=Paper=
{{Paper
|id=Vol-2393/paper_389
|storemode=property
|title=Mathematical Models and Methods of Supporting the Solution of the Geometry Tasks In Systems of Computer Mathematics for Educational Purposes
|pdfUrl=https://ceur-ws.org/Vol-2393/paper_389.pdf
|volume=Vol-2393
|authors=Michael Lvov,Ludmila Shishko,Irina Chernenko,Evgen Kozlovsky
|dblpUrl=https://dblp.org/rec/conf/icteri/LvovSCK19
}}
==Mathematical Models and Methods of Supporting the Solution of the Geometry Tasks In Systems of Computer Mathematics for Educational Purposes==
https://ceur-ws.org/Vol-2393/paper_389.pdf
Mathematical Models and Methods of Supporting the
Solution of the Geometry Tasks in Systems of Computer
Mathematics for Educational Purposes
Michael Lvov[0000-0002-0876-9928], Ludmila Shishko[0000-0001-9346-4338],
Irina Chernenko[0000-0003-2445-7366], Evgen Kozlovsky[0000-0001-6630-083X]
Kherson State University, 27, University St., 73000 Kherson, Ukraine
Lvov@ksu.ks.ua, Shishko@ksu.ks.ua, Chernenko@ksu.ks.ua,
Evgen@ksu.ks.ua
Abstract. The article is devoted to the problem of supporting the course of
solving tasks in geometry in systems of computer mathematics of educational
purposes. In the work: - the mathematical model of the learning geometric task
is defined; - the object-oriented approach to the description of mathematical
models of geometric training modules is presented; - the methods of supporting
step-by-step solving of learning geometric task are proposed; - the classification
of elementary transformations in geometric subject modules is proposed; - the
implementation of the concept of support for the solution of geometric tasks in
the systems of computer mathematics of educational purposes is illustrated. Ob-
ject-oriented analysis of the problem revealed three major classes of transfor-
mations of geometric objects. These are constructors, selectors, and converters
(elementary geometric tasks).
Keywords: Systems of computer mathematics for educational purposes, learn-
ing geometric task, computer software, support of learning processes.
1 Introduction
The quality of mastering mathematical knowledge largely depends on the student's
practical mathematical activity. This is the main form of educational activity in the
study of disciplines based on mathematical models and methods, and is to solve learn-
ing mathematical tasks.
Review most domestic and foreign software for educational purposes in mathemat-
ics (GRAN, DG, Geometer’s Sketchpad etc.) reveals the lecture part of the course is
the maximally advanced one from both methodical and technical points of view [16,
17]. But practical functionality of these software tools is limited. Educational purpose
practical mathematical activity is to construct the course of solving the learning math-
ematical task, but not to receive an answer [9].
The function of supporting the process of solving the learning mathematical task
(LMT) is realized in the concept of systems of computer mathematics for educational
purposes (SCMEP).
SCMEP is a programmed educational system for exact and natural educational dis-
ciplines that uses mathematical models and methods of subject areas based on tech-
nologies of symbolic transformations and methods of computer algebra.
The general theoretical and methodological foundations, the formulation of func-
tional requirements for SCMEP and the development of a model of SCMEP as a sys-
tem for supporting learning processes based on the analysis of actual forms and pecu-
liarities of learning processes in precise disciplines are described in [3-5].
2 The outline of the problem
The implementation of the tasks of supporting the solution of LMT requires the defini-
tion of a mathematical model of LMT in the framework of a mathematical model of
the training module and the construction of appropriate algorithms of computer alge-
bra.
The SD curriculum is defined by the quadrants SD = <, MM, ET, Task>, where
- own signature of the training module, MM - list of models of the training module,
ET - list of own elementary transformations of the training module, Task - a class of
learning tasks, which defines the content of the SD.
The educational task P is determined by the list of mathematical models of the
training module MM, the relation of the dependence between the models and their
elements (the condition of the task) and the questions of the task Q:
Р = . (1)
The scope of application of models of the module is mathematical discipline. Each
of the mathematical disciplines has its own class of LMT. Of course, mathematical
models of LMT, depending on the mathematical discipline, have their own peculiari-
ties.
Mathematical models and methods for solving algebraic tasks in computer science
mathematical systems are described in [6, 15]:
- functional requirements for activity environments supporting the solving of edu-
cational tasks in algebra are developed;
- definition of the concept of a training module in school algebra (signature, list of
mathematical models, list of own elementary transformations);
- definition of the concept of LMT in algebra, algebraic object, types of LMT in al-
gebra;
- the main specific tasks of supporting the step-by-step solution of the National
Academy of Sciences are described.
The problem of this study can be formulated as a study of the specifici6y of the
construction of mathematical models of learning geometric task, formal tasks support-
ing the process of solving the learning geometric task in the SCMEP and its imple-
mentation in the SCMEP.
3 Results
Model of educational geometric task. Under the learning geometric task (LGT), un-
like school algebraic tasks, we understand the task that is formulated in terms of geo-
metric objects, which is the subject of study and is supported by SCMEP. Learning
geometric task as well as LMT in algebra are determined and solved by analytical
methods. The peculiarity of LGT support is that the geometric object and the elemen-
tary transformations of the LGT can be interpreted geometrically, and therefore
should be reflected in the corresponding graphical interpretation.
In order to implement the support of the solution of the learning geometric task, it
is expedient to introduce the notion of a mathematical object. A mathematical object
is: an algebraic object (AO) and a geometric object (GО).
Algebraic objects are numbers, variables, numerical and symbolic expressions, de-
terminants, matrices, equality, inequalities, systems, sets of equalities or inequalities.
Primitive geometric objects (PGО) is the point of the plane and space, lines, curves
2-order curves in polar coordinates, surface 2nd order.
Primitive geometric object (PGО) is defined identifier (ID) and algebraic object for
this syntax: PGО:: = ().
Algebraic objects that determine the PGО are equations, inequalities, systems of
equations or inequalities.
General definitions of PGО, except for variables, include alphabetic designations
of AO coefficients - its general parameters. The general parameters of the PGО are
the alphanumeric coefficients of the algebraic object that determines it.
Mathematical models of geometric modules. The structure of the geometric learn-
ing module is defined in the framework of the object design paradigm. Each PGО is
an instance of the class.
For analytic geometry on a plane, these are the classes Point, Line, Curve2, Circle,
Ellipse, Parabola, Hyperbola. The Curve2 class defines the PGО curve of the 2nd
order. The PGО Hierarchy of Inheritance allows to distinguish general and specific
signatures, models, elementary transformations and standard learning tasks.
Curve2
Ellipse Parabola Hyperbola
Circle
Fig. 1. Fragment of the tree of the classes of the module "Curves of the 2nd order"
Definition of the PGО class contains, in particular, the AO, which determine it.
Class PrimitiveGObject(
CoordinateSpace Varset; // (x, y);
Variable ID; // l
AlgObject F(x,y);
...
Virtual CartesianSpace Draw();
...
);
Here is an example of a straight line class definition.
Class Line :: PrimitiveAnalGeomObject (
Canonicalforms (
Genequ a * x + b * y = c, // general equation
Canequ y = k * x + b, // canonical equation
Segmequ x / a + y / b = 1, // the equation in the segments
); . . .
);
Thus, the classes determine, in particular, various algebraic objects (general canon-
ical forms), which, in turn, determine the geometric objects of the class.
Definition of the class of PGО allows you to list the various forms of algebraic
representation of the GО, to indicate the parameters in letters, thereby defining the
functions of access to the parameters, as well as their geometrical predictions. In addi-
tion, the class defines the specifications of the functions of algebraic transformations
PGО.
A composite geometric object (СGО) is determined by the name, a set of PGО and
the relationships that determine them. СGО are, in particular, punctures, directed
segments, angles formed by rays, triangles formed by point-vertices, etc. The ray is
determined through straight line, point and inequality. Directional segment is an or-
dered pair of points. The angle is a pair of beams with a common point. A triangle is a
triple point.
From the point of view of the object-oriented programming paradigm [10], the
СGО is defined by the aggregation classes and, possibly, the relationships between
them and their parameters. Example:
Class Segment = (Point A, Point B);
Class SemiLine = (Line l, Point A)((A in l)&(x >= x A)).
Parameters of a composite object are marked by qualified identifiers. For example,
if D is a triangle, the coordinate x of its vertex A has the xD.A identifier.
Classes of the СGО contain the definition of class member functions (transfor-
mations) that characterize the corresponding СGО. For example, in the Segment
class, you should define the segment's characteristic as its length.
Length(A,B) = Sqrt(Sqr(xB – xA) + Sqr(yB – yA))
Apart from the classes of primitive and complex objects, the domain analytic ge-
ometry also defines elementary transformations - operations on objects. Example,
y yA x xA
LineAB(A( x A , y A ), B( x B , y B ), l( = ))
y B y A xB x A
is a transformation that defines a direct plane passing through two points.
The result of an elementary transformation may be several objects. So, the intersec-
tion of a circle and a straight line determines either two points, or one point, or none.
The names of the GО, whose mathematical models are defined, are used in the
СGО constructors as variables whose values are the corresponding mathematical
models.
Each specific LGT can be formulated in terms of model, condition and question in
the form (1):
Given: list of geometric objects; list of relations between them.
Find: a list of objects (geometric, algebraic, logical).
The LGT model is a GО or a set of GО and AO. The task ID is the service word
Task with the task number in parentheses. Example:
Task 1. The distance between the points A (-2; 5) and M (x; y) is equal to three
units of scale. Determine the coordinates of the point M if A and M are located on a
straight line parallel to the abscissa.
Given: Points A, M, with АМ = 3, xA = xM.
Find: Point M.
Hence the formal definition LGT: Task(1) = (MM, , Q), where MM = (A (-2; 5) &
M (x; y)) – mathematical model, = (АМ= 3)&(yА = yМ) - condition, Q = M - ques-
tion (denoted as ?M).
To solve this task, it is necessary to determine the length of the segment:
АB = Length(A, B) = ( x B x A ) 2 ( y B y A ) 2 .
Substituting an algebraic object instead of its name into a mathematical model of a
task defines a complete algebraic model of the task:
x A 2
yA 5
xM x
yM y
(x x )2 ( y y )2 3
M A M A
y A yM
The solution of the task can now be obtained by solving the systems of algebraic
equations.
That is, the solution of LGT is carried out in terms of the simplest tasks, which are
elementary transformations of analytic geometry, and algebraic transformations.
Support for a step-by-step solution to LGT. The introduced concept of the mathe-
matical model of LGT allows us to investigate the problem of supporting the course of
solving LGT in SCMEP.
LMTs are used to support the step-by-step solution in SCMEP using equivalence
inference - an example based on the application of the rules of rewriting [1,2], is
investigated in [6]. This type of inference naturally represents the course of the solu-
tion for LMT in algebra.
In order to support the course of solving LGT, it is proposed to use both an equiva-
lence inference and a logical inference, since the mathematical models of LGT essen-
tially use logical and algebraic means.
The course of the solution of LGT has two stages: the stage of compilation of the
mathematical model and the stage of transformation of the model.
At the first stage the user must enter into the program the condition of the task. For
LGT, the first stage plays a methodically important role. At this stage, support for user
actions is to verify the mathematical model of the task [14]. Technologically, this is
implemented in a separate window "Building mathematical model", which opens with
the Start Solution command (Fig. 2). The Verify command has the function of verify-
ing the correctness of the model.
Given: A(-2;5)
M(х; y)
АМ = 3
у=5
Find : х
Compose Verify
Fig. 2. Window "Building mathematical model" (schematic)
The implementation of this function requires:
1) the presence in the condition of the task of the correct model of this task and the
answer (hidden from the user of the formal model (1));
2) implementation of the algorithm for comparing the model constructed by the us-
er with the model or answer given in the condition of the task.
These requirements, in turn, require that the text of the task conditions all the sym-
bols needed to formulate the mathematical model of the task.
The presence in the condition of tasks hidden from the user model of the task al-
lows to automate the process of testing and debugging the text of the software module
"TaskBook", as well as implement the function of composing the task model in soft-
ware system (command Compose). Thus, if the user can not make a model of LGT
independently, the system will perform this action itself.
The second stage - the stage of step-by-step solving is to form the course of the so-
lution in the form of a sequence of transformations of the model of the task. The in-
ference is a sequence of triples
((M1, t1, M1), (M2, t2, M2), …, (Mj, tj, Mj), …, (Mn, tn, Mn) (2)
where Mj, Mj – mathematical object , tj - their transformation.
The problem we will discuss below is to define a complete, consistent and method-
ically correct list of transformations that support inference (2) and implemented as a
structure of commands (references) that form the contents of the software module
(SM) "Guide". This module, in turn, is used in the SM "Medium of Solving".
According to the definition of the GО, the SM "Guide" should contain both geo-
metric and algebraic transformations, that is, the section "Transformation of algebraic
objects", which contains equivalent algebraic transformations, the classification prob-
lem of which was investigated in [6,13], and the section "Transformation of geometric
objects", which, in fact, contains the transformation of geometric objects.
Elementary transformations of geometric objects. Analyzing the problem from
the point of view of the object-oriented programming paradigm, we distinguish trans-
formation-constructors and transformation-selectors.
Transformation-constructor PGО builds PGО for its algebraic definition. Thus, the
corresponding transformation has the specification t : AO PGО.
In sequence (2), this transformation is represented by a triple (AO, t, PGО). If the
AO conversion argument is allocated during the solution, the PGО is entered into the
solution as its last (new) row. In Fig. 3 shows the selected equation in the 5th line of
the solution and the transformation-constructor of a straight line by its equation, the
result of which is entered into the course of the solution as the 9th row.
The course of the solution Reference
… Construct a straight line for its
5. Convert the equation: y = 2 x + 1 equation
…
9. Construct a straight line l by the Highlight the equation of the line
equation (5): l(y = 2 x + 1) F(x, y) l(F(x, y))
Fig. 3. A fragment of the course of the solution of the LGT
Transformation-selector allocates one or more objects that are included in the defi-
nition of the GO. Constructors and selectors, as a rule, can be interpreted as one refer-
ence that contains mutually inverse transformations (Fig. 4).
Reference
Construct a circle according to its parameters
Highlight circle settings
a l a
l((x - a)2 + (y - b)2 = r2) bl b
r r
l
Fig. 4. Reference – constructor and selector circle parameters
Transformation-definition introduces AO - definition of GО. For example, in the
course of the solution of the LGT, we can include the definition of the tangent line to
the function y = f (x) at point A (Fig. 5).
Reference
Equation of tangent line L to function graph G (y = f (x)) at the point A
L (y - yA) = f '(xA) (x - xA)
Fig. 5. Transformation-definition of a tangent line to the function graph
The transformation of the СGО, whose classes contain the definition of additional
objects, are described by the reference-selectors of additional objects.
We separately note the necessity of the constructor and selectors of such algebraic
objects as the equation, the system of equations, and the aggregate of equations.
These transformations should be part of the Algebraic section of the SM "Guide".
In addition to the transformations of the classes of GО, one must identify elemen-
tary transformations - operations on objects, so-called elementary tasks of analytic
geometry. For example, a Point × Point Line transform, which defines a straight
line passing through two points (Fig. 6).
Reference
Calculate a straight line equation that passes through two points
x xA y yA
A, B l( x x = y y )
B A B A
Fig. 6. Conversion – basic task of analytical geometry
Logical transformations - the transformations of the type GО × GО Bool return
the value True, False. These transformations solve the tasks of the mutual position of
the GО - the parallelism (the perpendicularity) of the straight lines, etc.
A special particular type of transformation is the isomorphic transformation of GО.
They include: elementary transformations of the Cartesian plane or space, transfor-
mation of the transition to a polar coordinate system on a plane or transformation of
the transition to spherical or cylindrical coordinate systems in space, transformation
of the transition to vector algebra.
In the analytic geometry on the plane the following elementary transformations are
used: parallel transfer, turning to the angle, stretching / compressing.
Both methodically and technically, these transformations should be realized in two
forms: as an elementary transformation of a geometric object and as an elementary
transformation of a plane. The transformation of the GО is to construct a new GО in
the "old" xOy coordinate system. For example, a parallel transfer of a GО is defined
by the transformation
l(F(x, y) = 0) m(F(x + a, y + b) = 0).
The parallel transfer of the xOy plane is determined by the transformation
l(F(x, y) = 0) l(F(x + a, y + b) = 0) & x = x - a, y = y - b.
In the first form the result is a new object. Consequently, it is determined by the
new ID and the new AO. The coordinate system remains "old". The second form
changes the coordinate system and the AO. ID of geometric object is advisable to
leave. The graphic illustration of this transformation form is to create a new Cartesian
plane - the plane xOy and the reflection of the GО in a new coordinate system.
The transformation of the transition to the polar coordinate system associated with
the xOy system determines the transformation
l(F(x, y) = 0) l(F( cos(), sin()) = 0).
The inverse transform has the form
l(F(, ) = 0) l(F( x 2 y 2 , arctg(y/x) = 0).
The transformation of the transition to vector algebra. The solution of a LGT by a
vector method consists in constructing a task model in the form of a formula in the
signature of the Euclidean space and solving the task by algebraic transformations of
this formula. Conversion of the transition to the vector method consists in the applica-
tion the type of a reference (Fig. 7).
Reference
Consider the vector
A, B а = AB
Fig. 7. Reference - Transformation of the transition to vector algebra
The further course of the solution of the LGT should be based on the list of trans-
formations-definitions of vector geometric formulas and transformations-formulas in
the signature of the Euclidean space. The list of definitions of vector geometric ob-
jects determines the completeness of the “Guide” from the section "Vector method in
geometry". Here is an example of this section of the Guide:
Vector method in geometry / Vector properties of geometric shapes
• Axiom of a directed section: AB = -BA
• Axiom of the triangle ABC: AB + BC = AC, etc.
Transformation in the Euclidean Space
I. Axiom Euclidean vector space
II Theorems of geometry
• Definition of the scalar product: (a,b) = abcos(a,b).
• The cosine theorem: AB2 = BC2 + AC2 + 2BCACcos(BC,AC), etc.
A more detailed analysis can be found in [11]. The basis of the formation of sec-
tions of the “Guide” was chosen [12].
Inference in geometric modules. If the model of LGT is represented in the form of
one formula, the course of its solution, in principle, can be obtained as a result of the
equivalence inference (2). However, the submission of the terms in this form is not
accepted. It is generally acceptable to formulate the condition in the form of a list of
primitive conjuncts that specify the relationship between the PGО parameters of the
condition of the task. Here is an example of the application of the mathematical meth-
ods and models for a geometric task.
Task 2. Compose the equation of the tangent to the graph of the function y = x2 + 1
at the point A with abscissa x = 1.
Given: 1. Graph F(y = x2 + 1).
2. The point A(1, yA).
This condition must be supplemented by definitions by using the following ele-
mentary transformations-definitions:
3. Point A belongs to the graph F: yA = xA2 + 1.
4. Equation of tangent line L to graph F in point A: L(y - yA) = f '(xA)(x - xA).
Rows 1-4 of the solution are a model of the task:
F(y = x2 + 1)&A(1, yA)&(yA = xA2 + 1)&L(y - yA) = f '(xA)(x - xA).
Consequently, the step of inference (solving) depends on the data presented in the
previous steps. To execute it, the user must specify the appropriate lines, find the
transformation in the “Guide” and apply it.
Realization. The scientists of the Department of Informatics, Program Engineering
and Economic Cybernetics of the Kherson State University under the guidance of the
professor M. Lvov are engaged the implementation of the concept SCMEP.
Here are some SCMEP created by order and recommended by the Ministry of
Education and Science of Ukraine (copyright certificates [7-8]):
- Program-methodical complex «TerМ VII» of the support of a practical mathe-
matical learning activity;
- Software tool "Library of Electronic Visual Aids Algebra 7-9 grade for secondary
schools in Ukraine";
- Pedagogical software tool "Algebra, Grade 7";
- Software tool for educational purposes "Algebra, Grade 8";
- Integrated environment for the study of the course "Analytical geometry" for
higher education institutions; and others.
Features of the support of the course of solving LMT in algebra and the principles
of classification of elementary algebraic transformations for school algebra on the
example of the program module (PM) "Guide" of ТеrМ VII is considered in [6].
The concept of supporting the course of solving geometric tasks in SCMEP is par-
tially implemented in ТеrМ VII-ІХ, since the content of the course of school algebra
presents educational tasks that contain elements of analytic geometry and are formu-
lated in terms of geometric objects.
In ТеrМ VII-ІХ there is a program module "Graphs", which is intended for the so-
lution of LGT and LMT in algebra by graphic method. In it, it is possible to formulate
an algorithm for solving a task in the form of a sequence of commands defined by the
references of the PM "Guide" of the PM "Graphs". Its content is a structured list of
elementary transformations of this mathematical model of the training module.
The PM "Guide" contains the following references:
Section Formula-Graph
1. Construct a point A(x; y) by its coordinates,
2. Construct a straight line ax + by + c = 0, etc.
Section Graph-graph
1. Conduct a straight line passing through two given points,
2. Find the point of intersection of two straight lines, etc.
Section Graph -Formula
1. Find reproduction of the coordinates of the constructed point,
2. Find the equation of a constructed straight line, etc.
Section Converting Function Graphs
1. Parallel movement in the direction of the abscissa axis x => x – a,
2. Stretching (compression) from the abscissa axis x => kx, etc.
The use of the "Graphs" for solving a system of linear equations graphically and an
algorithm for solving it using references of the “Guide” are illustrated in Fig. 8.
Fig. 7. Graphical method of solving the system of linear equations
We note that often the course of the solution of the LGT can be described as the
language of algebra, as well as in the language of geometry ("find the intersection of
the GО" or "solve the system of equations").
The proposed methodology was verified experimentally during the introduction in-
to the educational process of “Integrated environment for the study of the course "An-
alytical geometry" in higher educational institutions of Ukraine. The experiment
showed that the study of the course "Analytical geometry" provides a higher level of
mathematical competence in the students of technical specialties [18, 19].
4 Conclusions
Practical mathematical activity of the student is the main form of educational activ-
ity in the study of mathematics. It consists in getting the course of the LMT solution.
In the work:
- the mathematical model of the LGT is defined within the framework of mathe-
matical model of the training module and construction of the corresponding algo-
rithms of computer algebra.
- the problem of forming a complete, non-contradictory and methodically correct
list of transformations, by which one can carry out logical derivation, as a step-by-
step solution to LGT, is solved.
- mathematical models of methods for supporting the solution of LGT in SCMEP
are constructed.
Object-oriented analysis of the problem revealed the main classes of transfor-
mations of geometric objects (constructors, selectors and converters (elementary
tasks)).
Obviously, in addition to algebra and geometry, this approach can be applied to
such subject areas, where the content of educational tasks is the formal properties of
interacting objects (physics, engineering, etc.).
References
1. Podkolzin A.S. Komp'juternoe modelirovanie logicheskih processov. Tom 1. Arhitektura i
jazyki reshatelja zadach. /A.Podkolzin. –M: Fizmatlit. –2008, 1024 s.
2. Lvov M.S. Меtodi proektuvannya system komp'juternoi pidtrimky matematichnoi osvity /
M.S. Lvov, О.V. Spivakovsky // Matematychnі modelі і suchasnі іnformacіjnі
tehnologіi:zb. nauk. prac NAN Ukraini. Kiev. 1998, S. 101-111.
3. Lvov M. Оsnovnie princypy postroenija pedagogicheskih programmnih sredstv poddergki
prakticheskih zanjatiy / Lvov М. // Upravljajushhie sistemy i mashiny. – 2006. – № 6. – S.
70-75.
4. Lvov M.S. Shkilna systema kompjuternoji algebry TerM 7-9. Pryntsypy pobudovy ta
osoblyvosti vykorystannia. / M.S. Lvov // Naukovyi chasopys NPU im. Drahomanova, se-
riia №2. Kompiuterno-oriientovani systemy navchannia: zb.nauk. prats/ redkol. – K.: NPU
im. Drahomanova. –#3(10) -2005. S. 160–168.
5. Lvov M., Shishko L., Chernenko I., Kozlovsky E. Systems of computer mathematics for
educational purposes as the means of reception of procedural knowledge. CEUR Work-
shop Proceedings of the 12th International Conference on ICT in Education, Research and
Industrial Applications. Integration, Knowledge and Harmonization Transfer, -K.:,
Ukraine, June 21-24, -2016, S. 215-130.
6. Lvov M., Shishko L., Chernenko I., Kozlovsky E. Mathematical Models of Supporting the
Solution of the Algebra Tasks in Systems of Computer Mathematics for Educational Pur-
poses CEUR Workshop Proceedings of the 14th International Conference on ICT in Edu-
cation, Research and Industrial Applications. Integration, Harmonization and Knowledge
Transfer, - Kyiv, Ukraine, May 14-17, - 2018, P. 472-487
7. A.s. 12440 MONU. Derzhavnyi departament intelektualnoi vlasnosti. Zbirka komput-
ernykh prohram Prohramno-metodychnyi kompleks «TerM VII» pidtrymky praktychnoi
navchalnoi matematychnoi diialnosti. Versiia 2.0, reliz 03.» osvitnoi haluzi «Matematyka»
dlia zahalnoosvitnikh navchalnykh zakladiv / Lvov M., Spivakovsky O., vs'ogo 9 osib;
zaiavl. 10.03.2005. Biul. №7. – 2005.
8. A.s. 32724 MONU. Derzhavnyi departament intelektualnoi vlasnosti. Komputerna
prohrama «Іntehrovane seredovische vivchennia kursu «Analitichna geometria» dlia vis-
chih navchalnih zakladiv» / Peschanenko V., Lvov M., Spivakovsky O.; vs'ogo 8 osib;
zaiavl. 06.04.10. Biul. №21. – 2010.
9. Lvov M.S. Koncepcija proghramnoji systemy pidtrymky matematychnoji dijaljnosti./
M.S.Lvov // Komp'juterno-orijentovani systemy navchannja: Zb. nauk. praci/ K.:NPU im.
M.P.Draghomanova.– Vyp. 7. – 2003. – S. 36–48.
10. Pol A. Ob'ektno-orientirovannoe programmirovanie na S / Pol A. – [2-e izd.].–Spb.;
M.«Nevskiy dialekt»–izd-vo «BINOM», 1999.– 462 s.:il
11. Bugrov Ya.S. Elementyi lineynoy algebryi i analiticheskoy geometri / Bugrov Ya.S. – M.:
Nauka, 1988. – 222 s.
12. Vyigodskiy M.Ya. Spravochnik po vyisshey matematike / Vyigodskiy M.Ya. - [8-e izd.]. –
M.: «Nauka», 1966. – 872 s.: il.
13. Lvov M.S. Term VII – shkilna systema kompiuternoi alhebry. / M.S. Lvov // Kompiuter u
shkoli ta simi. – 2004. – №7. – S. 27–30.
14. Lvov M.S. Proektirovanie logicheskogo vyvoda kak poshagovogo reshenija zadach v ma-
tematicheskih sistemah uchebnogo naznachenija / M. S. Lvov // Upravljajushhie sistemy i
mashiny. – 2008. – №1. – S.25–32.
15. Lvov M.S., Shishko L.S., Chernenko I.Е. Pro orhanizatsiiu kontroliu znan yak zvorotnoho
zviazku v systemakh kompiuternoi matematyky navchalnogo pryznachennia / L.S. Shish-
ko, I.Е. Chernenko // Informatsiini tekhnolohii v osviti: Zb. nauk. pr. Vyp 16. - Kherson:
Vyd. KhDU, 2013. – S. 29-42.
16. Windham Wyatt K. Geometry Activities for Middle School Students with The Geometer’s
Sketchpad /K. Windham Wyatt, A. Lawrence, G. M. Foletta// Key Curriculum Press.–
1998.– P. 221.
17. Lvov M. Software Module Representing a Geometry Tasks in Mathematical Systems of
Educational Appointment / M. Lvov, D. Melnyk //CEUR Workshop Proceedings of the
14th International Conference on ICT in Education, Research and Industrial Applications.
Integration, Harmonization and Knowledge Transfer, Volume II: Workshops Kyiv,
Ukraine, May 14-17, –2018, – P. 73-82.
18. Grigor’eva V.B. Komp’yuterno-orientovana metodika navchannya analitichnoyi geometri-
yi maybutnih programistiv: dis. kand. … ped. nauk: 13.00.02 / Grigor’eva Valentina Bori-
sivna. – Herson: HDU, 2017 – 270 s.
19. Zaytseva Т.: The Introduction of the Competence-based Approach in Educational Process
of Training of Skippers. In: 12th International Conference on ICT in Education, Research
and Industrial Applications: Integration, Harmonization and Knowledge Transfer, pp. 687-
699. Kyiv, Ukraine, CEUR-WS, 1614, (2016).