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First lesson in the topic "Ordinary fractions."

Tutorial N.Ya.Vilekina "Mathematics 5".

Objectives of the lesson: familiarization of students with the concept of circumference and circle; The formation of the ability to build a circle with a circulation along a given radius and diameter.

Training tasks aimed at achieving:

Personal Development:

• continue to develop the ability to clearly, accurately and competently express their thoughts in oral and written speech,
• develop creativity of thinking, initiative, resourcefulness, activity in solving mathematical problems.

MetaPered Development:

• expand the horizons, instilling the ability to work together (sense of partnership and responsibility for the results of their work);
• continue to develop the ability to understand and use mathematical means of visibility.

Subject development:

• form theoretical and practical idea of \u200b\u200bcircle and circle as geometric figures, their elements;
• continue the development of visual skills (teach the use of a circulation to build a circle of any radius);
• to form the ability to apply the studied concepts to solve a practical task.

Type of lesson: lesson for receiving new knowledge, skills and skills.

Forms of students:

• individual;
• frontal;
• independent work;
• work in pairs;
• test control.

Necessary equipment:

• Projector and screen.
• Presentation "Circle and Circle".
• Individual leaf to each student ( attachment 1).

Structure and course of lesson

	Stage lesson	Slide number	Teacher's activities	Activity student	Formation of Wood (Personal, MetaPered)	Time (in min)
1.	Organizing time	№1,2	welcomes students, configures to work, offers to check the readiness of the workplace, sets problems using the poem decorated in the presentation.	greet teachers check the readiness for the lesson, Express your opinion on the question, comparing the figures: a circle and a circle.	Cognitive (the ability to solve educational problems arising during front work).	2
2	Actualization of knowledge. Formulation of the problem.	№3	announces the objectives of the lesson, records the date and theme of the lesson - "Circle and Circle".	record the date and theme of the lesson in the notebook.	Regulatory (ability to volitional effort)	1
3.	"Opening" by children of new knowledge.	№4	Conducts a frontal survey according to the drawing on the slide. 1. Which of the drawn figures can be called lines?	Answer teacher's questions and write out the answers to individual sheets.	Cognitive (the ability to sensitively read, removing the necessary information; the ability to search and allocate the necessary information)	5
			2. Which of them are broken, what are the curves?	2. №2,4
			3. Divide the curves of the lines on closed and unlocked.	3. Required - 3,6,8, open -1,5,9
			4. In the closed curves of 3,6,8, the points are arranged, is it possible to assert that the distance from point о to the points A, B, C, D in each figure is the same? Measure the distance to these points using the ruler. Write down the answers.	4. Students measure the distance from T.OO to points A, B, C, D. Record the result in individual sheets.
			5. Create figures 6 and 8.	5. Similarity: These are closed lines curves, inside the point is noted, and points A, B, C, D are noted on the lines. Difference: The distance from the point of the point A, B, C, D in Figure 6 - different, in Figure 8 - the same
			6. How do you think why figure 8 is a circle, and figure 6 is not a circle?	6. What is in Figure 8, the distance from point O to points A, B, C, D is the same, and in the figure 6 - different
			7. Name the essential signs of the circle!	7. This is a closed line curve; The distance from the point of about all points on the circle is the same.
			8. Is it possible to call the circles of the figure 5, 7.9?	8. No! Figures 9 and 5 are not closed curves, and figure 7 does not have the center, the distance from which to all points on the circle is the same.
			9. What is the difference between the circles 3 and 8?	9. Distance from point about to dots on the circle!
			10. Mark any other point on the circle 8 and measure the distance from the center of the circle - to this point, make the output!	10. The distance from the center of the circle to any point on the circle is the same!
4		№5,6	Preparation of students to the next stage of the lesson. The riddle of the circula in verses. Safety work with a circulation. With the help of the presentation slides, the structure of the circular and its purpose shows.	Guess the riddle - "Circul" Find all elements on their circulation.	Communicative (ability to enter into dialogue)	2
5.	Studying a new material and its primary consolidation.	№7,8	The teacher offers students to build a circle of an arbitrary radius with him.	Perform the task of the teacher.	Cognitive(The ability to make a model and convert it if necessary). Communicative (ability to hear and listen) Regulatory (ability to analyze the course and method of action)	15
		№9	It suggests to remember what familiar items have a circle form, and what is the form of a circle?	List subjects
		№10, 11	Introduces new concepts "Circle Center", "Circle Radius"
		№12	It offers students, without disturbing patterns, to build radii in the recent circles in the research sheet. Then turns on the slide properly built radii.	Build radii and explain what pattern they revealed. Check the correctness.
		№13	Offers learners to make an independent study: Build a circle with a radius of 3 cm and mark its center. Connect two circumference points, so that this segment is passing through the center of the circle. It gives the definition of "circle diameter".	Perform a task in individual sheets, make a conclusion, then check and correct your errors using the presentation slides.
		№14	Write an expression by which you can find the length of this segment. Then asks for students to check the study performed using the presentation slide.	Students make appropriate records in the notebook.
		№15	Introduces the concept of "chord of the circle."	Students make appropriate records in the notebook.
		№16	Gives a task to students: list all the diameters, chords and circle radii.
		№17	Introduction of a new concept of "circle arc".	Students make appropriate records in the notebook.
		№18	Gives a task: Name all arc circumference.	Orally fulfill the task of the teacher.
		№19	It proposes to perform a practical task: using a circulation, build two circles in the notebook with the same radius, equal to 3 cm, flooding the inner area of \u200b\u200bone circle. Specifies the question: what can be explained that the first figure is called a circle, not a circle?	Perform the construction of figures in an individual sheet, and called the resulting figures. Answer the question: The first figure is painted, i.e. It belongs to all the points inside this figure, and it is called a circle.
		№20	Task: Name points lying in the internal (external) area.	Orally fulfill the task of the teacher.
6.	Research work in pairs.	№21	Gives a task and provides advisory assistance to students who have difficulty.	Perform work in pairs.	Communicative (skill to cooperate with other people in finding the necessary information)	10
7.	Test work with your interconnection.	№22	Offers students to check their knowledge using the test.	Students perform a test, followed by a multiple control.		2
8.	Outcome.	№23	Sums up the lesson. It proposes to describe his impressions of today's lesson and draw a smiley smile, depending on the mood of students. Specifies the task to the house:	They describe in individual sheets impressions of the conducted research activities, about their impressions and their emotional state. Record your homework in the diary.		3

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Signatures for slides:

Circle Presentation Prepared: Svetlana Svetlana Igor Teacher Mathematics MBOU SSh№2 лесковоо

Objectives and objectives: to systematize theoretical material on the topic "Circle". Improve the skills to solve problems. Prepare students to test. Prepare students for the successful solution of the "Geometry" module during the surrender of OGE.

the properties of a tangent C-tangency A-point of touch with OA O and C A B M A V

The theorem on the tangent and sequential C m and in the square of the tangential length is equal to the product of the sequential part of its external part. D C A b O The product of one section on its external part is equal to the product of another section on its external part M

Central and inscribed central corners inscribed in a o d a c b o

The inscribed angle is either equal to half of the corresponding central angle, or (2) complements half of this angle to 180 degrees. 12

Properties of inscribed angles about a in d c b k a c

Property intersecting chord with in to and d

Inserted circle, each point of bisector of an uniform angle is equidistant from its sides: each point lying inside the angle and equidistant from the side of the angle lies on its bisector about the intersection of the biscoming property of bisector A B with D The property of the described quadrilateral AB + CD \u003d BC + AD The sums of opposite sides are equal.

The described circle each point of the middle perpendicular to the segment is equal to the ends of this segment back: each point equidistant from the ends of the segment lies on the middle perpendicular to it to cross the middle perpendicular properties of the middle perpendicular A D C B property of the inscribed quadrilateer

Oral tasks on ready-made drawings 160 Answer: 80? Answer: 45 V and C C A D A B C M to P 5 6 3 Reply: 28?

And with in d 7 8 p \u003d? Answer: 30 m to T about 70 °? Answer: 20 ° about

Must be able to: apply when solving the definition problems, the properties of figures, various theorems. Be able to build a logical chain of reasoning. Apply the theory in the new situation.

120 ° 60 ° 120 ° 240 ° 115 ° 65 ° 230 ° 40 ° 140 ° 140 ° AC CB AB R KTP PK PT KPT - - 4 3 5 2, 5 30 ° 4 8 60 ° - - Replies:

2 Group 1 2 3 4 B and B A 1 Group 1 2 3 4 A V B G 3 Group 1 2 3 4 V ABB B

On the topic: Methodical development, presentations and abstracts

Mathematics lesson in the 6th grade on the topic "Circle. Circle. Circle Length" It is better to conduct in the form of practical work ....

The purpose of the lesson: repeat the concept of circle and circle; calculation of the value of the PI number; Enter the concept of the circumference length and formulas for calculating the circumference length ....

The first lesson on the topic of the circumference of the 6th grade. A practical work is carried out, during which the guys calculate the value of the PI number. There is a familiarity with the number of pi ....

Rodionova G. M. Numerical circle on the coordinate plane // Algebra and the beginning of the analysis of the 10th grade //. Presentation contains material: Number circle on the coordinate plane, the main ...

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Signatures for slides:

Name the figures to e t with in a x

For many parts divide the Figure Plane:

Circle and circle Circle - a closed line circle - a plane that lies inside the circle, together with a circle

Circle Circle divides the plane into two parts!

Building O 1) We celebrate the point of the center of the circle. 2) We set the radius of the circle with the help of a circulation and a ruler. 3) We set the feet of the circulation to the point O 4) we carry out a circle.

All points of the circle are removed from its center. O - Center of the circle and circle OA \u003d OS \u003d OE - Radius - R AB - Diameter - D AB \u003d O + s d \u003d 2R, R \u003d D: 2 O C A E in Radius - Cut connecting the center of the circle with a point lying on it. All circle radii are equal! The diameter is a segment connecting two circumference points, and passing through its center.

The diameter divides the circle for two semicircles, about with and in about with and in a circle on two semicircles.

Arc of the circle of the sv - arc of sv, the ends of the arc - the points of the C and V. Ace - the arc of the AU, the ends of the arc - points A and C. Av, be about with a e in

Examples of the circle and circle in life

Rooms for work: for fixing the material: No. 850 (orally) No. 851 No. 853 No. 855 for repetition: No. 871 (1) Independent work: No. 872 (1)

Homework: P.22, № 874, № 876, № 878 (A, G, E)

№ 853 O A in R \u003d 3 cm OA \u003d, OA R

No. 855 C d \u003d 3 cm, sv \u003d 3cm D a \u003d 4cm, in d \u003d 4cm b a

On the topic: Methodical development, presentations and abstracts

The image of a circle and its role in the story of V.Nabokova "Circle"

"9 CHA circles on Dante" Guide to the circles of hell from the "Divine Comedy" Dante Aligiery.

"Divine Comedy" (ITAL. La ComMedia, later La Divina Commedia) - Poem written by Dante Aligierey in the period from 1307 to 1321 and giving the widest synthesis of medieval cult ...

Test Find: sector, arc, radius, diameter, chord, segment

Three points A, B and C, which are not lying on one straight line (through the tops of ABC), it is possible to carry out a circle if there is such a fourth point. O, which is equally removed from points A, B and C. We prove that such a point exists and moreover only one. Any point, equally removed from the points A and B, should lie on the middle perpendicular Mn to the segment of AB, just as every point, the same remote from the points in and C should lie on the middle perpendicular of the PQ spent to the Sun side. It means that if there is a point equally removed from three points A, B and C, then it must lie on Mn, and on PQ, which is only possible when it coincides with the intersection point of these two straight lines. Straight Mn and PQ always intersect, as they are perpendicular to the intersecting direct AV and Sun. The point of their intersection and will be the point equally remote from A, from the C and from C, which means that we will take this point for the center, and for the radius we take the distance of OA (or OB, or OC), the circle will pass through points A, in and C. Since straight Mn and PQ can only cross at one point, the center of the circle can only be one and the length of its radius can be only one; So, the desired circumference is the only one.

Stronizing the drawing in the diameter of AB so that its left part fell to the right. Then the left semi-rareness is combined with the right semi-rapid and perpendicular COP will go on KD. It follows from this that the point C, which represents the crossing of a semicircle with the COP, will fall on D; Therefore, SC \u003d KD; BC \u003d BD, AC \u003d AD. BC \u003d BD AC \u003d AD

The properties of the diameter of the circle 1. The diameter spent through the middle of the chord is perpendicular to this chord and divides the arc that is consistent with it, and pamp. 2. The diameter spent through the middle of the arc, perpendicular to the chord, tightening this arc, and divides it in half.

1. Count the circle with the center of O. Av \u003d CD, P - middle chords Av, Q - middle CD. 2.Read ΔOar and ΔOCQ (rectangular): OA \u003d OS - radii, PA \u003d CQ - half of equal chore 3.ΔAr \u003d ΔOCQ (on hypotenuze and cathetu). From the equality of the triangles OP \u003d OQ (equal cathets), i.e. chords are equally removed from the center

Cases of the Mutual Layout of Direct and Circle D RD\u003e R RD\u003e R "\u003e RD\u003e R"\u003e RD\u003e R "Title \u003d" (! Lang: Cases of the Mutual Layout of Direct and Circle D RD\u003e R"> title="Cases of the Mutual Layout of Direct and Circle D RD\u003e R"> !}

D.

D\u003e R If the distance from the center of the circumference to a straight line is more than the circle radius, then the direct and circle do not have common points. O D\u003e R R R If the distance from the center of the circle to a straight line is more than the circle radius, then the direct and circle do not have common points. O D\u003e RR "\u003e R If the distance from the center of the circumference to a straight line is more than the circle radius, then the direct and circle do not have common points. O D\u003e RR"\u003e R If the distance from the center of the circle to a straight line is more than the circumference of the circle, then direct and circumference is not have common points. O d\u003e r "title \u003d" (! Lang: d\u003e r If the distance from the center of the circumference to a straight line is more than the circle radius, then the direct and circle do not have common points. O D\u003e R R"> title="d\u003e R If the distance from the center of the circumference to a straight line is more than the circle radius, then the direct and circle do not have common points. O D\u003e R R"> !}

Property tangent. Let the straight R relate to the circle at the point A, that is, and their only common point. Proof "From Nasty": 1. Double, that p is not perpendicular to the radius of OA. We will carry out perpendicular s on r. 2. We post on p segment Sun \u003d VA. 3. ov \u003d ABS (for two categories). Therefore, OS \u003d OA. 4. C lies on the circle. Consequently, P and the circle have two common points, which is impossible. So, R OA, as required

Take any point and the circumference f and carry out the radius of OA. Then we will spend direct p perpendicular to the OA radius. Any point in a straight line, different from point A, is removed from about more than a radius, since the inclined s is the perpendicular of OA. Therefore, the point B does not lie on F. So the point is the only total point P and F, i.e. R is facing F at point A.

Various cases of the relative position of two circles. D\u003e R + R 1D\u003e R + R 1 d \u003d R + R 1d \u003d R + R 1 D R + R 1D\u003e R + R 1 d \u003d R + R 1d \u003d R + R 1 D "\u003e R + R 1D\u003e R + R 1 D \u003d R + R 1d \u003d R + R 1 D"\u003e R + R 1D \u003e R + R 1 d \u003d R + R 1d \u003d R + R 1 D "title \u003d" (! Lang: various cases of the relative position of two circles. D\u003e R + R 1D\u003e R + R 1 d \u003d R + R 1d \u003d R + R 1 D"> title="Various cases of the relative position of two circles. D\u003e R + R 1D\u003e R + R 1 d \u003d R + R 1d \u003d R + R 1 D"> !}

1. Lightness lie one outside the other, not touching in this case, obviously, D\u003e R + R 1 R and R 1 - the radii of circles D - the distance between the centers of the circles R + R 1 R and R 1 - Radius of circles D - Distance between centers of circles "\u003e R + R 1 R and R 1 - Radius of circles D - Distance between centers"\u003e R + R 1 R and R 1 - Radius circles D - The distance between the cities of the circles "title \u003d" (! Lang: 1. Lightness is one outside the other, without touching in this case, obviously, D\u003e R + R 1 R and R 1 - Radius of circles D - distance between centers of circles"> title="1. Lightness lie one outside the other, not touching in this case, obviously, D\u003e R + R 1 R and R 1 - the radii of circles D - the distance between the centers of the circles"> !}

3. Circles intersect then D

5. One circle lies inside another, without touching, then, obviously, D

R + R 1, then the circumference is located one outside the other, without touching. 2. If D \u003d R + R 1, then the circle relate to the outside. 3. If D R - R 1, the circles intersect. 4. If D \u003d R - R 1, then the circle concern from the inside. 5. "Title \u003d" (! Lang: Reverse Proposition 1. If D\u003e R + R 1, then the circles are located one outside the other, not touching. 2. If D \u003d R + R 1, then the circle concern from the outside. 3. If D R - R 1, then the circles intersect. 4. If D \u003d R - R 1, then the circumference relate to from the inside. 5." class="link_thumb"> 59 !} Reverse proposals 1. If D\u003e R + R 1, then the circles are located one outside the other, without touching. 2. If D \u003d R + R 1, then the circle relate to the outside. 3. If D R - R 1, the circles intersect. 4. If D \u003d R - R 1, then the circle concern from the inside. 5. If D R + R 1, then the circle is located one outside the other, without touching. 2. If D \u003d R + R 1, then the circle relate to the outside. 3. If D R - R 1, the circles intersect. 4. If D \u003d R - R 1, then the circle concern from the inside. 5. "\u003e R + R 1, then the circles are located one outside the other, without touching. 2. If D \u003d R + R 1, then the circumference relate to the outside. 3. If D R - R 1, then the circles intersect. 4. If D \u003d R 1, then the circumferences are from the inside. 5. If D R + R 1, then the circle is located one outside the other, not touching. 2. If D \u003d R + R 1, then the circle concern from the outside. 3. If D R - R 1, then the circumference intersect. 4. If D \u003d R 1, then the circles are concerned from the inside. 5. " title \u003d "(! lang: Reverse Proposition 1. If D\u003e R + R 1, then the circles are located one outside the other, not touching. 2. If D \u003d R + R 1, then the circle relate to the outside. 3. If D R - R 1, then the circumference interspersed. 4. If D \u003d R 1, then the circle concern from the inside. 5."> title="Reverse proposals 1. If D\u003e R + R 1, then the circles are located one outside the other, without touching. 2. If D \u003d R + R 1, then the circle relate to the outside. 3. If D R - R 1, the circles intersect. 4. If D \u003d R - R 1, then the circle concern from the inside. five.">!}

Danar: Circle with the center O, ABC - inscribed to prove: ABC \u003d ½ AC Proof: Consider the case when the side of the aircraft passes through the center of 1.SU less semi-rapid, AOC \u003d AU (central) 2. Consider ΔAvo, JSC \u003d OS ( radius). ΔAvo iscerned 1 \u003d 2, AOC is an ΔAo, AOC \u003d 2 1, therefore ABC \u003d ½ AC 1 2

Danar: Circle with the center O, ABC - inscribed to prove: ABC \u003d ½ AC Proof: Consider the case when the center is lies inside the inscribed angle. 1. Additional construction: BD diameter 2. Bump into the ABC divides by two angle 3. The radiance crosses the arc of the AC at the point D 4. AC \u003d AD + DC, therefore ABD \u003d ½ AD and DBC \u003d ½ DC or ABD + DBC \u003d ½ AD + ½ DC or ABC \u003d ½ AS

Danar: Circle with the center O, ABC - inscribed to prove: ABC \u003d ½ AC Proof: Consider the case when the center is lied outside the inscribed angle. 1. Additional construction: BD diameter 2. The beam in does not divide ABC by two angle 3. Bescape does not cross the AC arc at point D 4. AC \u003d AD - CD, therefore ABD \u003d ½ AD and DBC \u003d ½ DC or ABD - DBC \u003d ½ AD - ½ DS or ABC \u003d ½ AS

72

Evidence. 1. Quality an arbitrary triangle ABC. Denote by the letter on the point of intersection of the middle perpendicular to its parties and carry out the segments of A, O B and OS. 2. Since the point is equidistant from the vertices of the ABC triangle, then OA \u003d OS \u003d OS, so the circle with the center of radius o and passes through all three vertices of the triangle and, it means described near the ABC triangle. Evidence. 1. Consider an arbitrary triangle ABC and denote the letter about the intersection point of its bisector. 2. We will spend from the point of perpendicular ok. OL and Ohm, respectively, to the sides of the AV, Sun and Ca. 3. Since the point is equidistant from the side of the ABC triangle, then OK \u003d OL \u003d OM. Therefore, the circle with the center of radius ok passes through points K, L and M. 4. The sides of the ABC triangle relate to this circumference at points K, L, M, since they are perpendicular to OK, OL radius. So, a circle with the center of radius OK is inscribed in the ABC triangle.