Some quick ways oral multiplication We’ve already figured it out, now let’s take a closer look at how to quickly multiply numbers in your head using various auxiliary methods. You may already know, and some of them are quite exotic, such as the ancient Chinese way of multiplying numbers.
Layout by ranks
It is the simplest technique for quickly multiplying two-digit numbers. Both factors need to be divided into tens and ones, and then all these new numbers must be multiplied by each other.
This method requires the ability to hold up to four numbers in memory at the same time, and to do calculations with these numbers.
For example, you need to multiply numbers 38 And 56 . We do it this way:
38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + 8 * 50 + 30 * 6 + 8 * 6 = 1500 + 400 + 180 + 48 = 2128 It will be even easier to do oral multiplication of two-digit numbers in three operations. First you need to multiply the tens, then add two products of ones by tens, and then add the product of ones by ones. It looks like this: 38 * 56 = (30 + 8) * (50 + 6) = 30 * 50 + (8 * 50 + 30 * 6) + 8 * 6 = 1500 + 580 + 48 = 2128 In order to successfully use this method, you need to know the multiplication table well, be able to quickly add two-digit and three-digit numbers, and switch between mathematical operations without forgetting intermediate results. The last skill is achieved through help and visualization.
This method is not the fastest and most effective, so it is worth exploring other methods of oral multiplication.
Fitting the numbers
You can try to bring the arithmetic calculation to a more convenient form. For example, the product of numbers 35
And 49
can be imagined this way: 35 * 49 = (35 * 100) / 2 - 35 = 1715
This method may be more effective than the previous one, but it is not universal and is not suitable for all cases. It is not always possible to find a suitable algorithm to simplify the problem.
On this topic, I remembered an anecdote about how a mathematician sailed along the river past a farm and told his interlocutors that he was able to quickly count the number of sheep in the pen, 1358 sheep. When asked how he did it, he said it was simple - you need to count the number of legs and divide by 4.
Visualization of columnar multiplication
This is one of the most universal ways of oral multiplication of numbers, developing spatial imagination and memory. First, you should learn to multiply two-digit numbers by single-digit numbers in a column in your head. After this, you can easily multiply two-digit numbers in three steps. First, a two-digit number must be multiplied by the tens of another number, then multiplied by the units of another number, and then sum the resulting numbers.
It looks like this: 38 * 56 = (38 * 5) * 10 + 38 * 6 = 1900 + 228 = 2128
Visualization with number arrangement
A very interesting way to multiply two-digit numbers is as follows. You need to sequentially multiply the digits in numbers to get hundreds, ones and tens.
Let's say you need to multiply 35 on 49 .
First you multiply 3 on 4 , you get 12 , then 5 And 9 , you get 45 . Recording 12 And 5 , with a space between them, and 4 remember.
You receive: 12 __ 5 (remember 4 ).
Now you multiply 3 on 9 , And 5 on 4 , and sum up: 3 * 9 + 5 * 4 = 27 + 20 = 47 .
Now we need to 47 add 4 which we remember. We get 51 .
We write 1 in the middle and 5 add to 12 , we get 17 .
In total, the number we were looking for is 1715 , it is the answer:
35 * 49 = 1715
Try multiplying in your head in the same way: 18 * 34, 45 * 91, 31 * 52
.
Chinese or Japanese multiplication
In Asian countries, it is customary to multiply numbers not in a column, but by drawing lines. For Eastern cultures, the desire for contemplation and visualization is important, which is probably why they came up with such a beautiful method that allows you to multiply any numbers. This method is complicated only at first glance. In fact, greater clarity allows you to use this method much more effectively than column multiplication.
In addition, knowledge of this ancient oriental method increases your erudition. Agree, not everyone can boast that they know the ancient multiplication system that the Chinese used 3000 years ago.
Video about how the Chinese multiply numbers
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There are three general methods: direct multiplication, reference number method and Trachtenberg method.
Master them all, as each may be preferable in a given situation.
You can practice your acquired skills using a training table.
Direct multiplication
This method is useful when one of the multipliers is in the range of 12-18 or ends in 1, and the other is significantly different from it.
One of the factors is mentally divided into tens and ones. Then they multiply the other factor by tens, then by units and add.
For example, 62×13 = 62×10 + 62×3 = 620 + 186 = 806.
Sometimes it is convenient to break the larger factor into tens and ones: 42×17 = 17×40 + 17×2 = 714.
Reference number method
The method requires a little practice to master, but it is very convenient when two factors are close numbers. In particular, this is the main method for squaring two-digit numbers.
The reference number is a round number close to both factors. It may be less than both factors, greater than both factors, or in between.
As a reference number, you should choose numbers that are easy to multiply by. For example, 50 or 100 if they are close to two factors.
Depending on how the reference number and the factors are related, the multiplication technique differs slightly.
A. The reference number is less than two factors. For example, you need to multiply 32 by 36.
- The reference number is 30. The multipliers are greater than the reference number by 2 and 6.
- Add 6 to the first factor and multiply by the reference number: 38 × 30 = 1140.
- Add the product of 2 and 6: 1140 + 2×6 = 1152.
b. The reference number is greater than two factors. For example, you need to multiply 43 by 48.
- The reference number is 50. The multipliers are 7 and 2 less than the reference number.
- Subtract 2 from the first factor and multiply by the reference number: 41 × 50 = 2050.
- Add the product of 7 and 2: 2050 + 7×2 = 2064.
V. The reference number is between the factors. For example, you need to multiply 37 by 42.
- The reference number is 40. The first factor is less by 3, the second is more by 2.
- Add 2 to the smaller factor and multiply by the reference number: 39 × 40 = 1560.
- Subtract the product of 3 and 2: 1440 − 3×2 = 1554.
Trachtenberg method
Since the Trachtenberg method is not entirely familiar, when mastering it it is better to have the multipliers before your eyes. In the future, practice without writing down the original numbers.
Let's look at the method using the example of multiplying 87 by 32.
- Present the numbers sequentially: 8732. Multiply the two inner numbers (7 and 3), the two outer numbers (8 and 2) and add. That turns out to be 37.
- Multiply the tens: 80x30 = 2400. Add 37x10. It turns out 2770.
- Add the product of ones (7 and 2). Total 2784.
Of all the sciences, mathematics enjoys special respect because its theorems are absolutely true and indisputable, while the laws of other sciences are to a certain extent controversial and there is always the danger of their refutation by new discoveries.
Primary school students should be able to perform simple arithmetic calculations in their heads. For example, children should be able to add and subtract two- and three-digit numbers mentally.
For adults, adding and subtracting two-digit and three-digit numbers does not cause difficulties, since an adult has independently developed for himself methods of basic mental calculation.
80 - 67 = 80 - 60 - 7 = 20 - 7 = 13 (separate the ones place when subtracting)
Combinations of different methods
79 - 50 (adding one to the numbers)
70 - 50 + 9 = 20 + 9 = 29 (units division)
80 + 67 (transfer of one from the number 68 to the number 79)
80 + 67 = 80 + 20 + 47 = 100 + 47 = 147
In similar ways, three-digit numbers can be easily added and subtracted in the mind.
300 + 57 (+3) + 38(-3) (transfer of three from 38 to 57)
287 (+1) - 29 (+1) (adding one to the minuend and to the subtrahend)
419-297(400-200), 219 (+3) - 97 (+3) (adding three to the minuend and to the subtrahend).
One of the techniques for accelerated multiplication is the technique of cross multiplication, which is very convenient when working with two-digit numbers. The method is not new; it goes back to the Greeks and Hindus and in ancient times was called the “lightning method” or “cross multiplication.”
"Multiplying with a cross."
Let’s say we need to multiply 2432. Mentally arrange the numbers according to the following scheme, one below the other:
Now we perform the following steps sequentially:
1) 42=8 is the last digit of the result;
2) 22=4; 43=12; 4+12=16; 6 is the average number of the result; we remember the unit;
3) 23 = 6 and also a unit retained in the mind, we have 7 - this is the first digit of the result.
We get all the digits of the product: 7, 6, 8=768
Another method, which consists in the use of so-called “supplements,” is conveniently used in those cases. when the numbers being multiplied are close to 100. The obtained result is correct, as can be clearly seen from the following transformations;
8896=88(100-4)=88100-884
496= 4(88+8)= 48+884
929 =8832+0
Multiplication table for "9".
There are a huge variety of techniques for speeding up the execution of arithmetic operations, techniques intended for everyday calculations.
Squaring numbers ending in "5".
To square a number, for example 65, you need to add 1 to the tens place (i.e. 6+1=7) and multiply 6*7=42, and 5*5=25. So =4225
| 35*35 | =1225 3*4=12 |
all answers end with the number 25. But how do you get the first two digits of the answer? They are obtained by multiplying the tens digit by the following natural number. To square a number, for example 65, you need to add 1 to the tens place (i.e. 6+1=7) and multiply 6*7=42, and 5*5=25. So =4225.
Memorizing a table of Sin, Cos, tg values for acute angles.
You see, the fingers of the left hand form angles:
little finger-0 (zero finger)
ring-30 (first finger)
middle-45 (second finger)
index - 60 (third finger)
thumb-90 (fourth finger)
Knowing the sines, you can fill in the cosines (vice versa), tangents and cotangents of acute angles.
Method for multiplying numbers close to 100
Example: 95 * 93
To get the last 2 digits of the answer (tens and ones), you need
To get the first 2 digits of the answer (thousands and hundreds), you need
4) 93 - 5 = 88 or (95 - 7 = 88)
We get 8835
Example 2: 98 * 92
We get 9016
Let's assume that you need to multiply 92 * 96. The addition for 92 to 100 will be 8, and for 86 - 4. The action is carried out according to the following scheme:
Multipliers: 92 and 96.
Additions: 8 and 4.
The first two digits of the result are obtained by simply subtracting the multiplicand from the “complement” of the multiplicand, or vice versa: i.e. 4 is subtracted from 92 or from 96-8. In both cases we have 88; the product of “additions” is added to this number: 8?4 = 32. We get the result 8832.
Another example - you need to multiply 78 by 77:
Multipliers: 78 and 77.
Additions: 22 and 23.
Numbers 1, 5 and 6
Probably everyone knows that multiplying a series of numbers ending in 1, 5 or 6 produces a number ending with the same digit.
46 = 2116; 46 = 97 336
Extraction from under the root
1). To extract a number from the root, for example, divide this number by two digits from right to left like this: = 568
1. Divide the number (5963364) into pairs from right to left (5`96`33`64)
2. Take the square root of the first group on the left (number 2). This is how we get the first digit of the number.
3. Find the square of the first digit (2 2 =4).
4. Find the difference between the first group and the square of the first digit (5-4=1).
5. We take down the next two digits (we get the number 196).
6. Double the first digit we found and write it on the left behind the line (2*2=4).
7. Now we need to find the second digit of the number: double the first digit we found becomes the tens digit of the number, when multiplied by the number of units, we need to get a number less than 196 (this is the number 4, 44*4=176). 4 is the second digit of the number.
8. Find the difference (196-176=20).
9. We demolish the next group (we get the number 2033).
10. Double the number 24, we get 48.
11. There are 48 tens in a number, when multiplied by the number of ones, we should get a number less than 2033 (484*4=1936). The units digit we found (4) is the third digit of the number.
The numbers 10, 11, 12, 13 and 14 have an amazing feature. Who would have thought that
10 2 + 11 2 + 12 2 = 13 2 + 14 2. Let's prove it: 100 + 121 +144 = 169 + 196
Addition of numbers close to each other in magnitude.
In the practice of technical and trading calculations, there are often cases when it is necessary to add columns of numbers that are close to each other in size. For example;
To add such numbers, the following technique is used
40*7=280, 3-2-1+5+1-1+2=7, 280+7=287.
We find the sum in the same way:
750*6+1=4501
The arithmetic mean of numbers that are close in magnitude
| Rub. |
| 465 |
| 473 |
| 475 |
| 467 |
| 478 |
| 474 |
| 468 |
| 472 |
They do the same thing when they find the arithmetic mean of numbers that are close in value. Let us find, for example, the average of the following prices:
We eyeball a round price close to the average, i.e. 470 rubles. We write down the deviations of all prices from the average: surpluses with a plus sign, deficiencies with a - sign.
We get: -5+3+5-3+8+4-2+2=12. Dividing the sum of deviations by their number. We have: 12:8 = 1.5.
Hence the required average price is 470 + 1.5 = 471.5 (471 rubles 50 kopecks).
Multiplication by numbers 5, 25, 125
Let's move on to multiplication.
Here, first of all, we point out that multiplication by the numbers 5, 25, 125 is significantly accelerated if we keep in mind the following:
Therefore, for example,
Multiply by 15.
When multiplying by 15, you can use the fact that
So it's easy to do mental calculations like this:
36*15=360*1=360+180=540,
Or simpler: 36*1*10=540;
Multiply by 11.
When multiplying by 11 there is no need to write five lines:
It is enough just to sign it again under the multiplied number, moving it one digit:
4213 or 4213 and add.
It is useful to remember the results of multiplying the first nine numbers by 12, 13, 14, 15. Then multiplying multi-digit numbers by such factors is significantly faster. Let it be required to multiply
Let's do it this way. We multiply each digit of the multiplicand in our minds immediately by 13:
7*13=91; 1 we write, 9 we remember;
8*13=104;104+9=113; 3 we write, 11 we remember;
5*13=65;65+11=76; 6 we write; 7 remember;
4*13=52; 52+7=59.
Total 59631.
After several exercises this technique is easy to remember.
A very convenient technique exists for multiplying two-digit numbers by 11: you need to move the digits of the multiplicand apart and enter their sum between them:
If the sum of digits is two-digit, then the number of its tens is added to the first digit of the multiplicand:
48*11=4(12)8, that is 528.
Division by 5; 25; 125.
Let us indicate some methods of accelerated division.
When dividing by 5, multiply the dividend and the divisor by 2:
3471:5=6942:10=694,2
When dividing by 25, multiply both numbers by 4:
3471;25=13884:100=138.84. Do the same when dividing by 1 (= 1.5) and 2 (= 2.5); 3471: 1=6942:3=2314; 3471: 2.5=13884:10=1388.4
Russian method of humiliation.
Here's an example:
32*13; 16*26; 8*52; 4*104; 2*208; 1*416
Dividing in half continues until the quotient reaches 1, while simultaneously doubling the other number. The last doubled number gives the desired result.
What should you do if you have to divide an odd number in half? If the number is odd, remove one and divide the remainder in half; but to the last number of the right column you will need to add all those numbers of this column that stand opposite the odd numbers of the left column: the sum will be the required product. 19 * 17; 9*34; 4*68; 2*136; 1*272. Adding the uncrossed numbers, we get the correct result: 17+34+272=323.
Multiplying numbers ending in 5.
When multiplying a pair of numbers in which the tens digits were even or odd, and the ones digit was 5, you need to multiply the tens digits and add half the sum of these digits to their product. We get the number of hundreds. To the number of hundreds you need to add the product 5*5=25.
For example:
85*45=(8*4+(8+4)/2)hundreds+5*5=38*100+25=3825
35*55=(3*5+(3+5)/2)hundreds+5*5=19*100+25=1925
Let's take an example that is familiar to us from 5th grade.
Find the sum of the first hundred natural numbers:
1+2+3+4+5+6+ : +94+95+96+97+98+99+100=?
How easy is it to calculate the following example:
34*48+18*12+23*24=34*2*24+9*24+23*24=24*(68+9+23)=24*100=2400
You can independently create examples for each rule and practice mental calculations. When creating examples and completing assignments, the children do not experience any difficulties.
Literature:
- Encyclopedia for children. Mathematics. M., Avanta, 2002.
- Ya.I.Perelman, Entertaining arithmetic. M., 1954.
- Magazine "Practical magazine for teachers and school administration". No. 9, 2004.
- J. "Mathematics", No. 4, 1994.
Multiplying two-digit numbers is a skill that is essential to our daily lives. People are constantly faced with the need to multiply something in their minds: the price tag in a store, the mass of products, or the size of a discount. But how to multiply two-digit numbers quickly and without problems? Let's figure it out.
How to multiply a two-digit number by a one-digit number?
Let's start with a simple problem - how to multiply two-digit numbers by single-digit numbers.
To begin with, a two-digit number is a number that consists of a certain number of tens and units.
In order to multiply a two-digit number by a single-digit number in a column, you need to write the desired two-digit number, and below it the corresponding single-digit number. Next, you should alternately multiply by a given number, first by units, and then by tens. If, when multiplying units, the result is a number greater than 10, then the number of tens must simply be transferred to the next digit by adding them.
Multiplying two-digit numbers by tens
Multiplying two-digit numbers by tens is not much more difficult than multiplying by single-digit numbers. The basic procedure remains the same:
- Write down the numbers one below the other in a column, with the zero supposed to be “on the side” so as not to interfere with arithmetic operations.
- Multiply a two-digit number by the number of tens, do not forget about transferring some digits to the next digits.
- The only thing that distinguishes this example from the previous one is that you need to add a zero at the end of the resulting answer, so that the tens that were omitted at the beginning become taken into account.
How to multiply two two-digit numbers?
Once you have fully understood the multiplication of two-digit and single-digit numbers, you can begin to think about how to multiply two-digit numbers by each other in a column. In fact, this action shouldn't require much effort from you either, since the principle is still the same.
- We write these numbers in a column - ones under units, tens under tens.
- We start multiplication from one in the same way as in the examples with single-digit numbers.
- After you have obtained the first number by multiplying the units by a given figure, you need to multiply the tens by the same figure in the same way. Attention: the answer must be written strictly under tens. The empty space below the units is an unaccounted zero. You can write it down if you prefer.
- Having multiplied both tens and ones and received two numbers written one under one, they need to be added into a column. The resulting value is the answer.
How to multiply two-digit numbers correctly? To do this, it is not enough to simply read or learn the instructions provided. Remember, in order to master the principle of how to multiply two-digit numbers, first of all you need to constantly practice - solve as many examples as possible, use the calculator as little as possible.
How to multiply in your head
After learning how to multiply brilliantly on paper, you may wonder how to quickly multiply two-digit numbers in your head.
Of course, this is not the easiest task. It requires some concentration, good memory, and the ability to retain a certain amount of information in your head. However, this can also be learned with enough effort, especially if you choose the right algorithm. Obviously, it's easiest to multiply by round numbers, so the easiest way is to factor the numbers.
- First, you need to divide one of these two-digit numbers into tens. For example, 48 = 4 × 10 + 8.
- Next, you need to sequentially multiply first the ones, and then the tens with the second number. These are quite difficult operations to perform mentally, since you need to simultaneously multiply numbers by each other and keep the resulting result in mind. You'll probably have a hard time getting this right the first time, but it's a skill that can be developed if you're diligent enough, because understanding how to correctly multiply two-digit numbers in your head is only possible with practice.
Some tricks for multiplying two-digit numbers
But is there an easier way to multiply two-digit numbers in your head, and how can you do it?
There are several tricks. They will help you multiply two-digit numbers quickly and easily.
- When multiplying by eleven, you simply put the sum of the tens and ones in the middle of the given two-digit number. For example, we needed to multiply 34 by 11.
We put 7 in the middle, 374. This is the answer.
If you add a number greater than 10, you should simply add one to the first number. For example, 79 × 11.
- Sometimes it's easier to factor a number and multiply them sequentially. For example, 16 = 2 × 2 × 2 × 2, so you can simply multiply the original number by 2 4 times.
14 = 2 × 7, so when doing math you can multiply first by 7 and then by 2.
- To multiply a number by multiples of 100, such as 50 or 25, you can multiply that number by 100 and then divide by 2 or 4, respectively.
- You also need to remember that sometimes when multiplying it is easier not to add, but to subtract numbers from each other.
For example, to multiply a number by 29, you can first multiply it by 30, and then subtract this number from the resulting number once. This rule is true for any tens.
How to quickly multiply large numbers, how to master such useful skills? Most people find it difficult to verbally multiply two-digit numbers by single-digit numbers. And there is nothing to say about complex arithmetic calculations. But if desired, the abilities inherent in each person can be developed. Regular training, a little effort and the use of effective techniques developed by scientists will allow you to achieve amazing results.
Choosing traditional methods
Methods of multiplying two-digit numbers that have been proven for decades do not lose their relevance. The simplest techniques help millions of ordinary schoolchildren, students of specialized universities and lyceums, as well as people engaged in self-development, improve their computing skills.
Multiplication using number expansion
The easiest way to quickly learn to multiply large numbers in your head is to multiply tens and units. First, the tens of two numbers are multiplied, then the ones and tens alternately. The four numbers received are summed up. To use this method, it is important to be able to remember the results of multiplication and add them in your head.
For example, to multiply 38 by 57 you need:
- factor the number into (30+8)*(50+7) ;
- 30*50 = 1500 – remember the result;
- 30*7 + 50*8 = 210 + 400 = 610 – remember;
- (1500 + 610) + 8*7 = 2110 + 56 = 2166
Multiplication by column in the mind
Many people use a visual representation of the usual columnar multiplication in calculations. This method is suitable for those who can memorize auxiliary numbers for a long time and perform arithmetic operations with them. But the process becomes much easier if you learn how to quickly multiply two-digit numbers by single-digit numbers. To multiply, for example, 47*81 you need:
- 47*1 = 47 – remember;
- 47*8 = 376 – remember;
- 376*10 + 47 = 3807.
The above multiplication methods are universal. But knowing more efficient algorithms for some numbers will greatly reduce the number of calculations.
Multiplying by 11
This is perhaps the simplest method that is used to multiply any two-digit numbers by 11.
It is enough to insert their sum between the digits of the multiplier:
13*11 = 1(1+3)3 = 143
If the number in brackets is greater than 10, then one is added to the first digit, and 10 is subtracted from the amount in brackets.
28*11 = 2 (2+8) 8 = 308
Multiplying large numbers
It is very convenient to multiply numbers close to 100 by decomposing them into their components. For example, you need to multiply 87 by 91.
- Each number must be represented as the difference between 100 and one more number:
(100 - 13)*(100 - 9)
The answer will consist of four digits, the first two of which are the difference between the first factor and the subtracted from the second bracket, or vice versa - the difference between the second factor and the subtracted from the first bracket.
87 – 9 = 78
91 – 13 = 78 - The second two digits of the answer are the result of multiplying those subtracted from two parentheses. 13*9 = 144
- As a result, the numbers 78 and 144 are obtained. If, when writing down the final result, a number of 5 digits is obtained, the second and third digits are summed. Result: 87*91 = 7944 .
And multiplication. The multiplication operation will be discussed in this article.
Multiplying numbers
Multiplication of numbers is mastered by children in the second grade, and there is nothing complicated about it. Now we will look at multiplication with examples.
Example 2*5. This means either 2+2+2+2+2 or 5+5. Take 5 twice or 2 five times. The answer, accordingly, is 10.
Example 4*3. Likewise, 4+4+4 or 3+3+3+3. Three times 4 or four times 3. Answer 12.
Example 5*3. We do the same as the previous examples. 5+5+5 or 3+3+3+3+3. Answer 15.
Multiplication formulas
Multiplication is the sum of identical numbers, for example, 2 * 5 = 2 + 2 + 2 + 2 + 2 or 2 * 5 = 5 + 5. Multiplication formula:
Where, a is any number, n is the number of terms of a. Let's say a=2, then 2+2+2=6, then n=3 multiplying 3 by 2, we get 6. Let's look at it in reverse order. For example, given: 3 * 3, that is. 3 multiplied by 3 means that three must be taken 3 times: 3 + 3 + 3 = 9. 3 * 3=9.
Abbreviated multiplication
Abbreviated multiplication is a shortening of the multiplication operation in certain cases, and abbreviated multiplication formulas have been derived specifically for this purpose. Which will help make calculations the most rational and fastest:
Abbreviated multiplication formulas
Let a, b belong to R, then:
The square of the sum of two expressions is equal to the square of the first expression plus twice the product of the first expression and the second plus the square of the second expression. Formula: (a+b)^2 = a^2 + 2ab + b^2
The square of the difference of two expressions is equal to the square of the first expression minus twice the product of the first expression and the second plus the square of the second expression. Formula: (a-b)^2 = a^2 - 2ab + b^2
Difference of squares two expressions is equal to the product of the difference of these expressions and their sum. Formula: a^2 - b^2 = (a - b)(a + b)
Cube of sum two expressions is equal to the cube of the first expression plus triple the product of the square of the first expression and the second plus triple the product of the first expression and the square of the second plus the cube of the second expression. Formula: (a + b)^3 = a^3 + 3a(^2)b + 3ab^2 + b^3
Difference cube two expressions is equal to the cube of the first expression minus triple the product of the square of the first expression and the second plus triple the product of the first expression and the square of the second minus the cube of the second expression. Formula: (a-b)^3 = a^3 - 3a(^2)b + 3ab^2 - b^3
Sum of cubes a^3 + b^3 = (a + b)(a^2 - ab + b^2)
Difference of cubes two expressions is equal to the product of the sum of the first and second expressions and the incomplete square of the difference of these expressions. Formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
Sign up for the course "Speed up mental arithmetic, NOT mental arithmetic" to learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even extract roots. In 30 days, you'll learn how to use easy tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.
Multiplying fractions
While looking at adding and subtracting fractions, the rule was brought up to bring fractions to a common denominator in order to complete the calculation. When multiplying this do No need! When multiplying two fractions, the denominator is multiplied by the denominator, and the numerator by the numerator.
For example, (2/5) * (3 * 4). Let's multiply two thirds by one quarter. We multiply the denominator by the denominator, and the numerator by the numerator: (2 * 3)/(5 * 4), then 6/20, make a reduction, we get 3/10.
Multiplication 2nd grade
The second grade is just the beginning of learning multiplication, so second graders solve simple problems to replace addition with multiplication, multiply numbers, and learn the multiplication table. Let's look at multiplication problems at the second grade level:
Oleg lives in a five-story building, on the top floor. The height of one floor is 2 meters. What is the height of the house?
The box contains 10 packages of cookies. There are 7 of them in each package. How many cookies are in the box?
Misha arranged his toy cars in a row. There are 7 of them in each row, but there are only 8 rows. How many cars does Misha have?
There are 6 tables in the dining room, and 5 chairs are pushed behind each table. How many chairs are there in the dining room?
Mom brought 3 bags of oranges from the store. The bags contain 22 oranges. How many oranges did mom bring?
There are 9 strawberry bushes in the garden, and each bush has 11 berries. How many berries grow on all the bushes?
Roma laid 8 pipe parts one after another, each of the same size, 2 meters each. What is the length of the complete pipe?
Parents brought their children to school on September 1st. 12 cars arrived, each with 2 children. How many children did their parents bring in these cars?
Multiplication 3rd grade
In third grade, more serious tasks are given. In addition to multiplication, Division will also be covered.
Multiplication tasks will include: multiplying two-digit numbers, multiplying by columns, replacing addition with multiplication and vice versa.
Column multiplication:
Column multiplication is the easiest way to multiply large numbers. Let's consider this method using the example of two numbers 427 * 36.
1 step. Let's write the numbers one below the other, so that 427 is at the top and 36 at the bottom, that is, 6 under 7, 3 under 2.
Step 2. We begin multiplication with the rightmost digit of the bottom number. That is, the order of multiplication is: 6 * 7, 6 * 2, 6 * 4, then the same with three: 3 * 7, 3 * 2, 3 * 4.
So, first we multiply 6 by 7, answer: 42. We write it this way: since it turned out 42, then 4 are tens, and 2 are units, the recording is similar to addition, which means we write 2 under the six, and 4 we add the number 427 to the two.
Step 3. Then we do the same with 6 * 2. Answer: 12. The first ten, which is added to the four of the number 427, and the second - ones. We add the resulting two with the four from the previous multiplication.
Step 4. Multiply 6 by 4. The answer is 24 and add 1 from the previous multiplication. We get 25.
So, multiplying 427 by 6, the answer is 2562
REMEMBER! The result of the second multiplication should begin to be written down SECOND number of the first result!
Step 5. We perform similar actions with the number 3. We get the multiplication answer 427 * 3=1281
Step 6. Then we add up the obtained answers during multiplication and get the final multiplication answer 427 * 36. Answer: 15372.
Multiplication 4th grade
The fourth class is already the multiplication of large numbers only. The calculation is performed using the column multiplication method. The method is described above in accessible language.
For example, find the product of the following pairs of numbers:
- 988 * 98 =
- 99 * 114 =
- 17 * 174 =
- 164 * 19 =
Presentation on multiplication
Download a presentation on multiplication with simple tasks for second graders. The presentation will help children better navigate this operation, because it is designed colorfully and in a playful style - the best way for a child to learn!
Multiplication table
Every student in the second grade learns the multiplication table. Everyone should know it!
Sign up for the course "Speed up mental arithmetic, NOT mental arithmetic" to learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even extract roots. In 30 days, you'll learn how to use easy tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.
Examples for multiplication
Multiplying by one digit
- 9 * 5 =
- 9 * 8 =
- 8 * 4 =
- 3 * 9 =
- 7 * 4 =
- 9 * 5 =
- 8 * 8 =
- 6 * 9 =
- 6 * 7 =
- 9 * 2 =
- 8 * 5 =
- 3 * 6 =
Multiplying by two digits
- 4 * 16 =
- 11 * 6 =
- 24 * 3 =
- 9 * 19 =
- 16 * 8 =
- 27 * 5 =
- 4 * 31 =
- 17 * 5 =
- 28 * 2 =
- 12 * 9 =
Multiplying two-digit by two-digit
- 24 * 16 =
- 14 * 17 =
- 19 * 31 =
- 18 * 18 =
- 10 * 15 =
- 15 * 40 =
- 31 * 27 =
- 23 * 25 =
- 17 * 13 =
Multiplying three-digit numbers
- 630 * 50 =
- 123 * 8 =
- 201 * 18 =
- 282 * 72 =
- 96 * 660 =
- 910 * 7 =
- 428 * 37 =
- 920 * 14 =
Games for developing mental arithmetic
Special educational games developed with the participation of Russian scientists from Skolkovo will help improve mental arithmetic skills in an interesting game form.
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 "Mathematical matrices"
"Mathematical Matrices" is great brain exercise for kids, which will help you develop his mental work, mental calculation, quick search for the necessary components, and attentiveness. The essence of the game is that the player has to find a pair from the proposed 16 numbers that will add up to a given number, for example in the picture below the given number is “29”, and the desired pair is “5” and “24”.
Game "Number Span"
The number span game will challenge your memory while practicing this exercise.
The essence of the game is to remember the number, which takes about three seconds to remember. Then you need to play it back. As you progress through the stages of the game, the number of numbers increases, starting with two and further.
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.
Game "Quick addition"
The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers whose sum is equal to a given number. In this game, a matrix from one to sixteen is given. A given number is written above the matrix; you need to select the numbers in the matrix so that the sum of these digits is equal to the given number. If you answered correctly, you score points and continue playing.
Visual Geometry Game
The game "Visual Geometry" develops thinking and memory. The main essence of the game is to quickly count the number of shaded objects and select it from the list of answers. In this game, blue squares are shown on the screen for a few seconds, you need to quickly count them, then they close. Below the table there are four numbers written, you need to select one correct number and click on it with the mouse. If you answered correctly, you score points and continue playing.
Game "Mathematical Comparisons"
The game "Mathematical Comparisons" develops thinking and memory. The main essence of the game is to compare numbers and mathematical operations. In this game you need to compare two numbers. At the top there is a question written, read it and answer the question correctly. You can answer using the buttons below. There are three buttons “left”, “equal” and “right”. If you answered correctly, you score points and continue playing.
Development of phenomenal mental arithmetic
We have looked at only the tip of the iceberg, to understand mathematics better - sign up for our course: Accelerating mental arithmetic.
From the course you will not only learn dozens of techniques for simplified and quick multiplication, addition, multiplication, division, and calculating percentages, but you will also practice them in special tasks and educational games! Mental arithmetic also requires a lot of attention and concentration, which are actively trained when solving interesting problems.
Speed reading in 30 days
Increase your reading speed by 2-3 times in 30 days. From 150-200 to 300-600 words per minute or from 400 to 800-1200 words per minute. The course uses traditional exercises for the development of speed reading, techniques that speed up brain function, methods for progressively increasing reading speed, the psychology of speed reading and questions from course participants. Suitable for children and adults reading up to 5000 words per minute.
Development of memory and attention in a child 5-10 years old
The course includes 30 lessons with useful tips and exercises for children's development. Each lesson contains useful advice, several interesting exercises, an assignment for the lesson and an additional bonus at the end: an educational mini-game from our partner. Course duration: 30 days. The course is useful not only for children, but also for their parents.
Super memory in 30 days
Remember the necessary information quickly and for a long time. Wondering how to open a door or wash your hair? I’m sure not, because this is part of our life. Easy and simple exercises for memory training can be made part of your life and done a little during the day. If you eat the daily amount of food at once, or you can eat in portions throughout the day.
Secrets of brain fitness, training memory, attention, thinking, counting
The brain, like the body, needs fitness. Physical exercise strengthens the body, mental exercise develops the brain. 30 days of useful exercises and educational games to develop memory, concentration, intelligence and speed reading will strengthen the brain, turning it into a tough nut to crack.
Money and the Millionaire Mindset
Why are there problems with money? In this course we will answer this question in detail, look deep into the problem, and consider our relationship with money from psychological, economic and emotional points of view. From the course you will learn what you need to do to solve all your financial problems, start saving money and invest it in the future.
Knowledge of the psychology of money and how to work with it makes a person a millionaire. 80% of people take out more loans as their income increases, becoming even poorer. On the other hand, self-made millionaires will earn millions again in 3-5 years if they start from scratch. This course teaches you how to properly distribute income and reduce expenses, motivates you to study and achieve goals, teaches you how to invest money and recognize a scam.
Don't like math? You just don't know how to use it! It's actually fascinating science. And our selection of unusual multiplication methods confirms this.
Multiply on your fingers like a merchant
This method allows you to multiply numbers from 6 to 9. To begin, bend both hands into fists. Then on your left hand, bend as many fingers as the first factor is greater than the number 5. On your right hand, do the same for the second factor. Count the number of extended fingers and multiply the sum by ten. Now multiply the sum of the bent fingers of the left and right hands. By adding both sums, you get the result.
Example. Let's multiply 6 by 7. Six is more than five by one, which means we bend one finger on our left hand. And seven is two, which means there are two fingers on the right. The total is three, and after multiplying by 10 it is 30. Now let’s multiply the four bent fingers of the left hand and three of the right. We get 12. The sum of 30 and 12 gives 42.
Actually, here we are talking about a simple multiplication table, which it would be good to know by heart. But this method is good for self-testing, and it’s also useful to stretch your fingers.
Multiply like Ferrol
This method was named after the German engineer who used it. Method allows you to quickly multiply numbers from 10 to 20. If you practice, you can do it even in your head.
The point is simple. The result will always be a three-digit number. So first we count units, then tens, then hundreds.
Example. Let's multiply 17 by 16. To get units, multiply 7 by 6, tens - add the product of 1 and 6 with the product of 7 and 1, hundreds - multiply 1 by 1. As a result, we get 42, 13 and 1. For convenience, we write them in a column and let's add it up That's the result!
Multiply like a Japanese
This graphic method, which is used by Japanese schoolchildren, makes it easy to multiply two- and even three-digit numbers. To try it out, have some paper and pen ready.
Example. Let's multiply 32 by 143. To do this, draw a grid: reflect the first number with three and two lines with a horizontal indent, and the second with one, four and three lines vertically. Place dots where the lines intersect. As a result, we should get a four-digit number, so we will conditionally divide the table into 4 sectors. And let's count the points that fall into each of them. We get 3, 14, 17 and 6. To get the answer, add the extra ones from 14 and 17 to the previous number. We get 4, 5 and 76 - 4576.
Multiply like an Italian
Another interesting graphic method is used in Italy. Perhaps it is simpler than the Japanese one: you definitely won’t get confused when transferring tens. To multiply large numbers using it, you need to draw a grid. We write down the first factor horizontally from above, and the second factor vertically to the right. In this case, there should be one cell for each number.
Now let's multiply the numbers in each row by the numbers in each column. We write the result in a cell (divided in two) at their intersection. If you get a single-digit number, then write 0 in the upper part of the cell, and the resulting result in the lower part.
All that remains is to add up all the numbers in the diagonal stripes. We start from the bottom right cell. In this case, we add tens to the ones in the adjacent column.
This is how we multiplied 639 by 12.
Fun, right? Have fun with mathematics! And remember that humanities specialists are also needed in IT!