Algebra. 8th grade

Teacher: Kuleshova Tatyana Nikolaevna

Topic: Converting Expressions Containing Square Roots

Lesson type: generalization and systematization of knowledge

The purpose of the lesson: the formation of students' skills to transform expressions containing square roots

Tasks:

Educational:know the properties of the arithmetic square root; learn how to transform such expressions containing square roots as taking a factor out of the root sign, putting a factor into the root sign and freeing from irrationality in the denominator of a fraction;

Developing: develop cognitive and creative abilities, thinking, observation, ingenuity and skills of independent activity; instilling an interest in mathematics;

Educational: ability to work in a team (group), desire to actively study with interest; clarity and organization in work; enable every student to succeed;

Equipment: School supplies, blackboard, chalk, textbook, handout.

Lesson plan

1. Organizing time
2. goal setting
3. Repetition
4. Independent work
5. Dictation
6. Test
7. Textbook work
8. Homework instruction
9. Lesson results. Reflection

Working process

1. Organizing time

Lesson motivation

“Close your eyes, sit back. Imagine something very pleasant for you. You are good, comfortable. There are many friends around you. Among them and integers with which we are well acquainted. The ranks of our friends are replenished and fractional numbers have joined them. And here they come negative numbers. And now you are going to meet rational and irrational numbers. Time will pass, and we will get to know you with new numbers, and as long as mathematics exists in the world, these numbers are endless.

“Knowledge is only then knowledge when it is acquired by the efforts of one’s thought, and not by memory.” L. N. Tolstoy.-These words of L. N. Tolstoy are important and relevant in the study of mathematics, because mathematics is one of the few sciences where you need to constantly think. Your task is to show your knowledge and skills in the process of oral work, testing, work at the blackboard.

Each of you has an evaluation sheet on the table, after each completed task, do not forget to grade, and at the end of the lesson, put the final grade.

1. goal setting

Solve the anagram (Group work)

ABOUT - ZO - RA - CONVERSION - NIE - VA CONVERSION

NIY - RA - SAME - YOU EXPRESSIONS

SHCHIKH - DER - ZHA - WITH CONTAINING

RAT - KV - NYE - AD SQUARE

NI - KO - R ROOTS

Having solved the anagram, students determine the topic of the lesson

What do you think we will do in class?

Let's formulate the purpose of our lesson together.

1. Repetition of previously studied material

A 1) Oral counting:

Testing the theory: Connect the corresponding parts of the definition with a line.

score -2 points

2). Finish approval.

a) The root of the product of non-negative factors is equal tothe product of the roots of these factors.(score -2 points)

b) Any infinite non-periodic decimal calledirrational number.(score -2 points)

c) The root of a fraction whose numerator is a non-negative number and the denominator is positive is equal toroot of the numerator divided by the root of the denominator.( score -2 points)

3) Set correspondence (2 points)

B. 3 students receive an algorithm for transforming expressions containing square roots. Task: depict, draw, write, show, etc. and protect (speaker).

3) Extract the root

1. Factorize the denominator of a fraction.
2. If the denominator isor contains a multiplier, then the numerator and denominator should be multiplied by or at .
3. Convert the numerator and denominator of the fraction, if possible, then reduce the resulting fraction.

1. Independent work

Take the factor out from under the root sign:

(2 points)

3)

Simplify the expression (4 points)

1. Test on a laptop (score is set automatically)

1) 6 =

a B C D) .

2) 5 =

3) 3 =

a B C D) .

1. Dictation:

Option 1	Answers:

For each correctly completed task 0.5 points.

1. Work according to the textbook - work on the board: each student receives a specific example, they solve it in turn on the board, they write everything down in a notebook. (1 point)
2. Homework Information
3. Summing up the lesson. Reflection

Evaluation

Evaluation paper. Full name of the student _______________________ Grade _____

Lesson stage	Points
Verbal counting
Independent work
Test
Dictation
Work according to the textbook - work on the board
Additional tasks
Total points for the lesson
My mood at the end of the lesson - after the assessment for the lesson

Converting points to grade

25 points or more - score "5"

24 - 18 points - score "4"

17 - 9 points - score "3"

0 - 8 points - score "2"

To evaluate the entire work for the lesson, use the "Transfer of points to grade" - on the back of the evaluation sheet.

Complete the assessment sheet. Lesson grades.

I want to finish the lessona poem by the great mathematician Sofia Kovalevskaya.

If in life you even for a moment

I felt the truth in my heart

If a ray of light through darkness and doubt

With a bright radiance your path lit up:

What would be in your decision unchanged

Rock has not appointed you ahead,

The memory of this sacred moment

Keep forever, like a shrine in your chest.

The clouds will gather in a discordant mass,

The sky will be covered with black mist,

With clear determination, with calm faith

Meet the storm and face the storm.

This poem expresses the desire for knowledge, the ability to overcome all obstacles that are encountered on the way. And how did we overcome obstacles today? What did we do in class?

- Today we have repeated the definition and properties of the arithmetic square root; taking the factor out of the sign of the root, entering the factor under the sign of the root, abbreviated multiplication formulas; got acquainted and fixed some ways of converting expressions containing square roots.

Everyone worked fruitfully, actively and collectively during the lesson.

The lesson is over. Thanks everyone for the lesson!

Enter a multiplier under the root sign:

1) 6 =

a B C D) .

2) 5 =

3) 3 =

a B C D) .

Test F.I.____________________

Enter a multiplier under the root sign:

1) 6 =

a B C D) .

2) 5 =

3) 3 =

a B C) - =

a B C D) .

2) 5 =

3) 3 =

a B C) - =

a B C D) .

2) 5 =

3) 3 =

a B C) - =

a B C D) .

2) 5 =

3) 3 =

a B C D) .

Algorithm for taking the multiplier out from under the sign of the root

1) We represent the root expression as a product of such factors so that one can extract the square root from one.

2) Let's apply the theorem about the root of the product.

3) Extract the root

Algorithm for adding a multiplier under the sign of the root

1) Let's represent the product as an arithmetic square root.

2) Let's transform the product square roots to the square root of the product of radical expressions.

3) Perform multiplication under the root sign.

Algorithm for getting rid of irrationality in the denominator of a fraction:

1) Factorize the denominator of a fraction.

The video lesson "Transformation of expressions containing the operation of extracting a square root" is a visual aid with which it is easier for a teacher to form skills and abilities in solving problems containing expressions with a square root. During the lesson, they are reminded theoretical basis, which serve as the basis for carrying out operations on numbers and variables present in the root expression, describes the solution of many types of problems that may require the ability to use formulas for converting expressions containing a square root, and provides methods for getting rid of irrationality in the denominator of a fraction.

The video tutorial starts by demonstrating the title of the topic. It is noted that earlier in the lessons transformations were performed rational expressions. At the same time, theoretical information about monomials and polynomials, methods for working with polynomials, algebraic fractions, as well as abbreviated multiplication formulas were used. This video tutorial covers the introduction of the square root operation for transforming expressions. Students are reminded of the properties of the square root operation. Among these properties, it is indicated that after extracting the square root from the square of the number, the number itself is obtained, the root of the product of two numbers is equal to the product of two roots of these numbers, the root of the quotient of two numbers is equal to the quotient of the roots of the members of the quotient. The last property considered is the extraction of the square root of a number raised to even degree√a 2 n , which as a result forms a number to the power of a n . The considered properties are valid for any non-negative numbers.

Examples are considered in which transformations of expressions containing a square root are required. It is indicated that in these examples it is provided that a and b are non-negative numbers. In the first example, it is necessary to simplify the expressions √16a 4 /9b 4 and √a 2 b 4 . In the first case, a property is applied that determines that the square root of the product of two numbers is equal to the product of the roots of them. As a result of the transformation, the expression ab 2 is obtained. The second expression uses the formula for converting the square root of a quotient to a quotient of roots. The result of the transformation is the expression 4a 2 /3b 3 .

In the second example, it is necessary to remove the factor from under the square root sign. The solution of expressions √81а, √32а 2 , √9а 7 b 5 is considered. Using the example of the transformation of four expressions, it is shown how the formula for transforming the root of the product of several numbers is used to solve such problems. At the same time, cases are noted separately when expressions contain numerical coefficients, parameters in an even, odd degree. As a result of the transformation, the expressions √81a=9√a, √32a 2 =4a√2, √9a 7 b 5 =3a 3 b 2 √ab are obtained.

In the third example, it is necessary to perform an operation opposite to that in the previous problem. To enter a factor under the square root sign, it is also necessary to be able to use the studied formulas. It is proposed in expressions 2√2 and 3a√b/√3a to introduce a multiplier before the brackets under the root sign. Using well-known formulas, the factor in front of the root sign is squared and placed as a factor in the product under the root sign. In the first expression, as a result of the transformation, the expression √8 is obtained. In the second expression, the formula of the horse of the product is first used to convert the numerator, and then the formula of the private root is used to convert the entire expression. After reducing the numerator and denominator in the radical expression, √3ab is obtained.

In example 4, you need to perform actions in the expressions (√a+√b)(√a-√b). For solutions given expression new variables are introduced that replace the monomials containing the sign of the root √a=x and √b=y. after substituting new variables, the possibility of using the abbreviated multiplication formula is obvious, after which the expression takes the form x 2 -y 2. Returning to the original variables, we get a-b. The second expression (√a+√b) 2 can also be converted using the reduced multiplication formula. After expanding the brackets, we get the result a+2√ab+b.

In example 5, the expressions 4a-4√ab+b and x√x+1 are factorized. To solve this problem, it is necessary to perform transformations, select common factors. After applying the properties of the square root to solve the first expression, the sum is converted into the square of the difference (2√а-√b) 2 . To solve the second expression, it is necessary to enter a multiplier under the root before the root sign, and then apply the formula for the sum of cubes. The result of the transformation is the expression (√x+1)(x 2 -√x+1).

Example 6 demonstrates the solution of a problem where it is necessary to simplify the expression (a√a+3√3)(√a-√3)/((√a-√3) 2 +√3a). The problem is solved in four steps. In the first step, the numerator is converted into a product using the abbreviated multiplication formula - the sum of the cubes of two numbers. In the second step, the denominator of the expression is transformed, which takes the form a-√3a+3. After the conversion, it becomes possible to reduce the fraction. In the last step, the reduced multiplication formula is also applied, which helps to get the final result a-3.

In the seventh example, it is necessary to get rid of the square root in the denominators of the fractions 1/√2 and 1/(√3-√2). When solving the task, the main property of the fraction is used. To get rid of the root in the denominator, the numerator and denominator are multiplied by the same number, which squares the root expression. As a result of calculations, we get 1/√2=√2/2 and 1/(√3-√2)=√3+√2.

Features are indicated mathematical language when working with expressions containing a root. It is noted that the content of the square root in the denominator of the fraction means the content of irrationality. And getting rid of the sign of the root in such a denominator is said to be getting rid of irrationality in the denominator. Methods are described on how to get rid of irrationality - to transform the denominator of the form √a, it is necessary to multiply the numerator simultaneously with the denominator by the number √a, and to eliminate irrationality for the denominator of the form √a-√b, the numerator and denominator are multiplied by the conjugate expression √a+√ b. It is noted that getting rid of irrationality in such a denominator greatly facilitates the solution of the problem.

At the end of the video tutorial, a simplification of the expression 7/√7-2/(√7-√5)+4/(√5+√3) is considered. To simplify the expression, the above methods of getting rid of irrationality in the denominator of fractions are applied. The resulting expressions are added, after which the simplified form of the expression looks like √5-2√3.

The video tutorial "Transformation of expressions containing the operation of extracting a square root" is recommended to be used on a traditional school lesson for the formation of skills for solving tasks that contain a square root. For the same purpose, the video can be used by the teacher during distance learning. Also, the material can be recommended to students for independent work Houses.

Sections: Maths

Lesson Objectives:

1. Repeat the definition of the arithmetic square root, the properties of the arithmetic square root.
2. Summarize and systematize students' knowledge on this topic.
3. Strengthen the skills and abilities of solving examples on identical transformations expressions containing arithmetic square roots.
4. To give each student the opportunity to develop their potential to the fullest extent possible.
5. Broaden their horizons and introduce students to the mathematicians of the Middle Ages.

Lesson type: practical lesson.

Lesson equipment: handouts, colored chalk, overhead projector, a portrait of Rene Descartes, posters with formulas.

During the classes

I.Organizing time.

The topic of our lesson is "Conversion of expressions containing arithmetic square roots." Today in the lesson we will repeat the rules for converting expressions containing square roots. This includes the transformation of roots from a product, fraction and degree, multiplication and division of roots, taking the factor out of the sign of the root, putting the factor into the sign of the root, bringing like terms and freeing from irrationality in the denominator of the fraction.

II. Oral survey on theory.

• Define an arithmetic square root. ( The arithmetic square root of a is called non-negative number, whose square is a).
• List the properties of the arithmetic square root. ( The arithmetic square root of the product of non-negative factors is equal to the product of the roots of these factors. The arithmetic square root of a fraction whose numerator is non-negative and whose denominator is positive is equal to the root of the numerator divided by the root of the denominator).
• What is the value of the arithmetic square root of x 2? ( |x| ).
• What is the value of the arithmetic square root of x 2 if x≥0? X<0? (X. -X).

III. oral work. (Written on the board).

Find the value of the root:

Find the value of the expression:

Enter the multiplier under the root sign:

Compare:

IV. Development of knowledge on the topic. (On the desks of each sheet with assignments).

1. Take action.

• How will we solve examples a and b? ( Open the brackets, give like terms).
• How will we solve examples c and d? ( Apply the difference of squares formula).
• How will we solve examples e and e? ( We take the factor out of the sign of the root and give like terms).

	2 + 0,3- 4 + 0,01
	3 + 0,5 - 2 + 0,01

(Students follow the options in their notebooks, 6 students solve 1 example at the back board).

– Checking through a graphic projector. Each answer corresponds to a certain letter. The result is the word: Descartes.

V. Historical reference.

The student gives a short presentation.

In 1626, the Dutch mathematician A. Shirar introduced the notation for the root V, close to the modern one. If the number 2 stood above this sign, then this meant a square root, if 3 - a cubic one. This designation began to replace the Rx sign. However, for a long time they wrote Va + b with a horizontal line above the sum. It was only in 1637 that Rene Descartes connected the root sign with a horizontal line, using the modern root sign in his Geometry. This sign came into general use only at the beginning of the 18th century. ( On the board - a portrait of Rene Descartes, drawing).

VI. Development of knowledge on the topic.

2. Factor out.

a and b - expand by the formula of the difference of squares, c and d - using the definition of the arithmetic square root, replace 7 and 13 with squares from square roots, and then take out the common factor).

a) a - 9, a≥0
b) 16 – c, c≥0

Students solve in notebooks according to options, 2 people (one from each option) decide at the blackboard.

- Examination.

3. Reduce the fraction.

How are we going to do this task? ( We factorize either the numerator or the denominator, and then reduce).

Students decide in notebooks according to options, 4 people decide at the blackboard. Examples e and f additionally decide who will be in time.

- Examination.

4. Get rid of the irrationality in the denominator of the fraction.

What are we going to do in this assignment? ( Let's transform the fraction so that the denominator does not contain a square root: a and b we will multiply both the numerator and the denominator by the square root written in the denominator; c and d we will multiply by the sum or difference of the expression written in the denominator in order to get the difference of squares).

Students decide by options, 2 people solve 2 examples at the blackboard.

- Examination.

VII. Test writing.

Everyone has a sheet with test tasks on their desks ( Attachment 1). They signed the sheet and completed the tasks in the same sheet. After writing the work, they handed it over, checked the answers and figured out why it was so, through a graph projector.

VIII. Homework. from. 109 No. 503 (a–d), 504.

"Secondary school No. 51"

For the competition "Teacher of the Year", school stage

Outline of a mathematics lesson for grade 8 "A"

Topic: Transformation of expressions containing the square root operation.

Performed:

Mathematic teacher

Aralbaeva Nurslu Erkagaleevna

MOBU "SOSH №51"

Orenburg, 2015

Lesson type: systematization and generalization of knowledge.

Teaching methods: problematic, verbal, visual, practical.

Classwork Forms: individual, pair.

Equipment:

chalk, blackboard

a computer

multimedia projector with screen

electronic version of the lesson - presentation

handout (cards with tasks of different levels)

Lesson Objectives:

Educational: generalize knowledge on all types of transformations of expressions containing the operation of extracting a square root, consolidate the ability to use the properties of a square root, learn to use the knowledge gained to prepare for the ROE.

Developing: development of a non-standard approach to solving the problem; development of thinking, competent mathematical speech, self-control skills; to form the ability to organize their activities.

Educational: to promote the development of interest in the subject, activity, to cultivate accuracy in work, the ability to express one's own opinion, to give recommendations.

Students should know:

Algorithm for adding a multiplier under the sign of the root.

Algorithm for taking the multiplier out from under the sign of the root.

Applying the properties of the square root.

Definition of the square root.

"The greatness of a man is in his ability to think."

Blaise Pascal.

I organizational moment

Introduction. Presentation of the topic and objectives of the lesson.

An outstanding French philosopher, scientist Blaise Pascal stated: "The greatness of man is in his ability to think." Today we will try to feel like great people by discovering knowledge for ourselves. The motto for today's lesson will be the words of the ancient Greek mathematician Thales:

What is the most in the world? - Space.

What is the fastest? - Mind.

What is the wisest? - Time.

What is the most enjoyable? - Achieve what you want.

I want each of you to achieve the desired result in today's lesson.

At the moment, they are knocking on the door and informing them that the school received a mail containing a package for the 8th “A” class. The teacher opens the package, which contains letters for each student. After receiving the envelopes, students get acquainted with the contents. One of the students reads a letter of recommendation aloud:

Dear Nurslu Erkagaleevna!

Orenburg State University invites you to take part in the international competition "Children are our future". The purpose of the competition is to identify gifted children in various regions of our country and provide them with the opportunity to study in higher educational institutions on a state basis.

Since our main subjects are mathematics, physics, computer science, in order to participate in the competition “Children are our future”, you must complete the task in the subject “Mathematics”. You will receive recommendations for other subjects later.

Remember, with positive results, you will have a chance to enter our university.

Good luck!

Teacher:

Guys, we are offered to take part in the competition “Children are our future” and you will have the opportunity to enter a university. To do this, you must complete the proposed tasks. However, before proceeding to the task, we will repeat the main points on the topic.

II Actualization of knowledge

Take out from under the root sign:

Enter a multiplier under the root sign:

Squaring:

Give like terms:

Get a drawing (work in pairs)

III Fizminutka

Physical education for the eyes

IV Test work.

Test from the tasks of the ROE

Find the value of an expression:

-2(
) 2

A. 9.6 B. 0 C. 0.38 D. 2.4

A. 42 B. 18 C. 60 D. 6

Find the value of an expression:

0,5
+ 3

A. 62.93 B. 0 C. 8.2 D. 1

Find the value of an expression:

- 0,5 (
) 2

A. 141 B. 9. C. 6 D. 0

A. 0 B. 0.7 C.1 D.0.1

Find the value of an expression:

-2(
) 2

A. 8.75 B. 0.1 C. 0.28 D. 3.6

A. 47 B. 8 C. 70 D. 16

Find the value of an expression:

0,5
+ 3

A. 0 B. 58.61 C. 8.1 D. 1

Find the value of an expression:

- 0,5 (
) 2

A. 7 B. 121 C. 6 D. 0

A. 0 B. 1 C. 0.3 D. 0.1

After filling out the table, students put the completed task into an envelope and hand it over to the teacher. The teacher gives grades, thanks the students for their work, and informs them that in the next lesson, students will receive envelopes with the result and learn about the chance of admission. VII Summary of the lesson.

Reflection

Our work is coming to an end and the moment of creativity comes. What holiday awaits us in the near future (New Year). We will decorate the "Christmas Tree of Mood". And let it combine your mood, your feelings and emotions from the lesson.

I am satisfied with my work in the lesson (corresponding emoticon)

I did well in class.

It was difficult for me in class.

Please choose an emoticon that matches your emotions, go to the blackboard and hang it on the Christmas tree.

What did we get? A very bright Christmas tree indicates that you worked with interest in the lesson, learned a lot of new things that made you think and change your attitude to algebra. Let me add a few touches:
- Let the snowflakes inspire us to success and creativity (I hang snowflakes).
- I hope that the lesson brought joy not only to me, but also to you, my dear students (Turn on the garland).
- And the knowledge that you have acquired today, let it remain with you forever.

VIII Homework:

Differentiated: level A - grade "3", level B - grade "4", level C - grade "5".

Grading

Literature:

Program: for educational institutions, edited by A.G. Mordkovich.

Lesson developments in algebra grade 8 O.V. Zanina, I.N. Dankov.