EDST8226 – ASSESSMENT 2 – UNIT OF
WORK
PART A – BACKGROUND
THE CLASS
The class is a year 10 class in a co-educational public school in Sydney’s Northern Beaches with 60-minute
lessons. The class has 24 students of which there are 13 boys and 12 girls. 7 of the students have a Language
Background Other Than English (LBOTE). The socio-educational advantage (SEA) is consistent with the school’s
which has an Index of Community Socio-Educational Advantage (ICSEA) of 1029 which puts it in the 60th
percentile. The class places somewhere in the middle in terms of ability but has a significant diversity in ability,
and the class is studying the Stage 5.2 mathematics syllabus (NSW Education Standards Authority[NESA], 2012,
p. 10).
SYLLABUS OUTCOME(S)
This unit of work introduces the topic of simultaneous equations (SE) to students. The relevant syllabus (NESA,
2012) outcomes are:
•
•
•
•
MA5.2-1WM
MA5.2-2WM
MA5.2-3WM
MA5.2-8NA
The relevant syllabus content is:
Solve linear SEs, using algebraic and graphical techniques, including with the use of digital technologies
(ACMNA237)
•
•
•
solve linear simultaneous equations by finding the point of intersection of their graphs, with and
without the use of digital technologies
solve linear simultaneous equations using appropriate algebraic techniques, including with the use of
3𝑎 + 𝑏 = 17
the 'substitution' and 'elimination' methods, e.g. solve {
2𝑎 − 𝑏 = 8
o select an appropriate technique to solve particular linear simultaneous equations by
observing the features of the equations (Problem Solving)
generate and solve linear simultaneous equations from word problems and interpret the results
As a prerequisite this topic requires understanding of equations and how to solve them through grouping of
like terms and simplification of algebraic expressions. Additionally, students need understanding of linear
equations, including fluency with: graphing, manipulation, finding gradient, and finding intercept. This topic
forms the basis of content in stage 5.3 where there is one linear equation and one non-linear equation. This
topic leads onto content under Functions in the Stage 6 Mathematics Advanced syllabus (NESA, 2017).
UNIT AIMS
The main aims of this unit are to introduced the methods for solving linear simultaneous equations including:
•
The graphical method
•
•
The substitution algebraic method
The elimination algebraic method
Additionally, this unit develops students’ inquiry and problem-solving skills, while also reinforcing students’
understanding of the real meaning of an equation rather than purely teaching the methods. Finally, this unit
aims to improve students’ strategic thinking when choosing an approach to a problem, particularly when
pertaining to real-world problems.
RATIONALE
The general approach that will be taken throughout this unit is generally consistent with traditional
mathematics teaching but with Inquiry-Learning (IL) / Problem-Based-Learning (PBL) used to introduce
students to the new understanding. First in the unit, there will be a review of prerequisite knowledge with
some informal diagnostic assessment. This is to address any misconceptions students may bring to the
classroom and to build a solid foundation for the new concepts.
The graphical method for solving simultaneous solutions will then be introduced through guided inquiry in a
PBL lesson. The graphical method is chosen for this as it is the most intuitive way of solving simultaneous
equations and it is generally the first in the sequence of lessons and in textbooks. The IL/PBL approach has
been chosen to encourage the development of more conceptual understanding of the topic by engaging higher
order thinking (Boaler, 1998), and to increase enjoyment and engagement (Boaler, 2015). In the many levels of
inquiry laid out by Banchi & Bell (2008), which range with increasing levels of inquiry from level 1 to level 4,
these IL lessons would fall somewhere between level 2 – structured inquiry and level 3 – guided inquiry. A
similar IL introduction will take place for each of the algebraic techniques in later lessons. As Cheeseman et al.
(2016) point out, care must be taken to present these problems in an accessible, rather than daunting manner.
To achieve this the lesson will be similar to the ‘ideal’ Japanese lesson observed by Jacobs & Morita (2002),
where the specific nature of the open-ended problem is carefully laid out and actively discussed by the class.
Following the IL introductions, a brief, more traditional, formal recapitulation of each method will be taught to
the class followed by practice problems. This is chosen to help cement the methods and learning for the
students, and to improve fluency with problem solving.
LIKELY MISCONCEPTIONS
Simultaneous equations take a lot of algebraic processes to solve and are considered to be a difficult and
demanding topic by students (Ugboduma, 2013). There are several likely misconceptions that students may
hold while they learn the topic. Students may bring misconceptions about what it means to solve equations
and the form that solutions can take. They may be used to solving equations with one variable and one
solution. Indeed, many students see an unknown variable in an equation just as an unqueen quantity rather
than a variable that can have multiple values that satisfy the equation (Küchemann, 1981). It is important that
students realise that there is an infinite number of value pairs of the two variables in these linear equations
can be solutions.
A misconception that may be encountered while solving the equations algebraically that is likely to be
encountered is that the modified equations that are created by manipulating the original equations are not
equivalent to one of the originals. This can lead to mistakes and confusion, particularly with the substitution
method. Another precaudal misconception observed by Johari & Shahrill (2020) is that students will
instinctively solve for 𝑥 even if this leads to more difficult subsequent workings, which are more error-prone.
PART B – UNIT OVERVIEW
Lesson
Number
1
Topic and basic outline
The first lesson in this sequence aims to build a solid foundation for an introduction to solving
simultaneous equations. This lesson focuses on reinforcing understanding of linear equations,
graphing, recognising gradient intercept form and solving equations. There will be a focus on
conceptual understanding rather than procedural methods. The purpose of this approach is to
minimise misconceptions in subsequent lessons and address pre-held misconceptions.
In the first two thirds of this lesson students will work in groups of 4 to answer prompt
questions and present them to the class. Each group will be asked to solve one of the following
questions:
1. What is an equation?
2. What does it mean to solve an equation?
3. What does a graph represent?
4. In gradient intercept form: 𝑦 = 𝑚𝑥 + 𝑏
a. Why does 𝑚 represent the gradient of a line?
b. Why does 𝑏 represent the intercept of a line with the y axis?
5. Can you find a single solution for 𝑥 in an equation which has both an 𝑥 and a 𝑦?
6. How many pairs of numbers are there that are solutions the equation 𝑦 = 10𝑥 + 7?
Students will be presented with all of the questions but their group will only be asked to
present their answer to one of the questions to the class.
The last third of the lesson will be spent completing worksheets individually with questions
relating to the prompt questions with a focus on solving equations with one pronumeral and
graphing linear equations.
2
This lesson aims to introduce students to solving simultaneous equations through a problembased guided inquiry approach that guides the students towards a graphical solution of solving
simultaneous equations. The teacher will first reiterate, sourcing contributions from the class,
the meaning of an equation, how to plot a linear equation in gradient-intercept form, and the
meaning of what a graph represents. These points will be left on the board for the duration of
the lesson.
In the same groups as the previous lesson, students will then begin attempting to answer the
guiding question:
𝑦 = 2𝑥 + 3 − ①
−②
“For the two equations 𝑦 = −𝑥 + 6
3
4
what pair values for 𝑥 and 𝑦 is a solution?”
Students will work in their groups with guidance from the teacher to come up with an answer.
Towards the end of the lesson the teacher will one or two students to explain their method.
Following this a formal explanation will be given by the teacher.
The aim of this lesson is to reinforce students’ understanding and fluency with the graphical
method of solving equations. A series of equation pairs of increasing difficulty will be given to
the students, most with one correct solution, some with no answers, and a couple of them will
have infinite answers. For the first 15 minutes the students will create plots by hand and for the
rest of the lesson, they will complete the problems using the free online tool called (GeoGebra,
n.d.). GeoGebra allows for students to easily plot multiple equations and displaying the
intercept. At the end of the class a couple of students will be asked to explain why some of the
questions have no solution and why some have infinite solutions.
This lesson will have a similar structure to the 2nd lesson but this time aiming for students to
discover the technique of solving simultaneous equations algebraically using the substitution
method. The guiding question will be:
𝑦 = 2𝑥 + 3 − ①
”
−②
“Without graphing, find the pair of values which is a solution for 𝑥 + 4 = 𝑦
To initiate the investigation the teacher will discuss with the class what an equation is,
reinforcing the concept of an equation being a statement of fact that may or may not be true
for certain values.
Again, at the end of the lesson the method will be presented by selected students, who have
developed good methods, and formally explained by the teacher.
5
This lesson is devoted to practise of the substitution method using problems from the textbook
(McSeveny et al., 2014). However, there will be some additional problems involving parallel
lines that have no solution such as
6
9
where the end point following the method
would result in 0 = 1 which is an incorrect statement. A student will be asked to explain to the
class why there is not correct solution for parallel lines.
Introduction to elimination method again through an inquiry-based introduction but with the
prompt using the equations:
7
8
𝑦 = 2𝑥 + 3 − ①
𝑦 = 2𝑥 + 2 − ②
5𝑥−3𝑦=20 − ①
2𝑥+3𝑦=15 − ②
Similar practice to lesson 5 but for the elimination method
This lesson will focus on practice with selecting the method that will produce the fastest result.
5 problems will be presented to the class and in the same groups as before students will work
on solving them with the least number of steps. At the end of the lesson, for each question the
group with the simplest method of solving the problem will present it to the class.
This lesson will focus on solving “real-world” problems. This class will include a section class
inquiry focusing on the Wait/Walk dilemma (Morton, 2008) with a set bus delay, asking: for up
to what travel distance is it better to walk?
PART C – THREE CONSECUTIVE LESSONS
LESSON 1
Name Alexander Murray
Date 04/10/2021
School Imaginary Secondary School
Unit of work Simultaneous Equations
Lesson topic Review of linear equations and solving single-variable equations
Lesson Length 60 minutes
Lesson begins 10:00| Lesson ends 11:00
Class size ~24
Syllabus Outcomes
•
selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-1WM
•
solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques MA5.2-8NA
Lesson Outcomes
Assessment (Diagnostic/Formative/Summative)
By the end of the lesson students will be able to:
-
Diagnostic: Class discussion relating to the nature of equations in general and
specifically linear-equations.
-
Solve single variable equations
-
-
Identify gradient and y-intercept from equations in gradientintercept form
Formative: Observing the progress that groups are making and the issues they
encounter.
-
-
Plot linear equations
Summative: The completed answers to the questions presented by the
groups and later submission of worksheets.
-
Recognise equations as statements of equality and plots are
representing all the points that are solutions for the equations.
The key/foundational ideas addressed in this lesson are
-
The nature of equations
-
The properties of linear equations
General Capabilities
Equipment /Resources
-
Literacy
-
Pens and large sheets of paper
-
Critical and creative thinking
-
Graphing paper
-
Personal and social capabilities
-
Calculators
-
Problem solving skills
Context of the lesson/ Students’ prior learning
-
This lesson reviews content that is pre-requisite for solving simultaneous equations.
Lesson Plan
Stages
/
Teacher Activity (Teacher:)
Student Activity (Students:)
Differentiation
S: Struggling Student
Timing
E: Excelling Student
Introduction (10 Minutes)
10
minute
s
1. Places students into preselected, mixed ability groups of
4.
2. Initiates class discussion on the nature of equations and
in particular linear equations.
Body of lesson (25 Minutes)
1. Move into their groups
S: If students do not recall
previous content, then more
time is spent on this.
2. Contribute the class discussion.
E: Goes further with deeper
explanations and prompts for
discussion.
15 min
3. Presents the inquiry questions to the students and
assigns one question to each group, the answer for which
that group will present to the class. Facilitates groups
answering the questions by pointing students in the right
direction, clarifying questions and addressing
misconceptions.
3. Produce answers to each question in their group with a
focus on an assigned question:
1.
2.
3.
4.
5.
6.
10 min
4. Facilitates students briefly presenting their explanations
for each question
What is an equation?
What does it mean to solve an equation?
What does a graph represent?
In gradient intercept form: 𝑦 = 𝑚𝑥 + 𝑏
a. Why does 𝑚 represent the gradient of a
line?
b. Why does 𝑏 represent the intercept of a
line with the y-axis?
Can you find a single solution for 𝑥 in an equation
which has both an 𝑥 and a 𝑦?
How many pairs of numbers are there that are
solutions the equation 𝑦 = 10𝑥 + 7?
4. One student from each group will briefly present their
explanation.
Review/Conclusion (15 Minutes) – if there is time remaining after group presentations
15
minute
s
5. Hand out worksheets and instruct students to complete
worksheets
5. Complete the problems on the worksheet with questions
related to equations and problems that specifically relate to
linear equations.
S: Assist students who are
struggling to start by pointing
them in the right direction,
resolving misconceptions, and
structuring inquiry.
E: Encourage more in-depth
explanations and the production
of alternative explanations for
their assigned question and the
other questions.
Homework:
Continuation of worksheets if not completed in class time – to be handed in next lesson
LESSON 2
Name Alexander Murray
Date 06/10/2021
School Imaginary Secondary School
Unit of work Simultaneous Equations
Lesson topic Inquiry based introduction to the graphical method of solving
simulations equations
Lesson Length 60 minutes
Lesson begins 10:00| Lesson ends 11:00
Class size ~24
Syllabus Outcomes
•
selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-1WM
•
solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques MA5.2-8NA
•
constructs arguments to prove and justify results MA5.2-3WM
Lesson Outcomes
Assessment (Diagnostic/Formative/Summative)
-
Diagnostic: Assessing the contributions at the beginning of the class
By the end of the lesson students will be able to:
-
Solve a pair of simultaneous equations using the graphical method
-
Interpret the intercept of two line as the point that solves both of
the equations for the lines
-
Formative: Observing the progress that groups are making and the issues they
encounter.
-
Summative: The completed solution to the guiding question presented by the
groups.
The key/foundational ideas addressed in this lesson are
-
The nature of equations
-
The properties of linear equations
-
Solving two equations simultaneously
General Capabilities
Equipment /Resources
-
Literacy
-
Pens and large sheets of paper
-
Critical and creative thinking
-
Graphing paper
-
Personal and social capabilities
-
Calculators
-
Problem solving skills
Context of the lesson/ Students’ prior learning
-
This lesson introduces students to methods of solving simultaneous graphical and needs strong foundational knowledge with linear equations
Lesson Plan
Stages
/
Teacher Activity (Teacher:)
Student Activity (Students:)
Differentiation
S: Struggling Student
Timing
E: Excelling Student
Introduction (10 Minutes)
10
minute
s
1. Places students into mixed ability groups of 4, the same
as the previous lesson and collect homework.
2. Reiterates content from the previous lesson sourcing
contributions from the class
Body of lesson (25 Minutes)
1. Move into their groups
S: If students do not recall
previous content, then more
time is spent on this.
2. Contribute the introduction in relation to the nature of
equations, linear equations, graphing and the meaning of
what graphs of equations represent.
E: Goes further with deeper
explanations and prompts for
discussion.
30 min
3. Presents the guiding question to the students:
𝑦 = 2𝑥 + 3 − ①
−②
“For the two equations 𝑦 = −𝑥 + 6
3. Attempt in groups to answer the guiding question.
what pair values for
𝑥 and 𝑦 is a solution?”
Assist students by guiding them in the right direction in
answering the question, pushing them towards graphing
both lines on the same graph.
10 min
4. Select one group with a good explanation to present to
the class.
E: Encourage more in-depth
explanations and the production
of alternative explanations.
4. One student from each group will briefly present their
explanation.
Review/Conclusion (10 Minutes)
10
minute
s
5. Formally explain the graphical method of solving
simultaneous equations with another example, this can be
done using ICT, such as GeoGebra.
Homework:
N/A
LESSON 3
S: Assist students who are
struggling to start by pointing
them in the right direction,
resolving misconceptions, and
structuring inquiry.
5. Listen to explanation and take notes
Name Alexander Murray
Date 08/10/2021
School Imaginary Secondary School
Unit of work Simultaneous Equations
Lesson topic Practice in solving simultaneous equations using the graphical
method
Lesson Length 60 minutes
Lesson begins 10:00| Lesson ends 11:00
Class size ~24
Syllabus Outcomes
•
selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-1WM
•
solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques MA5.2-8NA
•
constructs arguments to prove and justify results MA5.2-3WM
Lesson Outcomes
By the end of the lesson students will be able to:
-
Fluently solve simultaneous equations graphically
-
Use GeoGebra to plot linear equations
-
Understand that not all pairs of linear equations have solutions
The key/foundational ideas addressed in this lesson are
-
The nature of equations
-
The properties of linear equations
Assessment (Diagnostic/Formative/Summative)
-
Diagnostic: N/A
-
Formative: Observing the progress that students are making and the issues
they encounter.
-
Summative: The completed answers to the questions submitted teacher in a
later lesson
-
How to solve simultaneous equations graphically
General Capabilities
Equipment /Resources
-
Literacy
-
Pens and large sheets of paper
-
Critical and creative thinking
-
Calculators
-
Personal and social capabilities
-
A device that the student brings to the class, preferably a laptop.
-
Problem solving skills
-
ICT skills
Context of the lesson/ Students’ prior learning
-
This lesson builds fluency in solving simultaneous equations graphically and leads onto solving them algebraically.
Lesson Plan
Stages
/
Teacher Activity (Teacher:)
Student Activity (Students:)
Differentiation
S: Struggling Student
Timing
E: Excelling Student
Introduction (5 Minutes)
5
minute
s
1. Presents a brief recap of the graphical method
1. Listens to explanation
S: If students do not recall
previous content, then more
time is spent on this.
E: Goes further with deeper
explanations and prompts for
discussion.
Body of lesson (35 Minutes)
15 min
25 min
3. Hand out worksheets with problems for students to
complete.
4. Instruct students to work on the rest of the problems
on www.geogebra.com. With a shot explanation of how to
use the platform
3. Work on problems graphically by hand.
4. Work on problems graphically by using GeoGebra and
saving their plots.
S: Assist students who are
struggling to start by pointing
them in the right direction and
resolving misconceptions.
E: Encourage attempts to think
of alternative methods and
explanations for why parallel
lines do not have solutions.
Review/Conclusion (10 Minutes) – if there is time remaining after group presentations
15
minute
s
5. Select a student who has a good explanation
Homework:
Complete remaining problems as homework to bring to next lesson.
5. Selected student explains why parallel lines do not have
any common solutions.
REFERENCES
Banchi, H., & Bell, R. (2008). The Many Levels of Inquiry. Science and Children, 46(2), 26–29.
https://www.michiganseagrant.org/lessons/wp-content/uploads/sites/3/2019/04/The-Many-Levels-of-Inquiry-NSTA-article.pdf
Boaler, J. (1998). Open and Closed Mathematics: Student Experiences and Understandings. Journal for Research in Mathematics
Education, 29(1), 41–62. https://doi.org/10.2307/749717
Boaler, J. (2015). The Elephant in the Classroom: Helping Children Learn and Love Maths. Souvenir Press.
http://ebookcentral.proquest.com/lib/mqu/detail.action?docID=2069561
Cheeseman, J., Clarke, D., Roche, A., & Walker, N. (2016). Introducing Challenging Tasks: Inviting and Clarifying without Explaining and
Demonstrating. Australian Primary Mathematics Classroom, 21(3), 3–6.
GeoGebra. (n.d.). GeoGebra. Retrieved 3 October 2021, from https://www.geogebra.org/
Jacobs, J. K., & Morita, E. (2002). Japanese and American Teachers’ Evaluations of Videotaped Mathematics Lessons. Journal for Research
in Mathematics Education, 33(3), 154–175. https://doi.org/10.2307/749723
Johari, P. M. A. R. P., & Shahrill, M. (2020). The Common Errors in the Learning of the Simultaneous Equations. Infinity Journal, 9(2), 263–
274. https://doi.org/10.22460/infinity.v9i2.p263-274
Küchemann, D. (1981). Chapter 8: Algebra (pp. 102–119).
McSeveny, A., Conway, R., & Wilkes, S. (2014). Australian Signpost Mathematics New South Wales 10 (5.1-5.3).: Student book. Pearson
Education Australia.
Morton, A. B. (2008). A Note on Walking Versus Waiting. ArXiv:0802.3653 [Math]. http://arxiv.org/abs/0802.3653
NSW Education Standards Authority. (2012). Mathematics K–10 Syllabus. https://educationstandards.nsw.edu.au/wps/portal/nesa/k10/learning-areas/mathematics/mathematics-k-10
NSW Education Standards Authority. (2017). Mathematics Advanced Stage 6 Syllabus.
https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/stage-6-learning-areas/stage-6-mathematics/mathematics-advanced2017
Ugboduma, O. (2013). Students’ Preference of Method of Solving Simultaneous Equations. Global Journal of Educational Research, 11.
https://doi.org/10.4314/gjedr.v11i2.8