The main function of brackets is to change the order of actions when calculating values. For example, in the numerical expression \(5 3+7\) the multiplication will be calculated first, and then the addition: \(5 3+7 =15+7=22\). But in the expression \(5·(3+7)\), addition in brackets will be calculated first, and only then multiplication: \(5·(3+7)=5·10=50\).

Example. Expand the bracket: \(-(4m+3)\).
Solution : \(-(4m+3)=-4m-3\).

Example. Expand the bracket and give like terms \(5-(3x+2)+(2+3x)\).
Solution : \(5-(3x+2)+(2+3x)=5-3x-2+2+3x=5\).

Example. Expand the brackets \(5(3-x)\).
Solution : We have \(3\) and \(-x\) in the bracket, and five in front of the bracket. This means that each member of the bracket is multiplied by \ (5 \) - I remind you that the multiplication sign between a number and a bracket in mathematics is not written to reduce the size of records.

Example. Expand the brackets \(-2(-3x+5)\).
Solution : As in the previous example, the bracketed \(-3x\) and \(5\) are multiplied by \(-2\).

Example. Simplify the expression: \(5(x+y)-2(x-y)\).
Solution : \(5(x+y)-2(x-y)=5x+5y-2x+2y=3x+7y\).

It remains to consider the last situation.

When multiplying parenthesis by parenthesis, each term of the first parenthesis is multiplied with every term of the second:

\((c+d)(a-b)=c (a-b)+d (a-b)=ca-cb+da-db\)

Example. Expand the brackets \((2-x)(3x-1)\).
Solution : We have a product of brackets and it can be opened immediately using the formula above. But in order not to get confused, let's do everything step by step.
Step 1. Remove the first bracket - each of its members is multiplied by the second bracket:

Step 2. Expand the products of the bracket by the factor as described above:
- the first one first...

Then the second.

Step 3. Now we multiply and bring like terms:

It is not necessary to paint all the transformations in detail, you can immediately multiply. But if you are just learning to open brackets - write in detail, there will be less chance of making a mistake.

Note to the entire section. In fact, you don't need to remember all four rules, you only need to remember one, this one: \(c(a-b)=ca-cb\) . Why? Because if we substitute one instead of c, we get the rule \((a-b)=a-b\) . And if we substitute minus one, we get the rule \(-(a-b)=-a+b\) . Well, if you substitute another bracket instead of c, you can get the last rule.

parenthesis within parenthesis

Sometimes in practice there are problems with brackets nested inside other brackets. Here is an example of such a task: to simplify the expression \(7x+2(5-(3x+y))\).

To be successful in these tasks, you need to:
- carefully understand the nesting of brackets - which one is in which;
- open the brackets sequentially, starting, for example, with the innermost one.

It is important when opening one of the brackets don't touch the rest of the expression, just rewriting it as is.
Let's take the task above as an example.

Example. Open the brackets and give like terms \(7x+2(5-(3x+y))\).
Solution:

Example. Expand the brackets and give like terms \(-(x+3(2x-1+(x-5)))\).
Solution :

\(-(x+3(2x-1\)\(+(x-5)\) \())\)		This is a triple nesting of parentheses. We start with the innermost one (highlighted in green). There is a plus in front of the parenthesis, so it is simply removed.
\(-(x+3(2x-1\)\(+x-5\) \())\)		Now you need to open the second bracket, intermediate. But before that, we will simplify the expression by ghosting similar terms in this second bracket.
\(=-(x\)\(+3(3x-6)\) \()=\)		Now we open the second bracket (highlighted in blue). There is a multiplier in front of the parenthesis - so each term in the parenthesis is multiplied by it.
\(=-(x\)\(+9x-18\) \()=\)
		And open the last parenthesis. Before the bracket minus - so all the signs are reversed.

Bracket opening is a basic skill in mathematics. Without this skill, it is impossible to have a grade above three in grades 8 and 9. Therefore, I recommend a good understanding of this topic.

to form the ability to open brackets, taking into account the sign in front of the brackets;

developing:

develop logical thinking, attention, mathematical speech, the ability to analyze, compare, generalize, draw conclusions;

educators:

formation of responsibility, cognitive interest in the subject

During the classes

I. Organizational moment.

Check it out buddy
Are you ready for the lesson?
Is everything in place? Everything is fine?
Pen, book and notebook.
Is everyone seated correctly?
Is everyone watching closely?

I want to start the lesson with a question for you:

What do you think is the most valuable thing on earth? (Children's answers.)

This question has troubled mankind for thousands of years. Here is the answer given by the famous scientist Al-Biruni: “Knowledge is the most excellent possession. Everyone strives for it, but it does not come by itself.”

Let these words be the motto of our lesson.

II. Actualization of previous knowledge, skills, skills:

Verbal counting:

1.1. What is today's date?

2. What do you know about the number 20?

3. And where is this number located on the coordinate line?

4. Name the number of his reverse.

5. Name the number opposite to it.

6. What is the name of the number - 20?

7. What numbers are called opposites?

8. What numbers are called negative?

9. What is the modulus of the number 20? - 20?

10. What is the sum of opposite numbers?

2. Explain the following entries:

a) The ancient mathematician of genius Archimedes was born in 0 287 BC.

b) The brilliant Russian mathematician N.I. Lobachevsky was born in 1792.

c) The first Olympic Games took place in Greece in 776.

d) The first International Olympic Games took place in 1896.

e) The XXII Olympic Winter Games took place in 2014.

3. Find out what numbers are spinning on the “math carousel” (all actions are performed orally).

II. Formation of new knowledge, skills and abilities.

You have learned how to perform different operations with integers. What are we going to do next? How will we solve examples and equations?

Let's find the meaning of these expressions

7 + (3 + 4) = -7 + 7 = 0
-7 + 3 + 4 = 0

What is the procedure in 1 example? How much is in brackets? The order of actions in the second example? Result of the first action? What can be said about these expressions?

Of course, the results of the first and second expressions are the same, so you can put an equal sign between them: -7 + (3 + 4) = -7 + 3 + 4

What have we done with the brackets? (Lost.)

What do you think we will do in class today? (Children formulate the topic of the lesson.) In our example, what sign is in front of the brackets. (Plus.)

And so we come to the next rule:

If there is a + sign before the brackets, then you can omit the brackets and this + sign, preserving the signs of the terms in brackets. If the first term in brackets is written without a sign, then it must be written with a + sign.

But what if there is a minus sign in front of the brackets?

In this case, you need to reason in the same way as when subtracting: you need to add the number opposite to the one being subtracted:

7 – (3 + 4) = -7 + (-7) = -7 + (-3) + (-4) = -7 – 3 – 4 = -14

- So, we opened the brackets when there was a minus sign in front of them.

The rule for expanding brackets when there is a “-” sign in front of the brackets.

To open the brackets preceded by the - sign, you need to replace this sign with +, changing the signs of all the terms in the brackets to the opposite, and then open the brackets.

Let's listen to the rules for opening brackets in verses:

There is a plus in front of the parenthesis.
He talks about it
What are you dropping the brackets
Let all the signs out!
Before the parenthesis minus strict
Will block our way
To remove brackets
We need to change the signs!

Yes, guys, the minus sign is very insidious, it is a “watchman” at the gate (brackets), it releases numbers and variables only when they change their “passports”, that is, their signs.

Why do you need to open parentheses at all? (When there are parentheses, there is a moment of some element of incompleteness, some kind of mystery. It is like a closed door behind which something interesting is located.) Today we have experienced this mystery.

A small digression into history:

Curly brackets appear in the writings of Vieta (1593). Brackets were widely used only in the first half of the 18th century, thanks to Leibniz and even more so to Euler.

Fizkultminutka.

III. Consolidation of new knowledge, skills and abilities.

Textbook work:

No. 1234 (open brackets) - orally.

No. 1236 (open brackets) - orally.

No. 1235 (find the meaning of the expression) - in writing.

No. 1238 (simplify the expressions) - work in pairs.

IV. Summing up the lesson.

1. Scores are announced.

2. House. exercise. 39 No. 1254 (a, b, c), 1255 (a, b, c), 1259.

3. What have we learned today?

What have you learned?

And I want to end the lesson with wishes for each of you:

“Show the ability to mathematics,
Do not be lazy, but develop daily.
Multiply, divide, labor, think,
Don't forget to be friends with mathematics.

In this lesson, you will learn how to transform an expression that contains parentheses into an expression that does not contain parentheses. You will learn how to open brackets preceded by a plus sign and a minus sign. We will remember how to open brackets using the distributive law of multiplication. The considered examples will allow linking new and previously studied material into a single whole.

Topic: Equation Solving

Lesson: Parentheses expansion

How to open brackets preceded by a "+" sign. Use of the associative law of addition.

If you need to add the sum of two numbers to a number, then you can add the first term to this number, and then the second.

To the left of the equal sign is an expression with parentheses, and to the right is an expression without parentheses. This means that when passing from the left side of the equality to the right side, the brackets were opened.

Consider examples.

Example 1

Expanding the brackets, we changed the order of operations. Counting has become more convenient.

Example 2

Example 3

Note that in all three examples, we simply removed the parentheses. Let's formulate the rule:

Comment.

If the first term in brackets is unsigned, then it must be written with a plus sign.

You can follow the step by step example. First, add 445 to 889. This mental action can be performed, but it is not very easy. Let's open the brackets and see that the changed order of operations will greatly simplify the calculations.

If you follow the indicated order of actions, then you must first subtract 345 from 512, and then add 1345 to the result. By expanding the brackets, we will change the order of actions and greatly simplify the calculations.

Illustrative example and rule.

Consider an example: . You can find the value of the expression by adding 2 and 5, and then taking the resulting number with the opposite sign. We get -7.

On the other hand, the same result can be obtained by adding the opposite numbers.

Let's formulate the rule:

Example 1

Example 2

The rule does not change if there are not two, but three or more terms in brackets.

Example 3

Comment. Signs are reversed only in front of the terms.

In order to open the brackets, in this case, we need to recall the distributive property.

First, multiply the first bracket by 2 and the second by 3.

The first bracket is preceded by a “+” sign, which means that the signs must be left unchanged. The second is preceded by a “-” sign, therefore, all signs must be reversed

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6 - ZSH MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. Library of the teacher of mathematics. - Enlightenment, 1989.

1. Online math tests ().
2. You can download the ones specified in clause 1.2. books().

Homework

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M .: Mnemosyne, 2012. (see link 1.2)
2. Homework: No. 1254, No. 1255, No. 1256 (b, d)
3. Other assignments: No. 1258(c), No. 1248

In this article, we will consider in detail the basic rules for such an important topic in a mathematics course as opening brackets. You need to know the rules for opening brackets in order to correctly solve equations in which they are used.

How to properly open parentheses when adding

Expand the brackets preceded by the "+" sign

This is the simplest case, because if there is an addition sign in front of the brackets, when the brackets are opened, the signs inside them do not change. Example:

(9 + 3) + (1 - 6 + 9) = 9 + 3 + 1 - 6 + 9 = 16.

How to open brackets preceded by a "-" sign

In this case, you need to rewrite all the terms without brackets, but at the same time change all the signs inside them to the opposite ones. The signs change only for the terms from those brackets that were preceded by the “-” sign. Example:

(9 + 3) - (1 - 6 + 9) = 9 + 3 - 1 + 6 - 9 = 8.

How to open parentheses when multiplying

The parentheses are preceded by a multiplier

In this case, you need to multiply each term by a factor and open the brackets without changing signs. If the multiplier has the sign "-", then when multiplying, the signs of the terms are reversed. Example:

3 * (1 - 6 + 9) = 3 * 1 - 3 * 6 + 3 * 9 = 3 - 18 + 27 = 12.

How to open two brackets with a multiplication sign between them

In this case, you need to multiply each term from the first brackets with each term from the second brackets and then add the results. Example:

(9 + 3) * (1 - 6 + 9) = 9 * 1 + 9 * (- 6) + 9 * 9 + 3 * 1 + 3 * (- 6) + 3 * 9 = 9 - 54 + 81 + 3 - 18 + 27 = 48.

How to open brackets in a square

If the sum or difference of two terms is squared, the brackets should be expanded according to the following formula:

(x + y)^2 = x^2 + 2*x*y + y^2.

In the case of a minus inside the brackets, the formula does not change. Example:

(9 + 3) ^ 2 = 9 ^ 2 + 2 * 9 * 3 + 3 ^ 2 = 144.

How to open parentheses in a different degree

If the sum or difference of the terms is raised, for example, to the 3rd or 4th power, then you just need to break the degree of the bracket into “squares”. The powers of the same factors are added, and when dividing, the degree of the divisor is subtracted from the degree of the dividend. Example:

(9 + 3) ^ 3 = ((9 + 3) ^ 2) * (9 + 3) = (9 ^ 2 + 2 * 9 * 3 + 3 ^ 2) * 12 = 1728.

How to open 3 brackets

There are equations in which 3 brackets are multiplied at once. In this case, you must first multiply the terms of the first two brackets among themselves, and then multiply the sum of this multiplication by the terms of the third bracket. Example:

(1 + 2) * (3 + 4) * (5 - 6) = (3 + 4 + 6 + 8) * (5 - 6) = - 21.

These bracket opening rules apply equally to both linear and trigonometric equations.

In this lesson, you will learn how to transform an expression that contains parentheses into an expression that does not contain parentheses. You will learn how to open brackets preceded by a plus sign and a minus sign. We will remember how to open brackets using the distributive law of multiplication. The considered examples will allow linking new and previously studied material into a single whole.

Topic: Equation Solving

Lesson: Parentheses expansion

How to open brackets preceded by a "+" sign. Use of the associative law of addition.

If you need to add the sum of two numbers to a number, then you can add the first term to this number, and then the second.

To the left of the equal sign is an expression with parentheses, and to the right is an expression without parentheses. This means that when passing from the left side of the equality to the right side, the brackets were opened.

Consider examples.

Example 1

Expanding the brackets, we changed the order of operations. Counting has become more convenient.

Example 2

Example 3

Note that in all three examples, we simply removed the parentheses. Let's formulate the rule:

Comment.

If the first term in brackets is unsigned, then it must be written with a plus sign.

You can follow the step by step example. First, add 445 to 889. This mental action can be performed, but it is not very easy. Let's open the brackets and see that the changed order of operations will greatly simplify the calculations.

If you follow the indicated order of actions, then you must first subtract 345 from 512, and then add 1345 to the result. By expanding the brackets, we will change the order of actions and greatly simplify the calculations.

Illustrative example and rule.

Consider an example: . You can find the value of the expression by adding 2 and 5, and then taking the resulting number with the opposite sign. We get -7.

On the other hand, the same result can be obtained by adding the opposite numbers.

Let's formulate the rule:

Example 1

Example 2

The rule does not change if there are not two, but three or more terms in brackets.

Example 3

Comment. Signs are reversed only in front of the terms.

In order to open the brackets, in this case, we need to recall the distributive property.

First, multiply the first bracket by 2 and the second by 3.

The first bracket is preceded by a “+” sign, which means that the signs must be left unchanged. The second is preceded by a “-” sign, therefore, all signs must be reversed

Bibliography

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
2. Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
3. Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
4. Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6 - ZSH MEPhI, 2011.
5. Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. A manual for students of the 6th grade of the MEPhI correspondence school. - ZSH MEPhI, 2011.
6. Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Textbook-interlocutor for 5-6 grades of high school. Library of the teacher of mathematics. - Enlightenment, 1989.

1. Online math tests ().
2. You can download the ones specified in clause 1.2. books().

Homework

1. Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M .: Mnemosyne, 2012. (see link 1.2)
2. Homework: No. 1254, No. 1255, No. 1256 (b, d)
3. Other assignments: No. 1258(c), No. 1248