We encounter fractions in life much earlier than they begin to study at school. If you cut a whole apple in half, then we get a part of the fruit - ½. Cut it again - it will be ¼. This is what fractions are. And everything, it would seem, is simple. For an adult. For a child (and they begin to study this topic at the end of elementary school), abstract mathematical concepts are still frighteningly incomprehensible, and the teacher must explain in an accessible way what a proper fraction and improper, ordinary and decimal are, what operations can be performed with them and, most importantly, why all this is needed.
What are fractions
Introduction to a new topic at school begins with ordinary fractions. They are easy to recognize by the horizontal line separating the two numbers - above and below. The top is called the numerator, the bottom is called the denominator. There is also a lowercase version of writing improper and regular ordinary fractions - through a slash, for example: ½, 4/9, 384/183. This option is used when the line height is limited and it is not possible to apply the "two-story" form of the record. Why? Yes, because it is more convenient. A little later wewe will make sure of this.
Besides ordinary fractions, there are also decimal fractions. It is very easy to distinguish between them: if in one case a horizontal or slash is used, then in the other - a comma separating sequences of numbers. Let's see an example: 2, 9; 163, 34; 1, 953. We intentionally used a semicolon as a separator to delimit the numbers. The first of them will read like this: “two whole, nine tenths.”
New concepts
Let's get back to ordinary fractions. They come in two varieties.
The definition of a proper fraction is as follows: it is a fraction whose numerator is less than the denominator. Why is it important? We'll see now!
You have some apples cut into halves. In total - 5 parts. How do you say: you have "two and a half" or "five second" apples? Of course, the first option sounds more natural, and when talking with friends, we will use it. But if you need to calculate how many fruits each will get, if there are five people in the company, we will write down the number 5/2 and divide it by 5 - from the point of view of mathematics, this will be clearer.
So, for the naming of proper and improper fractions, the rule is as follows: if a fraction can have an integer part (14/5, 2/1, 173/16, 3/3), then it is incorrect. If this cannot be done, as in the case of ½, 13/16, 9/10, it will be correct.
Basic property of a fraction
If the numerator and denominator of a fraction are simultaneously multiplied ordivided by the same number, its value does not change. Imagine: the cake was cut into 4 equal parts and they gave you one. The same cake was cut into eight pieces and given you two. Isn't it all the same? After all, ¼ and 2/8 are the same thing!
Abbreviation
Authors of problems and examples in math textbooks often try to confuse students by offering cumbersome fractions that can actually be reduced. Here is an example of a proper fraction: 167/334, which, it would seem, looks very "scary". But in fact, we can write it as ½. The number 334 is divisible by 167 without a remainder - having done this operation, we get 2.
Mixed Numbers
An improper fraction can be represented as a mixed number. This is when the whole part is brought forward and written at the level of the horizontal line. In fact, the expression takes the form of a sum: 11/2=5 + ½; 13/6=2 + 1/6 and so on.
To take out the whole part, you need to divide the numerator by the denominator. Write the remainder of the division above, above the line, and the whole part before the expression. Thus, we get two structural parts: whole units + proper fraction.
You can also perform the reverse operation - for this you need to multiply the integer part by the denominator and add the resulting value to the numerator. Nothing complicated.
Multiplication and division
Oddly enough, multiplying fractions is easier than adding them. All that is required is to extend the horizontal line: (2/3)(3/5)=23 / 35=2/5.
Division is also everythingsimple: you need to multiply the fractions crosswise: (7/8) / (14/15)=715 / 814=15/16.
Adding fractions
What to do if you need to add or subtract fractions, and they have different numbers in the denominator? It will not work in the same way as with multiplication - here one should understand the definition of a proper fraction and its essence. It is necessary to reduce the terms to a common denominator, that is, the bottom of both fractions should have the same numbers.
To do this, you should use the basic property of a fraction: multiply both parts by the same number. For example, 2/5 + 1/10=(22)/(52) + 1/10=5/10=½.
How to choose which denominator to bring the terms to? This must be the smallest multiple of both denominators: for 1/3 and 1/9 it will be 9; for ½ and 1/7 - 14, because there is no smaller value that can be divided without a remainder by 2 and 7.
Use
What are improper fractions for? After all, it is much more convenient to immediately select the whole part, get a mixed number - and that's it! It turns out that if you need to multiply or divide two fractions, it is more profitable to use the wrong ones.
Take the following example: (2 + 3/17) / (37 / 68).
It would seem that there is nothing to cut at all. But what if we write the result of the addition in the first brackets as an improper fraction? Look at: (37/17) / (37/68)
Now everything falls into place!Let's write the example in such a way that everything becomes obvious: (3768) / (1737).
Let's reduce the 37 in the numerator and denominator and finally divide the top and bottom parts by 17. Do you remember the basic rule for proper and improper fractions? We can multiply and divide by any number as long as we do it for the numerator and denominator at the same time.
So, we get the answer: 4. The example looked complicated, and the answer contains only one digit. This often happens in mathematics. The main thing is not to be afraid and follow simple rules.
Common mistakes
When performing actions with fractions, a student can easily make one of the most popular mistakes. Usually they occur due to inattention, and sometimes due to the fact that the studied material has not yet been properly deposited in the head.
Often the sum of numbers in the numerator causes a desire to reduce its individual components. Suppose, in the example: (13 + 2) / 13, written without brackets (with a horizontal line), many students, due to inexperience, cross out 13 from above and below. But this should not be done in any case, because this is a gross mistake! If instead of addition there was a multiplication sign, we would get the number 2 in the answer. But when performing addition, no operations with one of the terms are allowed, only with the entire sum.
Also, guys often make mistakes when dividing fractions. Let's take two regular irreducible fractions and divide by each other: (5/6) / (25/33). The student can confuse and write the resulting expression as (525) / (633). But it wouldit turned out during multiplication, but in our case everything will be a little different: (533) / (625). We reduce what is possible, and in the answer we will see 11/10. We write the resulting improper fraction as a decimal - 1, 1.
Parentheses
Remember that in any mathematical expression, the order of operations is determined by the precedence of operation signs and the presence of parentheses. Other things being equal, the sequence of actions is counted from left to right. This is also true for fractions - the expression in the numerator or denominator is calculated strictly according to this rule.
After all, what is a proper fraction? It is the result of dividing one number by another. If they don't divide evenly, it's a fraction, and that's it.
How to write a fraction on a computer
Since standard tools do not always allow you to create a fraction consisting of two "tiers", students sometimes go for various tricks. For example, numerators and denominators are copied into the Paint editor and glued together, drawing a horizontal line between them. Of course, there is an easier option, which, by the way, also provides a lot of additional features that will be useful to you in the future.
Open Microsoft Word. One of the panels at the top of the screen is called "Insert" - click it. On the right, on the side where the icons for closing and minimizing the window are located, there is a Formula button. This is exactly what we need!
If you use this function, a rectangular area will appear on the screen in which you can use any mathematicalcharacters that are not on the keyboard, as well as write fractions in the classic form. That is, separating the numerator and denominator with a horizontal line. You may even be surprised that such a proper fraction is so easy to write.
Study math
If you are in grades 5-6, then soon the knowledge of mathematics (including the ability to work with fractions!) will be required in many school subjects. In almost any problem in physics, when measuring the mass of substances in chemistry, in geometry and trigonometry, fractions cannot be dispensed with. Soon you will learn to calculate everything in your mind, without even writing expressions on paper, but more and more complex examples will appear. Therefore, learn what a proper fraction is and how to work with it, keep up with the curriculum, do your homework on time, and then you will succeed.
