This development is intended for 8th grade students. The use of an electronic presentation contributes to the development of visual-figurative thinking and the formation of techniques and skills for working with drawing tools

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T.S. Frolova

Dividing a circle into equal parts

(8th grade)

Goals:

Educational: To give knowledge on the topic “Dividing a circle into equal parts. Show students the need to use geometric constructions when making drawings of parts; create conditions for the formation of skills

Educational : expand the horizons of students and increase cognitive interest in their subject; to cultivate accuracy, accuracy, attentiveness in graphic constructions.

Educational : formation of methods and skills of work, consolidation of acquired knowledge

Methods: graphic constructions, explanations with demonstrations, graphic constructions, non-standard learning situations for the application of knowledge.

Equipment for students: textbook, notebook, drawing tools.

Lesson plan: 1. Organizational part.

3. Explanation of new material.

4. Consolidation of what has been learned.

5. Summing up.

6. Homework

During the classes:

1. Organizational moment.

Checking the readiness of the class and students for the lesson (notebooks, drawing tools should be ready for the lesson)

2. Goal setting. Student motivation.

Students are encouraged to analyze the topic of this lesson, determine the purpose of the lesson.

The teacher motivates students to study this topic, gain knowledge and practice the acquired knowledge, skills and abilities in the future - the professional significance of knowledge on the topic.

Formulate the topic of this lesson.

Analyze and set the goal of the lesson.

The teacher explains new material using a presentation.

The construction of regular polygons is inextricably linked with the division of a circle. They are found in the most ancient ornaments of all peoples. People already appreciated their beauty back then. In addition, they saw these figures in nature. For example, the pentagon is found in the outlines of minerals, flowers, fruits, in the form of some marine animals, the hexagon is visible in honeycombs, etc. In the arts and crafts, designers and jewelers successfully used the division of the circle, creating beautiful works: orders, medals, coins, jewelry.

Techniques for dividing a circle into equal parts have been used by man since time immemorial. For example, the transformation of a wheel from a solid disk to a spoked rim has made it necessary for man to distribute the spokes evenly in the wheel. When drawing such a wheel, people looked for exact ways with the help of drawing tools.

To make drawings of parts, you must be able to divide the circle into the required number of equal parts ( slides 4-12).

Consolidation of the studied:

To consolidate the material, students are invited to independently perform one of the variants of the ornament, using the rules for dividing the circle into equal parts.(slide 13)

Summarizing.

5. Methodical materials / /http://www.pedagog.by/cherchur.html

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Slides captions:

Dividing a circle into equal parts Drawing teacher Frolova Tamara Serafimovna

Techniques for dividing a circle into equal parts have been used by man since time immemorial. For example, the transformation of a wheel from a solid disk to a spoked rim has made it necessary for man to distribute the spokes evenly in the wheel. When drawing such a wheel, people looked for exact ways with the help of drawing tools.

The construction of regular polygons is inextricably linked with the division of a circle. They are found in the most ancient ornaments of all peoples. People already appreciated their beauty back then. In addition, they saw these figures in nature. For example, the pentagon is found in the outlines of minerals, flowers, fruits, in the form of some marine animals, the hexagon is visible in honeycombs, etc. Polygons around us

Polygons around us

Dividing a circle into four equal parts Dash-dotted center lines drawn perpendicular to one another divide the circle into four equal parts. Consistently connecting their ends, we get a regular quadrilateral

Dividing the circle into eight equal parts Using a compass, arcs equal to the fourth part of the circle are divided in half. To do this, from two points limiting a quarter of the arc, as from the centers of the radii of the circle, notches are made outside it. The resulting points are connected to the center of the circles and at their intersection with the line of the circle, points are obtained that divide the quarter sections in half, that is, they receive eight equal sections of the circle. To divide the circle into eight equal parts, you need to draw two pairs of diameters, or by orienting an equilateral triangle, divide the fourth part of the circle in half.

Dividing the circle into three equal parts From point A, draw an arc BC equal to the radius of the circle AO. Connect points B and C with a chord. And points B and C with point D.

Dividing a circle into six equal parts To divide a circle into six equal parts, from points 1 and 4 of the intersection of the center line with the circle, make two notches on the circle with a radius R equal to the radius of the circle. Connecting the obtained points with line segments, we get a regular hexagon

Dividing a circle into twelve equal parts To divide a circle into twelve equal parts, it is necessary to divide the circle into four parts with mutually perpendicular diameters. Having taken the points of intersection of the diameters with the circle A, B, C, D as centers, four arcs are drawn with the radius value until they intersect with the circle. The resulting points 1, 2, 3, 4, 5, 6, 7, 8 and points A, B, C, D divide the circle into twelve equal parts

Dividing the circle into five equal parts From point A we draw an arc with the same radius as the radius of the circle before it intersects with the circle - we get point B. Dropping the perpendicular from this point - we get point C. From point C - the middle of the radius of the circle, as from the center, with an arc of radius CD we make a notch on the diameter, we get point E. The segment DE is equal to the length of the side of the inscribed regular pentagon. Having made notches on the circle with a radius DE, we obtain the points of dividing the circle into five equal parts

Dividing a circle into ten equal parts By dividing a circle into five equal parts, you can easily divide the circle into 10 equal parts. Drawing straight lines from the resulting points through the center of the circle to opposite sides of the circle - we get 5 more points

Dividing the circle into seven equal parts Connecting points B and C with a chord and taking its half GC, one obtains the side length of a regular heptagon.

Another way of dividing a circle of radius R into 7 equal parts: From the point of intersection of the center line with the circle (for example, from point A) describe how from the center an additional arc with the same radius R - get point B. Lowering the perpendicular from point B - we get point C. The segment BC is equal to the length of the side of the inscribed regular heptagon

Perform one of the ornament options using the rules for dividing the circle into equal parts. Come up with your own ornament that will contain regular polygons.

DEVELOPMENT OF A MATHEMATICS LESSON IN THE 4th CLASS OF THE MAOU SECONDARY EDUCATIONAL SCHOOL No. 111 FOR CHILDREN OF THE 8TH TYPE

OU name: MAOU "Secondary School No. 111"

OS address: Perm Territory, city of Perm, Lepishinskoy st., 43

Topic. Division into 8 equal parts.

Goals. Improve students' computing skills. Strengthen the ability to divide into 8 equal parts. Develop attention, imagination. Cultivate self-esteem, self-control, mutual control.

Lesson form: lesson - game "In the winter forest".

Equipment: painting (winter girl), pictures (winter forest, forest animals), cards (a minute of reading, individual tasks, reflection), drawing (snowflake), tablet (geometric task).

During the classes.

1. Organizational moment.

The math lesson begins. As usual, we will start it with a minute of reading. Outside the window, then rain, then snow, then frost, then thaw. These are the whims of winter. Winter this year is unusual, people have not seen such winter quirks for 50 years. But in our lesson, a real winter-winter will reign. (The painting “Winter Girl” opens).

2. A minute of reading.

Hey snowflakes, hurry up!

Whirlwind with snow

And send a sheet

To every student. (Students receive cards).

Read, memorize, repeat

And we will go to the world of mathematics.

Tasks on cards.

1) Numbers when multiplied are called as follows: 1 multiplier,

2 multiplier, product.

2) Numbers when dividing are called like this: dividend, divisor,

3) Numbers in addition are called as follows: 1 term, 2 term,

4) Numbers in subtraction are called as follows: reduced, subtracted, difference.

5) There are 100 centimeters in one meter.

6) To reduce the number by several times, you need to divide.

7) To increase the number several times, you need to multiply.

8) There are 10 millimeters in one centimeter.

3. Oral account.

Close your eyes and imagine that you are in a winter forest.

What did you see there? Who can you meet in the forest in winter?

(The image of the winter forest opens, the closed pictures are forest animals).

Here is a snow-covered forest.

It is covered with snow, there are many miracles in it.

If you solve my problems,

You will see all miracles.

48 chatty magpies

Came to the crow for a lesson.

They were divided into 8 teams.

How many did one team get?

24 kilos of meat

Wolf for 8 meals in store.

How much does he eat for lunch

Do you count or not?

32 kilograms of seeds

8 mice were dragged into the closet.

How many kilograms did you drag alone

Such tasty grain?

The squirrel had 40 nuts,

I ate 8 pieces a day with success.

How many days did she eat them

Until the closet is empty.

On a tall old spruce

16 sparrows were sitting.

8 branches they occupied,

How long did they sit on each?

As you solve problems, pictures open.

4. Work in notebooks.

Write down the number, great work.

What numbers do you see in the notebook? 2011

What do they mean? coming year.

In the Japanese calendar, each year is associated with the name of an animal. What animal is associated with this year? (rabbit)

And what is the name of his forest relative? (hare)

Write a problem using a picture and a short note.

A short note and a picture of a wolf appear on the board.

Wolf -40 kg

Z. -? 8 times less

What animal of the forest is written on the second line? Why do you think so? Make up a question so that the problem is solved in two steps.

Collectively compose the text of the problem and write down the solution

On the desk.

40:8=5 (kg) hare weighs.

40+5=45 (kg) weigh a wolf and a hare.

Group 1 students decide on their own.

Each student writes down the answer to the problem on their own.

5. Physical education minute.

a) For the eyes.

Stretch your right hand forward.

Snowflake fell on hand

The snowflake immediately sparkled.

I'll look at the snowflake

I'll take a look at the board.

Children look at the snowflake on their hand, look at the big snowflake on the board. Count up to 10.

b) Sitting exercises, in pairs.

From the snowflakes our hands became cold, let's warm them up.

The game "Claps".

6. Working with the book. Independent work.

I hear footsteps creaking in the snow,

Is it not the gait - these are the girlfriends of the blizzard?

She closed the task on the board,

Guess all of his numbers.

Name it quickly

What is colored,

Painted in bright color?

On the board on a large snowflake, the circle is highlighted in blue in red, the arc in green, the radius in black, and the diameter in yellow. When the children name them, the snowflake is removed, and under it is the task: p. 126, No. 17 (2.3 st.).

All students solve the examples on their own.

Group 3 students use an assistant card (multiplication table).

7. Geometric task.

Snow-covered trees, bushes,

But consider the tasks of winter.

The task is opened partially covered with tinsel.

Draw a line 4cm x 5mm long.

Turn it into a rectangle.

Pick up a pencil

Draw you now

Carefully, in order

Quickly everything in your notebook.

8. Summary, grades, homework. Examples in two actions on cards (multiplication and division by 8).

9. Minutes of reflection.

There are charts on the tables.

to solve a problem

solve examples

draw a line.

I need ... (to practice solving problems, repeat the table, draw segments more precisely).

A circle is a closed curved line, each point of which is located at the same distance from one point O, called the center.

Straight lines connecting any point of the circle with its center are called radii R.

A line AB connecting two points of a circle and passing through its center O is called diameter D.

The parts of the circles are called arcs.

A line CD joining two points on a circle is called chord.

A line MN that has only one point in common with a circle is called tangent.

The part of a circle bounded by a chord CD and an arc is called segment.

The part of a circle bounded by two radii and an arc is called sector.

Two mutually perpendicular horizontal and vertical lines intersecting at the center of a circle are called circle axes.

The angle formed by two radii of KOA is called central corner.

Two mutually perpendicular radius make an angle of 90 0 and limit 1/4 of the circle.

Division of a circle into parts

We draw a circle with horizontal and vertical axes that divide it into 4 equal parts. Drawn with a compass or square at 45 0, two mutually perpendicular lines divide the circle into 8 equal parts.

Division of a circle into 3 and 6 equal parts (multiples of 3 by three)

To divide the circle into 3, 6 and a multiple of them, we draw a circle of a given radius and the corresponding axes. The division can be started from the point of intersection of the horizontal or vertical axis with the circle. The specified radius of the circle is successively postponed 6 times. Then the obtained points on the circle are successively connected by straight lines and form a regular inscribed hexagon. Connecting points through one gives an equilateral triangle, and dividing the circle into three equal parts.

The construction of a regular pentagon is performed as follows. We draw two mutually perpendicular axes of the circle equal to the diameter of the circle. Divide the right half of the horizontal diameter in half using the arc R1. From the obtained point "a" in the middle of this segment with radius R2, we draw an arc of a circle until it intersects with the horizontal diameter at point "b". Radius R3 from the point "1" draw an arc of a circle to the intersection with a given circle (p. 5) and get the side of a regular pentagon. The "b-O" distance gives the side of a regular decagon.

Dividing a circle into N-th number of identical parts (building a regular polygon with N sides)

It is performed as follows. We draw horizontal and vertical mutually perpendicular axes of the circle. From the top point "1" of the circle we draw a straight line at an arbitrary angle to the vertical axis. On it we set aside equal segments of arbitrary length, the number of which is equal to the number of parts into which we divide the given circle, for example 9. We connect the end of the last segment with the lower point of the vertical diameter. We draw lines parallel to the obtained one from the ends of the segments set aside to the intersection with the vertical diameter, thus dividing the vertical diameter of the given circle into a given number of parts. With a radius equal to the diameter of the circle, from the lower point of the vertical axis we draw an arc MN until it intersects with the continuation of the horizontal axis of the circle. From points M and N we draw rays through even (or odd) division points of the vertical diameter until they intersect with the circle. The resulting segments of the circle will be the desired ones, because points 1, 2, …. 9 divide the circle into 9 (N) equal parts.

To find the center of an arc of a circle, you need to perform the following constructions: on this arc, mark four arbitrary points A, B, C, D and connect them in pairs with chords AB and CD. We divide each of the chords in half with the help of a compass, thus obtaining a perpendicular passing through the middle of the corresponding chord. The mutual intersection of these perpendiculars gives the center of the given arc and the circle corresponding to it.

Today in the post I post several pictures of ships and diagrams for them for embroidery with isothread (pictures are clickable).

Initially, the second sailboat was made on carnations. And since the carnation has a certain thickness, it turns out that two threads depart from each. Plus, layering one sail on the second. As a result, a certain effect of splitting the image appears in the eyes. If you embroider the ship on cardboard, I think it will look more attractive.
The second and third boats are somewhat easier to embroider than the first. Each of the sails has a central point (on the underside of the sail) from which rays extend to points along the perimeter of the sail.
Joke:
- Do you have threads?
- There is.
- And the harsh ones?
- It's just a nightmare! I'm afraid to come!

In December, in a couple of weeks, the blog turns a year old. It's scary to think - it's been a whole year already! When I started blogging, I had a good stock if I had a dozen topics for future posts, and there were no written posts in drafts at all, which, from the point of view of serious blogging, was no good. It turned out, I acted according to the principle - First we get involved, and then we'll see. And here's what happened. To date, my readership is represented by 58 countries. But I would really like to know more about who comes to my blog and for what purpose, how the blog materials are used. This is very important so that I can evaluate the usefulness of filling pages and, next year, at a new round of development, take into account the wishes of a respected audience (in zagnulJ). I developed a questionnaire consisting of 10 questions with a multi-choice, i. You must select one of the suggested answers. If there is something that you would like to express, but it was not included in the list of questions, write to me by e-mail or in the comments to this post ...

When performing graphic work, you have to solve many construction tasks. The most common tasks in this case are the division of line segments, angles and circles into equal parts, the construction of various conjugations.

Dividing a circle into equal parts using a compass

Using the radius, it is easy to divide the circle into 3, 5, 6, 7, 8, 12 equal sections.

Division of a circle into four equal parts.

Dash-dotted center lines drawn perpendicular to one another divide the circle into four equal parts. Consistently connecting their ends, we get a regular quadrilateral(Fig. 1) .

Fig.1 Division of a circle into 4 equal parts.

Division of a circle into eight equal parts.

To divide a circle into eight equal parts, arcs equal to the fourth part of the circle are divided in half. To do this, from two points limiting a quarter of the arc, as from the centers of the radii of the circle, notches are made outside it. The resulting points are connected to the center of the circles and at their intersection with the line of the circle, points are obtained that divide the quarter sections in half, i.e., eight equal sections of the circle are obtained (Fig. 2 ).

Fig.2. Division of a circle into 8 equal parts.

Division of a circle into sixteen equal parts.

Dividing an arc equal to 1/8 into two equal parts with a compass, we will put serifs on the circle. Connecting all serifs with straight line segments, we get a regular hexagon.

Fig.3. Division of a circle into 16 equal parts.

Division of a circle into three equal parts.

To divide a circle of radius R into 3 equal parts, from the point of intersection of the center line with the circle (for example, from point A), an additional arc of radius R is described as from the center. Points 2 and 3 are obtained. Points 1, 2, 3 divide the circle into three equal parts.

Rice. four. Division of a circle into 3 equal parts.

Division of a circle into six equal parts. The side of a regular hexagon inscribed in a circle is equal to the radius of the circle (Fig. 5.).

To divide a circle into six equal parts, it is necessary from points 1 and 4 intersection of the center line with the circle, make two serifs on the circle with a radius R equal to the radius of the circle. Connecting the obtained points with line segments, we get a regular hexagon.

Rice. 5. Dividing the circle into 6 equal parts

Division of a circle into twelve equal parts.

To divide a circle into twelve equal parts, it is necessary to divide the circle into four parts with mutually perpendicular diameters. Taking the points of intersection of the diameters with the circle BUT , AT, FROM, D beyond the centers, four arcs are drawn by the radius to the intersection with the circle. Received points 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 and points BUT , AT, FROM, D divide the circle into twelve equal parts (Fig. 6).

Rice. 6. Dividing the circle into 12 equal parts

Dividing a circle into five equal parts

From a point BUT draw an arc with the same radius as the radius of the circle before it intersects with the circle - we get a point AT. Lowering the perpendicular from this point - we get the point FROM.From point FROM- the midpoint of the radius of the circle, as from the center, by an arc of radius CD make a notch on the diameter, get a point E. Line segment DE equal to the length of the side of the inscribed regular pentagon. By making a radius DE serifs on the circle, we get the points of dividing the circle into five equal parts.

Rice. 7. Dividing the circle into 5 equal parts

Dividing a circle into ten equal parts

By dividing the circle into five equal parts, you can easily divide the circle into 10 equal parts. Having drawn straight lines from the resulting points through the center of the circle to the opposite sides of the circle, we get 5 more points.

Rice. 8. Dividing the circle into 10 equal parts

Dividing a circle into seven equal parts

To divide a circle of radius R into 7 equal parts, from the point of intersection of the center line with the circle (for example, from the point BUT) describe how from the center an additional arc the same radius R- get a point AT. Dropping a perpendicular from a point AT- get a point FROM.Line segment sun equal to the length of the side of the inscribed regular heptagon.

Rice. 9. Dividing the circle into 7 equal parts