1) a qualitative concept used in economics in the sense of "equality of income", "property equality", "equality of opportunity" to emphasize the existence of equality and inequality in the position of certain social groups; 2) mathematical identity, equation.
Great Definition
Incomplete definition ↓
EQUALITY
one of the principles of law. The concept of R. is a certain abstraction, i.e. the result of conscious (mental) abstraction from those differences that are inherent in equalized objects. Legal R. is not so abstract. The basis (and criterion) of the legal equation various people is the freedom of individuals in social relations, recognized and affirmed in the form of their legal capacity and legal personality. This is the specificity of legal R. and law in general. R. has a rational meaning, it is logically and practically possible in the social world that and only legal (formal-legal, formal) R. The history of law is the history of the progressive evolution of the content, volume, scale and measure of formal (legal) R. while maintaining this very principle as a principle of any system of law, law in general. Thus, the principle of formal R. is a principle permanently inherent in law with a historically changing content. On the whole, the historical evolution of the content, volume, and scope of the principle of formal R. does not refute, but, on the contrary, reinforces the significance of this principle as distinguishing feature law in its relation to other types of social regulation (moral, religious, etc.). The initial actual differences between people, considered and settled from the point of view of the legal principle of R. (equal measure), appear as a result in the form of inequality in already acquired rights (in their structure, content and scope of the rights of various subjects of law). Law as a form of relations according to the principle of R. does not destroy (and cannot destroy) the initial differences between different subjects of law, it only formalizes and arranges these differences on a single basis, transforms indefinite actual differences into formally defined rights of free, independent of each other, equal individuals. This, in essence, is the specificity, meaning and value of the legal form of mediation, regulation and ordering public relations. Legal R. and legal inequality are single-order legal definitions. The principle of legal R. of various subjects assumes that the real subjective rights acquired by them will be unequal. Thanks to law, the chaos of differences is transformed into a legal order of equalities and inequalities, agreed on a single basis and a common norm. The recognition of various individuals as formally equal means the recognition of their equal legal capacity, the possibility of acquiring certain rights to the corresponding goods, specific objects, etc. Formal law is only an ability, an abstract opportunity to acquire, in accordance with the general scale and equal measure of legal regulation, one's own, individually defined right to a given object. The difference in the rights acquired by different persons is a necessary result of precisely observing, and not violating, the principle of formal (legal) R. of these persons does not violate or cancel the principle of formal (legal) R. For all whose relations are mediated by a legal form, law acts as universal form, as universally significant and equal for all these persons (different in their actual, physical, mental, property status, etc.) the same scale and measure. R. itself consists in the fact that the behavior and position of the subjects of a given general range of relations and phenomena fall under the action of a single law for all, a single (common, equal) measure. Lit.: Nersesyants V.S. Law and law. From the history of legal doctrines. M, 1983; His own. Law is the mathematics of freedom. M, 1996; His own. The value of law as a trinity of freedom, equality and justice / / Problems of the value approach in law: traditions and renewal. M., 1996. V.S. Nersesyants
"Equality" is a topic that students go through as early as primary school. She also accompanies her "Inequalities". These two concepts are closely related. In addition, such terms as equations, identities are associated with them. So what is equality?
The concept of equality
This term is understood as statements, in the record of which there is a sign "=". Equality is divided into true and false. If in the entry instead of = stands<, >, then we are talking about inequalities. By the way, the first sign of equality indicates that both parts of the expression are identical in their result or record.
In addition to the concept of equality, the topic "Numeric Equality" is also studied at school. This statement is understood as two numerical expressions that stand on both sides of the = sign. For example, 2*5+7=17. Both parts of the record are equal to each other.
In numeric expressions of this type, parentheses can be used, affecting the order of operations. So, there are 4 rules that should be taken into account when calculating the results of numerical expressions.
- If there are no brackets in the entry, then the actions are performed with higher level: III→II→I. If there are multiple actions of the same category, then they are executed from left to right.
- If there are brackets in the entry, then the action is performed in brackets, and then taking into account the steps. Perhaps there will be several actions in brackets.
- If the expression is represented as a fraction, then you need to calculate the numerator first, then the denominator, then the numerator is divided by the denominator.
- If the entry contains nested parentheses, then the expression in the inner parentheses is evaluated first.
So, now it is clear what equality is. In the future, the concepts of equations, identities and methods for calculating them will be considered.
Properties of numerical equalities
What is equality? The study of this concept requires knowledge of the properties of numerical identities. The following text formulas allow you to better study this topic. Of course, these properties are more suitable for studying mathematics in high school.
1. Numerical equality will not be violated if the same number is added to the existing expression in both of its parts.
A = B↔ A + 5 = B + 5
2. The equation will not be violated if both parts of it are multiplied or divided by the same number or expression that is different from zero.
P = O↔ R ∙ 5 = O ∙ 5
P = O↔ R: 5 = O: 5
3. Adding to both parts of the identity the same function, which makes sense for any admissible values of the variable, we get a new equality that is equivalent to the original one.
F(X) = Ψ(X) ↔ F(X) + R(X) =Ψ (X) + R(X)
4. Any term or expression can be transferred to the other side of the equal sign, while you need to change the signs to the opposite.
X + 5 = Y - 20 ↔ X \u003d Y - 20 - 5 ↔ X \u003d Y - 25
5. By multiplying or dividing both sides of the equation by the same non-zero function that makes sense for each value of X from the ODZ, we get a new equation that is equivalent to the original one.
F(X) = Ψ(x)↔ F(X) ∙R(X) = Ψ(X) ∙R(x)
F(X) = Ψ(X)↔ F(X) : G(X) = Ψ(X) : G(X)
The above rules explicitly point to the principle of equality, which exists under certain conditions.
The concept of proportion
In mathematics, there is such a thing as equality of relations. In this case, the definition of proportion is implied. If you divide A by B, then the result will be the ratio of the number A to the number B. Proportion is the equality of two ratios:
Sometimes the proportion is written as follows: A:B=C:D. From this follows the main property of proportion: A*D=D*C, where A and D are the extreme members of the proportion, and B and C are the middle ones.
Identities
An identity is an equality that will be true for all valid values of those variables that are included in the task. Identities can be represented as literal or numerical equalities.
Equally equal are called expressions that contain an unknown variable in both parts of the equality, which is capable of equating two parts of one whole.
If we replace one expression with another, which will be equal to it, then we are talking about an identical transformation. In this case, you can use the formulas for abbreviated multiplication, the laws of arithmetic and other identities.
To reduce the fraction, you need to carry out identical transformations. For example, given a fraction. To get the result, you should use the formulas for abbreviated multiplication, factoring, simplifying expressions and reducing fractions.
It should be noted that this expression will be identical when the denominator is not equal to 3.
5 ways to prove identity
To prove the equality is identical, it is necessary to transform the expressions.
I way
It is necessary to carry out equivalent transformations on the left side. The result is right part, and we can say that the identity is proved.
II way
All actions to transform the expression occur on the right side. The result of the performed manipulations is the left side. If both parts are identical, then the identity is proved.
III way
"Transformations" occur in both parts of the expression. If the result is two identical parts, the identity is proved.
IV way
The right side is subtracted from the left side. As a result of equivalent transformations, zero should be obtained. Then we can talk about the identity of the expression.
5th way
The left side is subtracted from the right side. All equivalent transformations are reduced to the fact that the answer is zero. Only in this case can we speak of the identity of equality.
Basic properties of identities
In mathematics, the properties of equalities are often used to speed up the calculation process. Due to basic algebraic identities, the process of calculating some expressions will take a few minutes instead of long hours.
- X + Y = Y + X
- X + (Y + C) = (X + Y) + C
- X + 0 = X
- X + (-X) = 0
- X ∙ (Y + C) = X ∙ Y + X ∙ C
- X ∙ (Y - C) \u003d X ∙ Y - X ∙ C
- (X + Y) ∙ (C + E) = X ∙ C + X ∙ E + Y ∙ C + Y ∙ E
- X + (Y + C) = X + Y + C
- X + (Y - C) \u003d X + Y - C
- X - (Y + C) \u003d X - Y - C
- X - (Y - C) \u003d X - Y + C
- X ∙ Y = Y ∙ X
- X ∙ (Y ∙ C) = (X ∙ Y) ∙ C
- X ∙ 1 = X
- X ∙ 1/X = 1, where X ≠ 0
Abbreviated multiplication formulas
At their core, abbreviated multiplication formulas are equalities. They help solve many problems in mathematics due to their simplicity and ease of use.
- (A + B) 2 \u003d A 2 + 2 ∙ A ∙ B + B 2 - the square of the sum of a pair of numbers;
- (A - B) 2 \u003d A 2 - 2 ∙ A ∙ B + B 2 - the square of the difference between a pair of numbers;
- (C + B) ∙ (C - B) \u003d C 2 - B 2 - difference of squares;
- (A + B) 3 \u003d A 3 + 3 ∙ A 2 ∙ B + 3 ∙ A ∙ B 2 + B 3 - the cube of the sum;
- (A - B) 3 \u003d A 3 - 3 ∙ A 2 ∙ B + 3 ∙ A ∙ B 2 - B 3 - difference cube;
- (P + B) ∙ (P 2 - P ∙ B + B 2) \u003d P 3 + B 3 - the sum of cubes;
- (P - B) ∙ (P 2 + P ∙ B + B 2) \u003d P 3 - B 3 - the difference of cubes.
Abbreviated multiplication formulas are often used if it is necessary to bring the polynomial to its usual form, simplifying it in all possible ways. The presented formulas are proved simply: it is enough to open the brackets and bring like terms.
Equations
After studying the question of what equality is, you can proceed to the next point: An equation is understood as an equality in which there are unknown quantities. The solution of the equation is the finding of all values of the variable, in which both parts of the entire expression will be equal. There are also tasks in which finding solutions to the equation is impossible. In this case, we say that there are no roots.
As a rule, equalities with unknowns give integer numbers as solutions. However, there are cases when the root is a vector, a function, and other objects.
An equation is one of the most important concepts in mathematics. Most scientific and practical problems do not allow to measure or calculate any quantity. Therefore, it is necessary to draw up a ratio that will satisfy all the conditions of the task. In the process of compiling such a relationship, an equation or system of equations appears.
Usually, solving an equality with an unknown is reduced to transforming a complex equation and reducing it to simple forms. It must be remembered that the transformations must be carried out with respect to both parts, otherwise the output will be an incorrect result.
4 ways to solve an equation
By solving an equation, one understands the replacement of a given equality by another, which is equivalent to the first one. Such a change is known as identity transformation. To solve the equation, you must use one of the methods.
1. One expression is replaced by another, which in without fail will be identical to the first. Example: (3∙x+3) 2 =15∙x+10. This expression can be converted to 9∙x 2 +18∙x+9=15∙x+10.
2. Transferring the terms of equality with the unknown from one side to the other. In this case, it is necessary to change the signs correctly. The slightest mistake will ruin all the work done. Let's take the previous "sample" as an example.
9 x 2 + 12 x + 4 = 15 x + 10
9∙x 2 + 12∙x + 4 - 15∙x - 10 = 0
3. Multiplying both sides of the equality by an equal number or expression that does not equal 0. However, it is worth recalling that if the new equation is not equivalent to equality before transformations, then the number of roots may change significantly.
4. Squaring both sides of the equation. This method is simply wonderful, especially when there are irrational expressions in the equality, that is, the expression under it. There is one caveat: if you raise the equation to an even power, then extraneous roots may appear that will distort the essence of the task. And if it is wrong to extract the root, then the meaning of the question in the problem will be unclear. Example: │7∙х│=35 → 1) 7∙х = 35 and 2) - 7∙х = 35 → the equation will be solved correctly.
So, in this article, terms such as equations and identities are mentioned. All of them come from the concept of "equality". Thanks to various kinds of equivalent expressions, the solution of some problems is greatly facilitated.
The material of the article will allow you to get acquainted with the mathematical interpretation of the concept of equality. Let's talk about the essence of equality; consider its types and ways of recording it; we write down the properties of equality and illustrate the theory with examples.
The very concept of equality is closely intertwined with the concept of comparison, when we compare properties and features in order to identify similarities. The comparison process requires the presence of two objects, which are compared with each other. This reasoning suggests that the concept of equality cannot take place when there are not at least two objects to compare. In this case, of course, a larger number of objects can be taken: three or more, however, in the end, we will somehow come to a comparison of pairs collected from given objects.
The meaning of the concept of "equality" in a generalized interpretation is perfectly defined by the word "same". Two identical objects can be said to be “equal”. For example, squares and . But objects that differ from each other at least on some basis, we will call unequal.
Speaking of equality, we can mean both objects as a whole and their individual properties or features. Objects are equal in general when they are equal in all characteristics. For example, when we cited the equality of squares as an example, we meant their equality in all their inherent properties: shape, size, color. Also, objects may not be equal in general, but have the same individual features. For example: and . The specified objects are equal in shape (both circles), but different (unequal) in color and size.
Thus, it is necessary to understand in advance what kind of equality we have in mind.
Recording equalities, =
To record equality, use the equal sign (or equals sign), denoted as = . This notation is generally accepted.
Compiling equality, equal objects are placed side by side, writing an equal sign between them. For example, the equality of the numbers 5 and 5 will be written as 5 = 5 . Or, let's say we need to write down the equality of the perimeter of the triangle A B C to 6 meters: P A B C \u003d 6 m.
Definition 1
Equality- a record in which an equal sign is used, separating two mathematical objects (or numbers, or expressions, etc.).
When it becomes necessary to indicate in writing the inequality of objects, they use the not equal sign, denoted as ≠, i.e. essentially a crossed out equals sign.
True and false equalities
Compiled equalities may correspond to the essence of the concept of equality, or they may contradict it. On this basis, all equalities are classified into true equalities and false equalities. Let's give examples.
Let's make the equality 7 = 7 . The numbers 7 and 7, of course, are equal, and therefore 7 \u003d 7 is a true equality. Equality 7 = 2 , in turn, is incorrect, since the numbers 7 and 2 not equal.
Equality properties
We write down three main properties of equalities:
Definition 2
- the property of reflexivity, which says that an object is equal to itself;
- symmetry property: if the first object is equal to the second, then the second is equal to the first;
- property of transitivity: when the first object is equal to the second, and the second is equal to the third, then the first is equal to the third.
Let's write the literally formulated properties as follows:
- a = a;
- if a = b, then b = a;
- if a = b and b=c, then a = c.
We note the special use of the second and third properties of equalities - the properties of symmetry and transitivity - they make it possible to assert the equality of three or more objects through their pairwise equality.
Double, triple, etc. equality
Together with the standard notation of equality, an example of which we gave above, the so-called double equalities, triple equalities, etc. are also often compiled. Such records are, as it were, a chain of equalities. For example, the entry 2 + 2 + 2 = 4 + 2 = 6 - double equality, and | A B | = | B C | = | C D | = | D E | = | E F |- an example of a quarter equality.
With the help of such chains of equalities, it is optimal to compose the equality of three or more objects. Such entries in their meaning are the designation of the equality of any two objects that make up the original chain of equalities.
For example, the double equality written above 2 + 2 + 2 = 4 + 2 = 6 denotes equalities: 2 + 2 + 2 = 4 + 2 , and 4 + 2 = 6 , and 2 + 2 + 2 = 6 , and due to the symmetry property of the equalities and 4 + 2 = 2 + 2 + 2 , and 6 = 4 + 2 , and 6 = 2 + 2 + 2 .
Composing such chains, it is convenient to write down the sequence of solving examples and problems: such a solution becomes clear and reflects all the intermediate stages of calculations.
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After receiving general information about equalities in mathematics, we move on to narrower topics. The material of this article will give an idea of the properties of numerical equalities.
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What is numerical equality
The first time we encounter numerical equalities in elementary school, when we get acquainted with numbers and the concept of "the same". Those. the most primitive numerical equalities are: 2 = 2, 5 = 5, etc. And at that level of study, we called them simply equalities, without specifying "numerical", and laid in them a quantitative or ordinal meaning (which natural numbers carry). For example, the equation 2 = 2 will correspond to an image with two flowers and two bumblebees perched on each. Or, for example, two queues, where Vasya and Vanya are second in order.
As knowledge of arithmetic operations appears, numerical equalities become more complicated: 5 + 7 \u003d 12; 6 - 1 = 5; 2 1 = 2; 21: 7 = 3, etc. Then equalities begin to occur, in the recording of which numerical expressions of various kinds participate. For example, (2 + 2) + 5 = 2 + (5 + 2) ; 4 (4 − (1 + 2)) + 12: 4 − 1 = 4 1 + 3 − 1, etc. Then we get acquainted with other types of numbers, and numerical equalities become more and more interesting and diverse.
Definition 1
Numerical equality is an equality, both parts of which consist of numbers and/or numerical expressions.
Properties of numerical equalities
It is difficult to overestimate the importance of the properties of numerical equalities in mathematics: they are the basis for many things, determine the principle of working with numerical equalities, solution methods, rules for working with formulas, and much more. Obviously, there is a need for a detailed study of the properties of numerical equalities.
The properties of numerical equalities are absolutely consistent with how actions with numbers are defined, as well as with the definition of equal numbers through the difference: number a is equal to the number b only when the difference a-b there is zero. Further in the description of each property, we will trace this connection.
Basic properties of numerical equalities
Let's start studying the properties of numerical equalities with three basic properties that are inherent in all equalities. We list the main properties of numerical equalities:
- reflexivity property: a = a;
- symmetry property: if a = b, then b = a;
- transitivity property: if a = b and b=c, then a = c, where a , b and c are arbitrary numbers.
The property of reflexivity denotes the fact that a number is equal to itself: for example, 6 = 6, - 3 = - 3, 4 3 7 = 4 3 7, etc.
Proof 1
It is easy to demonstrate the validity of equality a − a = 0 for any number a: difference a - a can be written as a sum a + (− a), and the addition property of numbers gives us the opportunity to assert that any number a corresponds to the only opposite number − a, and their sum is zero.
Definition 3
According to the symmetry property of numerical equalities: if the number a is equal to the number b,
that number b is equal to the number a. For example, 4 3 = 64
, then 64 = 4 3
.
Proof 2
Justify given property possible through the difference of numbers. condition a = b corresponds to equality a − b = 0. Let's prove that b − a = 0.
Let's write the difference b - a as - (a - b), relying on the rule for opening brackets preceded by a minus sign. The new entry for the expression is - 0 , and the opposite of zero is zero. In this way, b − a = 0, Consequently: b = a.
Definition 4
The property of transitivity of numerical equalities states that two numbers are equal to each other if they are simultaneously equal to a third number. For example, if 81 = 9 and 9 = 3 2 , then 81 = 3 2 .
The property of transitivity also corresponds to the definition of equal numbers through the difference and properties of operations with numbers. Equalities a = b and b=c correspond to the equalities a − b = 0 and b − c = 0.
Proof 3
Let us prove the equality a − c = 0, from which the equality of numbers will follow a and c. Since adding a number to zero does not change the number itself, then a - c write in the form a + 0 − c. Instead of zero, we substitute the sum of opposite numbers −b and b, then the final expression becomes: a + (− b + b) − c. Let's group the terms: (a − b) + (b − c). The differences in brackets are equal to zero, then the sum (a − b) + (b − c) there is zero. This proves that when a − b = 0 and b − c = 0, the equality a − c = 0, where a = c.
Other important properties of numerical equalities
The main properties of numerical equalities discussed above are the basis for a number of additional properties that are quite valuable in the context of practice. Let's list them:
Definition 5
By adding to (or subtracting from) both parts of the numerical equality, which is true, the same number, we obtain the correct numerical equality. Let's write it literally: if a = b, where a and b are some numbers, then a + c = b + c for any c.
Proof 4
As a justification, we write the difference (a + c) − (b + c).
This expression can easily be converted to the form (a − b) + (c − c).
From a = b by condition it follows that a − b = 0 and c − c = 0, then (a - b) + (c - c) = 0 + 0 = 0. This proves that (a + c) − (b + c) = 0, Consequently, a + c = b + c;
Definition 6
If both parts of a true numerical equality are multiplied with any number or divided by a number, no zero, then we get the correct numerical equality.
Let's write it down literally: when a = b, then a c = b c for any number c. If c ≠ 0 then and a:c = b:c.
Proof 5
Equality is true: a c − b c = (a − b) c = 0 c = 0, and it implies the equality of the products a c and b c. And division by a non-zero number c can be written as a multiplication by the reciprocal of 1 c ;
Definition 7
At a and b, different from zero and equal to each other, their reciprocals are also equal.
Let's write: when a ≠ 0 , b ≠ 0 and a = b, then 1 a = 1 b. The extreme equality is not difficult to prove: for this purpose, we divide both sides of the equality a = b by a number equal to the product a b and not equal to zero.
We also point out a couple of properties that allow the addition and multiplication of the corresponding parts of the correct numerical equalities:
Definition 8
With term-by-term addition of the correct numerical equalities, the correct equality is obtained. This property is written as follows: if a = b and c = d, then a + c = b + d for any numbers a , b , c and d.
Proof 6
justify it useful property possibly based on the previously mentioned properties. We know that any number can be added to both sides of a true equality.
Towards equality a = b add the number c, and to equality c = d- number b, the result will be the correct numerical equalities: a + c = b + c and c + b = d + b. We write the last one in the form: b + c = b + d. From equalities a + c = b + c and b + c = b + d according to the property of transitivity, the equality follows a + c = b + d. Which is what needed to be proven.
It is necessary to clarify that term by term it is possible to add not only two true numerical equalities, but also three or more;
Definition 7
Finally, we describe such a property: term-by-term multiplication of two correct numerical equalities gives the correct equality. Let's write in letters: if a = b and c = d, then a c = b d.
Proof 7
The proof of this property is similar to the proof of the previous one. Multiply both sides of the equation by any number, multiply a = b on the c, a c = d on the b, we obtain the correct numerical equalities a c = b c and c b = d b. We write the last as b c = b d. The property of transitivity makes it possible from the equality a c = b c and b c = b d derive equality a c = b d which we needed to prove.
And again, we clarify that this property is applicable for two, three or more numerical equalities.
Thus, one can write: if a = b, then a n = b n for any numbers a and b, and any natural number n.
Let's finish this article by collecting all the considered properties for clarity:
If a = b , then b = a .
If a = b and b = c , then a = c .
If a = b , then a + c = b + c .
If a = b, then a c = b c.
If a = b and c ≠ 0, then a: c = b: c.
If a = b , a = b , a ≠ 0 and b ≠ 0 , then 1 a = 1 b .
If a = b and c = d, then a c = b d.
If a = b , then a n = b n .
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EQUALITY WITH QUANTITIES.
After the child gets acquainted with cards-quantities from 1 to 20, you can add the second stage to the first stage of training - equality with quantities.
What is equality? This is an arithmetic operation and its result.
You begin this learning phase with the topic Addition.
Addition.
To display two sets of quantity cards, you add equalities for addition.
This operation is very easy to learn. In fact, your child has been ready for this for several weeks. After all, every time you show him a new card, he sees that one additional point has appeared on it.
The kid does not yet know what it is called, but already has an idea of what it is and how it works.
You already have material for addition examples on the back of each card.
Equality Display Technology looks something like this: You want to give the child equality: 1 + 2 = 3. How can it be shown?
Before the lesson, place three cards on your knees, face down, one on top of the other. Raising the top card with one knuckle needle, say "one", then put it down, say "a plus", show a card with two bones, say "two", put it aside after the word "will be", show a card with three bones, saying "three".
On the day you conduct three classes with equalities and in each lesson you show three different equalities. In total, the baby sees nine different equalities per day.
The child understands without any explanation what the word means "a plus", he takes its meaning out of context. By performing actions, you thereby demonstrate the true meaning of addition faster than any explanations. When talking about equalities, always stick to the same manner of presentation, using the same terms. Having said "One plus two makes three" don't talk later "Add two to one makes three." When you teach a child facts, he himself draws conclusions and comprehends the rules. If you change the terms, then the child has every reason to think that the rules have also changed.
Prepare in advance all the cards needed for this or that equality. Don't expect your child to sit quietly and watch you rummage through a stack of cards, picking up the right ones. He will just run away and be right, because his time is worth as much as yours.
Try not to make equalities that would have something in common and would allow the child to predict them in advance (such equalities can be used later). Here is an example of such equalities:
It's much better to use these:
1 +2 = 3 5+6=11 4 + 8 = 12
The child must see the mathematical essence, he develops mathematical skills and ideas. After about two weeks, the baby discovers what addition is: after all, during this time you showed him 126 different equalities for addition.
Examination.
Check for this stage is a solution of examples.
How is an example different from equality?
Equality is an action with a result shown to the child.
An example is an action to be performed. In our case, you show the child two answers, and he chooses the correct one, i.e. solves the example.
You can lay out an example after the usual lesson with three equalities for addition. You show an example in the same way as you demonstrated equality before. That is, you shift the cards in your hands, saying each aloud. For example, "twenty plus ten is thirty or forty-five?" and show the baby two cards, one of which has the correct answer.
Answer cards should be kept at the same distance from the baby's eyes and no prompting actions should be allowed.
With the right choice of a child, you vigorously express your delight, kiss and praise him.
If you choose the wrong answer, without expressing disappointment, you push the card with the correct answer to the baby and ask the question: “There will be thirty, won’t it?”. To such a question, the child usually answers in the affirmative. Be sure to praise your child for this correct answer.
Well, if out of ten examples your kid correctly solves at least six, then it’s time for you to move on to equalities for subtraction!
If you do not consider it necessary to check the child (and rightly so!), Then after 10-14 days, you still go to subtraction equalities!
Let's consider subtraction.
You stop doing addition and switch completely to subtraction. Conduct three daily lessons with three different equalities each.
You voice the equalities for subtraction like this: "Twelve minus seven is five."
At the same time, you simultaneously continue to show quantity cards (two sets, five cards each) also three times a day. In total, you will have nine daily very short lessons. So you work no more than two weeks.
Examination
Verification, just as in the case of addition, can be a solution of examples with the choice of one answer out of two.
Consider Multiplication.
Multiplication is nothing more than repeated addition, so this operation will not be a big discovery for your child. As you continue to study the number cards (two sets of five cards each), you have the opportunity to make multiplication equalities.
You voice the equalities for multiplication like this: "Two times three is six."
The child will understand the word "multiply" as fast as he understood before that word "a plus" and "minus".
You still spend three lessons a day, each of which contains three different equalities for multiplication. Such work lasts no more than two weeks.
Keep avoiding predictable equalities. For example, such as:
It is necessary to constantly keep your child in a state of surprise and expectation of something new. The main question for him should be: "What's next?"- and at each lesson he should receive a new answer to it.
Examination
You solve the examples in the same way as in the topic "Addition" and "Subtraction". If your child enjoys the number-card-checking games, you can continue to play them, thus repeating new, larger numbers.
Adhering to the scheme we have proposed, by this time you can already complete the first stage of learning mathematics - study the quantities within 100. Now it's time to get acquainted with the card that children like the most.
Consider the concept of zero.
It is said that mathematicians have been studying the idea of zero for five hundred years. Whether this is true or not, children, as soon as they get to know the idea of quantity, immediately understand the meaning of its complete absence. They just love zero, and your journey into the world of numbers will not be complete if you do not show your baby a card that does not have any dots at all (i.e. it will be a completely empty card).
To make the baby’s acquaintance with zero fun and interesting, you can accompany the card display with a riddle:
At home - seven squirrels, On a plate - seven mushrooms. All the mushrooms ate the squirrels. What's left on the plate?
Saying the last phrase, we show the card "zero".
You will use it almost every day. It is useful for addition, subtraction and multiplication operations.
You can work with the "zero" card for one week. The child masters this topic quickly. As before, during the day, you spend three classes. At each lesson, you show your child three different equalities for addition, subtraction and multiplication with zero. In total, you will get nine equalities per day.
Examination
The solution of examples with zero goes according to the scheme familiar to you.
Consider -Division.
When you have gone through all the number cards from 0 to 100, you have all the necessary material for division examples with quantities.
The technology of displaying the equalities of this topic is the same. You have three classes every day. At each lesson, you show the baby three different equalities. Well, if the passage of this material will not exceed two weeks.
Examination
Checking is a solution of examples with the choice of one answer out of two.
When you have gone through all the quantities and are familiar with the four rules of arithmetic, you can diversify and complicate your studies in every possible way. First, show equalities where one arithmetic operation is used: only addition, subtraction, multiplication, or division.
Then - equalities, where addition and subtraction or multiplication and division are combined:
20 + 8-10=18 9-2 + 26 = 33 47+11-50 = 8
In order not to get confused in the cards, you can change the way of conducting classes. Now it is not necessary to show each card of knitting needles, you can only show the answer, and only pronounce the actions themselves. As a result, your classes will become shorter. You just tell the child: "Twenty-two divided by eleven, divided by two is one"- and show him the card "one".
In this topic, you can use equalities between which there is some pattern.
For example:
2*2*3= 12 2*2*6=24 2*2*8=32
When combining four arithmetic operations in equality, remember that multiplication and division must be moved to the beginning of the equality:
Do not be afraid to demonstrate equalities, of which there are more than a hundred, for example,
intermediate result in
42 * 3 - 36 = 90,
where the intermediate result is 126 (42 * 3 = 126)
Your little one will be great with them!
Checking is a solution of examples with the choice of one answer out of two. You can demonstrate an example by showing all the equality cards and two answer cards, or just say the whole equality by showing the baby only two answer cards.
Remember! The longer you study, the faster you need to introduce new topics. As soon as you notice the first signs of a child's inattention or boredom, move on to a new topic. After a while, you can return to the previous topic (but to get acquainted with equalities not yet shown).
Sequences
Sequences are the same equalities. The experience of parents with this topic has shown that sequences are very interesting for children.
Plus sequences are increasing sequences. Sequences with minus are decreasing.
The more varied the sequences, the more interesting they are for the baby.
Here are some examples of sequences:
3,6,9,12,15,18,2 (+3)
4, 8, 12, 16, 20, 24, 28 (+4)
5,10,15,20,25,30,35 (+5)
100,90,80,70,60,50,40 (-10)
72, 70, 68, 66, 64, 62, 60 (-2)
95,80,65,50,35,20,5 (-15)
Technology display sequences can be like this. You have prepared three plus sequences.
You announce the topic of the lesson to the child, lay out the cards of the first sequence one after another on the floor, voicing them.
Move with the child to another corner of the room and lay out the second sequence in the same way.
In the third corner of the room, you lay out the third sequence, while voicing it.
You can also lay out sequences under each other, leaving gaps between them.
Try to always go forward, moving from simple to complex. Vary the activities: sometimes saying aloud what you show, and sometimes show the cards silently. In any case, the child sees the sequence unfolded in front of him.
For each sequence, you need to use at least six cards, sometimes more, in order to make it easier for the child to determine the principle of the sequence itself.
As soon as you see the sparkle in the child's eyes, try adding an example to the three sequences (i.e. test his knowledge).
You show an example like this: first you lay out the whole sequence, as you usually do, and at the end you pick up two cards (one card is the one that comes next in the sequence, and the other is random) and ask the child: “Which is next?”
At first, lay out the cards in sequences one after another, then the laying out forms can be changed: put the cards in a circle, around the perimeter of the room, etc.
As you get better and better, don't be afraid to use multiplication and division in your sequences.
Sequence examples:
four; 6; eight; ten; 12; 14 - in this sequence, each next number increases by 2;
2; four; 7; fourteen; 17; 34 - in this sequence, multiplication and addition alternate (x 2; + 3);
2; four; eight; 16; 32; 64 - in this sequence, each next number increases by 2 times;
22; eighteen; fourteen; ten; 6; 2 - in this sequence, each next number decreases by 4;
84; 42; 40; twenty; eighteen; 9 - division and subtraction alternate in this sequence (: 2; - 2);
Signs "greater than", "less than"
These cards are part of 110 cards of numbers and signs (the second component of the ANASTA methodology).
The lessons of introducing the baby to the concepts of "more-less" will be very short. All you have to do is show three cards.
Display Technology
Sit on the floor and lay out each card in front of the child so that he can see all three cards at once. Name each card.
You can say it like this: "six more than three" or "six is more than three."
At each lesson, you show the child three different options inequalities with
cards "more" - "less". inequalities per day.
Thus, you demonstrate nine different
As before, you show each inequality only once.
After a few days, an example can be added to the three shows. It's already examination, and it is done like this:
Place cards prepared in advance on the floor, for example, a card with the number "68" and a card with a "more" sign. Ask your child: "Sixty-eight is greater than what number?" or "Sixty-eight more than fifty or ninety-five?" Ask your child to choose one of the two cards. The card correctly indicated by the baby, you (or he himself) puts after the “more” sign.
You can put two cards with quantities in front of the child and let him choose the sign that suits, that is, > or<.
Equalities and inequalities
Teaching equalities and inequalities is as easy as teaching more and less.
You will need six cards with arithmetic signs. You will also find them as part of 110 cards of numbers and signs (the second component of the ANASTA methodology).
Display Technology
You decide to show your child these two inequalities and one equality:
8-6<10 −7 11-3= 9 −1 55-12^50 −13
You lay them out on the floor sequentially so that the child can see each one at once. While you are talking, for example: "Eight minus six is not equal to ten minus seven."
In the same way, you pronounce the remaining equality and inequality while laying out.
At the initial stage of teaching this topic, all the cards are laid out.
Then it will be possible to show only cards “equal” and “not equal”.
One fine day you give the chance to the kid to show the knowledge. Lay out cards with quantities, and offer him to choose a card with which sign to put: “equal” or “not equal”.
Before you start learning algebra with a baby, you need to introduce him to the concept of a variable represented by a letter.
Usually the letter x is used in mathematics, but since it can be easily confused with the sign of multiplication, it is recommended to use y.
You put first a card with five beads - knuckles, then a + plus sign (+), after it with a y sign, then an equal sign, and finally a card with seven beads - knuckles. Then you ask the question: "What does u mean here?"
And you yourself answer it: “In this equation it means two”
Examination:
After about one to one and a half weeks of classes at this stage, you can let the baby choose the answer.
FOURTH STAGE OF EQUALITY WITH NUMBERS AND QUANTITIES
Once you've gone from 1 to 20, it's time to bridge the gap between numbers and numbers. There are many ways to do this. One of the simplest is the use of equalities and inequalities, greater than and less than relationships, demonstrated using cards with numbers and bones.
display technology.
Take the card with the number 12, put it on the floor, then put the “more” sign next to it, and then the card with the number 10, while saying: “Twelve is more than ten.”
The inequalities (equalities) might look like this:
Each (equal) day consists of three classes, and each lesson consists of three inequalities in numbers and numbers. The total number of daily equalities will be nine. At the same time, you simultaneously continue to study the numbers with the help of two sets of five cards each, also three times a day.
Examination.
You can give the child the opportunity to choose cards "greater than", "less than", "equal to" or compose an example in such a way that the baby himself can complete it. For example, we put a number card 7, then a “greater than” sign and give the child the opportunity to complete the example, that is, choose a number card, for example, 9, or a number card, for example, 5.
After the baby has understood the relationship between quantities and numbers, you can begin to solve equalities using cards with both numbers and quantities.
Equality with numbers and quantities.
Using cards with numbers and quantities, you go through already familiar topics: addition, subtraction, multiplication, division, sequences, equalities and inequalities, fractions, equations, equalities in two or more steps.
If you carefully look at the approximate scheme for teaching mathematics (p. 20), you will see that there is no end to the classes. Come up with your own examples for the development of the child’s mental counting, correlate the quantities with real objects (nuts, spoons for guests, pieces of chopped banana, bread, etc.) - in a word, dare, create, invent, try! And you will succeed!
