News: Isnt maths cool
The line at infinity
I foolishly said this on Twitter about a month ago:
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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The Sausage-Stacking Theorem
It's no secret that the powers of two are some of my favourite numbers. There are so many interesting things to say about them that often I don't know where to begin! (In case you're not au fait with the terminology, the powers of two are the numbers you can make by starting with 1 and multiplying by 2 over and over. The list starts out 1, 2, 4, 8, 16, 64, 128 etc.)
The square root of two
In first year maths, they briefly study the five families of number: the natural numbers N, the integers Z, the rational numbers Q, the real numbers R, and the complex numbers C. In particular, they focus on the distinction between the rational numbers and the real numbers. A classic proof they are given at this time is one that the number √2 is irrational. This blog post is about some alternative proofs.
The advent calendar function
In a previous post I discussed how we need ways to think about functions that are not curves on an x-y-plane. Well I have a seasonally-appropriate one for you: the Advent Calendar.
Splitting logs
In our bridging course (and indeed in Maths 1M and Maths 1A and several other courses) there is a section on differentiating logarithmic functions. One of the classic questions that we ask in such a section is to differentiate the log of some horrifying function, with the intention that the students use the log laws to simplify the original function first and then differentiate. There is something about this particular type of question has long bothered me and I only just figured out how to resolve my issue with it. I'm so excited I need to share it somewhere!
Archimedes's Integrals
One of my staff (thanks Fergus) told me ages ago about Archimedes' proof that the volume of a sphere is 4/3 π R3 (where R is the radius of the sphere). It is a very very cool proof and it's high time I shared it! One of the reasons it is so cool is that it uses the concept that a volume can be produced by stacking up a whole lot of thin slices. This is the idea behind integration, and Archimedes used this idea thousands of years before Newton or Leibniz.
Where's the t?
Once upon a time, I lectured Maths 1A calculus, and when I got to teaching hyperbolic trig functions I put a great deal of effort into making sure they were well-connected to other ideas the students knew. So I listed the properties of ordinary trig functions and alongside I listed the matching properties of hyperbolic trig functions.
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