Test sum and product permutation of factors. A tale about Russian mathematics, a kind farmer and stupid customers. Problems to find the sum

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 on 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 from a mathematical point of view, 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 adult life. 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 the physical meaning of the 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 when changing the grouping methods.

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 last two 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, an interconnected multiplication table for 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 in previous lessons, most 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, the difficulty in forming 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

Routing math lesson

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 learning 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 result and the result of your comrades, distinguish a correctly completed task from an incorrect one.

Communication 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 computational skills, mental operations of comparison, classification;

Methods and forms of training : Explanatory and illustrative; individual, frontal, steam room.

Techniques for organizing students' educational activities: searching for new knowledge through interviews and pair work; independent work with pedagogical support for those students who need it

During the classes:

(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 sequencing,

make one's 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.

You can organize 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)

Definition. Multiplication is the action of finding the sum of identical terms. Multiply number A per number b means find the sum b terms, each of which is equal to a.

The numbers that are multiplied are called factors (or factors), and the result of the multiplication is called a product.

At multiplication natural numbers the product is always a positive number. If one of the factors is equal to 0 (zero), then the product is equal to 0. If the product is equal to zero, then at least one of the factors is equal to 0.

If one of the two factors is equal to 1 (one), then work equal to the second factor.

• For example:
• 5 * 6 * 8 * 0 = 0
• 132 * 1 = 132

Multiplication laws

Combination law

Rule. To multiply the product of two factors by a third factor, you can multiply the first factor by the product of the second and third factors.

• For example:
• (7 * 6) * 5 = 7 * (6 * 5) = 210
• (a * b) * c = a * (b * c)

Travel law

Rule. Rearranging the factors does not change the product.

• For example:
• 7 * 6 * 5 = 5 * 6 * 7 = 210
• a * b * c = c * b * a

Distributive law

Rule. To multiply a number by a sum, you can multiply this number by each of the terms and add the resulting products.

• For example:
• 7 * (6 + 5) = 7 * 6 + 7 * 5 = 77
• a * (b + c) = ab + ac

The distributive law also applies to the action of subtraction.

• For example:
• 7 * (6 — 5) = 7 * 6 — 7 * 5 = 7

The laws of multiplication apply to any number of factors in numerical or alphabetic expression. The distributive law of multiplication is used to take the common factor out of brackets.

Rule. To convert a sum (difference) into a product, it is enough to take the same factor of the terms out of brackets, and write the remaining factors in brackets as the sum (difference).