The teaching objectives of this course are:
1. Review the meaning of quantity in direct proportion and inverse proportion.
2. Mastering the quantitative relationship and problem-solving ideas of the positive and negative proportional application problems can correctly solve the positive and negative proportional application problems.
3. Further cultivate students' thinking ability of analysis, reasoning and judgment.
Teaching focus:
Enable students to correctly answer application questions by using the meaning of positive and negative proportions.
Teaching difficulties:
By analyzing the known conditions and problems of application problems, students determine which quantities in the problems are what proportions, and list equations by using the meaning of positive and negative proportions.
This lesson is a review lesson of the unit, which aims to make students have a comprehensive understanding of the knowledge of this unit and make the knowledge they have learned organized and systematic. So teaching is divided into three levels: the first level is the ending. By sorting out, students can form a network of what they have learned. Only by forming network knowledge can students go deep into their minds and use it freely. The second level is review. Through review, students can master the concept of this unit, and experience the process of abstracting some practical problems into algebraic problems again, so as to further understand the connections and differences between things. The third level is layered practice. The design of exercises focuses on connecting with students' real life, choosing things close to students' life as far as possible, and cultivating students' interest in learning and applying mathematics in practice.
4.2 Algebraic evaluation
Number and algebra 1. Review of rational numbers and their operations II. Number axis 3. Subtraction of rational number 4. Subtraction of rational number 5. Scientific symbol 6. Real number and number axis 7. The number is represented by the letter 8. Algebraic formula 9. Factorize 1 0 with the multiplication formula (1). Equation 165438+ 13 in the calendar. Increase revenue and reduce expenditure 14. How wide is the lace (1) 15. Solve the linear inequality 16. An interesting seesaw problem-sidelights of problem inquiry activity 17. Linear function 18. 23. Parallel lines-characteristics of parallel lines 24. Pythagorean Theorem (1) 25. Identification of parallelogram (1) 26. Square 27. Square 28. Trapezoidal (1) 29. Trapezoidal 30. From perspective to view 3 1. Perspective view 32. A centrally symmetric figure 33. Pattern design. Determine position 33. 38. Geometry review courseware of statistics and probability 39. Importance of statistics-census and sample survey. Representation of data (2)4 1. Starting with the first gold medal in the Olympic Games. Mean, median and mode 43. Do you have to touch the red ball? Teaching design and comment of possibility 44. Study on frequency and probability. Learn from me. Let mathematics fly from here. "Make a cuboid as big as possible without a cover" teaching case and thinking 47. Review and reflection on the research of "making a cuboid with no cover as big as possible" Innovating the teaching process of mathematics.
An example of junior high school mathematics evaluation (image and essence of linear function)
The first plane rectangular coordinate system and its function
Plane rectangular coordinate system is one of the basic tools to study mathematical problems, and function is a very important concept in mathematics, which builds a bridge for the combination of numbers and shapes with the help of plane rectangular coordinate system. Correctly understanding the concept of function and mastering the function image and its properties play a key role in solving problems.
The concept of 1. function is abstract and difficult for junior high school students to understand. The key is to understand that the essence of our research function is to study the relationship between two variables. In the same problem, there is often a certain relationship between the variables, suggesting a certain law. When one quantity changes, the other quantity changes accordingly.
2. After the plane rectangular coordinate system is established, there is a one-to-one correspondence between points on the plane and ordered real number pairs. In the coordinate plane, finding points from the coordinates of points and finding coordinates from points is the most basic form of mutual transformation between "number" and "shape". The coordinates of points are the basis of solving function problems, and the resolution function is the key to solving function problems. Therefore, finding the coordinates of points and exploring the resolution function are two important topics in the study of functions.
3. The function reflects a changing process, which should have the following three points: (1) can only have two variables; (2) One variable changes with the value of another variable; (3) For each fixed value of the independent variable, the function has a unique value corresponding to it, allowing multiple X's to correspond to the same Y, but not one X's to correspond to multiple Y's.
4. The value range of function independent variables is an important content, which should not only ensure that the function relationship is meaningful, but also ensure that it conforms to the practical significance.
5. There are three ways to express functions: tables, images and analytical expressions, and each method has its own advantages and disadvantages.
6. In the plane rectangular coordinate system, if the value of the independent variable is taken as the abscissa and the corresponding function value as the ordinate, the graph composed of all such points is the image of this function. Generally, the image of drawing function is divided into three steps: list-tracking point-connecting line (smooth curve).
7. The relationship between function and image must be understood: the coordinates of points on the function image satisfy the function relationship; The point satisfying the function relation must be on the function image. It is what we often call purity and integrity.
8. Coordinate characteristics of points in coordinate plane: including points on coordinate axis, points on bisector of quadrant angle, points symmetrical about coordinate axis and origin, points on straight line parallel to coordinate axis, and translation transformation of points should be mastered skillfully.
Second block linear function
Linear function is a concrete form of junior middle school function. If the functional relationship between two variables x and y can be expressed as y=kx+b(k, b is constant and k is equal to 0), then y is a linear function of x, in which the independent variable x can take all real numbers. When b=0, y is also called the proportional function of x.
1. The proportional function is a linear function, but the linear function is not necessarily a proportional function. Only when b=0 is it a proportional function.
2. The image of a linear function is a straight line. When drawing a straight line y=kx+b, generally choose a point (0, b) and a point (-b/k, 0), which is exactly the intersection of the straight line with the Y axis and the X axis. When -b/k is not an integer, (-b/k, 0) is often replaced by a point whose abscissa and ordinate are integers. When b=0, the image passing through the origin, that is, the image of the proportional function y=kx is a straight line passing through the origin. When drawing a straight line y=kx, the origin (0,0) and the point (1, k) are generally selected.
3. In the linear function y=kx+b, the symbols of k and b are directly related to the increase and decrease of the function and the position of the straight line (referring to the quadrant that passes through), so it is necessary to master it skillfully. Generally speaking, k>0, the image passes through the first and third quadrants, and Y increases with the increase of X; K<0, the image passes through the second and fourth quadrants, and Y decreases with the increase of X; B>0, the image passes through the first and second quadrants; B<0, the image passes through the third and fourth quadrants; When b=0, the image passes through the origin.
4. To find the expression of linear function y=kx+b is actually to find the values of k and b. Generally, two conditions are needed. Use binary linear equations to find k and b, and then write the expression.
5. The coordinates of the intersection of the images of two linear functions are the solutions of the equations formed by two linear resolution functions.
Enable students to correctly judge the proportion of quantities involved in application problems and use them.
Teaching objectives
1. Enable students to correctly judge the proportional relationship between the quantities involved in application problems.
2. Enable students to use the meaning of positive and negative proportions to correctly answer application questions.
3. Cultivate students' ability of judgment, reasoning and analysis.
Teaching focus
Students can correctly judge what kind of proportional relationship exists between quantities in application problems, and list equations containing unknowns by using the meaning of positive and negative proportions, so as to correctly use proportional knowledge to solve application problems.
Teaching difficulties
Using the meaning of positive and negative proportion, the equation is listed correctly.
teaching process
First, review the preparation. (Courseware demonstration: the application of proportion)
(1) What is the proportional relationship between the two quantities in the following questions?
1. Constant speed, distance and time.
2. A certain distance, speed and time.
3. The unit price is fixed, the total price and quantity.
4. The number of hectares of cultivated land per hour is constant, and the total number of hectares of cultivated land is related to time.
5. The whole school students do exercises, the number of people standing in each line and the number of lines standing.
(B) the introduction of new courses
We have learned the meanings of proportion, positive proportion and inverse proportion, and also learned to understand proportion. Applying these proportions of knowledge can solve some practical problems. In this lesson, we will learn the application of proportion.
Teacher's blackboard writing: the application of proportion
Second, new teaching.
(a) teaching examples 1 (courseware demonstration: the application of proportion)
Example 1. A car travels for two hours 140 km. At this speed, it traveled from place A to place B for five hours. How long is the bus between A and B?
1. Students use the previous method to answer independently.
140÷2×5
=70×5
= 350 km
2. Answer with proportional knowledge.
(1) Thinking: Which three quantities are involved in this problem?
What kind of quantity is certain? How did you know?
What is the proportional relationship between the distance traveled and the time?
Teacher writes on the blackboard: the speed is certain, and the distance is proportional to the time.
The teacher asked: What is the equal distance and time between the two trips?
How to list the equations?
Solution: Set the expressway between Party A and Party B to be kilometers long.
=
2 = 140×5
=350
The expressway between the two places is 350 kilometers long.
3. How to check whether this problem is done correctly?
4. Different exercises
A car travels for two hours 140 km, and the road between A and B is 350 km long. At this rate, how many hours will it take to get from A to B?
(b) Teaching Example 2 (Courseware Demonstration: Application of Proportion)
Example 2. A car goes from A to B at a speed of 70km/h and takes 5 hours to arrive. If it takes 4 hours to get there, how many kilometers per hour will it take?
1. Students use the previous method to answer independently.
70×5÷4
=350÷4
= 87.5 km
2. So, how to solve this problem with proportional knowledge? Please think and discuss: (Projection)
The distance in this question is certain, and there are _ _ _ _ _ _ _ and _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.
Therefore, _ _ _ _ _ _ _ and _ _ _ _ _ _ _ _ _ refer to
3. If you need to drive kilometers per hour, who can list the equation according to the meaning of inverse proportion?
4 =70×5
=87.5
You need to drive at a speed of 87.5 kilometers per hour.
4. Different exercises
A car travels from A to B at a speed of 70km/h and arrives in 5 hours. How many hours does it take to drive at a speed of 87.5 kilometers per hour?
Third, class summary.
The key to solving application problems with proportional knowledge is to correctly find out two related quantities in the problem, judge what proportional relationship they are, and then list the equations according to the meaning of positive and negative proportions.
Fourth, classroom practice. (Courseware demonstration: the application of proportion)
(1) The canteen uses 780 yuan to buy 3 barrels of oil. According to this calculation, how much does it cost to buy 8 barrels of oil? (Answer with proportional knowledge)
(2) Students do broadcast exercises, with 20 people in each row, which is exactly 18 row. If there are 24 people in each row, how many rows can you stand?
(3) First think about the proportional relationship in the following questions, then fill in the conditions and questions and answer them with proportional knowledge.
1. Master Wang will produce a batch of parts, 50 pieces per hour, which will take 4 hours to complete. _ _ _ _ _ _ _ _ _ _ _ _?
Master Wang produced 200 parts in 4 hours. According to this calculation, _ _ _ _ _?
Homework after class.
1. A tractor can be cultivated in 2 hours 1.25 hectares. According to this calculation, how many hectares can be cultivated in 8 hours?
Bind a batch of paper into exercise books of the same size. If each book has 18 sheets, 200 volumes can be bound. If each book has 16, ......
Teaching and Research of Xiao Chun Middle School in Ningbo
20 14 On June 9th, jiangdong district's "learning-centered" mathematics teaching training class carried out teaching and research activities in Xiao Chun Middle School. This activity was taught by two teachers, Zhang Xiaotuan and Jing Yao, respectively, and we enjoyed a teaching feast. Teacher Zhang Xiaotuan's "Isosceles Rt△ in Inverse Proportional Function Images" is ingeniously introduced, and the experimental and inductive reasoning courseware designed by teacher Jing Yao is novel, eye-catching and quite bright. Finally, Wen Liming, a teacher in the teaching and research section of Yinzhou District, gave a lecture on the teaching strategy of "reading materials" in junior high school mathematics textbooks published by Zhejiang Education Publishing House, which showed the charm and significance of reading materials in mathematics textbooks, which should not be ignored in teaching. Before the end of the activity, teacher Pan Xiaomei, a district mathematics researcher, made a brief comment on this teaching and research activity, which made us realize that if we want to bring better classroom teaching to students, we must constantly improve teachers' skills and teaching level.
Selected Articles Catalogue of Junior High School Mathematics Evaluation in the New Curriculum
Number and algebra 1. Review of rational numbers and their operations II. Number axis 3. Subtraction of rational number 4. Subtraction of rational number 5. Scientific symbol 6. Real number and number axis 7. The number is represented by the letter 8. Algebraic formula 9. Factorize 1 0 with the multiplication formula (1). Equation 165438+ 13 in the calendar. Increase revenue and reduce expenditure 14. How wide is the lace (1) 15. Solve the linear inequality 16. An interesting seesaw problem-sidelights of problem inquiry activity 17. Linear function 18. 23. Parallel lines-characteristics of parallel lines 24. Pythagorean Theorem (1) 25. Identification of parallelogram (1) 26. Square 27. Square 28. Trapezoidal (1) 29. Trapezoidal 30. From perspective to view 3 1. Exploded view of perspective view 32. A centrally symmetric figure 33. Pattern design. Determine the location. 38. Geometry review courseware of statistics and probability 39. Importance of statistics-census and sample survey. Representation of data (2)4 1. Starting with the first gold medal in the Olympic Games. Mean, median and mode 43. Do you have to touch the red ball? Teaching design and comment of possibility 44. Study on frequency and probability. Learn from me. Let mathematics fly from here. "Make a cuboid as big as possible without a cover" teaching case and thinking 47. Review and reflection on the research of "making a cuboid with no cover as big as possible" Innovating the teaching process of mathematics.