The study of algebraic material in elementary school. Algebraic material in the initial course of mathematics

In "Mandatory minimum maintenance primary education" on educational field"Mathematics" - the study of algebraic material, as it was before, is not singled out as a separate didactic unit of the subject compulsory study. In this part of the document, it is briefly noted that it is necessary "to give knowledge about numerical and alphabetic expressions, their meanings and the differences between these expressions." In the "Requirements for the quality of graduate training" one can only find a short phrase indefinite meaning "to teach to calculate the unknown component of an arithmetic operation." The question of how to teach "calculate an unknown component" should be decided by the author of the program or learning technology.

Let's consider how the concepts of "expression", "equality", "inequality", "equation" are characterized and what is the methodology for studying them in various methodological training systems

7.1. Expressions and their types ...
in mathematics

elementary school

Expression call a mathematical notation consisting of numbers, denoted by letters or numbers, connected by signs arithmetic operations. A single number is also an expression. An expression in which all numbers are represented by digits is called numerical expression.

If we perform the indicated actions in a numerical expression, we will get a number that is called the value of the expression.

Expressions can be classified by the number of arithmetic operations that are used when writing expressions, and by the way numbers are denoted. According to the first basis, expressions are divided into groups: elementary (not containing an arithmetic operation sign), simple (one arithmetic operation sign) and compound (more than one arithmetic operation sign) expressions. According to the second base, numerical (numbers are written in numbers) and alphabetic (at least one number or all numbers are indicated by letters) expressions are distinguished.

Mathematical notation, which in mathematics is usually called an expression, must be distinguished from other types of notation.

An example or calculation exercise is called a record of an expression along with a requirement for its evaluation.

5+3 expression, 8- its value

5+3= calculation exercise (example),

8- result of the computational exercise (example)

Depending on the sign of the arithmetic operation, which is used in writing a simple expression, simple expressions are divided into groups of expressions with the sign "+,", "-", "", ":". These expressions have special names (2 + 3 - sum; 7 - 4 - difference; 7 × 2 - product; 6: 3 - private) and generally accepted reading methods that elementary school students are introduced to.

Ways to read expressions with a "+" sign:

25+17 - 25 plus 17

25 + 17 - add 17 to 25

25+17 - 25 yes 17

25 + 17 - 25 and 17 more.

25 + 17 - the sum of the numbers twenty-five and seventeen (the sum of 25 and 17)

25+17 - 25 increase by 17

25+17 - 1st term 25, 2nd term 17

With a record simple expressions children get to know each other as the corresponding mathematical action is introduced. For example, acquaintance with the action of addition is accompanied by writing an expression for addition 2 + 1, here are examples of the first forms of reading these expressions: “add one to two”, “two and one”, “two and one”, “two plus one”. Other formulations are introduced as children become familiar with the relevant concepts. By learning the names of action components and their results, children learn to read an expression using these names (the first term is 25, the second is 17, or the sum of 25 and 17). Acquaintance with the concepts "increase by ...", "decrease by ..." allows you to introduce a new formulation for reading expressions for addition and subtraction with these terms "twenty-five increase by seventeen", "twenty-five decrease by seventeen". Do the same with other types of simple expressions.

With the concepts of "expression", "meaning of expression" in a number of educational systems ("School of Russia" and "Harmony"), children get acquainted a little later than they learn to write, calculate and read them not in all, but in many formulations. In other programs and training systems (L.V. Zankov's system, "School 2000 ...", "School 2100"), these mathematical records are immediately called expressions and use this word in computational tasks.

Teaching children to read expressions various formulations, we introduce them into the world of mathematical terms, give them the opportunity to learn the mathematical language, work out the meaning of mathematical relations, which undoubtedly improves the mathematical culture of the student, contributes to conscious assimilation many mathematical concepts.

Ø Do as I do. Correct speech a teacher, after whom children repeat the wording, is the basis of a competent mathematical speech schoolchildren. A significant effect is obtained by using the method of comparing the wording that children pronounce with a given sample. It is useful to use a technique when the teacher specifically allows speech errors and the kids fix it.

Ø Give a few expressions and offer to read these expressions different ways. One student reads the expression, while others check. It is useful to give as many expressions as the children know by this time.

Ø The teacher dictates the expressions in different ways, and the children write down the expressions themselves without calculating their meaning. Such tasks are aimed at testing children's knowledge of mathematical terminology, namely: the ability to write down expressions or computational exercises read by different mathematical formulations.

If a task is set that involves checking the formation of a computational skill, it is useful to read expressions or computational exercises only with those formulations that are well learned, without caring about their diversity, and children are asked to write down only the results of calculations, the expressions themselves can not be written down.

An expression consisting of several simple ones is called composite.

Therefore, the essential feature of a compound expression is its composition from simple expressions. Compound expressions can be introduced as follows:

1. Give a simple expression and calculate its value

(7 + 2 = 9), call it first or given.

2. Compose the second expression so that the value of the first becomes a component of the second (9 - 3), call this expression a continuation of the first. Calculate the value of the second expression (9 - 3 = 6).

3. Illustrate the process of merging the first and second expressions, based on the manual.

The manual is a rectangular sheet of paper, which is divided into 5 parts and folded in the form of an accordion. On each part of the manual there are certain records:

7 + 2

— 3

= 6

Hiding the second and third parts of this manual (from the first expression we hide the requirement for its calculation and its value, and in the second we hide the answer to the question of the first), we get a compound expression and its value (7 + 2 -3 = 6). We give it a name - composite (composed of others).

We illustrate the process of merging other pairs of expressions or computational exercises by emphasizing:

ü You can combine into a composite only such a pair of expressions when the value of one of them is a component of the other;

ü The value of the continuation expression is the same as the value of the compound expression.

When reinforcing the concept of a compound expression, it is useful to perform tasks of two types.

1 view. Given a set of simple expressions, it is necessary to select pairs of them for which the relation "the value of one of them is a component of the other" is true. Compose one compound expression from each pair of simple expressions.

2nd view. Given a compound expression. It is necessary to write down the simple expressions from which it is composed.

This technique is useful for several reasons:

§ by analogy, we can introduce the concept of a composite problem;

§ the essential feature of a compound expression is highlighted more clearly;

§ Errors are prevented when calculating values compound expressions;

§ This technique allows us to illustrate the role of brackets in compound expressions.

Compound expressions containing the signs "+", "-" and brackets are studied from the first grade. Some educational systems ("School of Russia", "Harmony", "School 2000") do not provide for the study of brackets in the first grade. They are introduced in the second grade when studying the properties of arithmetic operations (the associative property of the sum). Parentheses are introduced as signs, with the help of which in mathematics one can show the order in which actions are performed in expressions containing more than one action. In the future, children get acquainted with compound expressions containing the actions of the first and second steps with and without brackets. The study of compound expressions is accompanied by the study of the rules for the order of actions in these expressions and how to read compound expressions.

Considerable attention in all programs is paid to the transformation of expressions, which are carried out on the basis of the combination property of the sum and product, the rules for subtracting a number from a sum and a sum from a number, multiplying a sum by a number and dividing a sum by a number. In our opinion, in separate programs, there are not enough exercises aimed at developing the ability to read compound expressions, which, of course, later affects the ability to solve equations in the second way (see below). AT latest editions educational and methodical complexes in mathematics for primary school for all programs great attention is given to tasks for compiling programs and calculation algorithms for compound expressions in three to nine actions.

Expressions, in which one number or all numbers are indicated by letters, are called alphabetic (a+ 6; (a+in)× with- literal expressions). Propaedeutics for the introduction of literal expressions are expressions where one of the numbers is replaced by dots or an empty square. This entry is called the expression “with a window” (+4 is an expression with a window).

Typical tasks containing literal expressions are tasks for finding the values of expressions, provided that the letter takes various meanings from a given list of values. (Compute expression values a+ in and a— in, if a= 42, in= 90 or a = 100, in= 230). To calculate the values of literal expressions, the given values of the variables are alternately substituted into the expressions and then they work as with numeric expressions.

Literal expressions can be used to introduce generalized records of the properties of arithmetic operations, form ideas about the possibility of variable values of action components and allow children to be brought to the central mathematical concept of "variable value". In addition, with the help of literal expressions, children are aware of the properties of the existence of the values of the sum, difference, product, quotient on the set of non-negative integers. So, in the expression a+ in for any values of variables a and in you can calculate the value of the sum, and the value of the expression a— in, on the indicated set can be calculated only if in less or equal a. By analyzing assignments aimed at establishing possible limits on the values a and in in expressions a in and a: in, children establish the properties of the existence of the value of the product and the value of the quotient in an age-adapted form.

Letter symbolism is used as a means of summarizing the knowledge and ideas of children about quantitative characteristics objects of the surrounding world and the properties of arithmetic operations. The generalizing role of alphabetic symbolism makes it a very powerful tool for the formation of generalized ideas and methods of action with mathematical content, which undoubtedly increases the possibilities of mathematics in the development and formation abstract shapes thinking.

7.2. Learning equalities and inequalities in the course

primary school mathematics

Comparison of numbers and/or expressions leads to the emergence of new mathematical concepts of "equality" and "inequality".

equality call a record containing two expressions connected by the sign "=" - equals (3 \u003d 1 + 2; 8 + 2 \u003d 7 + 3 - equals).

inequality name a record containing two expressions and a comparison sign indicating the relationship "greater than" or "less than" between these expressions

(3 < 5; 2+4 >2+3 are inequalities).

Equalities and inequalities are faithful and unfaithful. If the values of the expressions on the left and right sides of the equality are the same, then the equality is considered true, if not, then the equality will be false. Accordingly: if in the record of inequality the comparison sign correctly indicates the relationship between numbers (elementary expressions) or values of expressions, then the inequality is true, otherwise, the inequality is false.

Most tasks in mathematics are related to the calculation of the values of expressions. If the value of the expression is found, then the expression and its value can be connected with an "equal" sign, which is usually written as equality: 3+1=4. If the value of the expression was calculated correctly, then the equality is called true, if it is false, then the written equality is considered incorrect.

Children get acquainted with equalities in the first grade simultaneously with the concept of “expression” in the topic “Numbers of the first ten”. Mastering the symbolic model of education of subsequent and previous date, children write down the equalities 2 + 1 = 3 and 4 - 1 = 3. In the future, equalities are actively used in the study of the composition of single-digit numbers, and then the study of almost every topic in the elementary school mathematics course is connected with this concept.

The question of introducing the concepts of "true" and "false" equality in various programs is solved ambiguously. In the "School 2000 ..." system, this concept is introduced simultaneously with the recording of equality, in the "School of Russia" system - when studying the topic "Composition of single-digit numbers" in the records of equalities "with a window" (+3 \u003d 5; 3 + \u003d 5). By choosing a number that can be inserted into the box, the children are convinced that in some cases they are correct, and in others they are incorrect equalities. It should be noted that these mathematical records, on the one hand, allow you to consolidate the composition of numbers or other computational material on the topic of the lesson, on the other hand, form an idea of a variable and are a preparation for mastering the concept of "equation".

In all programs, two types of tasks are most often used, related to the concepts of equality and inequality, true and false equalities and inequalities:

· Numbers or expressions are given, you need to put a sign between them so that the record is correct. For example, "Put the signs:"<», «>"", "=" 7-5 ... 7-3; 6+4 … 6+3".

· Records are given with a comparison sign, it is necessary to substitute such numbers instead of the box to get the correct equality or inequality. For example, “Pick up the numbers so that the entries are correct: > ; or +2< +3».

If two numbers are compared, then the choice of sign is justified by the children, based on the principle of constructing a series natural numbers, the significance of a number or its composition. By comparing two numeric expressions or an expression with a number, children calculate the values of the expressions and then compare their values, that is, they reduce the comparison of expressions to a comparison of numbers. AT educational system"School of Russia" this method is given in the form of a rule: "To compare two expressions means to compare their meanings." Children perform the same set of actions to check the correctness of the comparison. "Check if the inequalities are true:

42 + 6 > 47; 47 - 5 > 47 - 4".

The tasks that require putting a comparison sign (or checking whether the comparison sign is set correctly) have the greatest developmental effect without calculating the values of data expressions in the left and right parts inequality (equality). In this case, children must put a comparison sign, based on the identified mathematical patterns.

The form of presentation of the task and the ways of registration of its implementation varies both within the same program and in different programs.

Traditionally, when deciding inequalities with variable Two methods were used: the method of selection and the method of reduction to equality.

First way called the method of selection, which fully reflects the actions performed by the child when using it. With this method, the value is not known number is selected either from an arbitrary set of numbers, or from a given set of them. After each choice of the value of the variable ( unknown number) the correctness of the choice is checked. To do this, the found value is substituted into the given inequality instead of the unknown number. The value of the left and right parts of the inequality is calculated (the value of one of the parts can be an elementary expression, i.e. a number), and then the value of the left and right parts of the resulting inequality is compared. All these actions can be performed orally or with a record of intermediate calculations.

Second way lies in the fact that in the record of inequality, instead of the sign "<» или «>» put an equal sign and solve the equality in a way known to children. Then, reasoning is carried out, which uses the knowledge of children about the change in the result of an action depending on the change in one of its components and determines allowed values variable.

For example, "Determine what values can take a in inequality 12 - a < 7». Решение и образец рассуждений:

Let's find the value a, if 12 - a= 7

I calculate using the rule for finding the unknown subtrahend: a= 12 — 7, a= 5.

I am clarifying my answer: a equal to 5 (“the root of the equation is 5” in the Zankov system and “School 2000 ...”) the value of the expression 12 - 5 is 7, and we need to find such values \u200b\u200bof this expression that would be less than 7, which means we need subtract numbers greater than five from 12. These can be numbers 6, 7, 8, 9, 10, 11, 12. (than more we subtract from the same number, so less value differences). Means, a= 6, 7, 8, 9, 10, 11, 12. Values greater than 12 variable a cannot accept, since a larger number cannot be subtracted from a smaller one (we do not know how, if negative numbers are not entered).

An example of a similar task from a 3rd grade textbook (1-4), authors: I.I. Arginskaya, E.I. Ivanovskaya:

No. 224. “Solve the inequalities using the solution of the corresponding equations:

to— 37 < 29, 75 — with > 48, a+ 44 < 91.

Check your solutions: substitute in each inequality several numbers greater and less than the root of the corresponding equation.

Make up your own inequalities with unknown numbers, solve them and check the solutions found.

Suggest your continuation of the task.

It should be noted that a number of technologies and training programs, enhancing the logical component and significantly exceeding the standard requirements for content mathematics education in primary school, introduce the concepts:

Ø variable value, variable value;

Ø the concept of "statement" (true and false statements are called statements (M3P)), "true and false statements";

Ø consider systems of equations (I.I. Arginskaya, E.I. Ivanovskaya).

7.3. Studying Equations in a Mathematics Course

primary school

An equality containing variable, called equation. To solve an equation means to find such a value of a variable (an unknown number) at which the equation is converted into a true numerical equality. The value of the variable at which the equation is converted into a true equality is called the root of the equation.

In some educational systems ("School of Russia" and "Harmony") the introduction of the concept of "variable" is not provided. In them, the equation is treated as an equality containing an unknown number. And further, to solve the equation means to find such a number, when substituting it, instead of the unknown, the correct equality is obtained. This number is called the value of the unknown or the solution of the equation. Thus, the term "solution of an equation" is used in two senses: as a number (root), when substituting which instead of an unknown number, the equation turns into a true equality, and as the process of solving the equation itself.

Most elementary school programs and systems consider two ways of solving equations.

First way called the method of selection, which fully reflects the actions performed by the child when using it. With this method, the value of an unknown number is selected either from an arbitrary set of numbers, or from a given set of them. After each choice of value, the correctness of the solution is checked. The essence of verification follows from the definition of the equation and is reduced to the performance of four interrelated actions:

1. In given equation the found value is substituted for the unknown number.

2. The value of the left and right parts of the equation is calculated (the value of one of the parts can be an elementary expression, i.e. a number).

3. The value of the left and right parts of the resulting equality is compared.

4. A conclusion is made about the correctness or incorrectness of the obtained equality and further, whether the found number is a solution (root) of the equation.

At first, only the first action is performed, and the rest are spoken out. This verification algorithm is saved for each way of solving the equation.

A number of training systems ("School 2000", the training system of D.B. Elkonin - V.V. Davydov) for solving simple equations use the relationship between the part and the whole.

8 + X=10; 8 and X - parts; 10 is an integer. To find a part, you can subtract the known part from the whole: X= 10 — 8; X= 2.

In these learning systems, even at the stage of solving equations by the selection method, the concept of “equation root” is introduced into speech practice, and the solution method itself is called solving the equation using “root selection”.

Second way solving the equation relies on the relationship between the result and the components of the action. From this dependence follows the rule for finding one of the components. For example, the relationship between the value of the sum and one of the terms sounds like this: “if one of them is subtracted from the value of the sum of two terms, then another term will be obtained.” From this dependence follows the rule for finding one of the terms: “to find unknown term, it is necessary to subtract the known term from the value of the sum. When solving the equation, the children reason like this:

Task: Solve the equation 8 + X= 11.

In this equation, the second term is unknown. We know that to find the second term, you need to subtract the first term from the value of the sum. So, it is necessary to subtract 8 from 11. I write down: X\u003d 11 - 8. I calculate, 11 minus 8 is 3, I write X= 3.

The complete record of the solution with verification will look like this:

8 + X = 11

X = 11 — 8

X = 3

The above method solves equations with two or more actions with and without brackets. In this case, you need to determine the order of actions in the compound expression and, naming the components in the compound expression according to the last action, you should highlight the unknown, which in turn can be an expression for addition, subtraction, multiplication or division (expressed as a sum, difference, product or quotient) . Then a rule is applied to find the unknown component, expressed as a sum, difference, product, or quotient, given the names of the components for the last action in the compound expression. By performing calculations in accordance with this rule, a simple equation is obtained (or again a compound one, if there were originally three or more action signs in the expression). Its solution is carried out according to the algorithm already described above. Consider the following task.

Solve the equation ( X + 2) : 3 = 8.

In this equation, the dividend is unknown, expressed as the sum of numbers X and 2. (According to the rules of the order of operations in the expression, the division operation is performed last).

To find the unknown dividend, you can multiply the quotient by the divisor: X+ 2 = 8 × 3

We calculate the value of the expression to the right of the equal sign, we get: X+ 2 = 24.

The full entry looks like: ( X+ 2) : 3 = 8

X+ 2 = 8 × 3

X+ 2 = 24

X = 24 — 2

Check: (22 + 2) : 3 = 8

In the educational system "School 2000 ..." due to the widespread use of algorithms and their types, an algorithm (block diagram) is given for solving such equations (see diagram 3).

The second way of solving equations is quite cumbersome, especially for compound equations, where the rule of the relationship between the components and the result of the action is applied repeatedly. In this regard, many authors of programs (the “School of Russia”, “Harmony” systems) do not include at all in the primary school curriculum an introduction to the equations complex structure or introduce them at the end of the fourth grade.

In these systems, they are mainly limited to the study of equations of the following types:

X+ 2 = 6; 5 + X= 8 - equations for finding the unknown term;

X – 2 = 6; 5 – X= 3 are equations for finding the unknown minuend and subtrahend, respectively;

X× 5 = 20.5 × X= 35 - equations for finding the unknown factor;

X: 3 = 8, 6: X= 2 are equations for finding the unknown dividend and divisor, respectively.

X× 3 \u003d 45 - 21; X× (63 - 58) = 20; (58 - 40) : X= (2 × 3) - equations where one or two numbers in the equation are represented by a numerical expression. The way to solve these equations is to calculate the values of these expressions, after which the equation takes the form of one of the simple equations of the above types.

A number of programs for teaching mathematics in primary grades (the educational system of L.V. Zankov and "School 2000 ...") practice introducing children to more complex equations, where the rule of the relationship between the components and the result of the action has to be applied repeatedly and, often, require the performance of actions to transform one of the parts of the equation based on the properties of mathematical actions. For example, in these programs, students in third grade are given the following equations to solve:

3× X — (20 + X) = 70 or 2 × X– 8 + 5 × X= 97.

In mathematics, there is third way solving equations, which is based on theorems on the equivalence of equations and consequences from them. For example, one of the theorems on the equivalence of equations in a simplified formulation reads as follows: “If both sides of the equation with the domain of definition X add the same expression with a variable, defined on the same set, then we get a new equation equivalent to the given one.

Consequences follow from this theorem, which are used in solving equations.

Corollary 1. If the same number is added to both parts of the equation, then we get a new equation equivalent to the given one.

Corollary 2. If in the equation one of the terms (a numerical expression or an expression with a variable) is transferred from one part to another, changing the sign of the term to the opposite, then we obtain an equation equivalent to the given one.

Thus, the process of solving the equation is reduced to the replacement given equation, equivalent, and this replacement (transformation) can be carried out only taking into account theorems on the equivalence of equations or consequences from them.

This method of solving equations is universal; children are introduced to it in the L.V. Zankov and in the senior classes.

In the methodology of working on equations, accumulated big number creative tasks :

the choice of equations according to a given attribute from a number of proposed ones;

· to compare equations and methods of their solutions;

· to draw up equations for given numbers;

· to change in the equation of one of the known numbers so that the value of the variable becomes more (less) than the originally found value;

selection of a known number in an equation;

drawing up solution algorithms based on block diagrams for solving equations or without them;

drawing up equations according to the texts of problems.

It should be noted that in modern textbooks there is a tendency to introduce material at the conceptual level. For example, each of the above concepts is given a detailed definition that reflects its essential features. However, not all encountered definitions meet the requirements of the scientific principle. For example, the concept of "expression" in one of the mathematics textbooks for elementary grades is interpreted as follows: "A mathematical notation from arithmetic operations that does not contain signs greater than, less than or equal to is called an expression" (educational system "School 2000"). Note that in this case the definition is written incorrectly, since it describes what is not in the record, but it is not known what is there. This is a fairly typical inaccuracy that is allowed in the definition.

Note that definitions of concepts are not given immediately, i.e. not during the initial acquaintance, but in a delayed time, after the children got acquainted with the corresponding mathematical notation and learned to operate with it. Definitions are given most often in an implicit form, descriptively.

For reference: In mathematics are found as explicit and implicit definitions of concepts. Among explicit definitions are the most common definitions through the nearest genus and specific difference. (An equation is an equality containing a variable.). Implicit definitions can be divided into two types: contextual and ostensive. In contextual definitions, the content of a new concept is revealed through a passage of text, through an analysis of a specific situation.

For example: 3+ X= 9. X is an unknown number to be found.

Ostensive definitions are used to introduce terms by demonstrating the objects that these terms denote. Therefore, these definitions are also called definitions by display. For example, in this way the concepts of equality and inequality are defined in elementary grades.

2 + 7 > 2 + 6 9 + 3 = 12

78 — 9 < 78 6 × 4 = 4 × 6

equality inequalities

7.4. Order of actions in expressions

Our observations and analysis student work shows that the study of this content line is accompanied by the following types student mistakes:

Cannot correctly apply the order of operations rule;

· Incorrectly select the numbers to perform the action.

For example, in the expression 62 + 30: (18 - 3) perform actions in the following order:

62 + 30 = 92 or so: 18 - 3 = 15

18 — 3 = 15 30: 15 = 2

30: 15 = 2 62 + 30 = 92

Based on data about common mistakes that arise in schoolchildren, two main actions can be distinguished that should be formed in the process of studying this content line:

1) an action to determine the order in which arithmetic operations are performed in numerical terms;

2) the action of selecting numbers to calculate the values of intermediate mathematical operations.

In the course of mathematics of elementary grades, traditionally, the rules for the order of actions are formulated in the following form.

Rule 1. In expressions without parentheses, containing only addition and subtraction or multiplication and division, the operations are performed in the order they are written: from left to right.

Rule 2 In expressions without parentheses, the multiplication or division is performed in order from left to right, and then the addition or subtraction.

Rule 3. In expressions with brackets, the value of the expressions in brackets is evaluated first. Then, in order from left to right, multiplication or division is performed, and then addition or subtraction.

Each of these rules is focused on a certain kind of expressions:

1) expressions without brackets, containing only actions of one stage;

2) expressions without brackets containing the actions of the first and second steps;

3) expressions with brackets containing actions of both the first and second stages.

With this logic of introducing the rules and the sequence of their study, the above actions will consist of the operations listed below, the mastery of which ensures the assimilation this material:

§ recognize the structure of the expression and name what type it belongs to;

§ correlate this expression with the rule that must be followed when calculating its value;

§ to establish the procedure for actions in accordance with the rule;

§ correctly select the numbers to perform the next action;

§ perform calculations.

These rules are introduced in the third class as a generalization for determining the order of actions in expressions of various structures. It should be noted that before getting acquainted with these rules, children have already met with expressions with brackets. In the first and second grades, when studying the properties of arithmetic operations (the associative property of addition, the distributive property of multiplication and division), they are able to calculate the values of expressions containing actions of one stage, i.e. they are familiar with rule number 1. Since three rules are introduced that reflect the order of actions in expressions of three types, it is necessary, first of all, to teach children to distinguish various expressions in terms of the signs that each rule is oriented to.

In the educational system "Harmony» the main role in the study of this topic is played by a system of appropriately selected exercises, through which children learn general way determining the order of actions in expressions of different structures. It should be noted that the author of the program in mathematics in this system very logically builds a methodology for introducing rules for the order of actions, consistently offers children exercises to practice the operations that are part of the above actions. The most common tasks are:

ü to compare expressions and then identify signs of similarity and difference in them (a sign of similarity reflects the type of expression, in terms of its orientation to the rule);

ü on the classification of expressions according to a given attribute;

ü the choice of expressions with given characteristics;

ü to construct expressions according to a given rule (condition);

ü on the application of the rule in various models of expressions (symbolic, schematic, graphic);

ü to draw up a plan or flowchart of the procedure for performing actions;

ü on setting brackets in an expression with a given value;

ü to determine the order of actions in the expression when its value is calculated.

AT systems "School 2000 ..." and « elementary School XXI century" a slightly different approach to studying the order of actions in compound expressions is proposed. This approach focuses on students' understanding of the structure of the expression. The most important learning action in this case, it is the selection of several parts in a compound expression (splitting the expression into parts). In the process of calculating the values of compound expressions, students use working rules:

1. If the expression contains brackets, then it is divided into parts so that one part is connected to the other by the actions of the first stage (plus and minus signs) that are not enclosed in brackets, the value of each part is found, and then the actions of the first stage performed in order from left to right.

2. If the expression does not contain actions of the first stage that are not enclosed in brackets, but there are operations of multiplication and division that are not enclosed in brackets, then the expression is divided into parts, focusing on these signs.

These rules allow you to calculate the values of expressions containing a large number of arithmetic operations.

Consider an example.

With plus and minus signs not enclosed in brackets, we divide the expression into parts: from the beginning to the first sign (minus) not enclosed in brackets, then from this sign to the next (plus) and from the plus sign to the end.

3 40 - 20 (60 - 55) + 81: (36: 4)

There were three parts:

1 part - 3 40

Part 2 - 20 (60 - 55)

and 3 part 81: (36:4).

Find the value of each part:

1) 3 40 = 120 2) 60 — 55 = 5 3) 36: 4 = 9 4) 120 -100 = 20

20 5 = 100 81: 9 = 9 20 + 9 = 29

Answer: the value of the expression is 29.

Purpose of the seminars along this content line

abstract and review articles (manuals) of didactic, pedagogical and psychological content;

compile a card file for the report, to study a specific topic;

perform logical and didactic analysis of school textbooks, training kits, as well as an analysis of the implementation in textbooks of a certain mathematical idea, line;

select tasks for teaching concepts, substantiating mathematical statements, forming a rule or building an algorithm.

Tasks for self-study

Topic of the lesson. Characteristics of the concepts of "expression", "equality", "inequality", "equation" and the methodology for their study in various methodological

Lecture 8. Methods of studying algebraic material.

Lecture 7

1. Methodology for considering elements of algebra.

2. Numerical equalities and inequalities.

3. Preparation for familiarization with the variable. Elements of alphabetic symbols.

4. Inequalities with a variable.

5. Equation

1. The introduction of elements of algebra into the initial course of mathematics allows from the very beginning of training to conduct systematic work aimed at developing in children such important mathematical concepts as: expression, equality, inequality, equation. Familiarization with the use of a letter as a symbol denoting any number from the area of \u200b\u200bnumbers known to children creates the conditions for generalizing many primary course questions arithmetic theory, is a good preparation for introducing children in the future with the concepts in function variable. An earlier acquaintance with the use of the algebraic method of solving problems makes it possible to make serious improvements in the entire system of teaching children to solve various text problems.

Tasks: 1. To form students' ability to read, write and compare numerical expressions.2. To acquaint students with the rules for performing the order of actions in numerical expressions and develop the ability to calculate the values of expressions in accordance with these rules.3. To form students' ability to read, write down literal expressions and calculate their values for given letter values.4. To acquaint students with equations of the 1st degree, containing the actions of the first and second stages, to form the ability to solve them by the selection method, as well as on the basis of knowledge of the relationship between the m / y components and the result of arithmetic operations.

The primary school program provides for the acquaintance of students with the use of alphabetic symbols, solutions elementary equations of the first degree with one unknown and their applications to problems in one action. These issues are being studied in close connection with arithmetic material, which contributes to the formation of numbers and arithmetic operations.

From the first days of training, work begins on the formation of the concepts of equality among students. Initially, children learn to compare many objects, equalize unequal groups, transform equal groups into unequal ones. Already when studying a dozen numbers, comparison exercises are introduced. First, they are performed based on objects.

The concept of expression is formed in junior schoolchildren in close connection with the concepts of arithmetic operations. There are two stages in the method of working on expressions. On 1-the concept of the simplest expressions is formed (sum, difference, product, quotient of two numbers), and on 2-of complex ones (the sum of a product and a number, the difference of two quotients, etc.). The terms ʼʼmathematical expressionʼʼ and ʼʼvalue of a mathematical expressionʼʼ are introduced (without definitions). After writing several examples in one action, the teacher reports that these examples are otherwise called metamathematical expressions. When studying arithmetic operations, exercises for comparing expressions are included, they are divided into 3 groups. Learning the rules of procedure. Aim for this stage- based on the practical skills of students, draw their attention to the procedure for performing actions in such expressions and formulate the appropriate rule. Students independently solve examples selected by the teacher and explain in what order they performed the actions in each example. Then they formulate the conclusion themselves or read the conclusion from the textbook. Identity transformation of an expression is the replacement of a given expression by another, the value of which is equal to the value of the given expression. Students perform such transformations of expressions, based on the properties of arithmetic operations and the consequences arising from them (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 property, students are convinced that in expressions of a certain type, actions can be performed in different ways, but the meaning of the expression does not change.

2. Numerical expressions from the very beginning are considered inextricably linked with numerical equals and unequals. Numerical equalities and inequalities are divided into ʼʼtrueʼʼ and ʼʼfalseʼʼ. Tasks: compare numbers, compare arithmetic expressions, solve simple inequalities with one unknown, move from inequality to equality and from equality to inequality

1. An exercise aimed at clarifying students' knowledge of arithmetic operations and their application. When introducing students to arithmetic operations, an expression of the form 5 + 3 and 5-3 is compared; 8*2 and 8/2. First, the expressions are compared by finding the values of each and comparing the resulting numbers. In the future, the task is performed on the basis that the sum of two numbers is greater than their difference, and the product is greater than their quotient; the calculation is only used to check the result. Comparison of expressions of the form 7 + 7 + 7 and 7 * 3 is carried out to consolidate students' knowledge of the relationship between addition and multiplication.

In the process of comparison, students get acquainted with the order in which arithmetic operations are performed. First, expressions are considered, the content of the bracket, of the form 16 - (1 + 6).

2. After that, the order of actions in expressions without brackets containing actions of one and two degrees is considered. Students learn these meanings in the process of performing examples. First, the order of actions in expressions containing actions of one stage is considered, for example: 23 + 7 - 4, 70: 7 * 3. At the same time, children must learn that if there are only addition and subtraction or only multiplication and division, then they are executed in the order in which they are written. Next, expressions containing the actions of both steps are introduced. Students are told that in such expressions, you must first perform multiplication and division in order, and then addition and subtraction, for example: 21/3+4*2=7+8=15; 16+5*4=16+20=36. To convince students of the importance of following the order of actions, it is useful to perform them in the same expression in a different sequence and compare the results.

3. Exercises, during which students learn and consolidate knowledge on the relationship between the components and the results of arithmetic operations. Οʜᴎ are already included when studying the numbers of ten.

In this group of exercises, students get acquainted with cases of changing the results of actions based on a change in one of the components. Expressions are compared in which one of the terms changes (6 + 3 and 6 + 4) or the reduced 8-2 and 9-2, etc. Similar tasks are also included in the study of tabular multiplication and division and are performed using calculations (5 * 3 and 6 * 3, 16:2 and 18:2), etc. In the future, you can compare these expressions without relying on calculations.

The considered exercises are closely related to the program material and contribute to its assimilation. Along with this, in the process of comparing numbers and expressions, students receive the first ideas about equality and inequality.

So, in grade 1, where the terms ʼʼequalityʼʼ and ʼʼinequalityʼʼ are not used yet, the teacher can ask questions in the following form when checking the correctness of the calculations performed by the children: ʼʼKolya added eight to six and got 15. Is this solution correct or incorrect?ʼʼ, or offer exercises for children in which you need to check the solution of these examples, find the correct entries, etc. Similarly, when considering numerical inequalities type 5<6,8>4 or more complex, the teacher can ask a question in this form: ʼʼAre these entries correct?ʼʼ, and after the introduction of an inequality - ʼʼAre these inequalities correct?ʼʼ.

Starting from grade 1, children get acquainted with transformations numeric expressions, performed on the basis of the application of the studied elements of the arithmetic theory (numbering, the meaning of actions, etc.). For example, on the basis of knowledge of numbering, the bit composition of numbers, students can represent any number as the sum of its bit terms. This skill is used when considering the transformation of expressions in connection with the expression of many computational tricks.

In connection with such transformations, already in the 1st grade, children encounter a ʼʼchainʼʼ of equalities.

Lecture 8. Methods of studying algebraic material. - concept and types. Classification and features of the category "Lecture 8. Methods of studying algebraic material." 2017, 2018.

1.1. General issues methods of studying algebraic material.

1.2. Methodology for studying numerical expressions.

1.3. The study of literal expressions.

1.4. The study of numerical equalities and inequalities.

1.5. Technique for studying equations.

1.6. Solving simple arithmetic problems by writing equations.

1.1. General questions of the methodology for studying algebraic material

The introduction of algebraic material into the initial course of mathematics makes it possible to prepare students for the study of the basic concepts of modern mathematics (variable, equation, equality, inequality, etc.), contributes to the generalization of arithmetic knowledge, and the formation of functional thinking in children.

Primary school students should receive initial information about mathematical expressions, numerical equalities and inequalities, learn how to solve the equations provided curriculum and simple arithmetic problems by formulating an equation ( theoretical background selection of an arithmetic operation in which the relationship between the components and the result of the corresponding arithmetic operation0.

The study of algebraic material is carried out in close connection with arithmetic material.

1.2. Methodology for studying numerical expressions

In mathematics, an expression is understood as a sequence of mathematical symbols built according to certain rules, denoting numbers and operations on them.

Expressions like: 6; 3+2; 8:4+(7-3) - numerical expressions; type: 8-a; 30:in; 5+(3+s) - literal expressions (expressions with a variable).

The tasks of studying the topic

2) To acquaint students with the rules for the order of performing arithmetic operations.

3) Learn to find numerical values expressions.

4) Familiarize yourself with identical transformations of expressions based on the properties of arithmetic operations.

The solution of the tasks set is carried out throughout all the years of education in the primary grades, starting from the first days of the child's stay at school.

The methodology for working on numerical expressions provides for three stages: at the first stage - the formation of concepts about the simplest expressions (sum, difference, product, quotient of two numbers); at the second stage - about expressions containing two or more arithmetic operations of one stage; at the third stage - about expressions containing two or more arithmetic operations of different levels.

With the simplest expressions - sum and difference - students are introduced in the first grade (according to the program 1-4) with the product and private - in the second grade (with the term "work" - in grade 2, with the term "private" - in the third grade).

Consider the method of studying numerical expressions.

When performing operations on sets, children, first of all, learn the specific meaning of addition and subtraction, therefore, in entries like 3 + 2, 7-1, action signs are perceived by them as short designation the words "add", "subtract" (add 2 to 3). In the future, the concepts of actions deepen: students learn that by adding (subtracting) a few units, we increase (decrease) the number by the same number of units (reading: 3 increase by 2), then the children will learn the name of the plus signs (reading: 3 plus 2), "minus".

In the topic “Addition and subtraction within 20”, children are introduced to the concepts of “sum”, “difference” as the names of mathematical expressions and as the name of the result of the arithmetic operations of addition and subtraction.

Consider a fragment of the lesson (grade 2).

Attach 4 red and 3 yellow circles to the board using water:

OOOO OOO

How many red circles? (Write down the number 4.)

How much yellow circles? (Write down the number 3.)

What action should be performed on the written numbers 3 and 4 in order to find out how many red and how many yellow circles are together? (record appears: 4+3).

Tell me, without counting how many circles are there?

Such an expression in mathematics, when there is a “+” sign between the numbers, is called the sum (Let's say together: sum) and read like this: the sum of four and three.

Now let's find out what the sum of the numbers 4 and 3 is equal to (we give a complete answer).

Likewise for the difference.

When studying addition and subtraction within 10, expressions consisting of 3 or more numbers connected by the same and different signs arithmetic operations: 3+1+2, 4-1-1, 7-4+3, etc. By revealing the meaning of such expressions, the teacher shows how to read them. Calculating the values of these expressions, children practically master the rule about the order of arithmetic operations in expressions without brackets, although they do not formulate it: 10-3+2=7+2=9. Such records are the first step in performing identical transformations.

The methodology for familiarizing yourself with expressions with brackets can be different (Describe a fragment of the lesson in your notebook, prepare for practical exercises).

The ability to compose and find the meaning of an expression is used by children in solving arithmetic problems, at the same time, further mastery of the concept of “expression” occurs here, the specific meaning of expressions in the records of solving problems is assimilated.

Of interest is the type of work proposed by the Latvian methodologist Ya.Ya. Mentzis.

A text is given, for example, like this: “The boy had 24 rubles, a cake costs 6 rubles, a candy 2 rubles”, it is proposed:

a) make all kinds of expressions on this text and explain what they show;

b) explain what the expressions show:

2 cells 3 cells

24-2 24-(6+2) 24:6 24-6 3

In grade 3, along with the expressions discussed earlier, they include expressions consisting of two simple expressions (37+6) - (42+1), as well as consisting of a number and a product or a quotient of two numbers. For example: 75-50:25+2. Where the order in which actions are performed does not match the order in which they are written, brackets are used: 16-6:(8-5). Children must learn to read and write these expressions correctly, to find their meanings.

The terms "expression", "expression value" are introduced without definitions. In order to make it easier for children to read and find the meaning of complex expressions, methodologists recommend using a scheme that is compiled collectively and used when reading expressions:

1) I will establish which action is performed last.

2) I'll think about how the numbers are called when performing this action.

3) I will read how these numbers are expressed.

The rules for the order of actions in complex expressions are studied in the 3rd grade, but children practically use some of them in the first and second grades.

The first is the rule on the order of performing actions in expressions without brackets, when numbers are either only addition and subtraction, or multiplication and division (3 cl.). The purpose of the work at this stage is, based on the practical skills of students acquired earlier, to pay attention to the order in which actions are performed in such expressions and formulate a rule.

Leading children to the formulation of the rule, understanding it can be different. The main reliance on existing experience, the maximum possible independence, the creation of a situation of search and discovery, evidence.

Can be used methodical technique Sh.A. Amonashvili "teacher's mistake".

For example. The teacher reports that when finding the meaning of the following expressions, he got answers, in the correctness of which he is sure (answers are closed).

36:2 6=6 etc.

Invites the children to find the meanings of the expressions themselves, and then compare the answers with the answers received by the teacher (at this point, the results of arithmetic operations are revealed). Children prove that the teacher made mistakes and, based on the study of particular facts, formulate a rule (see mathematics textbook, grade 3).

Similarly, you can introduce the rest of the rules for the order of actions: when expressions without brackets contain actions of the 1st and 2nd stage, in expressions with brackets. It is important that children realize that changing the order of performing arithmetic operations leads to a change in the result, in connection with which mathematicians decided to agree and formulate rules that must be strictly observed.

Expression conversion is the replacement of a given expression with another one with the same numerical value. Students perform such transformations of expressions, based on the properties of arithmetic operations and the consequences of them (, pp. 249-250).

When studying each property, students are convinced that in expressions a certain kind you can perform actions in different ways, but the value of the expression does not change. In the future, students apply knowledge of the properties of actions to transform given expressions into identical expressions. For example, tasks of the form are offered: continue recording so that the “=” sign is preserved:

76-(20 + 4) =76-20... (10 + 7) -5= 10-5...

60: (2 10) =60:10...

When completing the first task, the students reason as follows: on the left, the sum of the numbers 20 and 4 is subtracted from 76 , on the right, 20 was subtracted from 76; in order to get the same amount on the right as on the left, it is necessary to subtract 4 more on the right. Other expressions are similarly transformed, that is, after reading the expression, the student remembers the corresponding rule. And, performing actions according to the rule, it receives the transformed expression. To make sure the conversion is correct, the children calculate the values of the given and converted expressions and compare them.

Applying knowledge of the properties of actions to substantiate computational techniques, students I-IV classes perform transformations of expressions of the form:

72:3= (60+12):3 = 60:3+12:3 = 24 1830= 18(310) = (183) 10=540

It is also necessary here that students not only explain on the basis of what they receive each subsequent expression, but also understand that all these expressions are connected by the “=” sign, because they have the same meaning. To do this, occasionally you should offer children to calculate the values of expressions and compare them. This prevents errors like: 75 - 30 = 70 - 30 = 40+5 = 45, 24 12= (10 + 2) =24 10+24 2 = 288.

Students of grades II-IV perform the transformation of expressions not only on the basis of the properties of the action, but also on the basis of their specific meaning. For example, the sum of identical terms is replaced by the product: (6+ 6 + 6 = 6 3, and vice versa: 9 4 = = 9 + 9 + 9 + 9). Based also on the meaning of the action of multiplication, more complex expressions are converted: 8 4 + 8 = 8 5, 7 6-7 = 7 5.

Based on calculations and analysis of specially selected expressions, students of grade IV are led to the conclusion that if brackets in expressions with brackets do not affect the order of actions, then they can be omitted. In the future, using the learned properties of actions and the rules for the order of actions, students practice converting expressions with brackets into expressions that are identical to them without brackets. For example, it is proposed to write these expressions without brackets so that their values do not change:

(65 + 30)-20 (20 + 4) 3

96 - (16 + 30) (40 + 24): 4

So, the children replace the first of the given expressions with the expressions: 65 + 30-20, 65-20 + 30, explaining the order of performing actions in them. In this way, students ensure that the meaning of an expression does not change when changing the order of actions only if the properties of the actions are applied in the process.

Lecture 7

1. Methodology for considering elements of algebra.

2. Numerical equalities and inequalities.

3. Preparation for familiarization with the variable. Elements of alphabetic symbols.

4. Inequalities with a variable.

5. Equation

1. The introduction of elements of algebra into the initial course of mathematics allows from the very beginning of training to conduct systematic work aimed at developing in children such important mathematical concepts as: expression, equality, inequality, equation. Familiarization with the use of a letter as a symbol denoting any number from the area of \u200b\u200bnumbers known to children creates the conditions for generalizing many questions of arithmetic theory in the initial course, is a good preparation for introducing children in the future to concepts in variable functions. An earlier acquaintance with the use of the algebraic method of solving problems makes it possible to make serious improvements in the entire system of teaching children to solve various text problems.

The primary school program provides for the acquaintance of students with the use of alphabetic symbols, solutions of elementary equations of the first degree with one unknown and their applications to tasks in one action. These issues are studied in close connection with arithmetic material, which contributes to the formation of numbers and arithmetic operations.

The concept of expression is formed in younger students in close connection with the concepts of arithmetic operations. There are two stages in the method of working on expressions. On 1-the concept of the simplest expressions is formed (sum, difference, product, quotient of two numbers), and on 2-of complex ones (the sum of a product and a number, the difference of two quotients, etc.). The terms "mathematical expression" and "value of a mathematical expression" are introduced (without definitions). After writing several examples in one action, the teacher reports that these examples are otherwise called metamathematical expressions. When studying arithmetic operations, exercises for comparing expressions are included, they are divided into 3 groups. Learning the rules of procedure. The goal at this stage is, based on the practical skills of students, to draw their attention to the order in which actions are performed in such expressions and formulate the corresponding rule. Students independently solve the examples selected by the teacher and explain in what order they performed the actions in each example. Then they formulate the conclusion themselves or read the conclusion from the textbook. Identity transformation of an expression is the replacement of a given expression by another, the value of which is equal to the value of the given expression. Students perform such transformations of expressions, based on the properties of arithmetic operations and the consequences arising from them (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 property, students are convinced that in expressions of a certain type, actions can be performed in different ways, but the meaning of the expression does not change.

2. Numerical expressions from the very beginning are considered inextricably linked with numerical equals and unequals. Numerical equalities and inequalities are divided into "true" and "false". Tasks: compare numbers, compare arithmetic expressions, solve simple inequalities with one unknown, move from inequality to equality and from equality to inequality

2. After that, the order of actions in expressions without brackets containing actions of one and two degrees is considered. Students learn these meanings in the process of performing examples. First, the order of actions in expressions containing actions of one stage is considered, for example: 23 + 7 - 4, 70: 7 * 3. At the same time, children must learn that if there are only addition and subtraction or only multiplication and division, then they are performed in the order in which they are written. Then expressions containing the actions of both stages are introduced. Students are told that in such expressions, you must first perform multiplication and division in order, and then addition and subtraction, for example: 21/3+4*2=7+8=15; 16+5*4=16+20=36. To convince students of the need to follow the order of actions, it is useful to perform them in the same expression in a different sequence and compare the results.

3. Exercises, during which students learn and consolidate knowledge on the relationship between the components and the results of arithmetic operations. They are included already when studying the numbers of ten.

In this group of exercises, students get acquainted with cases of changing the results of actions depending on a change in one of the components. Expressions are compared in which one of the terms changes (6 + 3 and 6 + 4) or the reduced 8-2 and 9-2, etc. Similar tasks are also included in the study of tabular multiplication and division and are performed using calculations (5 * 3 and 6 * 3, 16:2 and 18:2), etc. In the future, you can compare these expressions without relying on calculations.

So, in grade 1, where the terms “equality” and “inequality” are still not used, the teacher can, when checking the correctness of the calculations performed by the children, ask questions in the following form: “Kolya added eight to six and got 15. Is this solution correct or incorrect?” , or offer children exercises in which you need to check the solution of these examples, find the correct entries, etc. Similarly, when considering numerical inequalities of the form 5<6,8>4 or more complex, the teacher can ask a question in this form: “Are these records correct?”, And after the introduction of an inequality, “Are these inequalities true?”.

Starting from grade 1, children also get acquainted with the transformations of numerical expressions, performed on the basis of the use of the studied elements of arithmetic theory (numbering, the meaning of actions, etc.). For example, based on the knowledge of numbering, the bit composition of numbers, students can represent any number as the sum of its bit terms. This skill is used when considering the transformation of expressions in connection with the expression of many computational tricks.

In connection with such transformations, already in the first grade, children encounter a "chain" of equalities.

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Methods of studying algebraic material

Lecture 1. Mathematical expressions

1.1 Learning the concept of "mathematical expression"

Algebraic material is studied starting from grade 1 in close connection with arithmetic and geometric material. The introduction of elements of algebra contributes to the communication of concepts about number, arithmetic operations, mathematical relations and at the same time prepares children for the study of algebra in the following classes.

Main algebraic concepts course are "equality", "inequality", "expression", equation". There are no definitions of these concepts in the course of mathematics of elementary grades. Students understand these concepts at the level of representations in the process of performing specially selected exercises.

The mathematics program in grades 1-4 provides for teaching children to read and write magmatic expressions: to familiarize them with the rules of the order in which actions are performed and teach them how to use them in calculations, to acquaint students with identical transformations of expressions.

When forming the concept of a mathematical expression in children, it must be taken into account that the action sign placed between numbers has a double meaning; on the one hand, it denotes an action to be performed on numbers (for example, 6 + 4 - add 4); on the other hand, the action sign serves to denote the expression (6 + 4 is the sum of the numbers 6 and 4).

There are two stages in the method of working on expressions. On the first of them, the concept of the simplest expressions (sum, difference, product, quotient of two numbers) is formed, and on the second - about complex ones (the sum of pro, products and numbers, the difference of two quotients, etc.).

Acquaintance with the first expression - the sum of two; numbers occurs in grade 1 when studying addition to subtraction within 10. Performing operations on sets, children, first of all, learn the specific meaning of addition and subtraction, therefore, in entries like 5 + 1, 6-2, action signs are perceived by them as a short designation of words "add", "subtract". This is reflected in the reading (adding 1 to 5 equals 6, subtracting 2 from 6 equals 4). In the future, the concepts of these actions are deepened. Students will learn that by adding a few units, we increase the number by the same number of units, and by subtracting, we decrease it by the same number of units. This is also reflected in new form reading records (4 increase by 2 equals 6, 7 decrease by 2 equals 5), Then the children learn the names of the action signs: "plus", "minus" and read examples, naming the action signs (4 + 2 = 6, 7-3 = 4),

Having become familiar with the names of the components and the result of the addition operation, students use the term "sum" to refer to the number that is the result of addition. Based on the children's knowledge of the names of numbers in addition, the teacher explains that in the addition examples, an entry consisting of two numbers connected by a plus sign is called the same as the number on the other side of the equals sign (9 sum "6 + 3 is also a sum). This is visually depicted as follows:

In order for children to learn the new meaning of the term "sum" as the name of the expression, the following exercises are given: "Write down the sum of the numbers 7 and 2; calculate what the sum of the numbers 3 and 4 is; read the entry (6 + 3), say what the sum is; replace number is the sum of numbers (9= ?+?); compare the sums of numbers (6+3 and 6+2), say which one is greater, write it down with a "greater than" sign and read the entry." In the process of such exercises, students gradually become aware of the double meaning of the term "sum": to write down the sum of numbers, you need to connect them with a plus sign; To find the value of the sum, you need to add the given numbers.

Approximately in the same way work in progress above the following expressions: difference, product and quotient of two numbers. However, now each of these terms is entered at once both as the name of the expression and as the name of the result of the action. The ability to read and write expressions, to find their meaning with the help of the corresponding action is developed in the process of repeated exercises, similar to exercises with the sum.

When studying addition and subtraction within 10, expressions consisting of three or more numbers connected by the same or various signs actions of the form: 3+1+1, 4-1-1, 2+2+2. By evaluating the values of these expressions, children in expressions learn the rule about the order in which Actions are performed in expressions without parentheses, although they do not formulate it. Somewhat later, children are taught to transform expressions in the process of calculations: for example: 7+5=3+5=8. Such records are the first step in performing identical transformations.

Acquaintance of first graders with expressions of the form: 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 the solution of compound problems, and also contribute to a deeper assimilation of the concept of an expression.

Methodology for introducing students to the expression of the form: 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 the solution of compound problems, and also contribute to a deeper assimilation of the concept of an expression.

The method of familiarizing students with the expression of the form: 10+(6-2), (5+3) -1 may be different. You can immediately learn to read ready-made expressions by analogy with the sample and calculate the values of expressions, explaining the sequence of actions. Another way of introducing children to expressions of this type is also possible - the compilation of these expressions by students from a given number and the simplest expression.

The ability to compose and find the meaning of expressions is used by students in solving compound problems, at the same time, further mastery of the concept of expression takes place, the specific meaning of expressions in the records of problem solutions is assimilated. An exercise is useful in this regard: the condition of the problem is given, for example, "The boy had 24 rubles. Ice cream costs 12 rubles, and candy costs 6 rubles." Children should explain what the following expressions show in this case:

In the second grade, the terms "mathematical expression" and "expression value" (without definition) are introduced. After recording several examples in one action, the teacher reports that these examples are otherwise called mathematical expressions.

On the instructions of the teacher, the children themselves make up various expressions. The teacher offers to calculate the results and explains that the results are otherwise called the values of mathematical expressions. Then more complex mathematical expressions are considered.

Later, when performing various exercises first the teacher, and then the children, use new terms (write down the expressions, find the meaning of the expression, compare the expressions, etc.).

In complex expressions, action signs connecting simple expressions also have a double meaning, which is gradually revealed by students. For example, in the expression 20+(34-8), the "+" sign denotes the action to be performed on the number 20 and the difference between the numbers 34 and 8 (add the difference between the numbers 34 and 8 to 20). In addition, the plus sign is used to denote the sum - this expression is the sum in which the first term is 20, and the second term is expressed as the difference between the numbers 34 and 8.

After the children get acquainted in the second grade with the order of performing actions in complex expressions, they begin to form the concepts of sum, difference, product, quotient, in which individual elements are given by expressions.

In the future, in the process of repeated exercises in reading, composing and writing expressions, students gradually master the ability to establish the type of a complex expression (in 2-3 actions).

The diagram, which is compiled collectively and used when reading expressions, greatly facilitates the work of children:

determine which action is performed last;

remember how the numbers are called when performing this action;

Reading and writing exercises complex actions, the simplest expressions, help children learn the rules of the order of actions.

1.2 Learning the rules of procedure

The rules for the order of actions in complex expressions are studied in grade 2, but almost some of them are used by children in grade 1.

First, we consider the rule about the order in which operations are performed in expressions without brackets, when numbers are either only added and subtracted, or only multiplied and divided. The need to introduce expressions containing two or more arithmetic operations of the same level arises when students become familiar with the computational methods of addition and subtraction within 10, namely:

Similarly: 6 - 1 - 1, 6 - 2 - 1, 6 - 2 - 2.

Since in order to find the values of these expressions, students turn to subject actions that are performed in a certain order, they easily learn the fact that arithmetic operations (addition and subtraction) that take place in expressions are performed sequentially from left to right.

With numerical expressions containing addition and subtraction operations, as well as brackets, students first meet in the topic "Addition and subtraction within 10". When children encounter such expressions in grade 1, for example: 7 - 2 + 4, 9 - 3 - 1, 4 +3 - 2; in the 2nd grade, for example: 70 - 36 +10, 80 - 10 - 15, 32 + 18 - 17; 4 * 10: 5, 60: 10 * 3, 36: 9 * 3, the teacher shows how to read and write such expressions and how to find their value (for example, 4 * 10: 5 read: 4 times 10 and divide the result by 5). By the time of studying the topic "Procedure of actions" in grade 2, students are able to find the meanings of expressions of this type. The purpose of the work at this stage is, based on the practical skills of students, to draw their attention to the order in which actions are performed in such expressions and formulate the corresponding rule. Students independently solve examples selected by the teacher and explain in what order they performed; actions in each example. Then they formulate the conclusion themselves or read the conclusion from the textbook: if only the operations of addition and subtraction (or only the operations of multiplication and division) are indicated in the expression without brackets, then they are performed in the order in which they are written (i.e. from left to right).

Despite the fact that in expressions of the form a + b + c, a + (b + c) and (a + c) + c, the presence of brackets does not affect the order of performing actions due to the associative law of addition, at this stage it is more expedient to orient students to that the action in parentheses is performed first. This is due to the fact that for expressions of the form a - (b + c) and a - (b - c) such a generalization is also unacceptable for students initial stage it will be quite difficult to navigate the assignment of brackets for various numerical expressions. The use of brackets in numerical expressions containing addition and subtraction is further developed, which is associated with the study of such rules as adding a sum to a number, a number to a sum, subtracting a sum from a number and a number from a sum. But when first introduced to brackets, it is important to direct students to the fact that the action in brackets is performed first.

The teacher draws the attention of the children to how important it is to follow this rule when calculating, otherwise you can get an incorrect equality. For example, students explain how the values of the expressions were obtained: 70 - 36 +10=24, 60:10 - 3 =2, why they are incorrect, what values these expressions actually have. Similarly, they study the order of actions in expressions with brackets of the form: 65 - (26 - 14), 50: (30 - 20), 90: (2 * 5). Students are also familiar with such expressions and are able to read, write and calculate their meaning. After explaining the order of performing actions in several such expressions, the children formulate a conclusion: in expressions with brackets, the first action is performed on the numbers written in brackets. Considering these expressions, it is easy to show that the actions in them are not performed in the order in which they are written; to show a different order of execution, and parentheses are used.

The next rule is the order of execution of actions in expressions without brackets when they contain actions of the first and second steps. Since the rules of the order of actions are adopted by agreement, the teacher communicates them to the children or the students get to know them from the textbook. In order for students to learn the introduced rules, along with training exercises include solving examples with an explanation of the order in which their actions are performed. Exercises in explaining errors in the order of performing actions are also effective. For example, from given pairs examples, it is proposed to write out only those where the calculations are performed according to the rules of the order of operations:

After explaining the errors, you can give the task: using brackets, change the order of actions so that the expression has a given value. For example, in order for the first of the given expressions to have a value equal to 10, you need to write it like this: (20+30):5=10.

Especially useful are exercises for calculating the value of an expression, when the student has to apply all the learned rules. For example, the expression 36:6 + 3 * 2 is written on the board or in notebooks. Students calculate its value. Then, on the instructions of the teacher, the children change the order of actions in the expression using brackets:

An interesting, but more difficult, exercise is the opposite: arrange the brackets so that the expression has the given value:

Also interesting are the exercises of the following type:

1. Arrange the brackets so that the equalities are true:

25-17:4=2 3*6-4=6

2. Replace the asterisks with "+" or "-" signs so that you get the correct equalities:

3. Replace the asterisks with signs of arithmetic operations so that the equalities are true:

By doing such exercises, students are convinced that the meaning of an expression can change if the order of actions changes.

To master the rules of the order of actions, it is necessary in grades 3 and 4 to include more and more complicated expressions, when calculating the values of which the student would apply each time not one, but two or three rules for the order of actions, for example:

90*8- (240+170)+190,

469148-148*9+(30 100 - 26909).

At the same time, the numbers should be selected so that they allow the execution of actions in any order, which creates conditions for the conscious application of the learned rules.

1.3 Understanding Expression Conversion

Expression conversion is the replacement of a given expression with another whose value is equal to the value of the given expression. Students perform such formations of expressions, relying on the properties of arithmetic operations and the consequences arising from them.

When studying each rule, students are convinced that in expressions of a certain type, actions can be performed in different ways, but the meaning of the expression does not change. In the future, students apply knowledge of the properties of actions to transform given expressions into expressions equal to them. For example, tasks of the form are offered: continue recording so that the "=" sign is preserved:

56- (20+1)=56-20...

(10+5) * 4=10*4...

60:(2*10)=60:10...

When completing the first task, students reason as follows: on the left, the sum of the numbers 20 and 1 is subtracted from 56; on the right, 20 is subtracted from 56; in order to get the same amount on the right as on the left, it is also necessary to subtract 1 on the right. Other expressions are similarly transformed, i.e., after reading the expression, the student remembers the corresponding rule and, performing actions according to the rule, receives the transformed expression. To make sure the conversion is correct, the children calculate the values of the given and converted expressions and compare them. Applying knowledge of the properties of actions to substantiate calculation methods, students in grades 2-4 perform transformations of expressions of the form:

54+30=(50+4)+20=(50+20)+4=70+4=74

72:3=(60+12):3=60:3+12:3=24

16 * 40=16 * (3 * 10)=(16 * 3) * 10=540

Here it is also necessary that students not only explain on the basis of what they receive each subsequent expression, but also understand that all these expressions are connected by the "=" sign, because they have the same meaning. To do this, sometimes you should invite children to calculate the values of expressions and compare them. This prevents errors like:

75-30=70-30=40+5=45,

24*12=(10+2)=24*10 +24*2=288.

Students in grades 2-3 perform expression transformations not only on the basis of action properties, but also on the basis of action definitions. For example, the sum of identical terms is replaced by the product: 6+6+6=6 * 3, and vice versa: 9 * 4=9+9+9+9. Based also on the meaning of the action of multiplication, they transform more complex expressions: 8 * 4+8=8 * 5, 7 * 6 - 7 =7 * 5.

Based on calculations and analysis of specially selected expressions, grade 3 students are led to the conclusion that if brackets in expressions with brackets do not affect the order of actions, then they can be omitted: (30 + 20) + 10 = 30 + 20 + 10, (10-6):4=10-6:4 etc. In the future, using the learned properties of actions and the rules for the order of actions, students practice converting expressions with brackets into expressions that are identical to them without brackets. For example, it is proposed to write these expressions without brackets so that their values do not change: (65+30) - 20 (20+4) * 3

Explaining the solution of the first of the given expressions based on the rule for subtracting a number from the sum, the children replace it with the expressions: 65 + 30 - 20, 65 - 20 + 30, 30 - 20 + 65, explaining the procedure for performing actions in them. By doing these exercises, students make sure that the meaning of an expression does not change when changing the order of actions only if the properties of the actions are applied in this case.

Thus, the acquaintance of primary school students with concept expression is closely related to the formation of computational skills and abilities. At the same time, the introduction of the concept of expression allows you to organize the appropriate work on the development of mathematical speech of students.

Lecture 2. Letter symbolism, equalities, inequalities, equations

2.1 Methodology for familiarization with alphabetic symbols

In accordance with the program in mathematics, letter symbols are introduced in grade 3.

Here, students get acquainted with the letter a, as a symbol for an unknown number or one of the components of the expression when solving expressions of the form: write the letter a instead of the "window". Find the values of the sum a+6 if a=8, a=7. Then, in subsequent lessons, they get acquainted with some letters. Latin alphabet, denoting one of the components in the expression. With the letter x, as a symbol for designating an unknown number when solving equations of the form: a + x \u003d b, x - c \u003d b - they get acquainted in 4 quarters in grade 3.

The introduction of a letter as a symbol to designate a variable makes it possible already in the primary grades to begin work on the formation of the concept of a variable, to introduce children to mathematical language characters.

Preparatory work for revealing the meaning of a letter as a symbol for denoting a variable is carried out at the beginning school year in 3rd grade. At this first stage, children are introduced to some letters of the Latin alphabet (a, b, c, d, k) to denote a variable, i.e. one of the components in the expression.

When introducing alphabetic symbols to denote a numeric variable important role in the system of exercises plays a skillful combination of inductive and deductive methods. In accordance with this, the exercises provide for transitions from numerical expressions to alphabetic ones and, vice versa, from alphabetic expressions to numerical ones. For example, a poster with three pockets is hung on the board, on which it is written: "1 term", "2 term", "sum".

In the process of talking with students, the teacher fills the pockets of the poster with cards with numbers and mathematical expressions written on them:

Next, it turns out whether it is still possible to compose expressions, how many such expressions can be composed. Children make up other expressions and find common things in them: the same action - addition and different - different terms. The teacher explains that, instead of writing different numbers, you can denote any number that can be a term by some letter, for example a, any number that can be a second term, for example, c. Then the amount can be denoted as follows: a + b (the corresponding cards are placed in the pockets of the poster).

The teacher explains that a + b is also a mathematical expression, only in it the terms are indicated by letters, each of the letters stands for any numbers. These numbers are called letter values.

Similarly, the difference of numbers is introduced as a generalized notation of numerical expressions. In order for students to realize that the letters included in the expression, for example, in + s, can take on many numerical values, and the literal expression itself is a generalized record of numerical expressions, exercises are provided for the transition from literal expressions to numeric ones.

Students are convinced that by giving letters personal numerical values, you can get as many numerical expressions as you like. In the same plan, work is underway to specify the literal expression - the difference of numbers.

Further, in connection with the work on expressions, the concept of a constant value is revealed. For this purpose, expressions are considered in which constant fixed with a number, for example: a±12, 8±s. Here, as in the first stage, exercises are provided for the transition from numerical expressions to expressions written using letters and numbers, and vice versa.

For this purpose, at first, a poster with three pockets is used.

Filling the pockets of the poster with cards with numbers and mathematical expressions written on them, students notice that the values of the first term change, and the second does not change.

The teacher explains that the second term can be written using numbers, then the sum of the numbers can be written as follows: m + 8, and the cards are inserted into the corresponding pockets of the poster.

In a similar way, one can obtain mathematical expressions of the form: 17 ± a, in ± 30, and later - expressions of the form: 7 * in, c * 4, a: 8, 48: in.

In grade 4, exercises of the form are carried out: Find the values of the expression a: b, if

a=3400 and b=2;

a=2800 and b=7.

When students understand the meaning of letter symbolism, letters can be used as a means of summarizing the knowledge they form.

The concrete basis for the use of alphabetic symbols as a generalization tool is the knowledge of arithmetic operations and the knowledge that is formed on their basis.

These include concepts of arithmetic operations, their properties, relationships between components and results of actions, changes in the results of arithmetic operations depending on a change in one of the components, etc.

Thus, the use of alphabetic symbols contributes to an increase in the level of generalization of knowledge acquired by primary school students and prepares them for the study of a systematic course of algebra in the next grades.

2.2 Numerical equalities, inequalities

The concept of equalities, inequalities and equations is revealed in interconnection. Work on them is carried out from grade 1, organically combined with the study of arithmetic material.

By new program the task is to teach children how to compare numbers, as well as compare expressions in order to establish relationships "greater than", "less than", "equal to"; teach how to write comparison results using the signs ">", "<", "=" и читать полученные равенства и неравенства.

Numerical equalities and inequalities are obtained by students by comparing given numbers or arithmetic expressions. Initially, younger Schoolchildren form concepts only about true Equalities and inequalities (5> 4, 6<7, 8=8).

Subsequently, when students gain experience working on expressions and inequalities with a variable, after considering the concepts of true and false (true and incorrect) statements, they move on to such a definition of the concepts of equality and inequality, according to which any two numbers, two expressions connected by one of the signs "greater than ", "less than" is called inequality. At the same time, true and false equalities and inequalities are distinguished. In grade 3, the following exercises are offered: check if the equality data is correct (4 quarter): 760 - 400 \u003d 90 * 4; 630:7=640:8.

But these exercises are not enough. In grade 4, similar exercises and others are offered, of the form: check if the inequalities are true: 478 * 24<478* (3*9); 356*10*6>356*16.

Familiarization with equalities and inequalities in elementary grades is directly related to the study of numbering and arithmetic operations. mathematical algebra equation

Comparison of numbers is carried out first on the basis of comparison of sets, which is performed, as you know, by establishing a one-to-one correspondence. This method of comparing sets is taught to children in the preparatory period and at the beginning of studying the numbering of the numbers of the first ten. Along the way, the elements of the sets are counted and the resulting numbers are compared. In the future, when comparing numbers, students rely on their place in the natural series: 9<10, потому что при счете число 9 называют перед числом 10, и т.д.

Established relationships are written using the signs ">", "<", "=", учащиеся упражняются в чтении и записи равенств и неравенств. Впоследствии при изучении нумерации чисел в пределах 100, 1000, а также нумерации многозначны: чисел сравнение чисел осуществляется либо на основе сопоставления их по месту в натуральном ряду, либо на основе разложения чисел по десятичному составу сравнения соответствующих разрядных чисел, начиная с высшего разряда.

The comparison of named numbers is first performed based on the comparison of the values of the quantities themselves, and then is carried out on the basis of a comparison of abstract numbers, for which the given named numbers are expressed in the same units of measurement.

Comparing named numbers causes great difficulties for students, therefore, in order to teach this operation, it is necessary to systematically offer various exercises in grades 2-4:

1 dm * 1 cm, 2 dm * 2 cm

Replace with an equal number: 7 km 500 m = _____ m

3) Choose the numbers so that the entry is correct: ____ h< ____ мин, ___ см=__ дм и т.д.

4) Check if the equalities are true or false, correct the sign if the equalities are false:

4 t 8 w = 480 kg, 100 min. = 1 h, 2 m 5 cm = 250 cm.

The transition to comparison of expressions is carried out gradually. First, in the process of studying addition and. subtractions within 10 children practice for a long time comparing expressions and numbers. The first inequalities of the form 3+1>3, 3 - 1<3 полезно получать из равенства (3=3), сопровождая преобразования соответствующими операциями над множествами. В дальнейшем выражение и число учащиеся сравнивают, не прибегая к операциям над множествами: находят значение выражения и сравнивают его с заданным числом, что отражается в записях:

After getting acquainted with the names of expressions, students read equalities and inequalities like this: the sum of the numbers 5 and 3 is greater than 5.

Based on operations on sets and comparison of sets, students practically learn the important properties of equalities and inequalities (if a = b, then b = a). To compare two expressions means to compare their values. Comparison of numbers and expressions is first included when studying numbers within 20, and then when studying actions in all concentrations, these exercises are systematically offered to children.

When studying actions in other concentrations, exercises for comparing expressions become more complicated: expressions become more complex, students are offered tasks to insert a suitable number into one of the expressions so as to obtain true equalities or inequalities, to compose true equalities or true inequalities from these expressions.

Thus, when studying all concentrations, exercises for comparing numbers and expressions, on the one hand, contribute to the formation of concepts of equalities and inequalities, and on the other hand, the assimilation of knowledge about numbering and arithmetic operations, as well as the development of computational skills.

2.3 Technique for familiarization with inequalities with a variable

Inequalities with a variable of the form: x + 3< 7, 10 - х >5 are introduced in 3rd grade. First, the variable is denoted not by a letter, but by a "window", then it is denoted by a letter.

The terms "solve inequality", "solution of inequality" are not introduced in elementary grades, since in many cases they are limited to selecting only a few values of the variable, which results in the correct inequality. Exercises are performed under the guidance of a teacher.

Exercises with inequalities reinforce computational skills, and also help to assimilate arithmetic knowledge. Selecting the values of the letter in inequalities and equalities of the form: 5 + x = 5, 5 - x =5 10 * x=10, 10* x<10, учащиеся закрепляют знания особых случаев действий. Но самым важным является то, что работая с неравенствами, учащиеся закрепляют представление о переменной и подготавливаются к решению неравенств в 5 классе. В соответствии с программой в 1-4 классах рассматриваются упражнения первой степени с одним неизвестным вида: 7+х=10, х* (17 - 10)=70.

Exercises in elementary grades are considered as true equalities, the solution of the equation is reduced to finding the value of the letter (unknown number), in which the given expression has the specified value. Finding an unknown number in such equalities is based on the knowledge of the relationship between the result and the components of arithmetic operations. These requirements of the program determine the methodology for working on equations,

2.4 Methodology for studying equations

At the preparatory stage for the introduction of the first equations in the study of addition and subtraction within 10, students learn the relationship between the sum and the terms. In addition, by this time, children have mastered the ability to compare an expression and a number and get their first ideas about numerical equalities of the form: 8=5+3, 6+4=40. Of great importance in terms of preparing for the introduction of equations are exercises for selecting a missing number in equalities of the form: 4 + * = 6, 5- * = 2. In the process of performing such exercises, children get used to the idea that not only the sum or difference can be unknown, but also one of the components.

The concept of an equation is introduced in grade 3. The equations are solved orally, by the method of selection, i.e. children are offered simple equations of the form: x + 3 \u003d 5. To solve such equations, children remember the composition of numbers within 10, in this case, the composition of the number 5 (3 and 2), which means x = 2.

In grade 4, the teacher shows a record of solving an equation, based on the children's knowledge of the relationships between the components and the result of arithmetic operations. For example, 6+x=15. We do not know the second term. To obtain the second term, we must subtract the first term from the sum.

Solution record:

Examination:

Students need to explain that when we check, it is necessary, after substituting the resulting number instead of x, to find the value of the resulting expression.

Later, at the next stage, the equations are solved based on the knowledge of the rules for finding the unknown component.

There is a separate lesson for each case.

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The study of algebraic material in elementary school. Algebraic material in the initial course of mathematics

1.1. General questions of the methodology for studying algebraic material

1.2. Methodology for studying numerical expressions

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