Numerical and algebraic expressions. Converting Expressions.

What is an expression in mathematics? Why do we need expression conversions?

The question, as they say, is interesting... The fact is that these concepts are the basis of all mathematics. All mathematics consists of expressions and their transformations. Not very clear? Let me explain.

Let's say you have an evil example in front of you. Very big and very complex. Let's say you're good at math and aren't afraid of anything! Can you give an answer right away?

You'll have to decide this example. Consistently, step by step, this example simplify. According to certain rules, of course. Those. do expression conversion. The more successfully you carry out these transformations, the stronger you are in mathematics. If you don't know how to do the right transformations, you won't be able to do them in math. Nothing...

To avoid such an uncomfortable future (or present...), it doesn’t hurt to understand this topic.)

First, let's find out what is an expression in mathematics. What's happened numeric expression and what is algebraic expression.

What is an expression in mathematics?

Expression in mathematics- this is very broad concept. Almost everything we deal with in mathematics is a set of mathematical expressions. Any examples, formulas, fractions, equations, and so on - it all consists of mathematical expressions.

3+2 is a mathematical expression. s 2 - d 2- this is also a mathematical expression. Both a healthy fraction and even one number are all mathematical expressions. For example, the equation is:

5x + 2 = 12

consists of two mathematical expressions connected by an equal sign. One expression is on the left, the other on the right.

IN general view term " mathematical expression"is used, most often, to avoid humming. They will ask you what an ordinary fraction is, for example? And how to answer?!

First answer: "This is... mmmmmm... such a thing... in which... Can I write a fraction better? Which one do you want?"

Second answer: " Common fraction- this is (cheerfully and joyfully!) mathematical expression , which consists of a numerator and a denominator!"

The second option will be somehow more impressive, right?)

This is the purpose of the phrase " mathematical expression "very good. Both correct and solid. But for practical application need to be well versed in specific types of expressions in mathematics .

The specific type is another matter. This It's a completely different matter! Each type of mathematical expression has mine a set of rules and techniques that must be used when making a decision. For working with fractions - one set. For working with trigonometric expressions - the second one. For working with logarithms - the third. And so on. Somewhere these rules coincide, somewhere they differ sharply. But don't be scared by these scary words. We will master logarithms, trigonometry and other mysterious things in the appropriate sections.

Here we will master (or - repeat, depending on who...) two main types of mathematical expressions. Numerical expressions and algebraic expressions.

Numeric expressions.

What's happened numeric expression? This is a very simple concept. The name itself hints that this is an expression with numbers. That is how it is. A mathematical expression made up of numbers, brackets and arithmetic symbols is called a numerical expression.

7-3 is a numerical expression.

(8+3.2) 5.4 is also a numerical expression.

And this monster:

also a numerical expression, yes...

An ordinary number, a fraction, any example of calculation without X's and other letters - all these are numerical expressions.

Main sign numerical expressions - in it no letters. None. Only numbers and mathematical symbols (if necessary). It's simple, right?

And what can you do with numerical expressions? Numeric expressions can usually be counted. To do this, it happens that you have to open the brackets, change signs, abbreviate, swap terms - i.e. do expression conversions. But more on that below.

Here we will deal with such a funny case when with a numerical expression you don't need to do anything. Well, nothing at all! This pleasant operation - To do nothing)- is executed when the expression doesn't make sense.

When does a numerical expression make no sense?

It’s clear that if we see some kind of abracadabra in front of us, like

then we won’t do anything. Because it’s not clear what to do about it. Some kind of nonsense. Maybe count the number of pluses...

But there are outwardly quite decent expressions. For example this:

(2+3) : (16 - 2 8)

However, this expression also doesn't make sense! For the simple reason that in the second brackets - if you count - you get zero. But you can’t divide by zero! This is a forbidden operation in mathematics. Therefore, there is no need to do anything with this expression either. For any task with such an expression, the answer will always be the same: "The expression has no meaning!"

To give such an answer, of course, I had to calculate what would be in brackets. And sometimes there’s a lot of stuff in parentheses... Well, there’s nothing you can do about it.

There are not so many forbidden operations in mathematics. There is only one in this topic. Division by zero. Additional restrictions arising in roots and logarithms are discussed in the corresponding topics.

So, an idea of ​​what it is numeric expression- got. Concept the numeric expression doesn't make sense- realized. Let's move on.

Algebraic expressions.

If letters appear in a numerical expression, this expression becomes... The expression becomes... Yes! It becomes algebraic expression. For example:

5a 2; 3x-2y; 3(z-2); 3.4m/n; x 2 +4x-4; (a+b) 2; ...

Such expressions are also called literal expressions. Or expressions with variables. It's practically the same thing. Expression 5a +c, for example, both literal and algebraic, and an expression with variables.

Concept algebraic expression - broader than numeric. It includes and all numerical expressions. Those. a numerical expression is also an algebraic expression, only without letters. Every herring is a fish, but not every fish is a herring...)

Why alphabetic- It's clear. Well, since there are letters... Phrase expression with variables It’s also not very puzzling. If you understand that numbers are hidden under the letters. All sorts of numbers can be hidden under letters... And 5, and -18, and whatever you want. That is, a letter can be replace on different numbers. That's why the letters are called variables.

In expression y+5, For example, at- variable value. Or they just say " variable", without the word "magnitude". Unlike five, which is a constant value. Or simply - constant.

Term algebraic expression means that to work with this expression you need to use laws and rules algebra. If arithmetic works with specific numbers, then algebra- with all the numbers at once. A simple example for clarification.

In arithmetic we can write that

But if we write such an equality through algebraic expressions:

a + b = b + a

we'll decide right away All questions. For all numbers stroke. For everything infinite. Because under the letters A And b implied All numbers. And not only numbers, but even other mathematical expressions. This is how algebra works.

When does an algebraic expression not make sense?

Everything about the numerical expression is clear. You can't divide by zero there. And with letters, is it possible to find out what we are dividing by?!

Let's take for example this expression with variables:

2: (A - 5)

Does it make sense? Who knows? A- any number...

Any, any... But there is one meaning A, for which this expression exactly doesn't make sense! And what is this number? Yes! This is 5! If the variable A replace (they say “substitute”) with the number 5, in brackets you get zero. Which cannot be divided. So it turns out that our expression doesn't make sense, If a = 5. But for other values A does it make sense? Can you substitute other numbers?

Certainly. In such cases they simply say that the expression

2: (A - 5)

makes sense for any values A, except a = 5 .

The whole set of numbers that Can substituting into a given expression is called range of acceptable values this expression.

As you can see, there is nothing tricky. Let's look at the expression with variables and figure out: at what value of the variable is the forbidden operation (division by zero) obtained?

And then be sure to look at the task question. What are they asking?

doesn't make sense, our forbidden meaning will be the answer.

If you ask at what value of a variable the expression has the meaning(feel the difference!), the answer will be all other numbers except for what is forbidden.

Why do we need the meaning of the expression? He is there, he is not... What's the difference?! The point is that this concept becomes very important in high school. Extremely important! This is the basis for such solid concepts as the domain of acceptable values ​​or the domain of a function. Without this, you will not be able to solve serious equations or inequalities at all. Like this.

Converting Expressions. Identity transformations.

We were introduced to numerical and algebraic expressions. We understood what the phrase “the expression has no meaning” means. Now we need to figure out what it is expression conversion. The answer is simple, to the point of disgrace.) This is any action with an expression. That's all. You have been doing these transformations since first grade.

Let's take the cool numerical expression 3+5. How can it be converted? Yes, very simple! Calculate:

This calculation will be the transformation of the expression. You can write the same expression differently:

Here we didn’t count anything at all. Just wrote down the expression in a different form. This will also be a transformation of the expression. You can write it like this:

And this too is a transformation of an expression. You can make as many such transformations as you want.

Any action on expression any writing it in another form is called transforming the expression. And that's all. Everything is very simple. But there is one thing here very important rule. So important that it can safely be called main rule all mathematics. Breaking this rule inevitably leads to errors. Are we getting into it?)

Let's say we transformed our expression haphazardly, like this:

Conversion? Certainly. We wrote the expression in a different form, what’s wrong here?

It's not like that.) The point is that transformations "at random" are not interested in mathematics at all.) All mathematics is built on transformations in which appearance, but the essence of the expression does not change. Three plus five can be written in any form, but it must be eight.

Transformations, expressions that do not change the essence are called identical.

Exactly identity transformations and allow us, step by step, to transform complex example into a simple expression, keeping the essence of the example. If we make a mistake in the chain of transformations, we make a NOT identical transformation, then we will decide another example. With other answers that are not related to the correct ones.)

This is the main rule for solving any tasks: maintaining the identity of transformations.

I gave an example with the numerical expression 3+5 for clarity. IN algebraic expressions Identical transformations are given by formulas and rules. Let's say in algebra there is a formula:

a(b+c) = ab + ac

This means that in any example we can instead of the expression a(b+c) feel free to write an expression ab + ac. And vice versa. This identical transformation. Mathematics gives us a choice between these two expressions. And which one to write - from concrete example depends.

Another example. One of the most important and necessary transformations is the basic property of a fraction. You can see more details at the link, but here I’ll just remind you of the rule: If the numerator and denominator of a fraction are multiplied (divided) by the same number, or an expression that is not equal to zero, the fraction will not change. Here is an example of identity transformations using this property:

As you probably guessed, this chain can be continued indefinitely...) A very important property. It is this that allows you to turn all sorts of example monsters into white and fluffy.)

There are many formulas defining identical transformations. But the most important ones are quite a reasonable number. One of the basic transformations is factorization. It is used in all mathematics - from elementary to advanced. Let's start with him. In the next lesson.)

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By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

Expressions are the basis of mathematics. This concept is quite broad. Most of what you deal with in mathematics - examples, equations, even fractions - are expressions. A distinctive feature of the expression is the presence of mathematical operations. It is indicated by certain signs (multiplication, division, subtraction or addition). The sequence of performing mathematical operations is corrected with brackets if necessary. Doing math means finding the meaning of an expression.

What is not an expression

Not every mathematical notation can be classified as an expression. Equalities are not expressions. Whether mathematical operations are present in the equality or not does not matter. For example, a=5 is an equality, not an expression, but 8+6*2=20 also cannot be considered an expression, although it contains multiplication and addition. This example also belongs to the category of equalities. The concepts of expression and equality are not mutually exclusive, the first is part of the second. The equal sign connects two expressions:
5+7=24:2 This equation can be simplified:
5+7=12An expression always assumes that the mathematical operations it represents can be performed. 9+:-7 is not an expression, although there are signs of mathematical operations here, because it is impossible to perform these actions. There are also mathematical examples that are formally expressions, but have no meaning. An example of such an expression:
46:(5-2-3)The number 46 must be divided by the result of the actions in brackets, and it is equal to zero. You cannot divide by zero; such an action is considered forbidden in mathematics.

Numeric and algebraic expressions

There are two types of mathematical expressions. If an expression contains only numbers and symbols of mathematical operations, such an expression is called a numeric expression. If, along with numbers, the expression contains variables denoted by letters, or there are no numbers at all, the expression consists only of variables and symbols of mathematical operations, it is called algebraic. The fundamental difference between a numerical value and an algebraic one is that a numerical expression has only one value. For example, the value of the numerical expression 56–2*3 will always be equal to 50; nothing can be changed. An algebraic expression can have many meanings, because any number can be substituted for a letter. So, if in the expression b–7 we substitute 9 for b, the value of the expression will be 2, and if 200, it will be 193.

In this lesson you will look at the topic “Numerical Expressions. Comparison of numerical expressions." This lesson will introduce you to defining numerical expressions. You will learn that numerical expressions can be read. You will also learn to find their meaning and compare. Several practical examples will help you reinforce what you have learned.

Lesson: Numerical expressions. Comparing Numeric Expressions

Look at these expressions and try to find the odd one out.

20 + a
s + 7
6 + 8
15 - (10 + 2)
18 > 9

The redundant entry is 18 > 9 (18 is greater than 9). Why do you think?

Correct answer: because only it uses a comparison sign. All others use action signs.

The written expressions can be divided into two groups:

Literal expressions Numeric expressions
20 + a 6 + 8
c + 7 15 - (10 + 2)

Literal expressions are expressions that use letters of the Latin alphabet.

Numeric Expressions- numbers connected by action signs. Numeric expressions can be read.

6 + 8…(sum of 6 and 8)

15 - (10 + 2)…(from 15 subtract the sum of 10 and 2)

Let's find the meanings of the expressions:

15 - (10 + 2) = …
First we perform the action written in parentheses. Add 2 to 10.
10 + 2 = 12
Now you need to subtract 12 from 15.
15 - 12 = 3
15 - (10 + 2) = 3

Now let's complete the task:

We reviewed what it means to find the value of a numerical expression.

Now we must learn to compare numerical expressions. Compare a numerical expression - find the value of each expression and compare them.

Let's compare the meanings of the two expressions. To do this, we will find the values ​​of each of them.

15 - 7 < 6 + 3

Now let’s compare the values ​​of two more expressions:

3. Festival of Pedagogical Ideas " Public lesson» (). Make it at home Solve numerical expressions: a) 20 +14 b) 56 - 22 c) 47 - 22 Compare expressions: a) 33 - 12 and 25 + 7 b) 45 - 5 and 19 + 21 c) 23 + 5 and 12 + 6

Formula

Addition, subtraction, multiplication, division - arithmetic operations (or arithmetic operations). These arithmetic operations correspond to the signs of arithmetic operations:

+ (read " plus") - sign of the addition operation,

- (read " minus") is the sign of the subtraction operation,

∙ (read " multiply") is the sign of the multiplication operation,

: (read " divide") is the sign of the division operation.

A record consisting of numbers interconnected by arithmetic signs is called numerical expression. A numeric expression may also contain parentheses. For example, the entry 1290 : 2 - (3 + 20 ∙ 15) is a numeric expression.

The result of performing actions on numbers in numerical expression is called the value of a numeric expression. Performing these actions is called calculating the value of a numeric expression. Before writing the value of a numerical expression, put equal sign"=". Table 1 shows examples of numerical expressions and their meanings.

A record consisting of numbers and small letters of the Latin alphabet interconnected by signs of arithmetic operations is called literal expression. This entry may contain parentheses. For example, record a+b - 3 ∙c is a literal expression. Instead of letters, you can substitute different numbers. In this case, the meaning of the letters may change, so the letters in the letter expression are also called variables.

By substituting numbers instead of letters into the literal expression and calculating the value of the resulting numerical expression, they find the meaning of a literal expression for given letter values(for given values ​​of variables). Table 2 shows examples of letter expressions.

A literal expression may have no meaning if, when substituting the values ​​of the letters, a numeric expression is obtained, the value of which for natural numbers could not be found. This numerical expression is called incorrect for natural numbers. It is also said that the meaning of such an expression is “ undefined" for natural numbers, and the expression itself "doesn't make sense". For example, the literal expression a-b does not matter when a = 10 and b = 17. Indeed, for natural numbers, the minuend cannot be less than the subtrahend. For example, if you have only 10 apples (a = 10), you cannot give away 17 of them (b = 17)!

Table 2 (column 2) shows an example of a literal expression. By analogy, fill out the table completely.

For natural numbers the expression is 10 -17 incorrect (does not make sense), i.e. the difference 10 -17 cannot be expressed as a natural number. Another example: you cannot divide by zero, so for any natural number b, the quotient b: 0 undefined.

Mathematical laws, properties, some rules and relationships are often written in literal form (i.e., in the form of a literal expression). In these cases, the literal expression is called formula. For example, if the sides of a heptagon are equal a,b,c,d,e,f,g, then the formula (literal expression) to calculate its perimeter p has the form:

p =a+b+c +d+e+f+g

With a = 1, b = 2, c = 4, d = 5, e = 5, f = 7, g = 9, the perimeter of the heptagon p = a + b + c + d + e + f + g = 1 + 2 + 4 + 5 +5 + 7 + 9 = 33.

With a = 12, b = 5, c = 20, d = 35, e = 4, f = 40, g = 18, the perimeter of the other heptagon p = a + b + c + d + e + f + g = 12 + 5 + 20 + 35 + 4 + 40 + 18 = 134.

Block 1. Vocabulary

Make a dictionary of new terms and definitions from the paragraph. To do this, write words from the list of terms below in the empty cells. In the table (at the end of the block), indicate the numbers of the terms in accordance with the numbers of the frames. It is recommended that you carefully review the paragraph again before filling in the cells of the dictionary.

1. Operations: addition, subtraction, multiplication, division.

2. Signs “+” (plus), “-” (minus), “∙” (multiply, “ : " (divide).

3. A record consisting of numbers that are interconnected by signs of arithmetic operations and which may also contain parentheses.

4. The result of performing actions on numbers in numerical expression.

5. The sign preceding the value of a numerical expression.

6. A record consisting of numbers and small letters of the Latin alphabet, interconnected by signs of arithmetic operations (brackets may also be present).

7. Common name letters in literal expression.

8. The value of a numeric expression, which is obtained by substituting variables into a literal expression.

9.A numerical expression whose value cannot be found for natural numbers.

10. A numerical expression whose value for natural numbers can be found.

11. Mathematical laws, properties, some rules and relationships, written in letter form.

12. An alphabet whose small letters are used to write alphabetic expressions.

Block 2. Match

Match the task in the left column with the solution in the right. Write your answer in the form: 1a, 2d, 3b...

Block 3. Facet test. Numeric and alphabetic expressions

Facet tests replace collections of problems in mathematics, but differ favorably from them in that they can be solved on a computer, the solutions can be checked, and the result of the work can be immediately found out. This test contains 70 problems. But you can solve problems by choice; for this there is an evaluation table, which indicates simple tasks and more difficult. Below is the test.

1. Given a triangle with sides c,d,m, expressed in cm
2. Given a quadrilateral with sides b,c,d,m, expressed in m
3. The speed of the car in km/h is b, travel time in hours is d
4. The distance traveled by the tourist in m hours is With km
5. The distance covered by the tourist, moving at speed m km/h is b km
6. The sum of two numbers is greater than the second number by 15
7. The difference is less than the one being reduced by 7
8. A passenger liner has two decks with the same number of passenger seats. In each of the rows of the deck m seats, rows on deck on n more than seats in a row
9. Petya is m years old, Masha is n years old, and Katya is k years younger than Petya and Masha together
10. m = 8, n = 10, k = 5
11. m = 6, n = 8, k = 15
12. t = 121, x = 1458

1. The meaning of this expression
2. The literal expression for the perimeter is
3. Perimeter expressed in centimeters
4. Formula for the distance s traveled by a car
5. Formula for speed v, tourist movement
6. Formula for time t, tourist movement
7. Distance traveled by the car in kilometers
8. Tourist speed in kilometers per hour
9. Tourist travel time in hours
10. The first number is...
11. The subtrahend is equal to...
12. Expression for the largest number passengers, which can transport the liner for k flights
13. The largest number of passengers that an aircraft can carry in k flights
14. Letter expression for Katya's age
15. Katya's age
16. The coordinate of point B, if the coordinate of point C is t
17. The coordinate of point D, if the coordinate of point C is t
18. The coordinate of point A, if the coordinate of point C is t
19. Length of segment BD on the number line
20. Length of segment CA on the number line
21. Length of segment DA on the number line

The concept of a mathematical expression (or just an expression) taught in primary school is important. Thus, this concept helps students master computational skills. Indeed, computational errors are often associated with a lack of understanding of the structure of expressions and uncertain knowledge of the order in which actions are performed in expressions. Mastering the concept of expression determines the formation of such important mathematical concepts as equality, inequality, equation. The ability to compose expressions for a problem is necessary for mastering the ability to solve problems algebraically, i.e. by writing equations.

Children become familiar with the first expressions – sum and difference – when studying addition and subtraction in the “Tens” concentration. Without using special terms, first-graders perform calculations, write down expressions, read them, replace a number with a sum, based on visual representations. In this case, they read the expression 4+3 as follows: “add three to four” or “increase 4 by 3.” By finding the values ​​of expressions consisting of three numbers that are connected by an addition and subtraction sign, students actually use the rule for the order of actions in an implicit form and perform the first identical transformations of expressions.

Having become familiar with expressions like a+c, first-graders first use the term “sum” to designate the number resulting from addition, i.e. the amount is treated as the value of the expression. Then, with the advent of more complex expressions, such as (a+c)-c, there is a need for a different understanding of the term “amount”. Expression a+c is called a sum, and its components are called terms. When introducing expressions like a-c, a·c, a:c do the same. First, the difference (product, quotient) is the meaning of the expression, and then the expression itself. At the same time, students are told the names of its components: minuend, subtrahend, factors, dividend and divisor. For example, in the equality 9-4=5 9 is the minuend, 4 is the subtrahend, 5 is the difference. The notation 9-4 is also called the difference. You can introduce these terms in a different order: have students write down Example 9-4, explain that the difference is written, and calculate what the written difference is. The teacher enters the name of the resulting number: 5 is also a difference. Other numbers when subtracting are called: 9 - minuend, 4 - subtrahend.

Memorization of new terms is facilitated by posters like

MINUS SUBTRACT DIFFERENCE DIFFERENCE (difference value)

To consolidate these terms, exercises like: “Calculate the sum of numbers; write down the sum of the numbers; compare the sums of numbers (insert > sign,< или = вместо · в запись 4 + 3 · 5 + 1 и прочтите полученную запись); замените число суммой одинаковых (разных) чисел; заполните таблицу; составьте по таблице примеры и решите их». Важно, чтобы дети поняли, что при вычислении суммы производится указанное действие (сложение), а при записи суммы получаем два числа, соединенных знаком плюс.

When studying addition and subtraction within 10, expressions consisting of three or more numbers connected by the same or different action signs of the form are included: 3+1+1, 4-1-1, 2+2+2+2, 7-4+ 2, 6+3-7. revealing the meaning of such expressions, the teacher shows how to read them (for example, add one to three and add one more to the resulting number). By calculating the meanings of these expressions, children practically master the rule about the order of actions in expressions without parentheses, although they do not formulate it. Somewhat later, children are taught to preform expressions in the process of calculations, for example: 10-7+5=3+5=8. such entries are the first step in performing identity transformations. Introducing first-graders to expressions like 10- (6+2), (7-4)+5, etc. prepares them to study the rules for adding a number to a sum, subtracting a number from a sum, etc., to write down solutions to compound problems, and also contributes to a deeper understanding of the concept of expression.

At the next stage of mastering the concept of expression, students become familiar with expressions that use brackets: (10-3)+4, (6-2)+5. they can be entered through word problems. The teacher suggests making up the sums and differences of the numbers 10 and 3 on a typesetting canvas, using cards on which these numbers and action signs are written. Then the teacher replaces the difference 10-3 compiled by the students with a card prepared in advance with this difference. Next task: create an expression (at this stage students talk about it as an example) using the difference, the number 4 and the + sign. When reading the resulting expression, attention is drawn to the fact that its components are a difference and a number. “To make it clear,” says the teacher, “that the difference is a term, it is enclosed in parentheses.”

By independently constructing expressions, children become aware of their structure, mastering the ability to read, write, and calculate their meanings.

The terms “mathematical expression” (or simply “expression”) and “meaning of expression” are introduced. These terms are not defined. Having written down several simple expressions: sums, differences, the teacher calls them mathematical expressions. After offering to evaluate these examples, he announces that the numbers resulting from the calculation are called the value of the expression. Further work on numerical expressions consists of children practicing reading, taking dictation, composing expressions, filling out tables, making extensive use of new terms.

Rules for the order of actions .

	Peculiarities numerical expression	execution actions
	Contains only + And – or just X And :	In order (from left to right)	65 - 20 + 5 - 8 = 42 24:4 · 2:3 = 4
	Contains not only + And - , but also X And :	First perform in order (from left to right) X And : , and then + And – (from left to right)	120 – 20: 4 6 = 90 460 + 40 – 50 4 = 300 1 3 4 2 360: 4 + 10 – 8 5 = 60 180: 2 - 90: 3 = 60
	Contains one or more pairs of parentheses	First, find the values ​​of the expressions in parentheses, and then perform actions according to rules 1 and 2	1000- (100 9 + 10) =90 5 (76 – 6 + 10) = 400 80+ (360 - 300) 5 = 380 3 1 4 2 99 · (24-23) –(12-4) =91

To calculate the value of an expression, you often have to convert it, especially if the expression contains a large number of operations and parentheses.

Converting an Expression is the replacement of a given expression with another whose value is equal to the value of the given expression. Transformations of expressions are performed based on the properties of arithmetic operations and the consequences that follow from them (rules: how to add a sum to a number, how to subtract a number from a sum, how to multiply a number by a product, etc.). When studying each rule, students are convinced that in expressions of a certain type they can perform actions in different ways, but the meaning of the expression does not change.

AND the use of conventional notation of numbers in teaching mathematics.

Bundles - tens of sticks and individual sticks are used to demonstrate the formation and decimal composition of two-digit numbers. For the same purpose, you can use strips with circles or triangles to illustrate tens (10 strips of 10 figures) and ones (strips with 1, 2, ..., 9 figures). Sometimes, instead of stripes, rectangular cards depicting numerical figures (dots) are used to illustrate units and triangle cards depicting tens.

Numbers obtained by counting tens and ones are considered. First, you can turn to your life situation. You can introduce tens and ones models in the form of triangles and individual points. Then they show a triangle filled with dots (circles) according to the same “rule”, which will denote a ten. On this lesson This manual can be used as a demonstration: children name the number, which is indicated by triangles and individual dots, or they themselves designate the number using this manual. In the future, when it will be difficult to work practically with bunches of sticks, drawings of triangles and individual dots will help children to understand the decimal composition of numbers well, while the triangles are no longer filled with dots, agreeing that the triangles drawn in one cell indicate tens, and the dots to the right of There are only a few of them. With this method, it is easy for children to draw drawings in notebooks:

In each lesson devoted to the study of numbering, work is done on problems. Simple problems are solved first. These are problems for finding the sum and remainder, for increasing and decreasing a number by several units, for difference comparisons. For the tasks, children draw “pictures with dots” or work with chips, explaining: there are 2 more boys than girls, which means we take as many circles as there are triangles, and 2 more; There are 2 fewer girls on the carousel than boys, which means there were the same number of them as boys, but without 2. The diagrams for these problems look like this.

An important place in lessons in grades 1-3 is occupied by typesetting canvases of various designs, made from cardboard, plywood, and fabric. Figure 4 shows a demo typesetting canvas, and Figure 5 shows an individual one.