G – 9th grade Lesson No. 2

Topic: The concept of a vector. Equality of vectors. Delaying a vector from a given point.

Goals:

introduce the concept of a vector, its length, collinear and equal vectors;

teach students to depict and designate vectors, to plot a vector equal to a given one from any point on the plane;

consolidate students’ knowledge while solving problems;

develop memory, attention, mathematical thinking;

develop diligence and the desire to achieve goals and objectives.

During the classes.

Organizational aspects.

Communicate the topic and objectives of the lesson.

Updating the knowledge and skills of students.

1. Checking homework completion. Analysis of unsolved tasks.

2. Checking theoretical information:

Isosceles triangle and its properties. Signs of equality of triangles.

Definition of the midline of a triangle and its properties.

Pythagorean theorem and its converse theorem.

Formula for calculating the area of ​​a triangle.

The concept of a parallelogram, properties and characteristics of a parallelogram, rhombus, rectangle.

Definition of trapezoid, types of trapezoids.

Area of ​​a parallelogram, area of ​​a trapezoid.

Learning new material.

Present the material in paragraphs 76–78 in the form of a short lecture using a variety of Vector presentations

1. The concept of vector quantities (or vectors for short).

2. Examples of vector quantities known to students from a physics course: force, displacement of a material point, speed and others (Fig. 240 of the textbook).

3. Determination of the vector (Fig. 241, 242).

4. Vector designation - two capital Latin letters with an arrow above them, for example,, or often denoted by a single lowercase Latin letter with an arrow above it:(Fig. 243, a, b).

5. The concept of a zero vector: any point on the plane is also a vector; in this case the vector is called zero; stand for:(Fig. 243, a).

6. Determining the length or modulus of a non-zero vector. Designation:. Zero vector length= 0.

7. Find the lengths of the vectors shown in Figures 243, a and 243, b.

8. Complete practical tasks No. 738, 739.

9. Consider an example of the movement of a body in which all its points move at the same speed and in the same direction (from paragraph 77 of the textbook), fig. 244.

10. Introduce the concept of collinear vectors (Fig. 245).

11. Definition of the concepts of co-directed vectors and oppositely directed vectors, their designation (Fig. 246).

12. The zero vector is codirectional with any vector.

13. Definition of equal vectors: ifAnd, That.

14. Explanation of the meaning of the expression: “Vectordelayed from point A” (Fig. 247).

15. Proof of the statement that from any point you can plot a vector equal to the given one, and only one (Fig. 248).

16. Completing practical task No. 743.

17. Orally solve problem No. 749 using the finished drawing on the board.

Problem solving.

1. Solve problem No. 740 (a) on the board and in notebooks.

2. Orally solve problem No. 744.

3. Solve problem No. 742.

4. Solve problem No. 745 (selectively).

5. Orally solve problem No. 746 using the prepared drawing.

6. Prove the direct statement in problem No. 750:

Proof

By condition, then AB || CD, therefore, according to the property of a parallelogram ABC is a parallelogram, and the diagonals of the parallelogram are divided in half by the intersection point, which means that the midpoints of the segments AD and BC coincide.

Organize repetition while solving the following problems - Tasks for repetition from the OGE (GIA)-2016 task bank:

№ 9, 10, 11, 12, 13 – from the “Geometry” module; No. 24 – from part 2 of the module “Geometry” Option No. 3

Lesson summary.

Summing up the lesson. Making marks.

As a result of studying § 1, students should know the definitions of a vector and equal vectors; be able to depict and designate vectors, plot a vector equal to a given one from a given point; solve problems like Nos. 741–743; 745–752.



Homework: study the material in paragraphs 76–78; answer questions 1–6, p. 213 textbooks; solve problems No. 747, 749, 751.

A vector is a directed segment of a straight line in Euclidean space, one end of which (point A) is called the beginning of the vector, and the other end (point B) the end of the vector (Fig. 1). Vectors are designated:

If the beginning and end of the vector coincide, then the vector is called zero vector and is designated 0 .

Example. Let the beginning of the vector in two-dimensional space have coordinates A(12.6) , and the end of the vector is the coordinates B(12.6). Then the vector is the zero vector.

Section length AB called module (length, the norm) vector and is denoted by | a|. A vector of length equal to one is called unit vector. In addition to the module, the vector is characterized by direction: the vector has a direction from A To B. A vector is called a vector, opposite vector.

The two vectors are called collinear, if they lie on the same line or on parallel lines. In the picture Fig. The 3 red vectors are collinear, because they lie on the same straight line, and the blue vectors are collinear, because they lie on parallel lines. Two collinear vectors are called equally directed, if their ends lie on the same side of the straight line connecting their beginnings. Two collinear vectors are called oppositely directed, if their ends lie on opposite sides of the straight line connecting their beginnings. If two collinear vectors lie on the same straight line, then they are called identically directed if one of the rays formed by one vector completely contains the ray formed by the other vector. Otherwise, the vectors are said to be oppositely directed. In Figure 3, the blue vectors are equally directed, and the red vectors are oppositely directed.

The two vectors are called equal if they have equal modules and the same directions. In Figure 2, the vectors are equal because their modules are equal and have the same direction.

The vectors are called coplanar, if they lie on the same plane or in parallel planes.

IN n In a dimensional vector space, consider the set of all vectors whose starting point coincides with the origin of coordinates. Then the vector can be written in the following form:

(1)

Where x 1 , x 2 , ..., x n vector end point coordinates x.

A vector written in the form (1) is called row vector, and the vector written in the form

(2)

called column vector.

Number n called dimension (in order) vector. If then the vector is called zero vector(since the starting point of the vector ). Two vectors x And y are equal if and only if their corresponding elements are equal.

The knowledge and skills acquired in this lesson will be useful to students not only in geometry lessons, but also in classes in other sciences. During the lesson, students will learn to plot a vector from a given point. This could be a regular geometry lesson, or an extracurricular or elective math class. This development will help the teacher save his time preparing for the lesson on the topic “Delaying a vector from a given point.” It will be enough for him to play the video lesson in class, and then reinforce the material with his own selection of exercises.

The duration of the lesson is only 1:44 minutes. But this is enough to teach schoolchildren to plot a vector from a given point.

The lesson begins with a demonstration of a vector, the beginning of which is at a certain point. They say that the vector is deferred from it. Then the author proposes to prove together with him the statement according to which from any point it is possible to plot a vector equal to the given one and, moreover, unique. During the proof, the author examines each case in detail. Firstly, it takes the situation when the given vector is zero, and secondly, when the vector is non-zero. During the proof, illustrations are used in the form of drawings and constructions, mathematical notation, which form mathematical literacy in schoolchildren. The author talks slowly, allowing students to take notes in parallel while commenting. The construction that the author carried out during the proof of the previously formulated statement shows how from a certain point one can construct a vector equal to the given one.

If students carefully watch the lesson and take notes at the same time, they will easily learn the material. Moreover, the author tells in detail, measuredly and quite completely. If for some reason you didn’t hear something, you can go back and watch the lesson again.

After watching the video lesson, it is advisable to begin consolidating the material. The teacher is recommended to select tasks on this topic in order to practice the skill of plotting a vector from a given point.

This lesson can be used for students to independently study the topic. But to consolidate it, you need to contact the teacher so that he can select the appropriate tasks. After all, without consolidating the material, it is difficult to achieve a positive result in learning.

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