CHAPTER 1
INTRODUCTION
1.1
Preview
In this chapter, the background of the study is presented as well as the statement of the
problem. In addition, the objectives and research questions are enumerated. Finally, the
purpose and significance as well as limitations and de-limitations are discussed.
1.2
Background of the Study
The world has now become a global village and trends in all human endeavours have
assumed new shapes and dimensions. It is therefore important that mathematics as a
subject should also be revolutionalized in its teaching and learning to meet the challenges
demanded of it by the ever-changing world. Teachers in particular, and educational
planners in general are therefore implored by society to design practical methods of
teaching and learning that are applicable to the learner’s environment and our everyday
life situations.
As core subjects in the educational system in Ghana and as far as the Basic and
Secondary levels are concerned, mathematics is not given the needed attention by
teachers and administrators. Some topics in mathematics are usually skipped and
relegated to the background by majority of those who teach it; especially the ones they
find uncomfortable (Ministry of Education & Sports, 2007).
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Those topics however may tend to be the bedrock or foundation upon which other
important topics could be built. Liner Equation is one of the most prominent ideas in
mathematics, and it is the foundation of Algebra (Dugopolski, 2000). Solving Linear
Equations as a topic, poses problems to students because of the improper way and
manner it is taught by some teachers. The basic skills and pre-requisite knowledge
needed to effect a fuller comprehension is normally lacking in the learners and sometimes
those who teach them. If well-taught rudiments, concepts and basic skills needed to solve
algebraic equations are well understood by students in the Basic and Junior High Schools,
it will enable students to subsequently apply their acquired knowledge to solve algebraic
equations using the appropriate methods at higher levels.
In Ghana, the topic Linear Equations in mathematics with one variable is prominent in
both the Core and Elective curriculum for Senior High Schools. The Researcher has
taught the topic Solving Linear Equations in one variable to students of the school where
he teaches and evaluated them with a test to measure their understanding of the topic. The
researcher realized that the students’ performance in the test was abysmally poor. When
solving linear equations in one variable, the students of Konadu Yiadom Senior High
School Two (SH2) applied algebraic concepts wrongly and used wrong ways of working
as well as inappropriate methods, displaying wrong solutions and eventually arriving at
wrong answers. According to Lima and Tall (2008), students also build their own ways of
working based on the embodied actions they perform on the symbols, picking them up
and moving them around with their own magic rules such as change sides, change signs.
The researcher observed similar problems with his students when they tried to solve
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linear equations. The students applied algebraic concepts wrongly and used wrong ways
of working and inappropriate methods of solving algebraic equations.
Most of the students responded to the general principle of doing the same thing to both
sides of the equation but in specific terms in which they shift the objects from one side to
the other side of the equation. Students did not solve Linear Equations in one variable by
using the general principles of maintaining balance of the equation by performing the
same operations on both sides. They are not able to negotiate shift from arithmetic to
algebra with the subtle change from embodied actions to symbolisms. Unlike arithmetic,
where addition of whole numbers correspond with physical actions on objects, algebra
works at a more general level and various embodiments that occur may prove to be more
problematic to students (Filloy & Rojano, 1989). Teaching strategies and curriculum
design for students should therefore emphasize on the importance of giving meaning to
algebraic symbols and using them in solving equations by applying real life situations.
Students’ conceptions of algebraic equations, what methods they use in solving equations
and what previous experiences interfere in their work with equations constitute the major
problems encountered by students of Konadu Yiadom Senior High School.
Onivehu and Ziggah(2004) observed that no nation can attain technological breakthrough
without a well planned and effectively implemented mathematics education programme,
since mathematics plays a leading role in all human endeavour. A well planned use of
balance model when implemented effectively in the mathematics classroom will help
students in solving linear equations in one variable. The interactive nature of the balance
model will enable students to apply what they learn to real life situations. Nabie(2004)
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argues that the traditional method of ‘talk and chalk’ does not enable learners to apply
what they learn to real life situations or solve problems.
The poor performance at Junior High School in mathematics and the need to use practical
activities, concrete materials, pictures and diagrams for learning linear equations in one
variable is reported in Trends in International Mathematics Science Study (TIMSS)
(Anamoah-Mensah, Mereku & Asabere-Ameyaw, 2008). The need to use balance model
as a concrete material for practical activities will help students to enhance their
understanding and performance in solving equations in one variable. Dugopolski (2002),
concluded that linear equations in one variable is one of the most important topics in
mathematics and also one of the first concepts students encounter in
pre-algebra. The
topic therefore needs to be taught using concrete methods to ensure application to real life
situations.
1. 3
Statement of the Problem
Effective knowledge in Science and Mathematics is the basis for human development in
all areas of life. In Ghana the Curriculum Research and Development Division (CRDD)
of the Ministry of Education and Sports (MOES) recommends the use of educational
resource materials for investigation and solving problems of real life situations (Ministry
of Education & Sports, 2007). The use of educational resource materials for investigating
and solving problems of real life situation help make teaching and learning more
effective. But significantly missing in the High School mathematics Curriculum is the
specified type of the educational materials to use and at what time and place to use them
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as well as the topics for which they are to be applied. Most teachers are at a loss at when
and where to use these educational materials for effective teaching and learning. What
they do therefore is to teach major topics in mathematics without the necessary material
inputs that could give better understanding of the topics.
The use of the Balance Model in teaching Linear Equations in the mathematics classroom
may prove unfamiliar to some teachers who are accustomed to certain routine of
instructions like lecturing. However, its use presents an alternative way to overcome
these instructional inadequacies and help students to understand key concepts in algebra
by making it real.
Solving Linear Equations in one variable posed problems to SHS 2 Home Economics
students at Konadu Yiadom Senior High School because they followed the use of rules
and formulae instead of understanding the principles involved. Some prescribed
textbooks used by the students also follow the change of sign rule when solving Linear
Equations in one variable. The balance model therefore is to help students to understand
Linear Equations in one variable by performing some operations on both sides of the
equation to maintain balance. Lack of concrete materials for teaching and learning,
hinders teachers’ effective teaching of linear equations in one variable and this results in
students’ low understanding of the concepts involved in solving linear equations in one
variable (Cockcroft, 1982).
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1. 4
Purpose of the Study
The purpose of the study was to help students improve their performance in solving
Linear Equations by the use of the Balance Model. The study also sought to advocate the
use of the Balance Model in classroom mathematics teaching and learning for effective
understanding of the concepts involved in solving linear equations in one variable, and to
link concrete material representation with abstract symbolic representation to enhance
conceptual understanding of mathematics (Capps & Pckreign, 1993).
1. 5
Objectives of the Study
The Research focuses on the use of the Balance Model to augment students’ achievement
in solving Linear Equations. The Specific Objectives therefore are to:
a) Determine the effects of inadequate teaching methods on students’ understanding of
the concepts in solving Linear Equations.
b) Improve students’ performance in solving linear equations in one variable.
c) Promote the use of the Balance Model in mathematics teaching and learning to:
(i) Improve students understanding of the concepts involved in solving linear
equations in one variable.
(ii) Determine the effects of using the Balance Model on students’ achievement in
solving Linear Equations.
(iii) Enhance students’ cultural and practical experience in solving Linear
Equations in one variable.
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1. 6
Research Questions
The Research Questions formulated for the study were:
1. How can the use of the balance model improve students understanding of the
concepts involved in solving linear equations with one variable?
2. In what ways can the use of the balance model improve students’ achievement
in solving linear equations in one variable?
3. How does the usage of the balance model affect students’ attitude towards
improving their performance in solving linear equations in one variable?
1. 7
Hypothesis
The hypothesis formulated for the study was that: There is significant difference in
students’ achievements in the test instrument for helping students overcome their
difficulties in solving Linear Equations in one variable.
1. 8
Significance of the Study
The significance of the study is to assist students to use the Balance Model to solve
Linear Equations in one variable and to improve the understanding of the topics. They
would gain confidence and be able to solve other related mathematical problems. It is
hoped that the use of the Balance Model may have a positive impact on achievement in
mathematics and positive students’ attitude towards learning mathematics will be greatly
enhanced.
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The use of the Balance Model, as well as other educational materials make instruction
more student-centered, encourages cooperative learning and stimulates student-teacher
relationship. The uses of appropriate classroom support materials go a long way to
encourage a new and a practical approach to the teaching of algebraic concepts.
New focus is being emphasized in the mathematics curriculum for students to gain
valuable insights in the classroom that can lead to success in the real world. The use of
real life situations, like the Balance Model, in the classroom will enable teachers to
respond to challenges of the changing curriculum.
1. 9
Limitations of the Study
The study was limited to Konadu Yiadom Senior High School at Asamang in the Ashanti
Region. It was also limited to the use of the Balance Model and not other models in
solving Linear Equations in one variable. The findings were therefore limited to students
of Konadu Yiadom Senior High School. In future, measures should be put in place to
allow teachers to interact more with a wider population than it is presently. This will
enable teachers to research into wider fields of a national outlook and make findings
more applicable to the Global Community.
1. 10
Delimitation
The emphasis was limited to solving Linear Equations whereas Algebra covers a lot more
than is derived in solving Linear Equations in one variable. The limited coverage is as a
result of students’ low performance as discussed in the background.
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1. 11
Organization of the Study
The study is made up of five (5) chapters. Chapter One comprises the introduction,
background to the study, statement of the problem, objective of the study, research
questions, purpose of the study, significance of the study, limitations of the study,
delimitation, organization of the study and hypothesis. The second chapter covers
literature review. Chapter Three discusses the research design, population and sampling,
instrumentation, intervention procedures, data collection procedure and data analysis
procedure all under methodology. The fourth chapter deals with the data analysis, results
and discussion of findings. Chapter Five includes the conclusions, implications and
recommendations.
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CHAPTER 2
LITERATURE REVIEW
This chapter reviews the literature that is relevant to solving linear equations in one
variable.
2. 1
Theoretical Framework
The performance of actions, via step by step procedures, then as input-output processes,
and as mental objects in their own right is well-represented in literature (Dubinsky, 1991;
Sfard, 1991). However, the actual shift from process to object seems to be a difficult one
for many students to accomplish. When symbols are first manipulated, the focus is
naturally on the sequence of steps performed, and there is initially considerable mental
effort involved in performing the steps. The phenomenon, where symbols represent both
a mental concept and also possibly different actions to attain the same effect is called a
procept (Gray & Tall, 1994). The idea of giving embodied meaning to procept in terms of
the effects of the actions was introduced in Watson (2000) and Watson, Spyrou and Tall
(2003). The shift from many actions to a single thinkable concept is an act of
compression that is an essential tool in making profound sense of mathematics. As
maintained by writers (Dubinsky, 1991; Sfard, 1991; Gray & Tall,1994; Watson, 2002;
Watson, Spyrou & Tall, 2003), the (sequence of) steps performed by the use of the
balance model enable the shift from many actions to the concept of maintaining balance
in solving linear equations in one variable.
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Lima and Tall (2008) suggest three different mental worlds of mathematics.

A conceptual embodied world of human perception and action (including pictures
mental images and the internal connections we make in our minds).

A proceptual symbolic world of mathematical symbols that operate flexibly either
as concepts to think about or as processes to make calculations and perform
symbolic manipulations.

The axiomatic formal world of mathematics expressed as axiomatic systems,
formal definitions and mathematical proof that is met in pure mathematics at the
tertiary level.
The interplay between the worlds of embodiment and symbolism proves to be crucial. A
most important aspect is how the symbolic development from process to object is
mirrored in a shift from the steps of an action to the effect of the action. For this reason,
the balance model seeks to reveal the symbolic nature of the concept involved in solving
linear equations in one variable in a more practical approach.
Most students find the solving of linear equations more cumbersome task according to
Geary (1994) and that solving of linear equations require extensive practice and Kieran
(1992) also refers to the skill of solving Linear Equations as not easily acquired by
students. The balance model is therefore a valuable educational tool for modelling
relationship between children’s practice in “seesaw” game and linear equations, thus
making connections between what is taught and children’s everyday life activities. A
mathematical equation can therefore be likened to a balance where the items on the left
hand side are equal in value to those on the right hand side of the equation (Kieran,
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1981). Using the balance model is therefore a way of presenting the concept of an
equation in concrete terms to students.
Tall and Thomas (2001) also described three kinds of algebra: (1) evaluation algebra, in
which algebraic expressions are evaluated; (2) manipulative algebra, in which
expressions are manipulated to simplify expressions and solve equations, and
(3) axiomatic algebra, in which algebraic systems are handled in the formal axiomatic
world of concept definition and formal proof on more complicated equations involving
embodiment symbolic manipulation. The evaluation algebra describes what kind of
balance model to use on the linear equation while the manipulative algebra describes how
to use the model. The axiomatic algebra also describes when to use the model to solve the
linear equation in one variable.
Filloy and Rojano (1989) christened the increased level of difficulty in solving linear
equations in one variable as the didactic cut. They reasoned that it would be more
difficult for students to find the required value because they now have to operate on the
unknown and not just on numbers. From the process-object viewpoint, it is not easy to
think of undoing both operations simultaneously; instead, it is easier to think of them as
mental objects (or group of mental objects) that can be manipulated mentally.
On symbol shifting as embodied action, Filloy and Rojano (1989) stipulated that the
wider range of students who focus naturally on detail, and practise the procedure face
difficulties in shifting from arithmetic to algebra, compounded by the limitations of
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undoing and for simple embodied actions for equations. Without meaningful embodiment
or compressed symbolic meaning, there is a natural tendency to build on what they
already know. To the authors, the embodied previous knowledge of shifting symbol
around in expressions fails when moving symbols over the equality sign.
Cortes and Pfaff (2000) found that the principle used by 17-year old students to solve
equations in their study were all based on the ‘movement’ of symbols from one side to
the other of an equation as if the symbols were physical entities that students ‘pass’ to the
other side of the equation with additional change of sign or a shift to place it below
(underneath) the number on the other side (in case of division). Thus the use of the
balance model will enable students to perform the same operations on both side to
maintain balance instead of shifting symbols around the equality sign.
Freistas (2000) found that procedures related to phrases such as change sides, change
sign, were usually meaningless to students and often resulted in mistakes. As procedures,
they are likely to become more complicated as mathematics gets more sophisticated and
the increasing complication will cause even greater problems at later stages. One can
deduce from the above literature that documents that guide the teaching strategies and
curriculum design for students should emphasize the importance of giving meaning to
algebraic symbols and using them to model real-life situations. It can be suggested that
this meaning comes from human interaction with the world. Things make sense because
they work in an expected way and if we look at them from a different perspective we can
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relate it to what we already know. Meaning therefore depends on our human perception
and action and how we build on this and on our earlier experiences.
Dogbe, Mereku and Quarcoe (2004) categorize students’ difficulties in solving linear
equations in terms of misunderstanding algebraic variables. The wrong manipulations of
algebraic letters by students seem to stem from the fact that students handle the letters as
objects and Umoren (2006) sees teacher factor as being the cause of students’ lack of
understanding of mathematics. Teachers do not often use concrete materials in their
instructions which could improve conceptual understanding of mathematics. The use of
the balance model therefore would help improve students understanding of the concepts
involved in solving linear equations in one variable.
Lima and Tall (2008) also researched into “procedural embodiment and algebraic
equations” and stated that algebra could be frightening for students. This is the reason
why for many years there have been large amounts of research in mathematics education,
specifically a search for why students face so many problems in learning equations and
methods of solving them. Sleeman (1984), Payne and Squibb (1990) and Freitas (2002)
diagnose students’ mistakes when solving equations while Dreyfus and Hock (2004),
discuss the understanding students have about an equation. Using the balance model,
Martin et al (1993) discussed the processes o methods employed by the model to ease
understanding and avoid mistake when solving linear equations in one variable.
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Apparently, students from many different countries make the same mistakes. Reasons
why such mistakes appear are more often related to the misinterpretations by students,
techniques used in solving equations and the lack of meaning attributed to the
mathematical symbols used. (Linchevski & Sfard, 1991; Cortes & Kavafian, 1999).
Evidence shows that students give meanings related to the procedures and embodiments
to equations and the solving methods they use. Students also do give meanings to the
ways of working they create (Lima & Tall, 2008). That is to say students take symbols as
physical entities that can be moved around, “putting them on the other side” of the
equation; with the magic of “changing signs”. According to Lima and Tall (2008),
procedural embodiments must be related to their underlying mathematical concepts and
algebraic principles. Attempts to minimize such students’ difficulties have involved the
use of concrete models like the balance or the geometric models (Vlassis, 2002; Filloy &
Rojano, 1989).
Vlassis (2002) using equations with the unknown on both sides showed that the balance
model was a helpful metaphor for almost all her students in giving meaning to the equal
sign as equality between the two sides of an equation. These models have been shown to
be very effective in helping students understand the equality between the two sides of an
equation but do not support situations in which negative and non-integer numbers are
involved. Care should therefore be taken to caution students when using such models of
their limitations and efforts should be made to introduce other models that overcome such
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limitations. Lima and Tall (2008), suggest that in the case of quadratic equation, the
geometric model can be used as an embodiment for solving such equations.
Finally, it is the belief of Sleeman (1984) and Payne and Squibb (1990), that it is not
simply a misinterpretation of techniques or lack of meaning for equations that prompt
students to use wrong rules. Students also give meanings to the ways of working they
create (Lima & Tall, 2008). Students use techniques to solve equations and procedures to
perform operations with symbols. In order to understand why students make mistakes in
solving equations, it is necessary to search for the roots of these meanings. Applying
some of the principles outlined in these research findings above would go a long way in
aiding the researcher to analyze and design appropriate lesson strategies to be applied in
using the balance model to help students overcome their difficulties in solving linear
equations.
2. 2
Inappropriate Procedure for Solving Algebraic Equations
In finding solutions to equations, Lima and Tall (2008) considered how students think
about Algebra. They considered a theoretical framework which builds from natural
human functioning in terms of embodiment, which involves perceiving the world, acting
on it and reflecting on the effects of the actions to shift to the use of symbolism to solve
linear equations.
The authors found out that the students involved in their study neither simplify algebraic
expressions from process to the object nor do they solve algebraic equations by undoing
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or reversing the operations or using general principles of maintaining the balance of the
equation by doing the same thing to both sides.
According to National Council of Teachers of Mathematics (NCTM. 2000), students
should be able to produce logical arguments and present formal proofs that effectively
explain their reasoning. For a student to produce proof he should use proper order of the
steps as exemplified with the use of the balance model in solving linear equations in one
variable. Otherwise that would be said to be lacking in reasoning skills, due to the fact
that they are unable to justify their taught process.
Wells (1988) is of the view that problem-solving strategy should be seen as a union of
discovery and direct presentation approaches. Hyde (1991) also agrees with Wells (1988)
when he stated that problem-solving strategy is a useful approach in assisting learners to
form mathematical concepts.
2. 3
Justification for Using the Balance Model
On using the Balance Model for solving Linear Equations, Martin et al (1993) explains
that one may add or subtract, as well as multiply or divide, objects (for numbers and
unknowns), as one wishes, but the scales must balance. That is to say one must perform
the same actions on both sides of the equation to maintain balance. Martin (1993)
continues to explain that though, obviously, it is not possible to have negative weights in
the real world it is possible to imagine them. For example, one can try to imagine (-2) as
a negative object. When a negative object is added to a positive object (or vice versa), the
two cancel each other out and one is left with nothing.
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There is evidence that earlier educational pioneers advocated the use of concrete
equipment in the teaching of elementary number (Orton, 2002). On the other hand, there
is some evidence that teachers in Britain who believed that they were under
overwhelming pressure from the National Curriculum are rejecting the use of apparatus
as being time-consuming (Trelfall, 1996). Whatever the pressure of teaching, the critical
questions remain, as they always have; namely, (1) Do students learn better with or
without the use of apparatus? and (2), how should apparatus be used in the classroom in
order to achieve greater understanding and enlightenment?
Cockcroft (1982), in recommending the use of apparatus to teach mathematics to
students, suggested that practical work provides the most effective means by which
understanding in mathematics can develop and McClung (1998) is of the view that using
diagrams make classroom an interesting and exciting place for both the teacher and
learner.
The case for using structural apparatus would be overwhelming if research evidence
showed that it was clearly beneficial. Unfortunately, and typically with education, it is
not as simple and clear cut as that. While some research certainly suggested immediate
gains, others say there were usually little long-term benefits (Orton, 2002). What many of
these studies also revealed very clearly are the difficulties inherent in carrying
educational research. For example if two different groups are being compared, how can
one ensure that the two groups are exactly compatible, and that either group is completely
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denied any input other than what they receive from their “official” lessons. Also, what
sort of test should be used at the end of the experiment to ensure a fair comparison
between the effects of two very different experiments (Orton, 2002)?
One additional well-known complication with research studies is that teachers who wish
to take part in an experiment in teaching and learning are often stimulated by the whole
idea and are very keen to see it succeed. Furthermore their enthusiasm is conveyed to
their students which makes them more enthusiastic in turn and so learning may be
enhanced through enthusiasm.
Under such circumstances, any measured gain is therefore likely to be the outcome of
unspeakable combination of the effects of new materials and methods and the total
involvement of the teacher and the students.
The effectiveness of the apparatus itself must also be addressed (Orton, 2002). The
balance model enables students to work both independently and in groups while the
teacher provides guidance to the students (Rosechelle, Pea, Hoadley, Gordon & Means,
2002). Working independently with the balance model enables every student derive
maximum benefit, while working in groups enables them to socialize and communicate
meaningfully at the children’s own level of understanding.
Rosechelle, Pea, Hoadley, Gordon and Means (2002) have also shown that learning is
most effective when four fundamental characteristics are present; (i) active engagement
(ii) participation in groups (iii) frequent interaction and feedback and (iv) connections
with real-world contexts. All these characteristics are facilitated when using the balance
model in solving linear equations in one variable.
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Using the balance model in teaching linear equations takes away the abstractness seen in
mathematical concepts (National Council of Teachers of Mathematics, 2000).
The balance model problem solving ability leads to discovery which is aesthetic and
favours both sexes thereby encouraging students’ participation in the mathematics
classroom (Austin & Vollrath, 1989). The mixed nature of sexes in the researcher’s
school of study is therefore suitable for use of the balance model in solving linear
equations in one variable.
2. 4
Attitude and Motivation
According to Devine & Meaghor (1989), research findings indicate that students attitude
have a corresponding implication on their performance in the class and that the success of
the individual student rest on his or her mastery of the skill.
Brookover, Beady, Flood, Schweitz & Wisenbake (1997) indicated that there exist
correlations between self concept of ability and school performance. Hewitt (2006) also
indicate the importance of concepts and diagrams in teaching and learning of
mathematical concepts. The use of the balance model enhances students’ ability and
confidence to solve linear equations and thereby improve their performance. The
activities employed by the model serve as flow charts or diagrams which aid
understanding of mathematical concepts.
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2.5
Summary
It must be noted that the balance model is not a panacea which if provided will solve
problems of solving linear equations. It must also be noted that the use of the balance
model for teaching and learning must be done in an appropriate classroom environment
and the teacher must discuss the use of the balance model with the students (Orton,
2002).
The conclusion one must draw is that far from providing the teacher with materials that
students must learn from without any teacher inputs, the provision of apparatus places
great demands on both lesson planning and classroom implementation.
Dickson, Brown and Gibson (1984) support the view that understanding is best facilitated
with the help of concrete materials,
Trelfall (1996) has opined that there are benefits which accrue from use of apparatus. At
the same time, he points out that the apparatus will not show full value if it is used
inappropriately.
In using the Balance Model to teach linear equations, the researcher will take cognizance
of the points raised in the review of literature above so that he can construct an
appropriate and well-structured methodology to undertake the study to benefit his
students.
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CHAPTER 3
METHODOLOGY
3.1
Introduction
In this chapter, researcher presents the research design of the study, provides details of
the context of the study and also the research procedures used. The chapter also seeks to
find strategies to assist students understand the concepts involved in solving Linear
Equation in one variable and apply them in their daily activities.
3. 2
Research Design
The study is an Action Research which seeks to identify the problem of solving linear
equations in one variable, suggest, and recommend the use of the balance model as its
intervention tool. The problem was a classroom situation in which the researcher aids
students overcome their difficulty in solving Linear Equations in one variable.
3. 3
Population And Sampling
The study was carried out at Konadu Yiadom Senior High School at Asamang in Ashanti
Region of Ghana. It was targeted at Senior High School Two (SHS2) students of the said
school in particular and in Ghana in general. The SHS2 student population was Two
Hundred and Six (206); comprising of Arts, Business and Home Economics departments.
There were Eighty (80) males and One Hundred and Twenty Six (126) females in the
SHS2. The sampling technique used was purposive because the problem of solving linear
equation was encountered in the class selected. All the fifty (50) SHS2 business students
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were selected and their results tabulated for further analysis of data. There were Thirty
(30) Females and Twenty (20) males which reflect the higher number of females in the
class than males.
3. 4
Instrumentation
Two test instruments were used in the study. A pre-test (see Appendix A) and post-test
(see Appendix C). The Pre-test was administered to the fifty (50) students sampled before
the intervention whereas the Post-test was administered after the intervention.
There was also an interview (see Appendix D) to determine the effectiveness of the use
of the Balance Model in helping the students solve linear equations in one variable.
3. 5
Interview
There was also an interview (see Appendix D) to determine the effectiveness of the use
of the Balance Model in helping the students solve linear equations in one variable.
3. 6
Intervention Procedures
An intervention which involved the use of the balance model was implemented. The
Balance Model is a tool used to covey the idea of balance when doing the same thing to
both sides of an equation (Wang, 1989; Vlassis, 2002).
Four (4) daily sections of teaching, two (2) hours thirty (30) minutes each day, were used
to implement the intervention in which the researcher used the balance model to
emphasize the idea of performing the same action on both sides of an equation, in an
attempt to solve it. An Intervention Implementation Plan (see Appendix B) was put in
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place, in which the students passed through the various steps in using the balance model
to solve some linear equations in one variable.
After the post-test, an interview (see Appendix D) was used to find out how the students
felt with their interaction in the use of the balance model in solving linear equations in
one variable.
Presentation
Students were guided to use the balance model in solving Linear Equation in one variable
as presented in the activities below.
Activities
Day 1
Activity 1
Solve the linear equation below using the balance model.
2x +3 = x – 1
Given: 2 x + 3 = x – 1
Required: to solve for x
Objective: the use of balance model in solving linear equations
Method:
Introduction: write down the equation
2x +3 = x – 1
Step 1: subtract 3 from both sides of the equation to maintain balance.
2x +3 – 3 = x – 1 – 3
Simplify;
2x = x -4
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Step 2: subtract x from both sides of the equation.
2x – x = x – 4 – x
Simplify;
x = - 4(answer).
ALTERNATIVELY
Step 1: add 1 to both sides of the equation 2 x + 3 = x – 1.
2 x + 3 +1 = x – 1 + 1
Simplify;
2x +4 = x
Step 2: subtract x from both sides of equation.
2x – x +4 = x – x
Simplify;
x +4=0
Step 3: subtract 4 from both sides of equation.
x +4–4=0–4
x = - 4 (answer).
NB: Set the students to check their answers by substituting x = - 4 into the original
equation.
Day 2
To solve
Given that
Activity 2
2
3
1
x  x .
3
5
3
2
3
1
x  x .
3
5
3
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Required: To solve for x by the balance model.
OBJECTIVES
(1) To clear fractions in the equation;
(2) To use of balance model to solve equations.
METHOD
Step 1: clear all the fractions by multiplying through the equation by 15(the Lowest
Common Denominator).
2  
3
1


15  x   15    15  x   15  
3  
5
3


Simplify
5  2x  3  3  15x  5 1
10 x – 9 = 15 x + 5
Step 2: add 9 to both sides of the equation.
10 x – 9 + 9 = 15 x + 5 + 9
Simplify
10 x = 15 x + 14
Step 3: subtract 15 x from both sides.
10 x – 15 x = 15 x – 15 x + 14
Simplify
-5 x = 14
Step 4: divide both sides of the equation by -5
 5 x 14

5
5
- 26 -
Simplify
x
14
(answer).
5
NB: Set the students to check their answers by substituting x  
14
into the original
5
equation for balance.
Day 3
Activity 3
Find the value of x in the equation below by the balance model.
3x  2 2
 x.
5
3
Given
3x  2 2
 x.
5
3
Required; to solve for x by balance model
Objectives;
(1) To clear all fractions.
(2) To use the balance model to solve equation.
METHOD
Step 1; clear the fractions by multiplying through the equation by 15 (the L.C.D).
15 
3x  2
2x
 15 
5
3
Simplify
3  (3x  2)  5  2 x
9x  6  10x
- 27 -
Step 2: subtract 6 from both sides of the equation.
9 x + 6 – 6 = 10 x – 6
Simplify
9 x = 10 x – 6
Step 3: subtract 10 x from both sides of equation.
9 x – 10 x = 10 x – 10 x – 6
Simplify
-x = - 6
Step 4: divide both sides of the equation by -1.
 x 6

1 1
Simplify:
x = 6 (answer).
NB: Set students to check their answer by substituting x = 6 into the original equation.
Day 4
Activity 4
Given that 5(3 x - 2) =
Given; 5(3 x - 2) =
4  3x
, find the value of x .
3
4  3x
3
Required: to find the value of x .
Objectives:
(1) To clear fractions in the equations.
(2) To expand the factors.
(3) To use the balance model to find the value of x
- 28 -
METHOD
Step 1; clear fractions by multiplying both sides of the equation by 3 (the L.C.D).
3 × {5(3 x - 2)} = 3 
4  3x
3
Expand
3 × (15 x - 10) = 4 + 3 x
45 x - 30 = 4 + 3 x
Step 2; add 30 to both sides of the equation.
45 x - 30 + 30 = 4 + 3 x + 30
Simplify
45 x = 3 x + 34
Step 3; subtract 3 x from both sides of the equation.
45 x - 3 x = 3 x - 3 x + 34
Simplify
42 x = 34
Step 4; divide both sides of the equation by 42
42 x 34

42
42
Simplify
x
17
21
NB: Set students to check their answers by substituting x 
- 29 -
17
into the original equation.
21
Exercise
Fill in the steps in the working below:
Solve for x in the equation
Given
5x
4
 .
3x  1 3
5x
4

3x  1 3
METHOD
Step 1; clear the fraction by multiplying both sides of the equation by 3
_______ 
5x
4
  _______
3x  1 3
Simplify
15 x
4
3x  1
Step 2; clear faction by multiplying both sides of the equation by 3 x + 1
________ 
15 x
 4  _______
3x  1
Simplify
15 x = 4(3 x + 1)
Step 3; expand
15 x = 12 x + 4
Step 4; subtract 12 x from both sides of the equation
15 x - ____ = 12 x + 4 – ____
Simplify
3x =4
- 30 -
Step 5; divide both sides of the equation by 3
3x 4

3
3
Simplify
x  _____ (answer)
NB. Set students to check answers by substituting the value of x into the original
equation.
3. 7
Data Collection Procedure
The results of the fifty (50) students sampled for the study from the pre-test (see
Appendix A) and post-test (see Appendix C) were collected, recorded, tabled and
analyzed for sampled means and standard deviation.
An intervention implementation plan (see Appendix B) in which the balance model was
used to help students overcome their difficulties in solving linear equations in one
variable was put in place.
An interview (see Appendix D) was conducted to ascertain the effectiveness of the
balance model in helping students to overcome their difficulties in solving linear
equations in one variable.
.
- 31 -
3. 8
Data Analysis Procedures
Data collected were analyzed both quantitatively and qualitatively. The pre-test and posttest were analyzed using both descriptive and inferential statistics using the SPSS.
In the descriptive statistics, the minimum and the maximum scores of the sample size, the
mean scores and standard deviations for both the pre-test and post-test were analyzed. For
further analysis of data, inferential statistics was used to project the paired sample scores
of the sample to determine whether there was a significant improvement in the students’
performances in solving linear equations in one variable.
On inferential statistics, the paired sample test for the sample was calculated to determine
whether there was any significant difference in the students’ performance in solving
linear equations
- 32 -
CHAPTER 4
DATA ANALYSIS, RESULTS AND DISCUSSIONS
The design of the research was to help students to overcome their difficulties in solving
linear equations using the balanced model. The test analysis used the pre-test and posttest scores in analyzing the data. The pre-test scores were used to determine the initial
difference in the means for the sample in the overall pre-test score. Similarly, the posttest scores were used as dependent variable to test the hypothesis for the study.
4. 1
Hypothesis
The design of the study was to determine improvement in the performance of the students
in solving linear equations based on student’s achievement in the test instrument. As a
result, the null hypothesis formulated was that: “There is no significant difference in
students’ achievements in the test instrument for helping students overcome their
difficulties in solving linear equations in one variable”.
4. 2.
Analysis of Pre – Test Results To Test Students Understanding
Table 1:
Pre – Test Results of Question 1
Response
Frequency
Percentage
Correct
21
42
Wrong
29
58
Total
50
100
- 33 -
In Question 1, the students who scored correctly were twenty one (21) representing forty
two percent (42%) whereas twenty nine (29) responded wrongly representing fifty eight
percent (58%). The twenty one (21) students applied correct principles whiles twenty
nine (29) applied wrong principles. It means that students had problems with answering
Question 1 involving linear equations in one variable.
Table 2:
Pre – Test Results of Question 2
Response
Frequency
Percentage
Correct
19
38
Wrong
31
62
Total
50
100
In Question 2, the students who scored correctly were nineteen (19) representing forty
two percent (38%) whereas thirty one (31) responded wrongly representing fifty eight
percent (62%). The nineteen (19) students applied correct principles whiles thirty one
(31) applied wrong principles. It means that students had problems with answering
Question 2 involving linear equations in one variable.
Table 3:
Pre – Test Results of Question 3
Response
Frequency
Percentage
Correct
22
44
Wrong
28
56
Total
50
100
- 34 -
In Question 3, the students who scored correctly were twenty two (22) representing forty
four percent (44%) whereas twenty eight (28) responded wrongly representing fifty six
percent (56%). The twenty two (22) students applied correct principles whiles twenty
eight (28) applied wrong principles. It means that students had problems with answering
Question 3 involving linear equations in one variable.
Table 4:
Pre – Test Results of Question 4
Response
Frequency
Percentage
Correct
18
36
Wrong
32
64
Total
50
100
In Question 4, the students who scored correctly were eighteen (18) representing thirty
six percent (36%) whereas thirty two (32) responded wrongly representing sixty four
percent (64%). The eighteen (18) students applied correct principles whiles thirty two
(32) applied wrong principles. It means that students had problems with answering
Question 4 involving linear equations in one variable.
Table 5:
Pre – Test Results of Question 5
Response
Frequency
Percentage
Correct
15
30
Wrong
35
70
Total
50
100
- 35 -
In Question 5, the students who scored correctly were fifteen (15) representing thirty
percent (30%) whereas thirty five (35) responded wrongly representing seventy percent
(70%). The fifteen (15) students applied correct principles whiles thirty five (35) applied
wrong principles. It means that students had problems with answering Question 5
involving linear equations in one variable.
4. 3
Analysis of Post-Test Results To Test Students Understanding
Table 6.
Post – Test Results of Question 1
Response
Frequency
Percentage
Correct
41
82
Wrong
9
18
Total
50
100
Results from Table 6 indicate that out of fifty (50) students sampled for the study, forty
one (41) representing 82% answered question one (1) correctly whereas nine (9) students
representing 18% answered wrongly. The forty one (41) students applied correct
principles whiles nine (9) applied wrong principles. This was due to the fact that the
balance model helped students to further understand the steps in answering the question
and mastering the concepts involved. There has therefore been an improvement in the
pre-test Question 1 scores (see Table 1).
- 36 -
Table 7.
Post – Test Results of Question 2
Response
Frequency
Percentage
Correct
39
78
Wrong
11
22
Total
50
100
Results from Table 7 also indicate that out of fifty (50) students sampled for the study,
thirty nine (39) representing 78% answered question two (2) correctly whereas eleven
(11) students representing 22% answered wrongly. The thirty nine (39) students applied
correct principles while nine (11) applied wrong principles. This was due to the fact that
the balance model helped students to further understand the steps in answering the
question and mastering the concepts involved. There has therefore been an improvement
in the pre-test Question 2 scores (see Table 2).
Table 8.
Post – Test Results of Question 3
Response
Frequency
Percentage
Correct
42
84
Wrong
8
16
Total
50
100
Results from Table 8 indicate that out of fifty (50) students sampled for the study, forty
two (42) representing 84% answered question three (3) correctly whereas eight (8)
students representing 16% answered wrongly. The forty two (42) students applied correct
- 37 -
principles whiles eight (8) applied wrong principles. This was due to the fact that the
balance model helped students to further understand the steps in answering the question
and mastering the concepts involved. There has therefore been an improvement in the
pre-test Question 3 scores (see Table 3).
Table 9.
Post – Test Results for Question 4
Response
Frequency
Percentage
Correct
38
76
Wrong
12
24
Total
50
100
Results from Table 9 also indicate that out of fifty (50) students sampled for the study,
thirty eight (38) representing 76% answered question four (4) correctly whereas twelve
(12) students representing 24% answered wrongly. The thirty eight (38) students applied
correct principles while twelve (12) applied wrong principles. This was due to the fact
that the balance model helped students to further understand the steps in answering the
question and mastering the concepts involved. There has therefore been an improvement
in the pre-test Question 4 scores (see Table 4).
- 38 -
Post – Test Results for Question 5
Table 10.
Response
Frequency
Percentage
Correct
43
86
Wrong
7
14
Total
50
100
Results from Table 10 indicate that out of fifty (50) students sampled for the study, forty
three (43) representing 86% answered question five (5) correctly whereas seven (7)
students representing 14% answered wrongly. The forty three (43) students applied
correct principles whiles seven (7) applied wrong principles. This was due to the fact that
the balance model helped students to further understand the steps in answering the
question and mastering the concepts involved. There has therefore been an improvement
in the pre-test question 5 scores (see Table 5).
4. 4
Summary of Test Results
Table 11.
Question
Summary of Results of Fifty (50) Students in Pre-Test and Post-Test
Wrong
Correct
Percentage(wrong
Percentage(correct
Answer
Answer
answer)
answer)
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
Pre-test
Post-test
1
29
9
21
41
58
18
42
82
2
31
11
19
39
62
22
38
78
3
28
8
22
42
56
16
44
84
4
32
12
18
38
64
24
36
76
5
35
7
15
43
70
14
30
86
- 39 -
In Question 1, the summary results Table 11 indicates that there was a reduction in the
wrong answers produced by students from twenty nine (29) in the pre-test to nine (9)
students in the post-test. There is equally an increase in the correct answers produced
from twenty one (21) in the pre-test to forty one (41) students in the post-test. This means
that for students who answered Question one (1), 82% had it correctly in the post-test
over 42% in the pre-test.
Similarly in Question 2, the summary results Table 11 indicates that there was a
reduction in the wrong answers produced by students from thirty one (31) in the pre-test
to eleven (11) students in the post-test. There is equally an increase in the correct answers
produced from nineteen (19) in the pre-test to thirty nine (39) students in the post-test.
This means that for students who answered question two (2), 78% had it correctly in the
post-test over 38% in the pre-test.
Again in Question 3, the summary results Table 11 indicates that there was a reduction in
the wrong answers produced by students from twenty eight (28) in the pre-test to eight
(8) students in the post-test. There is equally an increase in the correct answers produced
from twenty two (22) in the pre-test to forty two (42) students in the post-test. This means
that for students who answered question three (3), 84% had it correctly in the post-test
over 44% in the pre-test.
- 40 -
In Question 4, the summary results Table 11 indicates that there was a reduction in the
wrong answers produced by students from thirty two (32) in the pre-test to twelve (12)
students in the post-test. There is equally an increase in the correct answers produced
from eighteen (18) in the pre-test to thirty eight (38) students in the post-test. This means
that for students who answered Question four (4), 76% had it correctly in the post-test
over 36% in the pre-test.
Similarly in Question 5, the summary results Table 11 indicates that there was a
reduction in the wrong answers produced by students from thirty five (35) in the pre-test
to seven (7) students in the post-test. There is equally an increase in the correct answers
produced from fifteen (15) in the pre-test to forty three (43) students in the post-test. This
means that for students who answered question five (5), 86% had it correctly in the posttest over 30% in the pre-test.
4.5
Sample Statistics
Table 12
Pair 1
Paired Sample Statistics of Pre-test And Post-test Scores of 50 Students
Mean
N
Std Deviation
Std Error Mean
Pre-test score of respondents
19.00
5
2.74
1.22
Post-test score of respondents
40.60
5
2.07
0.93
From Table 12, the statistics indicate the mean score of post-test (40.6) greater than that
of the pre-test (19.00). The standard deviation for the post-test (2.07) which is lower than
that of pre-test (2.74) is an indication that there is an improvement in performance of the
post-test over the pre-test.
- 41 -
Table 13
Paired Differences of Pre-test And Post-test Scores of 50 Students
95% Confidence
Interval of the
Difference
Mean
Pair
Pre-test score of
1
respondents – Post-
-21.600
Std
Std Error
Deviation
Mean
lower
Upper
t
df
sig
3.5777
1.600
-26.0423
-17.157
-13.500
4
.000
test score of
respondents
From Table 13, the p-value of 0.000 which is less than 0.05 (95% confidence interval).
This indication shows that there is significant difference between the post-test scores and
the pre-test scores. The hypothesis that there is no significant difference in students’
achievement in the test instrument is therefore not valid.
Students’ Wrong and Correct Responses in the Pre-test
Number of students
Bar Chart Showing Students' Wrong and Correct
Answers to Questions in the Pre-test
40
35
30
25
20
15
10
5
0
Wrong Answer
Correct Answer
1
2
3
4
Question
Figure 4.1
- 42 -
5
The diagram represents the performance of students in the pre-test conducted. In
comparison, it was realized from figure 4.1 that students’ performance was low in all the
questions. All the questions indicate upward levels of wrong answers. This is an
indication that students did not understand linear equations in one variable very well.
Students’ Wrong and Correct Responses in the Post-test
Number of students
Bar Chart Showing Students' Wrong and Correct
Answers to Questions in the Post-test
50
40
30
Wrong Answer
20
Correct Answer
10
0
1
2
3
4
5
Question
Figure 4.2
The diagram above represents the performance of students in the post-test conducted. In
comparison, it was realized from figure 4.2 that students’ performance was high in all the
questions. All the questions indicate upward levels of correct answers. This is an
indication that students did understand linear equations in one variable very well after the
balance model had been introduced to students.
- 43 -
Students Wrong Responses in the Post-test
Pie Chart Showing Students' Wrong Answers in
the Post-test
Q5, 15%
Q1, 19%
1
2
3
Q4, 26%
Q2, 23%
4
5
Q3, 17%
Figure 4.3
In figure 4.3, it was observed that more than half of the sample (51%) had lower
percentage scores in wrong answers in the post-test. This indicates that a higher
percentage of students scored correctly in the post-test.
Students Correct Responses in the Post-test
Pie Chart Showing Students' Correct Answers in
the Post-test
Q5, 21%
Q1, 20%
1
2
3
Q2, 19%
Q4, 19%
4
5
Q3, 21%
Figure 4.4
- 44 -
In figure 4.4, it was observed more than half of the sample (62%) had higher percentage
scores in the correct answers in the post-test. This indicates that a higher of students
scored correctly in the post-test.
4.6
Interviews
An interview was conducted on the students to find out their understanding of concepts in
linear equations in one variable after their interactions with the balance model. They were
also interviewed to find out their attitudes towards the number of times they were given
class exercises and homework. The Tables 14 & 15 indicate students’ attitude towards the
use of the balance model and the number of times they were given class exercises and
homework.
Students’ Attitude towards Balance Model
Table 14
Percentage of Students Response to Use of Balance Model
Characteristics
Number of students Percentage
Accurate & Straightforward
40
80
Excitement Level
42
84
Interactive
41
82
Confidence Level
44
88
Table 14 shows that 80% of the sample of fifty (50) students, find the use of the balance
model accurate and straightforward. Over 80% of the students find the balance model
- 45 -
exciting, interactive and confidence. This indicates that more students are positively
affected by the use of the balance model.
Students’ Attitude towards Class Exercises and Homework
Table 15
Percentage of Students Response to Class Exercises and Homework
Class Exercise
Frequency
Homework
Number of Students Percentage Number of students Percentage
Not at all
0
00
0
00
Once a week
36
72
39
78
Once a month
39
78
40
80
Table 15 indicates percentage of students’ response to class exercises and homework.
This shows that over 70% of the students indicate that class exercises and homework
have been given once a week or once a month.
4.7
Discussions and Analysis of Findings
Under this section, the pre-test, post-test and interview conducted are discussed and the
results analyzed enabled the researcher to determine the level of improvement of
students’ performance in solving linear equations in one variable.
Research Question 1: How can the use of the balance model improve students’
understanding of the concepts involved in solving linear equations with one variable?
With the help of the balance, students were able to perform same operations on both sides
- 46 -
of the equal sign to maintain balance as in the activity 1(see Appendix B). The balance
model was able to help students clear fraction in the given equations by multiplying both
sides of the equal sign by the least common divisor (LCD) to maintain balance as in
activities 2, 3 & 5(see Appendix B). Students were able to expand the factors before
performing same operations on both sides of the equation as in activity 4 as stated by
Corte & Pfaff (2000) that the use of the balance model enables students to perform the
same operations on both side to maintain balance instead of shifting symbols around the
equality sign. Kieran (1992) also agrees that the use of the balance model is a way of
presenting the concept of an equation in concrete terms to students. These answer the
research question 1.
Research Question2: In what ways can the use of the balance model improve students’
achievement in solving linear equations in one variable? According to the summary
results of Question 1 (see Table 11), there was an increase in the correct responses of
students after the introduction of the balance model. The increase in correct responses
from 42% of the total students in the pre-test to 82% in the post-test is an indication of
improvement in students’ achievement results in solving linear equations in one variable.
Similarly, students’ achievement results of correct responses in Question 2 improved
from 38% of the total number of students on the pre-test to 78% in the post-test. Then
again in Question 3, the summary results (see Table 11) indicates that there was an
increase in the correct responses produced, an improvement of students’ achievement of
84% in the post-test over 44% in the pre-test. Additionally, the summary results (see
Table 11) for Question 4 indicates that there was an improvement in the correct responses
- 47 -
from 36% of the students population of fifty (50) to 76% in the pre-test and post-test
respectively. For Question 5 the summary results (see Table 11) shows 86% correct
responses in the post-test over 14% in the pre-test, an indication of much improvement
achievement results. As stated by Hewitt (2006), the use of diagrams in teaching
mathematics concepts enhances students’ understanding and performance, and that
answers the research question 2.
Research Question 3: How does the usage of the balance model affect students’ attitude
towards improving their performance in solving linear equations in one variable?
From the interviews conducted after the implementation of the intervention, thus, the
introduction of the balance model, forty (40) out of fifty (50) students representing 80%
of the total sample find the balance model accurate and straightforward for use. This is an
indication that the balance model positively affected the students in their quest to solving
linear equations in one variable. With regards to the excitement level, the interview
conducted revealed that forty two (42) out of fifty (50) students representing 84% of the
total sample claimed the balance model to be exciting. Forty one (41) and forty four (44)
out of fifty (50) students representing 82% and 88% of the total sample respectively also
find the balance model interactive and very confidence to use. This attitude towards the
use of the balance model affected the students positively and was reflected in the
improved achievement scores and as indicated by Devine & Meaghor (1989), students’
attitudes have a corresponding implication on their performance. Thus the research
question 3 has been answered.
- 48 -
The findings are indications that using teaching models like the balance model make the
classroom an interesting and exciting place for both the student and the teacher. When the
balance model is used in the teaching and learning of concepts involved in solving linear
equations in one variable it becomes self explanatory and clear to students.
Again the pre-test, post-test and interview findings indicate that students’ performance in
the solving of linear equations in one variable improved significantly after the
introduction of the intervention tool and as a result the balance model was effective.
- 49 -
CHAPTER 5
IMPLICATIONS, RECOMMENDATIONS AND CONCLUSION
5.0
General Overview
The purpose of the study was to use the Balance Model as a practical approach to help
students improve their performance in solving linear equations in one variable. The five
(5) weeks process of the intervention revealed statistical differences between mean
achievement scores of the students using the balance model to solve linear equations in
one variable. The statistical differences show the intervention tool (balance model) used
improved students’ performances in solving linear equations.
Moreover, the students exhibited more positive attitudes towards solving linear equations
and mathematics as a whole because they were more focused as they could easily answer
questions and reached the final solutions.
5. 1
Implications and Recommendations
When students were able to overcome their difficulties in solving linear equations, the
confidence gained could be applied in tackling similar difficulties in other topics in
mathematics thereby enhancing their problem solving abilities. Teachers are encouraged
to use appropriate concrete teaching and learning materials in the classroom and give
more classroom exercises and homework to students to ensure effective practice.
Educational planners should include the use of concrete classroom teaching materials in
the curriculum.
- 50 -
When students are positively affected by the use of concrete model in the teaching and
learning process, they discover the synthesis elements implicit in those actions (Filloy &
Rojano, 1989). Teachers and Educational Planners should ensure that the use of concrete
models in the teaching and learning of mathematics concepts bring out the discovery of
the syntax elements involved in those actions.
The balance model could be developed in collaboration with other suitable models for
use in solving major mathematical problems when effectively used.
5. 2
Suggestion For Future Research
Future efforts should be made to research into the use of other suitable models like the
geometric models in solving quadratics equations. The Pulley balance can be uses to
progress towards solution of a more extensive family of linear equations (Vlassis, 2002;
Filloy & Rojano, 1989).
5. 3
Conclusion
It is evident from the findings in this study that using the balance model improved
students achievement in mathematics by helping them overcome their difficulties in
solving linear equations. It is also evident that the balance model was effective in helping
students to use appropriate methods when solving linear equations. They were able to
gain confidence in solving linear equations and would be able to tackle similar
difficulties in solving mathematical problems.
- 51 -
The students develop cooperative learning skills when working in groups. They also had
real-world experiences in solving problems in mathematics when using the balance
model.
- 52 -
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APPENDIX A
PRE TEST
1. Solve the linear equation below for x
3x – 5 = x +1
2. Solve for x
2
6
1
x  x
5
7
5
3. Find the value of x
5x  1 x

2
3
4. Given that 2(4 x  3) 
6  7x
, find the value of x .
5
5. Solve for x in the equation
2x
1
 .
x 1 2
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APPENDIX B
Activities and Worksheets (Balance Model)
NB: Students should undertake these activities both individually and in groups.
ACTIVITY 1
Solve the linear equation below using the balance method.
2x +3 = x – 1
Given: 2 x + 3 = x – 1
Required: to solve for x
Objective: the use of balance method in solving linear equations
Method:
Introduction: write down the equation
2x +3 = x – 1
Step 1: subtract 3 from both sides of the equation to maintain balance.
2x +3 – 3 = x – 1 – 3
Simplify;
2x = x -4
Step 2: subtract x from both sides of the equation.
2x – x = x – 4 – x
Simplify;
x = - 4(answer).
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ALTERNATIVELY
Step 1: add 1 to both sides of the equation 2 x + 3 = x – 1.
2 x + 3 +1 = x – 1 + 1
Simplify;
2x +4 = x
Step 2: subtract x from both sides of equation.
2x – x +4 = x – x
Simplify;
x +4=0
Step 3: subtract 4 from both sides of equation.
x +4–4=0–4
x = - 4 (answer).
NB: Set the students to check their answers by substituting x = - 4 into the original
equation.
WORKSHEET FOR ACTIVITY 1
Fill in the empty spaces in the steps to solve the equation below.
Question 1
Solve the linear equation 3 x – 3 = 2 x + 1.
Using the balance method; we have:
Equation: 3 x – 3 = 2 x + 1
Step 1: add 3 to both sides of the equation.
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3 x – 3 + ___ = 2 x + 1 + ___
Simplify:
3x =2x +4
Step 2: subtract 2 x from both sides of the equation.
3 x – ___ = 2 x + 4 – ___
Simplify:
x = ___ (answer).
NB: Set students to check their answers by substituting the value of x into the original
equation.
ACTIVITY 2
Solve
2
3
1
x  x .
3
5
3
Given that
2
3
1
x  x .
3
5
3
Required: To solve for x by the balance method.
OBJECTIVES
(1) To clear fractions in the equation;
(2) To use of balance model to solve equations.
METHOD
Step 1: clear all the fractions by multiplying through the equation by 15(the Lowest
Common Denominator).
2  
3
1


15  x   15    15  x   15  
3  
5
3


- 62 -
Simplify
5  2x  3  3  15x  5 1
10 x – 9 = 15 x + 5
Step 2: Add 9 to both sides of the equation.
10 x – 9 + 9 = 15 x + 5 + 9
Simplify
10 x = 15 x + 14
Step 3: subtract 15 x from both sides.
10 x – 15 x = 15 x – 15 x + 14
Simplify
-5 x = 14
Step 4: divide both sides of the equation by -5
 5 x 14

5
5
Simplify
x
14
(answer).
5
NB: Set the students to check their answers by substituting x  
equation for balance.
WORKSHEET FOR ACTIVITY 2
Fill in the steps in the working below.
Question 2
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14
into the original
5
Use the balance method to solve the equation
Equation:
3x 5
1
  2x 
4 6
2
3x 5
1
  2x 
4 6
2
Step 1: clear all fractions by multiplying through the equation by 12 (the L.C.D).
 3x
 5

1

  __     __   2 x  __     __ 
 4
 6

2

Simplify
(3 x × 3) + (5 × 2) = 24 x – (1 × 6)
9 x + 10 = 24 x – 6
Step 2: subtract 10 from both sides of the equation.
9 x + 10 – __ = 24 x – 6 – __
Simplify
9 x = 24 x – 16
Step 3: subtract 24 x from both sides of the equation.
9 x – ___ = 24 x – 16 – ___
Simplify
- 15 x = - 16
Step 4: divide through both sides by -15
 15 x  16

 15
 15
Simplify
x = ____(answer).
NB: check your answer by substituting the value of x into the original equation.
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ACTIVITY 3
Find the value of x in the equation below by the balance method
3x  2 2
 x.
5
3
Given
3x  2 2
 x.
5
3
Required; to solve for x by balance method
Objectives;
(1) To clear all fractions.
(2) To use the balance method to solve equation.
METHOD
Step 1; clear the fractions by multiplying through the equation by 15 (the L.C.D).
15 
3x  2
2x
 15 
5
3
Simplify
3  (3x  2)  5  2 x
9x  6  10x
Step 2: subtract 6 from both sides of the equation.
9 x + 6 – 6 = 10 x – 6
Simplify
9 x = 10 x – 6
Step 3: subtract 10 x from both sides of equation.
9 x – 10 x = 10 x – 10 x – 6
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Simplify
-x = - 6
Step 4: divide both sides of the equation by -1.
 x 6

1 1
Simplify:
x = 6 (answer).
NB: Set students to check their answer by substituting x = 6 into the original equation.
WORKSHEET FOR ACTIVITY 3
Fill in the steps in the working below.
Question 3
Given that
2 x  7 3x

, find the value of x by using the balance method.
3
2
Equation:
2 x  7 3x

3
2
Step 1: clear the fractions by multiplying both sides of the equation by 6.
_____ 
2 x  7 3x

 _____
3
2
Simplify
2  ( 2 x  7)  3 x  3
4 x + 14 = 9 x
Step 2: subtract 14 from both sides of the equation.
4 x + 14 – ___ = 9 x – ___
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Simplify
4 x = 9 x – 14
Step 3 subtract 9 x from both sides of the equation
4 x – ___ = 9 x – 14 – ___
Simplify
-5 x = - 14
Step 4: divide both sides of the equation by -5
 5 x  14

5
5
Simplify
x = ____ (answer)
NB: check your answer by substituting the value of x into the original equation.
ACTIVITY 4
Given that 5(3 x - 2) =
Given; 5(3 x - 2) =
4  3x
, find the value of x .
3
4  3x
3
Required: to find the value of x .
Objectives:
(1) To clear fractions in the equations.
(2) To expand the factors.
(3) To use the balance method to find the value of x
- 67 -
METHOD
Step 1; clear fractions by multiplying both sides of the equation by 3 (the L.C.D).
3 × {5(3 x - 2)} = 3 
4  3x
3
Expand
3 × (15 x - 10) = 4 + 3 x
45 x - 30 = 4 + 3 x
Step 2; add 30 to both sides of the equation.
45 x - 30 + 30 = 4 + 3 x + 30
Simplify
45 x = 3 x + 34
Step 3; subtract 3 x from both sides of the equation.
45 x - 3 x = 3 x - 3 x + 34
Simplify
42 x = 34
Step 4; divide both sides of the equation by 42
42 x 34

42
42
Simplify
x
17
21
NB: Set students to check their answers by substituting x 
- 68 -
17
into the original equation.
21
WORKSHEET FOR ACTIVITY 4
Fill in the steps in the working below:
Equation 4
Given that 6 x  4  
Given: 6 x  4  
3x  7
, find the value of x .
5
3x  7
5
METHOD
Step 1; clear the fraction by multiplying both sides of the equation by 5 (the L.C.D).
__________  6 x  4 
3x  7
 _________
5
Step 2; expand
________  6 x  24 
3x  7
 ________
5
Simplify
30 x + 120 = 3 x + 7
Step 3; subtract 120 from both sides of the equation.
30 x + 120 - ________= 3 x + 7 - __________
Simplify
30 x = 3 x -113
Step 4; subtract 3 x from both sides of the equation.
30 x - ______ = 3 x - 113 - ________
Simplify
27 x = - 113
- 69 -
Step 5; divide both sides of the equation by 27.
27 x  113

27
27
Simplify
x = _______
NB: Set students to check their answers by substituting the value of x into the original
equation.
ACTIVITY 5
Solve for x in the equation
3x
3
 .
2x  1 5
Required; to solve for x .
Objectives:
(1) To clear the fractions in the equation.
(2) To solve for x .
METHOD
Step 1; clear the fraction by multiplying both sides of the equation by 5.
5
3x
3
 5
2x  1 5
Simplify
15 x
 3.
2x  1
- 70 -
Step 2; clear fraction by multiplying both sides of the equation by 2 x - 1.
(2 x  1) 
15 x
 3  (2 x  1)
2x  1
Simplify
15 x = 6 x - 3
Step 3; subtract 6 x from both sides of the equation
15 x - 6 x = 6 x - 6 x - 3
Simplify
9 x = -3
Step 4; divide both sides of the equation by 9
9x  3

9
9
Simplify
x
1
(answer).
3
NB: Set the students to check their answer by substituting x  
equation.
- 71 -
1
into the original
3
WORKSHEET FOR ACTIVITY 5
Fill in the steps in the working below:
Equation 5
Solve for x in the equation
Given
5x
4
 .
3x  1 3
5x
4

3x  1 3
METHOD
Step 1; clear the fraction by multiplying both sides of the equation by 3
_______ 
5x
4
  _______
3x  1 3
Simplify
15 x
4
3x  1
Step 2; clear faction by multiplying both sides of the equation by 3 x + 1
________ 
15 x
 4  _______
3x  1
Simplify
15 x = 4(3 x + 1)
Step 3; expand
15 x = 12 x + 4
Step 4; subtract 12 x from both sides of the equation
15 x - ____ = 12 x + 4 – ____
Simplify
- 72 -
3x =4
Step 5; divide both sides of the equation by 3
3x 4

3
3
Simplify
x  _____ (answer)
NB. Set students to check answers by substituting the value of x into the original
equation.
- 73 -
APPENDIX C
Post Test
Name __________________________
Date ________________________________
1. Solve the linear equation for x .
5 x – 1 = 3 x + 11
2. Solve for x in the equation
3x 3
1
  2x  .
7 6
3
3. Find the value of x in the equation
4. Given 5( x  2) 
2x  3 x
 .
5
2
1  3x
, find x .
6
5. Solve for x in the equation
3x
1
 .
2x  1 3
- 74 -
APPENDIX D
Interview Questions
1. How confident did you feel when using the balance model to solve linear
equations?
2. Was it helpful to use the balance model in order to solve linear equations?
3. Did you feel more exciting when using balance model to do the worksheet in
solving linear equation?
4. Was it easy to use balance model in solving linear equations?
5. What features of the balance model appealed to you for the enhancement of
solving linear equations?
- 75 -