Topic 1: algebraic fractions, arithmetic operations. Problems involving adding and subtracting fractions
In this lesson we will continue to consider the simplest operations with algebraic fractions - their addition and subtraction. Today we will focus on considering examples in which the most important part of the solution will be factoring the denominator in all the ways that we know: with the common factor, the grouping method, isolating the perfect square, using abbreviated multiplication formulas. During the lesson we will look at several fairly complex fraction problems.
Subject:Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson:Problems involving adding and subtracting fractions
During the lesson we will consider and generalize all cases of adding and subtracting fractions: with the same and with different denominators. In general, we will solve problems of the form:
We saw earlier that when adding or subtracting algebraic fractions, one of the most important operations is factoring the denominators. A similar procedure is performed in the case of ordinary fractions. Let's remember once again how to work with ordinary fractions.
Example 1. Calculate.
Solution. Let us use, as before, the basic theorem of arithmetic that any number can be factorized into prime factors: 
.
Let's determine the least common multiple of the denominators: - this will be the common denominator of the fractions, and, based on it, we will determine additional factors for each of the fractions: for the first fraction 
, for the second fraction 
, for the third fraction.
Answer..
In the above example, we used the fundamental theorem of arithmetic to factor numbers. Further, when polynomials act as denominators, they will need to be factored using the following methods known to us: taking out a common factor, grouping method, isolating a complete square, using abbreviated multiplication formulas.
Example 2. Add and subtract fractions .
Solution. The denominators of all three fractions are complex expressions that must be factored, then find the lowest common denominator for them and indicate additional factors for each of the fractions. Let's do all these steps separately, and then substitute the results into the original expression.
In the first denominator we take out the common factor: - after taking out the common factor, you can notice that the expression in brackets is folded according to the formula of the square of the sum.
In the second denominator we take out the common factor: - after taking out the common factor, we apply the formula for the difference of squares.
In the third denominator we take out the common factor: .
After factoring the third denominator, you can notice that in the second denominator you can select a factor for a more convenient search for the lowest common denominator of fractions, we will do this by placing the minus out of brackets, in the second bracket we have swapped the terms for a more convenient form of notation.
Let's define the least common denominator of fractions as an expression that is divided by all denominators at the same time, it will be equal to: .
Let us indicate additional factors: for the first fraction 
, for the second fraction 
- we do not take into account the minus in the denominator, since we will write it for the entire fraction, for the third fraction 
.
Now let’s perform actions with fractions, not forgetting to change the sign before the second fraction:
At the last stage of the solution, we brought similar terms and wrote them in descending order of powers of the variable.
Answer..
Using the above example, we once again, as in previous lessons, demonstrated the algorithm for adding/subtracting fractions, which is as follows: factor the denominators of the fractions, find the lowest common denominator, additional factors, perform the addition/subtraction procedure and, if possible, simplify expression and make a reduction. We will continue to use this algorithm in the future. Let's now look at simpler examples.
Example 3. Subtract fractions 
.
Solution. In this example, it is important to see the opportunity to reduce the first fraction before bringing it to a common denominator with the second fraction. To do this, we factorize the numerator and denominator of the first fraction.
Numerator: - in the first step we expanded part of the expression according to the difference of squares formula, and in the second step we took out the common factor.
Denominator: - in the first step we expanded part of the expression according to the formula of the square of the difference, and in the second step we took out the common factor. Substitute the resulting numerator and denominator into the original expression and reduce the first fraction by a common factor:
Answer:.
Example 4. Perform actions 
.
Solution. In this example, as in the previous one, it is important to notice and implement the reduction of the fraction before performing the actions. Let's factor the numerator and denominator.
This lesson covers the concept of an algebraic fraction. People encounter fractions in the simplest life situations: when it is necessary to divide an object into several parts, for example, to cut a cake equally into ten people. Obviously, everyone gets a piece of the cake. In this case, we are faced with the concept of a numerical fraction, but a situation is possible when an object is divided into an unknown number of parts, for example, by x. In this case, the concept of a fractional expression arises. You have already become acquainted with whole expressions (not containing division into expressions with variables) and their properties in 7th grade. Next we will look at the concept of a rational fraction, as well as acceptable values of variables.
Subject:Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson:Basic Concepts
1. Definition and examples of algebraic fractions
Rational expressions are divided into integer and fractional expressions.
Definition. Rational fraction is a fractional expression of the form , where are polynomials. - numerator denominator.
Examples rational expressions:
- fractional expressions; - whole expressions. In the first expression, for example, the numerator is , and the denominator is .
Meaning algebraic fraction, like anyone algebraic expression, depends on the numerical value of the variables that are included in it. In particular, in the first example the value of the fraction depends on the values of the variables and , and in the second example only on the value of the variable .
2. Calculating the value of an algebraic fraction and two basic fraction problems
Let's consider the first typical task: calculating the value rational fraction for different values of the variables included in it.
Example 1. Calculate the value of the fraction for a) , b) , c)
Solution. Let's substitute the values of the variables into the indicated fraction: a) , b) , c) - does not exist (since you cannot divide by zero).
Answer: 3; 1; does not exist.
As you can see, two typical problems arise for any fraction: 1) calculating the fraction, 2) finding valid and invalid values letter variables.
Definition. Valid Variable Values- values of variables at which the expression makes sense. The set of all possible values of variables is called ODZ or domain.
3. Acceptable (ADV) and unacceptable values of variables in fractions with one variable
The value of literal variables may be invalid if the denominator of the fraction at these values is zero. In all other cases, the values of the variables are valid, since the fraction can be calculated.
Example 2. Establish at what values of the variable the fraction does not make sense.
Solution. For this expression to make sense, it is necessary and sufficient that the denominator of the fraction does not equal zero. Thus, only those values of the variable will be invalid for which the denominator is equal to zero. The denominator of the fraction is , so we solve the linear equation:
Therefore, given the value of the variable, the fraction has no meaning.
From the solution of the example, the rule for finding invalid values of variables follows - the denominator of the fraction is equal to zero and the roots of the corresponding equation are found.
Let's look at several similar examples.
Example 3. Establish at what values of the variable the fraction does not make sense.
Solution. .
Example 4. Establish at what values of the variable the fraction does not make sense.
Solution..
There are other formulations of this problem - find domain or range of acceptable expression values (APV). This means finding all valid variable values. In our example, these are all values except . It is convenient to depict the domain of definition on a number axis.
To do this, we will cut out a point on it, as indicated in the figure:
Thus, fraction definition domain there will be all numbers except 3.
Example 5. Establish at what values of the variable the fraction does not make sense.
Solution..
Let us depict the resulting solution on the numerical axis:
4. Graphical representation of the area of acceptable (AP) and unacceptable values of variables in fractions
Example 6. Establish at what values of the variables the fraction does not make sense.
Solution.. We have obtained the equality of two variables, we will give numerical examples: or, etc.
Let us depict this solution on a graph in the Cartesian coordinate system:
Rice. 3. Function graph.
The coordinates of any point lying on this graph are not included in the range of acceptable fraction values.
5. Case of "division by zero" type
In the examples discussed, we encountered a situation where division by zero occurred. Now consider the case where a more interesting situation arises with type division.
Example 7. Establish at what values of the variables the fraction does not make sense.
Solution..
It turns out that the fraction makes no sense at . But one could argue that this is not the case because: 
.
It may seem that if the final expression is equal to 8 at , then the original one can also be calculated, and therefore makes sense at . However, if we substitute it into the original expression, we get - it makes no sense.
To understand this example in more detail, let’s solve the following problem: at what values does the indicated fraction equal zero?
(a fraction is zero when its numerator is zero) 
. But it is necessary to solve the original equation with a fraction, and it does not make sense for , since at this value of the variable the denominator is zero. This means that this equation has only one root.
6. Rule for finding ODZ
Thus, we can formulate an exact rule for finding the range of permissible values of a fraction: to find ODZfractions it is necessary and sufficient to equate its denominator to zero and find the roots of the resulting equation.
We considered two main tasks: calculating the value of a fraction for the specified values of the variables and finding the range of acceptable values of a fraction.
Let's now consider a few more problems that may arise when working with fractions.
7. Various tasks and conclusions
Example 8. Prove that for any values of the variable the fraction .
Proof. The numerator is a positive number. . As a result, both the numerator and the denominator are positive numbers, therefore the fraction is a positive number.
Proven.
Example 9. It is known that , find .
Solution. Let's divide the fraction term by term. We have the right to reduce by, taking into account what is an invalid variable value for a given fraction.
In this lesson we covered basic concepts related to fractions. In the next lesson we will look at main property of a fraction.
Bibliography
1. Bashmakov M.I. Algebra 8th grade. - M.: Education, 2004.
2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 8. - 5th ed. - M.: Education, 2010.
3. Nikolsky S. M., Potapov M. A., Reshetnikov N. N., Shevkin A. V. Algebra 8th grade. Textbook for general education institutions. - M.: Education, 2006.
1. Festival of pedagogical ideas.
2. Old school.
3. Internet portal lib2.podelise. ru.
Homework
1. No. 4, 7, 9, 12, 13, 14. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 8. - 5th ed. - M.: Education, 2010.
2. Write down a rational fraction whose domain of definition is: a) the set, b) the set, c) the entire number line.
3. Prove that for all possible values of the variable, the value of the fraction is non-negative.
4. Find the domain of expression. Instructions: consider separately two cases: when the denominator of the lower fraction is zero and when the denominator of the original fraction is zero.
Section of mathematics. Through line.
Numbers and calculations
Expressions and Transformations
Algebraic fraction.
Reducing fractions.
Operations with algebraic fractions.
| Program | ^ Number of hours | Control marks | |
| U-1. Combined lesson “Basic concepts” | 1 | Tasks for mental calculation. Exercise 1 "Numerical Expressions" |
|
| U-2. Lesson-lecture "The main property of an algebraic fraction. Reducing fractions" | 1 | Demonstration material "The main property of algebraic fractions" |
|
| U-3. Lesson - consolidation of what has been learned | 1 | Verbal counting Independent work 1.1 “The main property of a fraction. Reducing Fractions" | Tasks for mental calculation. Exercise 2 "Reducing Algebraic Fractions" |
| U-4. Combined lesson "Adding and subtracting fractions with like denominators" | 1 | |
|
| U-5. Lesson - problem solving | 1 | CD Mathematics 5-11 Exercises "Rational numbers". |
|
| U-6. Combined lesson "Adding and subtracting fractions with different denominators" | 1 | Demonstration material "Adding and subtracting algebraic fractions" |
|
| U-7. Lesson - problem solving | 1 | Verbal counting | Tasks for mental calculation. Exercise 3 “Adding and subtracting algebraic fractions” |
| U-8. Lesson - independent work | 1 | Independent work 1.2 "Adding and subtracting algebraic fractions" | |
| U-9. Lesson - problem solving | 1 | ||
| U-10. Lesson-test | 1 | Test No. 1 | |
| U-11. Combined lesson "Multiplication and division of algebraic fractions. Raising algebraic fractions to powers" | 1 | ||
| U-12. Lesson - problem solving | 2 | Independent work 1.3 "Multiplying and dividing fractions" | |
| U-13. Combined lesson "Transformation of rational expressions" | 1 | Verbal counting | Tasks for mental calculation. Exercise 4 “Multiplication and division of algebraic fractions” |
| U-14. Lesson - problem solving | 1 | ||
| U-15. Lesson - independent work | 1 | Independent work 1.4 "Transformation of rational expressions" | |
| U-16. Workshop lesson “First ideas about solving rational equations” | 1 | CD Mathematics 5-11 Virtual laboratory “Graph of a function”. |
|
| U-17. Lesson - problem solving | 1 | Test 1 "Algebraic fractions" | |
| U-18. Lesson - test. | 1 | Test No. 2 |
Be able to reduce algebraic fractions.
Be able to perform basic operations with algebraic fractions.
Be able to perform combined exercises on actions with algebraic fractions.
Topic 2. Quadratic function. Function . (18 hours)
Function
Mandatory minimum content of the educational field of mathematics
Program. Monitoring its implementation
| Program | Number per hour | Control marks | Computer software lesson |
| U-1. Combined lesson “Function , its properties and graph" | 1 | |
|
| | 1 | Verbal counting | Tasks for mental calculation. Exercise 5 "Function" Demonstration material “Parabola. Application in science and technology" |
| U-3. Problem solving lesson | 1 | Independent work 2.1 "Function y = kx 2
» | |
| U-4. Lesson-lecture "Function and its graph" | 1 | Demonstration material “Function, its properties and graph” |
|
| ^ U-5. Problem solving lesson | 3 | Verbal counting Independent work 2.2 "Function" | Tasks for mental calculation. Exercise 6 “Inverse proportionality” |
| U-6,7. Lessons-workshops “How to graph a function » | 2 | Practical work | |
| U-8,9. Lessons-workshops “How to graph a function , if the graph of the function is known » | 2 | CD “Mathematics 5-11 grades.” Virtual laboratory “Graphs of functions” |
|
| ^ U-10. Lesson-test | 1 | Test No. 3 | |
| U-11 Lessons-workshop “How to plot a function graph , if the graph of the function is known » | 1 | CD “Mathematics 5-11 grades.” Virtual laboratory “Graphs of functions” |
|
| U-12 Lesson-workshop “How to plot a function graph , if the graph of the function is known » | 1 | Independent work 2.3 "Function Graphs" | CD “Mathematics 5-11 grades.” Virtual laboratory “Graphs of functions” |
| U-13. Combined lesson “Function , its properties and graph" | 1 | Demonstration material “Properties of a quadratic function” |
|
| U-14. Lesson - consolidation of what has been learned.. | 1 | Verbal counting | Tasks for mental calculation. Exercise 7 “Quadratic function” |
| U-15. Problem solving lesson | 1 | Verbal counting Independent work 2.4 “Properties and graph of a quadratic function” | Tasks for mental calculation. Exercise 8 “Properties of a quadratic function” |
| U-16. Lesson-test | 1 | Test 2 "Quadratic function" | |
| ^ U-17. Workshop “Graphical solution of quadratic equations” | 1 | Demonstration material “Graphical solution of quadratic equations” |
|
| U-18. Lesson-test | 1 | Test No. 4 |
Requirements for mathematical preparation
Level of compulsory training of the student
Level of possible training of the student
Topic 3 Function . Properties of the square root (11 hours)
Section of mathematics. Through line
Numbers and calculations
Expressions and Transformations
Functions
Square root of a number. Arithmetic square root.
The concept of an irrational number. The irrationality of numbers.
Real numbers.
Properties of square roots and their applications in calculations.
Function.
Program. Monitoring its implementation
| Program | Number of hours | Control marks | Computer support for the lesson |
| ^ U-1. Lesson-lecture “The concept of the square root of a non-negative number” | 1 | Demonstration material “The concept of square root” |
|
| U-2. Lesson - problem solving | 1 | Independent work 3.1 "Arithmetic square root" | |
| U-3. Combined lesson “Function , its properties and graph" | 1 | Demonstration material “Function, its properties and graph” |
|
| ^ U-4. Lesson - problem solving | 1 | Verbal counting | Tasks for mental calculation. Exercise 9 “Arithmetic square root” |
| ^ U-5. Combined lesson “Properties of square roots” | 1 | Demonstration material “Application of the properties of arithmetic square root” |
|
| ^ U-6 Lesson - problem solving | 1 | Verbal counting Independent work 3.2 "Properties of the arithmetic square root" | Tasks for mental calculation. Exercise 10 “Square root of a product and a fraction” |
| ^ U-7.8. Workshops “Transforming expressions containing the operation of extracting a square root.” | 2 | Practical work | |
| ^ U-9. Lesson - problem solving | 1 | Verbal counting Independent work 3.3 "Applying the Properties of Arithmetic Square Root" | Tasks for mental calculation. Exercise 11 “Square root of a degree” |
| U-10. Lesson - problem solving | 1 | Test 3 "Square Roots" | |
| U-11. Lesson - test. | 1 | Test No. 5 |
^ Requirements for mathematical preparation
Level of compulsory training of the student
Find the meanings of roots in simple cases.
Know the definition and properties of a function , be able to build a schedule.
Be able to use the properties of arithmetic square roots to calculate values and simple transformations of numerical expressions containing square roots.
Level of possible training of the student
Know the concept of arithmetic square root.
Be able to apply the properties of arithmetic square roots when transforming expressions.
Be able to use the properties of a function when solving practical problems.
Have an understanding of irrational and real numbers.
^ Topic 4 Quadratic equations (21 hours)
Section of mathematics. Through line
Equations and inequalities
Mandatory minimum content of the educational field of mathematics
Quadratic equation: formula for the roots of a quadratic equation.
Solving rational equations.
Solving word problems using quadratic and fractional rational equations.
Program. Monitoring its implementation
| Program | Number of hours | Control marks | Computer software lesson |
| ^ U-1. Lesson-study of new material “Basic concepts”. | 1 | Demonstration material “Quadratic Equations” |
|
| U-2. Lesson - consolidation of what has been learned. | 1 | Verbal counting | Tasks for mental calculation. Exercise 12 “Quadratic equation and its roots” |
| U-3. Combined lesson “Formulas of roots of quadratic equations.” | 1 | Independent work 4.1 "Quadratic equation and its roots" | |
| U-4.5. Problem Solving Lessons | 2 | Verbal counting | Tasks for mental calculation. Exercise 11 “Solving quadratic equations” |
| U-6. Lesson - independent work | 1 | Independent work 4.2 “Solving quadratic equations using a formula” | |
| U-7. Combined lesson “Rational equations” | 1 | Practical work | |
| U-8,9. Problem Solving Lessons | 2 | Independent work 4.3 "Rational Equations" | |
| U-10,11. Workshops “Rational equations as mathematical models of real situations.” | 2 | ||
| U-12. Problem solving lesson | 1 | ||
| U-13. Lesson - independent work | 1 | Independent work 4.4 "Solving Problems Using Quadratic Equations" | |
| U-14. Combined lesson “Another formula for the roots of a quadratic equation.” | 1 | ||
| U-15. Lesson - problem solving | 1 | ||
| U-16. Combined lesson "Viete's Theorem". | 1 | Demonstration material “Vieta’s Theorem” |
|
| U-17. Lesson - problem solving | 1 | Verbal counting | Tasks for mental calculation. Exercise 14 “Viete’s Theorem” |
| U-18. Combined lesson “Irrational equations” | 1 | ||
| U-19. Lesson - problem solving | 1 | ||
| U-20. Problem solving lesson | 1 | Test 4 "Quadratic equations" | CD Mathematics 5-11. Virtual laboratory “Graphs of equations and inequalities” |
| U-21. Lesson - test. | 1 | Test No. 6 |
^ Requirements for mathematical preparation
Level of compulsory training of the student
Be able to solve quadratic equations, simple rational and irrational equations.
Be able to solve simple word problems using equations.
Level of possible training of the student
Understand that equations are a mathematical apparatus for solving various problems from mathematics, related fields of knowledge, and practice.
Be able to solve quadratic equations, rational and irrational equations that can be reduced to quadratic equations.
Be able to use quadratic equations and rational equations to solve problems.
Subject:
Lesson: Converting rational expressions
1. Rational expression and methods for simplifying it
Let us first recall the definition of a rational expression.
Definition. Rational expression- an algebraic expression that does not contain roots and includes only the operations of addition, subtraction, multiplication and division (raising to a power).
By the concept of “transforming a rational expression” we mean, first of all, its simplification. And this is carried out in the order of actions known to us: first the actions in brackets, then product of numbers(exponentiation), dividing numbers, and then adding/subtracting operations.
2. Simplification of rational expressions with sum/difference of fractions
The main goal of today's lesson will be to gain experience in solving more complex problems of simplifying rational expressions.
Example 1.
Solution. At first it may seem that these fractions can be reduced, since the expressions in the numerators of fractions are very similar to the formulas for the perfect squares of their corresponding denominators. In this case, it is important not to rush, but to separately check whether this is so.
Let's check the numerator of the first fraction: . Now the second numerator: .
As you can see, our expectations were not met, and the expressions in the numerators are not perfect squares, since they do not have doubling of the product. Such expressions, if you remember the 7th grade course, are called incomplete squares. You should be very careful in such cases, because confusing the formula of a complete square with an incomplete one is a very common mistake, and such examples test the student’s attentiveness.
Since reduction is impossible, we will perform the addition of fractions. The denominators do not have common factors, so they are simply multiplied to obtain the lowest common denominator, and the additional factor for each fraction is the denominator of the other fraction.
Of course, you can then open the brackets and then bring similar terms, however, in this case you can get by with less effort and notice in the numerator that the first term is the formula for the sum of cubes, and the second is the difference of cubes. For convenience, let us recall these formulas in general form:
In our case, the expressions in the numerator are collapsed as follows:
, the second expression is similar. We have:
Answer..
Example 2. Simplify rational expression 
.
Solution. This example is similar to the previous one, but here it is immediately clear that the numerators of the fractions contain partial squares, so reduction at the initial stage of the solution is impossible. Similarly to the previous example, we add the fractions:
Here, similarly to the method indicated above, we noticed and collapsed the expressions using the formulas for the sum and difference of cubes.
Answer..
Example 3. Simplify a rational expression.
Solution. You can notice that the denominator of the second fraction is factorized using the sum of cubes formula. As we already know, factoring denominators is useful for further finding the lowest common denominator of fractions.
Let's indicate the lowest common denominator of the fractions, it is equal to: https://pandia.ru/text/80/351/images/image016_27.gif" alt="http://d3mlntcv38ck9k.cloudfront.net/content/konspekt_image/ 23332/d6838ff258e40dc138ebee9552f3b9fb.png" width="624" height="70">.!}
Answer.
3. Simplification of rational expressions with complex “multi-story” fractions
Let's consider a more complex example with “multi-story” fractions.
Example 4. Prove identity https://pandia.ru/text/80/351/images/image019_25.gif" alt="http://d3mlntcv38ck9k.cloudfront.net/content/konspekt_image/23335/25bd4e84df065d130e03bf9d1738a99d.png" width="402" height="55">. Доказано при всех допустимых значениях переменной.!}
Proven.
In the next lesson we will look in detail at more complex examples of converting rational expressions.
Subject: Algebraic fractions. Arithmetic operations on algebraic fractions
Lesson: Converting More Complex Rational Expressions
1. An example of proving identity using transformations of rational expressions
In this lesson we will look at converting more complex rational expressions. The first example will be devoted to proving the identity.
Example 1
Prove the identity: .
Proof:
First of all, when transforming rational expressions, it is necessary to determine the order of actions. Let us remind you that the operations in parentheses are performed first, then multiplication and division, and then addition and subtraction. Therefore, in this example, the order of actions will be as follows: first we perform the action in the first brackets, then in the second brackets, then we divide the results obtained, and then we add a fraction to the resulting expression. As a result of these actions, as well as simplification, the expression should be obtained.
| № p/p | Content elements | Be able to solve problematic problems and situations | S-9 |
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| 26 | Power with negative integer exponent | Natural exponent, negative exponent, multiplication, division and exponentiation | Have an idea of a power with a natural exponent, a power with a negative exponent, multiplication, division and exponentiation of a number Be able to: – simplify expressions using the definition of a degree with a negative exponent and properties of a degree; – compose a text in a scientific style | S-10 |
| 29 | Test No. 2 “Transformation of rational expressions” | Be able to independently choose a rational way of transforming rational expressions, prove identities, solve rational equations by eliminating the denominators, creating a mathematical model of the real situation | K.R. No. 2 |
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Questions for testing
State the main property of a fraction.
Formulate
An algorithm for finding an additional factor for an algebraic fraction.
Rules for adding and subtracting algebraic fractions with like denominators.
Algorithm for finding the common denominator of several fractions
The rule for adding (subtracting) algebraic fractions with different denominators.
Rule for multiplying algebraic fractions
Rule for dividing algebraic fractions.
The rule for raising an algebraic fraction to a power.
