New People's Education Edition seventh grade second volume mathematics courseware: vertical line [teaching goal]
1. To understand the concepts of vertical line and vertical line segment, you will draw a vertical line with a known straight line with a triangular ruler or protractor.
2. Master the concept of distance from point to straight line and measure the distance from point to straight line.
3. Grasp the nature of the vertical line, and make simple reasoning by using the learned knowledge.
[Teaching Emphasis and Difficulties]
1. Teaching focus: the definition and nature of the vertical line.
2. Teaching difficulty: drawing vertical lines.
[Teaching process design]
First, review questions:
1, describing the definitions of adjacent complementary angle and vertex angle.
2. What is the essence of vertex angle?
2. New lesson:
Introduction:
Earlier we reviewed the angle formed by two intersecting lines. If two straight lines intersect at a special angle, what is the special positional relationship between them? Is there such an example in daily life? Let's take a look at this problem.
(A) the definition of vertical line
When one of the four angles where two straight lines intersect is a right angle, it is said that the two straight lines are perpendicular to each other, one of which is called the perpendicular of the other, and their intersection is called the vertical foot.
As shown in the figure, the straight lines AB and CD are perpendicular to each other, and the vertical foot is O. ..
Please give an example of two perpendicular straight lines in daily life.
note:
1. If line segments are perpendicular to each other, line segments and rays, rays and rays, line segments or rays are perpendicular to each other, indicating that their lines are perpendicular to each other.
2, master the following reasoning process:
On the contrary,
(b) vertical drawing
Explore:
1. Draw a vertical line with a known straight line L with a triangular ruler or protractor. How many vertical lines can you draw?
2. How many vertical lines can you draw by drawing a vertical line of L through point A on the straight line?
3. Draw a vertical line of L through a point B outside the straight line L. How many vertical lines can you draw?
Painting method:
Let a right-angle side of a triangle plate coincide with a known straight line, move the triangle plate left and right along the straight line so that the other right-angle side passes through a known point, and draw a straight line along this right-angle side, then this straight line is the perpendicular of the known straight line.
Note: if you draw a little perpendicular to a ray or line segment, it means drawing a perpendicular to a straight line where they are, and the vertical foot is sometimes on the extension line.
(3) the nature of the vertical line
After passing a point (on or outside the known straight line), a vertical line of the known straight line can be drawn, and only one vertical line can be drawn, namely:
The attribute 1 has one and only one straight line perpendicular to the known straight line.
Exercise: Page 7 of the textbook.
Explore:
As shown in the figure, connect a point P outside the straight line L with each point O on the straight line L,
Party A, Party B, Party C, ..., in which (we call PO the point P of a straight line.
L) of the vertical section. Compare the lengths of line segments PO, PA, PB and PC ................................................................................................................................................................
Property 2 Of all the line segments connecting a point outside a straight line with a point on a straight line, the vertical line segment is the shortest.
Simply put: the vertical line is the shortest.
(4) Distance from point to straight line
The length from a point outside a straight line to the vertical section of the straight line is called the distance from the point to the straight line.
As shown above, the length of PO is called the distance from point P to line L.
Example 1
(1)AB is perpendicular to AC;
(2)AD and AC are perpendicular to each other;
(3) The vertical line segment from point C to AB is line segment AB;
(4) The distance from point A to BC is line AD;
(5) The length of line AB is the distance from point B to AC;
(6) Line 6)AB is the distance from point B to AC.
The correct one is ()
A. 1 B.2
C.3 D. 4
Solution: a
Example 2 As shown in the figure, straight lines AB and CD intersect at point O,
Solution: Omit
Example 3 As shown in the picture, A is driving a car on the straight road AB.
Driving to b, m and n are villages on both sides of the road.
Let's assume that the car is closest to the village m when driving to point p,
When driving to point Q, it is closest to N village. Please draw two points, P and Q, on the AB highway in the picture.
Exercise:
Page 9 of textbooks 3 and 4
Textbook page 10 9, 10,1,12.
Summary:
1. To master the concepts of vertical line, vertical line segment and distance from point to straight line;
2. Make clear the special situation that the vertical line is the intersection line, contact the previous knowledge, and draw the standard figure correctly with tools;
3. The nature of the vertical line lays a foundation for future knowledge learning and should be mastered skillfully.
Homework: Pages 9, 5 and 6.
New People's Education Edition seventh grade second volume mathematics courseware: intersection [teaching goal]
1. Through hands-on, operation, reasoning, communication and other activities, further develop the concept of space, and cultivate the ability of map reading, reasoning and orderly expression.
2. Knowing the adjacent complementary angle and antipodal angle in specific cases, we can find out the adjacent complementary angle and antipodal angle of an angle in a graph, understand that antipodal angles are equal, and use it to solve some simple problems.
[Teaching Emphasis and Difficulties]
Emphasis: the concepts of adjacent complementary angle and antipodal angle, the properties and applications of antipodal angle.
Difficulty: the exploration of understanding the nature of equal vertex angle.
[Instructional design]
First, create a situation to stimulate curiosity to observe the process of scissors cutting cloth, and introduce the angle formed by the intersection of two straight lines.
In the world we live in, there are a lot of intersecting lines and parallel lines. This chapter will study the angle formed by intersecting lines and its characteristics.
Observe the process of scissors cutting cloth and introduce the angle formed by two intersecting straight lines.
Students observe, think and answer questions.
The teacher showed a piece of cloth and a pair of scissors, performed the process of cloth cutting, and asked: When cutting cloth, hold the handle tightly. What is the angle between the two handles? How does the opening of scissors change?
Teacher's comment: If the structure of scissors is regarded as two intersecting straight lines, then the above is related to the angle formed by the intersection of the two straight lines.
2. Understand the adjacent complementary angle and antipodal angle, and explore the nature of antipodal angle.
1. Students draw a straight line where AB and CD intersect at point O and name the four corners in the picture. They match each other.
How many diagonal angles can * * * form? How to classify according to different positions?
Students think and communicate in groups, and the whole class communicates.
When students intuitively perceive the relationship between "neighborhood" and "diagonal", teachers guide students to express it accurately in geometric language;
There is a vertex O with a common * * *, and its two sides are extension lines with opposite sides.
2. Students use a protractor to measure the degree of each angle and find out the relationship between the degrees of each angle.
Students come to the conclusion that the two corners of the adjacent relationship are complementary and the top two corners are equal.
3 Students complete the following table according to observation and measurement:
The angular classification, positional relationship and quantitative relationship formed by the intersection of two straight lines
The teacher asked: If you change the size, will it change its position and quantity relationship with other corners?
4. Summarize the concepts of adjacent complementary angle and antipodal angle and the properties of antipodal angle.
Three. Preliminary application
Exercise:
Is the following statement true?
(1) Adjacent complementary angles can be regarded as two angles divided by rays passing through their vertices.
(2) Adjacent complementary angles are two complementary angles, and these two complementary angles are adjacent complementary angles.
(3) The antipodal angles are equal, and two equal angles are antipodal angles.
Students use the property of equal vertex angle to explain the phenomenon seen in the process of scissors cutting cloth.
Fourth, examples of consolidation application: As shown in the figure, straight lines A and B intersect to find the degree.
[Consolidation exercise] (exercise on page 5 of the textbook) is known, as shown in the figure, and the degree of: is found.
[Abstract]
Adjacent complementary angles, relative vertex angles.
[Homework] Textbooks P9- 1, 2p10-7,8