Volume 02 Issue 09-2022
26
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
The article provides a classification, compilation and solution methods of descriptive mathematical problems in
secondary schools, and also shows the role of an illustrative issue in the development of logical thinking and spatial
imagination of students. With the help of illustrative issues, the stages of solving geometric problems have been
improved, and the possibilities of effective use of illustrative issues in the process of mathematical education have
been clarified.
KEYWORDS
Illustrated matter, geometric matter, algebraic matter, logical matter, mantl matter.
INTRODUCTION
Research Article
DRAWING AND IMAGE MODELS TOOL MATH LEARNING OPTIONS
Submission Date:
August 30, 2022,
Accepted Date:
September 06, 2022,
Published Date:
September 30, 2022
Crossref doi:
https://doi.org/10.37547/ajast/Volume02Issue09-05
Khonkulov Ulugbek Khursanalievich
Phd, Associate Professor Of Fergana State University, Uzbekistan
Tokhtasinov Tohirjon Shakirjon Ugli
Teacher Of Fergana State University, Uzbekistan
Madrakhimov Askarali
Associate Professor Of Fergana State University, Candidate Of Physical And Mathematical Sciences, Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 02 Issue 09-2022
27
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
In mathematics lessons, it is important to study the
didactic capabilities of students in the formation of
logical observation skills and practical knowledge by
classifying descriptive issues. One of the urgent
methodological problems is the improvement of
content based on the inclusion of requirements
(conceptual, reflexive, Technological) on issues of
creative and practical content in the principles of
structuring and solving illustrative issues in secondary
schools[1,2]. In this, it is desirable that the content of
the system of descriptive issues, aimed at ensuring the
continuity of gneseological, heuristic directions for the
synthesis of mathematical knowledge and skills, be in
accordance with the principles of didactic, logicality.
This process makes it possible to systematize the
mechanism and organizational and functional
capabilities of compiling, introducing a methodology
for solving illustrative issues aimed at teaching
students to logical, non-standard thinking.
Material and Methods. In general secondary schools,
mathematical issues can be classified as[3] (Table 1).
Table 1. Classification of mathematical problems in secondary schools.
Mathematical issue
Algebraic issue
Geometric matter
Logical
issue
Text
issue
Th
e
iss
u
e
o
f Nat
u
ral
d
ate
s
Em
p
lo
ym
en
t
iss
u
e
P
erc
entage
iss
u
e
Ac
ti
on
iss
u
e
Th
e
iss
u
e
o
f the
m
ix
tu
re
Th
e
iss
u
e
o
f p
rogr
es
sio
n
Calculation issue
The issue of proof
The issue of making
The issue to be
resolved with
practical exercises
Volume 02 Issue 09-2022
28
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
The following information can be cited about the
issues encountered in the geometry course and their
classification.
Issues to be resolved with practical exercises. Such
issues are initially used in the development of an idea
of geometric figures. Even without knowing the
definition or properties of a geometric figure or figure,
its elements can be measured using a ruler or
transporter. For example, it is possible to take issues of
comparing the lengths of the cuts, determining the
type of angle.
Issues concerning computing. Such issues are based on
theoretical knowledge learned in the course of
geometry, on the basis of the studied texture,
theorems and formulas, on finding the magnitude
necessary to be determined in geometric figures using
the magnitudes of the elements given in the condition
of the issue.
Issues concerning proof. Such issues are resolved by
inductive and deductive inference. The condition of the
issue and the conclusion part are distinguished, and the
conclusion part is formed on the basis of the given
conditions.
Issues related to making: such issues are solved only
with the help of a ruler and a circus. In this case, it is
necessary not only to make a geometric shape, but also
to indicate that the formed shape satisfies the
conditions of the issue, the Made shape is correct and
fully fulfilled.
Results. The development of students ' thinking
activity has been one of the urgent issues of teaching
mathematics. We divided the issues of general
secondary education schools in geometry into the
following three types, depending on the method of
submission:
theoretical
issues;
logical
issues;
descriptive issues[5].
An Illustrated issue refers to issues given by illustrative
means such as form, image, graph, diagram, table.
As a result of the analyzes, we classified the problems
with the image in the geometry course according to
the degree of complexity as follows (Table 2).
Table 2. Classification of descriptive issues in the geometry course.
Illustrated issue
Si
m
p
le
im
age
is
su
e
An
is
su
e
wh
ere
th
e
fo
rm
in
th
e
im
age
is
su
ffi
cie
n
t
Th
e
p
ro
b
lem
th
at
is
so
lve
d
b
y
fi
lli
n
g
o
u
t
th
e
fo
rm
in
th
e
An
ill
u
strated m
att
er
th
at
d
oe
s n
ot
d
epe
n
d
on
t
h
e
situ
ati
o
n
of
th
e
p
oi
n
t
Volume 02 Issue 09-2022
29
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
The school has various approaches to improving
geometry teaching, one of which we consider to rely
on the means of descriptive issues and to teach
students to draw up an illustrative issue. In the article,
it can be added about the importance of creating an
independent descriptive issue by readers with an
emphasis on compiling descriptive issues, that while
the reader himself creates an independent issue, has
complete knowledge of the concepts he is using in the
matter, he also conducts research on the existence of
ways to solve it, whether it is right or wrong. Drawing
up geometric issues is more likely to serve the
development of students ' thinking skills than
arithmetic problems. Below we will tell about the
methods of constructing geometric problems:
Teaching to draw up an issue on the method of
analogy. In doing so, the teacher will select an issue
suitable for the subject being studied. For example,
when the topic" volume of the pyramid "is passed, the
teacher reminds students of the issue in the planimetry
course" represent the face of an equilateral triangle
through its sides" and offers to draw up a stereometric
issue that is analogous to this issue. Considering that a
regular tetrahedron in space is an analogous figure to
an equilateral triangle in the plane, and an analogous
quantity to the face of a triangle is the volume of a
tetrahedron, the following issue can be formulated:
"express the volume of a regular tetrahedron through
its edge", "express the volume of a regular
tetrahedron through the face of its side fat".
Drawing up a new issue as a result of generalization. In
this, aspects of the commonality of a definition,
theorem or formulas on one topic with another are
found and applied.
For example, the geometric interpretation of
Pythagorean's theorem is as follows: the face of a
square made on the hypotenuse of a right-angled
triangle is equal to the sum of the faces of the squares
made on its cathets. Summarizing this theorem, one
can teach to draw up an issue as follows:
1) prove that the face edges of a cubic surface whose
edge is equal to the hypotenuse of a right-angled
triangle are equal to the sum of the faces of the
surfaces of cubes equal to its cathets;
2) prove that the face diameters of the surface of the
sphere, whose diameter is equal to the hypotenuse of
a right-angled triangle, will be equal to the sum of the
faces of the spheres equal to its cathets.
Sometimes readers can also formulate incorrect issues
regarding analogy and generalization. The reason for
their mistake in this will be the following:
- analogical figures or their incorrect selection of
analogical quantities (taking a sphere into a circle as an
analogical figure);
- incorrect application of generality (prove that the
volume of a sphere whose diameter is equal to the
hypotenuse of a right-angled triangle is equal to the
sum of the volumes of spheres whose diameters are
equal to its cathets).
Drawing up an issue by analogy with the studied one.
By analogy with the theorem of Sines, the following
question can be formulated: the ratio of the side
vertices of a triangular prism to the sine of two-sided
angles opposite these vertices will be equal among
themselves.
Drawing up a new issue. In doing so, it is possible to
train to create new issues on a voluntary subject.
Reflect the issue. The reflection of the issue is that the
compilation of such issues helps students to better
understand the structure of the issue, learn to
Volume 02 Issue 09-2022
30
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
02
I
SSUE
09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
formulate and prove inverse theorems to the concepts
studied in the lesson, to replace the adequate and
necessary conditions of the theorem. It should be
noted that the content of solutions to inverse
problems usually does not differ from solutions to
issues given, but the method differs from the point of
view. Let's see this in the following Issue[9].
Issue. If the diagonals of the rectangle are
perpendicular and are divided by two equal at the point
of intersection, then prove that all sides of this
rectangle are equal.
Reflected issue. Prove that the diagonals of an
equilateral rectangle are divided into equal halves
perpendicular and at the point of intersection.
We will improve the existing general requirements for
drawing up an issue as follows::
each Illustrated issue must be aimed at a specific
purpose;
The Illustrated issue should reveal the essence of the
given concept;
the descriptive issue should be in the educational,
educational, developmental movement;
the descriptive issue should form in the minds of
students knowledge, skills, competencies about the
new concept being studied and be aimed at developing
mathematical competence. In the formation of
knowledge in students, simple illustrative issues are
used, in the formation of skills, qualifications and
competencies, descriptive ones are used, in which the
form in the image is sufficient, which are solved with
the help of filling out the form in the image, which does
not depend on the situation of the point;
must serve to develop students ' creative thinking and
spatial imagination. In the development of students '
creative thinking, pictorial issues are used, which are
solved with the help of filling the form in the image,
which do not depend on the situation of the point, and
pictorial ones in all their forms in the development of
spatial representations;
The Illustrated issue must correspond to the types of
lessons. In elective classes, extracurricular activities,
circles, work with gifted students, descriptive issues
are used, which are solved with the help of filling out
the form in the image, which do not depend on the
situation of the point;
must be structured in accordance with specific stages
of the lesson process. In the statement of a new topic,
knowledge of the essence of the given theorem or
concept, simple descriptive issues are used, which are
solved with the help of generalization, repetition,
independent work of the lesson, when performing
homework, filling out the form in the image, which
does not depend on the situation of the point;
must be consistent with the curriculum and age-
specific;
must ensure continuity, consistency in the teaching
process. The use of issues in which the form in the
image is sufficient in ensuring continuity and
consistency in the teaching process gives an effective
result;
the mathematical magnitudes being entered in the
image must match the drawing;
in order to find a solution to an illustrated issue, its
condition must be complete, adequate and without
excessive additions. The condition of the issue should
not contradict mathematical concepts and laws.
Volume 02 Issue 09-2022
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American Journal Of Applied Science And Technology
(ISSN
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VOLUME
02
I
SSUE
09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
Discussion and Conclusions. In order for the teacher to
have a deeper knowledge of the concept initially given
to students, it is necessary to teach to compose simple
descriptive issues. Then, on the basis of the formed
knowledge, it is necessary to slowly teach students to
formulate issues in which the form in the image is
sufficient for the formation of skills, qualifications and
competencies, issues that are solved with the help of
image filling, and illustrative issues that do not depend
on the situation of the point. When compiling
illustrative issues, we also encounter some
problematic situations. Experiments show that the
mathematical magnitudes that represent the issue
when constructing descriptive issues are simply not
included. The magnitudes must correspond to the
drawing, the size being determined must be clear both
in terms of content and in terms of science. It is
advisable to check the issue with a structured image
and make sure that it is configured correctly.
Issue 1. ABCD is given trapeze. Find the cosine of angle
C (Figure 1).
Solution of the issue: we lower two heights from the
points D and C on the upper base of the trapezium and
mark its bases with E and F, respectively. The drawing
given without it will be in the form of the following
(Figure 2).
Image-1.
Image-2.
Triangle
АЕ
D and BCF for triangles We apply the Pythagorean theorem.
{AD
2
= AE
2
+ ED
2
BC
2
= BF
2
+ FC
2
Е
D = FC and EF = DC,
А
B =
АЕ
+ EF + BF according to the fact that it is and the issue is given,
{
4 = AE
2
+ ED
2
9 = (2 − AE)
2
+ ED
2
Volume 02 Issue 09-2022
32
American Journal Of Applied Science And Technology
(ISSN
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2771-2745)
VOLUME
02
I
SSUE
09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
is formed. Solving a given system, we get a value AE = - 0.25. The fact that the solution to the issue is negative
leads to two different results:
the issue is resolved incorrectly;
the terms of the issue are given incorrectly.
But none of these two conclusions is correct. The
reason for such a wrong solution to the issue is that the
trapeze that satisfies its condition is different from the
forms that usually remain a study. When the trapezium
that satisfies the condition of the matter is clearly
drawn, the small base will be non-projecting one value
onto the large base, that is, the perpendicular
transferred from the ends of the small base will lie
throughout the large base. This situation can result in a
result that is out of place for geometric magnitude
when solved in an algebraic way. In the above issue,
the length of the incision remained negative. The
reason for this is the cut sought in the drawing, the
direction of which has remained opposite with the cut,
which can be the solution to the issue. In order for the
student not to face such a problem, the teacher must
teach the definition of a trapezoid to draw various
drawings by its types. If a drawing is drawn from the
definition of trapeze to the given issue as follows, the
correct solution to the issue is formed (figure-3). The
teacher must explain to the students that the drawing
is trapeze and that it corresponds to the definition of
trapeze. Now let's come to a solution to the issue. We
draw two heights from the points D and C on the upper
base of the trapezium and mark its bases with E and F,
respectively (see Figure 4).
Image-3.
Image-4.
Solving the issue as above, E
А
=0,25 we will have a value. We will not face any conflict in this. From solving a
given issue FB = 2,25 sm comes from. Right-angled triangle from the definition of sine at BFC,
sinFCB =
FB
BC
=
2,25
3
=
3
4
.
Solving the issue as above
, EА=0,25
we will have a value. We will not face any conflict in this. When solving a given
problem, FB =2.25 CM will be equal in value. Right-angled triangle BFC at angle from the definition of sine,
sin < FCB =
FB
BC
=
2,25
3
=
3
4
we determine the equivalent of. Cosine of angle C according to the case Gart,
Volume 02 Issue 09-2022
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American Journal Of Applied Science And Technology
(ISSN
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2771-2745)
VOLUME
02
I
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09
Pages:
26-34
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
cos < C = cоs(90°+< FCB) = −sin < FCB = −
3
4
is equal to. Typically, solving geometric problems can
be done in four stages:
Stage 1.
Understanding the issue. At this stage, the
condition and conclusion of the issue are distinguished
separately. A drawing is drawn on the issue. All
information provided is marked on the drawing.
Stage 2.
Planning. At this stage, the method of solving
the issue is selected. It is determined what additional
information is needed for its application. Auxiliary lines
are drawn, the solution plan is determined.
Stage 3.
Solve. At this stage, the issue is resolved
according to the intended plan.
Stage 4.
Check. At this stage, The found solution to the
issue is checked directly. A critical look at the solution
process is taken. If an error is detected, it will be
corrected. If there is no way to fix, the issue will return
to the initial part of the solution, and all work will start
again.
When solving illustrative issues, it is necessary to pay
attention to:
Know and keep in mind the basic concepts of
mathematics, definitions, properties well.
Know different ways to prove theorems.
To understand the essence of the issue with a given
image and determine whether it is necessary to draw
additional lines.
Acknowledgement.
We will improve the solution of
geometric problems using illustrative problems as
follows:
Stage 1.
Understanding the issue. At this stage, the
issue is brought to the appearance of the text, when
the information can be read depending on the image
and conditions given, that is, the type of shape or div
referenced and the name of the magnitudes are
determined and the condition and conclusion of the
issue are distinguished separately.
Stage 2.
Planning. When the image given at this stage
is sufficient to solve the problem, the method of
solving it is selected. If the given image is not enough
to solve the problem, auxiliary forms are drawn into
the image or the image view without changing the
condition of the issue is adapted to solving it, and a
solution plan is drawn up, determining what additional
information is needed.
Stage 3.
Solve. At this stage, the issue is resolved
according to the intended plan.
Stage 4.
Check. At this stage, The found solution to the
issue is checked directly. If an error is detected or
defects are detected in the image, it is corrected. If
there is no way to fix, the issue will return to the initial
part of the solution, and all work will start again.
Usually the lesson begins not with the rule of solving a
new type of issue, but directly with the solution of an
unfamiliar issue. Readers will intuitively try to solve the
issue based on their previous knowledge and skills.
Together, they reason, make mistakes and correct it by
trying, making mistakes, and work on these mistakes
and, on the basis of this, gradually approach the
correct solution. At this point, students understand
that making mistakes is not such a bad thing, it is
precisely on the basis of their own experience that the
process of analyzing and verifying various error
Volume 02 Issue 09-2022
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American Journal Of Applied Science And Technology
(ISSN
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2771-2745)
VOLUME
02
I
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09
Pages:
26-34
SJIF
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FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
METADATA
IF
–
5.582
Publisher:
Oscar Publishing Services
Servi
solutions (small research) gradually leads them to the
right solution.
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