I. Formula derivation
The derivation of the distance formula between two points in the third section of the second chapter of the first volume of the textbook "Compulsory Mathematics" for ordinary senior high schools is different from the traditional textbook.
The new textbook focuses on the derivation of the distance formula between two points by vector method, and applies the vector knowledge learned by middle school students in this book to solve practical problems. This method is easy to understand. This reflects the connection and mastery of knowledge.
The right triangle is permeated in the "thinking" of the new textbook, and the distance formula between two points is deduced by Pythagorean theorem, which is consistent with the research idea of the old textbook.
Using Pythagorean theorem to derive the distance formula between two points needs to be discussed in different situations:
1. If both points are on the X axis (the straight line where the two points are located is parallel to the X axis), then the distance between the two points is equal to the absolute value of the difference between the abscissas of the two points.
Similarly, if both points are on the Y-axis (the straight line where the two points are located is parallel to the Y-axis), then the distance between the two points is equal to the absolute value of the vertical coordinate difference between the two points.
2. The straight line where the two points are located is not parallel to the coordinate axis.
Construct a right triangle by making two points as straight lines parallel to the coordinate axis, calculate the lengths of the two right sides of the right triangle, and calculate the length of the hypotenuse by pythagorean theorem, that is, the distance between two points.
Through analysis, the formula of the first case is consistent with the formula of the second case, and the formula of the distance between two points on the plane is derived.
Through the actual derivation process and comparative analysis, the vector method is simpler than Pythagorean theorem.
Second, the formula application
1. Directly apply the formula to solve the problem.
According to students' learning situation, the coordinates of two known points are given first, so that students can directly calculate the distance formula between two points by using the distance formula between two points.
Then, using example 3 on page 73 of the textbook, the coordinates of two known points A and B are given, and a point P is found on the X axis, so that the distance between P and the known two points is equal, and the coordinates of point P and the length of line segment PA are obtained.
This example involves solving equations, which is daunting for students with weak foundation.
When solving with distance formula, it is often necessary to solve quadratic equation, the number of roots of quadratic equation and the number of corresponding points. This examines the students' mathematical operation ability.
2. Deep exploration-study plane geometry problems with coordinate method.
Taking Example 4 on page 73 of the textbook as the carrier, let students experience, feel and appreciate the basic steps of solving plane geometry problems by coordinate method.
Title: Prove by coordinate method that "the sum of squares of two diagonals of parallelogram is equal to twice the sum of squares of two adjacent sides". (called "parallelogram pythagorean theorem" or "generalized pythagorean theorem")
Research steps:
Firstly, an appropriate rectangular coordinate system is established, and the geometric elements involved in the problem are expressed in coordinates, so that the plane problem is transformed into a coordinate problem. "Appropriate coordinate system" plays an important role in solving problems smoothly.
A plane rectangular coordinate system is established with the straight line on AB side as the X axis and the straight line passing through A and perpendicular to AB as the Y axis. The coordinates of points A, B and D are easy to write, and the acquisition of coordinates of point C is the key and difficult point to prove this problem, which needs to be combined with the properties of parallelogram.
Method 1: By geometric method, make a vertical line, find the similarity, and write the coordinates of point C; Method 2: Write the coordinates of point C by using the midpoint coordinate formula, and the midpoint of BD is also the midpoint of AC. Method 3: Using vector equation, vector AB is equal to vector DC (mobile phone input, vector symbol can't be typed out), and you can write the coordinates of point C (in solid geometry, writing the coordinates of point means looking and calculating).
Then the lengths of the corresponding line segments are calculated according to the subject conditions. Through coordinate operation, the relationship between geometric elements is studied.
Finally, the results of algebraic operation are transformed into geometric conclusions.
The above coordinate method for solving plane geometric problems embodies the general idea of studying geometric problems by algebraic method. This provides a good train of thought and method for the follow-up study of conic curve problems.
Third, classroom integration.
According to what you have learned in this lesson, lead the students to do three small questions in the exercise on page 74 of the textbook. These three small problems are completely consistent with the classroom teaching ideas, which play a role in reviewing and consolidating the knowledge learned.
Fourth, after-class summary
Let the students share the harvest and experience of this lesson: knowledge, core literacy of mathematics, moral education and so on.
Reflection on the Teaching of verb (abbreviation of verb)
1. Let students experience, participate in and experience the process of deriving the distance formula between two points, which is also a learning process, and cultivate students' ability to ask, analyze and solve problems.
2. Where students are prone to problems, teachers should give necessary guidance to help students tide over the difficulties.
3. Finding the coordinates of point C by various methods is of great benefit to the development of students' thinking breadth and depth.
4. The plane analytic geometry is studied by coordinate method, and a general research idea is finally summarized through a series of operations. It is easy for students to condense scattered operation processes into systematic thinking processes.