Lesson plan 1 – Explain it
Exploring
Extending
Assessment
 Generalise ideas (GI)
 Use generalisations (WS)
 Use words and symbols to describe patterns and generalisations (WS)
 Use appropriate vocabulary to explain ideas (V)
 Compare and contrast ideas (CC)
 Critically reflect (CR)
 Participate actively in a collaborative team or community (CT)
Mathematics Level 7
Number and Algebra Level 6

Form and use quadratic
 Form and solve quadratic equations.
equations.
 Relate graphs, tables, and equations to quadratic, and
 Display the graphs of linear and
relationships found in number and spatial patterns.
non-linear functions and
Key
Competencies
Essence
Statement
Number and Algebra Level 5

Form and solve simple
quadratic equations.

Relate tables, graphs, and
equations to simple quadratic
relationships found in number
and spatial patterns
Introducing
Use symbols, graphs, and
diagrams to help find and
communicate patterns and
relationships.
Curriculum
Objectives
This lesson explores ways of describing quadratic functions using graphs, symbols and words.
connect the structure of the
functions with their graphs.
Task
Use your personal parameters
to draw a parabola with the
equation given in the form
y = (x+b)(x+c) and describe it
for other members of your
group.
Pedagogy
How many different ways can
you describe the graph so that
someone else can create it?
What are the similarities and
differences within / across
groups?
Working in teams of 4 use the
sliders and animation feature
to manipulate ‘b’ and ‘c’ to
explore the effect of each on
the different representations.
What effect does changing b
have on the graph? On the
table of values? On the
equation?
What about c?
How can you find the vertex
if you know b and c?
How can you find the yintercept?
What impact does changing a
have on the graph?
Can you think of a reason
linked to the equation about
why this happens?
Add ‘a’ as a variable to
explore y = a(x+b)(x+c).
and describe a’s impact on
the graph, equation and table.


Learning
Linking language with
features: vertex, x-intercepts,
y-intercept, shape, variable,
parameter, equation, function,
table, x-coordinate, ycoordinate, parabola,
quadratic, representation.
(-b,0) and (–c,0) as xintercepts, y-intercept as bc,
or (0,y), vertex as midpoint of
b and c.
Technology
Geogebra – Explain it.ggb
Defining parameters by
entering b = group no. and
c = individual no. into input
bar.
Using algebra / graph and
spreadsheet views.
Use of sliders to explore
parameters.
Use of animation feature to
explore parameters.
Link factorised expression to
expanded version.
Use of factor and expand
commands.
a > 1, narrows graph as all y
values increase by factor a
0 < a < 1, graph widens as all
y values decrease by factor a
a = 0, then line y=0
a < 0, upside down graph
with same features as those
above
Redefining functions in
geogebra.
Setting up parameters.
Set up a VoiceThread file with an animation of each slider and comments from each of the group members about what you
notice.
View the VoiceThreads of at least one other group and add a comment agreeing or disagreeing with their statements, giving
an explanation or reason for your conclusion.
Lesson plan 2 – Key Ideas of it
Assessment
Extending
Exploring
Introducing
Curriculum
Objectives
Use symbols, graphs, and
diagrams to help find and
communicate patterns and
relationships.
Learn to structure and to
organise, to carry out
procedures flexibly and
accurately, to process and
communicate information,
and to enjoy intellectual
challenge.
Key Competencies
Essence Statement
This lesson explores converting within and between different representations of quadratic functions.







Generalise ideas (GI)
Use generalisations (WS)
Choose and use appropriate representation (Re)
Interpret information and results in context (In)
Compare and contrast ideas (CC)
Critically reflect (CR)
Participate actively in a collaborative team or community (CT)
Number and Algebra Level 5

Form and solve simple
quadratic equations.

Relate tables, graphs, and
equations to simple
quadratic relationships
found in number and
spatial patterns
Number and Algebra Level 6
 Form and solve quadratic equations.
 Relate graphs, tables, and equations to quadratic, and
relationships found in number and spatial patterns.
Mathematics Level 7

Form and use quadratic
equations.
 Display the graphs of linear and
non-linear functions and
connect the structure of the
functions with their graphs.
Task
Formulate a class
summary of the key ideas
from Lesson 1 via group
VoiceThreads.
Pedagogy
Identify common
misconceptions and have
other students’ address these,
refer back to geogebra as
appropriate.
Learning
Correct use of terminology to
label features.
Clear explanation of effect of
b and c.
Technology
Show VoiceThreads of
different groups, identify
common themes /
misconceptions.
Work in teams of 4 to
explore functions in
competed square form
y = a(x + b)2 + c.
Form and test conjectures
about the effect of each
parameter.
Explore how the
technology vs pen and
paper methods converts
between representations.
Set up a geogebra
worksheet to explore
y = ax2 + bx + c.
What effect does each
parameter have on the graph?
On the table of values? On
the equation?
How can you find the vertex,
y-intercept, x-intercepts from
each representation?
Can you predict different
representations before
pressing enter?
Conversions within
representations
 Expanding / factorising
 completing the square
Use of text tool
Conversions between
representations
Animation
What impact does changing a,
b and c have on the graph?
Can you think of a reason
linked to the equation about
why this happens?
Completing the square to find
quadratic formula.
Effect of a on shape, b and c
on location of graph.
Understand these a b and c
are not the same as those
above.
Setting up a geogebra sheet to
explore a function.
Trace function to determine
effect of b.

Expand
Factor
Add to your VoiceThread file, showing ways of finding the missing representation if given the other two, e.g. the graph
if given the equation or table of values etc.
Lesson Plan 3 – Connect it
Number and Algebra Level 6
 Form and solve quadratic equations.
 Relate graphs, tables, and equations to quadratic, and
relationships found in number and spatial patterns.
 Find optimal solutions, using numerical approaches.
Mathematics Level 7

Form and use quadratic
equations.
 Display the graphs of linear and
non-linear functions and
connect the structure of the
functions with their graphs.
Task
Area problem – what is
the maximum paddock
you can make with 50m of
fencing wire, if you only
have to fence three sides
of the paddock.
Pedagogy
Have students explore the
problem without using
geogebra, then make
predictions about how the
different representations in
geogebra would show their
findings.
Show VoiceThreads, identify
any common themes /
misconceptions.
What do you notice about the
maximum in each case?
Why does plotting the area
result in a parabola?
Why does the vertex give the
maximum area?
What does the x-coordinate of
the vertex tell you?
Learning
Linking area problems to
quadratics.
Linking features of side
length and area to geogebra
representations.
Linking this area problem to
factorised quadratic.
Technology
Use of sliders for lengths
Generalising finding the
vertex coordinates of
y = x (l – 2x) as a way of
finding the maximum.
Numerical solutions of
equations.
Using second difference
method to find formula for a
numeric quadratic pattern.
Multiplication of a variable
by itself leads to squared
variable which links to
quadratics.
Identifying terms of quadratic
pattern as c, a + b + c, 4a + 2b
+ c, … => first difference is a
+ b, 3a + b, 5a + b and second
difference is 2a.
Use of trace tool
Use of trace to spreadsheet
Use of spreadsheet functions
to find x, y, first difference
and second difference.
Exploring
Extending
Assess
Work in teams of 4 to
generalise the above
problem to any length of
wire.
Link numerical
representation to finding
formula.
Set up a worksheet to
explore another version of
this area (or
multiplication) problem
with different constraints.
Link ax2 + bx + c to
second difference method
of finding equation.



Key Competencies
Essence Statement
Number and Algebra Level 5

Form and solve simple
quadratic equations.

Relate tables, graphs, and
equations to simple
quadratic relationships
found in number and
spatial patterns
Introducing
Learn to create models
and predict outcomes, to
conjecture, to justify and
verify.
Generalising and
representing the patterns
and relationships found in
numbers, shapes, and
measures.
Curriculum
Objectives
This lesson links different representations to class of area problems.







Plan and carry out an investigation (PI)
Find, use, and justify a model (Mo)
Choose and use appropriate representation (Re)
Interpret information and results in context (In)
Compare and contrast ideas (CC)
Manage time effectively (MT)
Show awareness of the needs of others (AN)
What are the similarities and
differences between area /
multiplication problems?
How can the second
difference method be justified
using algebra?
Calculating lengths / areas
Trace to spreadsheet
Setting up a geogebra sheet to
explore a maxima or minima
problem.
Add to your VoiceThread file, clearly explaning your solution or solutions to the paddock area problem.
Upload an example with explanation of how to find the formula of a numeric quadratic pattern.
Upload a proof or explanation of the second difference method.
Lesson Plan 4 – Look at it another way
Assessment
Extending
Exploring
Introducing
Curriculum
Objectives
Create models to represent
both real-life and
hypothetical situations.
Develop the ability to
think creatively, critically,
strategically, and
logically.
Learn to create models
and predict outcomes.
Key Competencies
Essence Statement
This lesson links different representations to modelling problems.







Plan and carry out an investigation (PI)
Find, use, and justify a model (Mo)
Interpret information and results in context (In)
Compare and contrast ideas (CC)
Manage time effectively (MT)
Critically reflect (CR)
Participate actively in a collaborative team or community (CT)
Number and Algebra Level 5

Form and solve simple
quadratic equations.

Relate tables, graphs, and
equations to simple
quadratic relationships
found in number and
spatial patterns
Number and Algebra Level 6
 Form and solve quadratic equations.
 Relate graphs, tables, and equations to quadratic, and
relationships found in number and spatial patterns.
Mathematics Level 7

Form and use quadratic
equations.
 Display the graphs of linear and
non-linear functions and
connect the structure of the
functions with their graphs.
Task
Use geogebra to explore
the time lapse photo of
the motion of a bouncing
ball.
What questions could
your model be used to
answer?
Pedagogy
Identify what the x and y
represent in this context.
What do the parameters
represent?
What effect would changing
the parameters have on the
context?
What about scale?
Summarise all you have
learnt about quadratic
representations in this
particular context. What do
the vertex, intercepts etc
represent.
What are the advantages of
modelling motion with a
quadratic function?
Summarise all you have
learnt about quadratic
representations in this
particular context.
Link to limitations and
justifications of a
mathematical model.
Learning
Link context to word
problems to solving quadratic
equations.
Link representation to method
of solving equations.
Learning about modelling real
life events.
Technology
Scaling / enlarging images.
Connecting features of
different representations to
meaning in context .
Link to other pen and paper
solution methods.
Linking solving equations to
what is happening in the
different representations.
Time lapse photos from
internet (or create own).
Setting up a mathematical
model in geogebra.
Definition of variables
Solutions of quadratics in
context.
Quadratic formula.
Video clips
Work in teams of 4 to
explore one of the other
photos of your choice.
Write 5 questions for
other groups to solve.
Design an data collection
exercise, e.g. video and
find a model to
mathematically describe
the situation.



Setting up a mathematical
model in geogebra.
Use of geogebra to model.
Add to your VoiceThread file, either a time lapse photo, or a video of a natural phenomenon that can be modelled with a
quadratic function, clearly explaining your findings and summarising the key ideas from this unit.
Add at least 5 questions for other groups to answer.
Herbert Simon (1981) said "solving a problem simply means representing it so as to make the solution transparent." (p.
153) add your thoughts to this VoiceThread page about what he might have meant.