Interesting ways to count quickly. Quick mental arithmetic: teaching methods

Bibliographic description: Vladimirov A.I., Mikhailova V.V., Shmeleva S.P. Interesting ways to quickly count // Young scientist. 2016. No. 6.1. P. 15-17..02.2019).



Introduction

Mental arithmetic is mental gymnastics. Mental arithmetic is the oldest method of calculation. Mastering computational skills develops memory and helps to master science and mathematics subjects.

There are many techniques for simplifying arithmetic operations. Knowledge of simplified calculation techniques is especially important in cases where the calculator does not have tables and a calculator at his disposal.

We want to focus on methods of addition, subtraction, multiplication, division, for the production of which it is enough to count or use pen and paper.

The motivation for choosing the topic was the desire to continue developing computational skills, the ability to quickly and clearly find the result of mathematical operations.

The rules and methods of calculations do not depend on whether they are performed in writing or orally. However, mastering the skills of oral calculations is of great value not because they are used in everyday life more often than written calculations. This is also important because they speed up written calculations, gain experience in rational calculations, and provide benefits in computational work.

In mathematics lessons we have to do a lot of mental calculations, and when the teacher showed us a technique for quickly multiplying by numbers 11, we had an idea whether there were other methods for quick calculations. We set ourselves the task of finding and testing other methods of fast calculation.

b) to do well at school; (16%)

c) to decide quickly; (16%)

d) to be literate; (52%)

2. List, when studying, which school subjects you will need to count correctly ?

a) mathematics; (80%)

b) physics; (15%)

c) chemistry; (5%)

d) technology;

e) music;

3. Do you know quick counting techniques?

a) yes, a lot;

b) yes, several (85%);

c) no, I don’t know (15%).

4. Do you use quick counting techniques when making calculations?

b) no (85%)

5. Would you like to learn quick counting tricks to count quickly?

b) no (8%).

They say that if you want to learn to swim, you must get into the water, and if you want to be able to solve problems, you must start solving them. But first you need to master the basics of arithmetic. You can learn to count quickly and count in your head only with great desire and systematic training in solving problems.

But the techniques for quick mental counting have been known for a long time. The excellent mental arithmetic abilities of such brilliant mathematicians as Gauss, von Neumann, Euler or Wallis are a real delight. Much has been written about this. We want to tell and show some well-known computing secrets. And then a completely different kind of mathematics will open up before you. Lively, useful and understandable.

1.Methods for fast multiplication

1. COUNTING ON YOUR FINGERS

A way to quickly multiply numbers within the first ten by 9.

Let's say we need to multiply 7 by 9.

Let's turn our hands with our palms facing us and bend the seventh finger (starting from the thumb on the left).

The number of fingers to the left of the curved one will be equal to tens, and to the right - to units of the desired product.

Rice. 1. Counting on fingers

2. MULTIPLYING NUMBERS FROM 10 TO 20

You can multiply such numbers very simply.

To one of the numbers you need to add the number of units of the other, multiply by 10 and add the product of units of numbers.

Example 1. 16∙18=(16+8) ∙ 10+6 ∙ 8=288, or

Example 2. 17 ∙ 17=(17+7) ∙ 10+7 ∙ 7=289.

Task: Multiply quickly 19 ∙ 13. Answer 19 ∙13=(19+3) ∙10 +9 ∙3=247.

3. MULTIPLY BY 11

To multiply a two-digit number, the sum of its digits does not exceed 10, by 11, you need to move the digits of this number apart and put the sum of these digits between them.

72 ∙ 11 = 7 (7 + 2) 2 = 792;

35 ∙ 11 = 3 (3 + 5) 5 = 385.

To multiply a two-digit number by 11, the sum of the digits of which is 10 or more than 10, you need to mentally move apart the digits of this number, put the sum of these digits between them, and then add one to the first digit, and leave the second and last (third) unchanged.

Example .

94 ∙ 11 = 9 (9 + 4) 4 = 9 (13) 4 = (9 + 1) 34 = 1034.

Task: Multiply quickly 54 ∙ 11 (594)

Task: Multiply quickly 67∙11 (737)

4. MULTIPLY BY 22, 33, ..., 99

To multiply a two-digit number by 22, 33, ..., 99, this factor must be represented as the product of a single-digit number (from 2 to 9) by 11, that is, 44 = 4 11; 55 = 5 ∙ 11, etc. Then multiply the product of the first numbers by 11.

Example 1. 24 ∙ 22 = 24 ∙ 2 ∙ 11 = 48 ∙ 11 = 528

Example 2. 23 ∙ 33 = 23 ∙ 3 ∙ 11= 69 ∙ 11 = 759

Task: Multiply 18∙44

5. MULTIPLY BY 5, BY 50, BY 25, BY 125

When multiplying by these numbers, you can use the following expressions:

a ∙ 5=a ∙ 10:2 a ∙ 50=a ∙ 100:2

a ∙ 25=a ∙ 100:4 a ∙ 125=a ∙ 1000:8

Example 1. 17 ∙ 5=17 ∙ 10:2=170:2=85

Example 2. 43 ∙ 50=43 ∙ 100:2=4300:2=2150

Example 3. 27 ∙ 25=27 ∙ 100:4=2700:4=675

Example 4. 96 ∙ 125=96:8 ∙ 1000=12 ∙ 1000=12000

Task: multiply 824∙25

Task: multiply 348∙50

&2. Fast division methods

1. DIVISION BY 5, BY 50, BY 25

When dividing by 5, 50, or 25, you can use the following expressions:

a:5= a ∙ 2:10 a:50=a ∙ 2:100

a:25=a ∙ 4:100

35:5=35 ∙ 2:10=70:10=7

3750:50=3750 ∙ 2:100=7500:100=75

6400:25=6400 ∙ 4:100=25600:100=256

&3. Ways to quickly add and subtract natural numbers.

If one of the terms is increased by several units, then the same number of units must be subtracted from the resulting amount.

Example. 785+963=785+(963+7)-7=785+970-7= 1748

If one of the terms is increased by several units, and the second is decreased by the same number of units, then the sum will not change.

Example. 762+639=(762+8)+(639-8)=770 + 631=1401

If the subtrahend is reduced by several units and the minuend is increased by the same number of units, then the difference will not change.

Example. 529-435=(529-5)-(435+5)=524-440=84

Conclusion

There are ways to quickly add, subtract, multiply, divide, and exponentiate. We have looked at only a few ways to quickly count.

All the methods of mental calculation that we have considered indicate the long-term interest of scientists and ordinary people in playing with numbers. Using some of these methods in the classroom or at home, you can develop the speed of calculations and achieve success in studying all school subjects.

Multiplication without a calculator – training memory and mathematical thinking. Computer technology is improving to this day, but any machine does what people put into it, and we have learned some mental calculation techniques that will help us in life.

It was interesting for us to work on the project. So far we have only studied and analyzed already known methods of quick counting.

But who knows, perhaps in the future we ourselves will be able to discover new ways of fast computing.

Literature:

1. Harutyunyan E., Levitas G. Entertaining mathematics. - M.: AST - PRESS, 1999. - 368 p.
2. Gardner M. Mathematical miracles and secrets. – M., 1978.
3. Glazer G.I. History of mathematics at school. – M., 1981.
4. “First of September” Mathematics No. 3 (15), 2007.
5. Tatarchenko T.D. Ways to quickly count in circle classes, “Mathematics at School”, 2008, No. 7, p. 68.
6. Oral score / Comp. P.M. Kamaev. - M.: Chistye Prudy, 2007 - Library “First of September”, series “Mathematics”. Vol. 3(15).
7. http://portfolio.1september.ru/subject.php

The process of mental counting can be considered as a counting technology that combines human ideas and skills about numbers and mathematical arithmetic algorithms.

There are three types mental counting technologies, which use various physical capabilities of a person:

audiomotor counting technology;

visual counting technology.

Characteristic feature audiomotor mental counting is to accompany each action and each number with a verbal phrase like “twice two is four.” The traditional counting system is precisely an audiomotor technology. The disadvantages of the audiomotor method of calculations are:

absence in the memorized phrase of relationships with neighboring results,

the inability to separate tens and units of a product in phrases about the multiplication table without repeating the entire phrase;

the inability to reverse the phrase from the answer to the factors, which is important for performing division with a remainder;

slow speed of reproduction of a verbal phrase.

Supercomputers, demonstrating high speed of thinking, use their visual abilities and excellent visual memory. People who are good at speed calculations do not use words when solving an arithmetic example in their head. They demonstrate reality visual technology of mental counting, devoid of the main drawback - the slow speed of performing basic operations with numbers.

Perhaps our methods of multiplication are not perfect; Maybe an even faster and more reliable one will be invented.

Of course, it is impossible to know all the methods of quick counting, but the most accessible ones can be studied and applied.

Mental counting training.

There are people who can perform simple arithmetic operations in their heads. Multiply a two-digit number by a single-digit number, multiply within 20, multiply two small two-digit numbers, etc. - they can perform all these actions in their minds and quite quickly, faster than the average person. Often this skill is justified by the need for constant practical use. Typically, people who are good at mental arithmetic have a background in mathematics or at least experience solving numerous arithmetic problems.

Undoubtedly, experience and training play a vital role in the development of any ability. But the skill of mental calculation does not rely on experience alone. This is proven by people who, unlike those described above, are able to count much more complex examples in their minds. For example, such people can multiply and divide three-digit numbers, perform complex arithmetic operations that not every person can count in a column.

What does an ordinary person need to know and be able to do in order to master such a phenomenal ability? Today, there are various techniques that help you learn to count quickly in your head. Having studied many approaches to teaching the skill of counting orally, we can highlight3 main components of this skill:

1. Abilities. The ability to concentrate and the ability to hold several things in short-term memory at the same time. Predisposition to mathematics and logical thinking.

2. Algorithms. Knowledge of special algorithms and the ability to quickly select the necessary, most effective algorithm in each specific situation.

3. Training and experience, the importance of which for any skill has not been canceled. Constant training and gradual complication of solved problems and exercises will allow you to improve the speed and quality of mental calculation.

It should be noted that the third factor is of key importance. Without the necessary experience, you will not be able to surprise others with a quick score, even if you know the most convenient algorithm. However, do not underestimate the importance of the first two components, since having in your arsenal the abilities and a set of necessary algorithms, you can “outdo” even the most experienced “accountant”, provided that you have trained for the same amount of time.

Several ways to count mentally:

1. Multiply by 5 It’s more convenient to do this: first multiply by 10, and then divide by 2

2. Multiply by 9. In order to multiply a number by 9, you need to add 0 to the multiplicand and subtract the multiplicand from the resulting number, for example 45 9 = 450-45 = 405.

3. Multiply by 10. Add a zero to the right: 48 10 = 480

4. Multiply by 11. two-digit number. Spread the numbers N and A, enter the amount in the middle (N+A).

for example, 43 11 = = = 473.

5. Multiply by 12. is done in approximately the same way as for 11. We double each digit of the number and add to the result the neighbor of the original digit on the right.

Examples.Let's multiplyon.

Let's start with the rightmost number - this is. Let's double itand add a neighbor (he is not present in this case). We get. Let's write it downand remember.

Let's move left to the next number. Let's double it, we get, add a neighbor,, we get, add. Let's write it downand remember.

Let's move left to the next number,. Let's double it, we get. Let's add a neighborand we get. Let's add, which we remembered, we get. Let's write it downand remember.

Let's move to the left to a non-existent number - zero. Let's double it, get and add a neighbor, which will give us . Finally, we add , which we remembered, and we get . Let's write it down. Answer: .

6. Multiplication and division by 5, 50, 500, etc.

Multiplication by 5, 50, 500, etc. is replaced by multiplication by 10, 100, 1000, etc., followed by division by 2 of the resulting product (or division by 2 and multiplication by 10, 100, 1000, etc. ). (50 = 100: 2, etc.)

54 5=(54 10):2=540:2=270 (54 5 = (54:2) 10= 270).

To divide a number by 5.50, 500, etc., you need to divide this number by 10,100,1000, etc. and multiply by 2.

10800: 50 = 10800:100 2 =216

10800: 50 = 10800 2:100 =216

7. Multiplication and division by 25, 250, 2500, etc.

Multiplication by 25, 250, 2500, etc. is replaced by multiplication by 100, 1000, 10000, etc. and the resulting result is divided by 4. (25 = 100: 4)

542 25=(542 100):4=13550 (248 25=248: 4 100 = 6200)

(if the number is divisible by 4, then multiplication does not take time; any student can do it).

To divide a number by 25, 25,250,2500, etc., this number must be divided by 100,1000,10000, etc. and multiply by 4: 31200: 25 = 31200:100 4 = 1248.

8. Multiplication and division by 125, 1250, 12500, etc.

Multiplication by 125, 1250, etc. is replaced by multiplication by 1000, 10000, etc. and the resulting product must be divided by 8. (125 = 1000 : 8)

72 125=72 1000: 8=9000

If the number is divisible by 8, then first divide by 8, and then multiply by 1000, 10000, etc.

48 125 = 48: 8 1000 = 6000

To divide a number by 125, 1250, etc., you need to divide this number by 1000, 10000, etc. and multiply by 8.

7000: 125 = 7000: 10008 = 56.

9. Multiplication and division by 75, 750, etc.

To multiply a number by 75, 750, etc., you need to divide this number by 4 and multiply by 300, 3000, etc. (75 = 300:4)

4875 = 48:4300 = 3600

To divide a number by 75,750, etc., you need to divide this number by 300, 3000, etc. and multiply by 4

7200: 75 = 7200: 3004 = 96.

10. Multiply by 15, 150.

When multiplying by 15, if the number is odd, multiply it by 10 and add half of the resulting product:

23 15=23 (10+5)=230+115=345;

if the number is even, then we proceed even simpler - we add half of it to the number and multiply the result by 10:

18 15=(18+9) 10=27 10=270.

When multiplying a number by 150, we use the same technique and multiply the result by 10, since 150 = 15 10:

24 150=((24+12) 10) 10=(36 10) 10=3600.

In the same way, quickly multiply a two-digit number (especially an even one) by a two-digit number ending in 5:

24 35 = 24 (30 +5) = 24 30+24:2 10 = 720+120=840.

11. Multiplying two-digit numbers less than 20.

To one of the numbers you need to add the number of units of the other, multiply this amount by 10 and add to it the product of the units of these numbers:

18 16=(18+6) 10+8 6= 240+48=288.

Using the described method, you can multiply two-digit numbers less than 20, as well as numbers that have the same number of tens: 23 24 = (23+4) 20+4 6=27 20+12=540+12=562.

Explanation:

(10+a) (10+b) = 100 + 10a + 10b + a b = 10 (10+a+b) + a b = 10 ((10+a)+b) + a b .

12. Multiplying a two-digit number by 101 .

Perhaps the simplest rule: assign your number to yourself. Multiplication is complete.
Example: 57 101 = 5757 57 --> 5757

Explanation: (10a+b) 101 = 1010a + 101b = 1000a + 100b + 10a + b
Similarly, three-digit numbers are multiplied by 1001, four-digit numbers by 10001, etc.

13. Multiplication by 22, 33, ..., 99.

To multiply a two-digit number 22.33, ...,99, you need to represent this factor as the product of a single-digit number by 11. Multiply first by a single-digit number, and then by 11:

15 33= 15 3 11=45 11=495.

14. Multiplying two-digit numbers by 111 .

First, let’s take as a multiplicand a two-digit number whose sum of digits is less than 10. Let’s explain with numerical examples:

Since 111=100+10+1, then 45 111=45 (100+10+1). When multiplying a two-digit number, the sum of the digits of which is less than 10, by 111, it is necessary to insert twice the sum of the digits (i.e., the numbers represented by them) of its tens and units 4+5=9 in the middle between the digits. 4500+450+45=4995. Therefore, 45,111=4995. When the sum of the digits of a two-digit multiplicand is greater than or equal to 10, for example 68 11, you need to add the digits of the multiplicand (6+8) and insert 2 units of the resulting sum into the middle between the digits 6 and 8. Finally, add 1100 to the composed number 6448. Therefore, 68 111 = 7548.

15. Squaring numbers consisting of only 1.

11 x 11 =121

111 x 111 = 12321

1111 x 1111 = 1234321

11111 x 11111 =123454321

111111 x 111111 = 12345654321

1111111 x 1111111 = 1234567654321

11111111 x 11111111 = 123456787654321

111111111 x 111111111 = 12345678987654321

Some non-standard multiplication techniques.

Multiplying a number by a single-digit factor.

To multiply a number by a single-digit factor (for example, 34 9) orally, you must perform actions starting from the highest digit, sequentially adding the results (30 9=270, 4 9=36, 270+36=306).

For effective mental counting, it is useful to know the multiplication table up to 19*9. In this case, multiplication is 147 8 is performed in the mind like this: 147 8=140 8+7 8= 1120 + 56= 1176 . However, without knowing the multiplication table up to 19 9, in practice it is more convenient to calculate all such examples by reducing the multiplier to the base number: 147 8=(150-3) 8=150 8-3 8=1200-24=1176, with 150 8=(150 2) 4=300 4=1200.

If one of the multiplied items is decomposed into single-digit factors, it is convenient to perform the action by sequentially multiplying by these factors, for example, 225 6=225 2 3=450 3=1350. Also, it may be easier to use 225 6=(200+25) 6=200 6+25 6=1200+150=1350.

Multiplying two-digit numbers.

1. Multiply by 37.

When multiplying a number by 37, if the given number is a multiple of 3, it is divided by 3 and multiplied by 111.

27 37=(27:3) (37 3)=9 111=999

If the given number is not a multiple of 3, then 37 is subtracted from the product or 37 is added to the product.

23 37=(24-1) 37=(24:3) (37 3)-37=888-37=851.

It is easy to remember the product of some of them:

3 x 37 = 111 33 x 3367 = 111111

6 x 37 = 222 66 x 3367 = 222222

9 x 37 = 333 99 x 3367 = 333333

12 x 37 = 444 132 x 3367 = 444444

15 x 37 = 555 165 x 3367 = 555555

18 x 37 = 666 198 x 3367 = 666666

21 x 37 = 777 231 x 3367 = 777777

24 x 37 = 888 264 x 3367 = 888888

27 x 37 = 999 297 x 3367 = 99999

2. If tens of two-digit numbers begin with the same digit, and the sum of the ones is 10 , then when multiplying them we find the product in this order:

1) multiply the ten of the first number by the ten of the second larger number by one;

2) multiply the units:

8 3x 8 7= 7221 ( 8x9=72 , 3x7=21)

5 6x 5 4=3024 ( 5x6=30 , 6x4=24)

1. Algorithm for multiplying two-digit numbers close to 100

For example:97 x 96 = 9312

Here I use the following algorithm: if you want to multiply two

double-digit numbers close to 100, then do this:

1) find the disadvantages of factors up to a hundred;

2) subtract from one factor the deficiency of the second to a hundred;

3) add two digits to the result of the product of the shortcomings

factors up to hundreds.

The relevant literature mentions such methods of multiplication as “folding”, “lattice”, “back to front”, “diamond”, “triangle” and many others. I wanted to know what other non-standard multiplication techniques exist in mathematics? It turns out there are a lot of them. Here are some of these techniques.

Peasant method:

One of the multipliers is doubled, while the other is simultaneously decreased by the same amount. When the quotient becomes equal to one, the parallel product obtained is the desired answer.

If the quotient turns out to be an odd number, then one is removed from it and the remainder is divided. Then the products that stood opposite the odd quotients are added to the answer received

"Method of the Cross"

In this method, the factors are written one below the other and their numbers are multiplied in a straight line and crosswise.

3 1 = 3 – last digit.

2 1 + 3 3 = 11. The penultimate digit is 1, another 1 in the mind.

2 3 = 6; 6 + 1 = 7 is the first digit of the product

The required work is 713.

Sino-Japanese multiplication method.

It is no secret that teaching methods are different in different countries. It turns out that in Japan, first grade students can multiply three-digit numbers without knowing the multiplication table. For this it is used. The logic of the method is clear from the figure. After drawing, you just need to count the number of intersections in each area.

This method can be used to multiply even three-digit numbers. It is likely that when children later learn the multiplication tables, they will be able to multiply in a simpler and faster way, by columns. Moreover, the above method is too labor-intensive when multiplying numbers like 89 and 98, because you have to draw 34 stripes and count all the intersections. On the other hand, in such cases you can use a calculator. Many people will think that this method of Japanese or Chinese multiplication is too complicated and confusing, but this is only at first glance. It is visualization, that is, the image of all the points of intersection of lines (factors) on one plane, that gives us visual support, whereas the traditional method of multiplication involves a large number of arithmetic operations only in the mind. Chinese or Japanese multiplication not only helps you quickly and efficiently multiply two-digit and three-digit numbers by each other without a calculator, but also develops erudition. Agree, not everyone can boast that in practice they know the ancient Chinese method of multiplication (), which is relevant and works great in the modern world.

Multiplication can be done using a matrix table ts :

43219876=?

Lattice method.

A rectangle is drawn, divided into squares. Next are square cells, divided diagonally. In each line we will write the product of the numbers above this cell and to the right of it, while we will write the tens digit of the product above the slash, and the units digit below it. Now we add the numbers in each oblique strip, performing this operation, from right to left. If it turns out to be greater than 10, then we write only the units digit of the sum, and add the tens digit to the next sum.

5

2

4

1 7

3

7

7

We write the answer numbers from left to right: 4, 5, 17, 20, 7, 5. Starting from the right, we write, adding “extra” numbers to the “neighbor”: 469075.

Got: 725 x 647 = 469075.

Mental counting, like everything else, has its own tricks, and in order to learn to count faster you need to know these tricks and be able to apply them in practice.

Today we will do just that!

1. How to quickly add and subtract numbers

Let's look at three random examples:

1. 25 – 7 =
2. 34 – 8 =
3. 77 – 9 =

Like 25 – 7 = (20 + 5) – (5- 2) = 20 – 2 = (10 + 10) – 2 = 10 + 8 = 18

Agree that such operations are difficult to carry out in your head.

But there is an easier way:

25 – 7 = 25 – 10 + 3, since -7 = -10 + 3

It is much easier to subtract 10 from a number and add 3 than to make complicated calculations.

Let's return to our examples:

1. 25 – 7 =
2. 34 – 8 =
3. 77 – 9 =

Let's optimize the subtracted numbers:

1. Subtract 7 = subtract 10 add 3
2. Subtract 8 = subtract 10 add 2
3. Subtract 9 = subtract 10 add 1

In total we get:

1. 25 – 10 + 3 =
2. 34 – 10 + 2 =
3. 77 – 10 + 1 =

Now it’s much more interesting and easier!

Now calculate the examples below in this way:

1. 91 – 7 =
2. 23 – 6 =
3. 24 – 5 =
4. 46 – 8 =
5. 13 – 7 =
6. 64 – 6 =
7. 72 – 19 =
8. 83 – 56 =
9. 47 – 29 =

2. How to quickly multiply by 4, 8 and 16

In the case of multiplication, we also break numbers into simpler ones, for example:

If you remember the multiplication table, then everything is simple. And if not?

Then you need to simplify the operation:

We put the largest number first, and decompose the second into simpler ones:

8 * 4 = 8 * 2 * 2 = ?

Doubling numbers is much easier than quadrupling or octupling them.

We get:

8 * 4 = 8 * 2 * 2 = 16 * 2 = 32

Examples of decomposing numbers into simpler ones:

1. 4 = 2*2
2. 8 = 2*2 *2
3. 16 = 22 * 2 2

Practice this method using the following examples:

1. 3 * 8 =
2. 6 * 4 =
3. 5 * 16 =
4. 7 * 8 =
5. 9 * 4 =
6. 8 * 16 =

3. Dividing a number by 5

Let's take the following examples:

1. 780 / 5 = ?
2. 565 / 5 = ?
3. 235 / 5 = ?

Dividing and multiplying with the number 5 is always very simple and enjoyable, because five is half of ten.

And how to solve them quickly?

1. 780 / 10 * 2 = 78 * 2 = 156
2. 565 /10 * 2 = 56,5 * 2 = 113
3. 235 / 10 * 2 = 23,5 *2 = 47

To work through this method, solve the following examples:

1. 300 / 5 =
2. 120 / 5 =
3. 495 / 5 =
4. 145 / 5 =
5. 990 / 5 =
6. 555 / 5 =
7. 350 / 5 =
8. 760 / 5 =
9. 865 / 5 =
10. 1270 / 5 =
11. 2425 / 5 =
12. 9425 / 5 =

4. Multiplying by single digits

Multiplication is a little more difficult, but not much, how would you solve the following examples?

1. 56 * 3 = ?
2. 122 * 7 = ?
3. 523 * 6 = ?

Without special counters, solving them is not very pleasant, but thanks to the “Divide and Conquer” method we can count them much faster:

1. 56 * 3 = (50 + 6)3 = 50 3 + 6*3 = ?
2. 122 * 7 = (100 + 20 + 2)7 = 100 7 + 207 + 2 7 = ?
3. 523 * 6 = (500 + 20 + 3)6 = 500 6 + 206 + 3 6 =?

All we have to do is multiply single-digit numbers, some of which have zeros, and add the results.

To work through this technique, solve the following examples:

1. 123 * 4 =
2. 236 * 3 =
3. 154 * 4 =
4. 490 * 2 =
5. 145 * 5 =
6. 990 * 3 =
7. 555 * 5 =
8. 433 * 7 =
9. 132 * 9 =
10. 766 * 2 =
11. 865 * 5 =
12. 1270 * 4 =
13. 2425 * 3 =

Divisibility of a number by 2, 3, 4, 5, 6 and 9

Check the numbers: 523, 221, 232

A number is divisible by 3 if the sum of its digits is divisible by 3.

For example, take the number 732, represent it as 7 + 3 + 2 = 12. 12 is divisible by 3, which means the number 372 is divisible by 3.

Check which of the following numbers are divisible by 3:

12, 24, 71, 63, 234, 124, 123, 444, 2422, 4243, 53253, 4234, 657, 9754

A number is divisible by 4 if the number consisting of its last two digits is divisible by 4.

For example, 1729. The last two digits form 20, which is divisible by 4.

Check which of the following numbers are divisible by 4:

20, 24, 16, 34, 54, 45, 64, 124, 2024, 3056, 5432, 6872, 9865, 1242, 2354

A number is divisible by 5 if its last digit is 0 or 5.

Check which of the following numbers are divisible by 5 (the easiest exercise):

3, 5, 10, 15, 21, 23, 56, 25, 40, 655, 720, 4032, 14340, 42343, 2340, 243240

A number is divisible by 6 if it is divisible by both 2 and 3.

Check which of the following numbers are divisible by 6:

22, 36, 72, 12, 34, 24, 16, 26, 122, 76, 86, 56, 46, 126, 124

A number is divisible by 9 if the sum of its digits is divisible by 9.

For example, take the number 6732, represent it as 6 + 7 + 3 + 2 = 18. 18 is divisible by 9, which means the number 6732 is divisible by 9.

Check which of the following numbers are divisible by 9:

9, 16, 18, 21, 26, 29, 81, 63, 45, 27, 127, 99, 399, 699, 299, 49

Game "Quick addition"

1. Speeds up mental counting
2. Trains attention
3. Develops creative thinking

An excellent simulator for developing fast counting. A 4x4 table is given on the screen, and numbers are shown above it. The largest number must be collected in the table. To do this, click on two numbers whose sum is equal to this number. For example, 15+10 = 25.

Game "Quick Count"

The game "quick count" will help you improve your thinking. The essence of the game is that in the picture presented to you, you will need to choose the answer “yes” or “no” to the question “are there 5 identical fruits?” Follow your goal, and this game will help you with this.

Game "Guess the operation"

The game “Guess the Operation” develops thinking and memory. The main point of the game is to choose a mathematical sign for the equality to be true. Examples are given on the screen, look carefully and put the required “+” or “-” sign so that the equality is true. The “+” and “-” signs are located at the bottom of the picture, select the desired sign and click on the desired button. If you answered correctly, you score points and continue playing.

Game "Simplification"

The game “Simplification” develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical operation is given; the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need using the mouse. If you answered correctly, you score points and continue playing.

Today's task

Solve all examples and practice for at least 10 minutes in the game Quick Addition.

It is very important to work through all the tasks in this lesson. The better you complete the tasks, the more benefits you will receive. If you feel that you don’t have enough tasks, you can create examples for yourself and solve them and practice mathematical educational games.

Lesson taken from the course "Mal Calculus in 30 Days"

Learn to quickly and correctly add, subtract, multiply, divide, square numbers, and even take roots. I will teach you how to use easy techniques to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.

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This article is inspired by the topic “How and how quickly do you count in your head at an elementary level?” and is intended to spread the techniques of S.A. Rachinsky for oral counting.
Rachinsky was a wonderful teacher who taught in rural schools in the 19th century and showed from his own experience that it is possible to develop the skill of rapid mental calculation. For his students, it was not particularly difficult to calculate such an example in their heads:

Using round numbers

Because on 10 , 100 , 1000 etc. it’s faster to multiply round numbers; in your mind you need to reduce everything to such simple operations as 18 x 100 or 36 x 10. Accordingly, it is easier to add by “splitting off” a round number and then adding a “tail”: 1800 + 200 + 190 .
Another example:
31 x 29 = (30 + 1) x (30 - 1) = 30 x 30 - 1 x 1 = 900 - 1 = 899.

Let's simplify multiplication by division

Squaring a two-digit number

Rachinsky's technique is as follows. In order to find the square of any two-digit number, you need the difference between this number and 25 multiply by 100 and to the resulting product add the square of the complement of the given number to 50 or the square of its excess over 50 -Yu. For example,
37^2 = 12 x 100 + 13^2 = 1200 + 169 = 1369; 84^2 = 59 x 100 + 34^2 = 5900 + 9 x 100 + 16^2 = 6800 + 256 = 7056;
In general ( M- two-digit number):

Let's try to apply this trick when squaring a three-digit number, first breaking it into smaller terms:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 70 x 100 + 45^2 = 10000 + (90+5) x 2 x 100 + + 7000 + 20 x 100 + 5^2 = 17000 + 19000 + 2000 + 25 = 38025.
Hmm, I wouldn’t say that it’s much easier than erecting it in a column, but perhaps over time you can get used to it.
And, of course, you should start training by squaring two-digit numbers, and from there you can even get to disassembling in your mind.

Multiplying two-digit numbers

For example, let's calculate 77 x 13. The sum of the units of these numbers is equal to 10 , because 7 + 3 = 10 . First we put the smaller number before the larger one: 77 x 13 = 13 x 77.
To get round numbers, we take three units from 13 and add them to 77 . Now let's 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 a new number 10 :
13 x 77 = 10 x 80 + 3 x (77 - 10) = 800 + 3 x 67 = 800 + 3 x (60 + 7) = 800 + 3 x 60 + 3 x 7 = 800 + 180 + 21 = 800 + 201 = 1001.
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 number following it and the product of the units of these numbers is added to the resulting result. Let's see how elegant this technique is with an example.
48 x 42. Tens number 4 , next number: 5 ; 4 x 5 = 20 . Product of units: 8 x 2 = 16 . So 48 x 42 = 2016.
99 x 91. Tens number: 9 , next number: 10 ; 9 x 10 = 90 . Product of units: 9 x 1 = 09 . So 99 x 91 = 9009.
Yeah, that is, to multiply 95 x 95, just count 9 x 10 = 90 And 5 x 5 = 25 and the answer is ready:
95 x 95 = 9025.
Then the previous example can be calculated a little simpler:
195^2 = (100 + 95)^2 = 10000 + 2 x 100 x 95 + 95^2 = 10000 + 9500 x 2 + 9025 = 10000 + (90+5) x 2 x 100 + 9000 + 25 = 10000 + 19000 + 1000 + 8000 + 25 = 38025.

Instead of a conclusion

References:
“1001 problems for mental arithmetic at the school of S.A. Rachinsky".