Mathematics
Content Descriptors with Learning Goals /
Indicators and Proficiencies
Level 8
All Content Strands
Introduction
What is a Scope and Sequence?
scope

The breadth and depth of content to be covered in a curriculum at any
one time (e.g. week, term, year, over a student’s school life.) All that you
do in a given period.

sequence

The order in which content is presented to learners over time. The
order in which you do it.

Together a scope and sequence of learning
bring order to the delivery of content, supporting the maximising of student
learning and offering sustained opportunities for learning. Without a
considered scope and sequence there is the risk of ad hoc content delivery
and the missing of significant learning.
http://activated.act.edu.au/ectl/design/scope_and_sequence.htm

Why does a school need a scope and sequence?
An
agreed Scope and Sequence for a Learning Area, provides a sound basis for a
school being able to offer a guaranteed
and viable curriculum by addressing gaps in students’ leaning and
eliminating unnecessary repetition. A
shared Scope and Sequence within a school enables teachers to have clarity
about the knowledge, skills and dispositions that students will acquire in
their learning and what they need to learn next. A Scope and Sequence supports
teachers with effective unit and lesson planning and enables teachers to
maintain a developmental focus on student learning as students progress through
the school.
The Mathematics Scope and Sequence developed
by WMR
This document has been
developed to support schools with the transition to AusVELS Mathematics for
2013. While it provides examples of yearly overviews and learning sequences
based on the content descriptors in the Australian Curriculum, it is not a
complete curriculum. Each individual school can use the documents as a basis
for developing a guaranteed and viable
curriculum that caters for the needs of their school community.
Levels Foundation to 10A
each include a set of learning goals/ intentions for each content substrand
intended to provide a user friendly guide to the essential learnings around
which teachers and teams could base their unit and lesson development.
Proficiency strands are
listed next to each learning goal / intention as a guide only and teachers /
teams are encouraged to consider all proficiencies equally whilst planning
units and lessons. Where a particular proficiency is not listed for a content
substrand teachers and teams should endeavour to contextualise the learning
goals to address these proficiencies. Please note the following:
Sequence of teaching
The learning goals/intentions are listed adjacent to
the content descriptions to assist teachers when developing a teaching program.
They are not necessarily in the order to be taught – teachers /teams will make
their own decisions regarding this. The third column has been included to
assist teams to develop ideas for unit planning.
A sample Scope and Sequence Overview is also provided
for each of the year levels from F to 10A. The number of weeks given to each
unit in the overview acts as a guide and the total number of weeks allows for
the many interruptions in a typical school year.
Links between the Learning Goals/Intentions and the
proficiency strands
(a) The Learning Goals/Intentions have been identified to
relate most closely to one of the four proficiency strands (shown in 3 below).
This identification is shown in brackets at the end of each Learning
Goal/Intention:
·
Understanding is
identified by (U)
·
Fluency is identified
by (F)
·
Problem Solving is
identified by (PS)
·
Reasoning is
identified by (R)
(b) In this document there are less Problem Solving and
Reasoning proficiency strands identified than those for Understanding and
Fluency. Should teachers wish to include more of these proficiencies in their
curriculum, they are encouraged to emphasise them when teaching, and to develop
appropriate learning tasks.
Proficiency strands
The proficiency strands describe the actions
in which students can engage when learning and using the content. While not all
proficiency strands apply to every content description, they indicate the
breadth of mathematical actions that teachers can emphasise. The proficiencies listed
next to each learning goal / intention are examples of how students might
achieve the goal or what they have demonstrated by achieving the goal but are
dependent on the context in which the learning takes place.
Understanding
Students build a robust
knowledge of adaptable and transferable mathematical concepts. They make
connections between related concepts and progressively apply the familiar to
develop new ideas. They develop an understanding of the relationship between
the ‘why’ and the ‘how’ of mathematics. Students build understanding when they
connect related ideas, when they represent concepts in different ways, when
they identify commonalities and differences between aspects of content, when
they describe their thinking mathematically and when they interpret
mathematical information.
Fluency
Students develop
skills in choosing appropriate procedures, carrying out procedures flexibly,
accurately, efficiently and appropriately, and recalling factual knowledge and
concepts readily. Students are fluent when they calculate answers efficiently,
when they recognise robust ways of answering questions, when they choose
appropriate methods and approximations, when they recall definitions and
regularly use facts, and when they can manipulate expressions and equations to
find solutions.
Problem Solving
Students develop the
ability to make choices, interpret, formulate, model and investigate problem
situations, and communicate solutions effectively. Students formulate and solve
problems when they use mathematics to represent unfamiliar or meaningful
situations, when they design investigations and plan their approaches, when
they apply their existing strategies to seek solutions, and when they verify
that their answers are reasonable.
Reasoning
Students develop an
increasingly sophisticated capacity for logical thought and actions, such as
analysing, proving, evaluating, explaining, inferring, justifying and
generalising. Students are reasoning mathematically when they explain their
thinking, when they deduce and justify strategies used and conclusions reached,
when they adapt the known to the unknown, when they transfer learning from one
context to another, when they prove that something is true or false and when
they compare and contrast related ideas and explain their choices.
Useful references for teams and teachers to use when planning units of
work and lessons include the following:
·
Ultranet Design Space
– DEECD Big Ideas in Number Maps  128428217
·
Ultranet design Space
– Mathematics eBookboxes  66512121
·
Teaching Mathematics Foundations to Middle
Years
Dianne Siemon,
Kim Beswick, Kathy Brady, Julie Clark, Rhonda Faragher and Elizabeth Warren
·
Mathematics
Domain Page DEECD
·
Building
Numeracy – George Booker
·
Teaching
Primary Mathematics George Booker,
Denise Bond,
Len Sparrow,
Paul Swan
·
What
We Know About Mathematics Teaching and Learning MCREL
·
WMR
Numeracy Design Space 106126201
·
Acara
Scope and Sequence Documents http://www.australiancurriculum.edu.au/Download
·
VCAA – resources http://www.vcaa.vic.edu.au/Pages/foundation10/curriculum/index.aspx
Please note: Teachers will be
required to join each Ultranet design space before being able to access the
resource. The number associated with each space should be entered into the
search box in ‘available design spaces’ in order to find the space.
Level 8
Number and Algebra
Number and
Place Value
AusVELS Content Descriptors

Learning Goals/ Intentions and
Proficiencies
Essential Learning

Unit Development Ideas
How is the essential learning developed into units
of work? How do students make connections between the learning goals and
those included in other content strands or substrands? How do we ensure
students become proficient in fluency, understanding, reasoning and problem
solving?

·
Use index notation with numbers to establish the index laws with positive integral indices and the
zero index (ACMNA182)
·
Carry out the
four operations with rational numbers and integers, using efficient mental
and written strategies and appropriate digital technologies (ACMNA183)

Students will:
·
Evaluate numbers expressed
as powers of positive integers. (F)
·
Know that any number expressed to the power
of zero is 1 and why. (R)
·
Understand how and why we use index
notation. (U)
·
Add, subtract, multiply and divide positive
and negative numbers using written and digital technologies. (F)
·
Develop a range of mental strategies for
calculating involving the four operations. (F)


Achievement Standard:
By
the end of Year 8, students recognise index laws and apply them to whole
numbers. Students use efficient mental and written strategies to carry out the
four operations with integers.
eBookbox: Working with Numbers
Level
8 Number and Algebra
Real
Numbers and Financial Maths
AusVELS
Content Descriptors

Learning Goals/ Intentions and
Proficiencies
Essential
Learning

Unit Development Ideas
How is the essential learning developed
into units of work? How do students make connections between the learning
goals and those included in other content strands or substrands? How do we
ensure students become proficient in fluency, understanding, reasoning and
problem solving?

Real Numbers
·
Investigate terminating
and recurring decimals(ACMNA184)
·
Investigate the
concept of irrational numbers, including π (ACMNA186)
·
Solve problems
involving the use of percentages, including percentage increases and decreases, with and without
digital technologies (ACMNA187)
·
Solve a range of
problems involving rates and ratios, with and without digital
technologies (ACMNA188)
Financial Maths
·
Solve problems
involving profit and loss, with and without digital technologies (ACMNA189)

Students will:
·
Recognise
terminating, recurring and nonterminating decimals and choose their
appropriate representations. (F)
·
Give examples of
terminating, recurring and nonterminating decimals. (F)
·
Define and
identify rational and irrational numbers and give examples of each. (U)
·
Explain that the
real number system includes Irrational numbers. (U)
·
Explain why the
Real Number system includes Irrational numbers.(R)
·
Locate the
approximate position of an irrational number on a number line. (R)
·
Describe certain
subsets of the real number and explain their particular properties. Eg.
Square numbers, primes, etc (U)
·
Use percentages
to solve problems, including those involving markups, discounts, profit and
loss and GST. (F)
·
Develop mental strategies for calculating
percentage discounts using 10% as a reference. (F)
·
Solve rate and
ratio problems using fractions or percentages and chooses the most efficient
form to solve a particular problem.
(F)
·
Express profit
and loss as a percentage of cost or selling price, comparing the difference
eg. Investigate the methods used in retail stores to express discounts. (F)


Achievement Standard:
By
the end of Level 8, students solve everyday problems involving rates, ratios
and percentages. They describe rational and irrational numbers. Students solve
problems involving profit and loss.
eBookbox Sets and Real Numbers
Level 8
Number and Algebra
Patterns
and Algebra and Linear and NonLinear Relationships
AusVELS Content Descriptors

Learning Goals/ Intentions and
Proficiencies
Essential Learning

Unit Development Ideas
How is the essential learning developed into units
of work? How do students make connections between the learning goals and
those included in other content strands or substrands? How do we ensure
students become proficient in fluency, understanding, reasoning and problem
solving?

Patterns and Algebra
·
Extend and apply
the distributive law to the expansion of algebraic
expressions (ACMNA190)
·
Factorise algebraic expressions by identifying
numerical factors (ACMNA191)
·
Simplify
algebraic expressions involving the four operations (ACMNA192)
Linear and NonLinear
Relationships
·
Plot linear
relationships on the Cartesian plane with and without the use of digital
technologies(ACMNA193)
·
Solve linear
equations using algebraic and graphical techniques. Verify solutions by
substitution(ACMNA194)

Students will:
·
Expand and
simplify one bracket expressions eg 2(a +7) = 2a+14 (F)
·
Use the Area
Model to expand algebraic expressions. (F)
·
List all factors
of an algebraic term. (F)
·
Recognise that
factorising is the opposite of expanding. (U)
·
Identify the
highest common factor of algebraic expressions. (F)
·
Gather like
terms. (F)
·
Factorise an
expression by taking out the highest common factor. (F)
·
Use the Area
Model to factorise algebraic expressions. (F)
·
Plot points on
the Cartesian plane.(F)
·
Complete a table
of values, plot the data and discuss the resulting linear relationship. (U)
·
Plot points from
a linear relationship and describe the shape, steepness and where it cuts the
y axis.(F,R)
·
Find the rule
for a linear relationship. (R)
·
Use variables to
symbolise simple linear equations and use a variety of strategies to solve
them. (F)
·
Solve equations
using concrete materials, such as the balance model, and explain the need to
do the same thing to each side of the equation. (U)
·
Use strategies,
such as backtracking and guess, check and improve to solve equations. (F)
·
Apply solving
linear equations to real life problems and discuss the resultant findings.
(P)


Achievement Standard:
By
the end of Year 8, students make connections between expanding and factorising
algebraic expressions. They simplify a variety of algebraic expressions. They
solve linear equations and graph linear relationships on the Cartesian plane.
eBookbox: Linear and NonLinear
Functions
Level
8 Measurement and Geometry
Units
of Measurement
AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies
Essential
Learning

Unit
Development Ideas
How is the essential learning developed
into units of work? How do students make connections between the learning
goals and those included in other content strands or substrands? How do we
ensure students become proficient in fluency, understanding, reasoning and
problem solving?

·
Choose
appropriate units of measurement for area and volume and convert from one unit to another(ACMMG195)
·
Find perimeters
and areas of parallelograms, trapeziums, rhombuses and kites (ACMMG196)
·
Investigate the
relationship between features of circles such as circumference, area, radius
and diameter. Use formulas to solve problems involving circumference and
area (ACMMG197)
·
Develop the
formulas for volumes of rectangular and triangular prisms and prisms in
general. Use formulas to solve problems involving volume(ACMMG198)
·
Solve problems
involving duration, including using 12 and 24hour time within a single time
zone(ACMMG199)

Students will:
·
Distinguish between area and volume and
choose the appropriate units of measurement for each. (U)
·
Convert between units of area and between
units of volume. (F)
·
Name and determine the perimeter and area
of parallelograms, rhombuses and kites. (F)
·
Determine the circumference and area of a
circle by direct measurement. (R)
·
Demonstrate that by knowing circumference
of a circle (its perimeter) we can determine its radius which in turn, can
help me find its diameter and area. Or knowing its radius, I can find the
area, circumference and diameter. (U, F, R)
·
Explain how, what and why Pi is used in
equations related to circles. (U)
·
Know how the formulae for all 3D shapes are
related and variations of Length x Width x Height.(U)
·
Solve problems involving duration,
including using 12 and 24hour time within a single time zone. (PS)
·
Convert between 12 and 14 hour time and across
time zones. (U)
·
Determine the arrival time given a flight
time and time zones.(R)


Achievement Standard:
By
the end of Level 8, students convert between units of measurement for area and
volume. They perform calculations to determine perimeter and area of
parallelograms, rhombuses and kites. They name the features of circles and
calculate the areas and circumferences of circles. Students solve problems
relating to the volume of prisms. They make sense of time duration in real
applications.
eBookbox: Measurement: Circles and 3D objects
Level 8
Measurement and Geometry
Geometric
Reasoning
AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies
Essential Learning

Unit
Development Ideas
How is the essential learning developed
into units of work? How do students make connections between the learning
goals and those included in other content strands or substrands? How do we
ensure students become proficient in fluency, understanding, reasoning and
problem solving?

·
Define
congruence of plane shapes using transformations. (ACMMG200)
·
Develop the
conditions for congruence of triangles. (ACMMG201)
·
Establish
properties of quadrilaterals using congruent triangles and angle properties,
and solve related numerical problems using reasoning. (ACMMG202)

Students will:
·
Describe transformations including:
translations, rotations and reflections. (F)
·
Define congruence of plane shapes using
transformations. (R)
·
Use the conditions for congruence of
triangles including, congruence (SSS, SAS, ASA and RHS), and demonstrating
which conditions do not prescribe congruence (ASS, AAA). (R)
·
Use coordinates to describe the
transformation. (F)
·
Describe properties of quadrilaterals
including squares, rectangles, parallelograms, rhombuses, trapeziums and
kites. (U)
·
Determine the sum of internal angles of a
polygon, using triangles. (F)
·
Solve problems using the sum of internal
angles for triangles and other polygons. (F)
·
Determine lines of symmetry in a given
shape. (U)
·
Identify
properties related to side lengths, parallel sides, angles, diagonals and
symmetry. (R)


Achievement Standard:
By the end of Year 8, students identify conditions for the congruence
of triangles and deduce the properties of quadrilaterals.
eBookbox:
Angles, Shapes and
Transformations
Level 8 Statistics
and Probability
Chance and
Data Representations
AusVELS Content Descriptors

Learning Goals/ Intentions and Proficiencies
Essential Learning

Unit
Development Ideas
How is the essential learning developed into units
of work? How do students make connections between the learning goals and
those included in other content strands or substrands? How do we ensure
students become proficient in fluency, understanding, reasoning and problem
solving?

Chance
·
Identify
complementary events and use the sum of probabilities to solve problems. (ACMSP204)
·
Describe events
using language of 'at least', exclusive 'or' (A or B but not both), inclusive
'or' (A or B or both) and 'and'. (ACMSP205)
·
Represent events
in twoway tables and Venn diagrams and solve related problems. (ACMSP292)
Data Representations
·
Investigate
techniques for collecting data, including census, sampling and observation(ACMSP284)
·
Explore the
practicalities and implications of obtaining data through sampling using a
variety of investigative processes. (ACMSP206)
·
Explore the
variation of means and proportions of random samples drawn from the same
population. (ACMSP293)
·
Investigate the
effect of individual data values, including outliers, on the mean and median.
(ACMSP207)

Students will:
· Demonstrate
that probabilities range between 0 to 1 by convention and that calculating
the probability of an event allows the probability of its complement to be
identified. (R)
· Identify the complement of
familiar events (eg the complement of getting a head on a coin is getting a
tail, the complement of winning a game is not winning the game). (R)
· Calculate
probabilities for sample spaces
for singlestep experiments (eg drawing a marble from a bag with 2 black and
3 white marbles with replacement. (F)
· Pose
‘and’, ‘or’, ‘not’ probability questions about objects or people. (R)
·
Show
that representing data in Venn diagrams or twoway tables facilitates
the calculation of probabilities. (U)
· Use Venn
diagrams and twoway tables to calculate probabilities for events satisfying
‘and’, ‘or’, ‘given’ and ‘not’ conditions. (R)
· Collect
data to answer the questions using Venn diagrams or twoway tables. (P)
·
Know the
difference between a sample and a census and when each might be appropriate.
(U)
·
Be able to
create, implement and interpret survey data through sampling techniques.
(P,R)
·
Use sample
properties (for example mean, median, range) to predict characteristics of
the population acknowledging uncertainty. (U)
·
Use displays of
data to explore and investigate effects. (R)


Achievement Standard:
By
the end of Year 8, students choose appropriate language to describe events and
experiments. Students model authentic situations with twoway tables and Venn
diagrams. They explain issues related to the collection of data and the effect
of outliers on means and medians in that data.
eBookbox:
Sets and Real Numbers
Probability,
inferences and Conjecture
Sets and Logic