Lesson Objectives:
Educational:
- formulating rules for multiplying numbers with the same and different signs;
- mastering and improving the skills of multiplying numbers with different signs.
Developing:
- development of mental operations: comparison, generalization, analysis, analogy;
- development of independent work skills;
- expanding the horizons of students.
Educational:
- fostering a culture of record keeping;
- education of responsibility, attention;
- cultivating interest in the subject.
Lesson type: learning new material.
Equipment: computer, multimedia projector, cards for the game "Math Fight", tests, knowledge cards.
Posters on the walls:
- Knowledge is the most excellent possession. Everyone strives for it, but it does not come by itself.
Al-Biruni - I want to get to the bottom of everything...
B. Pasternak
Lesson plan
- Organizational moment (1 min).
- Introductory speech of the teacher (3 min).
- Oral work (10 min).
- Presentation of the material (15 min).
- Math chain (5 min).
- Homework (2 min).
- Test (6 min).
- Summary of the lesson (3 min).
During the classes
I. Organizational moment
student readiness for the lesson.
II. Introductory speech of the teacher
Guys, today we met not in vain, but for fruitful work: gaining knowledge.
Ever since the universe has existed,
There is no such thing, who would not need knowledge.
What we do not take the language and age,
Man has always striven for knowledge...
Rudaki
In the lesson, we will study new material, consolidate it, work independently, evaluate ourselves and our comrades. Everyone has a knowledge record card on the table, in which our lesson is divided into stages. You will enter the points you earn at different stages of the lesson into this card. Let's summarize at the end of the lesson. Put these cards in a conspicuous place.
III. Oral work (in the form of the game "Math fight")
Guys, before starting a new topic, we will repeat what we have learned so far. Everyone has a sheet with the game "Math Fight" on their desks. The vertical and horizontal columns contain the numbers to be added. These numbers are marked with dots. We write the answers in those cells on the field where the points are.
Three minutes to complete. We started work.
And now we exchanged works with a neighbor on our desk and check them with each other. If you think that the answer is wrong, then carefully cross it out and write the correct one next to it. We check.
And now check the answers with the screen ( correct answers are projected on the screen).
For correctly solved
5 tasks put 5 points;
4 tasks - 4 points;
3 tasks - 3 points;
2 tasks - 2 points;
1 task - 1 point.
Well done. They put everything aside. Guys, we will enter the number of points scored for the “Math Battle” into our knowledge records cards ( Appendix 1).
IV. Presentation of the material
Open workbooks. Write down the number, great job.
- What operations on positive and negative numbers do you know?
- How to add two negative numbers?
- How to add two numbers with different signs?
- How to subtract numbers with different signs?
- You always use the word "module". What is the modulus of a number a?
Today's topic of the lesson is also related to the action on numbers of different signs. But she hid in an anagram in which you need to swap the letters and get a familiar word. Let's try to figure it out.
ENOZHEUMNI
Write down the topic of the lesson: "Multiplication".
The purpose of our lesson: to get acquainted with the multiplication of positive and negative numbers and formulate the rules for multiplying numbers with both the same and different signs.
All eyes on the board. Before you is a table with tasks, by solving which we will formulate the rules for multiplying positive and negative numbers.
- 2*3 = 6°С;
- -2 * 3 \u003d -6 ° С;
- –2*(–3) = 6°С;
- 2*(–3) = –6°С;
1. The air temperature rises every hour by 2°C. Now the thermometer shows 0°C ( Annex 2– thermometer) (slide 1 on a computer).
- How much did you get?(6 ° WITH).
- Someone will write down the solution on the board, and we are all in notebooks.
- Let's look at the thermometer, did we get the right answer? (slide 2 on the computer).
2. The air temperature drops every hour by 2°C. Now the thermometer shows 0 ° C (slide 3 on a computer). What temperature will the thermometer show after 3 hours?
- How much did you get?(–6 ° WITH).
- We write the corresponding solution on the blackboard and in notebooks. Analogy with task 1.
- .(slide 4 on a computer).
3. The air temperature drops every hour by 2°C. Now the thermometer shows 0 ° C (slide 5 on a computer).
- How much did you get?(6 ° WITH).
- We write the corresponding solution on the blackboard and in notebooks. Analogy with tasks 1 and 2.
- Compare the result with the reading of the thermometer.(slide 6 on a computer).
4. The air temperature rises every hour by 2°C. Now the thermometer shows 0 ° C (slide 7 on a computer). What air temperature did the thermometer show 3 hours ago?
- How much did you get?(–6 ° WITH).
- We write the corresponding solution on the blackboard and in notebooks. Analogy with tasks 1-3.
- Compare the result with the reading of the thermometer.(slide 8 on a computer).
Look at your results. When multiplying numbers with the same signs (examples 1 and 3), what sign did you get the answer? (positive).
Good. But in example 3, both factors are negative, and the answer is positive. What mathematical concept allows you to move from negative numbers to positive ones? (module).
Attention rule: To multiply two numbers with the same sign, multiply their modulus and put a plus sign in front of the result. (2 people repeat).
Let's go back to example 3. What are the modules (-2) and (-3)? Let's multiply these modules. How much did you get? What sign?
When multiplying numbers with different signs (examples 2 and 4), what sign did you get the answer? (negative).
Formulate your own rule for multiplying numbers with different signs.
Rule: When multiplying numbers with different signs, you need to multiply their modules and put a minus sign in front of the result. (2 people repeat).
Let's go back to examples #2 and #4. What are the modules of their multipliers? Let's multiply these modules. How much did you get? What sign should be put in the result?
Using these two rules, you can also multiply fractions: decimal, mixed, ordinary.
Here are some examples on the board. We will decide three together with me, and the rest on our own. Pay attention to writing and formatting.
Well done. Let's open the textbooks and note the rules that need to be learned for the next lesson (page 190, §7(paragraph 35)). Knowing these rules will help in the future to quickly master the division of positive and negative numbers.
V. Mathematical chain
And now Dunno wants to check how you have learned the new material, and will ask you a few questions. Decisions and answers must be written down in notebooks ( Annex 3- Mathematical chain).
computer presentation
Hello guys. I see you are very smart and inquisitive, so I want to ask you a few questions. Be careful, especially with signs.
My first question is: multiply (-3) by (-13).
Second question: multiply what you got in the first task by (–0,1).
Third question: multiply the result of the second task by (-2).
Fourth question: multiply (-1/3) by the result of the third task.
And the last, fifth question: calculate the freezing point of mercury by multiplying the result of the fourth task by 15.
Thank you for your work. I wish you success.
Guys, let's check how we coped with the tasks. Everyone got up.
How much did you get in the first task?
Whoever has a different answer, sat down, and who sat down, put 0 points in the knowledge record card for the mathematical chain. The rest do nothing.
How much did you get in the second task?
Whoever has a different answer, sat down, and put 1 point on the knowledge record card for the mathematical chain.
How much did you get in the third task?
Whoever has a different answer, sat down, and put 2 points in the knowledge record card for the mathematical chain.
How much did you get in the fourth task?
Whoever has a different answer, sat down, and put ourselves in the knowledge record card for the mathematical chain of 3 points.
How much did you get in the fifth task?
Whoever has a different answer, sat down, and put ourselves in the knowledge record card for the mathematical chain of 4 points. The remaining children solved all 5 tasks correctly. Sit down, you put yourself in the knowledge record card 5 points for the mathematical chain.
What is the freezing point of mercury?(–39 °C).
VI. Homework
§7 (item 35, page 190), No. 1121 - textbook: Mathematics. Grade 6: [N.Ya. Vilenkin and others]
Creative task: Write a multiplication problem for positive and negative numbers.
VII. Test
Let's move on to the next step of the lesson: running the test ( Appendix 4).
You need to solve the tasks and circle the number of the correct answer. For the first two correctly completed tasks you will receive 1 point, for the 3rd task - 2 points, for the 4th task - 3 points. We started work.
Δ -1 point;
o -2 points;
-3 points.
And now we will write the numbers of the correct answers in the table under the test. Let's check the results. You should get the number 1418 in empty cells (writing on the board). Whoever received it puts 7 points in the knowledge record card. Who made mistakes, then puts in the knowledge record card the number of points scored only for correctly completed tasks.
It was 1418 days that the Great Patriotic War lasted, the victory in which the Russian people got at a heavy price. And on May 9, 2010 we will celebrate the 65th anniversary of the Victory over Nazi Germany.
VIII. Lesson summary
And now let's calculate the total number of points you scored for the lesson, and we will enter the results in the student knowledge record card. Then we hand over these cards.
15 - 17 points - score "5";
10 - 14 points - score "4";
less than 10 points - score "3".
Raise your hands, who got "5", "4", "3".
- What topic did we cover today?
- How to multiply numbers with the same sign; with different characters?
So our lesson has come to an end. I want to say THANK YOU for your work in class.
Now let's deal with multiplication and division.
Suppose we need to multiply +3 by -4. How to do it?
Let's consider such a case. Three people got into debt, and each has $4 in debt. What is the total debt? In order to find it, you need to add up all three debts: $4 + $4 + $4 = $12. We have decided that the addition of three numbers 4 is denoted as 3 × 4. Since in this case we are talking about debt, there is a “-” sign in front of 4. We know the total debt is $12, so now our problem is 3x(-4)=-12.
We will get the same result if, according to the condition of the problem, each of the four people has a debt of 3 dollars. In other words, (+4)x(-3)=-12. And since the order of the factors does not matter, we get (-4)x(+3)=-12 and (+4)x(-3)=-12.
Let's summarize the results. When multiplying one positive and one negative number, the result will always be a negative number. The numerical value of the answer will be the same as in the case of positive numbers. Product (+4)x(+3)=+12. The presence of the "-" sign only affects the sign, but does not affect the numerical value.
How do you multiply two negative numbers?
Unfortunately, it is very difficult to come up with a suitable example from life on this topic. It's easy to imagine $3 or $4 in debt, but it's completely impossible to imagine -4 or -3 people getting into debt.
Perhaps we will go the other way. In multiplication, changing the sign of one of the factors changes the sign of the product. If we change the signs of both factors, we must change the signs twice product mark, first from positive to negative, and then vice versa, from negative to positive, that is, the product will have its original sign.
Therefore, it is quite logical, although a bit strange, that (-3)x(-4)=+12.
Sign position when multiplied it changes like this:
- positive number x positive number = positive number;
- negative number x positive number = negative number;
- positive number x negative number = negative number;
- negative number x negative number = positive number.
In other words, multiplying two numbers with the same sign, we get a positive number. Multiplying two numbers with different signs, we get a negative number.
The same rule is true for the action opposite to multiplication - for.
You can easily verify this by running inverse multiplication operations. If in each of the examples above, you multiply the quotient by the divisor, you get the dividend, and make sure it has the same sign, like (-3)x(-4)=(+12).
Since winter is coming, it's time to think about what to change your iron horse into, so as not to slip on the ice and feel confident on winter roads. You can, for example, take Yokohama tires on the website: mvo.ru or some others, the main thing is that it would be of high quality, you can find more information and prices on the website Mvo.ru.
Educational:
- Activity education;
Lesson type
Equipment:
- Projector and computer.
Lesson plan
1. Organizational moment
2. Updating knowledge
3. Mathematical dictation
4.Performing the test
5. Solution of exercises
6. Summary of the lesson
7. Homework.
During the classes
1. Organizing moment
Today we will continue to work on multiplication and division of positive and negative numbers. The task of each of you is to figure out how he mastered this topic, and if necessary, to refine what is still not quite working out. In addition, you will learn a lot of interesting things about the first month of spring - March. (Slide1)
2. Actualization of knowledge.
3x=27; -5x=-45; x:(2,5)=5.
3.Mathematical dictation(slide 6.7)
Option 1
Option 2
4. Test execution ( slide 8)
Answer : Martius
5. Solution of exercises
(Slides 10 to 19)
March 4 -
2) y×(-2.5)=-15
March, 6
3) -50, 4:x=-4, 2
4) -0.25:5×(-260)
March 13
5) -29,12: (-2,08)
March 14th
6) (-6-3.6×2.5)×(-1)
7) -81.6:48×(-10)
March 17
8) 7.15×(-4): (-1.3)
March 22
9) -12.5×50: (-25)
10) 100+(-2,1:0,03)
30th of March
6. Summary of the lesson
7. Homework:
View document content
"Multiplication and division of numbers with different signs"
Lesson topic: “Multiplication and division of numbers with different signs”.
Lesson Objectives: repetition of the studied material on the topic “Multiplication and division of numbers with different signs”, practicing the skills of applying the operations of multiplying and dividing a positive number by a negative number and vice versa, as well as a negative number by a negative number.
Lesson objectives:
Educational:
Fixing the rules on this topic;
Formation of skills and abilities to work with operations of multiplication and division of numbers with different signs.
Developing:
Development of cognitive interest;
Development of logical thinking, memory, attention;
Educational:
Activity education;
Teaching students the skills of independent work;
Education of love for nature, instilling interest in folk signs.
Lesson type. Lesson-repetitions and generalizations.
Equipment:
Projector and computer.
Lesson plan
1. Organizational moment
2. Updating knowledge
3. Mathematical dictation
4.Performing the test
5. Solution of exercises
6. Summary of the lesson
7. Homework.
During the classes
1. Organizing moment
Hello guys! What did we do in previous lessons? (By multiplication and division of rational numbers.)
Today we will continue to work on multiplication and division of positive and negative numbers. The task of each of you is to figure out how he mastered this topic, and if necessary, to refine what is still not quite working out. In addition, you will learn a lot of interesting things about the first month of spring - March. (Slide1)
2. Actualization of knowledge.
Review the rules for multiplying and dividing positive and negative numbers.
Remember the mnemonic rule. (Slide 2)
Perform multiplication: (slide 3)
5×3; 9×(-4); -10×(-8); 36×(-0.1); -20×0.5; -13×(-0.2).
2. Perform division: (slide 4)
48:(-8); -24: (-2); -200:4; -4,9:7; -8,4: (-7); 15:(- 0,3).
3. Solve the equation: (slide 5)
3x=27; -5x=-45; x:(2,5)=5.
3.Mathematical dictation(slide 6.7)
Option 1
Option 2
Students exchange notebooks, check and grade.
4. Test execution ( slide 8)
Once upon a time in Russia, years were counted from March 1, from the beginning of the agricultural spring, from the first spring drop. March was the "beginner" of the year. The name of the month "March" comes from the Romans. They named this month in honor of one of their gods, to find out what kind of god it is, the test will help you.
Answer : Martius
The Romans named one month of the year in honor of Mars, the god of war, called Martius. In Russia, this name was simplified, taking only the first four letters. (Slide 9).
People say: "Mart is unfaithful, now he cries, now he laughs." There are many folk signs associated with March. Some of its days have their own names. Let's now all together we will make a folk calendar for March.
5. Solution of exercises
Students at the blackboard solve examples whose answers are the days of the month. An example appears on the board, and then the day of the month with the name and folk sign.
(Slides 10 to 19)
March 4 - Arkhip. On Arkhip, women were supposed to spend the whole day in the kitchen. The more she prepares any food, the richer the house will be.
2) y×(-2.5)=-15
March, 6- Timothy-spring. If on Timofeev's day there is snow with zadulina, then the harvest is for spring crops.
3) -50, 4:x=-4, 2
4) -0.25:5×(-260)
March 13- Vasily the dropper: drops from the roofs. Nest birds curl, and migratory birds fly from warm places.
5) -29,12: (-2,08)
March 14th- Evdokia (Avdotya-plushcha) - the snow flattens the infusion. The second meeting of spring (the first on Stretenie). What is Evdokia - such is the summer. Evdokia is red - and spring is red; snow on Evdokia - for the harvest.
6) (-6-3.6×2.5)×(-1)
7) -81.6:48×(-10)
March 17- Gerasim the rooker - drove the rooks. Rooks sit on arable land, and if they fly directly to the nests, there will be a friendly spring.
8) 7.15×(-4): (-1.3)
March 22- Magpies - day equals night. Winter ends, spring begins, larks arrive. According to an old custom, larks and waders are baked from dough.
9) -12.5×50: (-25)
10) 100+(-2,1:0,03)
30th of March- Alexey is warm. Water from the mountains, and fish from the camp (from the winter hut). What are the streams on this day (large or small), such is the floodplain (overflow).
6. Summary of the lesson
Guys, did you like today's lesson? What new did you learn today? What did we repeat? I suggest that you prepare the calendar for April yourself. You must find signs of April and make up examples with answers corresponding to the day of the month.
7. Homework: pp. 218 No. 1174, 1179(1) (Slide 20)
This lesson discusses the multiplication and division of rational numbers.
Lesson contentMultiplication of rational numbers
The rules for multiplying integers are also valid for rational numbers. In other words, to multiply rational numbers, you need to be able to
Also, you need to know the basic laws of multiplication, such as: the commutative law of multiplication, the associative law of multiplication, the distributive law of multiplication and multiplication by zero.
Example 1 Find the value of an expression
This is the multiplication of rational numbers with different signs. To multiply rational numbers with different signs, you need to multiply their modules and put a minus before the answer.
To see clearly that we are dealing with numbers that have different signs, we enclose each rational number in brackets along with its signs.
The modulus of a number is , and the modulus of a number is . Having multiplied the received modules as positive fractions, we got the answer, but before the answer we put a minus, as the rule required of us. To ensure this minus before the answer, the multiplication of modules was carried out in brackets, before which the minus is placed.
The short solution looks like this:
Example 2 Find the value of an expression 
Example 3 Find the value of an expression 
This is the multiplication of negative rational numbers. To multiply negative rational numbers, you need to multiply their modules and put a plus in front of the answer.
The solution for this example can be written shorter:
Example 4 Find the value of an expression 
The solution for this example can be written shorter:
Example 5 Find the value of an expression
This is the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus before the received answer
The short solution will look much simpler:
Example 6 Find the value of an expression
Convert the mixed number to an improper fraction. Rewrite the rest as is
We got the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus in front of the received answer. The entry with modules can be omitted so as not to clutter up the expression
The solution for this example can be written shorter
Example 7 Find the value of an expression
This is the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus before the received answer
At first, the answer turned out to be an improper fraction, but we singled out the whole part in it. Note that the integer part has been separated from the fraction modulus. The resulting mixed number was enclosed in brackets preceded by a minus. This is done in order to fulfill the requirement of the rule. And the rule required that the received answer be preceded by a minus sign.
The solution for this example can be written shorter:
Example 8 Find the value of an expression 
First, we multiply and and multiply the resulting number with the remaining number 5. We will skip the entry with modules so as not to clutter up the expression.
Answer: expression value equals −2.
Example 9 Find the value of an expression: 
Convert mixed numbers to improper fractions:
We got the multiplication of negative rational numbers. We multiply the modules of these numbers and put a plus in front of the received answer. The entry with modules can be omitted so as not to clutter up the expression
Example 10 Find the value of an expression
The expression consists of several factors. According to the associative law of multiplication, if an expression consists of several factors, then the product will not depend on the order of operations. This allows us to evaluate the given expression in any order.
We will not reinvent the wheel, but calculate this expression from left to right in the order of the factors. We skip the entry with modules so as not to clutter up the expression
Third action:
Fourth action:
Answer: the value of the expression is
Example 11. Find the value of an expression
Remember the law of multiplication by zero. This law states that the product is equal to zero if at least one of the factors is equal to zero.
In our example, one of the factors is equal to zero, therefore, without wasting time, we answer that the value of the expression is zero:
Example 12. Find the value of an expression 
The product is equal to zero if at least one of the factors is equal to zero.
In our example, one of the factors is equal to zero, therefore, without wasting time, we answer that the value of the expression equals zero:
Example 13 Find the value of an expression 
You can use the procedure and first calculate the expression in brackets and multiply the resulting answer with a fraction.
You can also use the distributive law of multiplication - multiply each term of the sum by a fraction and add the results. We will use this method.
According to the order of operations, if the expression contains addition and multiplication, then the first thing to do is to perform the multiplication. Therefore, in the resulting new expression, we take in brackets those parameters that must be multiplied. So we can clearly see which actions to perform earlier and which later:
Third action:
Answer: expression value equals
The solution for this example can be written much shorter. It will look like this:
It can be seen that this example could be solved even in the mind. Therefore, one should develop the skill of analyzing an expression before starting to solve it. It is likely that it can be solved in the mind and save a lot of time and nerves. And on control and exams, as you know, time is very expensive.
Example 14 Find the value of the expression −4.2 × 3.2
This is the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus before the received answer
Notice how the modules of rational numbers were multiplied. In this case, to multiply the modules of rational numbers, it took .
Example 15 Find the value of the expression −0.15 × 4
This is the multiplication of rational numbers with different signs. We multiply the modules of these numbers and put a minus before the received answer
Notice how the modules of rational numbers were multiplied. In this case, in order to multiply the modules of rational numbers, it took to be able to.
Example 16 Find the value of the expression −4.2 × (−7.5)
This is the multiplication of negative rational numbers. We multiply the modules of these numbers and put a plus in front of the received answer
Division of rational numbers
The rules for dividing integers are also valid for rational numbers. In other words, to be able to divide rational numbers, you need to be able to
Otherwise, the same methods for dividing ordinary and decimal fractions are used. To divide a common fraction by another fraction, you need to multiply the first fraction by the reciprocal of the second.
And in order to divide a decimal fraction into another decimal fraction, you need to move the comma to the right in as many digits in the dividend and in the divisor as there are after the decimal point in the divisor, then perform the division as for a regular number.
Example 1 Find the value of an expression:
This is the division of rational numbers with different signs. To calculate such an expression, you need to multiply the first fraction by the reciprocal of the second.
So let's multiply the first fraction by the reciprocal of the second.
We got the multiplication of rational numbers with different signs. And we already know how to calculate such expressions. To do this, you need to multiply the modules of these rational numbers and put a minus before the answer.
Let's complete this example. The entry with modules can be omitted so as not to clutter up the expression
Thus, the value of the expression is
The detailed solution is as follows:
A short solution would look like this:
Example 2 Find the value of an expression
This is the division of rational numbers with different signs. To calculate this expression, you need to multiply the first fraction by the reciprocal of the second.
The reciprocal of the second fraction is the fraction . We multiply the first fraction by it:
A short solution would look like this:
Example 3 Find the value of an expression
This is the division of negative rational numbers. To calculate this expression, again, you need to multiply the first fraction by the reciprocal of the second.
The reciprocal of the second fraction is the fraction . We multiply the first fraction by it:
We got the multiplication of negative rational numbers. We already know how such an expression is calculated. It is necessary to multiply the modules of rational numbers and put a plus in front of the answer.
Let's complete this example. You can skip the entry with modules to avoid cluttering up the expression:
Example 4 Find the value of an expression
To calculate this expression, you need to multiply the first number -3 by the reciprocal of the fraction.
The reciprocal of a fraction is a fraction. By it and multiply the first number −3
Example 6 Find the value of an expression
To calculate this expression, you need to multiply the first fraction by the reciprocal of 4.
The reciprocal of 4 is a fraction. We multiply the first fraction by it
Example 5 Find the value of an expression
To calculate this expression, you need to multiply the first fraction by the reciprocal of −3
The reciprocal of −3 is a fraction. We multiply the first fraction by it:
Example 6 Find the value of the expression −14.4: 1.8
This is the division of rational numbers with different signs. To calculate this expression, you need to divide the dividend modulus by the divisor modulus and put a minus before the received answer
Note how the modulus of the dividend has been divided into the modulus of the divisor. In this case, to do it right, it took to be able to.
If there is no desire to mess around with decimal fractions (and this happens often), then these, then convert these mixed numbers to improper fractions, and then go directly to division.
Let's calculate the previous expression -14.4: 1.8 in this way. Convert decimals to mixed numbers:
Now let's translate the resulting mixed numbers into improper fractions:
Now you can deal directly with division, namely divide a fraction by a fraction. To do this, you need to multiply the first fraction by the reciprocal of the second:
Example 7 Find the value of an expression 
Convert the decimal -2.06 to an improper fraction, and multiply this fraction by the reciprocal of the second:
Multistoried fractions
You can often find an expression in which the division of fractions is written using a fractional bar. For example, an expression could be written like this:
What is the difference between expressions and ? Actually there is no difference. These two expressions carry the same meaning and you can put an equal sign between them:
In the first case, the division sign is a colon and the expression is written on one line. In the second case, the division of fractions is written using a fractional line. The result is a fraction, which the people agreed to call multistory.
When encountering such multi-story expressions, you need to apply the same rules for dividing ordinary fractions. The first fraction must be multiplied by the reciprocal of the second.
It is extremely inconvenient to use such fractions in a solution, so you can write them in an understandable form, using not a fractional bar, but a colon as a division sign.
For example, let's write a multi-storey fraction in an understandable form. To do this, you first need to figure out where the first fraction is and where the second is, because it is not always possible to do this correctly. Multistoried fractions have several fractional features that can be confusing. The main fractional bar, which separates the first fraction from the second, is usually longer than the others.
After determining the main fractional line, you can easily understand where the first fraction is and where the second is:
Example 2
We find the main fractional line (it is the longest) and we see that the integer number −3 is divided by an ordinary fraction
And if we mistakenly took the second fractional line for the main one (the one that is shorter), then it would turn out that we divide the fraction by an integer 5 In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the divisible in this case is the number −3, and the divisor is a fraction.
Example 3 We write in an understandable form a multi-storey fraction
We find the main fractional line (it is the longest) and we see that the fraction is divided by an integer 2
And if we mistakenly took the first fractional line for the main one (the one that is shorter), then it would turn out that we divide the integer −5 by a fraction. In this case, even if this expression is calculated correctly, the problem will be solved incorrectly, since the divisible in this case is a fraction, and the divisor is an integer 2.
Despite the fact that multi-storey fractions are inconvenient in work, we will encounter them very often, especially when studying higher mathematics.
Naturally, the translation of a multi-story fraction into an understandable form takes additional time and space. Therefore, you can use a faster method. This method is convenient and at the output allows you to get a ready-made expression in which the first fraction has already been multiplied by the reciprocal of the second.
This method is implemented as follows:
If the fraction is four-story, for example, like, then the number located on the first floor is raised to the topmost floor. And the number located on the second floor is raised to the third floor. The resulting numbers must be connected with multiplication icons (×)
As a result, bypassing the intermediate notation, we get a new expression in which the first fraction has already been multiplied by the reciprocal of the second. Convenience and more!
To avoid mistakes when using this method, you can follow the following rule:
From first to fourth. From the second to the third.
The rule is about floors. The figure from the first floor must be raised to the fourth floor. And the figure from the second floor must be raised to the third floor.
Let's try to calculate a multi-storey fraction using the above rule.
So, the number located on the first floor is raised to the fourth floor, and the number located on the second floor is raised to the third floor.
As a result, bypassing the intermediate notation, we get a new expression in which the first fraction has already been multiplied by the reciprocal of the second. You can use what you already know:
Let's try to calculate a multi-storey fraction using a new scheme.
There is only the first, second and fourth floors. The third floor is missing. But we do not deviate from the main scheme: we raise the figure from the first floor to the fourth floor. And since there is no third floor, we leave the number located on the second floor as it is
As a result, bypassing the intermediate notation, we got a new expression , in which the first number −3 has already been multiplied by the fraction that is the reciprocal of the second. You can use what you already know:
Let's try to calculate a multi-storey fraction using a new scheme.
There are only the second, third and fourth floors. The first floor is missing. Since the first floor is missing, there is nothing to go up to the fourth floor, but we can raise the figure from the second floor to the third:
As a result, bypassing the intermediate notation, we got a new expression in which the first fraction has already been multiplied by the reciprocal of the divisor. You can use what you already know:
Using Variables
If the expression is complex and it seems to you that it will confuse you in the process of solving the problem, then part of the expression can be entered into a variable and then work with this variable.
Mathematicians often do this. A complex task is broken down into easier subtasks and solved. Then they collect the solved subtasks into one single whole. This is a creative process and this is learned over the years, training hard.
The use of variables is justified when working with multi-storey fractions. For example:
Find the value of an expression
So, there is a fractional expression in the numerator and in the denominator of which there are fractional expressions. In other words, we again have a multi-story fraction, which we do not like so much.
The expression in the numerator can be entered into a variable with any name, for example:
But in mathematics, in such a case, it is customary to give the name of the variables from capital Latin letters. Let's not break this tradition, and denote the first expression through a capital Latin letter A
And the expression in the denominator can be denoted by a capital Latin letter B
Now our original expression becomes . That is, we have made the replacement of a numeric expression with a letter, having previously entered the numerator and denominator in variables A and B.
Now we can separately calculate the values of variable A and the value of variable B. We will insert the finished values into the expression.
Find the value of a variable A
Find the value of a variable B
Now let's substitute in the main expression instead of variables A and B their values:
We got a multi-story fraction in which you can use the “from the first to the fourth, from the second to the third” scheme, that is, raise the number located on the first floor to the fourth floor, and raise the number located on the second floor to the third floor. Further calculation will not be difficult:
Thus, the value of the expression is −1.
Of course, we looked at the simplest example, but our goal was to find out how you can use variables to make your task easier, to minimize the possibility of errors.
Note also that the solution for this example can be written without using variables. It will look like
This solution is faster and shorter, and in this case it is more expedient to write it this way, but if the expression turns out to be complex, consisting of several parameters, brackets, roots and powers, then it is advisable to calculate it in several stages, putting some of its expressions into variables.
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