Formula for simplifying expressions with fractions. Types of fractions. Transformations. Solving complex problems

Rational expressions and fractions are the cornerstone of the entire algebra course. Those who learn to work with such expressions, simplify them and factor them, will essentially be able to solve any problem, since transforming expressions is an integral part of any serious equation, inequality, or even word problem.

In this video tutorial we will see how to correctly apply abbreviated multiplication formulas to simplify rational expressions and fractions. Let's learn to see these formulas where, at first glance, there is nothing. At the same time, we will repeat such a simple technique as factoring a quadratic trinomial through a discriminant.

As you probably already guessed from the formulas behind me, today we will study abbreviated multiplication formulas, or, more precisely, not the formulas themselves, but their use to simplify and reduce complex rational expressions. But, before moving on to solving examples, let's take a closer look at these formulas or remember them:

1. $((a)^(2))-((b)^(2))=\left(a-b \right)\left(a+b \right)$ — difference of squares;
2. $((\left(a+b \right))^(2))=((a)^(2))+2ab+((b)^(2))$ is the square of the sum;
3. $((\left(a-b \right))^(2))=((a)^(2))-2ab+((b)^(2))$ — squared difference;
4. $((a)^(3))+((b)^(3))=\left(a+b \right)\left(((a)^(2))-ab+((b)^( 2)) \right)$ is the sum of cubes;
5. $((a)^(3))-((b)^(3))=\left(a-b \right)\left(((a)^(2))+ab+((b)^(2) ) \right)$ is the difference of cubes.

I would also like to note that our school education system is structured in such a way that it is with the study of this topic, i.e. rational expressions, as well as roots, modules, all students have the same problem, which I will now explain.

The fact is that at the very beginning of studying abbreviated multiplication formulas and, accordingly, actions to reduce fractions (this is somewhere in the 8th grade), teachers say something like the following: “If something is not clear to you, then don’t worry, we will help you.” We will return to this topic more than once, in high school for sure. We'll look into this later." Well, then, at the turn of 9-10 grades, the same teachers explain to the same students who still don’t know how to solve rational fractions, something like this: “Where were you the previous two years? This was studied in algebra in 8th grade! What could be unclear here? It’s so obvious!”

However, such explanations do not make it any easier for ordinary students: they still had a mess in their heads, so right now we will look at two simple examples, on the basis of which we will see how to isolate these expressions in real problems, which will lead us to abbreviated multiplication formulas and how to then apply this to transform complex rational expressions.

Reducing simple rational fractions

Task No. 1

\[\frac(4x+3((y)^(2)))(9((y)^(4))-16((x)^(2)))\]

The first thing we need to learn is to select exact squares and more in the original expressions high degrees, on the basis of which we can then apply formulas. Let's get a look:

Let's rewrite our expression taking into account these facts:

\[\frac(4x+3((y)^(2)))(((\left(3((y)^(2)) \right))^(2))-((\left(4x \right))^(2)))=\frac(4x+3((y)^(2)))(\left(3((y)^(2))-4x \right)\left(3 ((y)^(2))+4x \right))=\frac(1)(3((y)^(2))-4x)\]

Answer: $\frac(1)(3((y)^(2))-4x)$.

Problem No. 2

Let's move on to the second task:

\[\frac(8)(((x)^(2))+5xy-6((y)^(2)))\]

There is nothing to simplify here, because the numerator contains a constant, but I proposed this problem precisely so that you learn how to factor polynomials containing two variables. If instead we had the polynomial below, how would we expand it?

\[((x)^(2))+5x-6=\left(x-... \right)\left(x-... \right)\]

Let's solve the equation and find the $x$ that we can put in place of the dots:

\[((x)^(2))+5x-6=0\]

\[((x)_(1))=\frac(-5+7)(2)=\frac(2)(2)=1\]

\[((x)_(2))=\frac(-5-7)(2)=\frac(-12)(2)=-6\]

We can rewrite the trinomial as follows:

\[((x)^(2))+5xy-6((y)^(2))=\left(x-1 \right)\left(x+6 \right)\]

We learned how to work with a quadratic trinomial - that's why we needed to record this video lesson. But what if, in addition to $x$ and a constant, there is also $y$? Let's consider them as another element of the coefficients, i.e. Let's rewrite our expression as follows:

\[((x)^(2))+5y\cdot x-6((y)^(2))\]

\[((x)_(1))=\frac(-5y+7y)(2)=y\]

\[((x)_(2))=\frac(-5y-7y)(2)=\frac(-12y)(2)=-6y\]

Let us write the expansion of our square construction:

\[\left(x-y \right)\left(x+6y \right)\]

So, if we return to the original expression and rewrite it taking into account the changes, we get the following:

\[\frac(8)(\left(x-y \right)\left(x+6y \right))\]

What does such a record give us? Nothing, because it cannot be reduced, it is not multiplied or divided by anything. However, as soon as this fraction turns out to be an integral part of more complex expression, such a decomposition will come in handy. Therefore, as soon as you see a quadratic trinomial (it does not matter whether it is burdened with additional parameters or not), always try to factor it.

Nuances of the solution

Remember the basic rules for converting rational expressions:

• All denominators and numerators must be factored either through abbreviated multiplication formulas or through a discriminant.
• You need to work according to the following algorithm: when we look and try to isolate the formula for abbreviated multiplication, then, first of all, we try to convert everything to the highest possible degree. After this, we take the overall degree out of the bracket.
• Very often you will encounter expressions with a parameter: other variables will appear as coefficients. We find them using the quadratic expansion formula.

So, once you see rational fractions, the first thing to do is factor both the numerator and denominator into linear expressions, using the abbreviated multiplication or discriminant formulas.

Let's look at a couple of these rational expressions and try to factor them.

Solving more complex examples

Task No. 1

\[\frac(4((x)^(2))-6xy+9((y)^(2)))(2x-3y)\cdot \frac(9((y)^(2))- 4((x)^(2)))(8((x)^(3))+27((y)^(3)))\]

We rewrite and try to decompose each term:

Let's rewrite our entire rational expression taking into account these facts:

\[\frac(((\left(2x \right))^(2))-2x\cdot 3y+((\left(3y \right))^(2)))(2x-3y)\cdot \frac (((\left(3y \right))^(2))-((\left(2x \right))^(2)))(((\left(2x \right))^(3))+ ((\left(3y \right))^(3)))=\]

\[=\frac(((\left(2x \right))^(2))-2x\cdot 3y+((\left(3y \right))^(2)))(2x-3y)\cdot \ frac(\left(3y-2x \right)\left(3y+2x \right))(\left(2x+3y \right)\left(((\left(2x \right))^(2))- 2x\cdot 3y+((\left(3y \right))^(2)) \right))=-1\]

Answer: $-1$.

Problem No. 2

\[\frac(3-6x)(2((x)^(2))+4x+8)\cdot \frac(2x+1)(((x)^(2))+4-4x)\ cdot \frac(8-((x)^(3)))(4((x)^(2))-1)\]

Let's look at all the fractions.

\[((x)^(2))+4-4x=((x)^(2))-4x+2=((x)^(2))-2\cdot 2x+((2)^( 2))=((\left(x-2 \right))^(2))\]

Let's rewrite the entire structure taking into account the changes:

\[\frac(3\left(1-2x \right))(2\left(((x)^(2))+2x+((2)^(2)) \right))\cdot \frac( 2x+1)(((\left(x-2 \right))^(2)))\cdot \frac(\left(2-x \right)\left(((2)^(2))+ 2x+((x)^(2)) \right))(\left(2x-1 \right)\left(2x+1 \right))=\]

\[=\frac(3\cdot \left(-1 \right))(2\cdot \left(x-2 \right)\cdot \left(-1 \right))=\frac(3)(2 \left(x-2 \right))\]

Answer: $\frac(3)(2\left(x-2 \right))$.

Nuances of the solution

So what we just learned:

• Not every square trinomial can be factorized; in particular, this applies to the incomplete square of the sum or difference, which are very often found as parts of sum or difference cubes.
• Constants, i.e. ordinary numbers that do not have variables can also act as active elements in the expansion process. Firstly, they can be taken out of brackets, and secondly, the constants themselves can be represented in the form of powers.
• Very often, after factoring all the elements, opposite constructions arise. These fractions must be reduced extremely carefully, because when crossing them out either above or below, an additional factor $-1$ appears - this is precisely a consequence of the fact that they are opposites.

Solving complex problems

\[\frac(27((a)^(3))-64((b)^(3)))(((b)^(2))-4):\frac(9((a)^ (2))+12ab+16((b)^(2)))(((b)^(2))+4b+4)\]

Let's consider each term separately.

First fraction:

\[((\left(3a \right))^(3))-((\left(4b \right))^(3))=\left(3a-4b \right)\left(((\left (3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2)) \right)\]

\[((b)^(2))-((2)^(2))=\left(b-2 \right)\left(b+2 \right)\]

We can rewrite the entire numerator of the second fraction as follows:

\[((\left(3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2))\]

Now let's look at the denominator:

\[((b)^(2))+4b+4=((b)^(2))+2\cdot 2b+((2)^(2))=((\left(b+2 \right ))^(2))\]

Let's rewrite the entire rational expression taking into account the above facts:

\[\frac(\left(3a-4b \right)\left(((\left(3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2 )) \right))(\left(b-2 \right)\left(b+2 \right))\cdot \frac(((\left(b+2 \right))^(2)))( ((\left(3a \right))^(2))+3a\cdot 4b+((\left(4b \right))^(2)))=\]

\[=\frac(\left(3a-4b \right)\left(b+2 \right))(\left(b-2 \right))\]

Answer: $\frac(\left(3a-4b \right)\left(b+2 \right))(\left(b-2 \right))$.

Nuances of the solution

As we have seen once again, incomplete squares of the sum or incomplete squares of the difference, which are often found in real rational expressions, however, do not be afraid of them, because after transforming each element they are almost always canceled. In addition, in no case should you be afraid of large constructions in the final answer - it is quite possible that this is not your mistake (especially if everything is factorized), but the author intended such an answer.

In conclusion, I would like to discuss one more complex example, which no longer directly relates to rational fractions, but it contains everything that awaits you on real tests and exams, namely: factorization, reduction to a common denominator, reduction of similar terms. This is exactly what we will do now.

Solving a complex problem of simplifying and transforming rational expressions

\[\left(\frac(x)(((x)^(2))+2x+4)+\frac(((x)^(2))+8)(((x)^(3) )-8)-\frac(1)(x-2) \right)\cdot \left(\frac(((x)^(2)))(((x)^(2))-4)- \frac(2)(2-x) \right)\]

First, let's look at and open the first bracket: in it we see three separate fractions with different denominators, so the first thing we need to do is bring all three fractions to a common denominator, and to do this, each of them should be factored:

\[((x)^(2))+2x+4=((x)^(2))+2\cdot x+((2)^(2))\]

\[((x)^(2))-8=((x)^(3))-((2)^(2))=\left(x-2 \right)\left(((x) ^(2))+2x+((2)^(2)) \right)\]

Let's rewrite our entire construction as follows:

\[\frac(x)(((x)^(2))+2x+((2)^(2)))+\frac(((x)^(2))+8)(\left(x -2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))-\frac(1)(x-2)=\]

\[=\frac(x\left(x-2 \right)+((x)^(3))+8-\left(((x)^(2))+2x+((2)^(2 )) \right))(\left(x-2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))=\]

\[=\frac(((x)^(2))-2x+((x)^(2))+8-((x)^(2))-2x-4)(\left(x-2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))=\frac(((x)^(2))-4x-4)(\ left(x-2 \right)\left(((x)^(2))+2x+((2)^(2)) \right))=\]

\[=\frac(((\left(x-2 \right))^(2)))(\left(x-2 \right)\left(((x)^(2))+2x+(( 2)^(2)) \right))=\frac(x-2)(((x)^(2))+2x+4)\]

This is the result of the calculations from the first bracket.

Let's deal with the second bracket:

\[((x)^(2))-4=((x)^(2))-((2)^(2))=\left(x-2 \right)\left(x+2 \ right)\]

Let's rewrite the second bracket taking into account the changes:

\[\frac(((x)^(2)))(\left(x-2 \right)\left(x+2 \right))+\frac(2)(x-2)=\frac( ((x)^(2))+2\left(x+2 \right))(\left(x-2 \right)\left(x+2 \right))=\frac(((x)^ (2))+2x+4)(\left(x-2 \right)\left(x+2 \right))\]

Now let's write down the entire original construction:

\[\frac(x-2)(((x)^(2))+2x+4)\cdot \frac(((x)^(2))+2x+4)(\left(x-2 \right)\left(x+2 \right))=\frac(1)(x+2)\]

Answer: $\frac(1)(x+2)$.

Nuances of the solution

As you can see, the answer turned out to be quite reasonable. However, please note: very often during such large-scale calculations, when the only variable appears only in the denominator, students forget that this is the denominator and it should be at the bottom of the fraction and write this expression in the numerator - this is a gross mistake.

In addition, I would like to draw your special attention to how such tasks are formalized. In any complex calculations, all steps are performed one by one: first we count the first bracket separately, then the second one separately, and only at the end do we combine all the parts and calculate the result. In this way, we insure ourselves against stupid mistakes, carefully write down all the calculations and at the same time do not waste any extra time, as it might seem at first glance.

From the algebra course school curriculum Let's get down to specifics. In this article we will study in detail special kind rational expressions – rational fractions, and also consider what characteristic identical conversions of rational fractions take place.

Let us immediately note that rational fractions in the sense in which we define them below are called algebraic fractions in some algebra textbooks. That is, in this article we will understand rational and algebraic fractions as the same thing.

As usual, let's start with a definition and examples. Next we’ll talk about bringing a rational fraction to a new denominator and changing the signs of the members of the fraction. After this, we will look at how to reduce fractions. Finally, let's look at representing a rational fraction as a sum of several fractions. We will provide all information with examples detailed descriptions decisions.

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Definition and examples of rational fractions

Rational fractions are studied in 8th grade algebra lessons. We will use the definition of a rational fraction, which is given in the algebra textbook for 8th grade by Yu. N. Makarychev et al.

IN this definition it is not specified whether the polynomials in the numerator and denominator of a rational fraction must be polynomials of the standard form or not. Therefore, we will assume that the notations for rational fractions can contain both standard and non-standard polynomials.

Here are a few examples of rational fractions. So, x/8 and - rational fractions. And fractions and do not fit the stated definition of a rational fraction, since in the first of them the numerator does not contain a polynomial, and in the second, both the numerator and the denominator contain expressions that are not polynomials.

Converting the numerator and denominator of a rational fraction

The numerator and denominator of any fraction are self-sufficient mathematical expressions; in the case of rational fractions, these are polynomials; in a particular case, monomials and numbers. Therefore, identical transformations can be carried out with the numerator and denominator of a rational fraction, as with any expression. In other words, the expression in the numerator of a rational fraction can be replaced by an identically equal expression, just like the denominator.

You can perform identical transformations in the numerator and denominator of a rational fraction. For example, in the numerator you can group and reduce similar terms, and in the denominator you can replace the product of several numbers with its value. And since the numerator and denominator of a rational fraction are polynomials, it is possible to perform transformations characteristic of polynomials with them, for example, reduction to a standard form or representation in the form of a product.

For clarity, let's consider solutions to several examples.

Example.

Convert rational fraction so that the numerator contains a polynomial of standard form, and the denominator contains the product of polynomials.

Solution.

Reducing rational fractions to a new denominator is mainly used in adding and subtracting rational fractions.

Changing signs in front of a fraction, as well as in its numerator and denominator

The main property of a fraction can be used to change the signs of the members of a fraction. Indeed, multiplying the numerator and denominator of a rational fraction by -1 is equivalent to changing their signs, and the result is a fraction identically equal to the given one. This transformation has to be used quite often when working with rational fractions.

Thus, if you simultaneously change the signs of the numerator and denominator of a fraction, you will get a fraction equal to the original one. This statement is answered by equality.

Let's give an example. A rational fraction can be replaced by an identically equal fraction with changed signs of the numerator and denominator of the form.

You can do one more thing with fractions: identity transformation, in which the sign of either the numerator or the denominator changes. Let us state the corresponding rule. If you replace the sign of a fraction together with the sign of the numerator or denominator, you get a fraction that is identically equal to the original one. The written statement corresponds to the equalities and .

Proving these equalities is not difficult. The proof is based on the properties of multiplication of numbers. Let's prove the first of them: . Using similar transformations, the equality is proved.

For example, a fraction can be replaced by the expression or.

To conclude this point, we present two more useful equalities and . That is, if you change the sign of only the numerator or only the denominator, the fraction will change its sign. For example, And .

The considered transformations, which allow changing the sign of the terms of a fraction, are often used when transforming fractional rational expressions.

Reducing rational fractions

The following transformation of rational fractions, called reduction of rational fractions, is based on the same basic property of a fraction. This transformation corresponds to the equality , where a, b and c are some polynomials, and b and c are non-zero.

From the above equality it becomes clear that reducing a rational fraction implies getting rid of the common factor in its numerator and denominator.

Example.

Cancel a rational fraction.

Solution.

The common factor 2 is immediately visible, let’s perform a reduction by it (when writing, it is convenient to cross out the common factors that are being reduced by). We have . Since x 2 =x x and y 7 =y 3 y 4 (see if necessary), it is clear that x is a common factor of the numerator and denominator of the resulting fraction, as is y 3. Let's reduce by these factors: . This completes the reduction.

Above we carried out the reduction of rational fractions sequentially. Or it was possible to perform the reduction in one step, immediately reducing the fraction by 2 x y 3. In this case, the solution would look like this: .

Answer:

.

When reducing rational fractions, the main problem is that the common factor of the numerator and denominator is not always visible. Moreover, it does not always exist. In order to find a common factor or verify its absence, you need to factor the numerator and denominator of a rational fraction. If there is no common factor, then the original rational fraction does not need to be reduced, otherwise, reduction is carried out.

Various nuances can arise in the process of reducing rational fractions. The main subtleties are discussed in the article reducing algebraic fractions using examples and in detail.

Concluding the conversation about the reduction of rational fractions, we note that this transformation is identical, and the main difficulty in its implementation lies in factoring the polynomials in the numerator and denominator.

Representation of a rational fraction as a sum of fractions

Quite specific, but in some cases very useful, is the transformation of a rational fraction, which consists in its representation as the sum of several fractions, or the sum of an entire expression and a fraction.

A rational fraction, the numerator of which contains a polynomial representing the sum of several monomials, can always be written as a sum of fractions with the same denominators, the numerators of which contain the corresponding monomials. For example, . This representation is explained by the rule for adding and subtracting algebraic fractions with like denominators.

In general, any rational fraction can be represented as a sum of fractions in many different ways. For example, the fraction a/b can be represented as the sum of two fractions - an arbitrary fraction c/d and a fraction equal to the difference between the fractions a/b and c/d. This statement is true, since the equality holds . For example, a rational fraction can be represented as a sum of fractions different ways: Let's imagine the original fraction as the sum of an integer expression and a fraction. By dividing the numerator by the denominator with a column, we get the equality . The value of the expression n 3 +4 for any integer n is an integer. And the value of a fraction is an integer if and only if its denominator is 1, −1, 3, or −3. These values ​​correspond to the values ​​n=3, n=1, n=5 and n=−1, respectively.

Answer:

−1 , 1 , 3 , 5 .

Bibliography.

• Algebra: textbook for 8th grade. general education institutions / [Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova]; edited by S. A. Telyakovsky. - 16th ed. - M.: Education, 2008. - 271 p. : ill. - ISBN 978-5-09-019243-9.
• Mordkovich A. G. Algebra. 7th grade. At 2 p.m. Part 1. Textbook for students educational institutions/ A. G. Mordkovich. - 13th ed., rev. - M.: Mnemosyne, 2009. - 160 pp.: ill. ISBN 978-5-346-01198-9.
• Mordkovich A. G. Algebra. 8th grade. In 2 hours. Part 1. Textbook for students of general education institutions / A. G. Mordkovich. - 11th ed., erased. - M.: Mnemosyne, 2009. - 215 p.: ill. ISBN 978-5-346-01155-2.
• Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Now that we have learned how to add and multiply individual fractions, we can look at more complex structures. For example, what if the same problem involves adding, subtracting, and multiplying fractions?

First of all, you need to convert all fractions to improper ones. Then we perform the required actions sequentially - in the same order as for ordinary numbers. Namely:

1. Exponentiation is done first - get rid of all expressions containing exponents;
2. Then - division and multiplication;
3. The last step is addition and subtraction.

Of course, if there are parentheses in the expression, the order of operations changes - everything that is inside the parentheses must be counted first. And remember about improper fractions: you need to highlight the whole part only when all other actions have already been completed.

Let's convert all the fractions from the first expression to improper ones, and then perform the following steps:

Now let's find the value of the second expression. Here fractions with whole part no, but there are parentheses, so we do the addition first, and only then the division. Note that 14 = 7 · 2. Then:

Finally, consider the third example. There are brackets and a degree here - it is better to count them separately. Considering that 9 = 3 3, we have:

Pay attention to the last example. To raise a fraction to a power, you must separately raise the numerator to this power, and separately, the denominator.

You can decide differently. If we recall the definition of a degree, the problem will be reduced to the usual multiplication of fractions:

Multistory fractions

So far we have considered only “pure” fractions, when the numerator and denominator are ordinary numbers. This is quite consistent with the definition of a number fraction given in the very first lesson.

But what if you put a more complex object in the numerator or denominator? For example, another numerical fraction? Such constructions arise quite often, especially when working with long expressions. Here are a couple of examples:

There is only one rule for working with multi-level fractions: you must get rid of them immediately. Removing “extra” floors is quite simple, if you remember that the slash means the standard division operation. Therefore, any fraction can be rewritten as follows:

Using this fact and following the procedure, we can easily reduce any multi-story fraction to an ordinary one. Take a look at the examples:

Task. Convert multistory fractions to ordinary ones:

In each case, we rewrite the main fraction, replacing the dividing line with a division sign. Also remember that any integer can be represented as a fraction with a denominator of 1. That is 12 = 12/1; 3 = 3/1. We get:

In the last example, the fractions were canceled before the final multiplication.

Specifics of working with multi-level fractions

There is one subtlety in multi-level fractions that must always be remembered, otherwise you can get the wrong answer, even if all the calculations were correct. Take a look:

1. The numerator contains the single number 7, and the denominator contains the fraction 12/5;
2. The numerator contains the fraction 7/12, and the denominator contains the separate number 5.

So, for one recording we got two completely different interpretations. If you count, the answers will also be different:

To ensure that the record is always read unambiguously, use a simple rule: the dividing line of the main fraction must be longer than the line of the nested fraction. Preferably several times.

If you follow this rule, then the above fractions should be written as follows:

Yes, it may be unsightly and takes up too much space. But you will count correctly. Finally, a couple of examples where multi-story fractions actually arise:

Task. Find the meanings of the expressions:

So, let's work with the first example. Let's convert all fractions to improper ones, and then perform addition and division operations:

Let's do the same with the second example. Let's convert all fractions to improper ones and perform the required operations. In order not to bore the reader, I will omit some obvious calculations. We have:

Due to the fact that the numerator and denominator of the basic fractions contain sums, the rule for writing multi-story fractions is observed automatically. Also, in the last example, we intentionally left 46/1 in fraction form to perform division.

I will also note that in both examples the fraction bar actually replaces the parentheses: first of all, we found the sum, and only then the quotient.

Some will say that the transition to improper fractions in the second example was clearly redundant. Perhaps this is true. But by doing this we insure ourselves against mistakes, because next time the example may turn out to be much more complicated. Choose for yourself what is more important: speed or reliability.

At the VIII type school, students become familiar with the following transformations of fractions: expressing fractions in larger fractions (6th grade), expression improper fraction whole or mixed numbers (6th grade), expressing fractions in equal parts (7th grade), expressing a mixed number with an improper fraction (7th grade).

Expressing an improper fraction with a whole or mixed number

The study of this material should begin with the task: take 2 equal circles and divide each of them into 4 equal shares, count the number of fourth shares (Fig. 25). Next, it is proposed to write this amount as a fraction. Then the fourth beats are

They are placed next to each other and the students are convinced that they have formed a whole circle. Therefore, to four quarters it adds -

sequentially again and the students write down:

The teacher draws the students' attention to the fact that in all the cases considered, they took an improper fraction, and as a result of the transformation they received either a whole or a mixed number, i.e., they expressed the improper fraction as a whole or mixed number. Next, we must strive to ensure that students independently determine what arithmetic operation this transformation can be performed. Vivid examples leading to the answer to the question are: Conclusion: to

To express an improper fraction as a whole or mixed number, you need to divide the numerator of the fraction by the denominator, write the quotient as an integer, write the remainder in the numerator, and leave the denominator the same. Since the rule is cumbersome, it is not at all necessary that students learn it by heart. They must be able to consistently communicate the steps involved in performing a given transformation.

Before introducing students to expressing an improper fraction with a whole or mixed number, it is advisable to review with them the division of a whole number by an integer with a remainder.

The consolidation of a new transformation for students is facilitated by solving problems of a practical nature, for example:

“There are nine quarters of an orange in a vase. How many whole oranges can be made from these parts? How many quarters will be left?”

Expressing whole and mixed numbers as improper fractions

Introducing students to this new transformation should be preceded by solving problems, for example:

“2 pieces of fabric of equal length, shaped like a square, were cut into 4 equal parts. A scarf was sewn from each such part. How many scarves did you get? .

Next, the teacher asks the students to complete the following task: “Take a whole circle and another half of a circle equal in size to the first one. Cut the whole circle in half. How many halves were there? Write down: it was a circle, it became a circle.

Thus, based on a visual and practical basis, we consider a number of more examples. In the examples under consideration, students are asked to compare the original number (mixed or integer) and the number that was obtained after transformation (an improper fraction).

To introduce students to the rule of expressing a whole number and a mixed number as an improper fraction, you need to draw their attention to comparing the denominators of the mixed number and the improper fraction, as well as how the numerator is obtained, for example:

will be 15/4. As a result, a rule is formulated: in order to express a mixed number as an improper fraction, you need to multiply the denominator by an integer, add the numerator to the product and write the sum as the numerator, leaving the denominator unchanged.

First, you need to train students in expressing unity as an improper fraction, then any other whole number indicating the denominator, and only then a mixed number -

Basic property of fraction 1

The concept of the immutability of a fraction while simultaneously increasing or decreasing its members, i.e., the numerator and denominator, is learned with great difficulty by students of the VIII type school. This concept must be introduced through visual and didactic material, and it is important that students not only observe the teacher’s activities, but also actively work with them. didactic material and based on observations and practical activities came to certain conclusions and generalizations.

For example, the teacher takes a whole turnip, divides it into 2 equal parts and asks: “What did you get when you divided a whole turnip in half? (2 halves.) Show the turnips. Cut (divide) half of the turnip into 2 more equal parts. What will we get? Let's write: Let's compare the numerators and denominators of these fractions. At what time

times did the numerator increase? How many times has the denominator increased? How many times have both the numerator and denominator increased? Has the fraction changed? Why hasn't it changed? How did the shares become: larger or smaller? Has the number of shares increased or decreased?

Then all students divide the circle into 2 equal parts, each half is divided into 2 more equal parts, each quarter into 2 more equal parts, etc. and write down: etc. Then

establish how many times the numerator and denominator of the fraction have increased, and whether the fraction has changed. Then draw a segment and divide it successively by 3, 6, 12 equal parts and write down:

When comparing fractions it turns out that

The numerator and denominator of the fraction are increased by the same number of times, but the fraction does not change.

After considering a number of examples, students should be asked to answer the question: “Will the fraction change if the numerator

Some knowledge on the topic "Ordinary fractions" is excluded from curricula in mathematics in correctional schools VIII type, but they are communicated to students in schools for children with mental retardation, in equalization classes for children who have difficulties in learning mathematics. In this textbook, paragraphs that provide methods for studying this material are indicated with an asterisk (*).

and multiply the denominator of the fraction by the same number (increase by the same number of times)?” In addition, you should ask students to give examples themselves.

Similar examples are given when considering decreasing the numerator and denominator by the same number of times (the numerator and denominator are divided by the same number). For example, a circle is divided into 8 equal parts, 4 eighths of the circle are taken,

Having enlarged the shares, they take the fourth ones, there will be 2 of them. Having enlarged the shares, they take the second ones. They will be compared sequentially

numerators and denominators of these fractions, answering the questions: “How many times do the numerator and denominator decrease? Will the fraction change?*.

A good guide is stripes divided into 12, 6, 3 equal parts (Fig. 26).

Based on the examples considered, students can conclude: the fraction will not change if the numerator and denominator of the fraction are divided by the same number (reduced by the same number of times). Then a generalized conclusion is given - the main property of a fraction: the fraction will not change if the numerator and denominator of the fraction are increased or decreased by the same number of times.

Reducing Fractions

It is first necessary to prepare students for this conversion of fractions. As you know, to reduce a fraction means dividing the numerator and denominator of the fraction by the same number. But the divisor must be a number that gives the answer an irreducible fraction.

A month to a month and a half before students are introduced to reducing fractions, preparatory work is carried out - they are asked to name two answers from the multiplication table that are divisible by the same number. For example: “Name two numbers that are divisible by 4.” (First, students look at 1 in the table, and then name these numbers from memory.) They name both the numbers and the results of dividing them by 4. Then the teacher offers students for fractions, 3

for example, select a divisor for the numerator and denominator (the basis for performing such an action is the multiplication table).

what table should I look at? What number can 5 and 15 be divided by?) It turns out that when the numerator and denominator of a fraction are divided by the same number, the size of the fraction has not changed (this can be shown on a strip, a segment, a circle), only the fractions have become larger: The type of fraction has become simpler . Students are led to the conclusion of the rules for reducing fractions.

Type VIII school students often find it difficult to choose greatest number, which divides both the numerator and denominator of the fraction. Therefore, errors of such a nature as 4/12 = 2/6 are often observed, i.e. the student did not find the greatest common

divisor for numbers 4 and 12. Therefore, at first you can allow gradual division, i.e., but at the same time ask by what number the numerator and denominator of the fraction were divided first, by what number then and then by what number the numerator and denominator could be immediately divided fractions Such questions help students gradually find the greatest common divisor numerator and denominator of the fraction.

Bringing fractions to lowest common denominator*

Reducing fractions to the lowest common denominator should not be viewed as an end in itself, but as a transformation necessary to compare fractions and then to perform the operations of adding and subtracting fractions with different denominators.

Students are already familiar with comparing fractions with the same numerators but different denominators and with the same denominators but different numerators. However, they do not yet know how to compare fractions with different numerators and different denominators.

Before explaining to students the meaning of the new transformation, it is necessary to repeat the material covered by completing, for example, the following tasks:

Compare fractions 2/5,2/7,2/3 Say the rule for comparing fractions with

identical numerators.

Compare fractions Say the rule for comparing fractions

with the same denominators.

Compare fractions It is difficult for students to compare fractions

are different because they have different numerators and different denominators. To compare these fractions, you need to make the numerators or denominators of these fractions equal. Usually the denominators are expressed in equal fractions, that is, they reduce the fractions to the lowest common denominator.

Students should be introduced to the way of expressing fractions in equal parts.

First, fractions with different denominators are considered, but those in which the denominator of one fraction is divisible without a remainder by the denominator of another fraction and, therefore, can also be the denominator of another fraction.

For example, in fractions the denominators are the numbers 8 and 2.

To express these fractions in equal parts, the teacher suggests multiplying the smaller denominator sequentially by the numbers 2, 3, 4, etc. and do this until you get a result equal to the denominator of the first fraction. For example, multiply 2 by 2 and get 4. The denominators of the two fractions are again different. Next, we multiply 2 by 3, we get 6. The number 6 is also not suitable. We multiply 2 by 4, we get 8. In this case, the denominators are the same. In order for the fraction not to change, the numerator of the fraction must also be multiplied by 4 (based on the basic property of the fraction). Let's get a fraction Now the fractions are expressed in equal fractions. Their

It’s easy to compare and perform actions with them.

You can find the number by which you need to multiply the smaller denominator of one of the fractions by dividing the larger denominator by the smaller one. For example, if you divide 8 by 2, you get the number 4. You need to multiply both the denominator and the numerator of the fraction by this number. This means that in order to express several fractions in equal parts, you need to divide the larger denominator by the smaller one, multiply the quotient by the denominator and numerator of the fraction with smaller denominators. For example, fractions are given. To bring these fractions

to the lowest common denominator, you need 12:6=2, 2x6=12, 306

2x1=2. The fraction will take the form . Then 12:3=4, 4x3=12, 4x2=8. The fraction will take the form Therefore, the fractions will take the form accordingly, i.e. they will be expressed

nymi in equal shares.

Exercises are conducted that allow you to develop the skills of reducing fractions to a common lowest denominator.

For example, you need to express it in equal parts of the fraction

So that students do not forget the quotient that is obtained from dividing a larger denominator by a smaller one, it is advisable.

write over a fraction with a smaller denominator. For example, and

Then we consider fractions in which the larger denominator is not divisible by the smaller and, therefore, is not

common to these fractions. For example, Denominator 8 is not

divisible by 6. In this case, the larger denominator 8 will be sequentially multiplied by the numbers number series, starting from 2, until we get a number that is divisible without a remainder by both denominators 8 and 6. In order for the fractions to remain equal to the data, the numerators must be multiplied accordingly by the same numbers. On the-

3 5 example, so that the fractions tg and * are expressed in equal proportions,

the larger denominator of 8 is multiplied by 2(8x2=16). 16 is not divisible by 6, which means we multiply 8 by the next number 3 (8x3=24). 24 is divisible by 6 and 8, which means 24 is the common denominator for these fractions. But in order for the fractions to remain equal, their numerators must be increased by the same number of times as the denominators are increased, 8 is increased by 3 times, which means that the numerator of this fraction 3 will be increased by 3 times.

The fraction will take the form Denominator 6 increased by 4 times. Accordingly, the numerator of the 5th fraction must be increased 4 times. The fractions will take the following form:

Thus, we bring students to general conclusion(rule) and introduce them to the algorithm for expressing fractions in equal parts. For example, given two fractions ¾ and 5/7

1. Find the lowest common denominator: 7x2=14, 7x3=21,
7x4=28. 28 is divisible by 4 and 7. 28 is the smallest common denominator
fraction holder

2. Find additional factors: 28:4=7,

3. Let's write them over fractions:

4. Multiply the numerators of fractions by additional factors:
3x7=21, 5x4=20.

We get fractions with the same denominators. This means

We have reduced the fractions to a common lowest denominator.

Experience shows that it is advisable to familiarize students with converting fractions before studying various arithmetic operations with fractions. For example, it is advisable to teach abbreviating fractions or replacing an improper fraction with a whole or mixed number before learning the addition and subtraction of fractions with like denominators, since the resulting sum or difference

You will have to do either one or both conversions.

It is best to study reducing a fraction to the lowest common denominator with students before the topic “Adding and subtracting fractions with different denominators,” and replacing a mixed number with an improper fraction before the topic “Multiplying and dividing fractions by whole numbers.”

Addition and subtraction ordinary fractions

1. Addition and subtraction of fractions with the same denominators.

A study conducted by Alysheva T.V. 1, indicates the advisability of using an analogy with addition and subtraction already known to students when studying the operations of addition and subtraction of ordinary fractions with the same denominators

numbers obtained as a result of measuring quantities, and study actions using the deductive method, i.e., “from the general to the specific.”

First, the addition and subtraction of numbers with the names of measures of value and length are repeated. For example, 8 rubles. 20 k. ± 4 r. 15 k. When performing oral addition and subtraction, you need to add (subtract) first rubles, and then kopecks.

3 m 45 cm ± 2 m 24 cm - meters are added (subtracted) first, and then centimeters.

When adding and subtracting fractions, consider general case: performing these actions with mixed numbers (the denominators are the same): In this case, you need to: “Add (subtract) the whole numbers, then the numerators, and the denominator remains the same.” This general rule applies to all cases of adding and subtracting fractions. Special cases are gradually introduced: adding a mixed number with a fraction, then a mixed number with a whole. After this, more difficult cases of subtraction are considered: 1) from a mixed number of a fraction: 2) from a mixed number of a whole:

After mastering these fairly simple cases of subtraction, students are introduced to more difficult cases where a transformation of the minuend is required: subtraction from one whole unit or from several units, for example:

In the first case, the unit must be represented as a fraction with a denominator equal to the denominator deductible. In the second case, we take one from a whole number and also write it in the form of an improper fraction with the denominator of the subtrahend, we get a mixed number in the minuend. Subtraction is performed according to the general rule.

Finally, the most difficult case of subtraction is considered: from a mixed number, and the numerator of the fractional part is less than the numerator in the subtrahend. In this case, it is necessary to change the minuend so that the general rule can be applied, i.e., in the minuend, take one unit from the whole and split it

in fifths, we get and also, we get an example

will take the following form: you can already apply to its solution

general rule.

Usage deductive method learning to add and subtract fractions will contribute to the development of students’ ability to generalize, compare, differentiate, and include individual cases of calculations in common system knowledge of operations with fractions.

Decimal numbers such as 0.2; 1.05; 3.017, etc. as they are heard, so they are written. Zero point two, we get a fraction. One point five hundredths, we get a fraction. Three point seventeen thousandths, we get the fraction. The numbers before the decimal point are the whole part of the fraction. The number after the decimal point is the numerator of the future fraction. If after the decimal point single digit number- the denominator will be 10, if two-digit - 100, three-digit - 1000, etc. Some resulting fractions can be reduced. In our examples

Converting a fraction to a decimal

This is the reverse of the previous transformation. What is the characteristic of a decimal fraction? Its denominator is always 10, or 100, or 1000, or 10000, and so on. If your common fraction has a denominator like this, there's no problem. For example, or

If the fraction is, for example . In this case, it is necessary to use the basic property of a fraction and convert the denominator to 10 or 100, or 1000... In our example, if we multiply the numerator and denominator by 4, we get a fraction that can be written as a decimal number 0.12.

Some fractions are easier to divide than to convert the denominator. For example,

Some fractions cannot be converted to decimals!
For example,

Converting a mixed fraction to an improper fraction

A mixed fraction, for example, can be easily converted to an improper fraction. To do this, you need to multiply the whole part by the denominator (bottom) and add it with the numerator (top), leaving the denominator (bottom) unchanged. That is

When converting mixed fraction to the wrong one, you can remember that you can use addition of fractions

Converting an improper fraction to a mixed fraction (highlighting the whole part)

An improper fraction can be converted to a mixed fraction by highlighting the whole part. Let's look at an example. We determine how many integer times “3” fits into “23”. Or divide 23 by 3 on a calculator, the whole number to the decimal point is the desired one. This is "7". Next, we determine the numerator of the future fraction: we multiply the resulting “7” by the denominator “3” and subtract the result from the numerator “23”. How do we find the extra that remains from the numerator “23” if we remove maximum amount"3". We leave the denominator unchanged. Everything is done, write down the result