If one and two factors is equal to 1, then the product is equal to the other factor.

III. Working on new material.

Students can explain the multiplication technique for cases where there are zeros in the middle of a multi-digit number entry: for example, the teacher suggests calculating the product of the numbers 907 and 3. The students write the solution in a column, reasoning: “I write the number 3 under units.

I multiply the number of units by 3: three times seven - 21, this is 2 des. and 1 unit; I write 1 under the units, and 2 dec. remember. I multiply tens: 0 times 3, you get 0, and even 2, you get 2 tens, I write 2 under the tens. I multiply hundreds: 9 times 3, it turns out 27, I write 27. I read the answer: 2,721.

To consolidate the material, students solve examples from task 361 with a detailed explanation. If the teacher sees that the children have understood the new material well, then he can offer a short commentary.

Teacher. We will explain the solution briefly, naming only the number of units of each digit of the first factor that you multiply, and the result, without naming which digit these units are. Multiply 4,019 by 7. I explain: I multiply 9 by 7, I get 63, I write 3, I remember 6. I multiply 1 by 7, it turns out 7, and even 6 is 13, I write 3, I remember 1. Multiply zero by 7, it turns out zero, and even 1, I get 1, I write 1. I multiply 4 by 7, I get 28, I write 28. I read the answer: 28 133.

P h i s c u l t m i n t k a

IV. Working on learned material.

1. Problem solving.

Problem 363 students solve with commenting. After reading the task, a brief condition is written.

The teacher can offer students to solve the problem in two ways.

Answer: 7,245 centners of grain removed in total.

Children solve problem 364 on their own (with subsequent verification).

1) 42 10 \u003d 420 (c) - wheat

2) 420: 3 = 140 (c) - barley

3) 420 - 140 \u003d 280 (c)

Answer: 280 quintals more wheat.

2. Solution of examples.

Children perform task 365 on their own: they write down expressions and find their meanings.

V. The results of the lesson.

Teacher. Guys, what did you learn at the lesson?

Children. We got acquainted with a new method of multiplication.

Teacher. What did you repeat in class?

Children. They solved problems, made expressions and found their meanings.

Homework: tasks 362, 368; notebook number 1, p. 52, nos. 5–8.

Lesson 58
Multiplication of numbers whose writing
ends in zeros

Goals: introduce the method of multiplication by a single number of multi-digit numbers ending in one or more zeros; consolidate the ability to solve problems, examples of division with a remainder; repeat the table of units of time.

What is it in appearance equations to determine whether this equation will incomplete quadratic equation? But as solve incomplete quadratic equations?

How to recognize "by sight" an incomplete quadratic equation

Left part of the equation is square trinomial, but right — number 0. Such equations are called complete quadratic equations.

At complete quadratic equation all odds, And not equal 0. There are special formulas for solving them, which we will get acquainted with later.

Most simple to solve are incomplete quadratic equations. These are quadratic equations in which some coefficients are zero.

Coefficient by definition cannot be zero, since otherwise the equation would not be quadratic. We talked about this. So, it turns out that to apply to zero can only odds or.

Depending on this, there three types of incomplete quadratic equations.

1) , where ;
2) , where ;
3) , where .

So, if we see a quadratic equation, on the left side of which instead of three members present two members or one member, then this equation will be incomplete quadratic equation.

Definition of an incomplete quadratic equation

Incomplete quadratic equation is called a quadratic equation in which at least one of the coefficients or zero.

This definition has a lot important phrase " at least one from coefficients... zero". It means that one or more coefficients can equal zero.

Based on this, it is possible three options: or one coefficient is zero, or another coefficient is zero, or both coefficients are simultaneously equal to zero. This is how three types of incomplete quadratic equation are obtained.

incomplete quadratic equations are the following equations:
1)
2)
3)

Equation solution

Let's outline solution plan this equation. left part of the equation can be easily factorize, since on the left side of the equation the terms and have common factor, it can be taken out of the bracket. Then the product of two factors will be obtained on the left, and zero on the right.

And then the rule “the product is equal to zero if and only if at least one of the factors is equal to zero, while the other makes sense” will work. Everything is very simple!

So, solution plan.
1) We factorize the left side.
2) We use the rule "the product is equal to zero ..."

I call equations of this type "a gift of fate". These are equations that the right side is zero, but left part can be split multipliers.

Solve the equation according to plan.

1) Let's decompose left side of the equation multipliers, for this we take out the common factor , we get the following equation .

2) In the equation we see that left costs work, but zero on the right.

Real a gift of fate! Here, of course, we will use the rule "the product is equal to zero if and only if at least one of the factors is equal to zero, while the other makes sense".

When translating this rule into the language of mathematics, we get two equations or .

We see that the equation fell apart for two simpler equations, the first of which has already been solved ().

Let's solve the second the equation . Move the unknown terms to the left and the known terms to the right. An unknown member is already on the left, we'll leave him there. And we move the known term to the right with the opposite sign. We get an equation.

We have found, and we need to find. To get rid of the factor , you need to divide both sides of the equation by .

Along with addition, important operations are multiplication and division. Let us recall at least the tasks of determining how many times Masha has more apples than Sasha, or finding the number of parts produced per year, if the number of parts produced per day is known.

Multiplication is one of four basic arithmetic operations, during which one number is multiplied by another. In other words, the entry 5 · 3 = 15 means that the number 5 was folded 3 times, i.e. 5 · 3 = 5 + 5 + 5 = 15.

Multiplication regulated by the system rules.

1. The product of two negative numbers is equal to a positive number. To find the modulus of the product, you need to multiply the modulus of these numbers.

(- 6) ( - 6) = 36; (- 17.5) ( - 17,4) = 304,5

2. The product of two numbers with different signs is equal to a negative number. To find the modulus of the product, you need to multiply the modulus of these numbers.

(- 5) 6 = - thirty; 0.7 ( - 8) = - 21

3. If one of the factors is equal to zero, then the product is equal to zero. The reverse is also true: the product is zero only if one of the factors is zero.

2.73 0 = 0; ( - 345.78) 0 = 0

Based on the above material, we will try to solve the equation 4 ∙ (x – 5) = 0.

1. Expand the brackets and get 4x - 20 = 0.

2. Move (-20) to the right side (do not forget to change the sign to the opposite) and
we get 4x = 20.

3. Find x by reducing both sides of the equation by 4.

4. Total: x = 5.

But knowing Rule #3, we can solve our equation much faster.

1. Our equation is 0, and by rule number 3, the product is 0 if one of the factors is 0.

2. We have two multipliers: 4 and (x - 5). 4 is not equal to 0, so x - 5 = 0.

3. We solve the resulting simple equation: x - 5 \u003d 0. Hence, x \u003d 5.

Multiplication relies on two laws - commutative and associative laws.

displacement law: for any numbers but And b true equality ab=ba:

(- 6) 1.2 = 1.2 ( - 6), i.e. = - 7,2.

Combination law: for any numbers a, b And c true equality (ab)c = a(bc).

(- 3) ( - 5) 2 = ( - 3) (2 ( - 5)) = (- 3) ( - 10) = 30.

The arithmetic operation inverse to multiplication is division. If the components of the multiplication are called multipliers, then in division the number that is divisible is called divisible, the number by which we divide, - divider, and the result is private.

12: 3 = 4, where 12 is the dividend, 3 is the divisor, 4 is the quotient.

Division, like multiplication, is regulated rules.

1. The quotient of two negative numbers is a positive number. To find the modulus of the quotient, you need to divide the modulus of the dividend by the modulus of the divisor.

- 12: (- 3) = 4

2. The quotient of two numbers with different signs is a negative number. To find the modulus of the quotient, you need to divide the modulus of the dividend by the modulus of the divisor.

- 12: 3 = - 4; 12: (- 3) = - 4.

3. Dividing zero by any non-zero number is zero. You can't divide by zero.

0:23=0; 23: 0 = XXXX

Based on the rules of division, let's try to solve an example - 4 x ( - 5) – (- 30) : 6 = ?

1. We perform the multiplication: -4 x (-5) \u003d 20. So, our example will take the form 20 - (-30): 6 \u003d?

2. Perform division (-30): 6 = -5. So, our example will take the form 20 - (-5) = ?.

3. Subtract 20 - (-5) = 20 + 5 = 25.

So our answer 25.

Knowledge of multiplication and division, along with addition and subtraction, allows us to solve various equations and problems, as well as to navigate perfectly in the world of numbers and operations around us.

Fix the material by deciding equation 3 ∙ (4x – 8) = 3x – 6.

1. Open the brackets 3 ∙ (4x - 8) and get 12x - 24. Our equation has become 12x - 24 \u003d 3x - 6.

2. We present similar ones. To do this, we move all components from x to the left, and all numbers to the right.
We get 12x - 24 \u003d 3x - 6 → 12x - 3x \u003d -6 + 24 → 9x \u003d 18.

When moving a component from one part of the equation to another, do not forget to change signs to opposite ones.

3. We solve the resulting equation 9x \u003d 18, from where x \u003d 18: 9 \u003d 2. So, our answer is 2.

4. To make sure that our decision is correct, let's check:

3 ∙ (4x - 8) = 3x - 6

3 (4 ∙ 2 - 8) = 3 ∙ 2 - 6

3 ∙ (8 – 8) = 6 – 6

0 = 0, so our answer is correct.

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