Let's say we have a line to which we need to set a perpendicular, i.e. another line at an angle of 90 degrees relative to the first. Or we have an angle (for example, the corner of a room) and we need to check if it is equal to 90 degrees.
All this can be done with just a tape measure and a pencil.
There are two great things like " egyptian triangle”and the Pythagorean theorem, which will help us with this.
When the causes and goals are found, the search for innovative knowledge will be a natural consequence. You have to be optimistic, but that's not enough. Beliefs must be translated into actions. If possible, not in isolated activities. If the class is the only space you need to have, you need to properly occupy it and make real what you once dreamed of.
The origin of geometry is somewhat obscure, as one of the many knowledge about mathematics, in which it is impossible to attribute its discovery to one person. However, it is believed to have its beginnings in Egypt and the earliest evidence of modern geometry dates from around 600 BC.
So, egyptian triangle is a right triangle with the ratio of all sides equal to 3:4:5 (leg 3: leg 4: hypotenuse 5).
The Egyptian triangle is directly related to the Pythagorean theorem - the sum of the squares of the legs is equal to the square of the hypotenuse (3*3 + 4*4 = 5*5).
How can this help us? Everything is very simple.
Task number 1. You need to draw a perpendicular to a straight line (for example, a line at 90 degrees to a wall).
Despite its importance in the historical and cultural context, geometry has not been sufficiently studied. At the same time, the skills that will be developed in students are outdated. According to the teaching proposal of Santa Catarina regarding the teaching of geometry and the competencies to be developed in the student, some factors must be taken into account.
The study or study of physical space and forms. Orientation and visualization and representation of physical space. Visualization and understanding of geometric shapes. Designation and recognition of forms according to their characteristics. Classification of objects according to their shapes.
Step 1. To do this, from point No. 1 (where our corner will be) you need to measure on this line any distance that is a multiple of three or four - this will be our first leg (equal to three or four parts, respectively), we get point No. 2.
For ease of calculation, you can take a distance, for example 2m (these are 4 parts of 50cm each).
The study of the properties of figures and the relationships between them. Construction of geometric figures and models. Construction and justification of relations and prepositions based on hypothetical deductive reasoning. For this, competencies related to geometry must be transferred from the second year elementary school taking into account the level of absorption of the content of the student.
In society, it is accepted and accepted, the principle "to do mathematics - to solve problems." In this regard, the solution of the problem is a subject for researchers and mathematicians. Understanding the difficulties that most students face in this vital activity faces great challenges. The first, of course, is an accurate understanding of the problem. For Lakatos and Marconi, "a problem is a difficulty, theoretical or practical, in knowing something of real value for which a solution must be found", and this understanding is of fundamental importance for students to work on solving the problem.
Step 2. Then, from the same point No. 1, we measure 1.5 m (3 parts of 50 cm each) up (set an approximate perpendicular), draw a line (green).
Step 3. Now from point number 2 you need to put a mark on the green line at a distance of 2.5m (5 parts of 50cm). The intersection of these marks will be our point number 3.
By connecting points No. 1 and No. 3, we get a line perpendicular to our first line.
First, it can be said that problem solving, as a strategy for the development of mathematics education, should get rid of this sense of "necessary evil" created by an endless list of "problems" that, as a rule, at the end of each unit of the program, the teacher presents to students.
The traditional use of problems, which is to apply and systematize knowledge, attracts the dislike and disinterest of the student, hindering their full intellectual development. The over-preparation of definitions, methods, and demonstrations becomes a routine and mechanical activity in which only the final product is evaluated. Failure to follow the stages of research and transmission of logical-mathematical ideas does not allow building concepts. Thus, "mathematical knowledge does not represent a student as a system of concepts, which allows him to solve many problems, but as an endless symbolic, abstract, incomprehensible speech."
Task number 2. Second situation- there is an angle and you need to check whether it is straight.
Here it is, our corner. It's much easier to check with a large square. And if he is not?
>>Geometry: Egyptian triangle. Complete Lessons
Mathematical knowledge has only evolved from many answers to many questions asked throughout history. Creativity, critical rewriting, curiosity and enjoyment were the fuel that fueled this process of discovery. According to Paul, a problem solving scheme.
The systematic use of this scheme helps the student to organize his thinking. The confrontation of his initial idea of a solution with that of a colleague or group promotes learning, thus re-evaluating the role of the teacher. The earliest evidence of the beginnings of trigonometry arose both in Egypt and in Babylon, from the calculation of the ratios between numbers and between the sides of similar triangles.
Lesson topic
Lesson Objectives
- Get acquainted with new definitions and recall some already studied.
- Deepen knowledge of geometry, study the history of origin.
- To consolidate the theoretical knowledge of students about triangles in practical activities.
- To introduce students to the Egyptian triangle and its application in construction.
- Learn to apply the properties of shapes in solving problems.
- Developing - to develop students' attention, perseverance, perseverance, logical thinking, mathematical speech.
- Educational - through a lesson, to cultivate an attentive attitude towards each other, to instill the ability to listen to comrades, mutual assistance, independence.
Lesson objectives
- Check students' ability to solve problems.
Lesson Plan
- Introduction.
- Good to remember.
- Triangle.
introduction
Did the ancient Egyptians know mathematics and geometry? They not only knew, but also constantly used it to create architectural masterpieces and even ... in the annual marking of fields on which water destroyed all the boundaries during a flood. There was even a special service of land surveyors who quickly restored the boundaries of the fields with the help of geometric techniques when the water subsided.
The Achem papyrus is the most extensive Egyptian document on mathematics that has come to this day. Who was in the power of the scribe Ahmes. The Babylonians took a great interest in astronomy, both for religious reasons and for its connection with the calendar and planting seasons. It is impossible to study the phases of the moon, the cardinal points and the seasons of the year without the use of triangles, a system of units and a scale.
This study is further subdivided into two parts: planar trigonometry and spherical trigonometry. The application of trigonometry in various fields of exact sciences is an indisputable fact. Knowing this truth is of fundamental importance for high school students, and it is the responsibility of the math teacher to bring this subject to the best of his ability, creating the necessary connection with regard to future professional choices. At present, trigonometry is not limited to the study of triangles. Its application extends to other areas of mathematics such as "Analysis" and other areas of human activity such as electricity, mechanics, acoustics, music, topography, civil engineering, etc.
It is not yet known what we will call our young generation, which grows up on computers that allow us not to memorize the multiplication table and not to perform other elementary mathematical calculations or geometric constructions in our minds. Maybe human robots or cyborgs. The Greeks, on the other hand, called those who could not prove a simple theorem without outside help, profane. Therefore, it is not surprising that the very theorem, which was widely used in applied sciences, including for marking fields or building pyramids, was called by the ancient Greeks the “bridge of donkeys”. And they knew Egyptian mathematics very well.
It is noted, however, that one of the biggest difficulties faced by high school students, as discussed in Trigonometry, has to do with the fact of memorizing formulas. However, non-memorization would take time to deduce during tests, which would make the situation impossible.
Here we present some of the basic relations and theorems related to geometry and more specifically trigonometry. Recall that the causes and respectively representing sine, cosine and tangent are valid for the previously discovered triangle and do not need to be embellished or taken as a rule, thus the concept is evaluated rather than memorization of the formula.
Good to remember
Triangle
Triangle rectilinear, part of the plane bounded by three line segments (sides of the Triangle (in geometry)), having in pairs one common end (vertices of the Triangle (in geometry)). A triangle in which the lengths of all sides are equal is called equilateral, or correct, Triangle with two equal sides - isosceles. The triangle is called acute-angled if all its angles are acute; rectangular- if one of its corners is right; obtuse- if one of its corners is obtuse. A triangle (in geometry) cannot have more than one right or obtuse angle, since the sum of all three angles is equal to two right angles (180° or, in radians, p). The area of a Triangle (in geometry) is equal to ah/2, where a is any of the sides of the Triangle taken as its base, and h is the corresponding height. The sides of the Triangle are subject to the condition: the length of each of them is less than the sum and greater than the difference in the lengths of the other two sides.
The main evolution of trigonometric concepts occurred after the use of the trigonometric cycle, formerly called the trigonometric circle. These are "coordinate axes that have as their unit of measure the radius of an oriented circle coinciding with the center of coordinates of the coordinate axes."
Euler, born in Basel, was one of the best and most productive mathematicians in history, and with his aforementioned contributions, he agreed to use one ray for the trigonometric cycle. Thus, "as the cycle is oriented, each degree measure will correspond to one point in the cycle."
Triangle- the simplest polygon, having 3 vertices (corners) and 3 sides; a part of a plane bounded by three points and three line segments connecting these points in pairs.
With this definition one can establish the same concepts for sine, cosine and tangent as follows. Consider the figure to the side, where a trigonometric circle is depicted. That is: the cosine of a right triangle is equal to the adjacent leg divided by its hypotenuse, the hypotenuse being the opposite of the right angle.
Recall that the radius of a trigonometric circle is 1, it is concluded that the sine and cosine of the arc are real numbers that vary in the real interval from -1 to. The scale adopted on the tangent axis is the same as for the abscissa and ordinate axes.
- Three points in space that do not lie on one straight line correspond to one and only one plane.
- Any polygon can be divided into triangles - this process is called triangulation.
- There is a section of mathematics entirely devoted to the study of the patterns of triangles - Trigonometry.
Triangle types
By the type of angles
Given the following representation for the law of breasts. The proportions related to the law of the mammary gland, indicated above, are determined by the following definition. Given the following representation for the law of cosine. According to the law of cosines, as indicated above, a triangle of any squared measure of one side is equal to the sum of the squared measures of the other two sides, minus twice the product of the measures of those sides and the cosine of the angle they form.
The purpose of this chapter is to develop a curriculum for the content of trigonometry, based on problematization, contextualization and historical research, in order to make learning by students. It is emphasized that it is understood that the training plan is a necessary condition for guiding the learning process by teaching any content, it emphasizes, as we will see below, the content, objectives, development of the plan, the materials to be And how to evaluate the content to be administered.
Since the sum of the angles of a triangle is 180°, at least two angles in a triangle must be acute (less than 90°). There are the following types of triangles:
- If all angles of a triangle are acute, then the triangle is called acute;
- If one of the angles of the triangle is obtuse (greater than 90°), then the triangle is called obtuse;
- If one of the angles of a triangle is right (equal to 90°), then the triangle is called a right triangle. The two sides forming a right angle are called the legs, and the side opposite the right angle is called the hypotenuse.
By the number of equal sides
Based on the thematic project, trigonometry arose: problematization and contextualization. Contextualize subject trigonometry using a historical approach and exploring physical space and forms present in the environment. Provide an environment for students to learn the basics of trigonometry.
Recognize in which areas it spreads and what impact it causes. Provide students with methods to facilitate understanding, interpretation and problem solving. The content of trigonometry will be applied according to the material designed to trace the content, which will follow the steps below.
- A triangle is called scalene if the lengths of three sides are pairwise different.
- An isosceles triangle is one in which two sides are equal. These sides are called the sides, the third side is called the base. In an isosceles triangle, the angles at the base are equal. The height, median and bisector of an isosceles triangle, lowered to the base, are the same.
- An equilateral triangle is one in which all three sides are equal. In an equilateral triangle, all angles are equal to 60 °, and the centers of the inscribed and circumscribed circles coincide.
As for the study, this can be done in groups and divided by topic. Socialization can be done through a presentation worthy of the creativity and interest of each group. After the presentation, the instructor can make their own placements, prioritizing the importance of the content.
Trigonometry is the branch of mathematics that studies triangles, especially triangles in the plane where one of the triangle's angles measures 90 degrees. He also specifically studies the relationships between the sides and angles of triangles; Trigonometric functions and calculations based on them. The trigonometric approach penetrates other areas of geometry, such as the study of spheres using spherical trigonometry.
- a right triangle with an aspect ratio of 3:4:5. The sum of these numbers (3+4+5=12) has been used since ancient times as a unit of multiplicity when constructing right angles using a rope marked with knots at 3/12 and 7/12 of its length. The Egyptian triangle was used in the architecture of the Middle Ages to build proportionality schemes.
The origin of trigonometry is unknown. A triangle is a geometric figure with three sides and three angles. To form a triangle, simply connect all three points with line segments if they are not aligned. Below are the triangles. The aperture obtained by two lines connected by the same point is called an angle, which has as its international measuring system radian, and degree is also very useful. In triangles, the sum of their interior angles is 180°.
A right angle is indicated by a symbol. In a right triangle, the opposite side of the right angle is called the hypotenuse. Some authors believe that Pythagoras was a student of the Tales, Eve, when he said that "he was fifty years younger than this and lived near Miletus, where Thales lived." Boyer says that "although some of the claims state that Pythagoras was a student of the Tales, this hardly makes a difference of half a century between his ages."
So where do you start? Is it from this: 3 + 5 = 8. and the number 4 is half of the number 8. Stop! The numbers 3, 5, 8... Don't they look very familiar? Well, of course, they are directly related to the golden ratio and are included in the so-called "golden row": 1, 1, 2, 3, 5, 8, 13, 21
... In this series, each subsequent term is equal to the sum of the two previous ones: 1 + 1= 2. 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8
and so on. It turns out that the Egyptian triangle is related to the golden ratio? And did the ancient Egyptians know what they were dealing with? But let's not jump to conclusions. It is necessary to find out the details more precisely.
Expression " golden ratio”, according to some, was first introduced in the 15th century Leonardo da Vinci
. But the “golden row” itself became known in 1202, when it was first published in his “Book of Accounts” by an Italian mathematician Leonardo of Pisa
. Nicknamed Fibonacci. However, almost two thousand years before them, the golden ratio was known Pythagoras and his students. True, it was called differently, as "division in the middle and extreme ratio." And here is the Egyptian triangle with its The "golden ratio" was known back in those distant times when the pyramids were built in Egypt when Atlantis flourished.
To prove the Egyptian triangle theorem, it is necessary to use a straight line segment of known length A-A1 (Fig.). It will serve as a scale, a unit of measurement, and will allow you to determine the length of all sides of the triangle. Three segments A-A1 are equal in length to the smallest of the sides of the triangle BC, whose ratio is 3. And four segments A-A1 are equal in length to the second side, whose ratio is expressed by the number 4. And, finally, the length of the third side is equal to five segments A-A1. And then, as they say, a matter of technology. On paper, draw a segment BC, which is the smallest side of the triangle. Then, from point B with a radius equal to the segment with a ratio of 5, we draw an arc of a circle with a compass, and from point C, an arc of a circle with a radius equal to the length of the segment with a ratio of 4. If now the intersection point of the arcs is connected by lines with points B and C, then we get a right-angled triangle with an aspect ratio of 3: 4: 5.
Q.E.D.
The Egyptian triangle was used in the architecture of the Middle Ages to build proportionality schemes and to build right angles by land surveyors and architects. The Egyptian triangle is the simplest (and first known) of the Heronian triangles - triangles with integer sides and areas.
Egyptian triangle - a mystery of antiquity
Each of you knows that Pythagoras was a great mathematician who made an invaluable contribution to the development of algebra and geometry, but he gained even more fame thanks to his theorem.
And Pythagoras discovered the Egyptian triangle theorem at the time when he happened to visit Egypt. While in this country, the scientist was fascinated by the splendor and beauty of the pyramids. Perhaps this was precisely the impetus that made him think that some definite pattern was clearly traced in the forms of the pyramids.
Discovery history
The name of the Egyptian triangle was due to the Hellenes and Pythagoras, who were frequent guests in Egypt. And it happened around the 7th-5th centuries BC. e.
The famous pyramid of Cheops, in fact, is a rectangular polygon, but the sacred Egyptian triangle is considered to be the pyramid of Khafre.
The inhabitants of Egypt compared the nature of the Egyptian triangle, as Plutarch wrote, with the family hearth. In their interpretations, one could hear that in this geometric figure, its vertical leg symbolized a man, the base of the figure belonged to the feminine, and the hypotenuse of the pyramid was assigned the role of a child.
And already from the topic studied, you are well aware that the aspect ratio of this figure is 3:4:5 and, therefore, that this leads us to the Pythagorean theorem, since 32 + 42 = 52.
And if we take into account that the Egyptian triangle lies at the base of the Khafre pyramid, then we can conclude that the people of the ancient world knew the famous theorem long before it was formulated by Pythagoras.
The main feature of the Egyptian triangle, most likely, was its peculiar ratio of sides, which was the first and simplest of the Heronian triangles, since both the sides and its area had integers.
Features of the Egyptian triangle
And now let's take a closer look at the distinctive features of the Egyptian triangle:
• Firstly, as we have already said, all its sides and area consist of integers;
• Secondly, by the Pythagorean theorem we know that the sum of the squares of the legs is equal to the square of the hypotenuse;
• Thirdly, with the help of such a triangle it is possible to measure right angles in space, which is very convenient and necessary in the construction of structures. And the convenience lies in the fact that we know that this triangle is a right triangle.
• Fourth, as we also already know that even if there are no appropriate measuring instruments, then this triangle can easily be built using a simple rope.
Application of the Egyptian triangle
In ancient times, the Egyptian triangle was very popular in architecture and construction. It was especially necessary if, to build right angle use rope or cord.
After all, it is known that laying a right angle in space is a rather difficult task, and therefore the enterprising Egyptians invented an interesting way to construct a right angle. For these purposes, they took a rope, on which twelve even parts were marked with knots, and then a triangle was folded from this rope, with sides that were equal to 3, 4 and 5 parts, and as a result, without any problems, they got a right triangle. Thanks to such an intricate tool, the Egyptians measured the land for agricultural work with great accuracy, built houses and pyramids.
This is how visiting Egypt and studying the features of the Egyptian pyramid prompted Pythagoras to discover his theorem, which, by the way, got into the Guinness Book of Records as the theorem that has the largest amount of evidence.
Reuleaux triangular wheels
Wheel- a round (as a rule), freely rotating or fixed on an axis disk, which allows the body placed on it to roll rather than slide. The wheel is widely used in various mechanisms and tools. Widely used for cargo transportation.
The wheel significantly reduces the energy costs for moving the load on a relatively flat surface. When using a wheel, work is done against the rolling friction force, which under artificial road conditions is significantly less than the sliding friction force. The wheels are solid (for example, the wheelset of a railway car) and consisting of quite a large number parts, for example, a car wheel includes a disk, a rim, a tire, sometimes a camera, mounting bolts, etc. Car tire wear is almost a solved problem (with properly set wheel angles). Modern tires travel over 100,000 km. An unresolved problem is tire wear on aircraft wheels. When a stationary wheel comes into contact with the concrete surface of the runway at a speed of several hundred kilometers per hour, tire wear is enormous.
- In July 2001, an innovative patent was obtained for the wheel with the following wording: "a round device used for transporting goods." This patent was issued to John Cao, a lawyer from Melbourne, who wanted to show the imperfection of the Australian patent law.
- The French company Michelin in 2009 developed a mass-produced Active Wheel with built-in electric motors that drive the wheel, spring, shock absorber and brake. Thus, these wheels make the following vehicle systems unnecessary: engine, clutch, gearbox, differential, drive and cardan shafts.
- In 1959, the American A. Sfredd received a patent for a square wheel. It easily walked through snow, sand, mud, overcame pits. Contrary to fears, the car on such wheels did not "limp" and developed a speed of up to 60 km / h.
Franz Relo(Franz Reuleaux, September 30, 1829 - August 20, 1905) - German mechanical engineer, lecturer at the Berlin Royal Academy of Technology, who later became its president. He was the first, in 1875, to develop and outline the main provisions of the structure and kinematics of mechanisms; dealt with the problems of aesthetics of technical objects, industrial design, in his designs he gave great importance external forms of machines. Reuleaux is often called the father of kinematics.
Questions
- What is a triangle?
- Types of triangles?
- What is the peculiarity of the Egyptian triangle?
- Where is the Egyptian triangle used? > Mathematics Grade 8
