Municipal educational institution "Parkanskaya secondary school No. 2 named after. DI. Mishchenko
Math lesson in 6th grade on the topic
"Mutually reciprocal numbers"
Conducted by the teacher
mathematics and computer science
I qualification category
Balan V.M.
Parkans 2011
P.S. Due to maximum file size restrictions (no more than 3MB), the presentation is divided into 2 parts. You must copy the slides sequentially into one presentation.
Math lesson in 6th grade on the topic "Reciprocal numbers"
Target:
- Introduce the concept of reciprocal numbers.
- Learn to identify pairs of reciprocal numbers.
- Review multiplication and reduction of fractions.
Lesson type : study and primary consolidation of new knowledge.
Equipment:
- computers;
- signal cards;
- workbooks, exercise books, textbook;
- drawing supplies;
- presentation for the lesson (seeApplication ).
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 )
I took the words as the epigraph for today’s lesson:
It is often said that numbers rule the world;
at least there is no doubt
that the numbers show how it handles.
And cheerful little men rush to my aid: Karandash and Samodelkin. They will help me teach 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 rearrangement of letters in a word to form another word. For example, “murmur” - “axe”).
(Children answer what an anagram is and solve the words.)
Well done! The topic of today's lesson: “Reciprocal numbers.”
We open the notebooks, write down the number, Classwork and the topic of the lesson. (Slide 4 )
Guys, please tell me what should you learn in class today?
(Children name the purpose of the lesson.)
The purpose of our lesson:
- Find out what numbers are called reciprocals.
- Learn to find pairs of mutually inverse numbers.
- Review the rules for multiplying and reducing fractions.
- Develop logical thinking students.
2. We work orally.(3 minutes)
Let's repeat the rule for multiplying fractions. (Slide 5 )
Assignment from Samodelkin (children read examples and perform multiplication):
What rule did we use?
Pencil has prepared a more difficult task (Slide 6 ):
What is the value of such a product?
Guys, we repeated the actions of multiplying and reducing fractions, which are essential 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. The resulting fraction is 17/8.
We write: the fraction 17/8 is called the reciprocal of the fraction 8/17.
Attention! The inverse of the fraction m/n is the fraction n/m. (Slide 8 )
Guys, how can we get the inverse of a given fraction?(Children answer.)
2. Assignment from Samodelkin:
Name the fraction that is the inverse of the given one.(Children call.)
Such fractions are said to be reciprocals of each other! (Slide 9 )
What then can be said about the fractions 8/17 and 17/8?
Answer: inverse to each other (we write it down).
3. What happens if you multiply two fractions that are their reciprocals?
(Working with slides. (Slide 10 ))
Guys! Look and tell me what m and n cannot be equal to?
I repeat once again that the product of any fractions that are 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 to us through the centuries from the Pythagorean school, which Boyanzhi Nadya will tell us about (short message).
5. We settled on the fact that the product of any numbers inverse to each other is equal to 1.
What are such numbers called?(Definition.)
Let's check whether the fractions are reciprocal numbers: 1.25 and 0.8. (Slide 12 )
You can check in another way whether the numbers are reciprocals (method 2).
Let's conclude, guys:
How to check if numbers are reciprocals?(Children answer.)
6. Now let’s look at several examples of finding reciprocal numbers (we consider two examples). (Slide 13)
4. Consolidation. (10 minutes)
1. Working with signal cards. You have signal cards on your 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 the unknown number (we solve the equations, the last 1/3 x = 1).
Attention question: When do two numbers in a product give 1?(Children answer.)
5. Physical education moment.(2 minutes)
Now take a break from the screen - let's relax a little!
- Close your eyes, close your eyes very tightly, open your eyes sharply. Do this 4 times.
- We keep our head straight, our eyes raised up, 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 new material [3], [4].(5 minutes)
We've rested, and now we'll consolidate the new material.
In textbook No. 563, No. 564 - at the blackboard. (Slide 16)
7. Lesson summary, homework. (3 minutes)
Our lesson is coming to an end. Tell me, guys, what new did we learn in class today?
- How to get numbers that are inverse to each other?
- What numbers are called reciprocals?
- How to find the reciprocal of a mixed number decimal?
Have we achieved the purpose of the lesson?
Let's open our diaries and write down our homework: No. 591(a), 592(a,c), 595(a), item 16.
And now, I ask you to solve this puzzle (if there is time left).
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.: Education, 1989.
- Mathematics 6th grade: lesson plans according to the textbook N.Ya. Vilenkina, V.I. Zhokhov. L.A. Tapilina, T.L. Afanasyeva. – Volgograd: Teacher, 2006.
- Mathematics: Textbook 6th grade. N.Ya.Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Shvartsburd.- M.: Mnemosyne, 1997.
- The journey of Pencil and Samodelkin. Yu. Druzhkov. – M.: Dragonfly Press, 2003.
Preview:
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Slide captions:
1 “It is often said that numbers rule the world; at least there is no doubt that the numbers show how it is managed." JOHANN WOLFGANG GOETHE
3 TO FIND OUT THE TOPIC OF TODAY'S LESSON, YOU NEED TO SOLVE ANAGRAMS! 1) ICHLAS NUMBERS 2) BDORB FRACTION 3) YTEANBOR REVERSE 4) INOMZAV HAVE YOU RESOLVED MUTUALLY? NOW REMOVE THE EXTRA WORD AND PLACE THE REST IN THE RIGHT ORDER!
4 REVERSIBLE NUMBERS
5 MULTIPLYING FRACTIONS CALCULATE ORAL: Well done!
6 AND NOW THE TASK IS MORE COMPLICATED! CALCULATE: WELL DONE!
1 What happens if you multiply two fractions that are their reciprocals? Let's take a look (write with me): ATTENTION! THE PRODUCT OF FRACTIONS THAT ARE THE REVERSE OF EACH OTHER IS EQUAL TO ONE! WHAT DO WE KNOW ABOUT UNIT? REMEMBER!
2 TWO NUMBERS, THE PRODUCT OF WHICH IS EQUAL TO ONE, ARE CALLED MUTUALLY REVERSIBLE NUMBERS. LET'S CHECK IF THE FRACTIONS ARE MUTUALLY REVERSIVE NUMBERS: 1.25 AND 0.8 WILL WRITE THEM IN THE FORM OF ORDINARY FRACTIONS: MUTUALLY REVERSIBLE NUMBERS Otherwise, can be checked by multiplication:
3 Let us prove that the reciprocal of the number is 0.75. We write: , and the inverse to it Let's find the number, the inverse of the number We write mixed number as improper fraction: The inverse of this number
4 WORKING WITH SIGNAL CARDS YES NO ARE THE NUMBERS INVERSE?
5 WORK ORAL: FIND AN UNKNOWN NUMBER:
6 WE WORK IN NOTEBOOKS. TEXTBOOK PAGE 8 9 No. 5 63
7 THANKS FOR THE LESSON?
Preview:
Analysis
math lesson in 6th grade
Municipal educational institution "Parkanskaya secondary school No. 2 named after. D.I. Mishchenko"
Teacher Balan V.M.
Lesson topic: “Reciprocal numbers.”
The lesson was built on previous lessons, students' knowledge was tested using various methods in order to find out how students learned the previous material, and how this lesson will “work” in the following 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. The assimilation of new material proceeded independently through the creation problematic situation and her decision. I believe that the chosen structure of the lesson is rational, because it allows us to implement all the goals and objectives of the lesson in a comprehensive manner.
Currently, the use of ICT in lessons is very actively used, so Balan V.M. used multimedia for greater clarity.
The lesson was held in 6th grade, where the level of performance, cognitive interest and memory are not very high, there are also guys who have gaps in factual knowledge. Therefore, at all stages of the lesson we used various methods activating students, 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 tried to intensify mental activity students using the following techniques and methods: anagram at the beginning of the lesson, conversation, students’ story “what do we know about the unit?”, visibility, working with signal cards.
Thus, I believe that the lesson is creative and represents whole system. The goals set during the lesson were achieved.
Category 1 mathematics teacher /Kurteva F.I./
Thanks to the fact that in almost all modern schools There is necessary equipment In order to show children videos and various electronic learning resources during lessons, it becomes possible to better interest students in a particular subject or topic. As a result, student achievement and the school's overall rating improve.
It's no secret that visual demonstration during a lesson helps to better remember and assimilate definitions, tasks and theory. If this is accompanied by voicing, then the student has both visual and auditory memory. Therefore, video tutorials are considered one of the most effective materials for training.
There are a number of rules and requirements that video lessons must meet 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 accordingly, the font size should not be too small so that the text can be read by visually impaired students, but not too large so as to irritate the eyesight and create inconvenience, etc. Special attention attention is also paid to illustrations - they should be kept in moderation and not distract from the main topic.
The video lesson “Reciprocal Numbers” is an excellent example of such a teaching resource. Thanks to it, a 6th grade student can fully understand what reciprocal numbers are, how to recognize them and how to work with them.
The lesson starts with simple example, in which two ordinary fractions 8/15 and 15/8 are multiplied by each other. It becomes possible to remember the rule according to which, as previously learned, fractions should be multiplied. That is, in the numerator you should write the product of the numerators, and in the denominator - the product of the denominators. As a result of the reduction, which is also worth remembering, we get one.
After this example, the announcer 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 reciprocals. The definition is very simple to remember, but it will be more firmly fixed in memory if you give some examples.
After defining the concept of reciprocal numbers, a series of products of numbers is displayed on the screen, which ultimately gives one.
To give a general example that will not depend on certain numerical 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 reciprocal numbers, one cannot do without placing these values in the denominator.
After deducing this formula and commenting on it, the speaker begins to consider the first task. The point is that you need to find the inverse of a given mixed fraction. To solve it, the fraction is written in in the wrong form, and the numerator and denominator are swapped. The result obtained is the answer. The student can check it independently, using the definition of reciprocal numbers.
The video tutorial is not limited to this example. Following the previous one, another task is displayed on the screen, in which you need to find the product of three fractions. If the student pays attention, he will discover that two of these fractions are reciprocals, therefore, their product will equal one. Based on the property of multiplication, you can first multiply mutually inverse fractions, and lastly, multiply the result, i.e. 1, by the first fraction. The announcer explains in detail, showing 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 your knowledge for sure, you should try to answer all the questions that will be presented at the end of the lesson.
Reciprocal - or mutually reciprocal - numbers are a pair of numbers that, when multiplied, give 1. In fact general view the reciprocals are numbers. Characteristic special case reciprocal numbers – a pair. The inverses are, say, numbers; .
How to find the reciprocal of a number
Rule: you need to divide 1 (one) by a given number.
Example No. 1.
The number 8 is given. Its inverse is 1:8 or (the second option is preferable, because this notation is mathematically more correct).
When looking for the inverse number for common fraction, then dividing it by 1 is not very convenient, because the recording is cumbersome. In this case, it is much easier to do things differently: the fraction is simply turned over, swapping the numerator and denominator. If given proper fraction, then after turning over the resulting fraction is improper, i.e. one from which a whole part can be isolated. Whether to do this or not, you need to decide in each specific case especially. 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 final result, then perhaps isolating the whole part is desirable.
Example No. 2.
Given a fraction. Reverse to it: .
If you need to find the reciprocal of a decimal fraction, you should use the first rule (dividing 1 by the number). In this situation, you can act in one of 2 ways. The first is to simply divide 1 by that number into a column. The second is to form a fraction with a 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 a 1 and as many zeros as necessary to get rid of decimal point in the denominator. The result will be an ordinary fraction, which is the result. If necessary, you may need to shorten it, select an entire part from it, or convert it to decimal form.
Example No. 3.
The number given is 0.82. The reciprocal number is: 
. Now let's reduce the fraction and select the whole part: .
How to check if two numbers are reciprocals
The verification principle is based on determining reciprocal numbers. That is, in order to make sure that the numbers are reciprocals of each other, you need to multiply them. If the result is one, then the numbers are mutually inverse.
Example No. 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, let's present these numbers in the form of ordinary fractions: (reduce the 1st fraction by 125). Conclusion: the numbers 0.125 and 8 are reciprocals.
Properties of reciprocal numbers
Property No. 1
A reciprocal exists for any number except 0.
This limitation is due to the fact that you cannot divide by 0, and when determining the reciprocal number for zero, it will have to be moved to the denominator, i.e. actually divide by it.
Property No. 2
The sum of a pair of reciprocal numbers is always no less than 2.
Mathematically, this property can be expressed by the inequality: .
Property No. 3
Multiplying a number by two reciprocal numbers is equivalent to multiplying by one. Let's express this property mathematically: .
Example No. 5.
Find the value of the expression: 3.4·0.125·8. Since the numbers 0.125 and 8 are reciprocals (see Example No. 4), there is no need to multiply 3.4 by 0.125 and then by 8. So, the answer here will be 3.4.
Let's give a definition and give examples of reciprocal numbers. Let's look at how to find the inverse of a natural number and the inverse of a common 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 numbers whose product equals one.
If a · b = 1, then we can say that the number a is the inverse of the number b, just as the number b is the inverse of the number a.
The simplest example of reciprocal numbers is two units. Indeed, 1 · 1 = 1, therefore 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 numbers above is equal to one. If this condition is not met, as for example for the numbers 2 and 2 3, then the numbers are not mutually inverse.
The definition of reciprocal numbers is valid for any number - natural, integer, real and complex.
How to find the inverse of a given number
Let's consider general case. If the original number is equal to a, then its inverse number will be written as 1 a, or a - 1. Indeed, a · 1 a = a · a - 1 = 1 .
For natural numbers and ordinary fractions, finding the reciprocal is quite simple. One might even say it's obvious. If you find a number that is the inverse of an irrational or complex number, you will have to make a series of calculations.
Let's consider the most common cases of finding the reciprocal number in practice.
The reciprocal of a common fraction
Obviously, the reciprocal of the common fraction a b is the fraction b a. So, to find the inverse of a fraction, you simply need to flip the fraction over. 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 number 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 inverse of any natural number in the same way as finding the inverse of a fraction. It is enough to represent the natural number a in the form of an ordinary fraction a 1. Then its inverse number will be the number 1 a. For natural number 3 its reciprocal is the fraction 1 3, for the number 666 the reciprocal is 1 666, and so on.
Special attention should be paid to the unit, since it singular, 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 looks like a b c. To find its inverse number, you need to represent the mixed number as an improper fraction, and then select the inverse number for the resulting fraction.
For example, let's find the reciprocal number for 7 2 5. First, let's imagine 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.
Reciprocal of a decimal
A decimal can also be represented as a fraction. Finding the reciprocal of a decimal number comes down to representing the decimal as a fraction and finding its reciprocal.
For example, there is a fraction 5, 128. Let's find its inverse number. First, convert the decimal fraction to an ordinary fraction: 5, 128 = 5 128 1000 = 5 32 250 = 5 16 125 = 641 125. For the resulting fraction, the reciprocal number will be the fraction 125 641.
Let's look at another example.
Example. Finding the reciprocal of a decimal
Let's find the reciprocal number for the periodic decimal fraction 2, (18).
Converting a decimal fraction to an ordinary fraction:
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 the reciprocal number for the fraction 24 11. This number will obviously be 11 24.
For an infinite and non-periodic decimal fraction, the reciprocal number 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 number will be 1 3, 6025635789. . . .
Likewise for irrational numbers, corresponding to non-periodic infinite fractions, reciprocal numbers are written as fractional expressions.
For example, the reciprocal for π + 3 3 80 will be 80 π + 3 3, and for the number 8 + e 2 + e the reciprocal will be the fraction 1 8 + e 2 + e.
Reciprocal numbers with roots
If the type of two numbers is different from a and 1 a, then it is not always easy to determine whether the numbers are reciprocals. 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 whether the numbers are reciprocals, let's calculate their product.
4 - 2 3 1 + 3 2 = 4 - 2 3 + 2 3 - 3 = 1
The product is equal to one, which means the numbers are reciprocal.
Let's look at another example.
Example. Reciprocal numbers with roots
Write down the reciprocal of 5 3 + 1.
We can immediately write that the reciprocal number 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
Let's say 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 the number a n is the number a - n . Let's check it out. Indeed: a n · a - n = a n 1 · 1 a n = 1 .
Example. Reciprocal numbers with powers
Let's find the reciprocal number for 5 - 3 + 4.
According to what was written above, the required number is 5 - - 3 + 4 = 5 3 - 4
Reciprocal numbers with logarithms
For the logarithm of a number to base b, the inverse is the number equal to logarithm numbers b to base a.
log a b and log b a are mutually inverse numbers.
Let's check it out. From the properties of the logarithm it follows that log a b = 1 log b a, which means log a b · log b a.
Example. Reciprocal numbers with logarithms
Find the reciprocal of log 3 5 - 2 3 .
In number, inverse logarithm the number 3 to base 3 5 - 2 is the logarithm of number 3 5 - 2 to base 3.
The inverse of a complex number
As noted earlier, the definition of reciprocal numbers is valid not only for real numbers, but also for complex ones.
Complex numbers are usually represented in algebraic form z = x + i y . The reciprocal of the given number is a fraction
1 x + i y . For convenience, you can shorten this expression by multiplying the numerator and denominator by x - i y.
Example. The inverse of a complex number
Let there be a complex number z = 4 + i. Let's find the inverse 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 = 4 - i 4 + i 4 - i = 4 - i 4 2 - i 2 = 4 - i 16 - (- 1) = 4 - i 17 .
Besides algebraic form, a complex number can be represented in trigonometric or demonstrative form in the following way:
z = r cos φ + i sin φ
z = r e i φ
Accordingly, the inverse 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
Let's look at examples with the representation complex numbers in trigonometric and exponential form.
Let's find the inverse number for 2 3 cos π 6 + i · sin π 6 .
Considering that r = 2 3, φ = π 6, we write the inverse number
3 2 cos - π 6 + i sin - π 6
Example. Find the inverse of a complex number
What number will be the reciprocal of 2 · e i · - 2 π 5 .
Answer: 1 2 e i 2 π 5
Sum of reciprocal numbers. Inequality
There is a theorem about the sum of two mutually inverse numbers.
Sum of reciprocal numbers
The sum of two positive and reciprocal numbers is always greater than or equal to 2.
Let us give a proof of the theorem. As is known, for any positive numbers a and b are the arithmetic mean 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 will take the form:
a + 1 a 2 ≥ a 1 a a + 1 a ≥ 2
Q.E.D.
Let's give practical example, illustrating this property.
Example. Find the sum of reciprocal numbers
Let's calculate the sum of the numbers 2 3 and its inverse.
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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