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Mathscape 9 Extension: Working Mathematically
March 2004 web update
Mathscape 9 Extension Syllabus Correlation Grid (Stage 5.1/5.2/5.3)
Highlight indicates Stage 4 review
Text Reference
Chapter 1 Rational numbers
1.1 Significant figures
1.2 The calculator
1.3 Estimation
Try this: Fermi problem
1.4 Recurring decimals
1.5 Rates
Try this: Desert walk
1.6 Solving problems with rates
Try this: Passing trains
Focus on working mathematically: A
number pattern from Galileo 1615
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
Substrand
Rational
Numbers
Outcome
NS5.2.1
Key Ideas
Round numbers to a
specified number of
significant figures
Express recurring
decimals as fractions
Convert rates from one
set of units to another
Knowledge and Skills
 identifying significant figures
 rounding numbers to a specified
number of significant figures
 using the language of estimation
appropriately, including:
 recognise that calculators show
approximations to recurring
decimals
e.g. 23 displayed as 0.666667
(Communicating)
.
– rounding
 justify that 0.9  1 (Reasoning)
– approximate
– level of accuracy
– using symbols for approximation
e.g. 
 determining the effect of truncating or
rounding during calculations on the
accuracy of the results
 writing recurring decimals in fraction
form using calculator and noncalculator methods
.
Working Mathematically
. .
.
e.g. 0. 2 , 0. 2 3 , 0.2 3
 converting rates from one set of units to
another
e.g. km/h to m/s, interest rate of 6% per
annum is 0.5% per month
 decide on an appropriate level of
accuracy for results of
calculations (Applying Strategies)
 assess the effect of truncating or
rounding during calculations on
the accuracy of the results
(Reasoning)
 appreciate the importance of the
number of significant figures in a
given measurement
(Communicating)
 use an appropriate level of
accuracy for a given situation or
problem solution (Applying
Strategies)
 solve problems involving rates
(Applying Strategies)
1
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 2 Algebra
2.1 Describing simple patterns
Try this: Flags
2.2 Substitution
2.3 Adding and subtracting algebraic
expressions
2.4 Multiplying and dividing algebraic
expressions
Try this: Overhanging the overhang
2.5 The order of operations
2.6 The distributive law
2.7 The highest common factor
2.8 Adding and subtracting algebraic
fractions
2.9 Multiplying and dividing algebraic
fractions
2.10 Generalised arithmetic
Try this: Railway tickets
2.11 Properties of numbers
2.12 Generalising solutions to
problems using patterns
2.13 Binomial products
2.14 Perfect squares
Try this: Proof
2.15 Difference of two squares
2.16 Miscellaneous expansions
Focus of working mathematically: A
number pattern from Blaise Pascal
1654
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
March 2004 web update
Substrand
Outcome
Key Ideas
Algebraic
Techniques
PAS5.2.1
Simplify, expand and
factorise algebraic
expressions including those
involving fractions
Knowledge and Skills
 simplifying algebraic expressions
involving fractions, such as
2x 2x

5
3
7 a 5a

8 12
2y y

3
6
2 ab 6

3
2b
 expanding, by removing grouping
symbols, and collecting like terms
where possible, algebraic
expressions such as
2 y ( y  5)  4( y  5)
4 x(3x  2)  ( x  1)
 3x 2 (5 x 2  2 xy )
 factorising, by determining common
factors, algebraic expressions such
as
3x  6 x
2
14ab  12a 2
21xy  3x  9 x 2
Working Mathematically
 describe relationships between
the algebraic symbol system and
number properties
(Reflecting, Communicating)
 link algebra with generalised
arithmetic
e.g. use the distributive property
of multiplication over addition to
determine that
a(b  c)  ab  ac
(Reflecting)
 determine and justify whether a
simplified expression is correct
by substituting numbers for
pronumerals (Applying
Strategies, Reasoning)
 generate a variety of equivalent
expressions that represent a
particular situation or problem
(Applying Strategies)
 check expansions and
factorisations by performing the
reverse process (Reasoning)
 interpret statements involving
algebraic symbols in other
contexts e.g. spreadsheets
(Communicating)
 explain why an algebraic
expansion or factorisation is
incorrect e.g. Why is the
following incorrect?
24 x 2 y  16 xy 2  8xy (3x  2)
(Reasoning, Communicating)
2
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Substrand
Chapter 2 Algebra
2.1 Describing simple patterns
Try this: Flags
2.2 Substitution
2.3 Adding and subtracting algebraic
expressions
2.4 Multiplying and dividing algebraic
expressions
Try this: Overhanging the overhang
2.5 The order of operations
2.6 The distributive law
2.7 The highest common factor
2.8 Adding and subtracting algebraic
fractions
2.9 Multiplying and dividing algebraic
fractions
2.10 Generalised arithmetic
Try this: Railway tickets
2.11 Properties of numbers
2.12 Generalising solutions to
problems using patterns
2.13 Binomial products
2.14 Perfect squares
Try this: Proof
2.15 Difference of two squares
2.16 Miscellaneous expansions
Focus of working mathematically: A
number pattern from Blaise Pascal
1654
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
Algebraic
Techniques
March 2004 web update
Outcome
Key Ideas
PAS5.3.1
Use algebraic techniques to
simplify expressions,
expand binomial products
Knowledge and Skills
 simplifying algebraic expressions,
including those involving fractions,
such as
 link algebra with generalised
arithmetic (Reflecting)
 expanding binomial products by
finding the area of rectangles
x
e.g.
8
x2
8x
3
3x
24
x  8x  3  x 2  8 x  3x  24
 x 2  11x  24
 using algebraic methods to expand a
variety of binomial products, such as
( x  2)( x  3)
( 2 y  1) 2
(3a  1)(3a  1)
 recognising and applying the special
products
2
 determine and justify whether a
simplified expression is correct
by substituting numbers for
pronumerals
(Applying Strategies, Reasoning)
 generate a variety of equivalent
expressions that represent a
particular situation or problem
(Applying Strategies)
hence
(a  b)(a b)  a  b
 describe relationships between
the algebraic symbol system and
number properties
(Reflecting, Communicating)
 develop facility with the algebraic
symbol system in order to apply
algebraic techniques to other
strands and substrands (Applying
Strategies, Communicating)
11x  2 y  7 x  8 y  5
4(3 x  2)  ( x  1)
2
7a
4a  3b  b 
3
2
x x 1

3
5
x
Working Mathematically
 check expansions and
factorisations by performing the
reverse process (Reasoning)
 interpret statements involving
algebraic symbols in other
contexts e.g. spreadsheets
(Communicating)
 solve problems, such as:
find a relationship that describes
the number of diagonals in a
polygon with n sides
(Applying Strategies)
2
(a  b)2  a 2  2ab  b 2
 factorising expressions:
3
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.


Mathscape 9 Extension: Working Mathematically
March 2004 web update
– common factors
NS5.3.1
 prove some general properties of
numbers such as
– the sum of two odd integers is
even
– the product of an odd and
even integer is even
– the sum of 3 consecutive
integers is divisible by 3
(Reasoning)
4
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 3 Consumer arithmetic
3.1 Salaries and wages
3.2 Other methods of payment
3.3 Overtime and other payments
3.4 Wage deductions
3.5 Taxation
3.6 Budgeting
Try this: Telephone charges
3.7 Best buys
3.8 Discounts
Try this: Progressive discounting
3.9 Profit and loss
Focus on working mathematically:
Sydney market prices in 1831
Language link with Macquarie
Chapter review
Substrand
Consumer
Arithmetic
March 2004 web update
Outcome
NS5.1.2
Key Ideas
Solve simple consumer
problems including those
involving earning and
spending money
Calculate simple interest
and find compound interest
using a calculator and tables
of values
Knowledge and Skills
Working Mathematically
 calculating earnings for various time
periods from different sources,
including:
 read and interpret pay slips
from part-time jobs when
questioning the details of their
own employment
(Questioning, Communicating)
– wage
– salary
– commission
– piecework
– overtime
– bonuses
– holiday loadings
– interest on investments
 calculating income earned in casual
and part-time jobs, considering
agreed rates and special rates for
Sundays and public holidays
 calculating weekly, fortnightly,
monthly and yearly incomes
 calculating net earnings considering
deductions such as taxation and
superannuation
Mathscape 9 and Mathscape 9
Extension School CD ROM
 prepare a budget for a given
income, considering such
expenses as rent, food,
transport etc
(Applying Strategies)
 interpret the different ways of
indicating wages or salary in
newspaper ‘positions vacant’
advertisements e.g. $20K
(Communicating)
 compare employment
conditions for different careers
where information is gathered
from a variety of mediums
including the Internet
e.g. employment rates,
payment (Applying Strategies)
 calculating a ‘best buy’
NS5.2.2
Solve consumer arithmetic
problems and successive
discounts
 calculating the result of successive
discounts
 explain why, for example, a
discount of 10% following a
discount of 15% is not the
same as a discount of 25%
(Applying Strategies,
Communicating, Reasoning)
5
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 4 Equations, inequations
and formulae
4.1 One- and two-step equations
4.2 Equations with pronumerals on
both sides
4.3 Equations with grouping symbols
4.4 Equations with one fraction
4.5 Equations with more than one
fraction
4.6 Inequations
4.7 Solving worded problems
Try this: A prince and a king
4.8 Evaluating the subject of a
formula
4.9 Equations arising from
substitution
Try this: Floodlighting by formula
4.10 Changing the subject of a
formula
Focus on working mathematically:
Splitting the atom
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
March 2004 web update
Substrand
Outcome
Algebraic
Techniques
PAS5.2.2
Key Ideas
Solve linear and simple
quadratic equations of the
form ax 2  c
Solve linear inequalities
Knowledge and Skills
Linear and Quadratic Equations
 solving linear equations such as
x x
 5
2 3
2y  3
 2
3
z 3
 6 1
2
3(a  2)  2(a  5)  10
3(2t  5)  2t  5
3r  1 2r  4

4
5
 solving word problems that result in
equations
 exploring the number of solutions
that satisfy simple quadratic
equations of the form x 2  c
 solving simple quadratic equations
of the form ax2  c
 solving equations arising from
substitution into formulae
Linear Inequalities
 solving inequalities such as
3x  1  9
2(a  4)  24
t4
 3
5
Working Mathematically
 compare and contrast different
methods of solving linear
equations and justify a choice for
a particular case (Applying
Strategies, Reasoning)
 use a number of strategies to
solve unfamiliar problems,
including:
- using a table
- drawing a diagram
- looking for patterns
- working backwards
- simplifying the problem and
- trial and error (Applying
Strategies, Communicating)
 solve non-routine problems using
algebraic methods
(Communicating, Applying
Strategies)
 explain why a particular value
could not be a solution to an
equation
(Applying Strategies,
Communicating, Reasoning)
 create equations to solve a
variety of problems and check
solutions
(Communicating, Applying
Strategies, Reasoning)
 write formulae for spreadsheets
(Applying Strategies,
Communicating)
6
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
March 2004 web update
 solve and interpret solutions to
equations arising from
substitution into formulae used in
other strands of the syllabus and
in other subjects. Formulae such
as the following could be used:
y 2  y1
x 2  x1
1 2
E  mv
2
4
V  r 3
3
SA  2r 2  2rh
m
(Applying Strategies,
Communicating, Reflecting)
 explain why quadratic equations
could be expected to have two
solutions (Communicating,
Reasoning)
 justify a range of solutions to an
inequality
(Applying Strategies,
Communicating, Reasoning)
7
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 4 Equations, inequations
and formulae
4.1 One- and two-step equations
4.2 Equations with pronumerals on both
sides
4.3 Equations with grouping symbols
4.4 Equations with one fraction
4.5 Equations with more than one fraction
4.6 Inequations
4.7 Solving worded problems
Try this: A prince and a king
4.8 Evaluating the subject of a formula
4.9 Equations arising from substitution
Try this: Floodlighting by formula
4.10 Changing the subject of a formula
Focus on working mathematically:
Splitting the atom
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Algebraic
Techniques
PAS5.3.2
Key Ideas
Solves a range of linear
equations
Knowledge and Skills
Working Mathematically
Linear, Quadratic and Simultaneous
Equations
 solve non-routine problems using
algebraic techniques (Applying
Strategies, Communicating)
 using analytical and graphical
methods to solve a range of linear
equations, including equations that
involve brackets and fractions such
as
3(2a  6)  5  (a  2)
2x  5 x  7

0
3
5
y 1 2y  3 1


4
3
2
 create equations to solve a
variety of problems and check
solutions
(Communicating, Applying
Strategies, Reasoning)
 explain why a particular value
could not be a solution to an
equation
 solving problems involving linear
equations
Mathscape 9 and Mathscape 9
Extension School CD ROM
8
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 5 Measurement
5.1 Length, mass, capacity and time
5.2 Accuracy and precision
5.3 Pythagoras’ theorem
Try this: Pythagorean proof by
Perigal
5.4 Perimeter
5.5 Circumference
Try this: Command module
5.6 Converting units of area
5.7 Calculating area
Try this: The area of a circle
5.8 Area of a circle
5.9 Composite areas
Try this: Area
5.10 Problems involving area
Focus on working mathematically:
The solar system
Language link with Macquarie
Chapter review
Substrand
Algebraic
Techniques
March 2004 web update
Outcome
MS5.1.1
Key Ideas
Uses formulae to calculate
the area of quadrilaterals
and finds areas and
perimeters of simple
composite figures
Knowledge and Skills
Working Mathematically
 developing and using formulae to
find the area of quadrilaterals
– for a kite or rhombus, Area  1 xy
2
where
x and y are the lengths of the
diagonals;
– for a trapezium, Area 
1
2
h( a  b)
where
h is the perpendicular height and a
and b the lengths of the parallel
sides
 calculating the area of simple
composite figures consisting of two
shapes including quadrants and
semicircles
 identify the perpendicular height
of a trapezium in different
orientations (Communicating)
 select and use the appropriate
formula to calculate the area of a
quadrilateral (Applying
Strategies)
 dissect composite shapes into
simpler shapes
(Applying Strategies)
 solve practical problems involving
area of quadrilaterals and simple
composite figures (Applying
Strategies)
 calculating the perimeter of simple
composite figures consisting of two
shapes including quadrants and
semicircles
Mathscape 9 and Mathscape 9
Extension School CD ROM
9
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
March 2004 web update
MS5.2.1
Finds areas and perimeters
of composite figures
 calculating the area and perimeter of
sectors
 calculating the perimeter and area of
composite figures by dissection into
triangles, special quadrilaterals,
semicircles and sectors
 solve problems involving
perimeter and area of composite
shapes (Applying Strategies)
 calculate the area of an annulus
(Applying Strategies)
 apply formulae and properties of
geometrical shapes to find
perimeters and areas e.g. find
the perimeter of a rhombus given
the lengths of the diagonals
(Applying Strategies)
 identify different possible
dissections for a given composite
figure and select an appropriate
dissection to facilitate calculation
of the area
(Applying Strategies, Reasoning)
10
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 6 Data representation
and analysis
6.1 Graphs
6.2 Organising data
6.3 Analysing data
6.4 Problems involving the mean
Try this: The English language
6.5 Cumulative frequency
6.6 Grouped data
Focus on working mathematically:
World health
Language link with Macquarie
Chapter review
Substrand
Data
Representation
and Analysis
March 2004 web update
Outcome
DS5.1.1
Key Ideas
Construct frequency
tables for grouped data
Find mean and modal
class for grouped data
Determine cumulative
frequency
Find median using a
cumulative frequency
table or polygon
Knowledge and Skills
 constructing a cumulative frequency
table for ungrouped data
 constructing a cumulative frequency
histogram and polygon (ogive)
 using a cumulative frequency
polygon to find the median
 grouping data into class intervals
 constructing a frequency table for
grouped data
 constructing a histogram for grouped
data
 finding the mean using the class
centre
 finding the modal class
Mathscape 9 and Mathscape 9
Extension School CD ROM
Working Mathematically
 construct frequency tables and
graphs from data obtained from
different sources (e.g. the Internet)
and discuss ethical issues that may
arise from the data (Applying
Strategies, Communicating,
Reflecting)
 read and interpret information from a
cumulative frequency table or graph
(Communicating)
 compare the effects of different
ways of grouping the same data
(Reasoning)
 use spreadsheets, databases,
statistics packages, or other
technology, to analyse collected
data, present graphical displays, and
discuss ethical issues that may arise
from the data
(Applying Strategies,
Communicating, Reflecting)
11
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 7 Probability
7.1 Probability and its language
7.2 Experimental probability
Try this: Two-up
7.3 Computer simulations
Try this: The game of craps
7.4 Theoretical probability
Try this: Winning chances
Focus on working mathematically:
Getting through traffic lights
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
Substrand
Probability
March 2004 web update
Outcome
NS5.1.3
Key Ideas
Knowledge and Skills
Determine relative
frequencies to estimate
probabilities
 repeating an experiment a number
of times to determine the relative
frequency of an event
Determine theoretical
probabilities
 estimating the probability of an event
from experimental data using
relative frequencies
 expressing the probability of an
event A given a finite number of
equally likely outcomes as
P( A) =
number of favourable outcomes
n
Working Mathematically
 recognise and explain differences
between relative frequency and
theoretical probability in a simple
experiment (Communicating,
Reasoning)
 apply relative frequency to predict
future experimental outcomes
(Applying Strategies)
 design a device to produce a
specified relative frequency e.g. a
four-coloured circular spinner
(Applying Strategies)
 where n is the total number of
outcomes in the sample space
 recognise that probability estimates
become more stable as the number
of trials increases (Reasoning)
 using the formula to calculate
probabilities for simple events
 recognise randomness in chance
situations (Communicating)
 simulating probability experiments
using random number generators
 apply the formula for calculating
probabilities to problems related to
card, dice and other games
(Applying Strategies)
12
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 8 Surds
8.1 Rational and irrational numbers
8.2 Simplifying surds
Try this: Greater number
8.3 Addition and subtraction of surds
8.4 Multiplication and division of
surds
Try this: Imaginary numbers
8.5 Binomial products with surds
8.6 Rationalising the denominator
Try this: Exact values
Focus on working mathematically:
Fibonacci numbers and the golden
mean
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Performs
operations
with surds
and indices
§ NS5.3.1
Key Ideas
Define the system
of real numbers
distinguishing
between rational
and irrational
numbers
Perform operations
with surds
Knowledge and Skills
Working Mathematically
 defining a rational number:
a
A rational number is the ratio
of two
b
integers where b ≠ 0.
 distinguishing between rational and irrational
numbers
 using a pair of compasses and a straight edge
to construct simple rationals and surds on the
number line
 defining real numbers:
Real numbers are represented by points on
the number line.
Irrational numbers are real numbers that are
not rational.
 demonstrating that
x is undefined for x < 0,
x  0 for x = 0, and x is the positive
square root of x when x  0
 explain why all integers and
recurring decimals are rational
numbers (Communicating,
Reasoning)
 explain why rational numbers
can be expressed in decimal
form (Communicating,
Reasoning)
 demonstrate that not all real
numbers are rational
(Communicating, Applying
Strategies, Reasoning)
 explain why a particular
sentence is incorrect
e.g.
3 5  8
(Communicating, Reasoning)
 using the following results for x, y > 0:
 x  x 
2
x2
xy  x . y
x

y
x
y
 using the four operations of addition,
subtraction, multiplication and division to
simplify expressions involving surds
 expanding expressions involving surds such
as
 3  5
2


or 2  3 2  3

 rationalising the denominators of surds of the
form
a b
c d
13
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 9 Indices
9.1 Index notation
9.2 Simplifying numerical
expressions using the index
laws
9.3 The index laws
9.4 Miscellaneous questions on
the index laws
9.5 The zero index
Try this: Smallest to largest
9.6 The negative index
9.7 Products and quotients with
negative indices
Try this: Digit patterns
9.8 The fraction index
9.9 Scientific notation
9.10 Scientific notation on the
calculator
Focus on working
mathematically: Mathematics is
at the heart of science
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
Substrand
Rational
Numbers
March 2004 web update
Outcome
NS5.1.1
Key Ideas
Define and use zero
index and negative
integral indices
Develop the index laws
arithmetically
Use index notation for
square and cube roots
Express numbers in
scientific notation
(positive and negative
powers of 10)
Knowledge and Skills
Working Mathematically
 describing numbers written in index form
using terms such as base, power, index,
exponent
 solve numerical problems involving
indices
(Applying Strategies)
 evaluating numbers expressed as
powers of positive whole numbers
 explain the incorrect use of index
laws
e.g. why
32  34  96 (Communicating,
Reasoning)
 establishing the meaning of the zero
index and negative indices e.g. by
patterns
32
31
30
31
32
9
3
1
1
3
1
9

 verify the index laws by using a
calculator
e.g. to compare the values of
1
32
,
2
 communicate and interpret technical
information using scientific notation
(Communicating)
 translating numbers to index form
(integral indices) and vice versa
 developing index laws arithmetically by
expressing each term in expanded form
e.g.
32  34  (3  3)  (3  3  3  3)  324  36
3 3 3 3 3
35  32 
 352  33
3 3
2 4
2
5 
  and 5 (Reasoning)
 
1
2
 writing reciprocals of powers using
negative indices
1
1
e.g. 34  4 
81
3
3   3  3 3  3 3  3 3  3  3
 5
24
 explain the difference between
numerical expressions such as
2104 and 2 4 , particularly with
reference to calculator displays
(Communicating, Reasoning)
 solve problems involving scientific
notation
(Applying Strategies)
 38
using index laws to simplify expressions
 using index laws to define fractional
indices for square and cube roots
e.g.
 9
2
2
 1
 9 and  9 2   9 , hence
 
1
9  92
14
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 9 Indices
9.1 Index notation
9.2 Simplifying numerical
expressions using the index laws
9.3 The index laws
9.4 Miscellaneous questions on the
index laws
9.5 The zero index
Try this: Smallest to largest
9.6 The negative index
9.7 Products and quotients with
negative indices
Try this: Digit patterns
9.8 The fraction index
9.9 Scientific notation
9.10 Scientific notation on the
calculator
Focus on working mathematically:
Mathematics is at the heart of
science
Language link with Macquarie
Chapter review
Substrand
Rational
Numbers
March 2004 web update
Outcome
NS5.1.1
Key Ideas
Knowledge and Skills
 writing square roots and cube roots in index
form
Working Mathematically
 Rational Numbers
1
e.g. 8 3  3 8  2
 recognising the need for a notation to express
very large or very small numbers
 expressing numbers in scientific notation
 entering and reading scientific notation on a
calculator
 using index laws to make order of magnitude
checks for numbers in scientific notation
e.g.
3.12  104  4.2  106  12  1010  1.2  1011



 converting numbers expressed in scientific
notation to decimal form
 ordering numbers expressed in scientific
notation
Mathscape 9 and Mathscape 9
Extension School CD ROM
15
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 9 Indices
9.1 Index notation
9.2 Simplifying numerical
expressions using the index laws
9.3 The index laws
9.4 Miscellaneous questions on
the index laws
9.5 The zero index
Try this: Smallest to largest
9.6 The negative index
9.7 Products and quotients with
negative indices
Try this: Digit patterns
9.8 The fraction index
9.9 Scientific notation
9.10 Scientific notation on the
calculator
Focus on working mathematically:
Mathematics is at the heart of
science
Language link with Macquarie
Chapter review
Substrand
Algebraic
Techniques
March 2004 web update
Outcome
PAS5.1.1
Key Ideas
Apply the index laws to
simplify algebraic
expressions (positive
integral indices only)
Knowledge and Skills
Working Mathematically
 using the index laws previously
established for numbers to develop the
index laws in algebraic form
2 2  2 3  2 2 3  2 5
e.g.
2 2  2
5
2
2   2
2 3
5 2
2
3
a m  a n  a mn
a a  a
m
n
mn
(a m ) n  a mn
6
establishing that a0  1 using the index laws
a a  a
3
e.g.
3
and
a3  a3  1

a0  1
33
a
0
 simplifying algebraic expressions that
include index notation
e.g.
5x 0  3  8
2 x 2  3x 3  6 x 5
12a 6  3a 2  4a 4
2m 3 (m 2  3)  2m 5  6m 3
 verify the index laws using a
calculator
e.g. use a calculator to
compare the values of (34 ) 2
and 38 (Reasoning)
 explain why x0  1 (Applying
Strategies, Reasoning,
Communicating)
 link use of indices in Number
with use of indices in Algebra
(Reflecting)
 explain why a particular
algebraic sentence is incorrect
e.g. explain why a 3  a 2  a 6 is
incorrect (Communicating,
Reasoning)
 examine and discuss the
difference between expressions
such as
3a 2  5a and 3a 2  5a
by substituting values for a
(Reasoning, Applying
Strategies, Communicating)
Mathscape 9 and Mathscape 9
Extension School CD ROM
16
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 9 Indices
9.1 Index notation
9.2 Simplifying numerical
expressions using the index
laws
9.3 The index laws
9.4 Miscellaneous questions
on the index laws
9.5 The zero index
Try this: Smallest to largest
9.6 The negative index
9.7 Products and quotients
with negative indices
Try this: Digit patterns
9.8 The fraction index
9.9 Scientific notation
9.10 Scientific notation on the
calculator
Focus on working
mathematically: Mathematics
is at the heart of science
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Key Ideas
Algebraic
Techniques
PAS5.2.1
Apply the index laws to
simplify algebraic
expressions involving
negative and fractional
indices
Knowledge and Skills
 applying the index laws to simplify expressions involving
pronumerals
 establishing that
 a 
2
a  a  a  a  a2  a
 using index laws to assist with the definition of the
fractional index for square root
given
2
2
 12 
a   a
 
and
 explain why finding the
square root of an
expression is the same
as raising the expression
to the power of a half
(Communicating,
Reasoning)
 state whether particular
equivalences are true or
false and give reasons
e.g. Are the following true
or false? Why?
1
 using index laws to assist with the definition of the
fractional index for cube root
 using index notation and the index laws to establish that
a 1 
1
1
1
, a 2  2 , a 3  3 , …
a
a
a
 applying the index laws to simplify algebraic expressions
such as
(3 y 2 ) 3
4b 5  8b 3
9 x 4  3x 3
1
3 x 2 5 x 2
1
5x0  1
a  a2
then
1
Mathscape 9 and Mathscape
9 Extension School CD ROM
 a  a
Working
Mathematically
9 x  3x 5  3x
5
a5  a7  a2
1
2c  4  4
2c
(Applying Strategies,
Reasoning,
Communicating)
 explain the difference
between particular pairs
of algebraic expressions,
such as x 2 and 2 x
(Reasoning,
Communicating)
1
6 y 3 4 y 3
17
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
§ Real
Numbers
March 2004 web update
NS5.3.1
Use integers and
fractions for index
notation
 using the index laws to demonstrate the reasonableness of
the definitions for fractional indices
1
x n n x
m
Convert between surd
and index form
x n  xm
n
 solve numerical
problems involving
surds and/or
fractional indices
(Applying Strategies)
 translating expressions in surd form to expressions in
index form and vice versa
 evaluating numerical expressions involving fractional
2
indices e.g. 27 3
1
 using the x y key on a calculator
 evaluating a fraction raised to the power of –1, leading to
a
 
b
1

b
a
18
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 10 Geometry
10.1 Angles
10.2 Parallel lines
10.3 Triangles
Try this: The badge of the
Pythagoreans
10.4 Angle sum of a
quadrilateral
10.5 Special quadrilaterals
Try this: Five shapes
10.6 Polygons
Try this: How many diagonals
in a polygon?
Try this: An investigation of
triangles
10.7 Tests for congruent
triangles
10.8 Congruent proofs
Try this: Triangle angles
10.9 Deductive reasoning and
congruent triangles
Focus on working
mathematically: Does a triangle
have a centre?
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Properties
of
Geometrical
Figures
SGS5.2.1
Key Ideas
Establish sum of
exterior angles
result and sum of
interior angles
result for polygons
Knowledge and Skills
Working Mathematically
 applying the result for the interior angle sum of a triangle to
find, by dissection, the interior angle sum of polygons with
4,5,6,7,8, … sides
 express in algebraic terms
the interior angle sum of a
polygon with n sides e.g.
(n–2)  180
(Communicating)
 defining the exterior angle of a convex polygon
 establishing that the sum of the exterior angles of any
convex polygon is 360
 applying angle sum results to find unknown angles
 find the size of the interior
and exterior angles of
regular polygons with
5,6,7,8, … sides
(Applying Strategies)
 solve problems using
angle sum of polygon
results (Applying
Strategies)
Properties
of
Geometric
Figures
SGS5.2.2
Apply tests for
congruent triangles
Use simple
deductive
reasoning in
numerical and nonnumerical problems
Congruent Triangles
 determining what information is needed to show that two
triangles are congruent using the 4 tests: SSS, SAS, AAS,
RHS
 applying the congruency tests to justify that two triangles
are congruent
 applying the 4 triangle congruency tests in numerical
exercises to find unknown sides and angles
 apply the properties of
congruent triangles to
solve problems justifying
the results
 apply simple deductive
reasoning in solving
numerical and nonnumerical problems
19
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 11 The linear
function
11.1 The number plane
11.2 Graphing straight lines 1
Try this: Size 8
11.3 Graphing straight lines 2
11.4 Gradient of a line
Try this: Hanging around
11.5 The linear equation
y=mx+b
Try this: Latitude and
temperature
Focus on working
mathematically: Television
advertising
Language link with Macquarie
Chapter review
Substrand
Coordinate
Geometry
March 2004 web update
Outcome
PAS5.1.2
Key Ideas
Graph linear and
simple non-linear
relationships from
equations
Knowledge and Skills
Midpoint, Length and Gradient
 using the right-angled triangle drawn between two points
on the number plane and the relationship
rise
gradient 
run
to find the gradient of the interval joining two points
 determining whether a line has a positive or negative
slope by following the line from left to right – if the line
goes up it has a positive slope and if it goes down it has
a negative slope
Working Mathematically
 explain the meaning of
gradient and how it can be
found for a line joining two
points
(Communicating, Applying
Strategies)
 distinguish between
positive and negative
gradients from a graph
(Communicating
 finding the gradient of a straight line from the graph by
drawing a right-angled triangle after joining two points on
the line
20
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Substrand
March 2004 web update
Outcome
Key Ideas
Knowledge and Skills
Graphs of Relationships
 constructing tables of values and using
coordinates to graph vertical and horizontal lines
such as
x  3, x  1
y  2, y  3
 identifying the x - and y -intercepts of graphs
 identifying the x -axis as the line y = 0
 identifying the y -axis as the line x = 0
 graphing a variety of linear relationships on the
number plane by constructing a table of values
and plotting coordinates using an appropriate
scale e.g. graph the following:
y  3 x
x 1
y
2
x y 5
x y 2
2
y x
3
 determining whether a point lies on a line by
substituting into the equation of the line
Working Mathematically
 describe horizontal and vertical
lines in general terms
(Communicating)
 explain why the x -axis has
equation y = 0
(Reasoning, Communicating)
 explain why the y -axis has
equation x = 0
(Reasoning, Communicating)
 determine the difference
between equations of lines that
have a negative gradient and
those that have a positive
gradient (Reasoning)
 use a graphics calculator and
spreadsheet software to graph,
compare and describe a range
of linear and simple non-linear
relationships
(Applying Strategies,
Communicating)
 apply ethical considerations
when using hardware and
software (Reflecting)
21
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
March 2004 web update
Text Reference
Substrand
Outcome
Chapter 12 Trigonometry 440
Applies
trigonometry
to solve
problems
(diagrams
given)
including
those
involving
angles of
elevation and
depression
MS5.1.2
12.1 Side ratios in right-angled
triangles
12.2 The trigonometric ratios
Try this: Height to base ratio
12.3 Trigonometric ratios using
a calculator
12.4 Finding the length of a
side
12.5 Problems involving finding
sides
Try this: Make a hypsometer
12.6 Finding the size of an
angle
12.7 Problems involving finding
angles
12.8 Angles of elevation and
depression
Try this: Pilot instructions
12.9 Bearings
Try this: The sine rule
Focus on working
mathematically: Finding your
latitude from the Sun
Language link with Macquarie
Chapter review
Key Ideas
Use trigonometry
to find sides and
angles in rightangled triangles
Solve problems
involving angles of
elevation and
angles of
depression from
diagrams
Knowledge and Skills
Trigonometric Ratios of Acute Angles
 identifying the hypotenuse, adjacent and opposite sides
with respect to a given angle in a right-angled triangle in
any orientation
 labelling the side lengths of a right-angled triangle in
relation to a given angle e.g. the side c is opposite angle C
 recognising that the ratio of matching sides in similar rightangled triangles is constant for equal angles
 defining the sine, cosine and tangent ratios for angles in
right-angled triangles
 using trigonometric notation e.g. sin A
 using a calculator to find approximations of the
trigonometric ratios of a given angle measured in degrees
 using a calculator to find an angle correct to the nearest
degree, given one of the trigonometric ratios of the angle
Working Mathematically
 label sides of right-angled
triangles in different
orientations in relation to a
given angle
(Applying Strategies,
Communicating)
 explain why the ratio of
matching sides in similar
right-angle triangles is
constant for equal angles
(Communicating,
Reasoning)
 solve problems in
practical situations
involving right-angled
triangles e.g. finding the
pitch of a roof
(Applying Strategies)
 selecting and using appropriate trigonometric ratios in
right-angled triangles to find unknown sides, including the
hypotenuse
 interpret diagrams in
questions involving angles
of elevation and
depression
(Communicating)
 selecting and using appropriate trigonometric ratios in
right-angled triangles to find unknown angles correct to the
nearest degree
 relate the tangent ratio to
gradient of a line
(Reflecting)
Trigonometry of Right-Angled Triangles
 identifying angles of elevation and depression
 solving problems involving angles of elevation and
depression when given a diagram
22
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
March 2004 web update
Substrand
Outcome
Applies trig to
solve problems
including those
involving
bearings
MS5.2.3
Key Ideas
Knowledge and Skills
Further Trigonometry of Right -Angled Triangles
 using three-figure bearings and compass bearings
 drawing diagrams and using them to solve word
problems which involve bearings or angles of elevation
and depression.
Working Mathematically
 Solve simple problems
involving three-figure
bearings (Applying
Strategies, Communication)
 Interpret directions given as
bearings (Communication)
23
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 13 Simultaneous
equations
13.1 Equations with two unknowns
13.2 The graphical method
13.3 The substitution method
Try this: Find the values
13.4 The elimination method
Try this: A Pythagorean problem
13.5 Solving problems using
simultaneous equations
Focus on working mathematically:
Exploring for water, oil and gas—
the density of air-filled porous rock
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Algebraic
Techniques
PAS5.2.2
Key Ideas
Solve
simultaneous
equations
Knowledge and Skills
Working Mathematically
 graph simultaneous equations using nonalgebraic methods, such as ‘guess and
check’, setting up tables of values and
looking for patterns
 using graphics calculators and
spreadsheet software to plot pairs of lines
and read off the point of intersection
(Applying Strategies)
 solving linear simultaneous equations by
finding the point of intersection of their
graphs
 solve linear simultaneous equations
resulting from problems and interpret the
results
 solving simple linear simultaneous
equations using an analytical method e.g.
solve the following
3a + b = 17
2a – b = 8
 generating simultaneous equations from
simple word problems
24
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 14 Co-ordinate geometry
14.1 The distance between two points
14.2 The midpoint of an interval
14.3 The gradient formula
Try this: A line with no integer coordinates
14.4 General form of the equation of a
line
14.5 The equation of a line given the
gradient and a point
14.6 The equation of a line given two
points
Try this: Car hire
14.7 Parallel lines
Try this: Temperature rising
14.8 Perpendicular lines
14.9 Regions in the number plane
14.10 Co-ordinate geometry problems
Focus on working mathematically:
Finding the gradient of a ski run
Language link with Macquarie
Chapter review
Substrand
Coordinate
Geometry
March 2004 web update
Outcome
PAS5.3.3
Key Ideas
Use and apply
various standard
forms of the
equation of a
straight line, and
graph regions on
the number plane
Knowledge and Skills
 describing the equation of a line as
the relationship between the x - and
y -coordinates of any point on the
line
 finding the equation of a line passing
through a point x1, y1  ,with a given
gradient m, using:
y  y1  m( x  x1 )
y  mx  b
 finding the equation of a line passing
through two points
 recognising and finding the equation
of a line in the general form:
ax  by  c  0
 rearranging equations from the
general form to the
gradient/intercept form and hence
graphing the line
 rearranging equations in the
gradient-intercept form to the
general form
 sketching the graph of a line by
finding the x -and y -intercepts from
its equation
 demonstrating that two lines are
perpendicular if the product of their
gradients is –1
 finding the equation of a line that is
parallel or perpendicular to a given
line
Working Mathematically
 recognise from a list of equations those
that result in straight line graphs
(Communicating)
 describe the conditions for a line to have
a negative gradient (Reasoning,
Communicating)
 describe conditions for lines to be parallel
or perpendicular (Reasoning,
Communicating)
 show that if two lines are perpendicular
then the product of their gradients is –1
(Applying Strategies, Reasoning,
Communicating)

discuss the equations of graphs that
can be mapped onto each other by a
translation or by reflection in the y -axis
e.g. consider the graphs
y  2x, y  2x, y  2x  1
and describe the transformation that would
map one graph onto the other
(Communicating)
 prove that a particular triangle drawn on
the number plane is right-angled
(Applying Strategies, Reasoning)
 use a graphics calculator and
spreadsheet software to graph, compare
and describe a range of linear
relationships (Applying Strategies,
Communicating)
 apply ethical considerations when using
hardware and software (Reflecting)
25
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
March 2004 web update
 find areas of shapes enclosed within a
set of lines on the number plane e.g. find
the area of the triangle enclosed by the
lines
y = 0, y = 2x, x + y = 6
(Applying Strategies)
 describe a region from a graph by
identifying the boundary lines and
determining the appropriate inequalities
for describing the enclosed region
(Applying Strategies, Communicating)
26
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9 Extension: Working Mathematically
Text Reference
Chapter 14 Co-ordinate geometry
14.1 The distance between two points
14.2 The midpoint of an interval
14.3 The gradient formula
Try this: A line with no integer coordinates
14.4 General form of the equation of a
line
14.5 The equation of a line given the
gradient and a point
14.6 The equation of a line given two
points
Try this: Car hire
14.7 Parallel lines
Try this: Temperature rising
14.8 Perpendicular lines
14.9 Regions in the number plane
14.10 Co-ordinate geometry problems
Focus on working mathematically:
Finding the gradient of a ski run
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Key Ideas
Coordinate
Geometry
PAS.5.3.4
Solve co-ordinate
geometry problem
Knowledge and Skills
Coordinate Geometry Problems
 solving a variety of problems by
applying coordinate geometry
formulae and reasoning
Working Mathematically
 derive the formula for the distance
between two points (Applying Strategies,
Reasoning)
 show that two intervals with equal
gradients and a common point form a
straight line and use this to show that
three points are collinear
(Applying Strategies, Reasoning)
 use coordinate geometry to investigate
and describe the properties of triangles
and quadrilaterals
(Applying Strategies, Reasoning,
Communicating)
 use coordinate geometry to investigate
the intersection of the perpendicular
bisectors of the sides of acute-angled
triangles
(Applying Strategies, Reasoning,
Communicating)
 show that four specified points form the
vertices of particular quadrilaterals
(Applying Strategies, Reasoning,
Communicating)
27
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.