Table 11.10 pyramid solution. Solving problems using ready-made drawings of a “regular triangular pyramid”. III. Formation of new knowledge
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Class: 10
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Lesson objectives:
- Educational:
- study mnemonic device;
- derive formulas for the transition of main angles in regular pyramids;
- learn to use mnemonic techniques to prove dependencies between angles in a regular pyramid and solve problems.
- Developmental:
- develop cognitive interest through the development of students’ research skills;
- develop figurative memory, abstract and logical thinking;
- develop students' computing skills.
- Educational:
- instill communication skills, skills in working with didactic material (handouts, electronic resource);
- to form clear execution of actions when performing practical work and when working in groups.
Equipment:
- computer,
- projector,
- screen,
- interactive whiteboard SMART Board,
- Handout
DURING THE CLASSES
I. Organizational moment
– Please open your notebooks and write down the date and topic of the lesson: Solving problems on the topic “Pyramid”. Today in the lesson we will learn when solving problems apply non-standard technique, which was called mnemonic, we will derive formulas for the transition of the main angles in regular pyramids and learn how to apply them when solving problems.
– To do this, we need to repeat some questions from the geometry course.
II. Updating of reference knowledge <Annex 1 >
Oral work (frontal questioning).
Given a right triangle ABC.
Let us recall the main elements of the Pyramid.
- Which polyhedron is called a pyramid?
- What is the top of the pyramid? Base?
- Which pyramid is called correct?
- Where is the height of a regular pyramid projected?
- Name the angle between the side edge of the pyramid and the base; Between the side edge and the base; angle between the side faces of the pyramid?
Let's consider solving a problem from a textbook. Attention to the board.
№ 255. In a regular triangular pyramid, the side of the base is 8 cm, and the plane angle at the apex is equal to find the height of the pyramid.
III. Formation of new knowledge
When solving the problem, we dealt with triangles that do not lie in the same plane, and, moreover, in each of them no more than two elements were known. Do you think there is an easier way to solve a problem in mathematics? I affirm that there is! Indeed, such a method exists. And it has a name: a mnemonic technique for solving geometric problems. It is with him that I will introduce you today. So …
Mnemonics (from Greek - memory) are various kinds of techniques that promote artificial memorization. In other words, it is the art of memorization. Already ancient peoples and savages already knew a whole series of techniques that provided support points for memory. You also know some mnemonic techniques, such as remembering the colors of the rainbow, determining the bisector, and others.
So, Mnemonic device <Appendix 3 >
for the relationship between angles in a regular pyramid:
Mnemonic trick:
1. Write down the names of the triangle in which the unknown angle is located.
2. From the three letters S, A, O we will make different pairs. We got three segments.
3. Cross out the one that is not common to triangles with known angles.
4. Add one letter at a time to get the name of the triangle that includes one of these angles:
5. Find a segment consisting of common letters.
6. To find the desired relationship, divide the numerator and denominator by the found segment.
– Now, using this mnemonic device, I will derive some dependencies between the angles in a regular pyramid.
1. The relationship between the flat angle at the vertex of a regular pyramid and the angle at the edge of the base (quadrangular pyramid)
2. The relationship between the flat angle at the vertex of a regular pyramid and the angle at the lateral edge
IV. Formation of primary skills
Dear 10th graders. Now, in practical work, you will explore the dependencies between the angles in a regular pyramid, as a result of which each group will have to obtain a transition formula. Each group has its own task. There are assignment sheets on your desk. <Appendix 2 > and mnemonic rule <Appendix 3 > , which will allow you to quickly find the necessary dependency.
Students work in groups. At the end of the work, the group representative enters the resulting transition formula into the table on the slide.
Each group received a red signal card on the table. <Appendix 4 > Using It, you can check the correctness of your reasoning.
As a result of practical work, we received a table of the relationships between angles in a regular pyramid. At the next stage of our lesson, we will apply the obtained formulas when solving problems, and along the way we will evaluate how significantly these formulas make our life easier.
Let's return to the problem that was solved at the beginning of the lesson. (On the screen is a slide with a solution and behind the curtain is a solution using transition formulas)
Another solution
It is obvious that with the help of transition formulas, difficulties in solving problems can be easily overcome. You have a table with transition formulas on your desk. <Appendix 5 > not only for triangular and quadrangular pyramids, but also for hexagonal and n-gonal ones. These formulas can and should be used when solving problems.
Transition formulas
Let's consider using formulas for another problem from the textbook
Isn't that a nice solution?
V. Reflection
– Today you became acquainted with a mnemonic technique that allows you to find dependencies between angles in regular pyramids, and using the mnemonic technique, you obtained several such dependencies and applied them when solving problems.
When solving complex stereometric problems, difficulties often arise. They can arise, in particular, because the linear elements given in the condition do not belong to the same plane, and, therefore, there is no right triangle from which to start solving. However, with the help of mnemonic devices and transition formulas, difficulties are easily overcome.
VI. Lesson summary
The following students receive grades for their work in class...
VII. Homework
As homework, I suggest you solve problem 254 (b, d, e) in two ways: traditional and using a mnemonic device (transition formulas).
- Thank you everyone for the lesson
Geometry. Tasks and exercises on ready-made drawings. 10-11 grades. Rabinovich E.M.
M.: 2014. - 80 p.
The manual is compiled in the form of tables and contains more than 350 tasks. The tasks of each table correspond to a specific topic of the school geometry course for grades 10-11 and are located inside the table in order of increasing complexity.
A high school mathematics teacher knows well how difficult it is to teach students to make visual and correct drawings for stereometric problems.
Due to a lack of spatial imagination, a stereometric task, for which you need to make a drawing yourself, often becomes overwhelming for the student.
That is why the use of ready-made drawings for stereometric problems significantly increases the volume of material covered in the lesson and increases its effectiveness.
The proposed manual is an additional collection of geometry problems for students in grades 10-11 of a general education school and is focused on the textbook by A.V. Pogorelov "Geometry 7-11". It is a continuation of a similar manual for students in grades 7-9.
Format: pdf(2014, 80 p.)
Size: 1.2 MB
Watch, download:drive.google ; Rghost
Format: djvu(2006, 80 p.)
Size: 1.3 MB
Download: drive.google
Table of contents
Preface 3
Repetition of planimetry course 5
Table 1. Solving triangles 5
Table 2. Area of triangle 6
Table 3. Area of quadrilateral 7
Table 4. Area of quadrilateral 8
Stereometry. 10th grade 9
Table 10.1. Axioms of stereometry and their simplest consequences... 9
Table 10.2. Axioms of stereometry and their simplest consequences. 10
Table 10.3. Parallelism of lines in space. Crossing lines 11
Table 10.4. Parallelism of lines and planes 12
Table 10.5. Sign of parallel planes 13
Table 10.6. Properties of parallel planes 14
Table 10.7. Image of spatial figures on a plane 15
Table 10.8. Image of spatial figures on a plane 16
Table 10.9. Perpendicularity of a line and a plane 17
Table 10.10. Perpendicularity of a straight line and a plane 18
Table 10.11. Perpendicular and oblique 19
Table 10.12. Perpendicular and oblique 20
Table 10.13. Theorem of three perpendiculars 21
Table 10.14. Theorem of three perpendiculars 22
Table 10.15. Theorem of three perpendiculars 23
Table 10.16. Perpendicularity of planes 24
Table 10.17. Perpendicularity of planes 25
Table 10.18. Distance between crossing lines 26
Table 10.19. Cartesian coordinates in space 27
Table 10.20. Angle between crossing lines 28
Table 10.21. Angle between straight line and plane 29
Table 10.22. Angle between planes 30
Table 10.23. Area of orthogonal projection of a polygon 31
Table 10.24. Vectors in space 32
Stereometry. 11th grade 33
Table 11.1. Dihedral angle. Triangular angle 33
Table 11.2. Straight prism 34
Table 11.3. Correct prism 35
Table 11.4. Correct prism 36
Table 11.5. Inclined prism 37
Table 11.6. Parallelepiped 38
Table 11.7. Constructing prism sections 39
Table 11.8. Regular pyramid 40
Table 11.9. Pyramid 41
Table 11.10. Pyramid 42
Table 11.11. Pyramid. Truncated pyramid 43
Table 11.12. Constructing pyramid sections 44
Table 11.13. Cylinder 45
Table 11.14. Cone 46
Table 11.15. Cone. Truncated cone 47
Table 11.16. Ball 48
Table 11.17. Inscribed and circumscribed ball 49
Table 11.18. Volume of parallelepiped 50
Table 11.19. Prism volume 51
Table 11.20. Pyramid volume 52
Table 11.21. Pyramid volume 53
Table 11.22. Volume of the pyramid. Volume of a truncated pyramid 54
Table 11.23. Volume and lateral surface area of the cylinder..55
Table 11.24. Volume and lateral surface area of the cone 56
Table 11.25. Cone volume. Volume of a truncated cone. The area of the lateral surface of the cone. Lateral surface area of a truncated cone 57
Table 11.26. Volume of the ball. Surface area of the ball 58
Answers, directions, solutions 59
Geometry. Tasks and exercises on ready-made drawings. Grades 10-11. Rabinovich E.M.
Table of contents
Preface 3
Repetition of planimetry course 5
Table 1. Solving triangles 5
Table 2. Area of triangle 6
Table 3. Area of quadrilateral 7
Table 4. Area of quadrilateral 8
Stereometry. 10th grade 9
Table 10.1. Axioms of stereometry and their simplest consequences... 9
Table 10.2. Axioms of stereometry and their simplest consequences. 10
Table 10.3. Parallelism of lines in space. Crossing lines 11
Table 10.4. Parallelism of lines and planes 12
Table 10.5. Sign of parallel planes 13
Table 10.6. Properties of parallel planes 14
Table 10.7. Image of spatial figures on a plane 15
Table 10.8. Image of spatial figures on a plane 16
Table 10.9. Perpendicularity of a line and a plane 17
Table 10.10. Perpendicularity of a straight line and a plane 18
Table 10.11. Perpendicular and oblique 19
Table 10.12. Perpendicular and oblique 20
Table 10.13. Theorem of three perpendiculars 21
Table 10.14. Theorem of three perpendiculars 22
Table 10.15. Theorem of three perpendiculars 23
Table 10.16. Perpendicularity of planes 24
Table 10.17. Perpendicularity of planes 25
Table 10.18. Distance between crossing lines 26
Table 10.19. Cartesian coordinates in space 27
Table 10.20. Angle between crossing lines 28
Table 10.21. Angle between straight line and plane 29
Table 10.22. Angle between planes 30
Table 10.23. Area of orthogonal projection of a polygon 31
Table 10.24. Vectors in space 32
Stereometry. 11th grade 33
Table 11.1. Dihedral angle. Triangular angle 33
Table 11.2. Straight prism 34
Table 11.3. Correct prism 35
Table 11.4. Correct prism 36
Table 11.5. Inclined prism 37
Table 11.6. Parallelepiped 38
Table 11.7. Constructing prism sections 39
Table 11.8. Regular pyramid 40
Table 11.9. Pyramid 41
Table 11.10. Pyramid 42
Table 11.11. Pyramid. Truncated pyramid 43
Table 11.12. Constructing pyramid sections 44
Table 11.13. Cylinder 45
Table 11.14. Cone 46
Table 11.15. Cone. Truncated cone 47
Table 11.16. Ball 48
Table 11.17. Inscribed and circumscribed ball 49
Table 11.18. Volume of parallelepiped 50
Table 11.19. Prism volume 51
Table 11.20. Pyramid volume 52
Table 11.21. Pyramid volume 53
Table 11.22. Volume of the pyramid. Volume of a truncated pyramid 54
Tasks and exercises on ready-made drawings, grades 10-11, Geometry, Rabinovich E. M., 2006.
Table of contents
Preface.
Repetition of the planimetry course.
Table 1. Solving triangles.
Table 2. Area of the triangle.
Table 3. Area of the quadrilateral.
Table 4. Area of the quadrilateral. Stereometry. Grade 10.
Table 10.1. Axioms of stereometry and their simplest consequences.
Table 10.2. Axioms of stereometry and their simplest consequences.
Table 10.3. Parallelism of lines in space. Crossing straight lines.
Table 10.4. Parallelism of straight lines and planes.
Table 10.5. Sign of parallel planes.
Table 10.6. Properties of parallel planes.
Table 10.7. Image of spatial figures on a plane
Table 10.8. Image of spatial figures on a plane
Table 10.9. Perpendicularity to a straight line and a plane.
Table 10.10. Perpendicularity to a straight line and a plane.
Table 10.11. Perpendicular and oblique.
Table 10.12. Perpendicular and oblique.
Table 10.13. Theorem of three perpendiculars.
Table 10.14. Theorem of three perpendiculars.
Table 10.15. Theorem of three perpendiculars.
Table 10.16. Perpendicularity of planes.
Table 10.17. Perpendicularity of planes.
Table 10.18. The distance between intersecting lines.
Table 10.19. Cartesian coordinates in space.
Table 10.20. The angle between intersecting lines.
Table 10.21. The angle between a straight line and a plane.
Table 10.22. Angle between planes.
Table 10.23. Area of orthogonal projection of a polygon
Table 10.24. Vectors in space. Stereometry. Grade 11.
Table 11.1. Dihedral angle. Triangular angle.
Table 11.2. Straight prism.
Table 11.3. Correct prism.
Table 11.4. Correct prism.
Table 11.5. Oblique prism.
Table 11.6. Parallelepiped.
Table 11.7. Construction of prism sections.
Table 11.8. Correct pyramid.
Table 11.9. Pyramid.
Table 11.10. Pyramid.
Table 11.11. Pyramid. Truncated pyramid.
Table 11.12. Construction of a cross section of a pyramid.
Table 11.13. Cylinder.
Table 11.14. Cone.
Table 11.15. Kohuc. Truncated kohyc.
Table 11.16. Ball.
Table 11.17. Inscribed and circumscribed sphere.
Table 11.18. Volume of a parallelepiped.
Table 11.19. Prism volume.
Table 11.20. Volume of the pyramid.
Table 11.21. Volume of the pyramid.
Table 11.22. Volume of the pyramid. Volume of a truncated pyramid.
Table 11.23. Volume and lateral surface area of the cylinder.
Table 11.24. Volume and lateral surface area of a cone.
Table 11.25. Cone volume. Volume of a truncated cone. The area of the lateral surface of the cone. The lateral surface area of a truncated cone.
Table 11.26. Volume of the ball. Surface area of the ball. Answers, directions, solutions
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Regular triangular pyramid Solving problems using ready-made drawings MBOU Verkhnyakovskaya Secondary School Mathematic teacher: Martynenko L.N. DABC is a regular pyramid, DO ┴ (ABC),CK ┴ AB AM ┴ BC BN ┴ AC. Task #1: Find DO
- Tips:
- Find DK
- Apply the property of medians of a triangle
- Apply the Pythagorean theorem to find DO
- Tips:
- Apply the law of cosines
- Tips:
- Consider triangles KDM and DO1O2
- Find KM
- Using the property of the midline of a triangle, find the side of the triangle
- Tips:
- Use the property of medians of a triangle
- Apply the Pythagorean theorem to find the height
- Tips:
- Apply the bisector property of a triangle
- Tips:
- Which element needs to be found to calculate DO?
- Use the property of medians of a triangle and ratios in a right triangle
- Tips:
- Apply the property of medians of a triangle to find OM
- Tips:
- Use the property of medians of a triangle and ratios in a right triangle
- Tips:
- Use the property of medians in a triangle and ratios in a right triangle
- Tips:
- Find DO
- Tips:
- Write down the formula for the area of a triangle
- Find PL from the similarity of triangles ABC and APL
- Find QL from the similarity of triangles ADC and AQL
- Find the height of triangle PQL using Pythagorean theorem
- Tips:
- Write down the formula for the area of a triangle
- Find CK
- Use the property of medians of a triangle to find CO
- Find the height of triangle CDK
