Minimal skills of counting, sense of number – the same element of human culture as literate writing and speech, mastering a foreign language, a basic understanding of art and the world around it.
In addition, when you easily count without any improvised means, you feel a completely different level of reality management – you know in advance how much change you will get in the store or whether it is worth stuffing everything in the elevator with a load capacity of 400 kilograms.
Think about the fact that the calculator and the actions in the column – it’s the same kind of magic. You probably don’t understand how it works, and you just have to trust them. And when you have a good understanding of how mathematical operations are arranged and can reproduce them with your “hands”, your sense of control (and self-confidence) gets a serious bonus.
Finally, oral counting develops your mental abilities: attention, memory, concentration, switching between several streams of thought, and can also serve as a tool for meditation or distraction from sad thoughts.
But where do you get training assignments? Come up with your own examples?
Of course not. The network is full of mobile applications that will offer you training of mathematical skills for every taste.
When choosing an app, please note that a good app, at least, should have enough flexible settings of complexity and keep statistics of tasks solved by you.
And how exactly should we train?
There are only four basic mathematical actions – addition, subtraction, multiplication and division. Each action has its own characteristics, but they are not complex. You have to figure it out once, and then practice for 5-10 minutes a day, and very soon you will feel that you think better. Most likely, in two or three months you will reach a decent enough level, which can be supported by episodic training.
So where do we start?
Start with the simplest level – adding up unambiguous numbers, and bring it to perfection: 99% of correct answers, for each answer 1-2 seconds. To solve the “jumping to 10” examples, try the following technique – “Relying on a decade”.
Let’s say you need to add up 8 and 7.
1) Ask yourself how many number 8 is missing up to 10 (this is 2).
2) Think of 7 as the sum of 2 and some second piece (this is 5).
3) Add to 8 first the part of number 7 that was missing up to 10, and then that second piece – you get 10 and 5, and that’s 15.
How do you add multi-digit numbers?
The most important principle here is to add up the same discharges with each other. Breaking down both numbers into “digit parts”, begin to add up with the senior digits – thousands with thousands, hundreds with hundreds, dozens with dozens, units with units. What you get, if necessary, enlarge and count all together again.
For example, how do you add up 456 and 789?
1) The 456 is made up of three digit parts – 400, 50 and 6.
789 is also broken down into three digit parts – 700, 80 and 9.
2) Add up hundreds with hundreds: 400+700 = 1100, dozens with dozens: 50+80 = 130, units with units: 6+9 = 15.
3) We enlarge, dividing into convenient parts, again group and add the same bits: 1100+130+15 is 1100+100+30+10+5, that is, 1200+40+5 = 1245.
What about the subtraction?
And here it is necessary to start from the basic level – subtracting a single-digit number from the numbers of the first and second ten – and bring this skill to perfection. As in the case of addition, problems usually arise with the subtraction “with passing through 10”. And here the similar method of “reliance on the ten” will help.
Let’s say we need to subtract 8 out of 12.
1) Let’s ask ourselves how much we need to subtract from 12 to get 10 (this is 2).
2) If we subtract 8 out of 12, we subtract that 2 first and then everything else. And the rest is how much? (that’s 6).
3) After subtracting 2 out of 12, we get 10, and we have to subtract 6 more, we get 4. That’s it!
What about the multi-digit numbers? Are they complicated?
Не особенно. Важно только не путать технику вычитания с техникой сложения. При сложении нам было удобно разбивать каждое из чисел на разрядные части, а здесь мы разбиваем только то число, которое вычитаем.
Итак, допустим, нам нужно вычесть 512−259.
1) Число 259, которое мы вычитаем, состоит из трех разрядных частей — 200, 50 и 9. Их-то по очереди мы и вычтем.
2) 512−200 — вычитание сотен никак не затрагивает десятков и единиц числа 512, влияет только на сотни, так что результат будет такой — 312.
3) Из того, что получилось после вычитания сотен, теперь вычтем десятки, 312−50.
Это похоже на вычитание через десяток. Вычтем из 312 сначала 10 до целых сотен (единицы не будут затронуты), получим 302. А потом вычтем все остальное (всего нужно было вычесть 50, 10 уже вычли, осталось вычесть 40), получается 262.
4) Осталось вычесть единицы: 262−9.
Чистый переход через десяток — вычитаем сначала 2, получим 260, а потом вычитаем остальную часть, 7, получаем 260−7 = 253. Вот и ответ.
How does multiplication work?
We’ll start by multiplying unambiguous numbers. The first thing to remember is that multiplication is when you add up the same number several times. For example, multiplying 4 by 7 means adding up four sevens. Using the addition technique, we can easily calculate – two sevens, 7 and 7, will be 14, if you add a third 7, you get 21, and by adding the last, fourth seven, the result is 28.
Gradually, as a result of training, you will memorize the multiplication reference values that are convenient for you and with their help you will be able to calculate neighboring values faster. For example, if you need to multiply 6 by 7 (that is, add six sevens), and you remember that 5 times 7 (that is, add five sevens) will be 35, then to get the final result, you just need to add the sixth seven – you get 42.
The most complex example in the multiplication table is 7∙8. To memorize it, there is a good mnemotechnical rule “five six seven eight”, which means 56 = 7∙8.
How do I multiply a multi-digit number by a single digit?
Let’s take an example. Let’s say we need to multiply 468 by 6.
1) 468 consists of 400, 60 and 8, all of which should be multiplied by 6. Well, separately, these tasks are as easy as multiplying unambiguous numbers.
2) Let’s go from senior to junior: 400∙6 = 2400 (since 400 is 100 times more than 4, the result 400∙6 will be 100 times more than the result 4∙6).
Accordingly, 60∙6 = 360, and 8∙6 = 48.
3) And now, as when adding up, we put it all together, grouping the same bits:
(2000+400)+(300+60)+(40+8) = [regroup] =
= 2000+(400+300)+(60+40)+8 = [add equal digits] =
= 2000+700+100+8 = [group and add the same bits] =
= 2000+800+8 = [nothing more to enlarge, we get the answer] = 2808.
How do you multiply two-digit numbers?
For an ordinary man it’s already aerobatics! If you have mastered multiplication of double digits, consider that you are accepted into the world of the elite of oral counting. But in fact, and there is nothing fundamentally difficult, just more load on short-term memory (at the same time and practice it).
So, for example, let’s multiply 78 by 56. It means that we need to add the number 78 by 56 times.
1) These 56 times can be broken down into stages – first we add 78 50 times, then 6 times and then combine the results.
2) The number 78 can be summed up 50 times – that’s 10 times more than sum it up 5 times. 78∙5 = 70∙5+8∙5 = 350+40 = 390. Which means 78∙50 = 3900, remember that number.
3) Now let’s count 78∙6 = 70∙6+8∙6 = 420+48 = 468.
4) Now add both results together: 3900+468 = 3000+900+400+60+8 = 3000+1300+60+8 = 4368. Voila!
What are the final recommendations?
Here, in general, and all the ways to know enough to train a confident account within 10,000 (and the ability to work in the mind with large numbers, perhaps, already goes beyond the necessary general development).
Surely you will also encounter other techniques, the so-called “tricks” of quick counting, but do not hurry to get carried away with them. In addition, remember that regularity is more important than intensity – try to exercise on the simulator every day for 5-10 minutes, no longer needed, otherwise there is a great risk of “burn out” and throw.
In the process of training do not hurry anywhere – catch your rhythm, focus on the correct answers, not on speed, speed will come later.
Be sure to try to speak out loud, especially at first – you will have a chance to feel how it all looks like poetry, and it will be easier to decide.
And don’t be upset if something doesn’t work out – the road will be overpowered by the one going, and sooner or later everything will work out for you.
