At Sixes and Sevens: How to Understand Numbers and Make Maths Easy

By Rachel Riley and Dr Gareth Moore

An engaging, accessible introduction into how numbers work and why we shouldn’t be afraid of them, from maths expert Rachel Riley.

Do you know your fractions from your percentages? Your adjacent to your hypotenuse? And who really knows how to do long division, anyway?

Puzzled already? Don’t blame you…

But fret not! You won’t be At Sixes and Sevens for long. In this brilliant, well-rounded guide, Countdown's Rachel Riley will take you back to the very basics, allow you to revisit what you learnt at school (and may have promptly forgotten, *ahem*), build your understanding of maths from the get-go and provide you with the essential toolkit to gain confidence in your numerical abilities.

Discover how to divide and conquer, make your decimal debut, become a pythagoras professional and so much more with these easy-to-learn tips and tricks. Packed full of working examples, fool-proof methods, quirky trivia and brainteasers to try from puzzle-pro Dr Gareth Moore, this book is an absolute must-read for anyone and everyone who ever thought maths was ‘above’ them. Because the truth is: you can do it. What’s more, it can be pretty fun too!

• Language: English
• Publisher: HarperCollins UK
• Release date: Oct 28, 2021
• ISBN: 9780008491086

Book preview

INTRODUCING YOUR

NEW FRIEND – MATHS!

This book is for all the people who were ever told they were bad at maths. It’s for parents who want to be able to help their kids with their schoolwork but don’t know how. It’s for anyone whose childhood teachers (almost) turned them off numbers for life. And it’s for everyone who wants to get better but just needs a friendly, helping hand.

I know I’m very lucky that my experiences with maths have always been good ones. As my Mum often tells me, my favourite toy as a four-year old was a primitive-style computer that I’d carry around and do sums on. These sums were part of a little game it had programmed in, where you raced against the clock to answer simple questions. You’d get praise and rewards when you got things right, and I really liked getting praise and rewards! I played it over and over, and the more I played, the better I got. The better I got, the more I wanted to play.

This was a very beneficial cycle and the start of a beautiful relationship. As the years would pass, I’d win maths prizes in competitions and get top marks in school, and in general it all came quite naturally to me. I didn’t have to try too hard to do well, so the effort/reward ratio was brilliant.

If you feel like slapping me in the face with a wet fish after reading that, I get it. I realise this isn’t the experience that most people have when it comes to maths. And as much as the praise and good marks that I got gave me encouragement and confidence in my maths skills, likewise, being told you’re no good and getting only red crosses through your work at an early age probably won’t give you the self-belief to aspire to become the next Einstein.

It’s a much-propagated myth that Einstein too was written off as a child. He wasn’t. Einstein was a maths prodigy. But in the case of Sir Roger Penrose, an award-winning long-term working partner and friend of Stephen Hawking, he was written off at a young age by one of his maths teachers. They moved him down a class for being too slow and always getting lost in lessons.

He’s since become an Emeritus Professor of Mathematics at Oxford University, he’s used maths to prove the existence of black holes and, among his large collection of prestigious awards and accolades (including a knighthood), in 2020, aged 89, he was the joint winner of the Nobel Prize for Physics. So never let anyone tell you you’re no good at doing something, especially when it comes to maths!

Positivity and self-belief that you can do maths are vital for you to be able to make progress; if you often find yourself thinking, ‘I just can’t do it’, then the first piece of homework handed down to you by Miss Riley, is to add a ‘yet’ to the end of that thought. Any time it creeps in, your new mantra is, ‘I just can’t do it, yet.’

I have a friend who considered herself a linguist (no, not Susie Dent), such that she’d written herself off in the maths department as ‘not being wired that way’. This is despite the fact she got an A in her maths GCSE and was far more competent than she gave herself credit for. She had what I call a ‘maths face’, and whenever she was asked to do something remotely related to maths or numbers, she’d grimace, break out in a cold sweat and give up before even trying.

Back in the days when VAT was 17.5%, she ashamedly confessed to me that she didn’t know how to work out what it would be for different values, which was something she was supposed to be doing at work. She then went on to explain that if she did have to do it, she’d take 10% of the price, then add half of that for another 5%, then half of that again for the final 2.5%, adding them all together to get 17.5%. What she’d done was tell me she couldn’t do something, then given me a perfectly good mathematical way of doing that exact thing. Her problem wasn’t her maths skills, it was her own confidence in them!

What my friend couldn’t do, was remember the formal method for working out percentages that she was taught in school. But she had enough understanding of how percentages work to be able to come up with her own method – and that’s exactly the type of skills that I want to help you to gain from reading this book.

One of the reasons that I always liked maths so much was because you didn’t really have to remember anything. I realised I was one of a minority who thought this way, as my friends were all busy panicking with last-minute cramming before our maths GCSE, while I was chilled out and listening to a World Cup match on my mini portable radio. What was there to cram?

Once you understand why things are in maths, then you never have to worry about remembering anything, as all the knowledge you need to be able to tackle a problem, is stored in your mind, not as a rigid formula or strict set of rules to blindly follow, but more as a way of thinking.

It’s a common thing to hear people declare they’re ‘bad at maths’, often as a badge of honour and shrugged off as inconsequential, in a way you’d never hear anyone say that they couldn’t read. I’ve heard this from all sorts of people, from teachers, to journalists, to politicians. I’d wager a lot of the time this isn’t even true, it may be self-deprecating or another case of selling themselves short in their own minds, but either way, I don’t like it when I hear it.

It’s said in a way that suggests it’s entirely out of their control, so why should they even bother trying? And when kids hear it, they pick up on this messaging.

The cold hard facts of the data show that maths abilities play a crucial role in life outcomes. I’m a strong supporter of the adult numeracy charity National Numeracy and, if you go on their website, you’ll read among other things that poor numeracy skills are linked with truancy, crime, and even physical and mental health problems. On the flip side, good numeracy skills correlate with higher levels of employment and earning potentials, better management of personal finances and higher levels of self-esteem.

I don’t say this to shame anyone who’s ever flippantly said they’re rubbish at maths, more to make them aware that they can improve their maths skills if they choose to, and that there is good reason to want to, so they should try to be more careful with the messaging they’re unwittingly passing down to kids the next time the topic crops up.

And it’s not just the kids who subconsciously absorb unhelpful ideas when it comes to maths. We’re all susceptible to them, and there are so many myths when it comes to this subject that if I had a pound for every time I’d heard one, even I couldn’t count up how much I’d have earned!

Lots of these myths revolve around gender stereotypes and ‘being wired for maths’ or about having (or not having) a fabled ‘maths brain’. When you dig down into them, as (often female) researchers have relatively recently started to do, you frequently find flaws in old studies that were said to ‘prove’ that men were more competent than women. And, actually, you can eradicate any difference in performance by sex, just by using some simple psychology and by tweaking your language somewhat.

One piece of research that fascinated me on this topic was published in 2016 by Margaret Tarampi, Nahal Heydari and Mary Hegarty. They gave 135 college students a test at visualising objects from different perspectives. In some cases they pitched their experiment as a spatial awareness test, explicitly stating within it that men naturally outperform women in this area. In others, students had the test explained to them as a social intelligence test, sometimes replacing the object in question with another person. This time it was explicitly stated that women tend to perform better than men at these types of social tasks. In the first group, the men outperformed the women. But in the second, the attainment gap between the sexes disappeared and the women performed much better.

Both formats of this experiment tested the same underlying concepts, but the latter was explained using a different approach and different language, and the contrasting ideas about gender stereotypes that were put in participants’ minds as they were performing these tasks had a significant effect on the women.

Other experiments into gender differences, particularly involving spatial reasoning in maths, have produced similar results. Data tells us that girls are more likely than boys to conform to gender stereotypes, so when we’re reminded that we’re female, and we’re not supposed to do as well as men in these tests, we’re fully compliant and we don’t! When different ideas are implanted, like getting us to imagine ourselves as men, or giving us information that positively reinforces female stereotypes in our minds, then our performance itself improves instantly.

This is one of the reasons I’m so keen to promote maths positivity and to choose our words carefully – the ideas we form about ourselves and others in relation to our maths abilities have a direct effect on how good those abilities will be. A simple change in psychology to a more optimistic and open mindset can have such a difference to what you can achieve, man or woman!

If you’ve been nervous or scared of maths up until now, let’s nip that in the bud. I want you to find pride and confidence in yourself every time you try something new, and every time a lightbulb turns on for you and you finally get whatever it is that you’ve been struggling with for so long. Don’t worry about how fast or slow you feel you’re progressing, it’s not a race, just take your time.

If learning maths is like building a tower of knowledge, then the fundamentals are the foundations on which to build from at the bottom, and more complex stuff gets added on as you build higher. If anything’s not that sturdy to start with, the higher up you go, the more wobbly the whole thing will get.

If you whizz through maths remembering formulae by rote and not really understanding what you are doing or why, then eventually it’ll catch up with you: you’ll forget bits, and the tower will end up more like a game of Jenga, with missing blocks here and there that you won’t want to go near for fear of it all crashing down.

This book will start with the very basics, so we can build upon a solid foundation of understanding that’s much more weatherproof. I’ll try to explain things in more than one way, so you can pick the one that’s right for you and stick with that. There may be parts of maths and parts of this book that seem trickier to you than others and there may be bits you just don’t get, and that’s fine. Don’t lose heart but do keep going. Just because one branch of a tree doesn’t have much fruit on it, doesn’t mean the rest can’t be covered in the stuff.

Speaking from my own experience, I can relate to not loving all of maths. There are two distinct types of mathematician – the pure mathematicians, who will marvel at the beauty of an equation and get excited by the most abstract of concepts, and the applied mathematicians, who prefer the kind of maths you can use in the real world to describe physical systems, or predict what might happen in the future.

I am firmly of the second camp. A short 14 years after dragging my 80s kids computer around the family home, in 2004 I went to Oriel College, Oxford to start my undergraduate Master’s degree in maths. As an Essex girl who’d never even met anyone who’d been to Oxford, I never imagined I’d get an interview, let alone get in, but I loved all of my four years there. The work got more and more interesting as the years went on and, among other things, I picked courses that involved modelling the human body, explaining climate change, exploring the way waves move and describing how particles behave on the tiniest of scales. But before I could do the fun stuff (yes, really!), I had to serve my time getting a grounding across everything, including from pure maths.

It was not for me, let me tell you! To this day I have never marvelled at the beauty of an equation and I have very little interest in how many digits of pi you can reel off, so I won’t be offended if certain sections of this book don’t float your boat either. If you don’t like stats, move on to algebra. If you don’t like algebra, well try my algebra section and see if I can change your mind (I love the stuff!). You can always come back to topics later on, so don’t fret about skipping a bit on occasion if you feel like it.

Just don’t fall into the trap of being scared of trying something because you think it sounds difficult, or because you’ve had trouble with it before, you think you still won’t be able to do it now either. Remember, we’re starting again with our positive, can-do attitude and I’m sure that now you have the right motivation to improve and maybe a less scary teacher/guide, you’ll even surprise yourself with what you’ll be able to nail down.

This book is designed to increase in complexity as we work through it, building our knowledge up as we go, but equally many parts of it stand alone (or at least very nearly do), so if there’s one topic you’re desperate to dive into in a later chapter, be my guest!

There are problems and puzzles in every section, to give you some practice and to help you get ideas straight in your mind, as doing maths is far better to develop your understanding than reading maths, so use my notes as a guide when trying to work things out for yourself. It’s hard to cater for all abilities at once but I’ve tried to be as thorough as possible with explanations to leave no stone unturned. For some of you it might be overkill, in which case plough on! For others, take your time and take from whichever parts of the explanations speak to you best.

I won’t be testing you with exam questions at the end of the book, but if you’ve bought this to be able to better help your kids with schoolwork, then they might well do that job for me when the time comes. If you understand the concepts here, you should be able to translate that knowledge into almost anything they throw at you up to GCSE level. I’ve left out some things that are as easily dealt with by Google, and tried to focus on mathematical concepts and ideas that I’m most asked for help with and are probably most misunderstood.

I’m a genuine, bona-fide puzzle geek at heart and I really do enjoy solving maths problems – I get a buzz from working out the answers! And 13 years into my job on Countdown, my favourite bit is still the 30 seconds when my mind is solely focused on getting to that target number. I find doing puzzles and maths problems relaxing, as I have to be totally engaged with what I’m doing. I’m entirely in the moment and at that point nothing else enters my mind.

So, when friends, family or people on social media send me puzzles from the web or questions from their kid’s homework they need a hand with, I’m happily at their service! I’ve noticed that since the dreaded pandemic hit and homeschooling kicked in, the level of questions that parents were asking for help with, seemed to be of a higher standard of difficulty and from a broader range of topics. Probably, for the first time in a long time, these parents were having to revisit the style of maths that they did at school and, however many years later, sometimes they simply didn’t know how to help.

So, it was those parents, and others like them quietly struggling by themselves, who were my inspiration for writing this book. And even if that’s not the reason that’s brought you to reading this, whatever your motivation, you wouldn’t be here if you didn’t want to improve your maths skills, and to do so you’ve come to the right place.

Maths at school was sometimes dry, sometimes dull, and sometimes the only reason your teacher gave you for why you were doing it was because they said so. It was in the curriculum so that was that. But it doesn’t need to be that way!

In the same way that when I was at school, the vocab we were taught in French class – exercise book, pencil case, rubber – is not particularly useful to me now, much of the stuff in maths books aimed at kids won’t be pitched quite right when revisiting maths as an adult.

When learning a language as an adult, you’re more likely to want to talk about business trips, arranging a meeting, or going to a bar – the vocab that will be most useful to you will have changed, and it’s no different in maths. This book will cover the same type of topics that you did in school, but is designed to be used by those who aren’t.

Maths can be fun, maths can give you fulfilment, and maths can be useful in the real world in ways you may never have realised before. By the end of this book, I want you to have confidence that you are a numbers person, I want you to feel the buzz of understanding what you’re doing and getting answers right, and most of all I want you to enjoy the process of learning again.

When your friends send you a puzzle that’s ‘dividing the internet’, or your kids ask you how to solve an equation, you’ll be far better placed to tell them and share your knowledge around.

And, after all of that, if you fancy trying out your new skills and angling to win yourself a Countdown teapot, you’ll have all the tips of the trade and you’ll know where to find me!

CHAPTER 1

TOOLS OF THE TRADE

ABSOLUTELY MENTAL ARITHMETIC

When most people think of maths, the first thing they think of is numbers. So, if we wanted to go all Julie Andrews à la The Sound of Music, and start at the very beginning, with the ABCs or do-re-mi’s of maths, we begin, of course, with one, two, three.

It would have been fitting for Julie to have sung about these counting numbers, as she did in the film, surrounded by the giant fir trees, babbling brooks and female deer of the Austrian Alps, as they are known officially as the ‘natural’ numbers, precisely because they appear in nature. So, you’ll have one dog, two flowers, three bees, four bottles of wine, etc – and these are all known as natural, counting or cardinal numbers.

Numbers have many other inherently natural properties too, and display clues that tell us about themselves just by looking at them. We use these different properties to separate numbers into different groups – such as into odd and even numbers, or into primes and non-primes.

Unlike almost everything else in life, numbers are unlimited. They are not something you can ever run out of. They go on forever, since you can just keep on adding 1 to make them bigger and bigger, until you get bored or fall asleep. In fact, even the maths symbol for infinity, a theoretical number that is always bigger than any other number, looks like an 8 that has given up the will and passed out under the tedium: ∞.

THE POWER OF PRACTICE

Everyone can add 1 to a number and quickly tell you the answer. But what about adding 10? Or 100? Or 5000?

When it comes to mental arithmetic it’s all about practice. When applying for my job on Countdown, I literally had to relearn my times tables from scratch. I’d just finished four years studying maths at Oxford, in which we’d barely seen numbers in lieu of algebra and Greek notation. In fact, numbers took on such a supporting role that my college friend once asked what that funny symbol on the board was that looked like a backwards epsilon (ε)? Reader, it was a 3.

Also, whereas practice might not always make perfect, when it comes to mental arithmetic, practice certainly does improve your confidence and speed of thought. And then, once you feel confident manipulating numbers at will, you can use those skills to solve real mathematical problems – rather than the mental arithmetic itself being the mathematical problem.

You should treat your brain like a muscle, in that exercising it will keep it in shape, so use the numbers that pop up all around you as your training tools. When you’re out shopping you can try and decide how much things are going to cost in your head before it’s worked out for you. When you’re hot and sweaty, staring down at the display on a treadmill, or bored at work and are digital clock-watching, or just walking down a street past all the shiny door numbers, use those digits to practise your tables, or add them up, or work out which numbers divide into them. Spend a bit of time thinking about that number and work out what you know about it just by looking, as per some tips coming later in this chapter. Not only will it help your maths skills, but it will also keep your brain active and make you a bit sharper to boot. If you have a family, get your children involved in practising along with you too!

IT ALL ADDS UP

Let’s say you want to add a couple of numbers together, such as 28 + 37. What’s the best way to do that? A sum like this isn’t something most people can do without a bit of thought.

To find the easiest method of doing this mentally, it helps to think literally about what the numbers actually mean. The number 28 means one plus one plus one plus one … with 28 ones added up. Likewise, 37 means 37 ones added up, so 28 + 37 is whatever 28 + 37 is, number of ones all added together. And the thing about addition is that you can add numbers in any order you like, and it will still give the same result, i.e. 28 + 37 = 37 + 28 = any grouping of the 37 and 28 ones you like, added in any order. So, why not regroup the ones you’re adding up to make an equivalent addition, with numbers that are more friendly to think about in your head.

For example, you can take 2 off 37 to move it over to the 28, to leave the much more manageable sum of 30 + 35.

Working with round numbers reduces the amount of brainpower required, which suits mathematicians perfectly – we are essentially the laziest group of people I know. The aim of the game is to do the minimum amount of work possible – the less thinking, the better!

Additions such as this can be made much easier, due to the commutative nature of addition. This is the fancy way of saying that you can put the numbers in any order and you still get the same result.

This is probably something you will have done with wooden blocks when you were a toddler, without realising that you were already thinking like a mathematician. If you had 10 blocks, for example, it was easy to see that 10 could be formed by adding 5 blocks + 5 blocks, or by adding 1 block 10 times. At school, the same concept is known in terms of ‘number bonds’, so children become familiar with breaking up numbers into multiple smaller numbers that add up to the same value – so, 2 and 8 are number bonds that make 10. There’s also the idea of ‘chunking’, which is when you divide a bigger number up into smaller, easier parts, such as breaking 56 into 50 + 6.

It’s always worth thinking about what the numbers on the page actually represent, since you can shift them around however you like to make it easier in your mind. You don’t have to go straight for adding two awkward numbers together.

In fact, I’m often asked how I ‘see’ numbers myself when I’m doing calculations, with some people expecting a kind of magical answer in terms of colours flying around or perhaps hearing numbers as notes on a musical scale, but the best way I’d describe how I ‘see’ sums in my head is like having jigsaw puzzle pieces that need to be fitted together. Smooth, regular pieces such as squares fit together much more easily than awkward shapes with lumps removed and blobs sticking out. So, I try to change sums to give me the simplest numbers to work with before I start, to speed things up and save myself as much work as possible.

LET’S GET NEGATIVE

Just as the natural numbers