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The line at infinity: coordinates

The story so far
I made a promise to someone on Twitter to talk about how conics relate to the line at infinity, but when I came to do that, I realised it’s a can of worms that will take a few blog posts to untangle. Last time, I talked about how I construct and think […]

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The line at infinity

Why I’m doing this

I foolishly said this on Twitter about a month ago:

@DavidKButlerUoA @MathguyArt on the line at infinity? Tell me about those!
— Tina Cardone (@crstn85) June 14, 2016

At the time I declared this was a bit of a can of worms and I promised to write something and post it later. Well, here it […]

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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.

Because of Quarter the Cross, I’ve also been interested in areas recently, so of course I had a desire to know more about the area […]

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A constant multiplied on will stay there

One of the most fundamental properties of the integral is usually presented as follows:
This means 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 […]

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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.
When you learn it at school, it usually goes […]

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There is only one kind of function that distributes over plus

There is a very common thing that students do that causes pain, distress, confusion and depression in any maths educator who witnesses it. Both the error itself and the educator’s response to it are very clearly described by this excellent picture from the blog “Math with Bad Drawings”:
Every single one of the statements in that […]

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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 […]

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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 […]

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A function is not a graph

When students learn about functions at school, we spend a lot of time forging the connection between functions and graphs. We plot individual points, and we find x-intercepts and y-intercepts. We use graphing software to investigate what the coefficients do to the graph, and discuss shifting along the x-axis and y-axis. We make reference to […]

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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 […]

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