Design of teaching plan for sine and cosine of junior high school mathematics
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.
Excellent rational number multiplication teaching plan in junior middle school mathematics
Teaching objectives
1. Understand the significance of rational number multiplication, master the symbolic rule and absolute value operation rule in rational number multiplication rule, and preliminarily understand the rationality of rational number multiplication rule;
2. Be able to perform rational number multiplication skillfully according to the law of rational number multiplication, so that students can master the symbolic law of the product of multiple rational numbers multiplication;
3. When three or more rational numbers that are not equal to 0 are multiplied, the multiplication exchange law, association law and distribution law can be correctly applied to simplify the operation process;
4. Cultivate students' operational ability through the application of rational number multiplication law and operation law in multiplication operation;
5. This lesson explains the rationality of the rules through trip questions, so that students can feel that mathematics knowledge comes from life and apply it to life.
Teaching suggestion
(A) Analysis of key points and difficulties
The teaching focus of this section is to be able to operate skillfully. Flexible multiplication of rational numbers according to the law and operation law is the basis for further learning division and power operation. Operation, like addition, includes two steps: symbol judgment and absolute value operation. In multiplication operations where the factor does not contain 0, the sign of the product depends on the number of negative signs contained in the factor. When the number of negative signs is odd, the sign of the product is negative; When the number of negative signs is even, the sign of the product is positive. The absolute value of the product is the product of the absolute value of each factor. The operation process can be simplified by using the multiplication and exchange law and appropriate combination factors.
The difficulty of this section is the understanding of the law. In the law, "the same sign is positive and the different sign is negative" is only for the multiplication of two factors. The multiplication rule gives a method to determine the sign of product and the absolute value of product. That is, the sign of the two factors is the same, and the sign of the product is positive; The signs of the two factors are different, and the sign of the product is negative. The absolute value of the product is the product of the absolute values of these two factors.
(B) knowledge structure
(3) Suggestions on teaching methods
1. The rational number multiplication rule is actually a regulation. The problem of travel is to understand the rationality of this regulation.
2. When two numbers are multiplied, the basis for judging the sign is "the same sign is positive and the different sign is negative". Absolute value multiplication is also the arithmetic multiplication in primary school.
3. Students with poor foundation should pay attention to the difference between the symbolic law of multiplication and quadrature and the symbolic law of addition and summation.
Multiply several numbers. If one factor is 0, the product is equal to 0. Conversely, if the product is 0, then at least one factor is 0.
5. The multiplication exchange law, association law and distribution law in primary school are still applicable to rational number multiplication. It should be noted that the letters A, B and C here can be positive rational numbers, 0 or negative rational numbers.
6. If the factor is a fraction, it is generally necessary to turn it into a false fraction to facilitate reduction.
Example of instructional design
(first class)
Teaching objectives
1. Make students understand the rational number multiplication rule on the basis of understanding the meaning, and initially understand the rationality of the rational number multiplication rule;
2. Cultivate students' computing ability through operation;
3. Through the trip question given in the textbook, we know that mathematics comes from practice and reacts to it.
Teaching emphases and difficulties
Key points: handle affairs according to law and be skilled in operation;
Difficulty: Understanding the multiplication rule of rational numbers.
Classroom teaching process design
First, ask questions from students' original cognitive structure
1. Calculate (-2)+(-2)+(-2).
2. What are the rational numbers? In what rational number range are the four operations in primary school? (Non-negative number)
3. What is the key problem of rational number addition and subtraction? What is the main difference between primary school and primary school? (Symbol problem)
4. According to the addition and subtraction of rational numbers, the new problem is mainly the addition and subtraction of negative numbers, and the key to the operation is to determine the symbol. Can you guess what new contents and key problems will be found in the multiplication of rational numbers and the division to be learned later? (Negative number problem, determination of symbol)
Second, teachers and students learn the rational number multiplication rule together.
Question 1 The water level of the reservoir rises by 3cm per hour, and by how many centimeters in two hours?
Solution: 3×2=6 (cm) ①
A: It has increased by 6 cm.
Question 2: The average water level of the reservoir drops by 3 centimeters per hour, and how many centimeters does it rise in 2 hours?
Solution: -3×2=-6 (cm) ②
Answer: -6 cm up (that is, 6 cm down).
Guide students to compare ① and ②, and draw the following conclusions:
Replace a factor with its reciprocal, and the product is the reciprocal of the original product.
This is a very important conclusion. Applying this conclusion, 3×(-2)=? (-3)×(-2)=? (Student answers)
Compare 3×(-2) with ①, where a factor "2" is replaced by its opposite number "-2", and the product obtained should be the opposite number "-6" of the original product "6", that is, 3×(-2)=-6.
Comparing (-3)×(-2) and ②, a factor "2" is replaced by its opposite number "-2", and the product obtained should be the opposite number "6" of the original product "-6", that is, (-3)×(-2)=6.
In addition, (-3)×0=0.
According to the above situation, guide students to summarize the law of rational number multiplication:
Multiply two numbers, the same sign is positive, the different sign is negative, and then multiply by the absolute value;
Any number multiplied by 0 is 0.
Four. abstract
Today, we mainly studied the multiplication rule of rational numbers. We should remember that multiplying a positive number by two negative numbers only means "negative is positive".
Verb (short for verb) homework
The nature of the bisector of junior middle school mathematics angle: an example teaching plan
(A) the creation of situations, the introduction of new courses
Without tools, please divide a corner made of paper into two equal corners. What can you do?
What should I do if I change the paper in front of me into an angle that can't be folded, such as a board or a steel plate?
Design purpose: to gather students' thinking and create a good teaching atmosphere for the development of new courses.
(B) explore cooperation and exchange of new knowledge
(Activity 1) Explore the principle of angular bisector. The specific process is as follows:
Play the video material of Obama's visit to China-draw an umbrella-observe its cross section, so that students can clearly understand the corner relationship-draw the bisector; And use the geometric drawing board to dynamically demonstrate the opening and closing of the umbrella, so that students can intuitively feel the relationship between the umbrella surface and the main pole-let students design and make an angle bisector; And use the knowledge learned before to find the theoretical basis and explain the principle of making this instrument.
Design purpose: to perceive with examples in life. Taking the recent events as the introduction and the most common things as the carrier, let students feel that there is mathematics everywhere in their lives and appreciate the value of mathematics. Among them, the design and production of the bisector can cultivate students' creativity and sense of accomplishment and their interest in learning mathematics. Let the students finish Activity 2 easily.
(Activity 2) Through the above exploration, can you sum up the general method of using a ruler to make the bisector of a known angle? Do it yourself, and then exchange operating experience with your partner.
Complete this activity in groups, let teachers participate in student activities, find problems in time, give inspiration and guidance, and make comments more targeted.
The discussion results show that: according to the students' narration, the teacher demonstrated the method of making the known bisector with multimedia courseware;
Known: ∠ ao B.
Find the bisector of ∠AOB.
Exercise:
(1) Make an arc with O as the center and appropriate length as the radius, so that OA and OB intersect at m and n respectively.
(2) Take m and n as the center and the length greater than 1/2MN as the radius. The two arcs intersect at point C in ∠AOB.
(3) Ray OC, which is what you want.
Design purpose: let students understand painting more intuitively and improve their interest in learning mathematics.
Discussion:
1. In the second step of the above method, can the condition that the length is greater than MN be removed?
2. Does the intersection of the two arcs made in the second step have to be in ∠AOB?
The purpose of designing these two questions is to deepen the understanding of the angular bisector and cultivate a good study habit of mathematical rigor.
Summary of student discussion results:
1. If the condition that the length is greater than MN is removed, the two arcs may not intersect, so the bisector of the angle cannot be found.
2. If two arcs are drawn with M and N as the center and the length greater than MN as the radius, then the intersection of the two arcs may be inside or outside of ∠AOB, and we are looking for the inner intersection of ∠AOB, otherwise the ray obtained by connecting the intersection of the two arcs with the vertex is not the bisector of ∠AOB.
The bisector of an angle is a ray. It is neither a line segment nor a straight line, so the two restrictions in the second step are indispensable.
The feasibility of this method can be proved by congruent triangles.
(Activity 3) Explore the nature of the angular bisector.
Thinking: It is known that an angle and its bisector plus auxiliary lines form a congruent triangles; Form an congruent right triangle. How many pairs of such triangles are there?
The purpose of this design is to deepen the understanding of congruence.
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