Positive and negative numbers, definition, examples. Lesson "Positive and negative numbers" (6th grade)

Numbers with a “+” sign are called positive, numbers with a “-” sign are called negative. A straight line with a reference point, a unit segment and a direction chosen on it is called a coordinate line. If the straight line is located horizontally, then the coordinates of points located to the right of point O are usually considered positive, and the coordinates of points located to the left of point O are considered negative. The positive direction is marked with an arrow. If the straight line is located vertically, then the coordinates of the points located above point O are considered positive, and the coordinates of the points located below point O are considered negative. The straight line with the origin, unit segment and direction selected on it is called the coordinate line.

MS 4 10 A coordinate ray is drawn on the highway. Number 4 is Cheburashka. To come to Gene, he must go 5 single segments to the right. What number is Gena on? The old woman Shapoklyak is at the same distance from Cheburashka as Gena, but only on the left side. Draw the drawing into a notebook and show where Shapoklyak stands. What do the point where it stands and the point with coordinate (1) have in common? What numbers are to the left of zero? Where else is it possible to “move” from zero in different directions?

Why to the question: “How many degrees?” - can you answer “20” both in winter and summer? Compare: winter - summer frost - heat minus - plus “debt” - “property” Compare the sayings: ( opposite words in meaning - antonyms, not numbers) Young for battle - and old for thought. A small deed is better than a big idleness A bad world is better than good fame An old friend is better than new friends Labor feeds, but laziness spoils the time for business, the hour for fun.

Solve the problems: A coordinate line is drawn along the highway. The length of one unit segment is 2 meters. All characters They move only along the coordinate line. 1. On the number 0 are Dunno and Toporyzhka. They went in different directions and walked equal distances. Dunno came on the number 4. What number did Dunno come on?? How many meters did Toporyzhka walk? 2. On the number 0 a dog and a cat met. The cat ran away from the dog and stopped at the number 24. The dog ran away from the cat in the other direction and ran twice as far. What number was the dog on? 3. On the number 9 are Malysh and Carlson. They went in different directions and walked equal distances. The baby arrived on the number 29. What number did Carlson arrive on? 4. On the number 4 are Stepashka and Filya. They went in different directions and walked equal distances. Stepashka came to the number -10. What date did Phil arrive? How many meters did Stepashka walk? How many meters did Filya walk?

5. On the number 4 are Gena and Cheburashka. They simultaneously flew in different directions and stopped at the same time. Gena walked 3 times the distance than Cheburashka and ended up on number 37. What number was Cheburashka on? Which of them walked faster and by how many times? 6. On the number 0 are Dunno and Toporyzhka. They went in different directions and walked equal distances. Dunno came to number a. On what date did Toporyzhka arrive? 7. On the number 5 are Dunno and Toporyzhka. They went in different directions and walked equal distances. Dunno came to number a. On what date did Toporyzhka arrive? 8. On number d are Dunno and Toporyzhka. They went in different directions and walked equal distances. Dunno came to number a. On what date did Toporyzhka arrive? A number line is drawn along the highway. The length of one unit segment is equal to half a meter. Everyone moves along the number line. Cipollino stood on number 4, then he walked 6 single segments to the left. On what date did Cipollino arrive? How many meters did he walk?

Now we'll figure it out positive and negative numbers. First, we will give definitions, introduce notations, and then give examples of positive and negative numbers. We will also dwell on the semantic load that positive and negative numbers carry.

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Positive and Negative Numbers - Definitions and Examples

Give identifying positive and negative numbers will help us. For convenience, we will assume that it is located horizontally and directed from left to right.

Definition.

Numbers that correspond to points of the coordinate line lying to the right of the origin are called positive.

Definition.

The numbers that correspond to the points of the coordinate line lying to the left of the origin are called negative.

The number zero, which corresponds to the origin, is neither a positive nor a negative number.

From the definition of negative and positive numbers it follows that the set of all negative numbers is the set of numbers opposite all positive numbers (if necessary, see the article opposite numbers). Therefore, negative numbers are always written with a minus sign.

Now, knowing the definitions of positive and negative numbers, we can easily give examples of positive and negative numbers. Examples of positive numbers are the natural numbers 5, 792 and 101,330, and indeed any natural number is positive. Examples of positive rational numbers are the numbers , 4.67 and 0,(12)=0.121212... , and negative ones are the numbers , −11 , −51.51 and −3,(3) . Examples of positive irrational numbers include the number pi, the number e, and the infinite non-periodic decimal fraction 809.030030003..., and examples of negative ir rational numbers are the numbers minus pi, minus e and the number equal to . It should be noted that in the last example it is not at all obvious that the value of the expression is a negative number. To find out for sure, you need to get the value of this expression in the form of a decimal fraction, and we will tell you how to do this in the article comparison of real numbers.

Sometimes positive numbers are preceded by a plus sign, just as negative numbers are preceded by a minus sign. In these cases, you should know that +5=5, etc. That is, +5 and 5, etc. - this is the same number, but designated differently. Moreover, you can find the definition of positive and negative numbers based on the plus or minus sign.

Definition.

Numbers with a plus sign are called positive, and with a minus sign – negative.

There is another definition of positive and negative numbers based on comparison of numbers. To give this definition, it is enough just to remember that the point on the coordinate line corresponding to the larger number lies to the right of the point corresponding to the smaller number.

Definition.

Positive numbers are numbers that are greater than zero, and negative numbers are numbers less than zero.

Thus, zero sort of separates positive numbers from negative ones.

Of course, we should also dwell on the rules for reading positive and negative numbers. If a number is written with a + or − sign, then pronounce the name of the sign, after which the number is pronounced. For example, +8 is read as plus eight, and - as minus one point two fifths. The names of the signs + and − are not declined by case. Example correct pronunciation is the phrase “a equals minus three” (not minus three).

Interpretation of positive and negative numbers

We have been describing positive and negative numbers for quite some time. However, it would be nice to know what meaning they carry? Let's look at this issue.

Positive numbers can be interpreted as an arrival, as an increase, as an increase in some value, and the like. Negative numbers, in turn, mean exactly the opposite - expense, deficiency, debt, reduction of some value, etc. Let's understand this with examples.

We can say that we have 3 items. Here the positive number 3 indicates the number of items we have. How can you interpret the negative number −3? For example, the number −3 could mean that we have to give someone 3 items that we don't even have in stock. Similarly, we can say that at the cash register we were given 3.45 thousand rubles. That is, the number 3.45 is associated with our arrival. In turn, a negative number -3.45 will indicate a decrease in money in the cash register that issued this money to us. That is, −3.45 is the expense. Another example: a temperature increase of 17.3 degrees can be described as a positive number +17.3, and a temperature decrease of 2.4 can be described using a negative number, as a temperature change of -2.4 degrees.

Positive and negative numbers are often used to describe the values of certain quantities in different measuring instruments. The most accessible example is a device for measuring temperatures - a thermometer - with a scale on which both positive and negative numbers are written. Often, negative numbers are depicted in blue (it symbolizes snow, ice, and at temperatures below zero degrees Celsius, water begins to freeze), and positive numbers are written in red (the color of fire, the sun; at temperatures above zero degrees, ice begins to melt). Writing positive and negative numbers in red and blue is also used in other cases when you need to highlight the sign of the numbers.

References.

Vilenkin N.Ya. and others. Mathematics. 6th grade: textbook for general education institutions.

North Kazakhstan region

Ayyrtau district

KSU "Vsevolodovskaya incomplete" high school»

Open lesson

mathematicians

"Positive

and negative numbers.

Coordinate line."

6th grade

Teacher

mathematics and physics

Brykina Larisa Vasilievna

Lesson type: lesson in the formation of new knowledge

Forms of student work: frontal, individual, group .

Objective of the lesson:

Formation of the concept of positive and negative numbers with the skill of working on a coordinate line .

Tasks:

- educational:

“discover” a set of negative numbers, determine their place on the coordinate line, introduce the designation of negative numbers, teach how to use them when solving problems of an interdisciplinary nature, analyze and systematize knowledge about the studied numbers

- developing:

learn to analyze your own skills, the causes of difficulties when performing a task, find new solutions, develop the ability to assess the productivity of your own activities

- educational:

develop students' creative activity and interest in the subject.

Used educational technologies, methods and techniques:

activity method, information and communication technologies, health-saving technologies.

Necessary technical equipment and didactic means: teacher's computer, presentation on this topic, thermometer model, signal cards, cards for individual work, math lotto, score sheets.

Progress of the lesson.

1. Organization of the educational process .

- Hello children! We have a holiday today. Guests came to us. And in what mood do we greet them? (signal cards)

2. Setting the topic and goals of the lesson.

The ancient Greek scientist Pythagoras said: “Numbers rule the world.” You and I live in this world of numbers, and in school years learning to work with different numbers. (Slide 2)

So today we are starting to study new numbers that are still unknown to you.

And in order to formulate the topic of our lesson, we will answer several questions and try to determine what is common in the answers to these questions? (Slide 3)

1) Name the heroes of Russian fairy tales.

Divide them into two groups. How can you name the heroes of each group? (positive and negative). (Slide 4)

What is the temperature outside today? (-10) (Slide 5)

What are these numbers called? (negative). What is the temperature in summer?

What is the topic of the lesson?

What lesson objectives should we solve when studying this topic? (What should we learn?)

Be able to recognize positive and negative numbers and write them.

Be able to represent positive and negative numbers on a coordinate line.

(Slide 6)

3. Updating new knowledge. (Slides 7-12)

Frontal work using signal cards.

(For each correct answer - a star.)

What numbers do you already know?

Natural numbers.

Ordinary fractions.

Decimals.

Mixed numbers

2) Find natural numbers from the following:

3) Find natural numbers from the following:

4) Find common fractions among these numbers:

5) Find ordinary fractions among the given numbers:

6) What numbers have you not encountered yet? (Slide 13)

1) 15 ; 2879; 15970;

2) -120; -5; -21

3) 8 𝟑/𝟒 ;𝟎,𝟐; 𝟕/𝟗

These are the numbers we will talk about today.

3. Studying new material.

Where is the concept of positive and negative numbers used in life?

When measuring air temperature. (Slides 14, 15, 16)

First task: recognize positive and negative numbers. How will we recognize them? Suggest your own methods.

If a number is preceded by a “-” sign, then the number is negative. And if there is a “+” sign in front of the number or there is no sign, then this number is positive.

Where else is the concept used? positive and negative numbers? (Slide 16)

The weather forecast is shown on TV.

Kokchetav
Petropavlovsk


Saumalkol
Karaganda

What does the entry say: Petropavlovsk – 9, Almaty + 13?

9 degrees below zero, 13 degrees warm.

What device is used to determine air temperature?

Using a thermometer.

Working with a thermometer layout

Mark on the thermometer - 20 degrees; - 10 degrees; - 5 degrees. Where are they located?

Below 0. Negative numbers on a thermometer are located below 0.

Show on the thermometer what the temperature is in Sochi - 15 degrees Celsius, in Almaty - 20.

What can you say about these numbers?

Positive numbers on a thermometer are located above 0.

What numbers do we classify 0 as?

The number 0 is neither positive nor negative. On a thermometer, 0 is the reference point.

Positive and negative numbers (Slide 18)

Where else is the concept used? “Positive and negative numbers” (Slide 19)

Guys, how are numbers represented in mathematics?

On a coordinate beam.

Do you remember how to depict numbers on a coordinate ray? Who can tell about this? (Slide 20)

We take a ray going from left to right. We denote the beginning of the ray as 0. From zero we plot unit segments. The length of a single segment can be any. For example, 1 cell of a notebook, 1 cm. How to mark the number 1, 3, 7?

How to represent the number – 1, -3, -7?

Let's extend the ray to a straight line. To the left of 0, we plot the segments equal to the unit segment and mark the negative numbers, starting from zero. To mark the number - 1, we count one unit segment from 0 to the left, put point B. We write - B (- 1).

What is the difference between a coordinate ray and a coordinate line?

A ray has a beginning but no end, and a straight line has neither a beginning nor an end.

Negative numbers can be marked on the coordinate line.

The coordinate ray has a direction, and for the coordinate line you need to choose a direction. Mark the positive direction with an arrow.

Guys, let's try to define coordinate line. Horizontal and vertical coordinate lines.

A straight line with a selected origin, a unit segment and a positive direction is called a coordinate line. (Slide 20, 21)

4) Physical exercise

The time has come to restore tone; with the help of physical education, we will not only prevent osteochondrosis, but also figure out where we use the concept of positive and negative numbers in life. A concept appears, if it is positive, then we nod our heads “Yes,” and if it is negative, “No.” All the backs were straightened. Started

River depth

mountain height

school grade -5

school grade-2

I hope that by new topic We will only have positive ratings!

5. Consolidation of the material covered.

1) Math lotto (for weak students)

Match.

5° below zero

income 132 rub.
consumption 2351 rub.
loss 5 points
win 10 points

For strong students.

Write using positive and negative numbers:

Lake depth -3m
mountain height -100 m
profit - 1000 tons.
income -2000 t.
loss - 10,000 tons.
heat - 40 degrees,
frost -30 degrees

For the weak. Work at the board and in a notebook.

Determine the coordinates of points A. B, C, D, E

Working with the dough. For the strong.





	c) profit d) loss
	b) profit

6. Working with the textbook.

No. 266 - at the blackboard;

7. Reflection. Summing up. Grading for the lesson.

– What new did you learn in the lesson?

– What was used to “discover” new knowledge?

– What difficulties did you encounter?

– Analyze your work in class. (signal cards)

8. HomeworkParagraph 9 page 55No. 267, 272, 277 (for strong students)

Make up a story about positive and negative numbers. (optional)

Card No. 1Vernigorova Augustina

Lake depth -3m
mountain height -100 m
profit - 1000 tons.
income -2000 t.
loss - 10,000 tons.
heat - 40 degrees,
frost -30 degrees


A1. Which numbers are positive?
A2.What is the coordinate of point C?
A3.Which of these points has coordinate -2?
A4.Values that can be said to be positive	c) profit d) loss
A5.Values that can be said to be negative	b) profit

Card No. 2Starkov Daniil.

Write using positive and negative numbers:

Lake depth -3m
mountain height -100 m
profit - 1000 tons.
income -2000 t.
loss - 10,000 tons.
heat - 40 degrees,
frost -30 degrees

Test. Mark the correct answer with a + sign


A1. Which numbers are positive?
A2.What is the coordinate of point C?
A3.Which of these points has coordinate -2?
A4.Values that can be said to be positive	c) profit d) loss
A5.Values that can be said to be negative	b) profit

Lake depth
mountain height 150 m
profit 1000 t.
winnings 20,000 t.
Loss 50,000 tons.
Heat 40 degrees
frost -30 degrees

Lake depth
mountain height 150 m
profit 1000 t.
winnings 20,000 t.
Loss 50,000 tons.
Heat 40 degrees
frost -30 degrees

Negative numbers are numbers with a minus sign (−), for example −1, −2, −3. Reads like: minus one, minus two, minus three.

Application example negative numbers is a thermometer that shows the temperature of the body, air, soil or water. IN winter time, when it is very cold outside, the temperature can be negative (or, as people say, “minus”).

For example, −10 degrees cold:

The ordinary numbers that we looked at earlier, such as 1, 2, 3, are called positive. Positive numbers are numbers with a plus sign (+).

When writing positive numbers, the + sign is not written down, which is why we see the numbers 1, 2, 3 that are familiar to us. But we should keep in mind that these positive numbers look like this: +1, +2, +3.

Lesson content

This is a straight line on which all numbers are located: both negative and positive. Looks like this:

The numbers shown here are from −5 to 5. In fact, the coordinate line is infinite. The figure shows only a small fragment of it.

Numbers on the coordinate line are marked as dots. Bold in the picture black dot is the starting point. The countdown starts from zero. Negative numbers are marked to the left of the origin, and positive numbers to the right.

The coordinate line continues indefinitely on both sides. Infinity in mathematics is symbolized by the symbol ∞. The negative direction will be indicated by the symbol −∞, and the positive direction by the symbol +∞. Then we can say that all numbers from minus infinity to plus infinity are located on the coordinate line:

Each point on the coordinate line has its own name and coordinate. Name is any Latin letter. Coordinate is a number that shows the position of a point on this line. Simply put, a coordinate is the very number that we want to mark on the coordinate line.

For example, point A(2) reads as "point A with coordinate 2" and will be denoted on the coordinate line as follows:

Here A is the name of the point, 2 is the coordinate of the point A.

Example 2. Point B(4) reads as "point B with coordinate 4"

Here B is the name of the point, 4 is the coordinate of the point B.

Example 3. Point M(−3) reads as "point M with coordinate minus three" and will be denoted on the coordinate line as follows:

Here M is the name of the point, −3 is the coordinate of point M .

Points can be designated by any letters. But it is generally accepted to denote them in capital Latin letters. Moreover, the beginning of the report, which is otherwise called origin usually means big Latin letter O

It is easy to notice that negative numbers lie to the left relative to the origin, and positive numbers lie to the right.

There are phrases such as “the further to the left, the less” And "the further to the right, the more". You probably already guessed what we are talking about. With each step to the left, the number will decrease downwards. And with each step to the right the number will increase. An arrow pointing to the right indicates a positive reference direction.

Comparing negative and positive numbers

Rule 1. Any negative number is less than any positive number.

For example, let's compare two numbers: −5 and 3. Minus five less than three, despite the fact that five strikes the eye first of all as a number greater than three.

This is due to the fact that −5 is a negative number, and 3 is positive. On the coordinate line you can see where the numbers −5 and 3 are located

It can be seen that −5 lies to the left, and 3 to the right. And we said that “the further to the left, the less” . And the rule says that any negative number is less than any positive number. It follows that

−5 < 3

"Minus five is less than three"

Rule 2. Of two negative numbers, the one that is located to the left on the coordinate line is smaller.

For example, let's compare the numbers −4 and −1. Minus four less, than minus one.

This is again due to the fact that on the coordinate line −4 is located to the left than −1

It can be seen that −4 lies to the left, and −1 to the right. And we said that “the further to the left, the less” . And the rule says that of two negative numbers, the one that is located to the left on the coordinate line is smaller. It follows that

Minus four is less than minus one

Rule 3. Zero is greater than any negative number.

For example, let's compare 0 and −3. Zero more than minus three. This is due to the fact that on the coordinate line 0 is located more to the right than −3

It can be seen that 0 lies to the right, and −3 to the left. And we said that "the further to the right, the more" . And the rule says that zero is greater than any negative number. It follows that

Zero is greater than minus three

Rule 4. Zero is less than any positive number.

For example, let's compare 0 and 4. Zero less, than 4. This is in principle clear and true. But we will try to see this with our own eyes, again on the coordinate line:

It can be seen that on the coordinate line 0 is located to the left, and 4 to the right. And we said that “the further to the left, the less” . And the rule says that zero is less than any positive number. It follows that

Zero is less than four

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Mathematics lesson in grade 6 "B" on the topic "Adding and subtracting positive and negative numbers"

Lesson objectives:

educational: consolidation of skills and abilities of addition and subtraction of numbers with different signs, skills to transfer your knowledge to a new non-standard situation, mastery of mathematical terminology;

developing: development of creative, speech, mental activity using various shapes work;

educational: fostering attentiveness, activity and perseverance in achieving goals, instilling independent work skills.

Lesson type: lesson of repetition and generalization.

Lesson format: lesson - solving cognitive problems.

Equipment: computer, multimedia projector, worksheets.

Progress of the lesson.

Message of the topic and statement of the problem.

In today's lesson we must consolidate the knowledge gained in adding and subtracting numbers with different signs and show the ability to apply them when performing various tasks.

The motto of the lesson will be the words “The one who walks can master the road, but the one who thinks about mathematics »

Updating students' knowledge.

Let's start the lessonfrom oral work .

Compare numbers

58 and 145 (<)

62.2 and -62.3 (>)

8.58 and -8.5 (<)

1\2 and -0.5 (=)

Answer the questions

How to compare two positive numbers?

How to compare two negative numbers?

How to compare numbers with different signs?

Calculate:

22+35=13

3,7+2,8=0,9

1,5+(-6,3)= - 4,8

8,2+(-8,2)=0

22-27= - 5

19 - (- 2)=21

27 – (- 3) = -24

35 + (- 9)= - 44

1,6 +(- 4,4)= - 6

Historical background

Your worksheets contain examples. A letter is written next to each example. The name of the mathematician of Ancient India who introduced negative numbers into use is encrypted here. Who is this mathematician? You can answer this question by solving the examples and writing the answers in the table in ascending order with the corresponding letters.

A) -5+9;

B) – 11 – 3

U) -10.5 + 20.5;

A) (-8.5) + 3.5;

D) - 4 – (- 10);

A) – 24 + 49;

T) – 10.7 + 30.7;

M) 2 + ;

P) – 19 + 10;

X) 6.9 + (- 6.9)

P) – (- 7) + 4.5.

11,5

You have been given the name of the Indian mathematician Brahmagupta.

Let's listen to a message about the history of the emergence of positive and negative numbers.

Indian mathematicians thought of positive numbers as “property” and negative numbers as “debts.”

The Indian mathematician Brahmagupta (7th century) set out the rules of addition and subtraction as follows:

“The sum of two properties is property”

"The sum of two debts is a debt"

“The sum of property and debt is equal to their difference”

The origin of the modern "+" and "-" signs is not entirely clear. In Italy, when lending money, moneylenders put the amount of the debt and a line in front of the debtor’s name, like our minus, and when the debtor returned the money, they crossed it out, so it looked like our plus.

The modern "+" and "-" signs appeared in Germany in the last decade of the 15th century in Widmann's book, which was a counting manual for merchants.

Consolidation of knowledge.

Find the meanings of the expressions:

Option 1 Option 2

76 – 59 - 1,3-2,5

41,5 + 55,6 -1+ 9,56

125 - (-37) 5 – 3,07

39,6 + (-15,9) 0,5+(-0,5)

31,25-(-8,75) -63-1,6

Solve equations :

1) x + 1.2 = - 0.17 x = - 1.37

2) 14 –x = -28 x=42

3) x – 9 = - 3.1 x = 5.9

4) -2.1 – x = -2 x = - 0.1

Fill in the blanks:

14 + … = -37 (- 23)

4,8 + … = -8,6 (- 2,8)

2,13 + … = -17 (- 14,87)

3,8 + … = -4,08 (- 0,28)

Find errors in calculations:

25+ (-17) = - 8 ( 8)

– 30,5 – 12,6 = 43,1 ( – 43,1)

15, 73 – 20,5= 4,77 (-4,77)

Replace * with signs

1) 3,9 * 7,4 * (- 9,3) = - 12,8 (-,+)

2)-6,1 * (-2,3) * 3,8= 0 (- ,+)

Answer the questions orally

Numbersa Andb have different signs. What sign will the sum of these numbers have if the larger module is negative? if the smaller modulus is negative? if the larger modulus is a positive number? if the smaller modulus is a positive number?

Lesson summary

Homework No. 601 (g-i), 602.

Worksheet

F.I.__________________________________________

1 task.

A) -5+9;

B) – 11 – 3

U) -10.5 + 20.5;

A) (-8.5) + 3.5;

D) - 4 – (- 10);

A) – 24 + 49;

T) – 10.7 + 30.7;