Goals:
formation of knowledge, skills, and skills in working with fractions;
development of memory, logical thinking, imagination, attention, speech, mathematical calculation skills;
nurturing a sense of responsibility, collectivism, mutual assistance, accuracy, independence, discipline, and observation.
Equipment: demonstration and handout models of fractions, blank circle, tangrams, problem diagrams, tables with fractions.
DURING THE CLASSES
I. Organizational moment.
II. Lesson topic message.
- The topic of our lesson... That's the problem. The topic has disappeared. Nobody has seen? You'll have to restore it. Let's solve the examples and write the answers in ascending order.
III. Verbal counting.
Arrange the examples in ascending order of answers and read the resulting word.
R 6300: 100: 7 x 9 = (81);
ABOUT 12000: 4000 x 7 x 10 = (210);
B 720: 90 x 10 x 8 = (640);
AND 90 x 30: 100 x 1000 = (27000);
D 16 x 100: 10:40 = (4).
The name of the topic appears on the board: “Fractions.”
IV. Setting a lesson goal
Sketch “Pinocchio at Malvina’s lesson.”
- What, guys, should we help Pinocchio?
V. Formation of knowledge, skills and abilities.
1) Division into shares.
Often in life we have to divide the whole into parts. Imagine that guests come to you, and you have 1 cake. What should I do? It must be divided equally. Take a “cake” model (circle) on the table.
The teacher shows and the children repeat.
Option 1 had 3 guests + the host. Divide into 4 parts. And for option II, 7 guests + the owner came. Divide into 8 parts. Cut along the fold line into pieces. We received the shares, but how to write this down? With the help of what kind of signs? We use letters to make sounds, numbers to write numbers, but how do we write beats? We will write the shares using fractions.
Fraction is one or more equal shares written using two natural numbers separated by a bar
Where m is the numerator and n is the denominator.
A note is posted on the board, and the children write it down in a notebook.
- Now let's write down the fractions.
- How many parts were they divided into? Write it below the line.
- How many of these parts did you take? We write above the line.
2) Writing fractions.
Ex. No. 1 p.78.
– How many equal parts is the figure divided into?
– How many parts are painted over?
– How many parts are unpainted?
– How to write using a fraction?
3) Coloring fractions.
Exercise No. 2 p. 79
– How many parts is the figure divided into?
– How much do you need to paint?
– What does this tell you? (Numerator and denominator)
4) Reading fractions.
Ex. No. 3 p. 79.
2/9,
4/5,
7/10,
11/24,
9/542,
37/9000.
– What does the numerator of a fraction indicate? (How many parts are taken.)
– What does the denominator of the fraction indicate? (How many parts did you divide into?)
5) Recording fractions using the "%" sign. Writing % using fractions.
6) Comparison of fractions.
Option 1: take 1/4 part;
Option 2: take 1/8 part;
– Who has more? What do we see?
Children compare in pairs using the overlapping method. Teacher on model
Conclusion: the larger the denominator, with the same numerator, the smaller the fraction; the smaller the denominator, with the same numerator, the larger the fraction.
VI. Competition in rows at the board.
Tables with fractions are posted on the board. Children are only asked to put a sign between a pair of fractions.
VII. Physical exercise.
7) Addition and subtraction of fractions.
– Take 3/8 and remove 1/8. How much is left? (2/8.)
– Take 1/4 and add 2/4, how much do you get? (3/4) .
Conclusion: With the same denominators, fractions are added and subtracted as natural numbers.
Tables with fractions are posted on the board. Children are only asked to write down the answer. Students come out from each row one by one and write down their answers. Examination.
VIII. Independent work in rows.
IX. Lesson summary.
– What new did you learn?
-What is a fraction?
-Which fraction is larger?
How do you add and subtract fractions?
– Today we received ratings of 20/4 and 20/5.
X. Additional material. Tangram.
– Determine how many parts of each color are in the drawing and make your own drawing.
Children are given cards on which a drawing is depicted using 8 multi-colored triangles, and 8 more multi-colored triangles are given separately so that the children can create their own drawing.
A CHALLENGE OF AWARENESS.
“A student came from school
And he says to mom and dad:
“We were given a task,
I solved it for an hour.
And it turned out in my answer
Two diggers and two thirds!”
– Did he solve the problem correctly? Why?
XI. Homework.
Create a problem with fractions.
This article is about common fractions. Here we will introduce the concept of a fraction of a whole, which will lead us to the definition of a common fraction. Next we will dwell on the accepted notation for ordinary fractions and give examples of fractions, let’s say about the numerator and denominator of a fraction. After this, we will give definitions of proper and improper, positive and negative fractions, and also consider the position of fractional numbers on the coordinate ray. In conclusion, we list the main operations with fractions.
Page navigation.
Shares of the whole
First we introduce concept of share.
Let's assume that we have some object made up of several absolutely identical (that is, equal) parts. For clarity, you can imagine, for example, an apple cut into several equal parts, or an orange consisting of several equal slices. Each of these equal parts that make up the whole object is called parts of the whole or simply shares.
Note that the shares are different. Let's explain this. Let us have two apples. Cut the first apple into two equal parts, and the second into 6 equal parts. It is clear that the share of the first apple will be different from the share of the second apple.
Depending on the number of shares that make up the whole object, these shares have their own names. Let's sort it out names of beats. If an object consists of two parts, any of them is called one second part of the whole object; if an object consists of three parts, then any of them is called one third part, and so on.
One second share has a special name - half. One third is called third, and one quarter part - a quarter.
For the sake of brevity, the following were introduced: beat symbols. One second share is designated as or 1/2, one third share is designated as or 1/3; one fourth share - like or 1/4, and so on. Note that the notation with a horizontal bar is used more often. To reinforce the material, let’s give one more example: the entry denotes one hundred and sixty-seventh part of the whole.
The concept of share naturally extends from objects to quantities. For example, one of the measures of length is the meter. To measure lengths shorter than a meter, fractions of a meter can be used. So you can use, for example, half a meter or a tenth or thousandth of a meter. The shares of other quantities are applied similarly.
Common fractions, definition and examples of fractions
To describe the number of shares we use common fractions. Let us give an example that will allow us to approach the definition of ordinary fractions.
Let the orange consist of 12 parts. Each share in this case represents one twelfth of a whole orange, that is, . We denote two beats as , three beats as , and so on, 12 beats we denote as . Each of the given entries is called an ordinary fraction.
Now let's give a general definition of common fractions.
The voiced definition of ordinary fractions allows us to give examples of common fractions: 5/10, , 21/1, 9/4, . And here are the records 
do not fit the stated definition of ordinary fractions, that is, they are not ordinary fractions.
Numerator and denominator
For convenience, ordinary fractions are distinguished numerator and denominator.
Definition.
Numerator ordinary fraction (m/n) is a natural number m.
Definition.
Denominator common fraction (m/n) is a natural number n.
So, the numerator is located above the fraction line (to the left of the slash), and the denominator is located below the fraction line (to the right of the slash). For example, let's take the common fraction 17/29, the numerator of this fraction is the number 17, and the denominator is the number 29.
It remains to discuss the meaning contained in the numerator and denominator of an ordinary fraction. The denominator of a fraction shows how many parts one object consists of, and the numerator, in turn, indicates the number of such shares. For example, the denominator 5 of the fraction 12/5 means that one object consists of five shares, and the numerator 12 means that 12 such shares are taken.
Natural number as a fraction with denominator 1
The denominator of a common fraction can be equal to one. In this case, we can consider that the object is indivisible, in other words, it represents something whole. The numerator of such a fraction indicates how many whole objects are taken. Thus, an ordinary fraction of the form m/1 has the meaning of a natural number m. This is how we substantiated the validity of the equality m/1=m.
Let's rewrite the last equality as follows: m=m/1. This equality allows us to represent any natural number m as an ordinary fraction. For example, the number 4 is the fraction 4/1, and the number 103,498 is equal to the fraction 103,498/1.
So, any natural number m can be represented as an ordinary fraction with a denominator of 1 as m/1, and any ordinary fraction of the form m/1 can be replaced by a natural number m.
Fraction bar as a division sign
Representing the original object in the form of n shares is nothing more than division into n equal parts. After an item is divided into n shares, we can divide it equally among n people - each will receive one share.
If we initially have m identical objects, each of which is divided into n shares, then we can equally divide these m objects between n people, giving each person one share from each of the m objects. In this case, each person will have m shares of 1/n, and m shares of 1/n gives the common fraction m/n. Thus, the common fraction m/n can be used to denote the division of m items between n people.
This is how we got an explicit connection between ordinary fractions and division (see the general idea of dividing natural numbers). This connection is expressed as follows: the fraction line can be understood as a division sign, that is, m/n=m:n.
Using an ordinary fraction, you can write the result of dividing two natural numbers for which a whole division cannot be performed. For example, the result of dividing 5 apples by 8 people can be written as 5/8, that is, everyone will get five-eighths of an apple: 5:8 = 5/8.
Equal and unequal fractions, comparison of fractions
A fairly natural action is comparing fractions, because it is clear that 1/12 of an orange is different from 5/12, and 1/6 of an apple is the same as another 1/6 of this apple.
As a result of comparing two ordinary fractions, one of the results is obtained: the fractions are either equal or unequal. In the first case we have equal common fractions, and in the second – unequal ordinary fractions. Let us give a definition of equal and unequal ordinary fractions.
Definition.
equal, if the equality a·d=b·c is true.
Definition.
Two common fractions a/b and c/d not equal, if the equality a·d=b·c is not satisfied.
Here are some examples of equal fractions. For example, the common fraction 1/2 is equal to the fraction 2/4, since 1·4=2·2 (if necessary, see the rules and examples of multiplying natural numbers). For clarity, you can imagine two identical apples, the first is cut in half, and the second is cut into 4 parts. It is obvious that two quarters of an apple equals 1/2 share. Other examples of equal common fractions are the fractions 4/7 and 36/63, and the pair of fractions 81/50 and 1,620/1,000.
But ordinary fractions 4/13 and 5/14 are not equal, since 4·14=56, and 13·5=65, that is, 4·14≠13·5. Other examples of unequal common fractions are the fractions 17/7 and 6/4.
If, when comparing two common fractions, it turns out that they are not equal, then you may need to find out which of these common fractions less different, and which one - more. To find out, the rule for comparing ordinary fractions is used, the essence of which is to bring the compared fractions to a common denominator and then compare the numerators. Detailed information on this topic is collected in the article comparison of fractions: rules, examples, solutions.
Fractional numbers
Each fraction is a notation fractional number. That is, a fraction is just the “shell” of a fractional number, its appearance, and all the semantic load is contained in the fractional number. However, for brevity and convenience, the concepts of fraction and fractional number are combined and simply called fraction. Here it is appropriate to paraphrase a well-known saying: we say a fraction - we mean a fractional number, we say a fractional number - we mean a fraction.
Fractions on a coordinate ray
All fractional numbers corresponding to ordinary fractions have their own unique place on, that is, there is a one-to-one correspondence between the fractions and the points of the coordinate ray.
In order to get to the point on the coordinate ray corresponding to the fraction m/n, you need to set aside m segments from the origin in the positive direction, the length of which is 1/n fraction of a unit segment. Such segments can be obtained by dividing a unit segment into n equal parts, which can always be done using a compass and a ruler.
For example, let's show point M on the coordinate ray, corresponding to the fraction 14/10. The length of a segment with ends at point O and the point closest to it, marked with a small dash, is 1/10 of a unit segment. The point with coordinate 14/10 is removed from the origin at a distance of 14 such segments.
Equal fractions correspond to the same fractional number, that is, equal fractions are the coordinates of the same point on the coordinate ray. For example, the coordinates 1/2, 2/4, 16/32, 55/110 correspond to one point on the coordinate ray, since all the written fractions are equal (it is located at a distance of half a unit segment laid out from the origin in the positive direction).
On a horizontal and right-directed coordinate ray, the point whose coordinate is the larger fraction is located to the right of the point whose coordinate is the smaller fraction. Similarly, a point with a smaller coordinate lies to the left of a point with a larger coordinate.
Proper and improper fractions, definitions, examples
Among ordinary fractions there are proper and improper fractions. This division is based on a comparison of the numerator and denominator.
Let us define proper and improper ordinary fractions.
Definition.
Proper fraction is an ordinary fraction whose numerator is less than the denominator, that is, if m Definition.
Improper fraction is an ordinary fraction in which the numerator is greater than or equal to the denominator, that is, if m≥n, then the ordinary fraction is improper. Here are some examples of proper fractions: 1/4, , 32,765/909,003. Indeed, in each of the written ordinary fractions the numerator is less than the denominator (if necessary, see the article comparing natural numbers), so they are correct by definition. Here are examples of improper fractions: 9/9, 23/4, . Indeed, the numerator of the first of the written ordinary fractions is equal to the denominator, and in the remaining fractions the numerator is greater than the denominator. There are also definitions of proper and improper fractions, based on comparison of fractions with one. Definition. correct, if it is less than one. Definition. An ordinary fraction is called wrong, if it is either equal to one or greater than 1. So the common fraction 7/11 is correct, since 7/11<1
, а обыкновенные дроби 14/3
и 27/27
– неправильные, так как 14/3>1, and 27/27=1. Let's think about how ordinary fractions with a numerator greater than or equal to the denominator deserve such a name - “improper”. For example, let's take the improper fraction 9/9. This fraction means that nine parts are taken of an object that consists of nine parts. That is, from the available nine parts we can make up a whole object. That is, the improper fraction 9/9 essentially gives the whole object, that is, 9/9 = 1. In general, improper fractions with a numerator equal to the denominator denote one whole object, and such a fraction can be replaced by the natural number 1. Now consider the improper fractions 7/3 and 12/4. It is quite obvious that from these seven third parts we can compose two whole objects (one whole object consists of 3 parts, then to compose two whole objects we will need 3 + 3 = 6 parts) and there will still be one third part left. That is, the improper fraction 7/3 essentially means 2 objects and also 1/3 of such an object. And from twelve quarter parts we can make three whole objects (three objects with four parts each). That is, the fraction 12/4 essentially means 3 whole objects. The considered examples lead us to the following conclusion: improper fractions can be replaced either by natural numbers, when the numerator is divided evenly by the denominator (for example, 9/9=1 and 12/4=3), or by the sum of a natural number and a proper fraction, when the numerator is not evenly divisible by the denominator (for example, 7/3=2+1/3). Perhaps this is precisely what earned improper fractions the name “irregular.” Of particular interest is the representation of an improper fraction as the sum of a natural number and a proper fraction (7/3=2+1/3). This process is called separating the whole part from an improper fraction, and deserves separate and more careful consideration. It's also worth noting that there is a very close relationship between improper fractions and mixed numbers. Each common fraction corresponds to a positive fractional number (see the article on positive and negative numbers). That is, ordinary fractions are positive fractions. For example, ordinary fractions 1/5, 56/18, 35/144 are positive fractions. When you need to highlight the positivity of a fraction, a plus sign is placed in front of it, for example, +3/4, +72/34. If you put a minus sign in front of a common fraction, then this entry will correspond to a negative fractional number. In this case we can talk about negative fractions. Here are some examples of negative fractions: −6/10, −65/13, −1/18. Positive and negative fractions m/n and −m/n are opposite numbers. For example, the fractions 5/7 and −5/7 are opposite fractions. Positive fractions, like positive numbers in general, denote an addition, income, an upward change in any value, etc. Negative fractions correspond to expense, debt, or a decrease in any quantity. For example, the negative fraction −3/4 can be interpreted as a debt whose value is equal to 3/4. On a horizontal and rightward direction, negative fractions are located to the left of the origin. The points of the coordinate line, the coordinates of which are the positive fraction m/n and the negative fraction −m/n, are located at the same distance from the origin, but on opposite sides of the point O. Here it is worth mentioning fractions of the form 0/n. These fractions are equal to the number zero, that is, 0/n=0. Positive fractions, negative fractions, and 0/n fractions combine to form rational numbers. We have already discussed one action with ordinary fractions - comparing fractions - above. Four more arithmetic functions are defined operations with fractions– adding, subtracting, multiplying and dividing fractions. Let's look at each of them. The general essence of operations with fractions is similar to the essence of the corresponding operations with natural numbers. Let's make an analogy. Multiplying fractions can be thought of as the action of finding a fraction from a fraction. To clarify, let's give an example. Let us have 1/6 of an apple and we need to take 2/3 of it. The part we need is the result of multiplying the fractions 1/6 and 2/3. The result of multiplying two ordinary fractions is an ordinary fraction (which in a special case is equal to a natural number). Next, we recommend that you study the information in the article Multiplying Fractions - Rules, Examples and Solutions. Bibliography. In the article we will show how to solve fractions using simple, understandable examples. Let's figure out what a fraction is and consider solving fractions! Concept fractions is introduced into mathematics courses starting from the 6th grade of secondary school. Fractions have the form: ±X/Y, where Y is the denominator, it tells how many parts the whole was divided into, and X is the numerator, it tells how many such parts were taken. For clarity, let's take an example with a cake: In the first case, the cake was cut equally and one half was taken, i.e. 1/2. In the second case, the cake was cut into 7 parts, of which 4 parts were taken, i.e. 4/7. If the part of dividing one number by another is not a whole number, it is written as a fraction. For example, the expression 4:2 = 2 gives an integer, but 4:7 is not divisible by a whole, so this expression is written as a fraction 4/7. In other words fraction is an expression that denotes the division of two numbers or expressions, and which is written using a fractional slash. If the numerator is less than the denominator, the fraction is proper; if vice versa, it is an improper fraction. A fraction can contain a whole number. For example, 5 whole 3/4. This entry means that in order to get the whole 6, one part of four is missing. If you want to remember, how to solve fractions for 6th grade, you need to understand that solving fractions, basically, comes down to understanding a few simple things. A wide variety of arithmetic operations are applicable to fractions. For example, you need to compare the fractions 3/4 and 4/5. To solve the problem, we first find the lowest common denominator, i.e. the smallest number that is divisible by each of the denominators of the fractions without leaving a remainder Least common denominator(4.5) = 20 Then the denominator of both fractions is reduced to the lowest common denominator Answer: 15/20 If it is necessary to calculate the sum of two fractions, they are first brought to a common denominator, then the numerators are added, while the denominator remains unchanged. The difference between fractions is calculated in the same way, the only difference is that the numerators are subtracted. For example, you need to find the sum of the fractions 1/2 and 1/3 Now let's find the difference between the fractions 1/2 and 1/4 Here solving fractions is not difficult, everything is quite simple here: For example: That's about it how to solve fractions, All. If you still have any questions about solving fractions, if something is unclear, write in the comments and we will definitely answer you.
If you are a teacher, then perhaps downloading a presentation for elementary school (http://school-box.ru/nachalnaya-shkola/prezentazii-po-matematike.html) will be useful for you. Vida 0.123 4 (\displaystyle 0(,)1234). In notation of a fraction of the form X / Y (\displaystyle X/Y) or X Y (\displaystyle (\frac (X)(Y))) the number before or above the line is called numerator, and the number after or below the line is denominator. The first plays the role of the dividend, the second - the divisor. Ordinary(or simple) fraction - writing a rational number in the form ± m n (\displaystyle \pm (\frac (m)(n))) or ± m / n , (\displaystyle \pm m/n,) Where n ≠ 0. (\displaystyle n\neq 0.) A horizontal or slash indicates a division sign, resulting in a quotient. The dividend is called numerator fractions, and the divisor is denominator. There are several types of writing ordinary fractions in printed form: Correct A fraction whose numerator is less than its denominator is called a fraction. A fraction that is not proper is called wrong, and represents a rational number with a modulus greater than or equal to one. For example, fractions 3 5 (\displaystyle (\frac (3)(5))), 7 8 (\displaystyle (\frac (7)(8))) and are proper fractions, while 8 3 (\displaystyle (\frac (8)(3))), 9 5 (\displaystyle (\frac (9)(5))), 2 1 (\displaystyle (\frac (2)(1))) And 1 1 (\displaystyle (\frac (1)(1)))- improper fractions. Any non-zero integer can be represented as an improper fraction with a denominator of 1. A fraction written as a whole number and a proper fraction is called mixed fraction and is understood as the sum of this number and a fraction. Any rational number can be written as a mixed fraction. In contrast to a mixed fraction, a fraction containing only a numerator and a denominator is called simple. For example, 2 3 7 = 2 + 3 7 = 14 7 + 3 7 = 17 7 (\displaystyle 2(\frac (3)(7))=2+(\frac (3)(7))=(\frac (14 )(7))+(\frac (3)(7))=(\frac (17)(7))). In strict mathematical literature, they prefer not to use such a notation because of the similarity of the notation for a mixed fraction with the notation for the product of an integer by a fraction, as well as because of the more cumbersome notation and less convenient calculations. A multi-story, or compound, fraction is an expression containing several horizontal (or, less commonly, oblique) lines: A decimal is a positional representation of a fraction. It looks like this: Example: 3.141 5926 (\displaystyle 3(,)1415926). The part of the record that comes before the positional decimal point is the integer part of the number (fraction), and the part that comes after the decimal point is the fractional part. Any ordinary fraction can be converted to a decimal, which in this case either has a finite number of decimal places or is a periodic fraction. Generally speaking, to write a number positionally, you can use not only the decimal number system, but also others (including specific ones, such as Fibonacci). A fraction is just a representation of a number. The same number can correspond to different fractions, both ordinary and decimal. This section covers operations on ordinary fractions. For operations on decimal fractions, see Decimal fraction. To compare, add and subtract fractions, they must be converted ( bring) to a form with the same denominator. Let two fractions be given: a b (\displaystyle (\frac (a)(b))) And c d (\displaystyle (\frac (c)(d))). Procedure: After this, the denominators of both fractions coincide (equal M). Instead of the least common multiple, in simple cases we can take as M any other common multiple, such as the product of denominators. For an example, see the Comparison section below. To compare two common fractions, you need to bring them to a common denominator and compare the numerators of the resulting fractions. A fraction with a larger numerator will be larger. Example. Let's compare 3 4 (\displaystyle (\frac (3)(4))) And 4 5 (\displaystyle (\frac (4)(5))). LCM(4, 5) = 20. We reduce the fractions to the denominator 20. Hence,
3
4
<
4
5
{\displaystyle {\frac {3}{4}}<{\frac {4}{5}}}
To add two ordinary fractions, you must reduce them to a common denominator. Then add the numerators and leave the denominator unchanged: The LCM of the denominators (here 2 and 3) is equal to 6. We give the fraction 1 2 (\displaystyle (\frac (1)(2))) to the denominator 6, for this the numerator and denominator must be multiplied by 3. The LCM of the denominators (here 2 and 4) is equal to 4. We present the fraction 1 2 (\displaystyle (\frac (1)(2))) to the denominator 4, for this you need to multiply the numerator and denominator by 2. We get 2 4 (\displaystyle (\frac (2)(4))). To multiply two ordinary fractions, you need to multiply their numerators and denominators: In particular, to multiply a fraction by a natural number, you need to multiply the numerator by the number, and leave the denominator the same: In general, the numerator and denominator of the resulting fraction may not be coprime, and the fraction may need to be reduced, for example: To divide one common fraction by another, you need to multiply the first fraction by the reciprocal of the second: For example: To convert a fraction to a decimal, divide the numerator by the denominator. The result can have a finite number of decimal places, but can also be an infinite periodic fraction. Examples: To convert a decimal to a common fraction, write the fractional part as a natural number divided by the appropriate power of 10. The signed integer part is then added to the result, forming a mixed fraction. Example: Russian term fraction, like its analogues in other languages, comes from Attention! 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?
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! 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 two 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? 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 finish here. 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). 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.Positive and negative fractions
Operations with fractions
How to solve fractions. Examples.
Reducing a fraction to a common denominator
Adding and subtracting fractions
Multiplying and dividing fractions
Types of fractions
Common fractions
Notation for common fractions
Proper and improper fractions
Mixed fractions
Compound fractions
Decimals
The meaning of a fraction and the main property of a fraction
Operations with fractions
Reduction to a common denominator
Comparison
Addition and subtraction
Happened 3 6 (\displaystyle (\frac (3)(6))). We give the fraction 1 3 (\displaystyle (\frac (1)(3))) to the same denominator, for this the numerator and denominator must be multiplied by 2. It turned out 2 6 (\displaystyle (\frac (2)(6))).
To get the difference between fractions, they also need to be brought to a common denominator, and then subtract the numerators, leaving the denominator unchanged: Multiplication and division
Convert between different recording formats
History and etymology of the term
Fractions
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)Types of fractions. Transformations.
The main property of a fraction.
How to convert fractions from one type to another.
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