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 question from a student: I am sitting my Higher Tier GCSE exam on Monday and I have tried every way to revise but it just doesn’t stay in my head! While I am reading the method, I just don’t seem to understand it - it feels like i am just staring at paint dry on the wall. I’m …
On a recent episode of everyone’s second-favourite maths podcast, Taking Maths Further, @stecks and @peterrowlett asked: You want to calculate the height of a tall building. You set up a device for measuring angles, on a 1m high tripod, which is 200m away from the building. The angle above …
This problem came via the lovely @realityminus3 and caused me no end of problems - although I got there in the end. I thought it’d be useful to look at not just the answer, but the mistakes I made on the way. Maths is usually presented as ‘here’s what you do, ta-da!’, without …
“So, the least common multiple of $52$ and $64$,” said the Mathematical Ninja, “is $13 \times 16 \times 4$, which is $832$.” “H-how did you do that?!” asked the student. The student was clearly new around here, so the Mathematical Ninja went easy on him. …
This post is a slight departure from the usual fare from Flying Colours HQ - more about personal development than about maths. But it’s still useful for your maths: today, children, we’ll be looking at effective ways of asking people to help you. In particular, if you’re asking for …
A student asks: How do I simplify horrible product and quotient rule expressions? The example they gave was differentiating: $f(x) = (2x - 3)^4 (x^2 + x + 1)^5$ First up, a careful bit of product rule: $u = (2x - 3)^4$, so $\diff{u}{x} = 8(2x- 3)^3$ $v = (x^2 + x + 1)^5$, so $\diff{v}{x} = …
A student asks: How do you remember the difference between Prim’s algorithm and Kruskal’s algorithm? In honesty, I don’t. This is one of the things that REALLY FRUSTRATES me about D1: it puts so much emphasis on remembering whose algorithm was whose1 ahead of figuring out how to program computers to …
Note: this post is only about arithmetic and quadratic sequences for GCSE. Geometric and other series, you’re on your own. Quite how the Mathematical Ninja had set up his classroom so that a boulder would roll through it at precisely that moment, the student didn’t have time to ponder. …
One of my favourite sources of puzzles at the moment is @WWMGT - What Would Martin Gardner Tweet? (Martin Gardner, in case you’re not up on the greats of popular maths writing, was one of the greats of popular maths writing - and is indirectly responsible for Big MathsJam.) Recently, it was decreed …
A very short episode of WBU, as @reflectivemaths is moving house.
Another guest NME cartoon from Dominika.
So, there you are, stuck on a desert island, you’ve played your eight pieces of music, burnt the Bible and Shakespeare, and now you’re kicking yourself for not bringing a calculator as your luxury item. An emergency has broken out and it’s vital for your to work out $3^{0.7}$ as …
This piece was entirely rewritten 2017-06-20 in response to a correction by Adam Atkinson. A friend of a friend stated: “… the planets exert an enormous influence on the tides…” … and that set my oh-no-they-don’t-o-meter. Let’s have a look, shall we? You …
A student asks: Why do you multiply by 1.07 if you’re adding 7%? I thought 7% was 0.07. You’re quite right - 0.07 is exactly the same thing as 7% (and, if you like, $\frac{7}{100}$). However, if you’re adding on 7%, you need to multiply by 1.07, and here’s why. Let’s …
A student asks: When you’ve got a value to the nearest whole number, why is the upper bound something $.5$ rather than $.4$? Doesn’t $.5$ round up? So I don’t have to keep writing something$.5$, let’s pick a number, and say we’ve got 12 to the nearest whole number. …
An occasional series highlighting common errors that refuse to die. “It just… won’t stay dead!” he said, as the Mathematical Zombie moved closer. “$(a+b)^2 = a^2 + b^2$”, it said. “Brains! $(a+b)^2 = a^2 + b^2$.” “But… it doesn’t!” he said. “You have to multiply out the brackets!” “$(a+b)^2 = a^2 + …
“Coincidence?” said the Mathematical Ninja. “I think not!” He looked at his whiteboard pen as if wondering how best to fashion a weapon out of it. He wrote: $10^3 = 1,000$ $2^{10} = 1,024 = 1.024 \times 10^3$ So $10 \ln (2) = 3 \ln(10) + \ln(1.024)$ But $\ln(1.024) \simeq 0.024$ $10 \ln (2) \simeq 3 …
It’s usually quite simple to spot the error in ‘proofs’ that $1=2$: either someone’s divided by 0 or glossed over inverting a multi-valued function (conveniently forgetting the second square root, for example). You sometimes (as with the sum of natural numbers being $-\frac{1}{12}$, if you throw out …
“Four points,” said the student. “On a circle.” The Mathematical Ninja nodded, impatiently. “$A(-5,5)$, $B(1,5)$, $C(-3,3)$ and $D(3,3)$,” he read from the book, for the third time. A slight crack of a smile. It may have been a snarl. You can never tell with the …
This month on Wrong, But Useful, @reflectivemaths and I (@icecolbeveridge) talk about… How close is it to Christmas? Who the hell are we and why the hell do we do this? Other maths podcasts: Taking Maths Further and Relatively Prime Colin offers a bounty for pictures of members of the …
Long ago on Wrong, But Useful, my co-host @reflectivemaths pointed out the ‘coincidence’ that $7\times8 = 56$ and $12 \times 13 = 156$ - a hundred more. In fact, it works for any pair of numbers that add up to 15: $x(15-x) = 15x - x^2$, and $(x+5)(20-x) = 100 + 15x - x^2$ - clearly, 100 more. This …
December! That means it’s time for CHOCOLATE! My dear friend Essbee showed me this: Free chocolate ahoy (and white chocolate, my favourite)! But surely there’s got to be a catch? Of course there’s a catch. You can’t just rearrange an area to end up with a bigger area - moving chocolate around …
At the 1939 World’s Fair, San Francisco Seals catcher Joe Sprinz tried to catch a baseball dropped from the Goodyear blimp 1,200 feet overhead. Sprinz knew baseball but he hadn’t studied physics — he lost five teeth and spent three months in the hospital with a fractured jaw. - from Futility Closet …
Over on reddit, noncognitivism posted a nice sequence s/he had come across: $4 + 1 = 5 = \sqrt{ (1)(2)(3)(4) + 1 }$ $(4+6) + 1 = 11 = \sqrt{ (2)(3)(4)(5) + 1}$ $(4+6+8) + 1 = 19 = \sqrt{ (3)(4)(5)(6) + 1}$ $…$ $(4+6+ … + (2k+2)) + 1 = \sqrt{ k(k+1)(k+2)(k+3) + 1 }$ Lovely. But why? Well, the left …
For the first time anyone could remember, the Mathematical Ninja trembled with fear. He’d pulled his scary face, and it hadn’t worked: Victoria Coren Mitchell had simply said “you don’t scare me” and he had no idea what to do next. “It’s time for the wall game,” she said. “Right,” thought the …
A reader asks: What’s the biggest lead a football team can have in the table after $n$ games? In a typical football league, teams get three points for a win, one for a draw, and none for getting beat. After, for example, one game, if one team wins and all of the other games are draws, the winners …
The brilliant @dragon_dodo has written a cartoon to explain - as if explanation were needed - of why radians are the correct way to measure angles.
There’s a legend, so well-known that it’s almost a cliche, about the wise man who invented chess. When asked by the great king what reward he wanted, he replied that he’d be satisfied by a chessboard full of rice: one grain on the first square, two on the second, four on the third, …
Recorded LIVE at Big MathsJam Apologies for the variable sound quality in this episode; the problems of recording live. The Wrong, But Useful tag team are joined by @peterrowlett (Peter Rowlett) Colin concedes that Dave’s talk was quite clever Dave mentions the 1, 3, 2, 6 sequence (a talk by …
A student asks: I’ve got to work out: $\int \cosec^2(x) \cot(x) \d x$. I did it letting $u = \cosec(x)$ and got an answer – but when I did it with $u = \cot(x)$, I got something else. What gives? Ah! A substitution question! My favourite – and it sounds like you’re off to a …