1. One of the pressing problems of modern teaching methods at school is the development of student motivation. The increase in mental load in mathematics lessons makes us think about how to maintain students’ interest in the material being studied and their activity throughout the lesson. We must ensure that every student works actively and enthusiastically during lessons. In this situation, gaming technologies come to the aid of the teacher - a modern and recognized method of teaching and upbringing, which has educational, developmental and nurturing functions that operate in organic unity. Game forms of teaching in mathematics lessons make it possible to effectively organize interaction between the teacher and students. Even the most passive students get involved in the game. Gaming activities motivate learning; during the game, each student gets the opportunity to think independently, develop creative thinking and solve various problems (that is, apply the acquired knowledge in a specific life situation).

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Municipal budgetary educational institution secondary school No. 24 with in-depth study of individual humanities subjects named after. I.S. Turgenev, Oryol

Methodological development of the lesson

Algebra and the beginnings of analysis

Grade 11

Textbook: Mordkovich A.G. Algebra and the beginnings of analysis. 10 -11 grades: Textbook. For general education institutions. – M.: Mnemosyne, 2013. – 336 p.: ill. (base)

Mathematics teacher: Moreva Oksana Vladimirovna

Abstract of the work: One of the pressing problems of modern teaching methods at school is the development of student motivation. The increase in mental load in mathematics lessons makes us think about how to maintain students’ interest in the material being studied and their activity throughout the lesson. We must ensure that every student works actively and enthusiastically during lessons. In this situation, gaming technologies come to the aid of the teacher - a modern and recognized method of teaching and upbringing, which has educational, developmental and nurturing functions that operate in organic unity. Game forms of teaching in mathematics lessons make it possible to effectively organize interaction between the teacher and students. Even the most passive students get involved in the game. Gaming activities motivate learning; during the game, each student gets the opportunity to think independently, develop creative thinking and solve various problems (that is, apply the acquired knowledge in a specific life situation).

Technological lesson map

Lesson Plan

1. Organizational moment (2-3 min.)
2. Updating basic knowledge (5 min.)
3. “Conquering the Peaks” (30 min.)

• First height (self-test)
• Second height (group work)
• Third height (individual differentiated work).

1. Summing up (4 - 5 min.)
2. Homework (2 – 3 min.)
3. Reflection on goal achievement (1 min.)

During the classes:

1. Organizing time

The lesson begins with listening to an excerpt from V.V. Vysotsky’s song “Only mountains can be better than mountains” (slide 2).

Teacher: Everyone in life has peaks that they strive to conquer. Someone wants to become a doctor, someone is an athlete, and someone might want to become a mountain climber. After all, heights have always attracted people. Remember Icarus, because his dream was to fly to the Sun. And he realized his dream. The essence of a person is to always achieve the intended goal. The epigraph to our lesson is the words from the song you listened to.

How it sparkles with eternal fire during the day
Top of emerald ice,
Which you never conquered.

V.V.Vysotsky

Today in class I invite you to an expedition to conquer mountain peaks. You have to transform into mountaineering athletes conquering the pinnacle of knowledge called “Degree with a fractional exponent” (slide 3).

Activities of students:Students write down the topic of the lesson in their workbook.

1. Updating of reference knowledge

Teacher: In front of each of you is a card - a counter, in which you will record your successes in conquering mountain peaks(Annex 1) . Enter your first and last name on the top line. On this card you will record the passage of each height in points. At the end of the lesson, you will independently calculate the points you scored for the lesson and find out whether you managed to conquer the “mountain height” or not.

Checking equipment: “What will we take with us on the road?”(slide4).

Teacher: As you know, an expedition is always preceded by careful preparation, so at the beginning, I suggest you check your readiness to conquer the mountain peak.

1) Continue the phrase: If is an ordinary fraction (q ≠1) and a ≥ 0, then under a p/q understand...

2) Calculate verbally: 16¼, 27 1/3, 81 ¼, 8 -1/3, (-144) ½ (Tasks can be written down on the board in advance or presented in the form of cards)

3) Continue with the following properties (Tasks can be written on the board in advance)

a s ∙ a t = …

a s : a t = …

(a s ) t = …

(ab)s = …

() s = ...

4) Calculate orally:(The task can be written on the board in advance)

Teacher: So, the equipment is collected. We go to the mountains to conquer mountain peaks.

1. Conquering the peaks

First height “Snow Avalanche”(Self-test)

Teacher: Any mountains are as beautiful as they are dangerous. Many dangers await climbers in the mountains. The first thing we will have to face in the mountains is an avalanche (slide 5). To get out from under the snow, you must complete the following task.

Activities of students:Students receive a task for two options and independently complete it in their workbooks. (Each student receives his task on a card.) Two students work from the back of the board. The task will take 5–7 minutes to complete.

Option 1

Option 2

1. Calculate: 27 1/3 -25 -1/2 +16 3/4 -27 4/3
2. Simplify the expression: a) (125x-6 ) -2/3 ; b) (a∙a -1/3 ) 1/6 ∙a 8/9

At the end of the work, the students who worked at the board turn the board away. Their work is checked by the teacher. Students who worked in notebooks carry out self-tests. That is, each student independently checks the correctness of their assignment, based on the solution on the board. Each correctly completed task is worth 2 points. The points scored for completing the “Snow Avalanche” are recorded in the counter card.

Physical education minute.

Teacher: Conquering mountain peaks is a very difficult task. We were all very tired freeing ourselves from the snowfall. I suggest you take a break.

Exercise “Come on, try it!”:

The teacher invites students to extend their hand forward with an open palm facing up. Press your thumb into your palm. The remaining fingers should be turned out. Now press your little finger. Happened? Not so!

Second height “Ice Crack”(work in groups)

Teacher: While we were resting, an ice crack appeared on our way (slide 6). Do you know how climbers act in such a situation?

Sample student answers:Climbers help each other... To lift a climber out of a crack, they throw him a rope... They work together... It’s very difficult to get out alone, you need a friend’s help…….

Teacher: From your answers it follows that in order to get out of an ice crack, you need to work as a team. So you and I will perform the next task in groups.

Activities of students:The class is divided into groups of 4–5 people. Each group receives a card with tasks in which they made mistakes. Students must find them and correct them. The task will take 5–7 minutes to complete.

Card 1

Find errors

1. (121 1/2 +128 5/7 -81 5/4 )∙125 -1/3 = (11+32-81∙3)∙(-5) = -200∙(-5) = 1000
2. p-q = (p 2/3 -q 2/3 )(p 2/3 +2p 1/3 q 1/3 + q 2/3 )

Card 2

Find errors

Card 3

Find errors

1. (x 1/4 +1) (x 1/4 -1)(x 1/2 -1) = (x 1/4 -1) 2 (x 1/2 -1) = (x 1/2 -1 )(x 1/2 -1) = (x 1/2 -1) 2
2. (-625) -1/4 = 625 1/4 = 5

Card 4

Find errors

At the end of the work, the teachers report to the teacher the errors they found and corrected. The teacher checks the correctness of the assignment. For each corrected error, 2 points are awarded to each group member. The points scored for completing the “Ice Crack” are recorded on the counter card.

Third height “Rockfall”(individual differentiated work).

Teacher: Before we had time to get out of the ice crack, a rockfall hit us (slide 7). The rubble needs to be cleared. All stones are different: big and small. Some will wear small stones, and some will wear large ones. Everyone will choose a task according to their strength.

Activities of students:Students receive a choice of differentiated tasks of varying difficulty levels.

Those who chose “big stones” receive higher-level tasks on individual cards. Based on the results of completing this task, they will be able to earn up to 8 points. Each correctly completed task is worth 2 points.

Option 1

Reduce the fraction:

A) ; b) ; c) ; d)

Option 2

Reduce the fraction:

At the end of the work, the teacher checks the correctness of the task.

And those who chose “small stones” perform basic level tasks in the form of a test (see the interactive test on the disk or in Appendix 2 ). Based on the results of completing this task, they can earn up to 5 points.

The points scored for completing the “Rockfall” are recorded in the counter card.

1. Summing up the game:

Teacher: Dear “climbers”! Let's calculate the points you scored based on the results of the three tests.

Activities of students:Students count the points they have scored and write them down in the “Overall Result” column.

Teacher: Let's summarize (slide 8). If you scored 18-20 points, then you have conquered the highest peak - well done (excellent mark)! If you scored 15 - 17 points, you conquered the second height, good ( mark good) . If 11 - 14 points means you have only overcome the first height, this is also not bad (mark satisfactory). If you scored less than 11 points, then you remained at the bottom of the peak. But don't be upset! You once again need to undergo training and repeat the climb, your peak is still ahead of you!

Activities of students:Students, according to the rating, give themselves a mark for the lesson in the “Mark” column and hand over their card - the counter to the teacher.

Teacher (at your discretion)transfers these marks to the journal.

1. Homework:§ 37; No. 37.28; No. 37.30ag; No. 37.39*b

No. 37.28. Reduce the fraction: a); b) ; V) ; G) .

No. 37.30ag. Simplify the expression: a) (1 +) 2 - 2 ; d) + - ( + ) 2

No. 37.39*b. Simplify the expression: b) ( + )

1. Reflection on goal achievement:

Teacher: Now I will ask you to continue one or more phrases (slide 9)

• it was interesting…
• it was difficult…
• I did the tasks...
• I managed …
• gave me a lesson for life...

Activities of students:Students continue one or more phrases as desired.

Teacher: Our lesson started with a song, and I want to end it with poetry(slide 10) . Reads a poem.

The heart's aspiration to the top is honorable,

It's nice to look down on the earth.

Ascended... You are a hero, a winner from now on

And it seems that the heavenly world is in our hands.

The top is a desert, only wise stones

Calmly watching the stars shine...

For them you are nothing, a lost wanderer,

Captive of illusions, dubious dreams...

The summit gives you the feeling of flying,

Freedom from the eternal bustle of the world,

The gates are open to a different knowledge...

The maturity of its purity is exciting...

Appendix to the lesson plan“Generalization of the concept of exponent”

Annex 1.

Card – counter __________________________ (Last name, first name)

Appendix 2.

Test

Choose one of the suggested answers.

1. Simplify the expression: (1 – s 1/2 )(1 + s 1/2 )

• (1 – with 1/2) 2
• 1 – s
• 1 – 2s 1/2 + s

1. Simplify the expression: (1 – a 1/2 ) 2

• 1 – a + a 2

• – 2a + a 2

• 1 – 2a 1/2 + a

1. Factor into: 3/4 – to 1/2

• in 3/4 (1 – in)
• in 1/2 (in 1/4 – 1)
• in 1/2 (in 1/2 – 1)
• cannot be decomposed

1. Factorize: a – b

• aw (a 1/2 – in 1/2)
• (a – in 1/2) (a + in 1/2)
• cannot be decomposed
• (a 1/2 – in 1/2) (a 1/2 + in 1/2)

Test scoring: 1 correct answer – 2 points; 2 correct answers – 3 points;3 correct answers – 4 points; 4 correct answer – 5 points.

Lesson and presentation on the topic: "Generalization of concepts about exponents"

Additional materials
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Teaching aids and simulators in the Integral online store for grade 11
Algebraic problems with parameters, grades 9–11
Software environment "1C: Mathematical Constructor 6.1"

Guys, in this lesson we will generalize knowledge about exponents. We can calculate powers with any integer exponent. What if the exponent is not an integer? And what is the connection between the roots and power functions of a non-integer exponent?

Let's repeat a little, consider a number of the form $a^n$.
1. If $n=0$, then $a^n=a^0=1$.
2. If $n=1$, then $a^n=a^1=a$.
3. If $n=2,3,4,5$… then $a^n=a*a*a…*a$ (n factors).
4. If $n=1,2,3,4,5$… and $a≠0$, then $a^(-n)=\frac(1)(a^n)$.
The above rules can also be used as a reminder!

In all the rules presented above, the exponent is an integer. What to do in the case of a fractional exponent?
What is the number $2^(\frac(2)(3))$ and how to work with it? When working with such powers, it is necessary that all properties for integer powers are preserved. For example, when raising a degree to a power, the indicators were multiplied.

For example: $((2^(\frac(2)(3))))^3=2^(\frac(2)(3)*3)=2^2$.
Let's introduce the following symbol replacement: $a=2^(\frac(2)(3))$.
Then: $a^3=2^2$.
We get: $a=\sqrt(2^2)$.
That is, we can present the original expression in this form: $2^(\frac(2)(3))=\sqrt(2^2)$.

Definition. Let us be given an ordinary fraction $\frac(a)(b)$, $b≠1$ and $x≥0$, then $x^(\frac(a)(b))=\sqrt[b](x ^a)$.

For example: $3^(\frac(1)(3))=\sqrt(3)$,
$5^(\frac(2)(5))=\sqrt(5^2)$.

Let's multiply two numbers with the same bases but different powers:
$a^(\frac(2)(3))*a^(\frac(1)(4))=\sqrt(a^2)*\sqrt(a)=\sqrt(a^8)*\ sqrt(a^3)=\sqrt(a^(11))=a^(\frac(11)(12))$.
But we also note: $\frac(2)(3)+\frac(1)(4)=\frac(8+3)(12)=\frac(11)(12)$.
That is: $a^(\frac(2)(3))*a^(\frac(1)(4))=a^(\frac(2)(3)+\frac(1)(4) )=a^(\frac(11)(12))$.
Adding fractions is much easier than working with radicals (you need to bring the exponents to the same form and then just multiply). Therefore, it is customary to switch to power functions with a fractional exponent.

Example.
Calculate:
a) $((27))^(\frac(1)(3))$.
b) $((32))^(\frac(3)(5))$.
c) $0^(\frac(5)(7))$.
d) $((-32))^(\frac(1)(5))$.
Solution.
a) $((27))^(\frac(1)(3))=\sqrt(27)=3$.

B) $((32))^(\frac(3)(5))=\sqrt((32)^3)=((\sqrt(32)))^3=2^3=8$.

B) $0^(\frac(5)(7))=\sqrt(0^5)=((\sqrt(0)))^5=0^5=0$.

D) We can only extract a root with a fractional exponent from a positive number, guys, look at our definition. Our expression makes no sense.
It seems $((-32))^(\frac(1)(5))=\sqrt(-32)=-2$ is the correct notation, but let's take a closer look at our expression: $((-32))^ (\frac(1)(5))$=$((-32))^(\frac(2)(10))$=$\sqrt(((-32))^2)$=$\sqrt (1024)=2$.
We received a contradictory expression, although all operations were performed correctly, according to the properties and definitions. Therefore, mathematicians forbade raising negative numbers to fractional powers.

Guys, remember: We can only raise positive numbers to fractional powers!

Definition. Let an ordinary fraction $\frac(a)(b)$, $b≠1$ and $х>0$ be given, then $x^(-\frac(p)(q))=\frac(1)(x ^(\frac(p)(q)))$.

For example: $2^(-\frac(1)(4))=\frac(1)(2^(\frac(1)(4)))=\frac(1)(\sqrt(2))$.
$3^(-\frac(3)(5))=\frac(1)(3^(\frac(3)(5)))=\frac(1)(\sqrt(3^3))=\ frac(1)(\sqrt(27))$.

All the properties that we encountered when working with power numbers are preserved in the case of rational powers, let's repeat the properties.

Let us be given positive numbers $a>0$ and $b>0$, x and y are arbitrary rational numbers, then the following 5 properties hold:
1. $a^x*a^y=a^(x+y)$.
2. $\frac(a^x)(a^y)=a^(x-y)$.
3. $((a^x)^y=a^(x*y)$.
4. $(a*b)^x=a^x*a^y$.
5. $((\frac(a)(b)))^x=\frac(a^x)(b^x)$.

Example.
Simplify the expression: $\frac(\sqrt(x))(x^(\frac(1)(2))+y^(\frac(1)(2)))+\frac(\sqrt(y)) (x^(\frac(1)(2))-y^(\frac(1)(2)))$.
Solution.
Let's rewrite the numerators in the form of power functions:
$\frac(x^(\frac(1)(2)))(x^(\frac(1)(2))+y^(\frac(1)(2)))+\frac(y^ (\frac(1)(2)))(x^(\frac(1)(2))-y^(\frac(1)(2)))$.
Let's bring it to a common denominator:
$\frac(x^(\frac(1)(2))(x^(\frac(1)(2))-y^(\frac(1)(2)))+y^(\frac( 1)(2))(x^(\frac(1)(2))+y^(\frac(1)(2))))((x^(\frac(1)(2))+y ^(\frac(1)(2)))(x^(\frac(1)(2))-y^(\frac(1)(2))))$ =$\frac(x-x^(\ frac(1)(2))*y^(\frac(1)(2))+y^(\frac(1)(2))*x^(\frac(1)(2))+y) (x-y)$=$\frac(x+y)(x-y)$.

Example.
Solve equations:
a) $\sqrt(x^4)=1$.
b) $x^(\frac(4)(5))=1$.
Solution.
a) Raise both sides of the equation to the fifth power:
$x^4=1$.
$x=±1$.

B) Our equation is very similar to the previous ones. If we move from writing roots to power functions, then the record will be identical, but it is worth considering that we are immediately given a power expression. By definition, the number x can only be positive, then we are left with one answer $x=1$.

Example.
Solve the equation: $x^(-\frac(2)(5))+x^(-\frac(1)(5))-12=0$.
Solution.
Let's introduce a new variable: $y=x^(-\frac(1)(5))$.
$y^2=((x^(-\frac(1)(5))))^2=x^(-\frac(2)(5))$.
Then our equation will take the form of an ordinary quadratic equation: $y^2+y-12=0$.
Having solved the equation, we get two roots: $y_1=-4$ and $y_2=3$.

We just have to solve two equations: $x^(-\frac(1)(5))=-4$ and $x^(-\frac(1)(5))=3$.
The first equation has no roots. Recall that power functions with a rational exponent are defined only for positive numbers.
Let's solve the second equation:
$x^(-\frac(1)(5))=3$.
$\frac(1)(x^(\frac(1)(5)))=3$.
$x^(\frac(1)(5))=\frac(1)(3)$.
$\sqrt(x)=\frac(1)(3)$.
$x=(\frac(1)(3))^5=\frac(1)(243)$.

Guys, we looked at two examples of solving irrational equations.

Let's list the main methods for solving irrational equations.
1) Raising both sides of an equation to the same power(when using this method, you need to check the solutions obtained, as extraneous solutions may arise).
2) Variable Replacement Method(introduction of new variables).
3) Plotting function graphs. We represent both sides of the equation as functions, construct their graphs and find the points of intersection of the graphs.

Problems to solve independently

The manual contains independent and test work on all the most important topics in the mathematics course for grades 10-11. The works consist of 6 options of three levels of difficulty. Didactic materials are intended for organizing differentiated independent work of students.

Examples.
There are 10 balls in a box, 3 of which are white. One ball at a time is sequentially removed from the box until a white ball appears. Find the probability of a white ball appearing.

Three shooters shoot at the same target 2 times each. It is known that the probability of a hit for each shooter is 0.5 and does not depend on the results of other shooters and previous shots. Is it possible to say
with a probability of 0.99 that at least one shot will hit the target?
with a probability of 0.5 that each shooter will hit the target at least once?

CONTENT
Trigonometry
S-1. Definition and properties of trigonometric functions. Degree and radian measures of angle
S-2. Trigonometric identities
S-3. Reduction formulas. Addition formulas
S-4. Double and half angle formulas
S-5. Trigonometric formulas for converting a sum into a product and a product into a sum
S-6*. Additional trigonometry problems (independent homework)
K-1. Converting trigonometric expressions
S-7. General properties of functions. Transformations of function graphs
S-8. Parity and periodicity of functions
S-9. Monotony of functions. Extremes C-10*. Research of functions. Harmonic oscillations (home practice work)
K-2. Trigonometric functions
S-11. Inverse trigonometric functions __
S-12*. Application of the properties of inverse trigonometric functions (independent homework)
S-13. The simplest trigonometric equations
S-14. Trigonometric equations
S-15. Selection of roots in trigonometric equations. Systems of trigonometric equations
S-16*. Methods for solving trigonometric equations (independent homework)
S-17*. Systems of trigonometric equations (independent homework)
S-18. The simplest trigonometric inequalities
S-19*. Methods for solving trigonometric inequalities (independent homework)
K-3. Trigonometric equations, inequalities, systems
Algebra
S-20. The nth root and its properties
S-21. Irrational equations
S-22. Irrational inequalities. Systems of irrational equations
S-23*. Methods for solving irrational equations, inequalities, systems (independent homework)
S-24. Generalization of the concept of degree
K-4. Powers and roots
S-25. Exponential equations. Systems of exponential equations
S-26. Exponential inequalities
S-27*. Methods for solving exponential equations and inequalities (independent homework)
S-28*. Exponential power equations and inequalities (independent homework)
K-5. Exponential function
S-29. Logarithm. Properties of logarithms
S-30. Logarithmic equations and systems
S-31*. Application of logarithms in solving transcendental equations and systems (independent homework)
S-32. Logarithmic inequalities
S-33*. Methods for solving logarithmic equations, inequalities, systems (independent homework)
K-6. Logarithmic function
S-34. Generalization of the concept of a module. Equations and inequalities with modulus
Start of analysis
S-35. Calculation of limits of number sequences and functions. Continuity of function
S-36. Definition of derivative. The simplest rules for calculating derivatives
S-37. Derivatives of trigonometric and complex functions
S-38. Geometric and mechanical meaning of derivative
K-7. Derivative
S-39. Studying a function for monotonicity and extrema
S-40*. Additional study of function (home independent work)
S-41*. Plotting graphs of functions (home practice)
S-42. The largest and smallest values ​​of a function. Extreme Challenges
S-43*. Selected problems of differential calculus (independent homework)
K-8. Application of derivative
S-44. Antiderivative. Calculation of antiderivatives
S-45. Definite integral. Calculating areas using a definite integral
S-46. Application of antiderivative and integral
S-47*. Selected problems of integral calculus (independent homework)
K-9. Antiderivative and integral
S-48. Derivative and antiderivative of an exponential function
S-49. Derivative and antiderivative of a logarithmic function
S-50. Power function
S-51*. Additional problems of mathematical analysis (independent homework)
K-10. Derivative and antiderivative of exponential, logarithmic and power functions
Complex numbers
S-52. The concept of a complex number. Operations with complex numbers in algebraic form
S-53. Modulus and argument of a complex number. Operations with complex numbers in geometric form
S-54. Trigonometric form of a complex number. Moivre's formula
S-55*. Additional problems with complex numbers (independent homework)
K-11. Complex numbers
Combinatorics
S-56. Multitudes. Set Operations
S-57. Basic formulas of combinatorics. The simplest combinatorial problems
S-58. Binomial theorem. Properties of binomial coefficients
S-59. Combinatorial problems. Sum rule and product rule
S-60*. Additional tasks in combinatorics (independent homework)
K-12. Elements of combinatorics
Probability theory
S-61. Classic probability. Using combinatorics formulas when calculating probability
S-62. Probability addition and multiplication theorems
S-63. The probability of at least one of the independent events occurring. Bernoulli scheme
S-64*. Additional chapters of probability theory (independent homework)
K-13. Elements of probability theory
ANSWERS
Answers to tests
Answers to home independent
work
LITERATURE.

Download the e-book for free in a convenient format, watch and read:
Download the book Independent and test work on algebra and principles of analysis, grades 10-11, Ershova A.P., Goloborodko V.V., 2013 - fileskachat.com, fast and free download.

The purpose of the lesson:

1. Generalization and systematization of knowledge, skills and abilities.
2. Updating basic knowledge in the conditions of passing the Unified State Exam.
3. Monitoring and self-control of knowledge, skills and abilities using tests.
4. Development of the ability to compare and generalize.

Lesson plan.

1. Statement of lesson purpose (1 min)
2. Oral work “I believe - I don’t believe!” (6 min)
3. Solving a series of examples to compare expressions (12 min)
4. Sophistry (4–5 min)
5. Solving an example to simplify an expression (from the Unified State Examination) with a discussion of the most “subtle” parts (15 min)
6. Independent work based on the demo version of the Unified State Exam (group A) (5 min)
7. Homework (on pieces of paper)

Equipment: projector.

1. Friends! Before your eyes is part of a statement by the English mathematician James Joseph Sylvester (1814–1897) about mathematics “Mathematics is the music of the mind.” How romantic isn't it?

Question. How do you think he defined music?

“Music is the mathematics of feelings.”

We can include various kinds of experiences as feelings. This year, one of the reasons for your and my worries is the successful passing of the Unified State Exam and, as a result, admission to a university. I really want positive emotions to prevail. There must be confidence, and this is our knowledge and skills. Today in class we will continue preparing for the Unified State Exam, repeating and generalizing the concept of degree.

So, the topic of today's lesson is “Generalization of the concept of degree.”

We have already repeated the basic properties and definitions, and I invite you to play the game “Believe it or not!”

Your task is to quickly (relying on your intuition, it will help in solving group A) answer the question affirmatively or negatively, and then explain your answer.

2. Oral work “I believe - I don’t believe!”

1. The expressions have meaning:

a) b) c) c) d)

3. The equation has three roots

(no, the root is one: 7, because)

4. Least root of equation 1

3. Solving a series of examples to compare fractions. Now I propose to draw your attention to a series of examples of comparing degrees.

Question. What ways of comparing degrees do you know?

Comparison of indicators with the same bases, comparison of bases with the same exponents.

1. Compare And .

2. Compare numbers And .

As you can see, the case is more complicated.

Question. What numbers are exponents?

Irrational.

Let's find rational numbers that are close to the given irrational numbers and try to compare the powers with the rational exponent.

Because the base of the degree is greater than 1, then by the property of degrees we have

Let us now compare and .

To do this, it is enough to compare and 2 or and.

But , A .

Now we get a chain of inequalities:

3. Compare numbers And .

Let's use the following property of radicals: if , then , where .

Let's compare and .

Let's evaluate their attitude:

Thus, .

Notes.

1) In this case, the degrees and are small, namely

, and they are not difficult to calculate “manually”, i.e. without a calculator. You can estimate the degrees without calculations:

That's why,

2) If the degrees really cannot be calculated (even on a calculator), for example, and , then you can use the inequality:

True for any , and do this:

with all natural.

You can prove it yourself

4. Sophistry. Well, let's switch to another job. Let's find an error in the following reasoning, refuting the statement:

“One is equal to an infinitely large degree to an arbitrary number.”

As is known, a unit raised to any power, including zero, is equal to one, i.e., where A– any number. Let's see, however, whether this is always the case.

Let X– arbitrary number. By simple multiplication it is easy to verify that expression (1) is an identity for any X. Then the identity that follows from (1) is also true, namely . (2)

For an arbitrary positive number A exists .

Equality (2) implies the equality

,

or, what is the same,

. (3)

Assuming in identity (3) x=3, we get

, (4)

and taking into account that , we get that .

So, the power of one, even when the exponent is equal to infinity, is equal to an arbitrary number, but by no means to one, as required by the rules of algebra.

Solution.

The error is as follows.

Equality (1) is indeed valid for all values X and therefore is an identity. The equality (2) obtained from it is no longer valid for all values X. So, X cannot be equal to 2. since the denominators on the left and right sides of (2) become zero, and X cannot be equal to 3, since the denominator on the right side of (2) also becomes zero. At x = 3 equality (2) takes the form , which makes no sense.

Relationship (4) is obtained from (3) precisely at x = 3, which led to an absurd result.

Well, now let's fast forward to 2004, when the following number was proposed in task C3.

5. Solution of the example (from the Unified State Examination).

Since f(x) is an increasing function, then .

Let's find which of these values ​​is closer to 0.7, for which we compare

And

Since , the value of f(26) lies closer to 0.7.

6. Independent work followed by checking on the board.

And now it’s time to practice: here are examples from the demo version, gr. A 2009.

You see them both on the board and on pieces of paper. Your task is to quickly solve and fill out the tables with answers. Match the letters and numbers in front of you. By correctly calculating or simplifying the expressions in the table, you will read what you need when passing the Unified State Exam.

Option 1 – luck, knowledge,

Option 2 – confidence.

So, today in class we saw how widely the concept of degree is used when passing the Unified State Exam. You can consolidate your acquired skills by doing homework.

7. Homework.

Pay attention to your homework, it will help you consolidate the material we covered in class.