Teaching plan of ninth grade mathematics circle 1, teaching material analysis.
(1) knowledge structure
(2) Analysis of key points and difficulties
Key points:
Three positional relationships between (1) point and circle, and related concepts of circle, because they are the basis of studying circle;
(2) Five common point trajectories, one is a profound understanding of geometric figures, and the other is an important preparation for studying solid geometry and analytic geometry in the future.
Difficulties:
① The set definition of circle makes it difficult for students to understand why two conditions must be met, and the content itself is difficult;
(2) the locus of points, because students have strong thinking in images and weak abstract thinking, this part of knowledge is abstract and difficult to understand.
2. Suggestions on teaching methods
This part takes 4 class hours.
Lesson 1: The definition of a circle and the positional relationship between a point and a circle.
(1) Let students draw their own circles, define their own circles, communicate, summarize and generalize, and mobilize students to actively participate in teaching activities; For high-level students, they can learn directly through the collection of points and the definition of circles;
(2) The relationship between points and circles, so that students can observe, classify and explore by themselves, and learn new knowledge in the process of "number and shape".
The second lesson: related concepts of circle.
(1) Let the students in (a) learn by themselves, while the students in (b) learn by themselves under the guidance of their teachers. In order to improve students' learning ability, especially the content with many concepts but little play, teachers don't need to talk about it;
(2) Classroom activities should grasp the main line of thinking from "number" to "shape" and from "shape" to "number".
The third and fourth lessons: the locus of points
Schools with better conditions can use computer animation to deepen and help students understand the trajectory of points. Ordinary schools can let students draw by hand, so that students can gradually change from image thinking to abstract thinking in the process of doing things, thinking, observing, thinking and understanding. But my point is that no matter how to organize teaching, we must follow the principle that students are the main body of learning.
The first lesson: Circle (1)
Teaching objectives:
1, understand the descriptive definition of circle, and understand the definition of circle from the perspective of set;
2. Understand the positional relationship between a point and a circle and the conditions for determining the circle;
3. Cultivate students' ability to find problems through hands-on practice;
4. Infiltrate the mathematical thinking method of "observation → analysis → induction → generalization".
Teaching emphasis: the relationship between point and circle
Teaching difficulties: two conditions for defining a circle with a set of points.
Teaching method: independent discussion.
Teaching process design (overall framework);
First, create a situation and carry out learning activities
1, let the students draw a circle, describe and communicate, and get the first definition of a circle:
Definition: 1: On a plane, the figure formed by the line segment OA rotating around its fixed endpoint O and the other endpoint A rotating with it is called a circle. The fixed endpoint o is called the center of the circle, and the line segment OA is called the radius. It is recorded as ⊙O and pronounced as "circle O".
2. Let students observe, think and communicate, and get the second definition of circle under the guidance of the teacher.
Discover new problems from old knowledge
Observe:
* * * Gender: The distance between these points and O point is equal.
Think about it: Are there any points on the plane that are equidistant from point O? What patterns do they form?
(1) The distance from each point on the circle to a fixed point (center o) is equal to a fixed length (length r of radius);
(2) All points whose distance to a fixed point is equal to a fixed length are on a circle.
Definition 2: A circle is a set of points whose distance from a fixed point is equal to a fixed length.
3. The positional relationship between a point and a circle
Question 3: What is the positional relationship between a point and a circle? (Students finish it by themselves and draw a conclusion)
If the radius of the circle is r and the distance from the point to the center of the circle is d, then:
Point d = r； on the circle;
Points in circle d
Outside the circle point d>r.
"number" and "shape"
Second, example analysis, variant exercises
Exercise: It is known that the radius ⊙O is 5cm, a is the midpoint of the line segment OP, and when OP=6cm, the point A is ⊙ o _ _ _ _ _ _ _ _; When OP= 10cm, point A is in ⊙O _ _ _ _ _ _ _ _ _ _; When OP= 18cm, point A is in ⊙ O _ _ _ _ _ _ _ _ _
Example 1 Prove that the four vertices of a rectangle are on the same circle with the diagonal intersection as the center.
Known (omitted)
Verification (omitted)
Analysis: quadrilateral ABCD is a rectangle
A=OC，OB = ODAC=BD
OA=OC=OB=OD
It is necessary to prove that four points A, B, C and D are on a circle with O as the center.
Prove: ∵ Quadrilateral ABCD is a rectangle.
∴ OA=OC，ob = odAC=BD
∴ OA=OC=OB=OD
∴ A, B, C and D are on a circle with O as the center and OA as the radius.
Application of the symbol ""(students are required to understand)
It is proved that quadrilateral ABCD is a rectangle.
OA=OC=OB=OD
A, B, C and D are on a circle with O as the center and OA as the radius.
Summary: It can be proved that several points are on the same circle, and the distances between these points and a fixed point are equal.
Problem expansion research: What vertices of the basic figures (parallelogram, rhombus, square, isosceles trapezoid) we have studied are on the same circle? (Let the students discuss)
Exercise 1 Prove that the midpoint of each side of the diamond is on the same circle.
Objective: To cultivate students' ability of analyzing problems and logical thinking. Level A independent completion)
Exercise 2 Let AB=3cm and draw a picture to show what a point set with the following properties looks like.
(1) A point set with a distance equal to 2cm;
(2) A point set whose distance to point B is equal to 2 cm;
(3) A point set with a distance of 2 cm from point A and point B;
(4) A point set with a distance of less than 2 cm from point A and point B; (Layer A is completed independently)
Third, the class summary
Q: What is the main content of this lesson? What problems should we pay attention to when studying? According to the students' answers, emphasize:
(1) mainly studies two different definitions of a circle and three positional relationships of a circle;
(2) When defining a circle with a point set, we must pay attention to two conditions, both of which are indispensable;
(3) Pay attention to the cultivation of mathematical ability.
Homework 82 pages 2, 3 and 4.
The second lesson: Circle (2)
Teaching objectives
1, so that students can understand the concepts of chord, arc, bow, concentric circle, equicircle and equisolitary; Initially, these concepts will be used to judge whether the proposition is true or not.
2. Gradually train students to read textbooks and practice by themselves. /Article/index . html & gt； Ability to summarize new concepts; Further guide students' ability to observe, compare, analyze and summarize knowledge.
3, through the whole process of hands-on and brain, stimulate students' initiative in learning, so that students can gain knowledge from initiative.
Teaching emphases, difficulties and doubts
1. key: understand the related concepts of circle.
2. Difficulties: Understand the characteristics of "mutual coincidence" in the definitions of "equal circle" and "equal arc".
3. Doubt: It is easy for students to regard two equal-length arcs as equal-length arcs. Let the students read the textbook and understand, communicate and solve problems in the dialogue with the teacher.
Teaching process design:
(1) Reading and understanding
Key concepts:
1. Chord: A line segment connecting any two points on a circle is called a chord.
2. Diameter: The chord passing through the center of the circle is the diameter.
3. Arc: The part between any two points on the circle is called arc.
Semi-arc: the two endpoints of a circle with any diameter are divided into two arcs, and each arc is called a semicircle;
Optimal arc: an arc larger than a semicircle is called an optimal arc;
Bad arc: An arc smaller than a semicircle is called a bad arc.
4. Bow: A figure composed of chords and their corresponding arcs is called a bow.
Concentric circles: Two circles with the same center and different radii are called concentric circles.
6. Equal circle: Two circles that can overlap are called equal circles.
7. Equal arcs: arcs that can overlap each other in the same circle or equal circle are called equal arcs.
(B) group communication, dialogue between teachers and students
Question:
1. How many strings are there in a circle? What is the longest string?
2. What kinds of arcs are there? How to express it?
3. What's the difference between bow and chord? How many bows can a string have in a circle?
4. What do you mean by equal circle and equal arc "overlapping"?
By asking questions, students and students, students and teachers can exchange learning, deepen their understanding of concepts and eliminate problems.
(3) Concept discrimination:
Judging the topic:
(1) diameter is chord () (2) chord is diameter ()
(3) A semicircle is an arc () (4) An arc is a semicircle ()
(5) Two equal-length arcs are equal-length arcs () (6) Equal-length arcs are equal-length arcs ()
(7) The sum of two lower arcs is equal to a semicircle () (8) Two semicircles with the same radius are equal arcs ().
(Mainly understand the following concepts: (1) chord and diameter; (2) Arc and semicircle; (3) Concentric circles and equal circles refer to two figures; (4) Equal circle and equal arc are superimposed on each other to obtain the conditional action of equal arc. )
(4) Application and practice
For example 1, it is known that AB and CB are two chords of ⊙ O as shown in the figure. Try to write all the arcs in the figure.
Solution: A * * * has six arcs. ,,,,,.
(Objective: Let students express arcs and deepen their understanding of the concepts of superior arcs and inferior arcs)
Example 2: As shown in the figure, in ⊙O, AB and CD are diameters. Verification: AD ∨ BC.
Students analyze, write the proof process, correct the existing problems, cultivate students' practical ability of speaking, thinking and hands-on, mobilize students' initiative in learning, and enable students to acquire knowledge from initiative.
Consolidation exercise:
Textbook P6