News: Thoughts about maths thinking
Problem strings and using the chain rule with functions defined as integrals
In Maths 1A here at the ³ÉÈË´óƬ, they learn that says that, given a function of x defined as the integral of an original function from a constant to x, when you differentiate it you get the original function back again. In short, differentiation undoes integration. And then they get questions on their assignments and they don't know what to do. They always say something like "I would know what to do if that was an x, but it's not just an x, so I don't know what to do".
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(Holding it together)
Last week, I helped quite a few students from International Financial Institutions and Markets with their annuity calculations, which involve quite detailed stuff. One of the more important problems was about how the calculator interprets what they type into it, which is really in essence about the order of operations.
Quadrilateral family tree
I have always loved the naming of quadrilaterals, right from when I first heard about it in high school. I'm not entirely sure why, but some of it has to do with the nested nature of the definitions – I like that a square is a kind of rectangle and a rectangle is a kind of parallelogram.
All dogs have tails
In maths, or at least university maths, there are a lot of statements that go like this: "If ...., then ..." or "Every ..., has ...." or "Every ..., is ...". For example, "Every rectangle has opposite sides parallel", "If two numbers are even, then their sum is even", "Every subspace contains the zero vector", "If a matrix has all distinct eigenvalues, then it is diagonalisable". Many students when faced with statements like these automatically and unconsciously assume that it works both ways, especially when the subject matter is new to them. This post is about a way of helping students see the problem.
Where the complex points are
When you first learn complex numbers, you find out that they give you ways to solve equations that were previously unsolvable. The classic example is the equation equation x^2 + 1 = 0, which if you're only using real numbers has no solutions, but with complex numbers has the solutions x=i and x=-i.
Brackets
I had a meeting with an international student in the MLC on Friday who has having a whole lot of language issues in her maths class.
David Butler and the Prisoner of Alhazen
Once upon a time, I did a PhD in projective geometry. It was all about objects called quadrals (a word I made up) - ovals, ovoids, conics, quadrics and their cones - and the lines associated with them - tangents, secants, external lines, generator lines. During the first two years, I did talks about my PhD research, which I could not resist calling "David Butler and the Philosopher's Cone" and "David Butler and the Chamber of Secants".
Four alternatives to the four fours
The "Four Fours" is a very well-known little problem that encourages some creative thinking and use of the order of operations. The purpose of this post is to show you four fourfoursesque puzzles I've created which have encouraged some great learning.
Things not sides
When doing algebra and solving equations, there is this move we often make which is usually called "doing the same thing to both sides". Quite recently this phrase of "both sides" has begun to bother me.
The right order for the fundamental trig identity
If you google "fundamental trig identity" you will get many many images and handouts which all list the fundamental trig identity as:
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