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.
Here’s a nice use of circle theorems: ever wondered why the sine rule works? [iframe http://market.flyingcoloursmaths.co.uk/geogebra/sineruleproof/m128342-Proving-the-sine-rule.html 500 800]
“Boom!” said the Mathematical Ninja, as the smoke cleared and the student jumped. Shell-shocked, he looked again at the whiteboard. “Wh-what just happened?” The Mathematical Ninja sighed. “OK, one more time. We’re trying to estimate the size of the shapes …
A friend asks for REASONS: A who to the what now? A twelve-letter word, a thaerrhugaL, representing a number somewhere in the region of twenty-three ninety-ninths of a sextillionth. It’s hardly unreasonable to demand REASONS. It’s not an ‘obvious’ number - a power of …
I don’t quite remember who asked this, but at some point someone asked me to list ten books that have stayed with me. 1066 And All That - Sellar and Yeatman I’ve a beat-up copy of this I borrowed (what do you mean stole?) from my parents, and it’s one of the few books my partner …
In this month’s episode of Wrong, But Useful, @reflectivemaths (Dave Gale) and I discuss: How Dave thinks Baby Bill should learn to count Decimal time McDonalds apple pies - Made 15:26. Ready 15:36 Discard 16:56 “School lunch could save you up to £437 a year. Only 1% of packed lunches …
The Mathematical Ninja, some time ago, pointed out a curiosity about Pascal’s Triangle and the Maclaurin1 (or Taylor2 ) series of a product: $\diffn{n}{(uv)}{x} = uv^{(n)} + n u’v^{(n-1)} + \frac{n(n-1)}{2} u’’ v^{(n-2)} + …$, where $v^{(n)}$ means the $n$th derivative …
A MathsJam classic question asks: Without using a calculator, which is bigger: $e^\pi$ or $\pi^e$? It’s one of those questions that looks perfectly straightforward: you just take logs and then… oh, but is $\pi$ bigger than $e\ln(\pi)$? The Mathematical Ninja says “$\ln(\pi)$ is …
A numerical curiosity today, all to do with $\i$th powers. Euler noticed, some centuries ago, that $13({2^\i + 2^{-\i}})$ is almost exactly $20$. As you would, of course. But why? And more to the point, how do you work out an $\i$th power? It’s all to do with the exponential form, of course …
A student asks: I know the method for finding the hyperbolic arcosine1 - but I get two roots out of my quadratic formula. Why is it just the positive one? A quick refresher, in case you don’t know the method Hyperbolic functions are the BEST FUNCTIONS IN THE WHOLE WIDE WORLD2 and I’ve …
Just for a change, an FP3 topic. I’ve been struggling to tutor complex mappings properly (mainly because I’ve been too lazy to look them up), but have finally seen - I think - how to solve them with minimal headache. A typical question gives you a mapping from the (complex) $z$-plane to …
“But I don’t liiiiike fractions,” said the student. He also didn’t like the look of the poker in the Mathematical Ninja’s hand, which was beginning to glow red. “Sure you do,” said the Mathematical Ninja. “Do I?” “How do you do …
You’ve got the formulas in the book, of course. $u_n = a + (n-1)d$ $S_n = \frac n2 \left(a + L\right) = \frac n2 \left(2a + (n-1)d\right)$ This is somewhere the book and I have a serious disagreement: as a mathematical document, it ought to define its terms. $a$ is the first term of the …
It’s the (slightly delayed) monthly chat between @reflectivemaths (Dave Gale) and me on whatever maths has caught our eyes. This month: Why protractors and set squares? You can find centres of rotation. Why constructions? Colin launches an impassioned defence and compares them to Killer Sudoku …
A student asks: We’ve just started integration and I don’t understand why there’s always a $+c$ - I understand it’s a constant, I just don’t understand why it’s there! Great question! The simple answer is, because constants vanish when you differentiate, they have …
“Don’t tell the Mathematical Ninja,” said the Mathematical Pirate. The student shook his head enthusiastically. “Narr!” “You’ve got $ \frac {7}{x} = 14$. Ask yourself: what would the Mathematical Ninja do?” “The Mathematical Ninja would do …
My own imagination asks: Why is an outlier defined as 1.5 interquartile ranges outside of each quartile? Great question, imagination! The simple answer, I think, is that it’s a nice and easy thing to work out, and 1.5 interquartile ranges is quite a long way from the central box (if …
“You can use BIDMAS,” said the student, and the Mathematical Ninja gave him a piece of paper and a marker. “Write BIDMAS on here, really big.” While the student complied, the Mathematical Ninja fired up the flamethrower. “Hold it up in front of your face.” …
A student asks: Hi, I am struggling with trying to revise for my GCSE maths calculator mock… I was wondering if you could give a few tips on how to revise for this exam in particular. There’s a commonly-held belief that the calculator paper is easier than the non-calc one, but I’m not …
“May I borrow some paper?” asked the student, meekly. He knew he should have come prepared; he feared for his safety, as the Mathematical Ninja’s reputation preceded him. “BORROW?!” hairdryered the Mathematical Ninja. “BORROW?! Why on earth would I want the paper …
OK, OK, stop screaming! I KNOW C1 is only a few days away, I know you’re underprepared, but panicking isn’t going to help anything. It’s a pretty common - and heartbreaking - thing for tutors and teachers to see: a student who’s waltzed through GCSE getting to the end of Year …
A question I don’t remember ever having been asked before: Why do you love maths so much? It’s a really hard question to answer - if you asked someone why they like music or football, they’d probably shrug and say ‘I just do!’ There are loads of things I love about …
It’s the 110th Carnival of Mathematics - and I fear I’ve drawn the short straw. I suppose, as the numbers climb higher, there are fewer interesting facts to go around. However, a few that jump out: it gives the minimal value (-5) of the Merten’s function ( …
The Mathematical Ninja stopped dead in their tracks. Not literally, of course: immortal beings don’t die. They turned their head slowly towards the student, unsure what was going on. They probably had in mind their recent OFSTED report, which had – while praising the results of the …
In the (very) late April episode of Wrong But Useful, with @reflectivemaths and @icecolbeveridge: Colin has laser eye surgery and doesn’t care about Dave’s health… … but does care about 20481 . Colin mistakenly credits @notonlyahatrack for getting to 2048, when he meant …
Towards the end of a GCSE paper, you’re quite frequently asked to simplify an algebraic fraction like: $\frac{4x^2 + 12x - 7}{2x^2 + 5x - 3}$ Hold back the tears, dear students, hold back the tears. These are easier than they look. There’s one thing you need to know: algebraic fractions …
Towards the end of a GCSE paper, you’re quite frequently asked to simplify an algebraic fraction like: $\frac{4x^2 + 12x - 7}{2x^2 + 5x - 3}$ Hold back the tears, dear students, hold back the tears. These are easier than they look. There’s one thing you need to know: algebraic fractions …
A student asks: The mark scheme says $Var(2 - 3X) = 9 Var(X)$. Where on earth does that come from? Great question, which I’m going to answer in two ways. Firstly, there’s the instinctive reasoning; secondly, there’s the maths behind it, just to make sure. Instinctively Well, …
I wanted - I really, really wanted - to like this book. On the surface, it’s exactly my cup of tea: a whole book of tricks to make mental arithmetic easy. Sadly, there’s so much about it that’s dreadful that the nuggets inside it are hardly worth the effort. The binding? Dreadful1. …
The student stared, blankly, at the sine rule problem in front of him. $\frac{15}{\sin(A)} = \frac{20}{\sin(50^º)}$ “I don’t know where to st,” he started whining as something flew past his head. He knew better than to turn and look at whatever implement of death and destruction he …
There’s nearly always a question on the non-calculator GCSE paper about Nasty Powers. I’m not talking about the Evil Empire or anything, I just mean powers that aren’t nice - we can all deal with positive integer powers, it’s the zeros, the negatives and the fractions that …