The "multiple" of one question refers to: designing basic training purposefully and emphatically by means of multiple solutions to one question and changeable questions, which is helpful to open up ideas, activate thinking and cultivate students' innovative ability. Now I will talk about my own ideas on the teaching of a changeable topic.
1.
The topic with multiple solutions to one question should be representative and can accommodate most knowledge points, not too complicated, but not too simple. It is too difficult to dampen students' enthusiasm for research and study, and it is too simple for students to be interested. This step is very important to stimulate students' interest in learning and exploring.
For example, there is a topic: three students, A, B and C, take taxis and go in the same direction. The three students agreed in advance to share the fare. Get off at 1/3, get off at 2/3 and get off at C. Ticket price ***54 yuan. Ask the students how much the fares of A, B and C are reasonable.
Students are very interested in this fare problem. Whether it is reasonable for students A, B and C to pay their own fares is quite different. After trying to design three schemes: the first scheme is shared by Party A, Party B and Party C, that is, each person pays 18 yuan; Scheme 2 is divided by distance: the travel distance ratio of Party A, Party B and Party C is 1∶2∶3, which is paid to 9 yuan, 18 yuan and 27 yuan respectively; Option 3, installment settlement: fare ***54 yuan. If the fare is calculated according to the first 1/3 distance, the middle 1/3 distance and the last 1/3 distance, then each car is 18 yuan, and the first 1/3 distance is 18 yuan. Then Party B and Party C each pay 9 yuan, and the final distance 1/3 needs to be paid 18 yuan, which shall be borne by Party C, so that Party A pays 6 yuan, Party B pays 15 yuan and Party C pays 33 yuan; As can be seen from the above examples, students are very interested in this topic, active in thinking, brave in exploration and have obvious learning effects.
2. One topic is changeable, which is conducive to cultivating innovative inquiry ability.
2. 1 Change the topic or conclusion, that is, by changing the topic or conclusion of the exercise, we can explore the same problem from multiple angles, which can not only enable students to comprehensively apply what they have learned to solve problems, enhance the adaptability of solving problems, but also cultivate the profundity and extensiveness of mathematical thinking, thus cultivating the good learning quality of innovative thinking.
For example, I also discussed the above problems in various ways, which broadened students' thinking and activated students' thinking.
Transformation (1): In trapezoidal abcd, ABCD, BC=AB+CD, and E is the midpoint of AD. Verification: CE⊥BE.
Transformation (2): In trapezoidal abcd, ABCD, CE⊥BE. and E are the midpoint of AD. Verification: BC=AB+CD.
Transformation (3): In trapezoidal abcd, is ABCD, BC=AB+CD, CE⊥BE.E the midpoint of AD? Why?
2.2 Change the question type, that is, change the original question type into a new one, change the monotonous and boring practice method, train students' comprehensive ability to solve various questions, and cultivate students' thinking flexibility, which is conducive to the cultivation of students' cooperative inquiry and innovation ability. For example, an exam in Grade 3: As shown in Figure 5 (omitted), it is known that in △ADE, ∠ DAE = 120, B and C are two points on DE respectively, and △ABC is a regular triangle, which is verified; BC is the middle value of BD and CE.
Analysis: This title is exploratory proof, which can guide students to find the conditions to prove △ABD∽△ECA from the conclusion, so that the problem can be solved easily. Based on this problem, the problem type is transformed as follows:
Conversion (1): fill in the blanks instead, as shown in Figure 5. It is known that in △ADE, B and C are two points on DE, ∠ DAE = 120, and △ABC is a regular triangle, then the quantitative relationship between line segments BC, BD and CE is.
On the surface, this problem is a simple form transformation of the original problem, but in essence it has the idea of inquiry, that is, BC needs to be replaced by AB and AC respectively, which boils down to the relationship between △ABD and △ECA.
Transformation (2): change to multiple-choice questions, as shown in Figure 5 (omitted). It is known that in △ADE, B and C are two points on DE, ∠ DAE = 120, and △ABC is a regular triangle, then the following relationship is wrong ().
It is called multiple-choice question, which really means finding out that there are three pairs of similar triangles in the picture, so as to know that options A, B and C are correct, and choose D. 。
Transformation (3): Turn it into a calculation problem, as shown in Figure 5 (omitted). It is known that in △ADE, B and C are two points on DE, ∠ DAE = 120, △ABC is a regular triangle with a side length of 4, and BD=2. Find the length of CE.
It is still necessary to explore the quantitative relationship between line segments BC, BD and CE, thus transforming it into the problem of "knowing two and seeking one".
Transformation (4): Change it to a true or false question, as shown in Figure 6 (omitted). If ∠ DAE = 135 and △ABC is an isosceles right triangle with right vertices, is the conclusion still valid?
Changing the conditions of the problem, using the same thinking method to explore and reaching the same conclusion further extends the thinking method of the original example and expands the thinking space of students.
Transformation (5): Open test, as shown in Figure 5 (omitted). It is known that in △ADE, ∠ DAE = 120, B and C are two points on DE respectively, and △ABC is a regular triangle, so which line segments in the graph are the proportional average of the other two line segments?
The openness of the conclusion gives students more thinking space and greatly exercises their open mathematical innovative thinking ability.
Transformation (6): change to comprehensive test, as shown in Figure 7 (omitted). In △ABC, AB=AC= 1, points D and E move on a straight line BC, and let BD=x and Ce = Y. 。
(1) If ∠ BAC = 30 and ∠ DAE = 105, try to determine the functional relationship between y and x;
(2) If the degree of ∠BAC is α, and the degree of ∠DAE is β, the functional relationship between y and x in (1) still holds, and the reasons are explained.
This transformation combines similarity with functional knowledge to cultivate students' comprehensive inquiry ability.
Through the transformation of the above six questions, the same mathematical thinking method is infiltrated into different questions, which not only exercises students' adaptability to different questions, but also deepens their understanding and application of mathematical thinking methods; It not only activates students' thinking, but also enlivens the classroom atmosphere; It seems to be a waste of time and energy, but in fact it touches the soul of thinking and inquiry, which can get twice the result with half the effort.
(4) How many diagonals does the N-polygon * * * have?
All these problems can be solved by establishing the same mathematical model, which not only cultivates students' ability of induction and arrangement, but also deepens students' awareness of modeling ideas and applying mathematical models, and stimulates students' interest in learning mathematics.
In short, in teaching practice, training a certain topic purposefully, planned and properly is conducive to activating thinking, exercising the flexibility of students' thinking, effectively opening up students' innovative thinking space, enabling students to integrate what they have learned, systematize knowledge, and use knowledge more flexibly, which is conducive to improving students' ability of induction, synthesis, innovation and inquiry, and enhancing their comprehensive quality and comprehensive application ability.