Methodical manual with electronic application / E.M. Savchenko. - M.: Planeta, 2011. - 256 p. -( modern school). ISBN978-5-91658-228-4
Given Toolkit is a collection of three parts. The first part of the book presents the methods and ways of using information technology by a mathematics teacher. The second part contains brief annotations and descriptions of the digital educational resources that are presented on the disc. The third part is the development of geometry lessons for students in grades 7-9, with a multimedia application for each lesson in the form of presentations. The material meets the requirements of the state educational standard and can be used by teachers working on any curriculum.
The electronic supplement to the book (CD-disk) contains: informative materials for explaining new material, tests, tasks for oral frontal work with students in the classroom. The presented multimedia material will help the teacher to make the lessons richer, more informative and visual. The CD-application can be used in any type of lesson: learning new material, repetition and generalization, during extracurricular activities by subject.
The teaching aid is intended for subject teachers, methodologists, students of advanced training courses for educators, students of pedagogical universities. .
CONTENT
Part I Application of multimedia presentations in geometry lessons
Introduction
- Organization of the subject teacher's media library
- Using presentations to illustrate definitions
- Using presentations to illustrate theorems
- Using Presentations to Illustrate Tasks
7th grade
- Initial geometric information
- Comparison of line segments and angles
- Section measurement. Blitz Poll
- Beam, angle, adjacent and vertical angles.
- Tests in Excel
- Perpendicular lines
- Adjacent and vertical corners
- The first sign of equality of triangles
- Medians, bisectors and heights of a triangle
- Isosceles triangle. Properties isosceles triangle
- Properties of an isosceles triangle. Problem solving
- The second sign of equality of triangles
- The third sign of equality of triangles
- Median, bisector, height, triangles.
- Tests in Excel
- Circle and circle
- Building tasks
- Parallel lines.
- Signs of parallel lines
- Parallel lines. Inverse theorems
- The sum of the angles of a triangle
- Signs of equality of right triangles
- Polygons.
- quadrilateral
- Parallelogram. Parallelogram Properties
- Parallelogram. Parallelogram features
- Trapeze
- Thales' theorem
- Rectangle, rhombus, square
- Rectangle area
- Parallelogram area
- Area of a triangle
- Areas of figures
- Trapezium area
- Pythagorean theorem
- Theorem converse to the Pythagorean theorem
- Similar triangles. Proportional segments
- The first sign of the similarity of triangles
- Collection of tasks. The first sign of the similarity of triangles
- The second and third signs of the similarity of triangles
- Middle line of the triangle
- Proportional segments in a right triangle.
- Practical applications of similar triangles
- Sine, cosine and tangent acute angle right triangle
- Tangent to a circle. Tangent Property
- Central and inscribed angles
- Collection of tasks. Central and inscribed angles
- Four wonderful points of the triangle
- Inscribed and circumscribed circles
- Vector concept
- Vector addition and subtraction
- Multiply a vector by a number
- Vector coordinates
- The simplest tasks in coordinates
- Circle equation
- Sine, cosine and tangent of an angle
- Triangle area theorem
- Sine theorem.
- Cosine theorem
- Dot product of vectors
- Dot product of vectors in coordinates
- Movement. Symmetry about a point
- Movement. Symmetry about a straight line
- Movement. Turn. Parallel transfer
- Crafts on the theme "Movement"
7th grade
- Day open doors in the gymnasium. Triangles. Signs of equality of triangles
- triangle inequality
- Final test(Specification of experimental examination work in geometry for students of grade 7 MOU gymnasium №1)
- Master class "Using PowerPoint presentations in geometry lessons" [ , 408.64 Kb] The master class was held as part of the international seminar "Organization of a developing space in the conditions of integrated education of children: from the experience of the education department of Polyarnye Zori on the implementation of an international project" Frontier Gymnasium.
- Vector addition
- Method of coordinates (Competitive materials "Teacher's Workshop". The competitive development includes 4 lessons on the topic)
- Lesson 1
- Lesson 2
- Lesson 3
- Lesson 4
Geometry
chapter 7
Prepared by Namazgulova Gulnaz, a student of grade 8b, SBEI RPLI, Kumertau
Teacher: Bayanova G.A.
The ratio of segments AB and CD is the ratio of their lengths, i.e. AB:CD
AB = 8 cm
CD = 11.5 cm
Segments AB and CD are proportional to segments A 1 AT 1 and C 1 D 1 , if:
CD= 8 cm
AB=4cm
FROM 1 D 1 = 6 cm
A1B1=3 cm
Two triangles are called similar , if their angles are respectively equal and the sides of one triangle are proportional to the corresponding sides of the other triangle
K- coefficient of similarity
The ratio of the areas of two similar triangles equal to the square of the similarity coefficient
Proof:
The similarity coefficient is K
S and S 1 are the areas of triangles, then
By the formula we have
The first sign of the similarity of triangles
If two angles of one triangle are respectively equal to two angles of another, then such triangles are similar
Prove:
Proof
1) According to the theorem on the sum of angles of a triangle
2) We prove that the sides of the triangles are proportional
Same with corners.
So the sides
proportional to similar sides
The second sign of the similarity of triangles
If two sides of one triangle are proportional to two sides of another triangle and the angles included between these sides are equal, then such triangles are similar
Prove:
Proof
The third sign of the similarity of triangles
If three sides of one triangle are proportional to three sides of another, then such triangles are similar
Prove:
Proof
middle line called a line segment that joins the midpoints of two of its sides
Theorem:
The midline of a triangle is parallel to one of its sides and equal to half of that side.
Prove:
Proof
Theorem:
The medians of a triangle intersect at one point, which divides each median in a ratio of 2:1, counting from the top
Prove:
Proof
Theorem:
Height of a right triangle drawn from a vertex right angle, divides the triangle into two similar right triangle, each of which is similar to a given triangle
Prove:
Proof
Theorem:
The height of a right triangle, drawn from the vertex of the right angle, is the average proportional for the segments into which the hypotenuse is divided by this height
Prove:
Proof
Sinus - attitude opposite leg to the hypotenuse in a right triangle
cosine - the ratio of the adjacent leg to the hypotenuse in a right triangle
Tangent- the ratio of the opposite leg to the adjacent leg in a right triangle
0 , 45 0 , 60 0
Value of sine, cosine and tangent for angles 30 0 , 45 0 , 60 0
Let's depict: a) two unequal circles; b) two unequal squares; c) two unequal isosceles right triangles; d) two unequal equilateral triangle. a) two unequal circles; b) two unequal squares; c) two unequal isosceles right triangles; d) two unequal equilateral triangles. What is the difference between the figures in each presented pair? What do they have in common? Why are they not equal?
In similar triangles ABC and A 1 B 1 C 1 AB \u003d 8 cm, BC \u003d 10 cm, A 1 B 1 \u003d 5.6 cm, A 1 C 1 \u003d 10.5 cm. Find AC and B 1 C 1. A B C A1A1 B1B1 C1C,6 10.5 similar,6 10.5 x y Answer: AC = 14 m, B 1 C 1 = 7 m.
Fizkultminutka: The lesson lasts a long time You decided a lot The call will not help here, Once your eyes are tired. We do everything at once. Repeat four times. - Go through the similarity sign with your eyes. - Close your eyes. - Relax your forehead muscles. – Slowly move your eyeballs to the extreme left position. Feel the tension in your eye muscles. - Fix the position - Now slowly with tension move your eyes to the right. – Repeat four times. - Open your eyes. - Go through the similarity sign with your eyes.
The first sign of similarity Theorem. (The first sign of similarity.) If two angles of one triangle are equal to two angles of another triangle, then such triangles are similar. A B C C1C1 B1B1 A1A1 C"C" B"
"Problems for similarity" - Similar triangles. Find x, y, z. Example No. 4. Solving problems in geometry on finished drawings. Problem condition: Given: ?ABC ~ ?A1B1C1. Task topics. Example No. 2. Author: Skurlatova G.N. MOU "Secondary School No. 62". The first sign of the similarity of triangles. End presentation. Example No. 1. The second and third signs of the similarity of triangles.
"Lesson Signs of similarity of triangles" - In such figures, the sides are proportional. A. A1. Geometry lesson "Similar triangles." IN 1. The purpose of the lesson: Generalization on the topic "Signs of similarity of triangles." When. B. In similar figures, the angles are equal. similar figures. Lesson Objectives: Are triangles similar?
"Practical applications of triangle similarity" - What are the ways to determine the height of an object? Question learning topic: Apply similar triangles. Presentation-abstract, booklet, newsletter on methods for determining the height of an object. How can you measure the height of an object using simple devices? Academic subjects Keywords: geometry, literature, physics.
"Tests of similarity" - A. Similar triangles. C. ABC and A1 B1C1 are triangles<А=А1; <В=<В1. C1. B. Дано. 4. Признаки подобия треугольников. 3. 1. 2. "Similarity of triangles grade 8" - 1 sign of the similarity of a triangle. Prepared by a student of class 8 "b" Dmitry Mikhalchenko. 3 sign of triangle similarity. Task number 1. 2 triangle similarity sign. Sides a and d, b and c are similar. Application of similarity in human life. "Application of the similarity of triangles" - Proportional segments in a right triangle. Division of a segment in a given ratio. Divide the segment in the ratio 2/3. Practical application of similar triangles. B. Application of triangle similarity in proving theorems. Measuring work on the ground. Theorem on the midline of a triangle. Similar triangles. Solving problems according to ready-made drawings Grade 8. Mathematics teacher of the 1st quarter of the RIOU Obskaya school Vodyanova E.A. Problem 1. Prove: ?XZR ~ ?RYZ Z Y 40° X 40° R. Problem 2. ABCD is a trapezoid Prove: ?BOC ~ ?DOA B C O A D. Problem 3. ABCD is a trapezoid Prove: ?ABC ~ ?ACD B C A D segments. Problem 4. BD || AF Find: AC; AB C 2 cm B D 3 cm A F 12 cm. Problem 5. KM || FH Find: FH H 4 cm K 7 cm 5 cm F M L. Task 6. Find: ABC 2 cm 1 cm D B 5 cm 10 cm A F. Task 7. Find: ВD В 2 cm F D 5.5 cm 2 cm A C. Problem 8. ABCD - parallelogram Find: BD B C 16 cm 12 cm 8 cm D A R F. - Similarity.ppt Similar triangles. proportional cuts. Definition of similar triangles. The number k, equal to the ratio of similar sides of triangles, is called the similarity coefficient. The ratio of the areas of similar triangles. The ratio of the areas of two similar triangles is equal to the square of the similarity coefficient. The bisector of a triangle divides the opposite side into segments proportional to the adjacent sides of the triangle. Signs of similarity of triangles. III sign of similarity of triangles If three sides of one triangle are proportional to three sides of another triangle, then such triangles are similar Given: ?ABC, ?A1B1C1, Prove: ?ABC ?A1B1C1. - Similarity of triangles.ppt Geometry. Triangle. Let's remember. similar figures. How are the figures similar? Form! Definition of similar triangles. Signs of similarity of triangles. The angles are equal. C1. Similar parties. Proportional. Similarity coefficient “k”. Name the similarities. Equality of relations of similar parties. What triangles are similar? Circles are always similar. Squares are always similar. Very interesting. The shadow of the pyramid. The shadow of the stick. A little more about triangles. Proportional segments in a triangle. The height of the triangle. The heights of a triangle intersect at one point O, called the orthocenter. - Similar Triangles.ppt Application of similarity in human life. 1 triangle similarity sign. 2 sign of triangle similarity. 3 sign of triangle similarity. Problem number 1. Sides a and d, b and c are similar. Task number 2. - Similarity of triangles Grade 8.ppt Similar triangles. Table of contents. proportional cuts. Segments. In everyday life there are objects of the same shape. Definition of similar triangles. A task. Similar parties. Two triangles are called similar. Similar triangles. The ratio of the areas of similar triangles. Theorem. similarity properties. Triangles have an equal angle. Signs of similarity of triangles. First sign. Similar sides are proportional. Second sign. General side. Third sign. The middle line of the triangle. Middle line. Medians in a triangle. O is the intersection of the medians. - "Similar Triangles" Grade 8.ppt The educational theme of the project. Similar triangles. Signs of similarity of triangles. Creative theme of the project: Annotation. The project was prepared outside school hours by students of the 8th grade. It is implemented within the framework of grade 8 geometry on the topic "signs of similarity of triangles". The project includes information and research part. Analytical work with information systematizes knowledge about similar figures. Didactic tasks will help to control the degree of assimilation of educational material. Reflection? Questions: What does the concept of "similar triangles" mean? How to measure the height of large buildings, trees...? - Geometry Similar Triangles.ppt Similar triangles. proportional cuts. property of the bisector of a triangle. Two triangles are called similar. Problem solving. Theorem on the ratio of the areas of similar triangles. The first sign of the similarity of triangles. The second sign of the similarity of triangles. Sides of a triangle. The third sign of the similarity of triangles. Mathematical dictation. Proportionality of the sides of the angle. Similar to right triangles. Continuation of the sides. The middle line of the triangle. The two sides of the triangle are connected by a segment that is not parallel to the third. Proportional segments in a right triangle. - Geometry "Similar Triangles".ppt Similar triangles. Use in life. Definition of similar triangles. Table of contents. proportional cuts. Two triangles are called similar. The ratio of the areas of similar triangles. The first sign of the similarity of triangles. The second sign of the similarity of triangles. The third sign of the similarity of triangles. Triangle ABC. The sides of triangle ABC are proportional. The sides of the triangle ABC are proportional to the corresponding sides. Consider triangle ABC. ABC. Triangles ABC and ABC have three equal sides. Practical applications of similar triangles. - Definition of Similar Triangles.ppt Similar triangles. Signs of similarity of triangles. Definition of similar triangles. The first sign of the similarity of triangles. Given. Prove: Proof: So, the sides of the triangle ABC are proportional to the similar sides of the triangle A1B1C1. The second sign of the similarity of triangles. 13. 16. The third sign of the similarity of triangles. Proof of the theorem. Theorem: Given: ?ABC, ?A1B1C1 AB/A1B1=BC/B1C1=CA/C1A1. Taking into account the second sign of similarity of triangles, it is enough to prove that Signs of similarity.ppt Signs of similarity of triangles. 1. A sign of the similarity of triangles in two angles. There are three signs of similarity: A in a1b1. 3. Sign of the similarity of triangles on three sides. Similar to right triangles. - Signs of similarity of triangles.ppt similarity in geometry. Theme "Similarities". proportional cuts. Two right triangles. Proportionality of segments. similar figures. Figures of the same shape are called similar figures. Similar triangles. Two triangles are said to be similar if their angles are respectively equal. Similarity coefficient. Additional properties. Perimeter ratio. common multiplier. Area ratio. property of the bisector of a triangle. Bisector. The equation. Signs of similarity of triangles. The first sign of the similarity of triangles. The angles of the triangles are respectively equal. Similar sides are proportional. - Three signs of similarity of triangles.ppt Geometry lesson "Similar triangles." The purpose of the lesson: Generalization on the topic "Signs of similarity of triangles." Lesson Objectives: Similar figures. In such figures, the angles are equal. In such figures, the sides are proportional. Are the triangles similar? When. The first sign of the similarity of triangles. If two sides of one triangle are proportional to two sides of another. So these triangles are similar. The second sign of the similarity of triangles. if the three sides of one triangle are proportional to the three sides of another, the third sign of the similarity of triangles. - Lesson Signs of similarity of triangles.ppt blue light. Similar triangles. The first sign of similarity. Let's depict: How do the figures in each presented pair differ? Definition. The coefficient of proportionality is called the coefficient of similarity. What does that mean? ABC is like a triangle? A1B1C1? The angles are equal. The sides are proportional. Similarity, resemblance. Specify proportional sides. The sides of the triangle are 5 cm, 8 cm and 10 cm. In similar triangles ABC and A1B1C1 AB = 8 cm, BC = 10 cm, A1B1 = 5.6 cm, A1C1 = 10.5 cm. . 2. Set aside: segment AB "= A1B1 (t. B" є AB) straight line B "C" || Sun. - The first sign of the similarity of triangles.ppt Similar triangles. Content. similar figures. In everyday life there are objects of the same shape, but different sizes. In geometry, figures of the same shape are called similar. The number k, equal to the ratio of similar sides of triangles, is called the similarity coefficient. The ratio of the perimeters of similar triangles. The ratio of the perimeters of two similar triangles is equal to the similarity coefficient. The ratio of the areas of similar triangles. The ratio of the areas of two similar triangles is equal to the square of the similarity coefficient. - Ratio of areas of similar triangles.ppt Application of similarity to problem solving. 8th grade. Pronunciation. Option 1 Definition of similar triangles. Formulate the third criterion for the similarity of triangles. State the property of the bisector of a triangle. Option 2 Determination of the midline of the triangle. Formulate the first criterion for the similarity of triangles. Formulate the property of the point of intersection of the medians of a triangle. oral work. What part of the area of triangle ABC is the area of trapezoid AMNC? Problem solving. Calculate the medians of a triangle with sides 25cm, 25cm and 14cm. O is the intersection point of the diagonals of the parallelogram ABCD, E and F are the midpoints of the sides AB and BC, OE=4 cm, OF=5 cm. - Application of similarity.ppt Practical application of similar triangles. Lesson plan. Application of triangle similarity in proving theorems. Building tasks. Measuring work on the ground. Theorem on the midline of a triangle. property of the medians of a triangle. Proportional segments in a right triangle. Division of a segment in a given ratio. Construction of triangles. Divide the segment in the ratio 2/3. Determining the height of an object. Determining the distance to an inaccessible point. Determining the height of an object using a mirror. - Application of similar triangles.ppt Practical application of similar triangles. Similarity in life. A bit of history. The rod is about the height of a man. Determining the height of an object. Determining the height of the pyramid. History reference. Tired foreigner. Thales. Thales method. The shadow of the stick. Determining the height of an object from a pole. Mysterious Island. Finding the fourth unknown term of the proportion. Determining the height of an object from a puddle. Determining the height of an object using a mirror. Advantages. Determining the distance to an inaccessible point. Finding the width of the lake. distance to the tree. Pin device for measurements. - Application of the similarity of triangles in life.ppt practical application of the similarity of triangles. Story. Shrek's birthday. Shrek came home. Geometry lessons. Similar triangles. Everything is decided right. Distance from one coast to another. You can apply the similarity of triangles. Solution. Rope of the required length. Idea. Bracelet. - Practical application of triangle similarity.pptx Topic: Practical applications of similar triangles. Creative title: Determining the height of an object. How can you measure the height of an object using simple devices? What are the ways to determine the height of an object? What instruments or fixtures are needed to measure the height of an object? What are the similarities and differences in determining the height of an object? Question of the educational topic: Application of the similarity of triangles. Subjects: geometry, literature, physics. Participants: 8th grade students. Presentation-abstract, booklet, newsletter on methods for determining the height of an object. - Practical Applications of Similar Triangles.ppt Solving geometry problems on ready-made drawings. Task topics. The first sign of the similarity of triangles. The second and third signs of the similarity of triangles. Similar triangles. Example No. 2. Example No. 1. Example No. 4. Example No. 3. Example No. 6. Example No. 7. Example No. 5. - Tasks for similarity.ppt Similar triangles. The first sign of similarity. What triangles are called similar. Formulate the first criterion for the similarity of triangles. The triangles shown in the figure. Draw a triangle. Triangle. Sides of a triangle. Rectangular Triangles. The two triangles are similar. sides of triangles. Perimeter. List all similar triangles. Side. Square. Vertex. Can a triangle be intersected by a line? Circle chords. Find similar triangles. Acute triangle. The product of segments. Circle radius. Circle. Two straight lines. - Tasks for the similarity of triangles.ppt Similar triangles. The concept of similarity is one of the most important in the course of planimetry. The study of the topic begins with the formation of the concepts of the ratio of segments and the similarity of triangles. Solving construction problems by the similarity method are considered with students interested in mathematics. This topic is designed for 8th grade students. 19 hours are allotted for the study of the material. Lesson topic: The first sign of the similarity of triangles. Checking homework. Solving problems in order to prepare students for the perception of new material. Learning new material. Statement 1 of the similarity criterion for triangles Proof of the theorem. - Similarity of triangles problem solving.ppt Similar triangles. Lesson motto. Individual card. Name similar triangles. Solution of practical problems. Determining the height of the pyramid. Thales method. The shadow of the stick. Measuring the height of large objects. Determining the height of an object. Determining the height of an object using a mirror. Determining the height of an object from a puddle. Solution of problems according to ready-made drawings. Gymnastics for the eyes. Independent work. -Similarity
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