Repetition lesson on decimal fractions. Lesson "all actions with decimal fractions." "Repetition: actions with decimals"

In this tutorial, we'll look at each of these operations one by one.

Lesson content

Adding decimals

As we know, a decimal fraction consists of an integer part and a fractional part. When adding decimals, the integer and fractional parts are added separately.

For example, let's add the decimals 3.2 and 5.3. It is more convenient to add decimal fractions in a column.

First, we write these two fractions in a column, while the integer parts must be under the integer parts, and the fractional ones under the fractional ones. In school, this requirement is called "comma under comma" .

Let's write the fractions in a column so that the comma is under the comma:

We add the fractional parts: 2 + 3 = 5. We write down the five in the fractional part of our answer:

Now we add up the integer parts: 3 + 5 = 8. We write the eight in the integer part of our answer:

Now we separate the integer part from the fractional part with a comma. To do this, we again follow the rule "comma under comma" :

Got the answer 8.5. So the expression 3.2 + 5.3 is equal to 8.5

3,2 + 5,3 = 8,5

In fact, not everything is as simple as it seems at first glance. Here, too, there are pitfalls, which we will now talk about.

Places in decimals

Decimals, like ordinary numbers, have their own digits. These are tenth places, hundredth places, thousandth places. In this case, the digits begin after the decimal point.

The first digit after the decimal point is responsible for the tenths place, the second digit after the decimal point for the hundredths place, the third digit after the decimal point for the thousandths place.

Decimal digits store some useful information. In particular, they report how many tenths, hundredths, and thousandths are in a decimal.

For example, consider the decimal 0.345

The position where the triple is located is called tenth place

The position where the four is located is called hundredths place

The position where the five is located is called thousandths

Let's look at this figure. We see that in the category of tenths there is a three. This suggests that there are three tenths in the decimal fraction 0.345.

If we add the fractions, and then we get the original decimal fraction 0.345

We first got the answer, but converted it to decimal and got 0.345.

Adding decimals follows the same rules as adding ordinary numbers. The addition of decimal fractions occurs by digits: tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.

Therefore, when adding decimal fractions, it is required to follow the rule "comma under comma". A comma under a comma provides the very order in which tenths are added to tenths, hundredths to hundredths, thousandths to thousandths.

Example 1 Find the value of the expression 1.5 + 3.4

First of all, we add the fractional parts 5 + 4 = 9. We write the nine in the fractional part of our answer:

Now we add up the integer parts 1 + 3 = 4. We write down the four in the integer part of our answer:

Now we separate the integer part from the fractional part with a comma. To do this, we again observe the rule "comma under a comma":

Got the answer 4.9. So the value of the expression 1.5 + 3.4 is 4.9

Example 2 Find the value of the expression: 3.51 + 1.22

We write this expression in a column, observing the rule "comma under a comma"

First of all, add the fractional part, namely the hundredths 1+2=3. We write the triple in the hundredth part of our answer:

Now add tenths of 5+2=7. We write down the seven in the tenth part of our answer:

Now add the whole parts 3+1=4. We write down the four in the whole part of our answer:

We separate the integer part from the fractional part with a comma, observing the “comma under the comma” rule:

Got the answer 4.73. So the value of the expression 3.51 + 1.22 is 4.73

3,51 + 1,22 = 4,73

As with ordinary numbers, when adding decimal fractions, . In this case, one digit is written in the answer, and the rest are transferred to the next digit.

Example 3 Find the value of the expression 2.65 + 3.27

We write this expression in a column:

Add hundredths of 5+7=12. The number 12 will not fit in the hundredth part of our answer. Therefore, in the hundredth part, we write the number 2, and transfer the unit to the next bit:

Now we add the tenths of 6+2=8 plus the unit that we got from the previous operation, we get 9. We write the number 9 in the tenth of our answer:

Now add the whole parts 2+3=5. We write the number 5 in the integer part of our answer:

Got the answer 5.92. So the value of the expression 2.65 + 3.27 is 5.92

2,65 + 3,27 = 5,92

Example 4 Find the value of the expression 9.5 + 2.8

Write this expression in a column

We add the fractional parts 5 + 8 = 13. The number 13 will not fit in the fractional part of our answer, so we first write down the number 3, and transfer the unit to the next digit, or rather transfer it to the integer part:

Now we add the integer parts 9+2=11 plus the unit that we got from the previous operation, we get 12. We write the number 12 in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 12.3. So the value of the expression 9.5 + 2.8 is 12.3

9,5 + 2,8 = 12,3

When adding decimal fractions, the number of digits after the decimal point in both fractions must be the same. If there are not enough digits, then these places in the fractional part are filled with zeros.

Example 5. Find the value of the expression: 12.725 + 1.7

Before writing this expression in a column, let's make the number of digits after the decimal point in both fractions the same. The decimal fraction 12.725 has three digits after the decimal point, while the fraction 1.7 has only one. So in the fraction 1.7 at the end you need to add two zeros. Then we get the fraction 1,700. Now you can write this expression in a column and start calculating:

Add thousandths of 5+0=5. We write the number 5 in the thousandth part of our answer:

Add hundredths of 2+0=2. We write the number 2 in the hundredth part of our answer:

Add tenths of 7+7=14. The number 14 will not fit in a tenth of our answer. Therefore, we first write down the number 4, and transfer the unit to the next bit:

Now we add the integer parts 12+1=13 plus the unit that we got from the previous operation, we get 14. We write the number 14 in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 14,425. So the value of the expression 12.725+1.700 is 14.425

12,725+ 1,700 = 14,425

Subtraction of decimals

When subtracting decimal fractions, you must follow the same rules as when adding: “a comma under a comma” and “an equal number of digits after a decimal point”.

Example 1 Find the value of the expression 2.5 − 2.2

We write this expression in a column, observing the “comma under comma” rule:

We calculate the fractional part 5−2=3. We write the number 3 in the tenth part of our answer:

Calculate the integer part 2−2=0. We write zero in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

We got the answer 0.3. So the value of the expression 2.5 − 2.2 is equal to 0.3

2,5 − 2,2 = 0,3

Example 2 Find the value of the expression 7.353 - 3.1

This expression has a different number of digits after the decimal point. In the fraction 7.353 there are three digits after the decimal point, and in the fraction 3.1 there is only one. This means that in the fraction 3.1, two zeros must be added at the end to make the number of digits in both fractions the same. Then we get 3,100.

Now you can write this expression in a column and calculate it:

Got the answer 4,253. So the value of the expression 7.353 − 3.1 is 4.253

7,353 — 3,1 = 4,253

As with ordinary numbers, sometimes you will have to borrow one from the adjacent bit if subtraction becomes impossible.

Example 3 Find the value of the expression 3.46 − 2.39

Subtract hundredths of 6−9. From the number 6 do not subtract the number 9. Therefore, you need to take a unit from the adjacent digit. Having borrowed one from the neighboring digit, the number 6 turns into the number 16. Now we can calculate the hundredths of 16−9=7. We write down the seven in the hundredth part of our answer:

Now subtract tenths. Since we took one unit in the category of tenths, the figure that was located there decreased by one unit. In other words, the tenth place is now not the number 4, but the number 3. Let's calculate the tenths of 3−3=0. We write zero in the tenth part of our answer:

Now subtract the integer parts 3−2=1. We write the unit in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 1.07. So the value of the expression 3.46−2.39 is equal to 1.07

3,46−2,39=1,07

Example 4. Find the value of the expression 3−1.2

This example subtracts a decimal from an integer. Let's write this expression in a column so that the integer part of the decimal fraction 1.23 is under the number 3

Now let's make the number of digits after the decimal point the same. To do this, after the number 3, put a comma and add one zero:

Now subtract tenths: 0−2. Do not subtract the number 2 from zero. Therefore, you need to take a unit from the adjacent digit. By borrowing one from the adjacent digit, 0 turns into the number 10. Now you can calculate the tenths of 10−2=8. We write down the eight in the tenth part of our answer:

Now subtract the whole parts. Previously, the number 3 was located in the integer, but we borrowed one unit from it. As a result, it turned into the number 2. Therefore, we subtract 1 from 2. 2−1=1. We write the unit in the integer part of our answer:

Separate the integer part from the fractional part with a comma:

Got the answer 1.8. So the value of the expression 3−1.2 is 1.8

Decimal multiplication

Multiplying decimals is easy and even fun. To multiply decimals, you need to multiply them like regular numbers, ignoring the commas.

Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in both fractions, then count the same number of digits on the right in the answer and put a comma.

Example 1 Find the value of the expression 2.5 × 1.5

We multiply these decimal fractions as ordinary numbers, ignoring the commas. To ignore the commas, you can temporarily imagine that they are absent altogether:

We got 375. In this number, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in fractions of 2.5 and 1.5. In the first fraction there is one digit after the decimal point, in the second fraction there is also one. A total of two numbers.

We return to the number 375 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 3.75. So the value of the expression 2.5 × 1.5 is 3.75

2.5 x 1.5 = 3.75

Example 2 Find the value of the expression 12.85 × 2.7

Let's multiply these decimals, ignoring the commas:

We got 34695. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 12.85 and 2.7. In the fraction 12.85 there are two digits after the decimal point, in the fraction 2.7 there is one digit - a total of three digits.

We return to the number 34695 and begin to move from right to left. We need to count three digits from the right and put a comma:

Got the answer 34,695. So the value of the expression 12.85 × 2.7 is 34.695

12.85 × 2.7 = 34.695

Multiplying a decimal by a regular number

Sometimes there are situations when you need to multiply a decimal fraction by a regular number.

To multiply a decimal and an ordinary number, you need to multiply them, regardless of the comma in the decimal. Having received the answer, it is necessary to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the decimal fraction, then in the answer, count the same number of digits to the right and put a comma.

For example, multiply 2.54 by 2

We multiply the decimal fraction 2.54 by the usual number 2, ignoring the comma:

We got the number 508. In this number, you need to separate the integer part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.54. The fraction 2.54 has two digits after the decimal point.

We return to the number 508 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 5.08. So the value of the expression 2.54 × 2 is 5.08

2.54 x 2 = 5.08

Multiplying decimals by 10, 100, 1000

Multiplying decimals by 10, 100, or 1000 is done in the same way as multiplying decimals by regular numbers. It is necessary to perform the multiplication, ignoring the comma in the decimal fraction, then in the answer, separate the integer part from the fractional part, counting the same number of digits on the right as there were digits after the decimal point in the decimal fraction.

For example, multiply 2.88 by 10

Let's multiply the decimal fraction 2.88 by 10, ignoring the comma in the decimal fraction:

We got 2880. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to count the number of digits after the decimal point in the fraction 2.88. We see that in the fraction 2.88 there are two digits after the decimal point.

We return to the number 2880 and begin to move from right to left. We need to count two digits from the right and put a comma:

Got the answer 28.80. We discard the last zero - we get 28.8. So the value of the expression 2.88 × 10 is 28.8

2.88 x 10 = 28.8

There is a second way to multiply decimal fractions by 10, 100, 1000. This method is much simpler and more convenient. It consists in the fact that the comma in the decimal fraction moves to the right by as many digits as there are zeros in the multiplier.

For example, let's solve the previous example 2.88×10 in this way. Without giving any calculations, we immediately look at the factor 10. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 2.88 we move the decimal point to the right by one digit, we get 28.8.

2.88 x 10 = 28.8

Let's try to multiply 2.88 by 100. We immediately look at the factor 100. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 2.88 we move the decimal point to the right by two digits, we get 288

2.88 x 100 = 288

Let's try to multiply 2.88 by 1000. We immediately look at the factor 1000. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 2.88 we move the decimal point to the right by three digits. The third digit is not there, so we add another zero. As a result, we get 2880.

2.88 x 1000 = 2880

Multiplying decimals by 0.1 0.01 and 0.001

Multiplying decimals by 0.1, 0.01, and 0.001 works in the same way as multiplying a decimal by a decimal. It is necessary to multiply fractions like ordinary numbers, and put a comma in the answer, counting as many digits on the right as there are digits after the decimal point in both fractions.

For example, multiply 3.25 by 0.1

We multiply these fractions like ordinary numbers, ignoring the commas:

We got 325. In this number, you need to separate the whole part from the fractional part with a comma. To do this, you need to calculate the number of digits after the decimal point in fractions of 3.25 and 0.1. In the fraction 3.25 there are two digits after the decimal point, in the fraction 0.1 there is one digit. A total of three numbers.

We return to the number 325 and begin to move from right to left. We need to count three digits on the right and put a comma. After counting three digits, we find that the numbers are over. In this case, you need to add one zero and put a comma:

We got the answer 0.325. So the value of the expression 3.25 × 0.1 is 0.325

3.25 x 0.1 = 0.325

There is a second way to multiply decimals by 0.1, 0.01 and 0.001. This method is much easier and more convenient. It consists in the fact that the comma in the decimal fraction moves to the left by as many digits as there are zeros in the multiplier.

For example, let's solve the previous example 3.25 × 0.1 in this way. Without giving any calculations, we immediately look at the factor 0.1. We are interested in how many zeros are in it. We see that it has one zero. Now in the fraction 3.25 we move the decimal point to the left by one digit. Moving the comma one digit to the left, we see that there are no more digits before the three. In this case, add one zero and put a comma. As a result, we get 0.325

3.25 x 0.1 = 0.325

Let's try multiplying 3.25 by 0.01. Immediately look at the multiplier of 0.01. We are interested in how many zeros are in it. We see that it has two zeros. Now in the fraction 3.25 we move the comma to the left by two digits, we get 0.0325

3.25 x 0.01 = 0.0325

Let's try multiplying 3.25 by 0.001. Immediately look at the multiplier of 0.001. We are interested in how many zeros are in it. We see that it has three zeros. Now in the fraction 3.25 we move the decimal point to the left by three digits, we get 0.00325

3.25 × 0.001 = 0.00325

Do not confuse multiplying decimals by 0.1, 0.001 and 0.001 with multiplying by 10, 100, 1000. A common mistake most people make.

When multiplying by 10, 100, 1000, the comma is moved to the right by as many digits as there are zeros in the multiplier.

And when multiplying by 0.1, 0.01 and 0.001, the comma is moved to the left by as many digits as there are zeros in the multiplier.

If at first it is difficult to remember, you can use the first method, in which the multiplication is performed as with ordinary numbers. In the answer, you will need to separate the integer part from the fractional part by counting as many digits on the right as there are digits after the decimal point in both fractions.

Dividing a smaller number by a larger one. Advanced level.

In one of the previous lessons, we said that when dividing a smaller number by a larger one, a fraction is obtained, in the numerator of which is the dividend, and in the denominator is the divisor.

For example, to divide one apple into two, you need to write 1 (one apple) in the numerator, and write 2 (two friends) in the denominator. The result is a fraction. So each friend will get an apple. In other words, half an apple. A fraction is the answer to a problem how to split one apple between two

It turns out that you can solve this problem further if you divide 1 by 2. After all, a fractional bar in any fraction means division, which means that this division is also allowed in a fraction. But how? We are used to the fact that the dividend is always greater than the divisor. And here, on the contrary, the dividend is less than the divisor.

Everything will become clear if we remember that a fraction means crushing, dividing, dividing. This means that the unit can be split into as many parts as you like, and not just into two parts.

When dividing a smaller number by a larger one, a decimal fraction is obtained, in which the integer part will be 0 (zero). The fractional part can be anything.

So, let's divide 1 by 2. Let's solve this example with a corner:

One cannot be divided into two just like that. If you ask a question "how many twos are in one" , then the answer will be 0. Therefore, in private we write 0 and put a comma:

Now, as usual, we multiply the quotient by the divisor to pull out the remainder:

The moment has come when the unit can be split into two parts. To do this, add another zero to the right of the received one:

We got 10. We divide 10 by 2, we get 5. We write down the five in the fractional part of our answer:

Now we take out the last remainder to complete the calculation. Multiply 5 by 2, we get 10

We got the answer 0.5. So the fraction is 0.5

Half an apple can also be written using the decimal fraction 0.5. If we add these two halves (0.5 and 0.5), we again get the original one whole apple:

This point can also be understood if we imagine how 1 cm is divided into two parts. If you divide 1 centimeter into 2 parts, you get 0.5 cm

Example 2 Find the value of expression 4:5

How many fives are in four? Not at all. We write in private 0 and put a comma:

We multiply 0 by 5, we get 0. We write zero under the four. Immediately subtract this zero from the dividend:

Now let's start splitting (dividing) the four into 5 parts. To do this, to the right of 4, we add zero and divide 40 by 5, we get 8. We write the eight in private.

We complete the example by multiplying 8 by 5, and get 40:

We got the answer 0.8. So the value of the expression 4: 5 is 0.8

Example 3 Find the value of expression 5: 125

How many numbers 125 are in five? Not at all. We write 0 in private and put a comma:

We multiply 0 by 5, we get 0. We write 0 under the five. Immediately subtract from the five 0

Now let's start splitting (dividing) the five into 125 parts. To do this, to the right of this five, we write zero:

Divide 50 by 125. How many numbers 125 are in 50? Not at all. So in the quotient we again write 0

We multiply 0 by 125, we get 0. We write this zero under 50. Immediately subtract 0 from 50

Now we divide the number 50 into 125 parts. To do this, to the right of 50, we write another zero:

Divide 500 by 125. How many numbers are 125 in the number 500. In the number 500 there are four numbers 125. We write the four in private:

We complete the example by multiplying 4 by 125, and get 500

We got the answer 0.04. So the value of the expression 5: 125 is 0.04

Division of numbers without a remainder

So, let's put a comma in the quotient after the unit, thereby indicating that the division of integer parts is over and we proceed to the fractional part:

Add zero to the remainder 4

Now we divide 40 by 5, we get 8. We write the eight in private:

40−40=0. Received 0 in the remainder. So the division is completely completed. Dividing 9 by 5 results in a decimal of 1.8:

9: 5 = 1,8

Example 2. Divide 84 by 5 without a remainder

First we divide 84 by 5 as usual with a remainder:

Received in private 16 and 4 more in the balance. Now we divide this remainder by 5. We put a comma in the private, and add 0 to the remainder 4

Now we divide 40 by 5, we get 8. We write the figure eight in the quotient after the decimal point:

and complete the example by checking if there is still a remainder:

Dividing a decimal by a regular number

A decimal fraction, as we know, consists of an integer and a fractional part. When dividing a decimal fraction by a regular number, first of all you need:

divide the integer part of the decimal fraction by this number;
after the integer part is divided, you need to immediately put a comma in the private part and continue the calculation, as in ordinary division.

For example, let's divide 4.8 by 2

Let's write this example as a corner:

Now let's divide the whole part by 2. Four divided by two is two. We write the deuce in private and immediately put a comma:

Now we multiply the quotient by the divisor and see if there is a remainder from the division:

4−4=0. The remainder is zero. We do not write zero yet, since the solution is not completed. Then we continue to calculate, as in ordinary division. Take down 8 and divide it by 2

8: 2 = 4. We write the four in the quotient and immediately multiply it by the divisor:

Got the answer 2.4. Expression value 4.8: 2 equals 2.4

Example 2 Find the value of the expression 8.43:3

We divide 8 by 3, we get 2. Immediately put a comma after the two:

Now we multiply the quotient by the divisor 2 × 3 = 6. We write the six under the eight and find the remainder:

We divide 24 by 3, we get 8. We write the eight in private. We immediately multiply it by the divisor to find the remainder of the division:

24−24=0. The remainder is zero. Zero is not recorded yet. Take the last three of the dividend and divide by 3, we get 1. Immediately multiply 1 by 3 to complete this example:

Got the answer 2.81. So the value of the expression 8.43: 3 is equal to 2.81

Dividing a decimal by a decimal

To divide a decimal fraction into a decimal fraction, in the dividend and in the divisor, move the comma to the right by the same number of digits as there are after the decimal point in the divisor, and then divide by a regular number.

For example, divide 5.95 by 1.7

Let's write this expression as a corner

Now, in the dividend and in the divisor, we move the comma to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we must move the comma to the right by one digit in the dividend and in the divisor. Transferring:

After moving the decimal point to the right by one digit, the decimal fraction 5.95 turned into a fraction 59.5. And the decimal fraction 1.7, after moving the decimal point to the right by one digit, turned into the usual number 17. And we already know how to divide the decimal fraction by the usual number. Further calculation is not difficult:

The comma is moved to the right to facilitate division. This is allowed due to the fact that when multiplying or dividing the dividend and the divisor by the same number, the quotient does not change. What does it mean?

This is one of the interesting features of division. It is called the private property. Consider expression 9: 3 = 3. If in this expression the dividend and the divisor are multiplied or divided by the same number, then the quotient 3 will not change.

Let's multiply the dividend and divisor by 2 and see what happens:

(9 × 2) : (3 × 2) = 18: 6 = 3

As can be seen from the example, the quotient has not changed.

The same thing happens when we carry a comma in the dividend and in the divisor. In the previous example, where we divided 5.91 by 1.7, we moved the comma one digit to the right in the dividend and divisor. After moving the comma, the fraction 5.91 was converted to the fraction 59.1 and the fraction 1.7 was converted to the usual number 17.

In fact, inside this process, multiplication by 10 took place. Here's what it looked like:

5.91 × 10 = 59.1

Therefore, the number of digits after the decimal point in the divisor depends on what the dividend and divisor will be multiplied by. In other words, the number of digits after the decimal point in the divisor will determine how many digits in the dividend and in the divisor the comma will be moved to the right.

Decimal division by 10, 100, 1000

Dividing a decimal by 10, 100, or 1000 is done in the same way as . For example, let's divide 2.1 by 10. Let's solve this example with a corner:

But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is moved to the left by as many digits as there are zeros in the divisor.

Let's solve the previous example in this way. 2.1: 10. We look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 2.1, you need to move the comma to the left by one digit. We move the comma to the left by one digit and see that there are no more digits left. In this case, we add one more zero before the number. As a result, we get 0.21

Let's try to divide 2.1 by 100. There are two zeros in the number 100. So in the divisible 2.1, you need to move the comma to the left by two digits:

2,1: 100 = 0,021

Let's try to divide 2.1 by 1000. There are three zeros in the number 1000. So in the divisible 2.1, you need to move the comma to the left by three digits:

2,1: 1000 = 0,0021

Decimal division by 0.1, 0.01 and 0.001

Dividing a decimal by 0.1, 0.01, and 0.001 is done in the same way as . In the dividend and in the divisor, you need to move the comma to the right by as many digits as there are after the decimal point in the divisor.

For example, let's divide 6.3 by 0.1. First of all, we move the commas in the dividend and in the divisor to the right by the same number of digits as there are after the decimal point in the divisor. The divisor has one digit after the decimal point. So we move the commas in the dividend and in the divisor to the right by one digit.

After moving the decimal point to the right by one digit, the decimal fraction 6.3 turns into the usual number 63, and the decimal fraction 0.1, after moving the decimal point to the right by one digit, turns into one. And dividing 63 by 1 is very simple:

So the value of the expression 6.3: 0.1 is equal to 63

But there is also a second way. It's lighter. The essence of this method is that the comma in the dividend is transferred to the right by as many digits as there are zeros in the divisor.

Let's solve the previous example in this way. 6.3:0.1. Let's look at the divider. We are interested in how many zeros are in it. We see that there is one zero. So in the divisible 6.3, you need to move the comma to the right by one digit. We move the comma to the right by one digit and get 63

Let's try to divide 6.3 by 0.01. Divisor 0.01 has two zeros. So in the divisible 6.3, you need to move the comma to the right by two digits. But in the dividend there is only one digit after the decimal point. In this case, one more zero must be added at the end. As a result, we get 630

Let's try dividing 6.3 by 0.001. The divisor of 0.001 has three zeros. So in the divisible 6.3, you need to move the comma to the right by three digits:

6,3: 0,001 = 6300

Tasks for independent solution

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Target

Consolidate and generalize students' knowledge on this topic;

Tasks:

1. develop computing skills, thinking; students' interest in mathematics and broaden their horizons;

2.formation of the values of a healthy lifestyle, the need for it.

Equipment: laptop, multimedia projector, textbook, workbooks.

During the classes.

Teacher: " If you want to participate in the big life, fill your head with math while you can. She will then be of great help to you in all your work. (M.I. Kalinin).

So let's not miss this opportunity and do the math. One of the wonderful qualities of mathematics is the development of curiosity.

But in order to determine what we will do, we will decide a small crossword

1. Surname of the Italian mathematician who was the first to use the fraction bar to write ordinary fractions.
2. Two numbers whose product is equal.
3. To add or subtract two fractions with different denominators, you need to bring them to a common ...
4. The product of any fraction and the number 0 is ...
5. A fraction written with a fraction bar is called ...
6. To divide one ordinary fraction into another, divisible enough ... by the reciprocal of the divisor.
7. What are the names of fractions whose denominators are 10, 100, 1000 …….

2. Message of the topic of the lesson

Goal setting (Children independently determine the topic and purpose of the lesson)

You already know how to perform all the actions with decimal fractions.

In today's lesson, we will not only solve problems and examples for applying the rules of action with decimal fractions, but also talk a little about health - one of the main values of human life, a source of joy. Still ancient - the Greek philosopher Socrates said, but what he said, we will now find out, but for this we need to complete the following task.

2. Oral account.

Calculate:

0,8 *2
3,4*10
0,6+0,4
40: 0,2
1,2*3
0,65+0,65
1,2: 2
23,8*10

1)1,6 2)34 3)1 4)200 5)3,6 6)1,3 7)0,6 8)238

« Health is not everything, but everything without health is nothing. Socrates.

front poll.

1. What fraction is called decimal?

2. Formulate a rule for adding and subtracting decimal fractions.

3. Formulate a rule for multiplying decimal fractions by 10,100,1000.

4. Formulate a rule for multiplying decimal fractions

5. Formulate the rule for dividing decimal fractions by a natural number, by a decimal fraction.

4. Fixing

Teacher:group work)

SLIDE #5

Exercise. Solve equations 1)x + 10.5 = 18.98

2) 34.5 - y \u003d 16.25

3) a * 1.9 = 3.8

4) 12.6: c = 12.6

Answers are letter coded. The answer to the first equation is the first letter of our word: TEETH

Teacher: Yes, today in the lesson we will talk about teeth. How beautiful it is when a person has white and even teeth, like precious pearls! After all, they are needed not only to bite and chew, but also so that a person can smile dazzlingly and everyone around can see that he is healthy, strong, cheerful and can work with pleasure.

After all, health is judged by the teeth. In our teeth, as in a mirror, the state of the body as a whole is reflected. No wonder the proverb says: "Judge health not by years, but by teeth."

Question: Guys, you probably know that nutrition is very important for good health. Do you want to know what foods affect the strengthening of teeth?

SLIDE #:6

5. Testing.

A- 24695 - mutton

B - 35636 - pork

A - 3474 - coffee

B - 3464 - tea

A - 3257 - potatoes

B - 3248 - fish

A - 1172 - black bread;

B - 1182 - white bread;

A - 17305 - kidneys

B - 428 - liver

A - 1682 - yolk

B - 168 - protein

B - 12345 - seafood

Teacher: If you want your teeth to be healthy, remember to eat these foods more often.

PHYSICAL MINUTE

Teacher: Guys, you did a good job, and now let's have a little rest, play a game. I call an ordinary fraction, if it can be converted to a decimal, then you get up, no - sit still.

Teacher: We rested, recharged with energy and continue to work.

Question: Guys, do you know how people brushed their teeth in the old days?

Let's solve the problems, and the answers will be the answers to my question.

Two dogs ran up to the owner at the same time. One ran 0.46s at 3.5m/min and the other 1.04s at 1.5m/min. Which dog was further from the owner and by how much (express the answer in cm)?
The area of the kitchen is 8.4 m², and the area of the rooms is 2.8 times larger. What is the total area of the apartment?

54 - honey, yolk, chalk, milk,

75 - sugar, citric acid ash, salt

Teacher: Why didn’t they brush their teeth in the old days! For this, ash, pieces of coal, table salt, milk, soda, chalk, crushed egg shells with honey were used. The Chinese used powder from soap tree beans, the inhabitants of Siberia and the Urals cooked chewing mastic (the so-called sulfur) from pine resin, which cleaned their teeth and strengthened the gums.

Probably, many of you have experienced a toothache and visited a doctor. Therefore, brushing your teeth should not be taken lightly.

Question: Guys, do you know what is the minimum number of brushes a person should change during the year?

And in order to find the correct answer to this question, let's do the following task. We write down the answers, add them up and divide by 4.5.

SLIDE №7

One corner of the table was sawn off. How many angles does it have now?(5)
There were three carrots and four apples on the plate. How many fruits were on the plate? (4)
The cat Murka had puppies: one black and two white. How many puppies does Murka have? (0)
Two siskins, two swifts and two snakes arrived. How many birds were in total near my house? (4)
One banana falls from the tree every 5 minutes. How many will fall in one hour? (0)
There were 5 glasses of berries on the table. Misha ate one and put it on the table. How many glasses are on the table? (5)

Teacher: Guys, you should change your toothbrush at least every 3 months. Do not use someone else's toothbrush.

REFLECTION

Teacher: Now let's sum up our lesson.

What have you learned?

What will you need in life?

What difficulties did you encounter in the lesson? (Students sum up the lesson).

Guys, this is the end of our lesson. It was my pleasure to work with you. I hope that the information that you heard in the lesson today will be useful to you in life, but for now, try to follow all the tips from today's lesson.

Helpful Hints: After eating, brush your teeth

Do this twice a day.

Prefer fruit over candy

Very important products.

We go to the dentist

Twice a year for admission.

And then smiles light

You will keep it for many years.

Homework.

Remember: It is very difficult to treat diseases,

It is easier to prevent disease.

Thank you for the lesson.

Feofelaktova Maria Stepanovna, 06.02.2017

1923 174

Development content

Math lesson in 6th grade

Topic "All actions with decimals"

lesson-consolidation of knowledge

Teacher: Feofelaktova M.S.

MBOU "Chendek secondary school"

With. Chandek

2014

Lesson-repetition "All actions with decimal fractions."

Target

Consolidate and generalize students' knowledge on this topic;

Tasks:

1. develop computing skills, thinking; students' interest in mathematics and broaden their horizons;

2.formation of the values of a healthy lifestyle, the need for it.

Equipment: laptop, multimedia projector,textbook, workbooks.

During the classes.

Organizational moment "Let's tune in to the lesson!".

Teacher: "If you want to participate in the big life, fill your head with math while you can. She will then be of great help to you in all your work. (M.I. Kalinin).

SLIDE #1

So let's not miss this opportunity and do the math. One of the wonderful qualities of mathematics is the development of curiosity.

But in order to determine what we will do, we will decide a smallcrossword

1. The surname of the Italian mathematician who was the first to use the fraction bar to write ordinary fractions.

2. Two numbers whose product is equal.

3. To add or subtract two fractions with different denominators, you need to bring them to a common ...

4. The product of any fraction and the number 0 is ...

5. A fraction written with a fraction bar is called ...

6. To divide one ordinary fraction into another, divisible enough ... by the reciprocal of the divisor.

7. What are the names of fractions whose denominators are 10, 100, 1000 ……..

2. Message of the topic of the lesson

Goal setting (Children independently determine the topic and purpose of the lesson)

You already know how to perform all the actions with decimal fractions.

SLIDE #3

2. Oral account.

Calculate:

0,8 *2

3,4*10

0,6+0,4

40: 0,2

1,2*3

0,65+0,65

1,2: 2

23,8*10

1)1,6 2)34 3)1 4)200 5)3,6 6)1,3 7)0,6 8)238

SLIDE #4

« Health is not everything, but everything without health is nothing. Socrates.

3.Updating basic knowledge

front poll.

1. What fraction is called decimal?

2. Formulate a rule for adding and subtracting decimal fractions.

3. Formulate a rule for multiplying decimal fractions by 10,100,1000.

4. Formulate a rule for multiplying decimal fractions

5. Formulate the rule for dividing decimal fractions by a natural number, by a decimal fraction.

4. Fixing

Teacher:Guys, after completing the following task, you will understand what exactly will be discussed next. (group work)

SLIDE #5

Exercise.Solve equations 1)x + 10.5 = 18.98

2) 34.5 - y \u003d 16.25

3) a * 1.9 = 3.8

4) 12.6: c = 12.6

Answers are letter coded. The answer to the first equation is the first letter of our word: TEETH

Teacher:Yes, today in the lesson we will talk about teeth. How beautiful it is when a person has white and even teeth, like precious pearls! After all, they are needed not only to bite and chew, but also so that a person can smile dazzlingly and everyone around can see that he is healthy, strong, cheerful and can work with pleasure.

After all, health is judged by the teeth. In our teeth, as in a mirror, the state of the body as a whole is reflected. No wonder the proverb says: "Judge health not by years, but by teeth."

Question:Guys, you probably know that nutrition is very important for good health. Do you want to know what foods affect the strengthening of teeth?

SLIDE #:6

5. Testing.

1) 23456 + 1239

A- 24695 - mutton

B - 35636 - pork

2) 4700 – 1236

A - 3474 - coffee

B - 3464 - tea

3) 232 * 14

A - 3257 - potatoes

B - 3248 - fish

4) 2344:2

A - 1172 - black bread;

B - 1182 - white bread;

5) 347 +(28+53)

A - 17305 - kidneys

B - 428 - liver

6) 456 + 1226

A - 1682 - yolk

B - 168 - protein

7) 12345 - 0

A - 0 - bow

B - 12345 - seafood

Teacher:If you want your teeth to be healthy, remember to eat these foods more often.

PHYSICAL MINUTE

Teacher:Guys, you did a good job, and now let's have a little rest, play a game. I call an ordinary fraction, if it can be converted to a decimal, then you get up, no - sit still.

Teacher:We rested, recharged with energy and continue to work.

Development of computing skills.

Question:Guys, do you know how people brushed their teeth in the old days?

Let's solve the problems, and the answers will be the answers to my question.

Two dogs ran up to the owner at the same time. One ran 0.46s at 3.5m/min and the other 1.04s at 1.5m/min. Which dog was further from the owner and by how much (express the answer in cm)?

The area of the kitchen is 8.4 m², and the area of the rooms is 2.8 times larger. What is the total area of the apartment?

54 - honey, yolk, chalk, milk,

75 - sugar, citric acid ash, salt

Teacher:Why didn’t they brush their teeth in the old days! For this, ash, pieces of coal, table salt, milk, soda, chalk, crushed egg shells with honey were used. The Chinese used powder from soap tree beans, the inhabitants of Siberia and the Urals cooked chewing mastic (the so-called sulfur) from pine resin, which cleaned their teeth and strengthened the gums.

Probably, many of you have experienced a toothache and visited a doctor. Therefore, brushing your teeth should not be taken lightly.

Question:Guys, do you know what is the minimum number of brushes a person should change during the year?

And to find the correct answer to this question, let's do the following task. We write down the answers, add them up and divide by 4.5.

SLIDE №7

One corner of the table was sawn off. How many angles does it have now?(5)

There were three carrots and four apples on the plate. How many fruits were on the plate? (4)

The cat Murka had puppies: one black and two white. How many puppies does Murka have? (0)

Two siskins, two swifts and two snakes arrived. How many birds were in total near my house? (4)

One banana falls from the tree every 5 minutes. How many will fall in one hour? (0)

There were 5 glasses of berries on the table. Misha ate one and put it on the table. How many glasses are on the table? (5)

Teacher:Guys, you should change your toothbrush at least every 3 months. Do not use someone else's toothbrush.

REFLECTION

Teacher:Now let's sum up our lesson.

What have you learned?

What will you need in life?

What difficulties did you encounter in the lesson? (Students sum up the lesson).

Helpful Hints: After eating, brush your teeth

Do this twice a day.

Prefer fruit over candy

Very important products.

We go to the dentist

Twice a year for admission.

And then smiles light

You will keep it for many years.

Homework.

Remember:It is very difficult to treat diseases,

It is easier to prevent disease.

Thank you for the lesson.

ACTION C DECIMAL FRACTIONS

The purpose of the lesson .

Summarize knowledge on the topic "Decimal Fractions".

LOGICAL DICTATION. 1,5; 33,7; 5/10; 11,12; 54,02; 17,143; 3/2; 0,0019; 5,305; 1/100.

1) -

CRITERIA FOR EVALUATION

6-7 tasks - "3"

8-9 tasks - "4"

10 tasks - "5"

Corrections are not allowed after grading!

GAME "YOU TO ME, I TO YOU". ( RULES OF THE GAME)

The leader is selected. He turns his back to the class, and at this time the guys pass the apple along the chain. After the leader's command, "stop", the transfer of the apple stops. The student who has an apple in his hands chooses a couple in the class to whom the question will be addressed. Having heard the answer, the leader gives a conclusion about his fidelity, if the answer is not correct, then the leader can ask anyone. Then, the answerer addresses his question to the opponent. The facilitator coordinates further actions. After the duel, the game continues.

FIND BUGS

Ι variant ΙΙ variant

a) 0.134 1000=13.4 a) 3.2 100=0.032

b) 16.12 ׃ 4 = 4.3 b) 27.18:3=9.6

c) 1.06+0.4=1.1 c) 2.7+0.03=3

d) 5.72-0.2=5.7 d) 3.61-0.1=3.6

e) 16.5:0.1=1.65 e) 5:100=0.5

THE SOLUTION OF THE PROBLEM (TRAFFIC ON THE RIVER)

υ boats =27.1 km/h

υ current=1.8 km/h

Ι variant ΙΙ variant

Find the path you have traveled Find the path you have traveled

against the current of the river downstream of the river

and round the result and round the result

to whole. to whole.

THE SOLUTION OF THE PROBLEM

Ι variant ΙΙ variant

1) 27.1-1.8=25.3(km/h) υ↓ 1) 27.1+1.8=28.9(km/h) υ

2) 25.3∙6=151.8(km) 2) 28.9∙6=173.4(km)

S≈152 km S≈173 km

INDEPENDENT WORK "RESTORE THE CHAIN" . (

Ι option

ΙΙ option

3,18-1,08 1,68:100

1,4575∙100 145,75-5,05

0,0168∙50 0,84+2,34

140,7-135 5,83:4

INDEPENDENT WORK "RESTORE THE CHAIN" . ( THE SOLUTION OF THE FIRST EXAMPLE IS THE BEGINNING OF THE SECOND. CONNECT THE EXAMPLES WITH ARROWS.)

Ι option

1,4575∙100 145,75-5,05

140,7-135 5,83:4