How to solve in the most rational way. Rational numbers, definition, examples

Class characteristic

5 “A” class is heterogeneous in composition, some of the children are quite strong in knowledge, but weak ones also stand out. In general, the class is energetic, students with interest and willingness pick up the initiatives of the teacher.

Topic: Rational methods of calculation (the lesson is a final lesson, conducted after the topic: “simplification of expressions” in the II quarter,? 3)

Lesson type: summarizing the material

a) educational

• repeat the properties of addition, subtraction, multiplication of natural numbers
• consolidate the theory of knowledge in practice
• show the advantage of rational ways to complete tasks, i.e. show that the creation of this project is necessary and significant for the children themselves
• improve the skill of applying methods in practice;

b) developing

• develop the ability to draw conclusions, systematize the material, compare methods with a specific building, clearly formulate thoughts
• develop the ability to reflect on their cognitive activity
• to form a creative consciousness, a true passion for business;

c) educational

• to cultivate independence, collectivism, the ability to listen to each other, respect the opinion of another, but also be able to prove one's own.

Equipment: magnetic board and magnets, felt-tip pens, tree leaves (album sheets), pictures of a cat Matroskin and Sharik, a slide screen.

Stage of the lesson, time	Tasks	Teacher activity	Student activities	Note
I Org. Moment	Relationship friendliness setting	- Hello guys! Check if everything is ready for the lesson. Smile at each other, now smile at me! I see you are in a good mood, you can start the lesson!	- smile general revival	- on the screen 1 slide with the text “Smile”
II Knowledge update	Intrigue children Unobtrusively lead to the topic of the lesson Summarize the stage	- Guys, the cat Matroskin and Sharik will work with us today. Children, you need to solve 2 examples, at the request of Sharik we solve the whole lesson! (I walk through the rows, I look at the solution) What are you? (surprised!) Well done! It's only been one minute! Let's see how the cat Matroskin and Sharik solved these examples. So the cat Matroskin decided, and Sharik finds it difficult. How did you decide? Who else? Cat Matroskin is interested in what this method is good for, why exactly was it used? This method is the property! And how is this property read! Specify what? Let's say again what this property allows us to do.	- Hurrah! (exclamations from the seat) (someone multiplies in a column!) I've already decided! Guys Answers Allows you to decide: Faster, More convenient, Easier, easier Saves time distributed law Additions, subtractions Simplify Expressions Decide faster Easier, easier	- drawing of the cat Matros-kin and Sharik on the board On the board 69*27+31*27=22*87-102*87= (in a column) 3) 27*(69+31) =2700 2 slide on screen
III Introduction of a new concept	Introduce a new concept	- All these words can be replaced by the word: rational, where in everyday life did you hear this word?	- on TV, in factories rationalizers, rational nutrition	3 slide
IV Definition of the topic	Define a theme	- Guys! The ball is trying to solve another example in the same way! I offer to help him. How to call this property? Is this the rational way? We only know these two ways? Okay, let's formulate the topic, and then list what other properties we know. What is the topic of the lesson? your assumptions. What word will the theme be associated with? Let's generalize! What happened?	- (students decide) (there is a drawing of the solution) Don't solve it the same way Associative property of multiplication Makes decisions easier, faster, easier. We don't know how yet! To the word "method" you can add "what" Calculation methods! Rational Rational methods of computing.	On the desk Lesson topic
V Goal-setting	Setting lesson goals	- Guys! If we replace the word “method! Will it be possible to apply the same concepts to “methods” to “methods”: “easier, faster, simpler”? What else can be said about the methods? Let's show it on the slide What did you notice special in the scheme? So what are the goals of each lesson? Let's summarize: Recall what methods we know and arrange these methods Recall expression simplification techniques Fix their application in practice Learn to compare a method with a specific example These are the goals or ideas of our lesson.	- Yes! And replace “what” with the word “what”! Where are they used? The word "what" with "?" Remember what methods we know, what properties, rules There may be new ways to learn. - (together with students)	6 slide
VI System-theme-tyzation of knowledge a) setting goals for stage 0. 5 min b) indie work 1.5 min c) work in pairs d) group work	Create a project Self-Execution Speak your notes Search for a general solution, conclusions	- Guys! Today we must create a project in which the methods known to you (at least 8) and everything we know about the methods will be recorded. The project will be in the form of a tree to which we will attach leaves. Sharik had an idea: to think on his own for 2 minutes, to remember ways to simplify expressions. Support the idea? Working in pairs And now we are seated in groups (4 people), Sharik with the cat Matroskin will work in pairs. Discuss your thoughts and decisions. You have leaflets on your desks, write down one way on each of them, then we will attach them to the tree Of course, with examples it will be even clearer. Choose who will answer	- what will this project look like? (students work independently, make notes) - (voice) (each student speaks their mind) (the representative of the group writes down the methods, the rest comment) It is possible with examples?	Groups are territorially isolated
VII Physical culture tour	Rest of students	“A flower was sleeping and suddenly woke up Didn't want to sleep anymore Moved, stretched Soared up and flew"	Conducted by one of the children	8 slide: "funny pictures"
VIII Project protection	Summarize the work of all groups	- representatives of each group are invited. . . (teacher directs work) This is how we got a tree, and now let's see the diagram that the cat Matroskin made after listening to your speeches	Student phrases: I agree with Pete... Our group wants to add... Can also be literal	On the desk: Tree trunk, children attach leaves on a magnetic board with a magnet (the same answers for one magnet)

Annex 1 presents the scheme of the project.

IX Testing	Check in practice the application of methods	- Guys! We remembered the theory, and now we will check how you will apply your knowledge in practice Now exchange notebooks with a neighbor and check his work. Grading norms: No errors: “5” 2 errors: "4" 3 errors: "3" and if more than 3, then you need to practice What could be the reason?	(students decide)	On the board slide 10
			Test
			B-I	B-2
			1) Run convenient way
			a) (30-4) *5= b) 85*137-75*137= G) 25*296*4= e) 633-(163+387) =	a) 7*(60-3) = b) 78*214-78*204= G) 4*268*25= e) (964+27) -464=
			2) Solve the equation
			x+3x+x=30	x+5x+x=98
			(appreciate each other) I did not make it in time Solved without using methods, performing columns	On the screen, slide 11 with the solution
X Summarizing 2min (self) 2min (voiceover)	Reflect on your work	- what did you remember? What did you remember? What new did you learn? What did you fix? What conclusion did you draw for yourself? Well done boys! And the cat Matroskin remembered many ways, but Sharik's thoughts got mixed up, let's repeat all the methods again	- fixed the use of properties when solving Learned to map a property to a specific example I remembered that the property is written using variables Learn what "rationality" is I realized that each example has its own approach I realized that the laws work in both lines I understood that rac. the most convenient ways These methods also allow you to save time, simplify the decision and your life. I realized that the methods allow you to solve orally, without columns
XI	Give instructions to d / z	- Guys! 1. talk at home with relatives, friends, maybe they know some other ways 2. make a project, with your own examples, it can be in the form of clouds, flowers, etc., you can use a computer 3. show younger sisters, brothers for their interest in mathematics 4. make a project report on the memo		- memo located on the stand
XII Conclusion		- cat Matroskin and Sharik say “thank you” to you and say goodbye to you guys! I also tell you “well done - for the lesson” and goodbye		slide12 Text "Well done"

The current level of development of computing automation tools has created the illusion for many that developing computing skills is not at all necessary. This affected the preparedness of the students. In the absence of a calculator, even simple computational tasks become a problem for many.

At the same time, exam tasks and materials for the exam contain many tasks, the solution of which requires the ability of the test subjects to rationally organize calculations.

In this article, we will consider some methods for optimizing calculations and their application for competitive tasks.

Most often, methods for optimizing calculations are associated with the application of the basic laws for performing arithmetic operations.

For example:

125 24 = 125 8 3 = 1000 3 = 3000; or

98 16(100 - 2) 16 = 100 16 - 2 16 = 1600 - 32 = 1568 etc.

Another direction - use of abbreviated multiplication formulas.

96 104 \u003d (100 - 4) (100 + 4) \u003d 100 2 - 4 2 \u003d 10000 - 16 \u003d 9984; or

115 2 = (100 + 15) 2 = 10000 + 2 15 100 + 225 = 10525.

The following example is interesting for calculations.

Calculate:

(197 · 203 + 298 · 302 + 13) / (1999 · 2001 + 2993 · 3007 + 50) =
= ((200 – 3) · (200 + 3) + (300 – 2) · (300 + 2) + 13) / ((2000 – 1) · (2000 + 1) + (3000 – 7) · (3000 + 7) + 50) =
= (200 2 – 3 2 + 300 2 – 2 2 + 13) / (2000 2 – 1 2 + 3000 2 – 7 2 – 50) =
= 130000 / 13000000 = 0,01

These are almost standard ways to optimize calculations. Sometimes more exotic ones are offered. As an example, consider the method of multiplying two-digit numbers, the sum of units of which is 10.

54 26 \u003d 50 30 + 4 (26 - 50) \u003d 1500 - 96 \u003d 1404 or

43 87 = 40 90 + 3 (87 - 40) = 3600 + 141 = 3741.

The multiplication scheme can be understood from the figure.

Where does such a multiplication scheme come from?

Our numbers by condition have the form: M = 10m + n, K = 10k + (10 – n). Let's create a work:

M K = (10m + n)(10k + (10 – n)) =
= 100mk + 100m - 10mn + 10nk + 10n - n 2 =
= m(k + 1) 100 + n(10k + 10 – n) =
= (10m) (10 (k + 1)) + n (K – 10m) and the method is justified.

There are many ingenious ways to turn rather complex calculations into mental tasks. But you can’t think that everyone needs to remember these and a bunch of other ingenious ways to simplify calculations. It is only important to learn some of the basic ones. Analysis of others makes sense only for developing skills in applying basic methods. It is their creative application that makes it possible to quickly and correctly solve computational problems.

Sometimes, when solving examples for calculation, it is convenient to switch from the transformation of an expression with numbers to the transformation of polynomials. Consider the following example.

Calculate in the most rational way:

3 1 / 117 4 1 / 110 -1 110 / 117 5 118 / 119 - 5 / 119

Decision.

Let a = 1/117 and b = 1/119. Then 3 1 / 117 = 3 + a, 4 1 / 119 = 4 + b, 1 116 / 117 = 2 - a, 5 118 / 119 = 6 - b.

Thus, the given expression can be written as (3 + a) (4 + b) - (2 - a) (6 - b) - 5b.

After performing simple transformations of the polynomial, we get 10a or 10 / 117 .

Here we have obtained that the value of our expression does not depend on b. And this means that we have calculated not only the value of this expression, but also any other one obtained from (3 + a) (4 + b) - (2 - a) (6 - b) - 5b by substituting the values ​​of a and b. If, for example, a = 5/329, then in the answer we get 50 / 329 , whatever b.

Let's consider another example, which is almost impossible to solve with a calculator, and the answer is quite simple if you know the approach to solving examples of this type.

Calculate

1 / 6 7 1024 – (7 512 + 1) (7 256 + 1) (7 128 + 1) (7 64 + 1) (7 32 + 1) (7 16 + 1) ( 7 8 + 1) (7 4 + 1) (7 2 + 1) (7 + 1)

Decision.

Let's transform the condition

1/6 7 1024 - 1/6 (7 512 + 1) (7 256 + 1) (7 128 + 1) (7 64 +1) (7 32 + 1) (7 16 + 1) (7 8 + 1) (7 4 + 1) (7 2 + 1) (7 + 1) (7 - 1) =

1/6 7 1024 - 1/6 (7 512 + 1) (7 256 + 1) (7 128 + 1) (7 64 + 1) (7 32 + 1) (7 16 + 1) ) (7 8 + 1) (7 4 + 1) (7 2 + 1) (7 2 – 1) =

1/6 7 1024 - 1/6 (7 512 + 1) (7 256 + 1) (7 128 + 1) (7 64 + 1) (7 32 + 1) (7 16 + 1) ) (7 8 + 1) (7 4 + 1) (7 4 – 1) =

1/6 7 1024 - 1/6 (7 512 + 1) (7 256 + 1) (7 128 + 1) (7 64 + 1) (7 32 + 1) (7 16 + 1) ) (7 8 + 1) (7 8 – 1) =

1/6 7 1024 - 1/6 (7 512 + 1) (7 256 +1) (7 128 + 1) (7 64 + 1) (7 32 + 1) (7 16 + 1) ) · (7 16 – 1) = … =

1/6 7 1024 - 1/6 (7 512 + 1) (7 512 - 1) = 1/6 7 1024 - 1/6 (7 1024 - 1) = 1/6

Consider one of the examples, which has already become textbook in the examination materials for the course of the basic school.

Calculate sum:

1/2 + 1 / (2 3) + 1 / (3 4) + 1 / (4 5) + ... + 1 / (120 121) =

= (1 – 1/2) + (1/2 – 1/3) + (1/3 – 1/4) + (1/4 – 1/5) + … + (1/120 – 1/121) =

= 1 – 1/121 = 120/121.

That is, the method of replacing each fraction with the difference of two fractions allowed us to solve this problem. The sum turned out to be pairs of opposite numbers to all but the first and last.

But this example can be generalized. Consider the sum:

k/(n (n + k)) + k/((n + k) (n + 2k)) + k/((n + 2k) (n + 3k)) + … + k/(( n + (m – 1)k) (n + mk))

For it, all the same reasonings that were carried out in the previous example are valid. Indeed:

1/n – 1/(n + k) = k/(n (n + k));

1/((n+k) – 1/(n + 2k) = k/((n + k) (n + 2k)) etc.

Then we construct the answer according to the same scheme: 1/n – 1/(n + mk) = mk/(n (n + mk))

And more about the "long" amounts.

Amount

X \u003d 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/1024

can be calculated as the sum of 11 members of a geometric progression with the denominator 1/2 and the first member 1. But the same sum can be calculated by a 5th grade student who has no idea about progressions. To do this, it is enough to successfully choose a number that we add to the sum X. This number will be 1/1024 here.

Compute

X + 1 / 1024 = 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + (1/1024 + 1 /1024) =
= 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/512 + 1/512 =
=1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + 1/256 = … = 1 + 1/2 + 1/2 = 2.

Now it is obvious that X = 2 – 1/1024 = 1 1023 / 1024 .

The second method is no less promising. With it, you can calculate the sum:

S = 9 + 99 + 999 + 9999 + ... + 99 999 999 999.

Here the "lucky" number is 11. Adding it to S and distribute it evenly among all 11 terms. Each of them will then get 1. Then we have:

S + 11 = 9 + 1 + 99 + 1 + 999 + 1 + 9999 + 1 + ... + 99 999 999 999 + 1 =
= 10 + 100 + 1000 + 10000 + ... + 100 000 000 000 = 111 111 111 110;

Therefore, S = 111 111 111 110 – 11 = 111 111 111 099.

1 + 11 + 111 + 1111 + ... + 1 111 111 111?

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In the distant past, when the calculus system had not yet been invented, people counted everything on their fingers. With the advent of arithmetic and the basics of mathematics, it has become much easier and more practical to keep records of goods, products, and household items. However, what does it look like modern system calculus: what types are the existing numbers divided into and what does the "rational form of numbers" mean? Let's figure it out.

How many types of numbers are there in mathematics?

The very concept of "number" denotes a certain unit of any object, which characterizes its quantitative, comparative or ordinal indicators. In order to correctly calculate the number of certain things or perform some mathematical operations with numbers (add, multiply, etc.), you should first of all familiarize yourself with the varieties of these same numbers.

So, the existing numbers can be divided into the following categories:

1. Natural numbers are those numbers with which we count the number of objects (the smallest natural number is 1, it is logical that the series of natural numbers is infinite, that is, there is no largest natural number). The set of natural numbers is usually denoted by the letter N.
2. Whole numbers. This set includes everything, while negative values ​​are added to it, including the number "zero". The designation of the set of integers is written in the form of the Latin letter Z.
3. Rational numbers are those that we can mentally convert into a fraction, the numerator of which will belong to the set of integers, and the denominator will belong to natural numbers. Below we will analyze in more detail what "rational number" means, and give a few examples.
4. - a set that includes all rational and This set is denoted by the letter R.
5. Complex numbers contain part of the real and part of the variable. They are used in solving various cubic equations, which, in turn, can have a negative expression in the formulas (i 2 = -1).

What does "rational" mean: we analyze it with examples

If rational numbers are those that we can represent as an ordinary fraction, then it turns out that all positive and negative integers are also included in the set of rational ones. After all, any integer, for example 3 or 15, can be represented as a fraction, where the denominator will be one.

Fractions: -9/3; 7/5, 6/55 are examples of rational numbers.

What does "rational expression" mean?

Move on. We have already discussed what the rational form of numbers means. Let's now imagine a mathematical expression that consists of a sum, a difference, a product, or a quotient various numbers and variables. Here is an example: a fraction, in the numerator of which is the sum of two or more integers, and the denominator contains both an integer and some variable. It is this expression that is called rational. Based on the rule "you cannot divide by zero", you can guess that the value of this variable cannot be such that the value of the denominator becomes zero. Therefore, when solving a rational expression, you must first determine the range of the variable. For example, if the denominator contains the following expression: x+5-2, then it turns out that "x" cannot be equal to -3. Indeed, in this case, the entire expression turns into zero, therefore, when solving, it is necessary to exclude the integer -3 for this variable.

How to solve rational equations correctly?

Rational expressions can contain quite a large number of numbers and even 2 variables, so sometimes their solution becomes difficult. To facilitate the solution of such an expression, it is recommended to perform certain operations in a rational way. So, what does "in a rational way" mean, and what rules should be applied when deciding?

1. The first type, when it is enough just to simplify the expression. To do this, you can resort to the operation of reducing the numerator and denominator to an irreducible value. For example, if the numerator contains the expression 18x, and the denominator 9x, then, reducing both indicators by 9x, we get just an integer equal to 2.
2. The second method is practical when we have a monomial in the numerator and a polynomial in the denominator. Let's look at an example: in the numerator we have 5x, and in the denominator - 5x + 20x 2 . In this case, it is best to take the variable in the denominator out of brackets, we get the following form of the denominator: 5x(1+4x). And now you can use the first rule and simplify the expression by reducing 5x in the numerator and denominator. As a result, we get a fraction of the form 1/1+4x.

What operations can be performed with rational numbers?

The set of rational numbers has a number of its own peculiarities. Many of them are very similar to the characteristic that is present in integers and natural numbers, in view of the fact that the latter are always included in the rational set. Here are a few properties of rational numbers, knowing which, you can easily solve any rational expression.

1. The commutativity property allows you to sum two or more numbers, regardless of their order. Simply put, the sum does not change from a change in the places of the terms.
2. The distributivity property allows solving problems using the distributive law.
3. And finally, the operations of addition and subtraction.

Even schoolchildren know what the "rational type of numbers" means and how to solve problems based on such expressions, so an educated adult simply needs to remember at least the basics of the set of rational numbers.

In this article, we will begin to study rational numbers. Here we give definitions of rational numbers, give the necessary explanations and give examples of rational numbers. After that, we will focus on how to determine whether a given number is rational or not.

Page navigation.

Definition and examples of rational numbers

In this subsection we give several definitions of rational numbers. Despite the differences in wording, all these definitions have the same meaning: rational numbers unite integers and fractional numbers, just as integers unite natural numbers, their opposite numbers, and the number zero. In other words, rational numbers generalize whole and fractional numbers.

Let's start with definitions of rational numbers which is perceived as the most natural.

From the sounded definition it follows that a rational number is:

• Any natural number n . Indeed, any natural number can be represented as an ordinary fraction, for example, 3=3/1.
• Any integer, in particular the number zero. Indeed, any integer can be written either as a positive common fraction, as a negative common fraction, or as zero. For example, 26=26/1 , .
• Any ordinary fraction (positive or negative). This is directly stated by the given definition of rational numbers.
• Any mixed number. Indeed, it is always possible to represent a mixed number as an improper common fraction. For example, and .
• Any finite decimal or infinite periodic fraction. This is so because the specified decimal fractions are converted to ordinary fractions. For example, , and 0,(3)=1/3 .

It is also clear that any infinite non-repeating decimal is NOT a rational number, since it cannot be represented as a common fraction.

Now we can easily bring examples of rational numbers. The numbers 4, 903, 100,321 are rational numbers, since they are natural numbers. The integers 58 , −72 , 0 , −833 333 333 are also examples of rational numbers. Ordinary fractions 4/9, 99/3, are also examples of rational numbers. Rational numbers are also numbers.

It can be seen from the above examples that there are both positive and negative rational numbers, and the rational number zero is neither positive nor negative.

The above definition of rational numbers can be formulated in a shorter form.

Definition.

Rational numbers call numbers that can be written as a fraction z/n, where z is an integer and n is a natural number.

Let us prove that this definition of rational numbers is equivalent to the previous definition. We know that we can consider the bar of a fraction as a sign of division, then from the properties of dividing integers and the rules for dividing integers, the following equalities follow and . Thus, which is the proof.

Let us give examples of rational numbers, based on this definition. The numbers −5 , 0 , 3 , and are rational numbers, since they can be written as fractions with an integer numerator and a natural denominator of the form and respectively.

The definition of rational numbers can also be given in the following formulation.

Definition.

Rational numbers are numbers that can be written as a finite or infinite periodic decimal fraction.

This definition is also equivalent to the first definition, since any ordinary fraction corresponds to a finite or periodic decimal fraction and vice versa, and any integer can be associated with a decimal fraction with zeros after the decimal point.

For example, the numbers 5 , 0 , −13 , are examples of rational numbers because they can be written as the following decimals 5.0 , 0.0 , −13.0 , 0.8 and −7,(18) .

We finish the theory of this section with the following statements:

• integer and fractional numbers (positive and negative) make up the set of rational numbers;
• each rational number can be represented as a fraction with an integer numerator and a natural denominator, and each such fraction is some rational number;
• every rational number can be represented as a finite or infinite periodic decimal fraction, and each such fraction represents some rational number.

Is this number rational?

In the previous paragraph, we found out that any natural number, any integer, any ordinary fraction, any mixed number, any final decimal fraction, and also any periodic decimal fraction is a rational number. This knowledge allows us to "recognize" rational numbers from the set of written numbers.

But what if the number is given as some , or as , etc., how to answer the question, is the given number rational? In many cases, it is very difficult to answer it. Let us point out some directions for the course of thought.

If a number is specified as a numeric expression that contains only rational numbers and arithmetic signs (+, −, · and:), then the value of this expression is a rational number. This follows from how operations on rational numbers are defined. For example, after performing all the operations in the expression, we get a rational number 18 .

Sometimes, after simplifying expressions and more complex type, it becomes possible to determine whether a given number is rational.

Let's go further. The number 2 is a rational number, since any natural number is rational. What about number? Is it rational? It turns out that no, it is not a rational number, it is an irrational number (the proof of this fact by contradiction is given in the 8th grade algebra textbook listed below in the list of references). It has also been proven that Square root from a natural number is a rational number only in those cases when the root is a number that is the perfect square of some natural number. For example, and are rational numbers, since 81=9 2 and 1 024=32 2 , and the numbers and are not rational, since the numbers 7 and 199 are not perfect squares of natural numbers.

Is the number rational or not? In this case, it is easy to see that, therefore, this number is rational. Is the number rational? It is proved that the kth root of an integer is a rational number only if the number under the root sign is the kth power of some integer. Therefore, it is not a rational number, since there is no integer whose fifth power is 121.

The method of contradiction allows us to prove that the logarithms of some numbers, for some reason, are not rational numbers. For example, let's prove that - is not a rational number.

Assume the opposite, that is, suppose that is a rational number and can be written as an ordinary fraction m/n. Then and give the following equalities: . The last equality is impossible, since on its left side there is odd number 5 n , and on the right side there is an even number 2 m . Therefore, our assumption is wrong, thus is not a rational number.

In conclusion, it is worth emphasizing that when clarifying the rationality or irrationality of numbers, one should refrain from sudden conclusions.

For example, one should not immediately assert that the product of irrational numbers π and e is an irrational number, this is “as if obvious”, but not proven. This raises the question: “Why would the product be a rational number”? And why not, because you can give an example of irrational numbers, the product of which gives a rational number:.

It is also unknown whether the numbers and many other numbers are rational or not. For example, there are irrational numbers whose irrational power is a rational number. To illustrate, let's give a degree of the form , the base of this degree and the exponent are not rational numbers, but , and 3 is a rational number.

Bibliography.

• Mathematics. Grade 6: textbook. for general education institutions / [N. Ya. Vilenkin and others]. - 22nd ed., Rev. - M.: Mnemosyne, 2008. - 288 p.: ill. ISBN 978-5-346-00897-2.
• Algebra: textbook for 8 cells. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; ed. S. A. Telyakovsky. - 16th ed. - M. : Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Kozhinova Anastasia

MUNICIPAL NON-TYPICAL BUDGET

GENERAL EDUCATIONAL INSTITUTION

"LYCEUM №76"

WHAT IS THE SECRET OF RATIONAL COUNTING?

Performed:

Student 5 "B" class

Kozhinova Anastasia

Supervisor:

Mathematic teacher

Shiklina Tatiana

Nikolaevna

Novokuznetsk 2013

Introduction………………………………………………………… 3

The main part....……………………………………….......... 5-13

Conclusion and Conclusions………………………………...................... 13-14

References………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

Applications……………………………………………………. 16-31

I. Introduction

Problem: finding the values ​​of numeric expressions

Objective: search, study of existing methods and techniques of rational counting, their application in practice.

Tasks:

1. Conduct a mini survey in the form of a questionnaire among parallel classes.

2. Analyze on the topic of research: the literature available in the school library, information in the textbook on mathematics for grade 5, on the Internet.

3.Choose the most effective methods and means of rational accounting.

4. Conduct a classification of existing methods of rapid oral and written counting.

5. Create memos containing rational counting techniques for use in parallel 5 classes.

Object of study: rational account.

Subject of study: ways of rational counting.

For efficiency research work I used the following techniques: analysis of information obtained from various resources, synthesis, generalization; opinion poll in the form of a questionnaire. The questionnaire was developed by me in accordance with the purpose and objectives of the study, the age of the respondents and is presented in the main part of the work.

In the course of the research work, issues related to the methods and techniques of rational counting were considered, and recommendations were given to eliminate problems with computational skills, to form a computational culture.

II. Main part

Formation of the computing culture of students

5-6 grades.

It is obvious that the methods of rational counting are a necessary element of the computational culture in the life of every person, primarily because of their practical significance, and students need it in almost every lesson.

Computational culture is the foundation for the study of mathematics and other academic disciplines, since, in addition to the fact that calculations activate memory, attention, help rationally organize activities and significantly affect human development.

In everyday life, in training sessions, when every minute is valued, it is very important to quickly and rationally carry out oral and written calculations without making mistakes and without using any additional computing tools.

We, schoolchildren, face this problem everywhere: in the classroom, at home, in the store, etc. In addition, after grades 9 and 11, we will have to take exams in the form of the IGA and the Unified State Examination, where the use of a microcalculator is not allowed. Therefore, the problem of the formation of a computational culture in each person, an element of which is the mastery of the methods of rational counting, becomes extremely important.

It is especially necessary to master the methods of rational counting.

in the study of such subjects as mathematics, history, technology, computer science, etc., that is, rational counting helps to master related subjects, to better navigate the material being studied, in life situations. So what are we waiting for? Let's go to the world of secrets of Rational methods of counting!!!

What problems do students have when doing calculations?

Often, peers of my age have problems when performing various tasks in which it is necessary to perform calculations in a quick and convenient way. . Why???

Here are some guesses:

1. The student did not master the topic studied well

2. The student does not repeat the material

3. Student has poor numeracy skills

4. The student does not want to study this topic

5. The student believes that it will not be useful to him.

I took all these assumptions from my experience and the experience of my classmates and peers. However, rational counting skills play an important role in computational exercises, so I have studied, applied and want to present you some rational counting techniques.

Rational methods of oral and written calculations.

At work and at home, there is a constant need different kind computing. Using the simplest methods of mental counting reduces fatigue, develops attention and memory. The use of rational calculation methods is necessary to increase labor, accuracy and speed of calculations. The speed and accuracy of calculations can only be achieved with rational use methods and means of mechanization of calculations, as well as with the correct use of methods of oral counting.

I. Simplified Number Addition Techniques

There are four methods of addition that allow you to speed up the calculations.

Sequential bitwise addition method used in mental calculations, as it simplifies and speeds up the summation of terms. When using this method, the addition begins with the highest digits: the corresponding digits of the second term are added to the first term.

Example. Let's find the sum of the numbers 5287 and 3564 using the method of sequential bitwise addition.

Decision. We will calculate in the following order:

5 287 + 3 000 = 8 287;

8 287 + 500 = 8 787;

8 787 + 60 = 8 847;

8 847 + 4 = 8 851.

Answer: 8 851

Another way of sequential bitwise addition consists in the fact that the highest rank of the second term is added to the highest digit of the first term, then the next digit of the second term is added to the next digit of the first term, and so on.

Let's consider this solution in the given example, we get:

5 000 + 3 000 = 8 000;

200 + 500 = 700;

Answer: 8851.

round number method . A number that has one significant digit and ends with one or more zeros is called a round number. This method is used when two or more terms can be chosen that can be completed to a round number. The difference between the round number and the number specified in the calculation condition is called the complement. For example, 1000 - 978 = 22. In this case, the number 22 is the arithmetic addition of the number 978 to 1000.

In order to add by the round number method, one or more terms close to round numbers must be rounded off, the addition of round numbers must be performed, and arithmetic additions must be subtracted from the resulting sum.

Example. Find the sum of the numbers 1238 and 193 using the round number method.

Decision. Round the number 193 to 200 and add as follows: 1 238 + 193 \u003d (1 238 + 200) - 7 \u003d 1 431. (associative law)

Method of grouping terms . This method is used when the terms, when grouped together, give round numbers, which are then added together.

Example. Find the sum of the numbers 74, 32, 67, 48, 33 and 26.

Decision. Let's sum the numbers grouped as follows: (74 + 26) + (32 + 48) + (67 + 33) = 280.

(associative-displacement law)

or, when grouping numbers results in equal sums:

Example: 1+2+3+4+5+…+97+98+99+100= (1+100)+(2+99)+(3+98)+…=101x50=5050

(associative-displacement law)

II. Techniques for simplified subtraction of numbers

The method of sequential bitwise subtraction. This method sequentially subtracts each digit subtracted from the reduced one. It is used when numbers cannot be rounded.

Example. Find the difference between the numbers 721 and 398.

Decision. Let's perform actions to find the difference of given numbers in the following sequence:

represent the number 398 as a sum: 300 + 90 + 8 = 398;

do bitwise subtraction:

721 - 300 = 421; 421 - 90 = 331; 331 - 8 = 323.

round number method . This method is used when the subtrahend is close to a round number. To calculate, it is necessary to subtract the subtrahend, taken as a round number, from the reduced, and add the arithmetic addition to the resulting difference.

Example. Let's calculate the difference between the numbers 235 and 197 using the round number method.

Decision. 235 - 197 = 235 - 200 + 3 = 38.

III. Techniques for simplified multiplication of numbers

Multiplication by one followed by zeros. When multiplying a number by a number that includes a unit followed by zeros (10; 100; 1,000, etc.), as many zeros are assigned to it on the right as there are in the multiplier after the unit.

Example. Find the product of the numbers 568 and 100.

Decision. 568 x 100 = 56,800.

bitwise multiplication method . This method is used when multiplying a number by any one-digit number. If you need to multiply a two-digit (three-, four-digit, etc.) number by a single-digit one, then first the single-digit multiplier is multiplied by tens of another factor, then by its units and the resulting products are summed up.

Example. Find the product of the numbers 39 and 7.

Decision. 39 x 7 = (30+9) x 7 = (30 x 7) + (9 x 7) = 210 + 63 = 273. ( distributive law multiplication with respect to addition)

round number method . This method is used only when one of the factors is close to a round number. The multiplier is multiplied by a round number, and then by the arithmetic addition, and at the end the second is subtracted from the first product.

Example. Find the product of the numbers 174 and 69.

174 x 69 \u003d 174 x (70-1) \u003d 174 x 70 - 174 x 1 \u003d 12 180 - 174 \u003d 12 006. (distributive law of multiplication with respect to subtraction)

A way to expand one of the factors. In this method, one of the factors is first decomposed into parts (terms), then the second factor is multiplied in turn by each part of the first factor, and the resulting products are summed up.

Example. Find the product of the numbers 13 and 325.

Let's decompose the number 13 into terms: 13 \u003d 10 + 3. Let's multiply each of the terms obtained by 325: 10 x 325 \u003d 3 250; 3 x 325 = 975. Summing up the resulting products: 3250 + 975 = 4225

Mastering the skills of rational mental counting will make your work more efficient. This is possible only with a good mastery of all the above arithmetic operations. The use of rational methods of counting speeds up calculations and provides the necessary accuracy. But not only you need to be able to calculate, but you also need to know the multiplication table, the laws of arithmetic operations, classes and digits.

There are mental counting systems that allow you to count quickly and rationally orally. We will look at some of the most commonly used techniques.

1. Multiplying a two-digit number by 11.

We have studied this method, but we have not studied it to the end. the secret of this method is that it can be considered the laws of arithmetic operations.

Examples:

23x11 \u003d 23x (10 + 1) \u003d 23x10 + 23x1 \u003d 253 (distributive law of multiplication with respect to addition)

23x11=(20+3)x 11= 20x11+3x11=253 (distributive law and round number method)

We studied this method, but we didn't know another one. The secret of multiplying two-digit numbers by 11.

By observing the results obtained when multiplying two-digit numbers by 11, I noticed that you can get the answer in a more convenient way. : when multiplying a two-digit number by 11, the digits of this number are moved apart and the sum of these digits is put in the middle.

a) 23 11=253, since 2+3=5;

b) 45 11=495, because 4+5=9;

c) 57 11=627, because 5+7=12, two was placed in the middle, and one was added to the hundreds place;

d) 78 11=858, since 7+8=15, then the number of tens will be equal to 5, and the number of hundreds will increase by one and will be equal to 8.

I found confirmation of this method on the Internet.

2) The product of two-digit numbers that have the same number of tens, and the sum of units is 10, i.e. 23 27; 34 36; 52 58 etc.

rule: the digit of tens is multiplied by the next digit in the natural series, the result is recorded and the product of units is attributed to it.

a) 23 27 = 621. How did you get 621? We multiply the number 2 by 3 (the “two” is followed by the “three”), it will be 6, and next we will assign the product of units: 3 7 \u003d 21, it turns out 621.

b) 34 36 = 1224, since 3 4 = 12, we attribute 24 to the number 12, this is the product of units of these numbers: 4 6.

c) 52 58 \u003d 3016, since we multiply the tens number 5 by 6, it will be 30, we attribute the product of 2 and 8, i.e. 16.

d) 61 69=4209. It is clear that 6 was multiplied by 7 and got 42. And where does the zero come from? We multiplied the units and got: 1 9 \u003d 9, but the result must be two-digit, so we take 09.

3) Dividing three-digit numbers that have the same digits by 37. The result is the sum of these identical digits of the three-digit number (or a number equal to three times the digit of the three-digit number).

Examples: a) 222:37=6. This is the sum of 2+2+2=6; b) 333:37=9, because 3+3+3=9.

c) 777:37=21, i.e. to 7+7+7=21.

d) 888:37=24, since 8+8+8=24.

We also take into account the fact that 888:24=37.

III. Conclusion

To unravel the main secret in the topic of my work, I had to work hard - to search, analyze information, question classmates, repeat the early known methods and find many unfamiliar methods of rational counting, and, finally, understand what is his secret? And I realized that the main thing is to know and be able to apply the known ones, find new rational methods of counting, the multiplication table, the composition of the number (classes and digits), the laws of arithmetic operations. Besides,

look for new ways to do this:

- Simplified Number Addition Techniques: (method of sequential bitwise addition; method of a round number; method of decomposing one of the factors into terms);

-Techniques for simplified subtraction of numbers(method of sequential bitwise subtraction; round number method);

-Techniques for simplified multiplication of numbers(multiplication by one followed by zeros; bitwise multiplication method; round number method; expansion method of one of the factors ;

- Secrets of fast mental counting(multiplying a two-digit number by 11: when multiplying a two-digit number by 11, the digits of this number are moved apart and the sum of these digits is put in the middle; the product of two-digit numbers that have the same number of tens, and the sum of units is 10; Division of three-digit numbers consisting of identical digits, on the number 37. There are probably many more such ways, so I will continue to work on this topic next year.

IV. Bibliography

1. Savin A. P. Mathematical miniatures / A. P. Savin. - M .: Children's literature, 1991

2. Zubareva I.I., Mathematics, grade 5: a textbook for students of educational institutions / I.I. Zubareva, A.G. Mordkovich. – M.: Mnemosyne, 2011

4. http:/ / www. xreferat.ru

5. http:/ / www. biografia.ru

6. http:/ / www. Mathematics-repetition. en

V. Applications

Mini study (survey in the form of a questionnaire)

To identify students' knowledge of rational counting, I conducted a survey in the form of a questionnaire on the following questions:

* Do you know what rational methods of counting are?

* If yes, where, and if not, why not?

* How many ways of rational counting do you know?

* Do you have difficulty in mental counting?

* How do you study math? a) on "5"; b) on "4"; c) on "3"

* What do you like most about math?

a) examples; b) tasks; c) fractions

* What do you think, where can mental counting be useful, except for mathematics? * Do you remember the laws of arithmetic operations, if so, which ones?

After conducting a survey, I realized that my classmates do not know enough the laws of arithmetic operations, most of them have problems with rational counting, many students count slowly and with errors, and everyone wants to learn how to count quickly, correctly and conveniently. Therefore, the topic of my research work is extremely important for all students and not only.

1. Interesting oral and written methods of calculations that we studied in mathematics lessons, using the examples of the textbook "mathematics, grade 5":

Here are some of them:

to quickly multiply a number by 5, it suffices to note that 5=10:2.

For example, 43x5=(43x10):2=430:2=215;

48x5=(48:2)x10=24x10=240.

To multiply a number by 50 , you can multiply it by 100 and divide by 2.

For example: 122x50=(122x100):2=12200:2=6100

To multiply a number by 25 , you can multiply it by 100 and divide by 4,

For example, 32x25=(32x100):4=3200:4=800

To multiply a number by 125 , you can multiply it by 1000 and divide by 8 ,

For example: 192x125=(192x1000):8=192000:8=24000

To make a round number ending with two 0's divided by 25 , you can divide it by 100 and multiply by 4.

For example: 2400:25=(2400:100) x 4=24 x 4=96

To divide a round number by 50 , can be divided by 100 and multiplied by 2

For example: 4500:50=(4500:100) x 2 =45 x 2 =90

But not only you need to be able to calculate, but you also need to know the multiplication table, the laws of arithmetic operations, the composition of the number (classes and digits) and have the skills to apply them

Laws of arithmetic operations.

a + b = b + a

Commutative law of addition

(a + b) + c = a + (b + c)

Associative law of addition

a · b = b · a

Commutative law of multiplication

(a · b) · c = a · (b · c)

Associative law of multiplication

(a = b) · c = a · c = b · c

Distributive law of multiplication (with respect to addition)

Multiplication table.

What is multiplication?

This is smart addition.

After all, it’s smarter to multiply times,

Than to add up everything for an hour.

Multiplication table

We all need it in life.

And not without reason named

MULTIPLY it!

Ranks and classes

In order to make it convenient to read, as well as memorize numbers with large values they should be divided into so-called "classes": starting from the right, the number is divided by a space into three digits "first class", then three more digits are selected, "second class" and so on. Depending on the meaning of the number, the last class can end with three, two or one digit.

For example, the number 35461298 is written as follows:

This number is divided into classes:

482 - first class (class of units)

630 - second class (class of thousands)

35 - third class (class of millions)

Discharge

Each of the digits that make up the class is called its category, the countdown of which also goes to the right.

For example, the number 35 630 482 can be decomposed into classes and digits:

482 - first class

2 - first digit (unit digit)

8 - second digit (tens digit)

4 - third digit (hundreds digit)

630 - second class

0 - first digit (thousands digit)

3 - second digit (digit of tens of thousands)

6 - third digit (hundred thousand digit)

35 - third grade

5 - first digit (digit of units of millions)

3 - second digit (digit of tens of millions)

The number 35 630 482 reads:

Thirty-five million six hundred thirty thousand four hundred and eighty-two.

Problems with rational counting and how to fix them

Rational methods of memorization.

As a result of the survey and observations from the lessons, I noticed that some students solve various problems and exercises poorly because they are not familiar with rational methods of computing.

1. One of the methods is to bring the studied material into a system that is convenient for memorization and storage in memory.

2. In order for the material to be remembered to be stored by memory in certain system need to do some work on its content.

3. Then you can start mastering each individual part of the text, rereading it and trying to immediately reproduce (repeat to yourself or aloud) what you read.

4. Of great importance for memorization is the repetition of material. This is also evidenced by the popular proverb: "Repetition is the mother of learning." But it must also be repeated reasonably and correctly.

The work of repetition must be revived by drawing on illustrations or examples that did not exist before or have already been forgotten.

Based on the foregoing, we can briefly formulate the following recommendations for the successful assimilation of educational material:

1. Set a task, quickly and firmly remember the educational material for a long time.

2. Focus on what needs to be learned.

3. Understand the study material well.

4. Make a plan of the memorized text, highlighting the main thoughts in it, break the text into parts.

5. If the material is large, sequentially assimilate one part after another, and then state everything as a whole.

6. After reading the material, it is necessary to reproduce it (tell what was read).

7. Repeat material until it is forgotten.

8. Distribute the repetition over a longer time.

9. Use when memorizing different types memory (primarily semantic) and some individual features of their memory (visual, auditory or motor).

10. Difficult material should be repeated before going to bed, and then in the morning, "for fresh memory."

11. Try to apply the acquired knowledge in practice. This is The best way their preservation in memory (not without reason they say: "The real mother of the doctrine is not repetition, but application").

12. It is necessary to acquire more knowledge, to learn something new.

Now you have learned how to quickly and correctly memorize the studied material.

An interesting technique of multiplying some numbers by 9 in combination with the addition of consecutive natural numbers from 2 to 10

12345x9+6=111111

123456x9+7=1111111

1234567x9+8=11111111

12345678x9+9=111111111

123456789x9+10=1111111111

Interesting game "Guess the number"

Have you played the Guess the Number game? This is a very simple game. Let's say I think of a natural number less than 100, write it down on paper (so that there is no way to cheat), and you try to guess it by asking questions that can only be answered with "yes" or "no". Then you guess the number, and I try to guess it. Whoever guesses in the least number of questions wins.

How many questions do you need to guess my number? Do not know? I undertake to guess your number by asking only seven questions. How? But, for example, how. Let you guess the number. I ask, "Is it less than 64?" - "Yes". – “Less than 32?” - "Yes". - "Less than 16?" - "Yes". – “Less than 8?” - "Not". - "Less than 12?" - "Not". - "Less than 14?" - "Yes". - "Less than 13?" - "Not". - "The number 13 is conceived."

Understandably? I divide the set of possible numbers in half, then the remaining half in half again, and so on, until the remainder is one number.

If you liked the game or, on the contrary, you want more, then go to the library and take the book “A. P. Savin (Mathematical miniatures). In this book you will find a lot of interesting and exciting things. Book picture:

Thank you all for your attention

And I wish you success!!!

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What is the secret of rational counting?

Purpose of the work: search for information, study of existing methods and techniques of rational counting, their application in practice.

Tasks: 1. Conduct a mini survey in the form of a questionnaire among parallel classes. 2. Analyze on the topic of research: the literature available in the school library, information in the textbook on mathematics for grade 5, as well as on the Internet. 3. Choose the most effective methods and means of rational counting. 4. Carry out a classification of existing methods of rapid oral and written counting. 5. Create Memos containing rational counting techniques for use in parallel 5 classes.

As I have already said, the topic of rational counting is relevant not only for students, but for every person, to make sure of this, I conducted a survey among 5th grade students. Questions and answers of the survey are presented to you in the application.

What is a rational account? A rational account is a convenient account (the word rational means convenient, correct)

Why do students have difficulty?

Here are some assumptions: The student: 1. did not master the studied topic well; 2. does not repeat the material; 3. has poor counting skills; 4 . thinks he won't need it.

Rational methods of oral and written calculations. In work and life, the need for various kinds of calculations constantly arises. Using the simplest methods of mental counting reduces fatigue, develops attention and memory.

There are four methods of addition that allow you to speed up the calculations. I. Techniques for simplified addition of numbers

The method of sequential bitwise addition is used in mental calculations, as it simplifies and speeds up the summation of terms. When using this method, the addition begins with the highest digits: the corresponding digits of the second term are added to the first term. Example. Find the sum of the numbers 5287 and 3564 using this method. Decision. We will calculate in the following sequence: 5,287 + 3,000 = 8,287; 8287 + 500 = 8787; 8787 + 60 = 8847; 8847 + 4 = 8851 . Answer: 8 851.

Another way of successive bitwise addition is that the highest digit of the second term is added to the highest digit of the first term, then the next digit of the second term is added to the next digit of the first term, and so on. Let's consider this solution in the given example, we get: 5,000 + 3,000 = 8,000; 200 + 500 = 700; 80 + 60 = 140; 7 + 4 = 11 Answer: 8851.

round number method. A number that ends in one or more zeros is called a round number. This method is used when two or more terms can be chosen that can be completed to a round number. The difference between the round number and the number specified in the calculation condition is called the complement. For example, 1000 - 978 = 22. In this case, the number 22 is the arithmetic complement of the number 978 to 1000. In order to add by the round number method, one or more terms close to round numbers must be rounded off, the addition of round numbers must be performed, and arithmetic additions must be subtracted from the resulting sum. Example. Find the sum of the numbers 1238 and 193 using the round number method. Decision. Round the number 193 to 200 and add as follows: 1238 + 193 = (1238 + 200) - 7 = 1431.

Method for grouping terms. This method is used when the terms, when grouped together, give round numbers, which are then added together. Example. Find the sum of the numbers 74, 32, 67, 48, 33 and 26. Solution. Let's sum the numbers grouped as follows: (74 + 26) + (32 + 48) + (67 + 33) = 280.

Addition method based on the grouping of terms. Example: 1+2+3+4+5+6+7+8+9+…….+97+98+99+100=(1+100)+(2+99)+(3+98)= 101x50=5050.

II. Techniques for simplified subtraction of numbers

The method of sequential bitwise subtraction. This method sequentially subtracts each digit subtracted from the reduced one. It is used when numbers cannot be rounded. Example. Find the difference between the numbers 721 and 398 . Let's perform actions to find the difference of given numbers in the following sequence: represent the number 398 as a sum: 300 + 90 + 8 = 398; perform a bitwise subtraction: 721 - 300 = 421; 421 - 90 = 331; 331 - 8 = 323.

round number method. This method is used when the subtrahend is close to a round number. To calculate, it is necessary to subtract the subtrahend, taken as a round number, from the reduced, and add the arithmetic addition to the resulting difference. Example. Let's calculate the difference between the numbers 235 and 197 using the round number method. Decision. 235 - 197 = 235 - 200 + 3 = 38.

III. Techniques for simplified multiplication of numbers

Multiplication by one followed by zeros. When multiplying a number by a number that includes a unit followed by zeros (10; 100; 1,000, etc.), as many zeros are assigned to it on the right as there are in the multiplier after the unit. Example. Find the product of the numbers 568 and 100. Solution. 568 x 100 = 56,800.

The method of sequential bitwise multiplication. This method is used when multiplying a number by any one-digit number. If you need to multiply a two-digit (three-, four-digit, etc.) number by a single-digit one, then first one of the factors is multiplied by tens of the other factor, then by its units and the resulting products are summed up. Example. Let's find the product of the numbers 39 and 7. Decision. 39 x 7 = (30 x 7) + (9 x 7) = 210 + 63 = 273.

round number method. This method is used only when one of the factors is close to a round number. The multiplier is multiplied by a round number, and then by the arithmetic addition, and at the end the second is subtracted from the first product. Example. Let's find the product of the numbers 174 and 69. Decision. 174 x 69 = (174 x 70) - (174 x 1) = 12,180 - 174 = 12,006.

A way to expand one of the factors. In this method, one of the factors is first decomposed into parts (terms), then the second factor is multiplied in turn by each part of the first factor, and the resulting products are summed up. Example. Let's find the product of the numbers 13 and 325. Decision. Let's decompose the number into terms: 13 \u003d 10 + 3. Let's multiply each of the terms obtained by 325: 10 x 325 \u003d 3 250; 3 x 325 = 975 We sum up the products obtained: 3,250 + 975 = 4,225.

Secrets of fast mental counting. There are mental counting systems that allow you to count quickly and rationally orally. We will look at some of the most commonly used techniques.

Multiplying a two-digit number by 11.

Examples: 23x11= 23x(10+1) = 23x10+23x1=253(distributive law of multiplication with respect to addition) 23x11=(20+3)x 11= 20x11+3x11=253 (distributive law and round number method) We studied this method , but we did not know one more secret of multiplying two-digit numbers by 11.

Observing the results obtained when multiplying two-digit numbers by 11, I noticed that you can get the answer in a more convenient way: when multiplying a two-digit number by 11, the digits of this number are moved apart and the sum of these digits is put in the middle. Examples. a) 23 11=253, since 2+3=5; b) 45 11=495, because 4+5=9; c) 57 11=627, because 5+7=12, two was placed in the middle, and one was added to the hundreds place; I found confirmation of this method on the Internet.

2) The product of two-digit numbers that have the same number of tens, and the sum of units is 10, i.e. 23 27; 34 36; 52 58, etc. Rule: the digit of tens is multiplied by the next digit in the natural series, the result is written down and the product of units is attributed to it. Examples. a) 23 27 = 621. How did you get 621? We multiply the number 2 by 3 (the “two” is followed by the “three”), it will be 6, and next we will assign the product of units: 3 7 \u003d 21, it turns out 621. b) 34 36 = 1224, since 3 4 = 12, we attribute 24 to the number 12, this is the product of units of these numbers: 4 6.

3) Dividing three-digit numbers consisting of the same digits by the number 37. The result is equal to the sum of these identical digits of the three-digit number (or a number equal to three times the digit of the three-digit number). Examples. a) 222:37=6. This is the sum of 2+2+2=6 . b) 333:37=9, because 3+3+3=9. c) 777:37=21, because 7+7+7=21. d) 888:37=24, since 8+8+8=24. We also take into account the fact that 888:24=37.

Mastering the skills of rational mental counting will make your work more efficient. This is possible only with a good mastery of all the above arithmetic operations. The use of rational methods of counting speeds up calculations and provides the necessary accuracy.

Conclusion To unravel the main secret in the topic of my work, I had to work hard - to search, analyze information, question classmates, repeat the early known methods and find many unfamiliar methods of rational counting, and, finally, understand what is its secret? And I realized that the main thing is to know and be able to apply the known ones, find new rational methods of counting, know the multiplication table, the composition of the number (classes and digits), the laws of arithmetic operations. Other than that, look for new ways to do this:

Techniques for simplified addition of numbers: (method of sequential bitwise addition; method of a round number; method of decomposing one of the factors into terms); - Techniques for simplified subtraction of numbers (method of sequential bitwise subtraction; method of a round number); - Techniques for simplified multiplication of numbers (multiplication by one followed by zeros; method of sequential bitwise multiplication; method of a round number; method of expanding one of the factors; - Secrets of quick mental counting (multiplying a two-digit number by 11: when multiplying a two-digit number by 11, the digits of this number are moved apart and in the middle they put the sum of these digits; the product of two-digit numbers that have the same number of tens, and the sum of units is 10; The division of three-digit numbers consisting of the same digits by the number 37. Probably, there are still a lot of such ways, so I will continue to work on this topic next year.

In conclusion, I would like to end my speech with the following words:

Thank you all for your attention, I wish you success!!!