Due to the fact that almost all modern schools have the necessary equipment to show children videos and various electronic learning resources during lessons, it becomes possible to better interest students in a particular subject or in a particular topic. As a result, student achievement and the overall rating of the school increase.
It's no secret that visual demonstration during the lesson helps to better remember and assimilate definitions, tasks and theory. If this is accompanied by voicing, then both visual and auditory memory work for the student at the same time. Therefore, video tutorials are considered one of the most effective learning materials.
There are a number of rules and requirements that video lessons must comply with in order to be as effective and useful as possible for students of the appropriate age. The background and color of the text should be chosen appropriately, the font size should not be too small so that visually impaired students can read the text, however, and not too large to irritate the eyesight and create inconvenience, etc. Particular attention is paid to illustrations - they should be contained in moderation and not distract from the main theme.
The video tutorial "Reciprocal Numbers" is a great example of such a learning resource. Thanks to him, a 6th grade student can fully understand what reciprocal numbers are, how to recognize them and how to work with them.
The lesson begins with a simple example in which two common fractions 8/15 and 15/8 are multiplied by each other. It becomes possible to recall the rule by which, as was studied earlier, fractions should be multiplied. That is, the numerator should be the product of the numerators, and the denominator should be the product of the denominators. As a result of the reduction, which is also worth remembering, a unit is obtained.
After this example, the speaker gives a generalized definition, which is displayed in parallel on the screen. It states that numbers that, when multiplied by each other, result in one, are called mutually inverse. The definition is very easy to remember, but it will stick more confidently in memory if you give some examples.
On the screen, after defining the concept of reciprocal numbers, a series of products of numbers is displayed, which as a result give out a unit.
To give a generalized example that will not depend on certain numeric values, the variables a and b are used, which are different from 0. Why? After all, schoolchildren in the 6th grade should be well aware that the denominator of any fraction cannot be equal to zero, and in order to show mutually reciprocal numbers, one cannot do without placing these values in the denominator.
After deriving this formula and commenting on it, the announcer begins to consider the first task. The bottom line is that you need to find the reciprocal of a given mixed fraction. To solve it, the fraction is written in the wrong form, and the numerator and denominator are reversed. The result obtained is the answer. The student can independently check it, using the definition of mutually reciprocal numbers.
The video tutorial is not limited to this example. Following the previous one, another task is displayed on the screen, in which it is necessary to find the product of three fractions. If the student is attentive, he will find that two of these fractions are reciprocals, therefore, their product will be equal to one. Based on the property of multiplication, one can first of all multiply mutually inverse fractions, and lastly, multiply the result, i.e. 1, by the first fraction. The speaker explains in detail, demonstrating the entire process step by step on the screen from start to finish. Finally, a theoretical generalized explanation is given for the property of multiplication, which was relied upon when solving the example.
To consolidate knowledge for sure, it is worth trying to answer all the questions that will be displayed at the end of the lesson.
We give a definition and give examples of reciprocal numbers. Consider how to find the reciprocal of a natural number and the reciprocal of an ordinary fraction. In addition, we write down and prove an inequality that reflects the property of the sum of reciprocal numbers.
Yandex.RTB R-A-339285-1
Reciprocal numbers. Definition
Definition. Reciprocal numbersReciprocal numbers are those numbers whose product gives one.
If a · b = 1 , then we can say that the number a is the reciprocal of the number b , just as the number b is the reciprocal of the number a .
The simplest example of reciprocal numbers is two ones. Indeed, 1 1 = 1, so a = 1 and b = 1 are mutually inverse numbers. Another example is the numbers 3 and 1 3 , - 2 3 and - 3 2 , 6 13 and 13 6 , log 3 17 and log 17 3 . The product of any pair of the above numbers is equal to one. If this condition is not met, as for example with the numbers 2 and 2 3 , then the numbers are not mutually inverse.
The definition of reciprocal numbers is valid for any numbers - natural, integer, real and complex.
How to find the reciprocal of a given number
Let's consider the general case. If the original number is equal to a , then its reciprocal number will be written as 1 a , or a - 1 . Indeed, a · 1 a = a · a - 1 = 1 .
For natural numbers and common fractions, finding the reciprocal is fairly easy. One might even say it's obvious. In the case of finding a number that is the inverse of an irrational or complex number, a number of calculations will have to be made.
Consider the most common cases in practice of finding the reciprocal.
The reciprocal of a common fraction
Obviously, the reciprocal of the common fraction a b is the fraction b a . So, to find the reciprocal of a fraction, you just need to flip the fraction. That is, swap the numerator and denominator.
According to this rule, you can write the reciprocal of any ordinary fraction almost immediately. So, for the fraction 28 57, the reciprocal will be the fraction 57 28, and for the fraction 789 256 - the number 256 789.
The reciprocal of a natural number
You can find the reciprocal of any natural number in the same way as the reciprocal of a fraction. It is enough to represent a natural number a as an ordinary fraction a 1 . Then its reciprocal will be 1 a . For the natural number 3, its reciprocal is 1 3 , for the number 666 the reciprocal is 1 666 , and so on.
Special attention should be paid to the unit, since this is the only number, the reciprocal of which is equal to itself.
There are no other pairs of reciprocal numbers where both components are equal.
The reciprocal of a mixed number
The mixed number is of the form a b c . To find its reciprocal, you need to present the mixed number in the side of an improper fraction, and choose the reciprocal for the resulting fraction.
For example, let's find the reciprocal of 7 2 5 . First, let's represent 7 2 5 as an improper fraction: 7 2 5 = 7 5 + 2 5 = 37 5 .
For the improper fraction 37 5 the reciprocal is 5 37 .
The reciprocal of a decimal
A decimal fraction can also be represented as a common fraction. Finding the reciprocal of a decimal fraction of a number comes down to representing the decimal fraction as a common fraction and finding the reciprocal of it.
For example, there is a fraction 5, 128. Let's find its reciprocal. First, we convert the decimal to a common fraction: 5, 128 = 5 128 1000 = 5 32 250 = 5 16 125 = 641 125. For the resulting fraction, the reciprocal will be the fraction 125641.
Let's consider one more example.
Example. Finding the reciprocal of a decimal
Find the reciprocal of the periodic decimal fraction 2 , (18) .
Convert decimal to ordinary:
2, 18 = 2 + 18 10 - 2 + 18 10 - 4 + . . . = 2 + 18 10 - 2 1 - 10 - 2 = 2 + 18 99 = 2 + 2 11 = 24 11
After the translation, we can easily write down the reciprocal of the fraction 24 11. This number will obviously be 11 24 .
For an infinite and non-recurring decimal fraction, the reciprocal is written as a fraction with a unit in the numerator and the fraction itself in the denominator. For example, for the infinite fraction 3 , 6025635789 . . . the reciprocal will be 1 3 , 6025635789 . . . .
Similarly, for irrational numbers corresponding to non-periodic infinite fractions, reciprocals are written as fractional expressions.
For example, the reciprocal of π + 3 3 80 is 80 π + 3 3 , and the reciprocal of 8 + e 2 + e is 1 8 + e 2 + e.
Reciprocal numbers with roots
If the form of two numbers is different from a and 1 a , then it is not always easy to determine whether the numbers are mutually inverse. This is especially true for numbers that have a root sign in their notation, since it is usually customary to get rid of the root in the denominator.
Let's turn to practice.
Let's answer the question: are the numbers 4 - 2 3 and 1 + 3 2 reciprocal.
To find out if the numbers are mutually inverse, we calculate their product.
4 - 2 3 1 + 3 2 = 4 - 2 3 + 2 3 - 3 = 1
The product is equal to one, which means that the numbers are mutually inverse.
Let's consider one more example.
Example. Reciprocal numbers with roots
Write down the reciprocal of 5 3 + 1 .
You can immediately write that the reciprocal is equal to the fraction 1 5 3 + 1. However, as we have already said, it is customary to get rid of the root in the denominator. To do this, multiply the numerator and denominator by 25 3 - 5 3 + 1 . We get:
1 5 3 + 1 = 25 3 - 5 3 + 1 5 3 + 1 25 3 - 5 3 + 1 = 25 3 - 5 3 + 1 5 3 3 + 1 3 = 25 3 - 5 3 + 1 6
Reciprocal numbers with powers
Suppose there is a number equal to some power of the number a . In other words, the number a raised to the power n. The reciprocal of a n is a - n . Let's check it out. Indeed: a n a - n = a n 1 1 a n = 1 .
Example. Reciprocal numbers with powers
Find the reciprocal of 5 - 3 + 4 .
According to the above, the desired number is 5 - - 3 + 4 = 5 3 - 4
Reciprocals with logarithms
For the logarithm of the number a to the base b, the reciprocal is the number equal to the logarithm of the number b to the base a.
log a b and log b a are mutually reciprocal numbers.
Let's check it out. It follows from the properties of the logarithm that log a b = 1 log b a , which means log a b · log b a .
Example. Reciprocals with logarithms
Find the reciprocal of log 3 5 - 2 3 .
The reciprocal of the logarithm of 3 to base 3 5 - 2 is the logarithm of 3 5 - 2 to base 3.
The reciprocal of a complex number
As noted earlier, the definition of reciprocal numbers is valid not only for real numbers, but also for complex ones.
Usually complex numbers are represented in algebraic form z = x + i y . The reciprocal of this will be a fraction
1 x + i y . For convenience, this expression can be shortened by multiplying the numerator and denominator by x - i y .
Example. The reciprocal of a complex number
Let there be a complex number z = 4 + i . Let's find the reciprocal of it.
The reciprocal of z = 4 + i will be equal to 1 4 + i .
Multiply the numerator and denominator by 4 - i and get:
1 4 + i \u003d 4 - i 4 + i 4 - i \u003d 4 - i 4 2 - i 2 \u003d 4 - i 16 - (- 1) \u003d 4 - i 17.
In addition to its algebraic form, a complex number can be represented in trigonometric or exponential form as follows:
z = r cos φ + i sin φ
z = r e i φ
Accordingly, the reciprocal number will look like:
1 r cos (- φ) + i sin (- φ)
Let's make sure of this:
r cos φ + i sin φ 1 r cos (- φ) + i sin (- φ) = r r cos 2 φ + sin 2 φ = 1 r e i φ 1 r e i (- φ) = r r e 0 = 1
Consider examples with the representation of complex numbers in trigonometric and exponential form.
Find the inverse of 2 3 cos π 6 + i · sin π 6 .
Considering that r = 2 3 , φ = π 6 , we write the reciprocal number
3 2 cos - π 6 + i sin - π 6
Example. Find the reciprocal of a complex number
What is the inverse of 2 · e i · - 2 π 5 .
Answer: 1 2 e i 2 π 5
The sum of reciprocal numbers. Inequality
There is a theorem on the sum of two reciprocal numbers.
Sum of mutually reciprocal numbers
The sum of two positive and reciprocal numbers is always greater than or equal to 2.
We present the proof of the theorem. As you know, for any positive numbers a and b, the arithmetic mean is greater than or equal to the geometric mean. This can be written as an inequality:
a + b 2 ≥ a b
If instead of the number b we take the inverse of a , the inequality takes the form:
a + 1 a 2 ≥ a 1 a a + 1 a ≥ 2
Q.E.D.
Let's give a practical example illustrating this property.
Example. Find the sum of reciprocal numbers
Let's calculate the sum of the numbers 2 3 and its reciprocal.
2 3 + 3 2 = 4 + 9 6 = 13 6 = 2 1 6
As the theorem says, the resulting number is greater than two.
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MOU "Parkanskaya school No. 2 named. DI. Mishchenko
Math lesson in 6th grade on the topic
"Reciprocal Numbers"
Spent teacher
mathematics and computer science
I qualification category
Balan V.M.
Parkans 2011
P.S. Due to the maximum file size limit (no more than 3MB), the presentation is divided into 2 parts. You need to copy the slides sequentially into one presentation.
Mathematics lesson in the 6th grade on the topic "Reciprocal numbers"
Target:
- Introduce the concept of reciprocal numbers.
- Learn to identify pairs of reciprocal numbers.
- Repeat multiplication and reduction of fractions.
Lesson type : study and primary consolidation of new knowledge.
Equipment:
- computers;
- signal cards;
- workbooks, notebooks, textbook;
- drawing accessories;
- presentation for the lessonAppendix ).
Individual task:unit message.
During the classes
1. Organizational moment.(3 minutes)
Hello guys, sit down! Let's start our lesson! Today you will need attention, concentration and, of course, discipline.(slide 1 )
As an epigraph for today's lesson, I took the words:
It is often said that numbers rule the world;
at least there is no doubt
that the numbers show how it is managed.
And funny little people hurry to help me: Pencil and Samodelkin. They will help me with this lesson.(slide 2 )
The first task from the pencil is to solve anagrams. (slide 3 )
Let's remember together what an anagram is? (An anagram is a permutation of letters in a word that forms another word. For example, "murmur" - "axe").
(Children answer what an anagram is and guess the words.)
Well done! The topic of today's lesson is "Reciprocal numbers."
Open notebooks, write down the number, class work and the topic of the lesson. (slide 4 )
Guys, tell me, please, what should you learn at the lesson today?
(Children name the purpose of the lesson.)
The purpose of our lesson:
- Find out what numbers are called mutually inverse.
- Learn to find pairs of reciprocal numbers.
- Review the rules for multiplying and reducing fractions.
- Develop students' logical thinking.
2. We work orally.(3 minutes)
Let's repeat the rule of multiplying fractions. (slide 5 )
Task from Samodelkin (children read examples and perform multiplication):
What rule did we use?
Pencil prepared a more difficult task (slide 6 ):
What is such a work?
Guys, we repeated the steps of multiplication and reduction of fractions, which are indispensable when studying a new topic.
3. Explanation of new material.(15 minutes) ( Slide 7 )
1. Take the fraction 8/17, put the denominator instead of the numerator and vice versa. You get a fraction 17/8.
We write: the fraction 17/8 is called the reciprocal of the fraction 8/17.
Attention! The reciprocal of the fraction m/n is called the fraction n/m. (Slide 8 )
Guys, how can you still get the reciprocal of this fraction from it?(Children answer.)
2. Task from Samodelkin:
Name the reciprocal of a given fraction.(Children call.)
They say about such fractions that they are inverse to each other! (Slide 9 )
What then can be said about the fractions 8/17 and 17/8?
Answer: inverse to each other (we write down).
3. What happens if you multiply two fractions that are inverse to each other?
(Working with slides. (Slide 10 ))
Guys! Look and tell me what cannot be equal to m and n?
I repeat once again that the product of any fractions reciprocal to each other is equal to 1. (slide 11 )
4. It turns out that one is a magic number!
What do we know about the unit?
Interesting judgments about the world of numbers have come down to us through the centuries from the Pythagorean school, which Boyanzhi Nadya will tell us about (a short message).
5. We settled on the fact that the product of any numbers reciprocal to each other is equal to 1.
What are such numbers called?(Definition.)
Let's check if the fractions are mutually reciprocal: 1.25 and 0.8. (slide 12 )
You can check in another way whether the numbers are mutually inverse (2nd way).
Let's conclude guys:
How to check if numbers are mutually inverse?(Children answer.)
6. Now let's look at a few examples of finding reciprocal numbers (we consider two examples). (Slide 13)
4. Fixing. (10 minutes)
1. Work with signal cards. You have signal cards on the table. (Slide 14)
Red - no. Green - yes.
(Last example 0,2 and 5.)
Well done! Know how to identify pairs of reciprocal numbers.
2. Attention to the screen! - we work orally. (Slide 15)
Find an unknown number (we solve equations, the last 1/3 x \u003d 1).
Attention to the question: When do two numbers in the product give 1?(Children answer.)
5. Physical education minute.(2 minutes)
Now take a break from the screen - let's get some rest!
- Close your eyes, close your eyes very tightly, open your eyes sharply. Do this 4 times.
- Keep your head straight, eyes raised up, lowered down, looked to the left, looked to the right (4 times).
- Tilt your head back, lower it forward so that your chin rests on your chest (2 times).
6. We continue to consolidate the new material [ 3], [ 4].(5 minutes)
We rested, and now the consolidation of new material.
In the textbook No. 563, No. 564 - at the blackboard. (Slide 16)
7. The result of the lesson, homework.(3 minutes)
Our lesson is coming to an end. Tell me, guys, what new did we learn at the lesson today?
- How to get reciprocal numbers?
- What are reciprocal numbers?
- How to find the reciprocal of a mixed number, to a decimal?
Did we achieve the purpose of the lesson?
Let's open the diaries, write down the homework: No. 591 (a), 592 (a, c), 595 (a), item 16.
And now, I ask you to solve this puzzle (if there is time).
Thank you for the lesson! (Slide 17)
Literature:
- Mathematics 5-6: textbook-interlocutor. L.N. Shevrin, A.G. Gein, I.O. Koryakov, M.V. Volkov, - M.: Enlightenment, 1989.
- Mathematics Grade 6: lesson plans according to the textbook by N.Ya. Vilenkina, V.I. Zhokhov. L.A. Tapilina, T.L. Afanasiev. - Volgograd: Teacher, 2006.
- Mathematics: Textbook Grade 6. N.Ya.Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd.- M.: Mnemosyne, 1997.
- Journey of Pencil and Samodelkin. Y. Druzhkov. - M .: Dragonfly press, 2003.
Preview:
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Slides captions:
1 “It is often said that numbers rule the world; at least there is no doubt that the figures show how it is managed.” JOHANN WOLFGANG GOETHE
3 TO LEARN THE TOPIC OF TODAY'S LESSON, YOU NEED TO SOLVE ANAGRAMS! 1) ICHLAS NUMBER 2) DORB FRACTION 3) YTEANBOR REVERSE 4) INOMZAV MUTUALLY GUESSED? NOW REMOVE AN EXCESSIVE WORD, ORDER THE REST!
4 REVERSE NUMBERS
5 MULTIPLICATION OF FRACTIONS CALCULATE ORALLY: Well done!
6 NOW THE MISSION IS HARDER! CALCULATE: GOOD FELLOWS!
1 What do you get when you multiply two fractions that are the reciprocals of each other? Let's see (write with me): ATTENTION! THE PRODUCT OF FRACTIONS REVERSE TO EACH OTHER IS EQUAL TO ONE! WHAT DO WE KNOW ABOUT THE UNIT? REMEMBER!
2 TWO NUMBERS, THE PRODUCT OF WHICH IS EQUAL TO ONE, ARE CALLED RECURRENT NUMBERS CHECK WHETHER THE FRACTIONS ARE RECURRENTLY NUMBERS: 1.25 AND 0.8 WRITE THEM IN THE FORM OF ORDINARY FRACTIONS: REVERSE NUMBERS: Otherwise, you can check by multiplication
3 Let's prove that the reciprocal of the number 0.75. We write: , and the reciprocal of it We find the number inverse to the number We write the mixed number as an improper fraction: The reciprocal of this number
4 WORKING WITH SIGNAL CARDS YES NO ARE THE NUMBERS REVERSE?
5 WORK ORALLY: FIND THE UNKNOWN NUMBER:
6 WE WORK IN NOTEBOOKS. TUTORIAL PAGE 8 9 №5 63
7 THANKS FOR THE LESSON?
Preview:
Analysis
math lesson in 6th grade
MOU "Parkanskaya OOSh No. 2 named after. D.I.Mishchenko
Teacher Balan V.M.
Lesson topic: "Reciprocal numbers."
The lesson is built on the basis of previous lessons, students' knowledge was tested by various methods in order to find out how students learned the previous material, and how this lesson will "work" in the next lessons.
The stages of the lesson are logically traced, a smooth transition from one to another. You can trace the integrity and completeness of the lesson. Assimilation of new material proceeded independently through the creation of a problem situation and its solution. I believe that the chosen structure of the lesson is rational, because it allows you to implement all the goals and objectives of the lesson in a complex.
Currently, the use of ICT is very actively used in the classroom, so Balan V.M. used multimedia for greater clarity.
The lesson was held in the 6th grade, where the level of working capacity, cognitive interest and memory are not very high, there are some guys who have gaps in factual knowledge. Therefore, at all stages of the lesson, various methods of activating students were used, which did not allow them to get tired of the monotony of the material.
To test and evaluate students' knowledge, slides with ready-made answers for self-testing and mutual testing were used.
During the lesson, the teacher sought to intensify the mental activity of students, using the following techniques and methods: an anagram at the beginning of the lesson, a conversation, a story of students "what do we know about the unit?, visibility, work with signal cards.
Thus, I think that the lesson is creative, it is an integral system. The objectives of the lesson have been achieved.
Mathematics teacher of the 1st category /Kurteva F.I./
Reverse - or reciprocal - numbers are a pair of numbers that, when multiplied, give 1. In the most general form, reciprocals are numbers. A characteristic special case of reciprocal numbers is a pair. The inverses are, say, the numbers ; .
How to find the reciprocal
Rule: you need to divide 1 (one) by the given number.
Example #1.
The number 8 is given. Its inverse is 1:8 or (the second option is preferable, because such a notation is mathematically more correct).
When looking for the reciprocal of an ordinary fraction, then dividing it by 1 is not very convenient, because recording becomes cumbersome. In this case, it is much easier to do otherwise: the fraction is simply turned over, swapping the numerator and denominator. If a correct fraction is given, then after turning it over, an improper fraction is obtained, i.e. one from which a whole part can be extracted. To do this or not, you need to decide on a case-by-case basis. So, if you then have to perform some actions with the resulting inverted fraction (for example, multiplication or division), then you should not select the whole part. If the resulting fraction is the final result, then perhaps the selection of the integer part is desirable.
Example #2.
Given a fraction. Reverse to it:.
If you want to find the reciprocal of a decimal fraction, then you should use the first rule (dividing 1 by a number). In this situation, you can act in one of 2 ways. The first is to simply divide 1 by this number into a column. The second is to form a fraction from 1 in the numerator and a decimal in the denominator, and then multiply the numerator and denominator by 10, 100, or another number consisting of 1 and as many zeros as necessary to get rid of the decimal point in the denominator. The result will be an ordinary fraction, which is the result. If necessary, you may need to shorten it, extract an integer part from it, or convert it to decimal form.
Example #3.
The number given is 0.82. Its reciprocal is: 
. Now let's reduce the fraction and select the integer part: .
How to check if two numbers are reciprocals
The principle of verification is based on the definition of reciprocals. That is, in order to make sure that the numbers are inverse to each other, you need to multiply them. If the result is one, then the numbers are mutually inverse.
Example number 4.
Given the numbers 0.125 and 8. Are they reciprocals?
Examination. It is necessary to find the product of 0.125 and 8. For clarity, we present these numbers as ordinary fractions: (let's reduce the 1st fraction by 125). Conclusion: the numbers 0.125 and 8 are inverse.
Properties of reciprocals
Property #1
The reciprocal exists for any number other than 0.
This limitation is due to the fact that you cannot divide by 0, and when determining the reciprocal of zero, it will just have to be moved to the denominator, i.e. actually divide by it.
Property #2
The sum of a pair of reciprocal numbers is never less than 2.
Mathematically, this property can be expressed by the inequality: .
Property #3
Multiplying a number by two reciprocal numbers is equivalent to multiplying by one. Let's express this property mathematically: .
Example number 5.
Find the value of the expression: 3.4 0.125 8. Since the numbers 0.125 and 8 are reciprocals (see Example #4), there is no need to multiply 3.4 by 0.125 and then by 8. So the answer here is 3.4.
