Fast multiplication method of two fingers. Multiplication on fingers
Training
Each finger on the left and on the right hand is assigned a certain number:
little finger - 6,
ring finger - 7,
average - 8,
index - 9
and big - 10.
At the beginning of mastering the method, these numbers can be drawn on your fingertips. When multiplying, the hands are arranged naturally, palms facing you.
Methodology
1. Multiply 7 by 8. Turn your hands with your palms facing you and touch the ring finger (7) of the left hand of the middle finger (8) of the right (see Fig.).
Let's pay attention to the fingers that turned out to be higher than the touching fingers 7 and 8. On the left hand above 7 there were three fingers (middle, index and thumb), on the right hand above 8 - two fingers (index and thumb).
We will call these fingers (three on the left hand and two on the right) upper. The remaining fingers (little and ring fingers on the left hand and the little finger, ring and middle fingers on the right) will be called lower. In this case (7 x 8) you get 5 upper fingers and 5 lower ones.
Now let's find the product of 7 x 8. To do this:
1) multiply the number of lower fingers by 10, we get 5 x 10 = 50;
2) multiply the number of upper fingers on the left and right hands, we get 3 x 2 = 6;
3) finally, add these two numbers, we get the final answer: 50 + 6 = 56.
We got that 7 x 8 = 56.
2. Multiply 6 by 6. Let's turn our hands with palms facing us and touch the little finger (6) of the left hand of the little finger (6) of the right (see Fig.).
Now there are 4 upper fingers on the left and right hands.
Let's find the product of 6 x 6:
1) multiply the number of lower fingers by 10: 2 x 10 = 20;
2) multiply the number of upper fingers on the left and right hands: 4 x 4 = 16;
3) add these two numbers: 20 + 16 = 36.
We got that 6 x 6 = 36.
3. Multiply 7 by 10. This will be a test of the multiplication rule by 10. Touch the ring finger (6) of the left hand thumb(10) right. There are 3 upper fingers on the left hand, 0 on the right (see fig.).
Let's find the product of 7 x 10:
1) multiply the number of lower fingers by 10: 7 x 10 = 70;
2) multiply the number of upper fingers on the left and right hands: 3 x 0 = 0;
3) add these two numbers: 70 + 0 = 70.
We got that 7 x 10 = 70.
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Multiply by 9
To do this, we put our hands palms down next to each other, fingers need to be straightened. Now, to multiply any number by 9, we simply bend our finger under the number of this number (counting from the left). The number of fingers before the bent one will be the tens of the answer, and after - the units.
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Then, with the ease of a magician, we "click" multiplication examples: 2 3, 3 5, 4 6 and so on. With age, however, we increasingly forget about factors closer to 9, especially if we haven’t known counting practice for a long time, why we surrender to the power of a calculator or hope for the freshness of a friend’s knowledge. However, having mastered one simple technique of "manual" multiplication, we can easily refuse the services of a calculator. But let's clarify right away that we are talking only about the school multiplication table, that is, for numbers from 2 to 9, multiplied by numbers from 1 to 10.
Multiplication for the number 9 - 9 1, 9 2 ... 9 10 - is easier to fade from memory and more difficult to manually recalculate by addition, but it is for the number 9 that multiplication is easily reproduced "on the fingers". Spread your fingers on both hands and turn your palms away from you. Mentally assign numbers from 1 to 10 to the fingers in sequence, starting with the little finger of the left hand and ending with the little finger right hand(this is shown in the figure).
Let's say we want to multiply 9 by 6. We bend a finger with a number equal to the number by which we will multiply the nine. In our example, you need to bend the finger with number 6. The number of fingers to the left of the bent finger shows us the number of tens in the answer, the number of fingers to the right - the number of ones. On the left, we have 5 fingers not bent, on the right - 4 fingers. Thus, 9 6=54. The figure below shows the whole "computation" principle in detail.
Another example: you need to calculate 9 8=?. Along the way, we will say that fingers may not necessarily act as a "calculating machine". Take, for example, 10 cells in a notebook. We cross out the 8th cell. There are 7 cells on the left, 2 cells on the right. So 9 8=72. Everything is very simple.
Now a few words to those inquisitive children who, in addition to the mechanical application of what has been said, want to understand why it works. Everything here is based on the observation that the number 9 is only one missing from the round number 10, in which the units place contains the number 0. Multiplication can be written as the sum of the same terms. For example, 9 3=9+9+9. Each time we add the next nine, we know that one more one in the answer will not reach the round number. Therefore, how many times a nine was added (or, in other words, what number x was multiplied by), the same number of ones will be missing in the answer. Since the unit digit calculates no more than 10 numbers (from 0 to 9), and when multiplying 9 x =? there are exactly x ones missing in the ones place, then the number in the ones place will be equal to 10-x. This is reflected in the example with hands: we bent the finger with number x and counted the remaining fingers on the right to place ones, but in fact, from 10 fingers, we simply excluded fingers with numbers from 1 to x, thus performing the operation 10-x.
At the same time, with each added nine, the number in the tens digit increases by 1, and initially this digit was empty (equal to zero). That is, for the first nine, the tens digit is zero, adding the second nine increases it by 1, the third nine - by another 1, and so on. This means that the number of tens is equal to x-1, since the tens count started from zero. In the hands example, we folded the finger numbered x, thus providing the "minus one" action, and counted the number of fingers to the left of the folded one, and there are exactly x-1 of them. This is the secret of this simple technique.
From this follow additional considerations. Not only that, the example 9 x=? it is easy to calculate through the number x (the tens digit is x-1, the units digit is 10-x), so another example can be calculated as x 10-x. In other words, we add one zero to the right of the number x and subtract the number x from the resulting number. For example, 9 5=50-5=45 or 9 6=60-6=54 or 9 7=70-7=63 or 9 8=80-8=72 or 9 9= 90-9=81. With this unusual step, we turn the multiplication example into a subtraction example, which is much easier to solve.
Multiplication for the number 8 - 8 1, 8 2 ... 8 10 - the actions here are similar to the multiplication for the number 9 with some changes. First, since the number 8 is already missing two to the round number 10, we need to bend two fingers at once each time - with the number x and the next finger with the number x + 1. Secondly, immediately after the bent fingers, we must bend as many more fingers as there are left unbent fingers on the left. Thirdly, this works directly when multiplying by a number from 1 to 5, and when multiplying by a number from 6 to 10, you need to subtract five from the number x and perform the calculation as for the number from 1 to 5, and then add the number 40 to the answer, because otherwise you will have to perform the transition through a dozen, which is not very convenient "on the fingers", although in principle it is not so difficult. In general, it should be noted that multiplication for numbers below 9 is the more inconvenient to perform "on the fingers", the lower the number is located from 9.
Now consider an example of multiplication for the number 8. Let's say we want to multiply 8 by 4. We bend the finger with number 4 and after it the finger with number 5 (4 + 1). On the left we have 3 unbent fingers, so we need to bend 3 more fingers after the finger with number 5 (these will be fingers with numbers 6, 7 and 8). There are 3 fingers not bent on the left and 2 fingers on the right. Therefore, 8 4=32.
Another example: calculate 8·7=?. As mentioned above, when multiplying by a number from 6 to 10, you need to subtract five from the number x, perform the calculation with the new number x-5, and then add the number 40 to the answer. We have x \u003d 7, which means we bend the finger with number 2 ( 7-5=2) and the next finger number 3 (2+1). On the left, one finger was not bent, so we bend another finger (with number 4). We get: 1 finger is not bent on the left and 6 fingers on the right, which means the number 16. But 40 must also be added to this number: 16+40=56. As a result, 8 7=56.
And just in case, let's look at an example with a transition through a dozen, where no fives need to be subtracted beforehand and no 40s need to be added after either. Suddenly it will be easier for you. Let's try to calculate 8·8=?. We bend two fingers with numbers 8 and 9 (8 + 1). On the left, there are 7 uncurved fingers. Remember that we already have 7 tens. Now we begin to bend 7 fingers on the right. Since there is only one unbent finger left, we bend it (6 more to bend), then go through a dozen (this means that we unbend all the fingers), and bend 6 unbent fingers from left to right. On the right, there are 4 fingers left not bent, which means that the answer will have the number 4 in the unit category. Earlier, we remembered that there were 7 tens, but since we had to go through a dozen, one ten must be discarded (7-1 \u003d 6 tens). As a result, 8 8=64.
Additional considerations: here it is also possible to calculate examples simply through the number x in the form of a subtraction expression x·10-x-x. That is, we add one zero to the right of the number x and subtract the number x twice from the resulting number. For example, 8 5=50-5-5=40, or 8 6=60-6-6=48, or 8 7=70-7-7=56, or 8 8=80-8-8 =64, or 8 9=90-9-9=72.
Multiplication for the number 7 - 7 1, 7 2 ... 7 10. Here you can not do without transitions through a dozen. The number 7 is enough for the triple to the round number 10, therefore, you will have to bend 3 fingers at once. We immediately remember the resulting number of tens by the number of fingers not bent on the left. Next, as many fingers are bent on the right as there are dozens. If during the bending of the fingers a transition through a dozen is required, we do it. Then the same number of fingers are bent a second time, that is, one operation is performed twice. And now the number of uncurved fingers remaining on the right is recorded in the category of units, the number of previously counted tens (minus the number of transitions through a dozen) - in the category of tens.
You see how it is already becoming more difficult to count "on the fingers" than to get this information out of memory. And then, for the numbers 7, 8 and 9, the forgetfulness of the elements of the multiplication table is somehow justified, but for the numbers below it is a sin not to remember. Therefore, at this point we will stop the story in the hope that you have grasped the very thread of "calculations" and, if it is absolutely necessary, you will be able to independently descend to numbers below 7, although a person who counts "on his fingers" something in the spirit of "five five "must look really stupid.
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Being able to multiply on your fingers is a valuable skill, and mankind has known how to count multiplication tables on your fingers since at least the 15th century. We may have mobile calculators, but in many cases, it's actually easier to keep your phone in your pocket and multiply on your fingers. This technique can also be helpful for toddlers who have trouble learning endless mathematical formulas.
You can start learning the multiplication table on the fingers after the child knows the multiplication from one to five. Already on the basis of this knowledge, it is possible to develop a skill in literally manual multiplication. So let's get started?
Multiplication table on fingers: nine
Hold your hands in front of you with palms up. Each of your ten fingers represents a number. Moving from the thumb of the left hand to the thumb of the right hand, count the numbers from one to ten.
Point the finger corresponding to the number you want to multiply by nine down towards your body. So, for example, if you want to decide how much 9x3 will be, you will need to hold the middle finger with your left hand. The middle finger represents the number three, because if you count your fingers from one to ten, starting with your left thumb, your middle finger is the third one.
We make a count
The problem is solved by counting fingers to the left and to the right. First, count the fingers to the left of your bent finger - in this case there will be two. Then count the fingers to the right of your bent finger - in this case it should be seven. The first digit of the answer is two, and the second digit is seven. So the answer is 27!
This is how the multiplication table for 9 works on the fingers. Try it with other multiples of nine. How would you multiply 9 by 2? How about 9 by 7? This method is incredibly simple and understandable even for kids. As practice shows, children are more willing and successful in studying mathematics, knowing this interesting way counting the product of two numbers!
Multiplication table on fingers for six, seven, eight and ten
Hold your hands so that your palms are facing your body and your fingers are facing each other. Again each finger will represent a number. Your little finger represents the number six. The ring finger will be seven, the middle finger eight. The index fingers of your hands will symbolize the nine, and your thumbs will symbolize the ten. So, how to learn the multiplication table on your fingers?
Calculation scheme
For example, if you want to calculate what 7 * 6 will be, you need to touch the ring finger of your left hand (since it represents the number on the left) with the little finger of your right hand, since it represents the number on the right. Again, remember that each finger represents a number, in which case your ring finger represents seven and your little finger represents six. Therefore, you need to connect them to solve this mathematical problem.
You may have to bend your wrist in weird ways to calculate the product of two numbers! Who said it would be easy?
In order to make sure that you have correctly understood the technique of the multiplication table on the fingers of six, seven, eight and ten, check yourself. If you need to calculate what the product of 9 and 7 will be, then which fingers would you connect? Think! The answer will be in the next sentence.
So, consider that you have learned the finger multiplication table for six, seven, eight and ten, if as an answer, which fingers you need to connect in order to calculate what the product of 9 and 7 is, you chose the index finger of your left hand and the ring finger finger of the right hand. It's a matter of small!
How to count?
The next step is to simply count the fingers that are touching as well as the fingers underneath. They will represent decimal numbers. In this case, you will count the ring finger on the left hand, the little finger on the left hand, and the little finger on the right hand. Each finger you count will equal 10. In this case, the total is 30.
Multiply the remaining fingers. The next step is to add up the number of fingers on each hand, not counting the fingers that touch each other. First count the number of fingers on the left hand that are above the touching fingers - in this case there will be 3. Then count the number of fingers of the right hand above the touching fingers - in this case there will be 4. 3 * 4 \u003d 12. Add the two numbers together, to find your answer. In this case, you need to add 30 to 12. The total will be 42. If 7 is multiplied by 6, then the answer will be 42!
The multiplication table on the fingers may seem complicated at first, however, if you carefully understand it, then learning it is much easier than the endless formulas in a real mathematical table.
Multiply by 10 using the same method. For example, if you want to find the answer, which is 10 times 7, then start by tapping thumb left hand ring finger of the right hand. Count the number of fingers under the connecting fingers, including fingers that touch each other. You should have a total of 7, which means 70. Then count the number of fingers above the touching fingers of the right and left hands. There should be 0 on your left and 3 on your right. Now multiply 3 by 0 = 0 and add 70 to 0 for your answer. The answer is 10 times 7 = 70!
Outcome
Try it with other multiples of six, seven, eight, and ten. How would you multiply 8 and 8 with your fingers? What about 8 and 10? If you are interested in the question of how to teach the multiplication table on the fingers of your child, then just try to include the practice of counting the product of various numbers in your daily routine. You will not even notice how the baby will begin not only to quickly count the product of two numbers, but also eventually remember the multiplication table.
That's the whole attraction. this method- he is cheerful, makes you think logically, turn on mathematical ability and at the same time develops memory. What could be better for a child? Let's finally calculate what the product of 6 and 10 will be? What about 8 and 9? What about 7 and 8? Here's some fun math.
Multiplication by 1 and 10
It’s worth starting with this to calm the child: multiplication by one is the number itself, and multiplication by 10, the number and zero after it. So he already knows the answers to the first and last examples in all columns.
Multiply by 2
Multiplying a number by two means adding two identical numbers.
Multiply by 3
To memorize this column, mnemonic techniques are suitable, for example, short poems. You can invent them with your child or look for "ready-made" ones on the net:
Well, my friend, look
What is three times three?
Nothing to do!
Well, of course, nine!
All the kids need to know
What is three times five
And don't be mistaken!
Three times five is fifteen!
If you are not strong in poetry, come up with prose stories whose heroes will be a two - a swan, a three - a snake, a four - an upside down chair, an eight - glasses, and so on - the children themselves will tell you who, in their opinion, the numbers look like .
Stories and rhymes can be invented not only for the triple, but also for any column of the Pythagorean table.
Multiply by 4
Multiplication by 4 can be represented as a multiplication by 2 and again by 2. This column for students who have mastered multiplication by two will not cause difficulties.
Multiply by 5
This is the easiest column to remember. All values in this column are located 5 units apart. 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.
Multiply by 6, 7, 8
These columns, as well as the multiplication by 9 column, traditionally cause difficulties for schoolchildren. You can reassure students by explaining that most they have already learned examples from these columns and the awesome 8x3 is the same as the already learned 3x8. 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. And you can learn to count on your fingers:
Similarly, you can multiply 7 by 8 or 8 by 9.
You can see the process of such multiplication with your own eyes on the video (note: in the video, the numbering is carried out in a similar way, but starting with the thumbs):
Multiply by 9
To begin with, you can remember that in the multiplication table for nine, the sum of tens and ones in the answer is always equal to 9. Namely: 9 × 2 = 18 (add the numbers of the answer: 1 + 8 = 9), the same in other examples: 9 ×6=54 (5+4=9).
In this case, the ten digit in the answer is always one less than the second factor in the example. In practice: 9 × 7 \u003d 63 (the second factor is 7, which means tens in the answer 6. If we now recall the first pattern that the sum of tens and ones in the answer should be 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 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 your 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 ring finger of the left hand is 2, and so on to the little finger of the right hand, which, accordingly, will be 10):
Name the number by which you want to multiply nine. Let's say it's number 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 indicate tens (in our case, this 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 ones (7) = 27. This is the product of 9 and 3.
Similarly, you can multiply 9 by 7 or 9 by 10.
Learning the multiplication table from any student will require perseverance and patience, but counting on the fingers, rhymes, cards with examples will help make memorization easier and make it interesting and fast.
Today in the lesson we will literally learn how to multiply numbers on our fingers. When you don’t have a notebook and a calculator at hand, pay attention to the hand itself - it has fingers on it. This multiplication method was shown to me by my grandmother, and I decided, since I myself will never become a grandmother, it's time to tell you about the possibilities of our fingers.
I hasten to warn you that the method talks about multiplying the numbers 6, 7, 8, 9. By default, it is assumed that you know how to multiply up to five.
So the counting rules are:
One bent finger is the number 6, two fingers is the number 7, three fingers is the number 8, four fingers is the number 9.
Example. We multiply 6x6. We bend the finger on both hands.
We multiply unbent fingers on each other. 4x4=16. We take the bent ones for tens, and add them up. This is 20. 20+16=36. Total 6x6=36
We multiply. 6x7.
We multiply unbent fingers on each other. 4x3=12. We take the bent ones for tens, and add them up. This is 30. 30+12=42. Total 6x7=42
Multiply 7x7
We multiply unbent fingers on each other. 3x3=9. We take the bent ones for tens, and add them up. This is 40. 40+9=49. Total 7x7=49
Multiply 7x8
We multiply unbent fingers on each other. 3x2=6. We take the bent ones for tens, and add them up. This is 50. 50+6=56. Total 7x8=56
Multiply 8x8
We multiply unbent fingers on each other. 2x2=4. We take the bent ones for tens, and add them up. This is 60. 60+4=42. Total 8x8=64
Multiply 8x9
We multiply unbent fingers on each other. 2x1=2. We take the bent ones for tens, and add them up. This is 70. 70+2=72. Total 8x9=72
And multiply 9x9
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