This article is dedicated to transformation of rational expressions, mostly fractionally rational, is one of the key issues in the 8th grade algebra course. First, we recall what type of expressions are called rational. Next we will focus on carrying out standard transformations with rational expressions, such as grouping terms, putting common factors out of brackets, bringing similar terms, etc. Finally, we will learn to represent fractional rational expressions as rational fractions.

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Definition and examples of rational expressions

Rational expressions are one of the types of expressions studied in algebra lessons at school. Let's give a definition.

Definition.

Expressions composed of numbers, variables, parentheses, powers with integer exponents, connected using arithmetic signs +, −, · and:, where division can be indicated by a fraction line, are called rational expressions.

Here are some examples of rational expressions: .

Rational expressions begin to be studied purposefully in the 7th grade. Moreover, in the 7th grade one learns the basics of working with the so-called whole rational expressions, that is, with rational expressions that do not contain division into expressions with variables. To do this, monomials and polynomials are sequentially studied, as well as the principles of performing actions with them. All this knowledge ultimately allows you to perform transformations of entire expressions.

In grade 8, they move on to studying rational expressions containing division by an expression with variables called fractional rational expressions. In this case, special attention is paid to the so-called rational fractions(they are also called algebraic fractions), that is, fractions whose numerator and denominator contain polynomials. This ultimately makes it possible to convert rational fractions.

The acquired skills allow you to move on to transforming rational expressions of any form. This is explained by the fact that any rational expression can be considered as an expression composed of rational fractions and integer expressions connected by signs of arithmetic operations. And we already know how to work with whole expressions and algebraic fractions.

Main types of transformations of rational expressions

With rational expressions, you can carry out any of the basic identity transformations, be it grouping terms or factors, bringing similar terms, performing operations with numbers, etc. Typically the purpose of performing these transformations is simplification of rational expression.

Example.

.

Solution.

It is clear that this rational expression is the difference between two expressions and , and these expressions are similar, since they have the same letter part. Thus, we can perform a reduction of similar terms:

Answer:

.

It is clear that when carrying out transformations with rational expressions, as well as with any other expressions, you need to remain within the accepted order of performing actions.

Example.

Perform a rational expression transformation.

Solution.

We know that the actions in parentheses are executed first. Therefore, first of all, we transform the expression in brackets: 3·x−x=2·x.

Now you can substitute the obtained result into the original rational expression: . So we came to an expression containing the actions of one stage - addition and multiplication.

Let's get rid of the parentheses at the end of the expression by applying the property of division by a product: .

Finally, we can group numeric factors and factors with the variable x, then perform the appropriate operations on the numbers and apply : .

This completes the transformation of the rational expression, and as a result we get a monomial.

Answer:

Example.

Convert rational expression .

Solution.

First we transform the numerator and denominator. This order of transformation of fractions is explained by the fact that the line of a fraction is essentially another designation for division, and the original rational expression is essentially a quotient of the form , and the actions in parentheses are performed first.

So, in the numerator we perform operations with polynomials, first multiplication, then subtraction, and in the denominator we group the numerical factors and calculate their product: .

Let's also imagine the numerator and denominator of the resulting fraction in the form of a product: suddenly it is possible to reduce an algebraic fraction. To do this, we will use in the numerator difference of squares formula, and in the denominator we take the two out of brackets, we have .

Answer:

.

So, the initial acquaintance with the transformation of rational expressions can be considered completed. Let's move on, so to speak, to the sweetest part.

Rational fraction representation

Most often, the ultimate goal of transforming expressions is to simplify their appearance. In this light, the simplest form to which a fractional rational expression can be converted is a rational (algebraic) fraction, and in the particular case a polynomial, monomial or number.

Is it possible to represent any rational expression as a rational fraction? The answer is yes. Let us explain why this is so.

As we have already said, every rational expression can be considered as polynomials and rational fractions connected by plus, minus, multiply and divide signs. All corresponding operations with polynomials yield a polynomial or rational fraction. In turn, any polynomial can be converted into an algebraic fraction by writing it with the denominator 1. And adding, subtracting, multiplying and dividing rational fractions results in a new rational fraction. Therefore, after performing all the operations with polynomials and rational fractions in a rational expression, we get a rational fraction.

Example.

Express as a rational fraction the expression .

Solution.

The original rational expression is the difference between a fraction and the product of fractions of the form . According to the order of operations, we must first perform multiplication, and only then addition.

We start with multiplying algebraic fractions:

We substitute the obtained result into the original rational expression: .

We came to the subtraction of algebraic fractions with different denominators:

So, having performed operations with rational fractions that make up the original rational expression, we presented it in the form of a rational fraction.

Answer:

.

To consolidate the material, we will analyze the solution to another example.

Example.

Express a rational expression as a rational fraction.

In the previous lesson, the concept of a rational expression was already introduced; in today's lesson we continue to work with rational expressions and focus on their transformations. Using specific examples, we will consider methods for solving problems involving transformations of rational expressions and proving the identities associated with them.

Subject:Algebraic fractions. Arithmetic operations on algebraic fractions

Lesson:Converting rational expressions

Let us first recall the definition of a rational expression.

Definition.Rationalexpression- 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.

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 recall 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 that in the numerator 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 us indicate the lowest common denominator of the fractions, it is equal to: , since it is divided by the denominator of the third fraction, and the first expression is generally an integer, and any denominator is suitable for it. Having indicated the obvious additional factors, we write:

Answer.

Let's consider a more complex example with “multi-story” fractions.

Example 4. Prove the identity for all admissible values ​​of the variable.

Proof. To prove this identity, we will try to simplify its left side (complex) to the simple form that is required of us. To do this, we will perform all the operations with fractions in the numerator and denominator, and then divide the fractions and simplify the result.

Proven for all permissible values ​​of the variable.

Proven.

In the next lesson we will look in detail at more complex examples of converting rational expressions.

Bibliography

1. Bashmakov M.I. Algebra 8th grade. - M.: Education, 2004.

2. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. 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.

2. Lesson developments, presentations, lesson notes ().

Homework

1. No. 96-101. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 8. - 5th ed. - M.: Education, 2010.

2. Simplify the expression .

3. Simplify the expression.

4. Prove the identity.

It would seem that converting a decimal fraction into a regular fraction is an elementary topic, but many students do not understand it! Therefore, today we will take a detailed look at several algorithms at once, with the help of which you will understand any fractions in just a second.

Let me remind you that there are at least two forms of writing the same fraction: common and decimal. Decimal fractions are all kinds of constructions of the form 0.75; 1.33; and even −7.41. Here are examples of ordinary fractions that express the same numbers:

Now let's figure it out: how to move from decimal notation to regular notation? And most importantly: how to do this as quickly as possible?

Basic algorithm

In fact, there are at least two algorithms. And we'll look at both now. Let's start with the first one - the simplest and most understandable.

To convert a decimal to a fraction, you need to follow three steps:

An important note about negative numbers. If in the original example there is a minus sign in front of the decimal fraction, then in the output there should also be a minus sign in front of the common fraction. Here are some more examples:

Examples of transition from decimal notation of fractions to ordinary ones

I would like to pay special attention to the last example. As you can see, the fraction 0.0025 contains many zeros after the decimal point. Because of this, you have to multiply the numerator and denominator by 10 as many as four times. Is it possible to somehow simplify the algorithm in this case?

Of course you can. And now we will look at an alternative algorithm - it is a little more difficult to understand, but after a little practice it works much faster than the standard one.

Faster way

This algorithm also has 3 steps. To get a fraction from a decimal, do the following:

1. Count how many digits are after the decimal point. For example, the fraction 1.75 has two such digits, and 0.0025 has four. Let's denote this quantity by the letter $n$.
2. Rewrite the original number as a fraction of the form $\frac(a)(((10)^(n)))$, where $a$ are all the digits of the original fraction (without the “starting” zeros on the left, if any), and $n$ is the same number of digits after the decimal point that we calculated in the first step. In other words, you need to divide the digits of the original fraction by one followed by $n$ zeros.
3. If possible, reduce the resulting fraction.

That's all! At first glance, this scheme is more complicated than the previous one. But in fact it is both simpler and faster. Judge for yourself:

As you can see, in the fraction 0.64 there are two digits after the decimal point - 6 and 4. Therefore $n=2$. If we remove the comma and zeros on the left (in this case, just one zero), we get the number 64. Let’s move on to the second step: $((10)^(n))=((10)^(2))=100$, Therefore, the denominator is exactly one hundred. Well, then all that remains is to reduce the numerator and denominator. :)

One more example:

Here everything is a little more complicated. Firstly, there are already 3 numbers after the decimal point, i.e. $n=3$, so you have to divide by $((10)^(n))=((10)^(3))=1000$. Secondly, if we remove the comma from the decimal notation, we get this: 0.004 → 0004. Remember that the zeros on the left must be removed, so in fact we have the number 4. Then everything is simple: divide, reduce and get the answer.

Finally, the last example:

The peculiarity of this fraction is the presence of a whole part. Therefore, the output we get is an improper fraction of 47/25. You can, of course, try to divide 47 by 25 with a remainder and thus again isolate the whole part. But why complicate your life if this can be done at the stage of transformation? Well, let's figure it out.

What to do with the whole part

In fact, everything is very simple: if we want to get a proper fraction, then we need to remove the whole part from it during the transformation, and then, when we get the result, add it again to the right before the fraction line.

For example, consider the same number: 1.88. Let's score by one (the whole part) and look at the fraction 0.88. It can be easily converted:

Then we remember about the “lost” unit and add it to the front:

\[\frac(22)(25)\to 1\frac(22)(25)\]

That's all! The answer turned out to be the same as after selecting the whole part last time. A couple more examples:

\[\begin(align)& 2.15\to 0.15=\frac(15)(100)=\frac(3)(20)\to 2\frac(3)(20); \\& 13.8\to 0.8=\frac(8)(10)=\frac(4)(5)\to 13\frac(4)(5). \\\end(align)\]

This is the beauty of mathematics: no matter which way you go, if all the calculations are done correctly, the answer will always be the same. :)

In conclusion, I would like to consider one more technique that helps many.

Transformations “by ear”

Let's think about what a decimal even is. More precisely, how we read it. For example, the number 0.64 - we read it as "zero point 64 hundredths", right? Well, or just “64 hundredths”. The key word here is “hundredths”, i.e. number 100.

What about 0.004? This is “zero point 4 thousandths” or simply “four thousandths”. One way or another, the key word is “thousands”, i.e. 1000.

So what's the big deal? And the fact is that it is these numbers that ultimately “pop up” in the denominators at the second stage of the algorithm. Those. 0.004 is “four thousandths” or “4 divided by 1000”:

Try to practice yourself - it's very simple. The main thing is to read the original fraction correctly. For example, 2.5 is “2 whole, 5 tenths”, so

And some 1.125 is “1 whole, 125 thousandths”, so

In the last example, of course, someone will object that it is not obvious to every student that 1000 is divisible by 125. But here you need to remember that 1000 = 10 3, and 10 = 2 ∙ 5, therefore

\[\begin(align)& 1000=10\cdot 10\cdot 10=2\cdot 5\cdot 2\cdot 5\cdot 2\cdot 5= \\& =2\cdot 2\cdot 2\cdot 5\ cdot 5\cdot 5=8\cdot 125\end(align)\]

Thus, any power of ten can be decomposed only into factors 2 and 5 - it is these factors that need to be looked for in the numerator so that in the end everything is reduced.

This concludes the lesson. Let's move on to a more complex reverse operation - see "

Fractions

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

Fractions are not much of a nuisance in high school. For the time being. Until you come across powers with rational exponents and logarithms. And there... You press and press the calculator, and it shows a full display of some numbers. You have to think with your head like in the third grade.

Let's finally figure out fractions! Well, how much can you get confused in them!? Moreover, it’s all simple and logical. So, what are the types of fractions?

Types of fractions. Transformations.

There are three types of fractions.

1. Common fractions , For example:

Sometimes instead of a horizontal line they put a slash: 1/2, 3/4, 19/5, well, and so on. Here we will often use this spelling. The top number is called numerator, lower - denominator. If you constantly confuse these names (it happens...), say to yourself the phrase: " Zzzzz remember! Zzzzz denominator - look zzzzz uh!" Look, everything will be zzzz remembered.)

The dash, either horizontal or inclined, means division the top number (numerator) to the bottom (denominator). That's all! Instead of a dash, it is quite possible to put a division sign - two dots.

When complete division is possible, this must be done. So, instead of the fraction “32/8” it is much more pleasant to write the number “4”. Those. 32 is simply divided by 8.

32/8 = 32: 8 = 4

I'm not even talking about the fraction "4/1". Which is also just "4". And if it’s not completely divisible, we leave it as a fraction. Sometimes you have to do the opposite operation. Convert a whole number into a fraction. But more on that later.

2. Decimals , For example:

It is in this form that you will need to write down the answers to tasks “B”.

3. Mixed numbers , For example:

Mixed numbers are practically not used in high school. In order to work with them, they must be converted into ordinary fractions. But you definitely need to be able to do this! Otherwise you will come across such a number in a problem and freeze... Out of nowhere. But we will remember this procedure! A little lower.

Most versatile common fractions. Let's start with them. By the way, if a fraction contains all sorts of logarithms, sines and other letters, this does not change anything. In the sense that everything actions with fractional expressions are no different from actions with ordinary fractions!

The main property of a fraction.

So, let's go! To begin with, I will surprise you. The whole variety of fraction transformations is provided by one single property! That's what it's called main property of a fraction. Remember: If the numerator and denominator of a fraction are multiplied (divided) by the same number, the fraction does not change. Those:

It is clear that you can continue to write until you are blue in the face. Don’t let sines and logarithms confuse you, we’ll deal with them further. The main thing is to understand that all these various expressions are the same fraction . 2/3.

Do we need it, all these transformations? And how! Now you will see for yourself. To begin with, let's use the basic property of a fraction for reducing fractions. It would seem like an elementary thing. Divide the numerator and denominator by the same number and that's it! It's impossible to make a mistake! But... man is a creative being. You can make a mistake anywhere! Especially if you have to reduce not a fraction like 5/10, but a fractional expression with all sorts of letters.

How to correctly and quickly reduce fractions without doing extra work can be read in the special Section 555.

A normal student doesn't bother dividing the numerator and denominator by the same number (or expression)! He simply crosses out everything that is the same above and below! This is where a typical mistake, a blunder, if you will, lurks.

For example, you need to simplify the expression:

There’s nothing to think about here, cross out the letter “a” on top and the “2” on the bottom! We get:

Everything is correct. But really you divided all numerator and all the denominator is "a". If you are used to just crossing out, then in a hurry you can cross out the “a” in the expression

and get it again

Which would be categorically untrue. Because here all the numerator on "a" is already not shared! This fraction cannot be reduced. By the way, such a reduction is, um... a serious challenge for the teacher. This is not forgiven! Do you remember? When reducing, you need to divide all numerator and all denominator!

Reducing fractions makes life a lot easier. You will get a fraction somewhere, for example 375/1000. How can I continue to work with her now? Without a calculator? Multiply, say, add, square!? And if you’re not too lazy, and carefully cut it down by five, and by another five, and even... while it’s being shortened, in short. Let's get 3/8! Much nicer, right?

The main property of a fraction allows you to convert ordinary fractions to decimals and vice versa without a calculator! This is important for the Unified State Exam, right?

How to convert fractions from one type to another.

With decimal fractions everything is simple. As it is heard, so it is written! Let's say 0.25. This is zero point twenty five hundredths. So we write: 25/100. We reduce (we divide the numerator and denominator by 25), we get the usual fraction: 1/4. All. It happens, and nothing is reduced. Like 0.3. This is three tenths, i.e. 3/10.

What if the integers are not zero? It's OK. We write down the whole fraction without any commas in the numerator, and in the denominator - what is heard. For example: 3.17. This is three point seventeen hundredths. We write 317 in the numerator and 100 in the denominator. We get 317/100. Nothing is reduced, that means everything. This is the answer. Elementary Watson! From all that has been said, a useful conclusion: any decimal fraction can be converted to a common fraction .

But some people cannot do the reverse conversion from ordinary to decimal without a calculator. And it is necessary! How will you write down the answer on the Unified State Exam!? Read carefully and master this process.

What is the characteristic of a decimal fraction? Her denominator is Always costs 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, 4/10 = 0.4. Or 7/100 = 0.07. Or 12/10 = 1.2. What if the answer to the task in section “B” turned out to be 1/2? What will we write in response? Decimals are required...

Let's remember main property of a fraction ! Mathematics favorably allows you to multiply the numerator and denominator by the same number. Anything, by the way! Except zero, of course. So let’s use this property to our advantage! What can the denominator be multiplied by, i.e. 2 so that it becomes 10, or 100, or 1000 (smaller is better, of course...)? At 5, obviously. Feel free to multiply the denominator (this is us necessary) by 5. But then the numerator must also be multiplied by 5. This is already mathematics demands! We get 1/2 = 1x5/2x5 = 5/10 = 0.5. That's all.

However, all sorts of denominators come across. You will come across, for example, the fraction 3/16. Try and figure out what to multiply 16 by to make 100, or 1000... Doesn’t it work? Then you can simply divide 3 by 16. In the absence of a calculator, you will have to divide with a corner, on a piece of paper, as they taught in elementary school. We get 0.1875.

And there are also very bad denominators. For example, there is no way to turn the fraction 1/3 into a good decimal. Both on the calculator and on a piece of paper, we get 0.3333333... This means that 1/3 is an exact decimal fraction does not translate. Same as 1/7, 5/6 and so on. There are many of them, untranslatable. This brings us to another useful conclusion. Not every fraction can be converted to a decimal !

By the way, this is useful information for self-testing. In section "B" you must write down a decimal fraction in your answer. And you got, for example, 4/3. This fraction does not convert to a decimal. This means you made a mistake somewhere along the way! Go back and check the solution.

So, we figured out ordinary and decimal fractions. All that remains is to deal with mixed numbers. To work with them, they must be converted into ordinary fractions. How to do it? You can catch a sixth grader and ask him. But a sixth grader won’t always be at hand... You’ll have to do it yourself. It is not difficult. You need to multiply the denominator of the fractional part by the whole part and add the numerator of the fractional part. This will be the numerator of the common fraction. What about the denominator? The denominator will remain the same. It sounds complicated, but in reality everything is simple. Let's look at an example.

Suppose you were horrified to see the number in the problem:

Calmly, without panic, we think. The whole part is 1. Unit. The fractional part is 3/7. Therefore, the denominator of the fractional part is 7. This denominator will be the denominator of the ordinary fraction. We count the numerator. We multiply 7 by 1 (the integer part) and add 3 (the numerator of the fractional part). We get 10. This will be the numerator of a common fraction. That's all. It looks even simpler in mathematical notation:

Is it clear? Then secure your success! Convert to ordinary fractions. You should get 10/7, 7/2, 23/10 and 21/4.

The reverse operation - converting an improper fraction to a mixed number - is rarely required in high school. Well, if so... And if you are not in high school, you can look into the special Section 555. By the way, you will also learn about improper fractions there.

Well, that's practically all. You remembered the types of fractions and understood How transfer them from one type to another. The question remains: For what do it? Where and when to apply this deep knowledge?

I answer. Any example itself suggests the necessary actions. If in the example ordinary fractions, decimals, and even mixed numbers are mixed together, we convert everything into ordinary fractions. It can always be done. Well, if it says something like 0.8 + 0.3, then we count it that way, without any translation. Why do we need extra work? We choose the solution that is convenient us !

If the task is all decimal fractions, but um... some kind of evil ones, go to ordinary ones and try it! Look, everything will work out. For example, you will have to square the number 0.125. It’s not so easy if you haven’t gotten used to using a calculator! Not only do you have to multiply numbers in a column, you also have to think about where to insert the comma! It definitely won’t work in your head! What if we move on to an ordinary fraction?

0.125 = 125/1000. We reduce it by 5 (this is for starters). We get 25/200. Once again by 5. We get 5/40. Oh, it's still shrinking! Back to 5! We get 1/8. We can easily square it (in our minds!) and get 1/64. All!

Let's summarize this lesson.

1. There are three types of fractions. Common, decimal and mixed numbers.

2. Decimals and mixed numbers Always can be converted to ordinary fractions. Reverse transfer not always available.

3. The choice of the type of fractions to work with a task depends on the task itself. If there are different types of fractions in one task, the most reliable thing is to switch to ordinary fractions.

Now you can practice. First, convert these decimal fractions to ordinary fractions:

3,8; 0,75; 0,15; 1,4; 0,725; 0,012

You should get answers like this (in a mess!):

Let's wrap this up. In this lesson we refreshed our memory on key points about fractions. It happens, however, that there is nothing special to refresh...) If someone has completely forgotten, or has not yet mastered it... Then you can go to a special Section 555. All the basics are covered in detail there. Many suddenly understand everything are starting. And they solve fractions on the fly).

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

At the VIII type school, students are introduced to the following transformations of fractions: expressing fractions in larger fractions (6th grade), expressing improper fractions as a whole or mixed number (6th grade), expressing fractions in like fractions (7th grade), expressing a mixed number as 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 using visual and didactic material, and it is important that students not only observe the teacher’s activities, but also actively work with the didactic material and, based on observations and practical activities, come 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 sequentially into 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 mathematics curricula in Type VIII correctional schools, but they are taught 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 find the largest number that divides both the numerator and the denominator of a 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 Questions like this help students gradually find the greatest common factor of the numerator and denominator of a 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

is divided by 6. In this case, the larger denominator 8 will be sequentially multiplied by numbers in the number series, starting with 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 multiply by the same numbers accordingly. 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 a 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.”

Adding and subtracting common 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 of the subtrahend. 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.

Using the deductive method of teaching addition and subtraction of fractions will help students develop the ability to generalize, compare, differentiate, and include individual cases of calculations into the general system of knowledge about operations with fractions.