Goals and objectives of the lesson:
- General lesson on mathematics in 6th grade "Addition and Subtraction positive and negative numbers"
- Summarize and systematize students’ knowledge on this topic.
- Develop subject and general academic skills and abilities, the ability to use acquired knowledge to achieve a goal; establish patterns of diversity of connections to achieve a level of systematic knowledge.
- Developing self-control and mutual control skills; develop desires and needs to generalize the facts received; develop independence and interest in the subject.
During the classes
Guys, we are traveling through the country of “Rational Numbers”, where positive, negative numbers and zero live. While traveling, we learn a lot of interesting things about them, get acquainted with the rules and laws by which they live. This means that we must follow these rules and obey their laws.
What rules and laws have we become familiar with? (rules for adding and subtracting rational numbers, laws of addition)
And so the topic of our lesson is “Adding and subtracting positive and negative numbers.”(Students write down the date and topic of the lesson in their notebooks)
II. Examination homework
III. Updating knowledge.
Let's start the lesson with oral work. There is a series of numbers in front of you.
8,6; 21,8; -0,5; 6,6; 4,7; 7; -18; 0.
Answer the questions:
Which number in the series is the largest?
What number has the greatest modulus?
Which number is the smallest in the series?
What number has the smallest modulus?
How to compare two positive numbers?
How to compare two negative numbers?
How to compare numbers with different signs?
Which numbers in the series are opposites?
List the numbers in ascending order.
IV. Find the mistake
a) -47 + 25+ (-18)= 30
c) - 7.2+(- 3.5) + 10.6= - 0.1
d) - 7.2+ (- 2.9) + 7.2= 2.4
V.Task “Guess the word”
In each group, I distributed tasks in which words were encrypted.
After completing all the tasks, you will guess the key words(flowers, gift, girls)
| 1 row | Answer | Letter |
|
| Answer | Letter |
||
| 54-(-74) | |||
| 2,5-3,6 | |||
| 23,7+23,7 | |||
| 11,2+10,3 | |||
| 3rd row | Answer | Letter |
|
| 2,03-7,99 | |||
| 67,34-45,08 | |||
| 10,02 | 112,42 | 50,94 | 50,4 |
VI. Fizminutka
Well done, you have worked hard, I think it’s time to relax, concentrate, relieve fatigue, and restore peace of mind with simple exercises
PHYSICAL MINUTE (If the statement is correct, clap your hands; if not, shake your head from side to side):
When adding two negative numbers, the modules of the terms must be subtracted -
The sums of two negative numbers are always negative +
When adding two opposite numbers, the result is always 0 +
When adding numbers with different signs, you need to add their modules -
The sum of two negative numbers is always less than each of the terms +
When adding numbers with different signs, you need to subtract the smaller module from the larger module +
VII.Solving tasks according to the textbook.
No. 1096(a,d,i)
VIII. Homework
Level 1 “3”-No. 1132
Level 2 - “4” - No. 1139, 1146
IX. Independent work according to options.
Level 1, "3"
| 1 option | Option 2 |
Level 2, “4”
| 1 option | Option 2 |
| 1 - (- 3 )+(- 2 ) |
Level 3, "5"
| 1 option | 2nd option |
| 4,2-3,25-(-0,6) | 2,4-1,75-(-2,6) |
Mutual check on the board, change desk neighbors
X. Summing up the lesson. Reflection
Let's remember the beginning of our lesson, guys.
What lesson goals did we set for ourselves?
Do you think we managed to achieve our goals?
Guys, now evaluate your work in class. In front of you is a card with a picture of a mountain. If you think you did a good job in class, you'll be fine.Obviously, then draw yourself on the top of the mountain. If anything is unclear, draw yourself below, and decide for yourself on the left or right.
Give me your drawings along with a score card, you will learn the final grade for your work in the next lesson.
In this article we will talk about adding negative numbers. First we give the rule for adding negative numbers and prove it. After that we'll sort it out typical examples adding negative numbers.
Page navigation.
Rule for adding negative numbers
Before formulating the rule for adding negative numbers, let us turn to the material in the article: positive and negative numbers. There we mentioned that negative numbers can be perceived as debt, and in this case determines the amount of this debt. Therefore, the addition of two negative numbers is the addition of two debts.
This conclusion allows us to understand rule for adding negative numbers. To add two negative numbers, you need:
- fold their modules;
- put a minus sign in front of the received amount.
Let's write down the rule for adding negative numbers −a and −b in letter form: (−a)+(−b)=−(a+b).
It is clear that the stated rule reduces the addition of negative numbers to the addition of positive numbers (the modulus of a negative number is a positive number). It is also clear that the result of adding two negative numbers is a negative number, as evidenced by the minus sign that is placed in front of the sum of the modules.
The rule for adding negative numbers can be proven based on properties of actions with real numbers (or the same properties of operations with rational or integer numbers). To do this, it is enough to show that the difference between the left and right sides of the equality (−a)+(−b)=−(a+b) is equal to zero.
Since subtracting a number is the same as adding the opposite number (see the rule for subtracting integers), then (−a)+(−b)−(−(a+b))=(−a)+(−b)+(a+b). Due to the commutative and associative properties of addition, we have (−a)+(−b)+(a+b)=(−a+a)+(−b+b). Since the sum of opposite numbers is equal to zero, then (−a+a)+(−b+b)=0+0, and 0+0=0 due to the property of adding a number with zero. This proves the equality (−a)+(−b)=−(a+b) , and hence the rule for adding negative numbers.
All that remains is to learn how to apply the rule of adding negative numbers in practice, which we will do in the next paragraph.
Examples of adding negative numbers
Let's sort it out examples of adding negative numbers. Let's start with the simplest case - the addition of negative integers; we will carry out the addition according to the rule discussed in the previous paragraph.
Example.
Add the negative numbers −304 and −18,007.
Solution.
Let's follow all the steps of the rule for adding negative numbers.
First we find the modules of the numbers being added: and 
. Now you need to add the resulting numbers; here it is convenient to perform column addition: 
Now we put a minus sign in front of the resulting number, as a result we have −18,311.
Let's write the entire solution in a short form: (−304)+(−18,007)= −(304+18,007)=−18,311.
Answer:
−18 311 .
The addition of negative rational numbers, depending on the numbers themselves, can be reduced either to the addition of natural numbers, or to the addition of ordinary fractions, or to the addition of decimal fractions.
Example.
Add a negative number and a negative number −4,(12) .
Solution.
According to the rule for adding negative numbers, you first need to calculate the sum of the modules. The modules of the negative numbers being added are equal to 2/5 and 4, (12) respectively. The addition of the resulting numbers can be reduced to the addition of ordinary fractions. To do this, we convert the periodic decimal fraction into an ordinary fraction: . Thus, 2/5+4,(12)=2/5+136/33. Now let's do it
REPINA KSENYA
an algorithm for adding and subtracting positive and negative numbers is given with examples and illustrations, independent tasks are given with subsequent verification.
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ADDING AND SUBTRACTING POSITIVE AND NEGATIVE NUMBERS Taisiya Alekseevna Ostrovskaya Mathematics teacher at Lyceum No. 15, student Repina Ksenia
About the general rule for adding and subtracting rational numbers.
DO YOU KNOW? 1. What is a positive and what is a negative number? 2. How are they located on the number line? 3. How to compare positive and negative numbers?
CHECK YOURSELF! Write down all positive and all negative numbers: - 7; 9.2; - 10.5; 73; - 55.99; - 0.056; 123; 41.9; - 0.4 Arrange them in ascending order. Arrange them in descending order.
ANSWERS: 9.2; 73; 123; 41.9; (+) -7; -10.5; - 55.99; - 0.056; - 0.4. (-) In ascending order: - 55.99; -10.5;-7;-0.4; - 0.056; 9, 2; 41.9;73; 123; In descending order: 123;73; 41.9;9.2; - 0.056; - 0.4;-7; - 10.5; -55.99.
Rules. 1. Numbers less than zero are called negative. And put a (-) sign. Numbers greater than zero are called positive. And put a (+) sign. The number 0 (zero) is neither a positive nor a negative number. │0│= 0; 2. The distance from the point representing the number to 0 is called the MODULE of the number and is always positive, like any distance. The module is designated by two dashes: │5│= 5; │-5│= 5; The moduli of opposite numbers are EQUAL: │-6│=│6 │The modulus of a positive number is equal to the number itself. │5│ = │5│
Rules. 3. The larger the number, the further to the right it lies on the number axis. 4. Of two negative numbers, the one with the smaller modulus is greater. 5. Numbers that have the same modules, but differ in sign, are called opposite.
ADDING NEGATIVE NUMBERS 1. To add negative numbers, you need: a). Put the immediately known result sign - “minus”; b). Add the modules of numbers: (- 3.5) + (- 4.8) = - (3.5 + 4.8) = - 8.3 Solve for yourself: (- 6.7) + (- 23.3) = ? (- 75.6) + (- 5.7) = ? (- 46.2) + (- 55) = ? 2. What happens if you add numbers with different signs? 6 + (- 2) = ... ; 1 + (- 3) = ... ?
Problem During heavy rain, 12 people stood at a bus stop. A bus pulled up and splashed mud on the five of them. The rest managed to jump into the thorny bushes. How many scratched passengers will travel on the bus if it is known that three were never able to get out of the thorny bushes?
When adding numbers with different signs, the sign of the result coincides with the sign of the number whose modulus is greater, and the answer itself is determined by the action of subtraction. Explain how the examples were solved: (- 17) + 7 = - (17 – 7) = - 10 12 + (- 20) = - (20 -12) = - 8 And now, using the rule, write down in detail the solutions of the following examples: 1). (-3) + 5 =... ; 2). 7 + (- 4) = … ; 3). (-10) + 3 = … ; 4). (-22) + 33 = … ; 5). (5) + (-9) = … ; 6). (1.7) + (- 3.9) = ... ; 7). 17 + (- 40) = ...?
CHECK YOUR DECISIONS! 1). 2 2). 3 3). - 7 4). 11 5). -4 6). - 2.2 7). - 23
PROBLEM During a game of hide and seek, 5 boys hid in a lime barrel, 7 in a green paint barrel, 4 in a red paint barrel and nine in a coal box. The boy who went to look for them accidentally fell into a barrel of yellow paint. How many colorful boys and how many black and white boys played hide and seek?
ADDITION ALGORITHM. YOU NEED TO REMEMBER: NUMBERS “are friends”? (SIGNS ARE THE SAME) Are the numbers “quarreling”? (DIFFERENT SIGNS) Put the same sign on the result and add the modules of numbers. 4 + 5=9 - 4 +(-5) = - 9 Solve the examples: 5 + 8 = …; (- 5) + (- 11) = ... (- 8.1) + (- 0.7) = ... (-2) + (-8) = ... (-49) + (-13) = ... Put a “winner” sign on the result and subtract the smaller one from the larger module. 3 +(-8) = - (8 -3)= -5 6 + (-4) = + (6-4) = 2 Solve the examples: (-2) + (8) = …; 3.5 +(-10) =... 18 + (-5.7) =... (-11) + 5 =...
SUBTRACTING RATIONAL NUMBERS. Subtraction can be replaced by addition with the Number opposite to the one being subtracted: 9 – (-3) = 9 + (+3) = 9 +3=12 We replaced subtraction with addition with the opposite number. Briefly it can be written as follows: 9 – (- 3) = 9 + 3 = 12; Two minuses before the number turned into a plus: -(- 3) = + 3 Let’s practice: 2 – (- 7) =... - 10 – (- 15 = - 10 + 15 = 15 – 10 = 5;- - 25 – (- 4) = - 25 + 4 = - 21
If a number is preceded by two identical signs (- -) or (+ +), then they change to (+). 3 – (-7) = 3 +7 = 10 12 – (+ 8) = 12 – 8 = … (-9) – (-5) =…. 6 + (- 10) = 6 – 10 = … 15 + (+10) = …. It can be seen that if a number is preceded by 2 different signs(+ -) or (- +), then they are replaced by minus (-)!
Check your solution 1. …. = 10 4. …. = - 4 2. …. = 4 5. …. = + 25 3. …. = - 4 CORRECT! Well done!
PROBLEM One grandfather was hunting cockroaches in the kitchen and killed five and wounded three times as many. Grandfather mortally wounded three cockroaches, and they died from their wounds, and the rest of the wounded cockroaches recovered, but were offended by grandfather and went to their neighbors forever. How many cockroaches have gone to their neighbors forever?
SOLVE THE EXAMPLES YOURSELF: 21 + (- 8) =…; -10 + (- 16) =…; - 7 – (-15) = …; 3 – (- 11) =... ; - 32 – (- 22) = …; 16 – (+ 5) = … ; 5 – (+ 15) = … ; 2 – (- 9) = … ; - 13 + (- 18) = ... ; - 49 + (- 10) = ... ; - 15 – (- 21) = … ; 6 – (+ 10) = … ;
Check your answers 1. = 13 2. = -26 3. = 8 4. = 14 5. = -10 6. = 11 Correct solution! 7. = 10 8. = 11 9. = 31 10. = -59 11. = 6 12. = -4 WELL DONE!
Let's complicate the problem and try to solve it long examples, using the same rules: 5 – (- 8)+ (-12) – (+ 5) +17 – 10 – (- 2) = = 5 +8 -12 – 5 + 17- 10 + 2= (8+ 17+2) + (-12-10)= = 27 + (- 22) 27 -22 = 5 Remember the calculation algorithm: Let’s discard the parentheses using the rule for converting “cat-dog” signs; The result is an algebraic sum. It is possible to mutually cancel the terms +5 and -5 that are opposite in sign; Let's group the (+) and (-) terms separately; Let's find the result.
PROBLEM Let's say that you decided to jump into water from a height of 8 meters and, after flying 5 meters, changed your mind. How many more meters will you have to fly against your will?
SUBTRACTION
Mathematics, 6th grade
(N.Ya.Vilenkin)
mathematics teacher of municipal educational institution "Upshinskaya basic"
comprehensive school" Orsha district of the Republic of Mari El
The meaning of subtraction
Task. A pedestrian walked 9 km in 2 hours. How many kilometers did he walk in the first hour if his distance in the second hour is 4 km?
In this problem the number 9 - amount two terms, one of which is equal 4 , and the other is unknown.
An action that uses the sum and one of the terms to find another term is called by subtraction.
The meaning of subtraction
Since 5 + 4 = 9,
then the required term is equal to 5.
They write 9 – 4 = 5
9 – 4 = 5
difference
subtrahend
minuend
The meaning of subtraction
– 5 + 14 = 9
9 – 14 = ?
? + 14 = 9
9 – 14 = –5
– 9 – 14 = ?
– 23 + 14 = –9
? + 14 = –9
– 9 – 14 = – 23
The meaning of subtraction
Subtracting negative numbers has the same meaning: The action by which the sum and one of the terms is used to find another term is called subtraction.
9 – (–14) = ?
23 + (–14) = 9
? + (–14) = 9
9 – (–14) = 23
Pick up unknown term
– 9 – (–14) = ?
5 + (–14) = –9
? + (–14) = –9
– 9 – (–14) = 5
9 – (–14) = 23
9 – 14 = –5
9 + (–14) = –5
9 + 14 = 23
– 9 – (–14) = 5
– 9 – 14 = – 23
– 9 + (–14) = – 23
– 9 + 14 = 5
Think about how to replace subtraction with addition.
RULE. To subtract another from a given number, you need to add to the minuend the number opposite to the subtracted one.
SUBTRACTION
A – b = a + ( –b )
15 – 18 = 15 + ( –18 ) =
15 – ( –18 ) = 15 + 18 =
SUBTRACTION
Replace subtraction with addition and find the value of the expression:
12 – 20 =
3,4 – 10 =
– 10 – ( –13 ) =
– 1,2 – ( –1,3 ) =
17 – ( –13 ) =
2,3 – ( –3,5 ) =
– 21 – 13 =
– 5,1 – 4,9 =
SUBTRACTION
5 – 10 = 5 + ( – 10 )
RULE. Any expression containing only addition and subtraction signs can be considered as a sum
Name each term in the sum:
5 – 10 + 7 –15 –23 =
– n + y – 9 + b – c – 1 =
CALCULATE:
– 10 + 7 – 15 =
12 – 17 – 11 =
12 + 23 – 41 =
– 2 – 33 + 20 =
24 – 75 + 20 =
6 – 2 –5 RULE. The difference between two numbers is positive if the minuend is greater than the subtrahend. "width="640" 8 – 6 =
2
minuend
subtrahend
difference
– 2 – ( –5 ) =
3
minuend
difference
subtrahend
When is the difference between two numbers positive?
8 6
– 2 –5
RULE. The difference of two numbers is positive if minuend is greater than subtrahend .
10 – 15 =
– 5
minuend
subtrahend
difference
– 8 – ( –6 ) =
– 2
minuend
difference
subtrahend
Compare minuend and subtrahend in the examples.
When is the difference between two numbers negative?
10 15
– 8 –6
RULE. The difference of two numbers is negative if minuend is less than subtrahend .
Think about when the difference of two numbers is 0. Give examples.
0
minuend
difference
subtrahend
Determine the sign of the difference without performing calculations:
– 12 – ( –13 ) =
3,4 – 10 =
15 – ( –11 ) =
2,3 – ( –3,5 ) =
– 5,1 – 4,9 =
– 31 – 23 =
Finding the length of a segment
X
A (–3)
– 3 + x = 4
x = 4 – (–3) = 7
AT 4)
AB - ?
AB = 7 units.
RULE.
Finding the length of a segment
A (–1)
AB = –1 – (–5) = 4 units.
AT 5)
AB - ?
AB = 4 units.
RULE. To find the length of a segment on a coordinate line, you need to subtract the coordinate of its left end from the coordinate of its right end.
Questions for consolidation:
- What does subtracting negative numbers mean?
- How to replace subtraction with addition?
- When is the difference between two numbers positive?
- When is the difference between two numbers negative?
- When is the difference between two numbers equal to zero?
- How to find the length of a segment on a coordinate line?
teacher primary classes MAOU Lyceum No. 21, Ivanovo
A LITTLE HISTORY
Indian mathematicians thought of positive numbers as "property" , and negative numbers are like "debts"
Rules for addition and subtraction as stated by Brahmagupta:
- “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”
Brahmagupta, Indian mathematician and astronomer.
Let's start with simple example. Let's determine what the expression 2-5 is equal to. From point +2 we will put down five divisions, two to zero and three below zero. Let's stop at point -3. That is, 2-5=-3. Now notice that 2-5 is not at all equal to 5-2. If in the case of adding numbers their order does not matter, then in the case of subtraction everything is different. The order of the numbers matters.
Now let's go to negative area scales. Suppose we need to add +5 to -2. (From now on, we will put "+" signs in front of positive numbers and enclose both positive and negative numbers in parentheses so as not to confuse the signs in front of numbers with addition and subtraction signs.) Now our problem can be written as (-2)+ (+5). To solve it, we go up five divisions from point -2 and end up at point +3.
Is there any practical meaning to this task? Of course have. Let's say you have $2 in debt and you earned $5. This way, after you pay off the debt, you will have $3 left.
You can also move down the negative area of the scale. Suppose you need to subtract 5 from -2, or (-2)-(+5). From point -2 on the scale, move down five divisions and end up at point -7. What is the practical meaning of this task? Let's say you owed $2 and had to borrow $5 more. You now owe $7.
We see that with negative numbers we can carry out the same addition and subtraction operations, as with the positive ones.
True, we have not yet mastered all operations. We only added to negative numbers and subtracted only positive ones from negative numbers. What should you do if you need to add negative numbers or subtract negative numbers from negative numbers?
In practice, this is similar to debt transactions. Let's say you were charged $5 in debt, it means the same thing as if you received $5. On the other hand, if I somehow force you to accept responsibility for someone else's $5 debt, that would be the same as taking that $5 away from you. That is, subtracting -5 is the same as adding +5. And adding -5 is the same as subtracting +5.
This allows us to get rid of the subtraction operation. Indeed, “5-2” is the same as (+5)-(+2) or according to our rule (+5)+(-2). In both cases we get the same result. From point +5 on the scale we need to go down two divisions and we get +3. In the case of 5-2 this is obvious, because subtraction is a downward movement.
In the case of (+5)+(-2) this is less obvious. We add a number, which means we move up the scale, but we add a negative number, which means we do the opposite, and these two factors taken together mean that we don't have to move up the scale, but in the opposite direction, that is down.
Thus, we again get the answer +3.
Why, exactly, is it necessary? replace subtraction with addition? Why move up “in the opposite sense”? Isn't it easier to just move down? The reason is that in the case of addition the order of the terms does not matter, but in the case of subtraction it is very important.
We already found out earlier that (+5)-(+2) is not at all the same as (+2)-(+5). In the first case the answer is +3, and in the second -3. On the other hand, (-2)+(+5) and (+5)+(-2) result in +3. Thus, by switching to addition and abandoning subtraction operations, we can avoid random errors associated with rearranging addends.
You can do the same when subtracting a negative. (+5)-(-2) is the same as (+5)+(+2). In both cases we get the answer +7. We start at point +5 and move “down in the opposite direction,” that is, up. We would act in exactly the same way when solving the expression (+5)+(+2).
Students actively use replacing subtraction with addition when they begin to study algebra, and therefore this operation is called "algebraic addition". In fact, this is not entirely fair, since such an operation is obviously arithmetic and not at all algebraic.
This knowledge is unchanged for everyone, so even if you receive education in Austria through www.salls.ru, although studying abroad is valued more highly, you will be able to apply these rules there too.