Currently, school is the most important factor in accelerating the socio-economic development of the country. The task is not limited to the formation of knowledge - the school is designed to teach young people to think creatively and act as society demands.
Primary school is the basis, the foundation. It is in elementary school the main part of the work on developing teaching skills must be completed
Therefore, we expect that for low- to intermediate-skilled participants, performance in simple multiplication and division should be significantly correlated. However, for high-skill participants, the revised identical items model assumes independent representations of multiplication and division facts. Thus, the performance of high-skilled participants in multiplication should be related less directly to performance in division, resulting in relatively lower correlations than the correlations for low- or average-skilled participants who use a division-by-mediation strategy.
Teachers at the center primary classes There should be work to improve the lesson through the introduction of forms and methods of active learning, improving methodological skills, overcoming stereotypes in the organization of the educational process, attracting technical and other visual aids, and wider use of new educational technologies.
Because of these relationships and assuming that the solution complex tasks division is based on solving simple division problems, we also expect relatively low correlations between simple multiplication and division difficulty for children with relatively higher skill levels than those with more low levels skills. As a consequence, this pattern should also be present for correlations between simple and complex division problems. Finally, we were also interested in whether the relationship between simple and complex multiplication varied by skill level.
List of used literature
However, the current state of research does not allow for a directional hypothesis on this issue. Moreover, these correlations may be influenced by other cognitive abilities. Because some children are cognitively more advanced than others, performance on many cognitive tasks tends to be correlated in children. To identify the unique contribution of performance on multiplication tasks to performance on division tasks, we controlled for the effects of nonverbal intelligence and short-term memory on these relationships.
Object of study: learning process junior schoolchildren table multiplication and division.
Subject of research: table multiplication and division.
Purpose of the study:
–research on teaching methods for tabular multiplication and division;
– development of table multiplication and division skills among students junior classes;
– using the developmental teaching method when studying table multiplication and division.
Moreover, significant correlations between multiplication and division may only suggest that arithmetic abilities are generally correlated. Therefore, we also used children's performance on addition and subtraction problems as a control variable to determine the specific partial variance shared only by multiplication and division.
We assessed the performance of 392 students in fifth and sixth grades in two different school types in the German educational system. Students were recruited from 20 classes from nine different schools located in urban and suburban areas surrounding the city of Tübingen, mainly in middle-class areas. We assessed the same number of classes for each school type. Because class sizes were smaller in comprehensive schools, this resulted in a lower overall student population than for secondary secondary schools.
Research hypothesis: if, when conducting lessons in primary school systematically organize tasks for visual perception of younger schoolchildren, their performance will be higher.
Compiling and mastering multiplication and division tables
In practice, it is quite often possible to observe that some students mechanically memorize the results of table multiplication, and having forgotten them, they cannot resort to known calculation methods. Therefore, in the process of compiling tables and mastering them, we must strive to develop in children the ability to use various computational techniques when multiplying and dividing and choose from them those that are suitable for this case are the most suitable. So, for example, when compiling a multiplication table by 4, the main computational technique is the method of typing equal terms, that is, multiplication is performed using addition. Let's assume that, having arranged the multiplication elements according to this method, we took 3 times 4 and got 12, then took 4 times 4 and got 16. Then there is no need to start the process of collecting fours from the very beginning. To make a sum of 5 fours, just add 4 to 16, etc. The process of sequential collection of quadruples is written by the following equalities:
In total, our sample consisted of 76 fifth grade and 75 sixth grade students who attended middle school and 112 fifth grade and 129 sixth grade students attending junior high school. Parental consent was obtained before the start of the study.
We observed significantly higher partial correlations between simple multiplication and simple division and between simple multiplication and complex multiplication for middle school students than for secondary school students. A similar model was presented for performance groups. Partial correlations between simple multiplication and simple division were higher for high-skilled students than for low-skilled students. However, we found no differences in partial correlation coefficients between simple multiplication and simple division across age groups.
4×4=4×4+4+4=16,
In cases where the multiplier is more than five, the technique of decomposing the multiplier into a summand is widely used, since here the result of multiplication using sequential addition is more difficult to find:
4×9 = 4×4 + 4×5 = 36
To make computational techniques completely understandable for children, you need to work through them carefully and slowly, specifying each such technique with the help of visual aids. For these purposes, subject didactic material is widely used - cards with images of objects in pairs, triplets and groups; rectangles divided into squares; drawings from the textbook. When compiling and mastering a table, each time attention is paid not only to the correctness of the answer received, but also to how it was obtained, what other ways there might be to calculate the same result, and which one is more rational. If a student finds it difficult to name the product of numbers, he is reminded of the previous line. Knowing the result of this line (or having received it from the teacher), he finds the given product using the method of compiling tables. In the process of calculations, students gradually memorize many tabular works, but this is achieved not through mechanical cramming, but through repeated use of a variety of computational techniques. Along with the methods of conscious assimilation of the table in the process of calculations, you need to use various means, promoting better assimilation and memory. For example, the main work on memorizing tables must be done in class. True, to consolidate the skills of table multiplication, long and varied training and a differentiated system of tasks are required. However, it should be borne in mind that when working on memorizing the multiplication table, you need to resort to computational techniques only in cases of errors.
Results for partial correlations between simple and complex operations varied by grade level, school type, and performance on multiplication and division problems. Partial correlations between simple multiplication and complex division and between simple division and complex division differed between middle school and junior high school students. As hypothesized, the partial correlation was higher for middle school than middle school students. For the performance groups, we found a significant difference for partial correlations between simple and complex multiplications, with higher correlations for high-skilled students than for low-skilled students.
I will give examples of some tasks on mastering and memorizing the multiplication tables. In practice, we have verified that for better memorization The table is useful for its visual perception. In my class, I widely used not only demonstration tables, but also individual ones, which are made during labor lessons. Reading the table by individual students and the whole class can also be used, since some works sound rhythmic and therefore are easy to remember when reading (five five - twenty five ). To better memorize the table, it is useful to present it in such a way that students can immediately cover all the material that they should know by heart. For this purpose, all tabular works are grouped into tens (this is done on a poster, and as they memorize it, it is hung in front of students in parts or in whole):
Other partial correlation coefficients did not differ. Moreover, we found no significant differences between age groups. In the present study, we sought to examine whether strong developmental parallels between multiplication and division persist through secondary school and to what extent this varies by proficiency level. We therefore assessed the performance of fifth and sixth graders in two secondary schools in the German educational system on issues of reproduction and separation. Consistent with our hypotheses, we found that sixth graders outperformed fifth graders, and middle school students outperformed high school students.
This poster is easy to use. The teacher shows one of the horizontal lines, and the students show the numbers from the multiplication of which these products are obtained. For example, they answer: “32 comes from multiplying 4 by 8; 36–6 times 6 or 4 times 9; 45–5 times 9” and so on. Noting in each row those cases that are difficult for children to remember, I try to return to them more often in the future in the process of calculations.
Thus, skill levels varied both between age groups and between school types. In the next section, we first discuss age differences before developing school type differences and finally discuss the specificity of the correlations between multiplication and division problems.
Our results showed that strong developmental parallels between multiplication and division persisted into the fifth and sixth grades of middle school. Specifically, simple multiplication and division were highly correlated, even after controlling for nonverbal intelligence, verbal short-term memory, and arithmetic performance on addition and subtraction tasks.
The next technique is an oral survey according to the table, first sequential, then selective (with presentation of the task by ear: 7 × 5 = ?) with a constant inverse question: “How much will it be if 35 is divided by 5?” – does not violate the value of conscious assimilation, since it does not precede calculations, but follows them.
Avoiding the monotony of techniques when testing mastery of the table, you can widely use games such as: “Who has more examples?” Contents of the game. Students are asked to compose and write down tabular cases multiplication with the following numbers: 35, 48, 81, etc. Examples are compiled in notebooks. The test is carried out as follows: one of the students reads examples with answer 35, the rest underline examples with this answer. The one who comes up with the most examples wins.
Moreover, simple and complex problems were reliably correlated, suggesting that knowledge about simple problems is indeed recruited to solve complex problems. More specifically, we found that students' knowledge of simple division contributed to their performance on complex division problems, even after controlling for performance on simple multiplication.
However, bivariate and partial correlations were lower for complex tasks, which may be due to children's overall poor performance on complex tasks. Thus, the lower correlations may be due to floor effects reducing variability. However, performance on complex problems varied over a fairly large range, suggesting that there is a general gender effect for complex problems. However, it would be interesting to evaluate the current results by giving children more time on complex problems, allowing them to solve more problems.
The topic “Multiplication and division of numbers within 100” is one of the main topics of the initial mathematics course. It is studied in 2nd and 3rd grades.
Knowledge of the multiplication table has always been given great importance. Modern methods require that children not only know the multiplication table, but also understand the principles of compiling a table, which makes it possible to find any product. Therefore, the student must not only learn and remember the results of tabular multiplication, but also be able, if necessary, to calculate the results in the shortest possible way.
According to the model, only students with low and intermediate qualifications should use the separation strategy of mediation. Contrary to this hypothesis, all students were quite proficient in solving multiplication and division problems, as evidenced by their fast solution times. For the breeding problems, five graders took about 26 s per problem, and the sixth grader took about 50 s per problem. For separation problems, fifth graders took about 35 s per problem, and sixth graders took about 13 s per problem.
Developing table multiplication and division skills in students is one of the main tasks of teaching mathematics. Solving this problem is possible by mastering systematic work to consolidate the skills of table multiplication. As a result of such work, students should learn to find the results of table multiplication and division not only correctly and consciously, but also quickly, and know the multiplication table by heart.
However, again this finding can be explained by the nature of the tasks we used. Most simple multiplication problems required students to write two digits, whereas all simple division problems required them to write only one digit. Thus, the slightly faster solution times for simple division problems may be due to the less time required to write solutions in simple division compared to simple multiplication problems. It is important to note that we did not find significant differences in the correlation coefficients between the fifth and sixth graders.
Therefore, when compiling tables and mastering them, you should try to develop in children the ability to use various computational cases that are most suitable when multiplying and dividing.
Compiling tables and assimilating them is a complex and lengthy process in which two stages can be distinguished. The first stage is associated with compiling tables, the second - with their assimilation, i.e., durable memorization. Since in modern elementary school we are talking about the formation of conscious computational skills, the compilation of multiplication (division) tables is preceded by the study of theoretical issues that are the basis of the computational techniques that students will use when compiling these tables.
Thus, we did not find age differences in skill level to modulate the association between multiplication and division. We hypothesize that the high correlations between the simple multiplication and division problems indicate that students relied on division by mediation strategy. Thus, strategy separation dependence on mediation cannot explain age differences in performance because the observed correlations appear to indicate that children in both age groups relied equally on strategy.
The questions on this topic are discussed in the following order: first, the specific meaning of the actions of multiplication and division is revealed and on this basis the first methods of multiplication and division are introduced, a table of multiplying two and dividing by two is compiled; then the commutative property of multiplication is studied, on the basis of which a multiplication table for 2 is compiled; further, the connections between the components and results of the actions of multiplication and division are studied, on their basis, tabular cases of division with the number 2, methods of multiplication and division with the numbers 1 and 10, as well as other tables of multiplication and division are considered; After this, the techniques of multiplication and division with the number zero are introduced.
Differences in performance between students from middle and middle high schools allowed us to look at how the relationship between multiplication and division was affected by differences in proficiency between types of high schools. We found that middle school students outperformed secondary school students. Consistent with the revised identical items theory, we hypothesized that correlations between simple multiplication and division problems should be lower for middle school students than for middle school children.
However, we found the complete opposite. Partial correlations were even higher for middle school students, suggesting that their multiplication and memory sharing networks were even more closely connected. This interpretation was supported by similar findings for partial correlations between simple multiplication and complex division. Solving complex separation problems involves dividing them into simple separation problems. If students who are more knowledgeable about reproduction problems use separation strategy mediation, they should also become better at solving complex separation problems.
Table multiplication and division include cases of multiplying single-digit natural numbers by single digit number and the corresponding division cases:
5·3 = 15; 15:3 = 5
7·4 = 28; 28:7 = 4, etc.
When learning this type of multiplication and division you need to:
1) introduce children to new operations of multiplication and division;
2) study the multiplication and division tables.
In summary, our results indicate that middle school students who performed better in simple and complex division were more likely to rely on or use conceptually or procedurally more consistently the feedback between multiplication and division.
The relationship between skill level and easy and difficult problems was less clear. For the correlation between simple multiplication and complex multiplication, we found that skill level, indexed by differences between school types, had no effect. However, significant correlations between simple and complex multiplication indicate that students from both middle and junior high schools rely on their knowledge of simple multiplications to solve complex multiplications. Thus, our finding shows that students different types schools rely on their knowledge of simple multiplication by a similar sum.
Thus, table multiplication and division are in turn broken down into two questions:
1) familiarity with the operations of multiplication and division;
2) study of multiplication and division tables.
Every teacher knows how difficult it is for children to learn the multiplication and division tables. Therefore, it should be noted that work on revealing the meaning of these actions begins in 1st grade.
Counting is done in groups;
The sums of several identical terms are calculated;
Simple problems are solved: finding the sum of several identical terms, dividing by content, and dividing into equal parts.
The following tasks are used:
1) How many legs do two tables have? What about two coffee tables?
2) How many legs do two geese have? Two roosters?
3) I see 12 bird legs. How many sparrows do I see? .
These tasks can only be solved practically (orally).
In 2nd grade this work receives its natural continuation. First, we become familiar with the action of multiplication. The meaning of this action is revealed through solving simple problems of finding the sum of several identical terms.
The following tasks are offered:
1) Each envelope has 2 stamps. How many stamps are there on 5 such envelopes?
2) There are 6 pencils in one box. How many pencils are in 4 of these boxes?
It is useful to illustrate such tasks (examples) with objects or drawings.
Exercises should be included: using these pictures, create problems (examples) for addition (Fig. 3)
When solving such problems and examples, students notice that there are sums with identical terms and count how many such terms there are.
By performing this operation, children become familiar with the action of multiplication, with the notation of multiplication, and learn the role of factors. The meaning of this action is revealed through solving simple problems of finding the sum of several terms.
Let's show how this can be done.
The teacher suggests solving the problem: “There are 3 pears on each plate. How many pears are there on 4 plates? .
After completing the illustrations, students write down the solution: 3+3+3+3=12.
Teacher. What can be said about the terms of this sum?
Children. Identical.
Teacher. How many are there?
Teacher. Here they took 3 4 times. If the terms are the same, then the sum can be written differently: 3·4=12. This entry is read like this: take 3 4 times, you get 12. (Children repeat.)
Teacher. Multiplication is denoted by a dot.
Teacher. What does the number 3 show in this entry?
Children. The number 3 is taken as a addend.
Teacher. What does the number 4 show?
Children. How many times did you take the number 3 as a term?
Then several exercises are performed to replace the sum with the product. At the same time, children establish what each number shows in the new entry.
It is very important that students understand under what conditions it is possible to replace a sum with a product and when it is not possible. This is helped by solving examples with identical and different terms.
Example on the board: 15+15+15.
Teacher. Replace the example on addition with an example on multiplication.
Children. 15·3.
Teacher. Is it possible to replace the example 22+22+28 with a multiplication example?
Children. It is forbidden.
Teacher. Why?
Children. The terms are different. The terms are not the same.
Teacher. Is it always possible to replace an example on addition with an example on multiplication?
Children. Not always.
Teacher. In what cases can this be done?
Children. When the terms are the same.
Next, the first computational technique for finding a product, based on the specific case of multiplication, is introduced - replacing the product with a sum and performing addition. For example, it is proposed to find the result: 6·4.
Teacher. Read the example.
Children. 6 times 4.
Teacher. What does the number 6 indicate in this entry?
Children. This number is taken as a addend.
Teacher. What does the number 4 mean?
Children. How many terms are taken?
Teacher. Let's replace the multiplication example with an addition example.
Entry: 6+6+6+6=24.
We must pay attention special attention consolidation of knowledge of this technique, since in the future it is used in compiling all multiplication tables. For this purpose, it is useful to teach children to reason when replacing a product with a sum according to a certain plan: name the first factor and say which number is taken as a summand; name the second factor and say how many such terms need to be taken; calculate the amount. For example, when calculating the product 5 3, children reason: the first number (first factor) is 3, therefore, there will be 3 terms; we calculate: 5+5+5=15.
Entry: .
When calculating some sums of identical terms, it is advisable to familiarize children with the technique of grouping terms (without introducing this term) and use this technique when it is convenient. For example, calculating the sum 2+2+2+2+2 +2+2 , we need to draw children's attention to the fact that the sum of five terms is equal to 10, and to 10 it is easy to add the sum of the remaining terms: 10+4=14. This technique is used later when compiling multiplication tables.
Such exercises help to consolidate knowledge of the specific meaning of the action of multiplication and a computational technique based on this knowledge.
1) Compare the expressions and replace the asterisks with the sign “>”, “< » или « = »:
2) Calculate the product, replacing multiplication with addition of identical terms.
9·2 2·3 1·5 0·4 12·2
3) In each column, find the value of the second expression using the value of the first.
9 2 = 18 2 6 = 12 7 4 = 28
9·3 = 2·7 = 7·5 =
4) Explain in different ways, how many cells the rectangle is divided into.
2) 4+4+4=4+4+4 =
The action of division is considered as the inverse of the action of multiplication. This provision is implemented during preparatory work to the study of division. Examples from practical life show the need for division to solve various problems.
The specific meaning of division is revealed in the process of solving simple problems of two types:
1) division by content;
2) division into equal parts.
The student must learn to perform, according to the conditions of the problem, the operation of dividing a given set into a number of equal subsets and associate this operation with the action of division, learn to write down the solution to problems using this action.
The first computational method of division is based on knowledge of the specific meaning of the action of division: students find the quotient by performing actions with objects. For example, to find the quotient of 8:4, they take 8 circles (sticks, etc.), lay them out in 4s and count how many times they got 4 circles, or lay out 8 circles into 4 equal parts and count how many circles they got in each part.
And for a more accurate assimilation of knowledge of the specific meaning of the action of division and a computational technique based on this knowledge, they use the solution of simple problems of division by content and into equal parts, as well as the solution of examples (problems) of division using actions with specific objects (circles, sticks, etc.).
Task. “They put 6 stamps on the envelopes: 2 stamps for each envelope. How many envelopes with stamps did you get? .
To solve this problem, it is necessary to perform practical actions with objects, both by the teacher and the students. The conversation could go like this:
Teacher. I have 6 stamps, and you put the same number of triangles. We will stick them on envelopes in groups of 2, I’m at the board, and you are on the desks. (Puts 2 stamps on the envelopes).
Teacher. How many envelopes did you have 2 stamps on?
Children. For 3 envelopes.
Teacher. Let's write down the solution to this problem. We pasted stamps, divided them, and we will write down the solution using a new action - division. It is written like this:
Answer: 3 envelopes.
":" is a division sign.
Problems involving division into equal parts are treated similarly. At the same time, a demonstration using substantive clarity is also necessary.
Example. “6 apples were divided equally on 3 plates. How many apples did you put on each plate? .
Here we need to show the principle of division into equal parts. The teacher puts out three plates.
Teacher. How many apples do I need to get to put 1 apple on the plates?
Children. 3 apples.
Teacher. How many more apples do I need to take to put 1 more apple on the plates?
Children. 3 apples.
Teacher. To solve the problem, you need to find out how many times 3 is contained in 6. Therefore, the problem is solved by division:
Answer: 2 apples.
At this time, students become familiar with the names of the components and results of multiplication and division: first factor, second factor, product, later - dividend, divisor, quotient. Here children learn that the terms “product” and “quotient” denote not only the result of an action, but also the corresponding expression, for example: 4·3 and 20:5. In connection with the introduction of terms, another way to read examples of multiplication and division is given, for example 4 3: the first factor is 4, the second factor is 3, find the product; 20:5: dividend 20, divisor 5, find the quotient. Children read the expression like this: the product of the numbers 4 and 3, the quotient of the numbers 20 and 5.
Next, we study the commutative property of multiplication. This property is necessary primarily for mastering the operation of multiplication, and in addition, knowledge of this property makes it possible to almost halve the number of cases that need to be memorized by heart. Instead of two cases (8·3 and 3·8), students remember only one.
Students can “discover” the commutative property of multiplication themselves, using visual aids in the form of rows of cells (circles, buttons, stars, etc.). For example, children draw a rectangle and divide it into squares.
It is proposed to find out in two ways how many squares there are (4·3=12 and 3·4=12). Having compared the examples received, students notice that the factors are the same, they have just swapped places, the products are equal.
After completing several similar exercises, students formulate the following properties: “The value of the product does not change by rearranging the factors.”
In order to consolidate knowledge of the commutative property of multiplication, the following exercises are offered:
1) Find the value of the expression in each pair, knowing the value of the first.
4·5=20 7·4=28 9·3=27
5·4=… 4·7=… 3·9=… .
2) Insert “>”, “” instead of asterisks<» или «=»:
8·22·8.
Having compared these expressions in the given exercises, children should notice that in the products the factors are rearranged, therefore, their values are equal.
3) Fill in the missing numbers so that the equations become true.
7·2 = 2·… 9·… =7·9 13·5=… ·13
3·5=…·3…·6=6·10…·18=18·2
When performing the last exercises, knowledge of the commutative property is also used.
After completing a sufficient number of exercises to consolidate, the commutative property is written in general form using letters: a·b=b·a.
Based on the commutative property of multiplication, a table of multiplication by 2 is compiled. Students are asked to compose this table themselves, using the multiplication table of two known to them. This results in the following entry:
2·4=8 4·2=8 etc.
Students reason: “2 multiplied by 3, you get 6, rearrange the factors and multiply 3 by 2, you also get 6,” etc. Here you should introduce another way to read the table: twice two is four, twice three is six, etc. etc., explaining the meaning of the words “twice”, “thrice”, etc. (two times, three times). In order for students to quickly reproduce the results of the multiplication table by 2, it is necessary to include appropriate cases of multiplication more often in oral exercises and in written work.
Based on the commutative property of multiplication, we need to consider the technique of rearranging factors. For this purpose, students are asked to find, using addition, the values of products that differ only in the order of the factors, for example: 2 6 and 6 2, 3 7 and 7 3, etc. Having compared the solutions, students come to the conclusion that it is easier to find the result of multiplication by addition, when we multiply a larger number by a smaller one, since there will be fewer terms. In the future, when compiling multiplication tables, students can, where convenient, rearrange the factors and find the result of the new product. Thus, they can replace the case 3 7 with the case 7 3 and add 3 terms, each of which is equal to 7, instead of adding 7 terms, each of which equals 3.
When considering the relationship between the components and the result of the multiplication action, we lead children to the conclusion: if the product is divided by the first factor, we get the second factor, etc.
The connection between the components and the result of the action is revealed with the help of visual aids. Students are asked to create an example of multiplication using a picture.
Students make up an example: 3·2=6.
Teacher. Name the first factor.
Teacher. Name the second factor.
Teacher. Name the work.
And as a consequence of this, we show that for each example of multiplication, two examples of division can be made.
This results in the following entry:
Teacher. Compare the division examples with the multiplication example. How did you get the second factor 2?
Children. The product 6 is divided by the first factor 3.
Teacher. How did you get the first factor of 3?
Children. The product 6 is divided by the second factor 2.
After completing several similar exercises, students conclude: if the product of two numbers is divided by the first factor, we get the second factor, and if the product of two numbers is divided by the second factor, we get the first factor.
Later, these two outputs are combined into one: if the product of two numbers is divided by one of the factors, you get the other factor.
To ensure that students understand the connection between a product and factors, the following exercises are offered:
1) Calculate the product and, using it, find the quotient.
2·3 6·2 2·7 4·2 9·2
2) Calculate the quotient and, using it, find the product:
16:8 14:2 18:9 10:5 .
3) Calculate the product and in each line, using it, find the quotient.
At the same stage, based on the connection between the product and the factors, tabular cases of division with the number 2 are considered. Students write down from memory the table they know for 2. Then, using knowledge of the connection between the components and the result of the multiplication action, they find the results of the corresponding cases of division.
This results in the following entry:
2·3=6 6:2=3 6:3=2
2·4=8 8:2=4 8:4=2, etc.
Students reason: the product of numbers 2 and 3 is equal to 6; if the product of 6 is divided by the first factor 2, then the second factor is 3, and if the product of 6 is divided by the second factor of 3, the first factor is 2, etc.
In order for students to learn the considered cases of division with the number 2, they must be included more often in oral exercises and in written work.
The connections between the components and the result of division are studied in a similar way: if the quotient is multiplied by the divisor, you get the dividend, and if the dividend is divided by the quotient, you get the divisor.
When consolidating the knowledge of these connections, it is necessary to familiarize students with the technique of selecting quotients. For example, you need to divide 18 by 6; to do this, select a number (quotient) that, when multiplied by the divisor 6, produces the dividend 18; this number is 3, since 6·3=18.
Based on the material studied, multiplication and division techniques with the numbers 1 and 10 are introduced.
First, we consider the technique of multiplying one.
Students solve the problem, find the result by addition: “5 horses each had 1 rider.”
Then, after comparing the result with the factors in each case, they come to the conclusion: when one is multiplied by any number, the result is the number that was multiplied by.
Then the rule of multiplication by 1 is introduced: when multiplying any number by 1, the result is the number that was multiplied, for example, 4·1=4, 12·1=12, a·1=a. Here it is necessary to use the technique of replacing the product with a sum; on the same basis, one cannot rely on rearranging factors. Therefore, it is necessary to inform children of this rule and use it in future calculations.
Division by a number equal to the dividend (3:3=1) is revealed based on the specific meaning of division: if, for example, 3 pencils are divided equally into 3 boxes, then each box will contain one pencil.
Reasoning in this way, students solve several similar examples: 4:4 = 1, 6:6 = 1, etc. At the same time, they notice that when divided by a number equal to the dividend, the quotient turns out to be 1.
Division by 1 is introduced based on the connection between the components and the result of the multiplication action: knowing that 1·4=4, we find that 4:1=4. Having thus solved a number of examples and compared them with each other, students come to the conclusion: when any number is divided by one, the quotient results in the same number. They use this conclusion later in their calculations.
When multiplying 10 by single-digit numbers, students use the following technique: to multiply 10 by 2, you can multiply 1 ten by 2, you get 2 tens, or 20. When multiplying by 10, children use the commutative property of multiplication: to multiply 2 by 10, you can multiply 10 by 2, you get 2 tens, or 20. When dividing, knowledge of the relationship between the components and the result of the division is used: in order to divide 20 by 10, you need to select a number that, when multiplied by 10, will result in 20; this is 2; that means 20:10=2. We also find that 20:2=10.
All of the above questions help when considering the next question, i.e. when studying the multiplication table. By looking at them, we prepared children to study the multiplication tables.
Studying the multiplication and division tables is a very important stage in learning the topic. The basic knowledge requirements for students in the program states: “Students must know the multiplication tables and the corresponding cases of division.” Studying the multiplication and division tables offers the following points:
· work on compiling a table;
· work that ensures its memorization.
When compiling and mastering a table, each time attention is paid not only to the correctness of the result obtained, but also to how the answer was obtained, what other ways there might be to calculate the same result, which of them are more rational.
Knowledge of the multiplication table has always been given great importance. Modern methods require that children not only know the multiplication table, but also understand the principles of compiling a table that makes it possible to find any product. Therefore, the student must not only learn and remember the results of tabular multiplication, but also be able, if necessary, to calculate the results in the shortest possible way.
In practice, it is quite often possible to observe that some students mechanically memorize the results of table multiplication, and, having forgotten them, cannot resort to known calculation methods. Therefore, in the process of compiling tables and mastering them, you need to try to develop in children the ability to use various computational techniques when multiplying and dividing and choose from them those that are most suitable for a given case.
Mastering the meaning of multiplication and the ability to apply this meaning in practice allows students to independently cope with compiling the multiplication table.
The commutative property of multiplication allows you to reduce the number of table cases that need to be memorized.
Mastery of the tabular case of multiplication is expected to provide knowledge of the tabular cases of multiplication.
Work begins on compiling the first multiplication and division tables at the preparation stage. When compiling, all the examples that the children have already learned in previous lessons are used.
Work begins on compiling the first multiplication and division tables at the preparation stage.
So, after revealing the meaning of the action of multiplication as the addition of identical terms, the first multiplication table for the number 2 is compiled. Here it is important to show children the principle of obtaining the result of an action.
2 6 2+2+2+2+2+2
2 7 2+2+2+2+2+2+2
2 8 2+2+2+2+2+2+2+2
2·9 2+2+2+2+2+2+2+2+2 .
However, here from the very beginning (starting with studying the multiplication table of two), it is useful to use the commutative property of a product to obtain the result. So, say, instead of adding 9 times 2, calculating the product 2·9, you can replace this example with another: 9·2 - and find the result like this: 9+9=18. Next, a table is compiled.
Here it is important to show children that if we know the corresponding result of the first table, then there is no need to calculate and write in the second.
Each multiplication table of a particular number compiled for the first time should appear in front of the children so that they understand the principle of its compilation. The table is written in a column, then in relation to each of the examples, an example is compiled for it, obtained by rearranging the factors, and two examples for division. When studying this issue, students rely on finding an unknown factor and show the principle of compiling reciprocal examples for multiplication and division:
8·3 3·8 24:8 24:3 .
On this basis, two division tables with the number 2 are compiled. This work must be duplicated on the board so that the correctly written multiplication tables and the corresponding division tables are in the notebooks.
Thus, already at the preparatory stage, before studying the multiplication and division tables, we introduced the children to the principle of compiling each of the four tables and how to use them.
We begin studying the multiplication and division tables with repetition and division with the number 2. We put all 4 tables compiled earlier together, remember the principle of compiling each of them, analyze in detail the rules for using them using specific examples, and guide children to memorize them.
Then we move on to studying tables with other numbers: 3, 4, 5, ..., 9. Each new table begins with the case of multiplying two identical numbers (for example, when studying the multiplication of four: 4 4), since all previous cases of multiplication of a given number are already known - they can be obtained in the tables discussed earlier if the factors are rearranged.
For each number, the teacher, together with the children, compiles all 4 tables in one lesson, continues to develop the children’s ability to work with them, and works to memorize them.
Work on memorizing the multiplication and division tables should begin in the same lesson where it was compiled. In this case, it is assumed that only the first of the four should be memorized, and children will quickly and confidently obtain the result in the rest based on the result of the first table and the corresponding rules of independence.
For example, if 3·4=12, then 4·3=12, because Rearranging the factors does not change the product. 12:3=4 and 12:4=3, because if we divide the product 12 by the first factor 3, we get the second factor 4, and if we divide it by the second factor 4, we get the first factor 3.
However, as practice and test results show, children quite often successfully master the first table, but find the results of the rest, especially division tables, with great difficulty.
This situation raises the problem of finding ways to improve the methodology for memorizing tabular cases of multiplication and division.
When working with a table, it is advisable to orient children to the obligatory memorization of the first column, teach them how, knowing the result of the first column, get the results of the rest in a given line, and even practice line-by-line memorization.
It should be noted that the teacher, in the process of memorizing the table, must systematically monitor and record how each child is progressing in mastering it. To do this, almost every lesson should include training work. The tasks offered to children should be varied and encourage the inclusion of all children in the class. It is necessary to use techniques, forms of work that help maintain children's interest, as well as various means of feedback.
At the same time, the teacher must provide the necessary practical assistance to children, especially at first. Some columns of the table, which have a large number of cases to remember, are difficult to learn in one go. In this case, you need to memorize it in parts, and determine exactly how many cases to learn today and how many tomorrow. It is also necessary to give practical advice on how to memorize (read, try to write down, forgetting - read and remember, close the answers, repeat, etc.).
To check the mastery of the table, it is advisable to use various forms of testing: frontal survey, mathematical dictation, punched cards, cards with mathematical tasks, games, etc.
As you master the table, you should take into account the level of its memorization when checking:
First, time is given for calculations;
Then exercises are given with a time limit (the automaticity of assimilation is checked).
After studying all the multiplication tables, cases of multiplication and division with zero are considered.
First, the case of multiplying zero by any number (0·5, 0·2, 0·7) is introduced. Students find the result by addition (0·2=0+0, 0·3=0+0+0=0). Having solved a number of similar examples, students notice that when zero is multiplied by any number, the result is zero. They are guided by this rule in the future.
If the second factor is zero, then the result cannot be found by addition, and the permutation of factors cannot be used, since this is a new area of numbers in which the commutative property of multiplication has not been revealed. Therefore, the second rule: “The product of any number and zero is considered equal to zero” - the teacher simply tells the children.
Both of these rules are then applied to various calculation exercises.
The division of zero by any number not equal to zero (0:6) is considered based on the relationship between the components and the result of the division. Students reason like this: in order to divide 0 by 6, you need to find a number that, when multiplied by 6, will result in 0. This is zero, since 0·6=0. So 0:6=0. As a result of solving a number of similar examples, students notice that when zero is divided by any number that is not equal to zero, the quotient is equal to zero. In the future, students use this rule.
As you know, you cannot divide by zero. This fact is communicated to children and explained with an example: 8 cannot be divided by 0, since there is no such number that, when multiplied by 0, produces 8.
It is necessary to include cases of multiplication and division with numbers 0 and 1 more often in training exercises, comparing the corresponding techniques (5·0 and 5·1) to prevent confusion.
