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Lesson topic: "Decimal fractions and actions with them."
The objectives of the lesson: to repeat and systematize the knowledge and skills of students on the topic “decimal fractions”, to determine the level of assimilation of knowledge on this topic, to check the degree of assimilation of the material; develop attention, memory, speech, logical thinking, independence; to cultivate the desire to achieve goals, a sense of responsibility, self-confidence, the ability to work in a team.
Lesson objectives: To show the importance of developing computational skills at this stage of learning. Stimulate the motivation of students to study mathematics;
Lesson type: lesson of generalization, systematization and correction of knowledge and skills on the topic: “Decimal fractions”
Forms of work of students: frontal, group, individual
Equipment: laptop, presentation, test on the topic “All actions with decimal fractions”, task cards, a set of signal cards for each student (red, green, yellow).
During the classes
Organizing time.
Hello guys!
Please take your seats.
Today is February 13th
Day of the week - Friday
Today we will
The lesson is
which will be dedicated
One interesting person.
Listen to me carefully
Answer the questions
Everyone, guys, notice
Don't forget anything
Please, don't let me down.
Come on, my young friend,
Are you ready to start the lesson?
Is everything okay on the table?
Is there order in your head?
To have knowledge
It will take patience and effort.
Lesson motivation.
We studied fractions for a long time,
Compared, rounded off,
added, subtracted,
Multiply and divide
The arithmetic mean was found.
And now the time has come
To check everything for you.
How do you solve problems
Multiply the fraction by ten
How do you solve equations?
Do you know many examples?
We will check everything for you
And at the end we will give an order:
Or give you a five, or learn to send you!
To be successful, we must:
Answer questions clearly and concisely;
Calculate proposed tasks quickly and correctly;
Provide assistance at work;
Be able to listen to others, etc.
Lesson motto: Have excellent knowledge on the topic “Decimal Fractions!”
3. Historical background slide 3-5
4. Actualization of basic knowledge
a) Chamomile game. The goal of the game is to repeat the rules that will be required when solving problems.
(A chamomile flower is attached to the board with a magnet, where questions are written on each petal. Opening the petal, the student answers the question posed):
Rules for adding and subtracting decimals
Rules for multiplying decimal fractions by 10, 100, 1000:.
Rules for dividing decimal fractions by 10, 100, 1000:.
Rules for multiplying decimals by natural number
Rules for multiplying decimals by decimals
Rules for dividing decimal fractions by a natural number
Rules for dividing decimals by decimals
Decimal Comparison Rules
b). And now we need a verbal account.
“Health is not everything, but everything without health is nothing.” Socrates
5. Consolidation of the acquired knowledge. Work in notebooks.
Assignment for students.
1. Mathematical dictation
On the sheets of paper, students write down only the answers.
1) 24,04: 2= 12,02
2) 1.3 1.5 + 1.5 1.7 = 4.5
3) 8,07 + 4,1 = 12,17
4) 1,28 +3,4 +1,72 -2,4 = 4
6) 0.7 * = 0.007 (instead of * put a number to get the correct equality) 0.01
7) 7.8 3.5 - 7.8 3.4 = 0.78
8) 2,54: * = 2540 = 0,001
9) 9,6: 100 =0,096
2). Right answers:
| 1) 12.02 W 4) 4 I 7) 0.78 P |
| 2) 4.5 K 5) 40 A 8) 0.001 C |
| 3) 12.17 Y 6) 0.01. 9) 0.096 N |
Students exchange papers and give grades.
Arrange the letters in the table according to the answers.
2) The clown came up with several examples of adding, subtracting and multiplying decimal fractions, and to make it funnier, he erased the commas in them. Here are the equations he got:
34 * 0,01 = 0034
Putting commas in the right place
3) Solve the problems:
1. On Monday, 37.6 tons of grain were threshed, on Tuesday - 3.8 tons more than on Monday, and on Wednesday - 1.5 times less than on Tuesday. How many tons of grain were harvested during these three days?
2. Tourists walked to the river at a speed of 6.6 km / h, and along the river bank at a speed of 4.2 km / h. In total, they walked 9.06 km. How long did the tourists walk along the shore if they walked 0.8 hours to the river.
6. Physical education
In class we wrote
Everything they knew was answered.
And now we'll rest
And let's start writing again!
We have relieved the tension that has accumulated during the solution of the problem and equations, we will continue to work in the notebook.
7. Test on the topic “Addition, subtraction, multiplication, division of decimal fractions”
Now let's test our knowledge with a quiz.
Option 1
1) Perform addition:
2) Perform the multiplication:
3) Find the value of the quotient:
Additionally:
Find the value of an expression:
4,36: (3,15 + 2,3)
Option 2
1) Perform addition:
2) Perform the multiplication:
3) Find the value of the quotient:
Additionally:
Find the value of an expression:
6,93: (0,028 + 1,512)
Test key:
1) 2) 3) Add.
I. option B) A) B) C)
II. option A) B) C) C)
We check the work ourselves. Next to each task we put a “+” or “-” sign.
Let's evaluate the result
Criteria for evaluation:
"5" - 5 tasks; "4" - 4 tasks; "3" -3 tasks.
Show with a signal card what grade you got: “5” - red, “4” - green, “3” - yellow.
8. Work in pairs
Take action. Cross out the answers and the letters corresponding to them in the table. The remaining letters will allow you to read the word.
1) 5,8 + 22,191=
2) 6.025 x 5.6 =
3) 1.15 x 0.4 =
5) 131,67: 5,7 =
1,4 23,1 0,46 2,11 0,14 0,4 27,991 3,4 33,74 27 8,22 2,6
M P Y O Z L O V D E C
Answer: the word YOUNG
9. Homework:
You are all great!
You are all daredevils!
And let for years beloved
For you, mathematics will always be!
Repeat steps 22-37. Solve tasks #1317, 1321, 1333
Come up with and beautifully arrange on a landscape sheet a problem that would be solved by adding and subtracting decimal fractions, write down the condition of the problem on the sheet and draw a picture according to this condition, and write down its solution in a notebook. Try to make the students of the class like your task so that the data in the condition corresponds to reality.
10. The result of the lesson. Reflection. Microphone principle. (Pupils take turns giving a reasoned answer to one of the questions).
I enjoyed my lesson today...
Today in class I did...
Today in class I fixed...
Today in class, I graded myself...
What types of work caused difficulties and require repetition ...
What knowledge do you have...
Did the lesson help to advance in knowledge, skills, skills in the subject ...
Who needs to work on what...
How effective was today's lesson...
In this tutorial, we'll look at each of these operations one by one.
Lesson contentAdding 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 parts. 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.
The digits in decimal fractions 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". The comma under the 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 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 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 integer 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 x 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 count the same number of digits to the right in the answer 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. Immediately look at the factor 1000. We are interested in how many zeros it contains. 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 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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Lesson-game on the topic: “Actions on decimal fractions” is carried out in the form of a “mathematical train”
Purpose: to test knowledge of the rules of addition, subtraction, multiplication and division from decimal fractions, the ability to apply them in action (in examples, tasks).
Mathematical train” consists of three cars: soft, compartment, reserved seat.
Rules for obtaining a ticket for travel.
"Cashier Hall".
Each student receives a boarding pass with tasks and six tokens.
1. Having solved all the tasks, the student applies for a ticket.
2. If the student cannot solve any task, then he turns to the help desk for help. Depending on the content of the certificate, the “fee” is determined.
"Inquiry Office".
1. Checking the correctness of the task solution and indicating the error is free of charge.
2. For a leading question that helps to find a way to solve the task, you should pay 1 token.
3. Fee for suggesting a solution path - 2 tokens.
4. Decision fee - 3 tokens.
"Conditions for obtaining a ticket."
1. A ticket for a soft car is issued if all tasks are correctly solved and more than 3 tokens are presented at the box office.
2. The correct solution of all problems and the presence of three tokens gives the right to receive a ticket to a compartment car.
3. One or two tokens are enough for a reserved seat car if all tasks are solved correctly.
During the classes.
"Cashier Hall".
Warm up
1. Recall the rules for adding and subtracting decimal fractions.
Orally (at the blackboard). Written (on paper).
3. Recall the rule for dividing by a decimal fraction.
Orally (at the blackboard) In writing (on leaflets).
After the check, “boarding tickets” are issued: a yellow flag for a soft car, a green flag for a compartment car, a red flag for a reserved seat.
Attention! Attention! The train “Decimal Fractions” departs from the station “ Gymnasium” to the station Decide." The announcer's voice about the departure of the train from the station and about its arrival at the station should be recorded on a tape recorder. This little touch improvises reality, makes the lesson serious and interesting.
Smart guys, true friends!”
Our train arrives at the Decide-ka station. You are met by the candidate of economic sciences “Arithmetic Mean”.
1. How to find the arithmetic mean of several numbers?
2. How to find the average speed?
3. How to find the average price of a product?
4. How to find the average daily earnings?
5. How to find the average yield?
(Codoscope)
1. In a volleyball team, 2 players are 21 years old, 3 players are 20 years old, and 1 player is 24 years old.
What average age team players? Answer: 21 years old.
2. Weight of 4 chickens - 5.5 kg, 6 chickens - 7.4 kg. Calculate average weight chicken. Answer: 1.29 kg.
3. The first number is 3 times less than the second number. The arithmetic mean of these numbers is 12. Find these numbers. Answer: 6 and 18.
Independent work.
1. Find the arithmetic mean of numbers 23.86; 22.7; 36.6. Answer: 27.72.
2. The boat traveled 22.7 km in 2 hours and covered 42.8 km in 3 hours. Determine the average speed. Answer: 13.1 km.
3. The arithmetic mean of two numbers is 0.48. One of them is 1.4 times larger than the other. Find these numbers. Answer: 0.4 and 0.56.
(Checking solutions through a codoscope).
The train leaves for Vesna station.
Task (the condition is written on the board). Finding a solution with a class.
Two starlings flew out of one birdhouse at the same time in opposite directions. After 0.15 hours, they were 16.5 km apart. The flight speed of one starling is 52.4 km/h. Find the speed of the other.
Independent problem solving.
Two bees flew out of the same hive in opposite directions at the same time. In 0.15 hours there were 6.3 km between them. One flew at a speed of 21.6 km / h. Find the flight speed of the other bee. Answer: 20.4 km/h.
Examination. Two students solve the problem on the back of the board: one using the arithmetic method, the other using the algebraic method.
Station Guess.
Dunno meets the guys at this station. Help Dunno quickly correct funny inequalities (put commas in the right place).
42 + 17 = 212 Correct solution: 4.2 +17 = 21.2
3 + 108 = 408 3 + 1,08 = 4,08
57 – 4 = 17 5,7 – 4 = 1,7.
Announcer: “Controllers work in the cars, present the colored cards received for the correct solutions to tasks during the trip. Our train is returning to the station.” Gymnasium"!
The result of the number of color cards. What did we repeat?
The journey is over.
Sandakova N.A.
Place of work, position:
Teacher of Physics and Mathematics, MBOU "Secondary School named after V.S. Arkhipov, Semenovka, Yoshkar-Ola"
Mari El Republic
Characteristics of the lesson (classes)
The level of education:
Basic general education
The target audience:
Teacher (teacher)
Class(es):
Item(s):
Maths
The purpose of the lesson:
Systematization of knowledge on the topic "Actions with decimal fractions": addition, subtraction, multiplication, division of decimal fractions.
Development of the ability to find errors in examples, analyze examples and tasks.
Development logical thinking, sociability, a sense of collectivism, the ability to evaluate their knowledge and skills.
Lesson type:
Lesson of generalization and systematization of knowledge
Students in the class (audience):
Used textbooks and study guides:
Mathematics 5th grade. Vilenkin.
Short description:
A lesson using a computer presentation to repeat and summarize actions with decimal fractions. Particular attention is paid to the ability to find and correct errors in examples, analyze the solution on the board, evaluate your knowledge on the topics being studied.
Lesson topic: These unusual decimal fractions.
Goals:
educational - generalization and systematization of students' knowledge on the topic "Actions with decimal fractions".
Educational - development of logical thinking of students, cognitive activity, development of independence, ability to self-control, self-esteem.
Educational - fostering a sense of collectivism, responsibility, interest in the subject.
Formation of UUD: communicative, cognitive, regulatory.
Lesson type: a lesson of generalization and systematization of knowledge and skills of students.
Form of organization: lesson-journey.
Equipment: multimedia computer, presentation.
Handout: stars of different colors (red-5, yellow-4, blue-3) for self-assessment.
The motto of the lesson: “Flight is mathematics” (V. Chkalov)
During the classes
1. Organizing time.
Let's remember what we learned in the fifth grade (slide 2): ordinary and decimal fractions, addition and subtraction, multiplication and division of fractions, comparison, finding a fraction of a number, percentages. What is the importance of studying mathematics. This is what the next verse tells us.
The teacher reads the poem:
The rocket crossed the sky
Her way into space is not new for a long time.
Can't hear the rumble and rumble
Already from under the cloudy carpets.
And the tamed peaceful atom
Obedient to the mind of people;
Over Padun, compressed by a dam-
The light of electric lights!
All this is the fruit of human quest,
All this was not created suddenly
By the mighty power of precise knowledge
And the skill of the workers!
And before that, notice by the way.
That rocket was given a sight,
Her route mathematician
I flew on the wings of formulas.
Dry lines of equations,
The power of reason poured into them,
They explain the phenomena
Things unraveled connection!
Without mathematics, there would not be many things that we used to use.
V. Chkalov said: "Flight is mathematics." Indeed, the conquest of space was not without mathematical calculations.
Today we also have to make a space journey from the math classroom to the various planets of our "School galaxy". We will travel by ship...
Guess the name of the ship if you arrange the numbers in ascending order: 0.81 (n), 1.81 (p), 0.081 (e), 3.51 (s), 3.15 (s), 2.44 (g) 0.82(e).
Assignment on the board. Frontal work with the class.
Answer: Energy.
Teacher: So, we are going to fly on the ship "Energy".
The purpose of our flight: to show our guests what knowledge and skills you have acquired on the topic "Decimal Fractions". During the flight, you need to draw up your own “star” map (each student has a set of stars of three colors red-5, yellow-4, blue-3). A self-assessment of knowledge or an assessment of the answer by the teacher is carried out.
The rocket is at launch. But before we go on a trip, we need to prepare for the flight.
Flight preparation:
1. Repetition of theoretical knowledge:
The teacher begins the sentence, the students continue (do not repeat after the teacher) ...
1. To add two decimals, ...
2. To subtract another from one decimal fraction, ...
3. To multiply a decimal by 10, ...
4. To multiply a decimal by 0.01….
5. To multiply a decimal by a decimal, ...
6. To divide a decimal fraction by a decimal fraction, ...
7. To find the arithmetic mean of two or more numbers,…
Teacher: Conduct a self-assessment of your knowledge and stick an asterisk on your star map.
2. Verbal counting(on task cards for all actions with decimal fractions).
Teacher: Conduct a self-assessment of your knowledge and stick an asterisk.
Planet "Mathematical"
Teacher: The first planet we landed on is Mathematical. You need to show how you can apply the learned rules in calculations. The first example goes to the board to solve ...., the second example at the board decides .... We write the third example in a notebook and solve it ourselves.
1) 296.2 - 2.7 * 6.6: 0.15 Answer: 318.38.
2) 135.2 * 2.1 - (0.083 + 0.841): 2.31. Answer: 283, 52.
3) 2.575: 2.5 - 4.25 * 0.16 + 0.03 Answer: 0.38.
We are testing the solution. Look at the slide and find answers from your examples. Whoever gets it right gets an asterisk.
Planet "Historical"
Teacher: We continue the flight. Our rocket landed on the Historical planet.
Students prepared reports on the history of decimal fractions at home. Self-assessment of their performances - an asterisk.
1st student: Decimal fractions were first used by the remarkable Uzbek scientist al-Kashi. At the beginning of the XV century. in Central Asia, a large observatory was created near the city of Samarkand. It made observations of the movement of stars, planets and the Sun, calculated the days of holidays, etc. The best scientists of that time worked at the observatory. The scientist Jamshid ib-Masud al-Kashi led the observatory.
2nd student: In 1427, al-Kashi completed the book “The Key to Arithmetic”. In this book, he used decimal fractions for the first time in the world, gave rules for working with them, explained these rules with examples, and described in detail the new system of writing fractions he discovered. To designate the categories, he used different options: he separated them with a vertical line, wrote with different ink, sometimes he wrote out the name of the category in full in words.
Planet "Cognitive".
Teacher: The next planet visited by our rocket is Cognitive.
Find out the answers to the equations and guess the word. The first and second equations are solved at the blackboard .... The third and fourth are decided on the spot.
Solve equations: (solving equations for verification - behind a closed board)
1) 9x + 3.9 \u003d 31.8 2) (y + 4.5): 7 \u003d 1.2. 3) (y - 8.48) + 2.16 = 3.9
x = 3.1. y = 3.9. y = 10.22.
4) 4y + 7y + 1.8 = 9.5
Answer: plus. Checking answers, finding answers in the table and guessing the word.. Self-assessment: those who guessed correctly get stars.
Planet "Entertaining".
Teacher: The next planet is "Entertaining".
Here you will find tasks of an unusual nature. Assignments are written on the board. Frontal work with the class.
1. Which example is wrong? Explain.
A) 3.7 + 1.2 = 4.9 _B) 7.34 + 10.1 = 17.35
C) 4.2 - 2.03 \u003d 2.17 _D) 8.95 - 0.6 \u003d 8.89
2. Arrange commas to get the correct equalities:
1) 42 + 17 = 212 3) 57 - 4 = 17 2) 63 - 27 = 603
3. Enter action signs:
a) 8.8 10 = 88; b) 3.3 100 = 0.033; c) 7.5 100 = 750.
4. Write down the missing number:
A) 42, 3 * = 423; b) 0.05 * = 50; c) 3800 * = 380.
Self-esteem.
Planet "Creative".
Teacher: The next planet we flew to is "Creative".
You have to draw up a text problem for movement according to the picture and solve it: (solution with commenting). The drawing is made on a poster. View various ways solution to this problem.
Rice: 15.4 km/h
km/h, 4 times >
In 3 hours Will it catch up in? h.
Evaluation of the answer by the teacher.
Planet "Theatrical".
Teacher: Our ship flew to the planet "Teatralnaya". Aliens offered you to perform a concert program and the jury assessed your performance with the following marks (marks are posted on the board): 4.2; 4.8; 5.0; 4.6; 4.3; 4.7; 4.9.
Find the arithmetic mean and round the result to tenths. Answer: 4.6. Self-esteem.
Planet Finish.
The last planet is "The Finish". Summing up the lesson.
And now let's see what kind of star map we got, who got how many stars. Let's do a self-assessment of our knowledge. Look at the slide: I read the statement to you, and you raise your hand if you agree.
I can multiply fractions.
I can divide a fraction by another fraction.
I can solve equations.
Learned how to find the percentage of a number.
I am learning to solve problems.
Basically everyone learned. It is more difficult for us only when solving problems. Thanks. Well done. We hand over notebooks with class work and our stars for verification.
D/z: Compose a fairy tale on the topic: "Journey to the Land of Decimal Fractions."
