Multiplication table for 10. Multiplication by four. Highlight identical values
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. She 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
psyh-olog.ru
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
ytimg.com
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
blogspot.com
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:
- When multiplied by 1, any number remains the same.
- 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.
- All examples of 10 end in 0 and begin with the number we are multiplying by.
- Examples with 5 are half as many as examples with 10 (10 × 5 = 50, and 5 × 5 = 25).
- 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.
- 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, giving 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 you should never do:
- Force the child if he doesn't want to. Instead, try to motivate him.
- Scold for mistakes and scare with bad grades.
- 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.
- Learn everything at once. A child can easily be frightened and tired by a large volume of material. Learn gradually.
- Ignore successes. Praise your child when he completes tasks. At such moments he has a desire to study further.
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:
- Children don't know what 7 x 8 is.
- They don’t see that the problem must be solved by multiplication (because it doesn’t directly say: “What is 8 times 4?”)
- 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.
- 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.
Multiply by seven.
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.
*******
*******
*******
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?
***
***
***
***
***
***
***
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, 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’s 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 a multiplication table for 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 from 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.
This lesson will look at how to perform multiplication and division by numbers of the form 10, 100, 0.1, 0.001. Various examples on this topic will also be solved.
Exercise. How to multiply the number 25.78 by 10?
The decimal notation of a given number is a shorthand notation for the amount. It is necessary to describe it in more detail:
Thus, you need to multiply the amount. To do this, you can simply multiply each term:
It turns out that...
We can conclude that multiplying a decimal fraction by 10 is very simple: you need to move the decimal point to the right one position.
Exercise. Multiply 25.486 by 100.
Multiplying by 100 is the same as multiplying by 10 twice. In other words, you need to move the decimal point to the right twice:
Exercise. Divide 25.78 by 10.
As in the previous case, you need to present the number 25.78 as a sum:
Since you need to divide the sum, this is equivalent to dividing each term:
It turns out that to divide by 10, you need to move the decimal point to the left one position. For example:
Exercise. Divide 124.478 by 100.
Dividing by 100 is the same as dividing by 10 twice, so the decimal point moves left 2 places:
If a decimal fraction needs to be multiplied by 10, 100, 1000, and so on, you need to move the decimal point to the right by as many positions as there are zeros in the multiplier.
Conversely, if a decimal fraction needs to be divided by 10, 100, 1000, and so on, you need to move the decimal point to the left by as many positions as there are zeros in the multiplier.
Example 1
Multiplying by 100 means moving the decimal place two places to the right.
After the shift, you can find that there are no more digits after the decimal point, which means that the fractional part is missing. Then there is no need for a comma, the number is an integer.
Example 2
You need to move 4 positions to the right. But there are only two digits after the decimal point. It's worth remembering that there is an equivalent notation for the fraction 56.14.
Now multiplying by 10,000 is easy:
If it is not very clear why you can add two zeros to the fraction in the previous example, then the additional video at the link can help with this.
Equivalent decimal notations
Entry 52 means the following:
If we put 0 in front, we get entry 052. These entries are equivalent.
Is it possible to put two zeros in front? Yes, these entries are equivalent.
Now let's look at the decimal fraction:
If you assign zero, you get:
These entries are equivalent. Similarly, you can assign multiple zeros.
Thus, any number can have several zeros after the fractional part and several zeros before the integer part. These will be equivalent entries of the same number.
Example 3
Since division by 100 occurs, it is necessary to move the decimal point 2 positions to the left. There are no numbers left to the left of the decimal point. A whole part is missing. This notation is often used by programmers. In mathematics, if there is no whole part, then they put a zero in its place.
Example 4
You need to move it to the left by three positions, but there are only two positions. If you write several zeros in front of a number, it will be an equivalent notation.
That is, when shifting to the left, if the numbers run out, you need to fill them with zeros.
Example 5
In this case, it is worth remembering that a comma always comes after the whole part. Then:
Multiplying and dividing by numbers 10, 100, 1000 is a very simple procedure. The situation is exactly the same with the numbers 0.1, 0.01, 0.001.
Example. Multiply 25.34 by 0.1.
Let's write the decimal fraction 0.1 as an ordinary fraction. But multiplying by is the same as dividing by 10. Therefore, you need to move the decimal point 1 position to the left:
Similarly, multiplying by 0.01 is dividing by 100:
Example. 5.235 divided by 0.1.
The solution to this example is constructed in a similar way: 0.1 is expressed as a common fraction, and dividing by is the same as multiplying by 10:
That is, to divide by 0.1, you need to move the decimal point to the right one position, which is equivalent to multiplying by 10.
Multiplying by 10 and dividing by 0.1 is the same thing. The comma must be moved to the right by 1 position.
Dividing by 10 and multiplying by 0.1 are the same thing. The comma needs to be moved to the right by 1 position:
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)
*× | 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.