Effective mental arithmetic or a brain workout
This article is inspired by the topic “How and how fast do you count in your mind at the elementary level?” and is intended to spread the techniques of S.A. Rachinsky for oral counting. Rachinsky was a remarkable teacher, who taught in rural schools in the 19th century and who showed from his own experience that it was possible to develop the skill of rapid oral counting. For his students it was no particular problem to calculate such an example in their heads:
Using round numbers
One of the most common methods of mental arithmetic is to represent any number as a sum or difference of numbers, one or more of which are “round”:
Since it is quicker to multiply a round number by 10, 100, 1000, etc., in your mind, you should reduce everything to such simple operations as 18 x 100 or 36 x 10. Accordingly, it is easier to add by “chipping off” a round number and then adding a “tail”: 1800 + 200 + 190. Another example:
Simplify multiplication by division
When counting orally it is sometimes more convenient to operate with a divisor and a divisor than with an integer (for example, 5 as 10:2 and 50 as 100:2):
Multiply or divide by 25 in the same way, because 25 = 100:4. For example,
Now it does not seem impossible to multiply 625 by 53 in the mind:
Squaring a two-digit number
It turns out that in order to simply square any two-digit number, it is sufficient to memorize the squares of all the numbers from 1 to 25. Fortunately, we already know the squares up to 10 from the multiplication table. The other squares can be found in the table below:
Rachinsky’s method is as follows. To find the square of any two-digit number, it is necessary to multiply the difference between this number and 25 by 100 and to the resulting product add the square of the addition of this number to 50 or the square of its excess over 50. For example,
In the general case ( M is a two-digit number):
Let’s try to apply this trick when squaring a three-digit number by first breaking it down into smaller summands:
Hmm, I wouldn’t say it’s much easier than columnar, but maybe you can get the hang of it over time. And you should start training, of course, with the square of two-digit numbers, and then you can get to disassembling in your mind.
Multiplication of two-digit numbers
This interesting method was invented by a 12-year-old student of Rachinsky and is one of the options for adding to a round number. Let two two two-digit numbers be given in which the sum of the units equals 10:
By making their product, we get:
For example, let’s calculate 77 x 13 . The sum of the units of these numbers is 10 , because 7 + 3 = 10. First we put the smaller number in front of the bigger one: 77 x 13 = 13 x 77 . To get round numbers we take three ones from 13 and add them to 77. Now multiply the new numbers 80 x 10 , and to the result we add the product of the selected 3 units by the difference of the old number 77 and the new number 10 :
This technique has a special case: everything is greatly simplified when two factors have the same number of tens. In this case, the number of tens is multiplied by the next number and the product of the units of these numbers is added to the result. Let us see how elegant this technique is using an example. 48 x 42 . The number of tens is 4 , the next number: 5 ; 4 x 5 = 20 . The product of the units: 8 x 2 = 16 . So 99 x 91 . Number of tens: 9 , followed by a number: 10 ; 9 x 10 = 90 . The product of units: 9 x 1 = 09 . So, Aha, that is, to multiply 95 x 95 , just count 9 x 10 = 90 and 5 x 5 = 25 and the answer is ready:
Then the previous example can be calculated a little easier:
In lieu of a conclusion
It would seem that why know how to calculate in your mind in the 21st century when you can just give a voice command to your smartphone? But if you think about it, what would happen to humanity if it were to outsource not only physical work, but any mental work as well? Wouldn’t it degenerate? Even if we do not consider oral calculation as an end in itself, it is quite suitable for tempering the mind.
Literature used : “1001 tasks for mental arithmetic in S. A. Rachinsky’s school” .
How to learn to count in math
Counting in your head, according to many people nowadays is not relevant, because there is a calculator in every smartphone, computer and laptop. However, the calculator will not accompany you at every step, and it is necessary to count constantly and a lot. The ability to count in the mind – a very necessary skill, even in the 21st century. And all the more it is needed for schoolchildren to solve examples in mathematics from a difficult school program. And it will be very useful for them to be able to count quickly, without resorting to electronic devices.
Experience and constant training are essential to the development of any ability, but the skill of oral arithmetic does not consist only of experience. People who can count in their heads for much more complex examples can prove this: for example, multiplying and dividing three- and four-digit numbers, and finding the sums and differences of huge examples.
What does a person need to know and do in order to repeat such a thing?
– Firstly, concentration or the ability to hold briefly in memory several things at the same time.
– Secondly, algorithms, special methods of calculation, and mathematical tricks that greatly facilitate the process of mental calculation.
– Third, practice . Constant training and a gradual increase in the complexity of the problems to be solved will improve the speed and quality of oral calculation.
It is important to note that it is practice makes the biggest difference .. Without sufficient experience, you will not be able to quickly apply convenient algorithms that are suitable for certain situations. And remember that the maximum effect will be achieved with the optimal use of all three components . You can train all aspects of this skill at once in the online oral arithmetic simulator.
Attention and Concentration
To count as quickly as possible in your mind, you need to be able to concentrate on a particular example. This skill is useful not only for performing mathematical operations, but also for solving any of life’s problems. There are several ways to improve your mindfulness and ability to concentrate:
When counting in your mind, it’s important to clearly visualize the example you’re solving – visualize it. It is necessary to memorize intermediate results not by ear, but as they look in the record, for example, on paper. You can train this kind of perception in different ways, and partly visualizing the solution comes with experience.
Always try to find something interesting in the routine, turning the activity into a game. This is what some parents do when they want their child to do some boring job.
A huge number of people always want to “be better” than their opponent. That’s why competitiveness is another way to develop your attentiveness. In verbal counting, you can find a rival and try to outdo them in it.
Another factor that creates the excitement of counting can be to fight with yourself when achieving a certain result, that is, personal records. They can be set, for example, in the speed of counting, in the number of examples solved and their accuracy of answers.
Finally, maximum concentration can be achieved by spontaneous fascination with the counting process. As an example, while reading, you stop thinking about the objects, people and situations around you, and fully immerse yourself in the book. It is a genuine interest in something that can make you acquire the greatest attentiveness in this case.
Of course, all these ways should be practiced. Various visual memory and attention exercises can help.
Simple arithmetic patterns.
Solving any complex task always comes down to the application of basic principles, and it is these principles and patterns that will allow you to quickly perform various operations. There is a specific set of such rules and patterns that you need to bring to automatism with the help of different online simulators in mathematics.
Subtracting 7, 8, 9 . To subtract 9 from any number, you must subtract 10 from it and add 1. To subtract 8 from any number, you must subtract 10 from it and add 2 . To subtract 7 from any number, subtract 10 from it and add 3. If you usually count differently, you need to get used to this new way of doing things to get better results.
Multiplication Table . To count quickly orally, it is a good idea to know the multiplication table, which is the basis of counting. If you’re still having trouble with this, you can use the online Multiplication Table Simulator.
Multiplication by 2 . For multiplying by 2 non-circular numbers, try rounding them up to the nearest more convenient number. So 139×2 is easier to count if you first multiply 140 by 2 (140×2=280) , and then subtract 1×2=2 (exactly 1 needs to be added to 139 to get 140). Total: 140×2 – 1×2 = 280 – 2 = 278 .
Dividing by 2 . Although many people find multiplication and division by 2 easy enough, try rounding up numbers in difficult cases as well. For example, to divide 198 by 2, you must first divide 200 (that’s 198 + 2) by 2 and subtract 2 divided by 2. Total: 198 : 2 = 200 :2- :2=100-1=99.
Division and multiplication by 4 and 8 . Division (or multiplication) by 4 and by 8 are two or three times division (or multiplication) by 2. It is convenient to perform these operations sequentially. For example, 46 × 4 = 46 × 2 × 2 = 92 × 2 =184.
Multiplication by 5 and 25 . Multiplying by 5 , and dividing by 2 are basically the same thing, so always multiply by 5 by dividing the number by 2 and multiplying it by 10: 88×5 =88: 2×10 =440. Multiplying by 25 corresponds to dividing by 4 (followed by multiplying by 100). So 120×25 = 120: 4×100 = 30 × 100 = 3000 .
Multiplication by 9 . You can quickly multiply any number by 9 as follows: first multiply that number by 10 , and then subtract the number itself from the result. For example: 89× 9 =89 0 -89=801 .
Multiplication by 11 . To multiply any two-digit number by 11, you need to write the sum of the first and second digits between the first and second digits of the number being multiplied. For example: 23×11= 2 (2+3) 3 = 2 5 3. Or if the sum of the numbers in the center gives a result greater than 10: 29×11 = 2 (2+9) 9 = 2 (11) 9 = 3 1 9.
Finally, it is useful to know the division of numbers divisible by 10 by numbers divisible by two: 1000 = 2 × 500 = 4 × 250 = 8 × 125 = 16 × 62.5 .
More complex techniques
The efficiency of multiplying some two-digit numbers in your mind can be higher with fewer steps if you use special algorithms. Below are three special techniques, including the introduction and use of a reference number.
Squaring the sum and squaring the difference
To square a two-digit number, you can use the formulas for the square of the sum or the square of the difference . For example:
23 2 = (20+3) 2 = 20 2 + 2×3×20 + 3 2 = 400+120+9 = 529
69 2 = (70-1) 2 = 70 2 – 70×2×1 + 1 2 = 4 900-140+1 = 4 761
Squaring numbers ending in 5
To square numbers ending in 5, multiply the number up to the last five by the sum of the same number and one. Add 25 to the result. Here are some examples:
25 2 = (2×(2+1)) 25 = 625
85 2 = (8×(8+1)) 25 = 7 225
155 2 = (15×(15+1)) 25 = (15×16)25 = 24 025
The most popular technique for multiplication of big numbers in mind is the use of the so-called reference number. A reference number for multiplication – This number is the number to which both multipliers are close and by which it is convenient to multiply. And the method of using this number depends on whether the multipliers are larger or smaller than itself.
Both multipliers are smaller than the reference . Let’s say we want to multiply 48 by 47 . These numbers are close enough to the number 50 , and therefore it is convenient to use 50 as the reference number. Then we proceed like this: subtract from 47 as much as 48 is missing to 50 (or subtract from 48 as much as 47 is missing to 50), multiply the result obtained by the reference number and add to it the product of differences of the survey number with each factor. An illustrative example:
( 48 -( 50 – 47 ))× 50 + ( 50 – 47 )×( 50 – 48 ) = 2250 + 6 = 2256
Both multipliers are larger than the reference . Proceed in exactly the same way, but do not subtract the deficiency, but add the excess:
( 51 +( 63 – 50 ))× 50 + ( 63 – 50 )×( 51 – 50 ) = 3200 + 13 = 3213
One multiplier is smaller, the other is larger than the reference . The scheme is the same, but the product of deficiency and excess must be subtracted:
( 45 +( 52 – 50 ))× 50 – ( 52 – 50 )×( 50 – 45 ) = 2350 – 10 = 2340
As we have said before, there are three components to verbal numeracy: the ability to concentrate on a particular example, the right choice of method for calculating quickly, and, of course, experience. Remember, even if you know by heart all the algorithms that make oral counting easier, you can’t count as fast without practice as you would if you had been doing it every day for years. It is constant practice on a variety of simulators verbal account will allow you to hone your skills in this business and acquire the invaluable skill of rapid oral calculus.