Introduction of Cartesian coordinates in presentation space. Cartesian coordinates in space. Organizing time. Good afternoon
Lesson #3
COORDINATE B METHOD
SPACE
Cartesian coordinates in space
René Decaert, French philosopher, mathematician, mechanic, physicist and physiologist
Height, width, depth.
Only three coordinates.
Where is the way past them? The bolt is closed.
Listen to the sonata of the spheres with Pythagoras,
Atoms can be counted like Democritus.
V. Bryusov.
1 Introduction of a rectangular coordinate system in space.
2 Location of points in the coordinate system.
3 Finding the coordinates of points in space.
4 Constructing a point in space using its coordinates.
5 The concept of radius vector.
6 Decomposition of a vector into coordinate vectors.
7 Finding the coordinates of the vector of the sum of vectors, vector
difference of vectors, a vector multiplied by a given number.
8 Problem solving.
9 Recording remote control. METHOD OF COORDINATES IN SPACE
Plane coordinate system
Y
y
Coordinate system in space
Z
z
M(x;y)
abscissa
ordinate
ABOUT
x
1) 2 straight
2) Point - NK
3) Direction of axes
4) Axes name
5) Point M
6) Title
coordinates
points M
X
X
1)
2)
3)
4)
x
applicate
y
Y
Abscissa axis
Y axis
Axis applicate
OX; OY; OZ
5) Coordinate planes
6) Point M
7) Title
coordinates
points M
ordinate
M(x;y;z)
ABOUT
3 straight
Tochka – NK
Axes direction
Axes name
abscissa
XOY; XOZ; YOZ Different locations of points in the coordinate system
Z
K
T
M
L
N
ABOUT
Y
P
X
Location of a point in the coordinate system
on the OX axis
in the XOY plane
on the OY axis
in the YOZ plane
on the OZ axis
in the XOZ plane 1) Finding the coordinates of points
2) Finding the coordinates of points
Given a cube with edge length 2
Z
C1
B1
A1
A
2
D1
B
Y
Given a rectangular parallelepiped
with 2 dimensions; 5; 7
2
X
Z
B1
A1
C
D
2
Find the coordinates of all the vertices of the cube
A
X
D1
5
2
B
7
C
D
Find the coordinates of all vertices
rectangular parallelepiped
3) Constructing a point using its coordinates
Plot the points in a rectangular
coordinate system:
M(3; 4; 5) and T(-2; 5; -7)
C1
Y Vector coordinates
Vector decomposition
by coordinate vectors
Z
WITH
OM OA OV OS
M
k
ABOUT
X
A
j
according to the parallelepiped rule
OM xi yj zk
In Y
i
R
OM (x; y; z)
radius - vector
M(x;y;z)
The coordinates of the radius vector are equal
end coordinates
given vector
Equal vectors have
same coordinates
р(x; y; z)
р xi yj zk a(x1;y1;z1)
Coordinates
vector sums
b(x2;y2;z2)
Coordinates
vector differences
(a+b)( )
(a-b)( )
fold
relevant
coordinates
Vector coordinates,
multiplied by the number
ka( )
every
coordinate
multiply by this
number
subtract
relevant
coordinates 4) Given the decomposition of the vector into unit vectors, write down the coordinates of the vector.
р 3i 2 j k , р j 6k , р k .
5) Given the coordinates of the vector, write down the decomposition of the vector into unit vectors.
p( 3;6;1), p( 2;5;0), p(0; 1;0). Homework from lesson 3:
paragraphs 46, 47 and notes, be able to compose a competent story,
№ 400, 402, 403, 404, 410
in the next lesson the simplest SR
Introduction of Cartesian coordinates in space. Distance between points. Coordinates of the midpoint of the segment. Prepared by teacher LSOSH No. 2 Besshabashnova L.f. I think - therefore I exist
- devoted himself to mathematics studies. He studied German algebra and French and Greek mathematics.
- After the capture of La Rochalie in 1628, Descartes left the army. He leads a solitary life in order to implement his extensive plans for scientific work. Descartes was the greatest philosopher and mathematician of his time. Descartes' most famous work is his Geometry. Descartes introduced a coordinate system that everyone uses today. He established a correspondence between numbers and line segments and thus introduced the algebraic method into geometry. These discoveries of Descartes gave a huge impetus to the development of both geometry and other branches of mathematics and optics. It became possible to display the dependence of quantities graphically on coordinate plane
Lesson topic
Introduction of Cartesian coordinates in space. Distance between points. Coordinates of the midpoint of the segment.- A coordinate system is a set of one, two, three or more intersecting coordinate axes, the point at which these axes intersect - the origin - and unit segments on each of the axes. Each point in the coordinate system is defined by an ordered set of several numbers - coordinates. In a particular non-degenerate coordinate system, each point corresponds to one and only one set of coordinates.
- If straight lines perpendicular to each other are taken as coordinate axes, then the coordinate system is called rectangular (or orthogonal). A rectangular coordinate system in which the units of measurement on all axes are equal to each other is called an orthonormal (Cartesian) coordinate system
- M (X;Y;Z)
|
On surface |
In space |
|
Definition. A coordinate system is a set of two intersecting coordinate axes, the point at which these axes intersect - the origin - and unit segments on each of the axes |
Definition. A coordinate system is a set of three coordinate axes, the point at which these axes intersect - the origin of coordinates - and unit segments on each of the axes |
|
OU-axis ordinate, OX - abscissa axis |
OX - abscissa axis, OU – ordinate axis, OZ - applicator axis. |
|
OX is perpendicular to OA |
OX is perpendicular to OU, OX is perpendicular to OZ, Op-amp is perpendicular to OZ |
|
Direction, single segment |
|
|
Distance between points. |
Distance between points |
|
Coordinates of the midpoint of the segment. |
Coordinates of the midpoint of the segment |
All the guys stood up together.
And they walked on the spot.
They stretched on their toes.
And now they’ve bent over backwards.
Like springs, we sat down.
And they sat down quietly at once.
Plot points
- A(9;5;10), B(4;-3;6), C (9;0;0), D(0;0;4), E(0;8;0), K(-2 ;4;6)
- P.23-25
- №7,№10(1)
Thank you for your attention!
Sections: Mathematics
Lesson objectives:
Educational: Consider the concept of a coordinate system and the coordinates of a point in space; derive the distance formula in coordinates; derive the formula for the coordinates of the midpoint of the segment.
Educational: To promote the development of students' spatial imagination; contribute to the development of solutions to problems and development logical thinking students.
Educational: Fostering cognitive activity, a sense of responsibility, a culture of communication, a culture of dialogue. Equipment: Drawing supplies, salt crystal lattice.
Lesson type: Lesson on learning new material (2 hours).
Lesson structure:
- Organizing time.
- Introduction.
- Communicate lesson objectives.
- Motivation.
- Updating.
- Learning new material.
- Understanding and awareness.
- Consolidation.
- Lesson summary.
Leading task: prepare proof of theorems and derivation of formulas, a report on Rene Descartes.
Training technology: Programmed learning technology (block learning).
During the classes
1. Organizational moment. Good afternoon.
2. Introduction.
Today in class we are starting to study the fourth block of the 10th grade geometry course “Cartesian coordinates and vectors in space.”
Introducing the table of the fourth block (the table is on each desk).
Grade 10. Cartesian coordinates and vectors in space. Block No. 4
Number of hours - 18 hours
| Name of topics | Theory (textbook) |
Workshop | Independent work | Theory test | Test papers |
| Introduction: Cartesian coordinates in space. Distance between points. Coordinates of the midpoint of the segment. |
P.152 | Practical work №6 | Independent work No. 5 | Geometric dictation. | Home test №4 Class test #4 |
| Symmetry. Parallel transfer. Movement. |
P.155, p.156 | Practical work No. 7 |
Independent work No. 6 |
Score card No. 3 | Home test No. 5 Class test #5 |
| Angle between: Crossing straight lines; Straight and flat; Planes. 9. The area of the orthogonal projection of a polygon. |
Practical work No. 8 | Score card No. 4 | |||
| Vectors in space. | P.164 | Practical work No. 9 | Score card No. 5 |
What topic is consistent with the topic of our lesson did we study in 8th grade? What keyword defines these two topics? (Coordinates). Plane and spatial coordinates can be entered in an infinite number of different ways.
When solving a geometric, physical, chemical problem, you can use various coordinate systems: rectangular, polar, cylindrical, spherical. (Showing models of the crystal lattice of table salt)
In the general education course, the rectangular coordinate system on the plane and in space is studied. Otherwise, it is called the Cartesian coordinate system after the French scientist philosopher Rene Descartes (1596 - 1650), who first introduced coordinates into geometry.
(Student's story about Rene Descartes.)
Rene Descartes was born in 1596 in the city of Lae in the south of France, into a noble family. My father wanted to make Rene an officer. To do this, in 1613 he sent Rene to Paris. Descartes had to spend many years in the army, participating in military campaigns in Holland, Germany, Hungary, the Czech Republic, Italy, and in the siege of the Huguenot fortress of La Rochalie. But Rene was interested in philosophy, physics and mathematics. Soon after his arrival in Paris, he met Vieta's student, a prominent mathematician of that time - Mersen, and then other mathematicians in France. While in the army, Descartes devoted all his free time to mathematics. He studied German algebra and French and Greek mathematics.
After the capture of La Rochalie in 1628, Descartes left the army. He leads a solitary life in order to implement his extensive plans for scientific work.
Descartes' philosophical views did not meet the requirements of the Catholic Church. Therefore, he moved to Holland, where he lived for 20 years, from 1629 to 1649, but due to the persecution of the Protestant Church in 1649 he moved to Stockholm. But the harsh northern climate of Sweden turned out to be disastrous for Descartes, and he died of a cold in 1650.
Descartes was the greatest philosopher and mathematician of his time. His philosophy was based on materialism. Descartes' most famous work is his Geometry. Descartes introduced a coordinate system that everyone uses today. He established a correspondence between numbers and line segments and thus introduced the algebraic method into geometry. These discoveries of Descartes gave a huge impetus to the development of both geometry and other branches of mathematics and optics. It became possible to depict the dependence of quantities graphically on the coordinate plane, numbers - as segments, and to perform arithmetic operations on segments and other geometric quantities, as well as various functions. It was a completely new method, distinguished by beauty, grace and simplicity.
R. Descartes - French scientist (1596-1650)
3. Communicate the purpose of the lesson.
Today in the lesson we will continue to study the Cartesian coordinate system, and show that coordinates in space are entered as simply as coordinates on a plane.
4. Motivation.
Rene Descartes once said: “… descendants will be grateful to me not only for what I said, but also for what I did not say and thereby gave them the opportunity and pleasure to figure it out on their own.” I will give you the opportunity and pleasure to understand the Cartesian coordinate system on your own.
5. Learning new material.
Explanation. Block study technology involves studying several topics in a lesson. The lesson will cover three topics. Each topic will contain the following structure:
- Study of new material (the study is based on a comparative analysis of the basic concepts and formulas discussed in planimetry and the proof of the necessary theorems);
- Awareness and comprehension.
Based on the material you know for grade 8, we will fill out the table. Let's make a comparative description.
(A table is drawn on the board, it must be filled out together with the students. Consider the basic concepts of Cartesian coordinates, the formula for the distance between points, the formula for the coordinates of the midpoint of a segment on a plane, and try for the students to formulate the basic concepts and formulas in space themselves)
| On surface | In space |
| Definition. | Definition. |
| 2 axles, OU - ordinate axis, OX - abscissa axis |
3 axles, OX - abscissa axis, OU – ordinate axis, OZ - applicator axis. |
| OX is perpendicular to OA | OX is perpendicular to OU, OX is perpendicular to OZ, OU is perpendicular to OZ. |
| (O;O) | (OOO) |
| Direction, single segment | |
| Distance between points. | Distance between points. d = v (x2 - x1)? + (y2 - y1)? + (z2 – z1)? |
| Coordinates of the midpoint of the segment.
|
Coordinates of the midpoint of the segment. |
Pictures used for conversation:
 |
 |
Questions to fill out the first part of the table.
1. Formulate the definition of a Cartesian coordinate system?
2. Try to formulate the definition of a Cartesian coordinate system in space?
3. What are the coordinate axes on the plane? What are the coordinate axes in space? Name, which axis have we not studied? (Introducing a new word “applicate”)
4. What planes are considered in planimetry (in space)?
5. What is the coordinate of the origin on the plane (in space)?
6. What other components should a coordinate system have on a plane and in space?
7. How is the coordinate of a point on the plane and in space determined?
Conclusion:
Tell us how the Cartesian coordinate system is introduced in space and what it consists of?
During a conversation, draw a drawing of the frontal-dimetric projection of the axes.
Consider the position of the axes in accordance with the drawing.
Construct a point with given coordinates A (2; - 3).
Construct a point with given coordinates A (1; 2; 3).
Consider the construction on the board. Work using cards (2 people at the board).
Work with the class: task No. 3 from the textbook, page 287, orally.
Questions to fill out the second part of the table.
1. Write down the formula for the distance between points on a plane.
2. How would you write the formula for the distance between points in space?
Let's prove its validity(derivation of the formula - paragraph 154, p. 273)
The advanced task is to display the formula on the board for students.
Work using cards: 2 people at the board.
Find the length of the segment:
- A (1;2;3;) and B (-1; 0; 5)
- A (1;2;3) and B (x; 2 ;-3)
Working with the class: Task No. 5 on page 288.
Questions to fill out the third part of the table.
1. How can we write the formula for the coordinates of the midpoint of a segment?
2. How would you write down the formula for the coordinates of the midpoint of a segment?
Let's prove its validity(derivation of the formula p. -154 p., 273).
The advanced task is to derive a formula for the coordinates of the midpoint of a segment near the board.
Working with the class. Orally.
Find the coordinates of point M - the middle of the segment
A(2;3;2), B (0;2;4) and C (4;1;0)
- Is point B the midpoint of segment AC?
Work with the class: Task No. 9 page 288.
Consolidation.
Workshop: Problem solving (Practical work).
While solving problems, students are surveyed on previous topics and newly learned material (proof of theorems).
Homework: study paragraphs 152, 153,154, questions 1 – 3, tasks 3, 4, 6, 10, prepare for the geometric dictation.
Lesson summary.
- How is the Cartesian coordinate system introduced? What does it consist of?
- How are the coordinates of a point in space determined?
- What is the coordinate of the origin equal to?
- What is the distance from the origin to given point?
- What is the formula for the coordinates of the middle of a segment and the distance between points in space?
Assessment(the teacher independently assigns grades for work in class and announces them to students).
Organizing time. Thank you for the lesson. Goodbye.
Literature.
- A.V. Pogorelov. Textbook 7-11. M. “Enlightenment”, 19992-2005.
- I.S. Petrakov. Math clubs in grades 8-10. M, “Enlightenment”, 1987
Slide 2
Lesson objectives 1. Show, using clarity as much as possible, that coordinates in space are entered as simply and naturally as coordinates on a plane. 2. Application of formulas to solve problems.
Slide 3
Lesson on the topic Cartesian coordinates in space
R. Descartes - French scientist (1596-1650) Descartes was the greatest philosopher and mathematician of his time. His philosophy was based on materialism. Descartes' most famous work is his Geometry. Descartes introduced a coordinate system that everyone uses today. He established a correspondence between numbers and line segments and thus introduced the algebraic method into geometry. These discoveries of Descartes gave a huge impetus to the development of both geometry and other branches of mathematics.
Slide 4
At one time, Rene Descartes said: “... descendants will be grateful to me not only for what I said, but also for what I did not say and thereby gave them the opportunity and pleasure to figure it out on their own.” Motivation
Slide 5
3. What are the coordinate axes on the plane? What are the coordinate axes in space? Name, which axis have we not studied? (Introduction to the new word “applicate”) 4. What planes are considered in planimetry (in space)? 5. What is the coordinate of the origin on the plane (in space)? 6. What other components should a coordinate system have on a plane and in space? Drawings are used for conversation
Slide 6
Tell us how the Cartesian coordinate system is introduced in space and what it consists of? During a conversation, draw a drawing of the frontal-dimetric projection of the axes. Consider the position of the axes in accordance with the drawing. Construct a point with given coordinates A (2; - 3). Construct a point with given coordinates A (1; 2; 3).
Slide 7
Basic concepts of Cartesian coordinates. . .
Slide 8
distance formula between points
Slide 9
Coordinates of the midpoint of the segment.
summary of other presentations“The condition of perpendicularity of a straight line and a plane” - Perpendicular and oblique. Perpendicularity of lines and planes. Theorem about two parallel lines. Construction plan. Straight line a is perpendicular to the ASM plane. Let us prove that line a is perpendicular to an arbitrary line m. Definition. Theorem about two lines perpendicular to a plane. A sign of perpendicularity of a line and a plane. A sign of perpendicularity of planes. Median. In plane b through point M we draw a straight line c.
“Subject of stereometry” - Indefinable concepts. Dots. Geometry. Regular polyhedra. Do you remember the Pythagorean theorem? Directions. Philosophical school. Stereometry. Axioms of stereometry. Invisible side. Pythagorean theorem. From the history. Egyptian pyramids. Pythagoras. Stereometry science concept. Visual representations. Universe. Today in class. Planimetry. Basic concepts of stereometry. Euclid. Spatial representations.
“Types of regular polyhedra” - Preparation of sulfuric acid. Plato. Tetrahedron. Stellated icosidodecahedron. Stellated icosahedron. Hexahedron. Hanging Gardens of Babylon. Halicarnassus Mausoleum. Polyhedra in nature. Dodecahedron. Squad. Regular polyhedra and nature. Regular polyhedra in Plato's philosophical picture of the world. Truncated icosahedron. Regular polyhedra. Mechanical puzzles. Stellated dodecahedron. Star polyhedra.
“Determination of dihedral angles” - Problem. The point on the edge can be arbitrary. Notes on problem solving. Construction of a linear angle. Find the distance. Problem solving. Half-planes forming a dihedral angle. Theorem of three perpendiculars. On one of the edges dihedral angle, equal to 30, the point M is located. Perpendicular, oblique and projection. Let's cast a beam. Point K is removed from each side. Degree measure corner. Find the angle.
“Basic axioms of stereometry” - Pyramid of Cheops. Axioms of stereometry. Axiom. Subject of stereometry. Corollaries from the axioms of stereometry. Images of spatial figures. Geometry. Plane. The planes have common point. Sources and links. The points of a straight line lie in a plane. Geometric bodies. Four equilateral triangles. Corollaries from the axioms. Basic figures in space. First lessons in stereometry. An ancient Chinese proverb.
“Parallelepiped” - Properties of the diagonals of a rectangular parallelepiped. Inclined parallelepiped. A line segment connecting two vertices. Basic elements of a parallelepiped. Derivation of the formula for the volume of a rectangular parallelepiped. Parallelepiped. "Salzburg parallelepiped". A prism whose base is a parallelogram. Volume of a parallelepiped. Surface area of a rectangular parallelepiped. Any pair of parallel faces can be taken as bases.