Multiplication. Commutative property of multiplication. The product does not change due to rearrangement of factors. At this time
3 4 = 12
WORK
ADDITION OF IDENTICAL TERMS CAN BE REPLACED BY MULTIPLICATION.
The multiplication sign is a dot (·).
2 3 = 6
3 2 = 6
2 3 = 3 2
COMPONENT NAMES
MULTIPLICATION OPERATIONS
Dividend Divisor Quotient
6: 3 = 2
PRIVATE
To find an unknown dividend, you need to multiply the quotient
To the divider.
2 3 = 6
To find the unknown
Divisor, you need to divide the dividend by the quotient.
6: 2 = 3
1. Division by content
12 apples were placed on plates, 3 apples on each plate. How many plates did you need?
In order to solve the problem, you need to answer the question - HOW MANY TIMES EVERY 12 IS THERE 3.
12: 3 = 4
2. Division into equal parts
12 apples were divided equally onto 4 plates. How many apples are on each plate?
We reason:
Take 4 apples, place one apple on each plate. Then take 4 more apples, place one more apple on a plate. And take 4 more apples, again place one apple at a time on a plate. Thus, in order to solve the problem, you need to answer the question - HOW MANY TIMES EVERY 12 IS THERE 4.
CONNECTION
BETWEEN RESULT AND
COMPONENTS OF MULTIPLICATION
4 2 = 8
8: 4 = 2
8: 2 = 4
If the product of two factors is divided by one of them, you get the other factor.
TASKS
CLASS
1. Analysis of the problem occurs according to plan:
Nastya collected a bouquet of daisies and cornflowers. The bouquet contains 6 daisies, and 3 more cornflowers. How many cornflowers are there in a bouquet?
- What is the problem talking about? What does the problem say?
- Repeat the problem statement.
- Problem question.
- What flowers did Nastya make the bouquet from?
- How many daisies were there?
- Do we know how many cornflowers there were?/ How many cornflowers there were. What do we know about cornflowers?
- What do you need to know?
At the end of the analysis, a short note is written down and a diagram or drawing is made.
2. An explanation is always written in the task for all actions except the last one.
3. In a problem with more than 1 action, an expression is written.
4. The answer is written strictly on the question of the task.
PROBLEMS TO FIND THE SUM
There were 7 blue cars and 10 red cars on the shelf. How many cars were there in total on the shelf?
How funny it is to watch the seething of shit in the heads of people who are far from mathematics, physics, natural sciences in general and from the methods of teaching them in secondary schools.
I’m talking about the widespread discussion of the teacher’s “unfair” assessment of this solution to a simple problem:
When people see such an assessment, a cognitive dissonance usually arises in their heads due to the fact that the majority, albeit intuitively, remember that the multiplication operation is communicative, i.e. Rearranging the places of the factors does not change the product, i.e. a*b = b*a.
But here you need to understand that the problem under discussion belongs to the category of the most basic ones, when the child not only does not know the properties of multiplication, but has just for the first time encountered the concept of multiplication, introduced as the addition of identical terms.
So with mathematical point In our opinion, the solution to the problem should look like this:
2l + 2l + 2l + 2l + 2l + 2l + 2l + 2l + 2l = 2l * 9 = 18l
And the order of the factors is really important for understanding the multiplication operation. And this is not a quirk of modern Russian methodologists. This is exactly what they wrote in mathematics textbooks 130 years ago: § 42. What is multiplication. Multiplication is the addition of identical terms. In this case, the number that is repeated as a addend is called multiplicand (it is multiplied), and the number showing how many such identical addends are taken is called a multiplier.(Kiselev, first edition 1884).
The same thing was written about in communist textbooks at the beginning of the last century (State pedagogical institute them. Herzen, I.N. Kavun, N.S. Popova, "Methods of teaching arithmetic. For primary school teachers and students of pedagogical colleges." Approved by the People's Commissariat of Education of the RSFSR, 1934):
It is obvious that the solution proposed by the student shows his lack of understanding of the essence of the multiplication operation, which was assessed accordingly by the teacher.
Even assuming that the student is a genius himself guessed (or even knew) about the communicative nature of the multiplication operation, his solution is still incorrect. The point is that if he had written in the decision:
then the answer would be correct. However, liters, as a dimension, are absent on the left side of the equation and appear out of nowhere on the right. The recording is
in this case, it is correct, despite the absence of dimension (l) on the left side, because this dimension is omitted based on the initial conditions of the problem, which imply that the dimension of the answer will be the same as the dimension of the multiplicand, which always comes first.
By the way, misunderstanding of dimensions leads to sad consequences in adulthood. Read the angry opus biglebowsky who, with a smug smile, writes outright nonsense, calculating the distance a car traveled in 2 hours at a speed of 60 kilometers per hour: S = 60km/h * 2h = 120 km/h. Next we remember physical meaning problem and discard the tail of the solution "/h".
And such illiterate people, who do not understand elementary mathematics and physics, consider it possible and acceptable to criticize the century and a half methods of teaching children the basics of mathematics.
Moreover, they themselves (and all of you too) studied multiplication in school in their time. In the USSR, there was one textbook for all schools, and in it the order of factors when studying the operation of multiplication was important. And in the same way, grades for rearranging factors were reduced, since this showed the student’s lack of understanding of the essence of the multiplication operation and indicated a simple selection of factors, without understanding the essence of the phenomena.
Another thing is that later, after studying the laws of multiplication and consolidating knowledge about the communicativeness of the multiplication option, the skill of correctly writing factors becomes unnecessary and is forgotten about. But we must not forget about the correct dimension. In the end, all further study of physics is built on this.
In general, I wanted to convey a simple idea. If a person does not understand what the teacher tells him, then, as a rule, it is not the teacher’s fault, but the person’s problem.
The way children are introduced to this rule (law) is determined by the previously introduced meaning of the action of multiplication. Using object models of sets, children calculate the results of grouping their elements in different ways, making sure that the results do not change if the grouping methods are changed.
Counting elements of a picture (set) in pairs horizontally coincides with counting elements in triplets vertically. Consideration of several variants of similar cases gives the teacher the basis to make an inductive generalization (i.e., a generalization of several special cases in a generalized rule) that rearranging factors does not change the value of the product.
Based on this rule, used as a method of counting, a multiplication table by 2 is compiled.
For example: Using the multiplication table for the number 2, calculate and remember the multiplication table for 2:
Based on the same technique, a multiplication table by 3 is compiled:
The compilation of the first two tables is distributed over two lessons, which accordingly increases the time allotted for memorizing them. Each of the two latest tables is compiled in one lesson, since it is assumed that children, knowing the original table, should not separately memorize the results of tables obtained by rearranging factors. In fact, many children learn each table separately, since the insufficient level of development of flexibility of thinking does not allow them to easily rebuild the model of the memorized table diagram in reverse order. When calculating cases of the form 9 2 or 8 3, children again return to sequential addition, which naturally takes time to obtain a result. This situation is most likely generated by the fact that for a significant number of children, such a separation in time of interconnected cases of multiplication (those connected by the rule of rearranging factors) does not allow the formation of an associative chain focused specifically on interconnection.
When compiling a multiplication table for the number 5 in grade 3, only the first product is obtained by adding identical terms: 5 5 = 5 + 5 + 5 + 5 + 5 = 25. The remaining cases are obtained by adding five to the previous result:
5 6 = 5 5+ 5 = 30 5 7 = 5 6+ 5 = 35 5 8 = 5 7 + 5 = 40 5 9 = 5 8 + 5 = 45
Simultaneously with this table, a multiplication table interconnected with it by 5 is compiled: 6 5; 7 5; 8 5; 9 5.
The multiplication table for the number 6 contains four cases: 6 6; 6 7; 6 8; 6 9.
The 6 multiplication table contains three cases: 7 6; 8 6; 9 6.
Theoretical approach to such a construction of a system for studying table multiplication assumes that it is in this correspondence that the child will remember cases of table multiplication.
The easiest to remember multiplication table for the number 2 contains the largest number of cases, and the most difficult to remember multiplication table for the number 9 contains only one case. In reality, considering each new “portion” of the multiplication table, the teacher usually restores the entire volume of each table (all cases). Even if the teacher draws the children’s attention to the fact that a new case in this lesson is, for example, only the case 9 9,a 9 8, 9 7it. items were studied on previous lessons, the majority of children perceive the entire proposed volume as material for new learning. Thus, in fact, for many children, the multiplication table for the number 9 is the largest and most complex (and this is indeed the case, if you keep in mind the list of all cases that relates to it).
A large amount of material that requires memorization, difficulty in education associative connections when memorizing interrelated cases, the need for all children to achieve solid memorization of all tabular cases by heart within the time limits established by the program - all this makes the topic of studying tabular multiplication in primary school one of the most methodologically complex. In this regard, issues related to how a child memorizes the multiplication tables are important.
Demonstration lesson in mathematics in 2nd grade
Mathematics lesson technological map
in 2nd grade on the topic “Permutation of factors”
Item: mathematics Class: 2-a
Lesson topic : Rearrangement of multipliers.
Target: creating conditions for students to achieve educational results:
- personal: 1) have a positive attitude towards school and learning; demonstrate cognitive needs and educational motives; Be organized and disciplined in class.
2) show attention and patience to the interlocutor, the ability to self-assess one’s activities.
- meta-subject:
Cognitive UUD:mine new knowledge, find the necessary information, process information (analysis, comparison) presented in different forms.
Regulatory UUD:together with the teacher, discover and formulate an educational problem,determine the purpose of your work, evaluate your own results and the results of your comrades, distinguish a correctly completed task from an incorrect one.
Communicative UUD:listen and engage in dialogue,defend one's position, express one's opinion, participate in group discussion,collaborate in pairs, speak in front of the class,
- subject: understand what the “commutative property of multiplication” is, be able to apply it, consolidate the meaning of the action of multiplication, and develop computational skills in mental calculation.
Lesson objectives:
introducing students to the commutative property of multiplication using specific examples;
develop the ability to apply it in practice; consolidate the meaning of multiplication;
development of mathematical speech based on the use of the studied pattern; develop computing skills, mental operations comparisons, classifications;
Methods and forms of training : Explanatory and illustrative; individual, frontal, steam room.
Organization techniques educational activities students: searching for new knowledge through interviews and pair work; independent work With pedagogical support those students who need it
During the classes:
Didactic structure lesson(lesson stages
Teacher activities
Activity
students
Planned results
1.Motivation for learning activities .
Reception: expressing good wishes to students
The bell called us all to class,
We have a math lesson.
Let's think and reason.
It's time for us to start our lesson.
Want to learn something new? (Yes)
So everyone can sit down!
Let's start our lesson.
Be attentive, active and diligent, everyone.
Open your notebooks and write down the number and class work.
Express good wishes to each other.
Write down the date and type of work.
Organizing time.
Be able to jointly agree on the rules of communication behavior at school and follow them.
Updating knowledge.
Look at numeric expressions
(Slide)
2 + 2 + 2 + 2
5 + 5 + 55 + 5
6 + 6 + 6
Find the extra expression.
Why did you choose the third expression?
What do all expressions have in common?
What action can be used to replace the sum of identical terms?
Present the sums as a product and find the values.
Checking from a slide(slide)
What does the work consist of?
What results from the action of multiplication?
What action do we continue to work with?
Find an unnecessary expression.
- the terms are not the same
-multiplication
2*4=8
6*3=18
-From multipliers.
-the meaning of the work
-With the action of multiplication
(Communicative UUD)
Be able to pronounce the sequence of actions,
make your guess.(Regulatory UUD)
Be able to verbally formulate your thoughts.(Communicative UUD)
Formulation of the problem. Lesson topic.
Goal setting
There are envelopes on your desks. (Envelope No. 1)
Analyze the contents of the envelope, what do you already know?
Whatis unknown and new to you.
What we have learned, we know, put it back in the envelope.
And leave what is new to you in front of you.
What topic will we work on?
How will this help us check the topic of the lesson?
Let's check and compare if we are right.
Let's define the goals of our lesson.
- What will we need to know?
- What will we learn then?
Let's try to assess our knowledge on the topic at the beginning of the lesson. And then we compare the result at the end of the lesson at the end of the lesson.
Complete the task in envelope No. 1
Check on slide
- textbook contents
What is permutation of factors?
Learn to apply the rule when performing various tasks
Be able to verbally formulate your thoughts.(Communicative UUD)
Be able to navigate your knowledge system: distinguish the new from the already known.(Cognitive UUD)
Initial assessment of knowledge on the topic
Let's try to assess our knowledge on the topic at the beginning of the lesson. And then we compare the result at the end of the lesson at the end of the lesson.
Knowledge is assessed at the beginning of the lesson.
(traffic lights)
(Personal UUD)
Discovery of new knowledge.
Now we're going to play soldiers a little. We will work in pairs.
There are little soldiers in envelopes on your tables. (envelope No. 2)
Try (in pairs) to arrange all the soldiers in a column of 2
What did you do7 Who can demonstrate at the board using the example of sailors?
(Option 2: If children find it difficult, open their textbooks)
Look at the illustration where Masha and Misha are playing soldiers and arguing.
Misha tells his sister that he arranged the soldiers in 2 ranks, each with 5 soldiers. But Masha believes that the soldiers are lined up in 5 rows. There are 2 soldiers in each row. Which child is right?
Write it down total number soldiers in the form of a worktwo ways.
- Is it possible to say that the values of the products will be equal?
What sign should we put between the works? Why?
5*2=2*5
How can you check that this equality is true?
What surprised you?
We are explorers! Let's check if this statement is true for other expressions?
Working in pairs with soldiers
I give you time to complete the task.
Explanation at the board.
Children explaining new material at the blackboard
We listen to the children’s opinions and suggest that they arrange the chips in the same way as the soldiers stand
Two children write two options at the board
We check orally and write on the board: 5 2 And 2 5
-Yes, since this is the same number of soldiers.
- The multipliers are the same, only they are swapped,
Replace multiplication with the sum of identical terms.
You can call two students to the board, asking one to calculate the value of the product 5 2, and the other to calculate 2 5 (5 2 = 5 + 5 = 10, 2 5 = 2 + 2 + 2 + 2 + 2 = 10).
The factors are swapped, but the value of the products is the same
Be able to pronounce the sequence of actions in the lesson.(Regulatory UUD)
Primary consolidation.
Application of knowledge
Let's check our assumptions (discoveries) once again.
Let's complete task No. 2
3 tbsp. - 1 row
4 st. - 2nd row.
5 st. - 3 row
What rule did you use to complete this task?
- Have our discoveries been confirmed?
What conclusion can be drawn?
- Let's compare our assumptions with the rule in the textbook on p. 109.
Do you know what rearranging factors is called in mathematics? Commutative property of multiplication or commutative law of multiplication.
Task No. 3 (oral)
2 8 = 8 2
9 4 = 4 9
5 3 = 3 5
8 4 = 4 8
5 9 = 9 5
3 7 = 7 3
Perform 1 and 2 columns - together at the board.
Swap notebooks with your neighbor and evaluate his work (mutual check).
rule for rearranging factors
They conclude: Rearranging the factors does not change the value of the product.
Read the rule
Be able to express your thoughts orally and in writing: listen and understand the speech of others ( Communicative UUD), (Regulatory UUD)
Be able to verbally formulate your thoughts. (Communicative UUD
Self-control
Evaluation of results
of their actions
Task No. 4 (U-1, p. 109)
Using the knowledge gained. Complete the task yourself.
- Let's read the wording of the task. (Find the values of the first product) How will we do it?(
We illustrate on the board a sample of the written form of an oral response.
Self-verification(answers on slide)
Who made two mistakes – 4
Who made 3 mistakes - 3
Independent work.
Can be arranged pair work,
If your children find it difficult, ask your neighbor!
-To find the value of the product 5 4 we used
equality 4 5 = 20.)
5 4 = 4 5 = 20.
Students independently find the remaining meanings of the works and make notes
Evaluate the completed task
Be able to pronounce the sequence of actions in class and express your guess. (Regulatory UUD)
Be able to evaluate your actions, your assumptions. (Regulatory UUD)
Reflection of activity. Lesson Summary
What task was given in the lesson?
Did you manage to achieve your goal?
Where will we use the new property of multiplication?
Whose results changed? Complete the sentences….
Thank you for the lesson!
Assessment using traffic lights.
The ability to self-assess based on the criterion of success in educational activities (Personal UUD)