When equality is possible. The concept of equality, the sign of equality associated with definitions. The main properties of identities
"Equality" is a topic that students take place in primary school. It also accompanies her "inequality". These two concepts are closely interrelated. In addition, such terms such as equations, identities are associated with them. So, what is equality?
The concept of equality
Under this term understand the statements, in which there is a sign "\u003d". Equality are divided into faithful and incorrect. If in the record instead \u003d<, >, then we are talking About inequalities. By the way, the first sign of equality says that both parts of the expression are identical on their result or record.
In addition to the concept of equality, the topic "Numerical equality" is also studied at school. Under this statement, there are two numerical expressions that are standing on both sides of the sign \u003d. For example, 2 * 5 + 7 \u003d 17. Both parts of the record are equal to each other.
In the numerical expressions of this type, brackets affecting the procedure may be used. 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 record, then the actions are performed from the highest level: III → II → I. If there are several actions of one category, then they are performed from left to right.
- If there are brackets in the recordings, 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 presented in the form of a fraction, then you need to calculate the numerator first, then the denominator, then the numerator is divided into the denominator.
- If there are nested brackets in the record, then the expression in internal brackets is calculated first.
So now it is clear what equality is. In the future, the concepts of equation, identities and methods for their calculation will be considered.
Properties of numerical equality
What is equality? The study of this concept requires knowledge of the properties of numerical identities. The text formulas below allow you to better explore this topic. Of course, these properties are more suitable for learning mathematics in high school.
1. Numerical equality will not be impaired if in both parts it is added to the same number to an existing expression.
A \u003d B.↔ a + 5 \u003d in + 5
2. The equation will not be broken if both parts of it are multiplied or divided into one and the same number or expression that are different from zero.
P \u003d O.↔ P ∙ 5 \u003d Oh 5
P \u003d O.↔ R: 5 \u003d A: 5
3. Adding the same function to both parts of the identity, which makes sense with any permissible values \u200b\u200bof the variable, we get new equality, equivalent to the original.
F (x) \u003d ψ(X) ↔ F (x) + r (x) \u003dΨ (X) + R (x)
4. Any category or expression can be transferred through the other side of the equality sign, while you need to change signs to opposite.
X + 5 \u003d y - 20 ↔ X \u003d y - 20 - 5 ↔ X \u003d y - 25
5. Multiplying or separating both parts of the equation on the same function other than zero and having a meaning for each value x from OTZ, we obtain a new equation, equivalent to the original.
F (X) \u003d ψ (X)↔ F (X) ∙R (X) \u003d ψ (X) ∙R (X)
F (x) \u003d ψ(X)↔ F (x): G (x) \u003d ψ(X): G (x)
The above rules indicate the principle of equality that exists under certain conditions.
The concept of proportion
In mathematics, there is such a concept as equality of relationships. In this case, the determination of proportion is meant. If divided by in, then the result will be the ratio of the number A to the number of V. The proportion is called the equality of two relations:
Sometimes the proportion is written as follows: A:B \u003d.C:D.Hence the main property of the proportion: A *D \u003dD *C., where a and d are extreme members of the proportion, and in and c - medium.
Identities
The identity is called equality that will be true with all valid values \u200b\u200bof the variables that are included in the task. Identities can be represented as alphabetic or numerical equality.
They are identically equal to the expressions containing an unknown variable in both parts of equality, which is capable of equating two parts of one whole.
If you replace one expression to others, which will be equal to it, then we are talking about identical conversion. In this case, you can take advantage of the formulas of abbreviated multiplication, the laws of arithmetic and other identities.
To reduce the fraction, you need to carry out identical transformations. For example, the fraction is given. To obtain the result, you should take advantage of the formulas of abbreviated multiplication, decomposition of multipliers, simplifying expressions and reduction of 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 equality identical, you need to transform expressions.
I method
Must be spent equipment transformations in the left side. The result is the right side, and we can say that the identity is proved.
II way
All expression conversion activities occur in the right part. The result of the manipulation done is the left part. If both parts are identical, then the identity is proved.
III way
"Transformation" occur in both parts of the expression. If the result is two identical parts, the identity is proved.
IV method
The right is deducted from the left side. As a result of equivalent transformations, zero should turn out. Then you can talk about the identity of the expression.
V method
From the right side, the left is deducted. All equivalent transformations are reduced to the answer stood zero. Only in this case can we talk about the identity of equality.
The main properties of identities
In mathematics, it is often used by the properties of equations to speed up the calculation process. Thanks to basic algebraic identities, the process of calculating some expressions will take a few minutes instead of long hours.
- X + y \u003d u + x
- X + (y + s) \u003d (x + y) + with
- X + 0 \u003d x
- X + (s) \u003d 0
- X ∙ (y + c) \u003d x ∙ u + x ∙ with
- X ∙ (y - c) \u003d x ∙ u - x ∙ with
- (X + y) ∙ (c + e) \u200b\u200b\u003d x ∙ s + x ∙ e + y ∙ c + y ∙ e
- X + (y + c) \u003d x + y + with
- X + (y - c) \u003d x + y - with
- X - (u + s) \u003d x - y - with
- X - (y - c) \u003d x - y + with
- X ∙ y \u003d y ∙ x
- X ∙ (y ∙ s) \u003d (x ∙ y) ∙ with
- X ∙ 1 \u003d x
- X ∙ 1 / x \u003d 1, where x ≠ 0
Formulas of abbreviated multiplication
In essence, the formula of abbreviated multiplication is equalities. They help solve many tasks in mathematics due to their simplicity and ease.
- (A + c) 2 \u003d a 2 + 2 ∙ A ∙ B + in 2 - the square of the amount of the pair of numbers;
- (A - c) 2 \u003d a 2 - 2 ∙ A ∙ B + in 2 - the square of the difference pair of numbers;
- (C + C) ∙ (C - B) \u003d C 2 - in 2 - the difference of squares;
- (A + c) 3 \u003d a 3 + 3 ∙ A 2 ∙ B + 3 ∙ A ∙ 2 + in 3 cube amounts;
- (A - c) 3 \u003d a 3 - 3 ∙ A 2 ∙ B + 3 ∙ A ∙ in 2 - in 3 cube difference;
- (P + c) ∙ (p 2 - p ∙ B + in 2) \u003d p 3 + in 3 - the sum of the cubes;
- (P - c) ∙ (p 2 + p ∙ B + in 2) \u003d p 3 - in 3 - the difference of cubes.
The formulas of abbreviated multiplication are often used if it is necessary to bring a polynomial to the usual mind, simplifying it by everyone possible methods. The presented formulas are simply proved: it is enough to reveal the brackets and bring similar terms.
Equations
After studying the issue, which such equality can be proceeding to the following item: the equation is understood as equality in which unknown values \u200b\u200bare present. The solution of the equation is called all the values \u200b\u200bof the variable, in which both parts of the entire expression will be equal. The tasks in which the finding of the equation solutions is impossible. In this case, they say that there are no roots.
As a rule, equality with unknown as a solution give out integers. However, there are cases when the root is the vector, function and other objects.
The equation is one of the most important concepts in mathematics. Most scientific I. practical tasks Do not allow you to measure or calculate any value. Therefore, it is necessary to make a relation that will satisfy all the conditions of the task. In the process of compiling this ratio, an equation or system of equations appear.
Usually, the decision of equality with the unknown is reduced to the transformation of the complex equation and the note to simple forms. It must be remembered that the transformation needs to be carried out relative to both parts, otherwise the output will be wrong.
4 ways to solve the equation
Under the solution of the equation, they understand the replacement of the specified equality to others, which is tantamount to the first. Such a substitution is known as identical conversion. To solve the equation, you must use one of the ways.
1. One expression is replaced by another, which is mandatory to be identically the first. Example: (3 ∙ x + 3) 2 \u003d 15 ∙ x + 10. This expression can be converted to 9 ∙ x 2 + 18 ∙ x + 9 \u003d 15 ∙ x + 10.
2. Transferring members of equality with an unknown one to another. In this case, it is necessary to change the signs correctly. The slightest mistake thwans the whole work done. As an example, take the previous "sample".
9 ∙ x 2 + 12 ∙ x + 4 \u003d 15 ∙ x + 10
9 ∙ x 2 + 12 ∙ x + 4 - 15 ∙ x - 10 \u003d 0
3. Multiplying both parts of equality per equal number or expression that are not equal to 0. However, it is necessary to recall that if the new equation is not equivalent to equality before the transformations, then the number of roots can change significantly.
4. Erend into the square of both parts of the equation. This method is simply wonderful, especially when in equality there are irrational expressions, that is, the expression under it. There is one nuance: if you build an equation to an even degree, then extraneous roots may appear, which are distorting the essence of the task. And if it is wrong to extract the root, then the meaning of the question in the task will be unclear. Example: │7 ∙ x│ \u003d 35 → 1) 7 ∙ x \u003d 35 and 2) - 7 ∙ x \u003d 35 → The equation will be solved correctly.
So, in this article, such terms are mentioned as equations and identities. All of them originate from the concept of "equality". Due to the various kinds of equivalent expressions, the solution of some tasks is largely facilitated.
Class: 3
Presentation to the lesson
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Type of lesson: Opening of new knowledge.
Technology: Technology for the development of critical thinking through reading and writing, game technology.
Objectives: Expand the knowledge of students on equals and inequalities, to acquaint with the concept of faithful and incorrect equalities and inequalities.
Didactic task: Organize joint, independent activities of students on the study of a new material.
Tasks lesson:
- Subject:
- introduce signs of equality and inequality; expand the ideas of students on equalities and inequalities;
- introduce the concept of faithful and incorrect equality and inequality;
- development of skills finding an expression value containing a variable;
- formation of computing skills.
- MetaPermet:
- Cognitive:
- promote the development of attention, memory, thinking;
- development of the ability to extract information, orient in their knowledge system and to realize the need for new knowledge;
- mastering the methods of selection and systematization of the material, the skills to compare and compare, convert information (into the scheme, table).
- Regulatory:
- development of visual perception;
- continue to work on the formation of the actions of self-control and self-assessment of students;
- Communicative:
- purchase over the interaction of children in pairs, make the necessary adjustments;
- relieve mutual assistance.
- Cognitive:
- Personal:
- raising training motivation students by using the Star Board interactive schoolboard at the lesson;
- improving work skills with Star Board.
Equipment:
- Tutorial "Mathematics" Grade 3, 2 / L.G. Peterson);
- individual distribution sheet ;
- cards for work in pairs;
- presentation to the lesson derived from the Star Board panel;
- computer, projector, Star Board panel.
During the classes
I. Organizational moment.
And so, friends, attention.
After all, the bell rang
Sit comfortably
Let's start the lesson!
II. Verbal counting.
- Today we will go with you to visit. Listening to the poem, you can call the name of the hostess. (Reading a pupil poem)
In the eyelids, mathematics was wanding with glory,
Light all the terrestrial shining.
Her queen is great
No wonder Gauss dubbed.
We are Slavim human mind,
Cases of his magic hands,
Hope of this century,
Queen of all earthly sciences.
- And so, we are waiting for mathematics. There are many principalities in her kingdom, but today we will visit one of them (slide 4)
- The name of the principality you will learn by deciding examples and putting the answers in ascending order. ( Statement)
| 7200: 90 = 80 | FROM | 280: 70 = 4 | AND | |
| 5400: 9 = 600 | S | 3500: 70 = 50 | Z. | |
| 2700: 300 = 9 | IN | 4900: 700 = 7 | BUT | |
| 4800: 80 = 60 | BUT | 1600: 40 = 40 | S | |
| 560: 8 = 70 | TO | 1800: 600 = 3 | E. | |
| 4200: 6 = 700 | IN | 350: 70 = 5 | N. |
- Let's remember what kind of saying? ( Statement)
- What could be a statement? (Faithful or wrong)
- We will work with mathematical statements today. What applies to them? (expression, equality, inequality, equations)
III. Stage 1. Call. Preparation for the study of the new.
(slide 5 cm. Note)
- Princess saying offered to you the first test.
- Before you card. Find an excess card show (A + 6 - 45 * 2).
- Why is it superfluous? (Expression)
- Is the expression of completed statement? (No, not, because it is not brought to a logical completion)
- And what is equality and inequality, can I call them a statement?
- Name faithful equality.
- How to call true equality in a different way? ( true)
- And wrong? (false)
- What kind of equalities can not be said that they are true? ( variable)
- Mathematics constantly teaches us to prove the truth or fart of our statements.
IV. Message Objectives lesson.
- And today we must learn what equality and inequality are and learn to determine their truth and falsity.
- Before you say. Read them carefully. If you think it is correct, then put "+" in the first column, if not - "-".
| Before reading | After reading | |
| Equality - these are two expressions connected by "\u003d" | ||
| Expressions may be numeric and letter. | ||
| If two numerical expressions, then equality is a statement. | ||
| Numeric equality can be true or false. | ||
| 6 * 3 \u003d 18 - faithful numerical equality | ||
| 16: 3 \u003d 8 - incorrect numerical equality | ||
| Two expressions connected by the sign "\u003e" or "<» - неравенство. | ||
| Numerical inequalities are statements. |
Collective verification with a justification for its assumption.
V. Stage 2. Comprehension. Studying new.
- How can we check whether our assumptions are true.
(tutorial p. 74.)
- What is equality?
- What is inequality?
- We fulfilled the task of the printed saying, and in the award she invites us for a holiday.
Vi. Fizkultminutka.
VII. Stage 3. Reflection reflection
1. p. 75, 5 (displayed) (slide 8)
- Read the task what needs to be done?
| 8 + 12 = 20 | a\u003e B. | |
| 8 + 12 + 20 | a - B. | |
| 8 + 12 > 20 | a + b \u003d with | |
| 20 = 8 + 12 | a + b * with |
- How many equalities stressed? Check.
- How much is inequal?
- What helped task? (signs "\u003d", "\u003e", "<»)
- Why not underlined records remained? (expressions)
2. Game "Molchanka" (slide 9)
(Students on narrow strips record equalities and show the teacher, then check themselves).
Write in the form of equality Saying:
- 5 more 3 to 2 (5 - 3 \u003d 2)
- 12 more 2nd 6 times (12: 2 \u003d 6)
- x less y 3 (y - x \u003d 3)
3. Solution of equations (slide 10)
- What about us? (equations, equality)
- Can we say faithful or false? (No, there is a variable)
- How to find, with what value of the variable is the right equality? (decide)
- 1 Column - 1 Stage
- 2 column - 2 columns
- 3 column - 3 columns
Change notebooks and check the work of your comrade. Rate.
VIII. The outcome of the lesson.
- What concepts did we work today?
- What can equality be? (false or true)
- What do you think, just on the lessons of mathematics you need to know how to distinguish false statements from the true? (A person in his life confronts a lot with various information, and we must be able to separate the true from false).
IX. Evaluating the work of students and setting marks.
- Why can we thank the Queen of Mathematics?
Note. If the teacher uses the online Star Board's interactive school board, this slide is replaced with cards scored on the board. When checking students work on the board.
Two numerical mathematical expressions, connected by the "\u003d" sign called equality.
For example: 3 + 7 \u003d 10 - equality.
Equality can be faithful and incorrect.
The meaning of solving any example is to find such an expression value that turns it into faithful equality.
For the formation of ideas about the faithful and incorrect equalities in a class textbook, examples with a window are used.
For example:
By selecting the child, the child finds suitable numbers and checks the loyalty of the equalization.
The process of comparing numbers and the designation of relations between them using comparison signs leads to inequalities.
For example: 5.< 7; б > 4 - Numerical inequalities
Inequalities can also be loyal and incorrect.
For example:
By selecting the child, the child finds suitable numbers and checks the loyalty of inequality.
Numerical inequalities are obtained when comparing numerical expressions and numbers.
For example:
When choosing a comparison mark, the child calculates the value of the expression and compares it with a given number, which is reflected in the choice of the corresponding mark:
10-2\u003e 7 5 + K7 7 + 3\u003e 9 6-3 \u003d 3
Another way to select a comparison sign is possible - without reference to the calculation of the expression value.
Capper:
The amount of numbers 7 and 2 will be obviously larger than the number 7, it means 7 + 2\u003e 7.
The difference between numbers 10 and 3 will be obviously smaller than the number 10, it means 10 - 3< 10.
Numerical inequalities are obtained when comparing two numeric expressions.
Compare two expressions - it means to compare their values. For example:
When choosing a comparison sign, the child calculates the values \u200b\u200bof 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 the calculation of the expression value. For example:
For comparison marks, you can conduct such arguments:
The sum of numbers 6 and 4 is larger than the amount of numbers 6 and 3, since 4\u003e 3, it means 6 + 4\u003e 6 \u200b\u200b+ 3.
The difference between numbers 7 and 5 is less than the difference between numbers 7 and 3, since 5\u003e 3, it means 7 - 5< 7 - 3.
The private numbers 90 and 5 are more than the private numbers 90 and 10, since when dividing the same number to the number more, the particular one turns out to be smaller, it means that 90: 5\u003e 90:10.
To form ideas about the faithful and incorrect equivals and inequalities in the new textbook version (2001), the tasks of the form are used:
To check, use the method for calculating the values \u200b\u200bof expressions and compare the numbers obtained.
Inequalities with a variable are practically not used in the last editions of the stable textbook of mathematics, although in earlier editions they attended. Inequalities with variables are actively used in alternative mathematics textbooks. These are inequalities of the form:
+ 7 < 10; 5 - > 2; \u206a\u003e 0; \u206a\u003e O.
After entering the letter to designate an unknown number, such inequalities acquire the usual type of inequality with a variable:
a + 7\u003e 10; 12-D.<7.
The values \u200b\u200bof unknown numbers in such inequalities are a selection method, and then the substitution is checked each selected number. The peculiarity of these inequalities is that several numbers suitable for them can be selected (giving faithful inequality).
For example: a + 7\u003e 10; a \u003d 4, a \u003d 5, a \u003d 6, etc. - - the number of values \u200b\u200bfor the letter and infinitely, for this inequality, any number A\u003e 3 is suitable; 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 set of solutions or a large number of solutions of inequality, the child is limited by the selection of several variable values \u200b\u200bin which the inequality is correct.
Equality with quantities.
After the child gets acquainted with the number cards from 1 to 20, you can add to the first step of learning the second stage - equality with quantities.
What is equality? This is an arithmetic effect and its result.
You begin this stage of learning from the topic "Addition".
Addition.
To the show of two sets of cards-quantity you add equal to add.
Teach this operation is very easy. In fact, your child has been ready for this several weeks. After all, every time you show it a new card, he sees that one extra point appeared on it.
The baby still does not know how it is called, but already has an idea of \u200b\u200bwhat it is and how it acts.
Material for examples for addition you already have on the reverse side of each card.
Technology showing equalities it looks like this: you want to give a child equality: 1 +2 \u003d 3. How can it be shown?
Before the start of the lesson, put yourself on your knees face down, one on the other, three cards. Raising the top card with one knuckle, say "one",then postpone her, say "a plus",show the card with two knuckles, pronounce "two",postpone her and after the word "will be",show the card with three knuckles, pronouncing "three".
On the day, you spend three classes with equalities and at each occupation show three different equality. Total, on the day, the kid sees nine different equalities.
A child without any explanation understands what the word means "a plus",its value it displays from the context. By producing action, you are the fastest of any explanation to demonstrate the authentic meaning of addition. Talking about the equalities, always adhere to the same manner of the presentation, using the same terms. Showing "One plus two will be three",do not speak later "To add two to the two".When you teach the child to the facts, he makes the findings and comprehends the rules. If you change the terms, the child has every reason to think that the rules also changed.
Prepare all the cards necessary for one or another equality. Do not think that your child will calmly sit and watch you drown in a stack of cards, picking up the right. He will just hold and will be right, because his time is worth not less than yours.
Try not to make equalities that would have something in common and allow the child to predict them in advance (such equivals can be used later). Here is an example of such equalities:
It is much better to use such:
1 +2 = 3 5+6=11 4 + 8 = 12
The child should see the mathematical essence, it produces mathematical skills and ideas. About two weeks later, the baby makes the discovery that such an addition: after all, during this time you showed it 126 different equal equals.
Check.
Check at this stage is a solution of examples.
What is the difference between an example from equality?
Equality is an action with the result shown by the child.
An example is an action that needs to be performed. In our case, you show the child two answers, and he chooses the right one, i.e. solves an example.
An example you can post after the usual classes with three equal equalities. An example you are shown just as the equality demonstrated before. That is, shift the cards in your hands, saying every loud. For example, "twenty plus ten will be thirty or forty five?" And show the baby two cards, one of which with the right answer.
Cards with answers need to keep at the same distance from the baby's eye and prevent any prompting actions.
With the right choice of the child, you are violently express your delight, kiss and praise it.
With an erroneous choice of a response, without expressing chagrins, you are making a card with the correct answer and ask the question: "Will be thirty, isn't it?" A child usually responds to a similar question. Be sure to praise the child for this correct answer.
Well, if from ten examples your baby correctly solves at least six, it means you exactly go to the equalities for subtraction!
If you do not consider it necessary to check the child (and right!), After 10-14 days, go to equivals to subtraction!
Consider - Specify.
You stop engaging and fully switched to subtraction. Cut three daily lessons with three different equalities in each.
Voicing equal to subtraction like this: "Twelve minus seven will be five."
At the same time, you at the same time continue to show the number of cards (two sets, five cards in each) also three times a day. Total, you will have nine daily very short lessons. So you work no more than two weeks.
Check
Checking the same as in the case of addiction can be a solution of examples with a choice of one response from two.
Consider multiplication.
Multiplication is nothing more than repeated addition, so this action will not be a big opening for your child. Since you continue to study cards - quantities (two sets of five cards in each), you have the possibility of drawing up the ability to multiply.
Voiced equal to multiplication like this: "Two multiplies three will be six."
Baby girl will understand the word "multiply"as quickly as he understood before the word "a plus"and "minus".
You still spend the three lessons per day, in each of which - three different equal equality on multiplication. Such work continues 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 waiting for something new. The main thing for him should be the question: "What's next?"-and at each lesson, he must receive a new answer for him.
Check
Solution of examples you spend the same way as in the topic "Addition" and "subtraction". If the baby liked the game-checking-pets with the quantities cards, you can continue to play them, repeating the new, large quantities.
By holding the scheme proposed by us, you can already complete the first stage of learning mathematics by this time - read the number within 100. Now it's time to get acquainted with the card that best like children.
Consider the concept of zero.
It is said that mathematicians have already been studying the idea of \u200b\u200bscratch five hundred years. True, this is or not, but the children, barely delight the idea of \u200b\u200bquantity, immediately understand the meaning of his complete absence. They simply adore zero, and your journey into the world of numbers will be incomplete if you do not show the baby card, on which there will be no points at all (i.e. it will be absolutely empty card).
To get acquainted with the baby with zero, it's fun and interesting, you can accompany the showing card to the mystery:
Houses - Seven Lachat, on a plate - seven oh. All mushrooms ate proteins. What remains on the plate?
Giving the last phrase, show the "zero" card.
You will use it almost every day. It will come in handy for you for addition, subtraction and multiplication operations.
You can work with a "zero" card one week. This topic is mastering this topic quickly. As before, during the day, you spend three lessons. At each lesson, you show the baby in three different equalities for addition, subtraction and multiplication with zero. Total you will have nine equals per day.
Check
Solution of the examples with zero runs on a familiar scheme.
Consider exploitation.
When you passed all the number cards from 0 to 100, you have all the necessary material for examples for division with quantities.
Technology showing equalities of this topic. Every day you spend three lessons. At each lesson, you show the baby in three different equals. Well, if the passage of this material will not exceed two weeks.
Check
The check is a solution of examples with a choice of one response from two.
When you passed all quantities and familiar with the four arithmetic rules, you can diversify everything in every way and complicate your classes. To begin with, show equality where one arithmetic action is used: only addition, subtraction, multiplication or division.
Then - equality 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 cards, you can change the way of holding classes. Now it is not necessary to show every card of the spikes, you can only show the answer, and the actions themselves only to pronounce. As a result, your classes will be shorter. You just say the child: "Twenty-two divided by eleven, divided into two will be one",- And show him the "one" card.
In this thread you can use equality between which there is any pattern.
For example:
2*2*3= 12 2*2*6=24 2*2*8=32
When combined in equality of four arithmetic actions, remember that multiplication and division must be taken to the beginning of equality:
Do not be afraid to demonstrate equalities that are more than one hundred, for example,
intermediate result B.
42 * 3 - 36 = 90,
where the intermediate result is 126 (42 * 3 \u003d 126)
Your baby can cope with them!
The check is a solution of examples with a choice of one response from two. You can demonstrate an example showing all equality cards and two cards to select a response or simply say all the equality, showing the baby only two cards for the answer.
Remember! The longer you do, the faster you need to introduce new topics. As soon as you noticed the first signs of the baby or boredom - go to the new topic. After time, you can return to the previous topic (but to get acquainted with the equalities not yet shown).
Sequences
Sequences are the same equality. The experience of parents with this topic showed that the sequences of children are very interesting.
Sequences on plus are increasing sequences. Sequences for minus - decreasing.
The more diverse there will be sequences, the more interesting to the baby.
We give a few 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)
Technologythe sequence shows may be such. You have prepared three sequences on plus.
Declare the child theme of the lesson, on the floor, lay out one after another card of the first sequence, voicing them.
Move with the child to another corner of the room and just also post the second sequence.
In the third corner of the room you post the third sequence, while voicing it.
It is possible to lay out the sequences from each other, leaving between them gaps.
Try to always go ahead, moving from simple to complex. Various classes: sometimes uttering what you show, and sometimes shown the cards silently. In any case, the child sees the sequence departed in front of him.
For each sequence, you need to use at least six cards, sometimes more, so that the child is easier to determine the principle of the sequence.
As soon as you saw the shine in the eyes of the child, try adding an example to three sequences (i.e. check its knowledge).
An example is shown like this: first lay out the entire sequence, as you usually do it, and at the end you raise two cards (one card is the one that goes next in the sequence, and the other is random) and ask the child: "What is the following?"
At first, lay out the card in sequences after the other, then the form of the lad out can be changed: put cards in a circle, around the perimeter of the room, etc.
When will get better and better, do not be afraid to use multiplication and division in sequences.
Examples of sequences:
four; 6; eight; 10; 12; 14 - in this sequence, each next number increases by 2;
2; four; 7; fourteen; 17; 34 - in this sequence alternates multiplication and addition (x 2; + 3);
2; four; eight; sixteen; 32; 64 - in this sequence, each next number increases by 2 times;
22; eighteen; fourteen; 10; 6; 2 - in this sequence, each next number decreases by 4;
84; 42; 40; twenty; eighteen; 9 - in this sequence alternates division and subtraction (: 2; - 2);
Signs "More", "Less"
These cards are in the composition of 110 cards of numbers and signs (the second component of the anasta technique).
Toddler dating lessons with the concepts of "more or less" will be very short. All you need is to show three cards.
Technology show
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 call each card.
You can sound like this: "Six more than three"or "Six more than three."
At each lesson, you show the child for three different options for inequalities with
"More" cards are "less." inequalities per day.
So you demonstrate nine different
As before, you show every inequality only once.
A few days later, you can add an example to three shows. This is already verificationand it is held like this:
Put on the floor prepared in advance cards, for example, a card with the number "68" and a card with a "more" sign. Ask the baby: "Sixty eight more than any number?"or "sixty-eight more than fifty or ninety-five?". Offer the child to choose from two cards you need. The card indicated by the baby, you (or he myself) put after the "more" sign.
You can put two cards with quantities in front of the child and give it the opportunity to choose a sign that is suitable, that is\u003e or<.
Equality and inequality
Training equalities and inequalities as simple as the concepts of "more" and "less".
You will need six cards with arithmetic signs. You, too, will also be found in 110 cards of numbers and signs (the second component of the anasta technique).
Technology show
You decide to show the child such two inequalities and one equality:
8-6<10 −7 11-3= 9 −1 55-12^50 −13
You post them on the floor sequentially so that the child could see each of them at once. At the same time, you all speak, for example: "Eight minus six is \u200b\u200bnot equal to ten minus seven."
Similarly, you are pronounced during the laid out of the remaining equality and inequality.
At the initial stage of learning this topic, all the cards are laid out.
You can then show only the cards "equal" and "not equal."
One day you give the baby to show your knowledge. Put the cards with quantities, and it is proposed to choose, a card with which sign must be put: "Equally" or "not equal."
Before starting to study the algebra with the baby, it is necessary to introduce it to the concept of a variable value represented by the letter.
Usually in mathematics, the letter X is used, but since it is easy to confuse with the multiplication sign, it is recommended to use Y.
You put first a card with five beads - knuckles, then a sign + plus (+), after it with a sign Y, then a sign of equality and, finally, a card with a seven beads. Then you put the question: "What does it mean here?"
And answer him themselves: "In this equation, two"
Check:
After about one - one and a half weeks of classes at this stage, you can give the baby to choose the answer.
Fourth stage of equality with numbers and quantities
When you passed the numbers from 1 to 20, it was time to "guide bridges" between numbers and quantities. There are many ways for this. One of the simplest is the use of equalities and inequalities, the relationship "more" and "less", demonstrated by the help of cards with numbers and knuckles.
Show technology.
Take a card with a number 12, put it on the floor, then put the "more" sign next to it, and then the card-quantity of 10, saying at the same time: "Twelve more than ten".
Inequality (equality) may look like this:
Each (equality) the day consists of three classes, and every occupation is from three inequalities with quantities and numbers. The total number of daily equations will be equal to nine. At the same time, you simultaneously continue to study the numbers with two sets of five cards in each, also three times a day.
Check.
You can provide a child with the possibility of choosing "More" cards, "less", "equal" or to make an example so that the kid himself could finish it. For example, put the card-number 7, then the "greater" sign and provide the child with the ability to complete an example, that is, to choose a card-quantity, for example, 9 or a digit card, for example, 5.
After the kid realized the relationship between quantities and numbers, you can proceed to solve equations using cards both with numbers and quantities.
Equality with numbers and quantities.
Using cards with numbers and quantities, you are already going to familiar topics: addition, subtraction, multiplication, division, sequences, equality and inequality, fractions, equations, equality in two or more action.
If you carefully look at the exemplary learning scheme of mathematics, (p. 20) then you will see that there is no end to occupation. Invent your examples for the development of the child's oral account, relate quantities with real objects (nuts, spoons for guests, pieces of a chopped banana, bread, etc.) - in a word, dare, create, invent, try! And you will succeed!
50. Properties of equations based on solutions of equations. Take some equation, not very complicated, for example:
7x - 24 \u003d 15 - 3x
x / 2 - (X - 3) / 3 - (x - 5) / 6 \u003d 1
We see in each equation sign of equality: all that is written to the left of the sign of equality is called the left or the first part of the equation (in the first equation 7x - 24 is the left or the first part, and in the second x / 2 - (x - 3) / 3 - (x - 5) / 6 is the first, or left, part); All that is written to the right of the sign of equality is called the government or second part of the equation (15-3x there is the right side of the first equation, 1 is the right, or the second, part of the 2nd equation).
Each part of any equation expresses some number. The numbers expressed by the left and perpetrators are part of the equation should be equal between themselves. It is clear to us: if we add to each of these numbers at the same number, or will be subtracted from them at the same number, or each of them will be multiplying on the same number, or finally divide the same number, the results of these actions should also To be equal between anyone. In other words: if a \u003d b, then A + C \u003d B + C, A - C \u003d B - C, AC \u003d BC and A / C \u003d B / C. Regarding the division, it should, however, keep in mind that in arithmetic there is no division to zero - we do not know how, for example, the number 5 is divided into zero. Therefore, in the equality A / C \u003d b / c, the number C cannot be zero.
- To both parts of the equation, you can add or subtract from them at the same number.
- Both parts of the equation can be multiplied or divided into one and the same number, eliminating the case when this number may be equal to zero.
Using these properties of the equation, we can find a convenient way to solve equations. Find out this case on the examples.
Example 1. Let it be necessary to solve the equation
5x - 7 \u003d 4x + 15.
We see that the first part of the equation contains two members; One of them is 5X, containing an unknown multiplier X, can be called an unknown member, and the other -7 - famous. In the second part of the equation also 2 members: unknown 4x and known +15. We will make it so that only unknown members would be in the left part of the equation (and a well-known member -7 would be destroyed), and only well-known members would be in the right part (and an unknown member of + 4x would be destroyed). For this purpose, we add to both parts of the equation the same numbers: 1) add +7 (so that the member -7) is destroyed) and 2) add -4x (so that the member + 4x is destroyed). Then we get:
5x - 7 + 7 - 4x \u003d 4x + 15 + 7 - 4x
By making such members in each part of the equation, we get
This is equality and is a solution to the equation, as it indicates that for x it is necessary to take the number 22.
Example 2. Solve equation:
8x + 11 \u003d 7 - 4x
Again we add to both parts of the Equation in -11 and + 4x, we get:
8x + 11 - 11 + 4x \u003d 7 - 4x - 11 + 4x
By making the creation of such members, we get:
Now we divide both part of the equation by +12, we get:
x \u003d -4/12 or x \u003d -1/3
(The first part of the 12x equation is divided by 12 - we obtain 12x / 12 or simply x; the second part of the equation is to divide by +12 - we get -4/12 or -1/3).
The last equality is the solution of the equation, as it indicates that for x it is necessary to take the number -1/3.
Example 3. To solve the equation
x - 23 \u003d 3 · (2x - 3)
We will reveal the brackets first, we get:
x - 23 \u003d 6x - 9
We add to both parts of the equation of +23 and in -6x, - we get:
x - 23 + 23 - 6x \u003d 6x - 9 + 23 - 6x.
Now, in order to accelerate the process of solving the equation subsequently, we will not immediately actuate all such members, but only note that members -23 and +23 in the left part of the equation are mutually destroyed, also members + 6x and -6x in the first part of mutually Destroy - we get:
x - 6x \u003d -9 + 23.
Compare this equation with the initial: initially there was an equation:
x - 23 \u003d 6x - 9
Now they received the equation:
x - 6x \u003d -9 + 23.
We see that in the end it turned out that a member of -23, which was first in the left part of the equation, now, as it were, he moved to the right side of the equation, and he changed a sign (in the left part of the initial equation was a member -23, now it is not there But in the right part of the equation there is a member of + 23, which was not there before). Also, exactly in the right part of the equation was a member of + 6X, now it is not there, but it appeared on the left side of the 9x member equation, which was not there before. Considering from this point of view, examples 1 and 2, we will come to a general conclusion:
You can transfer any member of the equation from one part to another, changing the sign from this member (In further examples, we will use it).
So, returning to our example, we got the equation
x - 6x \u003d -9 + 23
We divide both parts of the equation on -5. Then we get:
[-5x: (-5) We get x] - this is the solution of our equation.
Example 4. Solve equation:
We do so that the equation is not fractions. For this purpose, we find a common denominator for our fractions - the total denominator serves the number 24 - and multiply on it both parts of our equation (it is possible, because the equality does not violate, multiply by the same number of only both parts of the equation. In the first part of 3 members, each member is a fraction - it is necessary, therefore, every fraction is multiplied by 24: the second part of the equation is 0, and multiply by 24 - we get zero. So,
We see that each of our three fractions, due to the fact that it is multiplied by the total shortest multiple denominators of these fractions, will decrease and will be done by an entire expression, namely, we get:
(3x - 8) · 4 - (2x - 1) · 6 + (x - 7) · 3 \u003d 0
Of course, it is desirable to fulfill all this in the mind: it is necessary to imagine that, for example, the numerator of the first fraction lies in brackets and is multiplied by 24, after which the imagination will help us see the reduction of these fractions (by 6) and the final result, i.e. (3x - 8) · 4. also takes place for other fractions. Recall now in the resulting equation (in its left part) brackets:
12x - 32 - 12x + 6 + 3x - 21 \u003d 0
(We note that it took 2x 2x - 1 to multiply to 6 and the resulting product 12x - 6 deduct from the previous one, thanks to which the signs of the members of this work should change - above and written -12x + 6). We transfer well-known members (i.e. -32, +6 and -21) from the left side of the equation to its right part, and (as we already know) the signs of these members must change - we get:
12x - 12x + 3x \u003d 32 - 6 + 21.
Perform the creation of such members:
(When the skill should immediately be performed and transferring the necessary members from one part of the equation to another and bringing similar members), we divide, finally, both parts of the equation for 3 - we get:
x \u003d 15 (2/3) - this is the solution of the equation.
Example 5. Solve equation:
5 - (3x + 1) / 7 \u003d x + (2x - 3) / 5
Here are two fractions, and their overall denominator is 35. Multiply to release the equation from fractions, both parts of the equation on a general denominator 35. In each part of our equation 2 members. With multiplication of each part on 35, each member should multiply by 35 - we get:
The fraction will be reduced - we get:
175 - (3x + 1) · 5 \u003d 35x + (2x - 3) · 7
(Of course, it would be possible when writing this equation at once).
Perform all actions:
175 - 15X - 5 \u003d 35X + 14X - 21.
We transfer all unknown members from the right side (i.e. members + 35x and + 14x) to the left, and all known members from the left side (i.e. members +175 and -5) to the right - should not be forgotten from Portable Members Change Sign:
-15x - 35x - 14x \u003d -21 - 175 + 5
(A member -15x, as used to be in the left side, and now it remained in it - he should not be changed in it; there is no place to change the sign; the same place also for a member -21). By making the creation of such members, we get:
-64x \u003d -191.
[It is possible to make it so that there is no sign minus in both parts of the equation; To do this, you will multiply both parts of the equation on (-1), we obtain 64x \u003d 191, but this can not be done.]
We will then split both parts of the equation on (-64), we obtain the solution of our equation
[If both parts of the equation on (-1) were multiplied and the equation 64x \u003d 191 was obtained, then both part of the equation should be divided into 64.]
Based on what was performed in Examples 4 and 5, we can establish: it is possible to release the equation from fractions - for this it is necessary to find a common denominator for all fractions that are included in the equation (or the smallest general multiple denominators of all fractions) and multiply both Parts of the equation - then the fraction should disappear.
Example 6. Solve equation:
Member 4x from the right side of the equation to the left, we obtain:
5x - 4x \u003d 0 or x \u003d 0.
So, the solution is found: for x you need to take the number of zero. If we replace in this equation x by zero, we obtain 5 · 0 \u003d 4 · 0 or 0 \u003d 0, which indicates the fulfillment of the requirement expressed by this equation: to find such a number for x so that it is single-wing 5x to be equal to the same number as Single 4x.
If someone is noticeable from the very beginning that both parts of the equation 5x \u003d 4x can be divided into X and performs this division, then it turns out a clear incontripability of 5 \u003d 4! The reason for this is the fact that the 5x / x division in this case cannot be done, since we have seen above, the question expressed by our equation requires x \u003d 0, and the division to zero is not fulfilled.
Note also that the multiplication to zero requires some attentiveness: multiplying to zero and two unequal numbers, we will get equal works as a result of these multiplications, namely zeros.
If, for example, we have an equation
x - 3 \u003d 7 - x (its solution: x \u003d 5)
and if anyone wants to apply the property "both part of the equation can be multiplied to the same number" and multiply both parts on X, then will receive:
x 2 - 3x \u003d 7x - x 2.
After that, it may attract attention that all members of the equation contain a multiplier X, from which it is possible to conclude that to solve this equation you can take the number of zero, i.e. put x \u003d 0. And in fact, then we get:
0 2 - 3 · 0 \u003d 7 · 0 - 0 2 or 0 \u003d 0.
However, this solution x \u003d 0 is obviously not suitable for this equation x - 3 \u003d 7 - x; Replacing it x zero in it, we obtain explicit inconsistency: 3 \u003d 7!
