Lesson topic "Addition" decimals»

Teacher 1st qualification category MBOUSOSH s. Terbuny : Kirikova Marina Alexandrovna

Class: 5

Lesson type: learning new material

Goals and tasks training session:

Educational :

Repeat addition of ordinary fractions; read and write decimal number; comparison of decimal numbers

Introduce the algorithm for adding decimals

Show how this algorithm is used to add decimals

Teach students how to add decimals

Educational:

Develop verbal-logical thinking, mathematical speech

Teach the ability to generalize and draw conclusions, apply knowledge in a new situation

Expanding students' knowledge about the world around them

Increase ICT competence of students

Develop an environmental culture

Educational:

Promote the development of interest in the subject

Cultivate perseverance to achieve the final result

Ability to work in groups (pairs), team

Promote the development of cognitive activity and hard work

Bring up careful attitude to nature

Instill love for our small Motherland

Equipment:

computer, screen, projector

Progress of the training session:

Stage 1. Organizational moment.

Checking readiness for the lesson.Organization of students’ emotional mood for communication and interaction in the process of using existing knowledge and skills.

Stage 2. Motivation.

This legend came from the depths of the Middle Ages. A German merchant asked for advice on where to educate his son. They answered him. If you want your son to know addition, subtraction and multiplication, they can teach this here in Germany. But so that he also knows division, it is better to send him to Italy. The professors there studied this operation well. As we can see, even simple arithmetic operations were quite complex. From those times, the Germans still have the saying “in die Bruche kommen” (literally: “to fall into fractions”). This meant finding oneself in a difficult position, which one found oneself in when carrying out division. Nowadays, such operations based on a different, Arabic system of notation for numbers and other algorithms have become much easier.Today we will work not only with decimal fractions, we will study and learn how to apply one of the algorithms for working with decimal fractions, but we will also talk about one of global problems modernity. Which one do you think? Do you think environmental problems are relevant for our area?

Stage 3. Updating knowledge.

Frontal conversation.

1) What numbers are called decimal fractions? Answer: A decimal is a number whose fractional denominator is 10, 100, 1000, etc., which is written using a comma (the whole part is written first, and then, separated by a comma, the numerator of the fractional part).

2) How can you change the number of decimal places in a decimal fraction? Answer: If you add a zero or discard the zero at the end of a decimal fraction, you get a fraction equal to the given one.

3) Can a natural number be represented as a decimal fraction? Answer: Yes. To do this, you need to put a comma after the last digit in the number and add the required number of zeros

Oral exercises.

1.Read the fraction: 1925.2016.

2.a) Round to the nearest thousand? (1925.202)

b) Round to the nearest tenth? (1925.2)

c) Round to units? (1925)

1925. What happened this year? (Date of formation of our school).

3.Name a number between 0.3 and 0.4

4.What natural number is contained between 89.9 and 90.1? (90, how old is our school)

5. Arrange the fractions in ascending order: 20.01; 20.001;20.1(20.001; 20.01;20.1). Write down the date of the lesson - 20.01

6. Equalize the number of digits after the decimal point 0.2;0.02; 0.002. What needs to be done for this?(0.200;0.020;0.002)

4. Setting the topic, goals and objectives of the lesson.

Pollution problem environment in our area – one of the most relevant.

Harmful substances are constantly being released into the atmosphere. In the Lipetsk region, about

2012 322.9 thousand tons;

2013 353.1 thousand tons;

2014 330 thousand tons;

2015 330 thousand tons harmful substances. Is the emission of harmful substances increasing or decreasing? What measures are being taken to improve the environment?

How many tons of harmful substances were released in two last year? (660 thousand tons) What did they do with the numbers? How to add natural numbers?

Can we find out how many thousand tons have entered the atmosphere over these years?

What do you need to know? (Rule for adding decimals)

How do we record a lesson for him? (Adding decimals)

Lesson objectives? (Learn to add decimals, find the meaning of expressions, solve problems)

What plan will we work on? (Let's study the rule. Consider examples of adding decimal fractions. Find the value of the expression containing the sum of decimal fractions)

5. Studying new material.

Calculate 24+32=…(56) How did you perform the addition? (Bitwise)

And now 2.4+3.2=...(2 +3=5=5.6) Is it convenient to add decimals this way? (No)

How else can you add decimals? (Bitwise)

2,4

3,2

.....

5,6

If the number of digits after the decimal point in a decimal fraction is different, then what to do in this case? (Equalize the number of digits after the decimal point and perform addition one by one.

2. Write them one below the other so that the comma is under the comma.

3. Perform addition (subtraction) without paying attention to the comma.

4. Place a comma under the comma in the answer.

Consider example 5, 2 + 1.13

Add up the decimal fractions
Strictly write the number below the number,
And keep all the commas,
Write them in a row, don’t forget!

How to conveniently record an action?

It is convenient to add decimal fractions in a column. Read rule p.195 yourself.

6.Primary consolidation.

№705(a,c,e) at the board

№705 (g,f) independently

№706 (c-1 option, d-2nd) Who is faster? Checking at the board.

№717(Oral).

Physical education minute

Let's return to the environmental problem and find out how many tons of harmful substances have entered the atmosphere over the past 4 years in the Lipetsk region.

(322.9+353.1+330+330) thousand tons = 1336 thousand tons - harmful substances

Answer: 1336 thousand tons.

7.Independent work (training) Reconciliation against the standard.

Calculate and fill out the table. Having completed all the tasks correctly, you will receive the word “ecology” translated from Greek

5,8+22,191

3,99+0,06

8,9021+0,68

2,7+1,35

0,769+42,389

129+9,72

4.05-i;43.158-i;27.991-f;9.5821-l;138.72-i

Answer: dwelling (house)

8.Repetition. Inclusion in the knowledge system

Find the mistake. What is broken, what are the rules for adding decimal fractions?

1)0,2+0,15=0,17;

2)1,9+2,7=4,8;

3)5,48+4,52=100

Information about homework: P.42; No. 706 (e, f); No. 717 (v. g); No. 719

9.Reflection

1) What task was set in the lesson? Did you manage to solve it?

2) What else do you need to do to learn how to add decimals?

3) Complete the sentence: I was... I learned in class... I learned...

4) Image globe posted on the board. Everyone should attach a happy or sad emoticon, arguing why that particular one.

5) Should we take care of our planet? What do you need to do for this?

Arithmetic calculations such as addition And subtracting decimals, are necessary in order to obtain the desired result when operating with fractional numbers. The particular importance of carrying out these operations is that in many areas of human activity the measures of many entities are represented precisely decimals. Therefore, to carry out certain actions with many objects of the material world, it is required fold or subtract exactly decimals. It should be noted that in practice these operations are used almost everywhere.

Procedures adding and subtracting decimals in its mathematical essence it is carried out almost exactly in the same way as similar operations for integers. When implementing it, the value of each digit of one number must be written under the value of a similar digit of another number.

Subject to the following rules:

First, it is necessary to equalize the number of those signs that are located after the decimal point;

Then you need to write the decimal fractions one below the other in such a way that the commas contained in them are located strictly below each other;

Carry out the procedure subtracting decimals in full accordance with the rules that apply to subtracting integers. In this case, you do not need to pay any attention to commas;

After receiving the answer, the comma in it must be placed strictly under those that are in the original numbers.

Operation adding decimals carried out in accordance with the same rules and algorithm as described above for the subtraction procedure.

Example of adding decimals

Two point two plus one hundredth plus fourteen point ninety-five hundredths equals seventeen point sixteen hundredths.

2,2 + 0,01 + 14,95 = 17,16

Examples of adding and subtracting decimals

Mathematical operations addition And subtracting decimals in practice they are used extremely widely, and they often relate to many objects of the material world around us. Below are some examples of such calculations.

Example 1

According to design estimates, the construction of a small production facility requires ten point five cubic meters of concrete. Using modern technologies construction of buildings, contractors, without compromising the quality characteristics of the structure, managed to use only nine point nine cubic meters of concrete for all work. The savings amount is:

Ten point five minus nine point nine equals zero point six cubic meter of concrete.

10.5 – 9.9 = 0.6 m3

Example 2

Engine mounted on old model car, consumes eight point two liters of fuel per hundred kilometers in the urban cycle. For the new power unit, this figure is seven point five liters. The savings amount is:

Eight point two liters minus seven point five liters equals zero point seven liters per hundred kilometers in urban driving.

8.2 – 7.5 = 0.7 l

The operations of adding and subtracting decimal fractions are used extremely widely, and their implementation does not pose any problems. In modern mathematics, these procedures have been worked out almost perfectly, and almost everyone has mastered them well since school.

Adding Decimals is carried out according to the rules of column addition.

Decimal fractions are added in a column, like natural numbers, without paying attention to commas.

In the final result, a comma is placed under the commas as in the original fractions.

Pay attention! If the initial decimal fractions have a different number of signs (digits) after the decimal point, then to the fraction in which less number decimal places, you need to add the required number of zeros to equalize the number of decimal places in fractions.

If there are not enough digits of the fractional part to the right of the addend or minuend, then to the right in the fractional part you can add as many zeros (increase the digit of the fractional part) as there are digits in the other addend or minuend.

Let's look at an example. Determine the sum of decimal fractions:

0,678 + 13,7 =

We equalize the number of decimal places in decimal fractions. Add 2 zeros to the right of the decimal fraction 13,7 :

0,678 + 13,700 =

We write down the answer:

0,678 + 13,7 = 14,378

Basic rules for adding decimals:

• Equalize the number of decimal places.
• Write the decimal fractions one below the other so that the commas are below each other.
• Add decimal fractions, ignoring commas, according to the rules for adding natural numbers into a column.
• Put a comma under the commas in your answer.

In written addition and subtraction of decimal fractions, the comma that separates the whole part from the fractional part should be located next to the addends and the sum in the same column (a comma under the comma from writing the condition to the end of the calculation).

For example.Adding decimals into a string:

243,625 + 24,026 = 200 + 40 + 3 + 0,6 + 0,02 + 0,005 + 20 + 4 + 0,02 + 0,006 = 200 + (40 + 20) + (3 + 4)+ 0,6 + (0,02 + 0,02) + (0,005 + 0,006) = 200 + 60 + 7 + 0,6 + 0,04 + 0,011 = 200 + 60 + 7 + 0,6 + (0,04 + 0,01) + 0,001 = 200 + 60 + 7 + 0,6 + 0,05 + 0,001 = 267,651.

Is adding decimals. In this article we will look at the rules for adding finite decimal fractions, use examples to look at how to add finite decimal fractions in a column, and also dwell on the principles of adding infinite periodic and non-periodic decimal fractions. In conclusion, we will focus on adding decimals with natural numbers, ordinary fractions and mixed numbers.

Note that in this article we will only talk about adding positive decimals (see positive and negative numbers). The remaining options are covered by material from the articles addition of rational numbers and addition of real numbers.

Page navigation.

General principles of adding decimals

Example.

Add the decimal 0.43 and the decimal 3.7.

Solution.

The decimal fraction 0.43 corresponds to the common fraction 43/100, and the decimal fraction 3.7 corresponds to the common fraction 37/10 (if necessary, see the conversion of final decimal fractions to common ones). Thus, 0.43+3.7=43/100+37/10.

This completes the addition of finite decimal fractions.

Answer:

4,13 .

Now let's add periodic decimal fractions to our consideration.

Example.

Add the ending decimal 0.2 with the periodic decimal 0.(45) .

Solution.

Then .

Answer:

0,2+0,(45)=0,65(45) .

Now let us dwell on the principle of addition of infinite non-periodic decimal fractions.

Recall that infinite non-periodic decimal fractions, unlike finite and periodic decimal fractions, cannot be represented as ordinary fractions (they represent irrational numbers), therefore the addition of infinite non-periodic fractions cannot be reduced to the addition of ordinary fractions.

When performing the addition of infinite non-periodic fractions, they are replaced with approximate values, that is, they are first rounded (see rounding numbers) to a certain level. By increasing the accuracy with which approximations of the original infinite non-periodic decimal fractions are taken, more exact value the result of the addition. Thus, addition of infinite non-periodic decimal fractions comes down to adding finite decimal fractions.

Let's look at the example solution.

Example.

Add the infinite non-periodic decimal fractions 4.358... and 11.11002244....

Solution.

Let's round the added decimal fractions to hundredths (we will no longer be able to round the fraction 4.358... to thousandths, since the value of the ten-thousandth place is unknown), we have 4.358...≈4.36 and 11.11002244...≈11.11. Now all that remains is to add the final decimal fractions: .

Answer:

4,358…+11,11002244…≈15,47 .

To conclude this point, we will say that the addition of positive decimal fractions is characterized by all the properties of the addition of natural numbers. That is, the combinatory property of addition allows us to uniquely determine the addition of three and more decimal fractions, and the commutative property of addition allows you to rearrange the decimal fractions being added.

Adding decimal fractions in a column

It is quite convenient to perform column addition of finite decimal fractions. This method allows you to avoid converting added decimal fractions to ordinary fractions.

To execute column addition of decimal fractions, necessary:

• write one fraction under another so that the same digits are under each other, and the comma is under the comma (for convenience, you can equalize the number of decimal places by adding a certain number of zeros to one of the fractions on the right);
• then, without paying attention to the commas, perform the addition in the same way as adding a column of natural numbers;
• In the resulting amount, place a decimal point so that it is located under the decimal points of the terms.

For clarity, let's look at an example of adding decimal fractions in a column.

Example.

Add the decimals 30.265 and 1055.02597.

Solution.

Let's perform column addition of decimal fractions.

First, let's equalize the number of decimal places in the fractions being added. To do this, you need to add two zeros to the right in the fraction 30.265, which will result in an equal fraction 30.26500.

Now we write the fractions 30.26500 and 1 055.02597 in a column so that the corresponding digits are under each other:

We perform addition according to the rules of column addition, not paying attention to commas:

All that remains is to put a decimal point in the resulting number, after which the addition of decimal fractions in a column takes on the finished form:

Answer:

30,26500+1 055,02597=1 085,29097 .

Adding decimals with natural numbers

We'll announce it right away rule for adding decimals with natural numbers: to add a decimal fraction and a natural number, you need to add this natural number to the whole part of the decimal fraction, and leave the fractional part the same. This rule applies to both finite and infinite decimal fractions.

Let's look at an example of applying this rule.

Example.

Calculate the sum of the decimal fraction 6.36 and natural number 48 .

Solution.

Whole part The decimal fraction 6.36 is equal to 6, if we add the natural number 48 to it, we get the number 54. Thus, 6.36+48=54.36.

Answer:

6,36+48=54,36 .

Adding decimals with fractions and mixed numbers

The addition of a finite decimal or an infinite periodic decimal with a common fraction or mixed number can be reduced to the addition of common fractions or the addition of a common fraction and mixed number. To do this, it is enough to replace the decimal fraction with an equal ordinary fraction.

Example.

Add the decimal fraction 0.45 and the common fraction 3/8.

Solution.

Let's replace the decimal fraction 0.45 with an ordinary fraction: . After this, the addition of the decimal fraction 0.45 and the common fraction 3/8 is reduced to the addition of the common fractions 9/20 and 3/8. Let's finish the calculations: . Received if necessary common fraction can be converted to decimal.

Like addition, subtracting decimals depends on writing the numbers correctly.

Rule for subtracting decimals

1) COMMA UNDER THE COMMA!

This part of the rule is the most important. When subtracting decimal fractions, they should be written so that the commas of the minuend and subtrahend are strictly one below the other.

2) We equalize the number of digits after the decimal point. To do this, including where the number of digits after the decimal point is smaller, we add zeros after the decimal point.

3) Subtract the numbers, not paying attention to the comma.

4) Remove the comma under the commas.

Examples for subtracting decimals.

To find the difference between the decimal fractions 9.7 and 3.5, we write them so that the commas in both numbers are strictly one below the other. Then we subtract, ignoring the comma. In the resulting result, we remove the comma, that is, we write under the commas of the minuend and subtrahend:

2) 23,45 — 1,5

In order to subtract another from one decimal fraction, you need to write them so that the commas are located exactly one below the other. Since 23.45 has two digits after the decimal point, and 1.5 has only one, we add a zero to 1.5. After this, we carry out subtractions, not paying attention to the comma. As a result, we remove the comma under the commas:

23,45 — 1,5=21,95.

We begin subtracting decimal fractions by writing them so that the commas are located exactly one below the other. The first number has one digit after the decimal point, the second has three, so we write zeros in place of the missing two digits in the first number. Then we subtract the numbers, ignoring the comma. In the resulting result, remove the comma under the commas:

63,5-8,921=54,579.

4) 2,8703 — 0,507

To subtract these decimal fractions, we write them so that the decimal point of the second number is located exactly under the decimal point of the first. The first number has four digits after the decimal point, the second number has three, so we add a final zero after the decimal point to the second number. After this, we subtract these numbers like ordinary natural numbers, without taking into account the comma. In the resulting result, write a comma under the commas:

2,8703 — 0,507 = 2,3663.

5) 35,46 — 7,372

We begin subtracting decimal fractions by writing the numbers in such a way that the commas are one below the other. We add a zero after the decimal point to the first number so that both fractions have three digits after the decimal point. Then we subtract, ignoring the comma. In the answer we remove the comma under the commas:

35,46 — 7,372 = 28,088.

To subtract a decimal fraction from a natural number, put a comma at the end and add the required number of zeros after the decimal point. Why do we subtract without taking into account the comma? In response, we remove the comma exactly under the commas:

45 — 7,303 = 37,698.

7) 17,256 — 4,756

We perform this example on subtracting decimal fractions in the same way. The result is a number with zeros after the decimal point at the end. We do not write them in the answer: 17.256 - 4.756 = 12.5.