Over on Twitter, @missradders has sparked a really interesting discussion on subtracting fractions by offering the above method to subtract two fractions. (see the Twitter thread here)

I must admit, it is a method new to me and, from reading the replies to her tweet, it was unfamiliar to many other teachers, to.

But is it a valid method?

Again, reading through the comments on the thread, its taking a bit of a kicking. Before we come to a conclusion, perhaps its worth asking why, in the age of the scientific calculator, do we even bother to add or subtract fractions?

Beside me on my desk, I have the new(ish) Casio Classwiz fx-991 EX calculator, that will integrate, solve cubic inequalities, handle matrices – adding and subtracting fractions are well within its capability: what is the point of asking recalcitrant pupils to manually perform an algorithmic task that can be done electronically in moments?

The answer is algebra.

Being able to add or subtract – mentally, or on paper – straightforward fractions such as, say ^{3}/_{4 } – ^{1}/_{2 } has it’s merits, but I would be reaching for my aforementioned fancy calculator to do the subtraction at the top of the page: ^{8}/_{9 } – ^{3}/_{5 } but I need to know how to do it with numbers as it is the stepping stone to being able to work algebraically.

High end GCSE – and **all **A level – mathematicians need to be able to work with algebraic fractions and being secure with a method of adding and subtracting numeric fractions is crucial for this skill.

So the test of @missredders method is to see how well it works with algebraic fractions. So I tried it to subtract the fractions: ^{a}/_{x } – ^{b}/_{y }, my working is below.

It works. In fact, I think it helps to see the method in algebraic form as it explains why it works.

I don’t think that it is a method that I will be teaching (but happy to keep it in my armoury, just in case) but I don’t think that it is the dog’s dinner that some are claiming on Twitter.

As a method it works, and there is a clear route from using the method numerically to using the method algebraically, and that is important. I suspect that if a student is taught the “grid method” shown at the top of the page, masters it and goes on to use it algebraically for higher level maths, they will probably develop their own shortcuts for using the method (i.e. going straight to writing down the common denominator) and that – find short cuts – is what maths, and being mathematical, is all about.

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It was shortly after my standing tree pose had inadvertently become a fallen tree, and I lay on my mat, a crumpled collection of arms and legs listening to the calm, soothing words of my Yogi that it happened, that I had my epiphany, my moment of clarity, when I suddenly understood.

A mid-week evening in late September, the summer holidays a fast receding memory with only a damp autumn followed by a long, dark, bleak winter to look forward to. Back in the classroom for three weeks, and tiredness was already gnawing away at my very being. The obvious thing to do, the easy thing to do, would have been to flop on the sofa, fire up Netflix and nod-off in front of another American import. But no, it was Wednesday night, and Wednesday night means Yoga night, one of the highlights of my week (the other being my Friday night spinning class.)

Many, including myself, have asked me why I enjoy my yoga and spinning classes so much. Not too many people look forward to an hour of punishing pain pushing the pedals on a stationary bike every Friday night, but I do. I’ve tried to explain why I enjoy it so much – the virtues of physical exercise, the camaraderie of the class, the relief when its all over. When all you can think about is the burning pain in your thighs the problems of an unmarked pile of books and that difficult phone call earlier in the day shrink into insignificance, and that certainly cleanses the mind. But it still didn’t get to the nub of why I enjoyed it so much, why I would look forward to it all day long.

But then it came to me, on my yoga mat. Enlightenment.

As a teacher, in your classroom, you are the boss. It’s your domain, you are – you have to be – numero uno, top dog, the big cheese. You spend your day making decisions, giving instructions, controlling your environment. And when you’ve been doing the gig as long as I have (too long to count) you inevitability end up with various positions of responsibility which means more problem solving, telling others what to do, taking charge, and I love it. My siblings would describe me as bossy, I prefer organised and efficient. Regardless, I’ve found my niche and it’ll take a lot to prise me out.

I wasn’t looking for enlightenment, I only started doing yoga as I wasn’t very bendy and getting older wasn’t helping my suppleness. Spiritual awareness? Nah, thanks, but no thanks, not for me.

But my epiphany had been stalking me, and then: Wham! It stepped out of the shadows, taking me by surprise, and hit me in a moment of realisation. I suddenly understood why I do it, why I look forward so much to my yoga and spinning classes.

For an hour or so, twice a week, I’m no longer in charge. I’m on the other side of the fence, I’m the one listening to and following instructions and its hugely liberating. Letting someone else take control, doing what they say, surrendering to their direction frees the mind of all its cares and troubles. For a short while, a couple of times a week, you have no responsibilities other than to yourself.

Now, don’t get me wrong, this isn’t the birth of a new me – most of the time I definitely want to be in charge. For a megalomaniac like me, teaching is the perfect job: every hour a new group of subjects file into your domain to be inspired, nurtured, led on a pathway to discovery – a captive audience. But for a couple of hours each week to be the student, and not the teacher, to let someone else take the lead is a wonderful experience.

Maybe you should try it.

]]>**Possibly the best Venn Diagram, ever.**

Two words – absolute genius.

Witty, visually appealing but, most importantly, mathematically correct.

Sets and Venn Diagrams have been on the IGCSE syllabus for some time, and made it onto the new GCSE syllabus, so we’ve all got to teach them. Project this image onto your whiteboard, sit back and put your feet up – job done. You could spend hours telling your classes about intersections and unions, or you could just show them this and they’ll grasp it in a moment.

The image is just one of many great diagrams, charts and infographics to be found in Stephen Wildish’s fantastic book: Chartography – worth a tenner of anyone’s money.

]]>As most readers will know, this year sees GCSE maths grading change from A* to E to 9 – 1. There has been much speculation as to what the grade boundaries will be for each new level.

The first (that I have found) to publish the grade boundaries are Cambridge, for their **IGCSE** and can be seen above. Below, I have converted them into percentage scores (easier for all – and pupils & parents in particular to understand)

Higher Tier:

**9 – 80%, 8 – 68%, 7 – 58%, 6 – 47%, 5 – 36%, 4 – 27%, 3 – 22%**

Foundation Tier:

**5 – 73%, 4 – 58%, 3 – 43%, 2 – 29%, 1 – 14%**

The official table from Cambridge can be found here

**Important Note – These grade boundaries are for the Cambridge International Examinations board only, for their IGCSE exam.** The grade boundaries for different exam boards and for GCSE maths will be different. The information above is all that I have available at the time of writing.

However, the grade boundaries above will be of interest to those who sat Cambridge IGCSE and *might *also serve as a useful ball park figure to estimate grade boundaries for other boards.

It should also be noted that, unlike last year, grade boundaries for GCSE (and A Levels) will **not** be published the day before results day (although they will be made available to schools the day before). Grade boundaries will be published on the same day as the results (Thursday 17th August for A & AS Levels, Thursday 24th August GCSEs)

Ooops! A whole generation of youngsters could be scarred for life.

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There are two types of mathematicians (or 10 if you are into binary) – pure mathematicians and applied mathematicians. I am firmly in the latter camp. That’s not to say I don’t enjoy solving a tricky trig problem, just for the sake of it, but it is applying maths to the real world that really floats my boat.

And so it was with great delight that I stumbled across The Game of Trust , an interactive insight into game theory, and how it can be applied to building trust, whether that be in negotiations, the first world war trenches or pretty much any situation where risk and reward depends on your choices.

The site analyses different strategies – cheat, co-operate, copy etc. I won’t tell you what strategy to follow, visit the site now and invest 30 minutes or so playing the game, you will be rewarded (can you trust me when I make that statement? Visit the site and you can decide what category I’m in!)

But, without giving too much away, it does tell us that game theory says:

Without the non-zero sum game, trust cannot evolve

And in these increasingly turbulent times, we need trust more than ever.

So whether your negotiating with a teething toddler, a truculent teenager or a paranoid politician try and find an outcome that benefits you both – it’ll be in your best interests. Trust me, I’m a mathematician.

[Here’s that link again in case you missed it The Game of Trust – visit it now and invest 30 minutes of your time, it’s worth it]

]]>**Red sauce, brown sauce or no sauce at all – checking for bias at the BBC**

One of the great pleasures of a Saturday morning is being able to linger over your breakfast whilst listening to the wit and whimsy of Danny Baker on Radio 5. A highlight of his show is the sausage sandwich game in which two listeners compete to answer questions that only that day’s celebrity guest could possible know.

Today’s guest was Wayne Bridge, who will soon be moving house, and one question was how many letters separate the first letter of his new street and the first letter of the last foreign country he visited. So, for example, if he is moving to Filbert Street and the last country he visited was Japan, the answer would be 5, as there are five letters separating F and J.

The contestants were given three options:

- Less than 6 letters separating them
- Between 6 and 12
- More than 12

“What a great question” I thought, no use using Google – only Mr Bridge would know the answer to that. But is it a fair question? Can you use maths to help you decide if the three options are equally likely, or is this an example of bias at the BBC, the smoking gun its opponents have been hunting for? I fired up my spreadsheet, grabbed some paper and a sharpie and began to crunch the numbers …

I wont bore you with the full calculations now, but here is brief look at my methodology. (You could safely skip this bit if you wish.) Take the letter K as the first letter of the street name. The probability of K being the first letter is 1/26 (26 letters in the alphabet). To be separated by less than six letters it could have 5 letters above it (F to J), or 5 letters after it (L to P) or it (the country) could begin with K, so if K is the first letter of the street, there are 11 letters the country could start with to have a separation of less than six. So the probability is 1/26 x 11/26. For the second option (separation between 6 and 12 letters) the probability is 1/26 x 12/26 and for the third option, a separation of greater than 12 there are only 3 possibilities: X, Y and Z, so the probability is 1/26 x 3/26. Repeat this process for each letter of the alphabet and sum the probabilities.

And the results are in! (If you skipped the above paragraph, you need to rejoin now).

The chances of three options are below:

- Less than 6 letters separating them 38%
- Between 6 and 12 35%
- More than 12 27%

So not too much difference between the first two options, but picking a separation of more than 12 letters would not be a good choice.

I think we can conclude that the the BBC has done its best to be fair, although it has succumbed to a little negative bias against the more than 12 separation. It does, however, raise the fascinating question of whether the question could have been made more equal for all options. If the summer holiday weather remains as rubbish as it has begun today, I may just ponder that problem a little longer.

But what we can be sure of, is that The Danny Baker Show remains the broadcasting highlight of the week.

]]>A warm Sunday morning in May. I’m up (too) early and heading off to my local football ground. Its cup-final day for the local youth leagues and I shall be officiating in a couple of games, first as an assistant referee, then I shall swap with the man in the middle who will be my “Lino” as I referee the game.

I arrive early, park up and, as I step out of my car, another vehicle pulls up and a young man, also clad in black, hops out – my fellow referee. There is a glimmer of recognition as we shake hands and I begin to quiz him about his background so I can work out where I have encountered him before. It turns out he’s on the books of a local League 2 side and I’ve ref-ed some of their U16 trials games and will have seen him on the pitch there.

We chat some more as we wait for kick off. He’s currently in Year 11 at a local school, is quietly confident about his forthcoming GCSEs and is even more excited to tell me that he has been signed on as a “Sixth Form Scholar” at the aforementioned League 2 club. No mean achievement, and he is right to be proud.

I quiz him further (I’m a nosey old git!) He’ll be going to the sixth form college in the town of his club – he admits he probably wouldn’t have chosen that college if it weren’t for his football commitments, but it is a condition of signing on as a “scholar.”

He also tells me that none of the current “lower sixth scholars” at the club have been signed on for the “upper sixth.” They have all been “let go” – they will, of course, continue their studies at the local college but ties with the football club have been severed.

Another school, another Year 11, another Football Club (but also in League Two.) I found myself chatting to this student, asking what he would be doing next year. Again, it was with some pride that he told me he would be joining …… FC as one of their football scholars.

We chatted away, I was interested to see what this meant in practice. Like the young ref mentioned above, this lad will spend several days each week training with the club, and several days each week at the local college. Asking him what he would study he told me that all football scholars at the club followed the same academic course.

And that set a few alarm bells ringing. These boys are mortgaging their future on the possibility, and, as we will look at below, a slim possibility, of “making it” as a professional footballer. The football scholars will have different skills and strengths on the pitch, and it will be the same in the classroom. To shoe-horn them all onto the same academic course, to deny them the choice of studying the subjects that they want to study is doing them a disservice. But with the lure of of a glimmer of a pro-contract in two years time, heads are easily turned.

If they make it, does it really matter that they’ve missed out on A Level Maths, or English or Geography or whatever? No, it doesn’t. But most won’t make the grade.

The Independent recently published an article promoting a book: “No Hunger in Paradise: The Players. The Journey. The Dream” by Michael Calvin, who documents the sometimes seedy story from schoolboy to professional footballer, a story littered with shattered dreams, shady agents and broken promises. In it, he offers the staggering stat that only 180 out of the 1.5 million boys who play organised youth football in England will go on to play in the Premier League. That’s a success rate of 0.012%

I thought I’d do some of my own sums, and they back up the authors claims. I’ve made many generalisations, but I think I’m in the right ball park. Here are my calculations.

20 Teams in the Premier League each with, say, a playing staff of 30, so that’s 600 players playing in the Premier League.

Assume a career length of 15 years, so each year one fifteenth of the players will retire. To replace these we need ^{1}/_{15} x 600 new players each season, or 40 new footballers each season, 40 youngsters to take the place of those who have reached the end of their playing days.

There are 48 counties in England, so its not enough to simply be the best player in your county, you’ve got to be better than that. And that’s before we even factor in overseas players.

Another ballpark figure, let us say that half the players in the Premier League are English, half not. So we probably only need 20 new English lads each year to join the elite ranks of Premier League footballers. To succeed, statistically, you’ve got to be – at least – the best player in your county and the neighbouring county.

But of course, you don’t have to play in the Premier League to enjoy a successful, and financially rewarding, career in football. Those plying their trade in the Championship, League One and League Two will enjoy a good living and lifestyle. And as you drift further down the football pyramid their is still scope to make a significant additional income from playing the game. (Alas, my playing days were so far down in the roots of “grass roots” football, I was paying to play, a long way off being paid to play!)

I don’t want to deny anyone their dreams and I wish those two young men I mentioned above the very best of luck – it would cheer me enormously to cheer them on a professional football pitch – but I can’t help but worry that, because of choices that are being made for them by their clubs, doors are being closed to them, when education should be about opening as many doors as possible.

]]>Lawro – Mark Lawrenson – former player, football pundit and expert took a bit of a battering today, with a website castigating his weekly predictions of upcoming football fixtures:

## The table based on Mark Lawrenson’s predictions is actually ridiculous

Reading the article, it goes on to rubbish Mr Lawrenson’s weekley predictions for Premier League fixtures. Joe.co.uk has turned his predictions into a league table and the Give Me Sport article goes on to say

we’ve discovered how the Premier League table would look going into the final game week if all of Lawrenson’s predictions had come true.

To say it’s ridiculous would be an understatement.

Firstly, his former club Liverpool sit top on 89 points having not lost a single game. Not biased at all…

Also, Lawrenson clearly still rates Leicester with them sitting seventh with 71 points! After the Foxes, there’s then a bizarre 22-point gap to Everton in eighth.

Despite Sunderland being rock-bottom with 24 points in real-life, Lawrenson’s predictions has them in the dizzy heights of 14th on 34 points.

Later on the article describes Lawrenson’s predictions as

embarrassing

Harsh words, so do the numbers back up such criticism?

In a word, no.

Using Spearman’s Rank Correlation Coefficient we actually discover that Mark Lawrenson has done rather well.

Spearman’s Rank is a great tool to compare how closely two different variables compare. It is perfect for a case like this where we want to compare a predicted league rank with the actual league rank.

It is based on the difference between the square of the two different rankings.

For example, Lawrenson predictions would have Bournemouth in 13th place, in reality they are in 10th place, so the difference is 3, the square of the difference is 9 (we square the values so we don’t have to worry about negative numbers). This is the formula used to calculate Spearman’s Rank Correlation Coefficient:

where d is the difference and n is the number of pairs of data (in this case 20 as there are 20 teams we’ve compared.)

The value will always be between -1 and 1, where 1 tells us there is perfect agreement in ranking.

So how did he do?

Pretty well – he ended up with a coefficient of 0.890 (to 3 decimal places) which suggests that there is a strong positive correlation between his predictions and the actual league positions. In other words, the boy’s done good.

So ignore the sensationalist headline of the Give Me Sport piece and let the numbers do the talking. Mark Lawrenson has been proven, mathematically, to be a pretty good pundit and predictor.

]]>Regular followers of my blog know that I like my sport and some of you are aware that I am a qualified football referee and regularly ply my (alternate) trade as the “man in black.”

With my referee’s hat on, I am an active member of a forum that discusses the finer points of the offside law and whether or not any particular offence as seen on Match of the Day warranted a red or yellow card. All part of the learning process, everyday is a school day, and all that.

Today someone posted a problem that was a little left field for the average referee discussion – what is the area of the “D” on the edge of the penalty area.

What a great question!

Lets fill you in with a few dimensions. From the goal line to the edge of the penalty area is 18 yards. The penalty spot is 12 yards from the goal line. When a penalty is being taken all players (other than the penalty taker and the goal keeper) must be both outside the penalty area and at least 10 yards from the penalty spot. Hence the “D” on the edge of the penalty area – it’s arc marks the points that are 10 yards from the penalty spot.

Its not a simple question, but one that should be within the grasp of of a “good” GCSE student. It’s a problem involves trigonometry and Pythagoras, sectors and segments; the area of a triangle and fractions …

I got the answer to be 44.73 square yards (to two decimal places), do you?

If you’re not sure how to approach the problem you can see my back of a yellow card working below.

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