In this article, we will comprehensively analyze one of the main geometric shapes - the angle. Let's start with auxiliary concepts and definitions that will lead us to the definition of an angle. After that, we give the accepted methods for designating angles. Next, we will deal in detail with the process of measuring angles. In conclusion, we will show how you can mark the corners in the drawing. We provided all the theory with the necessary drawings and graphic illustrations for better memorization of the material.
Page navigation.
Angle definition.
Angle is one of the most important figures in geometry. The definition of an angle is given through the definition of a ray. In turn, the idea of a ray cannot be obtained without knowledge of such geometric figures as a point, a straight line and a plane. Therefore, before getting acquainted with the definition of the angle, we recommend refreshing the theory from sections and.
So, we will start from the concepts of a point, a straight line on a plane and a plane.
Let us first give the definition of a ray.
Let us be given some straight line on the plane. Let's denote it with the letter a. Let O be some point of the line a . The point O divides the line a into two parts. Each of these parts together with the point O is called beam, and the point O is called the beginning of the beam. You can also hear that the beam is called semidirect.
For brevity and convenience, the following notation for rays was introduced: a ray is denoted either by a small Latin letter (for example, ray p or ray k), or by two large Latin letters, the first of which corresponds to the beginning of the ray, and the second denotes some point of this ray (for example, ray OA or beam CD). Let's show the image and designation of the rays in the drawing.
Now we can give the first definition of an angle.
Definition.
Corner- this is a flat geometric figure (that is, lying entirely in a certain plane), which is made up of two mismatched rays with a common origin. Each of the rays is called corner side, the common beginning of the sides of the angle is called top corner.
It is possible that the sides of an angle form a straight line. This angle has its own name.
Definition.
If both sides of an angle lie on the same line, then the angle is called deployed.
We bring to your attention a graphic illustration of a developed angle.
An angle symbol is used to denote an angle. If the sides of the angle are indicated in small Latin letters (for example, one side of the angle is k, and the other is h), then to designate this angle, after the angle icon, letters corresponding to the sides are written in a row, and the order of recording does not matter (that is, or). If the sides of the angle are indicated by two large Latin letters (for example, one side of the angle OA, and the second side of the angle OB), then the angle is denoted as follows: after the angle sign, three letters are written that participate in the designation of the sides of the angle, and the letter corresponding to the vertex of the angle, located in the middle (in our case, the angle will be indicated as or ). If the vertex of an angle is not the vertex of some other angle, then such an angle can be denoted by the letter corresponding to the vertex of the angle (for example, ). Sometimes you can see that the corners in the drawings are marked with numbers (1, 2, etc.), these corners are denoted as and so on. For clarity, we present a figure in which the corners are shown and indicated.
Any angle divides the plane into two parts. Moreover, if the angle is not developed, then one part of the plane is called inner corner area, and the other outside corner area. The following image explains which part of the plane corresponds to the inside of the corner and which part to the outside.
Any of the two parts into which a flattened angle divides a plane can be considered an interior region of the flattened angle.
The definition of the interior of an angle leads us to the second definition of an angle.
Definition.
Corner- this is a geometric figure, which is made up of two mismatched rays with a common origin and the corresponding inner region of the angle.
It should be noted that the second definition of the angle is stricter than the first, since it contains more conditions. However, one should not dismiss the first definition of the angle, nor should one consider the first and second definitions of the angle separately. Let's explain this point. When it comes to an angle as a geometric figure, then an angle is understood as a figure composed of two rays with a common origin. If it becomes necessary to carry out any actions with this angle (for example, measuring an angle), then an angle should already be understood as two rays with a common origin and an internal region (otherwise a two-fold situation would arise due to the presence of both an internal and an external region of the angle ).
Let us give more definitions of adjacent and vertical angles.
Definition.
Adjacent corners- these are two angles in which one side is common, and the other two form a straight angle.
It follows from the definition that adjacent angles complement each other up to a straight angle.
Definition.
Vertical angles are two angles in which the sides of one angle are extensions of the sides of the other.
The figure shows vertical angles.
Obviously, two intersecting lines form four pairs of adjacent angles and two pairs of vertical angles.
Angle comparison.
In this paragraph of the article, we will deal with the definitions of equal and unequal angles, and also in the case of unequal angles, we will explain which angle is considered large and which is smaller.
Recall that two geometric figures are called equal if they can be superimposed.
Let us be given two angles. Let us give reasoning that will help us get an answer to the question: “Are these two angles equal or not”?
Obviously, we can always match the vertices of two corners, as well as one side of the first corner with any of the sides of the second corner. Let's combine the side of the first corner with that side of the second corner so that the remaining sides of the corners are on the same side of the straight line on which the combined sides of the corners lie. Then, if the other two sides of the corners are aligned, then the corners are called equal.
If the other two sides of the angles do not match, then the angles are called unequal, and smaller the angle is considered to be part of another ( big is the angle that completely contains another angle).
Obviously, the two straight angles are equal. It is also obvious that a developed angle is greater than any non-developed angle.
Angle measurement.
Angle measurement is based on comparing the measured angle with the angle taken as the unit of measure. The process of measuring angles looks like this: starting from one of the sides of the measured angle, its inner area is sequentially filled with single angles, tightly stacking them one to the other. At the same time, the number of stacked corners is remembered, which gives the measure of the measured angle.
In fact, any angle can be taken as the unit of measure for angles. However, there are many generally accepted units for measuring angles related to various fields of science and technology, they have received special names.
One of the units for measuring angles is degree.
Definition.
one degree is an angle equal to one hundred and eightieth of a straightened angle.
A degree is denoted by the symbol "", therefore, one degree is denoted as.
Thus, in a developed angle, we can fit 180 angles into one degree. It will look like half a round pie cut into 180 equal pieces. Very important: the "pieces of the pie" fit tightly together (that is, the sides of the corners are aligned), with the side of the first corner aligned with one side of the flattened corner, and the side of the last unit corner coincided with the other side of the flattened corner.
When measuring angles, it is found out how many times a degree (or other unit of measurement of angles) fits in the measured angle until the inner area of the measured angle is completely covered. As we have already seen, in a developed angle, the degree fits exactly 180 times. Below are examples of angles in which a one-degree angle fits exactly 30 times (such an angle is a sixth of a straight angle) and exactly 90 times (half a straight angle).
To measure angles less than one degree (or another unit of measurement of angles) and in cases where the angle cannot be measured by an integer number of degrees (taken units of measurement), you have to use parts of a degree (parts of taken units of measurement). Certain parts of the degree received special names. The most common are the so-called minutes and seconds.
Definition.
Minute is one sixtieth of a degree.
Definition.
Second is one sixtieth of a minute.
In other words, there are sixty seconds in a minute, and sixty minutes (3600 seconds) in a degree. The symbol "" is used to denote minutes, and the symbol "" is used to denote seconds (do not confuse with the signs of the derivative and the second derivative). Then, with the introduced definitions and notation, we have , and the angle in which 17 degrees 3 minutes and 59 seconds fit can be denoted as .
Definition.
Degree measure of an angle a positive number is called, which shows how many times a degree and its parts fit into a given angle.
For example, the degree measure of a straightened angle is one hundred and eighty, and the degree measure of an angle is 
.
To measure angles, there are special measuring instruments, the most famous of which is a protractor.
If both the designation of the angle (for example,) and its degree measure (let 110) are known, then use a short notation of the form 
and say: "The angle AOB is one hundred and ten degrees."
From the definitions of the angle and the degree measure of the angle, it follows that in geometry the measure of the angle in degrees is expressed by a real number from the interval (0, 180] (in trigonometry, angles with an arbitrary degree measure are considered, they are called). An angle of ninety degrees has a special name, it is called right angle. An angle less than 90 degrees is called acute angle. An angle greater than ninety degrees is called obtuse angle. So, the measure of an acute angle in degrees is expressed by a number from the interval (0, 90), the measure of an obtuse angle - by a number from the interval (90, 180), a right angle is equal to ninety degrees. Here are illustrations of an acute angle, an obtuse angle, and a right angle.
From the principle of measuring angles, it follows that the degree measures of equal angles are the same, the degree measure of a larger angle is greater than the degree measure of a smaller one, and the degree measure of an angle that consists of several angles is equal to the sum of the degree measures of the component angles. The figure below shows the angle AOB, which is made up of the angles AOC, COD and DOB, while .
In this way, sum of adjacent angles is one hundred and eighty degrees, since they form a straight angle.
It follows from this assertion that . Indeed, if the angles AOB and COD are vertical, then the angles AOB and BOC are adjacent and the angles COD and BOC are also adjacent, therefore, the equalities and are valid, from which the equality follows.
Along with the degree, a convenient unit for measuring angles is called radian. The radian measure is widely used in trigonometry. Let's define a radian.
Definition.
One radian angle- this is central corner, which corresponds to the length of the arc, equal to the length of the radius of the corresponding circle.
Let's give a graphical illustration of an angle of one radian. In the drawing, the length of the radius OA (as well as the radius OB ) is equal to the length of the arc AB , therefore, by definition, the angle AOB is equal to one radian.
The abbreviation "rad" is used to denote radians. For example, writing 5 rad means 5 radians. However, in writing, the designation "rad" is often omitted. For example, when it is written that the angle is equal to pi, it means pi rad.
It should be noted separately that the value of the angle, expressed in radians, does not depend on the length of the radius of the circle. This is due to the fact that the figures bounded by a given angle and an arc of a circle centered at the vertex of a given angle are similar to each other.
Measuring angles in radians can be done in the same way as measuring angles in degrees: find out how many times an angle of one radian (and its parts) fit into a given angle. And you can calculate the length of the arc of the corresponding central angle, and then divide it by the length of the radius.
For the needs of practice, it is useful to know how the degree and radian measures relate to each other, since quite a part has to be carried out. In this article, a relationship is established between the degree and radian measure of an angle, and examples of converting degrees to radians and vice versa are given.
Designation of corners in the drawing.
In the drawings, for convenience and clarity, corners can be marked with arcs, which are usually drawn in the inner region of the corner from one side of the corner to the other. Equal angles are marked with the same number of arcs, unequal angles with a different number of arcs. Right angles in the drawing are denoted by a symbol of the form "", which is depicted in the inner region of the right angle from one side of the corner to the other.
If the drawing has to mark many different angles (usually more than three), then when designating the angles, in addition to ordinary arcs, it is permissible to use arcs of some special type. For example, you can depict jagged arcs, or something similar.
It should be noted that you should not get carried away with the designation of angles in the drawings and do not clutter up the drawings. We recommend marking only those angles that are necessary in the process of solving or proving.
Bibliography.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Poznyak E.G., Yudina I.I. Geometry. Grades 7 - 9: a textbook for educational institutions.
- Atanasyan L.S., Butuzov V.F., Kadomtsev S.B., Kiseleva L.S., Poznyak E.G. Geometry. Textbook for 10-11 grades of high school.
- Pogorelov A.V., Geometry. Textbook for grades 7-11 of educational institutions.
- Let's remember the theme of the last previous lessons. (New units of area)
What new units of area did you learn? (hectare, are)
Was it difficult or easy to learn new units of area? Why?
Were you able to overcome difficulties?
What do you think, will we succeed in the study of the next new topic?
Let's see?
1. Mathematical dictation.
- Decrease 160 by 90.
- Increase 490 by 50.
- Reduce 560 by 80 times.
- Increase 70 by 9 times.
How much more is 820 than 290?
How many times is 400 less than 3600?
- Find the number whose sixth part is equal to 102.
- Find a quarter of 68.
(70, 540, 7, 630, 530, 9, 612, 17)
What groups can this series of numbers be divided into? (By the number of digits, by a multiplicity of 2, by a multiplicity of 10, by the sum of digits, numbers for writing numbers.)
Letters are placed on the board under the received numbers.
70, 540, 7, 630, 530, 9, 612, 17
G R F A U N L I
Arrange the resulting numbers in ascending order and read the resulting word. (FNIGURLA)
Does it make sense?
Cross out 2 letters to make a mathematical term. (FIGURE)
2. Working with geometric shapes.
What are the geometric shapes you see in the picture?
(On the image: point, straight line, circle, segment, angle, ray, quadrilateral, polyline)
What figures can be continued indefinitely? ( Straight line, beam, side angle)
If you draw a line segment connecting the center of the circle with a point on it, what happens? ( Radius)
What interesting things do you know about the radius? (All radii of one circle are equal. The radius is half the diameter.)
What is the relationship between a polygon and a polyline? (A polygon is a closed polyline.)
What other flat geometric shapes do you know? (Triangle, rectangle, square, oval, etc.)
What about space figures? (Sphere, cube, parallelepiped, cylinder, cone, pyramid.)
3. Working with an angle.
What are the sides of an angle? (Rays.)
If you continue the sides of the angle, will you get the same angle or a different one? (The same.)
What are the types of corners? (Straight, sharp, blunt.)
Show with pencils a model of an acute angle, an obtuse angle.
Imagine that your pencils are the hands of a clock. Lay them out on the desk so that they show 1h, 2h, 3h, 4h, 5h. What happens to the angle between them? (Increases.)
So we can say which angle between the hands of the clock is greater and which is smaller? (Yes.)
4. Practical work. Individual task.
Each student has a model of an acute angle (yellow), a model of an obtuse angle (blue) on the tables. Acute angle model by area much exceeds the obtuse angle model.
Compare angles with overlay.
(Some place the blue inside the yellow, based on area. Others based on the extension of the sides and that the angles should be compared based on the turn).
Problem situation:
Why, comparing the same angles, got a different result?
Where and why did the difficulty arise?
What task did you do? (compare angles)
Why didn't you justify your positions? (We don't know how to compare angles)
What do we need to do - put in front of you goal. (We need to build an angle comparison algorithm)
Formulate lesson topic. (Angle comparison)
1. Leading dialogue.
(Students choose a course of action, and then derive an algorithm based on it)
In what way do we compare something, for example, we say - one person knows more than another, or more number, fraction, fraction ...
(The lesser must be contained in the greater, be part of it)
So, how do we need to overlay the corners? (So that one corner is part of the other)
Why can't the blue corner be placed inside the yellow one? (The sides of the corner are rays. If you continue them, you can see that the blue corner is not inside the yellow one)
Children receive a blue corner model comparable in area to yellow.
Lay the blue corners on top of each other and make sure they are equal.
2. Work in groups.
Does this give you an idea of how to superimpose the blue and yellow corners to find out which one is larger?
Consult in groups.
(Children express their versions. If these versions are not correct, then the teacher or one of the children refute them. The correct way of imposing is spoken out and the algorithm is fixed.)
3. Algorithm.
1) Place the corners so that one of their sides coincides.
2) If the other one coincides, then the angles are equal; if not, then the smaller is the angle whose side is inside the other.
4. Scheme-support.
5. Comparison of the output with the text of the textbook. Page one.
- Did our conclusion match the text of the textbook?
Speak the algorithm for comparing angles.
1. Comparein pairs two arbitrary angles, pronouncing the algorithm.
2. Task number 4 on page 2.
Compare the angles using the support scheme.
What can you say about the OS beam? (He divided the corner into two corners)
What can you say about these rays? (Angle AOC is less than angle COB)
1. Task number 8 on page 2 (compare eye angles in the textbook) and unravel the name of the famous ruler of Ancient Egypt - Cheops. They remember what they know about him from the course of the world around him.
Is it possible to find corners at the pyramid of Cheops?
What have you learned about corners?
Problematic situation.
Do you think this is all known knowledge about angles or not?
1. Introduction of the concept of "bisector" using practical work.
Bend one of the corners lying on the table in half. Expand the corner.
What did you get? (A line that divides an angle into two equal angles)
What is this line called in mathematics? (Ray) Why?
For a ray drawn inside an angle from its vertex, which bisects the angle, there is a special name "bisector". (On the desk)
2. Drawing review in the textbook
There is a funny but helpful rhyme to remember a new concept:
“The bisector is such a ... that runs around the corners and divides the angle .... (Children finish the rhyme)
How did you cut the corner in half? (Bending over)
What new concept did you learn? (Bisector)
How would you explain to a classmate who skipped class what a bisector is?
1. Examples for finding a part of a number expressed as a fraction No. 10 p. 3.
(They decipher the name of the pharaoh, in whose honor the very first pyramid was built - Djoser)
2. Solving compound problems to find a part of a number, expressed as a fraction or as a percentage.
a) about Pharaoh Thutmose No. 11 on page 3.
b) about a camel, which is adapted for a long time to do without water and food to move through the desert No. 12 (a) at st. 3.
What is the topic of the lesson?
How are the angles compared?
How to find out which angle is larger and which is smaller?
What new concept did you learn?
How do you find the bisector of an angle? Why?
Who else needs help with the topic of the lesson?
Were we able to understand the new topic immediately? Why?
What new things did you learn while solving problems?
What knowledge will be useful to you in life? Where?
Homework: 1) basic level: repeat the algorithm for comparing angles, No. 5 - practical work on dividing an angle into parts and comparing parts by bending; No. 12 (b) - a problem for fractions;
2) advanced level: No. 7 - obtaining bisectors of the angles of a triangle and a rectangle by bending.
§ 28. Comparison of angles by imposition - Textbook on Mathematics Grade 5 (Zubareva, Mordkovich)
Short description:
Different geometric shapes can be compared with each other in various ways. One of these ways is the imposition of one figure on another. As well as other figures, you can compare angles with each other when necessary. Today you will learn about it from this paragraph of the textbook. One way to compare angles is overlay. Angles that coincide when superimposed are called equal. If the angles do not match, then you can easily determine which of the angles will be smaller and which will be larger than the other. To compare corners using overlay, you need to attach their vertices to each other. Then combine one side of one corner with the side of the other corner. If at the same time their second side also coincides, then such angles will be equal. The overlay method is the easiest graphical way to determine the equality of angles. In order to use this method, tracing paper or other translucent materials are suitable. Or you can use a protractor, measuring the value of one corner and transferring it to the second corner. Choose a convenient way for yourself to solve and depict various geometric problems, since in the future this knowledge will be useful in solving problems with shapes. Look through the textbook paragraph on this topic to better understand and remember the material!
§ 1 Comparison of angles
In this lesson, we will learn how to compare and measure angles.
Recall that an angle is a geometric figure formed by two rays (the sides of the angle) coming out of one point (which is called the vertex of the angle).
Let's compare two angles with an overlay and find out if the angles are equal or not.
Let's take two corners.
Paint one corner blue and the other red, and overlay the red corner on the blue.
The figure shows that the blue angle is larger than the red one, but we don't know by how much. To compare angles, you need to learn how to accurately measure them.
Angle is measured in the same way as any other value.
To do this, choose a unit of measurement (measurement) and find out how many times it is contained in the measured value.
Let's imagine the following situation: Seryozha, Petya and Kolya decided to measure the angle, but each decided to take the measurement himself.
What happened?
It turned out that the same angle for Seryozha is equal to three of his measurements, for Petya - for four measurements, and for Kolya - for six measurements.
Which of them is right?
What is this angle really?
In geometry, there is a generally accepted, common for all, measure - this is 1/90 of a right angle. This measure is called a degree and denoted: 1 °.
Thus, a right angle is 90°, and a straight angle is 180°.
Any acute angle will be less than 90° and any obtuse angle will be greater than 90°.
When adding angles, their degree measures are added, and when subtracting, they are subtracted, for example:
It must also be remembered that the sum of adjacent angles is always 180°.
§ 2 Protractor. Angle measurement
Let's try to solve the problem using our knowledge.
An angle OMR is given - it is a straight line, i.e. 90°, two beams divided it into three angles.
As you can see from the picture, one angle is 18 degrees and the other is 23 degrees.
We need to calculate what is the angle KMN?
To find the value of the angle KMN, it is necessary to subtract the degree measures of the angles KMR and NMO from the degree measure of the angle OMR:
∠KMN = ∠OMR - ∠KMR - ∠NMO = 90° - 18° - 23° = 49°
The KMN angle is 49°.
Let's solve one more problem.
In the figure, we see that ∠KOS is deployed, which means that it is equal to 180 °.
∠KOV = 60° and ∠AOC = 60°.
Let's find the value ∠BOA.
∠BOA = ∠KOS - ∠KOV - ∠AOC = 180° - 60° - 60° = 60°
∠BOA = 60°
To measure an angle in degrees, you need to know how many times it contains a measurement of 1 °. To measure angles in degrees, a special tool is used - a protractor.
The protractor consists of a ruler (rectilinear scale) and a semicircle (goniometric scale), divided into degrees from 0 to 180. In some models, for example, a circular protractor - from 0 to 360. The protractor scale is located on a semicircle.
The center of this semicircle is marked on the protractor with a dash, it is called the center of the protractor.
Let's measure ∠MKT.
To do this, we impose a protractor so that the center of the protractor coincides with point K, the beginning of the CT beam, and the CT beam itself passes through the origin of the protractor scale. The degree measure of an angle will be shown by a stroke on the protractor scale through which the other side of the angle passes.
So, ∠MKT is equal to 32°.
With the help of a protractor, you can not only measure, but also build angles.
Let's build an angle equal to 110°, one side of which is the ray OA.
Let's draw a ray OA first.
Then we put the protractor on our ray so that the center of the protractor coincides with the point O - the beginning of the ray OA, and the ray OA itself passes through the origin of the protractor scale.
Let's put point B against the stroke of the protractor scale with a mark of 110 ° and draw a beam of OB.
We get ∠AOB containing 110°.
For convenience, the reading of degrees on the protractor scale goes in two directions, and when we measure or build an angle, we must always remember that an acute angle is less than 90 °, and an obtuse one is more than 90 °.
§ 3 Summary of the lesson
Let's summarize our lesson:
1. Angles are measured with a protractor.
2. To measure the angle with a protractor, you need:
Attach the center of the protractor to the top of the corner;
Position the protractor so that one side of the angle passes through the origin of the protractor scale division 0;
see through which division of this scale the other side of the corner will pass;
When measuring, remember that an acute angle is less than 90°, and an obtuse angle is greater than 90°.
3. To build an angle of a certain size, you need:
hold a beam
· put a protractor on this beam so that the center of the protractor coincides with the beginning of the beam, and the beam itself passes through the origin of the protractor scale division 0;
· put a dot against the stroke of the protractor scale with a mark of the value we need and draw the second ray through this point from the beginning of the original ray.
4. A right angle is 90°, an acute angle is less than 90°, an obtuse angle is greater than 90°, a straight angle is 180°.
5. When adding angles, their degree measures are added, and when subtracting, they are subtracted.
6. The sum of adjacent angles is always 180°.
List of used literature:
- Peterson L.G. Maths. 4th grade. Part 1. / L.G. Peterson. – M.: Yuventa, 2014. – 96 p.: ill.
- Maths. 4th grade. Methodological recommendations for the mathematics textbook "Learning to learn" for grade 4. / L.G. Peterson. – M.: Yuventa, 2014. – 280 p.: ill.
- Zak S.M. All tasks for the mathematics textbook for grade 4 L.G. Peterson and a set of independent and control works. GEF. – M.: UNVES, 2014.
Class: 3
Subject: mathematics (Developing program of L.V. Zankov)
Topic: Types of angles and their comparison.
Lesson type: discovery of new knowledge
Goals:
Tutorials: Open ways to compare angles.
Developing:Develop attention, abstract thinking, observation, the ability to compare, independently analyze, draw conclusions.
Educators:To cultivate in students an interest in mathematics, cultural communication skills, an active personality.
Technology used: RKCHP
Formed UUD:
Regulatory: the ability to set a goal, a learning task; carry out pattern control.
Cognitive: the ability to compare and measure angles by eye and overlay method; build angles of a given value using measuring tools; the ability to choose the most effective ways to solve problems; searches for and highlights the necessary information to complete educational tasks; actions with sign-symbolic means (modeling); logical - comparison, identification, generalization.
Communicative: planning and implementation of educational cooperation with the teacher and peers; be able to listen to others, the ability to ask training questions; possession of monologue and dialogic forms of speech;
Personal: evaluating one's own learning activities according to criteria defined jointly with the teacher.
Equipment: computer, cards with angles and the game "Do you believe that ...", students' scissors, sticks and modeling clay
During the classes
Stages
Teacher activity
Student activities
Greetings
Call
Let's check the readiness. I wish you success.
I want to start today's lesson with the words of the French philosopher Jean Jacques Rousseau: “You are talented children! Someday you yourself will be pleasantly surprised how smart you are, how much and how well you know how, if you constantly work on yourself, set new goals to achieve them ... ".
I wish you today at the lesson to be convinced of the words of J. J. Rousseau.
Are you ready to go?
Then go.
Warm-up for the mind.
If you solve the expressions correctly, you will be able to formulate the topic of the lesson. Each correct answer is followed by a letter. If you arrange the answers in ascending order, you can read the topic of the lesson.
On the slide: 8x6, 9x5, 18:2, 7x4, 30:5, 42:6, 72:9, 4x6, 5x7
e i w c r a n n
500-200 900-2 733+100 580-40 806-6
u v o g l
And now I invite you to play a game with me "Do you believe that..."
1) the science that studies angles is called geometry;
2) angles are obtuse, straight and sharp;
3) two angles cannot be compared;
4) there are several ways to compare angles;
5) with the help of corners, animal figures can be modeled;
6) there is no tool for comparing angles;
7) from three sticks you can lay out three angles at once: straight, obtuse and acute
8) an acute angle is greater than an obtuse one
In what questions do you definitely have no doubts and think that you answered correctly?
Why are you sure the answers are correct?
check readiness
Calculate verbally
Topic: Comparison of angles
Answer questions on their own
Can answer in #1, 2, 6, 8
knew, read
Making sense
What questions do you doubt?
Then formulate, please, the purpose of the lesson.
(The goal is written on the board).
How will we achieve the goal?
I offer you task No. 148 p. 80 in the textbook.
We complete the task on our own.
We check according to the sample: (on the slide)
3, 2, 7, 1, 4, 5, 8, 6,
Was it easy to compare angles? What is the difficulty?
Who agrees, disagrees?
How were they compared? How?
Criteria:
"5" - 0 errors, "4" - 1-2 errors, "3" - 3-4 errors.
Practical work №1.
We complete task 3) of this number, draw in a notebook 2 corners that are easy to compare and 2 corners that are difficult to compare. (1 person - at the board)
Mutual check
We check, evaluate the ability to draw angles for comparison by eye.
And now, in order to confirm or refute other statements from the game "Do you believe that ...", I suggest that you get acquainted with a little information in which, if you read carefully, you can find answers to questions.
When reading, I suggest using the " Insert" for the convenience of capturing information. (+ knew, ! - new, ? did not understand)
Text for work:
So what did you already know?
And what new, interesting information on the topic of the lesson have you learned now?
In task No. 148, we compared the angles in what way?
What other way to compare angles did you learn about?
Practical work №2.
I propose to compare the two angles in this way.
Each child receives a sheet with two corners:
An algorithm for comparing angles with the help of an overlay is preliminarily compiled together with the children:
In order to compare angles, you need: Algorithm:
1) cut corner No. 1; 2) combine the tops of the corners and one of the sides of the corners; 3) on the second side of the angle, determine which angle is larger (smaller).
Children cut out one of the corners and put it on the other according to the algorithm.
How are the angles compared now?
Mathematics is an exact science. Which way do you think is more accurate?
Physical education minute
And now I will return to question number 7 of the game and complete this task to check it. Let's model the corners with plasticine and sticks.
Let's check the sample on the slide or on the board.
Estimate (the ability to model corners).
Recently, in a math lesson, they drew different angles. I suggest you solve the problem associated with this task. Slide
A task. Yulia in the drawing turned out 7 obtuse angles, 1 straight, and 11 acute, and Vali 5 obtuse angles, 2 straight and 14 acute. Who has more angles and by how many?
Which of the known methods of concise writing is more convenient to write it down? (table).
Let's make a table and solve the problem ourselves.
Examination. Assessment of the ability to solve problems.
Purpose: -Compare angles, -find ways to compare angles
Completing tasks
Check on a sample
Approximately
Work with the evaluation sheet
Draw angles in a notebook for eye comparison
Evaluate the work of a neighbor
Read the text, mark with icons
Children's statements
Protractor, 2 ways to compare angles, degrees, geometry
approximately
overlay
Together with the teacher, compose a comparison algorithm
Cut, impose, draw a conclusion
overlays
Model corners with sticks and plasticine
Appreciate
Read the task
Draw on the board and in a notebook
Check against the standard
Reflection
Let's get back to the game "Do you believe that...".
What questions did we not find answers to during the lesson?
Let's return to the goal set at the beginning of the lesson.
Have you achieved? Why? What was difficult? Have all questions been answered?
Let's take a look at the evaluation sheet. What skills did you develop in class?
Where can they be useful in life?
Homework (student's choice):
1) Crossword on the topic of the lesson
2) Draw the animal on the sheets, using only the corners.
3) Complete the tasks of the textbook p.80 No. 149, No. 150 (1)
Crossword:
Horizontally: 1. Two beams emanating from one point form ... .. 2. A device for measuring angles is called ... .. . Vertically: 1. The point connecting the two rays of the angle is called .... 2. The most accurate way to compare angles. 3. An angle greater than a right one is called ....
Complete the third column of the table.
Did not find the answer to question number 5
Answer.
Put an average mark for the lesson.
Cut, build, make crafts
Applications
Text for work:
The shape of objects and their dimensions are studied by geometry - part of the great science of mathematics. The main concept of geometry is a figure. The figures have their own name: ball, ray, line, point, segment, angle, triangle ....
Two rays emanating from the same starting point form an angle. The rays that form an angle are called the sides of the angle, and their starting point is called the vertex of the angle. Angles are different: obtuse, straight, sharp and deployed. Angle can be compared and measured. There are many ways to compare angles. You can compare by eye (approximately), or by superimposing corners on each other. Measure the angles with a special device - a protractor. The protractor shows the angle in degrees.
Assessment sheet
mark
mark
Outcome:
Outcome:
