1. Learn to identify acute and obtuse angles using the right angle model.
Educational:
1. Form an idea of flat geometric shapes as parts of a plane.
2. Continue working on the classification of geometric shapes.
Educational:
1. Cultivate accuracy and attentiveness.
Lesson type- introduction of new knowledge
Forms of student work - paired, individual, frontal work
Equipment: a circle with sectors, cards with geometric shapes, multi-level cards, wire, models of triangles, poems - reminders.
I Updating knowledge.
1. Organizational moment.
A student reads a poem.
There is a rumor about mathematics
That she puts her mind in order,
Because good words
People often talk about her.
Mathematics, you give us
Hardening is important for victory.
Young people study with you
Develop both will and ingenuity.
- So today in class we will continue to develop ingenuity, will, determination, accumulate knowledge, and practice skills.
During the lesson we will have to travel around the country of Mathematics. Here is our travel itinerary. There are 6 sectors on the map, 5 different areas of mathematics. Want to know them? Then let's open them in order. (Arithmetic, geometry, where we will get acquainted with a new topic, ecology and mathematics, folklore, logic.)
So, let's go! (Open the “Arithmetic” sector)
(Slide 1.)
A)
Game “Mathematical Basketball”.
Basketball- a team sports game, the goal of which is to throw a ball into a suspended basket with your hands.
Any of you will score a goal if you solve the example correctly. (Children solve examples in a chain).
8+ 7 9 + 5 12 – 4 6 + 5 13 – 7 14 – 6 8 – 8 5 + 7 15 – 9 9 + 9
b) Solve the problem in general form.
There are two expressions written on the board. Which expression is suitable for solving problems A+B A-B
- There was A candy on the plate, Masha ate B candy. How many candies are left?
- Olya solved A problems in mathematics, Misha solved B problems. How many problems did the guys solve in total?
- Lena has A pencils, and Olya B has pencils. How many more pencils does Lena have than Olya?
- There were A girls in the class, and there were B fewer boys. How many boys were there in the class?
c) Working with cards (image of geometric shapes)
What is shown on the leaves? (flat geometric shapes)
Divide them into groups, i.e. distribute into “bags” using colored pencils.
Let's check...
The first group included straight lines. Name them. Prove that these are straight lines.
The rays were separated into the second group. Name them. Prove that these are rays.
The third group was divided into segments. Name them. Prove it.
The fourth group is the corner.
II . “Discovery” of new knowledge by students
(Slide 2.)
1) - The crossword puzzle will tell you the topic of the lesson. Crossword “Geometric”.
1) Part of a line that has a beginning but no end. (Ray).
2) A geometric figure that has no corners. (Circle).
4) A geometric figure in the shape of an elongated circle. (Oval).
The topic of our lesson is hidden vertically. Find her. (Corner). (click, geometric shapes fly out).
Please formulate the topic of our lesson. (Sector “Geometry”)
Guys, why are we going to study angles?
Do you think this knowledge will be useful to you?
(Children's answers)
Angles surround us in everyday life. Give your own examples of where you can find angles around us.
Slide 3-4.
Look at the pictures: a connecting corner for pipes and a stationery corner for papers; carpenter's square and drafting square; corner table and corner sofa.
Guys, maybe someone knows what an angle is? (children's opinions are listened to)
We will check the correctness of our formulation a little later.
People of what professions are most likely to encounter angles? (constructor, engineer, designer, builder, architect, sailor, astronomer, architect, tailor, etc.)
Guys, now step back one cell from the red fields and place point O. Draw two rays from this point.
Draw point O (2) on the board in advance. I call 2 children to draw rays on the board.
What kind of figures did we get? (corner)
Look how different these angles are.
Guys, now try to define an angle.
Work in pairs.
(Conclusion: an angle is a geometric figure formed by two different rays
with a common beginning).
Guys, now look at the figure that I drew.
Is it an angle or not.
(The children say no, we return to the rule again, after which we conclude that this is also an angle - a reversed one)
Slide 6. (output by angle)
Poster on blackboard
Point O is the vertex of the angle. An angle can be called by one letter written near its vertex. Angle O. But there can be several angles that have the same vertex. What to do then? (There is a drawing of such angles on the board)
Children's answers.
In such cases, if you call different angles with the same letter, it will not be clear which angle you are talking about. If this does not happen, you can mark one point on each side of the angle, put a letter near it and designate the angle with three letters, while always writing in the middle the letter indicating the vertex of the angle. Angle AOB. Rays AO and OB are the sides of the angle.
Drawing on the board
Working with the textbook text in the orange frame p.52.
III . Primary consolidation.
Work in pairs. Task No. 2
- The angles are different. Here are different types of angles.
What is this angle called? (straight) How to prove that it is really straight?
- What are these angles called? (indirect)
- Today we will find out what they are called.
IV . Formulation of new knowledge.
(Slide 7 - 9)
It is not always convenient to determine a right angle by eye. To do this, use a ruler-square.
What color is used to highlight an angle greater than a right angle? (Blue).
Less direct? (Green).
Which of the three proposed angles is a straight line?
Why did you decide so? (The vertex and sides of the angle coincide with the right angle on the square ruler).
How to determine the type of angle?
CONCLUSION:
To determine the type of angle, you need to combine its vertex and side, respectively, with the vertex and side of the right angle on the square.
Each of the corners has its own name. An acute angle is an angle that is less than a right angle. An obtuse angle is an angle that is larger than a right angle.
(Tables with the names of the angles appear on the board)
Working with textbook text in an orange frame p. 53.
My mother took the piece of paper
And folded the corner
This is the angle for adults
It's called DIRECT.
If the corner is already SHARP,
If wider, then - DUMB.
V .Formulation of the topic and objectives of the lesson.
VI . Physical education minute.
How many mushrooms are there?
We squat a lot.
How many flowers are there?
We raise our hands.
We raise our hands,
We clear the clouds.
Brighter, sunshine, shine,
Ban the gloomy rain.
The long journey is over.
You can sit down and relax.
VII . Application of new knowledge.
Independent work. (Multi-level tasks)
Card No. 1.
1.Write the names of the angles
2.Divide the angles into groups:
Card No. 2
Circle all the figures for which the statement “The figure has an obtuse angle” is true.
Card No. 3
4.Write the names of acute, right and obtuse angles
Sharp corners: ___________________________________
Right angles:__________________________________________
Obtuse angles:__________________________________________
VIII. Mathematics and folklore.(Sector “Mathematics and folklore”)
- The creativity of the Russian people is closely connected with mathematics . People use the word with great pleasure corner in their proverbs and sayings. What proverbs and sayings did you find at home?
Now listen to my proverbs and sayings.
A house cannot be built without corners; speech cannot be said without a proverb.
The hut is red not in its corners, but in its pies.
If you say it from ear to ear, they will know from corner to corner.
Threshing - so from the edge, and at the table - so he climbed into the corner.
IX . Mathematics and ecology.(Sector “Mathematics and Ecology”)
Solving the problem. (Solve in different ways).
For the project “Mushrooms of the Bryansk Forest,” the children made 12 dummies of mushrooms. 4 of them were milk mushrooms, 5 were chanterelles, and the rest were porcini mushrooms. How many dummies of porcini mushrooms have the children made?
X . Logics.(Sector “Logic”)
The children put into boxes dummies of mushrooms brought to create a corner of the Bryansk forest. Find out where each mushroom is located if all the labels on the boxes are false.
Here Here Here
milk mushroom there is no russula. boletus
XI . Lesson summary. Reflection.
There is wire on your desks. Make a right angle out of it and test it with a square, then make it sharp and obtuse.
(Slide 10.)
Tell me, using a diagram, what did you learn from today's math lesson?
XII. Homework.(Sector “DZ”)
P. 53, No. 6, No. 7 – optional
An acute angle is an angle whose degree measure is up to 90 degrees.
A right angle is an angle whose degree measure is 90 degrees.
An obtuse angle is an angle whose degree measure is greater than 90 degrees. An acute angle is an angle less than 90°. An obtuse angle is an angle greater than 90° but less than 180°. A right angle is an angle = 90°.
20. What angles are called adjacent? What is their sum?
Adjacent angles- two angles with a common vertex, one of whose sides is common, and the remaining sides lie on the same straight line (not coinciding). The sum of adjacent angles is 180°. Or
Two angles are called adjacent, if they have one side in common, and the other sides are additional rays. the sum of adjacent angles is 180°. Each of these angles complements the other to the full angle.
21. What angles are called vertical? What property do they have?
Vertical angles - two angles in which the sides of one are continuations of the sides of the other. Vertical angles are equal. ( Angles are called vertical formed by intersecting straight lines and not adjacent to each other, that is, they do not have a common side, but vertical angles have a vertex at one point. Vertical angles are equal to each other).
22. Which lines are called perpendicular? Two intersecting lines are called perpendicular(or mutually perpendicular) if they form four right angles. Or Perpendicular lines These are straight lines intersecting at an angle of 90 degrees. Or Two straight lines forming right angles when intersecting, are called perpendicular.
23. Explain what segment is called a perpendicular drawn from a given point to a given line. What is the base of a perpendicular? A line segment perpendicular to a given line is called, which has one of its ends at their intersection point. This end of the segment is called the base of the perpendicular. Perpendicular to a given line A line segment perpendicular to a given line is called, which has one of its ends at their intersection point. The end of a segment lying on a given line , is called the base of the perpendicular.
24. What is a theorem and proof of the theorem? In mathematics, a statement whose validity is established by reasoning is called a theorem, and the reasoning itself is called a proof of the theorem.
Theorem- a statement for which there is a proof (in other words, a conclusion) in the theory under consideration. Unlike the theorems, axioms are statements that, within the framework of a particular theory, are accepted as true without any evidence or justification. Proof is a statement explaining the theorem. Theorem - a hypothesis that needs to be proven; A hypothesis always requires proof. Proof - arguments confirming the effectiveness and correctness of the theorem.
An angle is a geometric figure that consists of two different rays emanating from one point. In this case, these rays are called sides of the angle. The point that is the beginning of the rays is called the vertex of the angle. In the picture you can see the angle with the vertex at the point ABOUT, and the parties k And m.
Points A and C are marked on the sides of the angle. This angle can be designated as angle AOC. In the middle there must be the name of the point at which the vertex of the angle is located. There are also other designations, angle O or angle km. In geometry, instead of the word angle, a special symbol is often written.
Developed and non-expanded angle
If both sides of an angle lie on the same straight line, then such an angle is called expanded angle. That is, one side of the angle is a continuation of the other side of the angle. The figure below shows the expanded angle O.
It should be noted that any angle divides the plane into two parts. If the angle is not unfolded, then one of the parts is called the internal region of the angle, and the other is called the external region of this angle. The figure below shows an undeveloped angle and marks the outer and inner regions of this angle.
In the case of a developed angle, either of the two parts into which it divides the plane can be considered the outer region of the angle. We can talk about the position of a point relative to an angle. A point can lie outside the corner (in the outer region), can be located on one of its sides, or can lie inside the corner (in the inner region).
In the figure below, point A lies outside angle O, point B lies on one side of the angle, and point C lies inside the angle.
Measuring angles
To measure angles there is a device called a protractor. The unit of angle is degree. It should be noted that each angle has a certain degree measure, which is greater than zero.
Depending on the degree measure, angles are divided into several groups.
This article will discuss one of the basic geometric shapes - an angle. After a general introduction to this concept, we will focus on a specific type of such a figure. Straight angle is an important concept in geometry, which will be the main topic of this article.
Introduction to Geometric Angle
In geometry there are a number of objects that form the basis of all science. The angle refers to them and is defined using the concept of a ray, so let's start with it.
Also, before you begin to determine the angle itself, you need to remember several equally important objects in geometry - this is a point, a straight line on a plane, and the plane itself. A straight line is the simplest geometric figure that has neither beginning nor end. A plane is a surface that has two dimensions. Well, a ray (or half-line) in geometry is a part of a line that has a beginning, but no end.
Using these concepts, we can make a statement that an angle is a geometric figure that lies entirely in a certain plane and consists of two divergent rays with a common origin. Such rays are called sides of an angle, and the common beginning of the sides is its vertex.
Types of angles and geometry
We know that angles can be completely different. Therefore, a little below will be a small classification that will help you better understand the types of angles and their main features. So, there are several types of angles in geometry:
- Right angle. It is characterized by a value of 90 degrees, which means that its sides are always perpendicular to each other.
- Sharp corner. These angles include all their representatives that are less than 90 degrees in size.
- Obtuse angle. Here there can be all angles ranging from 90 to 180 degrees.
- Unfolded corner. It has a size of strictly 180 degrees and externally its sides form one straight line.
The concept of a straight angle
Now let's look at the rotated angle in more detail. This is the case when both sides lie on the same straight line, which can be clearly seen in the figure a little lower. This means that we can say with confidence that in a reversed angle, one of its sides is essentially a continuation of the other.
It is worth remembering the fact that such an angle can always be divided using a ray that emerges from its apex. As a result, we get two angles, which in geometry are called adjacent.
Also, the unfolded angle has several features. In order to talk about the first of them, you need to remember the concept of “angle bisector”. Recall that this is a ray that divides any angle exactly in half. As for the unfolded angle, its bisector divides it in such a way that two right angles of 90 degrees are formed. This is very easy to calculate mathematically: 180˚ (degree of the rotated angle): 2 = 90˚.
If we divide a rotated angle with a completely arbitrary ray, then as a result we always get two angles, one of which will be acute and the other obtuse.
Properties of rotated corners
It will be convenient to consider this angle, bringing together all its main properties, which is what we did in this list:
- The sides of the rotated angle are antiparallel and form a straight line.
- The rotated angle is always 180˚.
- Two adjacent angles together always form a straight angle.
- A full angle, which is 360˚, consists of two unfolded ones and is equal to their sum.
- Half of a straight angle is a right angle.
So, knowing all these characteristics of this type of angles, we can use them to solve a number of geometric problems.
Problems with rotated angles
To see if you have grasped the concept of a straight angle, try answering the following few questions.
- What is the magnitude of a straight angle if its sides form a vertical line?
- Will two angles be adjacent if the first is 72˚ and the other is 118˚?
- If a complete angle consists of two reverse angles, then how many right angles does it have?
- A straight angle is divided by a ray into two angles such that their degree measures are in the ratio 1:4. Calculate the resulting angles.
Solutions and answers:
- No matter how the rotated angle is located, it is always, by definition, equal to 180˚.
- Adjacent angles have one side in common. Therefore, to calculate the size of the angle they make together, you just need to add the value of their degree measures. This means 72 +118 = 190. But by definition, a reversed angle is 180˚, which means that two given angles cannot be adjacent.
- A straight angle contains two right angles. And since the complete one has two unfolded ones, it means there will be 4 straight lines.
- If we call the desired angles a and b, then let x be the coefficient of proportionality for them, which means that a=x, and accordingly b=4x. The rotated angle in degrees is 180˚. And according to its properties that the degree measure of an angle is always equal to the sum of the degree measures of those angles into which it is divided by any arbitrary ray that passes between its sides, we can conclude that x + 4x = 180˚, which means 5x = 180˚ . From here we find: x = a = 36˚ and b = 4x = 144˚. Answer: 36˚ and 144˚.
If you were able to answer all these questions without prompts and without peeking at the answers, then you are ready to move on to the next geometry lesson.