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Target:
- deepening and expanding children’s understanding of flat and three-dimensional objects; comparing them and identifying differences between them;
- identification and generalization of students’ knowledge about geometric figures and their properties;
- designing various flat figures;
- developing the skills to work in a group, following the rules, setting a goal, achieving it, analyzing your work and the work of the group.
Form: lesson-travel or group work in extracurricular activities.
Equipment: presentation for class; for each group: construction set, envelopes with tasks and figures, geometric bodies, rule cards.
Progress of the lesson
We came here to study, not to be lazy, but to work.
We work diligently and listen carefully.
Together, cheerfully and amicably we do everything we need.
Our work today takes place in groups. Let's repeat the rules of our work: (on the desks of each group there is a reminder card, remind each rule - the senior groups in turn). The rules are in the Appendix.
Did you know that in huge world There are many mathematicians interesting country with a beautiful name - Geometry. This country is not inhabited by numbers, but by various lines, figures and bodies. (Slide 2)
Today we will go on a journey through the country of Geometry and visit the cities where flat and three-dimensional figures live. Our task is to figure out what geometric shapes are flat and which are volumetric, and how do they differ?
We will travel in a hot air balloon. (Slide 3)
Why do you think? - Assembled from geometric shapes.
During the journey we will find out which group the parts of our balloon belong to.
II. Main part.
So, let's go!
We see the city ahead. What kind of city? Look!
1st stop - distribution stop.
Yes, not one city, but two. (Slide 4)
There are two cities in front of you. Read their names.
On the desks you also see various figures - these are city residents. Look at the figures in the envelope, name them, tell us about one.
Working in groups.
Now tell us what figures you populated in City of flat figures.
Children's answers. (Slide 4-left)
What do all flat figures have in common?
(They are laid entirely on a sheet or table, do not rise above the plane, they can be cut out of paper.)
Mathematicians say that plane – this is a two-dimensional space, i.e. it has two dimensions: length and width.
What other flat figures do you know?
Segments, straight lines, triangles, circles...
Now name the figures that settled in City of volumetric figures.
Children's answers. (Slide 4-right)
What do these figures have in common?
No matter how you place them, they will rise above the table or board.
What other three-dimensional figures do you know? Each group names its three-dimensional figures. Children's answers.
In geometry there is special name for volumetric figures – geometric body.
All bodies around us have three dimensions: length, width and height. True, not all geometric bodies can have length, width, and height. But at rectangular parallelepiped Can.
Demonstration by the teacher, children examine their parallelepipeds on the tables. All its faces are rectangular. Many objects have this shape. Name them. (Slide 6) Children's answers.
Let's return to our balloon. What shapes does it consist of, flat or three-dimensional? - The cylinder and the ball are three-dimensional figures, and the ribbon lines are flat. (Slide 7)
The sun has risen high and we are flying far away.
Stop 2 – scientific. Group No. 1.
Now guess what figure we are talking about.
Student 1: Three angles, three sides
Can be of different lengths. ( triangle). (Slide 8)
Student 2: This is a flat figure. It has 3 vertices, 3 corners, 3 sides. The sides may be the same or different lengths.
Student 3: A triangle is formed by three segments of a broken line.
What kind of figure is this, flat or three-dimensional? Children's answers.
(Slide 9) ENVELOPE with geometric shapes. Next figure...
Group No. 2.
Student 1: Trace the entire brick on the asphalt with chalk,
And you will get a figure - you, of course, are familiar with it.
This rectangle. (“click” on the slide )
Student 2: A rectangle has 4 corners, 4 vertices, 4 sides. Pairwise equal.
Student 3: The model is a closed broken line of 4 links. The links are equal in pairs.
Group No. 3.
Student 1: All four sides are the same length.
He is glad to introduce himself to you, but his name is...( square).
Student 2: A square has 4 vertices, 4 corners, 4 equal sides.
Student 3: model – a closed line of 4 links of the same length.
Group No. 4.
Student 1: Triangle stuck his nose into the jet vacuum cleaner.
And he has no nose - oh my God! – became like a skirt.
The most interesting thing is what his name is now. ( trapezoid)
Student 2: 4 corners, 4 vertices, 4 sides. The sides are all different or the sides are equal, but the bases are different.
Student 3: model – 4 closed lines, angles – 2 obtuse and 2 acute.
Group No. 5.
Student 1: if all the squares stood on the vertices at an angle,
What we saw, guys, was not squares, but... ( diamonds.)
Student 2: 4 corners, 4 vertices, 4 sides. The sides are equal opposite angles– are also equal.
Student 3: model – 4 closed lines, defined angles.
The sun has risen high and we are flying far away.
Stop Ahead. What is this? Look!
3rd stop - halt. Physical education lesson: “Dot, dot, comma...” Dance movements to music. (Video recording for class)
Stop 4 – design. (Slide 10) In front of you are containers with designer parts. Each group needs to assemble the figures according to the task. (See Appendix).
Find a task, sort out the details, discuss an action plan and get to work: assemble geometric shapes. Name them.
Work in pairs. The elders of the groups help and organize. Analysis of works.
III. Summary of the lesson. Reflection. So our first journey through the country of Geometry has ended. But you have to visit this amazing and wonderful country more than once and learn a lot of new things. Today you all worked great and that’s why you...well done.
Analysis of group work: whether the task was completed, quality of work, compliance with rules (cards for assessing work in groups).
Our lesson is over. Thank you for your attention. (slide 11)
APPLICATION:
Tasks to complete in group No. 1:
1. Look at the geometric shapes, name them and select TRIANGLES.
4. Make models of the figures.
Tasks to complete in group No. 2:
1. Consider geometric shapes, name them and select RECTANGLES.
2. Tell me what you know about this geometric figure.
3. Think about how to build a MODEL of this figure. Explain.
4. Make models of the figures.
Tasks to complete in group No. 3:
1. Look at the geometric shapes, name them and select SQUARE.
2. Tell me what you know about this geometric figure.
3. Think about how to build a MODEL of this figure. Explain.
4. Make models of the figures.
Tasks to complete in group No. 4:
1. Consider the geometric shapes, name them and select TRAPEZES.
2. Tell me what you know about this geometric figure.
3. Think about how to build a MODEL of this figure. Explain.
4. Make models of the figures.
Tasks to complete in group No. 5:
1. Look at the geometric shapes, name them and select Rhombuses.
2. Tell me what you know about this geometric figure.
3. Think about how to build a MODEL of this figure. Explain.
4. Make models of the figures.
Rules for working in a group.
- Respect your comrade.
- Know how to listen to everyone.
- Be responsible for your work and for the common cause.
- Be tolerant of criticism.
- If you don’t agree, suggest it!
Volumetric bodies Look around you, and you will find volumetric bodies everywhere. These are geometric shapes that have three dimensions: length, width and height. For example, to imagine a multi-story building, it is enough to say: “This house is three entrances long, two windows wide and six floors high.” Known to you from primary school cuboid and the cube are completely described by three dimensions. All objects around us have three dimensions, but not all of them can be named length, width and height. For example, for a tree we can specify only the height, for a rope - the length, for a hole - the depth. And for the ball? Does it also have three dimensions? We say that a body has three dimensions (is volumetric) if a cube or ball can be placed in it. Both the sphere, the cylinder, and the cone have three dimensions.
Polyhedra A body that is bounded by plane polygons is called a polyhedron. For example, a cube is bounded by equal squares. The polygons that form the surface of a polyhedron are called faces. The sides of these polygons are the edges of polyhedra. Vertices of polygons, vertices of polyhedra. For example, a cube has 6 faces (all of them equal squares), 12 edges and 8 vertices.
Polyhedra. Pyramid. The polyhedron on the right has a special name: regular quadrangular pyramid. This is exactly the shape of the famous Cheops pyramid: at its base there is a square, and side faces equal triangles. How many faces, edges and vertices does this polyhedron have? Some of the shapes in the picture are polyhedra and some are not. What numbers are the polyhedra shown under?
Convex and non-convex polygons Polygons, as we already know, can be convex and non-convex. A convex polygon lies on one side of any line containing any side of the polygon. And for a non-convex one, you can find a side such that the straight line containing it “cuts” the polygon into parts. In the figure, the yellow polygon is convex, and the blue one is non-convex. Polyhedra can also be convex or non-convex. A convex polyhedron lies on one side of any plane containing any of its faces. And for a non-convex polyhedron one can find such a face that a plane passing through it will “cut” it into pieces. The yellow polyhedron in the picture is convex. The blue polyhedron is non-convex. Which numbers in the figure show convex polyhedra, and which numbers show non-convex ones?
Answer the questions: 1. What is the face of a cube: a) a segment; b) a point; c) a square. 2.What is an edge of a cube: a) a segment; b) a point; c) a square. 3.What does the vertex of a cube represent: a) a segment; b) a point; c) a square. 4. How many faces does a rectangular parallelepiped have: a) 8b) 6c) 12 5. A polyhedron is a) any volumetric body b) a body that is limited by flat polygons
Answer the questions: 6. What lies at the base regular pyramid a) rectangleb) squarec) parallelogram 7. Which figure is the face of a regular pyramid a) rectangleb) squarec) regular triangle 8. A convex polyhedron a) lies on one side of any plane containing any of its faces b) any volumetric body c) lies on both sides of any plane containing any of its faces. 9.What numbers are shown in the figure for convex polyhedra?
Resources used: School website distance learning(Moscow) distance learning schools (Moscow) Online Encyclopedia Around the World OGRANNIK.html OGRANNIK.html Yandex / pictures %D0%B0%D0%B2%D0%B8%D0%BB%D1%8C%D0%BD% D0%B0%D 1%8F%20%D1%87%D0%B5%D1%82%D1%8B%D1%80%D1%91%D1 %85%D1%83%D0%B3%D0%BE %D0%BB%D1%8C%D0%BD%D0%B0 %D1%8F%20%D0%BF%D0%B8%D1%80%D0%B0%D0%BC%D0%B8 %D0%B4 %D0%B0&spsite= ru%3A8080%2For%2Fget_att.jsp%3Fatt_id%3D2493&rpt=simage Geometry textbook 6-9
Topic: “Flat figures and volumetric bodies»
Goals:
generalize ideas about flat geometric figures and volumetric geometric bodies;
create conditions under which students “discover” a way to obtain a three-dimensional figure.
Tasks:
consolidate knowledge on the classification of flat figures and three-dimensional bodies, their fundamental differences;
introduce the concepts of “bodies of revolution” and “polyhedra”;
establish a connection between the science of geometry and fine arts;
create a cube model using origami technique;
develop logical and spatial thinking, attention, memory, imagination, creativity;
cultivate accuracy and adherence to safety rules when working with tools.
Equipment: interactive whiteboard, presentation, models of volumetric geometric shapes, handout(individual cards).
Progress of the lesson.
Organizational moment. Creating a situation of success.
II . Updating basic knowledge.
Primary teacher: - Guys, today our lesson is dedicated to geometry.
Let's remember what geometry is? (Translated from Greek, the word “geometry” means “land surveying”. In mathematics, “geometry” is the science that studies geometric figures and their properties)
Primary teacher: - What geometric shapes do you know? (Square, rectangle, cube, ball, etc.)
Primary teacher: - What types can these geometric shapes be divided into? (Volume geometric bodies, flat geometric shapes, basic geometric concepts)
Primary teacher: - The topic of our lesson is “Flat figures and three-dimensional bodies.”
All objects are flat or three-dimensional.
How do flat figures differ from three-dimensional bodies? (Flat figures have only length and width, while solid figures have length, height and width.)
Art teacher: - Here you gofirst task (according to options):color flat shapes warm colors, and volumetric bodies are cold. Let's remember which colors are called warm and which are cold?
Primary teacher: - What is the structure of volumetric bodies? (Edges, faces, base, top).
- Who will show the listed parts of volumetric bodies on the model?
Primary teacher: - To consolidate, let’s dosecond task
(according to options):
1 option - Shade the front and top edge Cuba.
Option 2 - Draw the missing edges.
Option 3 - Count the number of vertices in a pentagonal prism.
Primary teacher: - Now let's play. Let's figure out who is “friends” with whom (Orange with a ball, carrot with a cone, lemon with an oval, box with a rectangle).
Art teacher:
- We can also find geometry in art. For example, monuments to geometric figures:
Sculpture Cube in Zabeel Park, Dubai UAE
Glowing cube in Beijing
Like thismarble ball installed on Bolshaya Sadovaya, the central street of the city of Rostov-on-Don. The amazingly precise shapes of this ball surprise all lovers of mathematics, and geometry in particular.
Monument to regular polyhedra in Germany
Irregular triangle in a Belgian village
Project of a monument to the artist Kazimir Malevich in the Moscow region
Kazemir Malevich was a Soviet artist who lived in the 20th century, who created non-figurative works consisting of geometric figures, where main role square plays.
Self-portrait of Kazimir Malevich
This art is called “suprematism” (superiority, supremacy). For example, one of his first paintings “Black Square”.
Woman carrying water
III . Discovery of something new.
1. Bodies of revolution and polyhedra.
Primary teacher: - Volumetric bodies are also divided into two groups: bodies of rotation and polyhedra.
Why do you thinkbodies of revolution ? (A cylinder can be considered as a body obtained by rotating a rectangle around its side as an axis. A cone can be considered as a body obtained by rotating right triangle around its side as an axis.)
Art teacher: - Look at the layout.
Primary teacher: - How to characterize polyhedra? ( A polyhedron is a geometric body bounded on all sides by faces. The sides of the faces are called the edges of the polyhedron, and the ends of the edges are called the vertices of the polyhedron.)
Art teacher: - How to portray volumetric figures?
Three-dimensional figures are depicted using chiaroscuro, otherwise it is impossible to show that they “rise” above the sheet of paper. And using a dotted line, an invisible contour is depicted. Let's try to show the volume of bodies of revolution and polyhedra using chiaroscuro.Third task :
Option 1 - cone;
Option 2 - pyramid;
Option 3 - cylinder.( Analysis of works.)
IV . Physical education minute. ( Performed to the song “Dot, dot, comma...”)
Period, period, comma.
They show with their hands while crouching.
It turned out to be a funny face.
Hands to ears, body turns.
Hands, legs, cucumber
Show arms, legs, draw an oval with hands
It turned out to be a little man.
Hands on the belt, turns the body to the left, to the right.
What will these dots see?
Blinking eyelashes - fingers
What will these pens build?
Hands forward to shoulders
How far are these legs?
They'll take him away
Steps in place
How will he live in the world -
We are not responsible for this:
Hands on the belt - body tilts left and right
We drew it
Sit down
That's all!
Got up
V . Practical work.
Art teacher: - One of the important spatial geometric figures is the cube.
Which flat figure is the face of a cube? (Square)
How many faces does a cube have? (6)
And now we will assemble a cube using the origami technique. Such a cube can be folded from identical parts. There should be as many of them as there are faces of the cube. Connect the parts according to the diagram. Sharp corners put it in your pockets. Remember: each corner must be inserted into a pocket. You will work in pairs. Each pair will solve their own cube. From the collected cubes we will create another geometric figure - a stepped pyramid.
VI . Exhibition and analysis of works.
VII . Lesson summary. - What groups can volumetric bodies be divided into? (Bodies of revolution and polyhedra)
Give examples of bodies of rotation. What flat figure underlies a cone, sphere, or cylinder?
Give examples of polyhedra. How many faces does a cube have?
VIII .Reflection.
VIII . Homework. G.s.46-47 (show the volume of a prism, cylinder, pyramid, write down visible and invisible edges and faces)
Geometric volumetric figures are solids, which occupy a non-zero volume in Euclidean (three-dimensional) space. These figures are studied by a branch of mathematics called “spatial geometry”. Knowledge about the properties of three-dimensional figures is used in engineering and the natural sciences. In the article we will consider the question of geometric three-dimensional figures and their names.
Geometric solids
Since these bodies have a finite dimension in three spatial directions, a system of three is used to describe them in geometry coordinate axes. These axes have the following properties:
- They are orthogonal to each other, that is, perpendicular.
- These axes are normalized, meaning the basis vectors of each axis are the same length.
- Any of the coordinate axes is the result vector product two others.
Speaking about geometric volumetric figures and their names, it should be noted that they all belong to one of 2 large classes:
- Class of polyhedra. These figures, based on the name of the class, have straight edges and flat faces. A face is a plane that limits a shape. The point where two faces join is called an edge, and the point where three faces join is called a vertex. Polyhedra include the geometric figure of a cube, tetrahedrons, prisms, and pyramids. For these figures, Euler's theorem is valid, which establishes a connection between the number of sides (C), edges (P) and vertices (B) for each polyhedron. Mathematically, this theorem is written as follows: C + B = P + 2.
- Class of round bodies or bodies of rotation. These figures have at least one surface forming them that is curved. For example, a ball, a cone, a cylinder, a torus.
As for the properties of volumetric figures, the two most important of them should be highlighted:
- The presence of a certain volume that a figure occupies in space.
- The presence of each three-dimensional figure
Both properties for each figure are described by specific mathematical formulas.
Let us consider below the simplest geometric volumetric figures and their names: cube, pyramid, prism, tetrahedron and ball.
Cube figure: description
The geometric figure cube is a three-dimensional body formed by 6 square planes or surfaces. This figure is also called a regular hexahedron, since it has 6 sides, or a rectangular parallelepiped, since it consists of 3 pairs parallel sides, which are mutually perpendicular to each other. They call it a cube whose base is a square and whose height is equal to the side of the base.
Since a cube is a polyhedron or polyhedron, Euler's theorem can be applied to it to determine the number of its edges. Knowing that the number of sides is 6, and the cube has 8 vertices, the number of edges is: P = C + B - 2 = 6 + 8 - 2 = 12.
If we denote the length of a side of a cube by the letter “a,” then the formulas for its volume and surface area will look like: V = a 3 and S = 6*a 2, respectively.
Pyramid figure
A pyramid is a polyhedron that consists of a simple polyhedron (the base of the pyramid) and triangles that connect to the base and have one common top(top of the pyramid). The triangles are called the lateral faces of the pyramid.
The geometric characteristics of a pyramid depend on which polygon lies at its base, as well as on whether the pyramid is straight or oblique. A straight pyramid is understood to be a pyramid for which a straight line perpendicular to the base, drawn through the top of the pyramid, intersects the base at its geometric center.
One of simple pyramids is a quadrangular straight pyramid, at the base of which lies a square with side “a”, the height of this pyramid is “h”. For this pyramid figure, the volume and surface area will be equal: V = a 2 *h/3 and S = 2*a*√(h 2 +a 2 /4) + a 2, respectively. Applying for it, taking into account the fact that the number of faces is 5, and the number of vertices is 5, we obtain the number of edges: P = 5 + 5 - 2 = 8.
Tetrahedron figure: description
The geometric figure tetrahedron is understood as a three-dimensional body formed by 4 faces. Based on the properties of space, such faces can only represent triangles. Thus, a tetrahedron is a special case of a pyramid, which has a triangle at its base.
If all 4 triangles forming the faces of a tetrahedron are equilateral and equal to each other, then such a tetrahedron is called regular. This tetrahedron has 4 faces and 4 vertices, the number of edges is 4 + 4 - 2 = 6. Using standard formulas from plane geometry for the figure in question, we obtain: V = a 3 * √2/12 and S = √3*a 2, where a is the length of the side of the equilateral triangle.
It is interesting to note that in nature some molecules have the form regular tetrahedron. For example, a methane molecule CH 4, in which the hydrogen atoms are located at the vertices of the tetrahedron and are connected to the carbon atom by covalent chemical bonds. The carbon atom is located at the geometric center of the tetrahedron.
The tetrahedron shape, which is easy to manufacture, is also used in engineering. For example, the tetrahedral shape is used in the manufacture of ship anchors. Note that space probe NASA's Mars Pathfinder, which landed on the surface of Mars on July 4, 1997, was also shaped like a tetrahedron.
Prism figure
This geometric figure can be obtained by taking two polyhedra, placing them parallel to each other in different planes of space, and connecting their vertices accordingly. The result will be a prism, two polyhedra are called its bases, and the surfaces connecting these polyhedra will have the shape of parallelograms. A prism is called straight if it sides(parallelograms) are rectangles.
A prism is a polyhedron, so Euler's theorem is true for it. For example, if a hexagon lies at the base of a prism, then the number of sides of the prism is 8, and the number of vertices is 12. The number of edges will be equal to: P = 8 + 12 - 2 = 18. For a straight prism of height h, at the base of which lies a regular hexagon with side a, volume equals: V = a 2 *h*√3/4, surface area equals: S = 3*a*(a*√3 + 2*h).
Speaking about simple geometric volumetric figures and their names, we should mention the ball. A volumetric body called a ball is understood as a body that is limited to a sphere. In turn, a sphere is a collection of points in space equidistant from one point, which is called the center of the sphere.
Since the ball belongs to the class of round bodies, there is no concept of sides, edges and vertices for it. The surface area of the sphere enclosing the ball is found by the formula: S = 4*pi*r 2, and the volume of the ball can be calculated by the formula: V = 4*pi*r 3 /3, where pi is the number pi (3.14), r is the radius of the sphere (ball).
Volumetric bodies. Look around you and you will find three-dimensional bodies everywhere. These are geometric shapes that have three dimensions: length, width and height. For example, to imagine a multi-story building, it is enough to say: “This house is three entrances long, two windows wide and six floors high.” The rectangular parallelepiped and cube you know from elementary school are completely described in three dimensions. All objects around us have three dimensions, but not all of them can be named length, width and height. For example, for a tree we can specify only the height, for a rope - the length, for a hole - the depth. And for the ball? Does it also have three dimensions? We say that a body has three dimensions (is volumetric) if a cube or ball can be placed in it.
Slide 2 from the presentation "Formula for the volume of a polyhedron". The size of the archive with the presentation is 1207 KB.Geometry 11th grade
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