Topic
Lesson type
- study and primary assimilation of new material
Lesson Objectives
Get to know the definitions of positive and negative, opposite numbers
Find opposite numbers when solving exercises, when solving equations
Developing - to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.
Educational - through the lesson to cultivate an attentive attitude towards each other, to instill the ability to listen to comrades, mutual assistance, independence.
Lesson objectives
Learn what opposite numbers are
Learn to use this concept when solving problems
Check students' ability to solve problems.
Lesson Plan
1. Introduction.
2. Theoretical part
3. Practical part.
4. Homework.
5. Interesting facts
Introduction
Look at the pictures and describe in one word what is the difference in them.
The pictures show opposites.
- these are two numbers that are equal in absolute value, but have different signs, for example. 5 and -5.
Theoretical part
First, let's remember what is negative numbers. Look video:
Points with coordinates 5 and -5 are the same distance from point O and are on opposite sides of it. To get from point O to these points, one must travel the same distances, but in opposite directions. The numbers 5 and -5 are called opposite numbers: 5 is the opposite of -5 and -5 is the opposite of 5.
Two numbers that differ from each other only in signs are called opposite numbers.
For example, 35 and -35 will be opposite numbers, since the number 35 \u003d +35, which means that the numbers 35 and -35 differ only in signs. The opposite numbers will also be 0.8 and -0.8, ¾ and -¾.
Properties of opposite numbers
one). For every number, there is only one opposite number.
2). The number 0 is the opposite of itself.
3). The opposite of a is called -a. If a = -7.8, then -a = 7.8; if a = 8.3, then -a = -8.3; if a = 0, then -a = 0.
4). The entry "-(-15)" means the opposite of -15. Since the opposite of -15 is 15, then -(-15) = 15. In general -(-a) = a.
The natural numbers, their opposite numbers and zero are called whole numbers.
opposite number n" in relation to the number n is the number that, when added to n, gives zero.
n + n" = 0
This equality can be rewritten as follows:
n + n" - n = 0 - n or n" = − n
In this way, opposite numbers have the same modules but opposite signs.
In accordance with this, the number opposite to the number n is denoted − n. When a number is positive, then its opposite number will be negative, and vice versa.
1. Give examples of opposite numbers.
2. Draw them on the coordinate line.
3. What is the opposite of -3.6; 7; 0; 8/9; -1/2
Practical part
Example
1) Mark points A(2), B(-2), C(+4), D(-3), E(-5.2), F(5.2), G(-6) on the coordinate line , H(7). 2) Among these points, find and indicate those that are symmetrical with respect to the point O (0). What can be said about the coordinates of symmetrical points?
Points symmetric with respect to point O(0): A(2) and B(-2), E(-5.2) and F(5.2)
Symmetric point coordinates are numbers that differ only in sign. Such numbers are called opposite.
Mark on the coordinate line points A (-3), B (+6), C (+4.2), D (+3), E (-4.2), F (-6) What can be said about these numbers ?
From the numbers 15; 2.5; - 2.5; - eighteen; 0; 45; - 45 choose: a) natural numbers; b) whole numbers; c) negative numbers; d) positive numbers; e) opposite numbers.
1) Write down the number opposite to number a.
2) Indicate the number opposite to the number a, if:
a=5, a=-3, a=0, a=-2/5;
A \u003d 6, -a \u003d - 2, -a \u003d 3.4.
1) Remember what the entry means: - (- a).
2) Replace * with such a number to get the correct equality: a) - (- 5) = *; b) 3 = - *.
Homework
one). Fill in the table:
2). Find: a) -m,
if m = -8,
if m = -16
if -k = 27
if -k = -35
if c = 41
if c = -3.6
3). How many pairs of opposite numbers are located between the numbers -7.2 and 3.6. Mark on the coordinate line.
4). Find out the name of an outstanding French scientist:
Do you know where in everyday life we encounter positive and negative numbers?
List of sources used
1. Mathematical encyclopedia (in 5 volumes). - M.: Soviet Encyclopedia, 2002. - T. 1.
2. "The latest schoolchildren's guide" "HOUSE XXI century" 2008
3. Summary of the lesson on the topic "Opposite numbers" Author: Petrova V.P., mathematics teacher (grades 5-9), Kiev
4. N.Ya. Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
5 and -5 (Fig. 61) are equally distant from point O and are on opposite sides of it. To get from point O to these points, one must travel the same distances, but in opposite directions. The numbers 5 and -5 are called opposite numbers: 5 is the opposite of 5, and -5 is the opposite of 5.
Two numbers that differ from each other only in signs are called opposite numbers.
For example, the opposite numbers will be 8 and -8, since the number 8 \u003d + 8, which means numbers 8 and - 8 differ only in signs. The opposite numbers will also be
For every number, there is only one opposite number.
The number 0 is the opposite of itself.
The opposite number of o is -a. If a \u003d -7.8, then -a \u003d 7.8; if a = 8.3, then - a = -8.3; if a \u003d 0, then -a \u003d 0. The entry "- (-15)" means the number opposite to the number -15. Since the number opposite to the number -15 is 15, then - (- 15) = 15. In general - (- a) \u003d a.
Natural numbers, their opposite numbers and zero are called integers.
? What are the opposite numbers?
The number b is opposite to the number a. What number is the opposite of b?
What is the opposite of zero?
Is there a number that has two opposite numbers?
What numbers are called integers?
TO 910. Find the opposite numbers:
911. Replace with such a number to get the correct equality:
912. Find the value of the expression:
913. Find the coordinates of points A, B and C (Fig. 62).
914. What number is -x if x:
a) negative; b) zero; c) positive?
915. Fill in the empty spaces in the table and mark on the coordinate straight points that have as their coordinates the numbers of the resulting table.
916. Solve the equation:
a) - x = 607; b) - a = 30.4; c) - y= -3
917. What integers are located on the coordinate line between the numbers:
P 918. Calculate orally:
919. Between which integers on the coordinate line is the number: 2.6; -thirty; -6; -eight
920. Find the numbers that are at a distance on the coordinate line: a) 6 units from the number -9; b) 10 units from the number 4; c) 10 units from the number -4; d) 100 units from the number 0.
921. Draw a coordinate line, taking as a unit section the length of 4 cells of the notebook, and mark on this straight line the points, F (2.25).
A 922. Mark on the "timeline" the following events from the history of mathematics:
a) The book "Beginnings" was written by Euclid in the 3rd century BC. BC e.
b) Number theory originated in ancient Greece in the 6th century. BC e.
c) Decimal fractions appeared in China in the 3rd century.
d) The theory of relations and proportions was developed in ancient Greece in the 4th century. BC e.
e) The positional decimal number system spread in the countries of the East in the 9th century. How many centuries ago did these events take place? Compare the "time line" and the coordinate line.
923. Specify pairs of mutually reciprocal numbers:
924. Victor bought 2.4 kg of carrots. How many carrots bought Kolya, if it is known that he bought:
a) 0.7 kg more than Vitya; f) what Vitya bought;
b) 0.9 kg less than Vitya; g) 0.5 of what Vitya bought;
c) 3 times more than Viti; h) 20% of what Vitya bought;
d) 1.2 times less than Viti; i) 120% of what Vitya bought;
e) what Vitya bought; j) 20% more than what Vitya bought?
925. Solve the problem:
1) The brick factory was supposed to produce 270 thousand bricks for the construction of the Palace of Culture. First
week he completed the tasks, in the second week he produced 10% more than in the first week. How many thousand bricks are left for the factory to produce?
2) The collective farm sold 434 tons of grain to the state in three days. On the first day he sold this amount, on the second day he sold 10% less than on the first day, and on the third day he sold the rest of the grain. How many tons of grain did the collective farm sell on the third day?
926. Notes differ in their duration. The sign denotes a whole note, a note half as long - half, sixteenth.
Check the equality of durations:
D 927. What numbers are opposite to numbers:
928. Write down all natural numbers less than 5 and their opposites.
929. Find the value:
930. On the second day, 2 times more wire was issued from the warehouse than on the first day, and on the third day 3 times more than on the first. How many kilograms of wire were given out during these three days, if on the first day they gave out 30 kg less than on the third?
931. On a collective farm, on irrigated lands, 60.8 centners of wheat were harvested per hectare. Replacing an old wheat variety with a new one gives a yield increase of 25%. How much wheat does the collective farm now harvest from 23 hectares of irrigated field?
932. Make an equation for each scheme and solve it:
933. Find the value of the expression:
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Schwarzburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
An interesting concept from a school course is opposite numbers, which can be considered both mathematically and geometrically. Understanding this topic simplifies the study of mathematics, allows you to quickly cope with some tasks - therefore, we will consider which numbers are called opposites, and what rules work for them.
What is the essence of the term?
To understand the meaning of opposite numbers, let's turn to geometry for a moment. Let's draw a coordinate line and mark the zero point on it, and then put two more marks on the line - for example, "2" on the right side and "-2" on the left side of zero. Of course, from both points the distance to the origin will be exactly the same - and this is easily verified by measurements. "2" and "-2" are the same distance from zero, but in different directions - respectively, they are completely opposite to each other.
This is the point. Numbers can be arbitrarily large or small, whole or fractional. However, each of them has a certain number that is its complete opposite. The definition can be given as follows - if on the line of coordinates from two points set on both sides of zero, an equal distance can be set aside to the origin - these points, or rather, the numbers corresponding to them, will be opposite.
What rules can be deduced from the definition?
It is worth remembering a few unconditional statements regarding the topic under consideration:
- The principle of opposites for two numbers works both ways. For example, the number 3 is opposite to the number -3 - and therefore the number -3 is opposite only to the number 3, and not to any other.
- A number cannot have two opposites - there is always only one.
- Opposite each other can be numbers with different signs. If the number is positive, then its opposite number will be with a minus sign - for example, 5 and -5. The same thing works in the opposite direction - for a number with a minus sign, the opposite will always be that with a plus sign - for example, -6 and 6.
- Two opposite numbers have the same absolute value, or modulus. In other words, if for the number 4
§ 1 The concept of a positive number
In this lesson, you will learn what numbers are called opposites, how to find the opposite number, and what are integer and rational numbers.
Let's start with practical work. On the coordinate line, mark the points A(2) and B(-2). They are symmetrical and the center of symmetry of these points is the origin O(0), since the distance OA=OB.
We see that the coordinates of points that are symmetrical about the origin are numbers that differ only in sign. Such numbers are called opposites.
There is another definition of opposite numbers. What are the modules of numbers 2 and -2? Equal to 2. Therefore, opposite numbers are numbers that have the same modules, but differ in sign.
To indicate the number opposite to a given number, use the minus sign, which is written in front of the given number. That is, the opposite of a is written as −a. For example, the number 0.24 is opposite to the number −0.24, the number -25 is opposite to the number −(−25), but the number -25 on the coordinate line is opposite to 25, which means -(-25) = 25. It follows from this that -( -a) = a and a = -(-a).
§ 2 Properties of opposite numbers
Let's single out some properties of opposite numbers.
The number opposite to a positive number is negative, and the number opposite to a negative number is positive. This is understandable, since the points of the coordinate line corresponding to opposite numbers are on opposite sides of the origin.
If the number a is opposite to the number b, then b is opposite to a - this follows from the symmetry property of points on the coordinate line.
Let's look at the coordinate line. How many points can be marked on a coordinate line that are symmetrical to the given one with respect to the origin? Only one. This means that for each number there is only one opposite number.
Only one number is opposite to itself - this is the number 0, since 0 \u003d -0 (therefore, it is not customary to write -0).
Numbers with a common feature form a set (or group), each set has its own name.
Recall that the numbers that we use in counting are called natural numbers, they form a set of natural numbers.
Every natural number has its opposite number. Natural numbers, their opposite numbers, and the number 0 are called integers.
Fractional numbers can also be positive or negative. All integers and all fractions are called rational numbers. They also say that together they form the set of rational numbers.
Let's single out two more groups of numbers. Let's take a coordinate line. If we remove the part of the straight line on which the negative numbers are located, there will be a ray with positive numbers and the reference number 0. The remaining numbers are called non-negative, that is, numbers that are greater than or equal to 0. Therefore, non-positive numbers are all negative numbers and the number 0, that is, numbers that are less than or equal to 0.
Today we learned what opposite, integer, rational, non-negative, non-positive numbers are, we learned how to find the number opposite to a given one.
List of used literature:
- Mathematics.6th grade: lesson plans for the textbook by I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilin. Mnemosyne 2009
- Mathematics. Grade 6: a textbook for students of educational institutions. I.I. Zubareva, A.G. Mordkovich.- M.: Mnemozina, 2013
- Mathematics. Grade 6: a textbook for students of educational institutions. /N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. – M.: Mnemosyne, 2013
- Mathematics Handbook - http://lyudmilanik.com.ua
- Handbook for students in secondary school http://shkolo.ru
Let's consider such an example. It is necessary to sequentially calculate: .
You can rearrange the numbers to be added, and then subtract the remaining ones: .
But this is not always convenient. For example, we can calculate the balance of things in some warehouse and we need to know the intermediate result.
You can perform actions in a row: .
We know that , which means that the result will be a subtraction from the number . This means that it is necessary to subtract, but not yet from anything. When there is something to subtract from, subtract:
But we can "cheat" and designate . Thus, we will introduce a new object - negative numbers.
We have already performed such an operation - in nature, for example, the number "" also did not exist, but we introduced such an object in order to facilitate the recording of actions.
Imagine that we were instructed to issue and receive balls in a sports warehouse. We need to keep records. You can write in words:
Issued , Accepted , Issued , Accepted , ... (See Fig. 1.)
Rice. 1. Accounting
Agree, if you need to issue and receive many times a day, then the recording is not very convenient.
You can divide the sheet into two columns, one - Accepted, the other - Issued. (See Figure 2.)
Rice. 2. Simplified notation
The entry got shorter. But here's the problem: how to understand how many balls were taken (or given away) at any particular moment in time?
The following consideration can be used for recording: when we issue balls from the warehouse, their number in the warehouse decreases, and when we receive, it increases.
But how to write "gave out the ball"? You can enter such an object: .
This object allows us to mathematically record the movement of the balls in the order in which they happened: 
Let's consider one more example.
On the account of your phone rubles. You went online, and it cost roubles. It turned out a debt of rubles. The operator could write down like this: "the client owes rubles." You have put rubles. The operator deducted the debt. It turned out on the account of rubles.
But it is convenient to record both transactions and money in the account using the signs "" and "". (See Figure 3.)
Rice. 3. Convenient recording
We enter a negative number to write down the result of subtracting a larger number from a smaller one: .
Adding a negative number is the same as subtracting: .
In order to distinguish negative numbers from the positive numbers that we dealt with earlier, we agreed to put a minus sign in front of it: .
Could you do without them? Yes, you can. In each specific situation, we would use the words “back”, “in debt”, and so on. But they, these words, would be different.
And so we have a universal convenient tool. One for all such cases.
We can draw an analogy with a car. It consists of a large number of parts, many of which are not needed individually, but together they allow you to ride. Similarly, negative numbers are a tool that, together with other mathematical tools, makes it easier to calculate and simplify the solution and recording of many problems.
So, we have introduced a new object - negative numbers. What are they used for in life?
First, let's recall the roles of positive numbers:
Quantity: e.g. wood, liters of milk. (See Figure 4.)
Rice. 4. Quantity
Ordering: For example, houses are numbered with positive numbers. (See Figure 5.)
Rice. 5. Ordering
Name: e.g. player number. (See Figure 6.)
Rice. 6. Number as a name
Now let's look at the functions of negative numbers:
Designation of the missing quantity. The number is not negative. But a negative number is used to show that the amount is being subtracted. For example, we can pour out of a bottle and write it as . (See Figure 7.)
Rice. 7. Designation of the missing quantity
Ordering. Sometimes zero is selected during numbering and you need to number objects on both sides of zero. For example, the floors located below the -th, in the basement. (See Figure 8.) Or a temperature that is below the selected zero. (See Figure 9.)
Rice. 8. Floor below th, in the basement
Rice. 9. Negative numbers on the thermometer scale
But still, the main purpose of negative numbers is a tool for simplifying mathematical calculations.
But for negative numbers to become such a handy tool, you need to:
A negative temperature is one that is below zero, below zero temperature. But what is zero temperature? To measure, record the temperature, you need to select the unit of measurement and the reference point. Both are an agreement. We use the Celsius scale named after the scientist who proposed it. (See Figure 10.)
Rice. 10. Anders Celsius
Here, the freezing point of water is chosen as the reference point. Anything below is indicated by a negative value. (See Figure 11.)
Rice. eleven.
But it is clear that if we take another reference point, another zero, then the negative temperature in Celsius can be positive in this other scale. And so it happens. In physics, the Kelvin scale is widely used. It is similar to the Celsius scale, only the value of the lowest possible temperature is chosen as zero (there is no lower). This value is called "absolute zero". In Celsius, this is approximately. (See Figure 12.)
Rice. 12. Two scales
That is, there are no negative values in the Kelvin scale at all.
Yes, our summer 
.
And frosty 
.
That is, a negative temperature is a convention, an agreement of people to call it that.
Let's start from scratch. Zero occupies a special position among numbers.
As we have already discussed, for our convenience, we can designate the subtraction of seven as a negative number. Since it means subtraction, we leave the sign "" as its sign. Let's call a new number.
That is, "" is a number that adds up to zero: . And in any order. This is the definition of a negative (or opposite) number.
For each number that we studied before, we introduce a new number, negative, whose sign is a minus sign in front of it. That is, for each previous number, its negative twin appeared. Such twins are called opposite numbers. (See Figure 13.)
Rice. 13. Opposite numbers
So, definition: two numbers are called opposite numbers, the sum of which is equal to zero.
Outwardly, they differ only in the sign "".
If a variable is preceded by the sign "", for example, what does this mean? This does not mean that this value is negative. The minus sign means that this value is opposite to the number: . Which of these numbers is positive, which is negative, we do not know.
If , then .
If (negative number), then (positive number).
What is the opposite of zero? We already know this.
If zero is added to any number, including zero, then the original number will not change. That is, the sum of two zeros is equal to zero: . But numbers whose sum is zero are opposite. Thus, zero is the opposite of itself.
So, we have given the definition of negative numbers, found out why they are needed.
Now let's spend some time on technology. For now, we need to learn how to find its opposite for any number:
In the last part of the lesson, we will talk about the new names and designations of sets that appear after the introduction of negative numbers.