Project volume polyhedron for college. Practical work "volumes of polyhedra". III. Solving problems for the development of the eye

Class: 11

Goals:

repeat the types of polyhedra, their elements and volume formulas; show the practical orientation of the topic being studied;
develop students' practical skills;
instill interest in the subject.

Equipment:

a set of all kinds of polyhedra;
drawings of polygons on the board;
a poster depicting any modern building;
projector.

I. Heuristic conversation

(repetition of theoretical material on the topic)

1. Name and write down the formulas for the volumes of a prism, a parallelepiped, a pyramid, a truncated pyramid.
(Vprisms = Sprim. h, Vpara. = abc or Vpara. = Sprim. h, Vpyram. = Sprim. h, V =

2. What quantities are repeated in all of the above formulas? (Height)
3. Show height on straight and oblique prisms.
4. Can a parallelepiped be called a prism? And the cube? (Yes, these are special cases of a prism)
5. Show the height on a straight and inclined pyramid.
6. What figures can be at the base of a prism and a pyramid? (Triangle, square, rhombus, rectangle, parallelogram, trapezium and other flat figures)
7. Can there be a trapezoid at the base of a parallelepiped? (No, because a parallelepiped is a prism at the base of which is a parallelogram)
8. Consider the polygons on the board. These polygons may lie at the base of the polyhedra we have considered.

On the cards, formulas with calculations of the areas of polygons ( Annex 1 ). Correlate these formulas with the figures shown on the board; What is the formula for calculating the area of each of these figures?
9. Which of these formulas is suitable for calculating the floor area of a room? ( a . b or a 2)

II. Solving problems with practical content

First option:"Service of experts of the sanitary and epidemiological station"

(a “senior expert” is selected who sets out the content of the problem and makes a conclusion based on the results of the solution).

Solution:

V = abc or V = Sbase h
V = 8.5 6 3.6 = 183.6( m 3)
183,6: 30 = 6,12(m 3) air is accounted for by one student.

Expert opinion:

Yes, 30 students can study in the classroom.

Second option:"Meteorological Service"

(a “senior meteorologist” is selected who sets out the content of the task and draws a conclusion based on the results of the solution)

Solution:

The flowerbed is a geometric figure - a straight triangular prism, where h = 20mm, then V = Sprim. h

1) Sosn. =
2) h = 20 mm, 1m = 1000mm, 1mm = 0,001m, then h = 0.02 m
3) V = 15.3 0.02 = 0.306( m 3) = 306(dm 3)
4) 1dm 3 = 1l(water), then 306 dm 3 = 306 liters of water

The conclusion of the "senior meteorologist":

During the day, 306 liters of precipitation fell on the flower bed.

III. Solving problems for the development of the eye

We often have to ask the question: is it a lot or a little? To learn how to answer such questions, you must constantly develop your eye. Now each of you will have the opportunity to check the quality of your eye.

1) How much do you think cm 3 colognes or lotions are included in this bottle? (The teacher shows the students a bottle in the form of a truncated pyramid or a rectangular parallelepiped).

While the students are giving their guesses, one of them goes to the blackboard, takes the appropriate measurements, and calculates the correct result. Students relate their guesses to this result, thereby testing the quality of their eye.

2) How much m 3 air in our office? (The teacher gives the parameters himself).

IV. "Time out" for the development of spatial imagination

1. A tablet with a drawing of a building is exhibited.

Question: What geometric shapes does this building consist of?
Answer: A rectangular parallelepiped, a regular quadrangular pyramid, and so on.

2. What geometric shapes are found in your workplace?

V. Laboratory and practical work

Everyone has a model of a polyhedron on the table.

Exercise: Take the necessary measurements, calculate the volume of this figure on a piece of paper.

(Pre-write on the piece of paper the number of the figure and its name).

VI. Crossword puzzle

Students who completed laboratory and practical work earlier than others are invited to solve the crossword puzzle "Polyhedrons".

1. Parallel faces of a prism (base);
2. One of the polyhedra (pyramid);
3. Perpendicular between the bases of the prism (height);
4. A plane intersecting a polyhedron (section);
5. Unit of measurement (meter).

VII. Homework

VIII. Lesson summary

The project on geometry in the 11th grade of the teacher of mathematics Nakonechnaya O.A. on the topic "Volumes and surfaces of polyhedra"

Lesson Plan

Lesson topic: "Volumes and Surfaces of Polyhedra"
The overall goal of the lesson.

Cognitive - to generalize and systematize the knowledge, skills and abilities of students obtained in the process of studying the topic “Surface areas of polyhedra. Volumes of polyhedra". To teach how to apply theoretical knowledge in solving practical problems.
Educational - to develop the logical thinking of students, practical skills in solving problems; develop spatial imagination, speech of students; develop practical problem solving skills.
Educational - to educate:

Interest in the subject

Skills of control and self-control,

Friendly attitude towards your classmates

Sense of responsibility,

The ability to express yourself

culture of speech,

Conscious attitude towards learning

Business qualities of students.

Lesson objectives:

Repeat the formulas for the surface areas of polyhedra and the volumes of polyhedra.
Compile a reference abstract-table for calculating the formulas for the areas and volumes of polyhedra.
Work out examples of solving problems using these formulas when testing.
To consolidate the ability to use formulas in solving problems of practical content.
The type of lesson is a lesson of generalization and systematization of knowledge.
Forms of organizing a training session:

Viewing the presentation and reviewing the material covered,

Conversation and compilation of a reference table on teacher issues (frontal work);

Testing;

Group work with multi-level tasks of a practical nature on the topic;

Summing up the results of group work using elements of mutual control;

Summing up the lesson.

Means of education:

- computer class

Multimedia presentations "Volumes and surfaces of polyhedra", "What does it cost us to build a house?",

LOCAL test system,

Test on the topic "Volumes and surface areas of polyhedra",

Multimedia overhead projector.

DURING THE CLASSES.

The topic of our lesson is "Volumes and surface areas of polyhedra".(1 slide included!)The purpose of the lesson is to generalize and systematize knowledge on this topic and learn how to apply it in solving problems of practical content. Let's check the readiness for the lesson. You have blanks of a reference table, a card with homework, a pen, a draft on your tables.

First, we need to remember all types of polyhedra and repeat the formulas for calculating the surface area and volume of each of them.

(Slide show No. 2-No. 10 with commenting and questioning students.)

Knowledge on the topics: “Surface areas of polyhedra” and “Volumes of polyhedra” are one of the most important in studying the geometry of a school course, but the most interesting thing is that they can be useful to you in various life situations.

Remember the phrase: “What does it cost us to build a house?” Yes, yes: "Let's draw, we will live!" I can see from your eyes that some of you dream of building a 3-storey mansion with a gym, someone dreams of a nice country house with a winter garden, and someone ... will ask: “what does geometry have to do with it?” Here's the thing: today in the lesson we will learn how to calculate the necessary costs for the construction of a house, cottage or other structure, using the knowledge of these formulas.

Slide #11

Before you is the village "Dreams 11" A ". A house in the center of the village is a design option. Our task: Calculate the cost of building this house from various materials:

from iron and concrete;
from slate and brick;
tiles, concrete and bricks.

1 brigade (this is 1 row) - calculates a house made of iron and concrete. Working at computers ## (presentation 1)

Brigade 2 (2nd row) - you are working on a house made of slate and brick at computers No. No. (presentation 2)

3rd brigade (3rd row) - you got a house made of tiles, concrete and bricks. Computers No. No. (presentation 3)

To save time, let's divide the house into its component parts: 1st floor - what figure? - a rectangular parallelepiped, it is considered at computers No. ______; 2nd floor - ? - rectangular parallelepiped, computers No. ______; roof - ? - quadrangular pyramid, computers No. ______. Responsible work will be done by experts - economists - their task, based on the results of the work of the groups, is to estimate the cost of the material for building a box at home. Previously, they need to: pass the test, get a list of experts, help their team with calculations and announce the results of the overall work.

The experts are: ____________________, your jobs are computers No. ______. We take our jobs. Bring a pen, a piece of paper and a spreadsheet with you.

(The teacher passes, distributes tasks and distributes students to computers, each desk works on the calculation of the necessary material for the construction of one of the parts of the house).

group work

1 group

Approximately how many sheets of iron 2x0.8 m in size (slate 1.5x1 in size) (tiles 0.4x0.4 in size) are needed to cover the roof? What is the cost of acquiring it?

2 group

How many cubic meters of concrete (brick size 12x10x30cm) must be poured to get the walls of the 1st floor. Wall thickness 50cm. The size of the window opening is 1.5x1.2m, the door opening is 2x1.7m.

3 group

How many bricks (cubic meters of concrete) are needed to lay down the walls of the 2nd floor. Wall thickness 50cm. The size of the window opening is 1.5x1.2m, the small one is 1x0.8m. Brick dimensions 12x30x10cm.

Summarizing.

We are finishing work. Who among the experts is ready to acquaint us with the results of the calculations? So WHAT DOES IT COST TO BUILD A HOUSE? House made of concrete and iron -? Brick and slate house - ? House made of concrete, brick, tiles - ? Now you can estimate how much money is needed to build such a small house. This, of course, does not take into account the cost of work, delivery of materials and other costs, but, nevertheless, you can now handle simple calculations on your own. At home, I suggest you do the following tasks:

calculate the cost of a brick and tile house according to the dimensions indicated on the cards.

2) creative nature. Try to realize your dream - come up with a house to your liking by choosing the appropriate building materials and calculate its cost. You can find out the prices for building materials from the relevant construction companies and trading organizations. Have questions? Dare!

Let's summarize the lesson:

Today we repeated the formulas for calculating the surfaces and volumes of polyhedra, while you showed good knowledge, your math teacher can be proud of you;

learned to apply these formulas in solving problems of practical content.

Thank you for your work!

Tasks for the presentation project No. 1, No. 2, No. 3

prism	Parallelepiped		Cube	pyramid	Truncated pyramid	Correct pyramid	Tetrahedron

S=Sside + 2Sbase	S=Sside + 2Sbase	S=Sside + 2Sbase 2H(a+b) + 2ab	S=Sside + 2Sbase 6a2	S=Sside + Sbase	S=Sside + Sbase1 + Sbase2	S=Sside + Sbase Anl/2 + Sbase	S=Sside + 2Sbase
V= Soch H	V= Soch H	V= Sobase H = a b H	V \u003d Soch H \u003d a 3

Formulas for surface areas and volumes of polyhedra

prism	Parallelepiped	Rectangular parallelepiped	Cube	pyramid	Truncated pyramid	Correct pyramid	Tetrahedron

Formulas for surface areas and volumes of polyhedra

prism	Parallelepiped	Rectangular parallelepiped	Cube	pyramid	Truncated pyramid	Correct pyramid	Tetrahedron

Formulas for surface areas and volumes of polyhedra

prism	Parallelepiped	Rectangular parallelepiped	Cube	pyramid	Truncated pyramid	Correct pyramid	Tetrahedron

slide 2

Polyhedron

A polyhedron is a body whose surface consists of a finite number of flat polygons.

slide 3

A polyhedron is called convex if it lies on one side of any plane containing its face. A polyhedron is called non-convex if there is such a face that the polyhedron is on both sides of the plane containing this face.

slide 4

What is in the everyday sense the volume of a body, in particular a polyhedron? This is how much liquid can be poured inside this polyhedron. Cut off the tops and pour water inside each polyhedron. A convex polyhedron has already been filled, but a non-convex one has not yet. But perhaps the water was poured at different speeds: in order to correctly compare the volumes, we pour the liquid from each polyhedron into identical glasses. The water level in the right glass is higher than in the left one, which means that the volume of a non-convex polyhedron is indeed greater than the volume of a convex one.

slide 5

Many significant achievements of the mathematicians of ancient Greece in solving problems of finding cubature (calculating volumes) of bodies are associated with the use of the exhaustion method proposed by Eudoxus of Cnidus (about 408-355 BC). A formula is known that makes it possible to find the volume of a polyhedron if only the lengths of its edges are known. The volume of an arbitrary polyhedron can be calculated by knowing only the lengths of its edges. However, the polyhedron must be of a special form.

slide 6

In the general case, it can be shown that the generalized volumes of polyhedra are the roots of polynomial equations with coefficients that do not depend on the location of the vertices of the polyhedron in space, but are polynomials in the squares of the lengths of its edges. The numerical coefficients of these polynomials are determined by the combinatorial structure of the polyhedron.

Slide 7

The volume of the pyramidTheorem. The volume of the pyramid is equal to one third of the product of the base area and the height.

Slide 8

Polyhedron volume

The volume of a polyhedron is equal to the sum of the volumes of the pyramids, which have the faces of the polyhedron as their bases, and the center of the sphere as their vertex. Since all pyramids have the same height, equal to the radius R of the sphere, then the volume of the polyhedron.

MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION

federal state budgetary educational institution
higher education

"ULYANOVSK STATE TECHNICAL UNIVERSITY"

Barysh College - branch

Ulyanovsk State Technical University

for the implementation of practical work

by discipline

« Mathematics: algebra and the beginnings of analysis, geometry»

for special students 02/09/03 Programming in computer systems, 02/38/01 Economics and accounting (by industry)

2018

Reviewed and approved

cyclic methodological commission

disciplines of the general natural and general professional cycle

Chairman _______ N.A. Zolina

I approve

Deputy Director of Education

I.I. Shmelkova

Lecturer at the Barysh College - a branch of UlSTU D.A. Sovetkin

EXPLANATORY NOTE

The purpose of conducting practical classes is to consolidate and deepen theoretical knowledge in the discipline, as well as the acquisition of practical skills by students.

Before performing each practical lesson, the student is obliged, using the materials of the literature specified in the assignment, to repeat the material covered related to the topic of the practical lesson. Checking the readiness of students is carried out through a survey.

When performing work, students should be given independence, and their creative attitude to work should be encouraged in every possible way.

At the end of the lesson, students draw up a report in which the material on the implementation of the practical lesson should be consecrated in the sequence indicated in the assignment.

After submitting the report, the student receives a credit for the work performed.

Rules for performing practical work:

When performing work, the student must independently study the methodological recommendations for carrying out a particular work; perform the relevant calculations; use reference and technical literature; prepare answers to control questions. Studying the theoretical justification, the student should keep in mind that the main goal of studying the theory is the ability to apply it in practice to solve practical problems.

After completing the work, the student must submit a report on the work done with the results and conclusions obtained and defend it orally. Reports on practical work are carried out on A4 sheets. The first page is designed according to the rules for the design of title pages. It is necessary to leave margins 25-30 mm wide for teacher's comments. All schemes and drawings accompanying the implementation of practical work are carried out in pencil in accordance with the requirements of GOST.

Inaccurate performance of practical work, non-compliance with accepted rules and poor design of drawings, graphs or diagrams may cause the work to be returned for revision.

The report must contain:

job title;

goal of the work;

work sequence;
answers to control questions;
conclusion about the work done.

PRACTICAL WORK

Topic " Volumes and surface areas of polyhedra and bodies of revolution »

Target: to consolidate the knowledge and skills of finding volumes and surface areas of polyhedra and bodies of revolution.

Time - 2 hours.

Guidelines

Before performing practical work, it is necessary to complete an individual project - to make a polyhedron or a body of revolution on the instructions of the teacher.

List of prisms

1. The figure is a parallelepiped.

Necessary measurements: measure the length, width, height with a ruler.

According to the measurements find:

parallelepiped diagonal

side surface area

total surface area

figure volume.

2. The figure is a right triangular prism ABCA 1 B 1 C 1 .

According to the measurements find:

side surface area

total surface area

figure volume

cross-sectional area through a side ribAA 1 and the middle of the edge of the baseBC

3. Figure - cube ABCDA 1 B 1 C 1 D 1.

Necessary measurements: measure all edges with a ruler.

According to the measurements find:

prism diagonals

side surface area

total surface area

figure volume

Control questions:

Definition of a polyhedron

Definition of a prism

Types of prisms, their definitions

Prism elements

Definition of a parallelepiped, its types and elements

Types of prism sections

Volume of the parallelepiped and prism

List of pyramids

The figure is a tetrahedron.

Necessary measurements: measure all edges with a ruler.

According to the measurements find:

the height of the pyramid

side surface area

total surface area

figure volume

sectional area passing through the lateral edge and apothem of the opposite face

The figure is a quadrangular pyramid.

Necessary measurements: measure all edges with a ruler.

According to the measurements find:

side surface area

total surface area

figure volume

sectional area passing through the diagonal of the base and the side edge

the angle between the side face and the base plane.

The figure is a truncated triangular pyramid.

Necessary measurements: measure all edges with a ruler.

According to the measurements find:

side surface area

total surface area

figure volume

the area of the section passing through the height of the base and the side edge.

The figure is a truncated quadrangular pyramid.

Required measurements: measure with a ruler.

According to the measurements find:

side surface area

total surface area

figure volume

sectional area passing through two opposite side ribs.

Control questions:

Definition of pyramid, truncated pyramid

Types of pyramids, their definitions

pyramid elements

Section types

Pyramid Volume

List of bodies of revolution

1. Cylinder

Necessary measurements: measure the diameter and height of the cylinder with a ruler.

According to the measurements find:

side surface area

total surface area

figure volume

find the area of a section drawn parallel to the axis of the cylinder at a distanceL(to ask each student individually) from her.

Questions:

Cylinder definition

Define right and equilateral cylinder

Cylinder elements

Section types

Cylinder volume

2. Cone

Necessary measurements: measure the generatrix and the diameter of the base with a ruler.

According to the measurements find:

side surface area

total surface area

figure volume

axial area

the angle of inclination of the generatrix to the plane of the base.

Questions:

Definition of cone, truncated cone

Cone elements

Section types

Area and volume of a cone, truncated cone

3. Ball and sphere

Necessary measurements: measure the length of the diametral circle.

According to the measurements find:

shape radius

surface area of a sphere

ball volume

find the cross-sectional area of a sphere or sphere by a plane drawn at a distanceX(set to each student individually) from the center.

Questions:

Definition of a ball, sphere

Types of sections of the ball and sphere

Sphere Equation

Definition of a plane tangent to a ball

Definition of spherical segment, spherical layer and spherical sector

Exercise:

1. Make the necessary measurements according to the figure

2. According to the measurement data, perform the necessary calculations

3. Complete the task in notebooks

4. Answer theoretical questions.

Design requirements: draw a figure, write down the given, write down what needs to be found, the full solution and the answer.

LIST OF SOURCES USED

1. Dadayan A.A. Collection of problems in mathematics: textbook. allowance / A.A. Dadayan. - M.: FORUM: INFRA-M, 2014. - 352 p.

2. Dadayan A.A. Mathematics: textbook. /A.A. Dadayan. - 2nd ed. - M.: FORUM, 2014. -544 p. _

3. Bogomolov N.V. Practical lessons in mathematics, - M .: Nauka, 2011. - 370 p.

4. Algebra and the beginnings of analysis. Mathematics for technical schools at 2 pm Ed. G.N. Yakovlev. – M.: Nauka, 2015. -1002 p.

5. Geometry: Proc. for 10-11 cells. general education institutions / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others - 6th ed. - M.: Education, 2013. - 207 p.

6. Alimov Sh. A. et al. Mathematics: algebra and principles of mathematical analysis, geometry. Algebra and the beginning of mathematical analysis (basic and advanced levels). Grades 10-11. - M., 2014.

Presentation for a geometry lesson in grade 11.

Topic: Solving problems on the topic "Areas and volumes of polyhedra".

Target: repetition, preparation for the exam 2016.

Volkova Nina Vitalievna

mathematic teacher

MBOU secondary school No. 3 of the municipality Timashevsky district

Classwork.

Preparation for the exam.

(Tasks B-8).

1. The volume of a cube is 8. Find its surface area.

Solution:

1.S P=6a

3. Find the edge, then the surface area.

2. The radius of the base of the cylinder is 2, the height is 3. Find the area of the side surface of the cylinder divided by.

S b=2 rh.

3. A rectangular parallelepiped is described about a cylinder whose base radius and height are are equal to 6. Find the volume of the parallelepiped.

1 3

4. The sides of the base of a regular quadrangular pyramid are 10, the side edges are 13.

Find the surface area of this pyramid.

5. The volume of the cone is 16. Through the middle of the height, a section is drawn parallel to the base of the cone, which is the base of a smaller cone with the same vertex. Find volume

smaller cone.

6. Water was poured into a vessel shaped like a regular triangular prism. The water level reaches 80 cm. At what height will the water level be if it is poured into another similar vessel, whose base side is 4 times larger than the first one?

7. The cylinder and the cone have a common base and a common height. Calculate the volume of the cylinder if the volume of the cone is 87.

8. Find the volume of the polyhedron shown in the figure (all dihedral angles of the polyhedron are right).

9. The two edges of a cuboid coming out of the same vertex are 3 and 4. The surface area of this cuboid is 94. Find the third edge outgoing from the same vertex.