Multiplication in a column. Multiplication and division by a column

In the third grade of primary school, children begin to study in-the-table cases of multiplication and division. Numbers within a thousand are the material on which the topic is mastered. The program recommends the operation of dividing and multiplying three-digit and two-digit numbers in the case of unambiguous ones. In the course of working 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, the training of skills continues, but a numeric material is used within a million. Division and multiplication in a column is performed on many-valued numbers.

What is the basis of multiplication

The main points on which the algorithm for multiplying a multivalued number by a multi-valued algorithm is constructed are the same as for operations on single-valued. There are several rules that children use. They were "uncovered" by schoolchildren in the third grade.

The first rule is the bit-order of operations. The second is to use the multiplication table in each digit.

It should be noted that these basic provisions are complicated when performing actions with multivalued numbers.

The example below will help you understand what is being said. Suppose you need 80 x 5 and 80 x 50.

In the first case, the student reasons like this: 8 dozens need to be repeated 5 times, dozens are also obtained, and there will be 40 of them, since 8 x 5 = 40, 40 tens is 400, hence 80 x 5 = 400. The reasoning algorithm is simple and understandable to kid. In case of difficulty, he can easily find the result by using the action of addition. The method of replacing multiplication with addition can also be used to verify the correctness of its own calculations.

To find the value of the second expression, it is also necessary to use the tabular case and 8 x 5. But to which category will the resulting 40 units belong? The question for most children remains open. The acceptance of replacing multiplication by the addition action in this case is not rational, since the sum will have 50 terms, so it is impossible to use it to find the result. It becomes clear that knowledge for solving the example is not enough. Apparently, there are still some rules for multiplying many-valued numbers. And they need to be identified.

As a result of the joint efforts of the teacher and children, it becomes clear that multiplication of a multivalued number by a multivalued one requires the ability to apply a combination law in which one of the factors is replaced by a product (80 x 50 = 80 x 5 x 10 = 400 x 10 = 4000)

In addition, a path is possible when the distributive law of multiplication with respect to addition or subtraction is used. In this case one of the factors must be replaced by the sum of two or more terms.

Research of children

Pupils are given a large number of examples of this kind. Children each time try to find a simpler and faster method of solution, but at the same time they all the time need a detailed record of the progress of the decision or detailed oral explanations.

The teacher does this, pursuing two goals. First, children realize, work out the main ways of performing multiplication by a multivalued number. Secondly, there comes an understanding that the way to write such expressions in a line is very inconvenient. There comes a time when the students themselves propose to write the multiplication in a column.

Steps of studying multiplication by a multivalued number.

In the 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 reveals to the teacher a general picture of the process of submitting material for children:

• Independent search by students for ways of finding the meaning of a product of multivalued factors;
• To solve the problem, a combining property is used, as well as multiplication by one with zeros;
• Working out the skill of multiplying by round numbers;
• Use in the calculation of the distribution property of multiplication with respect to addition and subtraction;
• Operations with multivalued numbers and multiplication by a column.

Following these steps, the teacher must constantly draw the children's attention to the close logical connections of the material studied earlier with what is being learned in the new topic. Students not only multiply, but also learn to compare, draw conclusions, make decisions.

The tasks of studying multiplication in an elementary school course

The teacher, teaching mathematics, knows for sure that there will come a time when the fourth graders will have a question about how to solve by multiplication multiplication of many-valued numbers. And if he and his students studied the concrete meaning of the multiplication and all the questions connected with this operation for 3 years, in 2, 3, and 4 classes, the problems in the development of the topic under consideration should not arise in children.

What tasks were previously solved by the students and their teacher?

1. Mastering tabular multiplication, that is, obtaining a result in one step. The mandatory requirement of the program is to bring the skill to automatism.
2. Multiplication of a multivalued number by a single-valued number. The result is obtained by repeating the step repeatedly, which the children already know perfectly.
3. The multiplication of a multivalued number by a multivalued number is accomplished by repeating the steps indicated in clauses 1 and 2. The final result will be obtained by combining the intermediate values and correlating the incomplete products with the bits.

Using multiplication properties

Before the examples of multiplication by a column begin to appear on the next pages of the textbooks, class 4 should learn very well how to use the combination and distribution property to rationalize the calculations.

Through observations and comparisons, students come to the conclusion that the combining property of multiplication for finding the product of many-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 a universal. Children notice that the multiplier can always be replaced by a sum or a difference, so the property is used to solve any example of multiplication of many-valued numbers.

The algorithm for writing the multiplication action into a column

The multiplication by a column is the most compact of all existing ones. Teaching children this form of design begins with the option of multiplying a multivalued number by two-digit.

Children are encouraged to independently compile a sequence of actions when performing multiplication. Knowing this algorithm will be the key to successful skill formation. Therefore, the teacher does not need to regret time, and try to make every effort to ensure that the order of performance of actions when multiplying in the column was learned by the children "perfectly".

Exercises to form a skill

First of all, it should be noted that the examples of multiplication in the column offered to children from the lesson to the lesson become more complicated. After learning to multiply by a two-digit number, children learn to perform actions with three-digit, four-digit numbers.

To work out the skill, examples are offered with a ready solution, but among them deliberately placed records with errors. The task of the students is to find inaccuracies, explain the reason for their appearance and correct the records.

Now when solving problems, equations and all other tasks where multiplication of many-valued numbers must be performed, students are required to write a record in a column.

Development of cognitive DAM 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 paid to the development of such cognitive actions as finding different ways of solving the task posed, choosing the most rational method.

The use of schemes for reasoning, the establishment of cause-effect relationships, the analysis of observable objects on the basis of the identified essential features is another group of cognitive skills that are formed when studying the topic "Multiplication in a column".

Teaching children how to divide multivalued numbers and write a column is done only after the children learn to multiply.