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International Journal of Mathematical Education in
Science and Technology
ISSN: 0020-739X (Print) 1464-5211 (Online) Journal homepage:
Various solution methods, accompanied by
dynamic investigation, for the same problem as a
means for enriching the mathematical toolbox
Victor Oxman & Moshe Stupel
To cite this article: Victor Oxman & Moshe Stupel (2017): Various solution methods, accompanied
by dynamic investigation, for the same problem as a means for enriching the mathematical
toolbox, International Journal of Mathematical Education in Science and Technology, DOI:
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Published online: 27 Oct 2017.
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Date: 29 October 2017, At: 00:31
Various solution methods, accompanied by dynamic
investigation, for the same problem as a means for enriching
the mathematical toolbox
Victor Oxmana,b and Moshe Stupelb,c
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Western Galilee College, Acre, Israel; b Shaanan College, Haifa, Israel; c Gordon College, Haifa, Israel
A geometrical task is presented with multiple solutions using different methods, in order to show the connection between various
branches of mathematics and to highlight the importance of providing the students with an extensive ‘mathematical toolbox’. Investigation of the property that appears in the task was carried out using a
computerized tool.
Received  January 
Conservation property;
multiple solutions/proofs;
geometric investigation
using dynamic software
1. Introduction
The methods of teaching and instruction in mathematics are derived from the programme
of studies and its goals. One of the principal goals of the studies of mathematics is to provide
the students with methods of reasoning that may assist them in other fields of study and
knowledge, and to develop such methods.
The meaning of ‘learning to think’ is: the teacher of mathematics should develop the
ability of students to apply information, perform analysis and synthesis at an adequate level,
of basic properties, rules and theorems that they studied at an earlier stage of the teaching
Solving different problems is one of the important means aimed at developing reasoning.
During their solution of problems in the class, the students are asked guidance questions,
such as: ‘can the conclusion of the previous exercise be implemented in solving the present
problem?’, ‘Is the problem a particular case of the previous problem?’, ‘Did you encounter a
problem of this type before, where?’, ‘Is the wording of the question similar?’. The purpose of
such questions is to stimulate and expand the field of reasoning and to prepare the student
for independent efforts, where he himself will have to ask these questions and answer them.
Impetus for developing reasoning is obtained from solving problems using different
methods. By using methods of solution and knowledge from other fields of mathematics,
one can obtain simpler and faster solutions.
Finding another method of solution by using the same mathematical field, and especially
from another field, contributes to the development of reasoning and raises it to a higher
CONTACT Victor Oxman,
©  Informa UK Limited, trading as Taylor & Francis Group
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Implementation of previous knowledge in a new situation that results in a shorter and
simpler solution or a more beautiful solution increases enjoyment and satisfaction from
studies of the subject.
Integration of fields in problem solution gives the students a wider outlook to mathematics as a comprehensive discipline, while creating connections between its different branches.
Successful development of reasoning of students and solution of diverse problems in
mathematics depend very much on the teacher’s ability to create and elicit appropriate stimuli in the teaching and instruction process, while accentuating the beauty of mathematics.
Solving a problem by a regular method usually leaves the students indifferent and without any particular reaction. However, different solutions of the same problem may cause
emotional enthusiasm. The special and the beautiful solution is unexpected, but tangible,
and in most cases – simpler and shorter.
The ordinary methods are usually technical solutions and they are obtained almost automatically. On the other hand, unorthodox methods permit new initiatives and develop additional skills. In many ways, this is similar to the difference between an algebraic method and
solution by trial and error of textual problems. The use of alternative methods highlights
mathematics as a magical field of surprising solutions.
Indeed, in recent times, one can see from several papers in this field that there is an
increasing tendency to make use of alternative methods to develop excellence and creativity
and to present mathematics as a specialty composed of a variety of intertwined fields.
Mathematics educators agree that linking mathematical ideas by using more than one
approach to solving the same problem (e.g. proving the same statement) is an essential
element in the development of mathematical reasoning [1–3]. Problem-solving in different
ways requires and develops mathematical knowledge [2], and encourages flexibility and
creativity in the individual’s mathematical thinking [4–7].
In addition to the specific roles of proof in mathematics, we suggest that attempts to
also prove a certain result (or solve a problem) using methods from several other different
areas of mathematics (geometry, trigonometry, analytic geometry, vectors, complex numbers, etc.) are very important in developing deeper mathematical understanding, creativity
and appreciating the value of argumentation and proof in learning different topics of mathematics. Our approach, that of presenting multiple proofs to the same problem, as a device
for constructing mathematical connections is supported by [1,2,8–11].
Very similar to our notion of ‘One problem, multiple solutions/proofs’ is the idea of multiple solution tasks (MSTs) presented by [7,11,12]. MSTs contain an explicit requirement
for proving a statement in multiple ways. Liekin [12] indicates that the differences between
the proofs are based on using (1) different representations of a mathematical concept; (2)
different properties (definitions or theorems) of mathematical concepts from a particular
mathematical topic; (3) different mathematics tools and theorems from different branches
of mathematics; or (4) different tools and theorems from different subjects (not necessarily mathematics). In our case, we apply the third-type of differences between the proofs;
we shall present various solutions to a problem using the tools and theorems of Euclidean
geometry, analytic geometry, trigonometry and vectors.
Adding the concept of multiple solutions/proofs for one problem into the curriculum of
mathematics studies, as well as MSTs, allows the development of connected mathematical
knowledge not only for students, but for their teachers as well.
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Based on many years of experience in teaching mathematics in high school, in academic courses and from the process of constructing and training pre-service teachers, we
have established the major importance of possessing comprehensive command of different
mathematical tools, which permit one to deal with complex problems and with challenges
that require the use of various solution strategies.
Of all branches of mathematics, geometry of the plane is one of the most spectacular
fields, due to the large number of methods of solution attainable in it. As a field based on
axioms and postulates from which theorems develop in a modular fashion, and in which
use is made of auxiliary constructions and alternative constructions, one can find in it many
methods of solution for the same task, some of which are standard and without any peculiarities – on the one hand, and where other, unorthodox, very short and impressive methods exist – on the other hand.
During the checking of final exams in mathematics of hundreds of high-school graduates, different methods of solution were found to the same questions, in particular questions
in geometry of the plane. These, both, point to the fact that different teachers teach using
various methods and ways, as well as different emphases with several mathematical tools,
and to the fact that the students could implement them in different forms.
In the framework of a course in a combination of subjects taught to pre-service teachers
in mathematics, at the end of the school year, we presented a geometric task to a group of 20
students, which they were asked to solve using a large number of methods. It was suggested
for them to solve it using: geometry of the plane, trigonometry, analytical geometry, vectors
and complex numbers. Each of them was required to solve it independently, and to try to
find several methods. Indeed, 10 proofs were found, 6 of which were completely different
and used different mathematical tools. After presenting the problem, we shall give some of
the solutions found by the students.
2. Technology in the classroom: the dynamic geometry environment
As is the custom in the modern world which is full of technological tools, today it is impossible to ignore the rapid development of technology and the way it affects almost every facet
of life. The education system is no exception, and one certainly cannot disregard the value
of technology in teaching mathematics.
Empirical/inductive arguments are the traditional methods that are usually used as the
first steps for deriving proofs of geometrical problems. However, the advent of dynamic
geometry environment (DGE) software serves as an intermediary tool that bridges the gap
between a mathematical problem or concept and its symbolic proof by providing a clear,
visual representation of the equation involved. It is well established today that DGE has
opened new frontiers by linking informal argumentation with formal proof [13,14]. Several researchers conducted extensive exploration into students’ behaviour when using DGE
software and its efficacy in connecting the processes of producing conjectures and proving
theorems or statements [15].
Introducing DGE software (such as GeoGebra) into classrooms creates a challenge to the
praxis of theorem acquisition and deductive proof in the study and teaching of Euclidean
geometry. Students/learners can experiment using different dragging modalities on
geometrical objects that they construct, and consequently they can infer properties,
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Figure . Presentation of the problem: x + y = z.
generalities and conjectures about the geometrical artifacts. The dragging operation on
a geometrical object enables students to apprehend a whole class of objects in which the
conjectured attribute is invariant, and hence, the students can become convinced that their
conjecture will always be true [16].
Nevertheless, because of the inductive nature of the DGE, we entitle this process ‘semi
proof’. Hence, following the employment of DGE, the experimental–theoretical gap that
exists in the acquisition and justification of geometrical knowledge becomes an important
pedagogical and epistemological concern [17,18].
Students must be aware that they still need to prove rather than rely on the virtual experiment.
A link to GeoGebra applet is given which allows the reader to have a dynamic experience
of investigating the property.
2.1. Presentation of the problem
Given is a square ABCD whose side length is a.
A straight line is drawn through the vertex A, which intersects the side DC at the point
E. The bisector of the angle ∠EAB is drawn through the vertex A, which intersects the side
BC at the point F. See Figure 1.
Prove that BF + DE = AE i.e. x + y = z .
At the first stage, the students were given an applet that allowed the point E to be dragged
along the side BC, including the possibility of dragging the point outside the square to the
right of the vertex C. During each stage of dragging, the lengths of the segments and the
sum of the relevant segment lengths appear on the screen. The experience with the applet
shows that at each stage the properties are conserved. It is possible to reach the applet using
the following link:
We denote : ∠EAB = 2α, BF = x, DE = y, AE = z.
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Figure . Proof using calculating areas.
2.2. Method A – proof using calculating areas
From point F, we drop a perpendicular FG to the segment AE (see Figure 2).
From the property of a point on an angle bisector, we have BF = FG = x.
We calculate the area of the square ABCD from the areas of the four triangles from which
it is composed:
a · x (a − x)(a − y) z · x a · y
a2 =
After simplification, we obtain: a2 = x (y + z).
By using the Pythagorean Theorem in the triangle ADE, we have: a2 = z2 − y2 .
By substituting this relation, we obtain: z2 − y2 = x (y + z).
After cancelling (y + z) from both sides, we obtain: x + y = z.
2.3. Method B – proof using similarity of triangles
From the point E, we drop a perpendicular EN to the side AB. We denote by M the point
of intersection of this perpendicular with the angle bisector AF. By using the angle bisector
theorem in the triangle AEN, we obtain:
The triangles AMN and AFB are similar; therefore, we can write the proportion:
EM = a − MN =
MN =
a2 − xy
(∗ ∗ ∗).
We substitute the relations (∗∗) and (∗∗∗) in (∗) to obtain: a2 = x (y + z).
Using the Pythagorean Theorem in the triangle ADE, we obtain: x + y = z.
z −a
E a−y C
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Figure . Proof using the properties of the angle bisector.
Figure . Proof using rotation of triangle.
2.4. Method C – proof using the properties of the angle bisector
From the properties of the angle bisector (see Figure 3):
AB = AG = a,
FB = FG = x.
From the Pythagorean Theorem in the triangle ECF: EF2 = (a − y)2 + (a − x)2 .
From the Pythagorean Theorem in the triangle EGF: EF2 = (z − a)2 + x2 .
From equating the values of EF2 , we obtain: (z − a)2 + x2 = (a − y)2 + (a − x)2 ,
z2 − 2az + a2 + x2 = 2a2 − 2ay − 2ax + y2 + x2 .
From the Pythagorean Theorem in the triangle EDA: a2 = z2 − y2 and by cancelling
terms, we obtain: z = x + y.
2.5. Method D – proof using rotation of triangle
We copy ABF in such a manner that the side AB coincides with the side AD, i.e. ABF
is rotated at 90° in anticlockwise direction around point A (see Figure 4).
The triangle F EA is an isosceles triangle since it has two equal angles:
∠EF A = ∠EAF = 90◦ − α.
Conclusion : EF = EA
x + y = z.
2.6. Method E1 – proof by trigonometry
From triangle AFB :
From triangle AED :
tgα = .
a2 = z 2 − y 2 .
cos 2α =
cos2 α =
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By substituting (1) and (3) in the trigonometric identity 1 + tg2 α =
cos2 α
we obtain:
and by substituting a2 from (2) in (4), we have:
(z − y)2 = x2
z = x + y.
2.7. Method E2 – proof by trigonometry
From triangle AFB :
x = atgα.
From triangle ADE :
y = actg2α.
sin 2α
It remains to prove that :
= actg2α + atgα.
sin 2α
We multiply both sides by sin 2α and obtain:
1 = cos 2α + sin 2αtgα = 1 − 2sin2 α +
2 sin α · cos α · sin α
= 1.
cos α
2.8. Method F – proof using analytic geometry
We use the rules of analytic geometry. We locate the square in a system of coordinates in
such a manner that the vertex A coincides with the origin and the sides AB and AD lie on
the axes x and y, respectively (see Figure 5).
The coordinates of the vertices are: A(0, 0), B(a, 0), C(a, a), D(0, a).
We select a point E on the side DC, whose coordinates are E(b, a), such that b < a.
The equation of the side AB is y = 0.
The equation of the line AE is y = ab x.
a x + b y + c = 0
When two straight lines are given by their general equations: 1
a 2 x + b 2 y + c2 = 0
1 x+b1 y+c1
2 x+b2 y+c2
The equation of the angle bisectors are: a√
= ± a√
a1 +b1
a2 +b2
D(0, a)
A (0, 0)
B(a, 0)
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Figure . Proof using analytic geometry.
Figure . Proof using vectors.
ax − by = 0
= ± y.
In accordance with the general equations of the lines AF and AB:
we obtain that the equations of their angle bisectors are: √ax−by
a +b
The explicit equation of the angle bisector ∠EAB (the one with the positive slope that
corresponds to the straight line AF) is: y = √a·x2 2 . Therefore, the coordinates of the
point F are: F(a,
a2 +b2
a +b
From the coordinates of the vertices A, B, C, D, E, F, we obtain the lengths of the segments:
BF =
, AE = a2 + b2 , DE = b.
b + a2 + b2
Using simple algebraic technique (including multiplying the numerator and the denom2
inator by the conjugate of the denominator), one can easily show that b + √a 2 2 =
b+ a +b
a2 + b2 , or in other words, BF + DE = AE.
2.9. Method G – proof using vectors
We denote: AD = u, AB = v, | u | = | v | = a (see Figure 6).
z = u + y, = v + x, z · = u · v + y · v + u · x + y · x.
u · v = y · x = 0 (since the vectors are perpendicular).
y · v = | y | · | v | = y · a (since the vectors are parallel).
D(0,a i )
E(y ,a i )
C(a,a i )
F(a, xi )
B(a, 0)
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A (0, 0)
Figure . Proof using complex numbers.
u · x = | u | · | x | = x · a (since the vectors are parallel).
Hence : z · = z · · cos α = z · · cos α = z · a;
Andhence : z · a = x · a + y · a
z = x + y.
2.10. Method H – proof using complex numbers
We place the square in a system of coordinates, where the horizontal axis is real and the
vertical axis is imaginary, and the vertex A is at the origin (see Figure 7).
z1 = a + xi,
z2 = y + ai, z1 2 = a2 − x2 + 2axi.
Since z2 and z1 2 both have an angle of 2α, one can write down: z1 2 = k · z2 , where k is real.
We equate the expressions: a2 − x2 + 2axi = k (y + ai) and obtain two equations:
a2 − x2 = ky,
2ax = ka ⇒ k = 2x.
We substitute k = 2x in the first equation and obtain:
a2 = x2 + 2xy.
Then, z = |z2 | =
y2 + a2 = y2 + x2 + 2xy = x + y.
3. The art of asking questions
Since the 80s of the last century, methodological aspects of solving mathematical problems moved towards adding a style of questions, such as: overloading data – what if more
(WIM)?, replacing data – what if instead (WII)?, removing data – what if not (WIN)? [1,19–
22], general case – is generalization possible? Therefore, as usual in investigation activities,
‘what if ’ questions were asked.
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Figure . The case when E is on the continuation of the side DC.
Figure . Replacement the square with a rectangle ABCD.
3.1. Question 1
What happens to the relation between the side lengths of the segments when the point E is
on the continuation of the side DC outside the square, as shown in Figure 8?
From Proof method D, we obtain that the triangle EF A is an isosceles triangle. Therefore, in this case as well, we obtain: BF + DE = AE.
3.2. Question 2
What will the relation between the segments DE, BF and AE be when we replace the square
with a rectangle ABCD? (See Figure 9).
We denote : AB = a,
AD = b.
We set a point G on the side AB so that AG = b, and drop a perpendicular H. We rotate the
triangle AHG by 90° about vertex A, as shown in Figure 9.
The triangle H EA is an isosceles triangle, therefore: AE = H D + DE = HG + DE.
From similarity of the triangles AHG and AFB , we have: HG = ab · FB.
Therefore, for a rectangle, the relation between the segments is: AE = DE + ab · FB.
4. Methodical discussion with the students
Towards the end of the activity, the students presented the different solutions to the entire
group. A methodical discussion was held on each solution from different aspects, such as:
the degree of difficulty of the solution method, the elegance and simplicity of the solution,
the creativeness of the solution method, integration between different fields in mathematics
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and their tools, the extent to which the solutions match the knowledge of the students and
the programme of studies.
Subsequently, we presented to the students the concept of ‘conservation’, based on the
conserved property of the sum of the lengths of two segments along a third segment by
shifting the point E on the side DC.
From this point on, a discussion was held concerning the importance of finding several methods of solution of the same task, the fact that in each mathematical theorem, and
especially in Euclidean geometry, there is a single property for several properties that are
conserved. A comprehensive discussion was held on the geometry conservation properties.
By applying GeoGebra applets, other examples of conservation properties were presented
and their contribution to solving problems in geometry has been emphasized [23].
In addition, a discussion was held on the importance of using the computerized technological tool for testing the correctness of different hypotheses.
4.1. Feedback of the students
• The different proofs found for the task reinforce the perception of mathematics as a
field built from different intertwined branches.
• At first glance, this seems to be a task in geometry, and without the requirement of
solving it using traditional methods, we would not have tried to solve it using analytical
geometry, vectors and complex numbers.
• The proof that the students liked the most, of those shown at the end of the activity, is
method D (by rotating a triangle), followed by method A (by calculating areas).
• There is much similarity between the proofs using vectors and the proof using complex
4.2. Concluding remarks and implication for teaching
Geometry is a goldmine for MST. Proofs may be derived by applying different methods
within the specific topic of geometry or within other mathematical areas such as analytic
geometry, trigonometry and vector spaces. The authors claim that multiple proofs foster
both better comprehension and increased creativity in mathematics for the student/learner.
The multiple solutions that were presented herein for one geometry problem demonstrate the connectivity between different areas of mathematics, and show how geometry can
serve as a base for topics such as trigonometry, analytic geometry and vector spaces. For
example, solving the problem by analytic geometry methods (besides the clear employment
of Euclidean geometry) reveals a connection to trigonometry by the slopes of the sides of
the triangle and the resulting need to solve trigonometric equations. When doing the proof
by vectors, the connection to trigonometry is shown by the requirement to apply the scalar
product formula and again, the need to solve trigonometric equations in order to arrive at
the conclusion.
This occupation with the connection between different mathematical domains builds
among students a vision of mathematics as a linked science and not as a collection of
discrete, isolated topics [24]. In most school textbooks worldwide, mathematics problems
are organized by specific topics that are presented in the curriculum: students tend to
understand that certain problems are connected to specific topics and hence assume that
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for each problem, there is one, and only one, method for its solution [25]. Also, while the
National Council of Teachers of Mathematics (NCTM) standards document emphasizes
that teachers should find tasks that exhibit connectivity between different mathematical
domains, it also indicates, nevertheless, that locating such problems is time-consuming
and calls for special initiative from the teachers. Certainly, our experience also shows
that while this mission is not an easy one, there is certainly a need to identify additional
problems that can be solved by various methods and demand the application of proofs
from different realms of mathematics. The authors see it their task to continue searching
for such appropriate problems and encourage their colleagues to do so, too. They believe
that mathematics teachers should present to their students problems that must be solved
in more than one way, requiring, as much as possible, the application of knowledge from
different domains in mathematics. In his study, Bingolbali [26] indicated that applying
multiple ways of solving problems has the potential to develop students’ ‘relational understanding’ (a term attributed to Skemp [27]) and contributes to the development of their
autonomy. Furthermore, when testing students, teachers should occasionally allow their
students to solve problems by employing proofs from any mathematical area, and not insist
on a proof from the specific subject under study.
In this day and age, it is impossible to ignore the rapid development of technology and
the way it affects almost every facet of life. The education system is no exception, and one
can certainly not disregard the value of technology in teaching mathematics. The authors
firmly believe in the advantages of incorporating DGS in mathematical problem-solving
and actually demonstrated our problem using GeoGebra software. In a special issue of The
International Group for the Psychology of Mathematics Education (PME), Jones et al. [28],
writing a guest editorial entitled ‘Proof in dynamic geometry environments’, stated that ‘there
is a range of evidence that working with dynamic geometry software affords students possibilities of access to theoretical mathematics, something that can be particularly elusive
with other pedagogical tools’ [23,p.3]. Inductive exploration with DGS can lead students to
develop their own conjectures about the solution of the problem and then to deal with the
deductive proof. This is in addition to the contribution of visualizing different graphical
representations of concepts and other related situations to the problem. We further recommend, therefore, that mathematics teachers initially allow their students to cope with
problem-solving by working in such an environment until they reach a solid conjecture for
deductive proof.
In conclusion, our efforts in coping with the presented problem afforded us a real feeling
of working as mathematicians who look for multiple solutions to a problem, especially those
that are short, elegant and mathematically aesthetic. By encouraging student/learners to do
the same, they, too, will learn to appreciate the connections between the various branches of
mathematics and discover how one problem can be tackled from different points of view. In
addition, the incorporation of DGE software will add a complementary technological tool
to assist the students in their investigations, while also providing teachers a means with
which to base pedagogical action and in-class discussion.
4.3. Summary
The authors believe that this activity suggests that students need to absorb the importance
of solving tasks using several different methods.
Disclosure statement
No potential conflict of interest was reported by the authors.
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