeatable engagement for something relatively long, narrow and thin it's going to need the thread cutting much deeper than for a conventional nut, I'm guessing you had to machine in a recess at least 1.5 mm into the worm thread?

In fact you have a twin-start thread and a pair of pins or lugs. They have your "rotational degree of freedom" in the sense of being axisymmetric. The problem is that this gives line contact rather than area contact, so it is easy to damage the material of the screw. Luckily the forces are not too high.

> Fascinating to hear about the level of design work that goes in to an epipen.  Having been shown how to use one I had no idea that it had such smarts in to to make it so intuitive to use.

Oh they are certainly the most complex assembly for their size that I have worked on. In some examples the "bolt" that we have been discussing is actually a sleeve with another, different pitch thread on the inside. And of course the manufacturing tolerances have to be really good to ensure that you get the correct dose every time.

> One of the key issues to me is learning style; be it mathematics or programming, some people are drawn in by the subject itself and thrive in the abstract; for others the abstract makes little-to-no sense until they apply it to a tangible, real world problem, at which point the theory underpinning the abstract goes "click".

That sounds worryingly similar the someone advocating revisiting VAK learning styles that we wasted a lot of time on  a couple of decades ago, during a dark time in educational research?

At that time I was running a large maths department, like many schools maths was the most popular A level but there were some students who struggled with the transition from GCSE to A level. A similar, but probably at a slightly simpler level, I felt the issue was similar to the OP, I think some students really enjoyed GCSE maths because they liked getting calculations correct, but really struggle with the more abstract sections of A level. My attempt to provide a more appropriate course for them, was to introduce an AS level in Accounting. The perception was that the results were good, but I'm not sure I would do it again.

I think educational research has improved significantly in the last ten or so years. Things like Robert Bjork's   desirable difficulties, batch teaching v interleaving would interest me far more than learning styles. I might be biased as his work showing that although students are better than senior leaders/inspectors, they are still pretty rubbish at assessing how effective their teacher is certainly resonated with me.

> In fact all these things are injection moulded, so the question becomes how do you get it out of the mould? That's not something a layperson could easily answer!

If it has its internal thread done as part of the moulding process, you have a 3-part mould with the positive for the internal thread as the third part; the two plates are then removed and its unscrewed from the constant pitch internal thread?

Insane that this is all made out of plastic - you mention tolerance and correct dose, but I imagine tollerancing is challenging enough just to get the part to work.

>> Surely this is not a solvable problem unless you further qualify it to "... to keep the applied torque constant for a given speed of rotation of the knob"?

> That would be true if there were a viscous drag term (which there probably is in reality), but neither spring rate nor friction is time/speed dependent.

Don't mind me, I'm having a brain failure at the interface of intuition and force and work and rate of work in rotational motion.  I was thinking about turning it faster being harder work, but that is not the same as applying a higher torque.  (Other than when changing speed and so building up KE in the system).

> Oh they are certainly the most complex assembly for their size that I have worked on. In some examples the "bolt" that we have been discussing is actually a sleeve with another, different pitch thread on the inside. And of course the manufacturing tolerances have to be really good to ensure that you get the correct dose every time.

Now I want to take an epipen apart, that won't make me any friends.  It sounds like it is in effect a limited purpose mechanical computer in some ways, great that they live on.  I just spend the last 20 minutes re-watching the videos here...

https://arstechnica.com/information-technology/2020/05/gears-of-war-when-mechanical-analog-computers-ruled-the-waves/2/

> That sounds worryingly similar the someone advocating revisiting VAK learning styles that we wasted a lot of time on  a couple of decades ago, during a dark time in educational research?

I'm suggesting some learn these things better by applying them to real world problems and some by studying them in abstract.  I wouldn't draw a bridge from that to VAK.  Perhaps its as much about what motivates as it is about learning styles.

> I think educational research has improved significantly in the last ten or so years.

I was going to launch a rant, but that part of my life is behind me now.

Data Centre engineer: Using observed failure rates of components calculate a) required redundancy to meet particular service level objectives b) minimise costs (e.g. is there a benefit from cooling). Would be underpinned by stats/probability  ...

> Perhaps its as much about what motivates as it is about learning styles.

I observed a PGCE science student last week. The main part of the lesson was fine, if someone else was observing, it would have ticked most of their boxes and she is going to be an asset to the teaching profession. It was the last ten minutes that stood out for me. Nothing to do with what she was supposed to be teaching, but prior to teacher training she had worked on the Farne Islands for two years, judging by the kids faces the enthusiasm with which she delivered a brief account/presentation of her work there was infectious.

Reading about VAK makes me throw up all over the internet. There is essentially no evidence for the existence of these types, and that everybody prefers and is more effective at learning specific topics in a specific manner is a banal truism.

What annoys me is the popular idea that the job of a teacher, in particular in higher education, is to make learning "fun". The motivation should come from the student and the topic, not the presentation. Much of learning, especially when starting out with some subject, is simply hard work and practise. This also goes for learning and instrument or training in sports!

That said, there are also bad lessons or lectures, there is no kudos in making a lesson deliberately confusing or boring.

CB

> The motivation should come from the student and the topic, not the presentation. Much of learning, especially when starting out with some subject, is simply hard work and practise. This also goes for learning and instrument or training in sports!

I think Steve Peters would agree in principle but want you to use "commitment" rather than "motivation" for training in sport and I see no reason why it doesn't apply in education.

Yes, "commitment" describes the required quality rather well!  I have estimated that during my 12 years of active Judo competition career I must have thrown my partners into a crash mat between 100 and 250 thousand times for each of my main attacking techniques (and will have been thrown just as often).

Repetitive and boring as hell, but necessary to get the techniques dialed into into your cerebellum....

CB