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Calculating 5th Roots - IN YOUR HEAD!

Multiplying a number by itself (e.g. $ 6 \times 6$, or $6^2 $) is relatively easy. Going back the other way - finding out what number has been squared to get a particular result (or finding the "square root", e.g. $ \sqrt{36} $) - is tougher: you need to either just know the answer or just guess it (and, of course, square it to see if you were right, and then if you're wrong have another guess, informed by the outcome of your first one).

Cubes (the cube of 6 would be written $ 6^3 $, or $6 \times 6 \times 6$, for example) are similarly relatively simple to calculate compared to cube roots (e.g. $ \sqrt[3]{216} $), which also require a bit of trial-and-improvement (if you're working with secondary-school level maths).

Raising numbers to the 5th power (i.e. $ 6^5 $, or $6 \times 6 \times 6 \times 6 \times 6 \times 6$) is, again, fairly easy (though you'll probably need a piece of paper), but working out a 5th root (so, figuring out that $ \sqrt[5]{7776} $ is actually 6) involves quite a bit of guessing, trying, failing, and guessing again.

Except... if you're trying to work out the 5th root of a number that is the result of raising any two-digit number to the power of 5 there's a rather nifty trick that you can use to do it in your head. Indeed, with relatively little practise you'll be able to calculate 5th roots quicker than someone who doesn't know the trick can bash it into a calculator.

I described this trick as part of a 24-hour maths marathon broadcast in October 2020. You can find out more about the event here, but if you'd like to find out how to calculate 5th roots in your head really quickly, watch my section in the player below:



If you'd like me to add a readable description of the trick to this post, please let me know in the comments!

Mathematics Teaching in Indonesia

My first assignment for my Master's course was to "look at a chosen aspect of education in one country and comment on the socio-economic, political and cultural influences on this aspect of education [using] an infographics tool to present [my] findings."

The country was a largely random choice: I didn't have a particular country in mind so I just sort of broadly read around until I landed on something that struck me as interesting. Could the aspect I chose really have been anything else?

I completed and submitted the assignment at the end of November and have just received my grade* so here it is:

Mathematics Teaching in Indonesia

The Third Reich Is Listening

As Bletchley Park's "Enigma Man*" I was often asked the question "... but what were the other side doing?" The museum doesn't cover this side of the story other than very fleetingly and in passing so it was always tricky to answer, but it is nevertheless a very pertinent - and thoroughly interesting - question.

This is my copy of The Third Reich is Listening and you can't have it (but you're welcome to buy your own, or borrow it from a library, of course)
This post is, essentially, a review of Christian Jennings' The Third Reich is Listening.

Its subtitle "Inside German Codebreaking 1939 - 1945" tells you the main focus of the book, but the story does start earlier than that, summarising work done by key characters in World War I and the inter-war years.

BP's story is utterly fascinating whether you have the briefest of overviews or a more in-depth understanding. As you wade deeper you find that every part of it is much more complicated and involved, from a historical perspective, than you had ever thought possible. Every aspect of allied operations in World War II has the site's work somewhere in its background and the story has an almost fractal nature to it: pick any part you like, zoom in, and you'll see it explode into so much detail you'll wonder why nobody has started a museum dedicated to just that bit.

You'll also notice, as you learn more and more, that almost every aspect of the story is steeped in myth and misconception. So many things that are common knowledge, routinely-told anecdotes when scratching the surface turn out to be simplifications, half-truths, or downright incorrect as you pick apart the evidence.

One such myth is that the codebreakers were working to reverse-engineer an almost completely unbreakable cipher. This, when dipping your toe into the story, is not incorrect: Enigma** is a tough nut to crack. But as you wade into deeper waters you find there's an unspoken caveat: if it's used properly. Bletchley's codebreakers had a helping hand, and not just from the Polish mathematicians who figured so much of it out before the war even started: Germany itself proves very helpful in the fight against Enigma. At the lowest level, mistakes were made by Enigma operators that were of great aid to their counterparts in the brick blocks and wooden huts of Bletchley Park; as were many features of the procedures put in place at much higher levels.

One key theme in favour of allied codebreaking was Germany's disbelief that its ciphers had been (or could be) broken. As Jennings puts it, "the Third Reich might have been listening but increasingly it had its fingers in its ears."

All this is just a very brief, very anglocentric view of what was going on in Germany.

Whilst there are still many and varied parts of allied codebreaking in World War II still to be uncovered, clarified or corrected in the public mind, the mirror within which we may see Germany's point of view is cloudy and smudged, and hasn't been dusted in a long time. Jennings' book takes a damp rag to it and does a surprisingly good job of clearing away some of the grime.

And what we can see is just as fascinating as the story we already know. There are parallel figures in that world, characters even more forgotten than the codebreakers of Bletchley Park, such as Wilhelm Tranow, and internal fights against an oppressive and militaristic regime to rival those depicted in The Imitation Game. There are revelations of mistakes made by British encryption procedures and communications staff that equal those made by Germany, and which similarly aided German cryptanalytical victories. There are disastrous decisions that reduced its effectiveness and eroded the viability of what could have been a genuinely effective rival to Bletchley Park.

If you're interested in secret messaging during World War II and, so far, you have only read about Bletchley Park you have restricted yourself to less than half of the story. The Third Reich is Listening is a surprisingly comprehensive introduction to the parallel world that is German codebreaking in World War II. It takes us out of the narrative that they're just the Bad Guys in the Bletchley Park story and shows us the human ingenuity, struggles, victories and defeats of a whole new, largely unexplored dimension of the history of cryptography and cryptanalysis.





* I used to visit schools (and other places) with an Enigma machine on behalf of the Bletchley Park Trust. I would often introduce myself at reception, with the response "oh, you're the Enigma man!"

** Of course, another myth is that Enigma was the only, or toughest, cipher being broken at BP.

The Most Important Thing in Your Pencil Case, or Calculators Ain't Bad

I have a friend who's starting secondary school in September. Her mum sent me this:

I need your opinion on something. B is laughing at me because I said that choosing a new pencil case is THE MOST IMPORTANT thing about a new school year. I messaged my English dept, but she said they’re bound to agree and I needed to message a Maths teacher. Thoughts?

I responded with:

I nearly agree... THE MOST IMPORTANT thing is assembling your pencil case kit, which includes choosing a new pencil case, but you have to make sure the right kit is in there - non-exploding pens, for example. The stuff that goes inside is important because that will help to inform which pencil case is the right one. It might feel right for a moment, but if it doesn't fit what you need, or allow you to arrange it how you want to then it's not the right one.

Two calculators that have stood the test of time from my own school days. I've just noticed that the batteries haven't, though.
Two calculators that have stood the test of time from my own school days. I've just noticed that the batteries haven't, though.


I then listed my essential pencil case kit. That's not really what this post is about, but of course I have included it at the bottom for all and sundry to disagree with. 

The final thing on the list, THE MOST IMPORTANT thing to go in your pencil case, is a calculator.

And that's something I really do believe.

Calculators are not the enemy

Teaching to the Void

Following a conversation sparked by one of their training events GEM, the professional network for people interested in learning through museums and heritage, asked me to write a guest post loosely based upon (and an expansion of) the ideas in this post. It's here:

    Teaching to the Void at GEM.org.uk

Is Zero Odd or Even?

It's been a while since I posted something mathsy, but I've seen a few people being wrong about this one lately, so here goes...

Let's start with the popular-but-wrong response; the one that would get the QI alarm awooga-ing: "it's neither!"

Wrong!

Now I've lit the blue touchpaper for a few ranty comments or tweets, let's look at the facts. A good place to start is to take a step back and ask...

What exactly is an even number?

The first port of call for almost all questions must surely be the great* Wikipedia, which says that "an even number is an integer that can be divided by two and remain an integer or has no remainder."

Wikipedia isn't the best source for dissuading ranty comments, though.

Wolfram|Alpha, possibly a more reliable source of mathematical information in many eyes, says "an even number is a number in the form $n = 2k$ where $k$ is an integer."

Does zero fit here?

Wikipedia first:
  • An integer is a number with no fractional part (i.e. no decimals; a whole number): ✓
  • If we divide zero by 2 we get zero (with no remainder): ✓
And Wolfram|Alpha:
  • If we can find an integer $k$ such that $0=2k$ that means $0$ is even:
    $k = \frac{0}{2}$, so $k = 0$. ✓
Both Wikipedia and Wolfram|Alpha agree:
$0$ is even.

Need some more?

  • Back at primary school you may have learnt a quick way of identifying whether a number of any size is even by glancing at its units digit: any whole number that ends in $0, 2, 4, 6$ or $8$ is even, $0$ ends in... $0$. So it's even.
  • Additionally, you might also recall the rule that if you add two even numbers together the result is always even, for example: \[2+2=4\] \[120 + 346  = 466\]
    Well, $-2 + 2 = 0$, so $0$ is even.
  • It's also true that two odd numbers added together ($3 + 7$, for example) give an even number ($10$).
    What's $-3 + 3$? Oh, it's $0$, so that's even.
  • Multiplicatively, any even number multipled by any integer results in another even number, e.g.:
    \[2 \times 3 = 6\] \[84 \times -37 = -3,108 \]
    $28 \times 0 = 0$, so $0$ is even.
  • We're all familiar with the number line, stretching from $ -\infty $ to $ + \infty$ . Let's look at a section of that line with $0$ in it:
    ... , -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, ...
    And let's colour in the even ones:
    ... , -5, -4, -3, -2, -1, 0, +1, +2, +3, +4, ...
    Without $0$ included there's a big hole and the pattern is broken. Why would we do that to a perfectly decent pattern?

Still not sure?

Many a maths fan's bible, the On-line Encyclopedia of Integer Sequences (OEIS) naturally includes an entry on the nonnegative even numbers and this begins with $0$. Are you going to argue with the OEIS?






* And often, I admit, fallible

Stony Stratford: Things to See and Do

 If you're looking for things to do in Milton Keynes I've recently discovered* Sophie etc, a blog covering just that topic. It focuses on food (she's pointed me towards some GREAT takeaways during lockdown) but also posts now and then about other adventures that are up for grabs in the area.

There's a post (this one, in fact) about all of the fantastic things you can do in Stony Stratford, a gorgeous former market town up in the furthest reaches of north-west Milton Keynes.

Actually, she doesn't quite cover all of the things that are worth seeing, though the one I'm going to tell you about is a bit niche.

Digital Skills for Teaching & Learning: How to Present with Google Slides or Powerpoint in its Own Window

A little background

EDIT (06/07/2020): Now with added PowerPoint!

I've taken part in a good number of online discussions, seminars, conferences and the like since our Lockdown period started and on the whole it's all been very good, with incredible use of broadly (and often freely) available use of resources to turn an experience that could be flat and impersonal into a very positive one. There are a few things that niggle me, however, and I've been looking into overcoming them, especially as I've been moving into this online teaching lark as well. I thought I'd post about some of the things I've found out and include a few "how to..." walkthroughs. This is largely so that I can find these things again but I realise there may be others out there who might benefit.

Thoughtful transitions can be used to make slides clearer and less cluttered. Losing them can make a big difference.

If that's you, fill your boots! Also ask questions if I'm at all unclear, miss out an important step, or say something you don't understand. If I get something wrong correct me; if there's an easier way let me know!

How to Use a Spreadsheet to Batch-Upload Events to Your Calendar

The other day I posted...
... and a couple of people asked me how to do it.

To some of you this may be the most obvious thing on the planet, but I only discovered it earlier this year and - my oh my - has it saved me some time. My day* job sees me visiting various places around the country and I keep track of it all in my calendar. As any self-respecting nerd is driven by unknown forces to do, I often put a bit of effort into finding out how I can get the tech around me to do some of the more boring jobs so that I have more time available for doing the interesting things. I'd list some potential use-cases for this spreadsheet-to-calendar trick but I figure that if you've read this far you already know what you want it for.

Before we get started, a disclaimer: Yes, there are plenty of other articles detailing how to do this and I visited many of them. I haven't referenced any of them because it was about six months ago so I've forgotten which ones I used, none of them gave me every piece of information I needed, and there were a few important stumbling blocks that weren't mentioned at all.

Another Puzzle for the Today Programme!

I set another puzzle for Radio 4's Today programme and it was featured this morning (Monday, 6th April 2020).

It goes like this...

"Doc note, I dissent. A fast never prevents a fatness. I diet on cod."

Bletchley Park Codebreaker Peter Hilton constructed this impressive 51-character palindromic sentence, allegedly in response to a challenge set by a fellow Codebreaker. Palindromic numbers work in the same way: the same digits in the same order whether you read them backwards or forwards.

What is the 51st Palindromic number?

You can find the puzzle (with a link to a solution) on the Today Programme's website, here.

As with many puzzles the solution is debatable depending on some starting assumptions. Feel free to discuss your assumptions (and how these might change the answer) in the comments. Teachers, I'd love to hear how you might use any of the ideas or topics featured (even tangentially) in your own teaching!

Also, please post your favourite palindromes!

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