Dividing algebraic fractions. Multiplying and dividing algebraic fractions. Topic: Multiplication and division of algebraic fractions

Video lesson “Multiplication and division of algebraic fractions. Raising an algebraic fraction to a power" is an auxiliary tool for teaching a mathematics lesson on this topic. With the help of a video lesson, it is easier for a teacher to develop in students the ability to multiply and divide algebraic fractions. The visual aid contains a detailed, understandable description of examples in which multiplication and division operations are performed. The material can be demonstrated during the teacher's explanation or become separate part lesson.

To develop the ability to solve problems on multiplication and division of algebraic fractions, important comments, points that require memorization and deep understanding stand out using color, bold font, and pointers. With the help of a video lesson, the teacher can increase the effectiveness of the lesson. This visual aid will help you quickly and effectively achieve your learning goals.

The video lesson begins by introducing the topic. After this, it is indicated that multiplication and division operations with algebraic fractions are performed similarly to operations with ordinary fractions. The screen demonstrates the rules for multiplying, dividing and exponentiating fractions. Multiplication of fractions is demonstrated using letter options. It is noted that when multiplying fractions, the numerators, as well as the denominators, are multiplied. This gives the resulting fraction a/b·c/d=ac/bd. The division of fractions is demonstrated using the expression a/b:c/d as an example. It is indicated that to perform the division operation it is necessary to write in the numerator the product of the numerator of the dividend and the denominator of the divisor. The denominator of a quotient is the product of the denominator of the dividend and the numerator of the divisor. Thus, the division operation turns into an operation of multiplying the fraction of the dividend and the reciprocal of the divisor. Raising a fraction to a power is equivalent to a fraction in which the numerator and denominator are raised to the assigned power.

The solution to the examples is discussed below. In example 1, it is necessary to perform the actions (5x-5y)/(x-y)·(x 2 -y 2)/10x. To solve this example, the numerator of the second fraction included in the product is factorized. Using abbreviated multiplication formulas, the transformation x 2 -y 2 = (x+y)(x-y) is made. Then the numerators of the fractions and denominators are multiplied. After carrying out the operations, it is clear that the numerator and denominator have factors that can be reduced using the basic property of a fraction. As a result of the transformations, the fraction (x+y) 2 /2x is obtained. Here we also consider the execution of actions 7a 3 b 5 /(3a-3b)·(6b 2 -12ab+6a 2)/49a 4 b 5. All numerators and denominators are considered for the possibility of factorization and identification of common factors. Then the numerators and denominators are multiplied. After multiplication, reductions are made. The result of the transformation is the fraction 2(a-b)/7a.

An example is considered in which it is necessary to perform the actions (x 3 -1)/8y:(x 2 +x+1)/16y 2. To solve the expression, it is proposed to transform the numerator of the first fraction using the abbreviated multiplication formula x 3 -1=(x-1)(x 2 +x+1). According to the rule for dividing fractions, the first fraction is multiplied by the reciprocal of the second. After multiplying the numerators and denominators, a fraction is obtained that contains the same factors in the numerator and denominator. They are shrinking. The result is the fraction (x-1)2y. The solution to the example (a 4 -b 4)/(ab+2b-3a-6):(b-a)(a+2) is also described here. Similar to the previous example, the abbreviated multiplication formula is used to convert the numerator. The denominator of the fraction is also converted. The first fraction is then multiplied with the reciprocal of the second fraction. After multiplication, transformations are performed, reducing the numerator and denominator by common factors. The result is the fraction -(a+b)(a 2 +b 2)/(b-3). Students' attention is drawn to how the signs of the numerator and denominator change when multiplying.

In the third example, you need to perform operations with fractions ((x+2)/(3x 2 -6x)) 3:((x 2 +4x+4)/(x 2 -4x+4)) 2 . In solving this example, the rule for raising a fraction to a power is applied. Both the first and second fractions are raised to a power. They are converted by raising the numerator and denominator of the fraction to a power. In addition, to convert the denominators of fractions, the abbreviated multiplication formula is used, highlighting common multiplier. To divide the first fraction by the second, you need to multiply the first fraction by the reciprocal of the second. The numerator and denominator form expressions that can be abbreviated. After the transformation, the fraction (x-2)/27x 3 (x+2) is obtained.

Video lesson “Multiplication and division of algebraic fractions. Raising an algebraic fraction to a power" is used to increase the effectiveness of a traditional mathematics lesson. The material may be useful to a teacher teaching remotely. A detailed, clear description of the solution to the examples will help students who are independently mastering the subject or require additional training.

This lesson will cover the rules for multiplying and dividing algebraic fractions, as well as examples of how to apply these rules. Multiplying and dividing algebraic fractions is no different from multiplying and dividing ordinary fractions. At the same time, the presence of variables leads to slightly more in complex ways simplification of the resulting expressions. Despite the fact that multiplying and dividing fractions is easier than adding and subtracting them, the study of this topic must be approached extremely responsibly, since there are many pitfalls in it that are usually not paid attention to. As part of the lesson, we will not only study the rules of multiplying and dividing fractions, but also analyze the nuances that may arise when using them.

Subject:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Multiplying and dividing algebraic fractions

The rules for multiplying and dividing algebraic fractions are absolutely similar to the rules for multiplying and dividing ordinary fractions. Let's remind them:

That is, in order to multiply fractions, it is necessary to multiply their numerators (this will be the numerator of the product), and multiply their denominators (this will be the denominator of the product).

Division by a fraction is multiplication by an inverted fraction, that is, in order to divide two fractions, it is necessary to multiply the first of them (the dividend) by the inverted second (divisor).

Despite the simplicity of these rules, many people make mistakes in a number of special cases when solving examples on this topic. Let's take a closer look at these special cases:

In all these rules we used the following fact: .

Let's solve a few examples of multiplying and dividing ordinary fractions to remember how to use these rules.

Example 1

Note: When reducing fractions, we used the decomposition of numbers into prime factors. Let us remind you that prime numbers these are called natural numbers, which are divisible only by and by itself. The remaining numbers are called composite . The number is neither prime nor composite. Examples of prime numbers: .

Example 2

Let us now consider one of the special cases with ordinary fractions.

Example 3

As you can see, multiplying and dividing ordinary fractions, if the rules are applied correctly, is not difficult.

Let's look at multiplication and division of algebraic fractions.

Example 4

Example 5

Note that it is possible and even necessary to reduce fractions after multiplication according to the same rules that we previously considered in the lessons devoted to reducing algebraic fractions. Let's look at a few simple examples for special cases.

Example 6

Example 7

Let us now consider a little more complex examples on multiplying and dividing fractions.

Example 8

Example 9

Example 10

Example 11

Example 12

Example 13

Previously, we looked at fractions in which both the numerator and denominator were monomials. However, in some cases it is necessary to multiply or divide fractions whose numerators and denominators are polynomials. In this case, the rules remain the same, but to reduce it is necessary to use abbreviated multiplication formulas and bracketing.

Example 14

Example 15

Example 16

Example 17

Example 18

We can multiply and divide arithmetic fractions, for example:

if the letters a, b, c and d stand for arithmetic integers.

The question arises whether these equalities remain in force if a, b, c and d denote: 1) some arithmetic numbers and 2) any relative numbers.

First of all, you will have to consider complex fractions, for example:

These examples are already enough to verify the validity of the equalities relating to the multiplication and division of fractions, when the numbers a, b, c and d are any (integer or fractional) arithmetic. Note that there are only 2 basic equalities, namely:

It now remains to consider whether these equalities will remain valid if some of the numbers a, b, c and d are assumed to be negative: if, for example, a negative number, b, c and d are positive, then the fraction is negative and the fraction is positive; therefore, for example, dividing by should result in a negative number, but we see that, according to our assumption, the expression should express a negative number, i.e. equality is justified in this case as well. It is also easy to consider other assumptions for the signs of a, b, c and d. The result of this consideration is the conviction of the validity of the equalities

and for the case when a, b, c and d express any relative numbers, i.e., for the multiplication and division of algebraic fractions, the same rules remain in force as for arithmetic ones.

We can now perform multiplication and division of algebraic fractions. The greatest difficulty here is the question of reducing fractions obtained after multiplication or division. If the algebraic fractions are monomial, then reducing the result obtained will not present any difficulties, but if the fractions are algebraic, then it is necessary to first factorize the numerator and denominator of each of these fractions.

Topic: Multiplication and division of algebraic fractions

Education is what remains when everything learned has already been forgotten.

Laue

Goals:

Educational:

pin ZUN on the topic

carry out primary current knowledge control

work on the gaps

Educational:

contribute to the development of communicative competence, i.e. the ability to collaborate effectively with other people.

promote the development of cooperative competence, i.e. ability to work in pairs.

contribute to the development of problem competence, i.e. the ability to understand the inevitability of difficulties arising in the course of any activity.

Educational:

instill the ability to adequately evaluate the work done by a friend;

When working in pairs, cultivate the qualities of mutual assistance and support.

Methodical:

creating conditions for the manifestation of individuality and cognitive activity of students;

show the methodology for conducting a lesson with designing the results educational activities and ways to study them based on a competency-based approach.

Equipment: board, colored chalk. Table "Multiplication and division of algebraic fractions"; cards for individual work, "reminder" cards. Task in a free minute.

Lesson progress

Organizational moment

The lesson plan is written on the board:

Oral warm-up.

Individual work.

Solving tasks.

Pair work.

Lesson summary.

Homework.

Teacher: In the old days in Rus' it was believed that if a person was versed in mathematics, this meant the highest degree of learning. And the ability to see and hear correctly is the first step to wisdom. I would like today all the students in your class to show how wise they are and how knowledgeable people are in 7th grade algebra.

So, the topic of the lesson is “Multiplication and division of algebraic fractions.” In the last lesson you began to study this topic, and we discussed why we were studying it. Let's remember where it will be useful to us in just a few lessons.

Students: For joint actions with algebraic fractions, for solving equations, and therefore problems.

Teacher: Even in the old days in Rus' they said that multiplication is torment, and division is trouble. Anyone who could quickly and accurately multiply and divide was considered a great mathematician.

What goals will you set for yourself?

Students: Continue studying the topic, learn how to quickly and accurately multiply and divide.

Teacher: To achieve our goals, we (opens the plan written on the board, speaks it out)

1. Oral warm-up: (at this time, 3 - 4 people solve the exercise for reducing fractions in pairs) factorize, filling in the blanks

1= (y-1) (...), 5a+5b=... (a+b), xy-x=x (...), 14-2x=...

reduce the fraction

Fractions, fractions, beat fractions, shorten them, don’t spare them.

find the mistake made when multiplying and dividing algebraic fractions

Teacher: Where was the mistake made? Why was the mistake made? What rule did the student not know? Which one did you know? How to do it right?

2. Work in a notebook, number from the textbook 488 (1) Analysis, decision, verification.

Teacher: And now you will have the opportunity to show your knowledge when completing the test, and in order to inspire you to work, I will read the poem “For the teacher to write down “5” in your diary, be able to multiply the numerator by the numerator in an instant, and so that the teacher is happy with you, you multiply the first denominator by the second "

Self-check, mutual check. According to the criteria (posted on the board) B-1 (321), B-2 (132) using the correct codes, assessment in pairs. Initial result. Ratings.

Working on mistakes in student-teacher pairs

If there are no mistakes in the pairs, do the task in a free minute.

Simplify the expression and find its value when

5. Lesson summary

At the end of the lesson, I would like to know from you, what types of work caused you difficulties? Why do you think? What new did you learn? How many of you are satisfied with your work in class? Do you think the goals set at the beginning of the lesson have been achieved?

Teacher: I would like to end the lesson with the words of the French engineer-physicist Laue: “Education is what remains when everything learned has already been forgotten.”

I hope that you will not forget this material, so that this does not happen, you must complete assignments No. 486,487,488 even.

Sections: Mathematics

Target: Learn to perform the operations of multiplication and division of algebraic fractions.

Lesson format: lesson of learning new material.

Teaching method: problematic, with an independent search for a solution.

Equipment: Computer, projector, lesson handouts, table.

Lesson progress

The lesson is taught using a computer presentation. (Appendix 1)

Ι. Lesson organization.

1. Preparation of the technical part.

2. Cards for working in pairs and independent work.

ΙΙ. Updating basic knowledge in order to prepare for studying a new topic.

Orally:

(Answers are displayed using a computer.)

1. Factorize:

2. Reduce a fraction:

3. Multiply fractions:

What are these numbers called? (Reciprocal numbers)

Find the inverse of a number

What two numbers are called reciprocals? (Two numbers are called reciprocals if their product is 1.)

Find the reciprocal fraction:

Divide fractions:

We discuss the rules for multiplying and dividing ordinary fractions. A poster with the rules is posted on the board.

ΙΙΙ. New topic

Addressing the poster, the teacher says: a, b, c, d- in this case, numbers. And if these are algebraic expressions, what are such fractions called? (Algebraic fractions)

The rules for their multiplication and division remain the same.

Follow these steps:

The first and second examples are given independently, followed by students writing down the solution on the board. The teacher shows the solution to the third example on the board.

ΙV. Consolidation

1) Work according to the problem book: No. 5.2 (b, c), No. 5.11 (a, b). Page 32

2) Work in pairs using cards:

(Solutions and answers are reflected through the projector.)

V. Lesson summary

Independent work.

Perform multiplication or division:

Ι Option		ΙΙ Option

Students hand in their workbooks.

VI. Homework

No. 5.8; No. 5.10; No. 5.13(a, b).