Chapters
- 01. Maths Is Everywhere
- 02. The Advantages of Fast Maths
- 03. Why Should I Learn How to do Fast Maths?
- 04. Are There Tricks for Fast Calculation?
- 05. Learn How To Calculate Quickly With Maths Tuition
- 06. Want to be the next Einstein? See these Fast Maths Tips
- 07. Further Advice on How to Improve your Calculating Skills
- 08. Using Games as Tricks For Fast Calculation
- 09. The Mental Arithmetic World Champions
From the second you start learning mathematics and taking maths tutorials, it requires a lot of concentration, persistence, and the ability to do basic mental arithmetic.
The work only gets harder by the time you do your SATs or GCSE maths revision at secondary school, A Levels at sixth form, and go on to university. Working out the square root, algebra, trigonometry and mental maths can be really tricky. But we have plenty of tips to help with maths.
Maths Is Everywhere
Whether you intend to enter a profession directly related to the Maths sector or not, you will still benefit from having these skills because Maths is fundamentally everywhere. If you have a part-time job at your local shop, you will know how Maths can come into play in just about any profession; when your till lets you down or spreadsheet gets corrupted.
It’s also really useful to know how to do quick and effective mental arithmetic in our everyday lives, whether it’s for budgeting when doing the shopping, working out measurements on a DIY project, observing patterns when sewing, etc...
You can't get away from Maths, no matter how hard you try!
And while we all have a mobile phone, we don’t always have it to hand when we need it and, when we do, we waste a lot of time just looking for the “Calculator” app, and then getting fed up with the numerous ads that pop up whilst we are trying to type in our calculation quickly.
Even if we did consult our digital calculators at every opportunity, how can we be sure that they are right? We have become so reliant on technology these days that it has led us to feel helpless when it fails us. Behind all the computers, there is always a human being checking that they are running as they should. While human error can and does happen, our conscience means that we are pretty reliable as beings.
Going back over the basics of mental arithmetic can help you get out a lot of trouble during your exams. However, it's not just for when you’re doing your A Levels. It can also help you make the most of other situations, such as during maths tutorials with a maths tutor.
So, how do you do it? How do you get better at maths? Are there exercises, techniques, or ways to improve your maths level?
The Advantages of Fast Maths
Whether you're a student or not, mental arithmetic has a lot of considerable advantages in our daily lives.
It allows you to work out figures quickly in any situation and, even better, means you don’t need to get your calculator out.
Put simply, you can learn to count quickly!
For anyone who wants to improve their level of maths, mental arithmetic allows you to work better with numbers and familiarise yourself with certain operations and their properties.
It may seem a little far-fetched but improving your mental arithmetic can also help to make you seem smarter (of course, brushing up on your maths skills does literally make you more intelligent!). It's more than just a bit annoying when we are put on the spot trying to split a bill in a restaurant, it's actually embarrassing. Even worse if the waiter or waitress is waiting with the card reader wanting to know how much you are paying on each card! Then working out how much tip to leave... what a nightmare!
Think of Carol Vorderman and Rachel Riley. The ability to add up quickly is no longer seen as something to shy away from. Men and women alike find strong mathematical skills sexy. So, all the more reason to dive into those numbers!
Get a maths tutor here on Superprof.
If It's All In Your Head, You Don't Need a Calculator
Mental arithmetic is used all the time in various situations: it allows you to predict a calculation’s order of magnitude, which could come in handy when studying decimals or proportionality, for example.
Find other mathematics classes online here on Superprof.
Finally, not forgetting the fact that mental arithmetic can sharpen your brain, change your way of thinking, and improve your memory, your analytical, and rational thinking. For the elderly and the young, practicing mental arithmetic can help to keep your brain functioning at its best as you age, enabling you to beat mental conditions head-on, like anxiety, panic attacks, or worse, Dementia or Alzheimer's.
There is a reason why sufferers of panic attacks are advised to breathe in for 4 and breathe out for 7, the combination of the two helps to settle your nerves and breathing in a way that fights off the growing panic attack or seizure.
Though it mightn’t seem it, it can also be a lot of fun: If you like doing “Magic Squares” or playing “Countdown”, mental arithmetic can help you get your neurons firing at full speed all while having fun, as well as doing maths without really noticing. And we all know that the best way of learning is whilst having fun!
Regardless of what you might think, you don't have to have an IQ over 100 to enjoy TV programmes like "Countdown". In fact, you can learn a lot by challenging yourself along with the contestants or by seeing how the pros do it.
Another sign that being good at Maths is now 'cool', is the rise in games like "Sudoku", and the popularity of the show '8 Out Of Ten Cats Does Countdown', which brings humour to the otherwise mathematical-based show.
Why Should I Learn How to do Fast Maths?
Reasons Why Mental Arithmetic Can Take You Far
There are several reasons for doing mental arithmetic and developing your ability to carry out calculations in your head. Here is an overview:
- Once this becomes second nature, it’ll free up your mind to focus on rational thinking,
You can save time and, in some cases, money (just ask either poker players or bankers!) - Travel light: you needn’t bring your heavy calculator with you. Fair enough, you’ve also got your smartphone but you won’t need to get it out to work out 8x7...
- Impress: Thanks to your training, the multiplication tables up to 11 and 12 pose no problem,
- You won’t get ripped off when it comes to money,
- You’ll need fewer maths tutorials!
Putting Things Into Perspective For You
Honing in on one of the above reasons for improving your mathematical skills, it is really important to know a bit about numbers if you don't want to risk getting ripped off.
Let's just say you are planning to go abroad and accidentally visit a non-legitimate currency exchange agency who takes your money in return for a fraction of what you should have received - it might have helped you to realise this error sooner if you could see the exchange rates on the board and work out roughly how much of the currency to expect.
Moreover, the idea of impressing others is more important than it sounds. Not only can a set of advanced Maths skills benefit you in your personal life - for instance, financial literacy is a vital skill that everyone needs.
Getting good at math, as we've mentioned, doesn't just serve to make you stand out at college or job interviews.
Is It True That Everyone Is Born A Mathematician?
According to statistics, there is a real lack in Mathematical skills among youngsters (said to be as a result of the way they are taught) and approximately 50% of adults can solve basic Maths problems. If you don't want to be subjected to this lack in crucial skills, then now is your time to work hard and to find ways of your own to teach yourself how to do Maths in a manner that suits you.
Some scientists truly believe that everyone has the ability to be an expert Mathematician. They argue that everybody has the ability to think logically but that we need to unlock that part of our brain. Have you ever heard of people having head injuries and then waking up speaking a language that was once completely foreign to them? Is it crazy to believe that we all have knowledge and skill in abundance just sitting behind a secured door in our brains, waiting to be unleashed?
If that is indeed the case, then training your brain with gentle mental arithmetic could be just the way to set out on the road to unlocking this logical part of your brain. As an example of it being totally possible to become a Maths genius later in life, take a look at this short article which tells the story of a man who became a Maths wiz after an accident, now seeing all aspects of his everyday life in shapes as if through a geometry lens.
This doesn't mean to say that you need to sustain an injury to wake these dormant skills. Increasing the activity in the left hemisphere of your brain could be all you need to trigger this change in the way you think.
Are There Tricks for Fast Calculation?
Now that you’ve got some good reasons to learn mental arithmetic, why are you hesitating?
Don't worry!
Learning to manipulate numbers is just a learned habit. I can assure anyone that thinks mathematical ability is innate that this is far from the truth.
If some people seem to have these abilities, it doesn’t mean that they were born with them. This is a skill they’ve learned. Or perhaps the skill is that they've learned how to let their logical brain develop.
Some scientists have been able to determine that mental arithmetic activates areas of the brain related to spatial awareness. The representation of numbers can, therefore, be compared to spatial awareness.
When we count, whether we add or subtract, it’s almost as if we move the numbers to one side or the other of a mental abacus to find the result. This is important to keep in mind when we find numeracy difficult.
Does Fast Calculation Come Easier When You're Young?
It’s also useful to develop mental arithmetic skills, in children especially, so that you can tackle various pedagogic approaches. Games using numbers and counters in a physical space stimulate the area of the brain concerned.
So make sure you play Ludo with your children, it’ll help them!
Still not convinced?
Other researchers focused on Rüdiger Gamm, a mental athlete, to understand how he manages to carry out such complex calculations in his head.
It emerged that, in this case, the mental calculator activates the frontal temporal brain regions generally concerned with long-term memory. He has a plethora of information at his disposal which helps him to break records in mental arithmetic.
Don’t despair! He was drawn to arithmetic late on at the age 20 and it’s thanks to daily training that he managed to develop his skills to such a level!
At any rate, you should remember that memory is your best ally when it comes to becoming good at mental arithmetic!
Learn How To Calculate Quickly With Maths Tuition
In any case, mental arithmetic practice has to be regular, around 10 minutes per day should be enough: your brain needs to develop reflex actions for this, repeating the same formulae over and over, the same calculations until they become obvious and natural, like riding a bike or simply breathing.
Mental arithmetic should be practiced orally and written down, using exercise books, maths worksheets, a slate, cards, dice, computer programmes, etc.
It’s important to keep a record of your mental arithmetic progress so you can see your mental development. This will help you out with maths problems and help you become a good maths student.
Having trouble finding a teacher near you? Try an online maths tutor.
Here are the basic tools to boost your mental arithmetic skills:
- Addition and multiplication tables,
- Knowledge of the complements of the number 10,
- Knowledge of squares up to 15² (225) as well as the powers of 2,
- Multiplying by powers of 10 with negative exponents (moving the decimal place to the left) and positive exponents (moving the decimal place to the right),
- Dividing by a number is the same as multiplying by its reciprocal,
- for example, divided by 0.25 is the same as multiplying by 4,
- Learning the special products: (a+b) ² = a²+2ab+b², (a-b) ² = a²-2ab+b², (a+b) (a-b) = a²-b²,
- Learning the rules of factorisation,
- Knowledge of the orders of magnitude for Pi (3.14159), the Golden Ratio (1.618), etc.
Want to be the next Einstein? See these Fast Maths Tips
Set Your Problem Out On Paper
Firstly, don't make life hard for yourself. No one expects you to be an extraordinary mathematician who can add, subtract, multiply or divide complicated numbers in your head, and you definitely won't reach a good level of mental arithmetic unless you take it easy.
Remember, you can write calculations down if you feel necessary. Some people find it easier to tackle a calculation or problem when they can see it on paper and therefore visualise it (or on a napkin, or on your hand... basically whatever you can find!). Others meanwhile, can retain a lot of information in their heads and can follow long calculations without the need to stop and reassess their workings.
Approximation Is Your Friend
Don't forget that approximation is also a good way to improve your mental arithmetic skills. By working out roughly what a figure should be, you can enter the process of working out with more confidence that you are on the right tracks. So, if you need to know what 60 x 120 is, think about multiplying the first number by 100 (a much simpler calculation to do there an then in an instant) and then work from there. At least you will know approximately how many digits the final number should have if nothing else!
Turn Numbers Into Shapes Or Objects
Moreover, try not to focus too much on the numbers. It's hard to get your head around, I know, but imagine the single digits as visual blocks or, for more advanced working out, think of the different parts of the calculations as building blocks. Even just allocating a picture in your head to the sets of numbers can change the whole way you approach that group of digits.
Looking beyond the numbers once more, consider using tips and tricks offered by others just like you (see below for some examples), or attempting to make your own fun techniques that work for you personally.
If all of these tricks aren't enough, then keep reading to find out how you can improve your mental arithmetic with these suggestions. Some you may never have heard of before and could be game-changers!
Look at the last digit
Whatever the number, look at how it ends.
If it’s even (0, 2, 4, 6, 8) or 5, it’ll be divisible by 2, 5, or 10.
For example, 22 ends with an even number so it’s divisible by 2, while 45 ends with a 5 and is divisible by 5.
150 ends with a 0 so it’s, therefore, divisible by 10.
A number divisible by 3 or 9? Add up the digits to find out if the number is divisible by 3 or 9.
In fact, a number is divisible by 3 if the sum of its digits is equal to a multiple of 3 (e.g.: 18 = 1 + 8 = 9, a multiple of 3).
A number is divisible by 9 if the sum of its digits is equal to a multiple of 9 (e.g.: 936, 9 + 3 + 6 = 18, which is 1 + 8 = 9, which is a multiple of 9).
It should be noted that, if the sum of the digits is divisible by 3 and an even number, it’s divisible by 6 too.
For Addition
Deconstruct the numbers to make things simpler.
For example, 72 + 29 is (70 + 2) + (20 + 9) = (70 + 20) + (2 + 9) = 90 + 11 = 101.
Or even: 13 + 48 is 13 + (50 - 2) = 63 - 2 = 61.
For Subtraction
Learn to simplify numbers.
For example: 1958 – 1907. The number 1900 is included in both numbers so just subtract the tens and units, 58 - 7 = 51.
The Japanese Method of Multiplication
Are you not ready to do complicated multiplication in your head, yet?
With the visual Japanese multiplication technique, you’ll see these operations much more clearly. All you have to do is draw lines and the result appears as if by magic.
Easy, right?
Once you’ve learned this technique, it’s much easier to carry out these operations in your head. In fact, by envisioning these lines in your head, you can visualise the result, without needing a sheet of paper to draw the lines.
Wisdom from the Land of the Rising Sun...
Butterfly Fractions
Not being able to solve fractions can give you a headache!
Adding or subtracting fractions requires a common denominator. To avoid this step, use the butterfly method. With a bit of training, you can do this in your head.
For example: 3/4 + 2/5
First do some cross multiplication: 3 x 5 =15 and 4 x 2 = 8
Then add the two results to get the final numerator, 15 + 8 = 23
To find the denominator, multiply the two denominators: 4 x 5 =20
Therefore: 3/4 + 2/5 = 23/20
You can also use this technique to subtract fractions.
How Do You Multiply by 11?
It couldn’t be simpler.
To work out 32 x 11, all you have to do is multiply 32 x 10, then add 32, so 320 + 32 = 352.
Do you know the second technique? It’s really simple for doing in your head.
For the same problem, 32 x 11, all you have to do is add the 2 digits from the number being multiplied by 11 and place them between the two numbers.
Therefore, 3 + 2 =5, putting the resulting 5 between the 3 and 2 and we get: 352!
If the addition results in a number with two digits like multiplying 56 x 11, the solution is simple. 5 + 6 = 11, we place the second 1 between the 5 and 6 and we add the second 1 to the 5. We, therefore, get: 616.
Further Advice on How to Improve your Calculating Skills
- Wisely group numbers together before working anything out. For example, 1.2 / 1.8 = 12 / 18 = 120 / 180,
- Put units that make 10 together in order to make your additions easier (1 + 9, 2 + 8, 3 + 7, etc.),
- Adding or subtracting by 9, 19, or 29. Just add or subtract by 10, 20, or 30, before adding or subtracting 1,
- Dividing by a number is the same as multiplying by its reciprocal,
- To add 2 fractions, make sure they have the same denominator.
Follow these maths tips to the letter!
Online Resources for Working on Your Mental Arithmetic
To get better at maths and mental arithmetic, the internet is an excellent resource that complements your tutorials with a maths tutor.
There are plenty of websites that have quizzes, texts, and exercises. Here are a select few:
- Khan Academy: This website features video lessons as well as practice questions to help you learn.
- BBC Bitesize: The BBC provides resources for students of all ages. Great school-aged children.
- TES: Resources for teachers in a variety of subject, including maths.
When you learn while having fun, you don’t even notice you’re learning.
We’ve realised that if you want to get good at maths, and mental arithmetic, in particular, memorising is hugely important: remember that you have to learn the principles off by heart and know your addition and multiplication tables. Do this repeatedly, again and again.
Train your brain to remember, everything will become second nature, and your ability in mental arithmetic will increase tenfold.
You have to train, using different strategies, in order to make these things come naturally to you.
This, of course, requires time and a personal commitment but the result is worth the cost because you’ll have these reflexes for the rest of your life.
Using Games as Tricks For Fast Calculation
Struggling maths students at school are a common issue. A lack of mental arithmetic can negatively impact students’ futures. A study carried out in 2014 highlighted that 40% of students at primary school have problems with maths.
It was also shown that these problems at primary school included multiplication and division tables, decimals, and representing large numbers.
Given these findings, new programmes tend to have a focus on mental arithmetic. After learning subtraction, addition, and multiplication tables, mental arithmetic also includes:
- Establishing reflexes,
- Refining logical thinking,
- Getting used to working with numbers,
- Learning the properties of numbers.
To help children get better at mental arithmetic, you should ensure that this practice is fun. Let them learn while they’re having fun! This is a great way to encourage them to enjoy numeracy.
Keep the learning fun
Learning has to be progressive. Games are a good way to learn to calculate without being subjected to the stricter aspects of tutorials with a maths tutor.
The best thing about this is that, now that we access to so much technology, playing games to enhance logical thinking is so easy.
Downloading apps on our mobile phones is like second nature to us, so why not use this to your advantage and download some of the amazing free games at your disposal. All you need to do is search 'Maths games' in your app finder to find a massive list of suitable games!
Otherwise, why not try out some of the websites and applications suggested by Tech Advisor, which has listed the best mathematical apps for children.
As a parent, you may be reluctant to let your child have a mobile phone at too young an age. But, why not tell them that they can have a phone if they use it for school work and revision? Or, introduce some quiet time at the same time each day whereby you allow your little ones to play games on a tablet for 20 minutes (and you don't even have to tell them that this is for educational purposes - they'll never know!).
The Mental Arithmetic World Champions
Even if such mathematical genius is rare, I still have to tell you about some of the best performances from the Mental Calculation World Cup Champions...
These sorts of performances show to what extent humans can train their brain and improve their reasoning and memory. These talented mathematicians are living proof.
What have they got in common? A high IQ, of course, but mainly incredible mental arithmetic records!
Rüdiger Gamm
This talented calculator only became interested in maths at the age of 20. Even though he has a good memory for numbers, he wasn’t very good at maths in school. As soon as he became interested in mental arithmetic, he memorised figures and techniques for specific calculations. He can also raise 2-digit numbers to the power of 15, divide 2 prime numbers, and carry out operations with complex numbers in just a few seconds.
Gert Mittring
Another incredible example! In 2016, Gert Mittring worked out the 89,247th root of a number in 6 minutes 1.4 seconds. This resulted in over a million digits over 154 pages. Pretty impressive, right? You can see why this maths-mad German psychologist has already 11 world champion titles.
Alexis Lemaire
France also has its own mental athlete star. Alexis Lemaire. In 2007, he worked out the 13th root of a 200-digit number chosen randomly by a computer in 72 seconds. To get this you have to multiply this number by itself 13 times. With this taking just over a minute, it’s safe to say this isn’t something everyone can do!
The world of mental arithmetic is really fascinating! So, as you can see, the human brain is capable of incredible things. You’ve got no excuse for not learning those times tables!
Have fun, get training, use all the resources available online and in print, challenge yourself (or ask someone to challenge you), but always train and revise regularly. This is the price you have to pay if you want to get good at mental arithmetic.
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