This development is intended for 8th grade students. The use of electronic presentation contributes to the development of visual-figurative thinking and the formation of techniques and skills for working with drawing tools
Download:
Preview:
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 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
Preview:
To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com
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, it is necessary from points 1 and 4 of the intersection of the center line with the circle to 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 a point E. Segment DE equal to length 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), they 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.
Formation of Elementary Mathematical Representations(preschool).
Topic: "Division into 8 parts."
Target: Teach children to divide the circle into 8 parts.
To form ideas about the relationship and dependence of the part and the whole: the whole is greater than the part, the part is less than the whole.
To consolidate knowledge of numbers from 1 to 7.
Develop attention, memory, fine motor skills hands.
Cultivate goodwill, perseverance.
Material:(demo) - cards with numbers, letters, geometric shapes of different colors, chips;
Handout: circles, scissors, pens, notebooks.
Lesson progress: Guys, we have guests today. They came to see how you can play and practice.
Turn to the guests. Smile and say hello. Now show me your kind, smart and beautiful eyes. Sit down.
What do you guys want to be when you grow up?
Very interesting and necessary professions for you, and they all require good mathematical knowledge.
What does it mean to know mathematics? (children's answers)
Without an account there will be no light on the street,
A rocket cannot take off without a score.
Without an account, a letter will not find an addressee,
And the guys will not be able to play hide and seek.
What else needs to be done?
Children: solve problems, know geometric shapes, be able to think, compare, analyze, etc.
To learn all this, what do you have to be?
Children: attentive, quick-witted ......
Are you attentive? Smart? Well, then I think this package has been delivered to the address.
The captain of one ship turned to us for help, unfortunately, he did not write his name. And we will know the name if we help him. Do you agree?
The sailors on his ship rebelled and encrypted the name of the ship. The captain asks us to help him complete the tasks of the sailors. He sent us a photo of his ship. (I'm posting a drawn ship).
So, the first task.
D / I "What has changed"
I put up cards on the board: 10-12 pieces, with an image geometric shapes, different colors, sizes, shapes).
Close your eyes, put your heads down on the table (changing cards)
Open your eyes. -What changed? (2-3 responses per ear, and then responses usually).
Well done guys, you were very attentive.
Close your eyes, put your heads down on the table (changing).
What changed?
Close your eyes again, lower your head. (I don't change anything this time)
What changed? (4-5 answers)
Well done guys, I'm very pleased with you. So you learned the first letter N
What is this letter? (I paste it on the drawing of the ship).
Let's move on to the second task. IN numerical series figures are lost. Which? 1 ... 3 ... 5 ... 7..9.10 (children put in the missing numbers).
Name the neighbors of the numbers 5,3,7.
Name the number 1 greater than 5, 1 less than 6.
Name the previous number 7, next 8, etc..
And in this task you were attentive, quick-witted. (open letter A). -What is this letter?
The doors on the ship are painted in different colors. Z,K, J.
What color is the door in the middle? This is the captain's cabin. What color is the door on the right? On the left? - These are the sailors' quarters.
Where is the captain's cabin? Sailors' quarters?
Well, I think if we get on the ship, we will find the captain's cabin, and even by chance we will not fall into the hands of rebellious sailors. (I open the third letter -U).
Name this letter. Cover your mouth with a "cup" and sing this letter.
Let's move on to the next task. There is a cook on the ship. Who do you think it is? He always bakes round bread, and the sailors argue when they divide it into pieces. Let's learn for ourselves and teach the sailors to divide the round shape into parts.
How to cut a circle in half? - One more time in half?
Fold in half again. Iron the fold lines.
How many times did you fold?
How many parts do you think?
Unfold the circle and cut along the fold lines. Count.
How many parts did it make? (3-4 answers)
Show one part out of eight.
How many parts are showing? (3-4 answers).
Show two parts. - How many parts? (3-4 answers).
Show four out of eight.
What can be said about these parts? (half).
Show eight out of eight. How else can you call 8 out of 8 (whole).
What is greater than one whole or 8 out of 8? (3-4 answers).
Well done! I think that now it will be easier for the sailors to divide the loaf. (I open the letters T).
What is this letter? "Put" her on the tongue, throw it to me.
In mathematics, there are also unusual “fun” tasks. Answers to these tasks will show on the fingers. Close your eyes, put your head down on the table.
How many corners are in the room?
How many legs do sparrows have?
How many eyes does a traffic light have?
How many tails do five donkeys have?
How many horns do two cows have?
Open your eyes. Sit nicely. Straighten your shoulders, straighten your backs.
Here is the next letter. Name it (H) - (3-4 answers).
Oh, what an unusual next assignment. "Rest", what does it mean?
Get up a little. Let's cheer up the captain of this ship with our song.
Captain, captain smile
After all, a smile is the flag of a ship.
Captain, captain pull up
Only the brave conquer the seas. (repeat 2 times).
Sit down. (I open the next letter). Guys, what is this letter? (L).
Well done, clever, almost deciphered the name of the ship. If someone has already guessed, keep the name a secret, because if we agreed to help the captain, then we must reach the end and complete all the tasks.
I have 8 chips. In the right hand - 2. How many chips are in the left hand?
There are 6 chips in the left hand, how many chips are in the right hand?
In the right - 0, how many in the left?
Now guess which hand has how much, but remember that there are -8 chips in total.
I'm very happy for you. (I open the letter U).
Guys, you noticed that there are no patterns on the cabin doors or on the ship. Let's draw a pattern and suggest to the captain and sailors.
I’ll open the notebook and put it in the right place, take a pen and start writing: one cell down, one to the right, one up, one to the right, one down, etc.
Finish the line to the end. The pattern turned out beautiful, you guys tried. I open the last letter (C).
Who read the name of the ship? Tell me in my ear. (2-3 answers)
What is the name of the ship? Who is the captain on the Nautilus?
Outcome: Captain Nemo thanks you for your help. You helped the sailors too. The team reconciled with the captain and set sail. And they left you gifts - mini steering wheels. -Did you like helping the captain and sailors? -What task did you like?
I thank you for being so attentive, thoughtful, diligent. Thank you.
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 on a 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.
The straight line MN, which has only one common point 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 point "1" draw an arc of a circle to the intersection with a given circle (point 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 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?
- Eat.
- 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 the 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 ...
Division of a circle into three equal parts. Install a square with angles of 30 and 60 ° with a large leg parallel to one of the center lines. Along the hypotenuse from a point 1 (first division) draw a chord (Fig. 2.11, A), getting the second division - point 2. Turning the square and drawing the second chord, get the third division - point 3 (Fig. 2.11, b). By connecting points 2 and 3; 3 And 1 straight lines form an equilateral triangle.
Rice. 2.11.
a, b - c using a square; V- using a circle
The same problem can be solved using a compass. By placing the support leg of the compass at the lower or upper end of the diameter (Fig. 2.11, V) describe an arc whose radius is equal to the radius of the circle. Get the first and second divisions. The third division is at the opposite end of the diameter.
Dividing a circle into six equal parts
The compass opening is set equal to the radius R circles. From the ends of one of the diameters of the circle (from the points 1, 4 ) describe arcs (Fig. 2.12, a, b). points 1, 2, 3, 4, 5, 6 divide the circle into six equal parts. By connecting them with straight lines, they get a regular hexagon (Fig. 2.12, b).
Rice. 2.12.
The same task can be performed using a ruler and a square with angles of 30 and 60 ° (Fig. 2.13). The hypotenuse of the square must pass through the center of the circle.
Rice. 2.13.
Dividing a circle into eight equal parts
points 1, 3, 5, 7 lie at the intersection of the center lines with the circle (Fig. 2.14). Four more points are found using a square with angles of 45 °. When receiving points 2, 4, 6, 8 the hypotenuse of a square passes through the center of the circle.
Rice. 2.14.
Dividing a circle into any number of equal parts
To divide a circle into any number of equal parts, use the coefficients given in Table. 2.1.
Length l chord, which is laid on a given circle, is determined by the formula l = dk, Where l- chord length; d is the diameter of the given circle; k- coefficient determined from Table. 1.2.
Table 2.1
Coefficients for dividing circles
To divide a circle of a given diameter of 90 mm, for example, into 14 parts, proceed as follows.
In the first column of Table. 2.1 find the number of divisions P, those. 14. From the second column write out the coefficient k, corresponding to the number of divisions P. In this case, it is equal to 0.22252. The diameter of a given circle is multiplied by a factor and the length of the chord is obtained l=dk= 90 0.22252 = 0.22 mm. The resulting length of the chord is set aside with a measuring compass 14 times on a given circle.
Finding the center of the arc and determining the size of the radius
An arc of a circle is given, the center and radius of which are unknown.
To determine them, you need to draw two non-parallel chords (Fig. 2.15, A) and set up perpendiculars to the midpoints of the chords (Fig. 2.15, b). Center ABOUT arc is at the intersection of these perpendiculars.
Rice. 2.15.
Pairings
When performing engineering drawings, as well as when marking workpieces in production, it is often necessary to smoothly connect straight lines with arcs of circles or an arc of a circle with arcs of other circles, i.e. perform pairing.
Pairing called a smooth transition of a straight line into an arc of a circle or one arc into another.
To build mates, you need to know the value of the radius of the mates, find the centers from which the arcs are drawn, i.e. interface centers(Fig. 2.16). Then you need to find the points at which one line passes into another, i.e. connection points. When constructing a drawing, mating lines must be brought exactly to these points. The point of conjugation of the arc of a circle and a straight line lies on a perpendicular lowered from the center of the arc to the mating line (Fig. 2.17, A), or on a line connecting the centers of mating arcs (Fig. 2.17, b). Therefore, to construct any conjugation by an arc of a given radius, you need to find interface center And point (points) conjugation.
Rice. 2.16.
Rice. 2.17.
The conjugation of two intersecting lines by an arc of a given radius. Given straight lines intersecting at right, acute and obtuse angles (Fig. 2.18, A). It is necessary to construct conjugations of these lines by an arc of a given radius R.
Rice. 2.18.
For all three cases, the following construction can be applied.
1. Find a point ABOUT- the center of the mate, which must lie at a distance R from the sides of the corner, i.e. at the point of intersection of lines passing parallel to the sides of the angle at a distance R from them (Fig. 2.18, b).
To carry out direct parallel sides angle, from arbitrary points taken on straight lines, with a compass solution equal to R, make serifs and draw tangents to them (Fig. 2.18, b).
- 2. Find the junction points (Fig. 2.18, c). For this, from the point ABOUT drop perpendiculars to given lines.
- 3. From point O, as from the center, describe an arc of a given radius R between junction points (Fig. 2.18, c).
