UKC

I have an interview for a post 16 teaching post, 25 minute lesson on "introduction to the modulus function, either algebraic or graphical".  The school is very keen on industrial applications in every lesson where possible so I'm keen to put something functional into the lesson.

Obviously the simple  way to go is to calculate some values to an quadratic curve, plot the modulus of it and see how it "bounces" on the x-axis, but I was looking for something more applied.

I can think of finding the difference between two values using Mod(x1-x2) but that seems a bit statistical for an algebra introduction.

I'm considering something along the lines of air pressure calculations for something like Bloodhound SSC - they probably don't care if the pressure is positive or negative on a panel, they just want to know how much pressure each panel is under but this feels a bit artificial and contrived.

Any other ideas?

http://mathforum.org/library/drmath/view/57177.html

"Do we have enough fuel to get there *and* back?" is a good one.

I guess you're not dealing with vectors yet as the idea of magnitude as distinct from direction is much clearer in that context.

Perhaps something based around speed/acceleration where for whatever reason it's useful to have negative values in some contexts but in others you are just interested in the magnitude (g force from an acceleration for example).

If they've come across standard deviation then I think mean deviation is a good non-contrived example and can lead to good discussion as to why we calculate sd as we do - to avoid the awkward algebra of the modulus function!

I'm not sure if they have - I had thought of this but 25 minute lesson doesn't give me much time to get side tracked if they don't remember what standard deviation is.  I also probably don't want to be saying "this is really awkward and you could just square it and square root it so you don't have to bother with it" as a way of inspiring them!

I suspect this is inappropriately difficult but frequenyly use the abs/mag when plotting / imaging imaginary functions.

velocity -> speed?

plotting grad or d/dx as a gradient in the lay term, (as in "how steep is this hill"). Often used when plotting 2-d gradient maps (and then you have another plot indicating the direction of the slope as an angle).

No - not looking at vectors.

I might look at the idea of the area under a velocity time graph to calculate distance traveled - but not sure I can get that far in 25 minutes.

The trouble is that I think the modulus function is best introduced after vectors as a natural 1 dimensional magnitude - your problem is that it's hard to come up with a non-contrived example whereas with vectors magnitude makes obvious sense!

Hi Blackett,

Probably not what you're looking for, but I'll mention it just in case. The modulus function is massively faster on a computer than the root-square (or even square) is, so the https://en.wikipedia.org/wiki/Sum_of_absolute_differences is used all the time in object recognition, video compression etc.

Cheers, i'm going with the sum of absolute differences on a 3x3 image to introduce the concept.

Interesting to hear how Robert Duran says that from an algebraic perspective the "square it, root it" idea is much easier and, from a computation perspective the modulus function is much easier.

I wonder why the standard deviation is so defined if it's much easier to compute the average absolute difference than the square root of the average squared difference?

Defining the variance as the RMS of the differences leads to the simple and important result Var(X+Y)=Var(X)+Var(Y) for independent variables and so to simple and important results about the variance of the mean of a sample and so on. As far as I know, no such simple results exist for the mean deviation.

Vectors are out, but are complex numbers in?  If so you use the divergence of the modulus of a complex number as the defining criteria for a number’s membership of many fractal sets such as the Mandelbrot set.  No idea how to link that to industrial applications but it gets some very compelling material in there, that may well engage students far more than calculating the total distance a car has travelled whilst driving backwards and forwards...

Another nice simple idea I came across today was using a bridge rectifier circuit to change AC to DC, Sin(x) to Abs (Sin(x)).  

I would love to look into Wintertrees suggestion, but I don't understand it and these students unfortunatly haven't come across complex numbers yet.

Even just one-dimensional distance works here, I think. Or something which is linked to distance (eg, as Deadeye said, fuel consumption).

I mean, I use modulus all the time in my work because I'm often interested in knowing what the size of the difference between two things without knowing or caring which one is bigger.

Lots of cryptography uses mod

E.g. Diffie helman key exchange.  Easy maths for that... 

A lot of quantum mechanics relies on the modulus of complex numbers, but as you say a bit tricky without knowing complex numbers. Might be one to mention as a throwaway "this lets you do cool things" comment

Diffie-Hellman (and RSA) use "mod" as in modular arithmetic, not as in the modulus function.

Ah yes ... Good point. I'll blame that on poor, quick reading.

Many thanks all, I got the job.  They liked the lesson, looking forward to the new job.

UKC comes up trumps on the most obscure topics again.