The Flying Colours Maths blog has been running posts twice weekly since 2012, covering maths from the basics to… well, the most advanced stuff I have a clue about.
Here they all are, sorted by date. Some day, other ways to filter them will be possible.
“$45 \cos($ thir… I mean $\frac{\pi}{6})$,” said the student, catching himself just before the axe reached his shoulder." “Thirty-nine,” said the Mathematical Ninja, without a pause. “A tiny bit less.” The student raised an eyebrow as a request to …
In a recent episode of Wrong, But Useful, I asked: A square is inscribed within a circle of radius $r$. A second square is inscribed within a semicircle of the same radius. What is the ratio of the areas of the squares? It’s easy enough to find the side length of the big square: the …
OK, this is a quick and dirty trick of the sort that I love and the Mathematical Ninja hates. He doesn’t have much time for stats at all, truth be told, least of all skewness. However, I’ve had several students struggle to remember ‘which way is which’ when it comes to …
“… which works out to be $\frac{13}{49}$,” said the student, carefully avoiding any calculator use. “Which is $0.265306122…$”, said the Mathematical Ninja, with the briefest of pauses after the 5. “I presume you could go on?” …
$y$ is directly proportional to $x^3$, you say? And when $x = 4$, $y = 72$? Well, then. The traditional method is to say: $y = kx^3$ and substitute in what you know. $72 = 64k$ $k = \frac{72}{64} = \frac{9}{8}$ That gives $y = \frac98 x^3$. Easy enough. But you can do it without the $k$ Rather than …
A special anniversary episode where @reflectivemaths (Dave Gale) and I pick the brains of two special guests: Abel prize nominee @samuel_hansen (Samuel Hansen) and @peterrowlett (Dr Peter Rowlett, surprisingly). “I think we should conclude that an argument has gone on long enough when Samuel …
@srcav wasn’t going to take that argument lying down! The Ninja looked smug. He thought that was it, game over. I thought it had been a sneaky trick he’d pulled with the Ninja Bread, but I couldn’t change it now. I finally pulled my mouth apart and took a big swig of tea to eliminate the last …
“Who DARES to challenge the Mathematical Ninja?” he bellowed. “It is I,” said the challenger. “@srcav, but Cav in real life.” “Oh!” said the Mathematical Ninja. “Hello there, old chap, I was expecting someone else. Come in, I’ll put the …
There’s a natural question, when you learn about the sine and cosine rules: “Is there a tan rule?” The answer to that is yes - yes, there is a tan rule. The natural follow-up is “Why don’t we learn it?” Let me explain why not! Here’s the tangent rule in all …
Every so often, a question that comes up that looks incredibly trivial, but - no matter how much each side protests that the preference doesn’t really matter - sets down clear divides in the maths community. My podcasting partner in crime-fighting @reflectivemaths (Dave Gale in real life) …
There are six minutes to play in the last Autumn international, and Australia are leading Wales by 30 points to 26. Australia, however, have just conceded a penalty in front of the posts, leaving the Welsh captain, Sam Warburton, with a dilemma: should he kick at goal (and take a …
A student asks: I’m currently preparing for my GCSE mocks. I started looking over quadratic expressions and factorisation and it just blew me away - I get stuck trying to work it out with negatives! First things first: thanks for asking for help. You’re not alone: factorising quadratics …
Since the dawn of time, two mysteries have plagued mathematicians: a) How do you find a centre of a 90º rotation? and b) What’s the 45º set square for? Imagine my surprise when I discovered that each is the answer to the other! Some facts about the centre of rotation Imagine you’ve got …
“$y$,” said the student, carelessly, “$=mx+c$,” and before he knew it, his wrists and ankles were secured to the table and a laser was slowly, but extremely surely, cutting the table in two. The Mathematical Ninja continued the lesson. “I think you mean $y - y_1 = …
A student asks: When I do inequalities with geometric series, I sometimes get my inequality sign the wrong way round - I can usually fudge it, but I’d like to be getting it right. OK, I confess, that student is me, circa 1995. And, in fact, I only figured it out once I started teaching. If I …
The dozenth episode of @reflectivemaths (Dave Gale in real life) and @icecolbeveridge (Colin Beveridge) chatting (and/or ranting) about the maths we’ve seen this month. Which includes: A brief discussion of dozenal Colin gives blood and gets woozy; Dave tried it once The big question: why is …
The student, by now, knew better than to pick up the calculator. “There’s got to be a way of doing that quickly.” The Mathematical Ninja looked a little disappointed; he’d built an elaborate electromagnetic pulse generator for the express purpose of killing calculators in a …
In a recent Wrong But Useful podcast, @reflectivemaths (who is Dave Gale in real life) asked the audience to: Prove that the sum of five consecutive square numbers is never a square. This one’s a bit easier than it looks: I chose to call the middle number of the five $n$. That makes my sum: …
If I had £35 every time a student said “I don’t get linear interpolation,” I’d have pretty much the same business model as I do right now. Everyone knows it’s something to do with finding medians and quartiles, and something to do with the class width and… stuff. …
Most of the students I help have a pretty good grasp of the three straightforward power laws: $(x^a)^b = x^{ab}$ $x^a \times x^b = x^{a+b}$ $x^a \div x^b = x^{a-b}$ So far, so dandy - and usually good enough if you’re hoping for a B at GCSE. The trouble comes when they start throwing strange …
“$1296$?!” said the student. “They want me to find the fourth root of $1296$?” “Evidently,” I said. The air turned, for a moment, blue. “Well, how about factorising it?” A different shade of blue. A whirring of pencil. A mutter of 648, a grumble of …
A student asks: I’m currently studying further maths A-level - and I love it! I want to obtain a degree in it. Could you provide an insight into what it’s like? I’m happy to share my experiences, although they’re probably specific to St Andrews, for me, in the late 90s - and …
“Where’s the Mathematical Ninja?” asked the student. “They’re… unavoidably detained,” I said. In fact, they were playing Candy Crush Saga. But sh. “What can I help you with today?” “Well, you know the binomial expansion…?” …
A few months ago, I wrote a post about replacing long division with a coefficient-matching process. That’s brilliant for C2, but what happens if you’re looking at a C4 question that wants a quotient and a remainder? Well, it gets a bit more complicated, that’s what happens. But …
The first non-trivial palindromic episode of Wrong, But Useful, in which Colin gets a touch of the Samuel Hansens and starts picking fights, and Dave does his best to calm things down. Fight #1, with loyal listener @srcav about which form of a straight line is better. The opposite of fighting is …
Need help with problem-solving? Fill out the short blue form on the left and get free tips on how to approach maths questions - delivered direct to your inbox twice a week → If you ever study GCSE vectors questions, you’ll spot a pattern: there’s normally a (relatively) straightforward …
If you ever study GCSE vectors questions, you’ll spot a pattern: there’s normally a (relatively) straightforward first part which involves writing down a few vectors, and then something like “show that points $O$, $X$ and $Y$ lie on a straight line.” Pretty much every student …
This is a follow-up to last week’s piece on the Numberphile video claiming that $1 + 2 + 3 + 4 + … = -\frac{1}{12}$. I mentioned something in the last article about certain1 infinite sums not being well-defined, and wanted to add some examples to show how they can be problematic. …
Thanks to Robert Anderson for the question. I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. - Srinivasa Ramanujan A recent Numberphile video made …
“So, $\sin(x) = 0.53$,” said the student. “32 degrees,” said the Mathematical Ninja. The student frowned - the Mathematical Ninja’s showing off was starting to wear her down - and typed it into the calculator to check. “$32.005^º$, actually.” …