Basic properties of addition and multiplication of numbers.
Commutative property of addition: rearranging the terms does not change the value of the sum. For any numbers a and b the equality is true
Combinative property of addition: in order to add a third number to the sum of two numbers, you can add the sum of the second and third to the first number. For any numbers a, b and c the equality is true
Commutative property of multiplication: rearranging the factors does not change the value of the product. For any numbers a, b and c the equality is true
Combinative property of multiplication: to multiply the product of two numbers by a third number, you can multiply the first number by the product of the second and third.
For any numbers a, b and c the equality is true
Distributive Property: To multiply a number by a sum, you can multiply that number by each term and add the results. For any numbers a, b and c the equality is true
From the commutative and combinative properties of addition it follows: in any sum you can rearrange the terms in any way you like and arbitrarily combine them into groups.
Example 1 Let's calculate the sum 1.23+13.5+4.27.
To do this, it is convenient to combine the first term with the third. We get:
1,23+13,5+4,27=(1,23+4,27)+13,5=5,5+13,5=19.
From the commutative and combinative properties of multiplication it follows: in any product you can rearrange the factors in any way and arbitrarily combine them into groups.
Example 2 Let's find the value of the product 1.8·0.25·64·0.5.
Combining the first factor with the fourth, and the second with the third, we have:
1.8·0.25·64·0.5=(1.8·0.5)·(0.25·64)=0.9·16=14.4.
The distributive property is also true when a number is multiplied by the sum of three or more terms.
For example, for any numbers a, b, c and d the equality is true
a(b+c+d)=ab+ac+ad.
We know that subtraction can be replaced by addition by adding to the minuend the opposite number of the subtrahend:
This allows numeric expression type a-b be considered the sum of numbers a and -b, a numerical expression of the form a+b-c-d be considered the sum of numbers a, b, -c, -d, etc. The considered properties of actions are also valid for such sums.
Example 3 Let's find the value of the expression 3.27-6.5-2.5+1.73.
This expression is the sum of the numbers 3.27, -6.5, -2.5 and 1.73. Applying the properties of addition, we get: 3.27-6.5-2.5+1.73=(3.27+1.73)+(-6.5-2.5)=5+(-9) = -4.
Example 4 Let's calculate the product 36·().
The multiplier can be thought of as the sum of the numbers and -. Using the distributive property of multiplication, we obtain:
36()=36·-36·=9-10=-1.
Identities
Definition. Two expressions whose corresponding values are equal for any values of the variables are called identically equal.
Definition. An equality that is true for any values of the variables is called an identity.
Let's find the values of the expressions 3(x+y) and 3x+3y for x=5, y=4:
3(x+y)=3(5+4)=3 9=27,
3x+3y=3·5+3·4=15+12=27.
We got the same result. From distributive properties it follows that in general, for any values of the variables, the corresponding values of the expressions 3(x+y) and 3x+3y are equal.
Let us now consider the expressions 2x+y and 2xy. When x=1, y=2 they take equal values:
However, you can specify values of x and y such that the values of these expressions are not equal. For example, if x=3, y=4, then
The expressions 3(x+y) and 3x+3y are identically equal, but the expressions 2x+y and 2xy are not identically equal.
The equality 3(x+y)=x+3y, true for any values of x and y, is an identity.
True numerical equalities are also considered identities.
Thus, identities are equalities that express the basic properties of operations on numbers:
a+b=b+a, (a+b)+c=a+(b+c),
ab=ba, (ab)c=a(bc), a(b+c)=ab+ac.
Other examples of identities can be given:
a+0=a, a+(-a)=0, a-b=a+(-b),
a·1=a, a·(-b)=-ab, (-a)(-b)=ab.
Identical transformations of expressions
Replacing one expression with another identically equal expression is called an identical transformation or simply a transformation of an expression.
Identical transformations of expressions with variables are performed based on the properties of operations on numbers.
To find the value of the expression xy-xz when given values x, y, z, you need to perform three actions. For example, with x=2.3, y=0.8, z=0.2 we get:
xy-xz=2.3·0.8-2.3·0.2=1.84-0.46=1.38.
This result can be obtained by performing only two steps, if you use the expression x(y-z), which is identically equal to the expression xy-xz:
xy-xz=2.3(0.8-0.2)=2.3·0.6=1.38.
We simplified the calculations by replacing the expression xy-xz identically equal expression x(y-z).
Identical transformations of expressions are widely used in calculating the values of expressions and solving other problems. Some identity transformations I already had to perform, for example, the reduction of similar terms and the expansion of parentheses. Let us recall the rules for performing these transformations:
in order to bring similar terms, you need to add up their coefficients and multiply the result by the common letter part;
if there is a plus sign before the brackets, then the brackets can be omitted, preserving the sign of each term enclosed in brackets;
If there is a minus sign before the parentheses, then the parentheses can be omitted by changing the sign of each term enclosed in the parentheses.
Example 1 Let us present similar terms in the sum 5x+2x-3x.
Let's use the rule for reducing similar terms:
5x+2x-3x=(5+2-3)x=4x.
This transformation is based on the distributive property of multiplication.
Example 2 Let's open the brackets in the expression 2a+(b-3c).
Using the rule for opening parentheses preceded by a plus sign:
2a+(b-3c)=2a+b-3c.
The transformation carried out is based on the combinatory property of addition.
Example 3 Let's open the brackets in the expression a-(4b-c).
Let's use the rule for opening parentheses preceded by a minus sign:
a-(4b-c)=a-4b+c.
The transformation performed is based on the distributive property of multiplication and the combinatory property of addition. Let's show it. Let's imagine in this expression the second term -(4b-c) in the form of a product (-1)(4b-c):
a-(4b-c)=a+(-1)(4b-c).
By applying specified properties actions, we get:
a-(4b-c)=a+(-1)(4b-c)=a+(-4b+c)=a-4b+c.
The numbers and expressions that make up the original expression can be replaced by identically equal expressions. Such a transformation of the original expression leads to an expression that is identically equal to it.
For example, in the expression 3+x, the number 3 can be replaced by the sum 1+2, which will result in the expression (1+2)+x, which is identically equal to the original expression. Another example: in the expression 1+a 5, the power a 5 can be replaced by an identically equal product, for example, of the form a·a 4. This will give us the expression 1+a·a 4 .
This transformation is undoubtedly artificial, and is usually a preparation for some further transformations. For example, in the sum 4 x 3 +2 x 2, taking into account the properties of the degree, the term 4 x 3 can be represented as a product 2 x 2 2 x. After this transformation, the original expression will take the form 2 x 2 2 x+2 x 2. Obviously, the terms in the resulting sum have common multiplier 2 x 2 , so we can perform the following transformation - bracketing. After it we come to the expression: 2 x 2 (2 x+1) .
Adding and subtracting the same number
Another artificial transformation of an expression is the addition and simultaneous subtraction of the same number or expression. This transformation is identical because it is essentially equivalent to adding zero, and adding zero does not change the value.
Let's look at an example. Let's take the expression x 2 +2·x. If you add one to it and subtract one, this will allow you to perform another identical transformation in the future - square the binomial: x 2 +2 x=x 2 +2 x+1−1=(x+1) 2 −1.
Bibliography.
- Algebra: textbook for 7th grade general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 17th ed. - M.: Education, 2008. - 240 p. : ill. - ISBN 978-5-09-019315-3.
- Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
- Mordkovich A. G. Algebra. 7th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 17th ed., add. - M.: Mnemosyne, 2013. - 175 p.: ill. ISBN 978-5-346-02432-3.
Lesson type: lesson of generalization and systematization of knowledge.
Lesson objectives:
- Improve the ability to apply previously acquired knowledge to prepare for the State Examination in 9th grade.
- Teach the ability to analyze and approach a task creatively.
- To cultivate a culture and efficiency of thinking, cognitive interest to mathematics.
- Help students prepare for the State Examination.
- Systematize theoretical knowledge students.
- Enhance practical orientation this topic in preparation for the State Examination.
- Develop mental work skills - searching for rational solutions.
Equipment: multimedia projector, worksheet, clock.
Lesson plan: 1. Organizational moment.
- Updating knowledge.
- Development of theoretical material.
- Lesson summary.
- Homework.
DURING THE CLASSES
I. Organizational moment.
1) Greeting from the teacher.
Cryptography is the science of ways to transform (encrypt) information in order to protect it from illegal users. One of these methods is called “grid”. It is one of the relatively simple ones and is closely related to arithmetic, but one that is not studied in school. A sample of the lattice is in front of you. Someone will figure out how to use it.
- the solution to the message.
“Everything that stops working out stops attracting.”
Francois Larachefoucauld.
2) Messages about the topic of the lesson, lesson objectives, lesson plan.
– slides in the presentation.
II. Updating knowledge.
1) Oral work.
1. Numbers. What numbers do you know?
– natural numbers are numbers 1,2,3,4... which are used when counting
– integers are numbers…-4,-3,-2,-1,0,1, 2… natural numbers, their opposites and the number 0.
– rational numbers are whole and fractional numbers
– irrational – these are infinite decimal non-periodic fractions
– real – these are rational and irrational.
2. Expressions. What expressions do you know?
– numerical are expressions consisting of numbers connected by arithmetic symbols.
– alphabetic – this is an expression containing some variables, numbers and action signs.
– Integers are expressions consisting of numbers and variables using the operations of addition, subtraction, multiplication and division by a number.
– fractional ones are whole expressions using division by an expression with a variable.
3. Transformations. What are the main properties used when performing transformations?
– commutative – for any numbers a and b it is true: a+b=b+a, ab=va
– associative – for any numbers a, b, c, the following is true: (a+b)+c=a+(b+c), (ab)c=a(c)
– distributive – for any numbers a, b, c it is true: a(b+c)=av+ac
4. Do:
– arrange the numbers in ascending order: 0.0157; 0.105; 0.07
– arrange the numbers in descending order: 0.0216; 0.12; 0.016
– one of the points marked on the coordinate line corresponds to the number v68. What point is this?
– what point do the numbers correspond to?
– the numbers a and b are marked on the coordinate line. Which of the following statements is true?
III. Development of theoretical material.
1. Work in notebooks, at the board.
Each teacher has a worksheet where tasks are written down for work in notebooks during the lesson. In the right column of this sheet there are assignments for work in class, and in the left column there is homework.
Students come out to work at the board.
Task No. 1. In which case is the expression converted to identically equal.
Task No. 3. Factor it out:
a 3 – av – a 2 c + a 2; x 2 y – x 2 -y + x 3.
2x + y + y 2 – 4x 2 ; a – 3c +9c 2 -a 2 .
2. Independent work.
On the worksheets you have independent work, below after the text there is a table in which you enter the number under the correct answer. It takes 7 minutes to complete the job.
Test “Numbers and Conversions”
1. Write 0.00019 in standard form.
1)0,019*10 -2 ; 2)0,19*10 -3 ; 3)1,9*10 -4 ; 4)19*10 -5
2. One of the points marked on the coordinate line corresponds to the number
3. About numbers a and b it is known that a>0, b>0, a>4b. Which of the following inequalities is false?
1) a-2a>-3b; 2) 2a>8b; 3) a/4>b-2; 4) a+3>b+1.
4.Find the value of the expression: (6x – 5y): (3x+y), if x=1.5 and y=0.5.
1) 1,5; 2) 1,3; 3) 1,33; 4) 2,5.
5.Which of the following expressions can be converted into (7 – x)(x – 4)?
1)– (7 – x)(4 – x); 2) (7 – x)(4 – x);
3) – (x – 7)(4 – x); 4) (x – 7)(x-4).
After completing the work, the check is carried out using the ASUOK program (automated training and control management system). The guys exchange notebooks with their deskmate and check the test together with the teacher.| exercise | |||||
| Answer: | 3 | 1 | 1 | 2 | 1 |
6. Lesson summary.
Today in class you solved tasks selected from collections to prepare for the State Examination. This is a small part of what you need to repeat to pass the exam perfectly.
- The lesson is over. What did you find useful from the lesson?
“An expert is a person who no longer thinks, he knows.” Frank Hubbard.
7. Homework
On the sheets of paper are tasks to complete at home.
Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8\)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2\)
The sum of monomials is called a polynomial. The terms in a polynomial are called terms of the polynomial. Monomials are also classified as polynomials, considering a monomial to be a polynomial consisting of one member.
For example, a polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.
Let's represent all terms in the form of monomials standard view:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16\)
Let us present similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
The result is a polynomial, all terms of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.
Behind degree of polynomial of a standard form take the highest of the powers of its members. Thus, the binomial \(12a^2b - 7b\) has the third degree, and the trinomial \(2b^2 -7b + 6\) has the second.
Typically, the terms of standard form polynomials containing one variable are arranged in descending order of exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1\)
The sum of several polynomials can be transformed (simplified) into a polynomial of standard form.
Sometimes the terms of a polynomial need to be divided into groups, enclosing each group in parentheses. Since enclosing parentheses is the inverse transformation of opening parentheses, it is easy to formulate rules for opening brackets:
If a “+” sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.
If a “-” sign is placed before the brackets, then the terms enclosed in the brackets are written with opposite signs.
Transformation (simplification) of the product of a monomial and a polynomial
Using the distributive property of multiplication, you can transform (simplify) the product of a monomial and a polynomial into a polynomial. For example:
\(9a^2b(7a^2 - 5ab - 4b^2) = \)
\(= 9a^2b \cdot 7a^2 + 9a^2b \cdot (-5ab) + 9a^2b \cdot (-4b^2) = \)
\(= 63a^4b - 45a^3b^2 - 36a^2b^3 \)
The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.
This result is usually formulated as a rule.
To multiply a monomial by a polynomial, you must multiply that monomial by each of the terms of the polynomial.
We have already used this rule several times to multiply by a sum.
Product of polynomials. Transformation (simplification) of the product of two polynomials
In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.
Usually the following rule is used.
To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.
Abbreviated multiplication formulas. Sum squares, differences and difference of squares
With some expressions in algebraic transformations have to deal with more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), i.e. the square of the sum, the square of the difference and difference of squares. You noticed that the names of these expressions seem to be incomplete, for example, \((a + b)^2 \) is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b does not occur very often; as a rule, instead of the letters a and b, it contains various, sometimes quite complex, expressions.
The expressions \((a + b)^2, \; (a - b)^2 \) can be easily converted (simplified) into polynomials of the standard form; in fact, you have already encountered this task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)
It is useful to remember the resulting identities and apply them without intermediate calculations. Brief verbal formulations help this.
\((a + b)^2 = a^2 + b^2 + 2ab \) - square of the sum equal to the sum squares and double the product.
\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is equal to the sum of squares without the doubled product.
\(a^2 - b^2 = (a - b)(a + b) \) - the difference of squares is equal to the product of the difference and the sum.
These three identities allow one to replace its left-hand parts with right-hand ones in transformations and vice versa - right-hand parts with left-hand ones. The most difficult thing is to see the corresponding expressions and understand how the variables a and b are replaced in them. Let's look at several examples of using abbreviated multiplication formulas.