Design and research work on the similarity of triangles in real life. Project incomparable likeness Incomparable likeness project

Project name

Brief summary of the project

The project was prepared using design technology. Implemented as part of the 8th grade geometry program on the topic “Signs of similarity of triangles.” The project includes an information and research part. Analytical work with information systematizes knowledge about such figures. Independent research of students, as well as acquired practical knowledge, skills and abilities teach them to see the importance of this theoretical material when applying it in practice. Didactic tasks will help monitor the degree of mastery of educational material.

Guiding Questions

The fundamental question is: “Does nature speak the language of similarity?”

“Is it possible to find examples of similarity around us?”, “How can I measure the height of my house?”, “Why are such triangles needed?”

Project plan

1.Brainstorming (formation of student research topics).

2. Formation of groups to conduct research, put forward hypotheses, discuss ways to solve problems.

3.Choice of a creative name for the project.

4. Discussion of the plan for theoretical and practical work of students in the group.

5. Discussion with students of possible sources of information.

6.Independent work of groups.

7. Students prepare presentations and reports on progress reports.

8. Presentation of research works.

Sections: Mathematics

Class: 8

An opportunity to introduce schoolchildren to educational activities of a creative nature is provided by mathematical tasks, as well as the project method, designed to develop curiosity, responsibility, the ability to work with information, the ability to work collectively - in a group, etc.

This project is proposed to be completed by 8th grade students. The project was developed within the framework of the topic “Similar figures”, for which 19 hours of teaching time are allocated. An educational project on this topic is perceived with great interest by students and makes it possible to create conditions under which students, on the one hand, can independently master new knowledge and methods of action, and on the other hand, apply previously acquired knowledge and skills in practice. In this case, the main emphasis is on the creative development of the individual.

Students work in groups; during the final discussion, the results of each group become the property of everyone else.

The project was prepared outside of school hours by 8th grade students.

The project includes an information and research part.

Based on the study of sources, students:

• learn the possibility of using signs of similarity of triangles in life;
• systematize knowledge about such figures.
• expand their horizons of knowledge;
• study the meaning of this topic in geometry lessons.

Independent research of students, as well as acquired practical knowledge, skills and abilities teach them to see the importance of this theoretical material when applying it in practice.

Didactic tasks will help monitor the degree of mastery of educational material.

Methodical presentation

1. Introduction.
2. Methodological passport of the educational project.
3. Project implementation stages
4. Implementation of the project.
5. Conclusions.
6. Student work as part of an educational project.

1. Introduction

“A project is a set of certain actions, documents, the creation of various kinds of theoretical products. This is always a creative activity. The project method is based on the development of students’ cognitive creative skills; the ability to independently construct one’s knowledge, the ability to navigate the information space, the development of critical thinking.” (E.S. Polat).

The teacher in this situation is not only an active participant in the educational process: he not only teaches, but understands and feels how the child learns himself.

The teacher helps students find sources; he himself is a source of information; coordinates the entire process; maintains continuous contact with children. Organizes the presentation of work results in various forms.

When analyzing an educational project, the teacher mentally imagines the children’s reaction, considers the form of the proposal to consider the problem, find a solution to the project problem, and plunge into the situation of the plot.

A project is the result of coordinated joint actions of a group or several groups of students.

2. Project passport

Project name : Matchless likeness

Project topic: Similar figures.

Type of project: educational.

Project typology: practice-oriented, individual-group.

Subject areas: mathematics.

Hypothesis: If a person knows the signs of similarity of triangles, will there be a need to apply them in life?

Problematic issues:

1. Where can the similarity of triangles be used in measurement?

2. Why do people make models to illustrate or explain certain objects or phenomena?

3. Why does a small negative make a large, high-quality photograph?

4. How to achieve what seems unattainable?

5. Why does similarity exist in the world?

7. Is it important in life to study the signs of similarity of triangles?

The goal of the project: to deepen and expand knowledge on the topic “Similar figures”.

Methodological objectives of the project:

• study the similarity characteristics of triangles;
• evaluate the importance of the topic “Similarity”
• develop the ability to apply theoretical material when solving practical problems;
• consolidate the acquired theoretical knowledge in practice;
• develop an interest in science and technology by searching for examples of the application of this topic in life;
• expand your mathematical horizons and explore new approaches to solving problems;
• acquire research skills.

Project participants: 8th grade students. Time spent on the project: February–March 2014.

Material, technical, educational and methodological equipment: educational and educational literature, additional literature, computer with Internet access.

3. Project implementation stages

Stage 1 – immersion in the project (updating knowledge; formulating topics; forming groups) (week);

Stage 2 – organization of activities (information collection; group discussion) (week);

Stage 3 – implementation of activities (research; conclusions (month);

Stage 4 – presentation of the project product (2 weeks).

4. Project implementation

Stage 1: Immersion in the project (preparatory stage)

Having chosen their research topics, students divided into groups, defined tasks and planned their activities.

5 project groups of 5 people were formed.

The following topics for future projects were selected:

1. From the history of similarity.

2. Similarity in GIA problems. (Real mathematics)

Similarities in our lives:

3. Determining the height of an object.

4. Similarity in nature.

5. Will the similarity of triangles help people of different professions?

The role of the teacher is to guide based on motivation.

Stage 2: search and research:

Students studied additional literature, collected information on their topic, distributed responsibilities in each group (depending on the selected individual research topic); made the necessary instruments for research, conducted research, and prepared a visual presentation of their research.

The role of the teacher is observational and consulting; students mostly worked independently.

Stage 3: results and conclusions:

Students analyzed the information they found and formulated conclusions. We compiled the results, prepared materials for defending the project, and created presentations

Stage 4: presentation and defense of the project:

During the conference, students publicly present the result of their project activities in the form of a multimedia presentation.

The role of the teacher is collaboration.

5. General conclusions. Conclusion

The implementation of this educational project allowed students to develop their skills in working not only with additional sources in mathematics, but also with a computer, to develop skills in working on the Internet, as well as students’ communication abilities.

Participation in the project allowed us to deepen our knowledge of the application of mathematics in various fields, as well as consolidate knowledge on this topic. It should be noted that the knowledge obtained during the implementation of the project is extracted for a specific purpose and is the object of interest of the student. This promotes their deep absorption.

In general, the work on the project was successful, almost all 8th grade students took part in it. Everyone was involved in mental activity on this issue and acquired new knowledge through independent work. Each member of the group spoke in defense of their project. At the final stage, practical work methods were tested and self-analysis was carried out in the form of a presentation.

Students' project activities contribute to true learning because... she:

1. Personally oriented.
2. Characterized by an increase in interest and involvement in the work as it is completed.
3. Allows you to realize pedagogical goals at all stages.
4. Allows you to learn from your own experience, from the implementation of a specific case.
5. Brings satisfaction to students who see the product of their own labor.

These valuable moments that participation in projects provide must be used more widely in the practice of developing the intellectual and creative abilities of schoolchildren. Thus, the use of the method of educational projects in pedagogical work is determined by the need to form a personality of the 21st century, a personality of a new era, when human intelligence and information will be the determining factors in the development of society.

The work is based on the study of the possibility of using the similarity of triangles in real life, experiments were carried out on measuring length using an altimeter.

"11Sushko-t.doc"

SIMILARITY OF TRIANGLES IN REAL LIFE

Sushko Daria Olegovna

8th grade student

KU "OSH"I - III steps No. 11, Yenakievo"

Ikaeva Marina Aleksandrovna

Mathematic teacher,II category

KU "OSH"I - III steps No. 11, Yenakievo"

[email protected]

Geometry originated in ancient times. The world we live in today is also filled with geometry. All objects around us have geometric shapes. These are buildings, streets, plants, household items. The relevance of my topic lies in the fact that without any tools, only relying on the similarity of triangles, you can measure the height of a pillar, bell tower, tree, the width of a river, lake, ravine, the length of an island, the depth of a pond, etc.

The goal of the work was to find areas of application of triangle similarity in real life.

The objectives of the work were

Objects and subjects of research : height: pillar; tree, pyramid model.

During the work, the following methods were used: literature review, practical work, comparison.

The work is practice-oriented in nature, since the practical significance of the work lies in the possibility of using the research results in geometry lessons and in everyday life.

As a result of the work, measurements were taken of the height of a pole, tree, and models made by the author.

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Content:

Introduction

The concept of similarity of figures. Signs of similarity.

4.1 Determining height by shadow

4.2. Measuring height using the Jules Verne method

4.3. Measuring height using an altimeter

5. Conclusions

Introduction.

Geometry originated in ancient times. Building dwellings and temples, decorating them with ornaments, marking the ground, measuring distances and areas, people applied their knowledge about the shape, size and relative position of objects, obtained from observations and experiments. The world we live in today is also filled with geometry. All objects around us have geometric shapes. These are buildings, streets, plants, household items. In everyday life, we often encounter figures of the same shape, but different sizes. Such figures in geometry are called similar. My work is devoted to the similarity of triangles, because, while studying this topic in mathematics lessons, I became interested in how the concept of similarity of triangles and signs of similarity are used in practice. The relevance of my topic is that without any tools, you can measure the height of a pillar, bell tower, tree, the width of a river, lake, ravine, the length of an island, the depth of a pond, etc.

The objectives of my work were

study literature on this topic;

study the history of the concept of similarity;

find out where the similarity of triangles is used;

measure the height of the pillar using the similarity of triangles in various ways;

2. The legend of Thales measuring the height of the pyramid.

There are many mysterious stories and legends associated with the pyramid. One hot day, Thales, together with the chief priest of the Temple of Isis, walked past the Pyramid of Cheops.

“Look,” continued Thales, “at this very time, no matter what object we take, its shadow, if we place it vertically, is exactly the same height as the object!” In order to use the shadow to solve the problem of the height of the pyramid, it was necessary to already know some geometric properties of the triangle, namely the following two (of which Thales discovered the first himself):

1. That the angles at the base of an isosceles triangle are equal, and vice versa - that the sides lying opposite the equal angles of the triangle are equal to each other; 2. That the sum of the angles of any triangle is equal to two right angles.

Only Thales, armed with this knowledge, had the right to conclude that when his own shadow is equal to his height, the sun's rays meet the level ground at an angle of half a straight line, and therefore the top of the pyramid, the middle of its base and the end of its shadow must mark an isosceles triangle. It would seem that this simple method is very convenient to use on a clear sunny day to measure lonely trees whose shadow does not merge with the shadow of neighboring ones. But in our latitudes it is not as easy as in Egypt to wait for the right moment for this: Our sun is low above the horizon, and shadows are equal to the height of the objects casting them only in the afternoon hours of the summer months. Therefore, Thales’ method in the indicated form is not always applicable.

The doctrine of the similarity of figures based on the theory of relations and proportions was created in Ancient Greece in the V-IV centuries. BC e. It is set out in Book VI of Euclid’s Elements (III century BC), which begins with the following definition: “Similar rectilinear figures are those that have respectively equal angles and proportional sides.”

3. The concept of similar figures.

In life, we encounter not only equal figures, but also those that have the same shape, but different sizes. Geometry calls such figures similar. Similar triangles are triangles in which the angles are respectively equal, and the sides of one are proportional to the similar sides of the other triangle. Triangle similarity features are geometric features that allow you to establish that two triangles are similar without using all the elements.

Signs of similarity of triangles.

4. Measuring work using similarity.

4.1. Determining height by shadow.

I decided to conduct an experiment to determine height by shadow.

For this I needed: a flashlight, a model of a pyramid, and a figurine. Making a miniature pyramid for experiments is not difficult. I needed: a sheet of paper; pencil; ruler; scissors; glue for paper. On a sheet of paper, I built a diagram of a pyramid, at the base of which is a square with a side of 7.6 cm, and the tank faces are equal isosceles triangles with a side side of 9.6 cm. The height of the resulting pyramid is 7.9 cm. The height of the figure is 8.1 cm. Let's try to measure the height of this pyramid by its shadow, also using the shadow of the figure. On a sunny day, I measured the shadow of the pyramid and the figure. I got it: 15 cm - the shadow of the figure, 13 cm - the shadow of the pyramid.

Let's build a geometric model of this problem:

, ∠ АСО= ∠ MLK as the angles of incidence of the sun's rays, which means at two angles.

Let us now find the height of the pyramid in another way to compare the results. Let's find the height of the side face: AB=

From this we find the height AO =

We got almost identical results. Having received these results, I decided to measure the height of the pole by going outside.

I chose a pillar from which a clear shadow fell and measured it. It was 21 m. Then I stood next to the pole and my assistant measured my shadow, it was 4.5 meters. My height, taking into account that I was wearing shoes and a hat, was 1.6.

Let's find the height of the pillar by creating a geometric model of the problem.

Let's consider KO - the length of my shadow, BC - the length of the pillar's shadow. AB – the desired one.

∠АВС=∠МКО= as the angles of incidence of the sun's rays.

4.2. Measuring the height of a pyramid using the Jules Verne method.

“The Mysterious Island” describes an interesting way of determining height: “The young man, trying to learn as much as possible, followed the engineer, who descended from the granite wall to the edge of the shore. Taking a straight pole, 12 feet long, the engineer measured it as accurately as possible, comparing it with his own height, which was well known to him. Herbert carried behind him the plumb line handed to him by the engineer: just a stone tied to the end of a rope. Not reaching 500 feet from the granite wall, which rose vertically, the engineer stuck a pole about two feet into the sand and, having firmly strengthened it, set it vertically with the help of a plumb line. Then he moved away from the pole to such a distance that, lying on the sand, he could lie in one straight line. lines to see both the end of the pole and the edge of the ridge. he carefully marked this point with a peg.

Are you familiar with the rudiments of geometry? - he asked Herbert, rising from the ground.

Do you remember the properties of similar triangles?

Their similar sides are proportional. - Right. So: now I will build two similar right triangles. The smaller one will have a vertical pole on one leg, and the distance from the peg to the base of the pole on the other; The hypotenuse is my line of sight. The legs of another triangle will be: a vertical wall, the height of which we want to determine, and the distance from the peg to the base of this wall; the hypotenuse is the line of sight that coincides with the direction of the hypotenuse of the first triangle.

Got it!” exclaimed the young man. “The distance from the peg to the pole is related to the distance from the peg to the base of the wall, as the height of the pole is to the height of the wall.” - Yes. And therefore, if we measure the first two distances, then, knowing the height of the pole, we can calculate the fourth, unknown term of the proportion, i.e. the height of the wall. We will thus do without directly measuring this height. Both horizontal distances were measured, the shorter being 15 feet and the longer being 500 feet. At the end of the measurements, the engineer made the following entry:

4.3 Determining altitude using an altimeter

Height can be measured with a special device - an altimeter. To make this device you will need: Thick white cardboard, ruler, pen, pencil, scissors, thread, weight, needle.

7. On it, we bend two rectangles measuring 3x5 cm from the sides and cut two holes with different diameters: one smaller one - near the eye, the other larger one - in order to point at the top of the tree. So, I decided to conduct an experiment and test this method of measuring the height of an object. As the object to be measured, I chose a tree growing near the school.

I moved 21 steps away from the object being measured, that is, EO = 6.3 m. I measured the readings of the device, it showed 0.7. My height is 1.6 m. I need to find the height of the tree.

To do this, we will build a geometric model of this problem:

=

Let's add my height to the resulting value and get: LV=LO+OB=3.71

1.6=5.31 – tree height.

Also, I could have made mistakes in using the device. Errors in using and manufacturing the device:

1.If you do not bend the upper rectangle from the base, then you will incorrectly determine the height.

2.When measuring the height of an object, the weight must be aimed at a specific marking value.

3.The distance from the object being measured must be accurate.

4. Accurately apply 1 cm markings.

The experiment showed that the method of determining the height of an object using a height meter is more accurate and convenient.

5. Conclusions.

Literature

5. Perelman Ya. I. Entertaining geometry. – M.: State Publishing House of Technical and Theoretical Literature, 1950
There are 3 ways to measure the height of a tree.

1. General explanatory dictionary of the Russian language [Electronic resource]. – Access mode: http://tolkslovar.ru/p22702.html

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"Title page"

Municipal institution “Comprehensive school of I-III levels No. 11 in Enakievo”

"Mathematics around us"

Creative work on the topic

"Similarity of triangles in real life"

Performed

8th grade student

Sushko Daria

Supervisor

mathematic teacher

Ikaeva Marina Aleksandrovna

Enakievo 2017

View presentation content
"Similarity of triangles in real life"

Institution "Comprehensive school of І-ІІІ levels No. 11, Enakievo"

Competition of student creative projects

"Mathematics around us"

Creative work on the topic

"Similarity of triangles in real life"

Performed

8th grade student

Sushko Daria

Supervisor

mathematic teacher

Ikaeva Marina Aleksandrovna

Enakievo 2017

The goal of my work was to find areas of application of triangle similarity in real life.

The objectives of my work were

• study literature on this topic;
• study the history of the concept of similarity;
• find out where the similarity of triangles is used;
• measure the height of the pillar using the similarity of triangles in various ways;

The legend of Thales measuring the height of the pyramid

One hot day, Thales, together with the chief priest of the Temple of Isis, walked past the Pyramid of Cheops.

Does anyone know what its height is? - he asked.

No, my son,” the priest answered him, “the ancient papyri did not preserve this for us.” “But you can determine the height of the pyramid very accurately and right now!” Thales exclaimed.

“Look,” continued Thales, “at this very time, no matter what object we take, its shadow, if we place it vertically, is exactly the same height as the object!”

Concept similarities figures

Similar triangles are triangles in which the angles are respectively equal, and the sides of one are proportional to the similar sides of the other triangle.

Two figures are called similar if they are converted into each other by a similarity transformation

Triangle similarity features are geometric features that allow you to establish that two triangles are similar without using all the elements.

If two angles of one triangle are respectively equal to two angles of another, then such triangles are similar.

If two sides of one triangle are proportional to two sides of another triangle and the angles between these sides are equal, then the triangles are similar.

If three sides of one triangle are proportional to three sides of another triangle, then the triangles are similar.

Measuring height by shadow

Initial data of the problem: The length of the shadow of the pyramid BC = 11 cm, the length of the shadow of the figurine KL = 15 cm, the height of the figurine KM = 8 cm, the base of the pyramid is a square with a side of 7.6 cm. The height of the pyramid AO is the required one.

Consider the right triangles AOS and MKL:

, ∠ АСО= ∠ МЛК as the angles of incidence of the sun's rays, which means at two angles.

Measuring the height of a pillar by its shadow

Let's consider, KO is the length of my shadow, BC is the length of the shadow of the pillar. AB – the desired one.

∠ ABC = ∠ MKO = as the angles of incidence of the sun's rays.

Thus, I got an approximate value of the pillar height of 7.46 m.

Measuring height using the Jules Verne method

This method involves driving a pole into the ground and lying on the ground so that the top end of the pole and the top of the object being measured are visible. Measure the distance from the pole to the object, measure the height of the pole and the distance from the top of the person’s head to the base of the pole.

In Jules Verne's novel The Mysterious Island, both horizontal distances were measured: the smaller was 15 feet, the larger was 500 feet. At the end of the measurements, the engineer made the following entry:

15: 500 = 10:x, 500 X 10 = 5000, 5000: 15 = 333.3.

Measuring height using an altimeter

1. Draw and cut out a square measuring 15x15cm from cardboard.

2. Divide the square into two rectangles: 5x15 cm, 10x15 cm.

3. Divide a 10x15 cm rectangle into two parts: 5 cm and 10 cm.

4. On the larger part with a length of 10 cm, we apply centimeter divisions and denote them with a decimal fraction, that is, 0.1;0.2, etc.

5. At point E, use a needle to make a hole and drag the thread and weight through, and then fasten the thread at the back.

6. To make it easier to watch, bend the upper rectangle from the base.

7. On it, we bend two rectangles measuring 3x5 cm from the sides and cut two holes with different diameters: one smaller one - near the eye, the other larger one - in order to point at the top of the tree.

Measuring height using an altimeter

To find the height of the LV, you need to add your height to the LO.

LV=LO+OV=3.71+1.6=5.31 – tree height.

Conclusions:

After completing my work, I learned that there are many different ways to determine the height of an object. I conducted an experiment to determine the height of an object by its shadow. I carried out the test at home on a model of a pyramid and a figurine, as well as on the street when measuring the height of a pillar. Also, I looked at Jules Verne's method for determining height. I studied the concept of an altimeter and made an altimeter device, which I used in practice to measure the height of a selected object. The most convenient way for me to measure height was to use an altimeter. Thus, the goals of my work have been achieved. We can safely say that the similarity of triangles is used in real life when measuring work on the ground.

Literature:

1. Glazer G.I. History of mathematics at school. – M.: Publishing House “Prosveshcheniye”, 1964.

2. Perelman Ya. I. Entertaining geometry. – M.: State Publishing House of Technical and Theoretical Literature, 1950.

3.J.Vern. Mysterious Island. - M: Children's Literature Publishing House, 1980.

4. Geometry, 7 – 9: textbook. for general education institutions / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev et al. – 18th ed. – M.: Education, 2010 Used materials and Internet resources.

5. Perelman Ya. I. Entertaining geometry. – M.: State Publishing House of Technical and Theoretical Literature, 1950 You can measure the height of a tree in 3 ways.

1. General explanatory dictionary of the Russian language [Electronic resource]. - Access mode: http://tolkslovar.ru/p22702.html

2. Figure 2 [Electronic resource]. – Access mode: http://www.dopinfo.ru

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