Mathematics teaching plan of the second volume of the ninth grade: the calculation of acute triangle function
First, the teaching objectives
1. Through observation, guessing, comparison, gymnastics and other mathematical activities, learn to find an acute trigonometric function value with a calculator.
2. Experience the process of using trigonometric function knowledge to solve practical problems and promote the development of observation, analysis, induction and communication ability.
3. Feel the close connection between mathematics and life, enrich the successful experience of mathematics learning, stimulate students' curiosity to continue learning, and cultivate students' awareness of cooperation and communication with others.
Second, teaching material analysis
In life, we often encounter such problems, such as measuring the height of buildings, measuring the width of rivers, positioning ships and so on. To solve this kind of problems, we often need to apply trigonometric function knowledge. Last lesson, we have learned the trigonometric function values of 30, 45 and 60, and we can do some calculations under certain circumstances, but it is impossible to solve the problems in life only by relying on the trigonometric function values of these three special angles. In this lesson, students can use a calculator to find trigonometric function values, so that they can get rid of heavy calculations and experience the process of finding problems, asking questions, analyzing problems, exploring solutions and finally solving problems.
Third, the analysis of the situation of schools and students.
The age of ninth grade students is generally around 15 years old. At this stage, students take abstract logical thinking as the main development trend, but to a great extent, students still rely on concrete empirical materials and operational activities to understand abstract logical relations. In addition, using calculators can greatly reduce the burden on students. Therefore, based on the background materials provided by textbooks and supplemented by the use of calculators, students can solve problems better.
Students have been using calculators since primary school and are familiar with the operation of calculators. At the same time, in the previous courses, students have learned the definition of acute trigonometric functions, the values of trigonometric functions 30, 45 and 60 and their simple calculations, and they have the knowledge and skills to learn this lesson.
Fourthly, teaching design.
(a) review of issues
1. The ladder leans against the wall. If the angle between the ladder and the ground is 60 and the length of the ladder is 3 meters, what is the distance from the bottom of the ladder to the wall?
Student activity: Find the value according to the meaning of the question.
2. In life, is the angle between the ladder and the ground always 60?
No, there can be various angles. 60 is just a special phenomenon.
Figure 1 (2) Create a situation and introduce a topic.
1? As shown in figure 1, when the hanging box of the mountaineering cable car passes through point A and reaches point B, it has already passed 200 m. Given that the included angle between the cable car route and the plane is ∠ a = 16, what is the vertical ascending distance of the cable car?
Which line segment represents the vertical distance that the cable car rises?
Line segment BC
Which right triangle can I use to find BC?
In Rt△ABC, BC = absin 16, so BC = 200sin 16.
Do you know what sin 16 is? We can calculate the trigonometric function value of acute triangle with the help of scientific calculator. So, how to find trigonometric function with scientific calculator?
To find the value of trigonometric function with scientific calculator, sin cos and tan keys should be used. Teacher activities: (1) Show the following table; (2) Dictation according to the table, so that students can learn to find the value of sin 16. The key sequence display result is sin16 sin16 = sin16 = 0? 275 637 355
Student activities: Find the values of sin 16 in the order listed in the table.
Can you calculate the values of cos 42, tan 85 and SIN 72 38' 25 "?
Student activities: By analogy with the method of finding sin 16, through guessing, discussing and learning from each other, we can find the corresponding trigonometric function value with a calculator (the operation steps are as follows):
Key sequence display result COS42 COS42 = COS42 = 0? 743 144 825 tan 85 tan 85 = tan 85 = 1 1？ 430 052 3 inch 72 38'25 inch sin 72d'M'S
38D ' M ' S2
5D′M′S = sin 72 38′25″→
0? 954 450 32 1
Teacher: Use a scientific calculator to solve the problem at the beginning of this section.
Health: BC = 200 sin 16 ≈ 52? 12 (m).
Description: Use students' interest in learning to consolidate the calculation method of finding trigonometric function value with calculator.
(3) think about it.
Teacher: In the question at the beginning of this section, when the cable car continues to reach point D from point B, it has passed 200 m, and the angle between the cable car's driving route from point B to point D and the horizontal plane is ∠ β = 42, so what else can be calculated?
Student activity: (1) You can find the vertical distance de of the second ascent, the sum of the vertical distances of the two ascends, the horizontal distance of the two passes, and so on. (2) Complement each other, and deepen the understanding of trigonometric functions in this process.
Classroom practice
1. If a person wants to climb from the foot of the mountain to the top of the mountain, he needs to climb a slope of 40 300 m first, and then climb a slope of 30100 m to find the height of the mountain (the result is accurate to 0. 1 m).
2. As shown in Figure 2, ∠ DAB = 56, ∠ CAB = 50, AB=20 m, find the length of the lightning rod CD in the figure (the result is accurate to 0.0 1 m).
Figure 2 Figure 3
(5) detection
As shown in Figure 3, Wuhua Building is 60 meters away from Xiao Wei's home. Xiao Wei looked at the building from his own window. The elevation angle at the top of the building was 45, while the depression angle at the bottom of the building was 37. Find the building height (the result is accurate to 0? 1 m).
Note: While students are practicing, teachers should patrol and guide, observe students' learning situation and give timely guidance to students' difficulties.
(6) Summary
Students talk about their feelings of learning this class, such as what new knowledge they have learned in this class, what difficulties they have encountered in the learning process, how to solve them and so on.
(7) homework
1. Use the calculator to find the following values:
(1) Tan 32; (2)cos 24？ 53 ; (3)sin 62 1 1′； (4) Tan 39 39' 39 ".
Figure 42? As shown in Figure 4, in order to measure the width of a river, the surveyor measured the position of a tree T on the other side of the river at two points, P and Q, which are 180 m apart. T is in the south direction of P and 50 in the southwest direction of Q, and the channel width is calculated (the result is accurate to 1 m).
Reflection on the Teaching of verb (abbreviation of verb)
1. This section is the content of learning to find trigonometric function value with calculator and applying it in practice. Through the study in this section, students can fully realize that trigonometric function knowledge is widely used in the real world. There are not many knowledge points in this course, but students have improved their ability to analyze and solve problems by actively participating in the classroom, and have developed well in willpower, self-confidence and rational spirit.
2. As the organizer, guide, collaborator and helper of students' learning, teachers create problem situations according to the characteristics of teaching materials to help students succeed from their existing knowledge background and activity experience.
Beijing normal university printing plate third grade mathematics second volume teaching plan
First, the goal of quality education
(A) the main points of knowledge teaching
Let the students know that when the acute angle of a right triangle is fixed, the ratio of its opposite side, adjacent side and hypotenuse is also fixed.
(2) Key points of ability training
Gradually cultivate students' logical thinking abilities such as observation, comparison, analysis and generalization.
(C) moral education penetration point
Guide students to explore and discover, thus cultivating students' independent thinking, innovative spirit and good study habits.
Second, the focus and difficulty of teaching
1. key: let students know that when the acute angle is fixed, the ratio of the opposite side, the adjacent side and the hypotenuse is also fixed.
2. Difficulties: It is difficult for students to think that the ratio of the opposite side, adjacent side and hypotenuse of any acute angle is also fixed. The key is that the teacher guides the students to make comparative analysis and draw a conclusion.
Third, the teaching steps
Clear goal
1. As shown in Figure 6- 1, if a ladder with a length of 5 meters is placed on a wall with a height of 3 meters, what is the distance between A and B?
The 2.5-meter-long ladder leans against the wall with an inclination of ∠CAB 30. What's the distance between a and b?
3. If a 5-meter-long ladder is installed on the wall with an inclination of 40, what is the distance between A and B?
4. If a ladder with a length of 5 meters leans against the wall, so the distance between A and B is 2 meters, what is the inclination angle ∠CAB?
The first two questions are easy for students to answer. These two questions are mainly designed to arouse students' memories and make them realize that this chapter needs this knowledge. However, the design of the latter two questions confuses students, which plays a role in stimulating students' interest in learning for those students who are curious and competitive in grade three. At the same time, make students have a preliminary understanding of the characteristics of the content to be studied in this chapter. Some problems can't be solved by Pythagoras theorem or the knowledge of right triangle and isosceles right triangle with an angle of 30. The key to solve these problems is to find a new method to find an edge or an unknown acute angle. As long as this is done, the unknown angles of all other right triangles can be found out with the knowledge learned.
Four examples lead to the topic.
(B) the overall perception
1. Please take out your own triangle and measure and calculate the ratio of the opposite side, adjacent side and hypotenuse at angles of 30, 45 and 60 respectively.
Students will soon answer the result: the proportion of the triangular ruler is a fixed value regardless of its size. Students with good degrees will also think that as long as they know one side of these special right-angled triangles in the future, they can calculate the lengths of other unknown sides.
2. Let the students draw a right triangle with an angle of 40, and measure and calculate the ratio of the opposite side, adjacent side and hypotenuse of 40. The students are happy to find that the required proportion is fixed regardless of the size of the triangle. Most students may wonder, when the acute angle takes other fixed values, is the ratio of the opposite side, the adjacent side and the hypotenuse also fixed?
Doing so not only cultivates students' practical ability, but also makes students have an overall perception of the knowledge to be learned in this class, stimulates students' thirst for knowledge and boldly explores new knowledge.
(C) the key and difficult learning and goal completion process
1. Through hands-on experiments, students guess that "no matter what the acute angle of a right triangle is, the ratio of its opposite side, adjacent side and hypotenuse is always fixed". But how to prove this proposition? Students' thinking is very active at this time. Some students may be able to solve this problem. Therefore, teachers should let students discuss and finish independently at this time.
Students may be able to solve this problem through research. If not, the teacher can guide them appropriately:
If a set of right triangles have equal acute angles, it can be
Vertices A 1, A2, A3 are marked as A, and right-angle sides AC 1, AC2, AC3…… ........................................................................................................................ Can students solve this problem? Instruct students to prove independence: Zhiyi, b 1c 1∨B2 C2∨B3 C3 ..., ∴△ ab1c/∽△ ab2c2 ∽△ ab3c3 ∽. ...
Formally, the ratio of the opposite side, adjacent side and hypotenuse of ∠A is a constant value.
Through guidance, students can master the key points independently, achieve the purpose of knowledge teaching, cultivate students' ability and infiltrate moral education.
The design of the hands-on experiment in the previous tutorial is actually to break through the difficulties and also plays a role in cultivating students' thinking ability.
This exercise is to let students know that the ratio of the opposite side to the hypotenuse of any acute angle can be found.
(4) Summary and expansion
1. Guide students to summarize knowledge: On the basis of reviewing Pythagorean theorem and the properties of a right triangle with an angle of 30, we find that as long as the acute angle of a right triangle is fixed, the ratio of its opposite side, adjacent side and hypotenuse is also fixed.
Teachers can add appropriately: after students' hands-on experiments, bold speculation and positive thinking, we have found new conclusions, and I believe that everyone's logical thinking ability has been improved. I hope everyone can carry forward this innovative spirit, change passive learning knowledge into active finding problems and cultivate their own innovative consciousness.
2. Extension: When the acute angle is 30, we know the ratio of the opposite side to the hypotenuse. Today, we find that when the acute angle is arbitrary, the ratio of the opposite side to the oblique side is also fixed. Knowing this ratio solves the problem of finding other unknown edges. It seems that this ratio is very important. We will focus on this "comparison" in the next class. Interested students can preview it in advance. Through this extension, we can
Fourth, homework
The content of this lesson is less, which lays the foundation for the concepts of sine and cosine, so students are required to preview the concepts of sine and cosine after class.
The ninth grade mathematics teaching plan of Beijing Normal University Volume II.
First, the goal of quality education
(A) knowledge teaching points
Let the students know that when the acute angle of a right triangle is fixed, the ratio of its opposite side, adjacent side and hypotenuse is also fixed.
(2) Key points of ability training
Gradually cultivate students' logical thinking abilities such as observation, comparison, analysis and generalization.
(C) moral education penetration point
Guide students to explore and discover, thus cultivating students' independent thinking, innovative spirit and good study habits.
Second, the focus and difficulty of teaching
1. key: let students know that when the acute angle is fixed, the ratio of the opposite side, the adjacent side and the hypotenuse is also fixed.
2. Difficulties: It is difficult for students to think that the ratio of the opposite side, adjacent side and hypotenuse of any acute angle is also fixed. The key is that the teacher guides the students to make comparative analysis and draw a conclusion.
Third, the teaching steps
Clear goal
1. As shown in Figure 6- 1, if a ladder with a length of 5 meters is placed on a wall with a height of 3 meters, what is the distance between A and B?
The 2.5-meter-long ladder leans against the wall with an inclination of ∠CAB 30. What's the distance between a and b?
3. If a 5-meter-long ladder is installed on the wall with an inclination of 40, what is the distance between A and B?
4. If a ladder with a length of 5 meters leans against the wall, so the distance between A and B is 2 meters, what is the inclination angle ∠CAB?
The first two questions are easy for students to answer. These two questions are mainly designed to arouse students' memories and make them realize that this chapter needs this knowledge. However, the design of the latter two questions confuses students, which plays a role in stimulating students' interest in learning for those students who are curious and competitive in grade three. At the same time, make students have a preliminary understanding of the characteristics of the content to be studied in this chapter. Some problems can't be solved by Pythagoras theorem or the knowledge of right triangle and isosceles right triangle with an angle of 30. The key to solve these problems is to find a new method to find an edge or an unknown acute angle. As long as this is done, the unknown angles of all other right triangles can be found out with the knowledge learned.
Four examples lead to the topic.
(B) the overall perception
1. Please take out your own triangle and measure and calculate the ratio of the opposite side, adjacent side and hypotenuse at angles of 30, 45 and 60 respectively.
Students will soon answer the result: the proportion of the triangular ruler is a fixed value regardless of its size. Students with good degrees will also think that as long as they know one side of these special right-angled triangles in the future, they can calculate the lengths of other unknown sides.
2. Let the students draw a right triangle with an angle of 40, and measure and calculate the ratio of the opposite side, adjacent side and hypotenuse of 40. The students are happy to find that the required proportion is fixed regardless of the size of the triangle. Most students may wonder, when the acute angle takes other fixed values, is the ratio of the opposite side, the adjacent side and the hypotenuse also fixed?
Doing so not only cultivates students' practical ability, but also makes students have an overall perception of the knowledge to be learned in this class, stimulates students' thirst for knowledge and boldly explores new knowledge.
(C) the key and difficult learning and goal completion process
1. Through hands-on experiments, students guess that "no matter what the acute angle of a right triangle is, the ratio of its opposite side, adjacent side and hypotenuse is always fixed". But how to prove this proposition? Students' thinking is very active at this time. Some students may be able to solve this problem. Therefore, teachers should let students discuss and finish independently at this time.
Students may be able to solve this problem through research. If not, the teacher can guide them appropriately:
If a set of right triangles have equal acute angles, it can be
Vertices A 1, A2, A3 are marked as A, and right-angle sides AC 1, AC2, AC3…… ........................................................................................................................ Can students solve this problem? Instruct students to prove independence: Zhiyi, b 1c 1∨B2 C2∨B3 C3 ..., ∴△ ab1c/∽△ ab2c2 ∽△ ab3c3 ∽. ...
Formally, the ratio of the opposite side, adjacent side and hypotenuse of ∠A is a constant value.
Through guidance, students can master the key points independently, achieve the purpose of knowledge teaching, cultivate students' ability and infiltrate moral education.
The design of the hands-on experiment in the previous tutorial is actually to break through the difficulties and also plays a role in cultivating students' thinking ability.
This exercise is to let students know that the ratio of the opposite side to the hypotenuse of any acute angle can be found.
(4) Summary and expansion
1. Guide students to summarize knowledge: On the basis of reviewing Pythagorean theorem and the properties of a right triangle with an angle of 30, we find that as long as the acute angle of a right triangle is fixed, the ratio of its opposite side, adjacent side and hypotenuse is also fixed.
Teachers can add appropriately: after students' hands-on experiments, bold speculation and positive thinking, we have found new conclusions, and I believe that everyone's logical thinking ability has been improved. I hope everyone can carry forward this innovative spirit, change passive learning knowledge into active finding problems and cultivate their own innovative consciousness.
2. Extension: When the acute angle is 30, we know the ratio of the opposite side to the hypotenuse. Today, we find that when the acute angle is arbitrary, the ratio of the opposite side to the oblique side is also fixed. Knowing this ratio solves the problem of finding other unknown edges. It seems that this ratio is very important. We will focus on this "comparison" in the next class. Interested students can preview it in advance. Through this extension, we can
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