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Advances in Social Sciences Research Journal – Vol. 8, No. 8
Publication Date: August 25, 2021
DOI:10.14738/assrj.88.10786. Nguyen, N., & Nguyen, T. H. (2021). Discovery Teaching About Calculating Side Lengths of Right Triangles Using Pythagoras Theorem
With the Help of Geogebra Software. Advances in Social Sciences Research Journal, 8(8). 690-710.
Services for Science and Education – United Kingdom
Discovery Teaching About Calculating Side Lengths of Right
Triangles Using Pythagoras Theorem With the Help of Geogebra
Software
Ngoc-Giang Nguyen
Faculty of Economic Mathematics
Banking University Ho Chi Minh City, Ho Chi Minh City, Vietnam
https://orcid.org/0000-0001-7560-7972
Thuy Hong-Anh Nguyen
Math Team, Dong Nai Pedagogical Practice High School Bien Hoa City
Dong Nai province, Vietnam
ABSTRACT
Discovery teaching is an active teaching method that encourages learners to create
their own knowledge. Students who participate in the discovery learning process
will remember longer than the conventional teaching method. This teaching
method is very important in the process of teaching innovation in Vietnam when
changing from a teaching method that focuses on knowledge transmission to one
that focuses on developing learners' capacity. Our study proposes an exploratory
teaching method for calculating the side lengths of right triangles using the
Pythagoras theorem with the help of GeoGebra software. By the method of empirical
investigation, we surveyed 51 students in Bien Hoa city and used descriptive
statistics with the help of SPSS software, the research has shown that the discovery
teaching of calculating side lengths of right triangles using Pythagoras theorem with
the help of GeoGebra software is needed. However, many teachers still do not fully
understand the discovery teaching method. The research results also show that
there are three groups of factors affecting the way of discovery teaching. First, the
effort of teachers in teaching discovery is huge. Second, the time spent on discovery
teaching is relatively large. Third, the current teaching environment has not yet met
the requirements of discovery teaching. The findings from the research results are
important for proposing a way of discovery teaching about calculating the side
lengths of right triangles using Pythagoras theorem with the help of GeoGebra
software.
Keywords: Discovery teaching, calculating the side lengths of a right triangle, Pythagoras
theorem, process, GeoGebra software.
INTRODUCTION
Learning only really makes sense, and learners only remember longer when learners
participate in the construction of knowledge themselves. Learners are more active and dynamic
in teamwork (Morgan K Williams, 2017). Teaching must encourage learners to discover
knowledge for themselves. Learning is an active process in which learners construct new ideas
or concepts on the basis of existing knowledge (Jerome S Bruner, 1960). Instead of just listening
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Nguyen, N., & Nguyen, T. H. (2021). Discovery Teaching About Calculating Side Lengths of Right Triangles Using Pythagoras Theorem With the Help
of Geogebra Software. Advances in Social Sciences Research Journal, 8(8). 690-710.
URL: http://dx.doi.org/10.14738/assrj.88.10786
to lectures, learners have the opportunity to apply different skills in activities and master their
learning. Learning is not a rigid process that cannot be changed, learners assimilate new
knowledge into their own knowledge base (Tracy Bicknell-Holmes Hoffman, 2000). Discovery
teaching focuses on active learning activities, creating opportunities for learners to explore and
solve problems on their own. The focus of learning is on learning how to analyze and synthesize
information to understand what's going on, rather than just coming up with exact answers from
rote memorization. It pushes students to a deeper level of understanding (Joyce Castronova,
2002).
In Vietnam, there are also many studies on discovery teaching. To use this teaching method, it
is necessary to first build lessons designed with questions, so that when answering or trying to
solve component questions, students gradually see how to solve the initial problem. Finding the
answer goes from easy to difficult, from things that are obvious, conspicuous, and explicit to
uncovering hidden rules and concepts (Tran Ba Hoanh, 2004). A discovery activity is a thinking
process including observation, analysis, evaluation, hypothesis and inference in order to
discover and acquire knowledge of the subject. It is usually done through a system of activities,
which will gradually appear a path to knowledge when students answer and do (Le Vo Binh,
2007). This teaching method is a process in which, under the role of the teacher's orientation,
learners take the initiative in their own learning, form questions posed in thinking, and expand
research and search. From there, new insights and knowledge are built. This knowledge helps
learners answer questions, find different methods of solving problems, prove a theorem or
point of view (Pho Đuc Hoa and Ngo Duy Son 2008). The teaching method encourages students
to ask questions and find answers on their own, or to draw principles from examples or
practical experiences. In addition, discovery teaching can be defined as a learning situation in
which the main content to be learned is not introduced first but must be discovered by the
students themselves, making the students an active participant in the learning process (Nguyen
Phu Loc, 2010).
GeoGebra software was invented by Austrian educator Markus Hohenwater, a lecturer at the
University of Salzburg in 2001. The software is a combination of Geometry, Algebra, Analysis,
and spreadsheet. GeoGebra is free, open-source, and multi-language software (can be used in
over 63 languages, including Vietnamese). GeoGebra's interface is friendly and easy to use, with
intuitive toolboxes that users can easily manipulate the software (Markus Hohenwarter and
Judith Hohenwarter, 2008). GeoGebra has the potential to promote active learning. Teaching
with the help of GeoGebra software is a learner-centered teaching method. GeoGebra creates
an environment that allows mathematical operations to be performed (Darius Majerek, 2014).
GeoGebra is software that can be embedded in an e-learning environment. GeoGebra promotes
discovery and self-knowledge (Preiner Judith, 2008). There is a positive relationship between
teachers' current perceptions and intention to use GeoGebra in teaching math. In addition, the
results confirmed that there was no significant difference between male and female teachers in
using GeoGebra in teaching mathematics (Soheila Belgheis and Rosemaliza Kamalludeen,
2018).
In Vietnam, there are also many authors doing research on GeoGebra software. GeoGebra is
effective software that supports the proof process, motivating learners to create new
knowledge in a sustainable way (Nguyen Danh Nam, 2012). The use of this software will bring
many opportunities for students to practice skills and master the content of math knowledge
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(Le Viet Minh Triet, 2018). The use of GeoGebra software in teaching mathematics may face a
number of difficulties such as limited equipment of some high schools, teachers' low ability to
use the software, teachers not being properly guided and trained on the effective exploitation
and use of software in teaching mathematics (Nguyen Thi Huong and Le Tuan Anh, 2018).
GeoGebra software has many outstanding advantages such as interoperability, allowing the
creation of new geometries, problems to be solved instantly, allowing verification of geometry
problem results, allowing geometry object movement, and illustrating theorems and properties
of animation. Thanks to its outstanding advantages, when combined with discovery learning
methods, teaching becomes better and more effective (Nguyen Ngoc Giang, 2016). Discovery
teaching using GeoGebra animation software not only helps students acquire knowledge
actively, but also creates opportunities for learners to develop self-study, self-discovery, and
group cooperation skills, enabling students to assert themselves, thereby improving the quality
of teaching and learning in high schools. (Tram Huy Khoi, 2019).
Although there are many studies on discovery teaching, discovery teaching with the support of
GeoGebra software, the topic of applying the discovery teaching method to calculate the side
lengths of a right triangle by using the Pythagorean theorem with the help of GeoGebra software
is still quite new and attractive and needs more research, especially in the current period when
the application of information technology in teaching and learning is developing strongly.
LITERATURE REVIEW
Advantages and disadvantages of discovery learning
Advantages
- Students participate actively and voluntarily in the process of learning new knowledge.
Exploratory activities stimulate students to ask questions about emerging problems. As
a result, students are more attracted and curious, and more eager to learn new
knowledge. In performing discovery learning, students also become speakers. They
must be able to present their findings on their own and defend them in front of the whole
class. By actively seeking information and solving problems, they can accumulate
knowledge and create their own capital of experience. (Vu Thi Lan Anh, 2020).
- This method is student-centered and teachers are actively involved in motivating
students to voice their ideas. Even the teacher can act as a student or a researcher in the
discussion. ( Ringgi Candraning Prawerti, 2014).
- This method promotes the development of thinking, skills, and experiences in the
process of discovery and learning. (Multi Efrini, 2016).
- Learners learn how to develop their own memory because, in discovery, learners have
to mobilize their own knowledge and experience and relate existing knowledge to
relationships with the problem to be learned, so they will remember the lesson longer.
It is even possible to reconstruct knowledge when relevant information is available.
(Novriana Rahma Siagian, 2018).
- Learners learn in interaction, form cooperative relationships, and solve learning tasks
together. Students participate in group activities to help solve problems of broader
content. Learners will develop social skills and communication skills through these
group activities. (Richard E Mayer, 2004).
- Lesson theories are suggested to explore. Students play an important role in learning
abstract mathematical concepts, and their will to learn becomes increasingly persistent
and creative (Ryan D Chen and Honomichl and Zhe, 2012).
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Nguyen, N., & Nguyen, T. H. (2021). Discovery Teaching About Calculating Side Lengths of Right Triangles Using Pythagoras Theorem With the Help
of Geogebra Software. Advances in Social Sciences Research Journal, 8(8). 690-710.
URL: http://dx.doi.org/10.14738/assrj.88.10786
- The class becomes lively, active and creates more learning motivation for students.
Student learning outcomes are better in the learning process. (Suphi and Yaratan, 2016).
- Discovery learning helps teachers assess students' learning development. This teaching
method is consistent with the current capacity development orientation. (Sobari and
Husnussalam, 2019).
Disadvantages
- To apply this method, students must have the necessary knowledge and skills to perform
exploratory tasks and find new knowledge. Average and weak students will have
difficulty learning this method. (Trinh Duc Toan, 2016).
- The time of the process of discovering new knowledge takes up a lot of the entire
learning process, so it depends on the content, the lesson objectives, and the way of
teaching time distribution so that we can apply it. Therefore, discovery teaching cannot
be applied to all lectures. (Emily Brown, 2006).
- The activity of discovering knowledge about geometry requires teachers to design many
models, symbols, images, etc. Therefore, the facilities of teaching and learning must meet
the desired results.
- The effort of teachers in discovery teaching is relatively large. Therefore, it is necessary
to have special treatment for teachers for this teaching method compared to other
teaching methods.
- This method is not effective to teach many students, because it will take a long time to
help them find a theory or solve other problems. It is necessary for students in different
regions to ask different discovery questions. ( Ringgi Candraning Prawerti, 2014).
- Teachers spend a lot of effort and money in preparing tests and assessment questions
for students by means of discovery teaching.
An example to illustrate the calculation of the side lengths of a right triangle using the
Pythagorean theorem with the help of GeoGebra software
Tika Karlina Rachmawati (2016) wrote the article “An Analysis of Students’ Difficulties in
Solving Story Based Problems and Its Alternative Solutions” in the Journal of Research and
Advances in Mathematics Education. The author analyzed the difficulties of students when
solving problems and practical problems based on the Pythagoras theorem to calculate the
length of the diagonal of the rectangle or the closest distance of the ship from the starting point.
(Tika Karlina Rachmawati, 2016).
Vu Huu Binh and colleagues in the book "Improving Math 7" used the Pythagorean theorem to
calculate the side lengths of a right triangle with exercises from simple to more complex. For
example, the exercises show the lengths of any two sides of a right triangle and calculate the
length of the other side; or practical related exercises using the Pythagorean theorem such as
roofing a roof in the form of an isosceles triangle when knowing the height of the roof and the
length of the ceiling. (Vu Huu Binh, 2014).
Fathiya Salsabila and colleagues (2017) presented “Students' Ability in Proving Pythagorean
Theorem through Discovery Learning Model Using Geogebra Software” during The 6th
International Conference on Multidisciplinary Research (ICMR) in conjunction with the
International Conference on Electrical Engineering and Informatics (ICELTICs) 2017. The
content is about training 8th graders the skills to use GeoGebra software to solve problems
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discovering knowledge of the 8th-grade geometry program in general and content of
Pythagorean theorem in particular. This has been tested in 2 classes; each class has 24 students,
corresponding to one class with the traditional teaching method, the other class applying the
discovery teaching method with the support of GeoGebra software. After two weeks of
experimentation, the evaluation results show that the discovery-based class with the support
of GeoGebra software has made great progress as shown in the test of math problems with
content from basic to advanced. Students apply the Pythagorean theorem masterfully to solve
real-life related problems in experiential activities quite well. (Fathiya Salsabila, 2017).
At the 3rd International Conference on Research of Educational Administration and
Management (ICREAM 2019), Nguyen Ngoc Giang and Tran Trung mentioned the use of
GeoGebra software in teaching geometry. For example, by teaching the Pythagoras theorem, we
can use the dynamism of the software to help students distinguish static and dynamic elements
in the lesson, thereby knowing how to construct and create moving elements by using the slide
bar to develop, discover new problems with immediate results. (Nguyen Ngoc Giang and Tran
Trung 2019).
From the above points of view, we think that to calculate the side lengths of a right triangle
using the Pythagorean theorem with the help of GeoGebra software, first of all, students need
to understand Pythagoras' theorem and the construction features as well as the elements of
motion to discover many new and more complex problems.
The process of organizing the implementation of the situation
In order to form students' skills in solving problems applying the Pythagorean theorem and the
ability to use GeoGebra dynamic geometry software in exercises to calculate the side lengths of
a right triangle, we provide a procedure as follows:
Step 1. Learn contents to be explored with the help of GeoGebra
From the given data of the problem, it is necessary to analyze and determine the relationship
between the initial data of the problem and the unknown quantity to be found. Learners
discover for themselves which knowledge is reasonable to apply to solve the problem posed.
After determining the data for the given problem, the teacher guides the students to bring the
problem back to the actual triangle image. Ask students to tell where the triangle is right, which
sides we already know the length of, and which theorem we use to find the length of the
remaining side of the right triangle.
Step 2. Build shapes with the help of GeoGebra software
The teacher instructs students to build shapes using GeoGebra software for the data of the
problem. Students need to discover relationships between unknown and known quantities, as
well as detect the moving or fixed quantity of the problem. From there, students explore and
discover how to build shapes with the help of this GeoGebra software.
Step 3. Solve the discovery task with the help of GeoGebra software
For the problem of applying the Pythagoras theorem, learners need to know the formula and
the factor to be calculated here is to calculate the right angle or hypotenuse, thereby using the
formula more accurately.
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Nguyen, N., & Nguyen, T. H. (2021). Discovery Teaching About Calculating Side Lengths of Right Triangles Using Pythagoras Theorem With the Help
of Geogebra Software. Advances in Social Sciences Research Journal, 8(8). 690-710.
URL: http://dx.doi.org/10.14738/assrj.88.10786
After completing the shape construction steps, the teacher will guide students to go deep into
applying the Pythagorean theorem to calculate the unknown side length, thereby discovering a
way to solve the problem and calculating the required result of the problem.
Step 4. Deepen the discovery quest with the help of GeoGebra software
When using the support of GeoGebra software, learners will immediately verify whether their
calculation is right or wrong, then correct it (if the calculation is wrong) to get the most accurate
answer.
The teacher will provide accurate results and generalized overview comments about the task
the students have discovered. After that, the advantages and limitations of students will be
summarized when solving the discovery task with the help of GeoGebra software. In addition,
based on the old problem, it is possible to exploit and develop the problem into a new problem.
This approach helps learners develop thinking and creativity in the process of discovery.
Step 5. Use GeoGebra software to illustrate Pythagoras theorem on right triangles to create
decorative works
Point out advantages and introduce outstanding features of GeoGebra software. In addition to
drawing and verifying, GeoGebra can also illustrate images of Pythagoras' right triangles using
the toolbar, especially the dynamic feature of the slider for students to refer, expand, and
explore knowledge. From there, students can apply these features to decorate their graphic
works.
Illustration
Example 1. A boy is holding a kite coil with his hand at a distance AD = 1.6m from the ground
and is standing and flying a kite at a distance AB = 8m from a tree (as shown in the figure). He
has released a kite string of length DG = 17m and the kite is hovering above the tree. Calculate
the distance GB of the kite from the ground.
Figure 1. A boy flying a kite (Source: Freepik).
In the table of activities of teachers and students below, from step 1 to step 4, we use the
operations of recognizing, applying, and discovering new problems. As for step 5, we use
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manipulations of creative thinking such as generalization and analogy to represent elements of
exploration, discovery, and shape building to create new decorative works with the help of tools
and means of learning mathematics in the general education program of mathematics 2018.
Figure 2. Building shapes from the problem of a boy flying a kite (Source: Personal Collection).
Activities of teachers and students:
Step 1. Learn contents to be explored with the help of GeoGebra
Teacher: Identify all the facts on the given problem.
Students:
- The distance from the boy's hand holding the kite coil relative to the ground is AD = 1.6m.
- The boy stands at a distance AB = 8m from the tree.
- The length of the kite string is DG = 17m.
Teacher: Find a relationship between known quantities and the desired quantity. Describe
how to calculate the distance of a kite from the ground.
Students: From the known quantities, the distance of the kite from the ground can be
calculated as follows:
+ Calculate the GC distance using the Pythagorean theorem on the right triangle GDC at C.
+ Then, sum GC + CB, we get the desired result.
Step 2. Build shapes with the help of GeoGebra software
Teacher: Instruct students to transfer the picture to a specific triangle case.
Use tools and math learning tools with the support of GeoGebra software to draw shapes
according to the following steps:
Open the GeoGebra software.
For unnecessary objects, we can hide.
Students: acquire and implement
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Nguyen, N., & Nguyen, T. H. (2021). Discovery Teaching About Calculating Side Lengths of Right Triangles Using Pythagoras Theorem With the Help
of Geogebra Software. Advances in Social Sciences Research Journal, 8(8). 690-710.
URL: http://dx.doi.org/10.14738/assrj.88.10786
Ordinal
number Shapes Steps
1
Line tool group. Click on the line tool group
2
Select the symbol: The line passes
through two points.
Then draw a line x, keep point B and
hide the other point.
3
Other tool group
Click on the other tool group
4
Select the icon: Slider
Then fill in:
Name: a = 8
Select: Number
Minimum: 0 – Maximum: 20
(Depending on the value of the given
topic, adjust accordingly)
Then click OK, we get the slider as
shown
5
Circle and arc tool group
Click on the circle and arc tool group
6
Select the icon: Circle when the
center and radius are known.
Then select point B and enter radius
as a and select OK.
7
Point creation tool group
Click on the point creation tool group.
8
Select the icon: New point
Then click to select the intersection of
the circle with center B, radius a =
8cm with the line x. We get point A.
(If you see the point with a different
name, you can right-click => select
rename.)
9
Relationship tool group
Click on the group of relationship
tools
10 Select the icon: Perpendicular line.
Then, select line AB and point A.
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Do the same, select line segment AB
and point B.
We have 2 lines perpendicular to AB
at points A and B.
11
Circle and arc tool group
Click on the circle and arc tool group
12
Select the icon: Circle when the
center and radius are known.
Then select point A and enter radius
as 1.6 and select OK. We have a circle
with center A and a radius of 1.6cm.
13
Point creation tool group
Click on the point creation tool group.
14
Select icon: New point
Then click the mouse to select the
intersection of the circle with center
A, radius 1.6cm with the line
perpendicular to AB at A. We get
point D. (If you see the point with a
different name, you can right-click =>
select rename.)
15
Relationship tool group
Click on the group of relationship
tools
16
Select icon: Parallel line.
Then, choose point D and line AB, we
get a line parallel to AB.
17
Point creation tool group
Click on the point creation tool group.
18
Select the icon: New point
Then click to select the intersection of
the line parallel to AB and the line
perpendicular to AB at B. We get
point C. (If the point is shown with a
different name, we can right-click =>
choose rename.)
19
Line tool group Click on the line tool group.
20
Select the icon: Slider
Then fill in:
Name: b = 17; Select: Number
Minimum: 0 – Maximum: 30
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Nguyen, N., & Nguyen, T. H. (2021). Discovery Teaching About Calculating Side Lengths of Right Triangles Using Pythagoras Theorem With the Help
of Geogebra Software. Advances in Social Sciences Research Journal, 8(8). 690-710.
URL: http://dx.doi.org/10.14738/assrj.88.10786
(Depending on the value of the given
topic, adjust accordingly)
Then click OK, we get the slider as
shown
21
Circle and arc tool group
Click on the circle and arc tool group
22
Select the icon: Circle when the
center and radius are known.
Then select point D and enter radius
as b and select OK. We get a circle
with center A and radius b = 17cm.
23
Point creation tool group
Click on the point creation tool group.
24
Select the icon: New point
Then click the mouse to select the
intersection of the circle with center
D, radius b = 17cm with the line
perpendicular to AB at B. We get the
point G. (If the point is shown with a
different name, we can right-click =>
select rename.
Hide unnecessary objects.
Select the line perpendicular to AB at A, right-click and select: Show object.
Do the same for the circle with center A and radius 1.6cm; the circle with center B and radius a =
8cm; the circle with center D and radius b = 17cm; the line parallel to AB at D; the line
perpendicular to AB at B
25
Line tool group Click on the line tool group
26
Select the icon: Line segment
Then, connect 2 points A and B; A and
D, D and C; B and C; D and G.
27
Select the icon: Angle with a given
magnitude
Then choose the points G - C - D in
turn, we get the angle GCD equal to
90o
Step 3: Solve the discovery task by finding the length of the remaining side of a right
triangle based on the Pythagorean theorem
Teacher: After guiding students to find out the problem to be explored and build the shape,
ask the students to proceed to solve the discovery problem with the help of GeoGebra
software.
Teacher: Based on the picture, what formula do you use to calculate the GC edge?
Student: Based on the Pythagorean theorem