News: Thoughts about maths thinking
Disjointed independence
There are two terminologies in probability which many students are confused about: "independent" and "disjoint". The other day I was working with a student listening to their thinking on this and I suddenly realised why.
Quarter the Cross
At the end of last year, the MTBoS (Math(s) Twitter Blog-o-Sphere) introduced me to this very interesting task: you have a cross made of four equal squares, and you are supposed to colour in exactly 1/4 of the cross and justify why you know it's a quarter. I call it "Quarter the Cross".
The crossed trapezium
Recently I started thinking about the properties of the following shape, which I like to call the "Crossed Trapezium". It has two parallel edges, which are joined by two crossing lines.
The trig functions are about multiplication
When I was taught trigonometry for the first time, I learned it as ratios of sides of right-angled triangles.
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A constant multiplied on will stay there
One of the most fundamental properties of the integral is that multiplying by a constant before doing the integral is the same as doing the integral and then multiplying by a constant. However, the way it's presented here makes it look like a rule for algebraic manipulation – I can move a constant multiple in and out of the integral sign. I do actually use it this way when I want to do algebraic manipulation – it comes in handy when I'm creating a reduction formula, for example. But most of the time when I do an integral, I don't use it that way at all.
Making sense of the effective population size formula
I was going to have a punchy title for this post, with a big moral to apply to the future, but I've decided I'm just going to describe to you what happened yesterday as I tried to learn some Genetics. You see what you can learn from my experience.
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Counting the story
Combinatorics is one of my favourite topics in discrete maths – that topic which is all about counting the number of ways there are to choose, arrange, allocate or combine things. I like the idea that I could theoretically find out the answer by writing down all the possibilities systematically and literally counting them, but that I can also come up with a quick calculation that produces the same answer by just applying some creative thought. It's this creativity in particular that appeals to me, so much so that I don't call it combinatorics, but "creative counting".
The reorder of operations
The community of maths users the world over agrees that when evaluating an expression or calculation, some operations should be done before others. Mostly it's to prevent us having to be needlessly specific about what order to do calculations in, mathematicians being very concerned with efficient communication.
Pure play
The other day I did a workshop with students from Advanced Mathematical Economics III, which is more or less a pure maths course for economics students. It covers such things as mathematical logic, analysis and topology – all a bit intimidating for students who started out the degree with almost no mathematical background!
Obscuring the GST by making it simple
I was helping out at Roseworthy Campus yesterday as the Vet Medicine students were learning about budgeting for a Vet Clinic as a business. One aspect of this was calculating the amount of the cost of goods and services that was GST (stands for "Goods and Services Tax" – in other countries it's known as VAT or Sales Tax). The Excel sheet they were working in already had the formula worked in and it was this: GST = (Total Price)/11.
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