A Christmas Number Adventure from Richard Phillips

The early 1980s were a pretty exciting time in education.  Money wasn’t too tight, governments let schools and teachers get on with things, and above all encouraged us to explore the use of the computer.  Indeed, computers had become affordable enough that in my own school I raised enough funds to acquire several at less than £100 a time.

A benign consequence of the introduction of computers was an absolute explosion in creatively designed software, most memorably exploratory and frequently open-ended.  (I had a particular interest in using mathematical adventure games closely related to games like Dungeons and Dragons – indeed, I still use Anita Straker’s Martello Tower today.)

One of the most prolific sources was the Shell Centre team at Nottingham, who in partnership with a group at Plymouth produced half a dozen collections of intriguing programs.  I particularly remember two or three of these – there was the delightful Sunflower, nominally about scientific method but actually about number magnitude and understanding of decimals.  There was a version of snakes and ladders played on a blank board where you had to deduce the rules governing the nature of the snakes and ladders from the moves the game allowed you to make.  This was so much fun that my own group of gamesplayers devised our own versions and played for many years.

The name I always associated with these packs was Richard Phillips.  His personal website – and since his sad death in 2015 this is maintained by his son Will – is  https://www.richardphillips.org.uk

What I’ve never seen elsewhere is the Christmas Number Adventure Richard developed from an original of Sam Costello.  To play it you’d better brush up on your knowledge of the Twelve Days of Christmas – “On the first day of Christmas my true love sent to me …”

I never met Richard, but I admired his work immensely, and this is a nice reminder.  Do give it a try.

https://badseypublications.co.uk/christmas/index.htm

Thanks, of course, to Will Phillips, for encouraging us to use it here.

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Top Triumphs

You’re always on the lookout for ways to enhance fluency in handling numbers and for ages I’ve wanted to come up with a game the captures the feel of Top Trumps. Not only is Top Trumps familiar to children; it’s very easy to grasp and it offers high-speed play.

Here’s a version that’s been such a rousing success my pupils have pleaded to include it in session after session. It’s all home-made, and you’ll want to adapt it to suit you and your pupils.

Children need about ten cards each, showing numbers in the range from 0 to about 25. There’s no need to have a complete set and it doesn’t matter if if some numbers are duplicated – so I let children choose their own numbers to start with. My contribution is a separate pile of cards of instruction cards such as ‘greatest total’, ‘least difference’, ‘product closest to 30’, etc. Soon I began to slip in one card challenges ‘highest’, ‘lowest’, ‘nearest to 10’, and three card challenges ‘lowest total’, ‘greatest product’, etc

Top Triumphs

The photograph probably makes it clear what happens. The first challenge card is shown and the players display the relevant number of cards from the top of their holding. All the number cards played are given to the winner, who puts them back at the bottom of their stack.

That’s basically all there is to it. We’ve tended to have games between two players but there’s no reason at all why more players shouldn’t take part.

What’s impressed me and surprised me – and it’s been a very big surprise – is just how good my pupils are at assessing the winner of each trick, even when relatively large numbers are involved – and they do so at pretty well lightning speed.

For example, Closest number to 15, when the displayed cards are 12 and 19.

Smallest difference, when the displayed pairs are 6 and 10, and 18 and 25.

Largest total, where the displayed cards are 17, 6, 10 and 11, 21, 9.

Highest product when one player displays 20 and 17 and the other 19 and 16.  (On the other hand when 19 and 16, and 17 and 18 appeared they recognised that the answers for the highest product would be so close that the only way by to tell was to carry out both long multiplications.)

The other thing to report is that my guinea pigs have been comfortable with more of the vocabulary than I expected, and I’ve made sure the instruction cards include not just ‘largest’, ‘smallest’, ‘highest’, ‘lowest’, but also ‘sum’, ‘total’, ‘difference’, and ‘product’.

My pupils have given me a lot to think about. These are children whose number fluency in recall and computational processes has great room for development. Yet they’re displaying an excellent insight into numbers and how they interact.   They rarely calculate the answers but come to a shared and largely unspoken evaluation.

They’re demonstrating a mature and sophisticated understanding, but unfortunately for them it’s an understanding that brings them little credit in most assessment formats – and for the rest of us, it seems to raise quite a few questions.   When they come to do their SATs papers in six months time my pupils are going to face dozens of questions which assess their ability to carry out written procedures, and they’re not likely to do all that well. And yet they’ve got a deep understanding of numbers and how they work, and sadly this knowledge is unlikely to be of little interest to those who set and mark the papers.

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A Conversation with Kitty

Of course, one of the great features offered by even the simplest programming and coding activities is that when a procedure doesn’t do what it should children respond positively, and see initial failure as a stimulus to find out what went wrong and do better next time.

But I’ve never known a child express this as strongly as Kitty.  She believes so strongly in this problem-solving aspect that she actually wants her programs to fail, because if a program works immediately she misses out on what she really enjoys, which is figuring out what went wrong and how to correct it.

Which raises the obvious question – how do I devise mathematical activities that children want to fail at in such a positive way?

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Evens Puzzles

You may have come across these puzzles before, but I certainly haven’t.  There’s lots of good arithmetic and oodles of potential for making harder or easier versions, and pupils can devise their own.

Evens_0001

Evens

Evens_0003

I’ve only found them in a newspaper I loathe and despise – and then only once a month – but I’ve got several ideas in the pipeline for variations.

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Tim Gowers and a Great Test Match Victory

I’ve no idea how many of us – but I’ll guess it’s a very substantial number – know for a sure and certain fact that if we don’t comport ourselves appropriately during an important sporting fixture it will be our fault should our team lose.   Non-believers will see this as ridiculous superstition, or even immense arrogance, to think that our own behaviour can affect the performance of 22 sportsmen a couple of hundred miles away – but sports fans know it’s foolhardy in the extreme to tamper with the universe.

I’m greatly comforted to know that Tim Gowers is one of us. Sir W T Gowers is a mathematician of such quality that he has a Fields medal to his name. Yet he too knows how he should behave when his country needs him – I spotted a succession of excited tweets at the climactic afternoon of what has been described as one of the greatest Test Matches of all time.

Tim Gowers

(Mind you Tim Gowers is probably not quite as severely afflicted as I am. Frequently I’ll record sporting events so I can watch them as live later that evening. Even then I still have to make sure I behave myself correctly in case my carelessness affects the result of a game that finished eight hours previously.)

Incidentally I can claim to have met Tim Gowers on one occasion. My wife and made the two-hour drive to Cambridge to hear him talk and demonstrate his considerable skills as a jazz pianist.   We left marvelling not only at his versatility and the clarity of exposition but also his ingenuity – I’ve never before come across a musician or writer who told us about “scrunchy” chords. Thanks, Tim.

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Prostate Cancer

Here’s the background to a health issue I alluded to in a recent post. It won’t apply directly to many readers, but it will clue you up on my own situation. There’s a distinct gender aspect to it, but I don’t see any way round that.

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Short version: at the very start of the year we were told I’ve developed prostate cancer. After lots and lots of reading and talking and thinking I’ve opted for a course of twenty sessions of radiotherapy later this year. This is expected not just to control the cancer but eliminate it completely.

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Much longer version:  to be honest it wasn’t really much of a surprise at the very start of the year to be told that I have prostate cancer. My PSA reading had been going up for eighteen months or so; its traditional level was around 3 and soon readings were coming in at 6, 8, 9, and eventually 12, so clearly something was going on. There were plenty of reassurances (“we don’t worry too much if it’s below 50”) but it was pretty obvious the hospital at Stoke Mandeville was sure there was something there and they weren’t going to stop until they found it. So I had an MRI scan, which was an unusual way of starting Sunday morning but at least meant there wasn’t a parking problem. The scan didn’t find anything but the PSA continued to rise and the next step was a biopsy (I’d been led to expect this would be painful but in practice it was no more than rather uncomfortable).

The whole investigation took nine months or so, sometimes slow (“see you in three months”), sometimes quick (for the MRI I saw the Head of Urology on the Tuesday, had a phone call the next day, fixing the scan for the weekend). There were various messages that looked encouraging: “did not reveal any significant abnormality”, “has not shown any clinically significant tumour”, “I feel his prostate is clinically benign but …”.   Nevertheless, however positively you read them there always seemed a little wiggle room. There was never any suggestion that they were satisfied and were going to stop looking; it was always a case of “we’ll see you again in three months”, or “next …”.

It was the biopsy that finally located the problem, and on January 2nd we had a session with the Macmillan nurse who broke the news. Not all the information sank in, but what I did take from the session was his enormous positivity. The approach was very much “You get ill, you get treatment, you get better”. One of the better questions I asked was “On a scale of 1 (dreadful) to 10 (excellent) how should I feel about this news?” “A definite 9” he said. That sent me away feeling much better than I might have done – even when my phone rang the instant we got out of the door and with exquisite timing the plumber said “Happy New Year, Alan, how are you?”

I left the hospital with an armful of booklets, and soon had plenty more, courtesy of the charities and a friend or two. From this point, things went downhill a bit. I was acquiring more and more information, but I couldn’t evaluate it and get it all into focus. This hit the low point when I had my first session with the oncologist. He presented me with three strategies for dealing with things, and scrupulously avoided offering advice (I think now that perhaps I should have asked him for a recommendation, but I didn’t, and spent the next two or three months in a total whirl, deciding first on one option, then another, and then back again).

Fortunately, prostate cancers tend to be very slow growing so there’s little urgency to commit yourself quickly to a treatment plan. Indeed, the first strategy option is simply to do nothing at all, keeping an eye on things and not intervening until it looks necessary.   This is the strategy of “watchful waiting” and for many men is perfectly viable not just long-term, but permanently.

The second strategy option is hormone therapy, administered as injections effective for three months at a time. One consultant said this stops the cancer “in its tracks”. He was right; my PSA reading had gone as high as 12 and the first treatment of hormone therapy brought it down to 0.33, lower than it’s ever been.

However, there are disadvantages to hormone therapy. To stop the cancer growing it blocks your testosterone and obviously for any male there are potentially going to be side-effects. It also only works for a limited period; the Head of Urology suggested perhaps three to four years; the oncologist thought this was pessimistic.   Actually, I found the father of one our friends has been on hormone therapy for best part of twenty years.   But perhaps most significant of all is that while hormone therapy may stop the cancer growing it will not shrink it or destroy it, and sooner or later it’s likely to start growing again.

The third option is to tackle the cancer head on and destroy it. Now this is where information overload really hit me. Prostate cancer is a high profile illness and there are many techniques and oodles of research projects. Cryotherapy, ultra-sound, brachytherapy, surgery, radiotherapy, ….   Of these, radiotherapy was the only one offered to me, but in that first visit the oncologist was so determined that I should make up my own mind that he emphasised the downside (including potentially awkward side-effects involving the bladder and bowels) and indeed the negative aspects of the other strategies as well. (Actually, in more recent meetings he’s been much less wary of side-effects than he was back in February.)

So as spring progressed I had a growing mountain of information – but information on its own wasn’t much use if I had little ability to assess and evaluate it all. Should I opt for the waiting, for the hormone therapy, or something more aggressive – and if so, should I take the radiotherapy or look into other methods, possibly involving private funding? The charity helplines were warmly supportive, but for what I wanted were completely useless; understandably they couldn’t advise on individual cases. I now know, but didn’t then, that my local surgery has a dedicated prostate nurse I could have talked to. My own GP was very hostile to the use of radiotherapy, but then he was clearly influenced by a patient he’d seen the previous day for whom radiotherapy had proved unsatisfactory.

What I desperately wanted, and with the wealth of information there must be it’s surprising this doesn’t exist, is an app of some kind into which I could plug all my details – PSA, age, cancer rating, etc – to give me an idea of the choices other men in my situation were making.   I wasn’t necessarily looking for how effective the treatments were, simply to get a feeling for how many people did what. Better still, of course, would be an indication of life-expectancies – though the overall survival rates were so good that I comforted myself with the fact that actually it looked rather difficult to make a really poor choice.

It took until the middle of May, but eventually there were two light-bulb moments. It suddenly hit me one evening during a jazz club concert that rather than attempting to compare and evaluate the three strategies I should be asking a different question – what was my fundamental goal? The answer was obvious – in recent years my wife has suffered vary considerably from arthritis which has greatly limited her mobility.   Once I focused on this it was obvious that my strategy can only be to make sure that I am around for as long as possible to make life easier for her. I put things in these terms to the oncologist and his reply was instant – it had to be radiotherapy.

But he did suggest I visit a colleague in Oxford who’s an expert in the alternative strategy of ultrasound. I’ve never previously used a private consultation, but this turned out to be a brilliant way of spending £250. In effect he was giving a second opinion and his views were identical to the Stoke Mandeville oncologist. Though we were discussing ultrasound he gave me the second light-bulb moment by volunteering his opinion that in practice radiotherapy was much the better option.

From the very start the survival figures had looked hugely encouraging. Even the most pessimistic figures suggest that around 80% of all prostate cancer sufferers survive at least ten years. Given that 78-year olds have plenty more things to die of, 10+ years survival looks pretty good. The Oxford specialist put things dramatically – “If you have radiotherapy then your chance of dying from prostate cancer in the next fifteen years is less than 1%”.

My Stoke Mandeville oncologist agreed with everything the Oxford expert had said, so though it’s taken a heck of a long time at least I’m now in no doubt about how I should go ahead, and we signed the forms a few days ago. I’m expecting a twenty-day course of radiotherapy at the Churchill Hospital in Oxford, perhaps in November. I’d have rather liked to have got things out of the way before October 31st, but the timescale makes this unlikely. Still, with luck I won’t be having to drive the 35 miles to Oxford (and both of the experts have said I’ll be able to drive myself) throughout January or February.

I’m a little bit bemused that it took me so long and very grateful I was able to take so much time to make my choice. As a cancer sufferer I’ve really felt a considerable fraud; others have ghastly versions of the disease while I have no symptoms, no suffering, and the expectation of a complete cure. And though I’d rather not be in this position, people are already treating me as an expert and beginning to ask me for advice.   Given the demographic profile of my friends and contacts it seems pretty likely that others will eventually find themselves in the same boat. For what it’s worth, here are some suggestions – some of these are things I did OK, others I should have done better:

[a] take wife / partner / relative to important sessions, not only for support, but to have a second pair of ears, (and take a notebook as well – I have a rather full A4 ringbinder),

[b] work with both the hospital and your local GP / surgery, and not only with doctors but with their specialist nurses,

[c] don’t simply accept information, but ask for opinions and recommendations – you may not get them, but ask anyway,

[d] you may want to consider a variety of treatments – there’s so much work going on in the field that things are changing all the time,

[e] the charities and support organisations are full of help – https://prostatecanceruk.org , https://www.macmillan.org.uk , https://www.cancerresearchuk.org

[f] welcome experiences from friends, relations, colleagues,

[g] don’t lose sight of how amazingly good the survival rates are – I’ve even seen it suggested that those with prostate cancer actually have longer life-expectancies than men without!

[h] and when you’ve got all this information from dozens of sources, you’ll need to evaluate it all and come up with your own decision.

Good luck!

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The Last Lessons Of The Year

If you started teaching in 1962 there have been a lot of occasions when you’ve taught your last lessons of the school year. This time there’s a possibility – hopefully rather slight – that these might be the last lessons of all.

 I’m due to have a course of radiotherapy later this year, and while it’s expected to clear up the problem once and for all there is some possibility that the side effects might make me an unsuitable person to have around in the classroom. With luck that won’t be the case – I’ll know more in the next month or two.

 All the same, the possibility did mean that this year’s last lessons had a slightly more sombre context, so I indulged myself and used some of my favourite activities.

 

****** First I used a couple of challenges from ATMs “We Can Work It Out!” booklet. (www.atm.org.uk ) Each of these presents children with a dozen or so statements on individual cards. I find these infinitely fascinating; children who usually have a fit of the vapours when presented with any question with words in it respond just as positively as anyone else. Typically, at first glance the information is totally confusing. There is no obvious way through the problem; perhaps even the starting point and the target have to be identified. There may be irrelevant information to be ignored, and the remaining statements need to be interpreted and assessed.

For example in The Great Race from the “We Can Work It Out” booklet they have to evaluate a whole series of statements such as ’the green car finished before the yellow car’.

I wrote a little more a couple of weeks back, and this included information about two activities of my own, Martian Kings and Queens and the Properties Of 2D Shapes, and you can find this at

https://established1962.wordpress.com/2019/08/06/collaborative-problem-solving-we-can-work-it-out-martians-and-shapes/

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****** I also returned to the Stephen Von Worley’s animated display of numbers by that you can find at http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/   So for example 3 and multiples of 3 have a display built around a triangle of three circles. 4 and multiples of 4 have a square array of four circles – so what do you do with 12?

I’ve written about this before (November 2016) at https://established1962.wordpress.com/2016/11/04/animated-factorisation/

One of the richest aspects of this is that you never know the path you’re going to follow. On one occasion we got into prime numbers, another looked at powers of 2, and this time we spent half an hour on estimation.

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****** Then there was Marilyn Burns’ lovely playing card challenge where a suit of cards is arranged so that when you deal them out in a particular fashion the cards arrive in sequence Ace, 2, 3, 4, …. The challenge for the pupil is to figure out just how the cards must be stacked to ensure this sequence happens for them. Everything you need, from introduction to months of extension and follow-up, is at http://www.marilynburnsmathblog.com/the-1-10-card-investigation/

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****** And I also included the simple simplest dice game of all, Pig. When it’s your turn you are allowed how to roll the die as many times as you like, keeping track of your total score. You can choose to stop rolling, and your score becomes safely banked. However, should you throw a 1 not only does your turn end, but your score for that turn is 0.

It’s wonderful fun, but Pig is not just fun, and it’s not just excellent mental arithmetic practice – every turn, and every throw, you need to make decisions, and the decisions change all of the time depending on the state of the game.

I wrote about Pig a few years back, but I never got around to publishing it until a few days ago.   It’s worth a look, because a couple of children taught me a version of their own that I’d never come across.

https://established1962.wordpress.com/2019/07/29/pig-or-bust/

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Collaborative Problem-Solving – We Can Work It Out!, Martians, and Shapes

I doubt they need any recommendation from me, but the “We Can Work It Out!” booklet is one of my favourite resources from the Association of Teachers of Mathematics ( www.atm.org.uk ). There are two collections, with lots of activities described as Collaborative Problem-Solving For The Mathematics Classroom, and they’re suitable for pupils in the upper primary and lower secondary age-range.

For each activity children are presented with a dozen or more cards, about the size of visiting cards.   Between them, the cards carry all the information about a situation – arranging a school trip, or the prices of a range of cakes.

These are four of the eleven cards in the first activity, The Great Race:

The Great Race

Children aren’t simply challenged to find the answer, they have to identify the path though the problem.  Indeed since they receive a pile of cards rather than a printed sheet they even have to locate the starting point for themselves. As the title and sub-title both suggest, these activities are particularly valuable for generating discussion and co-operative working with small groups of pupils.

Challenges like these can produce a positive response, even from children who are usually intimidated as soon as a couple of words creep into a question, and observing a group work through a problem gives you a vast amount of insight into their understanding, vocabulary, and every problem-solving skill you can think of.

As you’ll gather, I recommend “We Can Work It Out!” highly. The list price of each is £22, considerably cheaper for members; the books are spiral-bound to make photo-copying easier.  And if you opt for the e-book – which is sensible, because you need to print out activities – it’s cheaper still, at just over £8 for members.

You can find the current ATM catalogue, which includes both collections and is full to the brim with other great resources at http://bit.ly/ATM19cat

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A similar challenge of my own is called Martian Kings and Queens. I wrote about this in August 2015 (https://established1962.wordpress.com/2015/08/12/martian-kings-and-queens/ ) Have a go – you’ll find it works fine, though it’s a bit tough for my pupils and I really need to add an extra clue or two to soften things a bit.

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And here’s another one of mine, about two-dimensional shapes.  You need to find which shape goes in the central cell of a 3×3 square grid.

Here are the shapes:

Shapes 1

And here are the clues (which are deliberately in a not-very-helpful order):

Shapes 2

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Pig – Or Bust!

I fondly recall the time we took a group of children on a residential trip, and one evening after the children had gone to bed I introduced the adults to Pig.  Anyone who’s ever been away with a group of children knows that sooner or later you have to read the riot act after lights out to get them to stop talking and settle down.  This occasion reversed the natural order of things – it’s the only time in my life when I’ve been faced by a deputation of sleepy children demanding that the teachers behave themselves and stop making so much noise.

So let’s tell you about Pig.  It’s always been just about my favourite game.  You can play it in the classroom or at the pub, it needs nothing more than a single die and paper to keep score.  You can learn the rules in a minute and anyone – any age or any number – can play, and yet it’s so rich that a class of university maths students couldn’t agree whether it’s a game of luck or skill.

When it’s your turn to play you roll the die.  Indeed, you can roll it as many times as you like, keeping the total score as you go.  When you decide to stop throwing your total score is recorded.  So if I roll 3, 4, 2, 3, 6 I can stop throwing and bank the score of 18.  Alternatively I could make a couple more throws – 2 and 5, say –  in which case I bank 25 rather than 18.

There’s just one restriction to this – if I throw a 1, then my turn ends immediately.  All my score for that round is lost, and I score 0.

The advantage of banking your score is that banked scores cannot be lost.  If, therefore, I’ve banked 25 and next time I roll 6, 6, 6, 6, 6, 1 – then I’ll curse bitterly at the thought that I could have banked a further 30 points, but at least I’ll start my next turn with the 25 still in the bank.

The winner is of course the person with the highest score, or perhaps the first person to reach 100.

Alternatively, you can play to a time limit – and you get a whole new set of factors to take into account.

But when I started to tell Issa and Milly about the game they instantly recognised a game they play.  It’s called Bust.  It needs a pair of dice, and each player has a scoresheet.  Your scoresheet in divided into four columns, headed B, U, S, T.

Both dice are rolled and their score is totalled.  Each player scores the total and the dice are scored again, and again, just like Pig.  However, there are three differences; firstly, every player – not just the person who rolls – gets the score.  Secondly, each player makes their own decision when to stop scoring and bank the accrued total; the others keep scoring till they choose to stop (or of course Bust out).  Thirdly, the trigger that ends the round and (if you haven’t already opted to bank your score) gives you a score of zero is when both dice show the same number.  (Alternatively, the round must end when everyone has banked their score.)

In Bust play lasts for four rounds – your first score goes in column B, next round in column U, and then in columns S and T.  As you’ll guess, you win when the total of your four scores is higher than anyone else.

The arithmetic’s more challenging; everyone’s involved all the time, and with every roll of the dice you have a decision to make.

This was five years or more ago, and I’ve not till now got around to publishing it – and I’ve never come across anyone else who knows the game.

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Any Ideas?

I’d never seen one of these before.

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Here it is again, with a coin giving you an idea of the actual size:

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It certainly reminds you of the net for a cube, or possibly an open box.

It’s the backing for an adhesive plaster, as needed by anyone foolish enough to attempt a do-it-yourself thumbectomy while gardening.

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And here it is in action.  It’s certainly a lot neater than my original bandage, which was about the size of a peach.  It doesn’t look much now, and it’s all cleared up quite considerably, but it didn’t half hurt, and there was a vast amount of blood – I needed some quick improvisation to avoid a carpet crisis.

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