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1 MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION Federal State Budgetary Educational Institution of Higher Education "Togliatti State University" Institute of Mathematics, Physics and Information Technologies Department "Algebra and Geometry" L A V R S K A Y A B O T A Direction of preparation of the bachelor: Pedagogical education Orientation (profile): Mathematics and computer science Student V.V. Nazarov Scientific Supervisor: Ph.D., prof. R.A. Uteeva Admit for defense Head of the department: Doctor of Pedagogical Sciences, prof. R.A. Uteeva 016 Togliatti - 016
2 CONTENTS INTRODUCTION... CHAPTER I. METHODOLOGICAL SYSTEM OF TEACHING THE TOPIC "SQUARE ROOTS" IN THE COURSE OF ALGEBRA OF THE BASIC SCHOOL The main goals and objectives of teaching the topic "Square Roots" in the course of algebra of the main school schools Forms, methods and means of teaching the topic "Square Roots" in the course of algebra of the basic school ... 5 Conclusion on Chapter I ... CHAPTER II. METHODOLOGICAL RECOMMENDATIONS FOR THE ORGANIZATION OF TEACHING THE TOPIC "SQUARE ROOTS" IN THE COURSE OF ALGEBRA OF THE BASIC SCHOOL Tasks on the topic "Square Roots", focused on the basic level of knowledge and skills in the course of algebra of the main school Tasks on the topic "Square Roots", focused on preparing for the final certification and passing the OGE in mathematics Conclusion on Chapter II CONCLUSION LIST OF USED LITERATURE ... 58
3 INTRODUCTION Relevance of the study. The topic "Square Roots" is one of the traditional topics of the school algebra course of the basic school. Her study of the numbers obtained in is based on the knowledge and skills of students about the rational course of mathematics in the 6th grade. Improving the skills of performing operations on rational numbers occurs in the 7th grade algebra course. The significance and place of studying the topic "Square Roots" in the 8th grade algebra course is associated with the need to further expand the set of rational numbers and introduce irrational numbers. The well-known practical problem of finding the side (length of the side) of a square according to its given area can serve as a motivation for studying the topic, for which previously known numbers are not enough to solve. In addition, when solving many geometric problems, problems in physics, chemistry and biology, it becomes necessary to solve equations containing square roots. Therefore, it is important to know the rules of operations with square roots and learn how to transform expressions containing them. Let us turn to the history of the concept of the square root and its designation, compiled on the basis of the following sources. n The modern form of the square root sign for x and x did not appear immediately. The evolution of the radical sign lasted almost five centuries, starting from the 13th century, when Italian and some European mathematicians first called the square root the Latin word Radix (root) or abbreviated R. In the 15th century. N. Schücke wrote R 1 instead of 1. The modern root sign originated from the designation used by German mathematicians of the 15th-16th centuries, who called algebra the science of "Koss", and mathematicians-algebraists "kossists". (Mathematicians of the 12th-15th centuries wrote all their works exclusively in Latin. They called the unknown res (thing).
4 Italian mathematicians translated the word res as cosa. The last term was borrowed by the Germans, from whom the kossists and koss appeared.) In the 15th century. some German cossists used a dot in front of an expression or number to indicate the square root. In cursive writing, these dots were replaced by dashes, and later they turned into the symbol One such sign meant the usual square root. If it was necessary to designate the root of the fourth degree, then a double sign was used. It remains only to guess how exactly the root of the eighth degree was designated. If we take the analogy with the fourth degree, then this sign was supposed to identify the threefold extraction of the square root, that is, for this it was necessary to put three squares. However, this notation is taken by the cube root. Most likely, subsequently, from such designations, the sign V was formed, close in writing to the modern sign familiar to schoolchildren, but without the upper line. For the first time this sign was seen in German algebra "Beautiful and fast counting with the help of skillful rules of algebra." The author of this work was a mathematics teacher from Vienna, a native of the Czech Republic, Krishtof Rudolf. The book enjoyed great success and was constantly reprinted throughout the 16th century. and after right up to 1615. The sign of the root proposed by Krishtof was used by A. Girard, S. Stevin (he wrote the root exponent to the right of the radical sign in the circle: V () or V (). In 166, the Dutch mathematician A. Girard modified the sign of Rudolf's root and introduced a very close to the modern designation. This form of writing began to replace the previous sign R. However, for some time, the root sign was written breaking the upper line, namely: a + b. And only in 167, Rene Descartes connected the horizontal line with a tick, using a new designation in his book "Geometry". But even here there was no exact copy of the modern form. Descartes' record was somewhat different from the one we are used to, in one detail. He has 4
5 was written: C + 1 q qq p, where the letter C, placed immediately after the radical, indicated the notation of the cube root. In its modern form, this expression would look like this: C + 1 q q p. The closest to the modern spelling of the radical was used by Newton in his "Universal Arithmetic" (1685). Only some time after its writing, the mathematicians of the planet finally accepted a single and final form of recording the square root. Research problem: what are the methodological features of teaching the topic "Square Roots" at school? 5 in the course of algebra of the 8th grade the main The object of the research is the process of teaching algebra to students of the basic school. The subject of the research is the methodical system of teaching the topic "Square Roots". The purpose of the bachelor's work is to reveal the methodological system of teaching the topic "Square Roots". Research objectives: 1. To identify the main goals and objectives of teaching the topic "Square Roots" in the course of algebra of the main school (the target component of the methodological system). various forms, methods and means of teaching the topic "Square Roots" in the algebra course of the main school (organizational
6 components of the methodological system). 4. Formulate methodological recommendations for teaching the topic "Square Roots". Research methods: analysis of scientific and methodological literature, programs in mathematics, school textbooks on algebra on the topic of research, analysis, systematization and generalization of the material. The practical significance of this work lies in the fact that it presents a methodological system for teaching the topic "Square Roots" in the algebra course of the main school and formulates methodological recommendations that can be used by mathematics teachers, as well as bachelors during the period of teaching practice at school. The presented results and conclusions of the bachelor's work can be used as the basis for further development of the methodology for teaching students the topic "Square Roots". The following is submitted for defense: a methodological system for teaching the topic "Square Roots" in the course of algebra of the main school. Work structure. The bachelor's work consists of an introduction, two chapters, a conclusion, a list of references. 6
7 CHAPTER I. METHODOLOGICAL SYSTEM OF TEACHING THE TOPIC "SQUARE ROOTS" IN THE COURSE OF ALGEBRA OF THE BASIC SCHOOL 1. The main goals and objectives of teaching the topic "Square roots" in the course of algebra of the main school subject area "Mathematics" should provide: 1) the formation of ideas about mathematics as a method of cognition of reality, which allows you to describe and study real processes and phenomena;) the development of skills to work with an educational mathematical text (analyze, extract the necessary information), accurately and competently express your thoughts using mathematical terminology and symbolism, carry out classifications, rationalizations, proofs of mathematical statements;) development of ideas about number and number systems from natural to real numbers; mastering the skills of oral, written, instrumental calculations; 4) mastering the symbolic language of algebra, methods for performing identical transformations of expressions, solving equations, systems of equations, inequalities and systems of inequalities; the ability to model real situations in the language of algebra, to explore the constructed models using the apparatus of algebra, to interpret the result; In the program in mathematics, the author identifies the following goals and objectives for studying the topic "Square Roots": Expanding the set of rational numbers, introducing the concept of irrational and real numbers, studying square roots and actions with them. 7
8 As a result of studying the topic, students should know: 1. Definition of periodic and non-periodic infinite decimal fractions .. The function y \u003d x, its properties and graph .. The concept of a square root 4. Properties of arithmetic square roots. 5. The set of real, rational and irrational numbers. As a result of studying the topic, students should be able to: 1. Convert ordinary fractions to decimals and vice versa.. Compare real, rational and irrational numbers.. Be able to graph the function y=x. 4. Bring in and take out the factor from under the sign of the root. 5. Perform actions with square roots. In the mathematics program, the author identifies the following goals and objectives for studying the topic "Square Roots" (to Makarychev's textbook): Systematize information about rational numbers and give an idea of irrational numbers, thereby expanding the concept of number; develop the ability to perform simple transformations of expressions containing square roots. As a result of studying the topic, students should know: 1. Natural, integer, rational, irrational and real numbers. Modulus of a number a.. Arithmetic square root and its properties. 4. The function y= x, its properties and graph. As a result of studying the topic, students should be able to: 1. Solve the simplest quadratic equations. 8
9 . Bring in and take out the factor from under the sign of the root.. Find approximate values of square roots. 4. Extract the square root of the power of a number. 5. Transform irrational expressions. In the program in mathematics, the author identifies the following goals and objectives for studying the topic "Square Roots" in Alimov's textbook: Systematize information about rational numbers and give an idea of irrational numbers, thereby expanding the concept of number; develop the ability to perform simple transformations of expressions containing square roots. As a result of studying the topic, students should know: 1. The concept of an arithmetic square root .. Real numbers As a result of studying the topic, students should be able to find the square root of the degree, product and fraction. In the article by S. Minaeva [, P. 4-7] it is noted that the study of the section "Square roots" has the following goals: to teach how to perform transformations of expressions containing square roots; on the example of a square and cube root, form the initial ideas about the root of the n-th degree. The exemplary basic educational program of basic general education dated April 8, 015 says that a graduate must learn in the 8th grade (for use in everyday life, when studying other subjects and ensuring the possibility of successfully continuing education at a basic level): 1. Operate at a basic level concepts: natural number, integer, common fraction, decimal fraction, mixed fraction, rational number, arithmetic square root. Estimate the value of the square root of a positive integer. 9
10 . Recognize rational and irrational numbers. 4. Compare numbers. 5. Understand the meaning of writing a number in a standard form. 6. Solve quadratic equations using the formula of the roots of a quadratic equation. 7. Depict solutions of inequalities and their systems on the real line. The graduate will have the opportunity to learn in the 8th grade to ensure the possibility of successfully continuing education at the basic and advanced levels: 1. Operate with concepts: the set of natural numbers, the set of integers, the set of rational numbers, the irrational number, the square root, the set of real numbers, the geometric interpretation of natural numbers, integer, rational, real numbers.. Perform calculations, including using the methods of rational calculations.. Compare rational and irrational numbers. 4. Represent a rational number as a decimal fraction. 5. Perform transformations of expressions containing square roots. 6. Select the square of the sum or difference of a binomial in expressions containing square roots. Methodological analysis of the content of teaching the topic "Square roots" in the algebra course of the basic school Basic (known from the school course of mathematics 5-6 algebra 7 classes) knowledge: the concept of a rational number; the concept of the set of rational numbers and its notation; 10 and course
11 basic actions (operations) with rational numbers; function y = x. New (introduced) knowledge: the concept of the square root of a number; the concept of an arithmetic square root; properties of arithmetic square roots; insertion and removal of a factor from under the sign of the root; actions with square roots. An analysis of the content of the topic "Square Roots" in various 8th grade algebra textbooks is presented in Tables 1-4. In the textbook Yu.N. Makarycheva is allocated more than in other hours for studying the section "Square Roots", the entire section is divided into 4 paragraphs. The topic of studying the approximate finding of square roots is touched upon, but the topic of periodic decimal fractions is omitted. In the textbook G.K. Muravina and O.V. Muravina allocated a little less than 18 hours to the section "Square Roots", the section consists of paragraphs, the topic of periodic decimal fractions is touched upon, but there is no approximate finding of square roots. In Nikolsky's textbook, the section "Square Roots" consists of only one paragraph and 5 points, many topics and concepts are not presented. In the textbook G.V. Dorofeev included a topic dedicated to the Pythagorean theorem, which is absent in all of the above. The study of the cube root is also touched upon here. In all textbooks, the study of the section begins with real and irrational numbers, but each author has his own approach. Then comes the study of the square root itself and the arithmetic square root, properties and actions on them. eleven
12 The authors of the textbook Yu.N. Makarychev, N.G. Mindyuk, K.I. Neshkov, S.B. Suvorov Titles of chapters and paragraphs 4. Real numbers 9. Rational numbers 10. Irrational numbers 5. Arithmetic square root 11. Square roots. Arithmetic square root. 1. Equation x = a. 1. Finding approximate values of the square root. 14. Function y= x and its graph. 6. Properties of the arithmetic square root. 15. The square root of the product and the fraction. 16. The square root of a power. 7. Application of the properties of the arithmetic square root 17. Removal of the factor from under the sign of the root. Entering a multiplier under the sign of the root. 18. Transformation of expressions containing square roots. Table 1 Number of hours Total Table Authors of the textbook G.K. Muravin, K.S. Muravin, O.V. Muravina Titles of chapters and paragraphs 5. Real numbers 14. Rational and irrational numbers. 15. Periodic and non-periodic infinite decimal fractions. 6. Square roots. 16. Function y=x and its graph. 17. The concept of a square root. 18. Properties of arithmetic square roots. 19. Insertion and removal of a factor from under the sign of the root. 0. Actions with square roots. Number of hours Total 18 Table Textbook authors S.M. Nikolsky, M.K. Potapov, N.N. Reshetnikov, A.V. Shevkin Title of chapters and paragraphs. Square roots.1 The concept of a square root.. Arithmetic square root.. The square root of a natural number..4 Approximate calculation of square roots..5 Properties of arithmetic square roots. 1
13 Table 4 Authors of the textbook Title of chapters and paragraphs Number of hours G.V. Dorofeev .1 The problem of finding the side of a square. Irrational numbers. Pythagorean theorem.4 Square root (algebraic approach).5 Properties of square roots.6 Transformation of expressions containing square roots.7 Cube root Total 18 Studying the topic "Square roots" based on the algebra textbook for grade 8 of the authors Muravins. At the beginning, an extension of the set of rational numbers is given, the concepts of an irrational and a real number are introduced, the transition from an ordinary fraction to a decimal and vice versa is considered. Hours are allocated for rational and irrational numbers. In paragraph 14. “Rational and irrational numbers”, the history of their occurrence and the purpose of studying the topic are told. Definitions are given on the basis of examples, ratios of the lengths of segments. Definition 1: if two segments have a common measure that fits m times in one segment and n times in the other, then their ratio m, n is a rational number. The definition of an irrational number is given in example 1: Example 1: d = (m n) =. Hence (m n) =. The denominator of the fraction on the left side of the equality is different from one, therefore, in order for the fraction to be equal to an integer, it must be reduced by n. But the natural numbers m and n do not have common divisors, so their squares do not have common divisors either. Hence, the equality m = is false, i.e. the number d is not fractional. n 1
14 The example proved that the number d is not a rational number, which means that the diagonal of a square does not have a common measure with its side, the number d is an irrational number. The next paragraph is devoted to periodic and non-periodic fractions, the concept of a period is introduced and the inevitability of the appearance of a period in translation is substantiated. Part 1 is devoted to this point. In paragraph 15, periodic and non-periodic decimal fractions are being studied, the topic of finding an approximate root for rational and irrational numbers is considered. Then, using the considered examples, definitions of finite and infinite decimal fractions are given. Example: Let's translate 1 into a decimal fraction, it turns out: 0, A number that repeats endlessly in a record is called a period, and the fraction itself is called periodic. Property 1: any rational number can be represented as an infinite decimal periodic fraction (the converse is also true). Definition: Any irrational number is written as an infinite non-periodic decimal fraction, and any infinite non-periodic decimal fraction is an irrational number. Definition: An infinite periodic decimal is a rational number, and an infinite non-periodic decimal is an irrational number. After that, there is a transition directly to the study of the topic "Square Roots". The study begins with the paragraph "The function y=x and its graph". There is a repetition of material about functions and graphs. The hour is assigned. First, a graph of the function y=x is plotted by points in the Cartesian coordinate system, its study is carried out and the name of the graph is given: Definition 4: the graph of the function y=x is called a parabola. 14
15 The transition to the concept of a square root occurs through the solution of the quadratic equation x \u003d a, argues that this method allows us to explain the nature of the term. The hour is allotted. In the next 17 paragraph, the concept of a square root is introduced. Definition 5: The roots of the equation x = a are called the square roots of a. Definition 6: A non-negative number whose square is a is called the arithmetic square root of a and denoted by a. . The sign of the radical is introduced and the history of its origin is given. Then the authors proceed to the study of the properties of arithmetic square roots, at this point, the development of the ability to transform expressions with square roots begins. The hours are allotted. In paragraph 18, the properties of arithmetic square roots are given: Property: For any number a a \u003d a. Property: For any non-negative numbers a and b: ab= a b . After that, the topic of introducing and removing a multiplier from under the root sign is studied and worked out. Work continues with square roots. As the authors note, students may experience difficulties in converting literal expressions, because, in this situation, the modulus sign is essential when taking the factor out from under the root sign. The h is assigned. In the 19th paragraph, the introduction and removal of the factor from the modes of the sign of the root are studied, the property is given: Property 4: For non-negative values b a b= a * b= a * b . Next, the authors move on to operations with square roots, at this point, practicing mainly in the transformation of numerical expressions, 15
16 students deepen their knowledge on this topic. The study can be divided into two parts: 1. Working with square roots of numbers .. Converting literal expressions. Such a study model does not set the sequence of study, we can say that the second part, although useful at this stage, still carries out propaedeutics of the 9th grade material, where the transformation of literal expressions with radicals will be specially studied. Allotted 4 hours. In 0 and the final section, actions with square roots are studied. The previously studied properties are recalled and used in the transformation of numerical expressions. Such actions are considered as: liberation from irrationality in the denominator, factorization, simplification of the expression. In total, the authors allocate 19 hours to study the section, after each section there is a verification or independent work, at the end of the chapter there is a test. The study of the topic "Square Roots" on the basis of the textbook of algebra for grade 8 author Yu.N. Makarychev. The study of the chapter "Square Roots" begins with a repetition of real numbers. First, there is a reminder of the basic information about the set of natural numbers, the divisibility of natural numbers and consideration of typical problems on the topic. One hour is allotted. Then one lesson is given to repeat the basic information about integers and consider typical problems, and then a lesson on the set of rational numbers. This is followed by a lesson in which the concept of irrational numbers and the set of real numbers is given. After completing the lessons mentioned above, 16
17 direct study of square roots, a lesson is devoted to the concepts of square root and arithmetic square root. Then the lessons are devoted to solving the simplest quadratic equations x \u003d a and then 1 lesson, which studies finding approximate values of square roots. The following lessons are devoted to the consideration of the function y= x, its properties and graph. The following are lessons that can be attributed to the properties of the arithmetic square root. In lesson 1, the properties of the square root of the product and the fraction are considered, the next lesson is to extract the square root from the power of a number. At this stage, the author proposes to devote several lessons to the control work and its verification, and then move on to the lessons that relate to the application of the properties of the arithmetic square root. 1 lesson for reviewing and practicing the skills of adding and removing a multiplier from under the root sign. Then a lesson that discusses the basic techniques for transforming irrational expressions. In conclusion, the author proposes to conduct a final test on the topic "Square Roots". In total, one hour is allotted for studying this section. And now let's consider the recommendations of S. Minaeva on the introduction of the concept of a square root in algebra lessons according to the textbook by G.V. Dorofeev in the 8th grade: 1. The problem of finding the side of the square (lesson) To introduce the concept of a square root, a meaningful approach is used that is characteristic of this course, highlighting the motivational and semantic aspects. The material is presented as follows: students know the formula S = a, with the help of which, along the side of the square, a, its area S can be calculated; but in mathematics there is a formula for solving the inverse problem of finding the side of the square a according to 17
18 of a given area S, which is written as follows: a \u003d S. The symbol S denotes the side of the square, the area of \u200b\u200bwhich is equal to S. If, for example, S \u003d 100, then a \u003d 100. Since 100 \u003d 10, then a \u003d 100 \u003d 10. In order for students to learn a new symbol, several questions of the type can be offered: let the area of \u200b\u200bthe square be 81 m2: write down, using the symbol, an expression for the side of this square; what is the length of the side of the square? Passing from geometric language to algebraic, the meaning of the symbol S can be described as follows: S is a non-negative number, the square of which is equal to S. (After all, the length cannot be expressed by a negative number!) Thus, we come to a “working” formulation, which we will use when finding square roots. We draw the teacher's attention to how the symbol S is read: the square root of S. The adjective "arithmetic" is superfluous here, since at this point in the topic we work only with positive roots. However, the term will be used later.. Irrational numbers (lesson) In this paragraph, two aspects can be distinguished: ideological and practical. Ideological lies in the first acquaintance with irrational numbers; practical - in the formation of the ability to evaluate "non-extractable" roots, to find their approximate values both with the help of an estimate and with the help of a calculator. Students come to the need to introduce irrational numbers as a result of considering the already familiar problem of finding the side of a square by its area. In the textbook, figure 10 shows two squares. One of them is single, its area is 1 sq. units The second square has a side diagonal of the first, and its area is twice as large. (Indeed, the small square consists of two equal triangles, and the large one consists of four such 18
19 triangles.) So the area of the large square is square. units What is the length of the side of this square? Let's denote it by a. Using the square root sign, we can write that a=. Until now, students have dealt only with "extractable" roots. We need to give them a couple of minutes so that they try to extract the root in this case too, to make sure that the value a \u003d 1 is not enough, and if you take a \u003d, then this is already too much; would try to pick up a decimal fraction and see that 1.4<, а 1,5 >. Next, a fairly simple proof is given that there is neither an integer nor a fractional number whose square is equal (p. 7 of the textbook). Thus, there is no rational number that accurately expresses the length of the side of our square. I would like students to realize the amazing discovery that the mathematicians of antiquity came to (there is a segment, but it has no length!), And also that this fact gave impetus to the development of mathematics (it was necessary to introduce new numbers!). Students are told that the number expressing the length of the side of a square whose area is square. unit, belongs to the class of so-called irrational numbers: - this is a positive irrational number, the square of which is equal, that is, the equality () \u003d is true. They must be able to name other irrational numbers of the form a and perform transformations like (a) = a for specific positive values of a, and in both directions. So, the first acquaintance with irrational numbers is subject to a rather narrow goal: it occurs in connection with the study of square roots and provides, first of all, the needs of this topic. In addition to the information described above (namely: among rational numbers there is no number expressing the length of the side of a square whose area is equal; in addition to rational numbers, there are also so-called irrational numbers 19
20th; all numbers of the form a are irrational, if a is not the square of an integer or fractional number), students will learn that there are infinitely many irrational numbers of a different nature (an example is the number z), that irrational numbers can be negative, and that in practice they replaced by (approximately) decimal fractions. Students get more solid information about irrational and real numbers in the "second pass" in the 9th grade course. To demonstrate the fundamental possibility of finding a decimal approximation of an irrational number of the form a, the textbook uses an evaluation method: approximate values with a deficiency and an excess are found, expressed as consecutive integers (that is, with an accuracy of 1), consecutive decimal fractions with one decimal place (that is, with up to 0.1), etc. The basis of this method is the statement: if a and b are positive numbers and a 21 In connection with the application of the Pythagorean theorem to calculate the length of the hypotenuse of a right triangle by its legs (example 1 on p. 84), the textbook mentions “Pythagorean triples”. Note that although there are infinitely many of them, there is only one triple made up of consecutive natural numbers. It is desirable that the construction of segments with irrational lengths (or points on a coordinate line with irrational abscissas) using a compass and a ruler should not only be analyzed according to the text of the textbook (p. 85), but actually completed by each student in his notebook. Such work can be offered, for example, as homework. Students should be warned that the drawing should be neat, large enough and easy to "read". We will analyze the problem material in textbooks and highlight the main types of problems used in the topic "Square Roots". The article highlights exercises on the topic "Square Roots" from the textbook by G.V. Dorofeev, covering all the essential aspects of this introductory fragment of the topic. The main purpose of the exercises is to master a new concept, to develop the ability to use the radical sign. Attention is drawn to tasks, and 7. The ability to move from equality of the form a = b to equality b = a and vice versa is required very often. Exercises 8- - to calculate the values of numerical and alphabetic expressions containing square roots. Students should learn that the root sign, like brackets, is a grouping symbol. In Exercises 4-7, the previously started (and extremely important from the point of view of applications) work with formulas is further developed. Now these are formulas containing radicals or requiring the use of radicals when expressing some variable in terms of others. Such tasks often cause difficulties for students, so they can be partially completed by studying the following points. Except 1 22 moreover, it is useful to return to them. Tasks 8 and 9 from group B at this stage are among the difficult ones; They are definitely not for all students. In classes with a low level of preparation, you can complete tasks of group A, as well as, if possible, 41 and 44. Consider an example from a textbook: Find an approximate value of 60. Solution: The square of a number is enclosed between two "exact" squares - the numbers 49 and 64: 49<(60), то есть 7 <(60) <8. Значит как 7 <(60), то 7< 60; так как (60) <8, то 60<8. Значит 7< 60<8. Очевидно, что 60 ближе к 8, чем к 7, то есть Рассмотренный способ нахождения приближенного значения корня, в силу своей громоздкости, имеет, в основном, теоретическое значение; учащимся достаточно лишь уметь указывать два целых числа, между которыми заключено значение данного корня. А при решении задач предполагается широкое использование калькулятора. Упражнения к пункту предназначены, прежде всего, для осознанного восприятия этого сложного для учащихся материала (46-51). Кроме того, использование калькулятора позволяет, с помощью наблюдений, прийти к некоторым теоретическим обобщениям, например к выводу о возрастании значения корня с увеличением подкоренного выражения (5-54). В образце к упражнению 6 показаны приемы сравнения, которыми должны овладеть учащиеся. Упражнения 65 и 66 направлены на формирование умения выполнять преобразование выражений, содержащих квадратные корни, с использованием равенства (a) =а, где a 0. В классах с невысоким уровнем подготовки достаточно упражнений группы А. Вообще, в этом блоке дана представительная группа заданий, 23 which can be quite limited. Exercise 57. Method I. Using a calculator, we find the approximate values of the roots: 5.4; 6.45; 7.65. This means that each of these numbers belongs to a segment with ends at points and, and they are located on this segment in the following order: 5, 6, 7. Further: the number 5 belongs to a segment with ends at points, and,; the number 6 belongs to a segment with ends at points 4 and 5; the number 7 belongs to a segment with ends at points 6 and 7. Method II. We arrive at the same result with the help of an estimate. For example, for 5 we have:<(5) <, то есть < 5<;, <(5) <, то есть,< 5<,. Упражнение 59. Очевидно, что точке К соответствует число 0,4, так как 0< 0,4< 1. Точно так же легко понять, что число лишнее, так как на отрезке с концами х=1 и х точка не отмечена. Сложнее с точкой L: ведь каждое из оставшихся чисел 5 и 7 располагается на отрезке с концами и. Но точка L расположена в левой половине отрезка, значит число, которое ей соответствует, должно быть меньше,5. Ответ почти очевиден: это 5. Но все же какое-то объяснение необходимо. Можно обратиться к калькулятору, а можно рассуждать так:,5 = 6,5; так как (5) <,5, то 5 <,5. Упражнение 70. а) Сначала находим, что 8< 7,5 < 9, а затем сравниваем числа 8,5 и 7,5. Так как 8,5 = 7,5<7,5, то 8,5< 7,5, 24, which means that the number 7.5 is closer to 9. When solving problems, of course, the use of a calculator is expected. In classes with a low level of preparation from group A, you can limit yourself to tasks (they meet the level of mandatory requirements), and also consider a research task 91. Exercise 86. The task is solved based on figure 7 of the textbook. From visual considerations, it is clear that the segment of the greatest length is the diagonal of the parallelepiped. Compare the length of the diagonal with the length of the reed. First, find the length of the base diagonal l: l= a + b = = 700 (cm). Now let's find the length of the diagonal of the parallelepiped d: d= (700) + 50 = 9800 (cm). Since 9800< 10000=100, то трость данных размеров поместить в коробку нельзя. Упражнение 90. Геометрически выражение a + b + c можно истолковать как длину диагонали прямоугольного параллелепипеда, измерения которого a, b и с. В самом деле, из прямоугольного треугольника LMN имеем, что LM= a + b. А из прямоугольного треугольника KLN получим, что NM= c + (a + b) = a + b + c По неравенству треугольника a + b + c 25 10= + 1 ; 1= + ; 17= But there is another way. So, a segment of length 10 can be obtained according to the following algorithm: 10 = (5) + (5). In the textbook on algebra for grade 8, the authors of Muravina suggest starting the study of the section with the following exercises: Exercise 15. Does the graph of the function y \u003d x pass through the point: A (-; 4) B (-, 5; 1) C (; 59) D (-6.5;4.5). Answer: A-yes, B-no C-yes D-yes. Paragraph 17 provides exercises for calculating the square root. Example. Calculate The solution occurs through the decomposition of the number 1105 into prime factors. 1105= *5 * = 5 7 = (5 7) =*5*7=105 Answer: 105. Item 18 introduces the properties of arithmetic square roots and tasks for their application, simplification of radical expressions and their calculation. Example 4. (p. 100) Simplify (5). (5) = - 5 = 5-. Answer: 5-. Example 5. Calculate 0, = 0, =0.8*4*5=80. Answer 80. Example 6. Calculate = =4 7. Answer: 4 7. In paragraph 19, exercises for adding and removing a factor from under the root sign are considered, as well as problems for comparing the values of expressions. 5 26 Example 7. Take out the factor from under the root sign in the expression Let's decompose the numbers 10 and 90 into prime factors: 10= **5, 90=* *5. Hence 10*90= 4 * * = 4 5 = **5* =60. Answer: =,5 .. Answer:,5. Example 8. (p. 105) Simplify the expression = 5 = 5 = Example 9. (p. 105) Enter the factor under the root sign: 5 0.4. 5 0.4 \u003d 5 0.4 \u003d 5 0.4 \u003d 10. Answer: 10. Example 10. (p. 106) Compare the values of expressions and. = 9 = 18 and = 4 = 1. >. Answer: >. 0 point is devoted to operations with square roots, conversion of fractions with square roots, simplification of expressions, liberation from irrationality. Example 11. (p. 108) Convert the fraction 54 so that its denominator does not contain a radical. Answer: = 54 = 1 7 = 1 = 4= =. 9 Example 1. (p. 109) Simplify expression =5 6 96=4 6 = = = =() 6=1 6 Answer: 27 Example 1. (p. 109) Free the fraction from irrationality in the denominator. =(). Answer: (). 6 = 6(1+ 10) 1 10 (1 10)(1+ 10) =6(1+ 10) = 6(1+ 10) =. Forms, methods and means of teaching the topic "Square Roots" in the course of algebra of the basic school In this section, we will analyze the practical experience of studying the topic on the basis of published articles and teaching aids. In the article by S. Minaeva, it is noted that the concept of a square root “appears” in the course under study when discussing two problems - geometric (about finding the length of the side of a square by its area) and algebraic (about the number of roots of an equation of the form x = a, where a is an arbitrary number ). In connection with the consideration of the first task, students receive initial ideas about irrational numbers. In the content of the chapter, the author included a non-traditional question for algebra - the Pythagorean theorem. This is done to demonstrate the natural use of square roots for finding the lengths of segments, constructing segments with irrational lengths, and constructing points with irrational coordinates. At the same time, it does not matter where students first hear about the Pythagorean theorem - in the course of geometry or in the course of algebra. The author also claims that the most important result of learning, in addition to ideological aspects, is the ability to perform some transformations of expressions containing square roots (primarily numerical ones). Students are also introduced to the concept of a cube root; at the same time, they form initial ideas about the root 7 28 nth degree. Finally, through a system of exercises, students get an idea of the dependency graphs y = x and y = x. Throughout the topic, the author assumes the active use of the calculator, not only as a tool for extracting roots, but also as a means to illustrate some theoretical ideas. Due to the need to use a calculator to extract cube roots, another notation for the root of a positive number n is introduced: a = a 1 n. V. Olkhov's article draws attention to the fact that when studying the "Square Roots" section, special attention should be paid to the transformation of a complex radical. The author proposes the following methodology, giving an example of an individual form of work with a student when studying a topic: A student of a mathematical class was asked, using the Vieta theorem, to find by selection the roots in the equation x - 7x + 10 = 0, which he did without much difficulty: X 1 = 5, X = (even a little offended by the simplicity of the question). Then it was proposed to simplify the expression 7 ± 10. Here one should see a full square under the radical. Having previously written down the cumbersome formula A ± B= A A B ± A A B, (1) he substituted specific numerical values into it and obtained ± = 5 ±. But there is a direct analogy with the previous example 7=5 +, 10=5*, i.e. 7 ± 10= 5 + ± 5 = (5 ±) = 5 ± After that, the student has independently solved several examples: 8 29 6 ± 4 = 6 ± 8 = 4 + ± 4 =±, 1 ± 48= 1 ± 1 = ± 1 1= 1 ± 1, 18 ± 18= 18 ± = 16 + ± 16 =4±, 7 ± 4 = 7 ± 1 = 4 + ± 4 =± and said that now he understands how formula (1) arose, although it is not necessary to memorize it specially. A ± B= A ± B 4. Write the equation: X AX + B = 0, 4 X 1 = A+ A B, X = A A B ; A ± B \u003d A ± B 4 \u003d X 1 + X ± X 1 X \u003d X 1 ± X \u003d A + A B ± A A B, where A> 0, B> 0, A -B> 0, and the formula is simplified when A - In an exact square. Exercise 1. Show that = ; = 1 + The author of the article, V.I. Sedakova offers simple methods that allow you to quickly perform actions such as extracting square roots in your mind. These methods can increase productivity in the classroom, because oral and semi-oral exercises provide an opportunity to study a large amount of material in the lesson, allow the teacher to judge the readiness of the class 9 30 to learning new material. This material is useful for future teachers of mathematics. One of the main tasks of teaching a course of mathematics at school is the formation of conscious and strong computational skills in students. Computational skills are an important part of math skills. The topic of oral counting is especially relevant during the state final certification (OGE) and the unified state exam (USE), where the use of computing devices is not allowed. In combination with other forms of work, oral exercises make it possible to create conditions under which various types of students' activities are activated: thinking, speech, motor skills. That is why it is necessary to allocate up to 10 minutes for exercises with mental calculations in each lesson of mathematics. The formation of computational skills is a complex and systematic process. It consists of the following stages: The first stage of skill formation is mastering the skill. The second stage is the stage of skill automation. The automation of the skill is to get results when performing exercises orally, practically without making notes, notes, etc. Imagine the reception of oral counting in the topic "Square Roots" for students. Extracting the square root of a multivalued natural number. First, we write the algorithm for extracting the square root in a general form, which can be used when working with natural numbers. 1. Let's divide the number into groups (from right to left, starting from the last digit), including two adjacent digits in each group. In this case, one digit may appear in the last group (if the number of digits is odd) and two digits if the number of digits is even. The number of groups in such a number shows the number of digits of the result. 31 . We select the largest number, such that its square does not exceed the number in the last group (counting from right to left); this is the first digit of the result. Some number A will be obtained. Doubling the available part of the result, we obtain the number a. Now let's choose a digit x such that the product of the number a and x does not exceed the number A. The digit x is the second digit of the result. 4. We subtract the product of the number a by x from the number A, add the third group to the found difference on the right, we get some number B. Doubling the available part of the result, we get the number b. Now we choose the largest digit y so that the product of the number by and y does not exceed the number B. The digit y is the third digit of the result. 5. The next step of the rule repeats the 4th step. This continues until the very first group of number is used. Example 14. Let's demonstrate this algorithm using a simpler example, the result of which is obvious. We calculate 144. From the table of squares of natural numbers within two tens, we know that 144 = 1. In the number 144, from right to left, we separate two digits, 1/44. We got two groups of numbers, so the result is a two-digit number. We select a number whose square does not exceed the number in the second group (we count from right to left), this is the number 1. In our case, this number will be the number 1, because its square is equal to one. This means that in the answer in the category of tens there will be the number 1. From the number 144 we subtract the resulting number of tens, in the remainder we get the number 44. Let's determine the number of units in the answer. To do this, on the left, multiply the resulting figure of tens by, we get. Let's pick this 1 32 is a number, when multiplied by itself and by the resulting number, 44 is obtained. This number is, therefore, when extracting the square root of 144, we get the number 1. We select the numbers of the answer 1_. Answer: 144=1. Example 15. Consider the process of extracting square roots from a five-digit number. We select the numbers of the answer 4 Answer: 54756=4. Conclusions on the first chapter In chapter 1, the main goals and objectives of teaching the topic "Square Roots" in the algebra course of the primary school on the basis of the Federal State Educational Standards LLC and programs in mathematics were considered. An analysis of the theoretical and task material in algebra textbooks for grade 8 on the topic showed that the authors of the textbooks use different approaches to introducing both the concept of a square root itself and to a system of exercises focused on developing skills to calculate square roots and simplify numerical expressions. An analysis of the practical experience of studying the topic "Square Roots" on the basis of articles and teaching aids allows us to conclude that the topic is quite difficult for students. However, with the help of appropriate exercises and a special technique, one can achieve a solid assimilation of the concept of a square root and its basic properties. 33 CHAPTER II. METHODOLOGICAL RECOMMENDATIONS FOR THE ORGANIZATION OF TEACHING THE TOPIC "SQUARE ROOTS" IN THE COURSE OF ALGEBRA OF THE BASIC SCHOOL 4. Tasks on the topic "Square Roots", focused on the basic level of knowledge and skills in the course of algebra of the primary school All tasks on the topic "Square Roots" presented in algebra textbooks 8 classes can be conditionally combined into 4 groups: Group 1. Tasks for finding the value of expressions containing square roots. Group. Problems for solving quadratic equations using the arithmetic square root. Group. Tasks for simplifying and comparing expressions containing square roots. Group 4. Problems for extracting the square root. Consider examples of tasks: Group 1. Tasks for finding the value of expressions containing square roots. Example 1. Find the value of the expression: a) 1.5 0.1 0.5 b) 9 c) 16,. Solution: a) From the definition of the arithmetic root, it follows that 1.5=.5, because,5 > 0 and,5 = 1.5; 0.5= 0.5 because 0.5 > 0 and 0.5 = 0.5..5 0.1 0.5 = 7 0.05= 6.95 b) 9 = 9, because 9=9=9 34 c) This expression does not make sense, because the square of any number is a non-negative number. Answer: a) 6.95; b) 9; c) the expression does not make sense Example. Eliminate irrationality from the denominator: 1 a) 4 b) 7 c) Solution: 1 a) (1())() 4 1 b) 7 4 (7 4(7)() 7) 4(7 7) 4(7 4) 7 c) ((5 5 7)(7)(5 5 7) 7) (5 5 7) (5 6) 5 6 Answer: a) + b) 7 + c) 5 6 Group. Solving quadratic equations using the arithmetic square root Example. Find the value of x in the expression 10x 14 = 11. 4 35 10x 15 x 1.5 Solution: 10x x 14 x 15:10 10x x Check: , Answer: x = 1.5. 4 x x Example 4. Find the value of x in the expression 4 x = 1. Solution: x 1 Check: Answer: x Group. In this group, we combine tasks on simplifying expressions. Example 5 Simplify the expression: 5 36 6 Solution: To get rid of irrationality in the denominator of a fraction, you need to multiply the numerator and denominator of this fraction by the sum if the denominator contains the difference or the numerator and the denominator of this fraction by the difference if the denominator contains the sum) () ())(( ))(() ())(())(())(())((Answer: 4 6 Example 6. Simplify the expression: 8 4 Solution: Answer: 6 Example 7. Simplify the expression: ,5 8 Solution: ,5 8 (using the square root theorem) Answer: 5 37 Group 4. square root. In this group, we will offer extraction problems Example 8. Extract the root of the expression Solution: 5a 6 49 Let's use the theorem on extracting the arithmetic square root from a fraction. 5a a a a a 7 6 Let's use the theorem on extracting the arithmetic square root from the product. 5a a a a 6 Next, we use the following theorem: for any number a, the equality a a a 5a 5a 5 a 5 (a) 5 a 5 a is true Answer: If a 0, then If a< 0, то 5 a 7 5 a a 5 7 a Пример 9. Внести множитель под знак корня (буквами обозначены положительные числа): 1) a a 1) x x 5 7 38 1) a Solution: a a a a a a (The reciprocal square root theorem is used.) x x x x x x x x (The reciprocal square root theorem is used). Answer: 1) a) x 11x 4 1) 64 Example 10. Extract the root: x) 400 a, where a< 0 Решение: 1) 11x x x 4 11 (x 8) 11 x 8 11x 8 1 x 8) 400 a a a a 0 a (Используется теорема об извлечении арифметического квадратного корня из дроби, теорема об извлечении арифметического квадратного корня из произведения и теорема: a a). Ответ: 1) 1 x 8 0 a 8 39 The article offers multivariant didactic materials (Tasks in cards) aimed at simplifying numerical expressions with roots. They will undoubtedly assist the teacher of mathematics in organizing independent or test work. Let's give options. Option 1 1. Simplify: Simplify: Get rid of the irrationality in the denominator: Simplify the expression Calculate: 7 * Simplify the expression 6+4 4, Find the value of the expression 8. Calculate: * Find the value of the expression 10. Simplify the expression ()(75 7) and prove that the resulting number is the root of the equation x 0 = 0. Option 1. Simplify: WORKING PROGRAM ON ALGEBRA FOR 8 GRADES (general education level) Compiled by: Tikhonov VA, teacher of mathematics; Program implementation period: 1 year The work program is based on the federal MATHEMATICS EXPLANATORY NOTE This work program was developed on the basis of the Federal component of the State Educational Standard of Basic General Education and the Program of Basic General Work program for basic general education in mathematics at MBOU secondary school 30 Penza (grade 5) Explanatory note Status of the document Work program for basic general education in mathematics for grade 5 Annotation to the work program in mathematics in the 5th grade. Explanatory note The work program in mathematics for the 2016-2017 academic year in grade 5 is based on: 1. Federal Law 273 FZ 12/29/2012 Explanatory note The work program in mathematics is compiled on the basis of the following regulatory documents and guidelines: 1. Federal State Educational Standard of the Basic Explanatory note. This work program is aimed at students in grade 8 and is implemented on the basis of the following documents:. State standard of elementary general, basic general and secondary Working curriculum MATHEMATICS 5-6 grades 2017-2018 academic year ABSTRACT This working program has been developed in accordance with the main provisions of the Federal State Educational Municipal budgetary educational institution "Secondary school 17" of Belgorod "Agreed" Head of the ShMO N.A. Ilminskaya Protocol of 20 "Agreed" Deputy Director Mathematics work program for Grade 5 Considered at the Approving meeting of the Ministry of Defense, the director of teachers of cultural and MKOU LSOSH 1 technological activities M.M. Kostin and SPL service Order 109 Protocol 01 of September 01, 2017. dated September 01, 2017 Appendix to the basic educational program of basic general education order 488os dated 30.08.208. Tyumen Region Khanty-Mansi Autonomous Okrug Yugra Nizhnevartovsky District 1. Explanatory note The work program in algebra for grade 9 was compiled on the basis of regulatory documents and information and methodological materials: 1. On education in the Russian Federation: Federal Calendar-thematic planning of educational material in algebra for grade 8. Explanatory note Calendar-thematic planning in algebra for grade 8 is based on an exemplary program Requirements for the level of preparation of students of basic general education: Students must know/understand: - the importance of mathematics for solving problems that arise in theory and practice; latitude and at the same WORKING PROGRAM Class (level) at which the training course is studied 8 Subject area Mathematics and Computer Science Subject Mathematics (Algebra) Academic year 2017-2018 Number of hours per year 102 Municipal budgetary educational institution Gymnasium 4, Khimki APPROVED: Director of MBOU Gymnasium 4 / N.N. Kozelskaya / Order of 2015 Work program in algebra (basic level) Grade 8 Considered at the meeting of the school's pedagogical council in 2009. "Agreed" 2009 "Approved" Director of MBOSHI "KSHI" Taipova A.R. 2009 REVIEWED: at the meeting of the MO / ZYMurtazaeva Pr from AGREED: Deputy Director for Water Resource Management / EKKhairetdinova APPROVED: School Director / LMAmetova Pr from WORKING PROGRAM Algebra at 8 A MBOU "Starokrymskaya OSH" Content. Explanatory note 3 p. MUNICIPAL BUDGET GENERAL EDUCATIONAL INSTITUTION SECONDARY EDUCATIONAL SCHOOL 40 LIPETSK ALGEBRA WORKING PROGRAM for students with hearing disabilities Grade 8 Explanatory note This work program of the subject "Algebra" for students of the 8th grade of a general educational institution was developed on the basis of the author's program of basic general education Explanatory note This algebra program for the main general education school of the 8th grade is compiled on the basis of the federal component of the state standard for basic general education 1. Planned results of mastering the subject The subject results of studying the subject "Mathematics" in the 6th grade are the formation of the following skills: Appendix to the work program in mathematics Murmansk region, Kola district, p. Minkino State Regional Budgetary Educational Institution "Minkino Correctional Boarding School" Work program in mathematics Grade 6. 1. Calendar-thematic plan of the lesson Educational sections and topics Date Number of hours I quarter (42 lessons) 1. Divisibility of numbers (20 lessons) 1.09-28.09 1-3 Divisors PLANNED RESULTS Personal Metasubject Subject initial ideas about ideas and methods of mathematics as a universal language of science and technology, a means of modeling phenomena and processes; 1 EXPLANATORY NOTE The work program for the subject "Algebra" in the 9th grade is based on the federal component of the state standard for basic general education. This work program Explanatory note The work program for algebra grade 8 corresponds to the Federal component of the state educational standard of elementary general, basic general and secondary (complete) general Explanatory note The work program is based on: - The federal component of the state educational standard for basic general education in mathematics - Exemplary programs in mathematics. Chapter INTRODUCTION TO ALGEBRA .. SQUARE THREE-MEMBER ... The Babylonian problem of finding two numbers by their sum and product. One of the oldest problems in algebra was proposed in Babylon, where Explanatory note. This work program on the subject "Mathematics" for students of the 6th grade of a general education institution was developed on the basis of the author's program by S.M. Nikolsky, M.K. Potapov, EXPLANATORY NOTE. (Mathematics Grade 5) This work program is compiled in accordance with the State Program in Mathematics for General Educational Institutions of the Ministry of Education of the Russian Federation Explanatory note The work program in algebra was developed on the basis of the following regulatory legal documents: Federal Law No. 273-FZ of December 29, 2012 “On Education in the Russian Federation”; Order Explanatory note The work program in algebra for grade 8 was compiled in accordance with the provisions of the Federal State Educational Standard for Basic General Education of the second generation, Status of the document Explanatory note Considered Accepted Approved At the MoE of mathematics teachers at the meeting Director of the MoU SOSH Protocol 1 of 26.08. 2014. pedagogical p. Poima Head of the MO Praslova O.M. Council Rodionova O.I. Protocol 1 Private educational institution Lyceum 1 "Sputnik" CONSIDERED At a meeting of the Methodological Council of Lyceum 1 "Sputnik" Minutes of 2017 Chairman of the Methodological Council of the Lyceum 1 "Sputnik" Ursul Annotation to the work program grade 8, algebra 1. Explanatory note. The work program in algebra is compiled on the basis of the author's program "Algebra 8kl." ed. Makarychev and others in accordance with the content of the content of the educational subjects of the component of the state WORKING PROGRAM ON ALGEBRA AND GEOMETRY FOR 7 "A" CLASS FOR 2018 2019 ACADEMIC YEAR Municipal budgetary educational institution Secondary school 4 Considered at the pedagogical council Minutes 1 of 31.08. 2017 Order 162 dated 08/31/2017 APPROVED: Director Control and measuring materials for intermediate certification in mathematics in 2018 Grade 7 Explanatory note The content of the work is built in accordance with: with the Federal Law of the Russian Explanatory note Regulatory documents The work program is based on: federal law of the Russian Federation of 9..0 years 73-FZ "On Education in the Russian Federation" of the federal component Explanatory note The work program is based on: Order of the Ministry of Education of the Russian Federation dated 05.03.2004 1089 "On approval of the Federal component of state educational standards 1. PLANNED OUTCOMES OF MASTERING THE SUBJECT The study of mathematics in basic school enables students to achieve the following results: In the direction of personal development: - the ability to clearly, EXPLANATORY NOTE The work program in algebra for grade 8 was compiled in accordance with the provisions of the Federal State Educational Standard for Basic General Education of the second generation, APPENDIX to the educational program CALENDAR THEMATIC PLANNING in algebra in grade 8 Textbook "ALGEBRA 8", author Yu. N. Makarychev and others, edited by S. A. Telyakovsky Teacher: Dudnikova Explanatory note The algebra program for the basic school is compiled in accordance with the requirements of: - the federal component of the state educational standard for basic general education Explanatory note The work program in algebra for grade 8 (in-depth study) was compiled in accordance with the federal component of the state educational standard, the program in algebra MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION NOVOSIBIRSK STATE UNIVERSITY SPECIALIZED EDUCATIONAL AND SCIENTIFIC CENTER Mathematics Grade 8 Polynomials Novosibirsk Polynomials Rational The work program has been drawn up in accordance with the following regulatory documents: Federal Law No. 273-FZ of February 29, 202 "On Education in the Russian Federation", the requirements of the Federal State Educational Explanatory note This work program "Algebra" was developed on the basis of: - Federal Law of December 29, 2012 273-FZ (as amended on July 13, 2015) "On Education in the Russian Federation"; - based on copyright The work program was drawn up in accordance with the regulatory documents: Federal Law No. 273-FZ of February 29, 202 “On Education in the Russian Federation”. 2. Order of the Ministry of Education and Science of the Russian Ministry of General and Vocational Education RO state budget educational institution primary vocational education in the Rostov region vocational school no. 5 Practical work in the discipline of ODP. 01."Mathematics: Algebra and Beginnings
mathematical analysis; geometry"
on this topic:
"Conversions of expressions containing roots, powers and logarithms».
For students I course G. Rostov-on-Don 2017 Section number 1.
Algebra.
Topic 1.2.
Roots, powers and logarithms.
Practical lesson number 1.
Subject:
"Conversions of Expressions Containing Roots, Powers, and Logarithms". Target: know
properties of radicals, powers and logarithms; be able to apply them
performing transformations of expressions containing roots, degrees and logarithms. Number of hours
: 1 hour. theoretical material.
Roots.
The action by which the root is foundn-th degree, is called root extractionn-th degree. Definition.
Arithmetic root of natural degreen≥ 2 from a non-negative number a is called a non-negative number,nwhose th power is a. The arithmetic root of the second degree is also called the square root, and the root of the third degree is also called the cube root. For example. Calculate: arithmetic rootnth degree has the following properties: if a ≥ 0, b > 0 and n,
mare natural numbers, andn ≥ 2,
m≥ 2, then 1. 3.
2. 4.
Examples of applying the properties of the arithmetic root. Properties of a degree with a rational exponent.
For any rational numbers p and k and any a > 0 and b > 0, the equalities are true: 1. ; 2. ;
3. ; 4. ;
5. .
Examples of applying degree properties: 1). 7*
4). .
Logarithm of a number
Definition.
The logarithm of a positive numberbin base a, wherea > 0,
a≠ 1, called the exponent to which the number must be raiseda, To obtain b.
a
=
b
is the basic logarithmic identity. Properties of logarithms
Let a > 0,
a ≠ 1,
b>0, c >0, k is any real number. Then the formulas are valid: 1
.
log
(
bc
) =
logb
+
logc
, 4.
logb
=
,
2.
log = log - log
c, 5.log
a = 1
,
3.
log
b
= to *
logb
, 6.
log
0 = 1
.
Examples of applying formulas: log2 + log 18
=log( 2 * 18
)
=
log 36 = 2;
log 48
-log 4
= log=
log 12 = 1;
log 9 = *
log 9
= .
Decide on your own
.
Tasks. 1 option
1. Calculate:
1) ; 4)
log ;
2) ; 5) 0,5;
3) ; 6) 3
log 2 -
log 64.
2 if x = 7. 3. Compare numbers:log 11 and log 19.
4. Simplify: 1) ; 2). 5. Calculate: logloglog 3.
_________________________________________________________________
Option 2
1. Calculate:
1) ; 4)
log 64;
2) ; 5) ;
3) ; 6) 2
log 3 -
log 81.
2. Find the value of the expression: 3 if y = 2. 3. Compare numbers:log And log.
4. Simplify: 1) ; 2). 5. Calculate: logloglog 2.
__________________________________________________________________
Criteria for evaluation:
11 correct tasks - "5"; 9 - 10 correct tasks - "4"; 7 - 8 correct tasks - "3". Bashmakov. M. I. Mathematics: a textbook for NGOs and SPO. - M.: Publishing Center "Academy", 2013. Alimov Sh.A. et al. Algebra and the Beginnings of Analysis. 10 (11) class. – M.: 2012. Algebra. Grade 9: Textbook, task book for general education. institutions/ A.G. Mordkovich and others - M .: Mnemozina, 2009. Algebra. Grade 8: Textbook, task book for general education. institutions/ A.G. Mordkovich and others - M .: Mnemozina, 2008. Algebra. Grade 7: Textbook, task book for general education. institutions/ A.G. Mordkovich and others - M .: Mnemozina, 2007. Reporting form:
verification of assignments by the teacher To successfully use the operation of extracting the root in practice, you need to get acquainted with the properties of this operation. Theorem 1.
The nth root (n=2, 3, 4,...) of the product of two non-negative chipsets is equal to the product of the nth roots of these numbers:
Comment:
1.
Theorem 1 remains valid for the case when the radical expression is the product of more than two non-negative numbers. Theorem 2.If,
and n is a natural number greater than 1, then the equality Theorem 1 allows us to multiply m only roots of the same degree
, i.e. only roots with the same exponent. Theorem 3. If
,k is a natural number and n is a natural number greater than 1, then the equality In other words, to raise a root to a natural power, it is enough to raise the root expression to this power. Theorem 4. If
,k, n are natural numbers greater than 1, then the equality In other words, to extract a root from a root, it is enough to multiply the exponents of the roots. Be careful! We learned that four operations can be performed on roots: multiplication, division, exponentiation, and extracting the root (from the root). But what about the addition and subtraction of roots? No way. Theorem 5. If
the indicators of the root and the root expression are multiplied or divided by the same natural number, then the value of the root will not change, i.e. Examples of problem solving Example 2 Calculate Example 3 Calculate: Solution. Any formula in algebra, as you well know, is used not only "from left to right", but also "from right to left". So, the first property of the roots means that it can be represented in the form and, conversely, can be replaced by the expression. The same applies to the second property of roots. With this in mind, let's do the calculations. Congratulations: today we will analyze the roots - one of the most mind-blowing topics of the 8th grade. :) Many people get confused about the roots not because they are complex (which is complicated - a couple of definitions and a couple more properties), but because in most school textbooks the roots are defined through such wilds that only the authors of the textbooks themselves can understand this scribbling. And even then only with a bottle of good whiskey. :) Therefore, now I will give the most correct and most competent definition of the root - the only one that you really need to remember. And only then I will explain: why all this is necessary and how to apply it in practice. But first, remember one important point, which for some reason many compilers of textbooks “forget” about: Roots can be of even degree (our favorite $\sqrt(a)$, as well as any $\sqrt(a)$ and even $\sqrt(a)$) and odd degree (any $\sqrt(a)$, $\ sqrt(a)$ etc.). And the definition of the root of an odd degree is somewhat different from the even one. Here in this fucking “somewhat different” is hidden, probably, 95% of all errors and misunderstandings associated with the roots. So let's clear up the terminology once and for all: Definition. Even root n from the number $a$ is any non-negative a number $b$ such that $((b)^(n))=a$. And the root of an odd degree from the same number $a$ is generally any number $b$ for which the same equality holds: $((b)^(n))=a$. In any case, the root is denoted like this: \(a)\] The number $n$ in such a notation is called the root exponent, and the number $a$ is called the radical expression. In particular, for $n=2$ we get our “favorite” square root (by the way, this is a root of an even degree), and for $n=3$ we get a cubic root (an odd degree), which is also often found in problems and equations. Examples. Classic examples of square roots: \[\begin(align) & \sqrt(4)=2; \\ & \sqrt(81)=9; \\ & \sqrt(256)=16. \\ \end(align)\] By the way, $\sqrt(0)=0$ and $\sqrt(1)=1$. This is quite logical since $((0)^(2))=0$ and $((1)^(2))=1$. Cubic roots are also common - do not be afraid of them: \[\begin(align) & \sqrt(27)=3; \\ & \sqrt(-64)=-4; \\ & \sqrt(343)=7. \\ \end(align)\] Well, a couple of "exotic examples": \[\begin(align) & \sqrt(81)=3; \\ & \sqrt(-32)=-2. \\ \end(align)\] If you do not understand what is the difference between an even and an odd degree, reread the definition again. It is very important! In the meantime, we will consider one unpleasant feature of the roots, because of which we needed to introduce a separate definition for even and odd exponents. After reading the definition, many students will ask: “What did mathematicians smoke when they came up with this?” And really: why do we need all these roots? To answer this question, let's go back to elementary school for a moment. Remember: in those distant times, when the trees were greener and the dumplings were tastier, our main concern was to multiply the numbers correctly. Well, something in the spirit of "five by five - twenty-five", that's all. But after all, you can multiply numbers not in pairs, but in triplets, fours, and generally whole sets: \[\begin(align) & 5\cdot 5=25; \\ & 5\cdot 5\cdot 5=125; \\ & 5\cdot 5\cdot 5\cdot 5=625; \\ & 5\cdot 5\cdot 5\cdot 5\cdot 5=3125; \\ & 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5=15\ 625. \end(align)\] However, this is not the point. The trick is different: mathematicians are lazy people, so they had to write down the multiplication of ten fives like this: So they came up with degrees. Why not write the number of factors as a superscript instead of a long string? Like this one: It's very convenient! All calculations are reduced by several times, and you can not spend a bunch of parchment sheets of notebooks to write down some 5 183 . Such an entry was called the degree of a number, a bunch of properties were found in it, but happiness turned out to be short-lived. After a grandiose booze, which was organized just about the “discovery” of degrees, some especially stoned mathematician suddenly asked: “What if we know the degree of a number, but we don’t know the number itself?” Indeed, if we know that a certain number $b$, for example, gives 243 to the 5th power, then how can we guess what the number $b$ itself is equal to? This problem turned out to be much more global than it might seem at first glance. Because it turned out that for the majority of “ready-made” degrees there are no such “initial” numbers. Judge for yourself: \[\begin(align) & ((b)^(3))=27\Rightarrow b=3\cdot 3\cdot 3\Rightarrow b=3; \\ & ((b)^(3))=64\Rightarrow b=4\cdot 4\cdot 4\Rightarrow b=4. \\ \end(align)\] What if $((b)^(3))=50$? It turns out that you need to find a certain number, which, when multiplied by itself three times, will give us 50. But what is this number? It is clearly greater than 3 because 3 3 = 27< 50. С тем же успехом оно меньше 4, поскольку 4 3 = 64 >50. I.e. this number lies somewhere between three and four, but what it is equal to - FIG you will understand. This is exactly why mathematicians came up with $n$-th roots. That is why the radical icon $\sqrt(*)$ was introduced. To denote the same number $b$, which, to the specified power, will give us a previously known value \[\sqrt[n](a)=b\Rightarrow ((b)^(n))=a\] I do not argue: often these roots are easily considered - we saw several such examples above. But still, in most cases, if you think of an arbitrary number, and then try to extract the root of an arbitrary degree from it, you are in for a cruel bummer. What is there! Even the simplest and most familiar $\sqrt(2)$ cannot be represented in our usual form - as an integer or a fraction. And if you drive this number into a calculator, you will see this: \[\sqrt(2)=1.414213562...\] As you can see, after the decimal point there is an endless sequence of numbers that do not obey any logic. You can, of course, round this number to quickly compare with other numbers. For example: \[\sqrt(2)=1.4142...\approx 1.4 \lt 1.5\] Or here's another example: \[\sqrt(3)=1.73205...\approx 1.7 \gt 1.5\] But all these roundings are, firstly, rather rough; and secondly, you also need to be able to work with approximate values, otherwise you can catch a bunch of non-obvious errors (by the way, the skill of comparison and rounding is necessarily checked at the profile exam). Therefore, in serious mathematics, one cannot do without roots - they are the same equal representatives of the set of all real numbers $\mathbb(R)$, like fractions and integers that we have long known. The impossibility of representing the root as a fraction of the form $\frac(p)(q)$ means that this root is not a rational number. Such numbers are called irrational, and they cannot be accurately represented except with the help of a radical, or other constructions specially designed for this (logarithms, degrees, limits, etc.). But more on that another time. Consider a few examples where, after all the calculations, irrational numbers will still remain in the answer. \[\begin(align) & \sqrt(2+\sqrt(27))=\sqrt(2+3)=\sqrt(5)\approx 2,236... \\ & \sqrt(\sqrt(-32 ))=\sqrt(-2)\approx -1,2599... \\ \end(align)\] Naturally, by the appearance of the root, it is almost impossible to guess which numbers will come after the decimal point. However, it is possible to calculate on a calculator, but even the most advanced date calculator gives us only the first few digits of an irrational number. Therefore, it is much more correct to write the answers as $\sqrt(5)$ and $\sqrt(-2)$. That's what they were invented for. To make it easy to write down answers. The attentive reader has probably already noticed that all the square roots given in the examples are taken from positive numbers. Well, at least from zero. But cube roots are calmly extracted from absolutely any number - even positive, even negative. Why is this happening? Take a look at the graph of the function $y=((x)^(2))$: Let's try to calculate $\sqrt(4)$ using this graph. To do this, a horizontal line $y=4$ (marked in red) is drawn on the graph, which intersects the parabola at two points: $((x)_(1))=2$ and $((x)_(2)) =-2$. This is quite logical, since Everything is clear with the first number - it is positive, therefore it is the root: But then what to do with the second point? Does the 4 have two roots at once? After all, if we square the number −2, we also get 4. Why not write $\sqrt(4)=-2$ then? And why do teachers look at such records as if they want to eat you? :) The trouble is that if no additional conditions are imposed, then the four will have two square roots - positive and negative. And any positive number will also have two of them. But negative numbers will not have roots at all - this can be seen from the same graph, since the parabola never falls below the axis y, i.e. does not take negative values. A similar problem occurs for all roots with an even exponent: That is why the definition of an even root $n$ specifically stipulates that the answer must be a non-negative number. This is how we get rid of ambiguity. But for odd $n$ there is no such problem. To see this, let's take a look at the graph of the function $y=((x)^(3))$: Two conclusions can be drawn from this graph: It's a pity that these simple things are not explained in most textbooks. Instead, our brains begin to soar with all sorts of arithmetic roots and their properties. Yes, I do not argue: what is an arithmetic root - you also need to know. And I will talk about this in detail in a separate lesson. Today we will also talk about it, because without it, all reflections on the roots of the $n$-th multiplicity would be incomplete. But first you need to clearly understand the definition that I gave above. Otherwise, due to the abundance of terms, such a mess will begin in your head that in the end you will not understand anything at all. And all you need to understand is the difference between even and odd numbers. Therefore, once again we will collect everything that you really need to know about the roots: Is it difficult? No, it's not difficult. It's clear? Yes, it's obvious! Therefore, now we will practice a little with the calculations. Roots have a lot of strange properties and restrictions - this will be a separate lesson. Therefore, now we will consider only the most important "chip", which applies only to roots with an even exponent. We write this property in the form of a formula: \[\sqrt(((x)^(2n)))=\left| x\right|\] In other words, if we raise a number to an even power, and then extract the root of the same degree from this, we will get not the original number, but its modulus. This is a simple theorem that is easy to prove (it suffices to consider separately non-negative $x$, and then separately consider negative ones). Teachers constantly talk about it, it is given in every school textbook. But as soon as it comes to solving irrational equations (i.e. equations containing the sign of the radical), the students forget this formula together. To understand the issue in detail, let's forget all the formulas for a minute and try to count two numbers ahead: \[\sqrt(((3)^(4)))=?\quad \sqrt(((\left(-3 \right))^(4)))=?\] These are very simple examples. The first example will be solved by most of the people, but on the second, many stick. To solve any such crap without problems, always consider the procedure: Let's deal with the first expression: $\sqrt(((3)^(4)))$. Obviously, you first need to calculate the expression under the root: \[((3)^(4))=3\cdot 3\cdot 3\cdot 3=81\] Then we extract the fourth root of the number 81: Now let's do the same with the second expression. First, we raise the number −3 to the fourth power, for which we need to multiply it by itself 4 times: \[((\left(-3 \right))^(4))=\left(-3 \right)\cdot \left(-3 \right)\cdot \left(-3 \right)\cdot \ left(-3 \right)=81\] We got a positive number, since the total number of minuses in the product is 4 pieces, and they will all cancel each other out (after all, a minus by a minus gives a plus). Next, extract the root again: In principle, this line could not be written, since it is a no brainer that the answer will be the same. Those. an even root of the same even power "burns" the minuses, and in this sense the result is indistinguishable from the usual module: \[\begin(align) & \sqrt(((3)^(4)))=\left| 3\right|=3; \\ & \sqrt(((\left(-3 \right))^(4)))=\left| -3 \right|=3. \\ \end(align)\] These calculations are in good agreement with the definition of the root of an even degree: the result is always non-negative, and the radical sign is also always a non-negative number. Otherwise, the root is not defined. Thus, in no case should one thoughtlessly reduce the roots and degrees, thereby supposedly "simplifying" the original expression. Because if there is a negative number under the root, and its exponent is even, we will get a lot of problems. However, all these problems are relevant only for even indicators. Naturally, roots with odd exponents also have their own feature, which, in principle, does not exist for even ones. Namely: \[\sqrt(-a)=-\sqrt(a)\] In short, you can take out a minus from under the sign of the roots of an odd degree. This is a very useful property that allows you to "throw" all the minuses out: \[\begin(align) & \sqrt(-8)=-\sqrt(8)=-2; \\ & \sqrt(-27)\cdot \sqrt(-32)=-\sqrt(27)\cdot \left(-\sqrt(32) \right)= \\ & =\sqrt(27)\cdot \sqrt(32)= \\ & =3\cdot 2=6. \end(align)\] This simple property greatly simplifies many calculations. Now you don’t need to worry: what if a negative expression got under the root, and the degree at the root turned out to be even? It is enough to “throw out” all the minuses outside the roots, after which they can be multiplied by each other, divided and generally do many suspicious things, which in the case of “classic” roots are guaranteed to lead us to an error. And here another definition enters the scene - the very one with which most schools begin the study of irrational expressions. And without which our reasoning would be incomplete. Meet! Let's assume for a moment that only positive numbers or, in extreme cases, zero can be under the root sign. Let's score on even / odd indicators, score on all the definitions given above - we will work only with non-negative numbers. What then? And then we get the arithmetic root - it partially intersects with our "standard" definitions, but still differs from them. Definition. An arithmetic root of the $n$th degree of a non-negative number $a$ is a non-negative number $b$ such that $((b)^(n))=a$. As you can see, we are no longer interested in parity. Instead, a new restriction appeared: the radical expression is now always non-negative, and the root itself is also non-negative. To better understand how the arithmetic root differs from the usual one, take a look at the graphs of the square and cubic parabola already familiar to us: As you can see, from now on, we are only interested in those pieces of graphs that are located in the first coordinate quarter - where the coordinates $x$ and $y$ are positive (or at least zero). You no longer need to look at the indicator to understand whether we have the right to root a negative number or not. Because negative numbers are no longer considered in principle. You may ask: “Well, why do we need such a castrated definition?” Or: "Why can't we get by with the standard definition given above?" Well, I will give just one property, because of which the new definition becomes appropriate. For example, the exponentiation rule: \[\sqrt[n](a)=\sqrt(((a)^(k)))\] Please note: we can raise the radical expression to any power and at the same time multiply the root exponent by the same power - and the result will be the same number! Here are some examples: \[\begin(align) & \sqrt(5)=\sqrt(((5)^(2)))=\sqrt(25) \\ & \sqrt(2)=\sqrt(((2)^ (4)))=\sqrt(16) \\ \end(align)\] Well, what's wrong with that? Why couldn't we do it before? Here's why. Consider a simple expression: $\sqrt(-2)$ is a number that is quite normal in our classical sense, but absolutely unacceptable from the point of view of the arithmetic root. Let's try to convert it: $\begin(align) & \sqrt(-2)=-\sqrt(2)=-\sqrt(((2)^(2)))=-\sqrt(4) \lt 0; \\ & \sqrt(-2)=\sqrt(((\left(-2 \right))^(2)))=\sqrt(4) \gt 0. \\ \end(align)$ As you can see, in the first case, we took the minus out from under the radical (we have every right, because the indicator is odd), and in the second, we used the above formula. Those. from the point of view of mathematics, everything is done according to the rules. WTF?! How can the same number be both positive and negative? No way. It's just that the exponentiation formula, which works great for positive numbers and zero, starts to give complete heresy in the case of negative numbers. Here, in order to get rid of such ambiguity, they came up with arithmetic roots. A separate large lesson is devoted to them, where we consider in detail all their properties. So now we will not dwell on them - the lesson turned out to be too long anyway. I thought for a long time: to make this topic in a separate paragraph or not. In the end, I decided to leave here. This material is intended for those who want to understand the roots even better - no longer at the average “school” level, but at the level close to the Olympiad. So: in addition to the "classical" definition of the root of the $n$-th degree from a number and the associated division into even and odd indicators, there is a more "adult" definition, which does not depend on parity and other subtleties at all. This is called an algebraic root. Definition. An algebraic $n$-th root of any $a$ is the set of all numbers $b$ such that $((b)^(n))=a$. There is no well-established designation for such roots, so just put a dash on top: \[\overline(\sqrt[n](a))=\left\( b\left| b\in \mathbb(R);((b)^(n))=a \right. \right\) \] The fundamental difference from the standard definition given at the beginning of the lesson is that the algebraic root is not a specific number, but a set. And since we are working with real numbers, this set is of only three types: The last case deserves more detailed consideration. Let's count a couple of examples to understand the difference. Example. Compute expressions: \[\overline(\sqrt(4));\quad \overline(\sqrt(-27));\quad \overline(\sqrt(-16)).\] Solution. The first expression is simple: \[\overline(\sqrt(4))=\left\( 2;-2 \right\)\] It is two numbers that are part of the set. Because each of them squared gives a four. \[\overline(\sqrt(-27))=\left\( -3 \right\)\] Here we see a set consisting of only one number. This is quite logical, since the exponent of the root is odd. Finally, the last expression: \[\overline(\sqrt(-16))=\varnothing \] We got an empty set. Because there is not a single real number that, when raised to the fourth (that is, even!) Power, will give us a negative number −16. Final note. Please note: it was not by chance that I noted everywhere that we are working with real numbers. Because there are also complex numbers - it is quite possible to calculate $\sqrt(-16)$ and many other strange things there. However, in the modern school curriculum of mathematics, complex numbers are almost never found. They have been omitted from most textbooks because our officials consider the topic "too difficult to understand." That's all. In the next lesson, we will look at all the key properties of roots and finally learn how to simplify irrational expressions. :)
All properties are formulated and proved only for non-negative values of variables contained under root signs.
Brief(albeit inaccurate) formulation that is more convenient to use in practice: the root of the fraction is equal to the fraction of the roots.
This is a consequence of Theorem 1. Indeed, for example, for k = 3 we get
For example,
For example, you can’t write instead of Indeed, But it’s obvious that
Example 1 Calculate
Solution. Using the first property of the roots (Theorem 1), we get:
Solution. Convert the mixed number to an improper fraction.
We have Using the second property of the roots ( theorem 2
), we get:
Why do we need roots at all?
Why are two definitions needed?
The graph of a quadratic function gives two roots: positive and negative 
The cubic parabola takes on any value, so the cube root can be taken from any number
Basic properties and restrictions
Note on the order of operations
Removing a minus sign from under the root sign
arithmetic root
Root search area - non-negative numbers Algebraic root: for those who want to know more
