Chapter 16 score (about 13 class hours)

17 chapter 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.

I. Content analysis

[Chapter 16]

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. This chapter is the content of another chapter's function after Chapter 8 (1) "Linear Function in Chapter 1 1". 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, thus guessing 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.