What does it mean to write it as an equality? Equality concept, equal sign, related definitions

Class: 3

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Lesson type: discovery of new knowledge.

Technology: technology for developing critical thinking through reading and writing, gaming technology.

Goals: To expand students' knowledge about equalities and inequalities, to introduce the concept of true and false equalities and inequalities.

Didactic task: Organize joint, independent activities of students to study new material.

Lesson objectives:

1. Subject:
   • introduce the signs of equality and inequality; expand students' understanding of equalities and inequalities;
   • introduce the concept of true and false equality and inequality;
   • developing skills in finding the value of an expression containing a variable;
   • formation of computing skills.
2. Metasubject:
   1. Cognitive:
      • promote the development of attention, memory, thinking;
      • developing the ability to extract information, navigate one’s knowledge system and realize the need for new knowledge;
      • mastering the techniques of selecting and systematizing material, the ability to collate and compare, and converting information (into a diagram, table).
   2. Regulatory:
      • development of visual perception;
      • continue to work on the formation of self-control and self-esteem among students;
   3. Communicative:
      • observe the interaction of children in pairs and make the necessary adjustments;
      • foster mutual assistance.
3. Personal:
   • increasing students' learning motivation by using the Star Board interactive school board in the classroom;
   • improving skills in working with the Star Board.

Equipment:

• Textbook “Mathematics” 3rd grade, part 2 (L.G. Peterson);
• individual handout sheet ;
• cards for working in pairs;
• presentation for the lesson displayed on the Star Board panel;
• computer, projector, Star Board.

During the classes

I. Organizational moment.

And so, friends, attention.
After all, the bell rang
Sit back comfortably
Let's start the lesson soon!

II. Verbal counting.

– Today we will go with you to visit. After listening to the poem, you will be able to name the hostess. (Reading a poem by a student)

For centuries, mathematics has been covered in glory,
The luminary of all earthly luminaries.
Her majestic queen
No wonder Gauss christened it.
We praise the human mind,
The works of his magical hands,
The hope of this century,
Queen of all earthly sciences.

– And so, Mathematics awaits us. There are many principalities in her kingdom, but today we will visit one of them (slide 4)

– You will find out the name of the principality by solving the examples and arranging the answers in ascending order. ( Statement)

- Let's remember what a statement is? ( Statement)

– What could the statement be? (True or False)

– Today we will work with mathematical statements. What does this mean? (expression, equalities, inequalities, equations)

III. Stage 1. CHALLENGE. Preparing to learn new things.

(slide 5 see note)

– Princess Saying offers you the first test.

- There are cards in front of you. Find an extra card and show it (a + 6 – 45 * 2).

- Why is she superfluous? (Expression)

– Is the expression a complete statement? (No, it is not, because it has not been brought to its logical conclusion)

– What are equality and inequality? Can they be called statements?

– Name the correct equalities.

– What is another name for true equalities? ( true)

– What about the infidels? (false)

– What equations cannot be said to be true? ( with variable)

– Mathematics constantly teaches us to prove the truth or falsity of our statements.

IV. Communicate the purpose of the lesson.

– And today we must learn what equality and inequality are and learn to determine their truth and falsity.

- Here are statements before you. Read them carefully. If you think it is correct, then put “+” in the first column; if not, put “–”.

Collective verification with justification for your assumption.

V. Stage 2. REFLECTION. Learning new things.

– How can we check if our assumptions are correct?

(textbook p. 74.)

– What is equality?

– What is inequality?

“We have completed the task of Princess Saying, and as a reward she invites us to a holiday.”

VI. Physical education minute.

VII. Stage 3. REFLECTION-REFLECTION

1. p. 75.5 (displayed) (slide 8)

– Read the task, what needs to be done?

– How many equalities did you emphasize? Let's check.

– How many inequalities?

– What helped you complete the task? (signs “=”, “>”, “<»)

– Why were there ununderlined entries? (expressions)

2. Game “Silence” (slide 9)

(Students write down equalities on narrow strips and show them to the teacher, then check themselves).

Write the statement as an equality:

• 5 is more than 3 by 2 (5 – 3 = 2)
• 12 is 6 times greater than 2 (12: 2 = 6)
• x is less than y by 3 (y – x = 3)

3. Solving equations (slide 10)

– What is in front of us? (equations, equalities)

– Can we tell whether they are true or false? (no, there is a variable)

– How to find at what value of a variable the equalities are true? (decide)

• 1 column – 1 column
• Column 2 – Column 2
• 3 column – 3 column

Exchange notebooks and check your friend's work. Rate it.

VIII. Lesson summary.

– What concepts did we work with today?

– What kind of equality can there be? (false or true)

– Do you think it’s only in mathematics lessons that we need to be able to distinguish false statements from true ones? (A person encounters a lot of different information in his life, and one must be able to separate the true from the false).

IX. Assessing student work and assigning grades.

– What can Queen Mathematics thank us for?

Note. If the teacher is using the Star Board, this slide is replaced with cards typed on the board. When checking, students work on the board.

Two numerical mathematical expressions connected by the sign “=” are called equality.

For example: 3 + 7 = 10 - equality.

Equality can be true or false.

The point of solving any example is to find a value of the expression that turns it into a true equality.

To form ideas about true and false equalities, examples with a window are used in the 1st grade textbook.

For example:

Using the selection method, the child finds suitable numbers and checks the accuracy of the equality by calculation.

The process of comparing numbers and indicating the relationships between them using comparison signs leads to inequalities.

For example: 5< 7; б >4 - numerical inequalities

Inequalities can also be true or false.

For example:

Using the selection method, the child finds suitable numbers and checks the accuracy of the inequality.

Numerical inequalities are obtained by comparing numerical expressions and numbers.

For example:

When choosing a comparison sign, the child calculates the value of the expression and compares it with a given number, which is reflected in the choice of the corresponding sign:

10-2>7 5+K7 7 + 3>9 6-3 = 3

Another way to select a comparison sign is possible - without reference to calculating the value of the expression.

Nappimep:

The sum of the numbers 7 and 2 will obviously be greater than the number 7, which means 7 + 2 > 7.

The difference between the numbers 10 and 3 will obviously be less than the number 10, which means 10 - 3< 10.

Numerical inequalities are obtained by comparing two numerical expressions.

To compare two expressions means to compare their meanings. For example:

When choosing a comparison sign, the child calculates the meanings of expressions and compares them, which is reflected in the choice of the corresponding sign:

Another way to select a comparison sign is possible - without reference to calculating the value of the expression. For example:

To set comparison signs, you can carry out the following reasoning:

The sum of the numbers 6 and 4 is greater than the sum of the numbers 6 and 3, since 4 > 3, which means 6 + 4 > 6 + 3.

The difference between the numbers 7 and 5 is less than the difference between the numbers 7 and 3, since 5 > 3, which means 7 - 5< 7 - 3.

The quotient of 90 and 5 is greater than the quotient of 90 and 10, because when dividing the same number by a larger number, the quotient is smaller, which means 90: 5 > 90:10.

To form ideas about true and false equalities and inequalities, the new edition of the textbook (2001) uses tasks of the form:

To check, the method of calculating the meaning of expressions and comparing the resulting numbers is used.

Inequalities with a variable are practically not used in the latest editions of the stable mathematics textbook, although they were present in earlier editions. Inequalities with variables are actively used in alternative mathematics textbooks. These are inequalities of the form:

 + 7 < 10; 5 -  >2;  > 0;  > O

After introducing a letter to denote an unknown number, such inequalities take on the familiar form of inequalities with a variable:

a + 7>10; 12-d<7.

The values ​​of unknown numbers in such inequalities are found by selection, and then each selected number is checked by substitution. The peculiarity of these inequalities is that several numbers can be selected that fit them (giving the correct inequality).

For example: a + 7 > 10; a = 4, a = 5, a = 6, etc. - the number of values ​​for the letter a is infinite, any number a > 3 is suitable for this inequality; 12 - d< 7; d = 6, d = 7, d = 8, d = 9, d = 10, d = 11, d = 12 - количество значений для буквы d конечно, все значения могут быть перечислены. Ребенок подставляет каждое найденное значение переменной в выражение, вычисляет значение выражения и сравнивает его с заданным числом. Выбираются те значения переменной, при которых неравенство является верным.

In the case of an infinite number of solutions or a large number of solutions to an inequality, the child is limited to selecting several values ​​of the variable for which the inequality is true.

“Equality” is a topic that students are taught as early as primary school. It also goes hand in hand with “Inequalities.” These two concepts are closely interrelated. In addition, they are associated with terms such as equations and identities. So what is equality?

Concept of equality

This term refers to statements that contain the “=” sign. Equalities are divided into true and false. If in the entry instead of = there is<, >, 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 school also studies the topic “Numerical Equality.” This statement refers to two numerical expressions that stand on either side of the = sign. For example, 2*5+7=17. Both parts of the record are equal to each other.

Numeric expressions of this type may use parentheses that affect the order of operations. So, there are 4 rules that should be taken into account when calculating the results of numerical expressions.

1. If there are no brackets in the entry, then the actions are performed from the highest level: III→II→I. If there are several actions of the same category, then they are performed from left to right.
2. If there are parentheses in the entry, then the action is performed in parentheses and then in steps. There may be several actions in brackets.
3. If the expression is presented as a fraction, then you need to calculate the numerator first, then the denominator, then divide the numerator by the denominator.
4. If a record has 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? Studying this concept requires knowledge of the properties of numerical identities. The text formulas below 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 both parts of the existing expression.

A = B↔ A + 5 = B + 5

2. The equation will not be violated if both its parts are multiplied or divided by the same number or expression that is different from zero.

P = O↔ P ∙ 5 = O ∙ 5

P = O↔ P: 5 = O: 5

3. By adding to both sides of the identity the same function, which makes sense for any admissible values ​​of the variable, we obtain 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 moved to the other side of the equal sign, but the signs must be reversed.

X + 5 = Y - 20 ↔ X = Y - 20 - 5 ↔ X = Y - 25

5. By multiplying or dividing both sides of the equation by the same function, which is different from zero and has meaning for each value of X from the ODZ, we obtain 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 clearly indicate the principle of equality, which exists under certain conditions.

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, the result will be the ratio of the number A to the number B. A proportion is the equality of two ratios:

Sometimes the proportion is written as follows: A:B=C:D. This implies the basic property of proportion: A*D=D*C, where A and D are the extreme terms of the proportion, and B and C are the average.

Identities

An identity is an equality that will be true for all permissible values ​​of the variables included in the task. Identities can be represented as literal or numeric equalities.

Expressions that contain an unknown variable on both sides of the equality that can equate two parts of one whole are called identically equal.

If you 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 abbreviated multiplication formulas, the laws of arithmetic and other identities.

To reduce a fraction, you need to carry out identical transformations. For example, given a fraction. To get the result, you should use abbreviated multiplication formulas, factorization, simplifying expressions, and reducing fractions.

It is worth considering that this expression will be identical when the denominator is not equal to 3.

5 ways to prove identity

To prove the identity equality, you need to transform the expressions.

Method I

It is necessary to carry out equivalent transformations on the left side. The result is the right-hand side, and we can say that the identity is proven.

II method

All expression transformation actions occur on the right side. The result of the manipulations performed is the left side. If both parts are identical, then the identity is proven.

III method

“Transformations” occur in both parts of the expression. If the result is two identical parts, the identity is proven.

IV method

The right side is subtracted from the left side. As a result of equivalent transformations, the result should be zero. Then we can talk about the identity of the expression.

V method

The left side is subtracted from the right side. All equivalent transformations are reduced to ensuring that the answer contains zero. Only in this case can we talk about the identity of equality.

Basic properties of identities

In mathematics, the properties of equalities are often used to speed up the calculation process. Thanks to basic algebraic identities, the process of calculating some expressions will take a matter of 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) = 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) = X + Y - C
• X - (Y + C) = X - Y - C
• X - (Y - C) = 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 = A 2 + 2∙A∙B + B 2 - the square of the sum of a pair of numbers;

• (A - B) 2 = A 2 - 2∙A∙B + B 2 - squared difference of a pair of numbers;

• (C + B) ∙ (C - B) = C 2 - B 2 - difference of squares;

• (A + B) 3 = A 3 + 3∙A 2 ∙B + 3∙A∙B 2 + B 3 - cube of the sum;

• (A - B) 3 = A 3 - 3∙A 2 ∙B + 3∙A∙B 2 - B 3 - cube of the difference;

• (P + B) ∙ (P 2 - P∙B + B 2) = P 3 + B 3 - sum of cubes;

• (P - B) ∙ (P 2 + P∙B + B 2) = P 3 - B 3 - difference of cubes.

Abbreviated multiplication formulas are often used if it is necessary to bring a polynomial to its usual form, simplifying it in all possible ways. The presented formulas are easy to prove: just open the brackets and add similar 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 unknown quantities are present. Solving an equation is finding all values ​​of a variable such that both sides of the entire expression are equal. There are also tasks in which finding solutions to an equation is impossible. In this case they say that there are no roots.

As a rule, equalities with unknowns produce integers as solutions. However, there may be cases where the root is a vector, function, or other objects.

An equation is one of the most important concepts in mathematics. Most scientific and practical problems do not allow one 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.

Typically, solving an equality with an unknown comes down to transforming a complex equation and reducing it to simple forms. It must be remembered that transformations must be carried out on both sides, otherwise the output will be incorrect.

4 ways to solve an equation

By solving an equation we mean replacing a given equality with another, which is equivalent to the first. Such a substitution is known as an identity transformation. To solve the equation, you must use one of the methods.

1. One expression is replaced by another, which will necessarily 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. Transfer of 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 an equality by an equal number or expression that does not equal 0. However, it is worth remembering that if the new equation is not equivalent to the equality before the 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 nuance here: if you raise the equation to an even power, then extraneous roots may appear that will distort the essence of the task. And if you extract the root incorrectly, 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. They all come from the concept of “equality.” Thanks to various kinds of equivalent expressions, the solution of some problems is greatly facilitated.

EQUALITIES WITH QUANTITIES.

After the child becomes familiar with quantity cards from 1 to 20, you can add a second stage to the first stage of training - equalities with quantities.

What is equality? This is an arithmetic operation and its result.

You begin this stage of learning with the topic "Addition".

Addition.

By showing two sets of quantity cards, you add addition equations.

This operation is very easy to teach. 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 dot has appeared on it.

The baby doesn’t yet know what it’s called, but he 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.

Technology for showing equalities looks something like this: You want to give the child the equality: 1 +2 = 3. How can you show it?

Before starting the lesson, place three cards face down on your lap, one on top of the other. Picking up the top card with one knuckle spoke, say "one", then put it aside and say "plus", show a card with two dominoes, say "two", put it aside after the word "will", show a card with three dominoes, saying "three".

A day you conduct three classes with equalities and at each lesson you show three different equalities. In total, the baby sees nine different equalities a day.

The child understands without any explanation what the word means "plus", he himself deduces its meaning from the context. By performing actions, you thereby demonstrate the true meaning of addition faster than any explanation. When talking about equalities, always adhere to the same manner of presentation, using the same terms. Having said "One plus two equals three" don't talk later “Two added to one equals three.” When you teach a child facts, he draws his own conclusions and learns 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 a particular equality. Don’t think that your child will sit quietly and watch you rummage through a stack of cards, selecting the ones you need. He will simply run away and be right, since his time is worth no less than yours.

Try not to create equalities that 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 concepts. After about two weeks, the baby makes a discovery about what addition is: after all, during this time you showed him 126 different equations for addition.

Examination.

Checking at this stage is solving examples.

How is an example different from an 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 post an example after a regular lesson with three addition equations. You show the example in the same way as you demonstrated equality before. That is, you rearrange the cards in your hands, saying each one out loud. For example, “twenty plus ten is thirty or forty-five?” and show the child two cards, one of which has the correct answer.

Cards with answers should be kept at the same distance from the baby’s eyes and no prompting actions should be allowed.

When you choose the right child, you vigorously express your delight, kiss and praise him.

If you choose the wrong answer, without expressing disappointment, you move the card with the correct answer towards the baby and ask the question: “It 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 child solves at least six correctly, then it’s definitely time for you to move on to subtraction equations!

If you don’t think it’s necessary to check your child (and rightly so!), then after 10-14 days, still move on to subtraction equations!

Consider -Subtraction.

You stop doing addition and switch completely to subtraction. Conduct three daily lessons with three different equalities in each.

Express subtraction equations like this: “Twelve minus seven is five.”

At the same time, you 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 for no more than two weeks.

Examination

Testing, just as in the case of addition, can involve solving examples with choosing one answer out of two.

Consider-Multiplication.

Multiplication is nothing more than repeated addition, so this action will not be a big discovery for your child. As you continue to study quantity cards (two sets of five cards each), you have the opportunity to create multiplication equations.

Pronounce multiplication equalities like this: “Two times three equals six.”

The child will understand the word "multiply" as quickly as he understood this word before "plus" And "minus".

You still teach three lessons a day, each of which contains three different multiplication equations. This work lasts no more than two weeks.

Continue to avoid 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 liked the games of checking boxes with quantity cards, you can continue to play them, thus repeating new, larger quantities.

Adhering to the scheme we have proposed, by this time you can already complete the first stage of learning mathematics - study quantities within 100. Now it’s time to get acquainted with the card that children like best.

Let's consider the concept of zero.

They say that mathematicians have been studying the idea of ​​zero for five hundred years. Whether this is true or not, children, having barely learned the idea of ​​quantity, immediately understand the meaning of its complete absence. They simply adore zero, and your journey into the world of numbers will be incomplete if you do not show your baby a card that does not have any dots on it at all (i.e. it will be a completely blank card).

To make your child’s acquaintance with zero fun and interesting, you can accompany the display of the card with a riddle:

At home there are seven baby squirrels, On the plate there are seven honey mushrooms. All the mushrooms ate the squirrels. What's left on the plate?

When pronouncing the last phrase, we show the “zero” card.

You will use it almost every day. It will be 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 conduct 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

Solving examples with zero follows a familiar pattern.

Consider -Division.

When you have completed all the quantity cards from 0 to 100, you have all the necessary material for division examples with quantities.

The technology for displaying equalities for this topic is the same. Every day you conduct three classes. At each lesson, you show your child three different equalities. It is good if the passage of this material does not exceed two weeks.

Examination

The test consists of solving examples with choosing 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 you conduct classes. Now it is not necessary to show each knitting needle card; you can only show the answer, and only pronounce the actions themselves. As a result, your classes will become shorter. You simply tell the child: “Twenty two divided by eleven, divided by two equals one,”- and show him the “one” card.

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 an equality, remember that multiplication and division must be placed at the beginning of the equality:

Don't 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 baby will do great with them!

The test consists of solving examples with choosing one answer out of two. You can demonstrate an example by showing all the equality cards and two cards for choosing an answer, or simply say all the equality, showing your child only two cards for the answer.

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 the equalities that have not yet been shown).

Sequences

Sequences are the same equalities. Parents' experience with this topic has shown that children find sequences very interesting.

Plus sequences are increasing sequences. Sequences with a minus are decreasing.

The more varied the sequences are, 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 showing sequences can be like this. You have prepared three sequences for plus.

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 your 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.

Sequences can also be laid out one below the other, leaving gaps between them.

Try to always move forward, moving from simple to complex. Vary the activities: sometimes say out loud 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, 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 entire 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 one is next?”

At first, lay out the cards in sequences one after another, then you can change the layout forms: place 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.

Examples of sequences:

4; 6; 8; 10; 12; 14 - in this sequence, each subsequent number increases by 2;

2; 4; 7; 14; 17; 34 - in this sequence multiplication and addition alternate (x 2; + 3);

2; 4; 8; 16; 32; 64 - in this sequence, each subsequent number is increased by 2 times;

22; 18; 14; 10; 6; 2 - in this sequence, each subsequent number is reduced by 4;

84; 42; 40; 20; 18; 9 - in this sequence division and subtraction alternate (: 2; - 2);

Signs "greater than", "less than"

These cards are included in 110 cards of numbers and signs (the second component of the ANASTA method).

Lessons to introduce your child to the concepts of “more and less” will be very short. All you need 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. You name each card.

You can say it like this: "six is ​​more than three" or "six is ​​more than three."

At each lesson, you show your child three different versions of inequalities with

cards “more” - “less”. inequalities per day.

So you are showing nine different

As before, you show each inequality only once.

After a few days, you can add an example to the three shows. It's already examination, and it goes 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 baby: “Sixty-eight is greater than what number?” or “Is sixty-eight over fifty or ninety-five?” Invite your child to choose the one he needs from two cards. You (or he himself) place the correct card indicated by the child after the “more” sign.

You can put two cards with quantities in front of the child and give him the opportunity to choose the sign that fits, that is, > or<.

Equalities and inequalities

Equalities and inequalities are as easy to teach as the concepts of “more” and “less.”

You will need six arithmetic symbol cards. You will also find them as part of 110 cards of numbers and signs (the second component of the ANASTA method).

Display technology

You decided to show your child the following two inequalities and one equality:

8-6<10 −7 11-3= 9 −1 55-12^50 −13

You place them on the floor sequentially so that the child can see each of them at once. At the same time, you say everything, for example: "Eight minus six does not equal 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 you can only show "equal" and "not equal" cards.

One day you give your child the opportunity to show his knowledge. You lay out cards with quantities, and ask him to choose which card with which sign should be placed: “equal” or “not equal.”

Before you start learning algebra with your child, you need to introduce him to the concept of a variable represented by a letter.

The letter x is commonly used in mathematics, but since it can easily be confused with the multiplication sign, it is recommended to use y.

You first place a card with five domino beads, then a plus sign (+), followed by a y sign, then an equals sign, and finally a card with seven domino beads. Then you ask the question: “What does y mean here?”

And you answer it yourself: “In this equation it means two.”

Examination:

After about one to a week and a half of classes at this stage, you can give your child the opportunity to choose an answer.

FOURTH STAGE OF EQUALITY WITH NUMBERS AND QUANTITIES

When you have gone through the numbers 1 to 20, it is time to “build bridges” between numbers and quantities. There are many ways to do this. One of the simplest is the use of equalities and inequalities, the relationships of “more” and “less”, demonstrated using cards with numbers and dominoes.

Display technology.

Take a card with the number 12, place it on the floor, then place a “greater than” sign next to it, and then a card with the number 10, while saying: “Twelve is over ten.”

Inequalities (equalities) may look like this:

Each (equalities) day consists of three lessons, and each lesson consists of three inequalities in quantities and numbers. The total number of daily equalities will be nine. At the same time, you continue to study numbers using two sets of five cards each, also three times a day.

Examination.

You can give your child the opportunity to choose cards “more than”, “less than”, “equal to”, or create an example in such a way that the child can finish it himself. 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 child understands the connection between quantities and numbers, you can begin to solve equalities using cards with both numbers and quantities.

Equalities 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 operations.

If you carefully look at the approximate mathematics teaching scheme (p. 20), you will see that there is no end to the lessons. Come up with your own examples for developing a child’s mental counting, correlate 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!

50. Properties of equalities on which the solution of equations is based. Let's take some equation, not very complicated, for example:

7x – 24 = 15 – 3x

x/2 – (x – 3)/3 – (x – 5)/6 = 1

We see in each equation an equal sign: everything that is written to the left of the equal sign is called the left or first part of the equation (in the first equation 7x – 24 is the left or first part, and in the second x/2 – (x – 3)/ 3 – (x – 5)/6 is the first, or left, part); everything that is written to the right of the equal sign is called the right or second part of the equation (15 – 3x is the right side of the first equation, 1 is the right, or second, part of the 2nd equation).

Each part of any equation represents a number. The numbers expressed by the left and right sides of the equation must be equal to each other. It is clear to us: if we add the same number to each of these numbers, or subtract the same number from them, or multiply each of them by the same number, or, finally, divide by the same number, then the results of these actions should also be equal to each other. In other words: if a = b, then a + c = b + c, a – c = b – c, ac = bc and a/c = b/c. Regarding division, it should be borne in mind, however, that in arithmetic there is no division by zero - we cannot, for example, divide the number 5 by zero. Therefore, in the equality a/c = b/c, the number c cannot be equal to zero.

1. The same number can be added to or subtracted from both sides of the equation.
2. Both sides of an equation can be multiplied or divided by the same number, unless the number is zero.

Using these properties of the equation, we can find a convenient way to solve the equations. Let's clarify this case with examples.

Example 1. Suppose we need to solve the equation

5x – 7 = 4x + 15.

We see that the first part of the equation contains two terms; one of them 5x, containing the unknown factor x, can be called the unknown term, and the other -7 - known. The second part of the equation also has 2 terms: unknown 4x and known +15. Let's make sure that on the left side of the equation there are only unknown terms (and the known term –7 would be destroyed), and on the right side there would be only known terms (and the unknown term +4x would be destroyed). For this purpose, we add the same numbers to both sides of the equation: 1) add +7 each (so that the –7 term is destroyed) and 2) add –4x each (so that the +4x term is destroyed). Then we get:

5x – 7 + 7 – 4x = 4x + 15 + 7 – 4x

Having reduced similar terms in each part of the equation, we get

This equality is the solution to the equation, since it indicates that for x we ​​must take the number 22.

Example 2. Solve the equation:

8x + 11 = 7 – 4x

Again we add –11 and +4x to both sides of the equation, we get:

8x + 11 – 11 + 4x = 7 – 4x – 11 + 4x

Reducing similar terms, we get:

Now divide both sides of the equation by +12, we get:

x = –4/12 or x = –1/3

(the first part of the equation 12x divided by 12 - we get 12x/12 or just x; the second part of the equation -4 divided by +12 - we get -4/12 or -1/3).

The last equality is the solution to the equation, since it indicates that for x we ​​need to take the number –1/3.

Example 3. Solve with equation

x – 23 = 3 (2x – 3)

Let's first open the brackets and get:
x – 23 = 6x – 9

Add +23 and –6x to both sides of the equation, we get:

x – 23 + 23 – 6x = 6x – 9 + 23 – 6x.

Now, in order to subsequently speed up the process of solving the equation, we will not immediately perform the reduction of all similar terms, but only note that the terms –23 and +23 on the left side of the equation cancel each other out, and the terms +6x and –6x in the first part cancel each other out. are destroyed - we get:

x – 6x = –9 + 23.

Let's compare this equation with the initial one: in the beginning there was an equation:

x – 23 = 6x – 9

Now we have the equation:

x – 6x = –9 + 23.

We see that in the end it turned out that the term –23, which was initially on the left side of the equation, now seemed to move to the right side of the equation, and its sign changed (there was a term –23 on the left side of the initial equation, but now it is not there , but on the right side of the equation there is a term + 23, which was not there before). Similarly, on the right side of the equation there was a term +6x, now it is not there, but on the left side of the equation a term –6x has appeared, which was not there before. Considering examples 1 and 2 from this point of view, we come to a general conclusion:

You can transfer any term of the equation from one part to another by changing the sign of this term(we will use this in further examples).

So, returning to our example, we have the equation

x – 6x = –9 + 23

Divide both sides of the equation by –5. Then we get:

[–5x: (–5) we get x] – this is the solution to our equation.

Example 4. Solve the equation:

Let's make sure there are no fractions in the equation. For this purpose, we will find a common denominator for our fractions - the common denominator is the number 24 - and multiply both sides of our equation by it (it is possible, after all, to ensure equality is not violated, we can multiply only both sides of the equation by the same number). The first part has 3 terms, and each term is a fraction - it is necessary, therefore, to multiply each fraction by 24: the second part of the equation is 0, and multiply zero by 24 - we get zero. So,

We see that each of our three fractions, due to the fact that it is multiplied by the common least multiple of the denominators of these fractions, will be reduced and become a whole expression, namely we get:

(3x – 8) 4 – (2x – 1) 6 + (x – 7) 3 = 0

Of course, it is advisable to do all this in our minds: we need to imagine that, for example, the numerator of the first fraction is placed in parentheses and multiplied by 24, after which our imagination will help us see the reduction of this fraction (by 6) and the final result, i.e. (3x – 8) · 4. The same holds for other fractions. Let us now open the brackets in the resulting equation (on its left side):

12x – 32 – 12x + 6 + 3x – 21 = 0

(please note that here it was necessary to multiply the binomial 2x – 1 by 6 and subtract the resulting product 12x – 6 from the previous one, due to which the signs of the terms of this product should change - above it is written –12x + 6). Let's move the known terms (i.e. –32, +6 and –21) from the left side of the equation to its right side, and (as we already know) the signs of these terms should change - we get:

12x – 12x + 3x = 32 – 6 + 21.

Let's cast similar terms:

(with skill, you should immediately transfer the necessary terms from one part of the equation to another and bring similar terms), finally, divide both sides of the equation by 3 - we get:

x = 15(2/3) - this is the solution to the equation.

Example 5. Solve the equation:

5 – (3x + 1)/7 = x + (2x – 3)/5

There are two fractions here, and their common denominator is 35. To free the equation from fractions, we multiply both sides of the equation by the common denominator 35. Each part of our equation has 2 terms. When multiplying each part by 35, each term must be multiplied by 35 - we get:

The fractions are reduced and we get:

175 – (3x + 1) 5 = 35x + (2x – 3) 7

(of course, if you had the skill, you could write this equation right away).

Let's do all the steps:

175 – 15x – 5 = 35x + 14x – 21.

Let's move all unknown terms from the right side (i.e., terms +35x and +14x) to the left, and all known terms from the left side (i.e., terms +175 and –5) to the right - we should not forget transferred members change sign:

–15x – 35x – 14x = –21 – 175 + 5

(the term –15x, as it used to be on the left side, remains in it now - therefore, it should not change its sign at all; a similar thing occurs for the term –21). Having reduced similar terms, we get:

–64x = –191.

[It is possible to make sure that there is no minus sign on both sides of the equation; To do this, we multiply both sides of the equation by (–1), we get 64x = 191, but we don’t have to do this.]
We then divide both sides of the equation by (–64), and obtain a solution to our equation

[If we multiplied both sides of the equation by (–1) and got the equation 64x = 191, then now we need to divide both sides of the equation by 64.]

Based on what we had to do in examples 4 and 5, we can establish: it is possible to free the equation from fractions - to do this, we need to find the common denominator for all fractions included in the equation (or the least common multiple of the denominators of all fractions) and multiply both by it parts of the equation - then the fractions should disappear.

Example 6. Solve the equation:

Moving the 4x term from the right side of the equation to the left, we get:

5x – 4x = 0 or x = 0.

So, the solution has been found: for x we ​​need to take the number zero. If we replace x in this equation with zero, we get 5 0 = 4 0 or 0 = 0, which indicates that the requirement expressed by this equation is met: find a number for x such that the monomial 5x is equal to the same number as monomial 4x.

If one notices from the very beginning that both sides of the equation 5x = 4x can be divided by x and performs this division, the result is a clear inconsistency: 5 = 4! The reason for this is that dividing 5x/x cannot be done in this case, since, as we saw above, the question expressed by our equation requires that x = 0, and division by zero is not possible.

Let us also note that multiplying by zero requires some care: multiplying by zero and two unequal numbers, as a result of these multiplications we will obtain equal products, namely zeros.

If, for example, we have the equation

x – 3 = 7 – x (his solution: x = 5)

and if someone wants to apply to it the property “both sides of the equation can be multiplied by the same number” and multiply both sides by x, they will get:

x 2 – 3x = 7x – x 2.

After this, you may notice that all terms of the equation contain a factor x, from which we can conclude that to solve this equation we can take the number zero, that is, put x = 0. And indeed, then we get:
0 2 – 3 0 = 7 0 – 0 2 or 0 = 0.

However, this solution x = 0 is obviously not suitable for the given equation x – 3 = 7 – x; replacing x with zero, we get an obvious inconsistency: 3 = 7!