Mathematics teacher Nailya Rakhimovna Sarimova

MBOU Malobugulma comprehensive secondary school

Bugulminsky district of the Republic of Tatarstan

Lesson topic: Measuring work on the ground

(for students5-7 class)

Anyone who studies mathematics from childhood develops attention, trains their brain, their will, and develops perseverance and perseverance in achieving goals.(A. Markushevich)

For those who have at least once experienced the joyful feeling of solving a difficult problem, have known the joy of a small, but discovery, and every problem in mathematics is a problem that humanity has been working towards solving for many years, and children will strive to learn more and more and use , apply the acquired knowledge in life. This type of work will help the teacher to captivate students, develop the beginnings of mathematical and logical thinking, expand the student’s horizons, creative work, and awaken the desire to study one of the most interesting sciences. This desire depends not only on work in the classroom, but also on practical training.

The purpose of the lesson: To familiarize students with the methods of measuring work on the ground, to familiarize students with such tools as: tape measure, pole, plumb line, compass, eker, tell how to use them.

Tasks:

- educational: teach how to use and apply these tools when solving problems using measurement methods, improve independent work skills

-developing: develop logical thinking, memory, attention, the ability to draw up a solution plan and draw conclusions, develop cognitive interests, and self-control skills.

- educational: to cultivate accuracy, hard work, perseverance, the desire to complete the work started, a sense of mutual assistance and mutual support.

Lesson type: lesson on learning new material

Forms of student work: work in groups, in pairs

When selecting the content of each lesson on a given topic and forms of student activity, the following principles are used: the relationship of theory with practice, scientific character, and clarity.

taking into account the age and individual characteristics of students;

combinations of collective and individual activities of participants;

differentiated approach;

Criteria for assessing the achievement of expected results:

student activity;

students' independence in completing tasks;

practical applications of mathematical knowledge;

level of creative abilities of the participants.

Preparing and conducting such lessons allows you to:

connect, awaken and develop the potential abilities of students;

identify the most active and capable participants;

to cultivate the moral qualities of the individual: hard work, perseverance in achieving goals, responsibility and independence.

teach to apply mathematical knowledge in everyday practical life.

Lesson structure

Before carrying out measuring work on the ground, familiarize students with the following tools:

Roulette- a tool for measuring length. It is a metal or plastic tape with marked divisions, which is wound on a reel enclosed in a housing equipped with a special mechanism for winding up the tape. The winding mechanism can be one of two types: with a return spring - then the tape is wound when released, and is removed from the tape measure body with some force; with a rotating handle protruding outward and connected to a tape spool - then the tape is wound up when the handle rotates.

Veshka It is a straight wooden pole or light metal tube 1.5 - 3 m long with a pointed end for sticking into the ground. Poles are used for hanging lines, marking points and installing various devices when performing geodetic work. The simplest design poles for hanging lines and marking points. They can be temporary or permanent. Milestones (poles) are stakes that are driven into the ground.

Surveying compass(field compass - fathom) - an instrument in the shape of the letter A, 1.37 m high and 2 m wide, for measuring distances on the ground; for students it is more convenient to take the distance between the legs to be 1 meter.

Ecker consists of two bars located at right angles and mounted on a tripod. Nails are driven into the ends of the bars so that the straight lines passing through them are mutually perpendicular.

Plumb(cord plumb line) - a device consisting of a thin thread and a weight at the end of it, allowing one to judge the correct vertical position, serving for vertical adjustment of surfaces (walls, piers, masonry, etc.) and racks (pillars, etc.). ). Under the influence of gravity, the thread takes a constant direction (plumb line).

The tip of the weight must be exactly on the continuation of the tensioned thread; for this purpose, the weight is given the appearance of an overturned cone placed on a cylinder; a small cylinder is screwed into the base of the cylinder so that their centers coincide; a thread with a knot at the end is passed into the central hole of the latter.

The plumb line is used to install slats in a vertical position for vertical adjustment when leveling an uneven position, in the designs of scales, spirit levels and in goniometer tools for setting the center of the dial above a point in the terrain.

Review with students the following concepts: straight line, segment, rectangle, length, width, height, volume, plan, scale, area of ​​a square and rectangle, average step length, perimeter, rules for rounding numbers.

Then students are given tasks:

Draw a straight line on the ground. Measure the length of a line segment.

Draw a rectangular plot on the ground and calculate its area and perimeter, rounding the answer to whole numbers.

Determine the area of ​​the school site. Make the necessary measurements and calculations. Draw this area on the plan, plan scale 1:50000. Give your answer in hectares.

Determine the average length of your step and use this to find the distance from school to the nearest store; Round the answer to the nearest metre.

The class is divided into 4 groups, each receives a set of necessary tools. Each group can perform work starting from any number. The groups draw up a report describing the progress of work and submit it for inspection. The teacher evaluates the correctness of the progress of the work, the accuracy of the calculations and the aesthetics of the design, and gives an overall assessment to the whole group.

Solving field measurement problems

(approximate description)

№1. D To construct a straight line segment on the ground, you need to construct three poles on the expected segment.

To check the correctness of the construction of the straight line, you need to stand in front of the outer pole and look at it so that all the poles merge into one. If at least one pole peeks out, you need to move it so that it is not visible.

Measuring the length of a segment on the ground is carried out using a measuring tape or an earthen compass, or a tape measure; you can measure it approximately with your step if the average step length is known.

A compass is used to find the length and width of a field; the distance between its ends AB can vary, usually about 1.5m or 2m.

In order to measure the length of a segment on the ground with its help, you need to walk with it along the segment, constantly turning it over at point C. How many times its length AB fits, multiply this number by 1.5 m or 2 m. Let's get the length of the required segment.

For example: l= 1.5*10=15(m) or l=2*10=20(m). (You can then check the length with a tape measure).

№2. To build a right angle on the ground, use an eker. These are two mutually perpendicular strips, at the ends of which nails are driven vertically. All this is mounted on a special tripod (tripod), and there is a plumb line in the center so that the device is strictly perpendicular to the surface of the earth. We need two more poles.

At point O we install an ecker, and at points A and B we install poles. You need to stand at point O and look at the ecker bars so that two opposite nails on one bar merge with the pole at the point. A and B. If both poles have merged, then the angle BOA = 90 degrees, i.e. right angle. If not, then you need to move the poles until they merge completely.

This way you can build a rectangle or square on the ground. Then you can find the lengths of their sides. We calculate the perimeter and area. We round the answer to a whole number.

For example: a=12m6dm, b=34m8dm; 1) P=2(126dm+348dm)=2*474dm=948dm=94m 8dm. Р=95m. 2). S=AB*BC, S=126*348(dm) =3848(dm squared)=385 m squared.

The calculation for a square is similar, only all sides are equal.

№3 . We will measure the school site using a tape measure or compass.

For example: We get a length of 450m, width of 100m. If the scale is 1:5000, then we will convert these dimensions to build a plan.

450m= 45000cm;

45000:5000=9 (cm) - on the plan;

100m=10000cm-on the ground;

10000:5000-2(cm) - on the plan. We get rectangle ABCD. S = 450 * 100 m = 45000 sq m = 450 a = 45 hectares.

№4 Determining the average length of your step. To do this, we build a straight line segment on the ground. The student takes 10 steps and measures the length of the resulting segment. Then divide this length by 10, do this several times, add the resulting results and divide by the number of attempts.

For example:

Number of attempts	Number of steps	Total length	Length 1 step
Average stride length

Each member of the group determines the distance from the school to the nearest store using the length of their step. Then find the average length of the distance.

For example:

Participants	Step length	Total steps	Distances

L= (310+293+292):3=895:3=298.3(m)=298m.

Municipal educational institution

"Velikodvorskaya basic secondary school"

I've done the work:

Anfalov Sergey Vasilievich, 8

Class

Velikodvorskaya secondary school Babushkinsky

Date of birth: 06.16.1995

Home address: 161344, Vologda

region, Babushkinsky district, Velikiy village

Dvor, no. 76.

Supervisor:

Belyaeva Elena Vasilievna,

physics and mathematics teacher

MOU "Velikodvorskaya main

comprehensive school"

School address: 161344, Vologda

region Babushkinsky district, Velikiy village

Velikiy Dvor village

2009

INTRODUCTION

The basic school geometry course examines tasks related to the practical application of the knowledge learned: measuring work on the ground, measuring instruments. Practical work on the ground is one of the most active forms of connecting learning with life, theory with practice. We learn to use reference books, apply the necessary formulas, and master practical techniques of geometric measurements and constructions. Practical work using measuring instruments increases interest in mathematics, and solving problems of measuring the width of a river, the height of an object and determining the distance to an inaccessible point allows you to apply them in practical activities and see the scale of application of mathematics in human life. As you study the material, the methods for solving these problems change; the same problem can be solved in many ways. In this case, the following questions of geometry are used: equality and similarity of triangles, relations in a right triangle, the theorem of sines and the theorem of cosines (9th grade), the Pythagorean theorem, properties of right triangles, etc. At school, we do geometric constructions in quite detail using compasses and rulers and solve many problems. How to solve the same problems on the ground? After all, it is possible to imagine such a huge compass that could outline the circumference of a school stadium or a ruler for marking park paths. In practice, cartographers have to use special methods to draw maps and surveyors to mark areas on the ground, for example, to lay the foundation of a house.

The topic of our essay: On-site measurement work.
Target: study of some methods for solving geometric problems on the ground.

To achieve this goal, we have identified the followingtasks:

● Explore theoretical and methodological literature on this issue.

● Show relationships mathematics and basic life safety.

● Apply theoretical knowledge in practice.

The objects of my observations were:

● Determining the height of an object.

● Distance to an inaccessible point.

MAIN PART.

One of the most active forms of connection between learning and life, theory and practice is the implementation of practical work related to measurement, construction, and depiction during geometry lessons. The same issues are discussed in the course on the basics of life safety, but all measurements are taken without special instruments. The work is carried out both on the ground and solving problems in the classroom in various ways to find the height of an object and determine the distance to an inaccessible point. According to the program, the geometry course covers the following issues:
7th grade
● “Drawing a straight line on the ground” (item 2).
● “Measuring tools” (clause 8).
● “Measuring angles on the ground” (clause 10).
● “Construction of right angles on the ground” (p. 13) ● “Construction tasks. Circle" (clause 21).
● “Practical methods for constructing parallel lines” (p. 26).
● “Criminal reflector” (clause 36).
● “Distance between parallel straight lines” (clause 37 – surface planer).
● “Construction of a triangle using three elements” (p. 38).
8th grade
● “Practical applications of similarity of triangles” (item 64 – measuring the height of an object, determining the distance to an inaccessible point).
9th grade
● "Measuring work" (item 100 - measuring the height of an object, determining the distance to an inaccessible point).

Measuring instruments used for field measurements:

● ROULETTE – a tape with divisions printed on it, designed for constructing right angles on the ground.
● EKER – a device for measuring right angles on the ground.
● ASTROLABE – a device for measuring angles on the ground.
● MILESTONES (VESHKI) – stakes that are driven into the ground.
● EARTH COMPASSES (FIELD COMPASSES - SAZHEN) - a tool in the shape of the letter A, 1.37 m high and 2 m wide, for measuring on the ground.

EKER.

The ecker consists of two bars located at right angles and mounted on a tripod. Nails are driven into the ends of the bars so that the straight lines passing through them are mutually perpendicular.

ASTROLABE.

The astrolabe device consists of two parts: a disk (limbo), divided into degrees, and a ruler rotating around the center (alidade). When measuring an angle on the ground, it is aimed at objects lying on its sides. Aiming the alidade is called sighting. Diopters are used for sighting. These are metal plates with slots. There are two diopters: one with a slot in the form of a narrow slit, the other with a wide slot, in the middle of which a hair is stretched. When sighting, the observer's eye is applied to a narrow slit, therefore a diopter with such a slit is called an eye diopter. The diopter with a hair is directed towards the object lying on the side of the thing being measured; it is called subject. In the middle of the alidade there is a compass attached to it.

CONSTRUCTION OF A CIRCLE ON
TERRITORIES.

A peg is installed on the ground to which a rope is tied. By holding the free end of the rope and moving around the peg, you can describe a circle.




PRACTICAL WORK.

І. Measuring the height of an object.

Methods:

1 Measuring the height of a pillar using a flat mirror.

According to the laws of reflection (optics, physics), the angle of incidence of a solar ray is equal to the angle of reflection of this ray from the mirror.

∟3 = ∟4, where DK ┴ d, d – horizontal plane.

S – person; b – subject; a – mirror.

∟ADB=∟FDF, since the angles of incidence and reflection of the sun's ray are equal, and ∟1 = ∟2 = 90º-∟3, ∟A = ∟E = 90º, which means that triangles ABD and EFD are similar in two angles.

From the similarity of triangles it follows AB:AD = FE:DE EF = (AB·DE):AD, where AB is the “height” of a person - the distance from the ground to the eyes, EF is the measured height, AD and D E are respectively the distances from the person reflected in the mirror to the object being measured.

2. Measuring the height of an object using a shadow.

V M A

NE is the height of the telegraph pole.

MN – human height (1.6 m).

AM – human shadow (3.35m).

AB is the shadow of the pillar (15.3m).

The man stands in the area of ​​the pillar's shadow so that the shadow of the top of his head coincides with the end of the pillar's shadow.

Consider triangles ABC and AMN.

∟ABC =∟AMN = 90º. By two equal

∟YOU – common. corners.

Triangles ABC and AMN are similar.

You can write the aspect ratio AB:AM = CB:MN

CB = (AB·MN):AM

CB = (15.3 · 1.6) : 3.35

NE = 7.3m.

3. Measuring the height of an object using a pole.

We use a method based on measuring the shadow cast by an object.

● Measure the distance from the tree to the point where its shadow ends.

● Take a pole and, observing its shadow, move back to the tree until the point of complete overlap of their shadows.

● Place a pole in this place and measure the distance to it.

● From the similarity of the triangles it follows that the length of the pole is related to the length of its shadow in the same way as the height of the tree is to its own.

● We determine the height of the tree using the formula:

SE :BC = AD:AB, hence AD ​​= (CE·AB):BC.

4. Measuring the height of an object using the absence of a shadow.

In the absence of a shadow, the height of vertical objects is determined as follows.

Place a stick of known length vertically next to the object being measured and move away 25–30 steps. Hold a pencil or a straight stick vertically in front of your eyes with an outstretched hand. Mark the height of the vertical stick on a pencil and measure this distance. Mentally multiply this distance by the measured object. By multiplying the resulting number of times by the length of the stick, you can get the desired value. From this experiment, we determined that the height of the pillar is 6.89 m.

II. Measuring the distance to an inaccessible point.

Methods:

1. Measuring the distance to an inaccessible point using an eye meter.

Clearly visible:

● at a distance of 2 - 3 km - the outlines of large trees;

● at a distance of 1 km - tree trunks;

● at a distance of 0.5 km - large branches;

● at a distance of 300 m – you can distinguish leaves on the trees.

2. Measuring the distance to an inaccessible point using the similarity of triangles.

A) To measure the width of the river on the bank, measure the distance AC, use an astrolabe to set angle A = 90˚ (pointing at object B on the opposite bank), measure angle C. On a piece of paper, build a similar triangle on a scale of 1:1000 and calculate AB ( width of the river).

IN 1

A 1 C 1

Let's write down the ratio of the sides AB: A 1 B 1 = AC: A 1 C 1

AB = (AC AB 1): A 1 C 1

B) The width of the river can be determined this way: by considering two similar triangles ABC and AB 1 C 1 . Point A is selected on the river bank, B 1 and C at the edge of the water surface, BB 1 – river width.

3. Measuring the distance to an inaccessible point using the “cap” method.

To determine the width of a river (ravine), you need to stand on the bank and pull your cap over your forehead so that only the edge of the water on the opposite bank is visible from under the visor. Next, without changing the tilt of the head and the position of the cap, you should turn your head to the right (left), notice an object that is located on the same bank as the observer and is visible from under the edge of the visor. The distance to this object is equal to the width of the river. Based on experience, we determined that the width of the river is 6 m.

5. Measuring the distance to an inaccessible point using the equality of triangles.

One of the ways to determine the distance to an inaccessible point is related to the laws of geometry and is based on the equality of triangles.

● Stand in front of an object on the opposite bank of the river.

● Turning 90˚, walk along the shore 20 meters and place the milestone O.

● Go the same distance in the same direction.

● Turning 90˚, walk until the milestone O and the object on the opposite bank are on the same line.

● The distance CE is equal to the width of the river ВD.

BD is 5.78 m.

6. Measuring the distance to an inaccessible point using the “blade of grass” method.

The observer stands at point A and selects two stationary objects (landmarks) on the opposite bank near the water, then, holding in his hand a blade of grass (wire) that closes the gap between the landmarks, fold it in half and move away from the river until the distance between the landmarks will not fit into a blade of grass B folded in half. The distance from A to B is equal to the width of the river. AB is equal to 5.96 m.

CONCLUSION.

This abstract discusses the most pressing problems associated with geometric constructions on the ground - measuring the height of an object, determining the distance to an inaccessible point. The given problems are of significant practical interest, consolidate acquired knowledge in geometry and can be used for practical work.

Literature

Atanasyan L. S. Geometry 7-9. – M.: Education, 2003.

Yurchenko O. Methods of motivation and stimulation of student activity. // Mathematics at school, No. 1, 2005

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Indeed, the role of measurements in the life of modern man is very great. The popular encyclopedic dictionary defines measurement. Measurements are actions performed with the aim of finding numerical values, quantitative quantities in accepted units of measurement. The value can be measured using instruments. In everyday life, we can no longer do without a watch, ruler, measuring tape, measuring cup, thermometer, electric meter. We can say that we encounter devices at every step.





Organize research work to measure inaccessible distances on the ground and determine the height of a pole or tree. To promote the development of intellectual activity of students. Organize the work of project participants with a computer. Draw conclusions.











1) Statement of the problem. Defining the project goal. 2) Distribution into groups (measuring the height of a pole, measuring the height of a tree, measuring the length to an inaccessible point.) 2) Planning the project time. 3) Search for information on the project. Carrying out the necessary calculations when conducting research. 4) Creation of mini-projects for each project participant. Which includes: -Purpose. -Equipment. - Expected Result. -The solution of the problem. - Conclusion. 5) Draw a general conclusion about the project.









































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In its early stages, geometry was a set of useful but unrelated rules and formulas for solving problems that people encountered in everyday life. Only many centuries later, scientists of Ancient Greece created the theoretical basis of geometry.

In ancient times, the Egyptians, when starting to build a pyramid, palace or ordinary house, first noted the directions of the sides of the horizon (this is very important, since the illumination in the building depends on the position of its windows and doors in relation to the Sun). This is how they acted. They stuck a stick vertically and watched its shadow. When this shadow became the shortest, then its end pointed in the exact direction to the north.

Egyptian triangle

To measure area, the ancient Egyptians used a special triangle, which had fixed side lengths. The measurements were carried out by special specialists called “rope stretchers” (harpedonaptai). They took a long rope, divided it into 12 equal parts with knots, and tied the ends of the rope. In the north-south direction, they installed two stakes at a distance of four parts, marked on the rope. Then, using a third stake, they pulled the tied rope so that a triangle was formed, one side of which had three parts, the other four, and the third five parts. The result was a right triangle, the area of ​​which was taken as the standard.

Determining inaccessible distances

The history of geometry stores many techniques for solving problems of finding distances. One of these tasks is determining the distances to ships at sea.

The first method is based on one of the signs of equality of triangles

Let the ship be at point K, and the observer at point A. It is required to determine the distance of the spacecraft. Having constructed a right angle at point A, it is necessary to lay two equal segments on the shore:

AB = BC. At point C, again construct a right angle, and the observer must walk along the perpendicular until he reaches point D, from which ship K and point B would be visible lying on the same straight line. Right triangles BCD and BAK are equal, therefore, CD = AK, and the segment CD can be directly measured.

The second way is triangulation

With its help, distances to celestial bodies were measured. This method includes three steps:

□ Measure angles α, β and distance AB;

□ Construct triangle A1 B1K1 with angles α and β at vertices A1 and B1, respectively;

□ Considering the similarity of triangles ABC and A1 B1K1 and the equality

AK: AB = A1K1: A1 B1, using the known lengths of the segments AB, A1K1 and A1 B1, it is not difficult to find the length of the segment AK.

A technique used in Russian military instructions at the beginning of the 17th century.

Task. Find the distance from point A to point B.

At point A you need to select a rod approximately the size of a person. The upper end of the rod should be aligned with the top of the right angle of the square so that the extension of one of the legs passes through point B. Next, you need to mark the point C of the intersection of the extension of the other leg with the ground. Then, using the proportion

AB: AD = AD: AC, easy to calculate the length of AB; AB = AD2 / AC. In order to simplify calculations and measurements, it is recommended to divide the wand into 100 or 1000 equal parts.

An ancient Chinese technique for measuring the height of an inaccessible object.

The greatest Chinese mathematician of the 3rd century, Liu Hui, made a huge contribution to the development of applied geometry. He owns the treatise “Mathematics of a Sea Island,” which contains solutions to various problems of determining distances to objects located on a remote island and calculating inaccessible heights. These tasks are quite difficult. But they have practical value, so they are widely used not only in China, but also abroad.

Observe the sea island. To do this, they installed a pair of poles of the same height of 3 zhang at a distance of 1000 bu. The bases of both poles are in line with the island. If you move in a straight line from the first pole to 123 bu, then the eye of a person lying on the ground will observe the upper end of the pole coinciding with the top of the island. The same picture will appear if you move away from the second pole to 127 bu.

What is the height of the island?

In our usual notation, the solution to this problem is based on similarity properties.

Let EF = KD = 3 zhang = 5 bu, ED = 1000 bu, EM = 123 bu, CD = 127 bu.

Determine AB and AE.

Triangles ABM and EFM, ABC and DKS are similar. Therefore, EF:AB = EM:AM and KD:AB = DC:AC. We get: EM:AM = DC:AC, or EM: (AE + EM) = CD: (AE + ED + DC). As a result, we find AE = 123·1000: (127 – 123) = 30750 (bu). Triangles A1BF and EFM are similar, and AB = A1B + A1A. Hence AB = 5 1000(127 – 123) + 5 = 1255 (bu)

How to find the height of the island?

□ Multiply the height of the pole by the distance between the poles - this is the dividend.

□ The difference between the deviations will be the divisor, divide by it.

□ What happens, add the height of the pole.

□ Let's get the height of the island.

Recipe suggested by Liu Hui.

Distance to an inaccessible point.

❖ The deviation from the previous pole multiplied by the distance between the poles is the divisible.

❖ The difference between the wastes will be the divisor, divide by it.

❖ Let’s get the distance by which the island is distant from the pole.

Applied geometry was indispensable for land surveying, navigation and construction. Thus, geometry has accompanied humanity throughout the history of its existence. The solution to certain ancient problems of an applied nature can still be used today, and therefore deserve attention today.