7 by 9 multiplication. Useful materials to study

It is impossible to do without knowledge of the multiplication table, so it is included in the school curriculum already in the elementary grades. For a student, memorization is very difficult. Children get scared when the teacher points to the spread of the notebook, where examples that need to be learned are raised in columns.

Therefore, it is worth showing how to make memorization quick with the help of games, multiplication secrets and useful algorithms.

The multiplication table (a game to quickly learn the memorization algorithm is described below) was invented a very long time ago. There is an opinion that the Pythagorean table was developed by an ancient philosopher and mathematician. But no confirmation of this was found. But it is known that they counted using it already in ancient Japan, where during excavations they discovered wooden tablets with parts of a table (8th century).

Scientists suggest that the table came to Japan through China. In those days, the countries interacted closely. Japanese residents came to the Celestial Empire to study science. The assumption that the table was created in China is confirmed by a find at least 3 thousand years old in the form of a separate fragment of the table.

Some experts suggest that the table penetrated from China to India through trade caravans, and from there it spread to all other countries.

Another version says: tablets with numbers were found 5 thousand years ago in Mesopotamia. Perhaps that's where it was invented. In addition, it could have been invented in several countries at the same time, because already at that time the need for calculating large numbers was clear.

In which class do they teach multiplication tables?

In Russian schools they begin to study the table in 2nd grade. And by the beginning of 4th grade, teachers try to finish memorizing. However, the most commonly used standard method is memorization. It is ineffective, so some students peek at the answers on the back of the notebook until graduation.

How to teach a child multiplication

First of all, the child needs to be interested. Moreover, it is worth starting to study together, and not just playing the role of an inspector.

A few tips to help you memorize the Pythagorean table faster and better:

• It is worth preparing a printed multiplication table in advance and explaining how the action occurs (the answer to multiplication is at the intersection), explaining the basic terms: factors, product.
• Explain to the child how the table works (for example, explain that 3 x 2 is the same as 3 + 3).
• You should learn the material gradually, in small parts. You need to start with the simplest examples.
• You should explain to the child that changing the places of numbers in multiplication does not change the result (2 x 4 is the same as 4 x 2);
• Look for patterns in the table (for example, if any number is multiplied by 1, you get the same number).
• Do more reps. Gradually, the material covered may be forgotten, so it is worth systematically refreshing your memory.
• Use memorization techniques.
• Not every child is able to quickly memorize material, so parents should be calm and attentive. Even if a student fails to learn something, he should not be forced to sit through his lessons longer.

Where to begin

First you need to let your child understand the meaning of multiplication. This is done using the simplest examples: 3 x 4 - means that the number 3 must be taken 4 times. Once the meaning is clear, it will become much easier. On notebooks, the table is usually written in columns at the back. It is better to abandon it in favor of the Pythagorean one, where the result is at the intersection of multiplying numbers. The visual method works better.

Therefore, it is worth printing out the table and hanging it above the table or anywhere where the child will look at it. Let it be bright and large enough. They start learning with the smallest numbers: 2, then 3.

You should first think through all the ways of memorizing. A combination of methods will help you remember faster: using cards, games, poems, knowledge of patterns.

Useful materials to study

When memorizing the multiplication tables, it will not be a bad idea to use educational materials.

• Shklyarova T. “How I taught my girl the multiplication tables.” This book is small, smaller in volume than a notebook. The author begins it by crossing out half of the multiplication table - the law of displacement applies. The first part of the book talks about what multiplication and division are, even and odd numbers, the features of finding the product and quotient of 1, 10 and 0, how to find an unknown factor and more. The second part explains how to multiply and divide by all numbers, explains various nuances and simple ways to memorize.
• Samsonova L. “Tabular multiplication for all current textbooks”. This manual is built in a strict learning order: first, multiplying by 2, then by 3, by 4, and so on up to 9. The book contains many tests, problems, interesting examples, and coloring pages. After studying multiplication by certain numbers, independent work begins to check.

• Long Linnet "Magnificent Multiplication".

The latest tutorial is based on games, of which there are 28:

• multiplication on fingers;
• memorization using socks;
• pencils;
• corners of the room;
• playing cards;
• counting rhymes;
• inventing fairy tales;
• tic-tac-toe;
• digital lotto and much more.

The book is divided into 3 parts - according to difficulty level. Boys and girls will be surprised that to memorize complex examples you just need to decorate a teddy bear or learn a counting rhyme.

Cards

This is a simple but effective method. You should prepare cards with examples of multiplication without writing down the answers. They are mixed (you can use a lotto bag). The child draws one card at a time and tries to give an answer. If it is correct, then the card is removed to the side; if it is incorrect, it is returned to the bag.

You can diversify the game like this: give answers within a certain period of time. You should allocate 1 minute. (or more), then count how many cards were guessed. The child will want to improve his result, which will become an incentive to memorize.

Another option is to play until you run out of examples. For each incorrect answer, assign the child a task: collect toys, dance, do squats, recite a poem. If all the cards are guessed correctly, you can award a prize in the form of sweets.

All children love to play, so as soon as they learn that they can get points, win chocolate or beat someone (even themselves), it becomes much more interesting for them.

Therefore, you can offer your child to learn the material with the following phrase: “The multiplication table is a game. To learn faster, we will play “cards, sea battle, shop” (depending on what is taken). The goal is to score so many points, complete it in this amount of time, reach level 3, and much more.”

Poetic examples

With the help of short rhyming lines, you can remember the most difficult moments well.

You can compose simple rhymes that will speed up the child’s learning process. They don't have to be logical:

It is worth noting that some examples from the Pythagorean table are themselves rhyming phrases. These include:

• Six six is ​​thirty six;
• Seven five – thirty five and others.

Such variations do not cause much difficulty in remembering.

Counting on fingers

Children often resort to turning over a notebook with the multiplication table or even using a calculator when they forget complex examples. There is an easy way to count using your fingers.

This method makes it easy to multiply numbers by 9. Counting algorithm:

Example: you need to multiply 3 by 9. The hand is put forward, the third finger is bent. To the left of it are 2 fingers - indicating the number 20. To the right there are 7 fingers, which means 7 units. In the end there were 27.

Multiplying 9 by 9. The ninth finger is bent. There are 8 left on the left (which means 80), and 1 on the right. It turns out 81.

The following method will help you multiply numbers greater than 5. It's more complicated. Let's say you need to calculate how much 6 by 7 is. To do this, one finger on the left hand is extended (6 is more than 5 by 1), the others are bent. On the right, 2 fingers remain (7 is more than 5 by 2), the rest are hidden.

The visible fingers add up: 1 + 2 = 3 tens (that is, 30). The bent ones are multiplied: 4 x 3 = 12. It remains to add the resulting numbers: 30 + 12 = 42. The answer has been found.

A child should learn the multiplication tables with pleasure, so you should interest him.

You can do this in the form of a magic game: give the child a table and ask him to name any example from the desired column. He will ask, after which the parent will say that he can count with his own hand, and quickly show the whole sequence. It is important that the student himself wants to learn multiplication. This way the process will move much faster.

Using toys

The multiplication table (a game to quickly learn the algorithm can be offered to the child instead of boring memorization) is better learned with the help of additional “tools.” To do this, you will need your child's favorite toys.

The essence of the method is simple: take cars, construction parts, furniture from a dollhouse. Examples are built on this basis. Let’s say you need to multiply 5 by 3. Then 5 construction set pieces are laid out in front of the baby.

It is reported that you need to take 5 bricks 3 times to build a house. He takes these 5 parts, adds 5 more, and again the same number. Then he calculates that 15 bricks were brought to the construction site. With the help of such visual practice, the material is remembered faster.

If the method with toys is used, then you should not additionally burden the student with counting on fingers. It is better to choose one option so as not to confuse the child.

Mobile or online games

Modern assistants are computer games. The memorization process is fun and unnoticeable.

Examples of online games:

• "Multiplication tables: animal puzzles". The child needs to collect the picture. This can be done only by solving the examples - they are below. At the top of the screen there is a picture in the form of a table with numbers. After solving the example, the puzzle piece is dragged onto the square with the resulting number. As a result, an image of the animal should open. In the game you can select a mode with multiplication by any number.
• "Multiplication tables: adventures in the jungle." The player is transported to a world of dangerous forests where an ancient tribe lives. One of his representatives, the guy Jim, should get a mask. But before that, you need to cope with various challenges - fishing, drumming and target shooting. The child must help the little savage overcome obstacles. This can be done only by solving examples.
• "Multiplication tables: skateboard racing." A fun ride and memorization of examples will make learning unnoticeable. A skateboarder is rushing along the road, constantly having to overcome obstacles. To help him, you should quickly solve examples. In the game you can select a mode not only for multiplication, but also for other arithmetic operations.
• "Snake". A small worm crawls on a green meadow. There are numbers scattered everywhere - answers to examples that need to be guessed. The snake is only allowed to eat balls with the correct answer. The more eaten, the longer it is. The game has the ability to select a mode - the numbers by which the multiplication will be performed.

Many games are aimed not at learning, but at remembering, that is, it is assumed that the child has already memorized the material, and during the game he simply tests himself. Therefore, you can allow using this method after studying - as a cool-down.

Sea battle

The multiplication table (a game to learn faster, it can be used in the form of a sea battle) is easier to remember in a game form. According to the rules, a couple of players participate. Each player draws two fields on a checkered sheet of paper: one for their own ships, and the second for the opponent’s fleet. To memorize the table, instead of letters, numbers are written on one side - the field resembles a Pythagorean table.

Having drawn the ships, the battle begins. The game should be played like this: it is called a multiplication action, for example, 2 x 7. The child sees that the answer is 14 at the intersection, and looks to see if there are ships there. This is how all moves are made.

Another option is to call the answer, say, 32. And the child sees that this number is obtained by multiplying 4 by 8. The clarity of this method will simplify memorization.

How to learn multiplication tables by playing sea battle:

Examples from life

Learning will be easier and more enjoyable if you build the process on things that are close to the child. You can give examples from everyday life. Instead of the standard question: how much is 2 x 3, you can ask “how many cars are in the garage if 3 cars came in 2 times?” Any objects are used for explanations: matches, coins, cubes and felt-tip pens.

You can play in the store. Offer the child to buy 4 candies for 3 rubles. Let him count how much money he will need.

Multiplying by 1 and 10

Solving examples with these numbers is very simple. When any number is multiplied by 1, this number remains: 1 x 4 = 4 (that is, one was taken 4 times or four was taken 1 time). Same with all other cases: 1 x 6 = 6; 1 x 8 = 8 and so on. When increasing a number by 10 times, simply add 0 at the end: 5 x 10 = 50; 10 x 28 = 280 and so on with any number.

Multiply by 2

The product of numbers by 2 shows the doubling of the digit, that is, it is taken 2 times: 2 x 12 = 12 + 12; 5 x 2 = 5 + 5 and so on each. Therefore, all examples with doubling can be considered easy, because children already know addition.

Multiply by 3

The explanation of the table for the number 3 should begin with a simple one: 1 x 2 - this means that the number 1 is taken 2 times, that is, 1 + 1. If you take one more 1, you get 1 + 1 + 1. This example is replaced by multiplication: 1 x 3 or 3 x 1 (the result will not change if the factors are changed).

Then it turns out that all other examples of multiplication by 3 are built on the same principle: the number is doubled and added. For example, 6 x 3 - this means that 6 was multiplied by 2 and another 6 was added. It looks like this: 6 x 2 + 6 = 18, that is, 6 x 3 = 18.

Thus, it is worth doing multiplication with all numbers. You can also note that even results of multiplication by 3 in the table alternate with odd ones.

Multiply by 4

It is assumed that a number is taken 4 times, for example, 3 x 4 can be written like this: 3 + 3 + 3 + 3 = 12, that is, the three is taken 4 times. If you rearrange them, you get 4 x 3 = 4 + 4 + 4 - the four was taken 3 times. To explain, use the example of animal legs. The child is asked the question: how many legs does a cat have? He answers that 4 (it is better to use a picture of an animal for clarity).

If one cat has so many legs, then the example would be: 4 (legs) x 1 (cat) = 4. Next, you ask how many legs two cats have. The child will say that 8. The example is composed again: 2 (cats) x 4 (legs) = 8. In this way, you can write all examples of multiplication by 4.

Multiply by 5

Memorizing the multiplication table by 5 is quite easy, but if you need to multiply large numbers, difficulties arise. To multiply a number by 5, you need to divide it in half (by 2), then simply add 0 to the answer. If the result of dividing by 2 is a non-integer number, then simply add 5 at the end (ignoring the comma).

Example: 3258 x 5 = (3258 / 2) and a zero = 16290 is assigned.

Multiplying by 6,7,8

There is a simple way to count on your fingers. Until the child has memorized the entire multiplication table for large numbers, it will be useful for him to learn this simple version of multiplication. So, turn your hands with your palms facing you. Each finger on the hands is assigned numbers from 6 to 10 (you need to start with the little finger).

For example:

• calculate the product of 6 by 7. You should connect the finger under the sixth number on the left hand and the seventh on the right.

Now you need to count: the number of fingers under the connected ones and those that are connected. In this case, one finger is below and two are connected, resulting in 3 fingers. This will be 3 tens. The remaining fingers on top are multiplied: there are four on the left hand and three on the right - 3 x 4 = 12. Since the number is greater than 9, the tens and the resulting number are added: 30 + 12 = 42. This will be the answer.

One more example:

• 8 x 7. The eighth finger of the left hand is connected to the seventh finger of the right. Together with the connected ones, you get five fingers, that is, the number 50. If you multiply the remaining ones, you get 6 (units). The result is 56.

Multiply by 9

Multiplying by nine using fingers was discussed above. In fact, only one example is worth remembering here: 9 x 9 = 81. All the rest refer to other numbers, because changing places does not change the product. So, 9 x 8 = 8 x 9, and 9 x 5 = 5 x 9 and so on.

So that the child is not frightened by the huge number of examples on the spread of the math notebook, he should immediately be told that he can cross out more than half of the tablet, because it is not necessary to learn all this.

When multiplying by 1 and 10, you do not need to remember the answers. Multiplying by 2 is simply doubling a number; children are taught to add earlier. Changing the factors does not change the product, so not everything will have to be memorized.

So, out of 80 examples located on the back of the notebook, you will need to learn only 36 (the rest are crossed out).

Among the presented methods for working with the Pythagorean table and memorization tricks, one can highlight those that will help you learn the material quickly and effectively and will not turn mathematics into a boring and uninteresting subject, but will make it like a game.

With a little effort, your child won’t have to look anywhere for even the most complex examples.

Article format: Svetlana Ovsyanikova

Video on the topic: activities with a child: how to quickly learn the multiplication table

How to quickly learn the multiplication table:

If your child has trouble learning the multiplication tables, tell him about little tricks that will help him solve school problems and examples without problems! The easiest way to master multiplication is to multiply on your fingers.

Yes, yes, you can not only count on your fingers, but also multiply. And if the multiplication table by 1, 2, 3, 4 and 5, as a rule, is given to the child without difficulty, then in order to learn how to multiply by 6, 7, 8, and 9 he will need your help. Multiplication on fingers will help your child do math without difficulty.

Multiplication on fingers 6, 7 and 8

Multiplication on fingers by 6, 7, 8

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

Now try multiplying 7 by 8 in the same way. To do this, connect finger No. 7 on your left hand with finger No. 8 on your right.

Now count your fingers: the number of fingers under the joined fingers is tens.

Multiplication table: multiplication on fingers

Multiplication table: multiplication on fingers

Multiply the fingers of the left hand remaining on top by the fingers of the right hand - these will be units (3 x 2 = 6). The total is 56.

If, when multiplying “units”, the result is greater than 9, then both results must be added to the column.

For example, if you need to multiply 7 by 6.

In this case, the “units” are 12 (3 x 4). And tens are equal to 3.

3 (tens)
+
12 (units)
________
42

Multiplying by 9 on your fingers

Turn your hands with your palms facing you. Now the fingers will be numbered in order, from left to right, that is, from 1 to 10, as in the figure.

Multiplication table: multiplication on fingers

Try multiplying 2 by 9. 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.

In modern elementary schools, the multiplication tables begin to be taught in the second grade and end in the third, and learning the multiplication tables is often assigned for the summer. If you didn’t study in the summer, and your child is still “floating” in multiplication examples, we’ll tell you how to learn the multiplication table quickly and fun - with the help of drawings, games and even your fingers.

Problems that children often have in connection with the multiplication tables:

1. Children don't know what 7 x 8 is.
2. They don’t see that the problem must be solved by multiplication (because it doesn’t directly say: “What is 8 times 4?”)
3. They don't understand that if you know that 4 × 9 = 36, then you also know what 9 × 4, 36: 4 and 36: 9 are equal to.
4. They don’t know how to use their knowledge and use it to reconstruct a forgotten piece of the table.

How to quickly learn the multiplication table: the language of multiplication

Before you start teaching the multiplication table with your child, it’s worth stepping back a little and realizing that a simple multiplication example can be described in a surprising number of different ways. Take the 3×4 example. You can read it as:

• three times four (or four times three);
• three times four;
• three times four;
• product of three and four.

At first, it is far from obvious to the child that all these phrases mean multiplication. You can help your son or daughter if, instead of repeating yourself, you casually use different language when talking about multiplication. For example: “So how much is three times four? What do you get if you take three times four?”

In what order should I learn the multiplication tables?

The most natural way for children to learn multiplication tables is to start with the easiest ones and work their way up to the most difficult ones. The following sequence makes sense:

Multiplying by ten (10, 20, 30...), which children learn naturally as they learn to count.

Multiplying by five (after all, we all have five fingers and toes).

Multiplying by two. Pairs, even numbers and doubling are familiar even to young children.

Multiplying by four (after all, this is just doubling multiplying by two) and eight (doubling multiplying by four).

Multiplying by nine (there are quite convenient techniques for this, more on them below).

Multiplying by three and six.

Why is 3x7 equal to 7x3

When helping your child remember the multiplication tables, it is very important to explain to him that the order of the numbers does not matter: 3 × 7 gives the same answer as 7 × 3. One of the best ways to show this clearly is - use array. This is a special mathematical word that refers to a set of numbers or shapes enclosed in a rectangle. Here, for example, is an array of three rows and seven columns.

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*******
*******

Arrays are a simple and visual way to help your child understand how multiplication and fractions work. How many points are there in a 3 by 7 rectangle? Three rows of seven elements total 21 elements. In other words, arrays are an easy-to-understand way to visualize multiplication, in this case 3 × 7 = 21.

What if we draw the array in a different way?

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***
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***

Obviously, both arrays must have the same number of points (they do not have to be counted individually), since if the first array is rotated a quarter turn, it will look exactly like the second.

Look around, look nearby, in the house or on the street, for some arrays. Take a look at the brownies in the box, for example. The cakes are arranged in a 4 by 3 array. What if you rotate them? Then 3 by 4.

Now look at the windows of the high-rise building. Wow, this is also an array, 5 by 4! Or maybe 4 to 5, depending on how you look? Once you start paying attention to arrays, it turns out that they are everywhere.

If you've already taught your children the idea that 3 x 7 is the same as 7 x 3, then the number of multiplication facts you need to memorize decreases dramatically. Once you memorize 3 × 7, you get the answer to 7 × 3 as a bonus.

Knowing the commutative law of multiplication reduces the number of multiplication facts from 100 to 55 (not exactly half due to squaring cases such as 3×3 or 7×7, which have no pair).

Each of the numbers located above the dotted diagonal (for example, 5 × 8 = 40) is also present below it (8 × 5 = 40).

The table below contains one more hint. Children usually start learning their multiplication tables using counting algorithms. To figure out what 8 × 4 is equal to, they count like this: 4, 8, 12, 16, 20, 24, 28, 32. But if you know that eight is four is the same as four times eight, then 8, 16 , 24, 32 will be faster. In Japan, children are specifically taught to “put the lowest number first.” Seven times 3? Don't do this, count better 3 times 7.

Learning squares of numbers

The result of multiplying a number by itself (1 × 1, 2 × 2, 3 × 3, etc.) is known as square of the number. This is because graphically this multiplication corresponds to a square array. If you go back to the multiplication table and look at its diagonal, you will see that it is all made up of squares of numbers.

They have an interesting feature that you can explore with your child. When listing the squares of numbers, pay attention to how much they increase each time:

Squares of numbers 0 1 4 9 16 25 36 49...
Difference 1 3 5 7 9 11 13

This curious connection between squared numbers and odd numbers is a great example of how different kinds of numbers are related to each other in mathematics.

Multiplication table for 5 and 10

The first and easiest table to memorize is the 10 multiplication table: 10, 20, 30, 40...

In addition, children learn the multiplication table by five relatively easily, and they are helped in this by their arms and legs, which visually represent four fives.

It is also convenient that the numbers in the multiplication table for five always end in 5 or 0. (So, we know for sure that the number 3,451,254,947,815 is present in the multiplication table for five, although we cannot verify this using a calculator: on The device screen simply won’t fit such a number).

Children can easily double numbers. This is probably due to the fact that we have two hands with five fingers on each. However, children do not always associate doubling with multiplying by two. The child may know that if you double six you get 12, but when you ask him what six equals two, he has to count: 2, 4, 6, 8, 10, 12. In this case, you should remind him that six is ​​two - the same as twice six, and twice six is ​​double six.

So, if your child is good at doubling, then he essentially knows the two times table. At the same time, he is unlikely to immediately realize that with its help you can quickly imagine the multiplication table by four - for this you just need to double and double again.

Game: double adventure

Any game in which players roll dice can be adapted so that all rolls count as doubles. This gives several advantages: on the one hand, children like the idea of ​​going twice as far as the dice shows with each throw; on the other hand, they gradually master the multiplication table by two. In addition (which is important for parents busy with other things), the game ends in half the time.

Multiplication table by 9: compensation method

One way to master the nine times table is to take the result of multiplying by ten and subtracting the excess.

What is nine times seven? Ten times seven is 70, subtract seven to get 63.

7 × 9 = (7 × 10) - 7 = 63

Perhaps a quick sketch of an appropriate array will help cement this idea in the child's mind.

If you have only memorized the nine times table up to "nine ten", then nine 25 will baffle you. But ten times 25 is 250, subtract 25, we get 225. 9 × 25 = 225.

Test yourself

Can you solve the 9 × 78 example in your head using the compensation method (multiplying by 10 and subtracting 78)?

There is another convenient way to master the nine multiplication table. It uses fingers and kids love it.

Hold your hands in front of you, palms down. Imagine that your fingers (including your thumb) are numbered from 1 to 10. 1 is the little finger on your left hand (the outermost finger to your left), 10 is the little finger on your right (the outermost finger to your right).

To multiply a number by nine, bend the finger with the corresponding number. Let's say you are interested in nine 7. Bend the finger that you mentally designated as the seventh number.

Now look at your hands: the number of fingers to the left of the curled one will give you the number of tens in your answer; in this case it is 60. The number of fingers on the right will give the number of ones: three. Total: 9 × 7 = 63. Try it: This method works for all single-digit numbers.

Multiplication table for 3 and 6

For children, the multiplication table by three is one of the most difficult. In this case, there are practically no tricks, and the multiplication table by 3 will simply have to be memorized.

The multiplication table for six follows directly from the multiplication table for three; here, again, it all comes down to doubling. If you know how to multiply by three, just double the result - and you get a multiplication by six. So 3 × 7 = 21, 6 × 7 = 42.

Multiplication table for 7 - dice game

So all we have left is the seven times table. There is good news. If your child has successfully mastered the tables described above, there is no need to memorize anything at all: everything is already in the other tables.

But if your child wants to learn the 7 times table separately, we will introduce you to a game that will help speed up this process.

You will need as many dice as you can find. Ten, for example, is an excellent number. Tell your son or daughter that you want to see which of you can add the numbers on the dice the fastest. However, let the children decide how many dice to roll. And to increase your child’s chances of winning, you can agree that he must add the numbers indicated on the upper faces of the cubes, and you – those on both the top and bottom.

Have each child choose at least two dice and place them in a glass or mug (they are great for shaking the dice to create a random roll). All you need to know is how many cubes the child took.

As soon as the dice are rolled, you can immediately calculate the total of the numbers on the top and bottom faces! How? Very simply: multiply the number of dice by 7. Thus, if three dice were drawn, the sum of the top and bottom numbers would be 21. (The reason, of course, is that the numbers on opposite sides of the die always add up to seven.)

Children will be so amazed at the speed of your calculations that they will also want to master this method so that they can use it someday in a game with their friends.

In the era of the so-called British Imperial system of measures and "non-decimal" money, everyone needed to own an account up to 12 × 12 (then there were 12 pence in a shilling and 12 inches in a foot). But even today, 12 comes up every now and then in calculations: many people still measure and count in inches (in America this is the standard), and eggs are sold by dozens and half-dozens.

Little of. A child who can freely multiply numbers greater than ten begins to develop an understanding of how large numbers are multiplied. Knowing the 11 and 12 multiplication tables helps you spot interesting patterns. Here is the complete multiplication table for up to 12.

Note that the number eight, for example, appears four times in the table, while 36 appears five times. If you connect all the cells with the number eight, you get a smooth curve. The same can be said about cells with the number 36. In fact, if a certain number appears in the table more than twice, then all places where it appears can be connected by a smooth curve of approximately the same shape.

You can encourage your child to explore on his own, which will keep him busy for (maybe) half an hour or more. Print out several copies of the table for multiplying the first twelve numbers by 12, and then ask him to do the following:

• color all cells with even numbers red, and all cells with odd numbers blue;
• determine which numbers appear there most often;
• say how many different numbers are found in the table;
• answer the questions: “What is the smallest number not found in this table? What other numbers from 1 to 100 are missing in it?”

Focus with eleven

The 11 multiplication table is the easiest to construct.

1 × 11 = 11
2 × 11 = 22
3 × 11 = 33
4 × 11 = 44
5 × 11 = 55
6 × 11 = 66
7 × 11 = 77
8 × 11 = 88
9 × 11 = 99

• Take any number from ten to 99 - let it be, say, 26.
• Break it into two numbers and move them apart to create a space in the middle: 2 _ 6.
• Add the two digits of your number together. 2 + 6 = 8 and insert what you got into the middle: 2 8 6

This is the answer! 26 × 11 = 286.

But be careful. What do you get if you multiply 75 x 11?

• Breaking down the number: 7 _ 5
• Add: 7 + 5 = 12
• We insert the result in the middle and get 7125, which is obviously wrong!

What's the matter? There is a little trick in this example that needs to be used when the digits used to represent the number add up to ten or more (7 + 5 = 12). We add one to the first of our numbers. Therefore, 75 × 11 is not 7125, but (7 + 1)25, or 825. So the trick is actually not as simple as it might seem.

Game: beat the calculator

The purpose of this game is to develop the skill of quickly using the multiplication table. You will need a deck of playing cards without pictures and a calculator. Decide which player will be the first to use the calculator.

• The player with the calculator must multiply the two numbers drawn on the cards; he must use a calculator even if he knows the answer (yes, this can be very difficult).
• The other player must multiply the same two numbers in his head.
• The one who gets the answer first gets a point.
• After ten attempts, players change places.

In life, people who are able to do mental calculations look like “super smart people,” although there is nothing complicated about it. A calculator is a calculator, but counting in your head is useful!
How to help your child learn the multiplication tables
Below are some simple techniques

Multiplying by 2 or doubling. Doubling is quite easy, just add something to yourself. At first, I showed one, two, three, four, five fingers on my left and right hand at the same time - this is how we got 2, 4, 6, 8, 10. Together with my student’s fingers, we reached twenty, and then I pointed to various things in the room, and suggested counting and doubling - the number of letters in a poster, the number of symbols on a watch dial, counting the number of spokes on one side of a bicycle wheel, and checking whether the total number matches the double, and so on.

Multiplying by 4 and 8, 3 and 6

When you know how to multiply by two, this is mere nonsense. Multiplying by four is the same as doubling the answer for something that has already been doubled, for example, 7x4 is 7x2x2, and we already remembered well that 7x2 is 14 in the previous lesson about doubling, so turn 14 itself into 28 will not be difficult. Once you've figured out the four, it's not that hard to figure out the big eights. Along the way we noticed that, for example, 16 is both 2x8 and 4x4. So we learned that there are numbers consisting entirely of twos: 2, 4, 8, 16, 32, 64.

By multiplying by 3 and 6, we learned the old pirate method of "dividing by three." If you add the digits of a number multiplied by 3, 6, or any other number that is divisible by three, then the result of adding the digits of the answer is always a multiple of three. For example, 3x5 = 15, 1+5 = 6. Or 6x8 = 48, and 4+8 = 12, a multiple of three. And you can add the numbers into 12, you also get 3, so if you get to the end like this, you always get one of three numbers: 3, 6 or 9.

So we turned it into another game. I would ask a number, even a three- or four-digit one, and ask if it was divisible by 3. To answer, just add the numbers, which is quite simple. If the number was divisible by 3, then I asked - “and by 6?” – and then you just had to see if it was even. And then (in the special case of small numbers from the table) sometimes I also wanted to find out what would happen when dividing by 3 or 6. It was a very fun activity.

Multiplying by 5 and 7, prime numbers
And now we are left with multiplication by five, seven, and nine. This means that we learned to multiply them by many other numbers - by 1, 2, 3, 4, 6, 8 and 10. We figured out five very quickly - it’s easy to remember: at the end there is either a zero or five, just the same as a number to be multiplied: either even or odd. A clock dial is a great object to use with A's; you can come up with many problems about traveling in time and space. At the same time, I explained why there are sixty minutes in an hour, and we understood why this is convenient.

We saw that it is convenient to divide 60 by 1, 2, 3, 4, 5, 6, but it is inconvenient to divide by 7. So it was time to take a closer look at this number. From multiplication by seven, the only ones left to remember were 7x7 and 7x9. Now we knew almost everything we needed. I explained that seven is simply a very proud number - such numbers are called prime, they are divisible only by 1 and themselves.

Math can be fun and easy. Check out this cute table.
If you study it thoughtfully, there is not much to learn. There are 36 positions in total. The rest are either simple (1 x 10) or reversible (2 x 4 = 4 x 2). Minus 10 positions from the multiplication table by 9. It can be learned in 5 minutes. There is this trick:

So, let's go.

First, let's put our hands on the table and mentally number our fingers from left to right from 1 to 10. To perform the multiplication action, let's say 9 x 3 = ?, bend the third finger from the left. All! The answer is ready: the remaining uncurled fingers on the left form the number of tens in the answer, and the uncurled fingers on the right form the number of units. We count and say the answer: 27!

This way you can get the answer for any number. Here, for example, is an example 9 x 7 = 63

watch multiplication by 9 in the video: