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The World Is Maths

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January 2016

Conic Sections (Desmos)

EDITED:  Thanks to the wonderful folk at Desmos, who helped me solve my problem within minutes of tweeting it, I now have fully functional models.  The problem was making the black dotted distance lines only point to the relevant focus/directrix and not both.  Writing lines parametrically – that’s how to impose conditions on when they appear.

I’ve been trying to make some models to show the relationship between the curve, focus and directrix on conic sections, Continue reading “Conic Sections (Desmos)”

Continuous Random Variables (Desmos)

I dislike education acronyms, but I can make exceptions for mathematical ones.  One of my favourite topics in A-level Maths is full to bursting with them: DRVs, CRVs, PDF, CDF.  This is a visual representation of the CDF (cumulative distribution function) of a CRV (continuous random variable), which is the function for the area under the curve from x=-∞ to any other value, a, or more specifically, P(X<a).  Take note of the syntax for piecewise functions. Continue reading “Continuous Random Variables (Desmos)”

Projectile Motion (Desmos)

You can do a simple model of projectile motion in Desmos and create sliders to alter the angle and speed of projection in order to see how these affect the motion.  Make sure you have angles set to measure in degrees (settings, above zoom, right-hand side).

Graphical Linear Programming (Desmos)

Graphical inequalities aren’t quite how you’d want them to be on Desmos, simply because it shades the side of the line that produces true statements rather than false.  Of course, in linear programming with multiple inequalities, you really want the true sides left blank for clarity.  So you have to cheat and reverse your inequality signs to get Desmos to shade the way you want it to, but it’s still a lovely visual.

I’ve made up an example, and shown how you can use a slider to get the objective line to move within the region.  Since you can click on points of intersection, it’s easy to consider all the vertices of the region as well.

Beating the Odds

I was running a training session for new maths teachers last week and over lunch someone mentioned the weekend’s huge lottery draw, with the largest jackpot in UK lottery history of £66m, shared by two winners.  I remember when the National Lottery started, one of my first thoughts about it being, “if someone had enough money they could buy up every combination and guarantee the jackpot”.  We talked about this over our jacket potatoes with chilli and I started to consider the practicalities.  And, of course, the maths.  So here are my musings… Continue reading “Beating the Odds”

What is ‘e’? (Desmos)

Quite simply, demonstrate the value of e using the fact that

\frac{d}{dx}[e^x]=e^x

with the sliders on this model.

Flight and Fight

Back in 2010 I, and 12 million other people, became slightly addicted to a little game called Angry Birds.  I stopped playing after a couple of months, like I always do with these things, not sure what happened to it then, probably petered out like so many other overnight success stories.

The premise of the game was to get your little birds to destroy some nasty green piggies who were stealing your eggs by launching them from a slingshot at the pigs’ defences.  I can see why those birds were angry, I would be if I were repeatedly being launched, projectile-style, at towers of wood and ice and stone.  That’s gotta hurt.

The programmers use some fairly simple mathematics to get those birds to move in a realistic way.  You intuitively expect the birds to move through the air in the way they do because all objects move like that, unless some force causes them to change their path.  Picture it now – imagine throwing a tennis ball, what shape does the path of the ball take? What about a diver jumping off a diving board, or a bullet flying through the air, or water spouts from a fountain? Continue reading “Flight and Fight”

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