Ask Uncle Colin: A Separable Difficulty | Colin

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I have an equation $3y, \dydx =x$. When I separate and integrate both sides, I end up with $\frac{3}{2}y^2 = \frac{1}{2}x^2$, which reduces to $y = x\sqrt{\frac{1}{3}}+c$.

With the initial condition $y(3) = 11$, I get $y = x\sqrt{\frac{1}{3}}+11-3\sqrt{\frac{1}{3}}$, but apparently this is incorrect. What am I doing wrong?

–…

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A common problem: decimal division | Colin

I’m a big advocate of error logs: notebooks in which students analyse their mistakes. I recommend a three-column approach: in the first, write the question, in the second, what went wrong, and in the last, how to do it correctly. Oddly, that’s the format for this post, too.
The question

Decimal division: something like 14.4 ÷ 1.2
What went wrong

Got 1.2 instead of 12.
How to do it right

Approach 1: Estimation. 14 ÷ 1 is 14, so an answer of 1.2 is way off – 12 seems more reasonable.

Approach 2: Decimal fractions. (A little bit of commentary here: at this point, students normally…

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Ask Uncle Colin: 10,958 | Colin

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

There is a famous puzzle where you’re asked to form 100 by inserting basic mathematical operations at strategic points in the string of digits 123456789. This can be achieved, for example, by writing $1 + 2 + 3 – 4 + 5 + 6 + 78 + 9 = 100$.

Brazilian Mathematician Inder J Taneja has found a way to represent every natural…

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Wrong, But Useful: Episode 42 | Colin

On this month’s episode of Wrong, But Useful, @icecolbeveridge and @reflectivemaths are joined by special guest co-host @christianp. This time, we talk about:
Christian, who is involved in @mathsjam and the @aperiodical, and has a number of the podcast: 13. He dislikes it because of its times table; I like it because you can work out thirteenths in your head and look like a ninja, explained in three parts here: Part 1, Part 2, and Part 3. Dave likes 13 because of Belfigore’s prime.
Colin mistakes 91 for a prime (although immediately corrects himself). He would be lousy at Is This Prime?,…

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A Digital Root Puzzle | Colin

Every so often, a puzzle comes along and is just right for its time. Not so hard that you waste hours on it, but not so easy that it pops out straight away. I heard this from Simon at Big MathsJam last year and thought it’d be a good one to share and analyse. I’ve adopted (and slightly adapted) @colinthemathmo’s wording of it:

Apart from exactly one exception, the digital root of the product of twin primes is always 8. Why?

I’d recommend convincing yourself that it’s true, finding the exception, and having a go at a proof before reading on. Assuming you want to.

My proof is as…

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Ask Uncle Colin: another vile limit | Colin

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

Apparently, you can use L’Hôpital’s rule to find the limit of $\left(\tan(x)\right)^x$ as $x$ goes to 0 – but I can’t see how!

– Fractions Required, Example Given Excepted

Hi, FREGE, and thanks for your question!

As it stands, you can’t use L’Hôpital – but you can adjust it so you can!

If you take logs, you get $x \ln(…

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Ask Uncle Colin: Are normals… normal? | Colin

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions — and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I don’t understand why the normal gradient is the negative reciprocal of the tangent gradient. What’s the logic there?

— Pythagoras Is Blinding You To What’s Obvious

Hi, PIBYTWO, and thanks for your message!

My favourite way to think about perpendicular gradients is to imagine a line going across a chessboard. Let’s say…

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