Multiplication. Multiplication Systematized multiplication rules

A little theory

The multiplication table is most often presented in two versions: columns, in each of which the results of multiplication by a certain number are written (most often from 1 to 10) or the “Pythagorean table”, in which the factors (most often from 1 to 10 or up to 20) are written in a row in one row and in one column. The result of multiplying factors is written at the intersection of the column and row of factors. The site has a multiplication table for 1, a multiplication table for 2, a multiplication table for 3, a multiplication table for 4, a multiplication table for 5, a multiplication table for 6, a multiplication table for 7, a multiplication table for 8, a multiplication table for 9, a multiplication table on 10.

The easiest way is to learn the multiplication table by 5.

This simple calculator will allow you to create a multiplication table for the number you entered. The multiplication table calculator works with prime, fractional and negative numbers, and gives not one, not two answers, but a whole cycle from 1 to 20.

Three hundred years ago in England, a person who knew the multiplication table was already considered a learned person.

Multiplication table by 1 1 × 1 = 1 1 × 2 = 2 1 × 3 = 3 1 × 4 = 4 1 × 5 = 5 1 × 6 = 6 1 × 7 = 7 1 × 8 = 8 1 × 9 = 9 1 × 10 = 10	Multiplication table by 2 2 × 1= 2 2 × 2= 4 2 × 3= 6 2 × 4= 8 2 × 5= 10 2 × 6= 12 2 × 7= 14 2 × 8= 16 2 × 9= 18 2 × 10 =20	Multiplication table by 3 3 × 1= 3 3 × 2= 6 3 × 3= 9 3 × 4= 12 3 × 5= 15 3 × 6= 18 3 × 7= 21 3 × 8= 24 3 × 9= 27 3 × 10 =30
Multiplication table by 4 4 × 1= 4 4 × 2= 8 4 × 3= 12 4 × 4= 16 4 × 5= 20 4 × 6= 24 4 × 7= 28 4 × 8= 32 4 × 9= 36 4 × 10 =40	Multiplication table by 5 5 × 1= 5 5 × 2= 10 5 × 3= 15 5 × 4= 20 5 × 5= 25 5 × 6= 30 5 × 7= 35 5 × 8= 40 5 × 9= 45 5 × 10 =50	6 times table 6 × 1= 6 6 × 2= 12 6 × 3= 18 6 × 4= 24 6 × 5= 30 6 × 6= 36 6 × 7= 42 6 × 8= 48 6 × 9= 54 6 × 10 =60
Multiplication table for 7 7 × 1= 7 7 × 2= 14 7 × 3= 21 7 × 4= 28 7 × 5= 35 7 × 6= 42 7 × 7= 49 7 × 8= 56 7 × 9= 63 7 × 10 =70	8 multiplication table 8 × 1= 8 8 × 2= 16 8 × 3= 24 8 × 4= 32 8 × 5= 40 8 × 6= 48 8 × 7= 56 8 × 8= 64 8 × 9= 72 8 × 10 =80	Multiplication table by 9 9 × 1= 9 9 × 2= 18 9 × 3= 27 9 × 4= 36 9 × 5= 45 9 × 6= 54 9 × 7= 63 9 × 8= 72 9 × 9= 81 9 × 10 =90

The multiplication table can look like this

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History of the multiplication table.

The oldest known multiplication table was discovered in Ancient Babylon and is approximately 4,000 years old. It is based on the sexagesimal number system. The oldest decimal multiplication table was found in Ancient China and dates back to 305 BC. e. The invention of the multiplication table is sometimes credited to Pythagoras, after whom it is named in various languages, including French, Italian and Russian. In 493, Victoria of Aquitaine created a table of 98 columns that represented in Roman numerals the result of multiplying numbers from 2 to 50. John Leslie, in The Philosophy of Arithmetic (1820), published a table for multiplying numbers up to 99, which allowed numbers to be multiplied in pairs. He also recommended that students memorize the multiplication table up to 25. In Russian schools, the values ​​traditionally reach 10x10. In Great Britain up to 1212, which is also associated with units of the English system of measures of length (1 foot = 12 inches) and monetary circulation (which existed until 1971: 1 pound sterling = 20 shillings, 1 shilling = 12 pence).

Multiplication table without answers.

For many people, mathematics can be terrifying. This list will probably improve your general knowledge of math techniques and speed up mental math calculations.

1. Multiply by 11

We all know that multiplying by 10 adds 0 to the number, but did you know that there is an equally simple way to multiply a two-digit number by 11? Here he is:

Take the original number and represent the space between the two digits (in this example we use the number 52):
5_2

Now add the two numbers and write them in the middle:
5_(5+2)_2

So your answer is: 572.

If adding the numbers in brackets results in a two-digit number, simply remember the second digit and add one to the first number:
9_(9+9)_9
(9+1)_8_9
10_8_9
1089 - This always works.

2. Fast squaring

This trick will help you quickly square a two-digit number that ends in 5. Multiply the first digit by itself +1, and add 25 at the end. That's all!
252 = (2x(2+1)) & 25
2 x 3 = 6
625

3. Multiply by 5

Most people remember the 5 times table very easily, but when you have to deal with larger numbers, it becomes more difficult, or not? This technique is incredibly simple.

Take any number, divide by 2 (in other words, divide in half). If the result is a whole number, add a 0 at the end. If not, ignore the comma and add 5 at the end. This always works:
2682 x 5 = (2682 / 2) & 5 or 0
2682 / 2 = 1341 (integer, so add 0)
13410

Let's try another example:
5887 x 5
2943.5 (fractional number (skip the comma, add 5)
29435

4. Multiply by 9

It's simple. To multiply any number from 1 to 9 by 9, look at your hands. Bend the finger that corresponds to the number being multiplied (for example, 9x3 - bend the third finger), count the fingers before the bent finger (in the case of 9x3, this is 2), then count after the bent finger (in our case, 7). The answer is 27.

5. Multiply by 4

This is a very simple technique, although obvious only to some. The trick is to simply multiply by 2, and then multiply by 2 again:
58 x 4 = (58 x 2) + (58 x 2) = (116) + (116) = 232

6. Counting tips

If you need to leave a 15% tip, there is an easy way to do it. Calculate 10% (divide the number by 10), and then add the resulting number to half of it and get the answer:
15% of $25 = (10% of 25) + ((10% of 25) / 2)
$2.50 + $1.25 = $3.75

7. Complex multiplication

If you need to multiply large numbers and one of them is even, you can simply regroup them to get the answer:
32 x 125 is the same as:
16 x 250 is the same as:
8 x 500 is the same as:
4 x 1000 = 4,000

8. Division by 5

Dividing large numbers by 5 is actually very simple. All you need to do is simply multiply by 2 and move the decimal point: 195 / 5
Step1: 195 * 2 = 390
Step2: Move the comma: 39.0 or just 39.

2978 / 5
Step1: 2978 * 2 = 5956
Step2: 595.6

9. Subtraction from 1000

To subtract from 1000, you can use this simple rule: Subtract all digits from 9 except the last one. And subtract the last digit from 10: 1000
-648
Step1: subtract 6 from 9 = 3
Step2: subtract 4 = 5 from 9
Step3: subtract 8 from 10 = 2
Answer: 352

10. Systematized rules of multiplication

• Multiplying by 5: Multiply by 10 and divide by 2.
• Multiplying by 6: Sometimes it's easier to multiply by 3 and then by 2.
• Multiplying by 9: Multiply by 10 and subtract the original number.
• Multiplying by 12: Multiply by 10 and add the original number twice.
• Multiplying by 13: Multiply by 3 and add 10 times the original number.
• Multiplying by 14: Multiply by 7 and then by 2.
• Multiplying by 15: Multiply by 10 and add 5 times the original number, as in the previous example.
• Multiplying by 16: If you want, multiply by 2 4 times. Or multiply by 8 and then by 2.
• Multiplying by 17: Multiply by 7 and add 10 times the original number.
• Multiplying by 18: Multiply by 20 and subtract the original number twice.
• Multiplying by 19: Multiply by 20 and subtract the original number.
• Multiplying by 24: Multiply by 8, then by 3.
• Multiplying by 27: Multiply by 30 and subtract the original number 3 times.
• Multiplying by 45: Multiply by 50 and subtract the original number 5 times.
• Multiplying by 90: Multiply by 9 and add 0.
• Multiplying by 98: Multiply by 100 and subtract the original number twice.
• Multiplying by 99: Multiply by 100 and subtract the original number.

Bonus: Interest

Yanni in the 23rd comment gave excellent advice on how to calculate percentages. So I took the liberty of repeating it here:

Calculate 7% of 300. Seems difficult?

Interest: First you need to understand the meaning of the word Percent. The first part of the word is PRO (PER), like 10 points per page of the listverse site. PER = FOR EVERYONE. The second part is CENT, like 100. For example, CENTURY = 100 years. 100 CENTS in 1 dollar and so on. So PERCENTAGE = FOR EVERY HUNDRED.

So, it turns out that 7% of 100 will be 7. (7 for every hundred, only one hundred).
8% of 100 = 8.
35.73% of 100 = 35.73

But how can this be useful??
Let's return to the problem 7% of 300. 7% of
the first hundred is 7.7%, the second hundred is the same 7, and 7% of the third hundred is still the same 7. So, 7 + 7 + 7 = 21. If 8% of 100 = 8, then 8% of 50 = 4 (half of 8).

Fractionate each number if you need to calculate percentages out of 100, but if the number is less than 100, just move the decimal point to the left.

EXAMPLES:
8%200 = ? 8 + 8 = 16.
8%250 = ? 8 + 8 + 4 = 20,
8%25 = 2.0 (Move the decimal point to the left).
15%300 = 15+15+15 =45,
15%350 = 15+15+15+7,5 = 52,5

It's also useful to know that you can always reverse the numbers: 3% of 100 is the same as 100% of 3. 35% of 8 is the same as 8% of 35.

With the best free game you learn very quickly. Check it out for yourself!

Learn multiplication tables - game

Try our educational e-game. Using it, tomorrow you will be able to solve mathematical problems in class at the blackboard without answers, without resorting to a tablet to multiply numbers. You just have to start playing, and within 40 minutes you will have an excellent result. And to consolidate the results, train several times, not forgetting about breaks. Ideally, every day (save the page so as not to 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)

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Multiplication table (numbers from 1 to 20)

×	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	2	4	6	8	10	12	14	16	18	20	22	24	26	28	30	32	34	36	38	40
3	3	6	9	12	15	18	21	24	27	30	33	36	39	42	45	48	51	54	57	60
4	4	8	12	16	20	24	28	32	36	40	44	48	52	56	60	64	68	72	76	80
5	5	10	15	20	25	30	35	40	45	50	55	60	65	70	75	80	85	90	95	100
6	6	12	18	24	30	36	42	48	54	60	66	72	78	84	90	96	102	108	114	120
7	7	14	21	28	35	42	49	56	63	70	77	84	91	98	105	112	119	126	133	140
8	8	16	24	32	40	48	56	64	72	80	88	96	104	112	120	128	136	144	152	160
9	9	18	27	36	45	54	63	72	81	90	99	108	117	126	135	144	153	162	171	180
10	10	20	30	40	50	60	70	80	90	100	110	120	130	140	150	160	170	180	190	200
11	11	22	33	44	55	66	77	88	99	110	121	132	143	154	165	176	187	198	209	220
12	12	24	36	48	60	72	84	96	108	120	132	144	156	168	180	192	204	216	228	240
13	13	26	39	52	65	78	91	104	117	130	143	156	169	182	195	208	221	234	247	260
14	14	28	42	56	70	84	98	112	126	140	154	168	182	196	210	224	238	252	266	280
15	15	30	45	60	75	90	105	120	135	150	165	180	195	210	225	240	255	270	285	300
16	16	32	48	64	80	96	112	128	144	160	176	192	208	224	240	256	272	288	304	320
17	17	34	51	68	85	102	119	136	153	170	187	204	221	238	255	272	289	306	323	340
18	18	36	54	72	90	108	126	144	162	180	198	216	234	252	270	288	306	324	342	360
19	19	38	57	76	95	114	133	152	171	190	209	228	247	266	285	304	323	342	361	380
20	20	40	60	80	100	120	140	160	180	200	220	240	260	280	300	320	340	360	380	400

How to multiply numbers in a column (mathematics video)

To practice and learn quickly, you can also try multiplying numbers by column.

If my memory serves me correctly, the multiplication table up to 5 inclusive was quite easy. But with multiplication by 6, 7, 8 and 9, certain difficulties arose. If I had known this trick before, my homework would have been completed at least twice as fast ;)

Multiplying by 6, 7 and 8

Turn your hands with your palms facing you and assign numbers from 6 to 10 to each finger, starting with the little finger.

Now let's try to multiply, for example, 7x8. To do this, connect finger No. 7 on your left hand with finger No. 8 on your right.

Now we count the fingers: the number of fingers under the connected ones is tens.

(picture clickable)

And we multiply the fingers of the left hand remaining on top by the fingers of the right - these will be our units (3x2 = 6). The total is 56.

Sometimes it happens that when multiplying “units” the result is greater than 9. In such cases, you need to add both results into a column.

For example, 7x6. In this case, it turns out that the “units” are equal to 12 (3x4). Tens equal 3.

3 (tens)
+
12 (units)
________
42

Multiply by 9

Turn your hands again with your palms facing you, but now the numbering of your fingers will go in order from left to right, that is, from 1 to 10.

Now we multiply, for example, 2x9. Everything that goes up to finger No. 2 is tens (that is, 1 in this case). And all that remains after finger No. 2 is units (that is, 8). As a result we get 18.

First you need to do two things: print out the multiplication table itself and explain the principle of multiplication.

To work, we will need the Pythagorean table. Previously, it was published on the back of notebooks. It looks like this:

You can also see the multiplication table in this format:

Now, this is not a table. These are just columns of examples in which it is impossible to find logical connections and patterns, so the child has to learn everything by heart. To make his job easier, find or print the actual chart.

2. Explain the working principle

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When a child independently finds a pattern (for example, sees symmetry in the multiplication table), he remembers it forever, unlike what he has memorized or what someone else told him. Therefore, try to turn studying the table into an interesting game.

When starting to learn multiplication, children are already familiar with simple mathematical operations: addition and multiplication. You can explain to your child the principle of multiplication using a simple example: 2 × 3 is the same as 2 + 2 + 2, that is, 3 times 2.

Explain that multiplication is a short and quick way to do calculations.

Next you need to understand the structure of the table itself. Show that the numbers in the left column are multiplied by the numbers in the top row, and the correct answer is where they intersect. Finding the result is very simple: you just need to run your hand across the table.

3. Teach in small chunks

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There is no need to try to learn everything in one sitting. Start with columns 1, 2 and 3. This way you will gradually prepare your child to learn more complex information.

A good technique is to take a blank printed or drawn table and fill it out yourself. At this stage, the child will not remember, but count.

When he has figured it out and mastered the simplest columns well enough, move on to more complex numbers: first, multiplying by 4–7, and then by 8–10.

4. Explain the property of commutativity

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The same well-known rule: rearranging the factors does not change the product.

The child will understand that in fact he needs to learn not the whole, but only half of the table, and he already knows some examples. For example, 4×7 is the same as 7×4.

5. Find patterns in the table

secretwomans.ru

As we said earlier, in the multiplication table you can find many patterns that will simplify its memorization. Here are some of them:

1. When multiplied by 1, any number remains the same.
2. All examples of 5 end in 5 or 0: if the number is even, we assign 0 to half the number, if it is odd, 5.
3. All examples of 10 end in 0 and begin with the number we are multiplying by.
4. Examples with 5 are half as many as examples with 10 (10 × 5 = 50, and 5 × 5 = 25).
5. To multiply by 4, you can simply double the number twice. For example, to multiply 6 × 4, you need to double 6 twice: 6 + 6 = 12, 12 + 12 = 24.
6. To remember multiplication by 9, write down a series of answers in a column: 09, 18, 27, 36, 45, 54, 63, 72, 81, 90. You need to remember the first and last number. All the rest can be reproduced according to the rule: the first digit in a two-digit number increases by 1, and the second decreases by 1.

6. Repeat

medaboutme.ru

Practice repetition often. Ask in order first. When you notice that the answers have become confident, start asking randomly. Watch your pace too: give yourself more time to think at first, but gradually increase the pace.

7. Play

utahpubliceducation.org

Don't just use standard methods. Learning should captivate and interest the child. Therefore, use visual aids, play, use different techniques.

Cards

The game is simple: prepare cards with examples of multiplication without answers. Mix them, and the child should pull out one at a time. If he gives the correct answer, we put the card aside, if he gives the wrong answer, we return it to the pile.

The game can be varied. For example, give answers on time. And count the number of correct answers every day so that the child has a desire to break his yesterday’s record.

You can play not only for a while, but also until the entire stack of examples runs out. Then for every wrong answer you can assign the child a task: recite a poem or tidy things up on the table. When all the cards have been solved, give them a small gift.

From the reverse

The game is similar to the previous one, only instead of cards with examples, you prepare cards with answers. For example, the number 30 is written on the card. The child must name several examples that will result in 30 (for example, 3 × 10 and 6 × 5).

Examples from life

Learning becomes more interesting if you discuss with your child things that he likes. So, you can ask a boy how many wheels four cars need.

You can also use visual aids: counting sticks, pencils, cubes. For example, take two glasses, each containing four pencils. And clearly show that the number of pencils is equal to the number of pencils in one glass multiplied by the number of glasses.

Poetry

Rhyme will help you remember even complex examples that are difficult for a child. Come up with simple poems on your own. Choose the simplest words, because your goal is to simplify the memorization process. For example: “Eight bears were chopping wood. Eight nine is seventy two.”

8. Don't be nervous

Usually, in the process, some parents forget themselves and make the same mistakes. Here is a list of things that should never be done:

1. Force the child if he doesn't want to. Instead, try to motivate him.
2. Scold for mistakes and scare with bad grades.
3. Set your classmates as an example. When you are compared to someone, it is unpleasant. In addition, you need to remember that all children are different, so you need to find the right approach for each.
4. Learn everything at once. A child can easily be frightened and tired by a large volume of material. Learn gradually.
5. Ignore successes. Praise your child when he completes tasks. At such moments he has a desire to study further.