How to multiply 2 digit numbers in a column. How to quickly multiply two-digit numbers in your head? Visualization of multiplication in a column

The most popular technique for multiplying large numbers in your head is the technique of using the so-called reference number. In the last lesson, when we showed how to multiply numbers up to 20, in fact, we used the pivot number 10. It is also worth noting that you can read more about the methodology for using the pivot number in the book "" by Bill Handley.

General rules for using the reference number

The reference number is useful when multiplying close numbers and when squaring. You already understood how you can use the pivot number method from the last lesson, now let's summarize everything that has been said.

The reference number in multiplication is the number to which both factors are close and by which it is convenient to multiply. When multiplying numbers up to 100 with reference numbers, it is convenient to use all multiples of 10, and especially 10, 20, 50 and 100.

The technique for using the reference number depends on whether the factors are greater than or less than the reference number. There are three possible cases here. We will show all 3 methods with examples.

Both numbers are less than the reference (under the reference)

Let's say we want to multiply 48 by 47. These numbers are close enough to 50 that it's convenient to use 50 as a reference number.

To multiply 48 by 47 using the reference number 50, you need:

47*48

1. From 47, subtract as much as 48 is missing to 50, that is, 2. It turns out 45 (or subtract 3 from 48 - it's always the same)
2. Then 45 times 50 = 2250
3. Then we add 2*3 to this result and voila - 2256!

Schematically in the mind it is convenient to imagine the table below.

The reference number is written to the left of the product. If the numbers are less than the reference, then the difference between them and the reference is written below these numbers. To the right of 48 * 47 we write the calculation with the reference number, to the right of the remainders 2 and 3 we write their product.

If we use a simplified scheme, then the solution looks like this: 47*48=45*50 + 6= 2 256

Let's see other examples:

Multiply 18*19

Short entry: 18*19 = 20*17+2 = 342

Multiply 8*7

Short entry: 8*7 = 10*5+6 = 56

Multiply 98*95

Short entry: 98*95 = 100*93 + 10 = 9 310

Multiply 98*71

Short entry: 98*71 = 100*69 + 58 = 6 958

Both numbers are greater than the reference (above the reference)

Let's say we want to multiply 54 by 53. These numbers are close enough to 50 that it's convenient to use 50 as a reference number. But unlike the previous examples, these numbers are larger than the reference. In fact, the model of their multiplication does not change, but now you need not to subtract the remainders, but to add them.

1. Add to 54 as much as 53 exceeds 50, that is, 3. It turns out 57 (or add 4 to 53 - it's always the same)
2. Then multiply 57 by 50 = 2850 (multiplying by 50 is similar to dividing by 2)
3. Then we add 4*3 to this result. Answer: 2862

The short solution looks like this: 50*57+12 = 2862

For clarity, examples are given below:

Multiply 23*27

Short entry: Short entry: 23*27 = 20*30 + 21 = 621

Multiply 51*63

Short entry: Short entry: 51*63 = 64*50 + 13 = 3213

One number is below the pivot and the other is above

The third use of the reference number is when one number is greater than the reference number and the other is less. Such examples are not more difficult to solve than the previous ones.

Multiply 45*52

The product of 45 * 52 is considered as follows:

1. Subtract 5 from 52 or add 2 to 45. In either case, we get: 47
2. Then multiply 47 by 50 = 2350 (multiplying by 50 is similar to dividing by 2)
3. Then we subtract (and not add, as before!) 2 * 5. Answer: 2 340

Short entry: 45*52 = 47*50-10 = 2340

We also do with similar examples:

Multiply 91*103

Only one number is close to the reference, and the other is not

As you have already seen from the examples, it is convenient to use the pivot number even if only one number is close to the pivot. It is desirable that the difference between this number and the reference number should be no more than 2 or 3, or be equal to the number by which it is convenient to multiply (for example, 5, 10, 25 - see the second lesson)

Multiply 48*73

Short solution: 48*73 = 71*50 - 23*2 = 3 504

Multiply 23*69

Short entry: Short solution: 23*69 = 72*20 + 147 = 1587 - a bit more difficult

2

*

59

+118

Answer:

4018

Short entry: Short entry: 98*41 = 100*39 + 118 = 4018

Thus, by using one reference number, a large combination of two-digit numbers can be multiplied. If you are good at multiplying by 30, 40, 60, 70, or 80, then you can use this technique to multiply any number (up to 100 and even more).

Using Multiple Reference Numbers

The multiplication technique using reference numbers allows the use of 2 reference numbers. This is convenient when the reference number of one factor can be expressed in terms of the reference number of another. For example, in the product "23 * 88" it is convenient to use the reference number 20 for 23 and 80 for 88. Multiplying these numbers using two reference numbers is convenient because 20=80:4.

The technique of 2 reference numbers is that we first divide 88 by 4 and get 22, multiply 23 by 22 and multiply the product again by 4. That is, we first divide the product by 4, and then multiply by 4. It turns out: 23*22 = 250*2+6= 506 and 506*4 = 2024 is the answer!

For visualization, you can use the already familiar scheme. The product of 23 * 88 is calculated as follows:

1. We write down a convenient reference number "20" and next we attribute a factor of 4, with which you can express 80 through 20.
2. Then we do, as before, we write how much 23 exceeds 20 (3), and 88 exceeds 80 (8).
3. Above the triple we write the product 3 by 4 (that is, 3 by the reference multiplier).
4. To 88 we add the product of 3 by 4 and multiply by the reference (20), it turns out 100 * 20 \u003d 2000
5. We add to 2000 by the product of 3 and 8. Result: 2024

Short entry: 23*88 = (23-12:4)*100 + 24 = 2024

As you can see, the answer is the same.

The method using two reference numbers is somewhat more complicated and requires additional steps. First, you need to figure out which 2 base numbers are comfortable for you to use. Secondly, you need to perform an additional action to find the number that needs to be multiplied by the reference.

It is better to use this technique when you have already mastered multiplication with one reference number quite well.

Training

If you want to improve your skills on the topic of this lesson, you can use the following game. The points you receive are affected by the correctness of your answers and the time spent on passing. Please note that the numbers are different each time.

Habitual school mathematics can be very practical in everyday life, because it makes it possible to carry out serious arithmetic calculations in the mind. We will tell you a few tricks to help you multiply two-digit numbers quickly without using a calculator or a piece of paper and a pen.

How to multiply two digit numbers mentally?

It may seem that it is impossible to multiply such large numbers mentally, but it is not. There is a way that even schoolchildren will understand.

So, for example, take the numbers 96 and 97.

Calculate the difference between these numbers relative to 100. In our case, these are 3 and 4. Their product will be the second part of the solution for multiplying the numbers 97 and 96 (3*4=12).

The first part will be the difference between the first number and the difference between 100 and the second number. In our example, this is: 97-4=93.

Thus, we get 97*96 = 93 12

How to quickly multiply in your mind?

The essence of this simple and familiar method is to decompose factors into units and tens. Then their successive multiplication follows. This is easy to do, you will have to keep no more than 3 numbers in your mind at the same time.

Here is the standard way to do this multiplication:

64*86 = (60+4)*(80+6) = 60*80 + 60*6 + 4*80 + 4*6 = 4800 + 360 + 320 + 24 = 5504

And here is a method designed for only 3 steps.

1 ) Multiply tens of 60 and 80. The result is 4800, remember it.
2 ) Add the products 60 * 6 and 80 * 4. It turns out 680. Remember this number too.
3 ) Multiply units 4 * 6 = 24 and add all three numbers. 4800 + 680 +24 = 5504.

See how easy it is to multiply in your mind!

Don't like math? You just don't know how to use it! In fact, it is a 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. First, bend both hands into fists. Then, on the left hand, bend as many fingers as the first factor is greater than the number 5. On the right, do the same for the second factor. Count the number of extended fingers and multiply the amount by ten. Now multiply the sum of the bent fingers of the left and right hands. Adding both sums, you get the result.

Example. Multiply 6 by 7. Six is ​​more than five by one, which means we bend one finger on the left hand. And seven - two, so on the right - two fingers. In total, this is three, and after multiplying by 10 - 30. Now we multiply four bent fingers of the left hand and three - of the right. We get 12. The sum of 30 and 12 will give 42.

In fact, here we are talking about a simple multiplication table, which would be nice to know by heart. But this method is good for self-examination, and stretching your fingers is useful.

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 mind.

The point is simple. The result will always be a three-digit number. So first we count the ones, then the tens, then the hundreds.

Example. Multiply 17 by 16. To get units, we multiply 7 by 6, tens - we add the product of 1 and 6 with the product of 7 and 1, hundreds - we multiply 1 by 1. As a result, we get 42, 13 and 1. For convenience, we write them in a column and add up. Here is the result!

Multiply like a Japanese

This graphic method used by Japanese schoolchildren allows you to easily multiply two- and even three-digit numbers. Get some paper and a pen ready to try it out.

Example. Multiply 32 by 143. To do this, draw a grid: reflect the first number with three and two horizontally indented lines, and the second with one, four and three vertically indented lines. 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 recalculate the points that fall into each of them. We get 3, 14, 17 and 6. To get the answer, add the extra ones for 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 Japanese: you definitely won’t get confused when transferring dozens. To multiply large numbers with it, you need to draw a grid. We write the first multiplier horizontally from above, and the second one vertically to the right. In this case, there should be one cell for each digit.

Now 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 result obtained in the lower part.

It remains to add up all the numbers that are in the diagonal stripes. We start from the bottom right cell. At the same time, tens are added to the units in the next column.

Here's how we multiplied 639 by 12.

Fun, right? Have fun with mathematics! And remember that the humanities in IT are also needed!

With the best free game, learn very fast. Check it out yourself!

Learn multiplication table - game

Try our educational e-game. Using it, tomorrow you will be able to solve math problems in the classroom at the blackboard without answers, without resorting to a tablet to multiply numbers. One has only to start playing, and after 40 minutes there will be an excellent result. And to consolidate the result, train several times, not forgetting the breaks. Ideally, every day (save the page so you don't lose it). The game form of the simulator is suitable for both boys and girls.

See the full cheat sheet below.

Multiplication directly on the site (online)

How to multiply numbers by a column (mathematics video)

To practice and learn quickly, you can also try to multiply numbers by a column.

And multiplication. Just about the operation of multiplication and will be discussed in this article.

Number multiplication

Multiplication of numbers is mastered by children in the second grade, and there is nothing complicated about it. Now we will look at multiplication by examples.

Example 2*5. This means either 2+2+2+2+2 or 5+5. We take 5 two times or 2 five times. The answer is 10 respectively.

Example 4*3. Similarly, 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. The multiplication formula is:

Where, a is any number, n is the number of terms a. Let's say a=2, then 2+2+2=6, then n=3 multiplying 3 by 2, we get 6. Consider in reverse order. For example, given: 3 * 3, that is. 3 multiplied by 3 - this means that the three must be taken 3 times: 3 + 3 + 3 \u003d 9. 3 * 3 \u003d 9.

Abbreviated multiplication

Abbreviated multiplication is an abbreviation of the multiplication operation in certain cases, and formulas for abbreviated multiplication have been developed specifically for this. Which will help to make the calculations the most rational and fast:

Abbreviated multiplication formulas

Let a, b belong to R, then:

The square of the sum of two expressions is 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 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)

sum cube of two expressions is equal to the cube of the first expression plus three times the square of the first expression times the second plus three times the product of the first expression times 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 of two expressions is equal to the cube of the first expression minus three times the product of the square of the first expression and the second plus three times 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 by the incomplete square of the difference of these expressions. Formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

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Multiplication of fractions

Considering the addition and subtraction of fractions, the rule was voiced, bringing fractions to a common denominator in order to perform 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). 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, we make a reduction, we get 3/10.

Multiplication Grade 2

The second grade is just the beginning of learning multiplication, so second graders solve the simplest tasks to replace addition with multiplication, multiply numbers, learn the multiplication table. Let's look at multiplication tasks 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 packs of biscuits. Each pack contains 7 pieces. How many cookies are in the box?

Misha arranged his toy cars in a row. There are 7 of them in each row, and 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 in the dining room?

Mom brought 3 bags of oranges from the store. The packages contain 22 oranges. How many oranges did mom bring?

There are 9 strawberry bushes growing in the garden, and 11 berries grow on each bush. How many berries grow on all the bushes?

Roma put 8 pipe parts one after the other, the same size of 2 meters. What is the length of the full pipe?

Parents brought their children to school on the first of September. 12 cars arrived, each with 2 children. How many children did their parents bring in these cars?

Multiplication Grade 3

In the third grade, more serious tasks are given. In addition to multiplication, division will also be passed.

Among the tasks for multiplication will be: multiplication of two-digit numbers, multiplication by a column, replacement of addition by multiplication and vice versa.

Column multiplication:

Column multiplication is the easiest way to multiply large numbers. Consider this method using the example of two numbers 427 * 36.

1 step. Let's write the numbers one under the other, so that 427 is at the top and 36 is at the bottom, that is, 6 under 7, 3 under 2.

2 step. We start 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 the triple: 3 * 7, 3 * 2, 3 * 4.

So, first multiply 6 by 7, the answer is: 42. We write it down like this: since it turned out 42, then 4 are tens, and 2 are ones, the recording is similar to addition, which means we write 2 under the six, and 4 is added to the two of the number 427.

3 step. 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 - units. We add the resulting two with the four from the previous multiplication.

4 step. 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 be written down under SECOND number of the first result!

5 step. We perform similar actions with the number 3. We get the multiplication answer 427 * 3 = 1281

6 step. Then we add the received answers when multiplying and get the final answer of the multiplication 427 * 36. Answer: 15372.

Multiplication Grade 4

The fourth class is the multiplication of only large numbers. The calculation is performed by the multiplication method in a column. The method is described above in an accessible language.

For example, find the product of the following pairs of numbers:

1. 988 * 98 =
2. 99 * 114 =
3. 17 * 174 =
4. 164 * 19 =

Multiplication Presentation

Download a presentation on multiplication with the simplest tasks for second graders. The presentation will help children navigate this operation better, because it is presented in a colorful and playful way - in the best way for a child to learn!

Multiplication table

The multiplication table is studied by every student in the second grade. Everyone must know it!

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Multiplication Examples

Multiplication by unambiguous

1. 9 * 5 =
2. 9 * 8 =
3. 8 * 4 =
4. 3 * 9 =
5. 7 * 4 =
6. 9 * 5 =
7. 8 * 8 =
8. 6 * 9 =
9. 6 * 7 =
10. 9 * 2 =
11. 8 * 5 =
12. 3 * 6 =

Multiplication by two digits

1. 4 * 16 =
2. 11 * 6 =
3. 24 * 3 =
4. 9 * 19 =
5. 16 * 8 =
6. 27 * 5 =
7. 4 * 31 =
8. 17 * 5 =
9. 28 * 2 =
10. 12 * 9 =

Two-digit multiplication by two-digit

1. 24 * 16 =
2. 14 * 17 =
3. 19 * 31 =
4. 18 * 18 =
5. 10 * 15 =
6. 15 * 40 =
7. 31 * 27 =
8. 23 * 25 =
9. 17 * 13 =

Multiplication of three-digit numbers

1. 630 * 50 =
2. 123 * 8 =
3. 201 * 18 =
4. 282 * 72 =
5. 96 * 660 =
6. 910 * 7 =
7. 428 * 37 =
8. 920 * 14 =

Games for the development of mental counting

Special educational games developed with the participation of Russian scientists from Skolkovo will help improve oral counting skills in an interesting game form.

Game "Quick Score"

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" great brain exercise for kids, which will help you develop his mental work, mental counting, quick search for the right components, attentiveness. The essence of the game is that the player has to find a pair from the proposed 16 numbers that will give a given number in total, for example, in the picture below, this number is “29”, and the desired pair is “5” and “24”.

Game "Numerical coverage"

The game "number coverage" will load your memory while practicing with this exercise.

The essence of the game is to remember the number, which takes about three seconds to memorize. Then you need to play it. As you progress through the stages of the game, the number of numbers grows, start with two and go on.

Game "Guess the operation"

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

Game "Simplify"

The game "Simplify" 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 action is given, the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need with the mouse. If you answer correctly, you score points and continue playing.

Game "Fast Addition"

The game "Quick Addition" develops thinking and memory. The main essence of the game is to choose numbers, the sum of which is equal to a given number. This game is given a matrix from one to sixteen. A given number is written above the matrix, you must select the numbers in the matrix so that the sum of these numbers is equal to the given number. If you answer correctly, you score points and continue playing.

Game "Visual Geometry"

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, they must be quickly counted, then they close. Four numbers are written below the table, you must select one correct number and click on it with the mouse. If you answer 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 have to compare two numbers. At the top, a question is written, read it and answer correctly to the question posed. You can answer using the buttons below. There are three buttons "left", "equal" and "right". If you answer correctly, you score points and continue playing.

Development of phenomenal mental arithmetic

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