Lecture number 2

mathematics

Topic: "Mathematical concepts"

Mathematical concepts

Definition of concepts

Definition requirements

Some kinds of definitions

1. Mathematical concepts

The concepts that are studied in the elementary course of mathematics are usually presented in the form of four groups. The first includes concepts related to numbers and operations on them: number, addition, summand, greater, etc. The second includes algebraic concepts: expression, equality, equation, etc. The third is made up of geometric concepts: line, segment, triangle, etc. etc. The fourth group consists of concepts related to quantities and their measurement.

How can one study such an abundance of very different concepts?

First of all, one must have an idea of ​​the concept as a logical category and the features of mathematical concepts.

In logic, concepts are considered as a form of thought, reflecting objects (objects or phenomena) in their essential and general properties. The linguistic form of a concept is a word or a group of words.

Composing an idea of ​​an object means being able to distinguish it from other objects similar to it. Mathematical concepts have a number of peculiarities. The main point is that the mathematical objects about which it is necessary to formulate a concept do not exist in reality. Mathematical objects are created by the human mind. These are ideal objects that reflect real objects or phenomena. For example, in geometry, the shape and size of objects is studied, without taking into account their other properties: color, mass, hardness, etc. They are distracted from all this, abstracted away. Therefore, in geometry, instead of the word "object" they say " geometric figure».

Abstraction results in such mathematical concepts as "number" and "magnitude".

In general, mathematical objects exist only in a person's thinking and in those signs and symbols that form a mathematical language.

To what has been said, we can add that, studying the spatial forms and quantitative relations of the material world, mathematics not only uses various methods of abstraction, but abstraction itself acts as a multi-stage process. In mathematics, they consider not only the concepts that appeared in the study of real objects, but also the concepts that arose on the basis of the former. For example, general concept function as a correspondence is a generalization of the concepts of specific functions, i.e. abstraction from abstractions.

To master the general approaches to the study of concepts in the elementary course of mathematics, the teacher needs knowledge about the scope and content of the concept, about the relationship between concepts and about the types of definitions of concepts.

2. The scope and content of the concept. Relationships between concepts

Every mathematical object has certain properties. For example, a square has four sides, four right angles equal to the diagonal. You can also specify its other properties.

Among the properties of an object, essential and insignificant are distinguished. A property is considered essential for an object if it is inherent in this object and without it it cannot exist. For example, for a square, all the properties mentioned above are essential. The property "side AD is horizontal" is irrelevant for the square ABCD. If you turn the square, then the AD side will be located in a different way (Fig. 26).

Therefore, in order to understand what a given mathematical object is, one must know its essential properties.

When they talk about a mathematical concept, they usually mean a set of objects denoted by one term (word or group of words). So, speaking of a square, they mean all geometric shapes that are squares. It is believed that the set of all squares is the volume of the concept "square".

Generally the scope of a concept is the set of all objects designated by one term.

Any concept has not only volume, but also content.

Consider, for example, the concept of "rectangle".

The scope of a concept is a set of different rectangles, and its content includes such properties of rectangles as "have four right angles", "have equal opposite sides", "have equal diagonals", etc.

There is a relationship between the volume of a concept and its content: if the volume of a concept increases, then its content decreases, and vice versa. So, for example, the scope of the concept of "square" is part of the scope of the concept of "rectangle", and the content of the concept of "square" contains more properties than the content of the concept of "rectangle" ("all sides are equal", "diagonals are mutually perpendicular", etc. ).

Any concept cannot be learned without realizing its relationship with other concepts. Therefore, it is important to know in what relationships concepts can be, and to be able to establish these connections.

The relationship between concepts is closely related to the relationship between their volumes, i.e. sets.

Let us agree to denote concepts with lowercase letters of the Latin alphabet: a, b, c, ..., z.

Let two concepts a and b be given. Their volumes will be denoted by A and B, respectively.

If A B (A ≠ B), then they say that the concept a - specific in relation to the conceptb, and the concept b- generic in relation to the concept a.

For example, if a is a "rectangle", b is a "quadrangle", then their volumes A and B are in relation to inclusion (A B and A ≠ B), since every rectangle is a quadrangle. Therefore, it can be argued that the concept of "rectangle" is specific in relation to the concept of "quadrangle", and the concept of "quadrangle" is generic in relation to the concept of "rectangle".

If A = B, then they say that concepts a andbare identical.

For example, the concepts of "equilateral triangle" and "equilateral triangle" are identical, since their volumes coincide.

If the sets A and B are not connected by the inclusion relation, then they say that the concepts a and b are not in relation to genus and species and are not identical. For example, the concepts "triangle" and "rectangle" are not related by such relations.

Let us consider in more detail the relationship of genus and species between concepts. First, the concepts of genus and species are relative: the same concept can be generic in relation to one concept and specific in relation to another. For example, the concept of "rectangle" is generic in relation to the concept of "square" and specific in relation to the concept of "quadrangle".

Secondly, for of this concept often several generic concepts can be specified. So, for the concept of "rectangle" generic are the concepts of "quadrangle", "parallelogram", "polygon". Among them, you can indicate the closest one. For the concept of "rectangle" the closest is the concept of "parallelogram".

Third, a specific concept has all the properties of a generic concept. For example, a square, being a specific concept in relation to the concept of "rectangle", has all the properties inherent in a rectangle.

Since the scope of a concept is a set, it is convenient, when establishing relations between the scope of concepts, to depict them using Euler's circles.

Let us establish, for example, the relationship between the following pairs of concepts a and b, if:

1) a - "rectangle", b - "rhombus";

2) a - "polygon", b - "parallelogram";

3) a - "straight line", b - "segment".

In case 1) the volumes of concepts intersect, but not one set is not a subset of another (Fig. 27).

Therefore, it can be argued that these concepts a and b are not in relation to the genus and species.

In case 2) the data volumes of the concept are in relation to inclusion, but do not coincide - every parallelogram is a polygon, but not vice versa (Fig. 28). Therefore, it can be argued that the concept of "parallelogram" is specific in relation to the concept of "polygon", and the concept of "polygon" is generic in relation to the concept of "parallelogram".

In case 3) the volumes of concepts do not intersect, since no segment can be said to be a straight line, and no straight line can be called a segment (Fig. 29).

Consequently, these concepts are not in relation to the genus and species.

About the concepts of "line" and "segment" we can say that they are in relation to the whole and the part: a segment is a part of a straight line, not its kind. And if a specific concept has all the properties of a generic concept, then a part does not necessarily have all the properties of the whole. For example, a segment does not have the same property of a straight line as its infinity.

Formation of elementary mathematical concepts of a younger student

E.Yu. Togobetskaya, Master's student of the Department of Pedagogy and Teaching Methods

Togliatti Pedagogical University, Togliatti (Russia)

Keywords: mathematical concepts, absolute concepts, relative concepts, definitions.

Annotation: In school practice, many teachers require students to memorize the definitions of concepts and require knowledge of their basic properties to be proved. However, the results of such training are usually insignificant. This is because the majority of students, applying the concepts learned in school, rely on insignificant signs, while the essential features of the concepts are recognized and reproduced by the students only when answering questions that require the definition of the concept. Often, students accurately reproduce concepts, that is, they discover knowledge of its essential features, but they cannot apply this knowledge in practice, they rely on those random features identified through direct experience. The process of assimilating concepts can be controlled, formed with the given qualities.

Keywords: mathematical concepts, absolute concepts, relative concepts, definitions.

Annotation: In school practice many teachers achieve from pupils of learning of definitions of concepts and the knowledge of their basic proved properties demands. However results of such training are usually insignificant. It occurs because the majority of pupils, applying the concepts acquired at school, pupils lean against the unimportant signs, essential signs of concepts realise and reproduce only at the answer to the questions demanding definition of concept. Often pupils unmistakably reproduce concepts, that is find out knowledge of its essential signs, but put this knowledge into practice cannot, lean against those casual signs allocated thanks to a first-hand experience. Process of mastering of concepts it is possible to operate, form them with the set qualities.

In the assimilation of scientific knowledge, primary school students are faced with different types of concepts. The inability of the student to differentiate concepts leads to inadequate assimilation of them.

Logic in terms distinguishes between volume and content. The volume is understood as the class of objects that belong to this concept, are united by it. So, the scope of the concept of a triangle includes the entire set of triangles, regardless of their specific characteristics (types of angles, size of sides, etc.).

The content of concepts is understood as the system of essential properties, according to which these objects are combined into a single class. To reveal the content of a concept, it is necessary to establish by comparison what signs are necessary and sufficient to highlight its relationship to other objects. Until the content and signs are established, the essence of the object reflected by this concept is not clear, it is impossible to accurately and clearly distinguish this object from those adjacent to it, confusion of thinking occurs.

For example, the concept of a triangle, such properties include the following: a closed figure, consists of three line segments. The set of properties by which objects are combined into a single class are called necessary and sufficient features. In some concepts, these features complement each other, forming together the content, according to which objects are combined into a single class. Examples of such concepts are triangle, angle, bisector, and many others.

The collection of these objects, to which this concept applies, constitutes a logical class of objects. A logical class of objects is a collection of objects that have common features, as a result of which they are expressed by a general concept. The logical class of objects and the scope of the corresponding concept coincide. The concepts are divided into types in terms of content and volume, depending on the nature and number of objects to which they apply. In terms of volume, mathematical concepts are divided into singular and general ones. If the scope of a concept includes only one object, it is called single.

Examples of single concepts: "the smallest two-digit number", "number 5", "a square with a side length of 10 cm", "a circle with a radius of 5 cm". The general concept reflects the signs of a certain set of objects. The volume of such concepts will always be greater than the volume of one element. Examples of common concepts: "many two-digit numbers", "triangles", "equations", "inequalities", "multiples of 5", "mathematics textbooks for elementary school." In terms of content, they distinguish between the concepts of conjunctive and disjunctive, absolute and specific, non-relative and relative.

Concepts are called conjunctive if their features are interrelated and separately none of them allows you to identify objects of this class, features are linked by the union "and". For example, objects related to the concept of a triangle must necessarily consist of three line segments and be closed.

In other concepts, the relationship between necessary and sufficient features is different: they do not complement each other, but replace. This means that one feature is equivalent to another. An example of this type of relationship between signs can serve as signs of equality of segments, angles. It is known that the class of equal segments includes such segments that: a) or coincide when superimposed; b) or separately equal to the third; c) or consist of equal parts, etc.

In this case, the listed features are not required all at the same time, as is the case with the conjunctive type of concepts; here it is enough to have one of all the listed ones: each of them is equivalent to any of the others. Because of this, the signs are linked by the conjunction "or". Such a connection of features is called disjunction, and concepts, respectively, are called disjunctive. It is also important to take into account the division of concepts into absolute and relative.

Absolute concepts unite objects into classes according to certain characteristics that characterize the essence of these objects as such. So, the concept of an angle reflects the properties that characterize the essence of any angle as such. The situation is similar with many other geometric concepts: circle, ray, rhombus, etc.

Relative concepts combine objects into classes according to properties that characterize their relationship to other objects. So, in the concept of perpendicular straight lines, what characterizes the ratio of two straight lines to each other is fixed: the intersection, the formation at the same time right angle... Similarly, the concept of number reflects the ratio of the measured value and the accepted standard. Relative concepts cause more serious difficulties for students than absolute concepts. The essence of the difficulties lies precisely in the fact that schoolchildren do not take into account the relativity of concepts and operate with them as with absolute concepts. So, when the teacher asks students to draw a perpendicular, then some of them depict a vertical. Particular attention should be paid to the concept of number.

The number is the ratio of what is being quantified (length, weight, volume, etc.) to the standard that is used for this assessment. Obviously, the number depends on both the measured value and the standard. The larger the measured value, the larger the number will be with the same standard. On the contrary, the larger the standard (measure) is, the smaller the number will be when evaluating the same value. Therefore, students should understand from the very beginning that comparison of numbers in magnitude can only be made when the same standard is behind them. Indeed, if, for example, five is obtained when measuring length in centimeters, and three when measuring in meters, then three denote a greater value than five. If students do not grasp the relative nature of number, then they will experience serious difficulties in learning the number system. Difficulties in assimilating relative concepts persist among students in middle and even in high school. There is a relationship between the content and the volume of a concept: the smaller the volume of the concept, the greater its content.

For example, the concept of "square" has a smaller volume than the volume of the concept of "rectangle" since any square is a rectangle, but not every rectangle is a square. Therefore, the concept of "square" has more content than the concept of "rectangle": a square has all the properties of a rectangle and some others (all sides of a square are equal, the diagonals are mutually perpendicular).

In the process of thinking, each concept does not exist separately, but enters into certain connections and relationships with other concepts. In mathematics, an important form of communication is generic dependence.

For example, consider the concepts of "square" and "rectangle". The scope of the concept "square" is a part of the scope of the concept "rectangle". Therefore, the first is called specific, and the second - generic. In genus-specific relations, the concept of the closest genus and the following generic stages should be distinguished.

For example, for the type "square" the closest genus will be the genus "rectangle", for the rectangle the closest genus will be the genus "parallelogram", for the "parallelogram" - "quadrangle", for "quadrilateral" - "polygon", and for "polygon" - " flat figure ".

V primary grades for the first time, each concept is introduced visually, by observing specific objects or by practical operation (for example, when counting them). The teacher relies on the knowledge and experience of children that they acquired before school age... Acquaintance with mathematical concepts is recorded using a term or term and a symbol. This technique of working on mathematical concepts in primary school does not mean that different kinds of definitions are not used in this course.

To define a concept is to list all the essential features of objects that are included in this concept. The verbal definition of a concept is called a term. For example, “number”, “triangle”, “circle”, “equation” are terms.

The definition solves two problems: it singles out and dissociates a certain concept from all others and indicates those main features, without which the concept cannot exist and on which all other features depend.

The definition can be more or less deep. It depends on the level of knowledge about the concept that is meant. The better we know it, the more likely it is that we can better define it. In the practice of teaching younger students, explicit and implicit definitions are used. Explicit definitions take the form of equality or coincidence of two concepts.

For example: "Propedeutics is an introduction to any science." Here they equate one to one two concepts - "propaedeutics" and "entry into any science." In the definition “A square is a rectangle in which all sides are equal” we have a coincidence of concepts. In teaching younger schoolchildren, contextual and ostensive definitions are of particular interest among implicit definitions.

Any passage from the text, be it any context in which the concept that interests us occurs, is, in some sense, an implicit definition of it. The context puts a concept in connection with other concepts and thus reveals its content.

For example, using expressions such as “find the values ​​of an expression” in working with children, “compare the value of expressions 5 + a and (a - 3) 2, if a = 7”, “read expressions that are sums”, “read expressions , and then read the equations ", we reveal the concept of" mathematical expression "as a record that is made up of numbers or variables and signs of actions. Almost all definitions that we meet in Everyday life are contextual definitions. Having heard an unknown word, we try to establish its meaning ourselves on the basis of everything that has been said. The same is the case in the teaching of younger schoolchildren. Many math concepts in elementary school are defined through context. These are, for example, such concepts as "big - small", "any", "any", "one", "many", "number", "arithmetic operation", "equation", "task" and etc.

Contextual definitions remain for the most part incomplete and incomplete. They are used in connection with the unpreparedness of the younger student for the assimilation of a complete, and even more so scientific, definition.

Ostensive definitions are definitions by demonstration. They resemble ordinary contextual definitions, but the context here is not a passage of any text, but the situation in which the object designated by the concept finds itself. For example, the teacher shows a square (drawing or paper model) and says "Look - this is a square." This is a typical ostensive definition.

In elementary grades, ostensive definitions are used when considering concepts such as "red (white, black, etc.) color", "left - right", "left to right", "digit", "previous and next number", "signs arithmetic operations "," comparison signs "," triangle "," quadrilateral "," cube ", etc.

Based on the ostensive assimilation of the meanings of words, it is possible to introduce into the child's dictionary the verbal meaning of new words and phrases. Ostensive definitions - and only they - connect the word with things. Without them, language is just a verbal lace, which has no objective, objective content. Note that in elementary grades, acceptable definitions like "The word" pentagon "we will call a polygon with five sides." This is the so-called "nominal definition". Different explicit definitions are used in mathematics. The most common of these is the definition through the closest genus and species trait. The generic definition is also called classic.

Examples of definitions through genus and species: "A parallelogram is a quadrangle whose opposite sides are parallel", "A diamond is a parallelogram whose sides are equal", "A rectangle is a parallelogram whose angles are straight", equal "," A rhombus is called a square, which has right angles. "

Consider the definitions of a square. In the first definition, the closest genus is "rectangle", and the species feature is "all sides are equal." In the second definition, the closest genus is "rhombus", and the species character is "right angles". If we take not the closest genus ("parallelogram"), then there will be two species characteristics of a square. "A square is a parallelogram in which all sides are equal and all angles are straight."

In the generic relation are the concepts of "addition (subtraction, multiplication, division)" and "arithmetic operation", the concept of "acute (straight, obtuse) angle" and "angle". There are not so many examples of explicit generic relations among the many mathematical concepts that are considered in primary grades. But taking into account the importance of the definition through genus and species trait in further education, it is advisable to achieve understanding by the students of the essence of the definition of this species already in the elementary grades.

Separate definitions can consider the concept and by the way of its formation or occurrence. This type of definition is called genetic. Examples of genetic definitions: "An angle is rays that come out from one point", "A diagonal of a rectangle is a segment that connects opposite vertices of a rectangle." In the elementary grades, genetic definitions are used for concepts such as "segment", "broken line", "right angle", "circle". Definition through a list can also be attributed to genetic concepts.

For example, "The natural series of numbers are the numbers 1, 2, 3, 4, etc.". Some concepts in elementary grades are introduced only through the term. For example, the units of time are year, month, hour, minute. There are concepts in the elementary grades that are presented in symbolic language in the form of equality, for example, a 1 = a, and 0 = 0

From the above, we can conclude that in the elementary grades, many mathematical concepts are first mastered superficially, vaguely. At the first acquaintance, schoolchildren learn only about some properties of concepts, very narrowly represent their scope. And this is natural. Not all concepts are easy to learn. But it is indisputable that the teacher's understanding and timely use of certain types of definitions of mathematical concepts is one of the conditions for the formation of solid knowledge of these concepts in students.

Bibliography:

1. Bogdanovich M.V. Definition of mathematical concepts // Elementary school 2001. - № 4.

2. Gluzman N. A. Formation of generalized methods of mental activity in younger students. - Yalta: KGGI, 2001 .-- 34 p.

3. Drozd V.L. Urban M.A. From small problems to big discoveries. //Primary School. - 2000. - No. 5.

Ministry of Education of the Republic of Belarus

"Gomel State University them. F. Skaryna "

Faculty of Mathematics

Department of MPM

abstract

Mathematical concepts

Executor:

Student of group M-32

Molodtsova A.Yu.

Supervisor:

Cand. phys-mat. Sciences, Associate Professor

Lebedeva M.T.

Gomel 2007

Introduction

The formulations of many definitions (theorems, axioms) are clear to students, they are easily memorized after a small number of repetitions, so it is advisable to first offer them to remember, and then teach them how to apply them to solving problems.

separate.

1. Scope and content of the concept. Classification of concepts

Objects of reality have: a) uniform properties that express its distinctive properties (for example, an equation of the third degree with one variable - a cubic equation); b) general properties, which can be distinctive if they express the essential properties of an object (its features), distinguishing it from many other objects.

The term "concept" is used to denote a mental image of a certain class of objects, processes. Psychologists distinguish three forms of thinking:

1) concepts (for example, median is a segment connecting a vertex with the opposite side of a triangle);

2) judgments (for example, for the angles of an arbitrary triangle it is true :);

3) inferences (for example, if a> b and b> c, then a> c).

Characteristic of forms of thinking in concepts are: a) it is a product of highly organized matter; b) reflects the material world; c) appears in cognition as a means of generalization; d) means specifically human activity; e) its formation in consciousness is inseparable from its expression through speech, writing or symbol.

A mathematical concept reflects in our thinking certain forms and relations of reality, abstracted from real situations. Their formation occurs according to the scheme:

Each concept unites many objects or relationships, called the scope of the concept, and characteristic properties inherent to all elements of this set and only to them, expressing the content of the concept.

For example, a mathematical concept is a quadrangle. His volume: square, rectangle, parallelogram, rhombus, trapezoid, etc. Content: 4 sides, 4 corners, 4 vertices (characteristic properties).

The content of a concept rigidly determines its scope and, conversely, the scope of a concept completely determines its content. The transition from the sensory stage to the logical one occurs through generalizations: either by highlighting the general features of the object (parallelogram - quadrilateral - polygon); either through general features in combination with special or individual ones, which leads to a specific concept.

In the process of generalization, the volume expands, and the content narrows. In the process of specialization of the concept, the volume narrows, and the content expands.

For example:

polygons - parallelograms;

triangles are equilateral triangles.

If the volume of one concept is contained in the volume of another concept, then the second concept is called generic, in relation to the first; and the first is called specific in relation to the second. For example: parallelogram - rhombus (genus) (view).

The process of clarifying the scope of a concept is called classification, the scheme of which looks like this:

let a set and some property be given, and let there be elements in it, both possessing and not possessing this property. Let be:

Let's select a new property and split it by this property:

For example: 1) classification of numerical sets reflecting the development of the concept of number; 2) classification of triangles: a) by sides; b) in the corners.

Problem number 1. Draw a lot of triangles using the points of a square.

The property of isosceles;

Squareness property;

Are there triangles with these properties at the same time?

2. Mathematical definitions. Types of concept definition errors

The final stage in the formation of a concept is its definition, i.e. acceptance of a conditional agreement. A definition is understood as a listing of the necessary and sufficient features of a concept, reduced to a coherent sentence (speech or symbolic).

2.1 Methods for defining concepts

Initially, undefined concepts are distinguished, on the basis of which mathematical concepts are determined in the following ways:

1) through the closest genus and species difference: a) descriptive(clarifying the process by which the definition is built, or describing the internal structure, depending on the operations by which this definition was built from undefined concepts); b) constructive(or genetic) indicating the origin of the concept.

For example: a) a rectangle is a parallelogram with all angles straight; b) a circle is a figure that consists of all points of the plane equidistant from a given point. This point is called the center of the circle.

2) inductively. For example, the definition of an arithmetic progression:

3) through abstraction... For example, a natural number is a characteristic of classes of equivalent finite sets;

4) axiomatic (indirect definition)... For example, determining the area of ​​a figure in geometry: for simple figures, area is a positive value, numerical value which has the following properties: a) equal figures have equal areas; b) if the figure is divided into parts that are simple figures, then the area of ​​this figure is equal to the sum of the areas of its parts; c) the area of ​​a square with a side equal to the unit of measurement is equal to one.

2.2 Explicit and Implicit Definitions

Definitions are subdivided into:

a) explicit, in which the defined and defining concepts are clearly identified (for example, definition through the closest genus and species difference);

b) implicit, which are built on the principle of replacing one concept with another with a wider scope and the end of the chain is an undefined concept, i.e. formal logical definition (for example, a square is a rhombus with a right angle; a rhombus is a parallelogram with equal adjacent sides; a parallelogram is a quadrilateral, with pairwise parallel sides; a quadrilateral is a figure consisting of 4 corners, 4 vertices, 4 sides). V school definitions the first method is most often practiced, the scheme of which is as follows: we have sets and some property then

The main requirement when constructing definitions: the set being defined must be a subset of the minimum set. For example, let's compare two definitions: (1) A square is a rhombus with a right angle; (2) A square is a parallelogram with equal sides and a right angle (redundant).

Any definition is a solution to the problem of “proof of existence”. For example, a right-angled triangle is a right-angled triangle; its existence is construction.

2.3 Characteristics of the main types of errors

Note typical mistakes that are encountered by students when defining concepts:

1) the use of a non-minimal set as defining, the inclusion of logically dependent properties (typical when repeating material).

For example: a) a parallelogram is a quadrangle, in which the opposite sides are equal and parallel; b) a straight line is called perpendicular to the plane if it, intersecting with this plane, forms a right angle with each straight line drawn on the plane through the point of intersection, instead of: “a straight line is called perpendicular to the plane if it is perpendicular to all lines of this plane”;

2) the use of a defined concept and as a defining one.

For example, a right angle is defined not as one of equal adjacent angles, but as angles with mutually perpendicular sides;

3) tautology - a concept is defined through this concept itself.

For example, two figures are called similar if they are translated into one another by a similarity transformation;

4) sometimes the definition indicates the wrong defining set from which the defined subset is allocated.

For example, “the median is a straight line ...” instead of “the median is a segment connecting…”;

5)in the definitions given by students, sometimes the defined concept is completely absent, which is possible only when students are not trained to give complete answers.

The method of correcting errors in definitions involves, initially, clarifying the essence of the mistakes made, and then preventing their repetition.

3. Definition structure

1) Conjunctive structure: two points and are called symmetric with respect to the line p ( A(x)) if this line p is perpendicular to the segment and passes through its midpoint. We will also assume that each point of the line p is symmetric to itself with respect to the straight line p (the presence of the union “and”) (* - “The bisector of an angle is a ray that emanates from its vertex, passes between its sides and divides the angle in half”).

2)Constructive structure: “Let is a given figure and p is a fixed line. Take an arbitrary point of the figure and drop the perpendicular to the line p. On the continuation of the perpendicular beyond the point, set aside a segment equal to the segment. The transformation of a figure into a figure, in which each point goes to a point constructed in this way, is called symmetry about the straight line p. "

3) Disjunctive structure: set definition Z integers can be written in the language of properties as Z N or N or = 0, where N - set of numbers opposite to natural numbers.

4. Characteristics of the main stages in the study of mathematical concepts

The methodology for working on a definition assumes: 1) knowledge of the definition; 2) learning to recognize an object corresponding to this definition; 3) construction of various counterexamples. For example, the concept of "right-angled triangle" and the work on recognizing its constituent elements:

The study of mathematical definitions can be divided into three stages:

Stage 1 - introduction - creating a situation in the lesson where students either “discover” new things themselves, form definitions for them on their own, or simply prepare to understand them.

The 2nd stage - ensuring assimilation - boils down to ensuring that students:

a) learned to apply the definition;

b) quickly and accurately memorize them;

c) understood every word in their wording.

The third stage - consolidation - is carried out in subsequent lessons and comes down to repeating their formulations and processing the skills of applying them to solving problems.

Acquaintance with new concepts is carried out:

Method 1: Students prepare for independent formation of the definition.

Method 2: students prepare for conscious perception, understanding of a new mathematical sentence, the formulation of which is then communicated to them in a finished form.

Method 3: the teacher himself formulates a new definition without any preparation, and then focuses the efforts of students on their assimilation and consolidation.

1 and 2 represent the heuristic method, 3 the dogmatic. The use of any of the methods should be appropriate to the level of preparation of the class and the experience of the teacher.

5. Characteristics of the methods of introducing concepts

The following techniques are possible when introducing concepts:

1) Exercises can be designed to enable students to quickly formulate a definition of a new concept.

For example: a) Write out the first few members of the sequence (), in which = 2,. This sequence is called a geometric progression. Try to formulate its definition. You can limit yourself to preparing for the perception of a new concept.

b) Write out the first few members of the sequence (), in which = 4, Then the teacher informs that such a sequence is called an arithmetic progression and he himself reports its definition.

2) in the study of geometric concepts, exercises are formulated in such a way that the students themselves build the necessary figure and are able to highlight the signs of a new concept that are necessary to formulate a definition.

For example: construct an arbitrary triangle, connect its vertex with a segment to the middle of the opposite side. This segment is called the median. Formulate the definition of the median.

Sometimes it is proposed to draw up a model or, considering ready-made models and drawings, highlight the signs of a new concept and formulate its definition.

For example: the definition of a parallelepiped was introduced in grade 10. Using the proposed models of oblique, straight and rectangular parallelepipeds, identify the features by which these concepts differ. Formulate the appropriate definitions of straight and rectangular parallelepipeds.

3) Many algebraic concepts are introduced based on the consideration of particular examples.

For example: the graph of a linear function is a straight line.

4)The method of expedient tasks,(developed by S.I.Shokhor-Trotsky) With the help of a specially selected problem, students come to the conclusion that it is necessary to introduce a new concept and the expediency of giving it exactly the meaning that it already has in mathematics.

In grades 5-6, this method introduces the following concepts: equation, root of an equation, solution of inequalities, the concept of addition, subtraction, multiplication, division over natural numbers, decimal and common fractions, etc.

Specific inductive method

Essence:

a) specific examples are considered;

b) essential properties are highlighted;

c) a definition is formulated;

d) exercises are performed: for recognition; for construction;

e) work on properties not included in the definition;

f) application of properties.

For example: theme - parallelograms:

1, 3, 5 - parallelograms.

b) essential features: quadrilateral, pairwise parallelism of the sides.

c) recognition, construction:

d) find (build) the fourth vertex of the parallelogram (* - problem no. 3, article 96, Geometry grade 7-11: How many parallelograms can be built with vertices in three given points not lying on one straight line? Build them.).

e) other properties:

AC and BD intersect at point O and AO = OC, BO = OD; AB = CD, AD = BC.

f) A = C, B = D.

Consolidation: solving problems No. 4-23, p. 96-97, Geometry 7-11, Pogorelov.

Prospective value:

a) is used in the study and definition of a rectangle and a rhombus;

b) the principle of parallelism and equality of the segments enclosed between parallel lines in Thales' theorem;

c) the concept of parallel transfer (vector);

d) the parallelogram property is used when displaying the area of ​​a triangle;

e) parallelism and perpendicularity in space; parallelepiped; prism.

Abstract-deductive method

Essence:

a) definition of the concept: - quadratic equation;

b) highlighting essential properties: x - variable; a, b, c - numbers; a? 0 at

c) concretization of the concept: - given; examples of equations

d) exercises: for recognition, for construction;

e) study of properties not included in the definition: the roots of the equation and their properties;

f) problem solving.

At school, the abstract-deductive method is used when a new concept is fully prepared by the study of previous concepts, including the study of the closest generic concept, and the specific difference of the new concept is very simple and understandable to students.

For example: identifying a rhombus after examining a parallelogram.

In addition, the specified method is used:

1) when compiling a “pedigree” definition of the concept:

A square is a rectangle with all sides equal.

A rectangle is a parallelogram with all angles straight.

A parallelogram is a quadrilateral whose opposite sides are parallel.

A quadrilateral is a figure that consists of four points and four line segments connecting them in series.

In other words, a pedigree is a chain of concepts built through generalizations of the previous concept, the end of which is an undefined concept (recall that in the course of school geometry these include a point, figure, plane, distance (to lie between));

2) classification;

3) applies to theorem proving and problem solving;

4) is widely used in the process of updating knowledge.

Consider this process, represented by a system of tasks:

a) Given a right-angled triangle with sides 3cm and 4cm. Find the length of the median drawn to the hypotenuse.

b) Prove that the median drawn from the vertex of the right angle of the triangle is equal to half the hypotenuse.

c) Prove that in a right-angled triangle the bisector of the right angle bisects the angle between the median and the height drawn to the hypotenuse.

d) On the continuation of the largest side AC of the triangle ABC, the segment CM is plotted, which is equal to the side BC. Prove that AVM is dumb.

In most cases, a specific inductive method is used in school teaching. In particular, this method introduces concepts in the propaedeutic cycles of the beginnings of algebra and geometry in grades 1-6, and many defining concepts are introduced descriptively, without strict formulations.

The teacher's ignorance of various methods of introducing definitions leads to formalism, which manifests itself as follows:

a) students find it difficult to apply the definition in an unusual situation, although they remember its formulation.

For example: 1) consider the function - even, because “Cos” - even;

2) - they do not understand the connection between the monotonicity of the function and the solution of the inequality, i.e. cannot apply the corresponding definitions, in which the main research method is to assess the sign of the difference in the values ​​of the function, i.e. in solving inequality.

b) students have the skills to solve problems of any type, but cannot explain on the basis of what definitions, axioms, theorems they perform certain transformations.

For example: 1) - transform according to this formula and 2) imagine that there is a model of a quadrangular pyramid on the table. What polygon will be the base of this pyramid if the model is placed on the table with its side face? (quadrilateral).

The process of forming knowledge, skills and abilities is not limited to the communication of new knowledge.

This knowledge must be learned and consolidated.

6. Methods of ensuring the assimilation of mathematical concepts (sentences)

1. The formulations of many definitions (theorems, axioms) are clear to students, they are easily memorized after a small number of repetitions, so it is advisable at the beginning to offer them to be remembered, and then to teach them how to apply them to solving problems.

The method in which the processes of memorizing definitions and the formation of skills in their application occur in students at different times (separately) is called separate.

The separate method is used to study the definitions of a chord, trapezoid, even and odd function, Pythagorean theorems, signs of parallel lines, Vieta's theorem, properties of numerical inequalities, rules for multiplying ordinary fractions, adding fractions with the same denominator, etc.

Methodology:

a) the teacher formulates a new definition;

b) students in the class repeat it 1-3 times for memorization;

c) is practiced in exercises.

2. Compact method consists in the fact that students read in parts a mathematical definition or a sentence and, in the course of reading, simultaneously perform the exercise.

Reading the wording several times, they memorize it along the way.

Methodology:

a) preparation of a mathematical proposal for use. The definition is divided into parts according to features, the theorem - into a condition and a conclusion;

b) a sample of actions proposed by the teacher, which shows how to work with the prepared text: we read it in parts and at the same time perform the exercises;

c) students read the definition in parts and at the same time perform the exercises, guided by the prepared text and the teacher's model;

For example: the definition of a bisector in the fifth grade:

1) the concept is introduced by the method of expedient problems on the angle model;

2) the definition is written out: “A ray emerging from the top of the angle and dividing it into two equal parts is called the bisector of the angle”;

3) the task is being performed: indicate which of the lines in the drawings are the bisectors of the corners (equal angles are denoted by the same number of arcs).

In one of the drawings, the teacher shows the application of the definition (see below);

4) the work is continued by the students.

3. Combination of split and compact method : after the conclusion of a new rule, it is repeated 2-3 times, and then the teacher requires in the process of performing the exercises to formulate the rule in parts.

4. Algorithmic method used to develop skills in the application of mathematical sentences.

Methodology: mathematical sentences are replaced by an algorithm. By reading the instructions of the algorithm one by one, the student solves the problem. Thus, he develops the skill of applying a definition, an axiom and a theorem. In this case, either the subsequent memorization of the definition is allowed, or the reading, together with the algorithm, of the definition itself.

The main stages of the method:

a) preparing a list of instructions for work, which is either given ready-made, with subsequent clarification, or students are led to compile it on their own;

b) a sample of the teacher's answer;

c) students work in a similar way.

Separate and compact methods are used to study definitions. Algorithmic can be applied only when studying hard-to-digest definitions (for example, necessary and sufficient conditions). The most widely algorithmic method is used in the formation of problem solving skills.

7. The technique of consolidating mathematical concepts and sentences

1st appointment:

the teacher proposes to formulate and apply certain definitions, axioms, theorems that are encountered in the course of solving problems.

For example: build a graph of a function; definition of an even (odd) function; necessary and sufficient condition for existence.

2nd reception:

the teacher proposes to formulate a number of definitions, theorems, axioms during a frontal survey, in order to repeat them and at the same time check whether their students remember them. This technique is not effective outside of problem solving. It is possible to combine frontal polling with special exercises that require students to be able to apply definitions, theorems, axioms in different situations, the ability to quickly navigate in the condition of the problem.

Conclusion

Knowledge of the definition does not guarantee the assimilation of the concept. Methodological work with concepts should be aimed at overcoming formalism, which manifests itself in the fact that students cannot recognize the defined object in various situations where it occurs.

Recognition of an object corresponding to this definition and the construction of counterexamples is possible only with a clear understanding of the structures of the definition under consideration, by which in the definition scheme () is understood the structure of the right-hand side.

Literature

1. K.O. Ananchenko " General methodology teaching mathematics at school ", Minsk," Universitetskae ", 1997

2. N.M. Roganovsky "Teaching methodology in high school", Pl.," graduate School", 1990

3. G. Freudenthal “Mathematics as pedagogical task", M.," Education ", 1998

4. N.N. "Mathematical laboratory", M., "Education", 1997

5. Yu.M. Kolyagin "Methods of teaching mathematics in secondary school", M., "Education", 1999

6. A.A. Joiner "Logical problems of teaching mathematics", Minsk., "Higher school", 2000

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Features of studying mathematics in primary school according to the Federal State Educational Standard of Primary General Education. Course content. Analysis of basic mathematical concepts. The essence of an individual approach in didactics.

Lecture 5. Mathematical concepts

1. The scope and content of the concept. Relationships between concepts

2. Definition of concepts. Defined and undefined concepts.

3. Ways to define concepts.

4. Key findings

The concepts that are studied in the elementary mathematics course are usually presented in the form of four groups. The first includes concepts related to numbers and operations on them: number, addition, summand, greater, etc. The second includes algebraic concepts: expression, equality, equations, etc. The third group is made up of geometric concepts: line, segment, triangle, etc. .d. The fourth group consists of concepts related to quantities and their measurement.

To study all the variety of concepts, you need to have an idea of ​​the concept as a logical category and the features of mathematical concepts.

In logic concepts viewed as thought form reflecting objects (objects and phenomena) in their essential and general properties Oh. The linguistic form of the concept is word (term) or group of words.

To compose an idea of ​​an object - means to be able to distinguish it from other objects similar to it. Mathematical concepts have a number of peculiarities. The main point is that mathematical objects, about which it is extremely important to form a concept, do not exist in reality. Mathematical objects are created by the human mind. These are ideal objects that reflect real objects or phenomena. For example, in geometry, the shape and size of objects is studied, without taking into account other properties: color, mass, hardness, etc. All this is abstracted. For this reason, in geometry, instead of the word "object" they say "geometric figure".

Abstraction results in such mathematical concepts as "number" and "magnitude".

In general, mathematical objects exist only in a person's thinking and in those signs and symbols that form a mathematical language.

To what has been said, we can add that, studying spatial forms and quantitative relations of the material world, mathematics not only uses various methods of abstraction, but abstraction itself acts as a multi-stage process. In mathematics, they consider not only the concepts that appeared in the study of real objects, but also the concepts that arose on the basis of the former. For example, the general concept of a function as a correspondence is a generalization of the concepts of specific functions, ᴛ.ᴇ. abstraction from abstractions.

1. The scope and content of the concept. Relationships between concepts

Any mathematical object has certain properties. For example, a square has four sides, four right angles equal to the diagonal. You can also specify its other properties.

Among the properties of the object are significant and insignificant... Property consider essential for an object if it is inherent in this object and without it it cannot exist... For example, for a square, all the properties mentioned above are essential. The property "side AB is horizontal" is not essential for the square ABCD.

When they talk about a mathematical concept, they usually mean a set of objects, denoted by one term(by a word or group of words). So, speaking of a square, they mean all geometric shapes that are squares. It is believed that the set of all squares is the volume of the concept "square".

Generally, the scope of the concept - ϶ᴛᴏ the set of all objects designated by one term.

Any concept has not only volume, but also content.

Consider, for example, the concept of "rectangle".

The scope of the concept is ϶ᴛᴏ a set of different rectangles, and its content includes such properties of rectangles as “have four right angles”, “have equal opposite sides”, “have equal diagonals”, etc.

Between the scope of the concept and its content there is relationship: if the volume of a concept increases, then its content decreases, and vice versa... So, for example, the scope of the concept of "square" is part of the scope of the concept of "rectangle", and the content of the concept of "square" contains more properties than the content of the concept of "rectangle" ("all sides are equal", "diagonals are mutually perpendicular" and etc.).

Any concept cannot be learned without realizing its relationship with other concepts. For this reason, it is important to know in what relationships concepts can be, and to be able to establish these relationships.

The relationship between concepts is closely related to the relationship between their volumes, ᴛ.ᴇ. sets.

Let us agree to denote concepts with lowercase letters of the Latin alphabet: a, b, c, d,…, z.

Let two concepts a and b be given. Their volumes will be denoted by A and B, respectively.

If A ⊂ B (A ≠ B), then they say that the concept a is specific with respect to the concept b, and the concept b is generic with respect to the concept a.

For example, if a is a "rectangle", b is a "quadrangle", then their volumes A and B are in the relation of inclusion (A ⊂ B and A ≠ B), in this regard, any rectangle is a quadrangle. For this reason, it can be argued that the concept of "rectangle" is specific in relation to the concept of "quadrangle", and the concept of "quadrangle" is generic in relation to the concept of "rectangle".

If A = B, then they say that the concepts A and B are identical.

For example, the concepts of "equilateral triangle" and "isosceles triangle" are identical, since their volumes coincide.

Let us consider in more detail the relationship of genus and species between concepts.

1. First of all, the concepts of genus and species are relative: the same concept can be generic in relation to one concept and specific in relation to another. For example, the concept of "rectangle" is generic in relation to the concept of "square" and specific in relation to the concept of "quadrangle".

2. Secondly, for a given concept, it is often possible to indicate several generic concepts. So, for the concept of "rectangle" generic are the concepts of "quadrangle", "parallelogram", "polygon". Among the indicated, you can indicate the closest. For the concept of "rectangle" the closest is the concept of "parallelogram".

3. Third, a specific concept has all the properties of a generic concept. For example, a square, being a specific concept in relation to the concept of "rectangle", has all the properties inherent in a rectangle.

Since the scope of a concept is a set, it is convenient, when establishing relations between the scope of concepts, to depict them using Euler's circles.

Let us establish, for example, the relationship between the following pairs of concepts a and b, if:

1) a - "rectangle", b - "rhombus";

2) a - "polygon", b - "parallelogram";

3) a - "straight line", b - "segment".

The relationships between the sets are shown in the figure, respectively.

2. Definition of concepts. Defined and undefined concepts.

The appearance in mathematics of new concepts, and hence new terms denoting these concepts, presupposes their definition.

By definition usually called a sentence clarifying the essence of a new term (or designation). As a rule, they do this on the basis of previously introduced concepts. For example, a rectangle can be defined as follows: "A rectangle is usually called a quadrangle, in which all corners are straight." There are two parts to this definition - the concept being defined (rectangle) and the defining concept (a quadrangle with all corners right). If we denote the first concept by a and the second by b, then this definition can be represented in the following form:

a is (by definition) b.

The words “is (by definition)” are usually replaced by the symbol ⇔, and then the definition looks like this:

They read: "a is equal to b by definition." You can also read this entry like this: “but if and only if b.

Definitions with this structure are called explicit... Let's consider them in more detail.

Let's turn to the second part of the definition of “rectangle”.

It can be distinguished:

1) the concept of "quadrangle", ĸᴏᴛᴏᴩᴏᴇ is generic in relation to the concept of "rectangle".

2) the property “have all angles straight”, ĸᴏᴛᴏᴩᴏᴇ allows you to select one type of all possible quadrangles - rectangles; in this regard, it is called a species difference.

In general, a specific distinction is ϶ᴛᴏ properties (one or more) that make it possible to single out the defined objects from the scope of a generic concept.

The results of our analysis can be presented in the form of a diagram:

The "+" sign is used as a replacement for the "and" particle.

We know that any concept has volume. If the concept a is defined through the genus and species difference, then about its volume - the set A - we can say that it contains objects that belong to the set C (the volume of the generic concept c) and have the property P:

A = (x / x ∈ C and P (x)).

Since the definition of a concept through the genus and species difference is essentially a conditional agreement on the introduction of a new term to replace any set of known terms, it is impossible to say about the definition whether it is true or false; it is neither proven nor refuted. But, when formulating the definitions, they adhere to a number of rules. Let's call them.

1. The definition must be commensurate... This means that the volumes of the defined and defining concepts must coincide.

2. In the definition (or their system) there should be no vicious circle... This means that you cannot define a concept through itself.

3. The definition must be clear... It is required, for example, that the meanings of the terms included in the defining concept are known by the time the definition of the new concept is introduced.

4. One and the same concept is defined through the genus and species difference, observing the rules formulated above, can be different... So, a square can be defined as:

a) a rectangle whose adjacent sides are equal;

b) a rectangle whose diagonals are mutually perpendicular;

c) a rhombus that has a right angle;

d) a parallelogram in which all sides are equal and the corners are straight.

Different definitions of the same concept are possible because of the large number of properties included in the content of a concept, only a few are included in the definition. And then from the possible definitions one is chosen, proceeding from which of them is simpler and more expedient for the further construction of the theory.

Let's name the sequence of actions that we must follow if we want to reproduce the definition of a familiar concept or build a definition of a new one:

1. Name the defined concept (term).

2. Indicate the closest generic concept (in relation to the defined) concept.

3. List the properties that distinguish the defined objects from the scope of the generic, that is, formulate the species difference.

4. Check whether the rules for defining the concept have been followed (whether it is proportionate, whether there is a vicious circle, etc.).

Among the skills that mathematics teaches and that you all need to learn, the ability to classify concepts.

The fact is that mathematics, like many other sciences, studies not single objects or phenomena, but massive... So, when you study triangles, you study the properties of any triangles, and there are an infinite number of them. In general, the scope of any mathematical concept is, as a rule, infinite.

In order to distinguish objects of mathematical concepts, to study their properties, these concepts are usually divided into types, classes. Indeed, in addition to general properties, any mathematical concept has many more important properties that are not inherent in all objects of this concept, but only in objects of a certain kind. So, right-angled triangles, in addition to the general properties of any triangles, they have many properties that are very important for practice, for example the Pythagorean theorem, ratios between angles and sides, etc.

In the process of centuries-old study of mathematical concepts, in the process of their numerous applications in life, in other sciences, some special types having the most interesting properties, which are most often found and used in practice. So, there are infinitely many different quadrangles, but in practice, in technology, only certain types of them are most used: squares, rectangles, parallelograms, rhombuses, trapezoids.

Dividing the volume of a concept into parts is the classification of this concept. More precisely, classification is understood as the distribution of objects of a concept into interrelated classes (types, types) according to the most essential features(properties). The attribute (property) by which the classification (division) of the concept into types (classes) is made is called basis classification.

A correctly constructed classification of a concept reflects the most essential properties and connections between the objects of the concept, helps to better navigate in the set of these objects, makes it possible to establish such properties of these objects that are most important for the application of this concept in other sciences and everyday practice.

The classification of a concept is made on one or more of the most significant grounds.

So, triangles can be classified according to the magnitude of the angles. We get the following types: acute-angled (all angles are acute), rectangular (one corner is straight, the rest are acute), obtuse-angular (one corner is obtuse, the rest are sharp). If we take the relationship between the sides as the basis for dividing the triangles, then we get the following types: versatile, isosceles and regular (equilateral).

It is more difficult when you have to classify a concept on several grounds. So, if the convex quadrangles are classified according to the parallelism of the sides, then in essence we need to divide all convex quadrangles simultaneously according to two criteria: 1) one pair of opposite sides is parallel or not; 2) the second pair of opposite sides is parallel or not. As a result, we get three types of convex quadrangles: 1) quadrangles with non-parallel sides; 2) quadrangles with one pair of parallel sides - trapezoids; 3) quadrangles with two pairs of parallel sides - parallelograms.

Quite often, a concept is classified in stages: first, on one basis, then some species are divided into subspecies on a different basis, etc. An example is the classification of quadrangles. At the first stage, they are divided according to the bulge. Then the convex quadrangles are divided according to the parallelism of the opposite sides. In turn, parallelograms are divided according to the presence of right angles, etc.

When carrying out the classification, certain rules must be followed. Let us indicate the main ones.

1. As a basis for classification, one can take only a common feature of all objects of a given concept. So, for example, it is impossible to take the sign of the arrangement of terms by degrees of some variable as a basis for the classification of algebraic expressions. This feature is not common to all algebraic expressions; for example, it does not make sense for fractional expressions or monomials. Only polynomials have this feature, so polynomials can be classified according to the highest degree of the principal variable.
2. The basis for the classification must be taken the essential properties (attributes) of concepts. Consider again the concept of an algebraic expression. One of the properties of this concept is that the variables included in an algebraic expression are denoted by some letters. This property is general, but not essential, because the character of the expression does not depend on what letter this or that variable is designated. Thus, algebraic expressions x + y and a + b is essentially the same expression. Therefore, you should not classify expressions on the basis of the designation of variables by letters. It is another matter if we take as the basis for the classification of algebraic expressions the attribute of the type of actions by which the variables are connected, that is, the actions that are performed on the variables. This common feature is very essential, and a classification based on this feature will be correct and useful.
3. At each stage of the classification, only one kind of basis can be applied. You cannot simultaneously classify a concept on two different grounds. For example, it is impossible to classify triangles at once both in size and in the ratio between the sides, because as a result we get classes of triangles that have common elements (for example, acute-angled and isosceles or obtuse and isosceles, etc.). The following classification requirement is violated here: as a result of classification at each stage, the resulting classes (types) should not overlap.
4. In the same time classification for any reason must be exhaustive and each object of the concept must fall as a result of classification into one and only one class.

Therefore, the division of all integers into positive and negative is incorrect, because the integer zero did not fall into any of the classes. We should say this: integers are divided into three classes - positive, negative and the number zero.

Often, when classifying concepts, only some classes are clearly distinguished, and the rest are only implied. So, for example, in the study of algebraic expressions, only such types of them are usually distinguished: monomials, polynomials, fractional expressions, irrational. But these types do not exhaust all types of algebraic expressions, therefore such a classification is incomplete.

A complete correct classification of algebraic expressions can be done as follows.

At the first stage of the classification of algebraic expressions, they are divided into two classes: rational and irrational. At the second stage, rational expressions are divided into whole and fractional ones. In the third step, whole expressions are divided into monomials, polynomials, and complex whole expressions.

This classification can be represented as follows

Assignment 7

7.1. Why can't rational numbers be classified according to their evenness?

7.2. Establish whether the division of the concept is correct:

a) The values ​​can be equal or unequal.

b) Functions are increasing and decreasing.

c) Isosceles triangles can be acute-angled, rectangular and obtuse-angled.

d) Rectangles are squares and rhombuses.

7.3. Divide the concept of "geometric figure" by its property to occupy a part of the plane and give examples of each type.

7.4. Build possible classification schemes for rational numbers.

7.5. Build a classification scheme for the following concepts:

a) a quadrangle;

b) two corners.

7.6. Classify the following concepts:

a) triangle and circle;

b) angles in a circle;

c) two circles;

d) line and circle;

e) quadratic equations;

f) a system of two equations of the first degree with two unknowns.