Poisonous Plants The presentation was made by Student 3 "B" of the class of the Gymnasium No. 1504 Kostyukova Ekaterina.
Datura ordinary These are tall herbs with whole leaves bearing large teeth along the edges. Flowers are large, white. The box is studded with thorns. It grows mainly in damp places. The whole plant is highly poisonous, especially the seeds.
Black belena All parts of the plant are poisonous. In folk medicine, henbane is used as an analgesic for inflammation, sprains, joint pain, etc. Seed ointment is used for bone diseases. An infusion of henbane is used for acne, and rubbed with ointment for colds and coughs.
Digitalis purple It is considered one of the most poisonous plants. All parts of the plant are poisonous. It causes inflammation of the mucous membranes, and also has a harmful effect on the vascular system.
B ex The plant is poisonous in any form. The sweet stalk and the sweetish rhizome with a pleasant smell are especially poisonous. If you eat it, then after 15-20 minutes there is malaise, vomiting, abdominal pain, and then convulsive seizures, against which respiratory and cardiac arrest is possible. Milestone grows in damp, swampy places. The external resemblance to carrots and the pronounced carrot smell of the rhizome led to tragic poisoning of children.
In oronets The plant is very poisonous. The juice of the plant irritates the human skin, up to the formation of blisters. And even a small amount of berry pulp is enough to cause severe upset in the gastrointestinal tract. Don't be fooled by the fact that the berries are eaten by birds. For birds, unlike other warm-blooded ones, the crow is not dangerous.
In the eagle's eye, the berry and other parts of the plant are poisonous. The plant causes irritation of the gastrointestinal tract, diarrhea, nausea, vomiting, a sharp drop in heart rate and cardiac arrest.
Do not tear in nature unknown - plants you do not know! Often very beautiful plants are poisonous. Do not eat unfamiliar berries!
Help with poisoning by poisonous plants Rinse the stomach - let the patient drink 5-6 glasses of water or milk in a row. Then induce vomiting. This procedure can be repeated 3-5 times. Put the patient to bed. Apply warm heating pads to your arms and legs. Continuously give him a warm drink, and with a sharp weakness - strong tea. Call a doctor.
There is nothing superfluous in nature! Treat her with care!
Reading 11 min. Views 192 Published on 27.09.2018
Many people ask how to learn how to quickly count in the mind so that it looks imperceptible and not stupid. After all modern technologies allow less use of their memory and mental abilities. But sometimes these technologies are not at hand and sometimes it is easier and faster to calculate something in your mind. Many people have begun to count even elementary things on a calculator or phone, which is also not very good. The ability to count mentally remains a useful skill for modern man, despite the fact that he owns all sorts of devices capable of counting for him. The ability to do without special devices and at the right time to quickly solve the set arithmetic problem is not the only application of this skill. In addition to the utilitarian purpose, mental counting techniques will allow you to learn how to organize yourself in various life situations. In addition, the ability to count in your mind will undoubtedly have a positive effect on the image of your intellectual abilities and distinguish you from the surrounding “humanists”.
Quick counting methods
There is a certain set of simple arithmetic rules and patterns that you not only need to know for mental counting, but also constantly keep in mind in order to quickly apply the most effective algorithm at the right time. To do this, it is necessary to bring their use to automatism, fix it in machine memory, so that from solving the most simple examples successfully move on to more complex arithmetic operations. Here are the main algorithms that you need to know, remember and apply instantly, automatically:
Subtraction 7, 8, 9
To subtract 9 from any number, you need to subtract 10 from it and add 1. To subtract 8 from any number, you need to subtract 10 from it and add 2. To subtract 7 from any number, you need to subtract 10 from it and add 3. If usually If you think differently, then for the best result you need to get used to this new way.
Multiply by 9
You can quickly multiply any number by 9 with your fingers.
Division and multiplication by 4 and 8
Division (or multiplication) by 4 and by 8 are two or three division (or multiplication) by 2. It is convenient to perform these operations sequentially.
For example, 46*4=46*2*2 =92*2= 184.
Multiply by 5
Multiplying by 5 is very easy. Multiplying by 5 and dividing by 2 are basically the same thing. So 88*5=440 and 88/2=44, so always multiply by 5 by dividing the number by 2 and multiplying it by 10.
Multiply by 25
Multiplying by 25 corresponds to dividing by 4 (and then multiplying by 100). So 120*25 = 120/4*100=30*100=3000.
Multiplication by single digits
For example, let's multiply 83*7.
To do this, first multiply 8 by 7 (and add zero, since 8 is the tens place), and add the product of 3 and 7 to this number. Thus, 83*7=80*7 +3*7= 560+21=581 .
Let's take a more complex example: 236*3.
So, we multiply the complex number by 3 bit by bit: 200*3+30*3+6*3=600+90+18=708.
Defining Ranges
In order not to get confused in the algorithms and not to give a completely wrong answer by mistake, it is important to be able to build an approximate range of answers. So the multiplication of single-digit numbers by each other can give a result of no more than 90 (9*9=81), two-digit numbers - no more than 10,000 (99*99=9801), three-digit numbers - no more than 1,000,000 (999*999=998001).
Layout for tens and units
The method consists in splitting both factors into tens and ones, followed by multiplying the resulting four numbers. This method is quite simple, but requires the ability to keep up to three numbers in memory at the same time and at the same time perform arithmetic operations in parallel.
For example:
63*85 = (60+3)*(80+5) = 60*80 + 60*5 +3*80 +3*5=4800+300+240+15=5355
It is easier to solve such examples in 3 steps:
1. First, tens are multiplied by each other.
2. Then 2 products of units by tens are added.
3. Then the product of units is added.
Schematically, this can be described as follows:
- First action: 60 * 80 = 4800 - remember
- Second action: 60 * 5 + 3 * 80 \u003d 540 - remember
- Third action: (4800 + 540) + 3 * 5 \u003d 5355 - answer
For the fastest effect, you will need a good knowledge of the multiplication table of numbers up to 10, the ability to add numbers (up to three digits), as well as the ability to quickly switch attention from one action to another, keeping the previous result in mind. It is convenient to train the last skill by visualizing the arithmetic operations performed, when you have to imagine a picture of your solution, as well as intermediate results.
Mental visualization of multiplication in a column
56 * 67 - count in a column. Probably the column contains maximum amount actions and requires constantly keeping auxiliary numbers in mind.
But it can be simplified:
First action: 56*7 = 350+42=392
Second action: 56*6=300+36=336 (well, or 392-56)
Third action: 336*10+392=3360+392=3752
Private methods for multiplying two-digit numbers up to 30
The advantage of the three double-digit multiplication methods for mental counting is that they are universal for any numbers and, with good mental counting skill, they can allow you to quickly arrive at the correct answer. However, the efficiency of multiplying some two-digit numbers in the mind can be higher due to fewer steps when using special algorithms.
Multiply by 11
To multiply any two-digit number by 11, you need to enter the sum of the first and second digits between the first and second digits of the multiplied number.
For example: 23 * 11, we write 2 and 3, and between them we put the sum (2 + 3). Or in short, that 23*11= 2 (2+3) 3 = 253.
If the sum of the numbers in the center gives a result greater than 10, then we add one to the first digit, and instead of the second digit we write the sum of the digits of the multiplied number minus 10.
For example: 29*11 = 2 (2+9) 9 = 2 (11) 9 = 319.
You can quickly multiply by 11 verbally not only two-digit numbers, but also any other numbers.
For example: 324 * 11=3(3+2)(2+4)4=3564
The square of the sum, the square of the difference
In order to square a two-digit number, you can use the formulas of the square of the sum or the square of the difference. For example:
23²= (20+3)2 = 202 + 2*3*20 + 32 = 400+120+9 = 529
69² \u003d (70-1) 2 \u003d 702 - 70 * 2 * 1 + 12 \u003d 4 900-140 + 1 \u003d 4 761
Squaring numbers ending in 5. To square numbers ending in 5. The algorithm is simple. The number up to the last five, multiply by the same number plus one. Add 25 to the remaining number.
25² = (2*(2+1)) 25 = 625
85² = (8*(8+1)) 25 = 7225
This is true for more complex examples as well:
155² = (15*(15+1)) 25 = (15*16)25 = 24025
The technique for multiplying numbers up to 20 is very simple:
16*18 = (16+8)*10+6*8 = 288
Proving the correctness of this method is simple: 16*18 = (10+6)*(10+8) = 10*10+10*6+10*8+6*8 = 10*(10+6+8) +6*8. The last expression is a demonstration of the method described above. In fact, this method is a private way of using pivot numbers. In this case, the reference number is 10. In the last expression of the proof, it can be seen that it is by 10 that we multiply the bracket. But any other numbers can be used as a reference number, of which 20, 25, 50, 100 are the most convenient ...
reference number
Look at the essence of this method using the example of multiplying 15 and 18. Here it is convenient to use the reference number 10. 15 is greater than ten by 5, and 18 is greater than ten by 8.
In order to find out their product, you need to perform the following operations:
1. To any of the factors, add the number by which the second factor is greater than the reference one. That is, add 8 to 15, or 5 to 18. In the first and second cases, the same thing is obtained: 23.
2. Then we multiply 23 by the reference number, that is, by 10. Answer: 230
3. To 230 we add the product 5 * 8. Answer: 270.
Reference number when multiplying numbers up to 100. The most popular technique for multiplying large numbers in the mind is to use the so-called reference number.
Reference number in multiplication is a number to which both factors are close and by which it is convenient to multiply. When multiplying numbers up to 100 with reference numbers, it is convenient to use all multiples of 10, and especially 10, 20, 50 and 100.
The technique for using the reference number depends on whether the factors are greater than or less than the reference number. There are three possible cases here. We will show all 3 methods with examples.
Both numbers are less than the reference (under the reference). Let's say we want to multiply 48 by 47.
These numbers are close enough to 50 that it is convenient to use 50 as a reference number.
To multiply 48 by 47 using the reference number 50, you need:
1. From 47 subtract as much as 48 is missing to 50, that is, 2. It turns out 45 (or
subtract 3 from 48 - it's always the same)
2. Then multiply 45 by 50 = 2250
3. Then add 2*3 to this result - 2256
50 (reference number)
3(50-47) 2(50-48)
(47-2)*50+2*3=2250+6=2256
If the numbers are less than the reference number, then from the first factor we subtract the difference between the reference number and the second factor. If the numbers are greater than the reference number, then we add the difference between the reference number and the second factor to the first factor.
50(reference number)
(51+13)*50+(13*1)=3200+13=3213
One number is under the reference, and the other is above. The third use case for the reference number is when one number is greater than the reference number and the other is smaller. Such examples are not more difficult to solve than the previous ones. We increase the smaller factor by the difference between the second factor and the reference number, multiply the result by the reference number and subtract the product of the differences between the reference number and the factors. Or we reduce the larger factor by the difference between the second factor and the reference number, multiply the result by the reference number and subtract the product of the differences between the reference number and the factors.
50(reference number)
5(50-45) 2(52-50)
(52-5)*50-5*2=47*50-10=2340 or (45+2)*50-5*2=47*50-10=2340
When multiplying two-digit numbers from different tens, it is more convenient as a reference number
take a round number that is greater than the larger multiplier.
90(reference number)
63 (90-27) 1 (90-89)
(89-63)*90+63*1=2340+63=2403
Thus, by using one reference number, a large combination of two-digit numbers can be multiplied. The methods described above can be divided into universal (suitable for any numbers) and private (convenient for specific cases).
In extreme cases, you can use the "peasant" account. To multiply one number by another, let's say 21*75, we need to write the numbers in two columns. The first number of the left column is 21, the first number of the right column is 75. Then divide the numbers in the left column by 2 and discard the remainder until we get one, and multiply the numbers in the right column by 2. Cross out all lines that have even numbers in the left column, and the remaining numbers in the right column are added up, we get the exact result.
Conclusion
Like all calculation methods, these fast counting methods have their advantages and disadvantages:
PROS:
1.Using various methods fast calculations, even the most poorly educated person can count.
2. Quick counting methods can help get rid of a complex action by replacing it with several simpler ones.
3. Quick counting methods are useful in situations where multiplication by a column cannot be used.
4. Fast counting methods allow you to reduce the calculation time.
5.Mental count develops mental activity which helps to quickly navigate in difficult life situations.
6. The technique of mental counting makes the process of calculations more fun and interesting.
MINUSES:
1. Often, solving an example using quick counting methods turns out to be longer than just multiplying in a column, since you have to perform large quantity actions, each of which is simpler than the original.
2. There are situations when a person, out of excitement or something else, forgets the methods of quick counting or even gets confused in them; in such cases the answer is wrong and the methods are effectively useless.
3. Not for all cases, methods of fast counting have been developed.
4. When calculating using the quick counting technique, you need to keep a lot of answers in your head, which can get confused and come to an erroneous result.
Undoubtedly, practice plays a crucial role in the development of any abilities. But the skill of mental counting is not based on experience alone. This is proved by people who are able to count in their minds. complex examples. For example, such people can multiply and divide three-digit numbers, perform arithmetic operations that not every person can count in a column. What do you need to know and be able to ordinary person to master such a phenomenal ability? Today, there are various techniques that help you learn how to quickly count in your mind.
Having studied many approaches to teaching the skill of counting orally, we can distinguish 3 main components of this skill:
1. Ability. The ability to concentrate attention and the ability to keep several things in short-term memory at the same time. aptitude for mathematics and logical thinking.
2. Algorithms. Knowledge of special algorithms and the ability to quickly select the desired, most effective algorithm in each specific situation.
3. Training and experience, whose value for any skill has not been canceled. Constant training and the gradual complication of tasks and exercises will allow you to improve the speed and quality of mental arithmetic. It should be noted that the third factor is of key importance. Not possessing necessary experience, you will not be able to surprise others with a quick score, even if you know the most convenient algorithm. However, do not underestimate the importance of the first two components, because having the ability and a set of necessary algorithms in your arsenal, you can surprise even the most experienced "bookkeeper", provided that you have been training for the same time.
In mental counting, as elsewhere, there are tricks, and in order to learn how to count faster, you need to know these tricks and be able to put them into practice.
Today we will do this!
1. How to quickly add and subtract numbers
Consider three random examples:
- 25 – 7 =
- 34 – 8 =
- 77 – 9 =
Type 25 - 7 = (20 + 5) - (5- 2) = 20 - 2 = (10 + 10) - 2 = 10 + 8 = 18
Agree that such operations are difficult to turn in your head.
But there is an easier way:
25 - 7 \u003d 25 - 10 + 3, since -7 \u003d -10 + 3
It's much easier to subtract 10 from 10 and add 3 than it is to do complex calculations.
Let's go back to our examples:
- 25 – 7 =
- 34 – 8 =
- 77 – 9 =
Optimizing subtracted numbers:
- Subtract 7 = subtract 10 add 3
- Subtract 8 = subtract 10 add 2
- Subtract 9 = subtract 10 add 1
In total we get:
- 25 – 10 + 3 =
- 34 – 10 + 2 =
- 77 – 10 + 1 =
Now it's much more interesting and easier!
Now count the examples below in this way:
- 91 – 7 =
- 23 – 6 =
- 24 – 5 =
- 46 – 8 =
- 13 – 7 =
- 64 – 6 =
- 72 – 19 =
- 83 – 56 =
- 47 – 29 =
2. How to quickly multiply by 4, 8 and 16
In the case of multiplication, we also break numbers into simpler ones, for example:
If you remember the multiplication table, then everything is simple. And if not?
Then you need to simplify the operation:
We put the largest number first, and decompose the second into simpler ones:
8 * 4 = 8 * 2 * 2 = ?
It is much easier to double numbers than to quadruple or eight them.
We get:
8 * 4 = 8 * 2 * 2 = 16 * 2 = 32
Examples of decomposing numbers into simpler ones:
- 4 = 2*2
- 8 = 2*2 *2
- 16 = 22 * 2 2
Practice this with the following examples:
- 3 * 8 =
- 6 * 4 =
- 5 * 16 =
- 7 * 8 =
- 9 * 4 =
- 8 * 16 =
3. Divide a number by 5
Let's take the following examples:
- 780 / 5 = ?
- 565 / 5 = ?
- 235 / 5 = ?
Division and multiplication with the number 5 is always very simple and pleasant, because five is half of ten.
And how to solve them quickly?
- 780 / 10 * 2 = 78 * 2 = 156
- 565 /10 * 2 = 56,5 * 2 = 113
- 235 / 10 * 2 = 23,5 *2 = 47
In order to work out this method, solve the following examples:
- 300 / 5 =
- 120 / 5 =
- 495 / 5 =
- 145 / 5 =
- 990 / 5 =
- 555 / 5 =
- 350 / 5 =
- 760 / 5 =
- 865 / 5 =
- 1270 / 5 =
- 2425 / 5 =
- 9425 / 5 =
4. Multiplication by single digits
Multiplication is a little more difficult, but not much, how would you solve the following examples?
- 56 * 3 = ?
- 122 * 7 = ?
- 523 * 6 = ?
Without special counters, solving them is not very pleasant, but thanks to the Divide and Conquer method, we can count them much faster:
- 56 * 3 = (50 + 6)3 = 50 3 + 6*3 = ?
- 122 * 7 = (100 + 20 + 2)7 = 100 7 + 207 + 2 7 = ?
- 523 * 6 = (500 + 20 + 3)6 = 500 6 + 206 + 3 6 =?
We just have to multiply single-digit numbers, some of them with zeros, and add the results.
To work through this technique, solve the following examples:
- 123 * 4 =
- 236 * 3 =
- 154 * 4 =
- 490 * 2 =
- 145 * 5 =
- 990 * 3 =
- 555 * 5 =
- 433 * 7 =
- 132 * 9 =
- 766 * 2 =
- 865 * 5 =
- 1270 * 4 =
- 2425 * 3 =
Divisibility of a number by 2, 3, 4, 5, 6 and 9
Check the numbers: 523, 221, 232
A number is divisible by 3 if the sum of its digits is divisible by 3.
For example, let's take the number 732 and represent it as 7 + 3 + 2 = 12. 12 is divisible by 3, which means that the number 372 is divisible by 3.
Check which of the following numbers are divisible by 3:
12, 24, 71, 63, 234, 124, 123, 444, 2422, 4243, 53253, 4234, 657, 9754
A number is divisible by 4 if the number consisting of its last two digits is divisible by 4.
For example, 1729. The last two digits form 20, which is divisible by 4.
Check which of the following numbers are divisible by 4:
20, 24, 16, 34, 54, 45, 64, 124, 2024, 3056, 5432, 6872, 9865, 1242, 2354
A number is divisible by 5 if its last digit is 0 or 5.
Check which of the following numbers are divisible by 5 (the easiest exercise):
3, 5, 10, 15, 21, 23, 56, 25, 40, 655, 720, 4032, 14340, 42343, 2340, 243240
A number is divisible by 6 if it is divisible by both 2 and 3.
Check which of the following numbers are divisible by 6:
22, 36, 72, 12, 34, 24, 16, 26, 122, 76, 86, 56, 46, 126, 124
A number is divisible by 9 if the sum of its digits is divisible by 9.
For example, let's take the number 6732 and represent it as 6 + 7 + 3 + 2 = 18. 18 is divisible by 9, which means that the number 6732 is divisible by 9.
Check which of the following numbers are divisible by 9:
9, 16, 18, 21, 26, 29, 81, 63, 45, 27, 127, 99, 399, 699, 299, 49
Game "Fast Addition"
- Speeds up mental counting
- Trains attention
- Develops creative thinking
An excellent simulator for the development of fast counting. A 4x4 table is given on the screen, and numbers are shown above it. The largest number you need to collect in the table. To do this, click on two numbers with the mouse, the sum of which is equal to this number. For example, 15+10 = 25.
Game "Quick Score"
The game "quick count" will help you improve your thinking. The essence of the game is that in the picture presented to you, you will need to choose the answer "yes" or "no" to the question "are there 5 identical fruits?". Follow your goal, and this game will help you with this.
Game "Guess the operation"
The game "Guess the operation" develops thinking and memory. Main essence game, you need to choose a mathematical sign for the equality to be true. Examples are given on the screen, look carefully and put the desired “+” or “-” sign so that the equality is true. The sign "+" and "-" are located at the bottom of the picture, select the desired sign and click on the desired button. If you answer correctly, you score points and continue playing.
Game "Simplify"
The game "Simplify" develops thinking and memory. The main essence of the game is to quickly perform a mathematical operation. A student is drawn on the screen at the blackboard, and a mathematical action is given, the student needs to calculate this example and write the answer. Below are three answers, count and click the number you need with the mouse. If you answer correctly, you score points and continue playing.
Task for today
Solve all the examples and practice for at least 10 minutes in the Quick Addition game.
It is very important to work out all the tasks of this lesson. The better you perform the tasks, the more you will benefit. If you feel that there are not enough tasks for you, you can make up examples for yourself and solve them and train in mathematical educational games.
The lesson is taken from the course "Oral counting in 30 days"
Learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even take roots. I will teach you how to use easy tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.
Other developmental courses
Money and the mindset of a millionaire
Why are there money problems? In this course, we will answer this question in detail, look deep into the problem, consider our relationship with money from a psychological, economic and emotional point of view. From the course you will learn what you need to do to solve all your problems. financial difficulties, start accumulating money and invest it in the future.
Knowing the psychology of money and how to work with them makes a person a millionaire. 80% of people with an increase in income take out more loans, becoming even poorer. Self-made millionaires, on the other hand, will make millions again in 3-5 years if they start from scratch. This course teaches the proper distribution of income and cost reduction, motivates you to learn and achieve goals, teaches you to invest money and recognize a scam.
Speed reading in 30 days
Increase your reading speed by 2-3 times in 30 days. From 150-200 to 300-600 wpm or from 400 to 800-1200 wpm. The course uses traditional exercises for the development of speed reading, techniques that speed up the work of the brain, a method for progressively increasing the speed of reading, understands the psychology of speed reading and the questions of course participants. Suitable for children and adults reading up to 5,000 words per minute.
Development of memory and attention in a child 5-10 years old
The purpose of the course is to develop the child's memory and attention so that it is easier for him to study at school, so that he can remember better.
After completing the course, the child will be able to:
- 2-5 times better to remember texts, faces, numbers, words
- Learn to remember for longer
- The speed of remembering the necessary information will increase
Super memory in 30 days
Memorize the information you need quickly and permanently. Wondering how to open the door or wash your hair? I am sure not, because it is part of our life. Easy and simple memory training exercises can be made part of life and done little by little during the day. If you eat the daily norm of food at a time, or you can eat in portions throughout the day.
Quick Counting Techniques: Magic Available to All
In order to understand the role that numbers play in our lives, set up a simple experiment. Try to do without them for a while. No numbers, no calculations, no measurements... You will find yourself in a strange world where you will feel absolutely helpless, bound hand and foot. How to get to a meeting on time? Distinguish one bus from another? Make a phone call? Buy bread, sausage, tea? Cook soup or potatoes? Without numbers, and therefore, without counting, life is impossible. But how hard this science is sometimes given! Try to quickly multiply 65 by 23? Does not work? The hand itself reaches for a mobile phone with a calculator. Meanwhile, semi-literate Russian peasants 200 years ago calmly did this, using only the first column of the multiplication table - multiplication by two. Don't believe? But in vain. This is reality.
stone age computer
Even without knowing the numbers, people have already tried to count. If our ancestors, who lived in caves and wore skins, needed to exchange something with a neighboring tribe, they acted simply: they cleared the site and laid out, for example, an arrowhead. Near lay a fish or a handful of nuts. And so on until one of the exchanged goods ran out, or the head of the "trading mission" decided that enough was enough. Primitive, but in its own way very convenient: you won’t get confused, and you won’t be deceived.
With the development of cattle breeding, the tasks became more complicated. A large herd had to be counted somehow in order to know whether all the goats or cows were in place. The "calculating machine" of illiterate but smart shepherds was a dugout pumpkin with pebbles. As soon as the animal left the pen, the shepherd put a pebble in the gourd. In the evening, the herd returned, and the shepherd took out a stone with each animal that entered the pen. If the gourd was empty, he knew the flock was all right. If there were pebbles, he went to look for the loss.
When the numbers appeared, things got more fun. Although for a long time our ancestors used only three numerals: "one", "pair" and "many".
Can you count faster than a computer?
Outrun a device that performs hundreds of millions of operations per second? Impossible... But the one who says this is cruelly disingenuous, or simply deliberately overlooks something. A computer is just a set of chips in plastic; it doesn't count by itself.
Let's put the question in another way: can a person, calculating in his mind, overtake someone who performs calculations on a computer? And here the answer is yes. Indeed, in order to receive an answer from the "black suitcase", the data must first be entered into it. This will be done by a person with the help of fingers or voice. And all these actions have time limits. Insurmountable restrictions. Nature itself supplied them to the human body. Everything except one organ. Brain!
The calculator can only perform two operations: addition and subtraction. Multiplication for him is multiple addition and division is multiple subtraction.
Our brain behaves differently.
The class where the future king of mathematics, Carl Gauss, studied, somehow received the task: add up all the numbers from 1 to 100. Carl wrote the absolutely correct answer on his board as soon as the teacher finished explaining the task. He did not diligently add numbers in order, as any self-respecting computer would do. He applied the formula he discovered himself: 101 x 50 = 5050. And this is far from the only trick that speeds up mental calculations.
The simplest tricks for quick counting
They are taught at school. The simplest: if you need to add 9 to any number, add 10 and subtract 1, if 8 (+ 10 - 2), 7 (+ 10 - 3), etc.
54 + 9 = 54 + 10 - 1 = 63. Fast and convenient.
Two-digit numbers add up just as easily. If the last digit in the second term is greater than five, the number is rounded up to the next ten, and then the "excess" is subtracted. 22 + 47 = 22 + 50 – 3 = 69
With three-digit numbers, there are no difficulties in the same way. We add them, as we read, from left to right: 321 + 543 \u003d 300 + 500 + 20 + 40 + 1 + 3 \u003d 864. Much easier than in a column. And much faster.
What about subtraction? The principle is the same: we round the subtracted to the nearest integer and add the missing one: 57 - 8 = 57 - 10 + 2 = 49; 43 - 27 \u003d 43 - 30 + 3 \u003d 16. Faster than on a calculator - and no complaints from the teacher even during the test!
Do I need to learn the multiplication table?
Children usually hate this. And they do it right. No need to teach her! But do not rush to be outraged. No one claims that the table does not need to be known.
Its invention is attributed to Pythagoras, but, most likely, the great mathematician only gave a complete, concise form to what was already known. At the excavations of ancient Mesopotamia, archaeologists found clay tablets with the sacramental: "2 x 2". People have been using this highly convenient system of calculations for a long time and have discovered many ways that help to comprehend the internal logic and beauty of the table, to understand - and not stupidly, mechanically memorize.
IN ancient China they began to learn the table by multiplying by 9. It’s easier this way, and not least because you can multiply by 9 “on your fingers”.
Place both hands on the table, palms down. The first finger from the left is 1, the second is 2, and so on. Let's say you need to solve a 6 x 9 problem. Raise your sixth finger. Fingers on the left will show tens, on the right - units. Answer 54.
Example: 8 x 7. Left hand- the first multiplier, the right - the second. There are five fingers on the hand, and we need 8 and 7. We bend three fingers on the left hand (5 + 3 = 8), on the right 2 (5 + 2 = 7). We have five bent fingers, which means five dozen. Now multiply the rest: 2 x 3 = 6. These are units. Total 56.
This is just one of the simplest methods of "finger" multiplication. There are many of them. "On the fingers" you can operate with numbers up to 10,000!
The "finger" system has a bonus: the child perceives it as a fun game. He engages willingly, experiences a lot of positive emotions, and as a result, very soon begins to perform all operations in his mind, without the help of his fingers.
You can also divide with your fingers, but it's a little more complicated. Programmers still use their hands to convert numbers from decimal to binary - it's more convenient and much faster than on a computer. But within school curriculum you can learn to divide quickly even without fingers, in your mind.
Let's say you need to solve example 91: 13. Column? No need to mess up paper. The dividend ends with one. And the divisor is three. What is the very first thing in the multiplication table where the triple is involved, and ends with one? 3 x 7 = 21. Seven! That's it, we got her. Need 84: 14. Remember the table: 6 x 4 = 24. The answer is 6. Simple? Still would!
number magic
Most of the quick counting tricks are similar to magic tricks. Take at least the most famous example of multiplying by 11. To, for example, 32 x 11, you need to write 3 and 2 along the edges, and put their sum in the middle: 352.
To multiply a two-digit number by 101, simply write the number twice. 34 x 101 = 3434.
To multiply a number by 4, multiply it by 2 twice. To divide, divide by 2 twice.
Many witty and, most importantly, quick tricks help to raise a number to a power, extract Square root. The famous "30 tricks of Perelman" for mathematical thinking people will be cooler than the Copperfield show, because they also UNDERSTAND what is happening and how it happens. Well, the rest can just enjoy the beautiful focus. For example, you need to multiply 45 by 37. Let's write the numbers on a sheet and separate them with a vertical line. We divide the left number by 2, discarding the remainder, until we get one. Right - multiply until the number of lines in the column is equal. Then we cross out from the RIGHT column all those numbers opposite which an even result is obtained in the LEFT column. We add the remaining numbers from the right column. It turns out 1665. Multiply the numbers in the usual way. The answer will fit.
"Charging" for the mind
Quick counting techniques can make life easier for a child at school, for mom in a store or kitchen, and for dad at work or in the office. But we prefer the calculator. Why? We don't like to stress. It's hard for us to keep numbers, even two-digit ones, in our heads. For some reason they don't hold up.
Try to go to the middle of the room and sit on the twine. For some reason "does not sit down", right? And the gymnast does it quite calmly, without straining. Need to train!
The easiest way to train and, at the same time, warm up the brain: verbal counting aloud (mandatory!) through the number to one hundred and back. In the morning, standing in the shower, or preparing breakfast, count: 2.. 4.. 6.. 100... 98.. 96. You can count in three, in eight - the main thing is to do it out loud. After just a couple of weeks of regular practice, you will be surprised how EASIER it becomes to deal with numbers.
Why count in the mind, if you can solve any arithmetic problem on a calculator. Modern medicine and psychology prove that mental counting is an exercise for gray cells. Performing such gymnastics is necessary for the development of memory and mathematical abilities.
There are many tricks to simplify mental calculations. Everyone who has seen the famous painting by Bogdanov-Belsky "Mental Account" is always surprised - how do peasant children solve such a difficult task as dividing the sum of five numbers that must first be squared?
It turns out that these children are students of the famous teacher-mathematician Sergei Alexandrovich Rachitsky (he is also depicted in the picture). These are not geeks - students primary school rustic schools XIX century. But they all already know how to simplify arithmetic calculations and have learned the multiplication table! Therefore, it is quite possible for these kids to solve such a problem!
Secrets of mental counting
There are methods of oral counting - simple algorithms, which it is desirable to bring to automatism. After mastering simple techniques, you can move on to mastering more complex ones.
We add the numbers 7,8,9
To simplify the calculations, the numbers 7,8,9 must first be rounded up to 10, and then subtract the increase. For example, to add 9 to a two-digit number, you must first add 10 and then subtract 1, and so on.
Examples :
Add two digit numbers quickly
If the last digit of a two-digit number is greater than five, round it up. We perform the addition, subtract the “additive” from the resulting amount.
Examples :
54+39=54+40-1=93
26+38=26+40-2=64
If the last digit of a two-digit number is less than five, then add up by digits: first add tens, then ones.
Example :
57+32=57+30+2=89
If the terms are reversed, then you can first round the number 57 to 60, and then subtract 3 from the total:
32+57=32+60-3=89
Adding three-digit numbers in your mind
Quick counting and addition three-digit numbers- is it possible? Yes. To do this, you need to parse three-digit numbers into hundreds, tens, units and add them one by one.
Example :
249+533=(200+500)+(40+30)+(9+3)=782
Subtraction features: reduction to round numbers
Subtracted are rounded up to 10, up to 100. If you need to subtract a two-digit number, you need to round it up to 100, subtract, and then add an amendment to the remainder. This is true if the correction is small.
Examples :
576-88=576-100+12=488
Mind subtracting three-digit numbers
If at one time the composition of numbers from 1 to 10 was well mastered, then subtraction can be done in parts and in the indicated order: hundreds, tens, units.
Example :
843-596=843-500-90-6=343-90-6=253-6=247
Multiply and Divide
Instantly multiply and divide in your mind? It is possible, but one cannot do without knowledge of the multiplication table. is the golden key to quick mental counting! It applies to both multiplication and division. Let us remember that in primary school village school in the pre-revolutionary Smolensk province (picture "Mental count"), the children knew the continuation of the multiplication table - from 11 to 19!
Although in my opinion it is enough to know the table from 1 to 10 in order to be able to multiply larger numbers. For example:
15*16=15*10+(10*6+5*6)=150+60+30=240
Multiply and divide by 4, 6, 8, 9
Having mastered the multiplication table for 2 and 3 to automatism, making the rest of the calculations will be as easy as shelling pears.
For multiplication and division of two- and three-digit numbers, we use simple tricks:
multiplying by 4 is twice multiplying by 2;
to multiply by 6 means to multiply by 2 and then by 3;
multiplying by 8 is three times multiplying by 2;
multiplying by 9 is twice multiplying by 3.
For example :
37*4=(37*2)*2=74*2=148;
412*6=(412*2) 3=824 3=2472
Similarly:
divided by 4 is twice divided by 2;
divide by 6 is first divide by 2 and then by 3;
divided by 8 is three times divided by 2;
Divide by 9 is twice divided by 3.
For example :
412:4=(412:2):2=206:2=103
312:6=(312:2):3=156:3=52
How to multiply and divide by 5
The number 5 is half of 10 (10:2). Therefore, we first multiply by 10, then we divide the result in half.
Example :
326*5=(326*10):2=3260:2=1630
The rule of division by 5 is even simpler. First, we multiply by 2, and then we divide the result by 10.
326:5=(326 2):10=652:10=65.2.
Multiply by 9
To multiply a number by 9, it is not necessary to multiply it twice by 3. It is enough to multiply it by 10 and subtract the multiplied number from the resulting number. Compare which is faster:
37*9=(37*3)*3=111*3=333
37*9=37*10 - 37=370-37=333
Also, particular patterns have long been noticed that greatly simplify the multiplication of two-digit numbers by 11 or by 101. So, when multiplied by 11, a two-digit number seems to move apart. The numbers that make it up remain at the edges, and their sum is in the center. For example: 24*11=264. When multiplying by 101, it is enough to attribute the same to a two-digit number. 24*101= 2424. The simplicity and logic of such examples is admirable. Such tasks are very rare - these are entertaining examples, the so-called little tricks.
Counting on fingers
Today you can still meet many defenders of "finger gymnastics" and the method of mental counting on the fingers. We are convinced that learning to add and subtract by bending and unbending fingers is very visual and convenient. The range of such calculations is very limited. As soon as the calculations go beyond one operation, difficulties arise: it is necessary to master the next technique. Yes, and bending your fingers in the era of iPhones is somehow undignified.
For example, in defense of the "finger" technique, the technique of multiplying by 9 is given. The trick of the technique is as follows:
- To multiply any number within the first ten by 9, you need to turn your palms towards you.
- Counting from left to right, bend the finger corresponding to the number being multiplied. For example, to multiply 5 by 9, you need to bend the little finger on your left hand.
- The remaining number of fingers on the left will correspond to tens, on the right - units. In our example - 4 fingers on the left and 5 on the right. Answer: 45.
Yes, indeed, the solution is quick and visual! But this is from the field of tricks. The rule only works when multiplying by 9. Isn't it easier to learn the multiplication table to multiply 5 by 9? This trick will be forgotten, and a well-learned multiplication table will remain forever.
There are also many more similar tricks using fingers for some single mathematical operations, but this is relevant while you use it and is immediately forgotten when you stop using it. Therefore, it is better to learn standard algorithms that will remain for life.
Oral account on the machine
First, you need to know the composition of the number and the multiplication table well.
Secondly, you need to remember the methods of simplifying calculations. As it turned out, there are not so many such mathematical algorithms.
Thirdly, in order for the technique to turn into a convenient skill, it is necessary to constantly conduct brief “brainstorming sessions” - to practice oral calculations using one or another algorithm.
Workouts should be short: mentally solve 3-4 examples using the same technique, then move on to the next one. We must strive to use every free minute - and useful, and not boring. Thanks to simple training, all calculations over time will be done at lightning speed and without errors. This is very useful in life and will help out in difficult situations.
