During the classes
Before the start of the lesson, the teacher checks the readiness of the students for the lesson: readiness
blackboards, order on the desks, the presence of notebooks. Collecting exercise books is carried out before the lesson.
IMotivational - indicative stage
Home building check. Update. 10 min
Lesson readiness. Checking for the supplies needed for the lesson.
- What figure is shown in the picture? Rectangle ABCD.
- Name the elements of the rectangle ABCD.
- Vertices A, B, C, D; sides: AB, BC, CD, AD
- Are the following statements true:
1. The rectangle has 4 vertices and 4 sides.
2. Each side of the rectangle is a straight line, the apex is a point.
3. For a rectangle, all sides are equal.
What is the name of a rectangle in which all sides are equal?
4. In a rectangle, opposite sides are equal. Give an example of opposite sides.
- 1. Right.
2. Wrong. Each side of the rectangle is a line segment, and the vertex is a point.
3. Wrong. A special case.
Square.
4. Correct. Opposite sides: AB and DC, AD and BC.
- Create a drawing problem.
How do I calculate the area of a rectangle?
- To find the area of a rectangle, you need to multiply the length by the width.
- Write down the formula for calculating the area of a rectangle.
- S = ab
- Exercise. Find the unknown component verbally in the table. First row area
rectangle, the second and third lines are the sides of the rectangle. V
according to the key, substitute the required letter for each answer received.
The calculation is done frontally in the class. One by one, the students make calculations in the table and enter the correct answer.
- What word did we get?
- Parallelepiped.
- What is it?
- This is a three-dimensional geometric body.
- These bodies are divided into two groups: the upper 4 bodies and the lower ones. By what principle are they divided into two types? What is common among the bodies of each group?
 The upper bodies are polygons, and the lower bodies are round. In the upper group, each body consists of polygons, and in the lower group, one of the elements is a circle.
- In the world we are surrounded by many objects. They differ in shape, size, materials from which they are made, color ... People are interested in different qualities these items. Mathematicians are interested in their shape and size. Among the many geometric bodies, there are two large groups: polyhedrons and round bodies.
The word that we got - a parallelepiped, means a volumetric body, which is one of the types of polyhedra.
- Which of these polyhedra are parallelepipeds?
- Bodies A, B
- What makes them different from the remaining polyhedra?
- The edges are rectangles.
- Give examples of objects from the surrounding world that have the shape of a rectangular parallelepiped?
- Textbook, frame of the house, classroom, box.
- The study of spatial bodies occurs in the 10th grade, with you we will study the section of geometry - stereometry, but in the 5th grade
we can already give some initial information about volumetric figures,
get acquainted with its elements and some properties.
What is the purpose of today's lesson?
- Get acquainted with the elements that make up a rectangular parallelepiped.
Operationally - cognitive stage. 20 minutes
1. Let's write the topic of the lesson in notebooks.
Number, Classwork and the topic of the lesson.
2. Before us are several models of a rectangular parallelepiped: a model made of wood, as well as a wireframe model. On these models, the elements of the rectangular parallelepiped are clearly visible.
Show faces, edges, vertices of the box on the model.
There are a certain number of these components. Let's count how many there are. Let's fill in the table.
The teacher calls the students to the board to count the number of vertices, edges and faces.
The table is filled in parallel
(the first two columns are filled in):
- So, as we know any point in space and on a plane, we can designate the Latin letter of the alphabet.
Here is an image of a rectangular parallelepiped. Each vertex is marked with a Latin letter. By listing
Latin letters, we denote this parallelepiped. Who can tell me how this parallelepiped is designated?
- ABCDKLMN
- Exercise.
1. the first row writes out the designation of the vertices;
2. the second designation of the ribs;
3. third row - designation of faces.
To provide results, students go to the board in pairs. One reads out the elements, the second shows them in the drawing.
If a supplement is required, the teacher turns to other groups.
- Find equal edges for the box.
- AB = DC = MN = KL
AK = BL = CM = DN
AD = BC = LM = KN
Pupils write down in their notebook.
- Each group of equal edges has a name.
AB = DC = MN = KL - width
AK = BL = CM = DN - length
AD = BC = LM = KN - height
- Is the equality of all three dimensions possible?
- Yes.
- What figure do we get?
- Cube.
- From early childhood, we are familiar with such a figure as a cube.
What are the differences between a cube and the general view of a rectangular parallelepiped?
- All edges of a cube are equal. All faces are squares.
- Which faces will be equal for the parallelepiped ABCDEFGH.
At the same time, there is a slide show.
- ABСD = KLMN
ADNK = BCML
ABFE = DCGH
- How are these faces mutually located relative to each other?
- They lie opposite each other.
- Such faces are called opposite to each other.
What conclusion can be drawn from the above.
- The opposite faces of the rectangular parallelepiped are equal.
Both on the slide and on the model, equal faces are highlighted in the same color.
- Open the tutorial on page 121,№ 792.
What is the surface area of a rectangular parallelepiped?
- The sum of the areas of its edges.
- How many faces does a parallelepiped have?
- 6
- What geometric shapes are these faces?
- Rectangles.
- How to calculate the area of each face?
- Find the product of measurements for each face of the parallelepiped.
- What property do the faces of a parallelepiped have?
- Opposite faces are equal.
- Therefore, we will find the area only at three edges.
- What are the measurements of the first face?
- 5 cm and 6 cm 5∙6=30
cm2
- What are the measurements of the second face?
Calculate the area of this face.
- 5 cm and 3 cm 5∙3=15
cm2
- What are the measurements of the third face?
Calculate the area of this face.
- 3cm and 6cm 6∙3=18
cm2
- 2∙30+2∙15+2∙18=126
cm2
- How to write an expression for the surface area of a rectangular parallelepiped?
№ 796 (b) - Write a formula for calculating the area of the surface of a rectangular parallelepiped.
- Measurements of the first face a and b
S = a ∙ b
Measurements of the second face b and c
S = b ∙ c
Measurements of the third face a and c
S = a ∙ c
S = 2 ∙ ab + 2 ∙ bc + 2 ∙ ac
- So, we have derived a formula by which it is easy to find the surface area of a rectangular parallelepiped, knowing its measurements.
Task:
The boy wants to pack the gift prepared for his mother for the New Year in a box in the shape of a rectangular parallelepiped, the dimensions of which are 20 cm* 30cm× 40 cm
He decided to paste over this box on all sides with colored paper, 1 dm 2 of which costs 8 rubles. The boy expects to spend 450 rubles to buy the required amount of paper. Will he have enough money for this?
- First, we find the surface area of the parallelepiped.
S = 2 ∙ ab + 2 ∙ bc + 2 ∙ ac
1) 2∙20∙30+2∙30∙40+2∙20∙40=1200+2400+1600=5200
cm 2 - surface area of the parallelepiped.
2) 5200
cm2
=52
dm2
3) 52∙8=416
(rub) - required for purchase.
Answer: The boy can dare to go for colored paper.
III Reflexive - evaluation stage
We will first write down our homework, and then we will summarize our lesson.
§4,
paragraph 20, page 121 No.811,812, 814, 817.
Clear recommendations for the implementation of each number.
- What was the purpose of our lesson?
- Explore the components and properties of a rectangular parallelepiped.
- Have we achieved this goal?
- Yes, we did.
- Name the objects from the surrounding world that have the shape of a rectangular parallelepiped.
- Houses, classroom, brick, etc.
- What elements have we selected for the rectangular parallelepiped?
- Vertices, Edges and Faces.
- How many vertices, edges and faces does a rectangular parallelepiped have?
- Vertices - 8; ribs - 12; faces - 6.
- Name the dimensions of the parallelepiped.
- Length Width Height.
- Name the property for the faces of the parallelepiped.
- Opposite faces of the box are equal.
- How to find the area of the side surface of a rectangular parallelepiped.
- It is necessary to add the areas of the faces of the parallelepiped.
- Why do we need to find the area of the lateral surface of a parallelepiped?
- For practical purposes. For example, to paste over a box with paper, paint a room, glue wallpaper in the room.
- So, the lesson is over, but put a mark in your notebook for the work in the lesson, and add to this mark +
- if the lesson was interesting for you;
- if the lesson was boring.
Thank you for your attention!
The prism is called parallelepiped if its bases are parallelograms. Cm. Fig. 1.
Box properties:
Opposite faces of a parallelepiped are parallel (i.e., lie in parallel planes) and equal.
The diagonals of the parallelepiped intersect at one point and are halved by this point.
Adjacent faces of a parallelepiped- two faces that have a common edge.
Opposite faces of a parallelepiped- faces that do not have common edges.
Opposite vertices of a parallelepiped- two vertices that do not belong to the same face.
Diagonal of a parallelepiped- a segment that connects opposite vertices.
If the side edges are perpendicular to the planes of the bases, then the parallelepiped is called direct.
A straight parallelepiped, the bases of which are rectangles, is called rectangular... A prism, all faces of which are squares, is called cube.
Parallelepiped- a prism whose bases are parallelograms.
Straight parallelepiped- a parallelepiped with side edges perpendicular to the plane of the base.
Rectangular parallelepiped
Is a straight parallelepiped, the bases of which are rectangles.
Cube- rectangular parallelepiped with equal edges.
Parallelepiped called a prism, the base of which is a parallelogram; thus, a parallelepiped has six faces, and they are all parallelograms.
Opposite faces are pairwise equal and parallel. The parallelepiped has four diagonals; they all intersect at one point and are divided in half at it. Any facet can be taken as a base; the volume is equal to the product of the base area by the height: V = Sh.
A parallelepiped, the four side faces of which are rectangles, is called straight.
A straight parallelepiped, in which all six faces are rectangles, is called rectangular. Cm. Fig. 2.
The volume (V) of a straight parallelepiped is equal to the product of the base area (S) and the height (h): V = Sh .
For a rectangular parallelepiped, in addition, the formula V = abc, where a, b, c are edges.
The diagonal (d) of a rectangular parallelepiped is related to its edges by the relation d 2 = a 2 + b 2 + c 2 .
Rectangular parallelepiped- a parallelepiped in which the side edges are perpendicular to the bases, and the bases are rectangles.
Properties of a rectangular parallelepiped:
In a rectangular parallelepiped, all six faces are rectangles.
All dihedral corners of a rectangular parallelepiped are straight.
The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions (the lengths of three edges having a common vertex).
The diagonals of a rectangular parallelepiped are equal.
A rectangular parallelepiped, all of whose faces are squares, is called a cube. All the edges of a cube are equal; volume (V) of a cube is expressed by the formula V = a 3, where a is the edge of the cube.
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or (equivalently) a polyhedron with six parallelogram faces. Hexagon.
The parallelograms that make up the parallelepiped are facets of this parallelepiped, the sides of these parallelograms are edges of a parallelepiped, and the vertices of the parallelograms are peaks parallelepiped... For a parallelepiped, each face is parallelogram.
As a rule, any 2 opposite faces are distinguished and called the bases of the parallelepiped, and the remaining faces are side faces of the parallelepiped... The edges of the box that do not belong to the bases are lateral ribs.
2 faces of a box that have a common edge are adjacent, and those that do not have common edges - opposite.
The segment that connects 2 vertices that do not belong to the 1st face is by the diagonal of the parallelepiped.
The lengths of the edges of a rectangular parallelepiped that are not parallel are linear dimensions (measurements) of a parallelepiped. A rectangular parallelepiped has 3 linear dimensions.
Types of parallelepiped.
There are several types of parallelepipeds:
Direct is a parallelepiped with an edge perpendicular to the plane of the base.
A rectangular parallelepiped with all 3 dimensions of the same magnitude is cube... Each of the faces of the cube is equal squares .
Arbitrary parallelepiped. The volume and ratios in an oblique parallelepiped are mainly determined using vector algebra. The volume of the parallelepiped is equal to the absolute value of the mixed product of 3 vectors, which are determined by the 3 sides of the parallelepiped (which originate from one vertex). The ratio between the lengths of the sides of the parallelepiped and the angles between them shows the statement that the Gram determinant of these 3 vectors is equal to the square of their mixed product.
Box properties.
- The parallelepiped is symmetrical about the midpoint of its diagonal.
- Any segment with ends that belong to the surface of the parallelepiped and that passes through the middle of its diagonal is divided by it into two equal parts. All the diagonals of the parallelepiped intersect at the 1st point and are divided by it into two equal parts.
- Opposite faces of the box are parallel and equal in size.
- The square of the length of the diagonal of a rectangular parallelepiped is
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