→ Arithmetic operations

Arithmetic operations

Finding one new number from several given numbers is called arithmetic operation. There are six operations involved in arithmetic: addition, subtraction, multiplication, division, exponentiation, root extraction.

1. Addition. This action consists of using several numbers, called addends, to find a number called their sum.

Example: 4+3=7, where 4 and 3 are terms, and 7 is their sum.

2. Subtraction- an action by which the required term (difference) is found from a given sum (minuend) and a given term (subtrahend).
This is the reverse of addition.

Example: 7 – 3 = 4, where 7 is the minuend, 3 is the subtrahend, and 4 is the difference.

3. Multiplication. To multiply a certain number (multiplicand) by an integer (factor) means repeating the multiplicand as a summand as many times as there are units in the factor. The result of multiplication is called a product.

Example: 2 ∙ 3 = 6, where 2 is the multiplicand, 3 is the multiplier, and 6 is the product. (2 ∙ 3 = 2 + 2+ 2 = 6)

If the multiplier and the multiplicand change their roles, then the product remains the same. Therefore, the multiplier and the multiplicand are also called factors.

Example: 2 ∙ 3 = 3 ∙ 2, that is (2 + 2 + 2 = 3 + 3)

It is believed that if the factor is 1, then a ∙ 1 = a.

For example: 2 ∙ 1 = 2, 44 ∙ 1 = 44, 13 ∙ 1 = 13.

4. Division. By dividing by this work(divisible) and given factor (divisor) find the required factor (quotient).
This is the inverse of multiplication.

Example: 8: 2 = 4, where 8 is the dividend, 2 is the divisor, and 4 is the quotient.

Checking division: the product of divisor 2 and quotient 4 gives dividend 8. 2 ∙ 4 = 8

Division with remainder

If, when dividing an integer by an integer, the quotient results in an integer, then such division of integers is called accurate, or that the first number completely divided(or simply - divided) by the second.

For example: 35 is divisible (by an integer) by 5, the quotient is the integer 7.

The second number is called a divisor of the first, and the first is called a multiple of the second.

In many cases, you can find out without performing division Is it completely divisible? one integer divided by another (see signs of divisibility).

Exact division is not always possible. In this case, perform the so-called division with remainder. In this case they find this greatest number, which when multiplied by a divisor will give a product that does not exceed the dividend. This number is called incomplete private. The difference between the dividend and the product of the divisor and the partial quotient is called remainder of the division.
The dividend is equal to the divisor multiplied by the partial quotient plus the remainder. The remainder is always less than the divisor.

Example: The partial quotient of dividing the number 27 by 4 is 6, and the remainder is 3. Obviously, 27 = 4∙6 + 3 and 3˂4.

5. Exponentiation. Raising a certain number to an integer power (to the second, third, etc.) means taking this number as a factor two, three times, etc. In other words, exponentiation is accomplished by repeated multiplication.
The number that is taken as a factor is called degree basis; a number indicating how many times a base is repeated is called exponent; the result of raising a number to a power is called power of this number.

Example: 2∙2∙2 = 2³ = 8; where 2 is the base of the degree, 3 is the exponent, 8 is the degree.

The second power of a number is also called square, third degree – cube. The first power of a number is the number itself.

6. Root extraction is an action by which, according to a given degree ( radical number ) And this indicator degrees ( root exponent) find the desired base (root).
This is the opposite of raising to a power.

Example: ³√64 = 4; where 64 is the radical number, 3 is the root exponent, 4 is the root.

Root extraction check: 4³=64. Raising the number 4 to the 3rd power gives 64.

The root of the second degree is also called square; root of the third degree - cubic.
At the sign square root It is customary to omit the root exponent: √36 = 6 means ²√36 = 6.

Liter used:
Guide to elementary mathematics- Vygodsky M.Ya., “Science”, 1974
Handbook of Mathematics. Manual for students 9-11 grades. - Shakhno K.U., "Uchpedgiz", 1961

Lecture 7. Computational methods of addition and subtraction for numbers of the first and second ten

1. Basic concepts.

2. Computational techniques for numbers of the first ten.

3. Computational techniques for numbers of the second ten.

Basic Concepts

IN elementary school They study four arithmetic operations: in the 1st grade, children become familiar with addition and subtraction, in the 2nd grade - with multiplication and division.

Addition and subtraction are called first-stage operations. Multiplication and division are called second-stage operations.

The addition symbol is the “+” (plus) sign, the subtraction symbol is the “-” (minus) sign. The multiplication symbol is the “x” sign, which in writing is often replaced by a dot in the center of the “ ” cell. The division symbol is the “:” sign. In high school, a horizontal bar is also used as a division symbol (in printed texts, often replaced by a slash), considering a notation of the form 3 / 4, U 2 as a division notation.

From a set-theoretic point of view, addition corresponds to such objective actions with aggregates (sets, groups of objects) as combining and increasing by several elements either a given aggregate or an aggregate compared with a given one. In this regard, before becoming acquainted with the symbolism of recording actions and calculating the results of actions, the child must learn to model all these situations on objective aggregates, understand (i.e. correctly represent) them from the words of the teacher, be able to show with his hands both the process and and the result of an objective action, and then characterize them verbally.

Tasks that a child must learn to perform according to verbal description teacher before getting acquainted with the symbolism of the action of addition:

1. Take three carrots and two apples (visual). Put them in your cart. How to find out how many there are together? (We need to count.)

2. There are 2 cups and 4 glasses on the shelf. Label cups with circles and glasses with squares. Show how many there are together. Count it.

3. 4 candies and 1 wafer were taken from the vase. Label them with figures and show how many sweets were taken from the vase. Count it.

All three proposed situations below model the union of two sets.

1. Vanya has 3 badges. Mark the icons with circles. They gave him more and he got 2 more. What should you do to find out how many badges he now has? (You need to add 2.) Do it. Count the result.

2. Petya had 2 toy trucks. Mark the trucks with squares. And the same number of cars. Mark the cars with circles. How many circles did you put in? For his birthday, he was given three more cars. What cars are there more now? Mark them with circles. Show me how much more.

3. There are 6 pencils in one box, and 2 more in the other. Label the pencils from the first box with green sticks, and the pencils from the second box with red sticks. Show how many pencils are in the first box and how many are in the second. Which box has more pencils? Which one has less? How long?

These three situations model an increase of several units in a given population or a population being compared to a given one.

Symbolically, these situations are described using the action of addition: 6 + 2 = 8.

There are four types of subtraction action substantive actions:

a) removal of part of the population (set);

b) reducing this population by several units;

c) a decrease by several units in the population being compared with the given one;

d) difference comparison of two sets.

Here are the tasks that the child must learn to perform according to the teacher’s verbal description before becoming familiar with the symbolism of the action of subtraction:

1. A boa constrictor sniffed flowers in a clearing. There were 7 flowers in total. Mark the flowers with circles. The Baby Elephant came and accidentally stepped on 2 flowers. What needs to be done to show this? Show how many flowers the Baby Elephant can smell now.

2. The Monkey had 6 bananas. Mark them with circles. She ate a few bananas and lost 4 bananas. What needs to be done to show this? Why did you remove 4 bananas? (There are 4 fewer.) Show me the remaining bananas. How many are there?

3. The beetle has 6 legs. Indicate the number of beetle legs with red sticks. And an elephant has 2 fewer legs. Indicate the number of elephant legs with green sticks. Show who has fewer legs. Who has more legs? How long?

4. There are 5 cups on one shelf. Label the cups with circles. And on the other shelf there are 8 glasses. Mark the glasses with squares. Place them so that you can immediately see which is more - glasses or cups. Less of what? How long?

The following tasks are given in accordance with the types of subject actions indicated above.

Symbolically, these situations are described using the action of subtraction: 8-5 = 3.

After the child learns to understand by ear and model all the designated types of objective actions, he can be introduced to the signs of actions. At this stage, the sequence of instructions from the teacher is as follows:

1) indicate what is said in the task with circles (sticks, etc.);

2) designate specified number circles (sticks) with numbers;

3) put between them the right sign actions. For example:

There are 4 white and 3 pink tulips in a vase. Indicate the number of white tulips and the number of pink tulips. What sign should be put in the entry to show that all the tulips are in the same vase!

The entry is made: 4 + 3.

This notation is called a “mathematical expression.” She

characterizes the quantitative characteristics of the situation and the relationships of the populations under consideration.

The number 7 obtained in the answer is called the value of the expression.

A notation of the form 3 + 4 = 7 is called equality. You should not immediately direct your child to receive complete equality with writing the value of the expression:

expression\

expression value

equality

Before moving on to equality, it is useful to offer children tasks:

a) to correlate the situation and expression (choose an expression for a given situation or change the situation in accordance with the expression - the situation can be depicted in a picture, drawn on a board, modeled on a flannelgraph);

b) to compose expressions for situations (compose an expression in accordance with the situation).

After children learn to correctly choose the sign of an action and explain their choice, they can move on to drawing up an equation and recording the result of the action.

In a stable mathematics textbook, the operations of addition and subtraction are taught simultaneously. In some alternative textbooks (I.I. Arginskaya, N.B. Istomina), addition is first studied, and then subtraction.

An expression of the form 3 + 5 is called a sum.

The numbers 3 and 5 in this notation are called terms.

A notation of the form 3 + 5 = 8 is called equality. The number 8 is called the value of the expression. Since the number 8 is in this case obtained as a result of summation, it is also often called a sum.

For example:

Find the sum of the numbers 4 and 6. (Answer: the sum of the numbers 4 and 6 is 10.)

An expression of the form 8-3 is called a difference.

The number 8 is called the minuend, and the number 3 is called the subtrahend.

The value of the expression - the number 5 can also be called the difference.

For example:

Find the difference between the numbers 6 and 4. (Answer: the difference between the numbers 6 and 4 is 2.)

Since the names of the components of addition and subtraction actions are introduced by agreement (children are told these names and need to remember them), the teacher actively uses tasks that require recognizing the components of actions and using their names in speech. For example:

1. Among these expressions, find those in which the first term (minued, subtracted) is equal to 3:

3 + 2; 7 - 3; 6 + 3; 8 + 1; 3 + 5; 3 - 2; 7 - 3; 3 + 4; 3 - 1.

2. Make up an expression in which the second term (minued, subtracted) is equal to 5. Find its value.

3. Choose examples in which the sum is 6. Underline them in red. Choose examples where the difference is 2. Underline them in blue.

4. What is the number 4 called in the expression 5 - 4? What is the number 5 called? Find the difference. Make up another example in which the difference is equal to the same number.

5. Minuend 18, subtrahend 9. Find the difference.

6. Find the difference between the numbers 11 and 7. Name the minuend and subtrahend.

In grade 2, children become familiar with the rules for checking the results of addition and subtraction operations:

Addition can be checked by subtraction: 57 + 8 = 65. Check: 65-8 = 57.

Subtract one term from the sum and get another term. This means the addition was done correctly.

This rule applicable to checking the action of addition in any concentration (when checking calculations with any numbers).

Subtraction can be checked by addition: 63 - 9 =54. Check: 54 + 9 = 63.

We added the subtrahend to the difference and got the minuend. This means that the subtraction was performed correctly.

This rule also applies to testing the operation of subtraction with any numbers.

In grade 3, children become familiar with the rules for the relationship between the components of addition and subtraction, which are a generalization of the child’s ideas about how to check addition and subtraction: w

If you subtract one term from the sum, you get another term.

If you add the difference and the subtrahend, you get the minuend.

If you subtract the difference from the minuend, you get the subtrahend.

These rules are the basis for preparing for solving equations, which in elementary school are solved based on the rule for finding the corresponding unknown component of equality.

For example:

Solve equation 24 - x=19.

The subtrahend in the equation is unknown. To find the unknown subtrahend, you need to subtract the difference from the minuend: x = 24 - 19, x = 5.

For real numbers You can define arithmetic operations - addition, subtraction, multiplication and division. How this is done can be found in the fine print below. The reader who finds it necessary to become acquainted with these arguments will see that arithmetic operations on infinite fractions are associated with the need to perform some endless processes. In practice, arithmetic operations on real numbers are performed approximately.

Along this path it is possible formal definitions these actions. This will be discussed in § 1.8.

The next paragraph lists the properties of real numbers that follow from the definitions made. We formulate these properties. They can be proven, but we prove them only in in some cases (complete proof see, for example, in the textbook by S. M. Nikolsky “ Mathematical analysis", vol. I, ch. 2). These properties are collected in five groups (I – V). The first three of them contain elementary properties that guide us when arithmetic calculations and solving inequalities. Group IV constitutes one property (Archimedes). Finally, group V also consists of one property. This property is formulated in the language of limits. It will be proven, but later - in § 2.5.

Arithmetic operations

Objectives of studying the topic:

2) To acquaint students with the rules for the order of performing operations on numbers and, in accordance with them, develop the ability to find numeric values expressions.

3) Introduce students to identity transformations of expressions based on properties arithmetic operations.

In the works numerical expressions There are 2 main stages:

1) Studying the simplest expressions of the form: sum (2 + 3); difference(5 -1); product (3 4); private (12:4).

2) Studying complex expressions containing two or more actions, with and without parentheses.

1) When working with the simplest expressions in accordance with the requirements of the program, the teacher is faced with the task of developing in children the ability to read and write such expressions.

Students' first encounter with expressions occurs in the first grade in the topic "Numbers from 1 to 10", where children first become acquainted with the action signs "+" and "-". At this stage, children write down expressions and read them, focusing on the meaning of action signs, which they recognize as short designation the words "add" and "drop". This is reflected in the reading of expressions: 3 + 2 (3 yes 2); 3 - 1 (3 minus one).

Gradually, children’s ideas about these actions expand. Students will learn that adding a few units to a number increases it by the same number of units, and subtracting it decreases it. This is reflected when reading the expressions: 4 + 2 (4 increased by two units); 7 - 1 (7 decrease by one unit).

Then children learn the names of the plus and minus action signs. (When studying addition and subtraction of the first ten numbers). These expressions are read differently: 4 + 2 (4 "plus" 2); 7 - 1 (7 minus 1).

And only when familiarizing yourself with the names of the components and results of the action of addition, strict mathematical terminology is introduced, the name of this mathematical expression is given - “sum”, and a little later the term “difference” is similarly introduced.

The names of the next two mathematical expressions “product” and “quotient” are introduced similarly when studying the operations of multiplication and division in the second grade. Here, in the second grade, the terms “expression”, “meaning of expression” are introduced, which, like other mathematical terms, should be acquired by children naturally, just as they acquire other words that are new to them, if they are often used by others and find application in practice.

2) Along with the simplest mathematical expressions Complex expressions containing two or more actions, with and without parentheses, are also studied. Such expressions appear depending on the consideration of relevant issues in the mathematics course. However, their consideration is mainly subordinated to one didactic purpose– to develop the ability to find the meaning of an expression, and this is directly related to the rules for the order of performing arithmetic operations.

a) The first consideration is the rule about the order of operations in expressions without parentheses, when with numbers there is either only addition and subtraction, or only multiplication and division. The first such expressions of the form 5 + 1 + 1, 7 - 1 - 1 are found at the very beginning of the study of addition and subtraction of numbers within 10. Already here the main attention is paid to clarifying the question of how to reason when calculating the meaning of expressions. IN I-II grade there are exercises: 70 – 26 + 10, 90 – 20 – 15, 42 + 18 – 19; in grade II there are exercises: 4 · 10: 5, 60: 10 · 3, 36: 9: 2. Upon further examination of similar expressions, the conclusion is drawn: in expressions without parentheses, the actions of addition and subtraction (multiplication and division) are performed in the order how they are written: from left to right.

b) Then expressions containing brackets appear and again the main attention is paid to the rule about the order of actions in expressions with brackets. This way we actually introduce children to the second rule about the order of actions in expressions containing parentheses. Exercises: 80 – (34+13), 85 – (46 – 14), 60: (30 – 20), 90: (2 ·5).

In the second grade, when studying the operations of multiplication and division, we encounter expressions containing the actions of addition, subtraction, multiplication and division. To clarify the question of the order of execution of actions in such expressions, it is advisable for the first consideration to take the expression 3 · 5 + 3. Using the meaning of the multiplication action, we come to the conclusion that the value of this expression is 18. This implies the order of execution of actions. As a result, we actually get the third rule about the order of operations in expressions without parentheses containing the operations of addition, subtraction, multiplication and division: in expressions without parentheses, the operations of multiplication or division are performed first, and then the operations of addition or subtraction in the order in which they are written . At the same time, a sample of reasoning is given, where attention is drawn to pronunciation intermediate result, which allows you to warn possible errors children. Exercises: 21 + 9: 3, 34 – 12 2, 90: 30 – 2, 25 4 + 100.

Rules about the order of performing arithmetic operations deserve special attention. This is one of the complex and abstract questions initial course mathematics. Working on it requires numerous time-distributed training exercises. The ability to apply these rules in the practice of calculations is included in the basic requirements of the program at the end of each year, starting from the second grade and at the end of training in primary school.

Exercises:

1. From given pairs examples, select only those where calculations are performed according to the rules of the order of actions: 20 + 30: 5 = 10, 20 + 30: 5 = 26, 42 – 12: 6 = 40,

42 – 12: 6 = 5, 6 5 + 40: 2 = 50, 6 5 + 40: 2 = 35.

After explaining the errors, give the task: change the order of action so that the expression has set value.

2. Place parentheses so that the expression has the specified value:

72 – 24: 6 + 2 = 66, 72 – 24: 6 + 2 = 6, 72 – 24: 6 + 2 = 10, 72 – 24: 6 + 2 = 69

On last year teaching in primary school, the rules discussed are supplemented by new rules for children about the order of performing actions in expressions containing two pairs of brackets or two actions inside brackets. For example: 90 8 – (240 + 170) + 190, 469 148 – 148 9 + (30 100 – 26 909), 65 6500: (50 + (654 – 54)).

Familiarization with identical transformations of expressions. An identical transformation of an expression is a replacement given expression another whose value is equal to the value of the given expression. They perform such transformations of expressions based on the properties of arithmetic operations and the consequences arising from them (how to add a sum to a number, how to subtract a number from a sum, how to multiply a number by a product, etc.) For example: Continue writing so that the “=” sign is preserved :

76 – (20 + 4) = 76 – 20…

(10 + 7) 5 = 10 5…

60: (2 10) = 60: 10…

Using knowledge of the properties of actions to justify calculation methods, students perform transformations of expressions of the form:

36 + 20 + (30 + 6) =+ 20 = (30 + 20) + 6 = 56

72: 3 = (60 + 12) : 3 = 60: 3 + 12: 3 = 24

18 30 = 18 (3 10) = (18 3) 10 = 540

It is necessary to understand that all these expressions are connected by the “=” sign because they have the same meaning.

Identity transformations expressions are also performed on the basis specific meaning actions. For example, the sum of identical terms is replaced by the product: 6 + 6 + 6 + 6 = 6 4, and vice versa, 6 4 = 6 + 6 + 6 + 6. Also based on the meaning of the multiplication action, they transform more complex expressions: 8 4 + 8 = 8 5, 7 6 – 7 = 7 5.

If in expressions with brackets the brackets do not affect the order of actions, then they can be omitted: (30 + 20) + 10 = 30 + 20 + 10, (10 6) : 4 = 10 6: 4, etc.

Subsequently, using the studied properties of actions and rules for the order of actions, students practice transforming expressions with brackets into identical expressions without brackets. For example: write expressions without parentheses so that their values do not change: (65 + 30) – 20, (20 + 4) 3, 96 – (46 + 30)

Objectives of studying the topic:

2) To familiarize students with the rules for performing operations on numbers and, in accordance with them, develop the ability to find the numerical values of expressions.

3) Introduce students to identical transformations of expressions based on the properties of arithmetic operations.

There are 2 main stages in working on numerical expressions:

1) Studying the simplest expressions of the form: sum (2 + 3); difference(5 -1); product (3 4); private (12:4).

2) Studying complex expressions containing two or more actions, with and without parentheses.

Students' first encounter with expressions occurs in the first grade in the topic "Numbers from 1 to 10", where children first become acquainted with the action signs "+" and "-". At this stage, children write down expressions and read them, focusing on the meaning of action signs, which they recognize as shorthand for the words “add” and “drop.” This is reflected in the reading of expressions: 3 + 2 (3 yes 2); 3 - 1 (3 minus one).

2) Along with the simplest mathematical expressions, complex expressions containing two or more actions, with and without parentheses, are also studied. Such expressions appear depending on the consideration of relevant issues in the mathematics course. However, their consideration is mainly subordinated to one didactic goal - to develop the ability to find the meaning of an expression, and this is directly related to the rules for the order of performing arithmetic operations.

a) The first consideration is the rule about the order of operations in expressions without parentheses, when with numbers there is either only addition and subtraction, or only multiplication and division. The first such expressions of the form 5 + 1 + 1, 7 - 1 - 1 are found at the very beginning of the study of addition and subtraction of numbers within 10. Already here the main attention is paid to clarifying the question of how to reason when calculating the meaning of expressions. In grades I-II there are exercises: 70 – 26 + 10, 90 – 20 – 15, 42 + 18 – 19; in grade II there are exercises: 4 · 10: 5, 60: 10 · 3, 36: 9: 2. Upon further examination of similar expressions, the conclusion is drawn: in expressions without parentheses, the actions of addition and subtraction (multiplication and division) are performed in the order how they are written: from left to right.

In the second grade, when studying the operations of multiplication and division, we encounter expressions containing the actions of addition, subtraction, multiplication and division. To clarify the question of the order of execution of actions in such expressions, it is advisable for the first consideration to take the expression 3 · 5 + 3. Using the meaning of the multiplication action, we come to the conclusion that the value of this expression is 18. This implies the order of execution of actions. As a result, we actually get the third rule about the order of operations in expressions without parentheses containing the operations of addition, subtraction, multiplication and division: in expressions without parentheses, the operations of multiplication or division are performed first, and then the operations of addition or subtraction in the order in which they are written . At the same time, a sample of reasoning is also given, where attention is drawn to pronouncing the intermediate result, which makes it possible to prevent possible mistakes by children. Exercises: 21 + 9: 3, 34 – 12 2, 90: 30 – 2, 25 4 + 100.

Rules about the order of performing arithmetic operations deserve special attention. This is one of the complex and abstract questions of the initial mathematics course. Working on it requires numerous training exercises distributed over time. The ability to apply these rules in the practice of calculations is included in the basic requirements of the program at the end of each year, starting from the second grade and at the end of training in the primary grades.

Exercises:

1. From the given pairs of examples, select only those where the calculations are performed according to the rules of the order of actions: 20 + 30: 5 = 10, 20 + 30: 5 = 26, 42 – 12: 6 = 40,

42 – 12: 6 = 5, 6 5 + 40: 2 = 50, 6 5 + 40: 2 = 35.

After explaining the errors, give the task: change the order of action so that the expression has the specified value.

2. Place parentheses so that the expression has the specified value:

72 – 24: 6 + 2 = 66, 72 – 24: 6 + 2 = 6, 72 – 24: 6 + 2 = 10, 72 – 24: 6 + 2 = 69

In the last year of study in primary school, the rules discussed are supplemented by new rules for children about the order of performing actions in expressions containing two pairs of brackets or two actions inside brackets. For example: 90 8 – (240 + 170) + 190, 469 148 – 148 9 + (30 100 – 26 909), 65 6500: (50 + (654 – 54)).

Familiarization with identical transformations of expressions. An identical transformation of an expression is the replacement of a given expression with another whose value is equal to the value of the given expression. They perform such transformations of expressions based on the properties of arithmetic operations and the consequences arising from them (how to add a sum to a number, how to subtract a number from a sum, how to multiply a number by a product, etc.) For example: Continue writing so that the “=” sign is preserved :

76 – (20 + 4) = 76 – 20…

(10 + 7) 5 = 10 5…

60: (2 10) = 60: 10…

Using knowledge of the properties of actions to justify calculation methods, students perform transformations of expressions of the form:

36 + 20 + (30 + 6) =+ 20 = (30 + 20) + 6 = 56

72: 3 = (60 + 12) : 3 = 60: 3 + 12: 3 = 24

18 30 = 18 (3 10) = (18 3) 10 = 540

It is necessary to understand that all these expressions are connected by the “=” sign because they have the same meaning.

Identical transformations of expressions are also performed on the basis of the specific meaning of actions. For example, the sum of identical terms is replaced by a product: 6 + 6 + 6 + 6 = 6 4, and vice versa, 6 4 = 6 + 6 + 6 + 6. Also based on the meaning of the multiplication action, more complex expressions are transformed: 8 4 + 8 = 8 5, 7 6 – 7 = 7 5.

If in expressions with brackets the brackets do not affect the order of actions, then they can be omitted: (30 + 20) + 10 = 30 + 20 + 10, (10 6) : 4 = 10 6: 4, etc.

Methods for studying arithmetic operations. Revealing the specific meaning of the action “subtraction”