Mathematics presentation "Tetrahedron and parallelepiped. Construction of sections"

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Lesson Objectives:

• to teach how to build sections of a tetrahedron and a parallelepiped by a plane;
• to form the ability to analyze, compare, generalize, draw conclusions;
• develop the skills of independent activity among students, the ability to work in a group.

Equipment: projector, interactive whiteboard, handouts.

Lesson type: lesson learning new material.

Methods and techniques used in the lesson: visual, practical, problem-search, group, elements of research activity.

During the classes

I. Organizational moment.

The teacher tells the topic and purpose of the lesson ( slide 1).

II. Knowledge update.

Teacher: Doing your homework, you had to find the meeting points of lines and planes, the trace of the secant plane on the plane of the face of the polyhedron. Please comment on what needs to be done.

(Students comment on homework ( slides 2-3).

Teacher: To move on to the study of a new topic, let's repeat the theoretical material by answering the questions:

1. What is called a cutting plane ( slide 4)? (Students give the definition.)
2. What is called a section of a polyhedron ( slide 5)? (A definition is being formulated.)
3. What needs to be done in order to construct a section of a polyhedron by a plane?
   The construction of a section is reduced to the construction of lines of intersection of the cutting plane and the planes of the faces of the polyhedron.)
4. Does the cutting plane have to intersect the planes of all faces of the polyhedron?

Teacher: Let's do a little research and answer the question: "What figure can be obtained in a section of a tetrahedron or parallelepiped by a plane?"

(Students, working in groups, are looking for the answer to the question posed.)

(After a few minutes they formulate their assumptions, and there is a demonstration slides 6–7.)

Teacher: Let's repeat the rules that you need to remember when constructing sections of a polyhedron (students remember and formulate the necessary axioms, theorems, properties):

• If two points belong to the cutting plane and the plane of some face of the polyhedron, then the line passing through these points will be the trace of the cutting plane on the plane of the face.
• If a cutting plane is parallel to a straight line lying in a certain plane and intersects this plane, then the line of intersection of these planes is parallel to the given straight line.
• When two parallel planes are intersected by a cutting plane, parallel lines are obtained.
• If the cutting plane is parallel to some plane, then these two planes intersect the third plane along straight lines parallel to each other.
• If the cutting plane and the planes of two intersecting faces have a common point, then it lies on the line containing the common edge of these faces.

Teacher: Find errors in these drawings, justify your statement ( slides 8-9).

Teacher: So, guys, we have prepared a theoretical base to learn how to build sections of polyhedra by a plane, in particular, sections of a tetrahedron and a parallelepiped. You will perform most of the tasks on your own, working in groups, so each of you has worksheets with polyhedra drawings on which you will build sections. If necessary, you can seek advice from a teacher or a leader in the group.

So, we bring to your attention first task: (slide 10) construct a section of the tetrahedron by a plane passing through the given points M, N, K. (In the section, a triangle is obtained, check - slide 11.)

Teacher: Consider second task: Given a tetrahedron DABC. Construct a section of the tetrahedron by the MNK plane if M ∈DC, N∈AD, K∈AB. ( slide 12)

(To carry out the solution of the problem together with the class, commenting on the construction.)

(Task 3- independent work in groups slide 14). Examination - slide 15.)

Task 4: Construct a section of the tetrahedron by the MNK plane, where M and N are the midpoints of edges AB and BC ( slide 16). (Check for slide 17.)

Teacher: Let's move on to the next part of the lesson. Consider the problem of constructing sections of a parallelepiped by a plane. We found out that in the section of a parallelepiped by a plane, a triangle, quadrilateral, pentagon or hexagon can be obtained. The rules for constructing sections are the same. I propose to move on to the next problem, which you will solve on your own.

(Demonstrated slide 18)

Task 5

Construct a section of the parallelepiped ABCDA 1 B 1 C 1 D 1 by the plane MNK if M∈AA 1 , N ∈BB 1 , K∈CC 1 . (Check for slide 19).

Task 6: (Slide 20) Construct a section of the parallelepiped ABCDA 1 B 1 C 1 D 1 by the plane PTO if P, T, O belong respectively to the edges AA 1 , BB 1 , CC 1 .

(The solution is discussed, students build a section on individual sheets and record the construction progress ( slide 21).)

1. TO ∩ BC = M
2. TP ∩ AB = N
3. NM ∩ AD = L
4. NM ∩ CD = F
5. PL, FO
6. PTOFL is the required section.

Task 7: (slide 22) Construct a section of the parallelepiped by the plane KMN if K ∈ A 1 D 1 , N ∈BC , M ∈ AB.

Solution: ( slide 23)

1. MN∩AD=Q;
2. QK∩AA 1 =P;
3. NE || PC; KF || MN;

MPKFEN is the required section.

Creative tasks (cards by options):

1. In a regular triangular pyramid SABC through the vertex C and the middle of the edge SA draw a section of the pyramid parallel to SB. A point F is taken on the edge AB so that AF:FB=3:1. A straight line is drawn through the point F and the midpoint of the edge SC. Will this line be parallel to the plane of the section?
2. AB 1 C - section of a rectangular parallelepiped ABCD 1 B 1 C 1 D 1. Through the points E, F, K, which are respectively the midpoints of the edges DD 1 , A 1 D 1 , D 1 C 1, the second section is drawn. Prove that triangles EFK and AB 1 C are similar, and find out which angles of these triangles are equal to each other.

III. Summary lesson a.

So, we got acquainted with the rules for constructing sections of a tetrahedron and a parallelepiped, examined the types of sections, and solved the simplest tasks for constructing sections. In the next lesson, we will continue to study the topic, consider more complex tasks.

And now let's summarize the lesson by answering our traditional questions ( slide 24):

• “I liked (didn’t like) the lesson because….”
• “Today in class I learned….”
• “I want to….”
• “In this lesson, I would add…”

(Grading a lesson.)

IV. Homework assignment.

14 105, 106. ( slide 25)

Additional task to 105: Find the ratio in which plane MNK divides edge AB if CN: ND = 2:1, BM = MD and point K is the midpoint of median AL of triangle ABC.

(Finish the creative task.)

Construction of sections of a tetrahedron and a parallelepiped Victoria Viktorovna Tkacheva, teacher of mathematics at school No. 183 with in-depth study of the English language. St. Petersburg, 2011. Contents: 1. Goals and objectives 2. Introduction 3. The concept of a cutting plane 4. Definition of a section 5. Rules for constructing sections 6. Types of sections of a tetrahedron 7. Types of sections of a parallelepiped 8. The task of constructing a section of a tetrahedron with an explanation 9. The task of constructing a section of a tetrahedron with an explanation 10. The task of constructing a section of a tetrahedron on leading questions 11. The second solution to the previous problem 12. The task of constructing a section of a parallelepiped 13. The task of constructing a section of a parallelepiped 14. Sources of information 15. Wish to students Purpose of work: Development of spatial representations in students. Tasks: Introduce the rules for constructing sections. To develop the skills of constructing sections of a tetrahedron and a parallelepiped in various cases of setting a cutting plane. To form the ability to apply the rules for constructing sections when solving problems on the topics "Polyhedra". To solve many geometric problems, it is necessary to build their sections by different planes. The secant plane of a parallelepiped (tetrahedron) is any plane, on both sides of which there are points of this parallelepiped (tetrahedron). L The cutting plane intersects the faces of the tetrahedron (parallelepiped) along segments. L A polygon whose sides are these segments is called a section of a tetrahedron (parallelepiped). To build a section, you need to build the intersection points of the cutting plane with the edges and connect them with segments. In this case, the following must be taken into account: 1. Only two points that lie in the plane of one face can be connected. 2. The cutting plane intersects parallel faces along parallel segments. 3. If only one point belonging to the section plane is marked in the plane of the face, then an additional point must be constructed. To do this, it is necessary to find the points of intersection of the already constructed lines with other lines lying on the same faces. What polygons can be obtained in section? A tetrahedron has 4 faces In sections it can turn out: Triangles Quadagons A parallelepiped has 6 faces Triangles Pentagons In its sections you can get: Quadagons Hexagons Construct a section of the tetrahedron DABC by a plane passing through the points M,N,K D M AA 1. Draw a straight line through the points M and K, because they lie in the same face (ADC). N K BB C C they lie in the same face (CDB). 3. Arguing similarly, we draw the line MN. 4. Triangle MNK is the required section. Construct a section of the tetrahedron by a plane passing through the points E, F, K. 1. Draw KF. 2. We carry out FE. 3. Continue EF, continue AC. D F 4. EF  AC \u003d M 5. We carry out MK. E  M  C 6. MK AB=L A L K Rules B 7. Draw EL EFKL - the required section Construct a section of the tetrahedron by a plane passing through the points E, F, K. can continue to get the points lying in one connect? connect the resulting additional point? faces, name the section. extra point? D and E AC ELFK FSEK with point K, and FK F L C M A E K B Rules Second method Construct a section of a tetrahedron by a plane passing through points E, F, K. D F L C A E K B Rules First method O Method No. 1. Method number 2. Conclusion: regardless of the method of constructing the sections, they are the same. Construct a section of a parallelepiped by a plane passing through the points M,A,D. B1 D1 E A1 C1 B A 1. AD 2. MD 3. ME//AD, because (ABC)//(A1B1C1) 4. AE 5. AEMD - section. M D C Construct sections of a parallelepiped by a plane passing through points B1, M, N Rules B1 D1 C1 A1 P K B D A E N C O M 1. MN 3.MN ∩ BA=O 2. Continue 4. B1O MN,BA 5 B1O ∩ A1A=K 6. KM 7. We continue MN and BD. 8. MN ∩ BD=E 9. B1E 10. B1E ∩ D1D=P, PN Sources of information 1. Geometry 10-11: textbook for general education. institutions / L.S. Atanasyan, V.F. Butuzov et al., M. Enlightenment 2. Tasks for geometry lessons grades 7-11 / B.G. Ziv, St. Petersburg, NGO "World and Family", ed. - in "Acacia". 3. Mathematics: A great reference book for schoolchildren and applicants to universities / D.I. Averyanov, P.I. Altynov - M .: Bustard YOU LEARNED AND SAW A LOT! SO GO GUYS: GO AND BE CREATIVE! THANK YOU FOR YOUR ATTENTION.

• Targets and goals.
• Introduction.
• The concept of a cutting plane.
• Section definition.
• Rules for constructing sections.
• Types of sections of the tetrahedron.
• Types of sections of a parallelepiped.
• The task of constructing a section of a tetrahedron with an explanation.
• The task of constructing a section of a tetrahedron on leading questions.
• The second solution to the previous problem.
• The task of constructing a section of a parallelepiped.
• Wish for students.

Objective:

Tasks:

• Familiarize yourself with the rules for constructing sections.
• Develop skills in constructing sections of a tetrahedron and a parallelepiped in various cases of setting a cutting plane.
• To form the ability to apply the rules for constructing sections when solving problems on the topics "Polyhedra".

To solve many geometric problems, it is necessary to build them sections different planes.

cutting plane A parallelepiped (tetrahedron) is any plane, on both sides of which there are points of this parallelepiped (tetrahedron).

cutting plane intersects the faces of a tetrahedron (parallelepiped) along segments.

L

Polygon , whose sides are given segments, is called section tetrahedron (parallelepiped).

To build a section, you need to build the intersection points of the cutting plane with the edges and connect them with segments.

In doing so, the following must be taken into account:

1. You can connect only two points lying

in the plane of one side.

2. The cutting plane intersects parallel faces along parallel segments.

3. If only one point belonging to the section plane is marked in the plane of the face, then an additional point must be constructed. To do this, it is necessary to find the points of intersection of the already constructed lines with other lines lying on the same faces.

What polygons can be obtained in section?

A tetrahedron has 4 faces

In sections you can get:

• Quadrangles

• triangles

The parallelepiped has 6 faces

• triangles

• pentagons

In its sections

may get:

• Quadrangles

• Hexagons

Construct a section of a tetrahedron DABC plane passing through the points M , N , K

• Let's draw a line through

points M and K, because they are lying

in one face (A DC).

2. Let's draw a straight line through the points K and N, because they lie on the same face (C DB).

3. Arguing similarly, we draw the straight line MN .

4. Triangle MNK -

desired section.

passing through points E , F , K .

1. Draw To F .

2. We spend FE.

3. We continue EF , we continue AC .

5. We spend MK.

7. Conduct EL

EFKL - desired

Construct a section of a tetrahedron by a plane,

passing through points E , F , K .

F-point

F and K, E and K

Construct a section of a tetrahedron by a plane,

passing through points E , F , K .

Method number 2.

Method number 1.

Conclusion: regardless of the method of constructing the sections, they are the same.

Construct sections of a parallelepiped by a plane passing through points B 1, M, N

7. We continue MN and BD .

2. Continue MN ,BA

10. B 1 E ∩ D 1 D=P , PN

Construct a section of a parallelepiped by a plane,

passing through points M,A,D.

3. ME//AD , because (ABC)//(A 1 B 1 C 1)

5. AEMD- section.

YOU LEARNED A LOT

AND SEE MUCH!

SO GO GUYS:

GO AND BE CREATIVE!

THANK YOU FOR YOUR ATTENTION.