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.
If problems can be solved at the school boundary, the teacher will escort students to where they can be solved. A theodolite is an "optical instrument for accurately measuring horizontal and vertical angles" used in topography, surveying, and geodesy. It is basically a telescope with graduated movements vertically and horizontally and is mounted on a centered and vertical tripod, and may or may not have a built-in compass.
To solve problems created by the students themselves, they can build a homemade theodolite according to the following model. A homemade model of theodolite is available on the site. The lid of the cup will serve as the basis for the rotation of the theodolite and must be glued upside down so that its center is aligned with the center of the protractor, which will give the theodolite greater accuracy. To find the center of the cap, trace it with two diameters. And make a hole where they intersect. These types of lids usually have indentations on the edge that can help you find the right place.
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).
Use thin wire as a guide to align the center of the lid with the center of the protractor. The lid of the cup will serve as the basis for the rotation of the theodolite and must be glued upside down so that its center is aligned with the center of the protractor, which will give theodolite greater accuracy.
The thin wire will be the pointer of the theodolite, which will allow you to read in degrees into the protractor. To install it, drill two diametrically opposed holes in the side of the glass, next to your mouth, and thread the wire through the holes, leaving it through the glass.
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 antenna tube will be a crosshair where you will see the points to be measured. Glue the tube to the base of the glass so that it is parallel to the pointer. To improve this crosshair, glue two pieces of line at the end of the tube to form a cross.
Finish by attaching the cup to the lid. The homemade version works like the real thing. With it, you measure from your position the angle formed between two other points. Horizontally or vertically, simply align the 0° pointer of the protractor with one of the dots and rotate the sight until you see the other dot. The pointer will indicate how many degrees the deviation is.
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.
As far as the resolution of the exercises is concerned, it is worth remembering that they are aimed at practicing learned concepts, not at memorization for later use as standards for solving similar problems. Of the exercises in this teaching plan, all of them can be used only if the teacher considers it necessary, otherwise enough to heal the doubts and form the students' concepts. Excessive use of repetitive exercises can disrupt student interest in the content.
For better preparation The teacher is advised to have as much information as possible about the subject, so it is important that he knows the whole project developed here, as well as the topics: "Mathematics", "Teaching Mathematics", "Teaching Geometry", "Problem Solving" and "Trigonometry". ".
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?
The famous mathematician Pythagoras made many different discoveries, but for most people who do not have to regularly deal with algebra and geometry, he is known for his theorem. The scientist discovered it while in Egypt, where he was fascinated by the beauty and elegance of the pyramids, and this, in turn, led him to the idea that a certain pattern can be traced in their forms.
To develop the content of trigonometry, it will be necessary to use: a pencil, a pen, rubber, a protractor, a ruler, a notebook, a blackboard, chalk, a geoplan, a homemade theodolite and tape. The content assessment will take into account the participation and interest of students in the subject, which highlights the contribution of possible students to the development of concepts, as well as development at the moment of decision-making. All works can be assessed: historical research, development and problem solving, building a home theodolite and evaluation.
Students are expected to submit a report in the form of a portfolio after each activity in order to monitor the activities developed by each student, identify strengths and weaknesses in the learning process and the need for a formal written assessment.
Discovery history
The Egyptian triangle owes its name to the Hellenes, who often visited Egypt in the 7th-5th centuries BC. e., among them was Pythagoras. The basis of the pyramid of Cheops is a rectangular polygon, and the pyramids of Khafre are the so-called Egyptian triangle, which the ancients called sacred. Plutarch wrote that the inhabitants of Egypt correlated nature with this geometric figure: the vertical leg symbolized a man, the base - a woman, and the hypotenuse - a child. The aspect ratio in it is 3:4:5, and this leads to the Pythagorean theorem, since 3 2 x 4 2 \u003d 5 2. Therefore, the fact that the Egyptian triangle lies at the base of the Khafre pyramid allows us to assert that the famous theorem was known to the inhabitants ancient world even before Pythagoras formulated it. A feature of this figure is also considered to be that, due to this aspect ratio, it is the first and simplest of Heron's triangles, since its sides and area are integer.
Mathematics comes throughout its history, undergoing changes. For a long time the main concern with such advances, whether practical or theoretical, has been to apply them to advance the knowledge of mankind.
Over time, the concern about the need to spread this knowledge, giving everyone the opportunity to relate them, also begins to worry about how they are taught in school. That is, with processes adopted by teachers that guarantee everyone's right to know.
Today, when school mathematics is mostly treated formally and abstractly, it is of paramount importance that the teacher begins to reflect on what methodology or methodology might be more appropriate for a particular content. It is in the future not just to convey all the content, but rather to learn it.
Application
The Egyptian triangle has been popular in architecture and construction since antiquity.
It was mainly used when building right angles with a cord or rope divided into 12 parts. According to the marks on such a rope, it was possible to very accurately create a rectangular figure, the legs of which would serve as guides for installation right angle buildings. It is known that such properties of this geometric figure were used not only in ancient Egypt, but also, long before that, in China, Babylon and Mesopotamia. The Egyptian triangle was also used to create proportional structures in the Middle Ages.
These premises underlie the development of this work, which also aims to provide a tool for future discussions on the possibilities of improving the quality of teaching, especially in the field of geometry. During the construction of the curriculum, it was noted that it was possible to move away from traditional approaches, starting with a more dynamic and efficient classroom, generating interest and achieving student learning.
Learning content is inherently as important as enjoying it, and in the times we live, we as teachers can make a difference by looking for a more cultured, organized, and better organized society. When we start thinking a little like the geniuses of the past, we will understand what they were really looking for in their discoveries and will fully understand the content they write down, ensuring both the students' understanding of the content they are learning and for us that we are teaching.
corners 
The ratio of the sides of this triangle 3:4:5 leads to the fact that it is rectangular, i.e. one angle is 90 degrees, and the other two are 53.13 and 36.87 degrees. A right angle is an angle between sides whose ratio is 3:4.
Proof
With some simple calculations, you can prove that a triangle is a right triangle. If we follow the reverse theorem of the one that Pythagoras created, that is, if the sum of the squares of the two sides is equal to the square of the third, then it is rectangular, and since its sides lead to the equality 3 2 x 4 2 \u003d 5 2, therefore, it is rectangular.
Summing up, it should be noted that the Egyptian triangle, whose properties have been known to mankind for many centuries, continues to be used in architecture today. This is not at all surprising, because this method guarantees accuracy, which is very important in construction. In addition, it is very easy to use, which also makes the process much easier. All the benefits of using this method have been tested for centuries and remain popular to this day.
Difficulties arose throughout the work, but were seen as a challenge, and remuneration should include the possibility of an effective methodology aimed at teaching students. It should be emphasized that there was no claim to create "the most appropriate methodology to approach the proposed content", but rather to present a "way" of learning that met the expectations of teachers and students. In this sense, the realization that the subject has not been exhausted, but begun.
After creating this training plan, we feel the need to apply it and put into practice what was idealized at the time, which we are sure will raise new doubts and new research. Ministry of Education. National educational plans for primary education. Introductory Document: Preliminary Version.
