In our lives we often encounter a large number of various things, and with the advent and development of electronic computing technology, we also encounter a huge flow of fast-flowing information. All data received from the environment is actively processed by our mental activity, which is called thinking in the scientific language. This process includes various operations: analysis, synthesis, comparison, generalization, induction, deduction, systematization, and others. The significance of the above is complemented by the fact that the processes can be executed simultaneously. For example, during the comparison, we can also analyze the data. The operation of organizing information is no exception. It is also very actively used in everyday life and is one of the fundamental in thinking. Indeed, a lot of disparate information penetrates into our consciousness, for the perception of which at a normal level it must be somehow classified into homogeneous objects. This happens subconsciously, but if such manipulations of our brain are not enough, then you can resort toto conscious systematization. As a rule, to perform this work, people resort to the method of groupings that has long been proven by time and human experience. We should talk about him today.
Definition of concept
You've probably already read cumbersome and information overloaded definitions of terms written in scientific language. Of course, they meet all the necessary requirements in terms of their correct compilation. But because of this, such definitions are quite difficult to understand. This is especially true for the really smart ones. This is the concept of grouping. Therefore, to make it clearer, we will leave the classical scheme and "chew" everything to the smallest detail.
Grouping always refers to the systematization of information either received by us in a ready-made form (for example, when a report was read to us), or as a result of analysis, which is a mental breakdown of an object into parts (for example, when we analyze a conflict, then we necessarily divide it into several components: reasons, reason, participants, stages, completion, results). Systematization occurs on the basis of some criterion (fundamental feature). Let's say we have a spoon, a plate and a saucepan. Their main feature will be their kitchen tasks. People called such objects dishes. That is, from the above, we can conclude that a grouping is a combination of several items that are identical according to a common criterion into onegroup.
Applications
As mentioned above, the grouping method is used when it is necessary to "manually" divide various objects that fall into our perception into homogeneous classes of objects. This is necessary during the performance of scientific activities, the design of new tangible and intangible objects, the development of information technologies. Grouping is also very good at solving ordinary everyday tasks that are not related to the field of science. For example, it can be very useful while studying at school, when cleaning the room, or simply when it is necessary to rationally allocate time for the upcoming day. That is, from here we can derive the tasks of the grouping method: systematization and classification of information and heterogeneous objects in order to simplify working with them.
Group by quantitative and qualitative features
This is perhaps the most common type of grouping method.
In the case when a quantitative indicator is taken as a criterion, then, conditionally speaking, the numerical straight line denoting the range of changes in the state of the object taken for consideration is divided into several values, which can also form their own ranges with several more divisions.
In the case when a qualitative indicator is taken as a criterion, the initial data or data obtained as a result of the analysis are grouped in accordance with those characteristics that indicate the physical properties of the objects taken into consideration (such states are color, sound, smell, taste, state of aggregation)as well as morphological, chemical, psychological and other features. It must be remembered here that the criterion taken should not indicate the number of items.
Group method. Examples
For grouping by quantitative indicators, the age of a person is perfect as an example. We know that it is calculated in years, which can be grouped into several parts. Approximately, from 0 to 12 years old childhood flows, from 12 to 18 years of transition, etc. Please note that these two categories also have divisions. From 0 to 3 years old, a person experiences early childhood (divided into infancy and early childhood), from 3 to 7 years old - ordinary childhood (divided into preschool age and primary school age). Thus, grouping by quantitative characteristics is very well suited in the case of working with numerical data.
To group by quality, let's give an example. Before us are pears, apples, eggs. If pears and apples are green, then we will collect them together according to their common color, and we will remove the eggs separately (physical criterion). But according to the richness of useful substances for the body, we will group apples and eggs together, because it is known that they have organic matter necessary for humans (chemical criterion).
Types of grouping
Grouping is carried out not only on the basis of quantitative and qualitative indicators. There is a classification of this information processing technique based on other criteria. For example, one of the most commonis an indicator of direction (or purpose), i.e. what the grouping is used for.
Here we can highlight the method of analytical grouping. It is used to identify the relationship between various social phenomena, divided into factorial and resultative. Its goal is to study society with the help of a special algorithm. It assumes the dependence of the effective data on the factor data. For example, if a worker made more products at a factory (i.e., exceeded his quota), then he is likely to receive more money.
The group summary method also falls under the above criteria. It is used when it is necessary to compile statistics based on summarized (composed into a single whole) data. They may be heterogeneous. Therefore, in order to obtain correct and readable statistics, these data are grouped based on common features. For example, when a store has sold goods, it is necessary to divide these goods into groups and proceed to the following actions on this basis.
The indicator grouping method also fits the directional criterion. Obviously, it is used to classify data belonging to different classes of objects. This is a fundamental method, without which no method of grouping information can do. It makes no sense to give examples, since everything that was said above applies here.
As another criterion by whichyou can divide the grouping into separate types, you can select the scope or area of its application. Let's talk about it in more detail.
Group method in statistics
It is used in this field of scientific knowledge, which deals with the collection, processing, measurement of mass data (quantitative and qualitative). Naturally, the grouping method in statistics cannot but be relevant, since it needs to systematize information. There are several types of grouping in this science.
- Typological grouping. An array of information is taken, then divided into types determined by a person based on the necessary criteria. This view is very similar to the measure grouping method.
- Structural grouping. Produced in the same way as the previous one, it has a larger arsenal of actions due to additional actions: studying the structure of homogeneous data and their structural changes.
- The grouping is analytical. Has been reviewed above. Included in statistics because this science is somehow related to the study of society.
In Algebra
Knowing everything necessary that was stated above, we can talk about what the topic of today's conversation is devoted to. It's time to give a few words about the method of grouping in algebra. As you can see, this method of working with information is so common and necessary that it is included in the school curriculum.
The grouping method in algebra is the implementation of mathematical operations to decompose a polynomial intomultipliers.
That is, this method is used when working with polynomials, when they require simplification and implementation of their solution. This can be seen with an example, but first a little more about the steps that must be taken to get the correct answer.
Stages of factoring a polynomial
In fact, this is the grouping method in algebra. To start its implementation, you need to go through two stages:
- Stage 1. It is necessary to find such members of the polynomial that have common factors, then combine them into groups by "approach" (grouping).
- Stage 2. It is necessary to take the common factor of the "close" (grouped) members of the polynomial out of brackets, and then the resulting common factor for all groups.
At first glance it looks very complicated. But in fact, there is nothing difficult here. It is enough just to analyze one example.
Example of grouping solution
We have the following polynomial: 9a - 3y + 27 + ay. So, first we find terms with a common factor. We see that 9a and ay have a common factor a. Also, -3y and 27 have a common factor of 3. Now we need to make sure that these members are next to each other, that is, they need to be grouped in a certain way. This can be done by swapping them in the polynomial. The result is 9a + ay - 3y + 27. The first step is done, now it's time to move on to the second. We take out the common factors of the grouped terms out of brackets. Now the polynomial will take the following form a(9 + y) - 3(y + 9). We havea common factor appeared for all groups: y + 9. It also needs to be taken out of brackets. It turns out: (9 + y)(a - 3) Thus, the polynomial is greatly simplified and now it can be easily solved. To do this, you need to equate each group to zero and find the value of the unknown variables.
Where else in algebra can data be grouped?
As a rule, this method is very often used when solving polynomials. However, it is worth noting that in algebra, many mathematical models that are not "officially" called polynomials are, after all, such. Equations and inequalities can serve as a striking example. In their meaning, the first are equal to something, and the second, obviously, are not equal. But regardless of this, the presented models can also act as polynomials at the same time. Therefore, solving equations by the grouping method, as well as inequalities, often helps a lot when performing such tasks.
What to do if it doesn't work?
Please note: not all polynomials can be solved this way. If it is not possible to find common factors or there is only one common factor (at the first stage), then, obviously, the grouping method cannot be applied in this case. You should turn to other methods and then you can get the right answer.
A couple more moments
It is worth noting a few properties of the grouping method that are useful to know:
- After the second stage, if we swap the factors, the answers will still be the same (the general mathematical rule applies here: from a changeplaces of factors, their product does not change).
- In the case when the common factor is the same as one of the terms (members) of the polynomial (including also the sign), when grouping, the number 1 is written in place of this term with the corresponding sign.
- After taking out the common factor, the polynomial should have as many terms as there were before taking it out.
In closing
Thus, the solution by the grouping method in algebra is used quite widely. This method is one of the most common and universal. With a sufficient understanding of it, you can easily solve a large number of various mathematical models: polynomials, equations, inequalities, etc. This can be useful during a simple lesson at school, and when solving homework, and when passing the OGE or the Unified State Examination.
