Subject: mathematics
Grade: Grade 3
Textbook: "Mathematics" part 2.
Topic: Types of triangles
Lesson type: discovery of new knowledge
Target: Learn to identify the types of triangles by measuring the lengths of their sides.
Tasks :
1) Update knowledge about geometric shapes - rectangle, square, triangle.
2) Update the addition and subtraction of three-digit numbers, the division of a two-digit number into one-digit, two-digit and round; multiplying a two-digit number by a one-digit number.
3) Enter the terms: isosceles, equilateral, scalene triangle.
During the classes
1. Motivation for learning activities
Look, tell me what it is?
(pyramid)
Tell me, what does it consist of? (of parts, levels...)
Can this pyramid be compared with our knowledge? (Yes)
Every day you build more and more pyramids, each level of the pyramid is a new knowledge that you get in the lesson. And what will happen to the pyramid if we remove the blue level? (It will collapse, become smaller.)
And how can our pyramid of knowledge collapse because of what? (Due to unfulfilled d / s, missed lessons, do not listen carefully to the teacher.)
What needs to be done to make our pyramid stronger and grow? (To learn lessons, to work well in class, to do homework, not to skip school.)
Guys, you said everything right. Now let's imagine that our pyramid has cast a shadow. What geometric shape does the shadow look like?
(To the triangle.)
Today we will continue to work with such a geometric figure as a triangle.
2. Actualization of knowledge and fixation of difficulties in a problem situation
What geometric shapes are you familiar with? (square, rectangle, triangle).
There is a table on the board, fill it out based on your knowledge (each student has a card with such a table):
What are the names of the first two geometric figures? (rectangle and square, in a word, these are quadrilaterals.)
What types of quadrilaterals do you know? The image on the slide will help you answer this question.
The names of the quadrilaterals appear after the children's answers.
(rhombus, square, rectangle, trapezoid, parallelogram - they are called by the images on the slide or board.)
Can you tell what is a rectangle and what is a square?
(A rectangle is a quadrilateral with all right angles.
A square is a rectangle with all sides equal
Find an extra geometric figure based on the results of the table. (Triangle).
Okay, quadrilaterals are all very different, but what do you know about a triangle? (Triangles are: acute, obtuse, rectangular.)
What else do you know about the triangle? (Definition)
A triangle is a geometric figure that has 3 angles, 3 vertices, 3 sides.
Complete the following table based on your knowledge:
(The teacher fills in the table according to the children's answers. Different opinions appear in the "name" columns, and some children leave them blank.)
3. Identification of the place and cause of the difficulty.
What task did you do? (Fill in the table.)
Where did the difficulty arise? (When writing the names of triangles)
Why was there a problem? (We don't know what they are called)
What is the purpose of the lesson? (Find out what other types of triangles there are other than those studied (obtuse-angled, acute-angled, rectangular), learn to identify these types of triangles.)
What is the topic of our lesson? (Types of triangles)
4. Discovery of new knowledge.
Let's get back to the table.
Enter the dimensions of the sides of the triangles. (Enter.)
Okay, now look and tell me what you noticed? (The first triangle has all sides equal, the second has 2 equal sides, and the third has different sides.)
Right, but can you think of names for these triangles based on the explanation you just gave? (Yes)
What do you call a triangle with all sides equal? Think of an adjective consisting of 2 words: equal sides. (Equilateral)
What is the name of a triangle in which all sides are different? (Versatile)
What is the name of a triangle that has 2 equal sides? (Children have doubts, to answer this question they use the textbook p.73) (Isosceles) And what other triangle can we call isosceles? (Equilateral)
Complete the table yourself, based on new knowledge.
Can we now define the types of triangles? (Yes)
Equilateral A triangle with all three sides equal.
Isosceles A triangle that has at least two equal sides. An equilateral triangle is also an equilateral triangle.
Versatile A triangle with all sides different.
Check your definitions p.73 -tutorial. (Check.)
Are you correct in your definitions? (Yes.)
5. Primary consolidation with pronunciation in external speech
Complete the task from the textbook p.74 (under?)
1) Versatile: 2,3,5
2) Isosceles: 1,4 , 6, 7
(Students write in notebooks. Take turns saying answers, arguing. The sample is fixed on the board).
6. Independent work with self-checking according to the standard.
Completing the task on your own. At the end of the work - self-examination according to the model (on the board or on individual cards).
№1.Fill in the table , schematically depict triangles.
№2. Write down the numbers:
1) Scalene triangles.
2) Isosceles, from the numbers written out, underline the numbers of equilateral triangles.
Reference:
Task number 1:
Task number 2:
1) Scalene triangles: 2,3,4
2) Isosceles triangles (the number of an equilateral triangle is underlined): 1,5
7.Inclusion in the knowledge system and repetition
The boy drew triangles on the sand and encrypted the words, find the meanings of the expressions written in the triangles. First solve those that are written in scalene triangles, and then in isosceles triangles. And guess the encrypted words.
Hint: Write the numbers in ascending order and you will get words.
Card:
Solution:
Answer: Types of triangles
8. Reflection of educational activity.
Draw accordingly the pyramid of knowledge, consisting of 7 levels. Each level is the answer to a question.
Answer the questions:
1) Guys, what did you write down “types of triangles”? (the topic of our lesson)
2) What was our goal? (Learn how all 3 types of triangles are called, learn to identify these types by measuring the lengths of the sides.)
3) What types of triangles did you recognize? (scalene, isosceles, equilateral)
4) Why are they called that?
( Equilateral A triangle with all sides equal.
Isosceles - a triangle with at least two equal sides, including an equilateral triangle, because it has two equal sides.)
Versatile A triangle with all sides different.
5) Have you learned how to schematically depict all types of triangles? (Yes, on my own.)
6) What discoveries did you make today? (New types of triangles, their names.)
7) Guys, can you determine the type of triangle by its measurements? (Yes) I will now tell you the measurements, and you raise up a card with the name of the type of triangle (the cards were issued additionally - 3 cards each.)
1. 2 cm, 3 cm, 5 cm - versatile
2. 4cm, 4cm, 2cm - isosceles
3.6cm, 6cm,6cm - equilateral, isosceles
Raise your hands, who has reached the pinnacle of this knowledge today? (Raise)
And raise your hands, who lacked 1, 2 levels. (They raise.)
(The teacher analyzes the "pyramids of knowledge in children, draws conclusions - what level sinks and in the next lesson starts updating knowledge from this.)
When studying mathematics, students begin to get acquainted with various types of geometric shapes. Today we will talk about different types of triangles.
Definition
Geometric figures that consist of three points that are not on the same straight line are called triangles.
The line segments connecting the points are called sides, and the points are called vertices. Vertices are denoted by capital Latin letters, for example: A, B, C.
The sides are indicated by the names of the two points of which they consist - AB, BC, AC. Intersecting, the sides form angles. The bottom side is considered the base of the figure.
Rice. 1. Triangle ABC.
Types of triangles
Triangles are classified according to angles and sides. Each type of triangle has its own properties.
There are three types of triangles in the corners:
- acute-angled;
- rectangular;
- obtuse.
All angles acute-angled triangles are acute, that is, the degree measure of each is no more than 90 0.
Rectangular the triangle contains a right angle. The other two angles will always be acute, because otherwise the sum of the angles of the triangle will exceed 180 degrees, which is impossible. The side that is opposite the right angle is called the hypotenuse, and the other two legs. The hypotenuse is always greater than the leg.
obtuse the triangle contains an obtuse angle. That is, an angle greater than 90 degrees. The other two angles in such a triangle will be acute.
Rice. 2. Types of triangles in the corners.
A Pythagorean triangle is a rectangle whose sides are 3, 4, 5.
Moreover, the larger side is the hypotenuse.
Such triangles are often used to compose simple problems in geometry. Therefore, remember: if two sides of a triangle are 3, then the third one will definitely be 5. This will simplify the calculations.
Types of triangles on the sides:
- equilateral;
- isosceles;
- versatile.
Equilateral a triangle is a triangle in which all sides are equal. All angles of such a triangle are equal to 60 0, that is, it is always acute-angled.
Isosceles a triangle is a triangle with only two equal sides. These sides are called lateral, and the third - the base. In addition, the angles at the base of an isosceles triangle are equal and always acute.
Versatile or an arbitrary triangle is a triangle in which all lengths and all angles are not equal to each other.
If there are no clarifications about the figure in the problem, then it is generally accepted that we are talking about an arbitrary triangle.
Rice. 3. Types of triangles on the sides.
The sum of all the angles of a triangle, regardless of its type, is 1800.
Opposite the larger angle is the larger side. And also the length of any side is always less than the sum of its other two sides. These properties are confirmed by the triangle inequality theorem.
There is a concept of a golden triangle. This is an isosceles triangle, in which two sides are proportional to the base and equal to a certain number. In such a figure, the angles are proportional to the ratio 2:2:1.
A task:
Is there a triangle whose sides are 6 cm, 3 cm, 4 cm?
Solution:
To solve this task, you need to use the inequality a
What have we learned?
From this material from the 5th grade mathematics course, we learned that triangles are classified by sides and angles. Triangles have certain properties that can be used when solving problems.
Of all the polygons triangles have the least number of angles and sides.
Triangles can be distinguished by the shape of their angles.
If all angles of a triangle are acute, then it is called an acute triangle.(Fig. 113, a).
If one of the angles of a triangle is right, then it is called a right triangle.(Fig. 113, b).
If one of the angles of a triangle is obtuse, then it is called an obtuse triangle.(Fig. 113, c).
They say that we classified triangles according to their angles.
Triangles can be classified not only by the type of angles, but also by the number of equal sides.
If two sides of a triangle are equal, then it is called an isosceles triangle.
Figure 114, a shows an isosceles triangle ABC, in which AB \u003d BC. In the figure, equal sides are marked with an equal number of dashes. Equal sides AB and BC are called sides, and the side AC − basis isosceles triangle ABC.
If the sides of a triangle are equal, then it is called an equilateral triangle.
The triangle shown in Figure 114b is equilateral, it has MN = NE = EM.
A triangle with three sides of different lengths is called a scalene triangle.
The triangles shown in Figure 113 are scalene. If the side of an equilateral triangle is a, then its perimeter is calculated by the formula:
P = 3a
Example 1 . Using a ruler and a protractor, construct a triangle whose two sides are 3 cm and 2 cm and the angle between them is 50°.
Using a protractor, we will construct an angle A, the degree measure of which is 50 ° (Fig. 115). On the sides of this angle from its apex, using a ruler, set aside a segment AB 3 cm long and a segment AC 2 cm long ( fig. 116). Connecting points B and C with a segment, we get the desired triangle ABC ( fig. 117).
Example 2 . Using a ruler and a protractor, construct a triangle ABC whose side AB is 2 cm and whose angles CAB and CBA are respectively 40° and 110°.
Solution. Using a ruler, we build a segment AB 2 cm long ( fig. 118). From the beam AB with the help of a protractor we set aside an angle with a vertex at point A, the degree measure of which is 40 °. From the ray BA in the same direction from the straight line AB, in which the first angle was plotted, we lay off the angle with the vertex at point B, the degree measure of which is 110 ° (Fig. 119).
Having found the point C of the intersection of the sides of the angles A and B, we obtain the desired triangle ABC (Fig. 120).
Today we are going to the country of Geometry, where we will get acquainted with different types of triangles.
Examine the geometric shapes and find the “extra” among them (Fig. 1).
Rice. 1. Illustration for example
We see that figures No. 1, 2, 3, 5 are quadrangles. Each of them has its own name (Fig. 2).
Rice. 2. Quadrangles
This means that the "extra" figure is a triangle (Fig. 3).
Rice. 3. Illustration for example
A triangle is a figure that consists of three points that do not lie on the same straight line, and three segments connecting these points in pairs.
The points are called triangle vertices, segments - his parties. The sides of the triangle form There are three angles at the vertices of a triangle.
The main features of a triangle are three sides and three corners. Triangles are classified according to the angle acute, rectangular and obtuse.
A triangle is called acute-angled if all three of its angles are acute, that is, less than 90 ° (Fig. 4).
Rice. 4. Acute triangle
A triangle is called right-angled if one of its angles is 90° (Fig. 5).
Rice. 5. Right Triangle
A triangle is called obtuse if one of its angles is obtuse, i.e. greater than 90° (Fig. 6).
Rice. 6. Obtuse Triangle
According to the number of equal sides, triangles are equilateral, isosceles, scalene.
An isosceles triangle is a triangle in which two sides are equal (Fig. 7).
Rice. 7. Isosceles triangle
These sides are called lateral, Third side - basis. In an isosceles triangle, the angles at the base are equal.
Isosceles triangles are acute and obtuse(Fig. 8) .
Rice. 8. Acute and obtuse isosceles triangles
An equilateral triangle is called, in which all three sides are equal (Fig. 9).
Rice. 9. Equilateral triangle
In an equilateral triangle all angles are equal. Equilateral triangles always acute-angled.
A triangle is called versatile, in which all three sides have different lengths (Fig. 10).
Rice. 10. Scalene triangle
Complete the task. Divide these triangles into three groups (Fig. 11).
Rice. 11. Illustration for the task
First, let's distribute according to the size of the angles.
Acute triangles: No. 1, No. 3.
Right triangles: #2, #6.
Obtuse triangles: #4, #5.
These triangles are divided into groups according to the number of equal sides.
Scalene triangles: No. 4, No. 6.
Isosceles triangles: No. 2, No. 3, No. 5.
Equilateral Triangle: No. 1.
Review the drawings.
Think about what piece of wire each triangle is made of (fig. 12).
Rice. 12. Illustration for the task
You can argue like this.
The first piece of wire is divided into three equal parts, so you can make an equilateral triangle out of it. It is shown third in the figure.
The second piece of wire is divided into three different parts, so you can make a scalene triangle out of it. It is shown first in the picture.
The third piece of wire is divided into three parts, where the two parts are the same length, so it can be made into an isosceles triangle. It is shown second in the picture.
Today in the lesson we got acquainted with different types of triangles.
Bibliography
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 1. - M .: "Enlightenment", 2012.
- M.I. Moro, M.A. Bantova and others. Mathematics: Textbook. Grade 3: in 2 parts, part 2. - M .: "Enlightenment", 2012.
- M.I. Moreau. Mathematics lessons: Guidelines for teachers. Grade 3 - M.: Education, 2012.
- Regulatory document. Monitoring and evaluation of learning outcomes. - M.: "Enlightenment", 2011.
- "School of Russia": Programs for elementary school. - M.: "Enlightenment", 2011.
- S.I. Volkov. Mathematics: Testing work. Grade 3 - M.: Education, 2012.
- V.N. Rudnitskaya. Tests. - M.: "Exam", 2012.
- Nsportal.ru ().
- Prosv.ru ().
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Homework
1. Finish the phrases.
a) A triangle is a figure that consists of ..., not lying on the same straight line, and ..., connecting these points in pairs.
b) The points are called … , segments - his … . The sides of a triangle form at the vertices of a triangle ….
c) According to the size of the angle, triangles are ..., ..., ....
d) According to the number of equal sides, triangles are ..., ..., ....
2. Draw
a) a right triangle
b) an acute triangle;
c) an obtuse triangle;
d) an equilateral triangle;
e) scalene triangle;
e) an isosceles triangle.
3. Make a task on the topic of the lesson for your comrades.
A triangle (from the point of view of Euclid's space) is such a geometric figure, which is formed by three segments connecting three points that do not lie on one straight line. The three points that form a triangle are called its vertices, and the segments connecting the vertices are called sides of the triangle. What are triangles?
Equal Triangles
There are three signs of the equality of triangles. What triangles are called equal? These are the ones who:
- two sides and the angle between these sides are equal;
- one side and two angles adjacent to it are equal;
- all three sides are equal.
Right triangles have the following signs of equality:
- along an acute angle and hypotenuse;
- along an acute angle and leg;
- on two legs;
- along the hypotenuse and cathetus.
What are triangles
According to the number of equal sides, a triangle can be:
- Equilateral. It is a triangle with three equal sides. All angles in an equilateral triangle are 60 degrees. In addition, the centers of the circumscribed and inscribed circles coincide.
- Unequilateral. A triangle with no equal sides.
- Isosceles. It is a triangle with two equal sides. Two identical sides are the sides, and the third side is the base. In such a triangle, the bisector, median and height coincide if they are lowered to the base.
According to the size of the angles, a triangle can be:
- Obtuse - when one of the angles has a value of more than 90 degrees, that is, when it is obtuse.
- Acute-angled - if all three angles in the triangle are acute, that is, they have a value of less than 90 degrees.
- Which triangle is called a right triangle? This is one that has one right angle equal to 90 degrees. The legs in it will be called the two sides that form this angle, and the hypotenuse is the side opposite the right angle.
Basic properties of triangles
- A smaller angle always lies opposite the smaller side, and a larger angle always lies opposite the larger side.
- Equal angles always lie opposite equal sides, and opposite sides always lie different angles. In particular, in an equilateral triangle, all angles have the same value.
- In any triangle, the sum of the angles is 180 degrees.
- An external angle can be obtained by extending one of its sides to a triangle. The value of the outer angle will be equal to the sum of the inner angles not adjacent to it.
- The side of a triangle is greater than the difference of its other two sides, but less than their sum.
In the spatial geometry of Lobachevsky, the sum of the angles of a triangle will always be less than 180 degrees. On a sphere, this value is greater than 180 degrees. The difference between 180 degrees and the sum of the angles of a triangle is called a defect.