Method of coordinates in the school geometry course. Thesis: Studying the method of coordinates in the course of geometry of the basic school Methodological foundations of teaching the coordinate method
International University of Nature, Society and Man "Dubna"
Draft program for the course
Development of lessons on the topic:
"Distance from point to line"
"Distance between parallel lines"
Dmitrov, 2013
1. Introduction ……………………………………………………………………………… ......… 3
2. Draft program for the course
"Method of coordinates and foundations of analytical geometry on a plane" ……………………………………………………………………………… ....... 4
3. Lesson development:
Lesson-lecture "Distance from point to line" …………………….… ... 8
Lesson-lecture "Distance between parallel lines" ... ..17
4. Conclusion ………………………………………………………………………………… ..23
5. References ………………………………………………………………………… 23
6. Appendices ………………………………………………………………………………… .24
1. INTRODUCTION
The strategy for the development of modern society based on knowledge and highly effective technologies objectively requires significant adjustments to pedagogical theory and practice, an intensification of the search for new models of education.
The study of geometry at the level of basic general education is aimed at achieving the following goals:
- mastering the system of knowledge and skills necessary for application in practice, studying related disciplines, continuing education;
- intellectual development, the formation of personality qualities necessary for a person for a full life in modern society, inherent in mathematical activity: clarity and accuracy of thought, critical thinking, intuition, logical thinking, elements of algorithmic culture, spatial representations, the ability to overcome difficulties;
- formation of ideas on the ideas and methods of mathematics as a universal language of science and technology, a means of modeling phenomena and processes;
- upbringing culture of personality, attitude to mathematics as a part of human culture that plays a special role in social development.
In this project, the study of the basics of analytical geometry begins from grade 7, which will allow students to approach the solution of stereometric problems using the coordinate method at a more conscious and qualitative level.
2.MAIN PART
Draft program for the course
"Method of coordinates and foundations of analytical geometry on a plane"
for students in grades 7-8 of basic school
,
(International University of Nature, Society and Man "Dubna")
and students of the PC courses of the International University "Dubna
1. Course idea, goals and objectives –
Relevance This topic is due to the fact that the content and methods of teaching mathematics used in basic school in some part do not correspond to the modern needs of training specialists in technical areas.
Target: To bring the content and methods of teaching mathematics in basic school to the modern needs of a technological society.
Tasks:
1. Analyze the needs of a modern technological society and compare the apparatus of mathematics used in solving applied problems with the content of mathematics in basic school.
2. Creation of a program project for the course "Method of coordinates and the basics of analytical geometry on a plane"
3. Development of lessons on the topic "Distance from a point to a straight line", "Distance between parallel lines" R Section "Mutual arrangement of objects on a plane"
2. Place in the general school curriculum- 7-9 grade. Volume - 1 lesson per week, in parallel with the main course of traditional geometry, taught, for example, from a textbook Atanasyan (with co-authors). The total volume is 70 hours, which is 1/3 of the total volume of the geometry course for grades 7-9. Recommended terms for completing the course: beginning - the second half of the 7th grade, the end - the 1st half of the 9th grade. However, depending on the specific conditions for mastering the program in each particular school (curricula, work programs, basic textbooks, the presence of additional hours in the educational grid for geometry), other terms of its development are possible. For example, if additional hours are available, the learning curve can be shortened by increasing the number of hours per week.
3. Main sections and content.
Chapter | Watch |
|
Second half of grade 7 | ||
1. Introduction | Examples of tasks and applications. | 1 |
2. Vectors on a plane | Vector concept. Equality of vectors. Basic properties and operations on vectors (addition and subtraction of vectors, multiplication by a number). Zero vector. Vectors and geometric shapes. Independent work. | 4 |
3. Coordinate method | Cartesian rectangular coordinate system. Setting points. Distance between points (Pythagorean theorem). Algebraic description of a vector. Operations on vectors given in algebraic form. Algebraic description of polygons. Independent work. | 5 |
4. Dot product of vectors | Angle between vectors. Vector-to-vector projection. Scalar product (axioms). Algebraic rule for calculating the dot product. Determination of the cosine and sine of an angle on a circle. Sine and cosines of the simplest angles. Cosine of angle between vectors and dot product of vectors. Algebraic definition of the form of a triangle. Test. | 8 |
The first half of the 8th grade | 17 |
|
5. Equation of a straight line on a plane | Parametric equation of a straight line (two ways of setting). Division of a segment in a given ratio. Description of polygons. Particular cases of the straight line equation: canonical and explicit. General equation of the straight line. The geometric meaning of the coefficients in the general equation of the straight line. Equation of a straight line in segments. Direction cosines. Independent work. | 8 |
6. Mutual arrangement of straight lines on a plane | Parallelism of straight lines on a plane: formulation of the criterion depending on the method of defining straight lines. Constructs a straight line parallel to the given point and passing through the given point. Describes polygons with parallel sides. Perpendicularity of straight lines on a plane: formulation of the criterion depending on the method of defining straight lines. Creates a straight line perpendicular to a given point and passing through a given point. Test. | 9 |
Second half of grade 8 | 18 |
|
7. Mutual arrangement of plane objects | Determination of the type of a quadrilateral by coordinates. Finding points of intersection of lines. Distance from point to line. Distance between parallel lines. Independent work. | 7 |
8. Symmetries of the plane | Central symmetry. Definition and examples of symmetries in the simplest polygons. Construction of points and lines symmetric to the data about a given center of symmetry (geometric construction and algebraic description). Axial symmetry. Definition and examples of symmetries in the simplest polygons. Plotting points and lines, symmetric to data about the axis of symmetry (geometric construction and algebraic description). Test. | 11 |
1st semester 9th grade | 17 |
|
9. Singular points and line segments in the simplest polygons | Geometric construction of the point of intersection of medians and its algebraic finding. Calculation of coordinates of points of intersection of bisectors, heights and perpendiculars. Their special properties. Independent work. | 6 |
10. Solving polygons | Solving geometry problems using the coordinate method. Cosine theorem. Test. | 6 |
11. Movement *, Repetition | Parallel transfer, rotation | 5 |
3.LESSON DEVELOPMENT
Lesson-lecture: "Distance from a point to a straight line"
Goals:introduce the concept of distance from a point to a straight line, show how they are used in solving problems.
1. Explanation of the new material
Definition.
Distance from point to line Is the length of the perpendicular drawn from a given point to a given straight line
It should be noted that the distance from a point to a straight line is the smallest of the distances from this point to the points of a given straight line. Let's show it.
Let's take on a straight line a point Q not coinciding with the point M1... Section M1Q are called oblique drawn from point M1 to straight a... We need to show that the perpendicular drawn from the point M1 to straight a less than any oblique drawn from a point M1 to straight a... This is indeed the case: the triangle M1QH1 rectangular with hypotenuse M1Q, and the length of the hypotenuse is always greater than the length of any of the legs, therefore,font-size: 12.0pt; line-height: 115%; font-family: Verdana ">.
font-size: 12.0pt; line-height: 115%; font-family: Verdana "> If, when finding the distance from a point to a straight line, it is possible to enter a rectangular coordinate system, then you can use the coordinate method. In this lesson we will dwell on two ways of finding the distance from a point M1 to straight a, which are given in a rectangular Cartesian coordinate system Oxy on surface. In the first case, the distance from the point M1 to straight a we will search as a distance from a point M1 to the point H1, where H1- the base of the perpendicular dropped from the point M1 on a straight line a... In the second method of finding the distance from a point M1 to straight a we will use the normal equation of the line a.
So, let's set ourselves the following problem: let a rectangular coordinate system be fixed on the plane Oxy we can calculate using the formula for finding the distance from a point M1 to the point H1 by their coordinates:.
It remains to figure out how to find the coordinates of the point H1.
We know that a straight line in a rectangular coordinate system Oxy corresponds to some equation of a straight line on the plane. We will assume that the way of specifying the straight line a in the problem statement allows us to write the general equation of the straight line a or the equation of a straight line with a slope. After that, we can compose the equation of a straight line passing through a given point M1, perpendicular to the straight line a... Let us denote this straight line by the letter b... Then the point H1 Is the intersection point of the lines a a and b, solving the system of linear equationsfont-size: 12.0pt; line-height: 115%; font-family: Verdana; color: # 32322E "> or
;
4) calculate the required distance from the point M1 to straight a according to the formula.
Description of the presentation for individual slides:
1 slide
Slide Description:
Educational complex of the author's physics and mathematics school-lyceum No. 61. PROJECT "Method of coordinates in mathematics and geography" Completed by: students 7 B and 7 C of classes of the educational establishment of the School of Physics and Mathematics No. 61 Evlashkov Daniil Littau Roman Khegai Vladimir Supervisor: Gorborukova NV Bishkek - 2012
2 slide
Slide Description:
Determining the location of an object on the surface of the Earth or any point on a plane is the determination of their address. "Address" in geography is the geographic latitude; geographic longitude; absolute height. "Address" in mathematics - abscissa, ordinate of a point on a coordinate plane
3 slide
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Purpose of the project: To investigate and compare the ways of determining the "address" of an object in geography and mathematics.
4 slide
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Project objectives: To answer the following questions: Who, when and why first introduced the concept of "coordinates"? Is there a genetic connection between the concepts of "geographic coordinates" and "coordinate method" in mathematics? Or are they homonyms? The development of which sciences was influenced by the method of coordinates? What other types of coordinate systems, besides rectangular, exist and are used by humans at present in practice?
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Historical reference. In the II - III centuries BC. NS. meridians and parallels first appeared on the map of Eratosthenes. However, they did not yet represent a coordinate grid.
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In the II century. BC NS. For the first time, Hipparchus divided the circle into 360 parts and proposed to gird the globe with meridians and parallels on the map. He introduced the concept of the equator, drew parallels and drew meridians through the poles. Thus, a cartographic network was created and it became possible to map geographic objects.
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Completed the galaxy of great ancient astronomers and geographers Claudius Ptolemy (190 - 168 BC). In his work "Guide to Geography" in 8 books, he described over 8000 geographic objects, indicating their geographic coordinates: latitude and longitude.
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1. Geography: "geo" - Earth, "grafo" - I write. 2. Geometry: "geo" - earth, "metreo" - to measure. As you can see, these two sciences were closely related, their emergence is due to the practical activities of people of that time.
11 slide
Slide Description:
Why are latitude and longitude measured in degrees? Geographic latitude is the arc of the meridian from the equator to a given point. It is known from the geometry course that arcs are measured both in linear quantities and in angular ones: degrees and radians. Geographic longitude is the magnitude of the parallel arc from the prime meridian to a given point. It can be seen that geographic coordinates are a mathematical concept.
12 slide
Slide Description:
The emergence of algebra as a branch of mathematics. In the 9th century, the Uzbek mathematician and astronomer Muhammad al-Khwarizmi wrote a treatise "Kitab al-jabr wal-mukabala", where he gave general rules for solving equations of the 1st degree. The word "al-jabr" ("restoration") meant the transfer of negative terms of equations from one part of it to another with a change in sign. From him the new science got its name - algebra. For a long time, algebra and geometry developed in parallel and represented two branches of mathematics.
13 slide
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In the XIV century. French mathematician Nicola Oresme proposed to introduce, by analogy with geographical, coordinates on a plane. He proposed to cover the plane with a rectangular grid and call latitude and longitude what we now call abscissa and ordinate. This marked the beginning of the creation of the coordinate method and connected algebra and geometry.
14 slide
Slide Description:
Coordinate method Algebra A plane point is given by a pair of numbers M (x; y) - an algebraic object A straight line is given by the equation y = ax + c Geometry A plane point is a geometric object
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René Descartes (1596-1650) - French mathematician, philosopher, physicist and physiologist. Descartes is one of the founders of analytic geometry, modern algebraic symbolism, and the method of defining a curve using an equation was a decisive step towards the concept of a function. In mathematics, it was he who owed the main merit in creating the method of coordinates, which was the basis of analytical geometry.
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1. It should be noted that Descartes did not yet have what we today call the Cartesian coordinate system. Descartes began by translating construction problems with a compass and a ruler into algebraic language. 2. A considerable merit of Descartes was the introduction of convenient notation used today: x, y, z - for unknowns, a, b, c - for coefficients, as well as the designation of degrees. 3. Currently, Cartesian coordinates are orthogonal axes with the same scale in all directions, that is, O is the origin.
17 slide
Slide Description:
Let's compare coordinate systems in mathematics and geography. 1. To determine the position of an object on the surface of the Earth, 2 coordinates are required: longitude and latitude. 2. To determine the position of a point on a plane, 2 coordinates are required: abscissa and ordinate. 3. Parallels and meridians are mutually perpendicular. 4. Axes OX and OY are mutually perpendicular. 5. To determine a point in space, a 3rd coordinate is required: absolute height (in geography); applicate in mathematics. 6. Equator and prime meridian divide the surface of the globe into 4 parts. 7. Coordinate axes divide the plane into 4 parts, and the space into 8 parts.
18 slide
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Polar and spherical coordinates. The polar coordinate system includes the point O - the pole and the ray - the polar axis. Each point on the plane corresponds to a pair of numbers P (r; f), the angle between the direction to the object and the polar axis and the distance to the object In geography, the analogue of polar coordinates is azimuth. To determine the location of an object, you need to know the angle between the direction to the object and the direction to the north and the distance to the object.
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A spherical coordinate system is used if it is necessary to determine the position of a point in space. This method is used in air navigation. With the help of the radar, 3 coordinates are determined: the shortest distance in a straight line to the aircraft; the angle at which the plane is visible above the horizon; the angle between the direction of the plane and the direction of north
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CONCEPTUAL MAP Geography Cartography Coordinate system 1. Rectangular - geographic latitude - geographic longitude - absolute height 2. Polar - azimuth - distance to the object - absolute height Mathematics Algebra Geometry Method of coordinates 1. Rectangular - abscissa - ordinate - applicate 2. Polar - angle of rotation - distance from the origin to the point
21 slide
The essence of the coordinate method for solving geometric problems
The essence of solving problems using the coordinate method is to enter a coordinate system that is convenient for us in one case or another and rewrite all the data using it. After that, all unknown quantities or proofs are carried out using this system. How to enter the coordinates of points in any coordinate system was discussed by us in another article - we will not dwell on this here.
Let us introduce the main statements that are used in the coordinate method.
Statement 1: The coordinates of the vector will be determined by the difference between the corresponding coordinates of the end of this vector and its beginning.
Statement 2: The coordinates of the midpoint of the segment will be determined as the half-sum of the corresponding coordinates of its boundaries.
Statement 3: The length of any vector $ \ overline (δ) $ with the given coordinates $ (δ_1, δ_2, δ_3) $ will be determined by the formula
$ | \ overline (δ) | = \ sqrt (δ_1 ^ 2 + δ_2 ^ 2 + δ_3 ^ 2) $
Statement 4: The distance between any two points given by the coordinates $ (δ_1, δ_2, δ_3) $ and $ (β_1, β_2, β_3) $ will be determined by the formula
$ d = \ sqrt ((δ_1-β_1) ^ 2 + (δ_2-β_2) ^ 2 + (δ_3-β_3) ^ 2) $
Scheme for solving geometric problems using the coordinate method
To solve geometric problems using the coordinate method, it is best to use this scheme:
- Set the most suitable coordinate system for the task;
- The condition of the problem, the question of the problem, is mathematically written, a drawing is built for this problem.
Record all task data in the coordinates of the selected coordinate system.
- Make up the necessary relations from the condition of the problem, and also connect these relations with what needs to be found (prove in the problem).
- The result obtained is translated into the language of geometry.
Analyze what is given in the task:
Examples of problems solved by the coordinate method
The main tasks leading to the coordinate method are the following (we will not present their solutions here):
- Tasks for finding the coordinates of a vector by its end and beginning.
- Tasks related to dividing a segment in any relation.
- Proof that three points lie on the same straight line or that four points lie in the same plane.
- Tasks for finding the distance between two given points.
- Tasks for finding volumes and areas of geometric shapes.
The results of solving the first and fourth problems are given by us as the main statements above and are often used to solve other problems using the coordinate method.
Examples of tasks on the application of the method of coordinates
Example 1
Find the side of a regular pyramid with a height of $ 3 $ cm if the side of the base is $ 4 $ cm.
Let us be given a regular pyramid $ ABCDS $, the height of which is $ SO $. Let's introduce a coordinate system as in Figure 1.
Since the point $ A $ is the center of the coordinate system we have constructed, then
Since the points $ B $ and $ D $ belong to the axes $ Ox $ and $ Oy $, respectively, then
$ B = (4,0,0) $, $ D = (0,4,0) $
Since the point $ C $ belongs to the plane $ Oxy $, then
Since the pyramid is correct, $ O $ is the middle of the $$ segment. According to Statement 2, we get:
$ O = (\ frac (0 + 4) (2), \ frac (0 + 4) (2), \ frac (0 + 0) (2)) = (2,2,0) $
Since the height of $ SO $