Numbers with different signs Examples. Addition and subtraction of positive and negative numbers

formation of knowledge about the rules of addition of numbers with different signs, the ability to apply it in the simplest cases;

development of skills to compare, detect patterns, generalize;

education of a responsible attitude towards learning work.

Equipment: Multimedia projector, screen.

Type of lesson: Lesson studying a new material.

DURING THE CLASSES

1. The organizational moment.

Smoothly stood

Quietly sat down.

The call now called,

We start our lesson.

Guys! Today guests came to our lesson. Let's turn to them and smile to each other. So, we start our lesson.

Slide 2. - Epigraph of the lesson: "Who does not notice anything, he does not study anything.

Who does not study anything, he always hits and misses. "

Roman Sef ( children's writer)

Sweet 3 - I suggest play the game "On the contrary". Rules of the game: You need to share words into two groups: winnings, lie, warm, tried, truth, good, losing, took, evil, cold, positive, negative.

There are many contradictions in life. With their help, we define the surrounding reality. For our occupation, I need the latter: positive is negative.

What are we talking about in mathematics when we use these words? (About numbers.)

The great Pythagoras argued: "Numbers rule the world." I suggest talking about the most mysterious numbers In science - about numbers with different signs. - Negative numbers appeared in science, as opposite to positive. Their way to science was difficult, because even many scientists did not support ideas about their existence.

What concepts and values \u200b\u200bdo people measure positive and negative numbers? (charges elementary particles, temperature, losses, height and depth, etc.)

Slide 4- The words are opposite by value - antonyms (table).

2. Love the lesson theme.

Slide 5 (Working with Table) - What numbers studied at previous lessons?
- What tasks associated with positive and negative numbers do you know how to perform?
- Attention on the screen. (Slide 5)
- What numbers are presented in the table?
- Name the modules of the numbers recorded horizontally.
- specify the greatest number, Specify the number with the highest module.
- Answer the same questions for the numbers recorded vertically.
- Is there always the greatest number and the number with the largest module coincide?
- Find the amount of positive numbers, the amount negative numbers.
- Formulate the rule of addition of positive numbers and the rule of addition of negative numbers.
- What numbers remains to be folded?
- Do you know how to fold them?
- Do you know the magnitude of the addition of numbers with different signs?
- Word the topic of the lesson.
- What purpose do you put in front of ourselves? . Improve what we will do today? (Children's responses). Today we continue to get acquainted with positive and negative numbers. The topic of our lesson "Addition of numbers with different signs." And our goal: learn without mistakes, add numbers with different signs. Signed in the notebook the number and theme lesson.

3. Work on the lesson.

Slide 6. - Applying the concepts, find the results of the addition of numbers with different signs on the screen.
- What numbers are the result of the addition of positive numbers, negative numbers?
- What numbers are the result of adding numbers with different signs?
- What does the number of numbers with different signs depend on? (Slide 5)
- From the slant with the largest module.
- It's like when dragging the rope. The strongest wins.

Slide 7. - Let's play. Imagine that you tighten the rope. . Teacher. Rivals are usually found at competitions. And we will visit you today at several tournaments. The first thing is waiting for - this is the final of the contest to tighten the rope. Ivan minuses are found at number -7 and Peter Pluses at number +5. What do you think who will win? Why? So, Ivan the minuses won, he really turned out to be stronger than the opponent, and was able to drag him to his negative side Exactly two steps.

Slide 8.- . And now we will visit other competitions. Before you, Final Footage. The best in this form was minus Troikin with three balloons And plus chops, having four balloons in stock. And here guys, what do you think, who will become the winner?

Slide 9.- Competitions showed that they wins the strongest. So, when adding numbers with different signs: -7 + 5 \u003d -2 and -3 + 4 \u003d +1. Guys, how are the numbers with different signs? Students offer their own options.

The teacher formulates the rule, gives examples.

10 + 12 = +(12 – 10) = +2

4 + 3,6 = -(4 – 3,6) = -0,4

Students in the process of demonstration can comment on the solution appearing on the slide.

Slide 10.- Teacher- will play another game "Sea battle". The enemy ship is approaching our coast, it must be sinking and slaughter. For this we have a gun. But to get into the goal it is necessary to make accurate calculations. What are you seeing now. Ready? Then forward! Please do not distract, the examples are changing exactly 3 seconds. All ready?

Students in turn goes to the board and calculate the examples appearing on the slide. - Name the task execution steps.

Slide 11-Work on the textbook: p.180 p.33, read the rule of addition of numbers with different signs. Comments the rule.
- What is the difference between the rules proposed in the textbook, from the algorithm made by you? Consider examples in a tutorial with a comment.

Slide 12-Teacher and now guys let's spend experiment. But not chemical, and mathematical! Take the numbers 6 and 8, signs plus and minus and mix well. We get four examples experience. Do them in my notebook. (Two students decide on the wings of the board, then the answers are checked). What conclusions can be made from this experiment?(The role of signs). We will spend 2 more experiments But with your numbers (emerged by a person to the blackboard). Invent each other numbers and check the results of the experiment (mutual test).

Slide 13. .- The screen is displayed in poetic form .

4. Reflowing the lesson theme.

Slide 14 -Teacher- "Signs of all kinds need, all sorts of signs are important!" Now, guys, we will share with you for two teams. The boys will be in the team of Santa Claus, and the girls are a sun. Your task, without computing examples, determine in which of them there will be negative answers, and in what - positive and write down the letters of these examples in the notebook. Boys, respectively, are negative, and girls are positive (cards from the application are issued). Conducted self-test.

Well done! Anyone's signs are excellent. It will help you perform the following task.

Slide 15 - Fizkulminutka. -10, 0.15,18, -5,14,0, -8, -5, etc. (Negative numerics are squatted, positive numbers- Tightened up, bounce)

Slide 16.-New 9 examples independently (task on cards in the application). 1 SELLS at the board. Make self-test. Answers are displayed on the screen, the students' errors are corrected in the notebook. Raise your hands, who is true. (Marks are set only for a good and excellent result)

Slide 17. - Rules helps us correctly solve examples. Let them repeat them on the screen of the algorithm of addition of numbers with different signs.

5. Organization of independent work.

Slide 18 -F.rontal work through the game "Guess the word"(Task on cards in the application).

Slide 19 - It should be an estimate for the game - "Pyaterochka"

Slide 20 snow, attention. Homework. Homework should not cause any difficulties from you.

Slide 21 -Laws of addition B. physical phenomena. Come up with examples of adding numbers with different signs and ask them to each other. What new have you recognized? Have we achieved the goal?

Slide 22 -So the lesson ended, summarize the result. Reflection. The teacher commented and exposes estimates for the lesson.

Slide 23 - Thanks for attention!

I wish you so that in your life there was more positive and less negative, I want to tell you guys, thanks for your active work. I think you can easily apply the knowledge gained on subsequent lessons. The lesson is over. Thank you all very much. Bye!

Task 1. The player recorded winning the + and loss sign -. Find the result of each of the following records: a) +7 rub. +4 rub.; b) -3 rub. -6 rub.; C) -4 p. +4 r.; d) +8 p. -6 r.; E) -11 p. +7 r.; f) +2 p. +3 p. -5 r.; G) +6 p. -4 r. +3 p. -5 r. +2 p. -6 r.

Recording a) Indicates that the player first won 7 rubles. And then I also won 4 r. - Total won 11 r.; Recording C) Indicates that first player played 4 p. And then won 4 r. - Therefore, the overall result \u003d 0 (the player did nothing); Entry E) Indicates that the player first lost 11 rubles, then won 7 rubles, - Loss replaced the winnings for 4 rubles; Consequently, in general, the player lost 4 rubles. So, we have the right for these records to write down that

a) +7 p. +4 p. \u003d +11 r.; C) -4 p. +4 p. \u003d 0; E) -11 p. + 7 p. \u003d -4 rub.

The rest of the records are also easily disassembled.

In its sense, these tasks are similar to those that are solved in arithmetic with the help of the action of the addition, so we will assume that everywhere it is necessary to find the general result of the game to add relative numbers expressing the results of individual games, for example, in example c) the relative number -11 rub. It takes shape with a relative number +7 rubles.

Task 2. The cashier recorded the arrival of the box office sign +, and the expense is familiar -. Find the overall result of each of the following records: a) +16 p. +24 p.; b) -17 p. -48 r.; c) +26 p. -26 r.; d) -24 p. +56 r.; E) -24 p. +6 r.; f) -3 r. +25 p. -20 r. +35 r.; g) +17 p. -11 r. +14 p. -9 r. -18 r. +7 r.; h) -9 r -7 r. +15 p. -11 r. +4 p.

We will analyze, eg, record f): I count first the entire arrival of the box office: on this record it was 25 rubles. arrival, yes another 35 rubles. Come, in total, it was 60 rubles, and the flow was 3 rubles, and another 20 rubles, it was 23 rubles. consumption; The arrival exceeds consumption by 37 rubles. Track.,

- 3 rubles. + 25 rubles. - 20 rubles. + 35 rubles. \u003d +37 rubles.

Task 3. The point varies in a straight line, ranging from point A (damn 2).

Heck. 2.

Moving it to the right refer to the sign + and moving it to the left sign. Where the point will be after several oscillations recorded by one of the following records: a) +2 dm. -3 dm. +4 dm; b) -1 dm. +2 dm. +3 dm. +4 dm. -5 dm. +3 dm; c) +10 dm. -1 dm. +8 dm. -2 dm. +6 dm. -3 dm. +4 dm. -5 dm; d) -4 dm. +1 dm. -6 dm. +3 dm. -8 dm. +5 dm; e) +5 dm. -6 dm. +8 dm. -11 dm. In the drawing of the inches are designated by segments less than the real.

Last record (E) We will analyze: First, the oscillating point moved to the right from A to 5 dm., It was later moved to the left of 6 dm. - In general, it should be left from A to 1 dm, then moved to the right on 8 inches. , Next, now it is right from A to 7 dm., and then moved to the left of 11 dm., Therefore, it is left of A by 4 dm.

We provide other examples to disassemble the students themselves.

We have adopted that in all disassembled records you have to fold recorded relative numbers. Therefore, we agree:

If several relative numbers are written nearby (with their signs), then these numbers must be folded.

We now analyze the main cases encountered in addition, and take relative numbers without names (i.e., instead of talking, for example, 5 rubles. Win, yes another 3 rubles. Losing, or the point has moved to 5 dm. Right from A, yes, then another 3 DM. To the left, we will say 5 positive units, and even 3 negative units ...).

Here it is necessary to add numbers consisting of 8 positions. units, yes, from 5 position. units, we get a number consisting of 13 position. units.

So + 8 + 5 \u003d 13

Here it is necessary to fold the number consisting of 6 will deny. Units with a number consisting of 9 will deny. units, we get 15 will deny. units (compare: 6 rubles loss and 9 rubles. Losses - make up 15 rubles. Loss). So,

– 6 – 9 = – 15.

4 rubles win and then 4 rubles. Losses, in general, give zero (mutually destroyed); Also, if the point has advanced from A first to the right of 4 dm., And then to the left of 4 dm, then it will be again at the point a and, the next, the final distance from A is zero, and in general we should assume that 4 Position Units, and another 4 negative units, in general, will give zero, or mutually destroyed. So,

4 - 4 \u003d 0, also - 6 + 6 \u003d 0, etc.

Two relative numbers having the same absolute value, but different signs are mutually destroyed.

6 denied. Units are destroyed from 6 put. units, and there will still be 3 position. units. So,

– 6 + 9 = + 3.

7 Position Units will be destroyed with 7 denied. units, let it remain 4 will reverse. units. So,

7 – 11 = – 4.

Considering 1), 2), 4) and 5) cases

8 + 5 \u003d + 13; - 6 - 9 \u003d - 15; - 6 + 9 \u003d + 3 and
+ 7 – 11 = – 4.

From here we see that it is necessary to distinguish between two cases of addition of algebraic numbers: the case when the components have the same signs (1st and 2nd) and the incidence of numbers with different signs (4th and 5th).

It is not difficult now to see that

when the numbers are additioned with the same signs, their absolute values \u200b\u200bshould be added and write their overall sign, and when two numbers are addition, with different signs, it is necessary to calculate the arithmetic absolute values \u200b\u200b(from a greater smaller) and write a sign of the number that has an absolute value anymore.

Let it take to find the amount

6 – 7 – 3 + 5 – 4 – 8 + 7 + 9.

We can first fold all the positive numbers + 6 + 5 + 7 + 9 \u003d + 27, then all will deny. - 7 - 3 - 4 - 8 \u003d - 22 and then the results obtained between them + 27 - 22 \u003d + 5.

We can also take advantage of the fact that the numbers + 5 - 4 - 8 + 7 are mutually destroyed and then it remains to be addressed only the numbers + 6 - 7 - 3 + 9 \u003d + 5.

Another way of designation of addition

You can enter the brackets to write in the brackets and between the brackets. For example:

(+7) + (+9); (–3) + (–8); (+7) + (–11); (–4) + (+5);
(-3) + (+5) + (-7) + (+9) + (-11), etc.

We can, according to the previous one, immediately write the amount, for example. (-4) + (+5) \u003d +1 (the case of addition of numbers with different signs: it is necessary of the greater absolute value to deduct smaller and write a sign of the number that has an absolute value more), but we can also rewrite the same thing without brackets , using our condition that if numbers are written next to their signs, these numbers must be folded; track.,

to reveal brackets when adding positive and negative numbers, it is necessary to write the components next to their signs (addition sign and brackets).

For example: (+ 7) + (+ 9) \u003d + 7 + 9; (- 3) + (- 8) \u003d - 3 - 8; (+ 7) + (- 11) \u003d + 7 - 11; (- 4) + (+ 5) \u003d - 4 + 5; (- 3) + (+ 5) + (- 7) + (+ 9) + (- 11) \u003d - 3 + 5 - 7 + 9 - 11.

After that, you can fold the numbers.

The course of algebra should pay special attention to reducing disclosing brackets.

Exercises.

1) (– 7) + (+ 11) + (– 15) + (+ 8) + (– 1);

\u003e\u003e Mathematics: additions of numbers with different signs

33. Addition of numbers with different signs

If the air temperature was 9 ° C, and then it changed to 6 ° C (i.e. he dropped at 6 ° C), then it became equal to 9 + (- 6) degrees (Fig. 83).

To add numbers 9 and - 6 with the help of, it is necessary to move the point A (9) to the left of 6 single segments (Fig. 84). We get a point in (3).

It means 9 + (- 6) \u003d 3. The number 3 has the same sign as the term 9, and its module equal to the difference between the modules of the 3 and -6 modules.

Indeed, | 3 | \u003d 3 and | 9 | - | - 6 | \u003d \u003d 9 - 6 \u003d 3.

If the same air temperature of 9 ° C changed to -12 ° C (i.e. it dropped 12 ° C), then it became equal to 9 + (- 12) degrees (Fig. 85). After folding the number 9 and -12 using the coordinate straight (Fig. 86), we obtain 9 + (-12) \u003d -3. The number -3 has the same sign as the category -12, and its module is equal to the difference in the modules of the components -12 and 9.

Indeed, | - 3 | \u003d 3 and | -12 | - | -9 | \u003d 12 - 9 \u003d 3.

To fold two numbers with different signs, it is necessary:

1) from the larger module of the deduction smaller;

2) put in front of the number the sign of the term, whose module is greater.

Usually, first define and write the amount of the amount, and then find the difference in modules.

For example:

1) 6,1+(- 4,2)= +(6,1 - 4,2)= 1,9,
or shorter 6.1 + (- 4.2) \u003d 6.1 - 4.2 \u003d 1.9;

When adding positive and negative numbers, you can use microcalculator. To enter a negative number into a microcalculator, you need to enter the module of this number, then press the "Sign Change" key | / - / |. For example, to enter the number -56.81, you need to sequentially press the keys: | 5 |, | 6 |, | | |, | 8 |, | 1 |, | / - / |. Operations on the numbers of any sign are performed on the microcalculator in the same way as over positive numbers.

For example, the amount of -6.1 + 3.8 is calculated by Program

? Numbers a and b have different signs. What sign will have the amount of these numbers, if a larger module has a negative number?

if a smaller module has a negative number?

if a larger module has a positive number?

if a smaller module has a positive number?

Formulate the rule of addition of numbers with different signs. How to enter a negative number in a microcalculator?

TO 1045. The number 6 was changed to -10. Which side of the countdown is the resulting number? At what distance from the beginning of the countdown is it? What is equal to sum 6 and -10?

1046. The number 10 was changed to -6. Which side of the countdown is the resulting number? At what distance from the beginning of the countdown is it? What is the amount of 10 and -6?

1047. The number -10 changed to 3. Which parties from the beginning of the countdown are the resulting number? At what distance from the beginning of the countdown is it? What is the amount of -10 and 3?

1048. The number -10 has changed to 15. Which parties are the resulting number from the beginning of the reference? At what distance from the beginning of the countdown is it? What is the amount of -10 and 15?

1049. In the first half of the day, the temperature changed to - 4 ° C, and in the second - by + 12 ° C. How many degrees changed the temperature during the day?

1050. Perform addition:

1051. Add:

a) to the amount of -6 and -12 number 20;
b) to the number 2.6 amount -1.8 and 5.2;
c) to the sum of -10 and -1.3 amount 5 and 8.7;
d) to the amount of 11 and -6.5 amount -3.2 and -6.

1052. Which of the numbers 8; 7.1; -7.1; -7; -0.5 is the root equations - 6 + x \u003d -13.1?

1053. Guess the root of the equation and check:

a) x + (-3) \u003d -11; c) m + (-12) \u003d 2;
b) - 5 + y \u003d 15; d) 3 + n \u003d -10.

1054. Find the value of the expression:

1055. Perform actions using a microcalculator:

a) - 3,2579 + (-12,308); d) -3,8564+ (-0,8397) +7.84;
b) 7,8547+ (- 9,239); e) -0.083 + (-6,378) + 3,9834;
c) -0.00154 + 0.0837; e) -0.0085+ 0.00354+ (- 0.00921).

P 1056. Find the value of the amount:

1057. Find the value of the expression:

1058. How many integers are located between numbers:

a) 0 and 24; b) -12 and -3; in) -20 and 7?

1059. Imagine the number -10 as the sum of two negative terms so that:

a) both of the terms were integers;
b) both allegations were decimal fractions;
c) one of the components was the right ordinary fraction.

1060. What is the distance (in single segments) between the points of the coordinate direct coordinates:

a) 0 and a; b) -a and a; c) -a and 0; d) A and -Z?

M. 1061. Radius of geographic parallels ground surfacewhere the cities of Athens and Moscow are located, respectively, 5040 km and 3580 km are equal (Fig. 87). How many parallel in Moscow are shortly parallels of Athens?

1062. Make an equation to solve the problem: "The field with an area of \u200b\u200b2.4 hectares was divided into two sections. Find area Each site, if it is known that one of the sections:

a) by 0.8 hectare more than the other;
b) 0.2 hectares less than another;
c) 3 times more than the other;
d) 1.5 times less than the other;
e) is another;
e) is 0.2 different;
g) is 60% of the other;
h) is 140% of the other. "

1063. Decide the task:

1) On the first day, travelers drove 240 km, on the second day 140 km, on the third day they drove 3 times more than in the second, and on the fourth day they rested. How many kilometers they drove on the fifth day, if in 5 days they drove on average 230 km per day?

2) Father's earnings per month is 280 p. Daughter scholarship 4 times less. How much does mother earn in a month if there are 4 people in the family, the youngest son - a schoolboy and everyone accounts for an average of 135 r.?

1064. Perform the actions:

1) (2,35 + 4,65) 5,3:(40-2,9);

2) (7,63-5,13) 0,4:(3,17 + 6,83).

1066. Present in the form of the sum of two equal terms of the KDO from numbers:

1067. Find the value of A + B if:

a) a \u003d -1,6, b \u003d 3.2; b) a \u003d - 2.6, b \u003d 1.9; in)

1068. On one floor of a residential building there were 8 apartments. 2 Apartments had a living area of \u200b\u200b22.8 m 2, 3 apartments - 16.2 m 2, 2 apartments - 34 m 2. What kind of residential area had the eighth apartment, if on the floor on average for each apartment accounted for 24.7 m 2 living space?

1069.In the composition of the commercial train was 42 cars. Covered wagons were 1.2 times more than platforms, and the number of tanks was the number of platforms. How many wagons of each species were in the train?

1070. Find the value of the expression

N.Ya.Vilekin, A.S. Chesnokov, S.I. Schwarzburg, V.I.zhokhov, Mathematics for grade 6, tutorial for high School

Planning for mathematics, textbooks and books online, courses and math tasks for grade 6 download

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1 Slide

Mathematics teacher MOU SS No. 7 cities in Labinsk Krasnodar Territory Goncharova Irina Anatolyevna Nomination Physical and Mathematics Sciences Mathematics lesson in grade 6

2 Slide

Check homework № 1098 Teams Star Eagle Tractor Falcon Chaika The number of scored balls 49 37 17 21 6 Number of missed balls 16 28 23 35 28 The difference of scored and missed balls 33 9 -6 -14 -22

3 Slide

Let there be x Russian brands in the album, then 0.3x brands were foreign. In total, the album was (x + 0.3x) brands. Knowing that there was only 1105 stamps, and solve the equation. x + 0.3x \u003d 1105; 1.3x \u003d 1105; x \u003d 1105: 1.3; x \u003d 11050: 13; X \u003d 850. So, 850 stamps were Russian, then 850 0.3 \u003d 255 (mar.) There were foreign ones. Check: 850 + 255 \u003d 1105; 1105 \u003d 1105 - right. Answer: 255 brands; 850 brands. №1100 Foreign brands -? Russian stamps -? 1105 stamps Sost. thirty %

4 Slide

To fold two negative numbers, it is necessary: \u200b\u200b1. Init modules of these numbers. 2. Follow the result obtained to put a "minus" sign. -7 + (-9) i-7i + i-9i \u003d 7 + 9 \u003d 16 -7 + (-9) \u003d - 16 repeat the rule

5 Slide

Pick up such a number in order to get faithful equality: a) -6 + ... \u003d -8; b) ... + (-3.8) \u003d -4; c) -6,5 + ... \u003d - 10; d) ... + (-9,1) \u003d -10.1; d) ... + (-3.9) \u003d -13.9; e) - 0.2 + ... \u003d - 0.4. Task 1 (-2) (-0.2) (-3,5) (-1) (-10) (-0.2)

6 Slide

To fold two numbers with different signs, it is necessary: \u200b\u200bto find the modules of these numbers. From the larger module to deduct the smaller. Before the result obtained, put a sign of a number with a large module. -8 + 3 i-8i \u003d 8 i3i \u003d 3 because i-8i\u003e i3i, then -8 + 3 \u003d -5 because 8\u003e 3, then 8 - 3 \u003d 5 Repeat the rule

7 Slide

Perform addition: a) -7 + 11 \u003d b) -10 + 4 \u003d c) - 6 + 8 \u003d g) 7 + (-11) \u003d d) 10 + (- 4) \u003d e) - 8 + 6 \u003d ) -11 + 7 \u003d h) - 4 + 10 \u003d and) -24 + 24 \u003d task 2 4 -6 (-4) 6 -2 0 2 6 -4

8 Slide

To subtract different from this number, it is necessary: \u200b\u200b1. Find the number opposite to subtracted. 2. To reduced to add this number. 25 - 40 40 - subtractable, - 40 - to it opposite 25 + (- 40) \u003d \u003d - (40 - 25) \u003d - 15 Repeat the rule

9 Slide

Perform subtraction: a) 1.8 -3.6 \u003d b) 4 -10 \u003d c) 6 - 8 \u003d g) 7 - 11 \u003d d) 10 - 4 \u003d e) 2.18 - 4,18 \u003d g) 24 - 24 \u003d h) 1 - 41 \u003d and) -24 + 24 \u003d task 3 -1.8 -6 -2 (-4) 6 -2 0 -40 0

10 Slide

To find the length of the segment on the coordinate direct famous coordinates His ends, it is necessary to _________________________________ To complete the approval by selecting the desired phrase from the list: 1. Fold the coordinates of its left and right ends; 2. deny the coordinates of its ends in any order; 3. Subtract from the coordinate of the right end of the coordinate of the left end; 4. Calculate the coordinate of the middle of the segment, which will be equal to the length of the segment; 5. To the coordinate of the right end to add a number opposite coordinate Left end.

11 Slide

To find the length of the segment on the coordinate line according to the known coordinates of its ends, it is necessary to deduct from the coordinate of the right end of the coordinate of the left end. And in -3 0 4 x AV \u003d 4 - (-3) \u003d 4 + 3 \u003d 7 (one. OTR.) | | |

12 Slide

Sharely, the teacher's entertaining task suggested that the next task is to decide at home: "Find the sum of all integers from - 499 to 501." Dunno, as usual, sat down, but it was slow. Then Mom, Dad, Grandma came to the aid. Calculated so far from fatigue did not make eyes closed. And you guys, how would you solve this task?

13 Slide

Find an expression value: -499 + (- 498) + (- 497) + ... + 497 + 498 + 499 + 500 + 501. Solution: -499 + (- 498) + (- 497) + ... + 497 + 498 + 499 + 500 + 501 \u003d \u003d (- 499 + 499) + (- 498 + 498) + (- 497 + 497) + ... ... + (- 1 + 1) + 0 + 500 + 501 \u003d 500 + 501 \u003d \u003d 1001. Answer: The sum of all integers from - 499 to 501 is 1001. Solution of the problem

14 Slide

Work in notebooks No. 1123 No. 1124 (A, B) Find the distance in unit segments between points A (-9) and in (-2), C (5,6) and K (-3.8), E () and F ()

15 Slide

Independent work 1 Option 2 Option 1. 7.5 - (- 3.7) \u003d 1. -25.7-4.6 \u003d 2. -2.3-6.2 \u003d 2. 6.3 - (- 8.1 ) \u003d 3. 0.54 + (- 0.83) \u003d 3. -0.28 + (- 0.18) \u003d 4. -543 + 458 \u003d 4. 257 + (- 314) \u003d 5. -0, 48 + (- 0.76) \u003d 5. -0.37 + (- 0.84) \u003d

IN this lesson The addition and subtraction of rational numbers is considered. The topic refers to the category of complex. Here it is necessary to use the entire arsenal of the previously obtained knowledge.

Rules of addition and subtraction of integers are valid for rational numbers. Recall that rational is called numbers that can be represented as a fraction where a -this is a fraction numerator, b. - denominator of the fraci. Wherein, b. should not be zero.

In this lesson, the fractions and mixed numbers will increasingly be called one common phrase - rational numbers.

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Example 1. Find the value of the expression:

Let's conclude each rational number In brackets with their signs. We take into account that a plus which is given in the expression, is a sign of the operation and does not apply to the fraction. This fraction has a plus sign that is invisible due to the fact that it is not written. But we will write it for clarity:

This is the addition of rational numbers with different signs. To fold the rational numbers with different signs, it is necessary to subtract a smaller module from a larger module, and before the answer received to put a sign of that rational number, the module of which is greater. And in order to understand which module is more, and how less, you need to be able to compare the modules of these fractions before they are calculated:

The module of the rational number is greater than the rational module. Therefore, we are delayed. Received an answer. Then reducing this fraction to 2, they received the final answer.

Some primitive actions, such as: Conclusion Numbers in brackets and module stimulation, can be skipped. This example is quite possible to write down:

Example 2. Find the value of the expression:

We conclude every rational number in brackets along with your signs. We take into account that minus, standing between rational numbers and is a sign of the operation and does not apply to the fraction. This fraction has a plus sign that is invisible due to the fact that it is not written. But we will write it for clarity:

Replace subtraction by adding. Recall that for this you need to reduced to add a number opposite to subtract:

Received the addition of negative rational numbers. To fold the negative rational numbers, you need to add them modules and to put minus before the response received:

Note. To enter into brackets every rational number is not at all. It is done for convenience to see well what signs have rational numbers.

Example 3. Find the value of the expression:

In this expression, fractions are different denominators. To facilitate the task, we give these fractions to common denominator. Let's not dwell on how to do it. If you are experiencing difficulties, be sure to repeat the lesson.

After bringing fractions to the general denominator, the expression will take the following form:

This is the addition of rational numbers with different signs. We subtract a smaller module from a larger module, and before the response received, we put a sign of that rational number, the module of which is more:

We write the solution of this example shorter:

Example 4. Find an expression value

Calculate this expression in the following: Logging rational numbers and then from the result obtained subtracts the rational number.

First action:

Second Action:

Example 5.. Find the value of the expression:

Imagine an integer -1 in the form of a fraction, and a mixed number will be transferred to the wrong fraction:

We conclude every rational number in brackets together with your signs:

Received rational numbers with different signs. We subtract a smaller module from a larger module, and before the response received, we put a sign of that rational number, the module of which is more:

Received an answer.

There is a second solution. It consists in folding separately parts.

So, come back to the original expression:

We conclude every number in brackets. For this mixed number temporarily:

Calculate integers:

(−1) + (+2) = 1

In main expression, instead of (-1) + (+2), we write the resulting unit:

The resulting expression. To do this, write a unit and fraction together:

We write down the solution in this way.

Example 6. Find an expression value

Transfer a mixed number to the wrong fraction. The rest of the part is unchanged:

We conclude every rational number in brackets together with your signs:

Replace subtraction by adding:

We write the solution of this example shorter:

Example 7. Find an expression value

Imagine an integer -5 in the form of a fraction, and a mixed number will be transferred to the wrong fraction:

We give these fractions to the general denominator. After bringing them to a common denominator, they will take the following form:

We conclude every rational number in brackets together with your signs:

Replace subtraction by adding:

Received the addition of negative rational numbers. We show the modules of these numbers and in front of the response received minus:

Thus, the value of the expression is equal.

Decisive this example in the second way. Let's return to the original expression:

We write a mixed number in an expanded form. The rest will rewrite unchanged:

We conclude every rational number in brackets together with your signs:

Calculate integers:

In the main expression instead of writing the resulting number -7

The expression is a deployed form of a mixed number. We write the number -7 and fraction together, forming the final answer:

Write this solution shorter:

Example 8. Find an expression value

We conclude every rational number in brackets together with your signs:

Replace subtraction by adding:

Received the addition of negative rational numbers. We show the modules of these numbers and in front of the response received minus:

Thus, the value of the expression is equal

This example can be solved in the second way. It consists in folding the whole and fractional parts separately. Let's return to the original expression:

We conclude every rational number in brackets together with your signs:

Replace subtraction by adding:

Received the addition of negative rational numbers. We show the modules of these numbers and in front of the response received minus. But this time we are alone individually parts (-1 and -2), and fractional and

Write this solution shorter:

Example 9. Find expressions of expression

Transfer mixed numbers to incorrect fractions:

We conclude a rational number in brackets together with your sign. The rational number in the bracket is not necessary, since it is already in brackets:

Received the addition of negative rational numbers. We show the modules of these numbers and in front of the response received minus:

Thus, the value of the expression is equal

Now let's try to solve the same example by the second way, namely the addition of integers and fractional parts separately.

This time, in order to obtain a short solution, let's try to skip some actions, such as: recording a mixed number in deployment and replacement of subtraction by adding:

Please note that fractional parts were shown to a common denominator.

Example 10. Find an expression value

Replace subtraction by adding:

In the resulting expression there are no negative numbers that are the main cause of error assumptions. And since there are no negative numbers, we can remove the plus before subtractable, and also remove the brackets:

It turned out the simplest expression that is calculated easy. I calculate it in any way convenient for us:

Example 11. Find an expression value

This is the addition of rational numbers with different signs. The smaller module from a larger module, and before the response received, we will put a sign of that rational number, the module of which is more:

Example 12. Find an expression value

The expression consists of several rational numbers. According to, first of all it is necessary to perform actions in brackets.

First, we calculate the expression, then the expression obtained results are shown.

First action:

Second Action:

Third action:

Answer: The value of the expression equally

Example 13. Find an expression value

Transfer mixed numbers to incorrect fractions:

We conclude a rational number in brackets along with your sign. The rational number to enter into brackets is not necessary, since it is already in brackets:

We give these fractions in the general denominator. After bringing them to a common denominator, they will take the following form:

Replace subtraction by adding:

Received rational numbers with different signs. The smaller module from a larger module, and before the response received, we will put a sign of that rational number, the module of which is more:

Thus, the value of the expression equally

Consider the addition and subtraction of decimal fractions, which also relate to rational numbers and which can be both positive and negative.

Example 14. Find an expression value -3.2 + 4.3

We conclude every rational number in brackets along with your signs. We take into account that a plus which is given in expression, is a sign of the operation and does not apply to decimal fraction 4.3. This decimal fraction has a plus sign that is invisible due to the fact that it is not written. But we will write it for clarity:

(−3,2) + (+4,3)

This is the addition of rational numbers with different signs. To fold the rational numbers with different signs, it is necessary to subtract a smaller module from a larger module, and before the answer received to put a sign of that rational number, the module of which is greater. And in order to understand which module is more, and how less, you need to be able to compare the modules of these decimal fractions before they are calculated:

(−3,2) + (+4,3) = |+4,3| − |−3,2| = 1,1

The module of the number 4.3 is larger than the number -3.2 module therefore, therefore, we out of 4.3 detected 3.2. Received 1.1. The answer is positive, because before the answer should be a sign of that rational number, the module of which is greater. And the module of the number is 4.3 more than the module of the number -3.2

Thus, the value of expression is -3.2 + (+4.3) is 1.1

−3,2 + (+4,3) = 1,1

Example 15. Find an expression value 3.5 + (-8.3)

This is the addition of rational numbers with different signs. As in the last example, from a larger module, we subtract smaller and before the answer we put a sign of that rational number, the module of which is more:

3,5 + (−8,3) = −(|−8,3| − |3,5|) = −(8,3 − 3,5) = −(4,8) = −4,8

Thus, the value of expression is 3.5 + (-8.3) is --4.8

This example can be written shorter:

3,5 + (−8,3) = −4,8

Example 16. Find the value of expression -7.2 + (-3.11)

This is the addition of negative rational numbers. To fold the negative rational numbers, you need to add them modules and to put a minus before the response received.

Recording with modules can be skipped to not clutter expression:

−7,2 + (−3,11) = −7,20 + (−3,11) = −(7,20 + 3,11) = −(10,31) = −10,31

Thus, the value of expression is -7.2 + (-3.11) is - 10.31

This example can be written shorter:

−7,2 + (−3,11) = −10,31

Example 17. Find the expression value -0.48 + (-2.7)

This is the addition of negative rational numbers. We show their modules and before the response received will be minus. Recording with modules can be skipped to not clutter expression:

−0,48 + (−2,7) = (−0,48) + (−2,70) = −(0,48 + 2,70) = −(3,18) = −3,18

Example 18. Find an expression value -4,9 - 5.9

We conclude every rational number in brackets along with your signs. We take into account that minus which is located between rational numbers -4.9 and 5.9 is a sign of the operation and does not apply to the number 5.9. This rational number has its own sign of the plus, which is invisible due to the fact that it is not written. But we will write it for clarity:

(−4,9) − (+5,9)

Replace subtraction by adding:

(−4,9) + (−5,9)

Received the addition of negative rational numbers. We show their modules and before the response received by the answer.

(−4,9) + (−5,9) = −(4,9 + 5,9) = −(10,8) = −10,8

Thus, the value of expression is 4.9 - 5.9 is - 10.8

−4,9 − 5,9 = −10,8

Example 19. Find the value of expression 7 - 9.3

Enter into brackets every number together with your signs

(+7) − (+9,3)

Replace subtraction by adding

(+7) + (−9,3)

(+7) + (−9,3) = −(9,3 − 7) = −(2,3) = −2,3

Thus, the value of expression 7 - 9,3 is -2.3

We write the solution of this example shorter:

7 − 9,3 = −2,3

Example 20. Find an expression value -0.25 - (-1.2)

Replace subtraction by adding:

−0,25 + (+1,2)

Received rational numbers with different signs. The smaller module from the larger module, and before responding, we will put a sign of that number, the module of which is more:

−0,25 + (+1,2) = 1,2 − 0,25 = 0,95

We write the solution of this example shorter:

−0,25 − (−1,2) = 0,95

Example 21. Find the value of expression -3.5 + (4.1 - 7,1)

Perform actions in brackets, then show the resulting answer with a number -3.5

First action:

4,1 − 7,1 = (+4,1) − (+7,1) = (+4,1) + (−7,1) = −(7,1 − 4,1) = −(3,0) = −3,0

Second Action:

−3,5 + (−3,0) = −(3,5 + 3,0) = −(6,5) = −6,5

Answer: The value of the expression is -3.5 + (4.1 - 7,1) is - 6.5.

Example 22. Find an expression value (3.5 - 2.9) - (3.7 - 9.1)

Perform actions in brackets. Then, from among the first brackets resulting from the execution of the first brackets, will subtract the number that was obtained as a result of the execution of the second brackets:

First action:

3,5 − 2,9 = (+3,5) − (+2,9) = (+3,5) + (−2,9) = 3,5 − 2,9 = 0,6

Second Action:

3,7 − 9,1 = (+3,7) − (+9,1) = (+3,7) + (−9,1) = −(9,1 − 3,7) = −(5,4) = −5,4

Third action

0,6 − (−5,4) = (+0,6) + (+5,4) = 0,6 + 5,4 = 6,0 = 6

Answer: The value of the expression (3.5 - 2.9) - (3.7 - 9,1) is equal to 6.

Example 23. Find an expression value −3,8 + 17,15 − 6,2 − 6,15

We conclude in brackets every rational number together with your signs

(−3,8) + (+17,15) − (+6,2) − (+6,15)

Replace subtraction by adding where it can be:

(−3,8) + (+17,15) + (−6,2) + (−6,15)

The expression consists of several terms. According to the combination law of addition, if the expression consists of several terms, then the amount will not depend on the procedure. This means that the components can be folded in any order.

We will not be inventing the bike, and we turn all the components from left to right in the order of them:

First action:

(−3,8) + (+17,15) = 17,15 − 3,80 = 13,35

Second Action:

13,35 + (−6,2) = 13,35 − −6,20 = 7,15

Third action:

7,15 + (−6,15) = 7,15 − 6,15 = 1,00 = 1

Answer: The value of expression -3.8 + 17,15 - 6.2 - 6,15 is 1.

Example 24. Find an expression value

Translate decimal fraction -1.8 in a mixed number. The rest will rewrite without change: