Introduction to the second volume of mathematics, the standard experimental textbook of compulsory education curriculum?

Institute of Curriculum and Textbooks? Zuo Huailing?

The second volume of compulsory education curriculum standard experimental textbook "Mathematics" consists of five chapters, and it will take about 6 1 class hour to use in the next semester of grade eight. Details are as follows:

Chapter 16? Scores? (about 13 class hours)?

Chapter 17? Inverse proportional function? (About 8 class hours? )?

Chapter 18? Pythagorean theorem? (About 8 class hours? )?

Chapter 19? Quadrilateral? (about 17 class hours)?

Chapter 20? Data analysis? (about 15 class hours)?

The five chapters of this book cover four areas of mathematics curriculum standards: number and algebra, space and graphics, statistics and probability, practice and comprehensive application. For the content in the field of "practice and comprehensive application", this book arranges a special study in chapter 19 and chapter 20 respectively, and arranges 2 ~ 3 math activities at the end of each chapter, and implements the requirements of "practice and comprehensive application" through these special study and math activities. Generally speaking, these five chapters are arranged in a concentrated way. The first two chapters basically belong to the field of number and algebra, the last two chapters basically belong to the field of space and graphics, and the last chapter is the field of statistics and probability. This arrangement helps to strengthen the vertical connection between knowledge. In the preparation of the specific content of each chapter, special attention is paid to strengthening the horizontal connection between various fields. ?

First, content analysis?

"Chapter 16? Score "

This chapter mainly studies fractions and their basic properties, addition, subtraction, multiplication and division, Divison, fractional equations and so on. These contents are divided into three parts. ?

16. 1 Section gives the concept of fraction by analogy, discusses the basic properties of fraction, and introduces the general points and simplification points of the analogy method of fraction, which lays a theoretical foundation for the following two sections. Section 16.2 discusses four algorithms of fractions. Starting from practical problems, the textbook first studies the multiplication and division of fractions and discusses the multiplication and division algorithm of fractions by analogy. Next, the textbook also learns the addition and subtraction of fractions by analogy, obtains the operation rules, and learns the elementary arithmetic of fractions. Finally, the textbook combines the operation of fractions to study the exponential power of integers, and extends the operation properties of exponential power of positive integers to the range of integers, perfecting the scientific notation. The content of this section is the focus of the whole chapter, and the mixed operation of fractions is also the difficulty of the whole chapter. 16.3 discusses the concept and solution of fractional order equations, mainly involving fractional order equations that can be transformed into linear equations with one variable. Starting from practical problems, the textbook analyzes the quantitative relationship in the problem and lists the fractional equation, thus leading to the concept of fractional equation. Then, the solution of fractional equation is studied. Combining with the students' experience, the textbook discusses how to transform fractional equation into integral equation, so as to get the solution of fractional equation. If the basic properties of fraction are to be applied to solving fractional equations, it is necessary to test the roots, which is a problem that has not been encountered in previous equations. The textbook explains why the fractional equation needs root test with concrete examples. Fractional equation provides a mathematical model for solving practical problems, and it has a special function that the whole equation can't be replaced. Listing fractional equations according to practical problems is another difficulty in teaching this chapter. ?

"Chapter 17? Inverse proportional function "?

This chapter mainly includes the concept, image and properties of inverse proportional function, and the analysis and solution of practical problems by inverse proportional function. Chapter 8 (1) of this chapter "Chapter 1 1? The content of another chapter function after the first function. The whole chapter is divided into two sections: 17. 1 inverse proportional function, and 17.2 practical problems and inverse proportional function. The whole chapter revolves around practical problems, which is a main line running through the whole chapter. ?

17. 1 section mainly studies the concept, image and properties of inverse proportional function. Starting with several practical problems that students are familiar with, this section analyzes the corresponding relationship between variables in practical problems, lists the analytical formula of inverse proportional function, and introduces the concept of inverse proportional function, so that students' understanding of inverse proportional function has gone through a process from perceptual to rational; Next, the textbook draws the image of function sum by tracing points. By exploring the same characteristics of two function images, the fact that the inverse proportional function image belongs to hyperbola is given, and then the conclusion that the image of function sum is symmetrical about X axis and Y axis is drawn. Next, the textbook asks students to draw the images of function sum according to this conclusion, and further get the properties of inverse proportional function by analyzing the images drawn by these four functions. The content of section 17.2 is to use inverse proportional function analysis to solve practical problems. In this part, the textbook gives four practical problems by way of examples. These four problems are basically arranged in the order from simple to complex (cylinder bottom area and height, working time and speed, power arm, output power and resistance), which shows that inverse proportional function is an effective mathematical model to solve practical problems from different aspects. ?

"Chapter 18? Pythagorean theorem "

This chapter mainly studies Pythagorean theorem and the inverse theorem of Pythagorean theorem, including their discovery, proof and application. The whole chapter is divided into two sections, section 18. 1 is Pythagorean theorem, and section 18.2 is the inverse theorem of Pythagorean theorem. ?

In the section 18. 1, the textbook starts with Pythagoras' observation of the legend of Pythagorean theorem on the ground, and asks students to observe and calculate the relationship between the area of some small squares with two right-angled sides and the area of a square with a hypotenuse side, and finds that the sum of the areas of two small squares with right-angled sides is equal to the area of a square with a hypotenuse side, thus seeking Pythagorean theorem. At this time, the textbook took the 65438 proposition. There are many ways to prove Pythagorean theorem. The text of the textbook introduces the proof method of Zhao Shuang, an ancient man in China. After proving the correctness of the proposition 1 through reasoning, the textbook points out what a theorem is and makes it clear that the proposition 1 is Pythagorean theorem. Then, through three inquiry columns, the application of Pythagorean theorem in solving practical problems and mathematical problems (drawing unreasonable line segments, etc.) is discussed. ), so that students have a certain understanding of Pythagorean theorem. 18.2 is the inverse theorem of Pythagorean theorem. Starting with the method of drawing right angles by ancient Egyptians, the textbook gives the conclusion that a triangle is a right triangle when three sides of the triangle meet the requirements. Then, students draw some triangles with the sum of squares of sides equal to the square of the third side. By exploring the shapes of these triangles, we can find that all the drawn triangles are right-angled triangles, and guess that if the three sides of a triangle satisfy this relationship, then this triangle is a right-angled triangle. At this point, this inverse theorem is given in the form of proposition 2. By comparing the topics and conclusions of 1 proposition and 2 proposition, the textbook gives the concepts of original proposition and inverse proposition. Whether proposition 2 is correct needs to be proved. The textbook uses congruent triangles to prove Proposition 2, and obtains the inverse theorem of Pythagorean theorem. The inverse theorem of Pythagorean theorem gives a method to judge whether a triangle is a right triangle, which is widely used in mathematics and practice. Students can learn to solve problems in this way through two examples in the textbook. ?

"Chapter 19? Quadrilateral "?

This chapter mainly studies the concepts, properties and judgment methods of some special quadrangles. For special quadrangles, the teaching materials are divided into two categories according to the parallel relationship of opposite sides: two groups of quadrangles with parallel opposite sides-parallelogram, one group of quadrangles with parallel opposite sides, and the other group of quadrangles with non-parallel opposite sides-trapezoid. For parallelogram, besides the general parallelogram, several special parallelograms such as rectangle, diamond and square are also studied. ?

19. 1 section mainly studies the concept, properties and judgment of general parallelogram. Starting from the graphics in real life, the textbook abstractly summarizes the concept of parallelogram, and through a series of exploration activities, it obtains the nature and judgment method of parallelogram, and proves the conclusion appropriately through reasoning. As an application of judgment method, the textbook obtains the triangle midline theorem through examples. 19.2 mainly studies the concepts, properties and judgments of rectangle, rhombus and square. This section further studies these special parallelograms on the basis of the previous section. The textbook first studies rectangles and diamonds, both of which are parallelograms with special conditions. A rectangle is a parallelogram with right angles, and a diamond is a special parallelogram with a set of equal adjacent sides. On this basis, the textbook studies a parallelogram with two special conditions at the same time, namely a square, which is a special rhombus with a right angle and a group of special rectangles with equal adjacent sides. 19.3 section studies trapezoid, which is another special quadrangle juxtaposed with parallelogram, with one set of opposite sides parallel and the other set of opposite sides non-parallel. This section mainly introduces a special trapezoid-isosceles trapezoid, and discusses the properties and judgment methods of isosceles trapezoid. The last section of the textbook, namely 19.4, arranged a special study: the center of gravity. Through the activity of finding the center of gravity of geometric figures, it is known that the center of gravity of regular geometric figures is its geometric center, and the relationship between mathematics and physics is understood. ?

"Chapter 20? Data analysis "?

This chapter mainly studies the statistical significance of average (mainly weighted average), median, mode, range and variance. The whole chapter is divided into three sections. ?

20. Section1studies statistics that represent the trend of data sets: average, median and mode. In this section, the textbook first gives a practical problem, solves this practical problem through analysis, and introduces the concept of weighted average. In order to highlight the role and significance of "right", textbooks show the role of "right" from different aspects through two examples. Then the textbook expands the weighted average, including how to unify arithmetic average and weighted average, how to calculate the weighted average of interval grouping data, how to use the statistical function of calculator to calculate the average, how to estimate the overall average by sample average and so on. For median and mode, the textbook studies their statistical significance through several specific examples. At the end of this section, through a concrete example, the textbook studies the examples of comprehensively applying the average, median and mode to solve problems, and summarizes these three statistics, highlighting their respective statistical significance and characteristics. Section 20.2 will study statistics describing the degree of data fluctuation: range and variance. The textbook first uses the example of temperature difference to study the statistical significance of extreme difference. Variance is a statistic commonly used in statistics to describe the degree of data dispersion, and variance is studied in detail in textbooks. Firstly, the research on the fluctuation of two groups of data is put forward through a practical problem, and the scatter diagram is drawn to reflect the fluctuation of data intuitively. On this basis, the textbook introduces the method of describing the dispersion degree of data with variance, introduces the formula of variance, and analyzes how variance describes the fluctuation of data from the structure of variance formula. Then, the method of calculating variance by using the statistical function of calculator is introduced. At the end of this section, the textbook solves the problems raised in the preface of this chapter with what it has learned, and studies the problem of estimating population variance with sample variance. In the last section of the textbook, a comprehensive and practical "subject study" is arranged. This "research project" chooses physical health problems closely related to students' lives. Because this chapter is the last chapter in the statistics section, the comprehensive research on this topic is stronger than the previous two chapters. In order to facilitate the teaching operation, the textbook provides an example according to the Registration Form of Middle School Students' Physical Health. ?

Second, the writing characteristics of this book?

1. Strengthen the connection with practice and reflect the formation and application of knowledge?

Closely connecting with practice, reflecting the ins and outs of knowledge, and reflecting the formation and application process of knowledge are the characteristics of this set of teaching materials and a major feature of this book. When writing each chapter of this book, we should pay attention to the introduction of concepts and the formation of knowledge from practical problems, reflecting that mathematics comes from reality, and at the same time, we should pay attention to applying the obtained mathematical conclusions to practice, reflecting that mathematics serves reality by solving practical problems. For example, in the chapter of "fraction", the textbook arranges several practical problems for the introduction of the concept of fraction. By analyzing the quantitative relationship in practical problems, the concept of score is listed, which shows that the concept of score is produced because of objective practical needs. When discussing the fractional order equation, combined with practical problems, it is shown that the fractional order equation is a mathematical model to solve practical problems. In the chapter "Inverse Proportional Function", the concept of inverse proportional function is abstracted from several practical problems, and this chapter also specially arranges the section "Practical Problems and Inverse Proportional Function", highlighting that inverse proportional function is a mathematical model for studying practical problems. In the chapter of Pythagorean Theorem, the discovery of Pythagorean Theorem and its inverse theorem is combined with real life, and the application of these two theorems in solving practical problems is also written. In the chapter "Quadrilateral", the close relationship between quadrangles, especially parallelogram, rectangle, diamond, square and trapezoid and life is fully reflected. Because statistics are closely related to real life, we should pay attention to the role of typical cases in the chapter of "data analysis". The study of statistics such as weighted average, median, mode and variance is all carried out in the process of analyzing actual cases, and we can understand the concepts and principles of statistics in the process of solving practical problems. So when writing this book, I chose many practical problems that are full of the flavor of the times, typical, familiar to students or interested in. Some practical problems are used to create problem background, serve the derivation of concepts or the formation of knowledge, and some practical problems are designed for the application of mathematical knowledge and methods. ?

2. Focus on revealing the essence of mathematics?

Mathematics is a science that studies quantitative relations and spatial forms in the real world. Mathematics comes from the rich material world, and mathematics itself has a strict logical relationship. Only by profoundly revealing the essence of mathematical knowledge and clarifying the logical relationship between mathematical knowledge can we truly understand mathematics and better use mathematics to solve problems. In the process of writing this book, we pay full attention to respecting the internal system structure of mathematics, excavating the internal relations of mathematical knowledge and revealing the essence of mathematical knowledge. For example, when studying the concept and basic properties of fractions in the chapter of "Fractions", textbooks start with the relationship between fractions, and regard the relationship between fractions as concrete and abstract, special and general (that is, fractions are concrete and special basic objects compared with fractions), revealing that fractions are abstract representatives after summarizing specific fractions. According to this relationship between scores, the relevant conclusions of scores should correspond to the conclusions of scores, that is, they are consistent, which is what we often call the generality of numbers. Therefore, we can get the concept, basic properties and operation rules of fractions by analogy. For the problem of increasing roots in solving fractional equation, the textbook analyzes the reasons of increasing roots with concrete examples and reveals the essence of the problem. In the chapter "Inverse Proportional Function", when studying the definition, images and properties of inverse proportional function, the textbook fully penetrated the basic idea of "variation and correspondence", revealing that the essence of function concept is motion variation and contact correspondence. In the chapter of "Quadrilateral", the textbook pays attention to the concepts of parallelogram, rectangle, diamond and square, and adds some conditions (species difference) to expand the connotation of the concept and narrow the extension of the concept, thus leading to new concepts and revealing the relationship between these special parallelograms. In the chapter of "data analysis", the significance of statistics such as weighted average, median, mode and variance is emphasized, their calculation skills are diluted, the essential characteristics of each statistic are revealed, and statistical ideas are embodied. In a word, when writing this book, we should try our best to reflect the interrelation between knowledge, infiltrate mathematical thinking methods and reveal the essence of mathematical knowledge. ?

3. Create opportunities for students to explore and communicate, and increase the space for students to think?

It is a prominent feature of this book to advocate students' inquiry learning methods and leave enough space for students to explore and communicate. For the important concepts, properties and theorems in this book, the teaching materials are often set up with columns such as observation, thinking, discussion, exploration and induction, so that students can discover conclusions through exploration activities, experience the process of knowledge rediscovery, develop innovative thinking ability in the process of exploration activities, and change students' learning methods. ?

The two chapters "Fraction" and "Inverse Proportional Function" in this book belong to "Number and Algebra" and are also traditional contents. Compared with the original textbook, the contents of these two chapters increase the process of students drawing conclusions through exploration activities, that is, the elements of reasonable reasoning are added. For example, when discussing the basic nature of scores, textbooks set up a "thinking" column, requiring students to "compare the basic nature of scores". Can you think of any nature of the score? Through students' discussion and communication, it is concluded that the numerator and denominator of a fraction are multiplied (or divided) by a non-zero algebraic expression, and the value of the fraction remains unchanged, which cultivates students' inquiry ability and innovative consciousness. For another example, when discussing the properties of the inverse proportional function, the textbook sets up an "observation" column, which requires students to explore the properties of the inverse proportional function by observing the sum and the image of the sum, and finally sets up an "induction" column to summarize the properties of the inverse proportional function, so that students can experience a process of exploring and finding conclusions. ?

Pythagorean Theorem and Quadrilateral belong to the fields of space and graphics. Compared with the original textbook, a significant change in the content of these two chapters is to strengthen the composition of experimental geometry and organically combine experimental geometry with argumentation geometry. Demonstration geometry plays an important role in cultivating people's logical thinking ability, while experimental geometry is an effective tool to discover geometric propositions and theorems, which plays an important role in cultivating people's intuitive thinking and creative thinking. As for the conclusion in geometry, most textbooks let students explore and discover the geometric conclusion by drawing, origami, paper cutting, measuring or doing experiments, and then explain, explain or demonstrate the conclusion, paving the way for the transition from experimental geometry to demonstration geometry. For example, in the discovery of Pythagorean theorem, the teaching materials set up "observation" and "inquiry" columns respectively, requiring students to discover Pythagorean theorem by observing the properties of isosceles right triangle and calculating the area and other inquiry activities. Finally, Zhao Shuang's method of proving Pythagorean theorem is introduced, thus combining experimental geometry with demonstration geometry. For another example, the chapter "Quadrilateral" makes full use of graphic transformation when discussing the nature and judgment of special parallelogram. Taking the nature of the diamond as an example, the textbook has set up an "inquiry" column, requiring students to discover the axisymmetry of the diamond through activities such as folding in half and paper cutting. Then, using the axisymmetry of the diamond, we explore and find the properties of the diamond, such as the four sides are equal, the diagonals are perpendicular to each other, and the diagonals are equally divided, and ask questions in the boundary. This also enables students to experience a process of exploring and discovering the essence of graphics through observation, operation, transformation and other activities, and then proves the process of discovering the essence, so that intuitive operation and logical reasoning are organically combined. ?

"Data analysis" is the content of "Statistics and Probability". For the compilation of statistical content, the textbook emphasizes that students should go through the basic process of data processing through statistical investigation activities, learn relevant statistical knowledge and methods and establish statistical concepts in the statistical activities of collecting, sorting, describing and analyzing data. This provides students with a broad space for activities. ?

In addition, this textbook designs "subject learning" in the "quadrilateral" and "data analysis" chapters respectively, and designs 2 ~ 3 open and exploratory "mathematical activities" at the end of each chapter. These "subject learning" and "mathematical activities" are comprehensive and practical, which provide students with opportunities for practical activities and exploration and communication, and play a certain role in promoting students' inquiry learning methods. ?

Third, several issues worthy of attention.

1. Strengthen the relationship between knowledge and teaching on the basis of existing experience?

This book is the second volume of grade eight, and five chapters of it are inextricably linked with what students have learned. For example, in the chapter of "Fraction", the related concepts, properties and operation rules of fraction are closely related to the corresponding content of fraction, and the fractional equation can only be solved after it is finally transformed into an integral equation. In the thinking of compiling fractional equation, like integral equation, it also emphasizes that fractional equation is a mathematical model to solve practical problems. "Inverse proportional function" is the content of another chapter function after relaying elementary function in this set of textbooks. His writing ideas are similar to elementary functions in many places, emphasizing the idea of "variable correspondence" in functions and highlighting the idea that functions are mathematical models to solve single-value correspondence among variables. Knowledge of quadrangles, such as the concept of some special quadrangles, the calculation of the height and area of trapezoid, etc. , primary school students have learned. In the chapter "Triangle" in the second volume of the seventh grade, the students also learned the internal angles of quadrilaterals and so on. Therefore, in the "quadrangle" chapter, these contents are not repeated, but used directly; For the Pythagorean Theorem, seventh-grade students use the book "10 chapter? Real number "(for example, students can use Pythagorean theorem to make points representing irrational numbers on the number axis), and this chapter further improves understanding on this basis; As for the statistics that describe the trends in data sets: average, median and mode, students have learned them in the first two issues. In the chapter of "Data Analysis", the textbook improves the understanding of these statistical data on the basis of students' existing experience and under the background of studying the trend of data concentration. To sum up, we can combine the actual situation of students, review appropriately, strengthen the mutual connection and integration of knowledge, and teach on the basis of students' existing experience, so that students can form a positive migration. ?

2. The requirements of reasoning?

For the cultivation of reasoning ability, this set of teaching materials is gradually deepened according to different levels such as "speak frankly", "speak frankly", "simple reasoning" and "using symbols to express reasoning". The requirements for reasoning in this book are basically at the stage of further consolidation and improvement on the basis of students' initial mastery of reasoning and argumentation methods. For example, the chapter "Quadrilateral" is relatively simple in content and proof method, but the training of reasoning proof is still very important. In addition to requiring students to prove the conclusions drawn through observation, experiment and inquiry, some theorems are proved by exploratory methods. This method is not to prove it first, but to draw a conclusion through reasoning according to the topic and existing knowledge. In the chapter of Pythagorean Theorem, the proof method of Pythagorean Theorem and its inverse theorem is actually proved by calculation, which is different from some previous judgment methods. In addition, for the concepts of reciprocal proposition and reciprocal theorem, teaching materials are given in combination with Pythagorean theorem and its inverse theorem in order to let students have a perceptual understanding of these logical concepts. Students can write the proposition in the form of "If …… then ……", which is helpful to improve students' logical reasoning ability. Therefore, we should pay attention to guiding students in teaching, so that students can improve and develop their reasoning and argumentation ability on the basis of being familiar with the format of "normative proof". ?

3. Pay attention to cultural inheritance and humanistic education?

This set of teaching materials tries to be a mirror reflecting scientific development and cultural progress, which not only reflects the scientific and applied nature of mathematics, but also reflects the culture contained in mathematical science. This book not only involves the relationship between mathematics and practice, infiltration modeling, combination of numbers and shapes, transformation and other important mathematical ideas, but also involves the discovery of Pythagorean theorem and other important historical facts. For Pythagorean Theorem, there were many important achievements in ancient China, not only discovered Pythagorean Theorem, but also proved Pythagorean Theorem in many ingenious ways, especially in the application of Pythagorean Theorem, which had great influence on other countries. These are all important contributions of our people to mankind. In the chapter of Pythagorean Theorem, the textbook introduces the records of "Gou Suan, Gu Si and Xian Wu" in China's ancient mathematical work Zhou Bian Su Jing, and introduces Zhao Shuang's string diagram and Zhao Shuang's idea of proving Pythagorean Theorem by using string diagram. Zhao Shuang's String Diagram shows the spirit and wisdom of ancient people in China in learning mathematics, which is the pride of ancient mathematics in China. Because of this, this pattern was chosen as the emblem of the World Congress of Mathematicians held in Beijing in 2002. In addition, in the chapter "Pythagorean Theorem", the related research results abroad are also introduced. For example, the discovery of Pythagorean theorem was introduced from the legend related to Pythagorean, and the inverse theorem of Pythagorean theorem was introduced from the method of drawing right angles by ancient Egyptians. These are all good materials for cultivating students' culture and should be used in teaching.