How to teach a child to multiply quickly and easily, so that from elementary grades he can solve various mathematical problems well? If you want your child to receive all-round development, he cannot do without help.

In this age of information abundance, you can find many supporting materials - flashcards, playful courses, audio and video programs and much more, but none of the methods is universal. All children are unique in their own way, therefore, an individual approach to each is needed. In this article, we offer you several ways to master the multiplication table. After studying them, you can choose the most effective one for your child.

There are enough methods for studying the tablet now - you just need to choose the most suitable one for the child

An important preparatory moment

When children begin to study the multiplication table, they already have an idea of ​​simpler arithmetic operations - addition and subtraction. Now we need to explain to them what the essence of the multiplication action is. The previously learned skills will help you with this.

What is the principle of multiplication? This is multiple addition. For example, to multiply 4 by 3, add 4 (4 + 4 + 4) 3 times. Having mastered this, the child will make fewer mistakes in the further learning process.

In addition, children should also understand how to navigate in the device of the table. It is necessary to explain that the product is the number at the intersection of a row and a column.

Start

A large spreadsheet full of numbers can be discouraging for a child if not completely discouraged from learning. For this reason, it is best to start with the simplest examples. It doesn't take much effort to deal with them. In addition, the child will be able to complete them on his own, then part of the work will already be done:

1. Multiply by 1. Any number remains the same number.
2. What do you need to do to multiply by 10? It is enough just to put 0 at the end of the number.
3. Multiplication by 2 is the addition of two identical numbers. By at least, with prime numbers, children already know how to perform such actions when they begin to study multiplication.
4. Change of multipliers. This is the so-called displacement (commutative) multiplication law. That is, if you rearrange the factors, the product will not change. Thus, it turns out that you only need to learn half of the table.

As you can see, the picture is getting more optimistic. The child will also notice this and will continue to work with more enthusiasm than in the beginning.

The child must first of all understand that multiplication is a familiar addition to him, only multiple

Purposeful memorization

After mastering the simplest values, you can move on. To cope with more complex factors, you will need to connect other techniques - repeat, split into parts, build associations, apply knowledge in practice. Now, to memorize, you will need multiple repetition of actions and meanings.

The opinions of educators differ on the issue of the sequence of actions. Some adhere to a technique where the most difficult examples are mastered first, and then the simpler ones. Practice shows that this method is not suitable for everyone and can often even cause some stress for students. The best option is to teach them the simpler actions first and the most difficult ones at the end. How can this be explained? When multiplying small numbers (for example, 3 by 3), the child can test himself on his fingers - at the beginning of learning this technique is useful. If, however, children are immediately obliged to memorize an 8 by 9 product, then it will be just rote memorization without practical use. This technique can be very demotivating.

Squares of numbers

We begin a new stage in mastering the multiplication table with squares of numbers. To deduce the square of a number is to multiply it by itself. There are only 10 squares in the table, it is not so difficult to remember them (this is largely due to the fact that some of them are rhymed - for example, "five five - twenty five"). A 10 by 10 square is not worth remembering at all.

For the child to really understand, and not only remember the tablet, you need to start the study of each row with a square

Multiplication by 3

Here the situation is already a little more complicated. If you notice that the child is not able to memorize any actions, analyze his inclinations and include those auxiliary materials that are suitable in your particular case. For many children, flashcards are ideal. In the case of a humanitarian mindset, it is good to use lessons in poetic form (we will describe in detail the use of special verses for memorization in the section below).

Multiplication by 4

It will be a little easier here. Invite your child to try to logically construct that action himself, and he will probably guess that multiplying by 4 is the same as multiplying 2 times by 2. If he finds it difficult, you can easily explain this to him. Cards and poems will also be useful at this stage of mastering the material.

Multiplying by 5 is also easy, children usually enjoy this part of the learning process. First, all the values ​​of this multiplication are spaced from each other by 5 numbers. Secondly, they end in 5 or 0. At the end of even numbers multiplied by 5, there will be 0, and odd numbers - 5. As you can see, everything is simple.

If we consider the products multiplied by 5 numbers, you can see - they all end in 5 or 0

Multiplication by 6, 7, 8 and 9

The last stage in mastering the multiplication table is the most difficult, but it consists of memorizing only six products. To remember them well, you have to work hard, because even many adults get confused with the answers.

To make it easier for the child, use cards, and not 6, but 12. With this set of cards you can practice changing the places of the factors, and this will greatly facilitate memorization.

Playing with cards

Learning in the form of a game for children is a must. It performs the main function - it contributes to the emergence of interest. If a child becomes interested in the process, this will almost be a guarantee that he will successfully master it.

Despite the fact that now there are many more modern auxiliary devices and materials (programs, online games, sound posters, and others), ordinary cards do not lose their popularity. They are available to everyone and are easy to use. Even if you use different methods of learning the multiplication table, the flashcards will help you at any stage.

The first thing you need to do is print the cards or cut and fill by hand. It is advisable to stick them on cardboard for better safety during operation. On each card, you need to write an example from the multiplication table. You do not need to write an answer.

What is the game itself? Since you will be connecting the cards from any, even the earliest stage of training, for each lesson you need to select those examples that correspond to today's plan. Then the cards are shuffled, and the child randomly pulls out any of the piles. He needs to read the example and give the correct answer. After that, the card is set aside and another is pulled out. If the child answered incorrectly, the card is returned to the pile. In this case, be sure to voice the correct answer so that the child will remember it and answer correctly when he pulls out this card again.

For early education of the baby, just print a set of cards

The advantages of such a simple process:

1. Visual memory is connected. Children, especially visuals, will find it much easier to learn even the most difficult examples.
2. Memorization is much better with this approach. Multiple repetition is carried out in an interactive form, and not a simple cramming.
3. The child sees the result of the work done immediately. He has an incentive to finish the game faster and emerge victorious without leaving a single card in the pile. In this playful approach, you can arrange a competition by connecting another child.

Other study techniques

The more techniques you have in stock, the more successful the process of learning the multiplication table will progress with your child. Different methods can be used not only depending on the mentality of the children, but also on the level of complexity of a particular lesson. You just need to constantly analyze the situation and navigate in it, then you can explain even the most difficult example in an accessible way, and your child can quickly learn it. We present to your attention some of these techniques. They are no more difficult than playing with cards.

Case studies

You don't have to look far to find illustrative examples to teach you — there are many by your side in your normal daily environment. Show observation and fantasize a little, then your child will be able to learn the multiplication table not only with ease, but also with great interest.

How many wheels do you need for 3 cars? How many flowers do you need to plant on 3 flower beds, if each of them fits 8? How many legs do 4 teddy bears have? As you can see, there are many options. You can invite the child to find them on their own or set multiplication problems to a friend, taking examples from the home environment.

A great idea is to teach a child using his own toys, household items, sweets, and so on.

Examples of increased complexity

For more complex examples and those that are difficult for the child, pay maximum attention. At the same time, do not overload children's memory - alternate between simple and complex. When you see that the material is mastered, move on to another. Do not try to lay out all the information at once for memorization, divide it into several approaches.

Multiplication on fingers

Using this technique, you can master the entire multiplication table, but the most popular in this case is multiplication by 6, 7, 8 and 9. You can use it additionally in any lessons, but keep in mind that before showing such a game to a child, you need to understand well and learn its principle.

We put paper on the table, on top - hands with fingers horizontally to each other. We outline the contours of the hands and number the fingers in this way: thumb - 5, index - 6, middle - 7, ring - 8, little finger - 9. These contours will come in handy when we move our hands during the process. Now we choose the example that needs to be solved: let it be a multiplication of 7 by 8. The middle finger of the left hand will denote 7, and the ring finger of the right hand - 8. They need to be connected and move your hands to the edge of the table. The fingers in front of the connected ones, which will hang down at the same time, will denote tens, and all the other fingers that remain on the table - units. Now we count. There are 5 fingers at the bottom - that means, dozens of 5. The fingers that lie on the table must be multiplied. On the left hand there are 3, and on the right - 2. Now we multiply 3 by 2 - we get 6 units. The answer is 56.

Now multiply by 9. Place your hands side by side on the table so that your fingers are vertical. Each finger should be numbered from 1 to 10 from left to right. You can do this on paper to avoid confusion. The little finger of the left hand is 1, and the little finger of the right is 10. Now bend the finger with the number that we want to multiply by 9. For example, it will be 5. The fingers to the left of it will be tens, and the fingers to the right will be ones. The answer is 45.

Learning the multiplication table using rhyme (poems)

This memorization technique is a mnemonic technique. In the techniques of mnemonics, abstract concepts are replaced by representations based on some kind of sensory perception (in this case, auditory). That is, this technique is mostly psychological.

This way of memorizing information is loved by all children, regardless of their mindset and character. Why? The rhyme is well and quickly remembered, the poems vividly illustrate the content and learning short funny rhymes is much more interesting than mechanically cramming even simple examples.

However, the whole process should not be based on this technique, otherwise you risk overloading the child's memory with excessive memorization. We would recommend using it in the most difficult situations to relieve tension and add a gameplay element to the process. If you wish, you can even connect pictures illustrating an example in verse.

It is not difficult to find a poetic multiplication table, there are several options from different authors. We will give examples of tasks that usually cause difficulties for everyone. Some examples from the book by Alexander Usachev "Multiplication table in verse":

• 6 x 9: We do not mind the rolls. Open your mouth wider: Six nine will be Fifty four.
• 7 x 8: Once the deer asked the elk: - How much will the family be eight? - The moose did not
  climb into the textbook: - Fifty, of course, six!
• 8 x 9: Eight bears were chopping wood. Eight nine - seventy two.

Nicely two times seven times
The February holiday will help us
Valentine's Day, I remember
14th, friends!

How much is twice eight
We'll ask the tenth graders.
They will tell us the answer
After all, they are already SIXTEEN years old!

Musical chants are also possible, which will especially help a child with good auditory perception.

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

Learn multiplication table - game

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

See the full cheat sheet below.

Multiplication directly on the site (online)

*
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 with a column (video on math)

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

Monday, 09 Jun 2014

Many of us wondered why at school we memorized (crammed) the multiplication table without checking its correctness, and did not find an answer. Most of the students did not have this question, we were taught from the "cradle" to live on "faith" and this is what it led to. When you fill in the multiplication table line by line, it immediately becomes obvious that multiplication is just MULTIPLE addition, and accordingly division is MULTIPLE subtraction, therefore, it is easy to understand the fundamental difference between the expressions “how much more / less” and “how many times more / less”.

Continuing topics:

• Rybnikov Yuri Stepanovich - Unified System of Knowledge [video]

Zombification is a forced processing of a person's subconscious, thanks to which he is programmed to unconditionally obey the orders of his master. The zombie itself begins with kindergarten and continues throughout your life.

Practical methods of brainwashing: a lot of information is hammered into our heads.

All knowledge gained there is divided into:

• senseless
• useless
• harmful
• erroneous
• outdated

We must clearly know that all the words of the Rus are expressed in sentences. There is the concept of "grammar of the Russian language" and the concept of "root of the word". The root of a word carries the meaning of this expression and transfers it to functionals, i.e. to the verb.

We introduce two concepts:

1) addition;

2) multiplication.

Addition. To get the result of the addition, what do you need to do? Fold. LIVE WITH A LIE.

Multiplication. To get the result of the multiplication, what do you need to do? Multiply. MULTIPLY.

For many, mathematics at school was an incomprehensible and unloved subject. In most cases, the students are not to blame, they just were initially taught incorrectly and the further, the worse they teach. Let's consider the situation using the example of the well-known "multiplication table". There is an old anecdote: "A woman is indignant that it is very convenient 5x5 = 25, 6x6 = 36, but why 7x7 = 49, was it really difficult to make 47?" A very practical approach is to do what is convenient for her, not what is right. In elementary school, we all have "my first teacher", who very rarely goes against the standard, acts "as taught", "according to the textbook" and in accordance with the "methodological plans."

Creativity and innovation in this area is expressed in "feminine" approaches - with poems and songs, dances and tambourines, animals and tricks from the bottom of my heart with a naive desire to make it more attractive and "prettier", with the firm belief that "children, this is not understood, this will simply remember ":

There can be no talk of any abstract thinking here.- distracts everything, you have to strain even just to read. But let's not harshly condemn all the creators, they wanted the best, but it turned out as always.

Instead of malice, let's try to conjure a little over everything known, seemingly the simplest subject and consistently to cleanse the grains of truth from the chaff of insanity of improvement.

First, we remove unnecessary paints, pictures, distortions and get the usual columns of examples of multiplication:

Then, according to the principle of observance of necessary and sufficient conditions, we cut off the unnecessary as sculptors: all examples of multiplication by 1 and 10 as elementary and all repetitions. The latter is very important, because with rote memorization, a brisk answer follows 6x8 = 48, but 8x6 = already causes a hitch or an error. With the exclusion of repetitions, this is unrealistic, since the very system of material presentation makes you understand that it is one and the same. In addition, not only a decrease in the number of examples from 100 to 36, but also a sequential decrease in their number in the columns, makes it psychologically easier:

It was this abbreviated version (albeit with a column 1 x ... =) that could be seen on the covers of school notebooks until the 1970s. Undoubtedly, you can stop at this for the convenience of rote memorization, but it will not add understanding of mathematics. Therefore, we move on.

The attentive reader may have noticed that so far we have been talking about EXAMPLES of multiplication, and not about the TABLE of multiplication.

Let's see what a real, easy, easy-to-remember multiplication table looks like with a full and correct name: multiplication AND DIVISION table, since factors are also divisors at the same time. The symmetry of the table is clearly visible due to the selection of diagonal squares of numbers:

historical name "table of Pythagoras"

and this is how the multiplication table of the Sumerians looked like in antiquity:

We do the last conceptual transformation - we start the multiplication table not from the top, but from the bottom. Why? First, it is more intuitive: lower is less, higher is more, and the left-to-right direction is kept as the same direction as left is less, then more to the right.

Secondly ... we'll tell you a little later.

The correct multiplication table can be given to the student in finished form, but it is best if he compiles it himself. Yes Yes. It is quite affordable even for a first grader!

We draw a grid and number the rows and columns from 1 to 9 - this corresponds to the examples of multiplication by 1, they will also function as a factor / multiplicity / how many times.

Then the student fills in the row and column with 2 by adding the number 2 for each subsequent cell, then the row and column with 3 and so on, a simple multiplication table is obtained:

What does it do?

Already with primary school the student gets used to the tabular form, which he will then often have to meet, intuitively understands that tables are created as a convenient and concentrated reference material, some of which must be learned by heart for ease of use.

At first, for the convenience of using the table, it is better to use the "corner" to select rows and columns - we cut out a square from one corner of a blank notebook sheet. The habit of coordinate search is formed quickly enough.

With this approach, you do not need to stupidly mechanically memorize the columns of multiplication examples, but you can immediately let the entire table be used. Let it lie before your eyes to help solve examples and after a while of training, memorization will come by itself, the student will look into it less and less.

The table should become what it was originally - help in work. The emphasis always and everywhere should not be on memorization, but on understanding and knowing where to find reference material and how to use it.

When you fill out the multiplication table line by line, it immediately becomes obvious that multiplication is just MULTIPLE addition, and accordingly division is MULTIPLE subtraction, so it is easy to understand the fundamental difference between the expressions "how much more / less" and how many times more / less. " it is very important for the subsequent compilation of equations for the conditions of the problems.

Highlighting the diagonal (squares of numbers) with shading or color clearly shows the symmetry of the table, i.e. the equivalence of the sequence of factors, and here the redundancy of the material plays in the direction of consolidating it (repetition is the mother of learning) and independent identification of such a pattern.

Later, when it is required in the learning process, children will learn how much useful and interesting information is associated with a simple sign familiar from the first grade. Like Jourdain from Zh.B. Moliere, who was surprised to learn that he was speaking in prose, children only need to add new terminology and new conclusions.

For example, they will no longer be told just about the second factor or the multiplicity of addition, but will be called the coefficient.

Each row and column of the table is an arithmetic progression, from which we can easily move on to a geometric progression, factorials and other seemingly complexities.

If you select any rectangle on such a table, then the area (miracle!) Will be indicated in the upper right corner of it, i.e. thus, it is demonstrated that algebra and geometry are just different ways of displaying the general laws of the unified science of mathematics. In other words, it is clearly shown that the product of numbers corresponds to the area of ​​a rectangle, and the square of a number is really a square (accordingly, for a cube, you need to draw the third coordinate). And from here we can easily move on to solving geometric problems by algebraic methods and vice versa - whichever is more convenient.

Understanding the graphs with the X and Y axes, the names "abscissa" and "ordinate" will no longer cause any difficulties - this will be the usual form of presentation of the material from the elementary grades, you just need to draw the arrows. And ... explain how the cardinal numbers differ from the ordinal ones (they are also quantitative and ordinal, respectively).

In the end, the understanding of the integral as a sum of infinitesimal quantities comes precisely from the understanding of the essence of multiplication of natural numbers (and again geometric analogs - the area on a curved trapezoid on the graph of a function), otherwise the integration will be stupidly perceived as memorized mechanical actions when a tricky squiggle is found in the form long letter S.

So most of the problems are due to a misunderstanding of the basics.

Why haven't I seen this technique before ?!

And now I don’t understand why at school they force her to TEETH, for a long time and painfully, instead of just so easy and fun teaching children to use the multiplication table ?!

During the summer holidays, it is very convenient to learn the multiplication table. Simple and logical rules will help your child understand and remember the result for a long time.

Parents of schoolchildren often ask the question: How to quickly and easily learn the multiplication table? People study the chart for a variety of reasons, but most often simply because it is required for school. Why is this required?

The multiplication table is used:

• To carry out calculations with multidigit numbers in the head or on paper without a calculator. Example: to multiply 42 * 78, you need to use four "facts" from the multiplication table, plus knowledge of the decimal system


• To see deep connections in mathematics and develop your "mathematical intuition"

Both goals (but at a much higher level than the traditional memorization of the table allows) can be reached by pleasant, mathematically interesting and pedagogically grounded "roads". It is better, of course, to choose the speed of this journey individually. "Four days" from the content is a rough estimate calculated according to the following conditions:

• The student understands quantitative relationships within the first two hundred, knows how to add and subtract, and understands what multiplication is (for example, sees 3 * 4 as three groups of four subjects), but does not remember the table by heart


• Children play with a mentor individually or in small groups


• All students are interested in learning this topic.

If children do not yet know what multiplication is, or are just learning to operate with large numbers, our materials can be used, but it is better to modify the approach and speed.

From hundreds of existing tricks and methods related to the multiplication table, we chose according to two criteria. 1 - the trick is short, no more than two steps (because of this, for example, the Trachenberg system was eliminated); and 2 - there is a mathematically available proof-of-concept for the trick. What's left is easy to remember, easy to understand, and easy to use!

Problems are designed to be discussed with a mentor or with other students and with a mentor, rather than being solved independently. They can lead to fairly advanced mathematics, which the student himself may either not notice, or be unable to put into words.

Day 1

We start to learn the multiplication table. Free cells ... and 36 examples remain!

Here is a common multiplication table for integers from zero to ten:

It looks scary to learn by heart. One hundred separate facts! Cramming them for so long and boring ... But in fact, how many facts do you need to remember in order to know this whole table? Not a hundred, that's for sure. Carefully and for a long time, until you get bored, study the table, and you will find many interesting ideas for tricks and methods of quick memorization.

Problem 0... After examining the spreadsheet, find as many ways as possible to learn how to use the facts from it without cramming. Many mathematicians, and not only them, have worked to find such methods, so in fact you will have to cram less than a hundred facts. How much do you think? Remember your answer ...

We begin to look closely, and we see that the table is symmetrical. After all, 4 * 8 = 8 * 4, a 9 * 6 = 6 * 9, and so on. In order not to list everything, we will write this observation in the words:

If one number is multiplied by the second, then the answer is the same as if the second number is multiplied by the first.

That is, part of the table is given to us for free! What part? If you said half, you almost guessed it. In fact, symmetry gives us 45 free "facts".

Problem 1... Why exactly 45? Find 3 different ways to count. How many "free" facts will the symmetry of the multiplication table up to 20 * 20 give? Up to 30 * 30?

There are two more numbers that are easy to multiply by. These are 1 and 10.

Task 2... Why multiplying by 1 is easy, understandable, right? Why is it so easy to multiply by 10? Hint - think about other number systems, like hexadecimal.

Let's delete the multiplication by these numbers from the list of those that need to be memorized. These "free" facts are now shown in a very light gray on the table. And this is what will remain:

At the end of the first day, using one of the methods from Problem 1, we calculate how many facts we have left to learn. Well, isn't it so scary anymore? Then we look forward to the next day of multiplication!

Day 2

Twice two - four ... and 21 facts remain!

Doubling is easy. Scientists even believe that doubling is "programmed" in the brain of humans (and some animals), as well as distinguishing between the concepts of "big-small" or "one-many." Kids learn to double by dividing candy between two, counting shoes and gloves, looking at objects in the mirror ... To multiply by two - add the number to yourself! And to multiply by four? Multiplying by four is like multiplying by two by two. That is, to multiply by four, double the number (this is easy), and then double the result.

Problem 3... How do you use this same principle to multiply by 8, by 16, etc.? The numbers in this "etc." are called "powers of two". The first degree is 2, the second is 4, the third is 8 ... Continue this row until you get bored. And what power of two is 64? The answer to this question is called, in mathematical language, "finding the logo of the number 64 in base 2".

So you don't need to cram to multiply by two and four. As for multiplying by eight, this already takes three steps (because eight is the third power of two, see Problem 3), so we'll save the multiplication by 8 for another trick. For now, let's paint over the facts that doubling and multiplying by 4 using doubling saves us from cramming with light blue:

Look how few dark cells are left in the table - but there is a lot of interesting mathematics ahead. See you on the third day.

Day 3

The universal method and multiplication by 5 ... and 10 cells remain!

The results of multiplying by five can be learned to quickly extract without cramming, with several different ways... That is, you can choose the method that suits you the most for use.

Halving (equally) is almost as easy as doubling. Conclusion: to multiply by five, multiply by ten and then divide by two. For example, five times eight equals half of eighty. Five times four equals half of forty.

Task 4. And why, in fact, we "have the right" to do so? From a mathematical point of view ...

Another way to multiply a number by five: if the number is even, assign zero to half of the number. If the number is odd, assign five to half of the previous number. For example, to multiply eight by five, assign zero to half of eight. To multiply seven by five, assign five to half of six.

Task 5. Why does this method work? How does it differ from the first method? (Hint: nothing! Mathematically ...)

And here is the promised universal multiplication method. It works for all numbers without exception, but is too slow for most of them. We just count not one by one "One, two, three ..." but by the number that we multiply, as many times as we multiply. Try this for 7 * 8: “Seven, fourteen, twenty-one, twenty-eight, thirty-five, forty-two, forty-nine, fifty-six” Difficult, isn't it? And slowly ... Now try 5 * 8: "Five, ten, fifteen ... ... forty." Simple and fast!

Task 6, psychological. Why do you think people find it easy to count as fives?

By the way, it is also easy to count in three: three, six, nine ... (why, what do you think?). At the end of the third day, we will repaint the cells with light-violet, which now can not be crammed: all multiplication by five and multiplication by three. Here's what's left:

There are few cells left, but the most difficult ones, you say? The next day you will deal with them in one fell swoop!

Day 4

Tricks on the fingers ... And all the cells are painted over!

This very beautiful trick came from somewhere in the East, like many other great mathematical ideas (for example, the idea of ​​zero). It is assumed that you already know how to multiply numbers from two to five (you can use the ideas of the first three days to learn). On the fingers, we will multiply the numbers from six to nine.

Number the fingers of both hands: big - 5, index - 6, middle - 7, ring - 8, little fingers - 9. First, you can write the numbers on your nails with a felt-tip pen. Place your hands in front of you on the table, palm down, and your “analog computer” is ready! Let's say we multiply 7 * 8: pinch the number 7 finger on the left hand and the number 8 finger on the right, place those touching fingers along the edge. Hanging fingers (2 on the left hand and 3 on the right) are counted in tens - 50.

We multiply the fingers on the table: 3 from the left hand, multiplied by 2 from the right - it turns out 6, that's the answer: 7 * 8 = 56. Another example: 9 * 8. We touch with fingers number 9 on the left and number 8 on the right hands. There are 7 fingers left in front of the touching fingers (4 on the left, 3 on the right) - this is 70. The rest are multiplied: 1 on the left by 2 on the right - we get 2, and the answer is 72. That is, the fingers in front of the touching two are always counted in tens, and the rest we multiply the left hand by the right. After the third or fourth multiplication, it turns out very quickly and deftly.

Task 7. Why does this trick work? We know three different proofs - or maybe you can find not only them, but other proofs as well?

Let's now repaint the cells with the results that we can get from the last trick into light Orange color... Blimey! There is nothing left to cram - the whole thing is painted over! This means that we have finally learned the multiplication table.

Multiplication by 1 and 10

It's worth starting with this to reassure the child: multiplication by one is the number itself, and multiplication by 10, the number and zero after it. Now he already knows the answers to the first and last examples in all columns.

Multiplication by 2

Multiplying a number by two means adding two of the same number.

Multiplication by 3

To memorize this column, mnemonic techniques are suitable, for example, short rhymes. You can invent them together with your child or search for “ready-made” ones on the net:

Come on, my friend, look

What is three times three?

Nothing to do!

Well, of course, nine!

All the guys need to know

How much is three times five,

And don't be wrong!

Three times five - fifteen!

If you are not strong in poetry, come up with prosaic stories, the heroes of which will be a two - a swan, a three - a snake, a four - an inverted chair, an eight - glasses, and so on - the children themselves will tell you who, in their opinion, the numbers are like ...

Stories and rhymes can be invented not only for the troika, but also for any column of the Pythagorean table.

Multiplication by 4

Multiplication by 4 can be represented as multiplication by 2 and again by 2. This column will not cause difficulty for students who have mastered multiplication by two.

Multiplication by 5

This is the easiest column to memorize. All values ​​of this column are located 5 units from each other. Moreover, if an even number is multiplied by 5, the product will end with 0, and if it is odd, it will end with 5.

Multiplication by 6, 7, 8

These columns, as well as the column of multiplication by 9, traditionally cause difficulties for schoolchildren. You can reassure the students by explaining that they have already learned most of the examples from these columns and that the terrifying 8 × 3 is the same as the already learned 3 × 8. By swapping the factors, you can remember what the product is equal to.

This means that children will only have to remember 6 "unfamiliar" examples:

These examples can be written on cards, hung on the wall, and memorized mechanically. Or you can learn to count on your fingers:

Likewise, you can multiply 7 by 8 or 8 by 9.

You can personally see the process of such multiplication in the video (note: in the video, the numbering is carried out in a similar way, but starting with the thumbs):

Multiplication by 9

To begin with, you can remember that in the multiplication table by nine, the sum of tens and ones in the answer always equals 9. Namely: 9 × 2 = 18 (add the digits of the answer: 1 + 8 = 9), the same is also in other examples: 9 × 6 = 54 (5 + 4 = 9).

In this case, the number ten in the answer is always one less than the second factor in the example. In practice: 9 × 7 = 63 (the second factor is 7, which means tens in the answer 6. Now remember the first regularity that the sum of tens and ones in the answer should equal 9, we get the answer 63).

And one more "secret": if you have paper and a pencil at hand, it is fashionable to quickly write down the numbers from 0 to 9 in a column (these will be tens), and next to the second column from 9 to 0, you will get the answers of the multiplication table by 9.

You can quickly check multiplication by 9 on your fingers:

Place your hands on the table with palms;

Mentally number the fingers from the little finger of the left hand to the little finger of the right (the little finger of the left hand is 1, the nameless of the left hand is 2 and so on to the little finger of the right hand, which, accordingly, will be 10):

What is the number by which you want to multiply nine. Let's say this number is 3:

Bend the finger that was assigned the serial number 3 (this will be the middle finger of the left hand);

The fingers that remain to the left of the bent one represent tens (in our case it is the little finger and the ring finger - two fingers, that is, 2 tens, the number 20);

The fingers that remain to the right of the bent one are units. We have 2 fingers of the left hand on the right + all 5 fingers of the right hand - a total of 7 fingers, 7 units;

2 tens (20) + 7 units (7) = 27. This is the product of 9 and 3.

Likewise, you can multiply 9 by 7 or 9 by 10.

Studying the multiplication table from any student will require perseverance and patience, but counting on the fingers, rhymes, flashcards with examples will help facilitate memorization and make it interesting and fast.