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.
To open parentheses, this case remember 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
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6 - ZSH MEPhI, 2011.
- Rurukin A.N., Sochilov S.V., Tchaikovsky K.G. Mathematics 5-6. Allowance for 6th grade students correspondence school MEPhI. - ZSH MEPhI, 2011.
- Shevrin L.N., Gein A.G., Koryakov I.O., Volkov M.V. Mathematics: Interlocutor textbook for grades 5-6 high school. Library of the teacher of mathematics. - Enlightenment, 1989.
- Online math tests ().
- You can download the ones specified in clause 1.2. books().
Homework
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M .: Mnemosyne, 2012. (see link 1.2)
- Homework: No. 1254, No. 1255, No. 1256 (b, d)
- Other assignments: No. 1258(c), No. 1248
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
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M.: Mnemosyne, 2012.
- Merzlyak A.G., Polonsky V.V., Yakir M.S. Mathematics 6th grade. - Gymnasium, 2006.
- Depman I.Ya., Vilenkin N.Ya. Behind the pages of a mathematics textbook. - Enlightenment, 1989.
- Rurukin A.N., Tchaikovsky I.V. Tasks for the course of mathematics grade 5-6 - ZSH MEPhI, 2011.
- 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.
- 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.
- Online math tests ().
- You can download the ones specified in clause 1.2. books().
Homework
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics 6. - M .: Mnemosyne, 2012. (see link 1.2)
- Homework: No. 1254, No. 1255, No. 1256 (b, d)
- Other assignments: No. 1258(c), No. 1248
Bracket expansion is a type of expression transformation. In this section, we will describe the rules for expanding brackets, as well as consider the most common examples of tasks.
Yandex.RTB R-A-339285-1
What is parenthesis expansion?
Parentheses are used to indicate the order in which actions are performed in numeric and alphabetic expressions, as well as in expressions with variables. It is convenient to pass from an expression with brackets to identically equal expression without brackets. For example, replace the expression 2 (3 + 4) with an expression like 2 3 + 2 4 without brackets. This technique is called parenthesis opening.
Definition 1
Under the opening of brackets, we mean the methods of getting rid of brackets and are usually considered in relation to expressions that may contain:
- signs "+" or "-" in front of brackets that contain sums or differences;
- the product of a number, letter, or several letters, and the sum or difference, which is placed in brackets.
This is how we used to consider the process of expanding brackets in the course school curriculum. However, no one prevents us from looking at this action more broadly. We can call parenthesis expansion the transition from an expression that contains negative numbers in parentheses to an expression that does not have parentheses. For example, we can go from 5 + (− 3) − (− 7) to 5 − 3 + 7 . In fact, this is also parenthesis expansion.
In the same way, we can replace the product of expressions in brackets of the form (a + b) · (c + d) with the sum a · c + a · d + b · c + b · d . This technique also does not contradict the meaning of parentheses expansion.
Here is another example. We can assume that in expressions, instead of numbers and variables, any expressions can be used. For example, the expression x 2 1 a - x + sin (b) will correspond to an expression without brackets of the form x 2 1 a - x 2 x + x 2 sin (b) .
One more point deserves special attention, which concerns the peculiarities of writing solutions when opening brackets. We can write the initial expression with brackets and the result obtained after opening the brackets as equality. For example, after opening the parentheses, instead of the expression 3 − (5 − 7) we get the expression 3 − 5 + 7 . We can write both of these expressions as the equality 3 − (5 − 7) = 3 − 5 + 7 .
Performing actions with cumbersome expressions may require writing intermediate results. Then the solution will have the form of a chain of equalities. For example, 5 − (3 − (2 − 1)) = 5 − (3 − 2 + 1) = 5 − 3 + 2 − 1 or 5 − (3 − (2 − 1)) = 5 − 3 + (2 − 1) = 5 − 3 + 2 − 1 .
Rules for opening brackets, examples
Let's start with the rules for opening parentheses.
Single numbers in brackets
Negative numbers in parentheses often appear in expressions. For example, (− 4) and 3 + (− 4) . Positive numbers in brackets also take place.
Let us formulate the rule for opening brackets that contain single positive numbers. Suppose a is any positive number. Then we can replace (a) with a, + (a) with + a, - (a) with - a. If instead of a we take a specific number, then according to the rule: the number (5) will be written as 5 , the expression 3 + (5) without brackets will take the form 3 + 5 , since + (5) is replaced by + 5 , and the expression 3 + (− 5) is equivalent to the expression 3 − 5 , because + (− 5) is replaced by − 5 .
Positive numbers are usually written without using parentheses, since the parentheses are redundant in this case.
Now consider the rule for opening brackets that contain a single negative number. + (−a) we replace with − a, − (− a) is replaced by + a . If the expression starts with a negative number (-a), which is written in brackets, then the brackets are omitted and instead of (-a) remains − a.
Here are some examples: (− 5) can be written as − 5 , (− 3) + 0 , 5 becomes − 3 + 0 , 5 , 4 + (− 3) becomes 4 − 3 , and − (− 4) − (− 3) after opening the brackets takes the form 4 + 3 , since − (− 4) and − (− 3) is replaced by + 4 and + 3 .
It should be understood that the expression 3 · (− 5) cannot be written as 3 · − 5. This will be discussed in the following paragraphs.
Let's see what the parenthesis expansion rules are based on.
According to the rule, the difference a − b is equal to a + (− b) . Based on the properties of actions with numbers, we can make a chain of equalities (a + (− b)) + b = a + ((− b) + b) = a + 0 = a which will be fair. This chain of equalities, by virtue of the meaning of subtraction, proves that the expression a + (− b) is the difference a-b.
Based on properties opposite numbers and subtraction rules negative numbers we can assert that − (− a) = a , a − (− b) = a + b .
There are expressions that are made up of a number, minus signs and several pairs of brackets. Using the above rules allows you to sequentially get rid of brackets, moving from inner brackets to outer or into reverse direction. An example of such an expression would be − (− ((− (5)))) . Let's open the brackets, moving from the inside to the outside: − (− ((− (5)))) = − (− ((− 5))) = − (− (− 5)) = − (5) = − 5 . This example can also be parsed in reverse: − (− ((− (5)))) = ((− (5))) = (− (5)) = − (5) = − 5 .
Under a and b can be understood not only as numbers, but also as arbitrary numerical or literal expressions with a "+" in front that are not sums or differences. In all these cases, you can apply the rules in the same way as we did for single numbers in brackets.
For example, after opening the brackets, the expression − (− 2 x) − (x 2) + (− 1 x) − (2 x y 2: z) takes the form 2 x − x 2 − 1 x − 2 x y 2: z . How did we do it? We know that − (− 2 x) is + 2 x , and since this expression comes first, then + 2 x can be written as 2 x , - (x 2) = - x 2, + (− 1 x) = − 1 x and − (2 x y 2: z) = − 2 x y 2: z.
In the products of two numbers
Let's start with the rule for expanding brackets in the product of two numbers.
Let's pretend that a and b is two positive numbers. In this case, the product of two negative numbers − a and − b of the form (− a) (− b) can be replaced by (a b) , and the products of two numbers with opposite signs of the form (− a) b and a (− b) are replaced by (− a b). Multiplying a minus by a minus gives a plus, and multiplying a minus by a plus, like multiplying a plus by a minus, gives a minus.
The correctness of the first part of the written rule is confirmed by the rule for multiplying negative numbers. To confirm the second part of the rule, we can use the rules for multiplying numbers with different signs.
Let's look at a few examples.
Example 1
Consider the algorithm for opening brackets in the product of two negative numbers - 4 3 5 and - 2 , of the form (- 2) · - 4 3 5 . To do this, we replace the original expression with 2 · 4 3 5 . Let's expand the brackets and get 2 · 4 3 5 .
And if we take the quotient of negative numbers (− 4) : (− 2) , then the record after opening the brackets will look like 4: 2
Instead of negative numbers − a and − b can be any expressions with a leading minus sign that are not sums or differences. For example, these can be products, partials, fractions, degrees, roots, logarithms, trigonometric functions etc.
Let's open the brackets in the expression - 3 · x x 2 + 1 · x · (- ln 5) . According to the rule, we can make the following transformations: - 3 x x 2 + 1 x (- ln 5) = - 3 x x 2 + 1 x ln 5 = 3 x x 2 + 1 x ln 5 .
Expression (− 3) 2 can be converted to the expression (− 3 2) . After that, you can open the brackets: − 3 2.
2 3 - 4 5 = - 2 3 4 5 = - 2 3 4 5
Dividing numbers with different signs may also require the preliminary expansion of brackets: (− 5) : 2 = (− 5: 2) = − 5: 2 and 2 3 4: (- 3 , 5) = - 2 3 4: 3 , 5 = - 2 3 4: 3 , 5 .
The rule can be used to perform multiplication and division of expressions with different signs. Let's give two examples.
1 x + 1: x - 3 = - 1 x + 1: x - 3 = - 1 x + 1: x - 3
sin (x) (- x 2) \u003d (- sin (x) x 2) \u003d - sin (x) x 2
In the products of three or more numbers
Let's move on to the product and the quotients, which contain large quantity numbers. For expanding parentheses, here will act next rule. With an even number of negative numbers, you can omit the parentheses, replacing the numbers with their opposites. After that, you need to enclose the resulting expression in new brackets. For an odd number of negative numbers, omitting the brackets, replace the numbers with their opposites. After that, the resulting expression must be taken in new brackets and put a minus sign in front of it.
Example 2
For example, let's take the expression 5 · (− 3) · (− 2) , which is the product of three numbers. There are two negative numbers, so we can write the expression as (5 3 2) and then finally open the brackets, getting the expression 5 3 2 .
In the product (− 2 , 5) (− 3) : (− 2) 4: (− 1 , 25) : (− 1) five numbers are negative. so (− 2 , 5) (− 3) : (− 2) 4: (− 1 , 25) : (− 1) = (− 2 . 5 3: 2 4: 1 , 25: 1) . Finally opening the brackets, we get −2.5 3:2 4:1.25:1.
The above rule can be justified as follows. First, we can rewrite such expressions as a product, replacing by multiplication by reciprocal number division. We represent each negative number as the product of a multiplier and replace - 1 or - 1 with (− 1) a.
Using the commutative property of multiplication, we swap the factors and transfer all factors equal to − 1 , to the beginning of the expression. The product of an even number minus ones is equal to 1, and an odd number is equal to − 1 , which allows us to use the minus sign.
If we did not use the rule, then the chain of actions for opening brackets in the expression - 2 3: (- 2) 4: - 6 7 would look like this:
2 3: (- 2) 4: - 6 7 = - 2 3 - 1 2 4 - 7 6 = = (- 1) 2 3 (- 1) 1 2 4 (- 1 ) 7 6 = = (- 1) (- 1) (- 1) 2 3 1 2 4 7 6 = (- 1) 2 3 1 2 4 7 6 = = - 2 3 1 2 4 7 6
The above rule can be used when expanding brackets in expressions that are products and quotients with a minus sign that are not sums or differences. Take for example the expression
x 2 (- x) : (- 1 x) x - 3: 2 .
It can be reduced to an expression without brackets x 2 · x: 1 x · x - 3: 2 .
Opening parentheses preceded by a + sign
Consider a rule that can be applied to expand brackets that are preceded by a plus sign, and the "contents" of these brackets are not multiplied or divided by any number or expression.
According to the rule, brackets together with the sign in front of them are omitted, while the signs of all terms in brackets are preserved. If there is no sign in front of the first term in brackets, then you need to put a plus sign.
Example 3
For example, we give the expression (12 − 3 , 5) − 7 . By omitting the brackets, we keep the signs of the terms in the brackets and put a plus sign in front of the first term. The entry will look like (12 − 3 , 5) − 7 = + 12 − 3 , 5 − 7 . In the above example, it is not necessary to put a sign in front of the first term, since + 12 - 3, 5 - 7 = 12 - 3, 5 - 7.
Example 4
Let's consider one more example. Take the expression x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x and perform actions with it x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x = = x + 2 a - 3 x 2 + 1 - x 2 - 4 + 1 x
Here is another example of expanding parentheses:
Example 5
2 + x 2 + 1 x - x y z + 2 x - 1 + (- 1 + x - x 2) = = 2 + x 2 + 1 x - x y z + 2 x - 1 - 1 + x + x2
How to expand parentheses preceded by a minus sign
Consider cases where there is a minus sign in front of the brackets, and which are not multiplied (or divided) by any number or expression. According to the rule for opening brackets preceded by the “-” sign, the brackets with the “-” sign are omitted, while the signs of all terms inside the brackets are reversed.
Example 6
For example:
1 2 \u003d 1 2, - 1 x + 1 \u003d - 1 x + 1, - (- x 2) \u003d x 2
Variable expressions can be converted using the same rule:
X + x 3 - 3 - - 2 x 2 + 3 x 3 x + 1 x - 1 - x + 2,
we get x - x 3 - 3 + 2 x 2 - 3 x 3 x + 1 x - 1 - x + 2 .
Opening parentheses when multiplying a number by a parenthesis, expressions by a parenthesis
Here we will consider cases when it is necessary to open brackets that are multiplied or divided by any number or expression. Here formulas of the form (a 1 ± a 2 ± ... ± a n) b = (a 1 b ± a 2 b ± ... ± a n b) or b (a 1 ± a 2 ± … ± a n) = (b a 1 ± b a 2 ± … ± b a n), where a 1 , a 2 , … , a n and b are some numbers or expressions.
Example 7
For example, let's expand the brackets in the expression (3 − 7) 2. According to the rule, we can make the following transformations: (3 − 7) 2 = (3 2 − 7 2) . We get 3 · 2 − 7 · 2 .
Expanding the brackets in the expression 3 x 2 1 - x + 1 x + 2, we get 3 x 2 1 - 3 x 2 x + 3 x 2 1 x + 2.
Multiply a parenthesis by a parenthesis
Consider the product of two brackets of the form (a 1 + a 2) · (b 1 + b 2) . This will help us get a rule for expanding parentheses when multiplying a parenthesis by a parenthesis.
In order to solve the above example, we denote the expression (b 1 + b 2) like b. This will allow us to use the parenthesis-expression multiplication rule. We get (a 1 + a 2) (b 1 + b 2) = (a 1 + a 2) b = (a 1 b + a 2 b) = a 1 b + a 2 b . By doing a reverse substitution b on (b 1 + b 2), again apply the rule for multiplying the expression by the bracket: a 1 b + a 2 b = = a 1 (b 1 + b 2) + a 2 (b 1 + b 2) = = (a 1 b 1 + a 1 b 2) + (a 2 b 1 + a 2 b 2) = = a 1 b 1 + a 1 b 2 + a 2 b 1 + a 2 b 2
Thanks to a number of simple tricks, we can come to the sum of the products of each of the terms from the first bracket and each of the terms from the second bracket. The rule can be extended to any number of terms inside the brackets.
Let us formulate the rules for multiplying a bracket by a bracket: in order to multiply two sums among themselves, it is necessary to multiply each of the terms of the first sum by each of the terms of the second sum and add the results.
The formula will look like:
(a 1 + a 2 + . . . + a m) (b 1 + b 2 + . . . + b n) = = a 1 b 1 + a 1 b 2 + . . . + a 1 b n + + a 2 b 1 + a 2 b 2 + . . . + a 2 b n + + . . . + + a m b 1 + a m b 1 + . . . a m b n
Let's expand the brackets in the expression (1 + x) · (x 2 + x + 6) It is a product of two sums. Let's write the solution: (1 + x) (x 2 + x + 6) = = (1 x 2 + 1 x + 1 6 + x x 2 + x x + x 6) = = 1 x 2 + 1 x + 1 6 + x x 2 + x x + x 6
Separately, it is worth dwelling on those cases when there is a minus sign in brackets along with plus signs. For example, let's take the expression (1 − x) · (3 · x · y − 2 · x · y 3) .
First, we represent the expressions in brackets as sums: (1 + (− x)) (3 x y + (− 2 x y 3)). Now we can apply the rule: (1 + (− x)) (3 x y + (− 2 x y 3)) = = (1 3 x y + 1 (− 2 x y 3) + (− x) 3 x y + (− x) (− 2 x y 3))
Let's expand the brackets: 1 3 x y − 1 2 x y 3 − x 3 x y + x 2 x y 3 .
Parentheses expansion in products of several brackets and expressions
If there are three or more expressions in brackets in the expression, it is necessary to expand the brackets sequentially. It is necessary to start the transformation with the fact that the first two factors are taken in brackets. Inside these brackets, we can perform transformations according to the rules discussed above. For example, the parentheses in the expression (2 + 4) 3 (5 + 7 8) .
The expression contains three factors at once (2 + 4) , 3 and (5 + 7 8) . We will expand the brackets sequentially. We enclose the first two factors in one more brackets, which we will make red for clarity: (2 + 4) 3 (5 + 7 8) = ((2 + 4) 3) (5 + 7 8).
In accordance with the rule of multiplying a bracket by a number, we can carry out the following actions: ((2 + 4) 3) (5 + 7 8) = (2 3 + 4 3) (5 + 7 8) .
Multiply bracket by bracket: (2 3 + 4 3) (5 + 7 8) = 2 3 5 + 2 3 7 8 + 4 3 5 + 4 3 7 8 .
Parenthesis in kind
Powers whose bases are some expressions written in brackets, with natural indicators can be thought of as a product of several parentheses. Moreover, according to the rules from the two previous paragraphs, they can be written without these brackets.
Consider the process of transforming the expression (a + b + c) 2 . It can be written as a product of two brackets (a + b + c) (a + b + c). We multiply bracket by bracket and get a a + a b + a c + b a + b b + b c + c a + c b + c c .
Let's take another example:
Example 8
1 x + 2 3 = 1 x + 2 1 x + 2 1 x + 2 = = 1 x 1 x + 1 x 2 + 2 1 x + 2 2 1 x + 2 = = 1 x 1 x 1 x + 1 x 2 1 x + 2 1 x 1 x + 2 2 1 x + 1 x 1 x 2 + + 1 x 2 2 + 2 1 x 2 + 2 2 2
Dividing a parenthesis by a number and a parenthesis by a parenthesis
Dividing a parenthesis by a number suggests that you must divide by the number all the terms enclosed in brackets. For example, (x 2 - x) : 4 = x 2: 4 - x: 4 .
Division can be previously replaced by multiplication, after which you can use the appropriate rule for opening brackets in the product. The same rule applies when dividing a parenthesis by a parenthesis.
For example, we need to open the brackets in the expression (x + 2) : 2 3 . To do this, first replace the division by multiplying by the reciprocal of (x + 2) : 2 3 = (x + 2) · 2 3 . Multiply the bracket by the number (x + 2) 2 3 = x 2 3 + 2 2 3 .
Here is another example of parenthesis division:
Example 9
1 x + x + 1: (x + 2) .
Let's replace division with multiplication: 1 x + x + 1 1 x + 2 .
Let's do the multiplication: 1 x + x + 1 1 x + 2 = 1 x 1 x + 2 + x 1 x + 2 + 1 1 x + 2 .
Bracket expansion order
Now consider the order of application of the rules discussed above in the expressions general view, i.e. in expressions that contain sums with differences, products with quotients, brackets in kind.
The order of actions:
- the first step is to raise the parentheses to a natural power;
- at the second stage, brackets are opened in works and private;
- the final step is to open the brackets in the sums and differences.
Let's consider the order of actions using the example of the expression (− 5) + 3 · (− 2) : (− 4) − 6 · (− 7) . Let us transform from the expressions 3 (− 2) : (− 4) and 6 (− 7) , which should take the form (3 2:4) and (− 6 7) . Substituting the obtained results into the original expression, we obtain: (− 5) + 3 (− 2) : (− 4) − 6 (− 7) = (− 5) + (3 2: 4) − (− 6 7). Expand the brackets: − 5 + 3 2: 4 + 6 7 .
When dealing with expressions that contain parentheses within parentheses, it is convenient to perform transformations from the inside out.
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I continue a series of methodological articles on the topic of teaching. It's time to consider the features individual work math tutor with 7th grade students. With great pleasure I will share my thoughts on the forms of submission of one of the major topics algebra course in grade 7 - "opening brackets." In order not to try to embrace the immensity, let's focus on her elementary school and analyze the methodology of the tutor with the multiplication of a polynomial by a polynomial. How math tutor valid in difficult situations, when weak student does not perceive classic shape explanations? What tasks should be prepared for a strong seventh grader? Let's consider these and other questions.
It would seem, well, what's so difficult? “Parentheses are easy,” any good student will say. “There is a distributive law and properties of degrees for working with monomials, a general algorithm for any number of terms. Multiply each by each and bring the like. However, not everything is so simple in working with the lagging behind. Despite the efforts of a math tutor, students manage to make mistakes of various calibers even in the simplest transformations. The nature of the errors is striking in its diversity: from small omissions of letters and signs, to serious dead-end "stop errors".
What prevents the student from correctly performing the transformations? Why is there misunderstanding?
There are individual problems great multitude and one of the main obstacles to the assimilation and consolidation of the material is the difficulty in timely and quick switching of attention, the difficulty in processing a large amount of information. It may seem strange to some that I am talking about large volume, but a weak student of grade 7 may not have enough memory and attention resources even for four terms. Coefficients, variables, degrees (indicators) interfere. The student confuses the sequence of operations, forgets which monomials have already been multiplied and which have remained untouched, cannot remember how they are multiplied, etc.
Math tutor's numerical approach
Of course, you need to start with an explanation of the logic of building the algorithm itself. How to do it? We need to set the task: how to change the order of actions in the expression 
without changing the result? I quite often give examples explaining the operation of certain rules on specific numbers. And then I replace them with letters. Technique of use numerical approach will be described below.
Problems of motivation.
At the beginning of the lesson, it is difficult for a math tutor to gather a student if he does not understand the relevance of what is being studied. Within the framework of the program for grades 6-7, it is difficult to find examples of using the polynomial multiplication rule. I would emphasize the need to learn change the order of actions in expressions The fact that this helps to solve problems, the student should know from the experience of addition. similar terms. He also had to add them in when solving equations. For example, in 2x+5x+13=34 he uses that 2x+5x=7x. A math tutor just needs to focus the attention of the student on this.
Mathematics teachers often call the parenthesis opening technique fountain rule.
This image is well remembered and must be used. But how is this rule proven? Recall the classical form using obvious identity transformations:
(a+b)(c+d)=(a+b) c+(a+b) d=ac+bc+ad+bd
It is difficult for a math tutor to comment on anything here. The letters speak for themselves. Yes, and not needed by a strong student of grade 7 detailed explanations. However, what to do with the weak, who point-blank does not see any content in this "alphabetic mishmash"?
The main problem that hinders the perception of the classical mathematical justification of the "fountain" is the unusual form of writing the first factor. Neither in the 5th grade nor in the 6th grade did the student have to drag the first bracket to each term of the second. Children dealt only with numbers (coefficients), located, most often, to the left of the brackets, for example:
By the end of the 6th grade, the student develops visual image object - a certain combination of signs (actions) associated with brackets. And any deviation from the usual look towards something new can disorient a seventh grader. It is the visual image of the “number + bracket” pair that the math tutor takes into circulation when explaining.
The following explanation can be offered. The tutor argues: “If there was some number in front of the bracket, for example 5, then we could change the course of action in this expression? Of course. Let's do it then 
. Think about whether its result will change if instead of the number 5 we enter the sum of 2 + 3 enclosed in brackets? Any student will tell the tutor: "What difference does it make how to write: 5 or 2 + 3." Wonderful. Get a record. The math tutor takes a short pause so that the student visually remembers the picture-image of the object. Then he draws his attention to the fact that the bracket, like the number, "distributed" or "jumped" to each term. What does this mean? It means that this operation can be performed not only with a number, but also with a bracket. We got two pairs of factors and . With them most of students can easily cope on their own and write out the result to the tutor. It is important to compare the resulting pairs with the content of brackets 2+3 and 6+4 and it will become clear how they open.
If necessary, after the example with numbers, the math tutor conducts a literal proof. It turns out to be a cakewalk through the same parts of the previous algorithm.
Formation of the skill of opening brackets
The formation of the skill of multiplying brackets is one of milestones work of a tutor in mathematics with a theme. And even more important than the stage of explaining the logic of the “fountain” rule. Why? The justifications for the transformations will be forgotten the very next day, and the skill, if it is formed and fixed in time, will remain. Students perform the operation mechanically, as if extracting the multiplication table from memory. This is what needs to be achieved. Why? If every time the student opens the brackets, he will remember why he opens it this way and not otherwise, he will forget about the problem he is solving. That is why the math tutor spends the rest of the lesson on transforming understanding into rote memorization. This strategy is often used in other topics as well.
How can a tutor develop the skill of opening brackets in a student? To do this, a 7th grade student must perform a series of exercises in sufficient quantity to consolidate. This raises another problem. A weak seventh grader cannot cope with the increased number of transformations. Even small ones. And mistakes keep coming in one after another. What should a math tutor do? First, it is necessary to recommend painting arrows from each term to each. If the student is very weak and is not able to quickly switch from one type of work to another, loses concentration when following simple commands from the teacher, then the math tutor draws these arrows himself. And not all at once. First, the tutor connects the first term of the left bracket with each term of the right bracket and asks to perform the appropriate multiplication. Only after that do the arrows go from the second term to the same right bracket. In other words, the tutor divides the process into two stages. It is better to maintain a small temporary pause (5-7 seconds) between the first and second operation.
1) One set of arrows should be drawn above the expressions and another set below them.
2) It is important to skip between lines at least couple of cells. Otherwise, the record will be very dense, and the arrows will not only climb to the previous line, but will also mix with the arrows from the next exercise.
3) In the case of multiplying brackets in the format 3 by 2, arrows are drawn from the short bracket to the long one. Otherwise, these "fountains" will be not two, but three. The implementation of the third is noticeably more complicated due to the lack of free space for the arrows.
4) the arrows are always directed from one point. One of my students kept trying to put them side by side and this is what he did:
Such an arrangement does not allow to single out and fix the current term, with which the student works at each of the stages.
The work of the tutor's fingers
4) To keep attention on a separate couple multiplied terms, the math tutor puts two fingers on them. This must be done in such a way as not to block the student's view. For the most inattentive students, you can use the "pulsation" method. The math tutor brings the first finger to the beginning of the arrow (to one of the terms) and fixes it, and with the second “knocks” on its end (on the second term). Pulsation helps to focus attention on the term by which the student multiplies. After the first multiplication by the right bracket is done, the math tutor says: “Now we work with another term.” The tutor moves a “fixed finger” to it, and “pulsating” runs over the terms from another bracket. The pulsation works like a "turn signal" in a car and allows you to collect the attention of an absent-minded student on the operation he is conducting. If the child writes small, then two pencils are used instead of fingers.
Repetition optimization
As in the study of any other topic in the course of algebra, the multiplication of polynomials can and should be integrated with previously covered material. To do this, the math tutor uses special bridge tasks that allow you to find the application of the studied in various mathematical objects. They not only connect the topics into a single whole, but also very effectively organize the repetition of the entire course of mathematics. And the more bridges the tutor builds, the better.
Traditionally, in algebra textbooks for grade 7, the opening of brackets is integrated with the solution linear equations. At the end of the list of numbers there are always tasks of the following order: solve the equation. When opening the brackets, the squares are reduced and the equation is easily solved by means of class 7. However, for some reason, the authors of textbooks safely forget about plotting a graph of a linear function. In order to correct this shortcoming, I would advise math tutors to include brackets in analytic expressions linear functions, for example . In such exercises, the student not only trains the skills of conducting identical transformations, but also repeats the graphs. You can ask to find the intersection point of two "monsters", determine mutual arrangement lines, find the points of their intersection with the axes, etc.
Kolpakov A.N. Mathematics tutor in Strogino. Moscow
A + (b + c) can be written without brackets: a + (b + c) \u003d a + b + c. This operation is called parenthesis expansion.
Example 1 Let's open the brackets in the expression a + (- b + c).
Solution. a + (-b + c) = a + ((-b) + c) = a + (-b) + c = a-b + c.
If there is a “+” sign before the brackets, then you can omit the brackets and this “+” sign, retaining 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.
Example 2 Let's find the value expressions -2.87+ (2.87-7.639).
Solution. Opening the brackets, we get - 2.87 + (2.87 - 7.639) \u003d - - 2.87 + 2.87 - 7.639 \u003d 0 - 7.639 \u003d - 7.639.
To find the value of the expression - (- 9 + 5), you need to add numbers-9 and 5 and find the number opposite to the amount received: -(- 9 + 5)= -(- 4) = 4.
The same value can be obtained in a different way: first write down the numbers opposite to these terms (i.e. change their signs), and then add: 9 + (- 5) = 4. Thus, - (- 9 + 5) = 9 - 5 = 4.
To write the sum opposite to the sum of several terms, it is necessary to change the signs of these terms.
So - (a + b) \u003d - a - b.
Example 3 Find the value of the expression 16 - (10 -18 + 12).
Solution. 16-(10 -18 + 12) = 16 + (-(10 -18 + 12)) = = 16 + (-10 +18-12) = 16-10 +18-12 = 12.
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 ones, and then open the brackets.
Example 4 Let's find the value of the expression 9.36-(9.36 - 5.48).
Solution. 9.36 - (9.36 - 5.48) = 9.36 + (- 9.36 + 5.48) == 9.36 - 9.36 + 5.48 = 0 -f 5.48 = 5 .48.
Opening parentheses and the use of commutative and associative properties additions make calculations easier.
Example 5 Find the value of the expression (-4-20)+(6+13)-(7-8)-5.
Solution. First, we open the brackets, and then we find separately the sum of all positive and separately the sum of all negative numbers, and, finally, add the results:
(- 4 - 20)+(6+ 13)-(7 - 8) - 5 = -4-20 + 6 + 13-7 + 8-5 = = (6 + 13 + 8)+(- 4 - 20 - 7 - 5)= 27-36=-9.
Example 6 Find the value of the expression
Solution. First, we represent each term as the sum of their integer and fractional parts, then open the brackets, then add the whole and separately fractional parts and finally sum up the results:
How do you open parentheses that are preceded by a "+" sign? How can you find the value of an expression that is the opposite of the sum of several numbers? How to open brackets preceded by a "-" sign?
1218. Expand the brackets:
a) 3.4+(2.6+ 8.3); c) m+(n-k);
b) 4.57+(2.6 - 4.57); d) c+(-a + b).
1219. Find the value of the expression:
1220. Expand the brackets:
a) 85+(7.8+ 98); d) -(80-16) + 84; g) a-(b-k-n);
b) (4.7 -17) + 7.5; e) -a + (m-2.6); h) - (a-b + c);
c) 64-(90 + 100); e) c+(-a-b); i) (m-n)-(p-k).
1221. Expand the brackets and find the value of the expression:
1222. Simplify the expression:
1223. Write amount two expressions and simplify it:
a) - 4 - m and m + 6.4; d) a + b and p - b
b) 1.1+a and -26-a; e) - m + n and -k - n;
c) a + 13 and -13 + b; e)m - n and n - m.
1224. Write the difference of two expressions and simplify it:
1226. Use the equation to solve the problem:
a) There are 42 books on one shelf, and 34 on the other. Several books were removed from the second shelf, and as many as were left on the second from the first. After that, 12 books remained on the first shelf. How many books were taken off the second shelf?
b) There are 42 students in the first class, 3 students less in the second than in the third. How many students are in the third grade if there are 125 students in these three grades?
1227. Find the value of the expression:
1228. Calculate orally:
1229. Find highest value expressions:
1230. Enter 4 consecutive integers if:
a) the smaller of them is equal to -12; c) the smaller of them is equal to n;
b) the greater of them is equal to -18; d) the larger of them is equal to k.
