In our life we often encounter a large number of various things, and with the advent and development of electronic computing technology - also with a huge stream of fast-flowing information. All data obtained from the environment is actively processed by our mental activity, which is called thinking in a 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 processes can be performed simultaneously. For example, during the comparison, we can also analyze the data. The operation to systematize 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 scattered information penetrates our consciousness, for the perception of which at a normal level it needs to be somehow classified into homogeneous objects. This happens subconsciously, but if such manipulations of our brain are not enough, then we can resort to conscious systematization. As a rule, to perform this work, people resort to the method of grouping, long time-tested by human experience. We should talk about him today.
Definition of a concept
Probably, you have already read the bulky and information overloaded definitions of terms written in scientific language. Of course, they meet all the necessary requirements in terms of their proper preparation. But because of this, such definitions are understood quite difficult. Especially it concerns absolutely abstruse. This is the concept of grouping. Therefore, in order to make it clearer, we will get away from the classical and circuitry and “develop” everything to the smallest detail.
Grouping always refers to the systematization of information either that we have received in a ready-made form (for example, when a report has been read to us), or as a result of an analysis, which is a mental disintegration of an object into parts (for example, when we analyze a conflict, we must separate it into several components: reasons, reason, participants, stages, completion, results). Systematization takes place on the basis of some criterion (fundamental attribute). Say we have a spoon, plate and pan. Their main feature will be their fulfillment of kitchen tasks. People called such items dishes. That is, from the above, we can conclude that the grouping is a combination of several objects that are identical by the general criterion into one group.
Areas of use
As already mentioned above, the grouping method is used when it is necessary to "manually" break up various objects into our perceptions into homogeneous classes of objects. This is necessary during the implementation of scientific activities, the design of new tangible and intangible objects, the development of information technology. The grouping is very good in solving ordinary everyday tasks that are not related to the field of science. For example, it can be very useful when studying at school, when cleaning the room, or just when it is necessary to rationally distribute the time for the coming 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 the work with them.
Grouping by quantitative and qualitative characteristics
These are perhaps the most common types of grouping method.
In the case when a quantitative indicator is taken as a criterion, then, relatively speaking, a numerical line denoting a range of changes in the state of an object taken into consideration is divided into several values, which can also form their ranges having a few 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 accepted for consideration (such states are color, sound, smell, taste, aggregate condition), as well as morphological, chemical, psychological and other signs. It must be remembered here that the criterion taken should not indicate the number of items.
Grouping Method. Examples
To group by quantitative indicators, as an example, the age of a person is perfect. We know that it is calculated in years, which can be grouped into several parts. Approximately, from 0 to 12 years of childhood, from 12 to 18 is a transitional age, etc. Note that these two categories also have divisions. From 0 to 3 years, a person experiences young childhood (divided by infancy and early age), from 3 to 7 years old - ordinary childhood (divided by preschool age and primary school age). Thus, grouping by quantitative characteristics is very well suited for working with numerical data.
To group by quality indicators, we give an example. Before us are pears, apples, eggs. If pears and apples are green, then we will collect them together according to the general color, and we will remove the eggs separately (physical criterion). But according to the wealth of useful substances for the body, we group apples and eggs together, because it is known that they possess the necessary organics 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 common is the directivity indicator (or target), i.e., for what the group is used.
Here we can distinguish the method of analytical grouping. It is used to identify the relationship between various social phenomena, is divided into factor and effective. Her goal is to study society using a special algorithm. It assumes the dependence of productive data on factor data. For example, if a worker manufactured more products at a factory (i.e., exceeded his norm), then he would most likely receive more money.
The grouping summary method also falls under the above criterion. It is used when it is necessary to compile statistics based on summarized (arranged into a single whole) data. They can be heterogeneous. Therefore, in order to get correct and readable statistics, these data are grouped on the basis of common features. For example, when a store has sold goods, it is necessary to divide these goods into groups and, on this basis, proceed to the following actions.
The method of grouping indicators is also suitable for the directional criterion. Obviously, it is used to classify data related to different classes of objects. This is a fundamental method, without which there is no way to group information. There is no point in giving examples, since all that has been said above applies here.
As another criterion by which it is possible to divide the grouping into separate species, one can single out the scope or area of its application. Let's talk about this in more detail.
The grouping 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 method of grouping in statistics cannot but be relevant, since it needs to be systematized information. There are several types of grouping in this science.
- The typological grouping. An array of information is taken, then it is divided into types determined by a person based on the necessary criteria. This view is very similar to the method of grouping indicators.
- The grouping is structural. Produced in the same way as the previous one, it has a greater 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. It is included in statistics, since this science, in one way or another, is connected with the study of society.
In algebra
Knowing everything you need that was stated above, you 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 technique of working with information is so widespread and necessary that it is included in the school curriculum.
The method of grouping in algebra is the implementation of mathematical operations for factoring a polynomial.
That is, this method is used when working with polynomials, when they require simplification and implementation of their solutions. This can be considered as an example, but first a little more about the actions that must be performed in order to get the right answer.
Factoring steps of a polynomial
In fact, this is the method of groupings in algebra. To begin its implementation, you need to go through two stages:
- Stage number 1. It is necessary to find such members of the polynomial that have common factors, then combine them into groups by "convergence" (grouping).
- Stage number 2. It is necessary to take out the common factor of the "converged" (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. It is enough to make out one example.
Grouping Solution Example
We have a polynomial of the following form: 9a - 3y + 27 + ay. So, first we find the 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 stage is completed, now it's time to go to the second. Factor the grouped members out of brackets. Now the polynomial takes the following form a (9 + y) - 3 (y + 9). We have a common factor for all groups: y + 9. It also needs to be put out of brackets. It turns out: (9 + y) (a - 3) Thus, the polynomial has been greatly simplified and now it can be easily solved. To do this, equate each group to zero and find the value of 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 mathematical algebra many mathematical models that are not “officially” called polynomials are still such. A striking example is equations and inequalities. In terms of their meaning, the former are equal to something, and the latter are obviously 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 in this way. If it is not possible to detect common factors or if 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 points
It is worth noting several properties of the grouping method that are useful to know:
- After completing the second stage, if we swap the factors, the answers will still be the same (there is a general mathematical rule: their product does not change from changing the places of the factors).
- 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 in place of this term, the number 1 is written with the corresponding sign.
- After making the common factor, the polynomial should have as many terms as there were before making it.
Finally
Thus, the solution by the method of grouping in algebra is widely used. This method is one of the most common and universal. With its sufficient understanding, it is easy to 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 exam or the exam.