Fractional rational equations 9. Fractional rational equations

SOLVING FRACTIONAL RATIONAL EQUATIONS

Lesson objectives:Educational:

formation of the concept of a fractional rational equation; consider various ways to solve fractional rational equations; consider an algorithm for solving fractional rational equations. teach solving fractional rational equations using an algorithm; checking the level of mastery of the topic by conducting a test.

Developmental:

developing the ability to correctly operate with acquired knowledge and think logically; development of intellectual skills and mental operations - analysis, synthesis, comparison and generalization; development of initiative, the ability to make decisions, and not stop there;

Educating:

fostering cognitive interest in the subject; fostering independence in solving educational problems; nurturing will and perseverance to achieve final results.

Lesson type: lesson - explanation of new material.

During the classes

1. Organizational moment. Hello guys! There are equations written on the board, look at them carefully. Do we know how to solve all of these equations? Which ones are not and why?

8.
What are the names of the expressions from which equations 5, 6, 7 and 8 are made? (fractional rational) Equations in which the left and right sides are fractional rational expressions are called fractional rational equations. What do you think we will study in class today? Try to formulate the topic of our lesson. So, open the notebooks and write down the topic of the lesson “Solving fractional rational equations.” Let's formulate the goals of our lesson (children independently formulate the goals of the lesson) Now we will repeat the main theoretical material, which we need to study a new topic. Please answer the following questions: What is an equation? ( Equality with a variable or variables.)

What is the name of equation number 1? ( Linear.) A method for solving linear equations. ( Move everything with the unknown to the left side of the equation, all numbers to the right. Give similar terms. Find unknown factor). What is the name of equation number 3? ( Square.) Methods for solving quadratic equations. (P about formulas using Vieta’s theorem and its consequences.)
What properties are used when solving equations? ( 1. If you move a term in an equation from one part to another, changing its sign, you will get an equation equivalent to the given one. 2. If both sides of the equation are multiplied or divided by the same non-zero number, you get an equation equivalent to the given one.)

3. Explanation of new material.So, in our lesson you are not just 9th grade students, but representatives of one of three tribes. Why do you think I called them that? (correct, because when solving equations you will use certain rules. What are these rules? Try to formulate them for me:

The Proportion tribe will look for a solution by applying the property of proportion. What is proportion? ( Equality of two ratios.) Formulate the basic property of proportion. ( If the proportion is correct, then the product of its extreme terms is equal to the product of the middle terms.)

Card 1:

USING THE PROPERTY OF PROPORTION

The product of the middle terms is equal to the product

extreme terms of the proportion.

Tribe "Fraction" - applying the property of a fraction being equal to zero. Answer when does a fraction equal zero? ( A fraction is equal to zero when the numerator is zero and the denominator is not zero..)

Card 2: SOLVE FRACTIONAL RATIONAL EQUATION,

for the same thing non-zero number.

The “Denominator” tribe solves by multiplying by a common, non-zero denominator.

Card 3: SOLVE FRACTIONAL RATIONAL EQUATION,

MULTIPLYING BY A COMMON DENOMINATOR

Both sides of the equation can be multiplied or divided

for the same thing non-zero number.

After solving and discussing in groups, one representative from each group comes to the board and writes the solution to the equation on the board. / *4x ODZ: x≠0 x²-4=6x-4 2x³-8x=12x²-8x
x²-6x=0 2x³-12x²=0
x=0 or x=6 2x²(x-6)=0
Answer: x=0, x=6 x=0, x=6 x²-6x=0 x=0, x=6 Answer: x=0, x=6 4x≠ 0 x ≠0 Answer: x=6 If I get different answers, I ask guiding questions: Let's compare the answers. Explain why this happened? Why are there two roots in one case and one in the other? What numbers are the roots of this fractional rational equation? (Until now, students have not encountered the concept of an extraneous root; it is indeed very difficult for them to understand why this happened. If no one in the class can give a clear explanation of this situation, then the teacher asks leading questions.)

How do equations No. 2 and 4 differ from equations No. 5,6,7,8? ( In equations No. 2 and 4 there are numbers in the denominator, No. 5-8 are expressions with a variable.) What is the root of an equation? ( The value of the variable at which the equation becomes true.) How to find out whether a number is the root of an equation? ( Make a check.)

When testing, some students notice that they have to divide by zero. They conclude that the number 0 is not the root of this equation. The question arises: what needs to be added to each of these methods to eliminate this error? (exclude extraneous roots) ------ add on the board the inequality of the denominator to zero or ODZ). Here we are faced with the concept foreign root, i.e. this is the value of a variable that is not included in the scope of definition of a fractional rational expression. Let's try to formulate an algorithm for solving fractional rational equations using these methods. Let's consider the first method: equality of a fraction to zero. Children formulate the algorithm themselves 1. Algorithm for solving fractional rational equations:

Move everything to the left side. Reduce fractions to a common denominator. Create a system: a fraction is equal to zero when the numerator is equal to zero and the denominator is not equal to zero. Solve the equation. Check inequality to exclude extraneous roots. Write down the answer.

How to formalize a solution if the basic property of proportion is used? Algorithm for solving fractional rational equations. 1. Use the property of proportion: in the correct proportion, the product of the extreme terms is equal to the product of the middle terms. 2. Solve the resulting whole equation. 3. Eliminate from the roots those that make the common denominator vanish. 4. Write down the answer. How to formalize the solution if both sides of the equation are multiplied by a common denominator? 3. Algorithm for solving fractional rational equations:

Find the common denominator of the fractions in the equation. Multiply both sides of the equation by a common denominator that is not zero. Solve the resulting whole equation. Eliminate from the roots those that make the common denominator vanish. 5. Write down the answer.

Name the ODZ for each equation. We looked at three ways to solve fractional rational equations. (Work in groups. Students choose how to solve the equation independently depending on the type of equation). The teacher monitors the completion of the task, answers any questions that arise, and provides assistance to the students. Self-test: answers are written on the board a) Answer: x = 1, x = b) Answer: a = 3.5 c) Answer: x = -3, x = 2 d) -5 is an extraneous root. Answer: x = 5; 5. Summing up the lesson. So, today in the lesson we got acquainted with fractional rational equations, learned to solve these equations in various ways, and tested our knowledge through independent work. Which method of solving fractional rational equations, in your opinion, is easier, more accessible, and more rational? But, regardless of the method for solving fractional rational equations, what should you remember? What is the “cunning” of fractional rational equations?

Setting homework.

Class 9.

Lesson topic:"Fractional rational equations"

Lesson type: combined.

Goals:

1. Educational: give a definition of “fractional rational equations”, show ways to solve such equations.

2. Developmental: development of skills and abilities to solve examples with this type of equations, find the roots of fractional rational equations.

3. Educators: cultivate attention, attentiveness, activity, accuracy; respectful attitude towards the mother.

Tasks:to interest students in the subject, to show the importance of the ability to solve various equations and problems.

Material and technical equipment:

Multimedia projector, screen, presentation for the lesson “Fractional rational equations”

Time: 45 minutes

Lesson plan.

Lesson steps	Teacher activities	Student activity
I. Organizing time. (1 min.)	Greets students and checks their readiness for the lesson.	Greetings from the teachers.
II. Communicate the topic and objectives of the lesson. (2 minutes)	Informs the topic and purpose of the lesson.	Write down the topic in your notebook.
III. Repetition of the covered topic. (2 minutes)	Asks questions to review the topic covered.	Answer questions.
IV. Learning new material. (15 minutes.)	Shows slides and narrates. Listens, asks targeted questions in the role of an ordinary participant	They discuss the subject with the teacher and receive information if necessary, set goals, and plan the trajectory of work. Develop an action plan and formulate tasks. They search for information, collect data and historical facts, initially research the information received, and solve intermediate problems.
V. Physical education minute. (1 min.)	Performs a physical exercise	Perform physical education
VI. Fixing the material. (20 minutes.)	Problem solving, offers questions for consolidation.	They solve problems in notebooks, at the blackboard, and ask questions to the teacher.
VIII. Summing up the lesson. (4 min)	Evaluates student work.	They talk about what they learned in class. Workplaces are being removed.

DURING THE CLASSES

I. Reflection on the beginning of the lesson(music; presentation about mother).

Checking readiness for the lesson.

II. Communication of a new topic, goal and task:

Teacher: Hello! Please look at each other and smile from your heart.

I would like to start today’s lesson with the words of M. Gorky:

Slide 1
Without the sun, flowers don't bloom,

without love there is no happiness,

without women there is no love,

without a mother there is neither a poet nor a hero.

All the pride in the world comes from mothers.
(M. Gorky)

Teacher:

– What could be more sacred in the world than the name of a mother! ...

A person who has not yet taken a single step on the ground and is just beginning to “burb”, hesitantly and diligently spells out “mama” syllable by syllable and, feeling his luck, laughs, happy...

When does a baby cry for the first time?

And his mother will touch him carefully,

Her love... Oh, how disturbing she is.

Anxious every day and hour.

Guys, Mother's Day is coming soon, so I want to connect today's lesson with this topic. In previous lessons, we learned how to solve, find the roots of various equations, today we will continue to get acquainted with one of the types of equations - these are fractional rational equations, we will find out the importance of equations, and remember how to solve problems using equations. We will try not to let our mother down, we will decide carefully and without distractions to prepare for the State Examination. The mother of each of you wants her child to be the best. So today we have a lesson on learning a new topic. (slide 2).

III. Repetition of the covered topic.

1. Checking homework(slide 3).

No. 925(a, b), No. 935(a, b), No. 936.

2. We repeat orally(slide 3 ,4,5,6 ).

Let's repeat:

What is the name of this equation? How many roots does this equation have?

IV . Learning new material.(slide 7).

Teacher: The equation y (x ) =0 called fractional rational equation if expression y (x ) is fractional(i.e. contains division into an expression with variables).

To solve a rational equation, it is necessary to transform it into a linear or quadratic equation, solve this equation and discard those roots that are not included in the permissible value range of the original rational equation.

Open the textbook on page 78 and read the rule. You already worked on this topic in 8th grade.

Algorithm for solving fractional rational equations: ( slide 8).

(Annex 1)

Teacher: Now, together with me, let's solve the fractional-rational equation using the algorithm (slide 9).

VI . Independent work(slide 10).

… Your letter. Your native lines.

Your last maternal command:

“The laws of life are wise and cruel.

Live. Work hard. Don't ruin your eyes with tears.

My love is always with you. Forever.

You love life. She's really good.

Love people. And remember - in a person

what's important? High soul."

Let us also try to have a “high soul.” And for this you need to respect and love your parents, of course, try to study and pass the state test well. exams. Let's start preparing for certification.

Independent work. Self-control – 4 options. Testing your integrity. The work is done in notebooks. While completing the work, students determine for themselves an algorithm for solving fractional rational equations. On each desk there is a table - a reminder “Algorithm for solving fractional rational equations.” Annex 1.

Option 1.	Option 2.
Option 3.	Option 4.

ANSWERS:

Option I:
,
(
;
).

Option II:
(
;
)

Option III:
(

)

IV option:
,
(
;
).

VII . Physical education minute(slide 11).

Teacher: Now for the warm-up.

Turn to me. I speak out sentences. If it is fair, you stand up; if not, then you remain sitting.

1) 5x = 7 has a single root.

2) 0x = 0 has no roots.
3) If D 0, then the quadratic equation has two roots.
4) If D
5) The number of roots is not greater than the degree of the equation.

VIII . Reinforcement and repetition of material.(slide 12).

Teacher. Men want to look only courageous, only strong, only unbending in front of their loved ones. Perhaps this is what makes them men. And only in front of their own mother are they not afraid to expose their weaknesses and failures, to admit mistakes and losses, because no matter how far they have gone in their age and development, in front of her they are still gray-haired - still children. And she understands in her heart that the poor and the offended, first of all, need a mother more than anyone else. Today everyone will have good grades, so I think there will be no offended people.

Solving the problem No. 942 from the textbook. (Algebra – 9th grade / Yu.N. Makarychev) (slide 13).

1st car		x -20 km/h	h
2nd car		x km/h	h

Solve the example on the board.(slide 14).

No. 289(a)

VII . Summing up the lesson.

What new did you learn in the lesson?

What did you learn in the lesson?

2. Algorithm for solving fractional rational equations:

The teacher evaluates the students' work and assigns grades.

Teacher. Acquiring the features of a symbol and fulfilling a huge social mission, the mother never lost her usual human features, remaining a hospitable hostess and an intelligent interlocutor, a diligent worker and a born songwriter, broad in the feast and courageous in grief, open in joy and restrained in sadness, and always kind, understanding and feminine! I really want your parents' dreams to come true, may you be worthy people (slide 15).

VIII . Homework. No. 943, No. 940 (a, b), No. 290 (slide 16).

Annex 1.

Algorithm for solving fractional rational equations:

Find acceptable values ​​of the fractions included in the equation.

Find the common denominator of the fractions in the equation.

Multiply both sides of the equation by the common denominator.

Solve the resulting equation.

Eliminate roots that are not included in the acceptable values ​​of fractions of the equation .

Novoselitskaya Municipal educational institution secondary school No. 1

"Solving fractional rational equations."

Open lesson in 9A class

Mathematic teacherDemidenko N.Yu.

S. Novoselitskoye 2015.

Lesson topic : Solving fractional rational equations .(Slide 1)

Goals and objectives lesson:

Educational:

consolidation of the concept of a fractional rational equation;

continue to develop the skills to solve fractional rational equations;

repeat solving linear equations;

repeat solving quadratic equations.

Educational:

development of students' memory;

development of skills to overcome difficulties in solving mathematical problems;

development of curiosity;

development of logical thinking, attention, skills to analyze, compare and draw conclusions;

develop interest in the subject.

Educational:

formation of such personality qualities as responsibility, organization, discipline, decency, truthfulness;

promote the formation of a system of knowledge, ideas, concepts;

fostering cognitive interest in the subject;

fostering independence in solving educational problems;

nurturing will and perseverance to achieve final results.

Lesson type: consolidation of the studied material.

Form: workshop lesson.

Lesson equipment: PC, projector, fileMSExcelcontaining test tasks, presentation.

Checking homework

ANSWER THE QUESTIONS (Slide 2)

How many modules are there in the OGE test? What modules are these?

How many points do you need to score to successfully pass the exam?

Formulate the topic of our lesson.

"Solving Equations" (Slide 3)

continue the sentence:

the equation is called...

The root of the equation is...

Verbal counting (Slide 4)

1) x+3=0;

2) 3(x-7)=0;

3) x(x-1)(x+3)(x-9)=0;

4) x³-9x=0;

5) 7x²=0;

6) x²-5=0;

7) -7x²=28.

LET'S REPEAT (Slide 5)

1. What is the name of this equation? How many roots does this equation have?

2. Tell me, what degree is this equation? How many roots does this equation have?

3. Tell me, what degree is this equation? How many roots does this equation have?(X 3 – 1) 2 + x 5 - X 6 = 2

4. What is the name of this equation?

5. How to find the degree of an entire equation? (X 3 – 3) 2 + 5x 2 = 0

CONTINUE PHRASE (Slide 6)

A quadratic equation has 2 roots if......

A quadratic equation has 2 equal roots (or one root) if......

A quadratic equation has no roots if......

The range of acceptable values ​​of a fractional rational equation is.....

SPECIFY ODZ OF EQUATIONS (Slide 7)

a) 2(1-x²) +3x -4 =0;

b)x - 3 = x² - x +1 ;

4 2

c) x² -x - 7 = x +8;

G)2x - 4 = 3__;

x² +1 x +1

e)3x + 1 = x;

x -1

Remember the algorithms for solving equations! (Slide 8)

The equationy ( x ) =0 calledfractional rational equation , Ifexpression y ( x ) isfractional

(i.e. contains division into an expression with variables).(Slide 9)

Algorithms for solving fractional rational equations! (Slide 10)

Find acceptable values ​​of the fractions included in the equation.

Find the common denominator of the fractions in the equation.

Multiply both sides of the equation by the common denominator.

Solve the resulting equation.

5. Eliminate roots that are not included in the permissible values ​​of the fractions of the equation

Example #1: (Slide 11,12)

(Slide 13) Example #2: Kim Option No. 6, task No. 21

(x-2)(x 2 +8x+16) = 7(x+4)

(Slide 14) PHYSICAL MINUTE for the eyes

(Slide 15-19) Independent test work

1. Among these equations, choose the one that is not fractional rational:

1) ;

2)

3) .

(3)

2. At what values ​​of the variableX the equation doesn't make sense:

1) -2;

2) -2 and -1;

3) always makes sense.

(Slide 20) Teacher: Check your result (a table with the correct answers is displayed on the screen).

Let's check the answers with the answers on the board. We put “+” or “-” on the pieces of paper, depending on the correctness of execution. Rate yourself:

everything done correctly – “5”;

one error – “4”;

two mistakes made - “3”;

less than 3 tasks completed – “2”.

(Slide 21) Homework

Tests

Option 20-30 No. 4 (equations)

And I would like to end our lesson with the words of the great scientist A. Einstein:“I have to divide my time between politics and equations. However, equations, in my opinion, are much more important, because politics exists only for this moment, and equations will exist forever.”

(Slide 22) Independent work

"Solving fractional rational equations."

Open lesson in 9A class

Mathematic teacher Demidenko N.Yu.

S. Novoselitskoye 2015.

Lesson topic : Solving fractional rational equations.(Slide 1)

Goals and objectives lesson:

Educational:

• consolidation of the concept of a fractional rational equation;
• continue to develop the skills to solve fractional rational equations;
• repeat solving linear equations;
• repeat solving quadratic equations.

Educational:

• development of students' memory;
• development of skills to overcome difficulties in solving mathematical problems;
• development of curiosity;
• development of logical thinking, attention, skills to analyze, compare and draw conclusions;
• develop interest in the subject.

Educational:

• formation of such personality qualities as responsibility, organization, discipline, decency, truthfulness;
• promote the formation of a system of knowledge, ideas, concepts;
• fostering cognitive interest in the subject;
• fostering independence in solving educational problems;
• nurturing will and perseverance to achieve final results.

Lesson type: consolidation of the studied material.

Form: workshop lesson.

Lesson equipment: PC, projector, MS Excel file containing test tasks, presentation.

Checking homework

ANSWER THE QUESTIONS(Slide 2)

• How many modules are there in the OGE test? What modules are these?
• - How many points do you need to score to successfully pass the exam?
• - formulate the topic of our lesson.

"Solving Equations"(Slide 3)

continue the sentence:

• the equation is called...
• The root of the equation is...

Verbal counting(Slide 4)

3) x(x-1)(x+3)(x-9)=0;

LET'S REPEAT(Slide 5)

1. What is the name of this equation? How many roots does this equation have?

2. Tell me, what degree is this equation? How many roots does this equation have?

3. Tell me, what degree is this equation? How many roots does this equation have? (x 3- 1) 2 + x 5 - x 6 = 2

4. What is the name of this equation?

5. How to find the degree of an entire equation? (x 3 - 3) 2 + 5x 2 = 0

CONTINUE PHRASE(Slide 6)

1. A quadratic equation has 2 roots if......
2. A quadratic equation has 2 equal roots (or one root) if......
3. A quadratic equation has no roots if......
4. The range of acceptable values ​​of a fractional rational equation is.....

SPECIFY ODZ OF EQUATIONS(Slide 7)

a) 2(1-x²) +3x -4 =0;

b) x - 3= x² - x +1;

c) x² - x - 7= x +8;

G) 2x - 4= 3__;

e) 3x + 1= x;

Remember the algorithms for solving equations!(Slide 8)

The equation y(x) =0 called fractional rational equation , If expression y(x) is fractional

(i.e. contains division into an expression with variables). (Slide 9)

Algorithms for solving fractional rational equations!(Slide 10)

1. Find acceptable values ​​of the fractions included in the equation.
2. Find the common denominator of the fractions in the equation.
3. Multiply both sides of the equation by the common denominator.
4. Solve the resulting equation.

5. Eliminate roots that are not included in the permissible values ​​of the fractions of the equation

Example #1:(Slide 11,12)

(Slide 13)Example #2: Kim Option No. 6, task No. 21

(x-2)(x 2 +8x+16) = 7(x+4)

(Slide 14) PHYSICAL MINUTE for the eyes

(Slide 15-19)Independent test work

1. Among these equations, choose the one that is not fractional rational:

2. At what values ​​of the variable X the equation doesn't make sense: 1) -2; 2) -2 and -1; 3) always makes sense. (-2)

3. How many roots does the equation have? 1) 1 root; 2) has no roots; 3) 2 roots. (has no roots )

4. Find the roots of the equation 1) x=-?; 2) x=? or x=-3; 3) x=-? or x=3. (x=- )

5.Indicate the common denominator: 1) x-3; 2) x(x-3); 3) (5x-7)(4x-3). (X(x-3))

(Slide 20)Teacher: Check your result (a table with the correct answers is displayed on the screen).

Let's check the answers with the answers on the board. On the pieces of paper we put “+” or “-”, depending on the correctness of execution. Rate yourself:

everything done correctly - “5”;

one error - “4”;

two mistakes made - “3”;

less than 3 tasks completed - “2”.

(Slide 21)Homework

Option 20-30 No. 4 (equations)

And I would like to end our lesson with the words of the great scientist A. Einstein: “I have to divide my time between politics and equations. However, equations, in my opinion, are much more important, because politics exists only for this moment, and equations will exist forever.”

(Slide 22)Independent work