Properties of a degree with a real exponent - theoretical information. Written exam. Properties of the function arcctg

This lesson is part of the topic "Transformations of expressions containing powers and roots."

The summary is a detailed development of a lesson on the properties of a degree with a rational and real exponent. Computer, group and game learning technologies are used.

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Methodological development of an algebra lesson

Mathematics teacher of State Autonomous Institution KO ON KST

Pekhova Nadezhda Yurievna

on the topic: “Properties of degrees with rational and real exponents.”

Lesson objectives:

educational: consolidation and deepening of knowledge of the properties of the degree with rational indicator and their use in exercises; improving knowledge on the history of degree development;
developing: developing the skill of self- and mutual control; development of intellectual abilities, thinking skills,
educating: fostering cognitive interest in the subject, instilling responsibility for the work performed, promoting the creation of an atmosphere of active creative work.

Lesson type: Lessons to improve knowledge, skills and abilities.

Methods of conducting: verbal - visual.

Educational technologies: computer, group and game learning technologies.

Lesson equipment: projection equipment, computer, lesson presentation, workers

notebooks, textbooks, cards with the text of a crossword puzzle and a reflective test.

Lesson time: 1 hour 20 minutes.

Main stages of the lesson:

1. Organizational moment. Message about the topic and objectives of the lesson.

2. Update background knowledge. Repetition of properties of degree with rational exponent.

3. Mathematical dictation on properties of degrees with rational exponent.

4. Student reports using a computer presentation.

5. Work in groups.

6. Solving the crossword puzzle.

7. Summing up, grading. Reflection.

8. Homework.

Lesson progress:

1. Org. moment. Communicate the topic, lesson objectives, lesson plan. Slides 1, 2.

2. Updating basic knowledge.

1) Repetition of the properties of a degree with a rational indicator: students must continue the written properties - frontal survey. Slide 3.

2) Students at the blackboard - analysis of exercises from the textbook (Alimov Sh.A.): a) No. 74, b) No. 77.

C) No. 82-a;b;c.

No. 74: a) = = a ;

B) + = ;

B) : = = = b .

No. 77: a) = = ;

B) = = = b .

No. 82: a) = = = ;

B) = = y;

B) () () = .

3. Mathematical dictation with mutual verification. Students exchange work, compare answers and give grades.

Slides 4 - 5

4. Messages from some students historical facts on the topic being studied.

Slides 6 – 12:

First student: Slide 6

The concept of a degree with a natural indicator was formed among ancient peoples. Square and cubenumbers were used to calculate areas and volumes. The powers of some numbers were used by scientists to solve certain problems Ancient Egypt and Babylon.

In the 3rd century, a book by the Greek scientist Diophantus was published“Arithmetic”, in which the introduction of letter symbols was laid. Diophantus introduces symbols for the first six powers of the unknown and their reciprocals. In this book, a square is denoted by a sign and a subscript; for example, a cube - sign k with index r, etc.

Second student: Slide 7

The ancient Greek scientist Pythagoras made a great contribution to the development of the concept of degree. He had a whole school, and all his students were called Pythagoreans. They came up with the idea that each number can be represented as figures. For example, they represented the numbers 4, 9 and 16 as squares.

First student: Slides 8-9

Slide 8

Slide 9

XVI century. In this century, the concept of degree has expanded: it began to be referred not only to a specific number, but also to a variable. As they said then “to numbers in general” English mathematician S. Stevin invented a notation to denote the degree: the notation 3(3)+5(2)–4 denoted such a modern notation 3 3 + 5 2 – 4.

Second student: Slide 10

Later, fractional and negative exponents are found in “Complete Arithmetic” (1544) by the German mathematician M. Stiefel and in S. Stevin.

S. Stevin suggested that by degree with an exponent of the form root, i.e. .

First student: Slide 11

At the end of the 16th century, François Vietintroduced letters to denote not only variables, but also their coefficients. He used abbreviations: N, Q, C - for the first, second and third degrees.

But modern designations (such as, ) was introduced in the 17th century by Rene Descartes.

Second student: Slide 12

Modern definitionsand notations for degrees with zero, negative and fractional exponents originate from the work of English mathematicians John Wallis (1616–1703) and Isaac Newton.

5. Crossword solution.

Students receive crossword puzzle sheets. They decide in pairs. The pair that solves it first gets the mark. Slides 13-15.

6. Working in groups. Slide 16.

Students do independent work, working in groups of 4, consulting each other. Then the work is submitted for inspection.

7. Summing up, grading.

Reflection.

Students complete a reflective test. Mark “+” if you agree, and “-” otherwise.

Reflective test:

1. I learned a lot of new things.

2. This will be useful to me in the future.

3. There was a lot to think about during the lesson.

4. I received answers to all the questions I had during the lesson.

5. I worked conscientiously during the lesson and achieved the goal of the lesson.

8. Homework: Slide 17.

1) № 76 (1; 3); № 70 (1; 2)

2) Optional: create a crossword puzzle with the basic concepts of the topic studied.

Used literature:

Alimov Sh.A. algebra and beginnings of analysis grades 10-11, textbook - M.: Prosveshchenie, 2010.
Algebra and beginning of analysis grade 10. Didactic materials. Enlightenment, 2012.

Internet resources:

Educational site - RusCopyBook.Com - Electronic textbooks and GDZ
Website Educational Resources Internet - for schoolchildren and students. http://www.alleng.ru/edu/educ.htm
Website Teacher's portal - http://www.uchportal.ru/

Lesson topic: Degree with rational and real exponents.

Goals:

Educational :

generalize the concept of degree;
practice the ability to find the value of a degree with a real exponent;
consolidate the ability to use the properties of degrees when simplifying expressions;
develop the skill of using the properties of degrees in calculations.

Developmental :

intellectual, emotional, personal development student;
develop the ability to generalize, systematize based on comparison, and draw conclusions;
intensify independent activity;
develop cognitive interest.

Educational :

nurturing the communicative and information culture of students;
Aesthetic education is carried out through the formation of the ability to rationally and accurately draw up a task on the board and in a notebook.

Students should know: definition and properties of degree with real exponent

Students should be able to:

determine whether an expression with a degree makes sense;

use the properties of degrees in calculations and simplification of expressions;

solve examples containing degrees;

compare, find similarities and differences.

Lesson format: seminar - workshop, with elements of research. Computer support.

Form of training organization: individual, group.

Educational technologies : problem-based learning, collaborative learning, student-centered learning, communicative.

Lesson type: lesson of research and practical work.

Lesson visuals and handouts:

presentation

formulas and tables (Appendix 1.2)

assignment for independent work (Appendix 3)

Lesson Plan

№

Lesson stage

Purpose of the stage

Time, min.

Start of the lesson

Reporting the topic of the lesson, setting lesson goals.

1-2 min

Oral work

Repeat the power formulas.

Properties of degrees.

4-5 min.

Front solution

boards from textbook No. 57 (1,3,5)

№58(1,3,5) with detailed adherence to the solution plan.

Formation of skills and abilities

students apply properties

degrees when finding the values of an expression.

8-10 min.

Work in micro groups.

Identifying knowledge gaps

students, creating conditions for

individual development student

in class.

15-20 min.

Summing up the work.

Track the success of work

Students at independent decision tasks on the topic, find out

the nature of the difficulties, their causes,

collectively indicate solutions.

5-6 min.

Homework

Introduce students to homework assignments. Give the necessary explanations.

1-2 min.

PROGRESS OF THE LESSON

Organizational moment

Hello guys! Write down the date and topic of the lesson in your notebooks.

They say that the inventor of chess, as a reward for his invention, asked the Raja for some rice: on the first square of the board he asked to put one grain, on the second - 2 times more, i.e. 2 grains, on the third - 2 times more, i.e. 4 grains, etc. up to 64 cells.

His request seemed too modest to the rajah, but it soon became clear that it was impossible to fulfill. The number of grains that had to be given to the inventor of chess as a reward is expressed by the sum

1+2+2 2 +2 3 +…+2 63 .

This amount is equal to a huge number

18446744073709551615

And it is so large that this amount of grain could cover the entire surface of our planet, including the world’s oceans, with a layer of 1 cm.

Powers are used when writing numbers and expressions, which makes them more compact and convenient for performing actions.

Degrees are often used when measuring physical quantities, which can be "very large" or "very small".

The mass of the Earth 6000000000000000000000t is written as a product 6.10 21 T

The diameter of a water molecule 0.0000000003 m is written as the product

3.10 -10 m.

1. What mathematical concept are the words associated with:

Base
Indicator(Degree)

What words can be used to combine the words:
Rational number
Integer
Natural number
Irrational number(real number)
Formulate the topic of the lesson.(Degree with real exponent)

2. So a x,Wherex is a real number. Select from expressions

With natural indicator

With an integer indicator

With a rational exponent

With an irrational indicator

3. What is our goal?(USE)
Whichgoals of our lesson ?
– Generalize the concept of degree.

Tasks:

– repeat properties of degree
– consider the use of degree properties in calculations and simplifications of expressions
– development of computing skills

4 . Power with rational exponent

Base

degrees

Degree with indicatorr, base a (nN, mn

r= n

r= - n

r= 0

r =0

a n= a. a. … . a

a -n=

a 0 =1

a n=a.a. ….a

a -n=

Doesn't exist

a 0 =1

a=0

0 n=0

Doesn't exist

5 . From these expressions, choose those that do not make sense:

6 . Definition

If the numberr- natural, then a rthere is a workrnumbers, each of which is equal to a:

a r= a. a. … . a

If the numberr- fractional and positive, that is, wheremAndn- natural

numbers, then

If the indicatorris rational and negative, then the expressiona r

is defined as the reciprocal ofa - r

7 . For example

8 . Powers of positive numbers have the following basic properties:

9 . Calculate

10. What operations (mathematical operations) can be performed with degrees?

Match:

A) When multiplying powers with equal bases

1) The bases are multiplied, but the indicator remains the same

B) When dividing powers with equal bases

2) The bases are divided, but the indicator remains the same

B) When raising a power to a power

3) The base remains the same, but the indicators are multiplied

D) When multiplying powers with equal exponents

4) The base remains the same, but the indicators are subtracted

D) When dividing degrees with equal exponents

5) The basis remains the same, but the indicators add up

11 . From the textbook (at the blackboard)

To solve in class:

№57 (1,3,5)

№58 (1, 3, 5)

№59 (1, 3)

№60 (1,3)

12 . By Unified State Exam materials

(independent work) on pieces of paper

XIVcentury.

Answer: Orezma. 13. Additionally (individually) for those who complete the tasks faster:

14. Homework

§ 5 (know definitions, formulas)

№57 (2, 4, 6)

№58 (2,4)

№59 (2,4)

№60 (2,4) .

At the end of the lesson:

“Mathematics must be taught later because it puts the mind in order”

– So said the great Russian mathematician Mikhail Lomonosov.

- Thanks for the lesson!

Appendix 1

1.Degrees. Basic properties

Indicator

a 1 =a

a n=a.a. ….a

a R n

3 5 =3 . 3 . 3 . 3 . 3 . 3=243,

(-2) 3 =(-2) . (-2) . (-2)= - 8

Degree with an integer exponent

a 0 =1,

where a

0 0 - not defined.

Degree with rational

Indicator

Wherea

m n

Degree with irrational exponent

Answer: ==25.9...

1. a x. a y=a x+y

2.a x: a y==a x-y

3. .(a x) y=a x.y

4.(a.b) n=a n.b n

5. (=

6. (

Appendix 2

2. Degree with rational exponent

Base

degrees

Degree with indicatorr, base a (nN, mn

r= n

r= - n

r= 0

r =0

a n= a. a. … . a

a -n=

a 0 =1

a n=a.a. ….a

a -n=

Doesn't exist

a 0 =1

a=0

0 n=0

Doesn't exist

Appendix 3

3. Independent work

Operations on powers were first used by a French mathematicianXIVcentury.

Decipher the name of the French scientist.

For any angle α such that α ≠ πk/2 (k belongs to the set Z), the following holds:

For any angle α the equalities are valid:

For any angle α such that α ≠ πk (k belongs to the set Z), the following holds:

Reduction formulas

The table provides reduction formulas for trigonometric functions.

Function (angle in º)	90º - α	90º + α	180º - α	180º + α	270º - α	270º + α	360º - α	360º + α
sin	cos α	cos α	sinα	-sin α	-cos α	-cos α	-sin α	sinα
cos	sinα	-sin α	-cos α	-cos α	-sin α	sinα	cos α	cos α
tg	ctg α	-ctg α	-tg α	tan α	ctg α	-ctg α	-tg α	tan α
ctg	tan α	-tg α	-ctg α	ctg α	tan α	-tg α	-ctg α	ctg α
Function (angle in rad.)	π/2 – α	π/2 + α	π – α	π + α	3π/2 – α	3π/2 + α	2π – α	2π + α

Parity of trigonometric functions. Angles φ and -φ are formed when the beam is rotated in two mutually opposite directions (clockwise and counterclockwise).
Therefore, the end sides OA 1 and OA 2 of these angles are symmetrical about the abscissa axis. Coordinates of vectors of unit length OA 1 = ( X 1 , at 1) and OA 2 = ( X 2 , y 2) satisfy the following relations: X 2 = X 1 y 2 = -at 1 Therefore cos(-φ) = cosφ, sin (- φ) = -sin φ, Therefore, sine is odd, and cosine is even function corner.
Next we have:
That's why tangent and cotangent are odd functions of angle.

8)Inverse trigonometric functions - mathematical functions, which are the inverse of trigonometric functions. Six functions are usually classified as inverse trigonometric functions:

§ arcsine(symbol: arcsin)

§ arc cosine(symbol: arccos)

§ arctangent(designation: arctg; in foreign literature arctan)

§ arccotangent(designation: arcctg; in foreign literature arccotan)

§ arcsecant(symbol: arcsec)

§ arccosecant(designation: arccosec; in foreign literature arccsc)

Back title trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix “arc-” (from Lat. arc- arc). This is due to the fact that geometrically the value of the inverse trigonometric function can be related to the length of the arc unit circle(or an angle subtending this arc) corresponding to one or another segment. Occasionally in foreign literature, notations like sin −1 are used for arcsine, etc.; this is considered unjustified, since there may be confusion with raising a function to the power −1.

Properties of the arcsin function

(the function is odd). at .

Properties of the function arccos[

· (the function is centrally symmetric with respect to the point) is indifferent.

Properties of the arctg function

· , for x > 0.

Properties of the function arcctg

· (the graph of the function is centrally symmetrical with respect to the point

· for any

12) The power of a number a > 0 with a rational exponent is a power whose exponent can be represented as an ordinary irreducible fraction x = m/n, where m is an integer and n natural number, and n > 1 (x is the exponent).

Degree with real exponent

Let a positive number and an arbitrary real number be given. The number is called the power, the number is the base of the power, and the number is the exponent.

By definition they believe:

If and are positive numbers and are any real numbers, then the following properties hold:

14)Logarithm of a number to the base(from the Greek λόγος - “word”, “relation” and ἀριθμός - “number”) is defined as an indicator of the power to which the base must be raised to obtain a number. Designation: , pronounced: " base logarithm".

Properties of logarithms:

1° is the basic logarithmic identity.

The logarithm of one to any positive base other than 1 is zero. This is possible because any real number can only be converted to 1 by raising it to the zero power.

4° is the logarithm of the product.

The logarithm of the product is equal to the sum of the logarithms of the factors.

5° - logarithm of the quotient.

The logarithm of the quotient (fraction) is equal to the difference between the logarithms of the factors.

6° is the logarithm of the degree.

The logarithm of a power is equal to the product of the exponent and the logarithm of its base.

7°

8°

9° - transition to a new foundation.

15) Real number - (real number), any positive, negative number or zero. Through real numbers the results of measurements of all physical quantities are expressed. ;

16)Imaginary unit- usually a complex number whose square is equal to negative one. However, other options are also possible: in the construction of doubling according to Cayley-Dixon or within the framework of algebra according to Clifford.

Complex numbers(obsolete imaginary numbers) - numbers of the form , where and are real numbers, - an imaginary unit; that is . Plenty of everyone complex numbers usually denoted from Lat. complex- closely related.

Lesson topic: Degree with a real exponent.

Tasks:

Educational:
- generalize the concept of degree;
- practice the ability to find the value of a degree with a real exponent;
- consolidate the ability to use the properties of degrees when simplifying expressions;
- develop the skill of using the properties of degrees in calculations.
Developmental:
- intellectual, emotional, personal development of the student;
- develop the ability to generalize, systematize based on comparison, and draw conclusions;
- intensify independent activity;
- develop cognitive interest.
Educational:
- nurturing the communicative and information culture of students;
- Aesthetic education is carried out through the formation of the ability to rationally and accurately draw up a task on the board and in a notebook.

Students should know: definition and properties of a degree with a real exponent.

Students should be able to:

determine whether an expression with a degree makes sense;
use the properties of degrees in calculations and simplification of expressions;
solve examples containing degrees;
compare, find similarities and differences.

Lesson format: seminar - workshop, with elements of research. Computer support.

Form of training organization: individual, group.

Lesson type: lesson of research and practical work.

PROGRESS OF THE LESSON

Organizational moment

“One day the king decided to choose a first assistant from among his courtiers. He led everyone to a huge castle. “Whoever opens it first will be the first assistant.” No one even touched the lock. Only one vizier came up and pushed the lock, which opened. It was not locked.
Then the king said: “You will receive this position because you rely not only on what you see and hear, but rely on your own strength and are not afraid to try.”
And today we will try and try to come to the right decision.

1. What mathematical concept are the words associated with:

Base
Indicator (Degree)
What words can be used to combine the words:
Rational number
Integer
Natural number
Irrational number (real number)
Formulate the topic of the lesson. (Degree with real exponent)

2. What is our strategic goal? (USE)
Which goals of our lesson?
– Generalize the concept of degree.

Tasks:

– repeat the properties of the degree
– consider the use of degree properties in calculations and simplifications of expressions
– development of computing skills.

3. So, a p, where p is a real number.
Give examples (choose from the expressions 5 –2, 43, ) degrees

– with natural indicator
– with an integer indicator
– with a rational indicator
– with an irrational indicator

4. At what values A the expression makes sense

аn, where n (а – any)
аm, where m (а 0) How to move from a degree with a negative exponent to a degree with a positive exponent?
, where (a0)

5. From these expressions, choose those that do not make sense:
(–3) 2 , , , 0 –3 , , (–3) –1 , .
6. Calculate. The answers in each column have one common property. Please indicate an extra answer (one that does not have this property)

2 = =
= 6 = (incorrect others) = (cannot write dec. others)
= (fraction) = =

7. What operations (mathematical operations) can be performed with degrees?

Match:

One student writes formulas (properties) in general form.

8. Add the degrees from step 3 so that the properties of the degree can be applied to the resulting example.

(One person works at the board, the rest in notebooks. To check, exchange notebooks, and another one performs actions on the board)

9. On the board (student working):

Calculate : =

Independently (with checking on sheets)

Which answer cannot be obtained in part “B” of the Unified State Exam? If the answer turned out to be , then how to write such an answer in part “B”?

10. Independent completion of the task (with checking at the board - several people)

Multiple choice task