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Lesson Objectives:

1. To repeat and deepen the ability of students to find the values ​​of numerical expressions made up of rational numbers using the signs of addition, subtraction, multiplication and division;
2. Students should be aware that an expression containing the action division by zero does not make sense.
3. To develop students' cognitive interest in learning a new subject.
4. Develop thinking, memory, speech, improve students' computational skills, the ability to work at an optimal pace.

Equipment: PC, multimedia installation; homework cards (Appendix 1)

Lesson type: lesson of repetition and generalization of knowledge obtained in the course of mathematics grades 5-6.

Forms of work: frontal, collective, independent work.

During the classes

1. Organizational moment (2-4 minutes)

Congratulate students on the start of the new school year.

***
And again in the gilding of poplar,
And the school is like a ship at the pier,
Where teachers wait for students
To start a new life.

***
Let happiness knock on your door
Open it up wider.
The path of life is shrouded in mystery,
But it's so beautiful in this world!
And let there always be light in the window,
Mom's smile - from the threshold.
May there be many good years
And life is easy!

***
Autumn motives
This gorgeous woman is AUTUMN
I gave myself to the dissolute wind,
And whatever he says, whatever he asks,
She gave it to him without feeling the measure.
Foliage multi-colored large armfuls
Threw at his feet with a wedding bouquet,
And violent colors, and the remains of the sun,
And tears of rain, and fog before dawn.
And the wind is a dissolute roamer around the world,
Loving only yourself, your whim,
And even this gorgeous woman
Tried to hurt as much as possible
To rip off her dress with a cheeky impulse,
So that she would stand naked until the winter ...
AUTUMN forgave, only with a quiet anguish
Already doomed tears dropped.
In winter's arms she dies,
And gray hair now, not blue.
Under the snow cape no one will know
This gorgeous woman is AUTUMN.
<slide 1>

2. What does algebra study?

U.: What subject did we study last year?

Students: Mathematics.

There is a rumor about mathematics
That she puts her mind in order.
So good words
People often talk about her.

W.: What do we do in math class?

Students: They carried out calculations with integer and fractional numbers, solved equations, problems, built figures in the coordinate plane.

<slide 2>

W.: All this was the content of the subject "Mathematics". This subject is divided into a huge number of independent disciplines: algebra, geometry, probability theory, mathematical analysis, game theory, etc. We begin the study of algebra. You have already read the textbook at home. How does it differ, for example, from a literature textbook?

<slide 3>

Students: It has a lot of numbers and letters, and Latin letters.

W.: You and I remember that letters help us write down the properties of actions on numbers in a form that is easy to remember. They say: "The stated statement is written in mathematical language." For example, the commutative property of multiplication: the product does not change from a permutation of factors ( a · b = b · a). Remember how to find the distance, knowing the time and speed.

<slide 4>

Students: To find the distance, you need to multiply the time by the speed.

W.: Let's write it short: s = v · t. That is, the letters help to write down in the form of formulas the rules for finding the values ​​of the quantities of interest to us. How else is algebra different, for example, from arithmetic? In arithmetic problems, according to known rules, an unknown number is found. In algebra, an unknown quantity is denoted by a letter. This unknown quantity and the data in the condition of the problem are interconnected by an equation, from the solution of which the unknown quantity is found. Separate algebraic concepts and methods for solving problems arose several thousand years ago in the ancient states - Babylon and Egypt. The state of mathematical knowledge in those centuries can be judged by ancient manuscripts (papyri) found on the sites of ancient cities.<slide 5>

About 4000 years ago, in Babylon and Egypt, scientists already knew how to write linear equations, with the help of which they solved a wide variety of problems in land surveying, building art and military science. For example, in the British Museum there is a task from the Rhinda papyrus (it was also called the Ahmes papyrus) dating back to the period 2000-1700. BC e .: "Find a number if it is known that by adding 2/3 of it to it and subtracting from the resulting sum of its third, the number 10 is obtained." The solution of this problem is reduced to the solution of a linear equation:

<slide 6, 7>

In the 7th century BC e. the Greeks learned the achievements of the Egyptians in mathematics. At the beginning of the ninth century (830) Khorezmian scholar Muhammad-ben-Musa al-Khwarizmi wrote the book "Hisab al-jabr val-Mukabala" ("Method of restoration and opposition") - this was the first book on algebra. It is of particular importance in the history of mathematics as a manual that has long taught all of Europe. In it, he first considered the methods and techniques of algebra.

Al Jabr
(transfer of terms)

When solving the equation,
If in part one,
no matter what,
There will be a negative term,
We to both parts,
With this member can be compared.
Let's give an equal term,
Only with a sign to others, -
And we will find the result we want!

wal-mukabala
(bringing like)

<Slide 8>

Since the writing of this book, algebra has become an independent science. The word "algebra" itself probably comes from the word "al jebr", which means "restoration". The word "algebra" in Arabic was the art of a doctor to restore a broken arm or leg. The Arabs called the surgeon an algebraist. Thus, mathematics borrowed this word from medicine.

<Slide 8>

The further development of algebra took place mainly in India (until the 12th century) and in Central Asia (until the 15th century). Algebra until the 17th century. conventionally called rhetorical (verbal). The fact is that then there were no single conventional signs "+", "-", "a 2" and many others that we use. The condition of the problem, all actions and the answer were written down completely in words. For ease of memorization, sometimes this entry was made in verse. Mathematical symbols were introduced gradually. So the equal sign "=" was introduced by the English scientist R. Ricord in 1557, the signs ":" and "*" - by the German mathematician Leibniz at the end of the 17th century. , brackets - XVI century. Mathematical symbols made it possible for scientists from different countries to understand each other. In the formation of algebra as a science, great merits belong to the French scientists Francois Vieta and Rene Descartes. During the XVIII-XX centuries. new mathematical sciences grew out of algebra: polynomial algebra, vector algebra. These sciences are studied in higher education.

In school algebra, problems are solved by compiling equations, they study the equations themselves, the relationships between quantities (some of these relationships are called functions). In this case, letters are used, expressions with letters are subjected to various transformations (identical transformations). But behind all these letters, numbers are most often hidden.

<Slide 9>

Sometimes they say: “Algebra rests on four pillars: an equation, a number, an identity, a function.” Algebra, which we are starting to study, gives a person the opportunity not only to perform various calculations, but also teaches him to do it as quickly and rationally as possible.

<Slide 10>

3. Oral exercises.

1. Find the sum of the numbers -3.7 and 6.7 (answer 3); find the product of numbers find the difference between the numbers Repeat the rules for performing arithmetic operations with ordinary fractions and rational numbers.

2. I thought of three numbers. Find the first if you know that the number opposite to it is 6. Find the second if the number of its opposite is 3. Find the third, if you know that, by multiplying it by

3. Calculate:

<slide 11, 12>

4. Learning a new topic.

When solving many problems, it is necessary to perform arithmetic operations on given numbers: addition, subtraction, multiplication and division. But often, before completing each of these actions, it is convenient to indicate in advance the order (plan) following which these actions should be performed. This plan boils down to the fact that, according to the task data, using numbers, action signs and brackets, a numeric expression.

Examples:

If you perform all the actions indicated in a numerical expression, then as a result we get a number about which they say that it is equal to a given numerical expression.

So the first numerical expression is equal to 2, the second is also equal to 2, and the third is equal to 0.

Definition 1: A record composed of numbers using arithmetic operations (addition, subtraction, multiplication, division, exponentiation) is called a numerical (arithmetic) expression.

A numeric expression can consist of a single number.

Definition 2: The value of a numeric expression is the number obtained as a result of performing the actions specified in the numeric expression.

<slide 13>

Examples: The train moved first for 50 minutes at a speed of sixty kilometers per hour, then stopped at the station for ten minutes, then moved for another hour at a speed of 40 km/h. Find the average speed of the train.

Solution: By definition, the average speed of movement is equal to the ratio of the distance traveled to the time spent on this path. Let's calculate the distance and time of motion. First of all, we take into account that (switched to the same time units). At the beginning of the movement, the path at the end was passed - the path 40 1 (km).

The total distance traveled is described by a numerical expression:

The time spent on this path (including the time spent on stopping) is described by a numerical expression: Then the average speed of movement is described by the expression: If we calculate this expression, we get: .

Definition 3: Two numeric expressions connected by the "=" sign form a numerical equality. If the values ​​of the left and right parts of the numerical equality are the same, then the equality is called true, otherwise it is false.

Examples: - correct numerical equality;

6 + 12 3 \u003d (6 + 12) 3 - incorrect numerical equality, since 42 ≠ 54.

<Slide 14>

Parentheses help establish the order of operations. It is assumed that all actions can be carried out. It is always possible to perform addition, subtraction and multiplication of any numbers. But you can divide one number by another only if the divisor is not equal to zero: you cannot divide by zero. If in this expression at some stage of the calculation it is required to divide by zero, then this expression does not make sense.

Examples: These expressions don't make sense .

<slide 15>

Repeat the order of operations in numerical terms. Repeat the rules for performing operations with fractions.

5. Consolidation of the studied material.

Etc. #1 Decide which of the following expressions make sense and which do not. For those that make sense, find the numbers they are equal to.

<slide 16>

Etc. #2 Write as an equality and check if it is true:

a) 20% of the number 240 is equal to 62 (240 0.2 = 62 is not correct);

b) the number 18 is 3% of the number 600 (18 = 0.03 600 is not correct);

c) the product of numbers and 5 is 11% of the number 700 right;

d) the fourth part of the number 18 is 5% of the number 90 right;

e) the number 111:3 is equal to 10% of the number 370 (111:3 = 0.1 370, right);

f) 650% of the number 12 is equal to 77 (6.5 12 = 77 78 ≠ 77, not true).

<Slide 17>

Etc. #3 Calculate:

<slide 18, 19>

6. Homework: abstract, 10 (A)

<Slide 20>

7. Summing up the lesson

<slide 21, 22>

Literature:

1. Mathematics No. 12, 2004
2. Algebra: Grade 7. Control, independent, rating work / V. A. Goldich. – M.: Eksmo, 2008. – 144 p. – (Master class for the teacher).
3. Internet resources.

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Slides captions:

And again in the gilding of poplar, And the school is like a ship at the pier, Where the teacher's students are waiting, To start a new life. Let happiness knock on your door, Open it wider as soon as possible. The path of life is shrouded in mystery, But it is so beautiful in this world! And let there always be light in the window, Mom's smile - from the threshold. May there be many good years And an easy road in life!

There is a rumor about mathematics, That it puts the mind in order. Therefore, good words are often said about her among the people.

S = v t a b = b a

Babylon Egypt

About 4000 years ago, in Babylon and Egypt, scientists already knew how to write linear equations, with the help of which they solved a wide variety of problems in land surveying, building art and military science. The British Museum has a task from the Rhind papyrus (it was also called the Ahmes papyrus)

The task from the Rhind papyrus (it was also called the Ahmes papyrus) is kept in the British Museum. Find a number if it is known that by adding 2/3 of it to it and subtracting its third from the resulting amount, the number 10 is obtained.

"Hisab Al-jabr Wal-muqabala" ("Method of restoration and opposition") - this was the first book on algebra. Al-jabr When solving an equation, If in one part, no matter which one, There is a negative member, We are to both parts, We are comparable with this member. We will give an equal member, Only with a sign to others, - And we will find the result that we desire! Val-mukabala Then we look at the equation, Is it possible to make a ghost, If the members are similar, It is convenient to compare them. By subtracting an equal term from them, we reduce them to one.

Algebra equation number identity function Algebra, which we are starting to study, gives a person the opportunity not only to perform various calculations, but also teaches him to do it as quickly and rationally as possible.

Theme of the lesson: "Numeric expressions" To repeat and deepen the ability of students to find the values ​​of numerical expressions; Remember that an expression containing the action division by zero does not make sense; To develop students' cognitive interest in learning a new subject. Lesson Objectives:

orally Calculate: 6 7 10 80 289 72 8 5 8100 170

A record composed of numbers using arithmetic operations (addition, subtraction, multiplication, division, exponentiation) is called a numerical (arithmetic) expression. 2 2 0 The value of a numeric expression is the number obtained as a result of performing the actions specified in the numeric expression. Exploring the topic

Two numeric expressions connected by the "=" sign form a numerical equality. If the values ​​of the left and right parts of the numerical equality are the same, then the equality is called true, otherwise it is false. correct incorrect Exploring the topic

If in this expression at some stage of the calculation it is required to divide by zero, then this expression does not make sense. Exploring the topic

Task Kiosk #1 Determine which of the following expressions make sense and which do not. For those that make sense, find the numbers they are equal to. a) b) c) doesn't make sense -3/7 54/95

Task kiosk No. 1 (first, second lines), No. 3, No. 4 (e - h), No. 5, No. 6 (first, third lines), No. 7 (a, b), No. 13

Homework P.1 (study, learn definitions), No. 2, No. 4 (a - d), No. 6 (b, e, h)

Lesson summary What expressions did we talk about today? What is a numeric expression? What is the value of a numeric expression? What is numerical equality? What kinds of equalities do you know? When does a numeric expression not make sense?

Thank you for the lesson, Children Creative success to you in the new school year!

Presentation in mathematics on the topic "Algebraic expressions" (Grade 7). This presentation is designed to cover a new 7th grade math topic, Algebraic Expressions. Examples of algebraic expressions are given, a definition of algebraic expressions is given. The difference between an algebraic expression and a numerical expression is shown. Examples are given for what you need to be able to compose algebraic expressions, that is, where they are used. Examples for composing algebraic expressions are considered.

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Slides captions:

Algebraic expressions.

Checking homework. What information from mathematics did you have to remember in the process of doing your homework?

The order of arithmetic operations. Commutative law of addition: a + b = b + a Commutative law of multiplication: a * b = b * a : abc = (ab)c = a(bc) Concept of common fraction, decimal fraction, negative number. Arithmetic operations with decimal fractions. Arithmetic operations with ordinary fractions. The main property of an ordinary fraction: Rules for actions with decimal fractions.

Example 1 One refrigerator costs $350. Then two refrigerators cost twice as much, i.e. 350 2=700$; five refrigerators cost five times as much, i.e. 350 5=1750 $ . It is easy to figure out that refrigerators cost a times more, i.e. 350· a $ Using the expression 350· a, you can find the cost of a different number a of refrigerators by substituting different values ​​of a and performing multiplication. Since the letter a can take on various natural values, then a is a variable 350 a is an algebraic expression (or an expression with a variable)

Example 2. Let the length of one side of the rectangle be a cm, the other - b cm. Find the perimeter of the rectangle. b a P = 2 a + 2 b a , b – variables 2 a + 2 b – algebraic expression

Example 3. Record 2a - 3b + 5 - algebraic expression with variables a and b. - algebraic expression with variables x and y .

Example 4. Find the value of the expression for a = 3 , b = 4 and c = 2 In this algebraic expression, substitute the values ​​of the variables a = 3 , b = 4 , c = 2 . We get a numeric expression. Having performed the actions, we will find its value: = = = 9 The number 9 is the value of the algebraic expression for the given values ​​of the variables. The value of a numerical expression, which is obtained by substituting the selected values ​​of the variables into an algebraic expression, is called the value of the algebraic expression.

We can write some mathematical expressions in different ways. Depending on our goals, whether we have enough data, etc. Numeric and Algebraic Expressions differ in that we write the first only as numbers combined with the help of signs of arithmetic operations (addition, subtraction, multiplication, division) and brackets.

If instead of numbers you enter Latin letters (variables) into the expression, it will become algebraic. Algebraic expressions use letters, numbers, signs of addition and subtraction, multiplication and division. And also the sign of the root, degree, brackets can be used.

In any case, whether this expression is numerical or algebraic, it cannot be just a random set of characters, numbers and letters - it must have a meaning. This means that letters, numbers, signs must be connected by some kind of relationship. Correct example: 7x + 2: (y + 1). Bad example): + 7x - * 1.

The word "variable" was mentioned above - what does it mean? This is a Latin letter, instead of which you can substitute a number. And if we are talking about variables, in this case, algebraic expressions can be called an algebraic function.

The variable can take on different values. And substituting some number in its place, we can find the value of the algebraic expression for this particular value of the variable. When the value of the variable is different, the value of the expression will also be different.

How to solve algebraic expressions?

To calculate the values ​​you need to do transformation of algebraic expressions. And for this you still need to consider a few rules.

First, the domain of an algebraic expression is all possible values ​​of a variable for which the expression can make sense. What is meant? For example, you cannot substitute a value for a variable that would require you to divide by zero. In the expression 1 / (x - 2), 2 must be excluded from the domain of definition.

Secondly, remember how to simplify expressions: factorize, bracket identical variables, etc. For example: if you swap the terms, the sum will not change (y + x = x + y). Similarly, the product will not change if the factors are interchanged (x * y \u003d y * x).

In general, they are excellent for simplifying algebraic expressions. abbreviated multiplication formulas. Those who have not yet learned them should definitely do this - they will still come in handy more than once:

we find the difference of the variables squared: x 2 - y 2 \u003d (x - y) (x + y);

we find the sum squared: (x + y) 2 \u003d x 2 + 2xy + y 2;

we calculate the difference squared: (x - y) 2 \u003d x 2 - 2xy + y 2;

we cube the sum: (x + y) 3 \u003d x 3 + 3x 2 y + 3xy 2 + y 3 or (x + y) 3 \u003d x 3 + y 3 + 3xy (x + y);

cube the difference: (x - y) 3 \u003d x 3 - 3x 2 y + 3xy 2 - y 3 or (x - y) 3 \u003d x 3 - y 3 - 3xy (x - y);

we find the sum of the variables cubed: x 3 + y 3 \u003d (x + y) (x 2 - xy + y 2);

we calculate the difference of the variables cubed: x 3 - y 3 \u003d (x - y) (x 2 + xy + y 2);

we use the roots: xa 2 + ya + z \u003d x (a - a 1) (a - a 2), and 1 and a 2 are the roots of the expression xa 2 + ya + z.

You should also have an idea about the types of algebraic expressions. They are:

rational, and those in turn are divided into:

integers (they do not have division into variables, there is no extraction of roots from variables and there is no raising to a fractional power): 3a 3 b + 4a 2 b * (a - b). The scope is all possible values ​​​​of variables;

fractional (except for other mathematical operations, such as addition, subtraction, multiplication, in these expressions they divide by a variable and raise to a power (with a natural exponent): (2 / b - 3 / a + c / 4) 2. Domain of definition - all values variables for which the expression is not equal to zero;

irrational - in order for an algebraic expression to be considered as such, it must contain the exponentiation of variables to a power with a fractional exponent and / or the extraction of roots from variables: √a + b 3/4. The domain of definition is all values ​​of the variables, excluding those in which the expression under the root of an even degree or under a fractional degree becomes a negative number.

Identity transformations of algebraic expressions is another useful technique for solving them. An identity is an expression that will be true for any variables included in the domain of definition that are substituted into it.

An expression that depends on some variables can be identically equal to another expression if it depends on the same variables and if the values ​​of both expressions are equal, whichever values ​​of the variables are chosen. In other words, if an expression can be expressed in two different ways (expressions) whose values ​​are the same, these expressions are identically equal. For example: y + y \u003d 2y, or x 7 \u003d x 4 * x 3, or x + y + z \u003d z + x + y.

When performing tasks with algebraic expressions, the identical transformation serves to ensure that one expression can be replaced by another, identical to it. For example, replace x 9 with the product x 5 * x 4.

Solution examples

To make it clearer, let's look at a few examples. transformations of algebraic expressions. Tasks of this level can be found in KIMs for the Unified State Examination.

Task 1: Find the value of the expression ((12x) 2 - 12x) / (12x 2 -1).

Solution: ((12x) 2 - 12x) / (12x 2 - 1) \u003d (12x (12x -1)) / x * (12x - 1) \u003d 12.

Task 2: Find the value of the expression (4x 2 - 9) * (1 / (2x - 3) - 1 / (2x +3).

Solution: (4x 2 - 9) * (1 / (2x - 3) - 1 / (2x + 3) \u003d (2x - 3) (2x + 3) (2x + 3 - 2x + 3) / (2x - 3 )(2x + 3) = 6.

Conclusion

When preparing for school tests, USE and GIA exams, you can always use this material as a hint. Keep in mind that an algebraic expression is a combination of numbers and variables expressed in Latin letters. And also signs of arithmetic operations (addition, subtraction, multiplication, division), brackets, degrees, roots.

Use short multiplication formulas and knowledge of identity equations to transform algebraic expressions.

Write us your comments and wishes in the comments - it is important for us to know that you are reading us.

blog.site, with full or partial copying of the material, a link to the source is required.

LESSON #3 Chapter 1. Expressions, identities, equations(22 hours)

Topic. Numeric expressions.

Target. introduce the concepts of a numerical expression, the value of a numerical expression; to form the ability to find the value of a numerical expression by performing operations on numbers and using brackets.

During the classes.

Organizing time.

Analysis of diagnostic work.

Updating of basic knowledge.

Example 1 Calculate. (Orally).

a) 13 - 18.5 = -5.5; b) –19 + 21.3 = 2.3; c) -14 - 71.03 = -85.03;

d) 17 - (-21.3) = 38.3; e) - (-3 - 2.8) = 5.8; f) 3 ∙ 15 - 7 = 38;

g) (15 - 2) ∙ (-3) = - 39; h) ; to) .

Explanation of new material.

1. When solving many problems, it is necessary to perform arithmetic operations on given numbers: addition, subtraction, multiplication and division.

Definition . Numeric expressions - expressions consisting of numbers and action signs.

But often, before completing each of these actions, it is convenient to indicate in advance the order (plan) following which these actions should be performed. This plan boils down to the fact that, according to the task data, using numbers, action signs and brackets, a numeric expression.

2. Examples of numeric expressions:

3. If all the actions indicated in it are performed in a numerical expression, then as a result we get a real number, about which they say that it is equal to a given numerical expression and is called expression value .

Definition . To find the value of a numeric expression means to perform all the actions in it.

Example 2. Find the value of a numeric expression:

4. We, of course, assume that all activities are feasible. Let's explain these words. It is always possible to perform addition, subtraction and multiplication of any numbers. But dividing numbers one by another is possible only if the divisor is not equal to zero: you cannot divide by zero. If in a given expression at some stage it is required to divide by zero, then this requirement is not feasible. Such an expression doesn't make sense.

Example 3 Does the expression make sense:

These expressions do not make sense, because when performing the actions indicated in it, it becomes necessary to divide by zero.

5. Let's remember how to find a fraction of a number.

Definition. To find a fraction of a number, you need to multiply that number by the fraction.

Example 4 Find from 34.

6. Let's remember how to find a number by its fraction.

Definition. In order for a number to be given the known value of its fraction, it is necessary to divide this value by the given fraction.

Example 5 Find the number that equals 45.

7. Let's remember what a percentage is.

Definition. One hundredth of any value or number is called a percentage.

8. Recall how to find the percentage of a given number?

Definition. To find the percentage of a given number, write the percentage as a fraction and multiply that number by the fraction.

Example 6 Find 8% of 400.

2) 400 ∙ 0,08 = 32.

9. Recall how to find a number by its percentage?

Definition. To find a number by its percentage, you need to write the percentage as a fraction and divide this value by a fraction.

Example 7 Find the number if 16% of that number is 80,

Formation of skills and abilities.

Uch.s.6 No. 5 (1st page).

Uch.s.6 No. 6 (1st page).

Uch.s.7 No. 8. The package of milk says that milk contains 3.2% fat, 2.5% protein and 4.7% carbohydrates. How much of each of these substances is contained in a glass (200 g) of milk?

Milk - 200 g

Fat - ? d, 3.2% of total

Protein - ? g, 2.5% of total

Carbohydrates - ? d, 4.7% of total

2) 200 ∙ 0.032 = 6.4 (g) - fats;

4) 200 ∙ 0.025 = 5 (g) - protein;

6) 200 ∙ 0.047 = 9.4 (g) - carbohydrates. Answer: 6.4g, 5g, 9.4g

4. The price of the product first increased by 20%, and then decreased by the same percentage. How and by what percentage has the price changed compared to the original?

Solution.

1) ,

2) 1a 0 - 0.96a 0 = 0.04a 0 ;

3) 0,04 = 4%. Answer : decreased by 4%.

Summing up the lesson.

Why are there parentheses in a numeric expression?

When does a numeric expression make sense? Give an example of such an expression.

When does a numeric expression not make sense? Give an example of such an expression.

What is the value of a numeric expression?

What is the order of operations when finding the value of a numeric expression?

How to express 15% as a common and decimal fraction?

Homework.item 1 (learn the theory). No. 5(2str), 6(2str), 10, 13(2.4), 15.