Example of instructional design 1 1. Teaching objectives 1. Understand the format of reasoning and proof, and master the judgment axiom and the first judgment theorem of parallel line. 2. Will use judgment axiom and the first judgment theorem for simple reasoning and argumentation. 3. Demonstrate through the model, that is, the application of the mathematical thinking method of "movement-change". Cultivate students' abilities of "observation-analysis" and "induction-summary". Second, the guidance of learning methods 1. Teachers' teaching method is heuristic guidance and discovery. Second, students' learning method is independent thinking. Active discovery. 3. Key points, difficulties and solutions (1) focus on summing up axioms and deducing theorems on the basis of observing experiments. (2) Logical reasoning and writing format in the process of forming difficult judgment theorems. (3) Solution 1. Ask questions skillfully and solve the key points through observation experiments. 2. By guiding correct thinking, strictly display the reasoning writing format. Clear solutions to difficulties and doubts. Fourth, arrange class hours. Fifth, prepare teaching AIDS, learning tools, triangles, projection films, projectors and computers. Sixth, design interactive activities between teachers and students. Review old knowledge and introduce new knowledge through two groups of questions. Second, guide thinking through experimental observation, sum up the derivation of axioms and theorems, and consolidate them through practice. Third, ask the teacher questions. Students answer and complete the summary. Seven. Teaching steps (-) Clarify the goal Teaching suggestion 1, teaching material analysis (1) Knowledge structure: The drawing of parallel lines leads to the judgment axiom of parallel lines (the same angle is equal, and two straight lines are parallel). It is deduced from axioms that the internal dislocation angles are equal and the two straight lines are parallel. These two theorems. (2) Analysis of key points and difficulties: The key points of this section are: the judgment axiom and two judgment theorems of parallel lines. The general definition is equivalent to the first judgment theorem. Both can be used as judgment methods. But the definition of parallel lines is not easy to judge whether two straight lines intersect. This needs to be judged by the angle of the third straight line cutting. Therefore, this judgment axiom and two judgment theorems are particularly important. They are the basis for judging the parallelism of two straight lines, and also lay the foundation for learning the properties of parallel lines in the next section. The difficulty of this section is to understand the proof process of judgment theorem derived from judgment axiom. Students have just been exposed to proving the properties of geometric theorems or figures by deductive reasoning, and they still don't quite understand the significance of geometric proof. Some students even think that attributes can be identified from intuitive graphics. There's no need to prove it. All these bring difficulties to the introduction teaching of geometry. Therefore, there should be both intuitive demonstration and operation in teaching and blackboard demonstration with strict reasoning proof. Create a situation, constantly infiltrate, so that students can initially understand the steps and basic methods of proof, and fill in appropriate axioms or theorems in brackets according to what they have learned. 2. Teaching suggestion In the teaching of parallel line judgment axiom, we should fully reflect a main clue: "Full experiment-careful observation-forming conjecture-actual test-clarifying conditions and conclusions." The teacher can demonstrate the teaching AIDS shown in the textbook, or let each student draw parallel lines with a triangle and a ruler. In this process, pay attention to the change of angle. The facts are sufficient and students can understand that if the same angle is equal, then two straight lines are parallel. After judging the axiom of parallel lines, some students may realize that "two straight lines will be parallel if the internal angles are equal". Teachers can organize students' discussions according to the given figures. How to use the axioms and theorems of known geometry to prove this obvious fact? Can also let more students retell, let students gradually realize the rigor of mathematical proof. The process of discovering and proving another theorem is similar.
Example of instructional design 1 1. Teaching objectives 1. Understand the format of reasoning and proof, and master the judgment axiom and the first judgment theorem of parallel line. 2. Will use judgment axiom and the first judgment theorem for simple reasoning and argumentation. 3. Demonstrate through the model, that is, the application of the mathematical thinking method of "movement-change". Cultivate students' abilities of "observation-analysis" and "induction-summary". Second, the guidance of learning methods 1. Teachers' teaching method is heuristic guidance and discovery. Second, students' learning method is independent thinking. Active discovery. 3. Key points, difficulties and solutions (1) focus on summing up axioms and deducing theorems on the basis of observing experiments. (2) Logical reasoning and writing format in the process of forming difficult judgment theorems. (3) Solution 1. Ask questions skillfully and solve the key points through observation experiments. 2. By guiding correct thinking, strictly display the reasoning writing format. Clear solutions to difficulties and doubts. Fourth, arrange class hours. Fifth, prepare teaching AIDS, learning tools, triangles, projection films, projectors and computers. Sixth, design interactive activities between teachers and students. Review old knowledge and introduce new knowledge through two groups of questions. Second, guide thinking through experimental observation, sum up the derivation of axioms and theorems, and consolidate them through practice. Third, ask the teacher questions. Students answer and complete the summary. Seven. The teaching step (-) defines the goal, grasps the judgment axiom and the first judgment theorem of parallel lines, and uses them for simple reasoning and demonstration. (2) The overall perception is to design the situation, introduce topics, demonstrate with models, guide students to observe, analyze and summarize, impart new knowledge, and consolidate new knowledge with variant training. In the whole class, logical reasoning is fully embodied. (3) The teaching process creates situations and leads to the theme teacher: Last class, we learned parallel lines, parallel axioms and inferences. Please judge whether the following statement is correct and explain the reason (show the projection). 1. If two straight lines do not intersect, they are called parallel lines. 2. There is only one straight line parallel to the straight line. 3. If all straight lines are parallel, then they are parallel. Student activities: Students answer the above three questions orally. The teaching method shows that students can review what they learned in the last section through three true-false questions. 1 question is a prerequisite for strengthening the definition of parallel lines "in the same plane". The second question not only reviews the parallel axiom, but also lets students know how to learn geometry. Language must be accurate and standardized, and the same question will have different conclusions under different conditions. Question 3: It is also a method to review the inference of consolidating the parallel axiom and remind students to judge that two straight lines are parallel. Teacher: measure the intersection of two straight lines, and the angle formed is a right angle. Can you judge that these two straight lines are perpendicular? According to what? Student: According to the definition of verticality, two straight lines that do not intersect the same plane are parallel lines. Is there any way to determine whether two straight lines are parallel? Student activity: Students think about how to determine whether two straight lines are parallel. The teacher pointed out that we can't directly use the definition of hand line to judge whether two straight lines are parallel, so we must find other methods. What are the methods? Student activities: Students think. After reviewing the previous parallel axiom inference, some classmates suggest drawing another straight line, so that we can see whether it is parallel to it. Teacher: That's a good idea. So, how to make it parallel? If it is made, how to judge whether it is parallel imports? Student activities: students think about the teacher's question and realize that the answer just now is specious and can't solve the problem. Teacher: Obviously, our problem has not been solved, so let's find some other judgment methods, that is, the judgment of parallel lines (blackboard writing topic) that we are going to learn today. [blackboard writing] 2.5 judgment of parallel lines (1). The teaching method shows that whether two straight lines are vertical or not can be judged by the definition of vertical lines. Students naturally want to judge by the definition of parallel lines, but we can't be sure whether straight lines don't intersect, so we can't judge by definition. At this time, students will consider parallel axiomatic reasoning. At this time, teachers only need to simply ask questions, so that students can find that the problems have not been solved, thus introducing the content of the new lesson. The teacher who explores new knowledge and teaches new lessons gives the model that two straight lines shown in Figure 2-20 on page 78 of the textbook are cut and rotated by the third straight line. Let the students observe whether the size changes when they rotate to different positions, then let it change from small to large, and tell the changing law of the position relationship between straight line and. Teaching guidance allows students to fully observe, analyze, think and summarize conclusions under the heuristic questions of teachers.
Figure 1 Student activities: There are also changes when moving to different locations. When growing up, the straight line intersects with the original straight line from right to left. Teacher: In this process, there is a position that does not intersect with parallel lines. So how big is the straight line? In other words, if two straight lines are judged to be parallel, we need to find the relationship between the angles. Teacher: Let the students recall the drawing method of parallel lines first, and draw parallel lines outside the straight line. Student activities: Students finish in exercise books and the teacher demonstrates on the blackboard (see figure 1). Teacher: From the demonstration just now, please consider what the process of drawing parallel lines actually guarantees.
Figure 2 Student: Make sure that two congruence angles are equal. Teacher: What can you guess from it? Student: Two straight lines are cut by a third straight line. If the same angle is equal, two straight lines are parallel. Teacher: Is our guess right? Will there be a moment when two straight lines are parallel, even if the angles are different? The teacher used a computer to demonstrate the process of movement change. Before observing the experiment, let the students see the angle and angle clearly (as shown in Figure 2), and then start the experiment, let the students fully observe and discuss what conclusions can be drawn. Student activities: students observe, discuss and analyze. It is concluded that when it is appropriate, it is not parallel, no matter what value it takes, it is parallel as long as it is. Figure 3 Teachers guide students to express their own conclusions. And tell the students that this conclusion is called the parallel line axiom. [Blackboard] Two straight lines are cut by the third line. If the isosceles angles are equal, two straight lines are parallel. Simply put, the isosceles angles are equal and two straight lines are parallel. That is, ∫ (as shown in Figure 3) and ∴ (The isosceles angles are equal and the two straight lines are parallel). The teaching method shows that through actual drawing and demonstration, sports and computer-2. If it is timely, it can be done. The teaching method explains these two topics in order to consolidate the learned judgment axioms. Because the second question is a known conclusion, find out the topic that makes it effective. This is a kind of thinking method that should be mastered when proving problems, which requires students to gradually learn the thinking method of citation and citation, and teachers should pay attention to gradually cultivating students' mathematical thinking in teaching. (Show projection) A straight line is cut by a straight line.
Figure 5 1. See figure 5. If so, what does it matter? 2. What does it matter? 3. What is the diagonal of the positional relationship? Student activities: students observe, think, analyze and give answers: time, time and equality, and sum is an interior angle. Teacher: What conditions can be obtained? Why? Student activity: Because you can get it by equivalent substitution. Teacher: What conclusion can you draw from it? Student Activity: Teacher: What correct conclusion can you draw from it? Student activities: the inner angles are equal and the two straight lines are parallel. Teacher: That is to say, we get another way to judge that two straight lines are parallel: [blackboard writing] Two straight lines are cut by a third straight line. If the internal angles are equal, two straight lines are parallel. Simply put, the internal angles are equal and the two straight lines are parallel. The teaching method explains that through the teacher's inspiration and guidance of questioning method, students are guided to discover the relationship between angles themselves, and then summarized. Mainly through discussing problems, students can cultivate good study habits of positive thinking and good brain analysis. Teacher: The above reasoning process can be written as: (known), (equal to the vertex angle), ∴. [∫ (proof)], ∴ (equal to the same angle, two straight lines are parallel). The teaching method explains the reasoning process here. Cultivate their enterprising spirit. The teacher pointed out that the "⊙" in square brackets is the "∴" just got above. In this case, this step in square brackets can be omitted. Try feedback consolidation exercise (show projection) 1. As shown in figure 1, the line is cut by wire. (65438 (2) Through measurement, what can you judge its basis? 2. As shown in Figure 2, it is. (1) From which can we determine which two straight lines are parallel? What is the basis? (2) From, which two straight lines are parallel can be determined? What is the basis?
Figure 1 Figure 2 Student activities: students' oral answers. The purpose of teaching method to explain this group of questions is to consolidate the judgment axioms and methods of parallel lines and make students familiar with and use them to solve simple reasoning problems. Variant training can cultivate their ability (display projection) 1. As shown in Figure 3, which two straight lines are parallel can be judged? From which two straight lines are parallel? 2. As shown in Figure 4, you know? Why?
Figure 3 Figure 4 Student activities: Students answer questions after thinking. The teacher corrected, inspired and guided them to get the answer. The teaching method shows that this group of questions can not only make students understand variant graphics and strengthen their reading ability, but also cultivate students' divergent thinking, that is, cultivate students to consider problems from multiple angles and in all directions. Thus, we can get multiple solutions to one problem and improve students' problem-solving ability. (4) Summarize and expand. 2. Combined with the proof process of judging a theorem, familiar with the requirements of expressing reasoning proof, and initially understand the format of reasoning proof. 8. Assign exercise 2.2A( 1) in groups 4, 5 and 6 on page 97 of the homework textbook. (2) problems. Homework answer 4. At the right time, you can ... deduce that two straight lines are parallel at the same angle. (2) Deduce that two straight lines are parallel at the same angle. 6.( 1) According to the same angle, two straight lines are parallel. (2) According to the same angle, it can be concluded that two straight lines are parallel.