Subject
Lesson type
- studying and primary assimilation of new material
Lesson Objectives
Learn the definitions of positive, negative and opposite numbers.
Find opposite numbers when solving exercises, when solving equations
Developmental – to develop students’ attention, perseverance, perseverance, logical thinking, mathematical speech.
Educational - through the lesson, cultivate an attentive attitude towards each other, instill the ability to listen to comrades, mutual assistance, and independence.
Lesson Objectives
Find out what opposite numbers are
Learn to use this concept when solving problems
Test students' problem-solving skills.
Lesson Plan
1. Introduction.
2. Theoretical part
3. Practical part.
4. Homework.
Introduction
Look at the pictures and describe in one word what is different about them.
The pictures show opposites.
- these are two numbers equal in absolute value, but having different signs, eg. 5 and -5.
Theoretical part
First, let's remember what it is negative numbers. Look video:
Points with coordinates 5 and -5 are equally distant from point O and are located on opposite sides of it. To get from point O to these points you need to 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 by signs are called opposite numbers.
For example, opposite numbers would be 35 and -35, since the number 35 = +35, which means that the numbers 35 and -35 differ only in signs. Opposite numbers will also be 0.8 and -0.8, ¾ and -¾.
Properties of opposite numbers
1). For every number there is only one opposite number.
2). The number 0 is the opposite of itself.
3). The opposite number of a is denoted -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 notation "-(-15)" means the opposite number of -15. Since the opposite of -15 is 15, then -(-15) = 15. In general -(-a) = a.
The natural numbers, their opposites and zero are called integers.
Opposite number n" in relation to the number n is a 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
Thus, opposite numbers have the same modules, but opposite signs.
Accordingly, the opposite number of n is denoted − n. When a number is positive, its opposite number will be negative, and vice versa.
1. Give examples of opposite numbers.
2. Draw them on a coordinate line.
3. Name the number opposite -3.6; 7; 0; 8/9; -1/2
Practical part
Example
1) Mark on the coordinate line points A(2), B(-2), C(+4), D(-3), E(-5.2), F(5.2), G(-6) , 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 symmetrical with respect to point O(0): A(2) and B(-2), E(- 5.2) and F(5.2)
Coordinates of symmetrical points are numbers that differ only in sign. Such numbers are called opposite.
Mark the points A(-3), B(+6), C(+4.2), D(+3), E(-4.2), F(-6) on the coordinate line. What can you say about these numbers? ?
Of the numbers 15; 2.5; – 2.5; - 18; 0; 45; – 45 select: a) integers; b) integers; c) negative numbers; d) positive numbers; d) opposite numbers.
1) Write down the opposite number of a.
2) Indicate the number opposite to number a if:
a=5, a=-3, a=0, a=-2/5;
A = 6, -a = - 2, -a = 3.4.
1) Remember what the entry means: - (- a).
2) Place a number instead of * to obtain the correct equality: a) - (- 5) = *; b) 3 = – *.
Homework
1). Fill out 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 the outstanding French scientist:
Do you know where in Everyday life do we encounter positive and negative numbers?
List of sources used
1. Mathematical encyclopedia (in 5 volumes). - M.: Soviet Encyclopedia, 2002. - T. 1.
2. “The newest schoolchildren’s reference book” “HOUSE XXI century” 2008
3. Lesson summary on the topic " Opposite numbers" Author: Petrova V.P., mathematics teacher (grades 5-9), Kiev
4. N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school
Let's consider this example. You need to count sequentially: .
You can rearrange the numbers that need 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, therefore, the result will be a subtraction from the number. This means that we need to subtract , but not from anything yet. When we have something to subtract from, we subtract:
But we can “cheat” and designate . Thus we will introduce 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 to make it easier to record actions.
Imagine that at a sports warehouse we were tasked with issuing and receiving balls. 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 recording is not very convenient.
You can divide the sheet into two columns, one - Accepted, the other - Issued. (See Figure 2.)
Rice. 2. Simplified recording
The recording has become shorter. But here's the problem: how to understand how many balls were taken (or given away) at any particular moment in time?
You can use the following consideration for recording: when we issue balls from the warehouse, their quantity in the warehouse decreases, and when we accept them, it increases.
But how to write “gave the ball out”? You can enter the following object: .
This object allows us to do mathematical notation the movements of the balls in the order they happened: 
Let's look at another example.
There are rubles in your phone account. You went online and it cost rubles. The result was a debt of rubles. The operator could have written down: “the client owes rubles.” You put in 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 the result of subtracting a larger number from a smaller number: .
Adding a negative number is equivalent to subtracting: .
In order to distinguish negative numbers from the positive numbers with which we dealt earlier, we agreed to put a minus sign in front of it: .
Could you do without them? Yes you can. In any given situation, we would use the words “back”, “borrow” 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 large quantity parts, many of which are not needed individually, but all together allow you to drive. Likewise, negative numbers are a tool that, together with other mathematical tools, makes it easier to calculate and simplify the solution and writing of many problems.
So, we have introduced a new object - negative numbers. What are they used for in life?
First, let's remember the roles of positive numbers:
Quantity: for example wood, liter of milk. (See Figure 4.)
Rice. 4. Quantity
Ordering: For example, houses are numbered with positive numbers. (See Figure 5.)
Rice. 5. Organize
Name: for example, football player number. (See Figure 6.)
Rice. 6. Number as a name
Now let's look at the functions negative numbers:
Indication of the missing quantity. Quantity is never negative. But a negative number is used to show that a quantity is being subtracted. For example, we can pour from a bottle and write it as . (See Figure 7.)
Rice. 7. Indication of missing quantity
Arranging. Sometimes, when numbering, zero is selected 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 located below the th, in the basement
Rice. 9. Negative numbers on the thermometer scale
But still, the main purpose of negative numbers is as a tool to simplify mathematical calculations.
But for negative numbers to become like this convenient tool, need to:
A negative temperature is one that is below zero, below zero temperature. But what is zero temperature? To measure and record temperature, you need to select a unit of measurement and a reference point. Both are agreements. We use the Celsius scale after the scientist who proposed it. (See Fig. 10.)
Rice. 10. Anders Celsius
The freezing point of water is chosen as the reference point here. 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 negative temperature Celsius can be positive on this other scale. This is what happens. The Kelvin scale is widely used in physics. It is similar to the Celsius scale, only the value of the lowest possible temperature is selected as zero (it cannot be 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.
So, our summer 
.
And the frosty ones 
.
That is, negative temperature is a convention, an agreement among 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 denote the subtraction of seven as a negative number. Since it means subtraction, we leave the “” sign as its sign. Let's name 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 earlier, we will introduce a new number, negative, the sign of which is the minus sign in front of it. That is, for each previous number its negative twin appeared. We call such twins opposite numbers. (See Figure 13.)
Rice. 13. Opposite numbers
So, the definition: opposite numbers are two numbers whose sum is equal to zero.
Externally, they differ only in the “” sign.
If a variable is preceded by a "" sign, for example, what does that mean? This does not mean that this value is negative. The minus sign means that this value is the opposite of the number: . We don’t know which of these numbers is positive and which is negative.
If, then.
If (negative number), then (positive number).
What number is opposite to 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 zero: . But numbers whose sum is zero are opposites. Thus, zero is opposite to itself.
So, we have given the definition of negative numbers and found out why they are needed.
Now let's spend a little 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 new names and notations for sets that appear after the introduction of negative numbers.
In this article we will explore opposite numbers. Here we will answer the question of what numbers are called opposites, show how the opposite of a given number is designated, and give examples. We will also list the main results characteristic of opposite numbers.
Page navigation.
Determining opposite numbers
It will help us to get an idea of opposite numbers.
Let us mark some point M on the coordinate line, different from the origin. We can get to point M by sequentially laying off a unit segment, as well as its tenth, hundredth, and so on, from the origin in the direction of point M. If we plot the same number of unit segments and its shares in the opposite direction, then we will get to another point, denoted by the letter N. Let's give an example to illustrate our actions (see figure below). To get to point M on the coordinate line, we laid off two unit segments and 4 segments, constituting a tenth of a unit, in the negative direction. Now let's put two unit segments and 4 segments, constituting a tenth of a unit, in the positive direction. This will give us point N.
We are almost ready to understand the definition of opposite numbers; all that remains is to discuss a couple of nuances.
We know that each point on the coordinate line corresponds to a single real number, therefore, both point M and point N correspond to some real numbers. So the numbers corresponding to points M and N are called opposite.
Separately, it is necessary to say about point O - the origin. Point O corresponds to the number 0. The number zero is considered to be the opposite of itself.
Now we can voice determining opposite numbers.
Definition.
Two numbers are called opposite if the points on the coordinate line corresponding to these numbers can be reached by laying off the same number of unit segments from the origin in opposite directions, as well as fractions of a unit segment, the number 0 is opposite to itself.
Notation of opposite numbers and examples
It's time to enter symbols of opposite numbers.
To indicate the opposite of a given number, use the minus sign, which is written in front of the given number. That is, the number opposite to the number a is written as −a. For example, the opposite number 0.24 is −0.24, and the opposite number −25 is −(−25).
Let's give examples of opposite numbers. The pair of numbers 17 and −17 (or −17 and 17) is an example of opposite integers. The numbers and are opposite rational numbers. Other examples of the opposite rational numbers are the pairs of numbers 5.126 and −5.126. as well as 0,(1201) and −0,(1201) . It remains to give a few examples of the opposite
5 and -5 (Fig. 61) are equally distant from point O and are located on opposite sides of it. To get from point O to these points, you need to 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, opposite numbers would be 8 and -8, since the number 8 = + 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 o is denoted -a. If a = -7.8, then -a = 7.8; if a = 8.3, then - a = -8.3; if a = 0, then -a = 0. The entry “- (-15)” means the number opposite to the number -15. Since the opposite number of -15 is 15, then -(- 15) = 15. In general - (- a) = a.
The natural numbers, their opposites and zero are called integers.
? What numbers are called opposites?
Number b is opposite to number a. What number is the opposite of b?
What number is opposite to zero?
Is there a number that has two opposite numbers?
What numbers are called integers?
TO 910. Find the opposite numbers:
911. Substitute a number to get the correct equality:
912. Find the meaning 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 blanks 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 conventionally:
919. Between what integers on the coordinate line is the number located: 2.6; -thirty; -6; -8
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 unit line segment the length of 4 notebook cells, and mark the point on this straight line, F (2,25).
A 922. Mark on the “time line” the following events from the history of mathematics:
a) The book “Elements” was written by Euclid in the 3rd century. BC e.
b) Number theory originated in Ancient Greece in the 6th century BC e.
V) Decimals 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 to 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 inverse numbers:
924. Vitya bought 2.4 kg of carrots. How many carrots bought Kolya, if you know what he bought:
a) 0.7 kg more than Viti; f) what Vitya bought;
b) 0.9 kg less than Viti; 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 had to produce 270 thousand bricks for the construction of the Palace of Culture. First
week he produced the tasks, in the second week he produced 10% more than in the first week. How many thousand bricks does the plant have left 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 - 10% less than on the first day, and on the third day - the rest of the grain. How many tons of grain did the collective farm sell on the third day?
926. Notes differ in the duration of their sound. The sign denotes a whole note, a note half as long - a half note, a sixteenth note.
Check for equality of durations:
D 927. What numbers are opposite numbers:
928. Write down all the natural numbers less than 5 and their opposites.
929. Find the value:
930. On the second day, 2 times more wire was released 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 issued in these three days, if on the first day they were issued 30 kg less than on the third?
931. On the collective farm, on irrigated lands, 60.8 centners of wheat were collected per hectare. Replacing an old wheat variety with a new one gives a 25% increase in yield. How much wheat does the collective farm now collect from 23 hectares of irrigated field?
932. Make up an equation for each diagram and solve it:
933. Find the meaning of the expression:
N.Ya.Vilenkin, A.S. Chesnokov, S.I. Shvartsburd, V.I. Zhokhov, Mathematics for grade 6, Textbook for high school