Routing lesson
Item:mathematics
Class: 2
The name of the educational and methodological set (TMC): " Promising Primary School»
Lesson topic:"Rearranging multipliers"
Lesson type: discovery of new knowledge
Place of the lesson in the lesson system 1
Target:
to acquaint students with the movable property of multiplication; to form the ability to apply it in practice; consolidate the meaning of multiplication;
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Tasks:Educational:
Developing:
Educational:
to form the ability to apply it in practice; consolidate the meaning of multiplication;
develop computing skills, thought operations comparison, classification;
fostering 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 define and formulate the goals of the lesson with the help of a teacher; pronounce the sequence in the lesson; work according to a collectively drawn up plan; evaluate the correctness of the action at the level of an adequate assessment;
plan your action in accordance with the task at hand; make the necessary adjustments to the action after its completion, based on its assessment and taking into account the nature of the mistakes made; make a guess
skill
listen to and understand the speech of others; jointly agree on and follow the rules of conduct and communication at school
the ability to navigate in your knowledge system:
to distinguish the new from the already known with the help of the teacher; gain new knowledge: find answers to questions using a textbook, your life experience and information from the lesson.
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Planned results:
Subject results:
Subject results in the field of ICT:
Metasubject results:
Personal results:
understand what the "transposable property of multiplication" is. To consolidate the meaning of multiplication .
Be able to solve word problems. To be able to solve combinatorial problems to establish the number of pairs composed of elements of two sets. Finding whole or parts, read mathematical expressions, inequalities, equality.
be able to define and formulate a goal in the lesson with the help of a teacher; to pronounce the sequence of actions in the lesson; work according to a collectively drawn up plan; evaluate the correctness of the action at the level of an adequate assessment; plan your action in accordance with the task at hand; make the necessary adjustments to the action after its completion, based on its assessment and taking into account the nature of the mistakes made; express your assumption ( Regulatory UUD); be able to
formalize your thoughts orally;
listen to 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 the teacher; gain new knowledge: find answers to questions using the textbook, your life experience and information received in the lesson (Cognitive UUD).
be able to self-assess
based on success criterion learning activities.
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Basic concepts:
Concepts:
Introduction to the travel property of multiplication
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Interdisciplinary connections:
Mathematics
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Resources:
additional
Educational complex "Perspective primary school" "Mathematics" 2nd grade AL Chekin, interactive environment PeroLogo, tsor, handouts.
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Didactic
structure
lesson
(steps of the lesson)
Planned results
Student assignments that will lead to achievement of planned results
Activity
pupils
Activity
teachers
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Stage 1. Organizing time.
Target: student activation
Creation of conditions for inclusion in educational activities (motivation)
Stage 1. Organizing time.
To be able to jointly agree on the rules of behavior 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 already rung? (Yes)
Are we having a math lesson? (Yes)
Are you ready for the lesson? (Yes)
Will you listen to the lesson attentively? (Yes)
Want to know something new? (Yes)
So everyone can sit down!
We begin our lesson. Let's recall the rules of conduct in the lesson.
Why these rules need to be observed by each of us.
We have math
Hence, with new topic get to know the whole class.
Today we will open it without a doubt.
The property of multiplication is very important for us.
Be all attentive, active and diligent.
Do you want to get acquainted with a new topic?
Formulate and argue rules of conduct in the lesson.
Listen and watch.
Conducts instruction, sets students up for work. Creates conditions of the internal need for inclusion in educational activities.
Motivates
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2.Knowledge update.
Target: organize the actualization of the skills of finding the whole or parts;
Organize learners with trial activities; to organize by students. individual difficulties.
2.Knowledge update
(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 guess ... (Regulatory UUD)
(Personal UUD)
Frontal work
1. Make a note of 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 do you get the number 12 using two single-digit terms?
Can you replace addition with multiplication? Why7
Read the expression in different ways.
1. What does each factor in the number record 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 legs. How many paws do 2 kittens have? (eight)
How many ears does 4 dogs have? (8)
How many times is 5 in 15? (3)
Which 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 records. How can you call them? (amounts)
12+12+12+12+12 22+22+22
Can you replace the action of addition with multiplication? Why? (Yes, in expressions all terms are the same)
Individual work.
Replace addition with multiplication and compute 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 in a discussion of problematic issues.
Organizes and provides control over the implementation 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 the factors.
Stage 3. Formulation of the problem.
Be able to verbally formulate your thoughts. (Communicative UUD)
To be able to navigate in your system of knowledge: to distinguish the new from the already known. (Cognitive UUD)
Cognitive UUD
Regulatory UUD
Cognitive UUD
Regulatory UUD
Open the tutorial and read the topic of the lesson. ("Swap Multipliers")
What is the goal of the lesson? (Get to know the multiplier permutation property)
1. Learn the property of multiplication
2. Be able to apply the displacement law of multiplication
3. Exercise in calculations
What will help us achieve our lesson goal.
I can tell you;
Or will you work in pairs and pull yourself out? (themselves)
Let's compare and find the result of the two tasks?
Well, a physical education lesson, the boys lined up in two lines of 4 people each. How many boys are lined up in two lines?
2. Girls for a tennis lesson lined up in 2 columns of 4 people each. How many girls are lined up?
Do you think these tasks are different or the same? Can we answer the problem question?
What will help us answer the question?
(Creation of an illustration for the problem will help us.) Where can we create an illustration? (PervoLogo program) What should we remember? (Let's remember the rules for working with a computer.)
Rules for working with a computer
1) Start work strictly,
With the permission of the teacher,
And consider: you are in charge,
For order in the office.
2) If somewhere sparkles,
Or smokes something.
Don't waste your time -
You need to call the teacher.
3) Loves the mouse to be
Hands are clean and dry.
It's better not to drink here, not to eat,
So as not to break the order.
4) Do not enter wet clothes,
Do not wet your hands either.
5) Cords, sockets, wires
Never touch.
6) Keep your back exactly,
At a distance of 60 cm
You sit from the screen.
7) You sit at the computer,
You watch the display.
No unnecessary items
Can't be on the table.
8) I have worked, read,
I wrote down everything I needed.
Turn off your computer
Take everything off the table.
Turn on your computer.
On your desktop, find the PervoLogo folder
.
Open it.
1.Choose a 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 tab of the turtle's costumes among the command tabs:
5. Click on the suit you want. (we need boys and girls) The turtle on the leaf will turn into a boy, then into a girl
6. Copy as many items as you need to solve these problems. while choosing the command stamp
7.Select new text in the tools (letter A)
8. Make a note of the expression you want.
9.Highlight the expression in italics and select the desired font (20)
10.Choose the color you want (blue)
11.Click on the letter A in the lower right corner.
12.Check the work.
Now, on your own, depict in the upper left corner, first, the boys, who stand in two rows of 4 people each, and in the upper right corner, depict the girls.
Work in pairs.
Compare the illustrations.
Write down the result as a multiplication action. 2 * 4 = 8 (m) and 4 * 2 = 8 (d)
What conclusion can be drawn? (the value of the product does not change from permutation of factors)
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 instructing students.
Teach work in the PervoLogo program
Estimate the correctness of the task
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Stage 4.Physical education.
Communicative UUD
Let's leave the desks. Watch and repeat movements (music sounds)
Perform movement, mobilize strength and energy
Organizes physical education.
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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 page 108
Open the tutorial on page 108.
Read the dialogue between Masha and Misha.
- How did Misha build the soldiers?
- What did Masha say?
- Prove which one of them is right.
On the board: 5 2 2 5
Can it be argued that the meanings of these works 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. Product values are equal)
What is your conclusion?
(From rearranging the factors, the value of the product does not change)
Work with information presented in the form of a picture.
Realize mutual control
Render in cooperation mutual assistance
Formulate and argue own opinion
Organizes individual performance, exchange of views
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Stage 5. Primary anchoring.
Find the value of expressions, first relying on the formulated property, and then calculations (replacing products with sums)
Develop mathematical speech and logical thinking building chains of reasoning
Be able to formulate your thoughts verbally and in writing: listen and understand the speech of others ( Communicative UUD), (Regulatory UUD)
Let's check our assumptions (discoveries) again.
No. 2, s109 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 against the rule in the textbook on page 109.
Have our discoveries been confirmed?
Fulfill tasks
Organizes assimilation by students of a new way of action
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Stage 7. Systematization and repetition of what was previously learned.
The ability for self-assessment based on the criterion of the success of educational activities (Personal UUD)
Working with a computer (TB)
task 2.
Group work(3 persons)()
Fulfill tasks
Independent application information. Perform self-test
Recall
group rules
Organizes self-work, self-test
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Stage 8. Reflection of activity
Target: fix the new content of the lesson; summarize the work done in the lesson.
Be able to pronounce the sequence of actions in the lesson (Regulatory UUD)
The ability for self-assessment based on the criterion of the success of educational activities (Personal UUD)
What new did you learn in the lesson?
Have you coped with all the tasks?
Where will we use the new multiplication property?
Thank you for the lesson.
Formulate the end result of your work
Organizes reflection
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Maths lesson project
Subject and teaching materials: 1st grade mathematics, teaching materials "Perspective primary school".
Lesson topic: Addition with the number 10.
Place of the lesson in the topic: 1 lesson
Lesson type: discovery of new knowledge.
Purpose and expected result: Open new trick addition and use it in assignments different kind.
Lesson objectives (teacher activity):
1. Create problem situation to discover new knowledge.
2. Facilitate the discovery of a new addition technique for students.
3. To promote the conscious assimilation and application of new knowledge when adding to the number 10.
4. Organize self-assessment of students' work in the classroom ..
Equipment for the lesson: mathematics textbook 1st grade (A.L. Chekin), workbook"Mathematics in questions and tasks" No. 2 (OA Zakharova, EP Yudina), cards
Stages of the lesson, tasks and activities of students
Teacher activity
Student activities
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Study
problem 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 what expressions he could not cope with.
Let's help him.
How are these expressions similar?
How do they differ?
Find an extra expression? Why do you think that it is superfluous?
Cover with a card the expression that you think is superfluous.
He had already dealt with such expressions with Masha.
Children answer:
they are similar in that all expressions are addition.
They differ in that not all expressions have the same second term.
The second expression is superfluous, because the first term single digit.
Communicative
(children's sayings)
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2. Goal-setting.
Determine the topic of the lesson, set a goal, educational tasks.
The teacher removes this expression and a note remains on the board:
Open the tutorial 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 action plan in the lesson:
(the plan is posted on the board)
Tasks: 1) 10 + 2
Physical minute.
Children read the topic of the lesson.
Addition with the number 10.
Open a new addition technique and learn how to write down its result.
Open the method of addition with the number 10.
Learn to correctly write down the result of addition with the number 10
Practice solving these examples.
Rate your work.
Search and retrieval of information)
Regulatory (adopting a goal and setting lesson objectives)
Regulatory (action planning)
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3. Discovery of new knowledge
Learn how to add single-digit numbers to 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 assignment:
Once Misha said: "Masha, I noticed that if you add the number 10 with the single-digit number 2, then you get the number 12, in which there is 1 dozen and 2 more units."
Tell me how to solve this example using a model?
What can you say?
How many tens and how many units are in the number 1
Who wants to execute 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 the textbook.
Write the rest of the addition in a notebook.
Can you complete the new circuit by adding any single digit to 10?
Finish the output:
Adding the number 10 to any single-digit number results in a two-digit number, which has ...
Check our output against the textbook output.
Let's summarize the work. Read 1 problem.
Have we coped with this task? (we put v in front of the completed task)
Well done boys.
Open the method of addition with the number 10.
Children lay out the circles on the board and in the notebook. (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 ones.
Children do a similar job.
The result is two-digit numbers.
They are similar in that the answer in the place of tens is the number 1, and the difference is 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 a single-digit number 2 was added, and in the second example 5 was added.
in the tens place is the number 1, and in the ones place the figure of this single-digit number.
Children read:
When the number 10 is added to a single-digit number, a two-digit number is obtained, which has the digit 1 in the tens place, and the digit of this single-digit number in the ones place.
Regulatory (holding the purpose of the lesson)
Communicative (monologic statements of children)
Cognitive
(logical observations, comparisons, inferences)
Cognitive (informational)
Cognitive
(modeling)
Cognitive (informational)
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4. Formation
primary skills based on self-control
Be able to perform addition with the number 10.
Learn to complete difficult tasks.
Let's move on to task 2 of the lesson.
Task number 2.
Work in pairs.
Read the assignment.
Take the chips and cover the correct amounts.
Write down the amounts in a notebook. What task still need to be completed?
Have you solved all the examples for addition with the number 10?
Perform simulation.
Make a conclusion.
Read lesson objective 2.
Have we coped with task 2? (we put v in front of the completed task)
Tell us why you rated yourself so?
What task have we not yet completed?
Task number 2 in the notebook on page 31
Read the assignment.
1 option-1 column (1-4 examples)
Option 2 1 column (5-8 examples)
We carry out the task ourselves.
Look closely at the examples in the second column. What needs to be done to make the records correct?
Tell us how to control yourself when recording the missing terms?
Option 1 - Column 2 (1-4 examples)
Option 2 - Column 2 (5-8 examples)
Can we say that we have coped with task 3.
(we put v in front of the completed task)
Examples are written on a hidden board. After finishing work, the children independently check their work.
1 criterion: I know the conclusion when adding with the number 10
Criterion 2: I can write missing terms
Who will tell you how they rated themselves?
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.
The children conclude that with this addition, the result is the same.
Yes. (Children joined hands)
Several children talk about their work results.
Practice solving these examples.
Fill in the blanks so that the entries are correct.
Mutual verification
Write down either the first term or the second.
By the value of the sum, based on the rule, determine which summand is 10, and which summand is a single-valued summand.
Children evaluate themselves according to criteria.
Communicative (children's statements)
Communicative (communication)
Cognitive
(modeling)
Regulatory (control)
Cognitive (sign-symbolic and alphabetic)
Regulatory (control)
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5. Reflection
Learn to evaluate your work in the classroom.
What was the goal we set at the beginning of the lesson?
Have you coped with all the tasks (clearly visible)
1. I can teach another student a new method of addition.
2. I know and can add with the number 10.
3. I know, but I doubt the solution to these examples.
Children talk.
Self-assessment of learners using statements.
Regulatory
(goal hold)
Personal
(the ability to self-esteem based on the criterion of the success of educational activities)
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The way children get to know this rule (law) is due to the previously introduced meaning of the multiplication action. Using object models of sets, children count the results of grouping their elements in different ways, making sure that the results do not change as the methods of grouping change.
The count of elements of the picture (set) in pairs horizontally coincides with the count of elements in triplets vertically. Consideration of several options for 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 permutation of 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 of the number 2, calculate and remember the multiplication table by 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, do not have to separately memorize the results of the tables obtained by rearranging the factors. In fact, many children learn each table separately, because the lack of development of thinking flexibility does not allow them to easily rebuild the model of the memorized table case schema into reverse order... When calculating cases of the form 9 2 or 8 3, the children again return to the method of successive addition, which naturally takes time to get the result. This situation is most likely generated by the fact that for a significant number of children, such a time spacing of interrelated cases of multiplication (those connected by the rule of permutation of factors) does not allow the formation of an associative chain focused specifically on the relationship.
When compiling the multiplication table for the number 5 in grade 3, only the first product is obtained by adding the same 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 by 5: 6 5 is compiled; 7 5; 8 5; 9 5.
The multiplication table of the number 6 contains four cases: 6 6; 6 7; 6 8; 6 9.
The multiplication table by 6 contains three cases: 7 6; 8 6; 9 6.
Theoretical approach to a similar construction of the system for studying table multiplication assumes that it is in this correspondence that the child will remember the cases of table multiplication.
The largest number cases contains the most easy-to-remember multiplication table for 2, and the most difficult-to-remember multiplication table for 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 the new case is this lesson is, for example, only the case 9 9, and 9 8, 9 7it. 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 of the number 9 is the largest and most complex (and this is really so, if we bear in mind the list of all cases that refers to it).
A large amount of material that requires memorization by heart, the difficulty in the formation of associative links when memorizing interrelated cases, the need for all children to achieve a strong memorization of all table cases by heart within the time frame established by the program - all this makes the topic of studying table multiplication in primary grades one of the most methodically difficult. In this regard, important are the issues related to the methods of memorizing the multiplication table by the child.
Definition. Multiplication is an action that results in finding the sum of the same terms. Multiply number but by the number B means to find the amount 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 the product.
At multiplication natural numbers, the 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 1 (one), then work is equal to the second factor.
- For example:
- 5 * 6 * 8 * 0 = 0
- 132 * 1 = 132
The laws of multiplication
Combination 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)
Travel law
Rule. The product does not change from the rearrangement of the factors.
- 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
Distributive law also applies to the action of deduction.
- For example:
- 7 * (6 — 5) = 7 * 6 — 7 * 5 = 7
The laws of multiplication apply to any number of factors in numerical or literal expression. The distributional multiplication law is used to factor out the common factor.
Rule. To convert the sum (difference) into a product, it is enough to take out the same multiplier of the terms outside the brackets, and write the remaining factors in the brackets as the sum (difference).
3 4 = 12
WORK
THE ADDITION OF THE SAME ADDITIONS CAN BE REPLACED BY MULTIPLICATION.
The multiplication sign is a dot (·).
2 3 = 6
3 2 = 6
2 3 = 3 2
NAMES OF COMPONENTS
MULTIPLICATION ACTIONS
SEPARATE DIVIDER PRIVATE
6: 3 = 2
PRIVATE
To find the unknown dividend, you need to multiply the quotient
Into the divisor.
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 laid out 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 IN 12 CONTAINS ONE 3.
12: 3 = 4
2. Division into equal parts
12 apples were divided equally on 4 plates. How many apples are on each plate?
We argue:
We take 4 apples, put one apple on each plate. Then we take 4 more apples, put one more apple on a plate. And we take 4 more apples, put again one apple in a plate. Thus, in order to solve the problem, you need to answer the question - HOW MANY TIMES IN 12 CONTAINS BY 4.
CONNECTION
BETWEEN THE 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.
Z A G A C H I I X V I D S
CLASS
1. Analysis of the problem occurs according to the plan:
Nastya has collected a bouquet of daisies and cornflowers. There are 6 daisies in the bouquet, and there are 3 more cornflowers. How many cornflowers are in the bouquet?
- Who is the problem talking about? What does the problem say?
- Repeat the problem statement.
- Task 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 were there. What do we know about cornflowers?
- What do you need to find out?
At the end of the parsing, a short note is recorded, a diagram or drawing is made.
2. In the task, an explanation is always written for 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 question of the problem.
TASKS FOR FINDING THE SUM
There were 7 blue typewriters and 10 red typewriters on the shelf. How many typewriters were there on the shelf?
Draw a rectangle with sides 5 cm and 3 cm on a piece of paper in a cage. Divide it into squares with a side of 1 cm (Fig. 140). How do you count the number of these squares?
You can, for example, reason like this. The rectangle is divided into three rows, each of which contains five squares. Therefore, the required number is 5 + 5 + 5 = 15. On the left side of the written equality is the sum of equal terms. As you know, this amount is written using the product 5 * 3. We have: 5 * 3 = 15.
In the equality a * b = c the numbers a and b are called multipliers, and the number c and the notation a * b - product.
So, 5 * 3 = 5 + 5 + 5.
Similarly:
3
* 5
= 3
+ 3
+ 3
+ 3
+ 3
;
7
* 4
= 7
+ 7
+ 7
+ 7
;
1
* 6
= 1
+ 1
+ 1
+ 1
+ 1
+ 1
;
0
* 5
= 0
+ 0
+ 0
+ 0
+ 0
.
Written in literal form as follows:
$$ a * b = \ underbrace (a + a + a + ... + a) _ (b-terms) $$
By multiplying the number a by natural number b, not equal to 1, is the sum of b terms, each of which is equal to a.
What if b = 1? Then you will have to consider a sum consisting of one term. And this is not accepted in mathematics. Therefore, we agreed that:
a * 1 = a.
If b = 0, then we have agreed to consider that:
a * 0 = 0.
In particular,
0
* 0
= 0
.
Consider the products 1 * a and 0 * a, where a is a natural number other than 1.
$$ 1 * a = \ underbrace (1 + 1 + 1 + ... + 1) _ (a-terms) = a, $$
$$ 0 * a = \ underbrace (0 + 0 + 0 + ... + 0) _ (a-terms) = 0. $$
Now the following conclusions can be drawn.
If one of the two factors is 1, then the product is equal to the other factor:
a * 1 = 1 * a = a
If one of the two factors is zero, then the product is zero:
a * 0 = 0 * a = 0
The product of two numbers other than zero cannot be zero.
If the product is zero, then at least one of the factors is zero.
We calculated the number of squares in Figure 140 as follows:
5 + 5 + 5 = 5 * 3 = 15. However, this half-count could have been done in another way. The rectangle is divided into five columns, each with three squares. Therefore, the given number of squares is
3
+ 3
+ 3
+ 3
+ 3
= 3
* 5
= 15
.
Counting the squares in Figure 140 in two ways illustrates travel property of multiplication.
Permutation of the factors does not change the product.
This property is written in literal form as follows:
ab = ba
Can you multiply in writing (column) multi-digit number to two-digit. The multiplication of any two multidigit numbers is performed in a similar way.
For example:
This method is convenient because only single-digit numbers have to be orally multiplied.
Consider problems in the solution of which the action of multiplication is used.
Example 1
.
The garden was full of cherries, apples and pears. There were 24 cherries, which is 6 times less than apple trees, and 18 trees less than pears. How many trees were there in the garden?
1) 24 * 6 = 144 (trees) - made up apple trees.
2) 24 + 18 = 42 (wood) - made up of pears.
3) 24 + 144 + 42 = 210 (trees) - grew in the garden.
Answer: 210 trees.
Example 2
.
A truck drove out of one city at the same time in one direction at a speed of 48 km / h and a car at a speed of 64 km / h. What is the distance between them 3 hours after the start of the movement?
1) 64 - 48 = 16 (km) - this is how much the distance between cars increases every hour.
2) 16 * 3 = 48 (km) - the distance between cars after 3 hours.
Answer: 48 km.
Example 3
.
A rider at a speed of 14 km / h and a pedestrian at a speed of 4 km / h simultaneously set off from one village in opposite directions. What is the distance between them in 4 hours after the start of the movement?
1) 14 + 4 = 18 (km) - this is how much the distance between the rider and the pedestrian increases every hour.
2) 18 * 4 = 72 (km) - the distance between the rider and the pedestrian in 4 hours.
Answer: 72 km.
Example 4
.
Two boats departed from the two piers at the same time towards each other, which met 5 hours after the start of the movement. One of the boats moved at a speed of 28 km / h, and the second at 36 km / h. Find the distance between the marinas.
1) 28 + 36 = 64 (km) - this is how much the boats approached each hour.
2) 64 * 5 = 320 (km) - the distance between the piers.
The answer is 320 km.
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