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Requirements for knowledge, skills and abilities 3 As a result of studying the lecture, the student should know: The concept of natural, integer and rational numbers. The concept of an irrational number. The concept of real numbers. As a result of studying the lecture, the student should be able to: * Perform transformations with real numbers.
Natural. N Naturalis To count objects, numbers are used, which are called naturals. To denote the set of natural numbers, the letter N is used - the first letter of the Latin word Naturalis, “natural”, “natural”. number".
Negative numbers were introduced into mathematical use by Michael Stiefel Michael Stiefel () in the book "Complete Arithmetic" (1544), Nicola Schuecke and Nicola Schuecke () - his work was discovered in 1848.
Natural numbers Numbers, their opposites Integers
A rational number (Latin ratio ratio, division, fraction) is a number represented by an ordinary fraction, where the numerator m is an integer and the denominator n is a natural number. Such a fraction should be understood as the result of dividing m by n, even if it cannot be completely divided. In real life, rational numbers are used to count the parts of some integer but divisible objects, such as cakes or other products that are cut into several parts.
Integers Fractional numbers,13,20,(2) 0.1 2/7 Rational
Decimal fractions Decimal fractions were introduced in the 15th century by Samarkand scientist al-Kashi. Knowing nothing about the discovery of al-Koshi, decimal fractions were discovered a second time, approximately 150 years later, by the Flemish mathematician and engineer Simon Stevin Simon Stevin in his work Decimal (1585).
The set of rational numbers Q=m:n The set of rational numbers is denoted and can be written as: Q=m:n You need to understand that numerically equal fractions, such as, for example, 3/4 and 9/12, are included in this set as one number. Since by dividing the numerator and denominator of a fraction by their greatest common divisor, one can obtain the only irreducible representation of a rational number, one can speak of their set as a set of irreducible fractions with coprime integer numerator and natural denominator:
To turn a purely periodic fraction, the numerator, into an ordinary one, you need to put in the numerator of an ordinary fraction the number in the period formed from the digits in the period, the denominator 9 as many digits in the period and in the denominator - write the number 9 as many times as there are digits in the period. 0,(2)=2 9 1 digit 0,(81)=81 2 digits 99
To turn a mixed periodic fraction in the numerator into an ordinary one, in the numerator of the ordinary fraction of the difference between the beginning of the second period of the beginning of the first period, put a number equal to the difference between the number formed by the digits after the decimal point before the beginning of the second period and the number formed from the digits after the decimal point before the beginning of the first period; 9 digits in a period, with zeros comma at the beginning of the period and in the denominator write the number 9 as many times as there are digits in the period, and with as many zeros as there are digits between the comma and the beginning of the period. 0.4(6)=464 1 digit 9 0
Rational Numbers as Infinite Decimals The same notation can be used for all rational numbers. Consider 1. Integer 5 5, Common fraction 0, 3(18) 3. Decimal 8.377 8.3(7)
The presentation for the lesson "Rational numbers" has a clear structure, the presentation of the material corresponds to the logic of the presentation and explanation of this topic. In order to maximize the interest of students in the study of this educational material, we suggest using the proposed educational presentation.
slides 1-2 (Presentation topic "Rational numbers", definition)
The explanation goes sequentially, clearly, supported by relevant examples, so the teacher does not need to write everything on the board (as a result, there is a time saving, which is better spent on consolidating the material received), and the attention of the students, attracted by the relevant animation, will be completely focused on the demonstrated information.
slides 3-4 (rational numbers)
The explanation begins with the introduction of the definition of rational numbers. In order to demonstrate to students that all integer and mixed numbers (including negative ones), as well as decimal fractions, are rational numbers, the presentation provides a number of examples that prove that all these numbers can be represented as ordinary fractions.
slides 5-6 (periodic fractions)
Since a rational number, in its essence, is an ordinary fraction, students easily learn the rule that the sum, difference, and product of rational numbers are also rational numbers. To reinforce this statement, a number of examples are considered in which it is necessary to perform the voiced actions. In addition, students are shown by example that the quotient of two rational numbers is also rational. However, attention is focused on the fact that the divisor must be different from zero.
slides 7-8 (properties of rational numbers)
Since not all common fractions can be represented as a decimal, the next step in this educational presentation "Rational Numbers" is devoted to familiarity with periodic fractions. Students are shown (using division into a column) how an ordinary fraction is “transformed” into a periodic one, how to write down a period, how to find an approximate value.
slides 9-10 (examples, questions)
Having considered all the above transformations, students come to the conclusion that any rational number can be written as a decimal (in particular, an integer) or a periodic fraction.
Answering the questions presented in the presentation at the end of the presentation of the educational material (last slide), students demonstrate the level of understanding of a new topic, learn to analyze, reproduce what they have just heard and seen, and correctly formulate their thoughts.
Using the presentation "Rational Numbers" is advisable not only during classroom lessons, but also for independent study of this topic at home. Study material submitted to accessible form Therefore, the student can master it both collectively, with a teacher, with parents, and independently.
Math lesson
in 6th grade.
Mathematical relay race
Option 1.
Option 2.
Distribute in groups of numbers.
Math lesson in 6th grade
on this topic
"Rational Numbers"
Lesson Objectives:
- Introduce the concept of a rational number;
- Learn to write numbers as rational numbers;
- To generalize students' knowledge on the topic "Actions with rational numbers";
- Develop activity, ability to work independently.
rational number
__
a
Whole
number
n
Natural
number
Q(rational) numbers include the set Z(whole) and N(natural) numbers
A bunch of
rational number
Z(integer) numbers are the natural numbers, their opposite numbers, and the number zero.
Q(rational) numbers
… , -1, -0,5, 0, 1/2, 1 …
N(natural) numbers are numbers that are used to count objects
Z(whole numbers
… , -3, -2, -1, 0, 1, 2, 3 …
N(integers
- The sum, difference, and product of rational numbers are also rational numbers.
- If the divisor is nonzero, then the quotient of two rational numbers is also a rational number.
Why does the second property only hold if the divisor is non-zero?
Take action. Write the result as a ratio, where a is an integer, n is a natural number.
Right answers:
Option 1 Option 2
Show that the numbers are rational
Homework:
Study item 37, learn the definition and properties of rational numbers, solve No. 1191, 1196, 1200 (a).
Thank you
for the lesson!
Purpose: To know what a natural, integer, rational number, a periodic fraction is; be able to write an infinite decimal fraction in the form of an ordinary, be able to perform actions with decimal and ordinary fractions.
1. To consolidate the studied material, changing the types of work, on this topic “Integer and rational numbers”.
2.
Develop skills and abilities in performing actions with decimal and ordinary fractions, develop logical thinking, correct and competent mathematical speech, develop independence and confidence in their knowledge and skills when performing different types works.
3.
Raise interest in mathematics by introducing different types of consolidation of the material: oral work, work with a textbook, work at the blackboard, answering questions and the ability to do introspection, independent work; stimulating and encouraging the activities of students.
I. Organizing time.
II. New topic:
"Integers and Rational Numbers".
1.Theoretical part.
2. Practical part.
3. Work according to the textbook and at the blackboard.
4. Independent work on options.
III. Outcome.
1. For questions.
IV. Homework.
During the classes
I. Organizational moment.
Emotional mood and readiness of the teacher and students for the lesson. Communication of goals and objectives.
II. New topic: “Integers and rational numbers”:
Theoretical part.
1. Initially, the number was understood only as natural numbers. Which is enough to count individual items.
Set N = (1; 2; 3...) natural numbers is closed under the operations of addition and multiplication. This means that the sum and product of natural numbers are natural numbers.
2. However, the difference of two natural numbers is no longer always a natural number.
(Give examples: 5 - 5 = 0; 5 - 7 = - 2, the numbers 0 and - 2 are not natural).
Thus, the result of subtracting two identical natural numbers leads to the concept of zero and the introduction sets of non-negative integers
Z0 = (0; 1; 2;...).
3. To make the subtraction operation feasible, enter negative integers, that is, numbers opposite to natural ones. Thus, a set of integers is obtained Z={...; -3; -2; -1; 0; 1; 2;...}.
To make the operation of dividing by any number not equal to zero feasible, it is necessary to add the set of all positive and negative fractions to the set of all integers. The result is set of rational numbers Q=.
When performing four arithmetic operations (except division by zero) on rational numbers, rational numbers are always obtained.
4. Every rational number can be represented as a periodic decimal fraction.
Let's remember what is periodic fraction. This is an infinite decimal fraction, in which, starting from a certain decimal place, the same digit or several digits are repeated - the period of the fraction. For example, 0.3333…= 0,(3);
1,057373…=1,05(73).
These fractions are read like this: “0 whole and 3 in the period”, “1 whole, 5 hundredths and 73 in the period”.
We write rational numbers as an infinite periodic decimal fraction:
natural number 25 = 25.00…= 25,(0);
integer -7 = -7.00…= -7,(0);
(we use the corner division algorithm).
5. The converse statement is also true: each infinite periodic decimal fraction is a rational number, since it can be represented as a fraction, where m is an integer, n is a natural number.
Consider an example:
1) Let x \u003d 0.2 (18) multiplying by 10, we get 10x \u003d 2.1818 ... (You need to multiply the fraction by 10 n, where n is the number of decimal places contained in the record of this fraction up to the period: x10 n).
2) Multiplying both sides of the last equality by 100, we find
1000x = 218.1818…(Multiplying by 10 k , where k is the number of digits in the period x10 n 10 k = x10 n+k).
3) Subtracting from equality (2) equality (1), we obtain 990x = 216, x = .
Practical part.
1) - on the board;
3) - at the blackboard one student writes down the decision, the rest decide on the ground, then check each other;
4) - under dictation, everyone performs the task, and one speaks out loud.
1) - on the board;
3) - under dictation, everyone performs the task, and one speaks out loud;
5) - independently with subsequent verification.
6) -2.3(82) - the teacher shows the solution on the board, based on the algorithm:
X \u003d -2.3 (82) \u003d -2.3828282 ...
10x = -23.828282…
1000x = -2382.8282…
1000x – 10x = -2382.8282…– (23.828282…)
1) 0,(6); 3) 0.1(2); 5) -3, (27) - on the board, students go out in turn.
4. Calculate:
(Do it yourself according to the options.)
1) (20,88: 18 + 45: 0,36) : (19,59 + 11,95);
2)
5. Calculate:
- independently with subsequent verification.
III. Outcome.
- What sets of numbers do you know? Give examples.
- What is a periodic fraction?
- How to write a periodic fraction as a common fraction?
- Conduct a self-analysis: “What have you learned and what have you learned?”
IV. Homework.
1. Write as a decimal fraction:
2)
2. Perform actions and write the result as a decimal fraction:
2)
3. Write an infinite decimal fraction in the form of an ordinary fraction:
2) 1,(55); 4) -0,(8).
5. Calculate:
2)
