Adding and Subtracting Fractions

Subtracting Fractions

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 3/1/ has it’s merits, but I would be reaching for my aforementioned fancy calculator to do the subtraction at the top of the page: 8/3/ 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/b/, my working is below.

Subtracting Algebraic Fractions

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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Epiphany on a yoga mat

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.

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Holy Venn Diagram

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.

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9 – 1 Grade Boundaries

9 - 1 IGCSE Maths Grade Boundaries

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)

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Maths Fail #5

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



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