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.
A good friend, a brilliant maths teacher, was recently dissed on Twitter. And if there’s one thing I don’t stand for, it’s good friends and brilliant maths teachers being dissed on Twitter. There was an implication — perhaps inadvertent; I know I’ve fired off short messages too quickly and implied …
Dear Uncle Colin, I have to revise for an upcoming test – and my teacher is demanding proof! What’s the best way to convince my teacher I’ve revised? -- Totally Evidence-based Study Time Hello, TEST - great question! Depending on the kind of test, you have several options that …
An interesting tweet, some time ago, from @RJS2212: And of course, you wonder two things: a) why does it work, and b) can all Pythagorean triples be written that way? The first one is an easier proposition, and I’ll set it up like so: we have $\frac{1}{n-1} + \frac{1}{n+1} = \frac{2n}{n^2-1}$. Are …
Dear Uncle Colin, I clumsily dropped a particle of mass $m$! Luckily, it’s attached to a light elastic string with a modulus of elasticity of $3mg$ and natural length $a$. The other end of the string is attached to the point where I dropped the weight from. When I say ‘dropped’, I …
“Forty-two degrees,” said the Mathematical Ninja, as smugly as possible while still using degrees. The student’s hand had barely twitched towards the calculator. “Go ahead, punk,” said the Mathematical Ninja. “Make my day.” “Righto,” said the student, and tapped in $\tan^{-1} \left( 0.9 \right)$, …
Dear Uncle Colin, Inspired by a recent XKCD cartoon, I want to start measuring temperatures in radians celsius. How can I quickly convert between the two? Made Up Nonsense? Réaumur’s Octogesimal First up, MUNRO, that’s a really bad idea. I’ve said elsewhere that I don’t like …
It’s encouraging to see a few less-predictable questions coming up in the new GCSE and A-level specifications. @mathsjem highlighted an especially nice GCSE one: This is unusual more than it is tricky: $x = 1.0\dot2 = \frac{92}{90}$ and $y = 0.\dot 4 = \frac 49$, so $x-y = \frac{92-40}{90} = …
Dear Uncle Colin, You know how sometimes $\sin(2x)$ is rational and $\sin(5x)$ is rational and $\sin(7x)$ is rational, right? Would that necessarily mean that $\sin(12x)$ is rational? Asking for a friend. — Perhaps You THink All Geometry’s On Right Angled Stuff Hi, PYTHAGORAS, I believe it does! (In …
Uncle Colin recently explained how he would prove the identity $\sin(2x) \equiv 2 \sin(x)\cos(x)$. Naturally, that isn’t the only proof. @traumath pointed me at an especially elegant one involving the unit circle. Suppose we have an isosceles triangle set up like this: The vertical …
Dear Uncle Colin, In Statistics, we were shown a picture of the standardised normal distribution curve, and the base stops at +4 and -4. Why is it not $\pm 5$, $\pm 10$, or anything else? Is there something special about 4? -- Got An Unanswered Statistics Struggle Dear GAUSS, The standard normal …
Over at @onthisdayinmath, Pat highlights a @jamestanton question about squares: $2^2$ ends with 4 and $12^2$ ends with 44. Is there a square than ends 444? How about one that ends 4444? Pat’s answer (yes to the first – $38^2 = 1444$ is the smallest – and probably not to the second) …
In this month’s episode, @reflectivemaths and @icecolbeveridge discuss: Dave is baffled by the idea of the Anniversary Games but happy to have a reduced workload We agree on the name and age of the podcast Number of the podcast: 36, the smallest non-trivial square triangle number. Colin has a …
Dear Uncle Colin, I’m OK at multiplying simple fractions by numbers and fractions by each other, but I don’t understand how to multiply mixed fractions together. Help! -- Variations In Numerators Can Upset Learners Understanding Maths Hello VINCULUM1 ! I think I’m on record as …
“Look,” said the student, “we all know how this goes down. A nasty-looking fraction comes out of the sum, I reach for the calculator, you commit some act of exaggerated violence and tell me how you, o wondrous one, can do it in your head.” “You’re not as dumb as …
Dear Uncle Colin, I’m struggling with this STEP question. The first two parts are fine – equality holds when there is some constant $k$ for which $a = kx$, $b = ky$ and $c=kz$, and part (i) follows directly from the original inequality. I can get an answer to part (ii) – …
This article is one of those ‘half-finished thoughts’ put together late at night. Details are missing, and – in a spirit of collaboration – I’d be glad if you wanted to fill them in for me. The estimable @onthisdayinmath (Pat in real life) recently posted about …
The cadets are at it again. Thanks to @dragon_dodo and @FennekLyra. Agent Lyra and Agent Dodo were bored. After several weeks of suspending Gale’s desk in bizarre positions, fencing on the office chairs and sneakily pilfering Beveridge’s ginger beer, they had quite run out of things to do. Draped …
Dear Uncle Colin, I find it easier to remember trigonometric identities if I can ‘see’ how they fit together. I’m expected to know that $\sin(2x) \equiv 2\sin(x)\cos(x)$, but haven’t been able to prove it. Any ideas? -- Geometry? Right Angles? How About Medians? Hi, GRAHAM! …
I recently became aware of the IYGB papers, available from Madas Maths. Like the Solomon papers, they’re intended to stretch you a bit – they’re ranked by difficulty from standard to extremely hard. My student, being my student, demanded we go through one of the extremely hard …
Dear Uncle Colin, I’ve got a line with equation $10y+36x=16.5$. That equation has no negative numbers in it, yet its gradient is apparently negative. I don’t understand why. -- Silly Line, Only Positive Equation Dear SLOPE, It looks like we’re in misconception-land! In fact, you …
The estimable @solvemymaths tweeted, some time back: A sensible option? Perhaps. But Wolfram Alpha is being a bit odd here: that’s something that can be simplified significantly. (One aside: I’m not convinced that actually is $\sin(22º)$, because I get a different answer in the first …
Dear Uncle Colin, I get $-\frac{\ln(0.02)}{0.03}$ as my answer to a question. They have $\frac{100\ln(50)}{3}$. Numerically, they seem to be the same, but they look completely different. What gives? -- Polishing Off Weird Exponents, Really Stuck Dear POWERS, What you need here are the log laws (to …
Some while back, Ben Orlin of the brilliant Maths With Bad Drawings blog posted a puzzle he’d set for some eleven-year-olds: Which is larger, $\frac{3997}{4001}$ or $\frac{4996}{5001}$? Hint: they differ by less than 0.000 000 05. He goes on to explain how he solved it (by considering the …
Dear Uncle Colin, I have to solve the inequality $x^2 - \left|5x-3\right| \lt 2+x$. I rearranged to make it $x^2 - x - 2 \lt \left|5x-3\right|$ , but the final answer is eluding me. -- Put Right Inequality Muddle Hello, PRIM! You’re off to a good start; the next thing I would do would be to …
While I’m no Mathematical Ninja, it does amuse me to come up with mental approximations to numbers, largely to convince my students I know what I’m doing. One number I’ve not looked at much1 is $\sqrt{\frac{1}{3}}$, which comes up fairly frequently, as it’s …
Dear Uncle Colin, My maths mock went terribly, and I got a U. Since then I’ve done some real revision and got a good grade on a paper I did off my own bat. However, I’m a long way behind on the new material and I feel like it’s too late to fix because I’m not intelligent …
The Mathematical Ninja played an implausible trick shot, not only removing himself from a cleverly-plotted snooker, but potting a red his student had presumed safe and setting himself up on the black. Again. “One!” he said, brightly, and put some chalk on the end of his cue. The student …
In this month’s Wrong, But Useful, Dave and Colin discuss: Colin gets his plug for Cracking Mathematics in early Colin is upset by a missing apostrophe Dave teases us with the number of the podcast and asks about the kinds of things it’s reasonable to expect students to know, and asserts …
Dear Uncle Colin, Could you please tell me how to solve simultaneous equations? I have a rough idea, but I get confused about it. -- Stuck In Mathematical Examinations/Qualifications Hello, SIMEQ! Here’s how I attack linear simultaneous equations, such as: $5x + 6y = -34$ (A) $7x + 2y = 10$ (B) …
There’s something neat about an identity or result that seems completely unexpected, and this one is an especially nice one: $$ e^{2\pi \sin \left( i \ln(\phi)\right) }= -1$$ (where $\phi$ is the golden ratio.) It’s one of those that just begs, “prove me!” So, here goes! …