Multiplication and division are mutually inverse operations. If you divide the product by one factor, you get another factor

Multiplication is an arithmetic operation in which the first number is repeated as a term as many times as the second number shows.

A number that repeats as a term is called multiplyable(it is multiplied), the number that shows how many times to repeat the term is called multiplier. The number resulting from multiplication is called work.

For example, multiplying the natural number 2 by the natural number 5 means finding the sum of five terms, each of which is equal to 2:

2 + 2 + 2 + 2 + 2 = 10

In this example, we find the sum by ordinary addition. But when the number of identical terms is large, finding the sum by adding all the terms becomes too tedious.

To write multiplication, use the sign × (slash) or · (dot). It is placed between the multiplicand and the multiplier, with the multiplicand written to the left of the multiplication sign, and the multiplier to the right. For example, the notation 2 · 5 means that the number 2 is multiplied by the number 5. To the right of the notation of multiplication, put an = (equal) sign, after which the result of the multiplication is written. Thus, the complete multiplication entry looks like this:

This entry reads like this: the product of two and five equals ten or two times five equals ten.

Thus, we see that multiplication is simply a short form of adding like terms.

Multiplication check

To check multiplication, you can divide the product by the factor. If the result of division is a number equal to the multiplicand, then the multiplication is performed correctly.

Consider the expression:

where 4 is the multiplicand, 3 is the multiplier, and 12 is the product. Now let's perform a multiplication test by dividing the product by the factor.

Task 2. How many strawberries? How many cherries? Write using multiplication. 3 · 5 = 15 (z.); 3 6 = 18 (in.).

– How many children can the strawberries be divided between? (15:3 = 5 or 15:5 = 3.)

– How many children can the cherries be divided between? (18:3 = 6 or 18:6 = 3.)

Task 3. Several rings were divided equally into three pins. There were 4 rings on each pin. How many rings did you take? (4 3 = 12 (k.))

– Divide the 12 rings equally into 4 pins. How much will it be for each? Write down the equality. (12: 4 = 3 (k.))

Task 4. Students perform multiplication and write the corresponding equalities with the division sign.

6 4 = 24 5 6 = 30 7 4 = 28 8 3 = 24

4 6 = 24 6 5 = 30 4 7 = 28 3 8 = 24

24: 4 = 6 30: 6 = 5 28: 4 = 7 24: 3 = 8

24: 6 = 4 30: 5 = 6 28: 7 = 4 24: 8 = 3

Task 5. Remember the fairy tale “Turnip”. Name the heroes of this fairy tale. How many were there? (6 heroes.) Grandfather cut the turnip into 18 pieces. Will he be able to distribute them equally to all the heroes of the fairy tale? How many pieces will each person get? (18: 3 = 6 (k.))

Task 6. Students perform calculations:

15 2 – 16 = 30 – 16 = 14 5 5 – 19 = 25 – 19 = 6

6 3 + 27 = 18 + 27 = 45 40: 2 – 9 = 20 – 9 = 11

60: 2 + 36 = 30 + 36 = 66 20 2 + 48 = 40 + 48 = 88

34 2 – 26 = 68 – 26 = 42 9 3 + 18 = 27 + 18 = 45

Task 7. Make up equalities from the numbers 2, 8 and 16. And let your neighbor at the desk make up equalities from the numbers 6, 3 and 18.

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 16 3 + 3 + 3 + 3 + 3 + 3 = 18

8 + 8 = 16 6 + 6 + 6 = 18

2 8 = 16 3 6 = 18

8 2 = 16 6 3 = 18

16: 2 = 8 18: 3 = 6

16: 8 = 2 18: 6 = 3

IV. Lesson summary.

– What are the operations of multiplication and division called?

Lesson 74
The meaning of arithmetic operations

Goals of the teacher: help consolidate ideas about the meaning of four arithmetic operations; to promote the development of the ability to formulate rules for multiplying numbers by 1 and 0, solve word problems, and perform calculations with 0 and 1.

Subject:have ideas know how

Personal UUD: perceive the speech of the teacher (classmates) not directly addressed to the student; independently evaluate the reasons for their successes (failures); express a positive attitude towards the learning process.

regulatory: evaluate (compare with a standard) the results of activities (others’ and their own); educational: use diagrams to obtain information; compare different objects; explore the properties of numbers; solve non-standard problems; communicative: convey their position to all participants in the educational process - formalize their thoughts in oral speech; listen and understand the speech of others (classmates, teachers); solve the problem.

During the classes

I. Oral counting.

1. Fill in the empty cells so that the sum of the numbers in each rectangle made up of three cells is equal to 98.

2. Solve the short-notation problem.

a) How much does a pike weigh?

b) How many kilograms do carp and pike weigh?

c) How much do two carp weigh? How much do two pikes weigh?

3. Compare, without calculating, using the signs “>”, “<», «=».

4. Make up all possible examples from groups of numbers.

a) 26, 2, 28; b) 80, 4, 76; c) 50, 3, 47.

II. Lesson topic message.

– Today in class we will make up equalities using drawings and diagrams.

III. Work according to the textbook.

Task 1. What arithmetic operation does the first picture represent? (Addition.) Write down the equality. (5 + 7 = 12.)

– What is the name of the “+” sign?

– What arithmetic operation does the second picture represent? (Subtraction.) Write down the equality. (9 – 5 = 4.)

– What is the name of the “–” sign?

– What arithmetic operation does the third picture represent? (Multiplication.) Write down the equality. (3 4 = 12.)

– What is the name of the sign “·”?

– What arithmetic operation does the fourth picture represent? (Division.)

– Write down the equality. (9: 3 = 3.)

– What is the name of the “:” sign?

Task 2. Students match the drawing and equality.

Task 3. Do the calculations.

1 3 = 1 + 1 + 1 = 3

1 10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10

4 1 = 1 4 = 1 + 1 + 1 + 1 = 4

100 1 = 1 100 = 100

– What conclusion can be drawn? (If you multiply any number by 1, you get the same number.)

– Carry out the calculations.

0 3 = 0 + 0 + 0 = 0

5 0 = 0 5 = 0 + 0 + 0 + 0 + 0 = 0

100 0 = 0 100 = 0

– What conclusion can be drawn? (If you multiply any number by 0, you get 0.)

Task 4. Students perform calculations according to the model.

Task 5. There are 4 corners in the room. There is a cat in every corner. Each cat has 4 kittens. Each kitten has 4 mice.

– How many cats are there in the room?

4 · 4 = 16 (alive) – kittens in the room.

16 + 4 = 20 (alive) – cats and kittens.

- How many mice?

16 · 4 = 16 + 16 + 16 + 16 = 32 + 32 = 64 (living) – mice.

– How many animals are there in total?

64 + 20 = 84 (living) – total.

– How many fewer cats than mice?

64 – 20 = 44 (alive) – there are fewer cats than mice.

Task 6. Do the calculations.

– Write down expressions from different columns for which the calculation results are the same.

Task 7. Work in pairs.

35 – 5 = 30 20 – 5 = 15 10 – 5 = 5

30 – 5 = 25 15 – 5 = 10 5 – 5 = 0

– How many people will get the potatoes? (to seven people.)

IV. Working with cards.

1. Compare.

5 2 … 5 3 2 5 … 2 4

2 7 … 8 2 3 7 … 6 3

3 6 … 3 5 4 8 … 4 7

2. solve examples.

2 4 = 2 3 = 2 8 =

4 2 = 3 2 = 8 2 =

3. Calculate by replacing multiplication with addition:

8 5 = 7 4 = 16 3 =

4. Fill in the missing numbers:

5. Make up division examples:

V. Lesson summary.

– What new did you learn in the lesson? Name arithmetic operations. What do we get if we multiply a number by 1? What do we get if we multiply a number by 0?

Lesson 75
Solving multiplication and division problems

Goals of the teacher: promote the development of the ability to solve word problems on multiplication and division; help improve the ability to choose an arithmetic operation in accordance with the meaning of a word problem, and restore correct equalities.

Planned educational outcomes.

Subject:have ideas about the properties of the numbers 0 and 1 (if you increase one factor by 2 times and decrease the other by 2 times, the result will not change); know how increase/decrease numbers by a factor of 2, perform multiplications with numbers 0 and 1, find a product using addition, perform calculations in two steps, solve problems involving increasing/decreasing by a factor of 2, finding a product (using addition, division into parts and in content (selection).

Personal UUD: evaluate their own educational activities: their achievements, independence, initiative, responsibility, reasons for failures.

Meta-subject (criteria for the formation / assessment of components of universal learning activities - UUD):regulatory: adjust activities: make changes to the process taking into account difficulties and errors encountered; outline ways to eliminate them; analyze the emotional state obtained from successful (unsuccessful) activities; educational: search for essential information; give examples as evidence of the proposed provisions; draw conclusions; navigate their knowledge system; communicative: accept a different opinion and position, allow the existence of different points of view; adequately use speech means to solve various communicative tasks; construct monologue statements and master the dialogical form of speech.

During the classes

I. Oral counting.

1. Compare without calculating.

2. Solve the problem.

A duck requires 7 kg of feed per day, a chicken needs 3 kg less than a duck, and a goose needs 5 kg more than a chicken. How many kilograms of feed does a goose need per day?

3. Fill in the missing numbers:

4. In the picture you see two trees: birch and spruce. The distance between them is 15 meters. A boy is standing between the trees. It is 3 meters closer to birch than to spruce.

– What is the distance between the birch tree and the boy? (6 m.)

II. Lesson topic message.

– Today in class we will solve problems on multiplication and division.

III. Work according to the textbook.

– Read task 1. What is known? What do you need to know? Write down expressions to solve each problem.

– Find the meaning of each expression.

Formulate answers to the task questions.

a) 1 time – 3 r. Solution:

4 times - ? R. 3 · 4 = 12 (r.).

b) 1 row – 9 k. Solution:

4 rows – ? k. 9 · 4 = 36 (k.).

c) 1 time – 8 points each Solution:

3 times – 9 points each 8 2 + 9 3 = 16 + 27 = 43 (points).

Total - ? points

d) 3 piles – 12 b. Solution:

1 pile – ? b. 12: 3 = 4 (b.).

It was 12 points. Solution:

Divided equally 4 alive. - By? b. 12: 4 = 3 (b.).

d) 3 people - By? R. Solution:

Total – 60 rub. 60: 3 = 20 (r.).

Task 2. Determine who made how many blades. Who forged the largest number of blades?

1) 7 + 2 = 9 (cl.) forged by Dili;

2) 9 · 2 = 18 (cl.) – forged by Kili;

3) 9 · 2 = 18 (cl.) – forged by Balin;

4) 18: 2 = 9 (cl.) – forged by Dwalin;

5) 9 – 2 = 7 (cl.) forged by Bombur.

Task 3. How many balls must be placed on the second cup to balance the scales?

Task 4. How many legs does a centipede have? (40 legs.)
The goose? (2.) The pig? (4.) A beetle? (6.)

– Write an expression to count the legs of all these animals.

IV. Frontal work.

– Based on the picture, make up a multiplication problem and two division problems.

Lesson 76
Solving non-standard problems

Teacher's goals: promote the consideration of a graphical method for solving non-standard problems (combinatorial) and presenting data in a table; promote the development of the ability to solve combinatorial problems using multiplication, form two-digit numbers from given numbers, make sums and differences, carry out oral and written calculations with natural numbers; to promote the development of the ability to check the correctness of calculations, the ability to classify and divide into groups.

Planned educational outcomes.

Personal UUD: evaluate their own educational activities; apply the rules of business cooperation; compare different points of view.

Meta-subject (criteria for the formation / assessment of components of universal learning activities - UUD):regulatory: control their actions for accurate and operational orientation in the textbook; determine and formulate the purpose of the activity in the lesson with the help of the teacher; educational: navigate their knowledge system, complement and expand it; communicative: enter into collective educational cooperation, convey their position to all participants in the educational process - formalize their thoughts in oral and written speech; listen and understand the speech of others (classmates, teachers); solve the problem.

During the classes

I. Oral counting.

1. Fill in the missing terms so that the sum of the numbers along each side of the triangle is equal to the number written inside the triangle.

2. Use an arrow to indicate which box each pencil comes from.

3. Coffee, juice and tea were poured into a glass, cup and jug. There is no coffee in the glass. There is no juice or tea in the cup. There is no tea in the jug. What container is it in?

II. Work according to the textbook.

– Today in class we will solve tasks in different ways.

Task 1. How many boys were there? Girls? How many different pairs did you get? Make different pairs using the diagram.

– Write down the total number of pairs using addition and then multiplication.

3 + 3 + 3 = 9 (p.). 3 · 3 = 9 (p.).

Task 2. Solve a combinatorial problem using a table.

- How many pairs did you get? (20 pairs)

- Count in different ways.

4 5 = 20 5 4 = 20

Task 3. Working in pairs, compose all possible products according to the scheme ○ · □, where ○ is an odd number, □ is an even number (including 0).

– Calculate all these products.

– How many works can you compose?

Task 4. The flag consists of two stripes of different colors. How many of these flags can be made from paper of four different colors? (24 checkboxes.)

– How many three-color flags can you make? (6 checkboxes.)

– How many more three-color flags will there be than two-color ones? (6 – 2 = 4.)

Task 5. Make a table to solve a combinatorial problem.

Answer: 20 options.

Task 6 (work in pairs).

– Make two-digit numbers from the numbers 2, 4, 7, 5.

Entry: 24, 25, 27, 22.

– Make sums and differences from these pairs of numbers. Find their meanings.

Task 7. The menu in the dining room has three first courses and six second courses. How many ways are there to choose a two-course meal? (6 3 = 18.)

Students fill out the table.

– In addition to the first and second, you can also choose one of three desserts. Write down the number of three-course meal options using multiplication. (18 · 3.)

- Calculate this number by addition.

18 · 3 = 18 + 18 + 18 = 36 + 18 = 54.

Lesson 77
Getting to know new activities
(repetition)

Goals of the teacher: create conditions for successful repetition of addition, subtraction, multiplication, division, and use of appropriate terms; contribute to the formation of ideas about the use of multiplication in Ancient Egypt.

Planned educational outcomes.

Personal UUD: motivate their actions; express readiness in any situation to act in accordance with the rules of behavior; show kindness, trust, attentiveness, and help in specific situations.

Meta-subject (criteria for the formation / assessment of components of universal learning activities - UUD):regulatory: know how to evaluate their work in class; analyze the emotional state obtained from successful (unsuccessful) activities in the lesson; educational: compare different objects - select from a set one or more objects that have common properties; give examples as evidence of the proposed provisions; communicative: accept a different opinion and position, allow the existence of different points of view; adequately use speech means to solve various communicative tasks.

During the classes

I. Oral counting.

1. Sasha and Petya each fired 3 shots at the shooting range, after which their targets looked like this:

- name the winner.

– Find the third term.

2. The girl read the book in three days. On the first day she read 9 pages, and on each subsequent day she read 3 more pages than the previous day. How many pages are in the book?

All other division tables are obtained in a similar way.

TECHNIQUES FOR MEMORIZING THE DIVISION TABLE

Techniques for memorizing tabular division cases are associated with methods for obtaining a division table from the corresponding tabular multiplication cases.

1. A technique related to the meaning of the action of division

With small values of the dividend and divisor, the child can either perform objective actions to directly obtain the result of division, or perform these actions mentally, or use a finger model.

For example: 10 flower pots were placed equally on two windows. How many pots are there on each window?

To obtain the result, the child can use any of the models mentioned above.

For large values of the dividend and divisor, this technique is inconvenient. For example: 72 pots of flowers were placed on 8 windows. How many pots are there on each window?

Finding the result using a domain model in this case is inconvenient.

2. A technique associated with the rule for the relationship between the components of multiplication and division

In this case, the child is oriented. To memorize an interconnected trio of cases, for example:

If a child manages to remember one of these cases well (usually the reference case is the case of multiplication) or he can get it using any of the techniques for memorizing the multiplication table, then using the rule “if the product is divided by one of the factors, you get the second factor,” it is easy get the second and third table cases.

№ 13 Methodology for studying the technique of dividing a two-digit number by a single-digit number

When studying the technique of dividing a two-digit number by a single-digit number, use the rule of dividing the sum by the number. Groups of examples are considered:

1) 46: 2 = "(40 + 6) : 2=40: 2 +-"6: 2=20 + 3=23 (replace the dividend with the sum of the bit terms)

2) 50: 2= (40 + 10) : 2=40: 2 + 10: 2=20 + 5=25 (the dividend is replaced by the sum of convenient terms - round numbers)

3) 72: 6= (60 +12) : 6=60: 6+ 12: 6= 10 + 2= 12 (the dividend is replaced by the sum of two numbers: a round number and a two-digit number)

In all examples, these terms will be convenient if, when dividing them by a given divisor, the digit terms of the quotient are obtained.

During the preparatory period, exercises are used: highlight round numbers up to 100 that are divisible by 2 (10, 20, 40, 60, 80), by 3 (30, 60, 90), by 4 (40, 80), etc.; imagine numbers in different ways as the sum of two terms, each of which is divisible by a given number without a remainder: 24 can be replaced by a sum, each term of which is divisible by 2: 20 + 4, 12 + 12, 10 + 14, etc.; Solve examples of the form: (18 + 45) : 9 in different ways.

After the preparatory work, examples of three groups are considered, with great attention paid to replacing the dividend with the sum of convenient terms and choosing the most convenient method:

42: 3= (30+12) : 3=30: 3+12: 3= 14

42:3=(27+15) :3=27: 3+15: 3=14 42:3= (24+1&) : 3 = 24: 3+18:3=14

42: 3= (36 + 6) : 3=36:3+6: 3=14, etc.

The most convenient method is the first method, since when dividing the convenient terms (30 and 12), the digit terms of the quotient (10 + 4 = 14) are obtained.

Difficult examples are: 96:4. In such cases, it is advisable to replace the dividend with a sum of convenient terms, the first of which expresses the largest number of tens divisible by the divisor: 96: 4 = (80+16): 4.

1. Bit composition of the number

2. property of dividing a sum by a number

3. Divide a number ending in 0

4. Tabular division cases

5. “Convenient” number composition.

Division with remainder.

Division with a remainder is studied in grade II after completing work on non-table cases of multiplication and division.

Working on division with a remainder within 100 expands students' knowledge of the operation of division, creates new conditions for applying knowledge of tabular results of multiplication and division, for applying computational techniques for non-tabular multiplication and division, and also prepares students in a timely manner to study written division techniques.

A special feature of division with a remainder compared to operations known to children is the fact that here, using two given numbers - the dividend and the divisor - two numbers are found: the quotient and the remainder.

In their experience, children have repeatedly encountered cases of division with a remainder when dividing objects (candies, apples, nuts, etc.). Therefore, when studying division with a remainder, it is important to rely on this experience of children and at the same time enrich it. It is useful to begin work by solving vitally practical problems. For example: “Distribute 15 notebooks to students, 2 notebooks each. How many students received notebooks and how many notebooks are left?”

Students distribute, arrange objects and orally answer the questions posed.

Along with these tasks, work is carried out with didactic material and drawings.

We divide 14 circles into 3 circles. How many times are there 3 mugs in 14 mugs? (4 times.) How many circles are left? (2.) Enter division with remainder: 14:3=4 (remainder 2). Students solve several similar examples and problems using objects or drawings. Let's take the problem: "Mom brought 11 apples and distributed them to the children, 2 apples to each. How many children received these apples and how many apples were left?" Students solve the problem using circles.

The solution and answer to the problem are written as follows: 11:2=5 (remaining 1).

Answer: 5 children and 1 apple remains.

Then the relationship between the divisor and the remainder is revealed, i.e., students establish: if a division produces a remainder, then it is always less than the divisor. To do this, first solve examples of dividing consecutive numbers by 2, then by 3 (4, 5). For example:

10:2=5 12:3 = 4 16:4 = 4
11:2=5(remaining 1) 13:3 = 4 (remaining 1) 17:4 = 4(rest 1)
12:2=6 14:3 = 4(remaining 2) 18:4 = 4 (remaining 2)

13:2=6(remaining 1) 15:3 = 5 19:4 = 4 (remaining 3)

Students compare the remainder with the divisor and notice that when divided by 2, the remainder only produces the number 1 and cannot be 2 (3, 4, etc.). In the same way, it turns out that when divided by 3, the remainder can be the number 1 or 2, when divided by 4, only the numbers 1, 2, 3, etc. Having compared the remainder and the divisor, children conclude that the remainder is always less than the divisor.

In order for this ratio to be learned, it is advisable to offer exercises similar to the following:

What numbers can be left as a remainder when divided by 5, 7, 10? How many different remainders can there be when dividing by 8, 11, 14? What is the largest remainder that can be obtained when dividing by 9, 15, 18? Can the remainder be 8, 3, 10 when divided by 7?

To prepare students for mastering division with a remainder, it is useful to offer the following tasks:

What numbers from 6 to 60 are divisible by b, 7, 9 without a remainder? What is the smallest number closest to 47 (52, 61) that is divisible by 8, 9, 6 without a remainder?

Revealing the general technique of division with a remainder, it is better to take examples in pairs: one of them is for division without a remainder, and the other is for division with a remainder, but the examples must have the same divisors and quotients.

Next, examples of division with a remainder are solved without a helping example. -Let us divide 37 by 8. The student must understand the following reasoning: “37 cannot be divided by 8 without a remainder. The largest number that is less than 37 and divisible by 8 without a remainder is 32. 32 divided by 8 equals 4; from 37 we subtract 32, we get 5, the remainder is 5. So, divide 37 by 8, we get 4 and the remainder is 5.”

The skill of division with a remainder is developed through practice, so it is necessary to include more examples of division with a remainder in both oral exercises and written work.

When doing division with a remainder, students sometimes get a remainder greater than the divisor, for example: 47:5=8 (rest. 7). To prevent such errors, it is useful to offer children incorrectly solved examples, let them find the error, explain the reason for its occurrence and solve the example correctly.

1. choose a number close to the dividend, which is less than it and is divisible without a remainder;

2. divide this number;

3. find the remainder;

4. check whether the remainder is less than the divisor;

5. write down an example

In grades II and III, it is necessary to include as many different exercises as possible for all studied cases of multiplication and division: examples in one and several actions, comparing expressions, filling out tables, solving equations, etc.

№ 14. The concept of a compound task.

A compound problem includes a number of simple problems interconnected in such a way that the required values of some simple problems serve as data for others. Solving a compound problem comes down to breaking it down into a number of simple problems and solving them sequentially. Thus, To solve a compound problem, it is necessary to establish a number of connections between the data and the required one, in accordance with which to select and then perform arithmetic operations.

In solving a compound problem, something essentially new has appeared compared to solving a simple problem: here not one connection is established, but several, in accordance with which arithmetic operations are selected. Therefore, special work is carried out to familiarize children with a compound problem, as well as to develop their skills in solving compound problems.

Preparatory work for familiarization with component tasks should help students understand the main difference between a compound problem and a simple one - it cannot be solved immediately, that is, in one action, but to solve it it is necessary to isolate simple problems, establishing appropriate connections between the data and what is being sought. For this purpose, special Exercises are provided.