Actions with rational numbers: rules, examples, solutions. Topic: "laws of arithmetic operations" - Document Work 7 rectangle formulas laws of arithmetic operations

In the course of historical development, of course, they added and multiplied for a long time, without realizing the laws that govern these operations. Only in the 20s and 30s of the previous century, mainly French and English mathematicians clarified the basic properties of these operations. Whoever wants to get acquainted with the history of this issue in more detail, I can recommend here, as I will do it repeatedly below, a large "Encyclopedia of Mathematical Sciences".

Returning to our topic, I mean now to really list those five basic laws to which addition is reduced:

1) always represents a number, in other words, the action of addition is always feasible without any exceptions (as opposed to subtraction, which is not always feasible in the region of positive numbers);

2) the amount is always uniquely determined;

3) there is a combination, or associative law: so that the brackets can be omitted altogether;

4) there is a transposable, or commutative law:

5) the law of monotony takes place: if, then.

These properties are understandable without further explanation if we have before our eyes a visual representation of number as a quantity. But they must be expressed strictly formally, so that one can rely on them in the further strictly logical development of the theory.

As for multiplication, first of all, there are five laws that are similar to those just listed:

1) there is always a number;

2) the product is unique,

3) the law of combination:

4) the law of transposition:

5) the law of monotony: if, then

Finally, the connection between addition and multiplication is established by the sixth law:

6) the law of distribution, or distribution:

It is easy to see that all calculations are based solely on these 11 laws. I will confine myself to a simple example, say, multiplying the number 7 by 12;

according to the law of distribution

In this short discussion, you will, of course, recognize the individual steps that we take when calculating in the decimal system. I leave it to you to parse the examples yourself. We will only state the summary result here: our digital calculations consist in the repeated application of the above eleven basic provisions, as well as in the application of memorized results of actions on single-digit numbers (addition table and multiplication table).

However, where do the laws of monotony find their application? In ordinary, formal calculations, we do not really rely on them, but they turn out to be necessary in problems of a slightly different kind. Let me remind you here about the method, which in decimal counting is called the estimation of the value of the product and the quotient. This is a technique of the greatest practical importance, which, unfortunately, is far from being well known at school and among students, although on occasion it is spoken about in the second grade; I will limit myself here to just an example. Let's say we need to multiply 567 by 134, and in these numbers the unit digits are set - say, through physical measurements - only very inexactly. In this case, it would be completely useless to calculate the product with full accuracy, since such a calculation still does not guarantee us the exact value of the number of interest to us. But what is really important for us is to know the order of magnitude of the product, that is, to determine within what number of tens or hundreds the number is contained. But this, estimate, the law of monotony really gives you directly, because it follows from it that the required number is contained between 560-130 and 570-140. Again, I leave the further development of these considerations to you.

In any case, you see that in "evaluative calculations" you have to constantly use the laws of monotony.

As for the actual application of all these things in school teaching, there can be no question of a systematic presentation of all these basic laws of addition and multiplication. The teacher can dwell only on the laws of combination, displacement and distributive, and then only in the transition to literal calculations, heuristically deriving them from simple and clear numerical examples.

The approach to the addition of non-negative integers makes it possible to substantiate the well-known laws of addition: displacement and combinational.

Let us first prove the displacement law, that is, we will prove that for any non-negative integers a and b the equality a + b \u003d b + a holds.

Let a be the number of elements in the set A, b - the number of elements in the set B and AB \u003d 0. Then, by definition, the sum of non-negative integers a + b is the number of elements of the union of the sets A and B: a + b \u003d n (A // B). But the set A B is equal to the set B A according to the displacement property of the union of sets, and, Hence, n (AU B) \u003d n (B U A). By the definition of the sum n (BiA) \u003d b + a, therefore a + b \u003d b + a for any non-negative integers a and b.

We now prove the combination law, that is, we will prove that for any non-negative integers a, b, c the equality (a + b) + c \u003d a + (b + c) holds.

Let a \u003d n (A), b \u003d n (B), c \u003d n (C), and AUB \u003d 0, BUC \u003d 0 Then, by the definition of the sum of two numbers, we can write (a + b) + c \u003d n (A / /) B) + n (C) \u003d n ((AUBUC).

Since the union of sets obeys the combination law, then n ((AUB) U C) \u003d n (A U (BUC)). Whence, by the definition of the sum of two numbers, we have n (A J (BUC)) \u003d n (A) + n (BU C) \u003d a + (b + c). Therefore, (a + b) + c - a + (b + c) for any non-negative integers a, b and c.

What is the purpose of the combination law of addition? He explains how you can find the sum of three terms: for this, it is enough to add the first term to the second and add the third term to the resulting number or add the first term to the sum of the second and third. Note that the combination law does not imply a permutation of terms.

Both the displacement and combination laws of addition can be generalized to any number of terms. In this case, the displacement law will mean that the sum does not change with any permutation of the terms, and the combinational law means that the sum does not change with any grouping of terms (without changing their order).

It follows from the transpositional and combination laws of addition that the sum of several terms will not change if they are rearranged in any way and if any group of them is enclosed in brackets.

Let's calculate, using the laws of addition, the value of the expression 109 + 36+ 191 +64 + 27.

On the basis of the transposition law, we rearrange the terms 36 and 191. Then 109 + 36 + 191 + 64 + 27 \u003d 109 + 191 + 36 + 64 + 27.

We will use the combination law, grouping the terms, and then find the sums in brackets: 109+ 191 +36 + 64 + 27 \u003d\u003d (109 + 191) + (36 + 64) + 27 \u003d 300 + 100 + 27.

Let's apply the combination law once again, enclosing the sum of numbers 300 and 100 in parentheses: 300+ 100 + 27 \u003d (300+ 100) + 27.

Let's make the calculations: (300+ 100) + 27 \u003d 400+ 27 \u003d 427.

Pupils with displaceable addition primary grades get acquainted when studying the numbers of the first ten. It is first used to compile a single-digit addition table and then to rationalize various calculations.

The combination law of addition is not explicitly studied in the elementary course of mathematics, but is constantly used. So, it is the basis for the method of adding a number in parts: 3 + 2 \u003d 3 + (1 + 1) \u003d (3+ 1) + 1 \u003d 4 + 1 \u003d 5. In addition, in cases where it is necessary to add a number to the amount, the amount to the number, the amount to the amount, the combination law is used in combination with the transpositional law. For example, adding the sum 2 + 1 to the number 4 is proposed in the following ways:

1) 4 + (2+1) = 4 + 3 = 7;

4+2+ 1 = 6+1 =7;

4 + (2+1) = 5 + 2 = 7.

Let's analyze these methods. In case 1, the calculations were performed in accordance with the indicated procedure. In case 2, the combination property of addition is applied. Calculations in the latter case are based on the displacement and combination laws of addition, and the intermediate transformations are omitted. They are like that. First, on the basis of the displacement law, the terms 1 and 2 were rearranged: 4+ (2-1) \u003d 4 + (1 + 2). Then we used the combination law: 4 + (1 + 2) \u003d (4+ 1) + 2. And, finally, we made calculations according to the order of actions (4 +1) + 2 \u003d 5 + 2 \u003d 7.

Rules for subtracting a number from a sum and an amount from a number

Let us prove the well-known rules for subtracting a number from a sum and a sum from a number.

The rule for subtracting a number from a sum. To subtract a number from the sum, it is enough to subtract this number from one of the summands in the sum and add another summand to the result.

We write this rule using the symbols: If a, b, c are non-negative integers, then:

a) for a\u003e c we have that (a + b) - c \u003d (a - c) + b;

b) for b\u003e c we have that (a + b) - c \u003d a + (b - c);

c) for a\u003e c and b\u003e c, any of these formulas can be used.

Let a\u003e c, then the difference a - c exists. Let us denote it by p: a - c \u003d p. Hence a \u003d p + c. Substitute the sum p + -c instead of a into the expression (a + b) - c and transform it: (a + 6) - c \u003d (p + c + b) - c \u003d p + b + -c - c \u003d p + b

But the letter p denotes the difference a - c, which means that we have (a + b) - - c \u003d (a - c) + b, which is what we had to prove.

Reasoning is carried out in a similar way for other cases. Let us now give an illustration of this rule (case "a") using Euler circles. Take three finite sets A, B, and C such that n (A) \u003d a, n (B) \u003d b, n (C) \u003d c and AUB \u003d 0, CUA. Then (a + b) - c is the number of elements of the set (AUB) C, and the number (a - c) + b is the number of elements of the set (AC) UB. On Euler's circles, the set (AUB) C is represented by the shaded area shown in the figure.

It is easy to make sure that the set (AC) UB will be represented by exactly the same area. Hence, (AUB) C \u003d (AC) UB for data

sets A, B, and C. Therefore, n ((AUB) C) \u003d n ((AC) UB) and (a + b) - c - (a - c) + b.

Case b can be illustrated in a similar way.

The rule for subtracting the amount from the number. To subtract the sum of numbers from the number, it is sufficient to subtract from this number successively each term one after the other, i.e. if a, b, c are non-negative integers, then for a\u003e b + c we have a - (b + c ) \u003d (a - b) - c.

The rationale for this rule and its set-theoretic illustration are carried out in the same way as for the rule for subtracting a number from a sum.

These rules are discussed in primary school on specific examples, illustrative images are used for justification. These rules allow you to perform calculations efficiently. For example, the rule for subtracting a sum from a number is the basis for subtracting a number in parts:

5-2 = 5-(1 + 1) = (5-1)-1=4-1=3.

The meaning of the above rules is well revealed when solving arithmetic problems in various ways. For example, the problem “20 small and 8 large fishing boats went out to sea in the morning. 6 boats have returned. How many boats with fishermen still have to return? " can be solved in three ways:

/ way. 1.20 + 8 \u003d 28 2.28 - 6 \u003d 22

// way. 1.20 - 6 \u003d 14 2.14 + 8 \u003d 22

Method III. 1.8 - 6 \u003d 2 2.20 + 2 \u003d 22

Multiplication laws

Let us prove the laws of multiplication, proceeding from the definition of a product in terms of the Cartesian product of sets.

1. The displacement law: for any non-negative integers a and b, the equality a * b \u003d b * a is true.

Let a \u003d n (A), b \u003d n (B). Then, by the definition of the product, a * b \u003d n (A * B). But the sets A * B and B * A are of equal power: each pair (a, b) from the set AXB can be associated with a single pair (b, a) from the set BxA, and vice versa. Hence, n (AXB) \u003d n (BxA), and therefore a-b \u003d n (AXB) \u003d n (BXA) \u003d b-a.

2. Combination law: for any non-negative integers a, b, c, the equality (a * b) * c \u003d a * (b * c) is true.

Let a \u003d n (A), b \u003d n (B), c \u003d n (C). Then, by the definition of the product (ab) -c \u003d n ((AXB) XQ, a- (b -c) \u003d n (AX (BXQ). The sets (AxB) XC and A X (BX Q are different: the first consists of pairs of the form ((a, b), c), and the second - from pairs of the form (a, (b, c)), where aЈA, bЈB, cЈC. But the sets (AXB) XC and AX (BXC) are of equal power, since there is a one-to-one mapping from one set to another, therefore n ((AXB) * \u200b\u200bC) \u003d n (A * (B * C)), and, therefore, (a * b) * c \u003d a * (b * c).

3. Distributive law of multiplication with respect to addition: for any non-negative integers a, b, c the equality (a + b) x c \u003d ac + be is true.

Let a - n (A), b \u003d n (B), c \u003d n (C) and AUB \u003d 0. Then, by the definition of the product, we have (a + b) xc \u003d n ((AUB) * C. Whence, based on equality (*) we obtain n ((A UB) * C) \u003d n ((A * C) U (B * C)), and further, by the definition of the sum and product n ((A * C) U (B * C) ) - \u003d n (A * C) + n (B * C) \u003d ac + bc.

4. Distributive law of multiplication with respect to subtraction: for any non-negative integers a, b and c and a ^ b the equality (a - b) c \u003d \u003d ac - bc is true.

This law is derived from the equality (AB) * C \u003d (A * C) (B * C) and is proved similarly to the previous one.

The movable and combinational laws of multiplication can be extended to any number of factors. As with addition, these laws are often used together, that is, the product of several factors does not change if they are rearranged in any way and if any group of them is enclosed in parentheses.

Distributional laws establish the connection between multiplication and addition and subtraction. On the basis of these laws, the brackets are expanded in expressions like (a + b) c and (a - b) c, as well as the factor is taken out of the brackets if the expression has the form ac - be or

In the elementary course of mathematics, the transposable property of multiplication is studied, it is formulated as follows: "The product will not change from the permutation of the factors" - and is widely used in compiling the multiplication table for single-digit numbers. The combination law in elementary school is not explicitly considered, but is used together with the proxy law when multiplying a number by a product. It happens as follows: students are invited to consider various ways of finding the value of the expression 3 * (5 * 2) and compare the results.

Cases are given:

1) 3* (5*2) = 3*10 = 30;

2) 3* (5*2) = (3*5) *2 = 15*2 = 30;

3) 3* (5*2) = (3*2) *5 = 6*5 = 30.

The first of them is based on the rule of the order of actions, the second - on the combination law of multiplication, the third - on the transposable and combination laws of multiplication.

The distributive law of multiplication with respect to addition is considered at school with specific examples and is called the rules for multiplying a number by a sum and a sum by a number. Consideration of these two rules is dictated by methodological considerations.

Rules for dividing sum by number and numbers by product

Let's get acquainted with some properties of division of natural numbers. The choice of these rules is determined by the content of the elementary mathematics course.

The rule for dividing a sum by a number. If the numbers a and b are divisible by the number c, then their sum a + b is divisible by c; the quotient obtained by dividing the sum a + b by the number c is equal to the sum of the quotients obtained by dividing a by c and b by c, i.e.

(a + b): c \u003d a: c + b: c.

Evidence. Since a is divisible by c, there exists a natural number m \u003d a: c such that a \u003d c-m. Similarly, there exists a natural number n - b: c such that b \u003d c-n. Then a + b \u003d \u003d c-m + c- / 2 \u003d c- (m + n). Hence it follows that a + b is divided by c and the quotient obtained by dividing a + b by the number c is equal to m + n, that is, a: c + b: c.

The proven rule can be interpreted from a set-theoretic standpoint.

Let a \u003d n (A), b \u003d n (B), and AGB \u003d 0. If each of the sets A and B can be partitioned into c equally powerful subsets, then the union of these sets admits the same partition.

Moreover, if each subset of the partition of the set A contains a: c elements, and each subset of the set B contains b: c elements, then each subset of the set A [) B contains a: c + b: c elements. This means that (a + b): c \u003d a: c + b: c.

The rule for dividing a number by a product. If a natural number a is divisible by integers B and c, then in order to divide a by the product of the numbers b and c, it is enough to divide the number a by b (c) and divide the resulting quotient by c (b): a: (b * c) - (a: b): c \u003d (a: c): b Proof. We put (a: b): c \u003d x. Then, by the definition of the quotient, a: b \u003d c-x, hence it is analogous to a - b- (cx). Based on the combination law of multiplication a \u003d (bc) -x. The resulting equality means that a: (bc) \u003d x. So a: (bc) \u003d (a: b): c.

The rule for multiplying a number by a quotient of two numbers. To multiply a number by a quotient of two numbers, it is enough to multiply this number by the dividend and divide the resulting product by the divisor, i.e.

a- (b: c) \u003d (a-b): c.

Application of the formulated rules allows simplifying calculations.

For example, to find the value of the expression (720+ 600): 24, it is enough to divide by 24 terms 720 and 600 and add the resulting quotients:

(720+ 600): 24 \u003d 720: 24 + 600: 24 \u003d 30 + 25 \u003d 55. The value of the expression 1440: (12 * 15) can be found by dividing first 1440 by 12, and then the resulting quotient is divided by 15:

1440: (12 * 15) = (1440:12): 15 = 120:15 = 8.

These rules are considered in the initial course of mathematics with specific examples. At the first acquaintance with the rule of dividing the sum 6 + 4 by the number 2, illustrative material is used. In the future, this rule is used to rationalize calculations. The rule of dividing a number by a product is widely used when dividing numbers ending in zeros.

18-19.10.2010

Theme: "LAWS OF ARITHMETIC ACTIONS"

Goal: to acquaint students with the laws of arithmetic operations.

Lesson Objectives:

to reveal the displacement and combination laws of addition and multiplication using specific examples; to teach them to apply when simplifying expressions;

to form the ability to simplify expressions;

work on the development of logical thinking and speech of children;

educate independence, curiosity, interest in the subject.

UUD: the ability to act with symbolic symbols,

the ability to choose the grounds, criteria for comparison, comparison, assessment and classification of objects.

Equipment: textbook, TVE, presentation

Figure: Fig. 30 31

Using Figure 30, explain why the equality is true

a + b \u003d b + a.

This equality expresses the property of addition that you know. Try to remember which one.

Check yourself:

The sum does not change from the change of places of the terms

This property - travel law of addition.

What equality can be written from Figure 31? What property of addition expresses this equality?

Check yourself.

From Figure 31 it follows that (a + b) + c \u003d a + (b + c): if you add the third term to the sum of two terms, you get the same number as from adding the sum of the second and third terms to the first term.

Instead of (a + b) + c, as well as | instead of a + (b + c), you can simply write a + b + c.

This property - combination law of addition.

In mathematics, the laws of arithmetic operations are written as in | verbal form, and in the form of equalities using letters:

Explain how, using the laws of addition, you can simplify the following calculations and perform them:

212. a) 48 + 56 + 52; e) 25 + 65 + 75;

b) 34 + 17 + 83; f) 35 + 17 + 65 + 33;

c) 56 + 24 + 38 + 62; g) 27 + 123 + 16 + 234;

d) 88 + 19 + 21 + 12; h) 156 + 79 + 21 + 44.

213. Using Figure 32, explain why the equality is true ab = b a.

Can you guess which law illustrates this equality? Can it be argued that for

multiplication are the same laws as for addition? Try to formulate them,

and then check yourself:

Using the laws of multiplication, calculate the values \u200b\u200bof the following expressions orally:

214. a) 76 · 5 · 2; c) 69 * 125 * 8; e) 8 941 125; B C

b) 465 * 25 * 4; d) 4 * 213 * 5 * 5; f) 2 5 126 4 25.

215. Find the area of \u200b\u200ba rectangle ABCD (fig. 33) in two ways.

216. Using Figure 34, explain why the equality holds: a (b + c) \u003d ab + ac.

Figure: 34 What property of arithmetic operations does it express?

Check yourself. This equality illustrates the following property: when multiplying a number by a sum, you can multiply this number by each term and add the results obtained.

This property can be formulated in another way: the sum of two or more products containing the same factor can be replaced by the product of this factor by the sum of the remaining factors.

This property is another law of arithmetic operations - distribution... As you can see, the verbal formulation of this law is very cumbersome, and the mathematical language is the means that makes it concise and understandable:

Think about how to verbally perform the calculations in tasks No. 217 - 220 and complete them.

217. a) 15 13; b) 26 22; c) 34 12; d) 27 21.

218. a) 44 52; b) 16 42; c) 35 · 33; d) 36 26.

219. a) 43 16 + 43 84; e) 62 16 + 38 16;

b) 85 47 + 53 85; f) 85 44 + 44 15;

c) 54 60 + 460 6. g) 240 710 + 7100 76;

d) 23 320 + 230 68; h) 38 5800 + 380 520.

220. a) 4 63 + 4 79 + 142 6; c) 17 27 + 23 17 + 50 19;

b) 7 125 + 3 62 + 63 3; d) 38 46 + 62 46 + 100 54.

221. Draw a drawing in your notebook to prove equality a ( b - c) \u003d a b - ace

222. Calculate orally, applying the distribution law: a) 6 · 28; b) 18 21; c) 17 · 63; d) 19 98.

223. Calculate orally: a) 34 · 84 - 24 · 84; c) 51 78 - 51 58;

b) 45 40 - 40 25; d) 63 7 - 7 33

224 Calculate: a) 560 188 - 880 56; c) 490 730 - 73 900;

b) 84 670 - 640 67; d) 36 3400 - 360 140.

Calculate orally using the techniques you know:

225. a) 13 5 + 71 5; c) 87 5 - 23 5; e) 43 25 + 25 17;

b) 58 5 - 36 5; d) 48 5 + 54 5; f) 25 67 - 39 25.

226. Without doing any calculations, compare the values \u200b\u200bof the expressions:

a) 258 * (764 + 548) and 258 * 764 + 258 * 545; c) 532 · (618 - 436) and 532 · 618 –532 · 436;

b) 751 * (339 + 564) and 751 * 340 + 751 * 564; d) 496 · (862 - 715) and 496 · 860 - 496 · 715.

227. Fill the table:

Did you have to do the calculations to fill in the second line?

228. How will this product change if the factors are changed as follows:

229. Write down which natural numbers are located on the coordinate ray:

a) to the left of the number 7; c) between the numbers 2895 and 2901;

b) between the numbers 128 and 132; d) to the right of the number 487, but to the left of the number 493.

230. Insert action signs to get the correct equality: a) 40 + 15? 17 \u003d 72; c) 40? 15 ? 17 \u003d 8;

b) 40? 15 ? 17 \u003d 42; d) 120? 60? 60 \u003d 0.

231 ... The socks are blue in one box and white in the other. There are 20 more pairs of blue socks than white socks, and there are only two boxes of 84 Lara of socks. How many pairs of socks of each color?

232 ... There are three types of groats in the store: buckwheat, pearl barley and rice, only 580 kg. If we sold 44 kg of buckwheat, 18 kg of barley and 29 kg of rice, then the mass of cereals of all types would become the same. How many kilograms of each type of cereal are in the store.

To use the preview of presentations, create yourself an account ( account) Google and log into it: https://accounts.google.com

Slide captions:

10/22/15 Cool work

Find the length of the segment AB a b A B b a B A AB \u003d a + b AB \u003d b + a

11 + 16 \u003d 27 (fruits) 16 + 11 \u003d 27 (fruits) Will the total number of fruits change from the rearrangement of the terms? Masha collected 11 apples and 16 pears. How many fruits did Masha have in her basket?

Make up a literal expression to write a verbal statement: "the sum will not change from the permutation of the terms"

(5 + 7) + 3 \u003d 15 (toys) Which way of counting is easier? Masha was decorating the Christmas tree. She hung up 5 Christmas balls, 7 cones and 3 stars. How many toys did Masha hang up? (7 + 3) + 5 \u003d 15 (toys)

Make up a literal expression for writing a verbal statement: "To add the third term to the sum of two terms, you can add the sum of the second and third terms to the first term" (a + b) + c \u003d a + (b + c) Combination law of addition

Let's count: 27+ 148 + 13 \u003d (27 + 13) + 148 \u003d 188 124 + 371 + 429 + 346 \u003d \u003d (124 + 346) + (371 + 429) \u003d \u003d 470 + 800 \u003d 1270 Learning to count fast!

Are the same laws true for multiplication as for addition? a b \u003d b a (a b) c \u003d a (b c)

b \u003d 15 a \u003d 12 c \u003d 2 V \u003d (a b) c \u003d a (b c) V \u003d (12 15) 2 \u003d \u003d 12 (15 2) \u003d 360 S \u003d a b \u003d b a S \u003d 12 15 \u003d \u003d 15 12 \u003d 180

a b \u003d b a (a b) c \u003d a (b c) Displacement law of multiplication Combination law of multiplication

Let's count: 25 756 4 \u003d (25 4) 756 \u003d 75600 8 (956 125) \u003d \u003d (8 125) 956 \u003d \u003d 1000 956 \u003d 956000 Learning to count fast!

LESSON TOPIC: What are we working with today in the lesson? Formulate the topic of the lesson.

212 (1 column), 214 (a, b, c), 231, 230 In the classroom Homework 212 (2 columns), 214 (d, e, f), 253

On the subject: methodological developments, presentations and notes

The development of a lesson in mathematics in grade 5 "Laws of arithmetic operations" includes a text file and a presentation for the lesson. In this lesson, the displacement and combinational laws are repeated, ...

Arithmetic laws

This presentation is prepared for a lesson in mathematics in grade 5 on the topic "The laws of arithmetic operations" (textbook by II Zubarev, AG Mordkovich) ....

A lesson in learning new material using EOR ...

Arithmetic laws

The presentation was created to visually accompany a lesson in grade 5 on the topic "Arithmetic operations with integers". It presents a selection of tasks for both general and independent solutions ...

lesson development Mathematics Grade 5 Laws of arithmetic operations

lesson development Mathematics Grade 5 Laws of arithmetic operations № p / p The structure of the annotation The content of the annotation 1231 FIO Malyasova Lyudmila Gennadievna 2 Position, taught subject Teacher ma ...