First, let's remember which shape is called a rectangle (Fig. 1).
Figure: 1. Defining a rectangle
Look at the figures shown (Fig. 2).
Figure: 2. Shapes
We need to determine if there is a rectangle among them.
For this we need a square. Let's find a right angle at the square and apply it to each of the corners of our figures. By applying a square to all corners of the first figure, we see that it coincides with all corners. This means that shape number 1 is a rectangle.
We apply the right angle of the square to figure 2 and see that the angle does not coincide with the right angle. This means that shape # 2 is not a rectangle.
We apply the right angle of the square to figure 3. The first angle of the right. The second corner of the figure is straight. The third corner of the figure is also straight. And the fourth corner is also right. The third shape is a rectangle.
Figure 4. We apply the right angle of the square, and it coincides with the corner of the figure. We apply it to the second corner of the figure, and it also matches. We apply the right angle of the square to the third corner. The third corner also matches. The fourth corner also matches. This means that shape # 4 is a rectangle.
Figure # 5. We apply the right angle of the square to the first corner. This angle does not coincide with the right angle of the square. This means that shape # 5 is not a rectangle.
It turns out that the rectangles are figures numbered 1, 3, 4 (Fig. 4).
Figure: 3. Rectangles
We have established that shapes 1, 3 and 4 have right angles.
The square is a drawing tool for drawing corners. Squares are made of metal, plastic or wood (Fig. 3).
Figure: 4. Square
Figures 1 and 3 have equal sides that lie opposite each other. And figure 4 has all sides equal. Such figures have a special name.
A quadrilateral whose sides are pairwise equal is called a rectangle.
A rectangle with all sides equal is called a square.
Let's build a rectangle using a square and a ruler.
To do this, first put a point on the plane. Then we find the angle on the gon and apply it so that the point is the apex of the angle (Fig. 5).
Figure: 5. Point - apex of the corner
Now we outline the sides of the corner (Fig. 6).
Figure: 6. Sides of the corner
We do the same with the second corner of the rectangle (Fig. 7).
Figure: 7. Sides of two corners
Now we will take a ruler and use it to measure the segments of a given length. Using the same ruler, we will draw the fourth side (Fig. 8).
Figure: 8. Drawing the sides of the figure
We now have a geometric shape. Let's call it. Let's name each vertex of our rectangle (Fig. 9).
Figure: 9. Designation of the vertices of the rectangle
We have built a rectangle ABCD using a ruler and a square.
In the lesson, we learned how to distinguish a rectangle from other quadrangles. We also learned how to draw a rectangle on a piece of paper using a square and a ruler.
Bibliography
- Alexandrova E.I. Maths. Grade 2. - M .: Bustard - 2004.
- Bashmakov M.I., Nefedova M.G. Maths. Grade 2. - M .: Astrel - 2006.
- Dorofeev G.V., Mirakova T.I. Maths. Grade 2. - M .: Education - 2012.
- Proshkolu.ru ().
- Social network of educators Nsportal.ru ().
- Illagodigardarivista.com ().
Homework
- Select rectangles from the suggested shapes (Fig. 10):
Figure: 10. Drawing for the task
- Prove that the figure shown in Figure 11 is a rectangle.
Figure: 11. Drawing for the task
- Build a 5 cm and 8 cm rectangle yourself using a square and a ruler.
MBOU "Okskaya secondary school"
Abstract of an open lesson in mathematics
in the 4th grade on the topic:
"Constructing a rectangle on unlined paper."
Primary school teacher: Yashina Tatyana Vasilievna
year 2013
Lesson "Building a rectangle on unlined paper" grade 4
Lesson objectives: Teach how to build a rectangle and square on unlined paper using a compass and a ruler.
Tasks:
1. Educational:
update previous knowledge about rectangle and square;
to form practical skills in constructing geometric shapes, using knowledge about them;
to consolidate the skills of solving word problems, comparing named numbers;
develop computational skills, logical thinking.
2. Developing:
develop the spatial imagination of students;
to develop the communication skills of students in the course of pair work, the ability for mutual control and self-control.
3. Educational:
instill a love of mathematics;
bring up accuracy when performing constructions;
to awaken in the student a sense of pride in their personal achievements and the successes of their peers.
Lesson type:
combined
Lesson form:
practical work.
Equipment:
for students: textbook, square, sheet of unlined white paper, pencil, compasses
for the teacher: textbook, laptop, TV, presentation.
During the classes .
1. Organizational moment.
2. Motivation for activity.
Oh, how many wonderful discoveries we have
Prepares the spirit of enlightenment.
And experience, son of difficult mistakes,
And a genius, a friend of paradoxes.
And chance, god is an inventor.
I hope that this lesson in mathematics will become another confirmation of our motto "Mathematics is the queen of sciences", and the great people of the past and present will help us with this.
3. Verbal account.
Test (Slide) We will evaluate each task.
1. Given numbers: 713754, 713654, 713554, ... Choose the next number :
a) 713854
b) 713554
c) 713454
2. What is the deductible if the deductible is 73 and the difference is 600?
a) 527
b) 673
c) 763
3. Find the smallest of the numbers:
a) 18215
b) 18152
c) 18125
d) 18521
4. How many tens are contained in the number 387 560?
a) 6
b) 38
c) 38 756
5. How many digits there will be in the quotient 64 080: 9
a) 1
b) 2
in 3
d) 4
6. Complete the sentence "To find an unknown dividend, you need the value of the quotient ..."
a) multiply by the divisor;
b) divide by a divisor;
c) divide by the dividend.
4. Updating basic knowledge.
1. Guess the riddle:
This important science
Examines everything around:
Dots, lines, squares,
Triangles and a circle ...
For her, a ruler, compasses
They are best friends.
But for you this science
You can't forget!
That's right, this science is called GEOMETRY.
What does this word mean?
Translated from Greek, this word means "surveying" ("geo" - land, "metrio" - to measure). This name is explained by the fact that the origin of geometry was associated with various measuring work, which had to be performed when marking land plots, conducting roads, constructing buildings and other structures. As a result of this activity, various rules related to geometric measurements appeared and gradually accumulated. Thus, geometry arose on the basis of the practical activities of people and at the beginning of its development served primarily practical purposes.
Subsequently, geometry was formed as an independent science, in which geometric figures and their properties are studied.
The world around us is the world of geometry. HELL. Alexandrov(Slide)
2. Guys, look closely at the drawing.
How many triangles are there? (9)
How many quadrangles are in the drawing? (2).
How do they differ from each other?
(One is a rectangle and the other is not.)
- What do you know about a rectangle?
All corners in a rectangle are straight.
The opposite sides of the rectangle are equal.
The diagonals at the intersection are halved
The diagonal of the rectangle divides it into two equal triangles.
3. Well done! You've talked a lot about the rectangle.
Now solve the problem: (Slide)
A diagonal is drawn in the rectangle. The area of \u200b\u200bone of the resulting triangles is 25 cm 2 ... What is the area of \u200b\u200bthe rectangle?
Solve the problem.
How did you find the area of \u200b\u200ba rectangle?
(We know that the diagonal of a rectangle divides it into two identical triangles. The area of \u200b\u200bone triangle is 25 square cm, so the area of \u200b\u200bthe whole rectangle will be 25 * 2 \u003d 50 cm 2 ).
That's right, well done! ANDhow to draw rectangle if we only know its area?
What do you need to know for this? (Its length and width).
How do I know the dimensions of a rectangle?
(By the selection method. Knowing that the area is found by multiplying the length by the width, 50 sq. Cm can be obtained by multiplying 5 cm by 10 cm or 25 cm multiplied by 2 cm).
Right. Choose which rectangle is more convenient to draw in a notebook (it is more convenient to draw a rectangle with sides of 5 cm and 10 cm.).
Right. Draw a rectangle like this.
5. Goal setting.
–Guys, tell me, was it easy for you to draw a rectangle in a notebook? (Yes Easy).
Why? (there are cells)
In the last lesson, we learned how to draw a rectangle on unlined paper using a square, and I asked you to draw at homepattern ... Let's check what you got, and one person at the board will draw a rectangle using a square.
(Exhibition of works, checking the student at the blackboard - construction algorithm)
What do you think, is it easy to draw a rectangle on unlined paper, for example, on a scrapbook sheet, if you don't have a square? (difficult)
This means there is a way to build with other tools. Today in the lesson we need a compass and a ruler.
What do you thinklesson topic ? ( Draw a rectangle on unlined paper using a compass and a ruler) (Slide)
Whatthe purpose of the lesson can be put in connection with the topic? (Learn to draw a rectangle on unlined paper using a compass and a ruler) (Slide)
– Where in our life can the ability to construct a rectangle or square be useful on unlined paper?
Tasks:
1) To develop practical skills in constructing geometric shapes, using knowledge about them.
2) Develop spatial imagination.
3) Cultivate accuracy when performing constructions.
The topic has been determined, the goals have been set - on the road for new knowledge!
6.Discovery of new knowledge
For work we need a compass and a ruler.
To use these tools safely, you need to remember
safety regulations:
You cannot bring the compass to your face, there is a needle at the end, you can inject yourself.
You cannot pass the compass with the needle forward, you can prick your friend.
The desktop must be neat.
Maybe someone guessed what to do?
If not, look at the board.
B FROMKM
AD
Figure: Fig. 1 2
What do we do first? (You need to draw a circle).
What is “diameter”? (This is a line segment connecting two points on a circle and passing through its center).
Let's compose an algorithm for constructing a rectangle. (Slide)
Draw a circle.
Draw two diameters in it.
Connect the ends of the diameters with segments. The result is a rectangle.
7 practical work
Take the album sheet.
We draw a circle, the radius of which is 5 cm.
We draw two diameters.
We connect the ends of the diameters.
Let's denote the vertices of the rectangle
How can I check that the result is a rectangle? (You can measure the sides of a figure, the opposite sides should be the same, you can measure the angles using a right angle, the corners should be right).
Check if you have a rectangle.
Was it interesting for you to build?
"Inspiration is needed in geometry no less than in poetry" A.S. Pushkin
(Slide)
Rememberproperties of the diagonals of a square
The diagonals of the square are equal,
they form right angles when crossing,
the intersection point of the diagonals divides them into equal segments.
Where do we start building? (Let's draw a circle).
We found only two vertices of the square, how can we find two more? (We willperpendicular to the diameter, another diameter is obtained ... These lines intersect at right angles like a square. Thus, we found two more vertices of the square).
Let's compose an algorithm for constructing a square. (Slide)
Draw a circle.
Draw one diameter.
Draw a perpendicular line to this diameter.
Connect the points of intersection with the circle with segments. It turned out to be a square.
8. Practical work on the algorithm.
9. Physical education for a minute.
10.Inclusion in the knowledge system .
Choose your level. (Slide)
1. Find the area and perimeter of the rectangle and square.
R etc. \u003d (6 + 8) * 2 \u003d 24 (cm)
S etc \u003d 6 * 8 \u003d 48 (cm 2 )
R sq. \u003d 7 * 4 \u003d 28 (cm)
S sq. \u003d 7 * 7 \u003d 49 (cm 2 )
2. The Ivanov family has a dacha plot measuring 20 meters by 40 meters, and the Sidorov family has 30 meters by 30 meters. Whose fence is longer?
P \u003d (20 + 40) * 2 \u003d 120 (m.)
P \u003d 30 * 4 \u003d 120 (m)
Answer: their fences have the same length, which means they are equal.
3. Look at the plan of the school garden, in which 1 cm represents 10 m. Find the area of \u200b\u200bthis garden in macaws (page 7)(Choosing the best option).
moving the triangle;
measuring the sides of the resulting rectangle;
finding the area in m 2 ;
express in macaws.
S\u003d 60 * 30 \u003d 1800 (m 2 .) \u003d 18 amu.
Were all constructions and calculations easy for you?
- "There is no royal way in geometry" Euclid. (Slide)
Well done! You have done well on this assignment. You have proven that you can rightfully call yourself friends of GEOMETRY.
11. Consolidation of the passed material.
1) Geometry struck me as a very interesting and magical science. I.K. Andronov(Slide)
and) Find equal values.
b) What is the excess?
at) Continue the pattern:
Well done, now you can easily handle No. 33 bldg. 7
Let's check the solution. (Slide)
(6 km 5 m \u003d 6 km 50 dm
2 days 20 h \u003d 68 h
3 t 1 c\u003e 3 t 10 kg
90 cm 2< 9 дм 2 )
2) Solution of the problem.
Solving a difficult math problem can be compared to taking a fortress. N. Ya Vilenkin(Slide)
Read problem number 31. Let's make a short note
How many boys were in the club?
How many girls?
How tall are all the boys?
How tall are all the girls?
What does the problem ask? (The table is filled in the process of work).
Make a plan for solving the problem:
express height in centimeters
find the average height of the boys;
find the average height of girls;
compare
Solve the problem yourself.
11m04cm \u003d 1104cm
12m60cm \u003d 1260cm
1) 1104: 8 \u003d 138 (cm) -the average height of boys
2) 1260: 9 \u003d 140 (cm) -the average height of girls
3) 140-138 \u003d 2 (cm) -more
Answer: On average, boys are 2 cm more tall than girls.
Let's check the solution. Well done, we took another mathematical fortress!Rate your work.
3) Work on computational skills.
Solve 1 example # 34 on page 7.
Let's recall the procedure. What action do we perform first?
After completion - mutual check.
(100 000 - 62 600) : 4 + 3 * 108 = 9 674
37 400
9 350
324
9674
- Rate the work.
12) Summing up the lesson and reflection.
1) -What was the topic of our lesson?
What goals and objectives did you set for yourself?
Have we reached them?
– What tools can you use to draw a rectangle on unlined paper? (Using a compass and a ruler, using a square)
- Let's repeat the algorithm for constructing a rectangle and a square.
-What was left unclear?
2 ) Let's go back to the rectangle that we built at the beginning of the lesson. On it, paint over the part of the assignments that you completed and evaluate your work in the lesson.
GOOD MEN !!!
13) Homework.
Optional: (Slide)
Construct a rectangle and a square on unlined paper, find and compare their areas.
Create a geometric pattern using your new knowledge.
Literature.
MI Moro et al. Textbook "Mathematics, grade 4", M. "Education" 2011.
LISemakina "To help the teacher", M., "Wako", 2011
Class: 4
Lesson presentation
Back forward
Attention! The slide preview is used for informational purposes only and may not represent all the presentation options. If you are interested in this work, please download the full version.
The purpose of the lesson: To teach how to build a rectangle on unlined paper using a square.
1. Educational:
- update previous knowledge about rectangle and square;
- to form practical skills in constructing geometric shapes, using knowledge about them;
- to consolidate the skills of solving word problems for proportional division, comparing named numbers.
2. Developing:
- develop the spatial imagination of students;
- to develop the communication skills of students in the course of pair work, the ability for mutual control and self-control.
3. Educational:
- educate accuracy when performing constructions;
- to awaken in the student a sense of pride in their personal achievements and the successes of their peers.
Lesson type: learning new material.
Lesson form: practical work.
Equipment:
for students:textbook, square, sheet of unlined white paper, simple pencil;
for a teacher: a textbook,computer, multimedia projector, screen.
During the classes
1. Organizational moment.
2. Verbal counting.
– Find errors in calculations on the board.
Correct answers: 100,024; 12,548; 6 504.
3. Checking homework.Checking squares on unlined paper. (Show on the board how to construct a square using a compass and a ruler.)
- What knowledge about the square helped to cope with the construction? (The diagonals of the square are equal, intersect, forming four right angles.)
4. Actualization of students' knowledge about the rectangle.- In the last lesson, you and I learned how to build a rectangle using a compass and a ruler. Remember, please, what kind of geometric figure it is - a rectangle. (A rectangle is a quadrilateral with all corners straight.)
- What else do you know about the rectangle? (Opposite sides are equal. The diagonals are equal.)
- This knowledge will be useful to us today.
5. Demonstration of presentation. Explanation of the new material.SLIDE 1. Announcement of the topic of the lesson: "Constructing a rectangle on unlined paper."
- What tools are needed for practical work? (Square, pencil)
SLIDE 2. Objective: To learn how to draw a rectangle on unlined paper using a square.
SLIDE 3. Objectives: 1. To form practical skills in constructing geometric shapes, using knowledge about them.
2. Develop spatial imagination.
3. To cultivate accuracy when performing constructions.
SLIDE 4. Algorithm for constructing a rectangle using a gon.
SLIDE 5. Drew an arbitrary beam of HELL. One of the sides of the square was applied to the beam so that the vertex of the right angle coincided with the beginning of the beam at point A. Draw a pencil along the second side of the square, beam AB. Received one right angle VAD.
SLIDE 6. One of the sides of the square was applied to the beam AB so that the vertex of the right angle coincided with point B. Draw a pencil along the second side of the square beam BC. Received the second right angle ABC.
SLIDE 7. One of the sides of the square was applied to the BP beam so that the vertex of the right angle coincided with point D. Draw the DS beam along the second side of the square. Received the third right angle ADS.
SLIDE 8. The problematic question is posed to the students - is it a rectangle?
Pupils express their assumptions and suggest ways to solve this problem.
SLIDE 9. Testing student assumptions.
It is necessary to find out whether the angle of the VSD is right. If yes, then the rectangle turned out (since, by definition, a rectangle is a quadrilateral with all corners straight). If not, then the AVSD figure is not a rectangle.
Checking is carried out using a square. One of its sides must be applied to the BC beam so that the vertex of the right angle coincides with point C. Next, we see if the LED beam coincides with the second side of the square. In our case, this happened, that is, we can conclude that the angle of the VSD is a straight line and the quadrilateral AVSD is a rectangle.
Further independent work of students to build a rectangle on unlined paper using a square based on the presentation algorithm assumes returning to slides 4-9 (using a hyperlink).
The teacher at this time controls the building process and provides individual assistance to students.
6. Physical education for the eyes(using SLIDES 10-12 of the presentation)7. Working with the textbook.
- Open the tutorial on page 7. Task number 33. (Work on options. The board has 2 students.)
- What values \u200b\u200bwill we need to remember? (Mass and time.)
Compare named numbers.
| (6 km 5 m \u003d 6 km 50 dm | 2 days 20 h \u003d 68 h |
| 3 t 1 c\u003e 3 t 10 kg | 90 cm 2< 9 дм 2) |
2 students are checking. At the desks - a mutual check.
- Task 34. Calculate the value of the first expression. The blackboard has 1 student.
(100 000 – 62 600) : 4 + 3 108 = 9 674
1 student checks.
- Task 30. A table is prepared on the board for a short note. We fill everything together. What do we call the columns of the table? (Per page / Number of pages / Total)
1 student solves the problem on the blackboard.
1) 90: 6 \u003d 15 (p.) - on one page
2) 75: 15 \u003d 5 (p.)
Answer: 5 pages are required.
1 student checks.
* Additional task - №31.
8. Lesson summary.
- What new have you learned?
- What have you learned?
- What tools can you use to build a rectangle on unlined paper? (Using a compass and a ruler, using a square)
- Where in our life can the ability to construct a rectangle or square be useful on unlined paper?
What was left unclear?
Marking students who are actively working in the lesson.
9. Homework.
1. Construct a square on unlined paper using a square and a ruler.
- What is a square? (A rectangle with all sides equal.)
Use this definition in your homework.
- How do you make a short entry? (In the form of a table.)
- How many days were the jackets made in the atelier? (Two days.)
- What would you call the columns of your table? (Consumption for 1 jacket / number of jackets / total meters)
The concepts of "perpendicular lines", "perpendicular". Draw a right angle on unlined paper (using a compass).
Create symmetrical shapes using a square, ruler and compass.
Construction of symmetrical line segments, shapes using drawing tools on checkered and unlined paper.
Parallelism of straight lines.
Draw parallel lines using a square and a ruler.
Constructing rectangles.
Repetition of the basic properties of opposite sides of a rectangle and a square. Create drawings with a ruler and square on unlined paper.
Measuring time.
Time units. The relationship between units of time. Time measuring devices.
Project "How time was measured in antiquity"
Examples of subtopics: ancient calendar, sundial, water clock, flower clock, measuring instruments in ancient times.
Solving logical problems. Encryption of text.
Logic problems related to measures of length, area, time. Graphic models, diagrams, maps. Modeling from paper supported on a graphic card with instructions.
Project "Encryption of location" (or "Transmission of secret messages")
Examples of subtopics: methods of encrypting texts, devices for encryption, encryption of a location, signs in encryption, the game "Treasure hunt", a decryption competition, creating an encryption device.
Class (34 h)
Decimal number system.
The value of the digit depending on the place in the number entry. Decimal number system: why is it called so? (study)
Numeral Systems Project
Examples of subtopics: decimal number system, binary number system, computers and number system, number systems in different professions.
Coordinate angle.
Familiarity with the coordinate angle, ordinate and abscissa. Introduce the concept of image transfer, the ability to navigate by the coordinates of points on a plane. Creation of a coordinate angle. Reading, writing named coordinate points, designating coordinate ray points using a pair of numbers.
Graphs. Diagrams. Tables. Building charts, graphs, tables using MS Office.
Use of charts, tables, diagrams in reference books and mass media. Collection of information on tables, graphs, diagrams. Types of charts (bar, pie). Building charts, graphs, tables using MS Office.
Draft "Strategy".
Examples of subtopics: games with winning strategies, strategies in games, strategies in sports, strategies in computer games, strategies in life (behavior strategies), combat strategies, strategies in antiquity, strategy in advertising, championship in a computer game in the genre of "Strategy", a collection of games with winning strategies, an album of battle patterns won by correctly chosen strategies, sports team games, commercials and posters.
Polyhedron.
The concept of a "polyhedron" as a figure whose surface consists of polygons. Faces, edges, vertices of a polyhedron.
Rectangular parallelepiped.
Determination of the number of vertices, corners, polyhedron faces. Acquaintance with a rectangular parallelepiped. Surface area of \u200b\u200ba rectangular parallelepiped.
Cube Unfolding a cube.
A cube is a rectangular parallelepiped, all faces of which are squares. We build a scan of a geometric body (parallelepiped and cube) from paper. Surface area of \u200b\u200brectangular parallelepiped and cube.
Wireframe model of a parallelepiped.
Making a wireframe model of a rectangular parallelepiped and a cube from wire. Solution of practical problems (material calculation).
Dice. Dice games.
Making a dice for board games. Collection of games with a cube.
The volume of a rectangular parallelepiped.
The concept of "volume of a geometric body". Cubic centimeter. Making a model of a cubic centimeter. Cubic decimeter. Cubic meter. Two ways to find the area of \u200b\u200ba rectangular parallelepiped.
Grids. Game "Sea Battle", "Tic-Tac-Toe" (including on an endless board)
A new kind of visual relationship between quantities. Plotting coordinates on a ray, on a plane. Organization of games "Sea Battle", "Tic-Tac-Toe" on an endless board.
13. Dividing the segment into 2, 4, 8, ... equal parts using a compass and a ruler.
Practical task: how to divide a segment into 2 (4, 8, ...) equal parts, using only a compass and a ruler (no scale)?
Angle and its magnitude. Protractor. Comparison of angles.
Repetition and generalization of knowledge about the angle as a geometric figure. Angle value (degree measure). Measuring the angle in degrees using a protractor. Different ways to compare angles. Plotting angles of a given value.
Types of angles.
Classification of angles depending on the size of the angle. Acute, straight, obtuse, unfolded angle. Construction and measurement.
Classification of triangles.
Classification of triangles depending on the size of the angles and the length of the sides. An acute-angled, rectangular, obtuse-angled triangle. Versatile, isosceles, equilateral triangle.
Draws a rectangle using a ruler and protractor.
Practical task: how you can build a rectangle with given sides using a protractor and a ruler. Repetition of methods for finding the area and perimeter of a rectangle.
Plan and scale.
Plan. The concept of "scale". Reading the scale, determining the ratio of the length on the plan and the terrain. Plan scale recording. A drawing of the plan of a classroom, one of the rooms of his apartment (optional). Observance of scale.
3. Finish the definitions: "A rectangle is called ...", "A square ...", "An isosceles triangle ...", "Parallelogram ...".
Name at least three educational games that use geometric shapes as play material. State the main goal of each of these games.
5. Give concrete and convincing examples of different types of assignments (at least 5) using geometric material, but aimed at achieving goals related to the study of arithmetic.
6. Give at least three examples of tasks related to dividing polygons into parts.
Indicate equipment that would benefit from providing a lesson in familiarization with corner types.
8. Name the types of practical work of students in the course of which children identify:
a) essential features of the concept of "right angle";
b) the property of the sides of the rectangle.
9. Connect by arrows or write using pairs of the form ( and;and), (and, b) those concepts, in the formation of which it is useful to use the method of their comparison (juxtaposition or opposition):
Create an algorithm for constructing a rectangle with given sides using a compass, a ruler, and a square.
Formulate (in a generalized form) construction tasks that primary school students should confidently perform.
Construct a convex and non-convex heptagon. Are there nonconvex quadrangles? What features of polygon models should vary, and which ones should remain unchanged when forming the concept of "heptagon"?
13. Come up with at least 5 examples of tasks for the recognition of geometric shapes.
Provide three geometric proof problems available to primary school students. When can younger students be offered proof problems? Why?
Ticket number 24
Solving problems using equations
In solving problems using equations, the following must be observed: first, write down the condition of the problem in algebraic language, i.e. so as to get the equation; second, to simplify this equation to such a form in which the unknown quantity will be on one side, and all known quantities - on the opposite side. The ways of this have already been discussed earlier. One of the basic principles of algebraic solutions is that magnitude must be present in the equation. This will allow us to write the conditions as if the problem had already been solved. After that, only decide equation and find the common value of all known quantities. Since these values \u200b\u200bare equal unknownvalue on the other side of the equation, then the value of all known values \u200b\u200bwill mean that the problem is solved.
Problem 1. When asked how much he paid for the watch, the man replied: "If you multiply the price by 4 and add 70 to the result, and subtract 50 from this amount, the remainder will be $ 220." How much did he pay for the watch? To solve this problem, we must first write the problem condition as an algebraic expression, that is, as an equation. Let the price of the watch be xx
This price has been multiplied by 4, so we get 4x4x
70 was added to the product, that is, 4x + 704x + 70
We subtract 50 from this, that is, 4x + 70-504x + 70-50 So, we wrote down the condition of the problem using numbers in algebraic form, but we don't have equations... However, according to the last condition of the problem, all the previous actions eventually led to a result that is equal 220220 So this equation looks like this: 4x + 70-50 \u003d 2204x + 70-50 \u003d 220
After performing operations with the equation, we get that x \u003d 50x \u003d 50.
That is, xx is $ 50, which is the target price for the watch. verify, that we got the correct value of the desired value, we must substitute this value instead of xx in the equation that we wrote down according to the condition of the problem. If, as a result of this substitution, the sides are equal, we have performed the calculation correctly.
The problem equation was 4x + 70-50 \u003d 2204x + 70-50 \u003d 220
Substituting 50 for xx, we get 4-50 + 70-50 \u003d 2204-50 + 70-50 \u003d 220
Hence, 220 \u003d 220 220 \u003d 220.
2) VALUE is a special property of real objects or phenomena, and the peculiarity lies in the fact that this property can be measured, that is, to name the quantity of quantities that express the same property of objects are called quantities one kind or homogeneous quantities... For example, table length and room length are uniform quantities. Quantities - length, area, mass and others have a number of properties. The method of studying the area of \u200b\u200ba geometric figure
The method of working on the area of \u200b\u200bthe figure has a lot in common with working on the length of a segment.
First of all, the area stands out as a property of flat objects among their other properties. Already preschoolers compare objects in terms of area and correctly establish the relationship "more", "less", "equal" if the objects being compared differ sharply from each other or are completely identical. In this case, children use the imposition of objects or compare them by eye, comparing objects according to the occupied place on the table, on the ground, on a sheet of paper, etc. however, when comparing objects in which the shape is different, and the difference in area is not very clearly expressed, children have difficulty. In this case, they replace the comparison by area with comparison by the length or width of the objects, i.e. go over to a linear extent, especially in those cases when in one of the dimensions objects differ greatly from each other.
In the process of studying geometric material in grades I-II, children clarify their ideas about area as a property of flat geometric shapes. The understanding becomes clearer that the figures can be different and the same in area. This is facilitated by exercises for cutting out figures from paper, drawing and coloring them in notebooks, etc. In the process of solving problems with geometric content, students get acquainted with some properties of the area. They make sure that the area does not change when the position of the figure on the plane changes (the figure does not get any larger or smaller). Children repeatedly observe the relationship between the entire figure and its parts (a part is less than a whole), they exercise in composing figures of different shapes from the same given parts (i.e., the construction of equal parts). Students gradually accumulate the idea of \u200b\u200bdividing figures into unequal equal parts, comparing the superimposed parts obtained, comparing the superposed parts obtained. Children acquire all this knowledge and skills in a practical way along the way with the study of the figures themselves.
Acquaintance with the area can be done as follows:
"Look at the pieces attached to the board and tell me which one takes up the most space on the board (square AMKD takes up the most space). In this case, the area of \u200b\u200ba square is said to be larger than the area of \u200b\u200beach triangle and CDMB square. Compare" area of \u200b\u200btriangle ABC and square AMKD (the area of \u200b\u200bthe triangle is less than the area of \u200b\u200bthe square).
These figures are compared by superposition - the triangle occupies only part of the square, which means that its area is really less than the area of \u200b\u200bthe square. Compare by eye the area of \u200b\u200bthe FVS triangle and the area of \u200b\u200bthe DOE triangle (they have the same areas, they occupy the same place on the board, although they are located differently). Check overlay.
Similarly, other figures are compared in area, as well as objects of the environment.
Ticket number 25
LESSON 1. SUBJECT "MATHEMATICS". OBJECT COUNTING
Lesson objectives: to acquaint students with the subject "Mathematics"; to acquaint with the educational set "Mathematics"; to reveal the ability of students to count subjects.
During the classes
I. Organizational moment.
II. Acquaintance with the subject "Mathematics" and the educational set "Mathematics".
The teacher, talking with children, tells them in an accessible form that they are studying the subject "Mathematics", what they will learn, what "discoveries" they will make in mathematics lessons.
Teacher. What do you guys think is the subject "Mathematics" for?
Further, the teacher informs the children that a textbook consisting of two books will help them in mastering mathematics, it was written for first-graders M.I. Moro, S.I. Volkov and S.V. Stepanov, and also two notebooks will be needed in which the students will be able to draw, paint, write, but only in specially designated places.
