In the third grade of elementary school, children begin to learn extra-table cases of multiplication and division. Numbers within a thousand are the material on which the topic is mastered. The program recommends dividing and multiplying three-digit and two-digit numbers using single-digit numbers as an example. 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 development continues, but numerical material within a million is used. Division and multiplication in a column are performed on multi-digit numbers.
What is the basis of multiplication
The main provisions on which the algorithm for multiplying a multi-valued number by a multi-valued one is based are the same as for operations on a single-valued number. There are several rules that children use. They were "revealed" by students in the third grade.
The first rule is the bitwise operation. The second is to use the multiplication table in each digit.
Be aware that these basics become more complicated when performing operations with multi-digit 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 tens must be repeated 5 times, there will also be tens, and there will be 40, since 8 x 5=40, 40 tens is 400, which means 80 x 5=400. The reasoning algorithm is simple and understandable to the child. 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 check the correctness of your own calculations.
To find the value of the second expression, you also need to use the table case and 8 x 5. But what category will the resulting 40 units belong to? The question remains open for most children. The method of replacing multiplication by the action of addition 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 knowledge is not enough to solve the example. Apparently, there are some other rules for multiplying multi-valued numbers. And they need to be identified.
As a result of the joint efforts of the teacher and children, it becomes clear that in order to multiply a multi-digit number by a multi-digit one, it is necessary to be able to apply the combination law, in which one of the factors is replaced by the product (80 x 50 \u003d 80 x 5 x 10 \u003d 400 x 10 \u003d4000)
In addition, a way 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.
Children's research work
Students are offered a fairly large number of examples of this kind. Children each time try to find an easier and faster way to solve, but at the same time they are constantly required to write down the detailed solution of the solution or detailed verbal explanations.
The 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, the understanding comes that the way of writing such expressions in a line is very inconvenient. There comes a moment when the students themselves suggest writing the multiplication in a column.
Steps in learning multiplication by a multi-digit number
In the guidelines, the study of this topic takes place in several stages. They should follow one after another, enabling students to understand the whole meaning of the studied action. The list of stages gives the teacher an overall picture of the process of presenting material to children:
- independent search by students for ways to find the value of the product of multivalued factors;
- to solve the problem, the combination property is used, as well as multiplication by one with zeros;
- practice the skill of multiplying by round numbers;
- use in calculations of the distributive property of multiplication with respect to addition and subtraction;
- operations with multi-digit numbers and multiplication in a column.
Following these steps, the teacher must constantly draw the attention of children to the close logical connections of previously studied material with what is being mastered in a new topic. Schoolchildren not only do multiplication, but also learn to compare, draw conclusions, and make decisions.
Problems of learning multiplication in elementary school course
A 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-digit numbers in a column. And if he, together with his students over the course of three years of study - in grades 2, 3, and 4 - purposefully and thoughtfully studied the specific meaning of multiplication and all the issues that are associated with this operation, then children should not have difficulties in mastering the topic under consideration.
What problems were previously solved by the students and their teacher?
- Mastering tabular cases of multiplication, that is, getting the result in one step. A mandatory requirement of the program is to bring the skill to automatism.
- Multiplying a multi-digit number by a single-digit number. The result is obtained by repeatedly repeating a step that children already master perfectly.
- Multiplication of a multi-digit number by a multi-digit one is carried out by repeating the steps indicated in paragraphs 1 and 2. The final result will be obtained bycombining intermediate values and matching incomplete products with digits.
Using the properties of multiplication
Before examples of column multiplication begin to appear on subsequent pages of textbooks, grade 4 should learn very well how to use the associative and distributive property to rationalize calculations.
By observing and comparing, students come to the conclusion that the associative property of multiplication for finding the product of multi-digit numbers is used only when one of the factors can be replaced by a product of single-digit numbers. And this is not always possible.
The distributive property of multiplication in this case acts as a universal one. Children notice that the multiplier can always be replaced by the sum or difference, so the property is used to solve any multi-digit multiplication problem.
Algorithm for recording the action of multiplication in a column
The record of multiplication by a column is the most compact of all existing ones. 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 order of performing actions when multiplying in a column is learned by the children as “excellent”.
Skill building exercises
First of all, it should be noted that the examples of multiplication in a column offered to children become more complicated from lesson to lesson. After being introduced to two-digit multiplication, children learn to perform operations with three-digit, four-digit numbers.
To practice the skill, examples with ready-made solutions are offered, but among them, entries with errors are deliberately placed. The task of the students is to find inaccuracies, explain the reason for their occurrence and correct the entries.
Now when solving problems, equations and all other tasks where it is necessary to perform multiplication of multi-digit numbers, students are required to write a column.
Development of cognitive UUD when studying 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 to solve the problem, choosing the most rational method.
Using schemes for reasoning, establishing cause-and-effect relationships, analyzing observed objects based on the identified essential features - another group of formed cognitive skills when studying the topic "Multiplication in a column".
Teaching children how to divide multi-digit numbers and how to write in a column is carried out only after the children learn how to multiply.
