In the third grade of primary school, children begin to study extra-tabular cases of multiplication and division. Numbers within a thousand - the material on which the mastery of the topic takes place. The program recommends the operation of division and multiplication of three-digit and two-digit numbers to produce on the example of single-digit. In the course of work on the topic, the teacher begins to form in children such an important skill as multiplication and division by a column. In the fourth grade, skill development continues, but numerical material is used within a million. Division and multiplication in a column is carried out by multi-digit numbers.
What is the basis of multiplication
The main provisions on which the algorithm for multiplying a multi-digit number by a multi-digit number is based are the same as when acting on a single-digit number. There are several rules that children use. They were "uncovered" by students in the third grade.
The first rule is bitwise operations. The second is to use a multiplication table in each rank.
It should be noted that these basic provisions are complicated when performing actions with multi-valued numbers.
The example below will help you understand what is at stake. Let's say you need 80 x 5 and 80 x 50.
In the first case, the student argues as follows: 8 dozens must be repeated 5 times, dozens will also be obtained, and there will be 40, since 8 x 5 = 40, 40 dozens is 400, which means 80 x 5 = 400. The reasoning algorithm is simple and clear to kid. In case of difficulty, he can easily find the result using the action of addition. The method of replacing multiplication by addition can also be used to verify the correctness of their own calculations.
To find the meaning of the second expression, it is also necessary to use the table case and 8 x 5. But to what category will the received 40 units belong? The question for most children remains open. The method of replacing the multiplication by the addition action in this case is irrational, since the sum will have 50 terms, so it is impossible to use it to find the result. It becomes clear that the knowledge to solve the example is not enough. Apparently, there are some other rules for the multiplication of multivalued numbers. And they need to be identified.
As a result of the joint efforts of the teacher and the children, it becomes clear that in order to multiply a multi-digit number by a multi-digit number, you need to be able to apply the combination law, in which one of the factors is replaced by the product (80 x 50 = 80 x 5 x 10 = 400 x 10 = 4000)
In addition, a way is possible when the distribution law of multiplication is used with respect to addition or subtraction. In this case, one of the factors must be replaced by the sum of two or more terms.
Children's research
Students are offered a fairly large number of examples of this kind. Each time, children try to find a simpler and faster way to solve, but at the same time, they are required to keep a detailed record of the course of the decision or detailed oral explanations.
Teacher does this for two purposes. Firstly, children realize, work out the main ways of performing the operation of multiplication by a multi-digit number. Secondly, it comes to the understanding that the way of writing such expressions on a line is very inconvenient. There comes a time when the students themselves suggest writing the multiplication in a column.
Stages of the study of multiplication by a multi-digit number.
In methodological recommendations, the study of this topic occurs in several stages. They must follow one after another, giving students the opportunity to understand the whole meaning of the action being studied. The list of stages gives the teacher a general picture of the process of presenting material to children:
- students' independent search for ways to find the value of the product of multi-valued factors;
- to solve the problem, a combination property is used, as well as multiplication by one with zeros;
- development of the skill of multiplication by round numbers;
- use in calculations of the distribution property of multiplication relative to addition and subtraction;
- multi-digit operations and column multiplication.
Following these steps, the teacher must constantly draw the attention of children to the close logical connections of the previously studied material with what is being mastered in a new topic. Schoolchildren not only engage in multiplication, but also learn to compare, draw conclusions, make decisions.
Objectives of Studying Multiplication in an Elementary School Course
The teacher, teaching mathematics, knows for sure that the moment will come when fourth-graders will have a question about how to solve the multiplication of multi-valued numbers in a column. And if he, along with the students during the three years of study - in grades 2, 3, and 4 - purposefully and thoughtfully studied the specific meaning of multiplication and all the questions that are associated with this operation, then children should not have difficulties in mastering the topic under discussion.
What tasks were previously solved by students and their teacher?
- Mastering tabular cases of multiplication, that is, obtaining the result in one step. A mandatory requirement of the program is to bring the skill to automatism.
- Multiplication of a multi-digit number by a single-digit. The result is obtained by repeatedly repeating a step that children already know perfectly.
- Multiplication of a multi-digit number by a multi-digit number is carried out by repeating the steps indicated in paragraphs 1 and 2. The final result will be obtained by combining intermediate values and correlating incomplete products with digits.
Using Multiplication Properties
Before examples of multiplication by a column begin to appear on subsequent pages of textbooks, Grade 4 should learn very well to use the combination and distributive property to rationalize computations.
Through observations and comparisons, students come to the conclusion that the combined property of multiplication to find the product of multi-valued numbers is used only when one of the factors can be replaced by the product of single-valued numbers. And this is not always possible.
The distributive property of multiplication in this case acts as universal. Children notice that the factor can always be replaced by a sum or a difference, therefore the property is used to solve any example by multiplying multivalued numbers.
The algorithm for writing the multiplication action in the column
The column multiplication record is the most compact of all. Teaching children this type of design begins with the option of multiplying a multi-digit number by a two-digit number.
Children are invited to independently compose a sequence of actions when performing multiplication. Knowledge of this algorithm will be the key to successful skill formation. Therefore, the teacher does not need to spare time, but try to make every effort to ensure that the procedure for performing actions when multiplying in a column was adopted by the children to "excellent".
Skill Exercises
First of all, it should be noted that the examples of column multiplication offered to children are complicated from lesson to lesson. After becoming acquainted with double-digit multiplication, children learn to perform actions with three-digit, four-digit numbers.
To develop the skill, examples are offered with a ready-made solution, but among them intentionally post entries with errors. The task of the students is to detect inaccuracies, explain the reason for their appearance and correct the notes.
Now, when solving problems, equations and all other tasks, where it is necessary to multiply multi-valued numbers, students are required to make a record in a column.
The development of cognitive UUD in the study of the topic "Multiplication of numbers in a column"
Much attention in the lessons devoted to the study of this topic is given to the development of such cognitive actions as finding different ways to solve the problem, choosing the most rational technique.
The use of schemes for conducting reasoning, the establishment of causal relationships, the analysis of observed objects on the basis of the identified essential features is another group of cognitive skills formed when studying the topic “Multiplication in a column”.
Children are taught how to divide the multi-digit numbers and design a column with a column only after the children learn to multiply.