In the third grade of primary school children startto study in-tabular 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 by the example of single-valued. 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, skill training continues, but the numeric material is used within a million. Division and multiplication in a column is performed on multivalued numbers.
The main provisions on which the algorithm is constructedmultiplication of a multivalued number by a multivalued number, are the same as for actions on single-valued. There are several rules that children use. They were "uncovered" by students 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 taken into account that these basic provisions are complicated when performing actions with multivalued numbers.
The example below will help you understand what is at stake. Suppose you need 80 x 5 and 80 x 50.
In the first case, the student reasons like this:8 dozens must be repeated 5 times, tens will also be 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 for the child. In case of difficulty, he can easily find the result by using the action of addition. The method of replacing multiplication by addition can also be used to verify the correctness of its own calculations.
To find the value of the second expression, tooit is necessary to use the tabular case and 8 x 5. But what kind of 40 units will belong to which category? The question for most children remains open. Accepting the change of 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 joint efforts of the teacher and childrenit becomes clear that for the multiplication of a multivalued number by a multivalued one, it is necessary to use 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 path is possible when useddistributive law of multiplication with respect to addition or subtraction. In this case one of the factors must be replaced by the sum of two or more terms.
Pupils are offered a fairly largenumber of examples of this kind. Children each time try to find a simpler and faster way to solve, 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 the operation of 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 record the multiplication in a column.
In the methodological recommendations, the study oftopic 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:
Following these steps, the teacher must constantlyTo draw the children's attention to the close logical links of the previously studied material with what is being mastered in a new topic. Students not only multiply, but also learn to compare, draw conclusions, make decisions.
A teacher, teaching mathematics, knows for sure thatthere 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 specific meaning of multiplication in a purposeful and thoughtful way during the three years of study - in 2, 3, and 4 classes - and all the questions connected with this operation, then the children should not have difficulties in mastering the topic under consideration.
What tasks were previously solved by the students and their teacher?
Before the next pages of the textbooksthere will appear examples of multiplication by a column, class 4 should learn very well to use for combining computational and distributive properties.
Through observations and comparisons, students come toto 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 thiscase appears as a universal. Children notice that the factor 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 record of 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 a two-digit number.
Children are asked to make up their ownsequence 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, and try to make every effort to ensure that the order of performing actions when multiplying in the column was learned by the children "perfectly".
First of all, it should be noted that examplesmultiplication in the column, offered to children, from the lesson to the lesson are complicated. After learning to multiply by a two-digit number, children learn to perform actions with three-digit, four-digit numbers.
For practicing the skill, examples are offered withready-made solution, but among them deliberately placed records with errors. The task of the students is to discover 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.
A great deal of attention is paid to the lessons devoted to the study ofThis topic is given to the development of such cognitive actions as finding different ways of solving the task posed, choosing the most rational method.
Use of schemes for reasoning,the establishment of cause-effect relationships, the analysis of observed objects on the basis of the identified essential features is another group of cognitive skills that are being 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.
