Mathematics is a science that studies the relationship between spatial form and quantity. The combination of numbers and shapes is one of the important mathematical ideas. It not only analyzes the algebraic meaning of the research object, but also reveals its geometric meaning according to the internal relationship between the conditions and conclusions of mathematical problems, so as to combine the relationship between quantity and spatial form skillfully and harmoniously, and make full use of this combination to find the solution to the problem. Its essence is to combine abstract mathematical language with intuitive graphics, and find a way to solve problems by combining algebra and geometry. It includes two aspects: "helping numbers with shapes", that is, clarifying the relationship between numbers with the help of the visualization and intuition of shapes; "Using numbers to help shape" is to clarify some properties of shape with the help of the accuracy and rigor of numbers. As Mr. Hua, a famous mathematician in China, said, "The number is not so intuitive without it, and it is difficult to be nuanced without it." The combination of numbers and shapes is good in all aspects, and everything is over. "
1. Understand abstract concepts with intuitive diagrams and experience the idea of combining numbers with shapes.
In the teaching of the first chapter of PEP B Compulsory 1, it is difficult for students to understand the relationship between sets because they are unfamiliar with the concept of sets, so I have done the following in the teaching process.
Firstly, I introduced Venn diagram, another representation of set, that is, a set is represented by the interior of a closed curve on a plane. Then I asked the students to discuss how many different positional relationships two closed curves can have and let them draw them. After discussion, the students drew four different positional relationships (pictured).
Next, I asked them to observe the similarities and differences of these four relationships, and guided them to describe them in set language, and found that (1) has no common parts, that is, the set has no common elements; (2) There are common parts, that is, there are * * * identical elements in one set, but some elements are not in another set; (3) completely inside, (4) and overlapping, that is, any element in the set is an element of the set, and we call the set a subset of the set (). After further analysis, it is found that some elements in the set in (3) do not belong to the set, while the elements in the set in (4) are completely the same, so the subsets are divided into two categories: proper subset, that is, the set is a subset of the set B, and at least one element in the set does not belong to the set; Set equality means that every element of a set is an element of a set, and conversely, every element of a set is also an element of a set. Through venn diagram's intuitive expression, students quickly understood abstract concepts such as "subset", "proper subset" and "set equality", and realized the idea of combining numbers with shapes.
When talking about the operation of sets, I first let students try to understand the meaning of intersection, union and complement literally, then intuitively feel the meaning of intersection, union and complement with venn diagram, and finally explain it in set language, so that students can experience the intersection of sets from different angles.
In order to examine whether students can use the idea of combining numbers and shapes to solve the problems related to sets, at the end of this chapter, I came up with such an exercise: "There are 50 students in a class, 32 of whom take part in the computer drawing competition first, and then 24 take part in the computer typesetting competition. If three students did not participate in these two competitions, how many students in this class participated in these two competitions at the same time? " Judging from the answer results, most students can use venn diagram to help shape numbers, get correct answers, and get a preliminary understanding of the mathematical thought of combining numbers with shapes.
2. Through the algebraic analysis of analytic function, draw the image of the function, study the properties of the function, and initially form the idea of combining numbers with shapes.
In the teaching of the required 1 version of the second chapter function of People's Education Edition B, although students have a preliminary understanding of the function in junior high school, it is still difficult to describe the concept of the function with set language and study the monotonicity and parity of the function with algebraic method, so I have done the following processing in teaching.
After talking about the concept of function, I did an exercise: The image that cannot be used as a function in the following images is ().
Let students further understand the concept of function from the perspective of form; When learning the properties and images of linear function and quadratic function, because students have already made images of linear function and quadratic function by tracing points in junior high school, I will start with students' existing knowledge, let students list, trace points and connect lines to make images of linear function and quadratic function, and guide them to understand monotonicity, parity and symmetry from the perspective of numbers first, and then intuitively feel monotonicity, parity and symmetry through images. In this way,
3. With the help of the intuition of the unit circle, we use the trigonometric function line related to the unit circle and the idea of combining numbers and shapes to solve related problems.
In the teaching of Basic Elementary Function (Ⅱ) in the first chapter of Compulsory 4 of People's Education Press, because the idea of combining numbers and shapes has been effectively infiltrated in the compulsory 1, I want to try to let students use the idea of combining numbers and shapes to solve related problems. Let me take the section "Unit Circle and trigonometric function line" as an example to talk about how I use the unit circle and trigonometric function line related to the unit circle to guide students to use the idea of combining numbers with shapes.
Before the section "Unit Circle and trigonometric function line", I learned the definition of trigonometric function, and revealed that the value of trigonometric function is a "ratio" from the angle of algebra. I asked students to analyze the symbols of trigonometric functions in each quadrant in algebraic form, and also asked them to find some trigonometric functions with axis angles, and analyze the definition fields of sine functions, cosine functions and tangent functions. Students can get answers, but it is difficult for them to remember these conclusions. Therefore, after completing the teaching of unit circle and trigonometric function line, I ask students to re-analyze the above problems from the perspective of geometry. Because the trigonometric function line represents the absolute value of trigonometric function with the length of the vector on the axis and the sign of trigonometric function value with the direction, the sign of trigonometric function in each quadrant can be directly seen through the direction of trigonometric function line. For the trigonometric function values of these axes and the definition fields of sine function, cosine function and tangent function, I made a geometric sketchpad courseware myself, so that students can get the answer directly from the perspective of shape. And in the process of changing the angle, some students found that the sine value gradually increased from 0 to 1, and then gradually decreased. When the terminal edge of the angle falls on the non-positive semi-axis of the shaft, the sine value is 0, and then it continues to rotate counterclockwise, and the sine value gradually decreases until it gradually increases. When the terminal edge of the angle falls on the non-negative semi-axis of the shaft, the sine value is 0. But the cosine value gradually decreases from 1. When the terminal edge of the angle falls on the non-negative semi-axis of the shaft, the cosine value is 0, and then continues to rotate counterclockwise, and then slowly decreases until it gradually increases. When the terminal edge of the angle falls on the non-positive semi-axis of the shaft, the cosine value is 0, and then continues to increase until 1. Further observation also shows that sine line and cosine line will appear repeatedly every time the angle rotates, thus the relationship between angle and trigonometric function is obtained, which lays the foundation for understanding monotonicity and periodicity of trigonometric function in the future. After class, I left two questions to do, one is to compare the value of trigonometric function with non-special angle, and the other is the known value. Judging from the feedback after class, some students can still solve it by combining trigonometric function lines with numbers and shapes.
Teachers should carefully study the teaching materials, focus on the overall development of mathematics, and gradually infiltrate the idea of combining numbers and shapes from the specific teaching process, so that students can develop the good habit of combining numbers and shapes and become tools for analyzing and solving problems. This is the goal that all mathematics educators should pursue.