3 4 = 12
WORK
ADDITION OF THE SAME 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 ACTIONS
DIVISION DIVISOR PARTICIPANT
6: 3 = 2
PRIVATE
To find the unknown dividend, you need to multiply the quotient
For a divider.
2 3 = 6
To find the unknown
The divisor is to divide the dividend by the quotient.
6: 2 = 3
1. Division by content
12 apples were arranged 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 IS 3 IN 12.
12: 3 = 4
2. Division into equal parts
12 apples are divided equally among 4 plates. How many apples are on each plate?
We argue:
We take 4 apples, lay out one apple on each plate. Then we take 4 more apples, lay out one more apple on a plate. And we take 4 more apples, lay out again one apple on a plate. Thus, in order to solve the problem, you need to answer the question - HOW MANY TIMES IS 4 IN 12?
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, then another factor is obtained.
CHALLENGES
CLASS
1. The task is analyzed according to the plan:
Nastya collected a bouquet of daisies and cornflowers. There are 6 daisies in the bouquet, and 3 more cornflowers. How many cornflowers are in the bouquet?
- What is the task about? What is the task about?
- Repeat the task.
- Task question.
- What flowers did Nastya make a bouquet of?
- How many daisies were there?
- Do we know how many cornflowers were? / How many cornflowers were. What do we know about cornflowers?
- What do you need to know?
At the end of the analysis, a brief note is recorded, a diagram or drawing is made.
2. In the task, an explanation is always written in all actions, except for the last one.
3. In a task with more than 1 action, an expression is written.
4. The answer is written strictly on the issue of the problem.
PROBLEM TO FIND THE SUM
There were 7 blue cars and 10 red cars on the shelf. How many cars were on the shelf in total?
The way children get acquainted with this rule (law) is due to the previously introduced meaning of the action of multiplication. Using object models of sets, children count the results of grouping their elements different ways, making sure the results don't change from changing the grouping methods.
The counting of elements of a picture (set) in pairs horizontally coincides with the counting of elements in triples vertically. Considering several variants of such cases gives the teacher a reason to make an inductive generalization (that is, a generalization of several special cases in a generalized rule) that rearranging the factors does not change the value of the product.
Based on this rule, used as a counting method, a multiplication table by 2 is compiled.
For example: Using the multiplication table for the number 2, calculate and memorize 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 the tables obtained by rearranging the factors. In fact, many children learn each table separately, because the insufficient level of development of the flexibility of thinking does not allow them to easily rebuild the model of the learned table case scheme in the reverse order. When calculating cases of the form 9 2 or 8 3, the children again return to the method of 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 that are connected by the rule of permutation of factors) does not allow the formation of an associative chain focused specifically on the relationship.
When compiling a multiplication table for the number 5 in grade 3, only the first product is obtained by adding the same terms: 5 5 \u003d 5 + 5 + 5 + 5 + 5 \u003d 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 table of multiplication 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.
The theoretical approach to such a construction of a system for studying tabular multiplication suggests that it is in this correspondence that the child will memorize cases of tabular multiplication.
The most easy 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 attention of the children to the fact that a new case in this lesson is, for example, only the case 9 9, and 9 8, 9 7 etc. items were studied in previous lessons, most of the children perceive the entire proposed volume as material for new memorization. Thus, in fact, for many children, the multiplication table for the number 9 is the largest and most complex (and this is true, if we keep in mind the list of all cases that relate to it).
A large amount of material that requires memorization by heart, the difficulty in forming associative links when memorizing interconnected cases, the need for all children to achieve a solid memorization of all tabular cases by heart within the time limits set by the program - all this makes the topic of studying tabular multiplication in primary school one of the most methodologically difficult. In this regard, questions related to the methods of memorizing the child's multiplication table are important.
Definition. Multiplication is an operation that results in finding the sum of identical terms. Multiply number A per number b means to find the sum b terms, each of which is equal to a.
The numbers that are multiplied are called factors (or multipliers), and the result of multiplication is called a product.
At multiplication natural numbers product is always a positive number. If one of the factors is 0 (zero), then the product is 0. If the product is zero, then at least one of the factors is 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
Laws of multiplication
associative law
Rule. To multiply the product of two factors by the 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)
displacement law
Rule. By rearranging the factors, the product does not change.
- 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 subtraction.
- For example:
- 7 * (6 — 5) = 7 * 6 — 7 * 5 = 7
The laws of multiplication apply to any number of factors in numerical or literal terms. The distributive law of multiplication is used to take the common factor out of brackets.
Rule. To convert the sum (difference) into a product, it is enough to bracket the same factor of the terms, and write the remaining factors in brackets as the sum (difference).
Technological map of the lesson
Item:mathematics
Class: 2
Name of the educational and methodical kit (EMC): “ Promising Primary School»
Lesson topic:"Permutation of factors"
Lesson type: discovery of new knowledge
The place of the lesson in the system of lessons 1
Target:
introduce students to the commutative property of multiplication; to form the ability to apply it in practice; reinforce the meaning of multiplication;
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Tasks:Educational:
Developing:
Educational:
to form the ability to apply it in practice; reinforce the meaning of multiplication;
develop computational skills, mental operations of comparison, classification;
education of interest in the study of the subject, the ability to work in groups.
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Subject UUD:
Regulatory UUD:
Communicative UUD:
Cognitive UUD:
Personal UUD:
the ability to determine and formulate the objectives of the lesson with the help of a teacher; pronounce the sequence in the lesson; work according to a collective plan; evaluate the correctness of the performance of an action at the level of an adequate assessment;
plan your action in accordance with the task; make the necessary adjustments to the action after its completion, based on its assessment and taking into account the nature of the errors made; make one's guess
skill
listen and understand the speech of others; jointly agree on the rules of behavior and communication at school and follow them
ability to navigate in your knowledge system:
to distinguish the new from the already known with the help of a teacher; acquire new knowledge: find answers to questions using a textbook, your life experience and information received in the lesson.
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Planned results:
Subject Results:
Subject Results in ICT:
Metasubject results:
Personal results:
understand what the “commutative property of multiplication” is. Fix the meaning of multiplication .
Be able to solve word problems. To be able to solve combinatorial problems to establish the number of pairs made up of elements of two sets. Finding the whole or parts, read mathematical expressions, inequalities, equalities.
be able to determine and formulate the goal in the lesson with the help of a teacher; pronounce the sequence of actions in the lesson; work according to a collective plan; evaluate the correctness of the performance of an action at the level of an adequate assessment; plan your action in accordance with the task; make the necessary adjustments to the action after its completion, based on its assessment and taking into account the nature of the errors made; make one's guess Regulatory UUD); be able to
formulate your thoughts orally;
listen and understand the speech of others; jointly agree on the rules of behavior and communication at school and follow them ( Communicative UUD); be able to navigate in your knowledge system:
to distinguish the new from the already known with the help of a teacher; acquire new knowledge: find answers to questions using a textbook, your life experience and information received in the lesson (Cognitive UUD).
be able to self-assess
based on success criteria learning activities.
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Basic concepts:
Concepts:
Introduction to the commutative property of multiplication
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Interdisciplinary connections:
Mathematics
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Resources:
additional
EMC "Perspective Primary School" "Mathematics" Grade 2 A.L. Chekin, interactive environment PeroLogo, dzor, handout.
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Didactic
structure
lesson
(lesson stages)
Planned results
Tasks for students, the implementation of which will lead to the achievement of planned results
Activity
students
Activity
teachers
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Stage 1. Organizing time.
Target: student activation
Creating conditions for inclusion in educational activities (motivation)
Stage 1. Organizing time.
Be able to jointly agree on the rules of conduct for communication at school and follow them. (Communicative UUD)
Be able to verbally express your thoughts. (Communicative UUD)
To be able to find the difference between the new and the already known with the help of a teacher .(Cognitive UUD)
Be able to listen and understand the speech of others. (Communicative UUD)
Has the bell rung already? (Yes)
Are we having a math lesson? (Yes)
Are you ready for the lesson? (Yes)
Will you listen carefully to the lesson? (Yes)
Do you want to know something new? (Yes)
So everyone can sit down!
Let's start our lesson. Let's remember the rules of conduct in the classroom.
Why these rules must be followed by each of us.
We have mathematics
So with new topic meet the whole class.
Today we will open without a doubt.
A very important property of multiplication for us.
Everyone, be careful, active and diligent.
Do you want to know a new topic?
Formulate And argue rules of conduct in the classroom.
Listen and watch.
Conducts instruction, prepares students for work. Creates conditions of the internal need for inclusion in educational activities.
Motivates
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2.Updating knowledge.
Target: organize the actualization of the skills of finding the whole or parts;
Organize trial activities for students; arrange by students.individual difficulty.
2.Updating knowledge
(Communicative UUD)
.(Regulatory UUD)
Be able to verbally formulate your thoughts. (Communicative UUD)
Be able to pronounce the sequence of actions in the lesson to express your assumption . (Regulatory UUD)
(Personal UUD)
Front work
1. Write down today's date.
What can you say about the number 12? (natural, two-digit, odd, consists of 1 dec. and 2 units, neighbors 11 and 13)
How to get the number 12 using two single-valued terms?
Can you replace addition with multiplication? Why7
Read the expression in different ways.
1. What does each factor in the number entry mean?
2. Read the words: term, multiplier, product value, sum value, term, multiplier.
What two groups can these words be divided into? (Group 1 - components of the addition action, group 2 - components of the multiplication action)
3. Let's count orally.
The kitten has 4 paws. How many paws does 2 kittens have? (8)
How many ears do 4 dogs have?(8)
How many times does 5 go into 15? (3)
What term must be taken 3 times to get the number to get the number 12? (4)
The goose has 2 wings. How many wings does 7 geese have?
4. Review the notes. How can you name them? (sums)
12+12+12+12+12 22+22+22
Is it possible to replace the operation of addition with multiplication? Why? (Yes, in expressions all terms are the same)
Individual work.
Replace addition with multiplication and calculate the result.
Work with information
Participate in discussion of problematic issues.
own opinion.
Work on one's own
Organizes frontal work, offers tasks for practicing oral calculations
Includes students to discuss problematic issues.
Organize and provide control over the execution of the task.
Organizes individual work
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Stage 3. Formulation of the problem. Target- make the initial assumption that the value of the product does not depend on the permutation of factors.
Stage 3. Formulation of the problem.
Be able to verbally formulate your thoughts. (Communicative UUD)
Be able to navigate your system of knowledge: to distinguish the new from the already known. (Cognitive UUD)
Cognitive UUD
Regulatory UUD
Cognitive UUD
Regulatory UUD
Open the textbook and read the topic of the lesson. ("Permutation of factors")
What is the goal of the lesson? (To get acquainted with the permutation property of the factor)
1. Learn the property of multiplication
2. Be able to apply the commutative law of multiplication
3. Practice math
What will help us achieve our goal of the lesson.
I can tell you;
Or will you work in pairs and bring out yourself? (themselves)
Let's compare and find the result of two tasks?
Well, a physical education lesson, the boys lined up in two lines of 4 people each. How many boys lined up in two lines?
2. The girls lined up for a tennis lesson in 2 columns of 4 people each. How many girls lined up?
Do you think these tasks are different or the same? Can we answer the question of the problem?
What will help us answer the question?
(It will help us to create an illustration for the problem.) Where can we create an illustration? (In the program Pervologo) What should we remember? (Remember the rules of working with a computer.)
Rules for working with a computer
1) Start work strictly,
With the permission of the teacher,
And remember: you are the answer,
For order in the office.
2) If it sparkles somewhere,
Or something smokes.
Don't waste time -
You need to call the teacher.
3) The mouse loves to be
Hands are clean and dry.
It's better not to drink here, not to eat,
So as not to disturb the order.
4) Do not enter in wet clothes,
Don't get your hands wet either.
5) Cords, sockets, wires
You should never touch.
6) Keep your back straight
At a distance of 60 cm
From the screen you sit.
7) You sit at the computer,
You are watching the display.
No extra items
It can't be on the table.
8) Worked, read,
Everything you need is written down.
You turn off your computer
Take everything off the table.
Turn on your computer.
Find the Pervologo folder on your desktop
.
Open it.
1.Select the drawing tool in the tools.
2. Then select backgrounds.
3. Select the Newborn Turtle from the toolbox and place it on the sheet.
4. Select the turtle costume tab from the command tabs:
5. Click on the desired suit. (we need boys and girls) The turtle on the sheet will turn into a boy, then into a girl
6. Copy as many items as you need to solve these problems. while choosing the stamp command
7.Select in tools new text(letter A)
8. Write down the desired expression.
9.Italicize the expression and select the desired font (20)
10.Select the desired color (blue)
11.Click on the letter A in the lower right corner.
12.Check the work.
And now independently depict in the upper left corner first the boys who stand in two lines of 4 people, and in the upper right corner depict the girls.
Work in pairs.
Compare illustrations.
Write down the result by multiplying. 2*4=8(m) and 4*2=8(d)
What conclusion can be drawn? (permutation of the factors does not change the value of the product)
Participate in research and practical work
Fulfill work according to the algorithm proposed by the teacher
Work in pairs
Implement and provide mutual control in cooperation, the necessary mutual assistance
Organize research work
Conducts student instruction.
Learn work in the program Pervologo
Estimate the correctness of the task
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Stage 4.Fizkultminutka.
Communicative UUD
Let's leave the desks. Watch and repeat the movements (music sounds)
Perform movements, mobilize strength and energy
Organizes physical education minute.
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Stage 5 Discovery of new knowledge Purpose: carry out their assumptions that the product does not depend on the order of the factors.
Regulatory UUD
Cognitive UUD
Regulatory UUD
Be able to pronounce the sequence of actions in the lesson. (Regulatory UUD)
Working with the textbook on p.108
Open the textbook on p.108.
Read the dialogue between Masha and Misha.
- How did Misha build the soldiers?
What did Masha say?
- Which of them is right, prove it.
On the board: 5 2 2 5
Can it be argued that the values of these products are equal? Why?
Open your notebooks and write down the corresponding equality of the two expressions.
5 2 = 2 5
Check the validity of this equality by calculating the value of each of the products using addition.
5 2 = 5 + 5 = 10
2 5 = 2 + 2 + 2 + 2 + 2 = 10
Who is right: Masha or Misha? Why? (both are right. The values of the product are equal)
What conclusion did you draw?
(The value of the product does not change from the rearrangement of factors)
Work with information presented in the form of a drawing.
Realize mutual control
Render in cooperation mutual assistance
Formulate and justify own opinion
Organizes individual presentation, exchange of opinions
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Stage 5 Primary fastening.
Find the value of expressions, first based on the formulated property, and then calculations (replacing products with sums)
Develop math skills and logical thinking, building chains of inference
Be able to formulate your thoughts orally and in writing: listen and understand the speech of others ( Communicative UUD), (Regulatory UUD)
Let's once again be convinced of our assumptions (discoveries).
#2, p109 in writing (we make 2-3 columns).
Calculate the values of the products in the column.
1 row-2 column
2 row-3 column
What conclusion can be drawn?
-
Let's check our assumptions with the rule in the textbook on p.109.
Were our discoveries confirmed?
Fulfill tasks
Organizes students learning a new mode of action
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Stage 7. Systematization and repetition of previously studied.
Ability for self-assessment based on the criterion of success of educational activities (Personal UUD)
Working with a computer (TB)
task 2.
Group work (3 people)()
Fulfill tasks
Independent application information. Perform self-test
Recall
group work rules
Organizes doing self-work, self-checking
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Stage 8. Reflection of activity
Target: fix the new content of the lesson; Summarize the work done in class.
Be able to pronounce the sequence of actions in the lesson (Regulatory UUD)
Ability for self-assessment based on the criterion of success of educational activities (Personal UUD)
What new did you learn in the lesson?
Have you completed all the tasks?
Where will we use the new property of multiplication?
Thank you for the lesson.
Formulate end result of your work
Organizes reflection
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Project training session mathematics
Subject and teaching materials: mathematics grade 1, teaching materials "Perspective elementary school".
Topic of the lesson: Addition with the number 10.
Place of the lesson in the topic: 1 lesson
Type of lesson: discovery of new knowledge.
Purpose and expected result: Open new trick addition and use it in assignments different kind.
Lesson objectives (teacher activities):
1. Create a problem situation for the discovery of new knowledge.
2. Contribute to the discovery of students of a new method of addition.
3. Promote the conscious assimilation and application of new knowledge when adding to the number 10.
4. Organize self-assessment of the work of students in the lesson.
Equipment for the lesson: mathematics textbook grade 1 (A.L. Chekin), workbook"Mathematics in questions and assignments" No. 2 (O.A. Zakharova, E.P. Yudina), cards
Stages of the lesson, tasks and activities of students
Teacher activity
Student activities
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Studying
problematic situation.
Learn to see the problem and find ways out of it.
Expressions are written on the board.
Guys, Misha got confused in solving expressions, he was able to solve only one expression. Which?
And with what expressions he could not cope.
Let's help him.
How are these expressions similar?
How are they different?
Find an extra expression? Why do you think it's redundant?
Close with a card the expression that you think is superfluous.
He had already solved such expressions with Masha.
Children answer:
they are similar in that all expressions involve addition.
They differ in that not all expressions have the same second term.
The second expression is superfluous, because the first term is a single-digit number.
Communicative
(children's statements)
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2. Goal setting.
Determine the topic of the lesson, set a goal, learning objectives.
The teacher removes this expression and a note remains on the board:
Open the textbook and read the topic of the lesson. (the topic is posted on the board)
What should be done to find the meaning of these expressions?
I propose to discuss the following course of action in the lesson:
(the plan is posted on the board)
Tasks: 1) 10+2
Fizminutka.
Children read the topic of the lesson.
Addition with the number 10.
Discover a new method of addition and learn how to write down its result.
Open the addition trick with the number 10.
Learn how to correctly write the result of addition with the number 10
Practice solving these examples.
Rate your work.
Search and extraction of information)
Regulatory (goal acceptance and lesson setting)
Regulatory (action planning)
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3. Discovery of new knowledge
Learn how to add single digit numbers with the number 10.
Develop the ability to generalize observations, draw conclusions.
What is the first task of the lesson?
Working with the tutorial on page 32
The teacher reads the task:
Once Misha said: “Masha, I noticed that if you add the number 10 with the single-digit number 2, you get the number 12, in which there is 1 ten and 2 more units.”
Can you tell me how to solve this example using the model?
What can be said?
How many tens and how many units in the number 1
Who wants to run the second model and tell how the expression 10 + 5 is solved
What did you notice as a result of the addition action?
How are these examples similar and different, and why?
Compare your rule with what is in the textbook.
Write down the rest of the addition steps in your notebook.
Can you complete the new scheme by adding any one-digit number to the number 10?
Finish the output:
When adding the number 10 to any one-digit number, a two-digit number is obtained, which has ...
Check our conclusion with the conclusion in the textbook.
Let's summarize the work. Read 1 problem.
Are we up to the task? (put v opposite the completed task)
Well done boys.
Open the addition trick with the number 10.
Children lay out mugs on the board and in notebooks. (10 green and 2 red)
1 term - 10 is denoted in green, the second term - 2 is denoted in red
There are 12 circles in total.
In the number 12 = 1 ten and 2 units.
Children perform similar work.
The result is two-digit numbers.
They are similar in that in the answer the number 1 is in the place of tens, but they differ in that in the place of units in the first example there is the number 2, and in the second -5, because in the first example they added a single-digit number 2, and in the second example they added 5.
in the tens place is the number 1, and in the units place is the digit of this single-digit number.
Children read:
When adding the number 10 to a single-digit number, a two-digit number is obtained, in which the digit 1 is in the tens place, and the digit of this single-digit number is in the ones place.
Regulatory (holding the goal of the lesson)
Communicative (monologic statements of children)
cognitive
(logical observations, comparisons, inferences)
Cognitive (informational)
cognitive
(modeling)
Cognitive (informational)
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4. Shaping
primary skills based on self-control
Learn how to add 10.
Learn to do difficult tasks.
Let's move on to lesson 2.
Task number 2.
Work in pairs.
Read the task.
Take the chips and close the correct amounts.
Write down the amounts in your notebook. What task still needs to be completed?
Have you solved all the addition examples with the number 10?
Run the simulation.
Make a conclusion.
Read lesson 2.
Did we get it right on task 2? (put v opposite the completed task)
Tell us why you value yourself so much?
What task have we not completed yet?
Task number 2 in the notebook on page 31
Read the task.
1 option-1 column (1-4 examples)
Option 2 1 column (5-8 examples)
We do the task ourselves.
Look carefully at the examples of the second column. What needs to be done to make the records correct?
Tell us how to control yourself when writing missing terms?
Option 1 - 2 columns (1-4 examples)
Option 2 - 2 column (5-8 examples)
Can we say that we coped with the 3rd task.
(put v opposite the completed task)
Examples are written on a hidden blackboard. After finishing the work, the children independently check their work.
1 criterion: I know the output when adding to the number 10
Criterion 2: I can write missing terms
Who will tell you how you value yourself?
Write down in a notebook all the sums in which the first term is -10, and the second is a single-digit number.
Children discuss and complete the task in pairs.
Find the value of the sum.
10+1=11, 10+7=17, 10+9=19, 10+4=14
No, there are 2 examples left:
Children draw 2 red circles and 10 green ones.
Children conclude that with this addition, the same result is obtained.
Yes. (Children hold hands)
Several children talk about their work results.
Practice solving these examples.
Fill in the blanks so that the entries are correct.
Mutual check
Write either the first term or the second.
Based on the value of the sum, based on the rule, determine which term is the number 10, and which term is a single-valued term.
Children evaluate themselves according to criteria.
Communicative (statements of children)
Communicative (communication)
cognitive
(modeling)
Regulatory (control)
Cognitive (sign-symbolic and literal)
Regulatory (control)
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5. Reflection
Learn to evaluate your work in class.
What was our goal at the beginning of the lesson?
Did they cope with all the tasks (clearly visible)
1. I can teach another student a new addition trick.
2. I know and can add with the number 10.
3. I know, but I doubt the solution of these examples.
Children are talking.
Self-assessment of students with the help of statements.
Regulatory
(target hold)
Personal
(ability for self-assessment based on the criterion of success in educational activities)
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How funny it is to watch the seething of shit in the minds of people who are far from mathematics, physics, natural sciences in general and on the methods of their teaching in secondary schools.
This is me about the widespread discussion of the "unfair" assessment by the teacher of such a solution to a simple problem:
When people see such an assessment in their heads, as a rule, there is a cognitive dissonance associated with the fact that the majority, albeit intuitively, remember that the multiplication operation is communicative, i.e. from the permutation of the places of the factors, the product does not change, i.e. a*b = b*a.
But here you need to understand that the problem under discussion belongs to the category of the most elementary, when the child not only does not know the properties of multiplication, but has just met for the first time 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 operation of multiplication. And this is not a whim of contemporary Russian Methodists. This is what they wrote in mathematics textbooks 130 years ago: § 42. What is multiplication. Multiplication is the addition of like terms. In this case, the number that is repeated as a term is called a multiplier (it is multiplied), and the number showing how many such identical terms are taken is called a multiplier.(Kiselev, first edition 1884).
The same was written about in the communist textbooks of the beginning of the last century (State pedagogical institute them. Herzen, I.N. Kavun, N.S. Popova, "Methods of teaching arithmetic. For teachers elementary school and students of pedagogical colleges". Approved by the People's Commissariat for Education of the RSFSR, 1934):
It is obvious that the solution proposed by the student shows that he did not understand the essence of the multiplication operation, which was appropriately assessed by the teacher.
Even assuming that the genius student himself guessed (or even knew) about the communicativeness of the multiplication operation, his decision is still wrong. The point is that if he wrote in the solution:
then the answer would be correct. However, liters, as a dimension, are missing on the left side of the equation and appear out of nowhere on the right side. Record same
moreover, is correct, despite the lack of dimension (n) in the left part, because this dimension is omitted, based on the initial conditions of the problem, implying 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 an angry opus biglebowsky
who, with a smug smile, writes frank nonsense, calculating the distance that the car traveled in 2 hours at a speed of 60 kilometers per hour: S = 60km/h * 2h = 120km/h. Next, we remember physical meaning problem and discard the tail of the solution "/h".
And such illiterate people, who are not versed in elementary mathematics and physics, consider it possible and acceptable to scold the one and a half century methods of teaching children the basics of mathematics.
Moreover, they themselves (yes, all of you too) studied multiplication at school in due time. In the USSR, there was one textbook for all schools, and in it the order of factors in the study of the multiplication operation was important. And in the same way, they lowered the marks for rearranging the factors, since this showed the student's misunderstanding of the essence of the multiplication operation and testified to 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 operation, the skill of correctly writing multipliers becomes unnecessary and is forgotten about. But at the same time, one should not forget about the correct dimension. In the end, all further study of physics is based 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 has problems.
