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Mathscape 9: Working Mathematically
March 2004 web update
Mathscape 9 Syllabus Correlation Grid (Stage 5.1/5.2)
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
Outcome
Rational
Numbers
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 non-calculator
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: Working Mathematically
Text Reference
Chapter 2 Algebra
2.1 Describing simple patterns
2.2 Finite differences
Try this: Flags
2.3 Substitution
2.4 Adding and subtracting algebraic
expressions
2.5 Multiplying and dividing algebraic
expressions
Try this: Overhanging the overhang
2.6 The order of operations
2.7 The distributive law
2.8 The highest common factor
Try this: Proof
2.9 Adding and subtracting algebraic
fractions
2.10 Multiplying and dividing
algebraic
fractions
2.11 Generalised arithmetic
Try this: Railway tickets
Focus of working mathematically:
Party magic
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Algebraic
Techniques
PAS5.2.1
Key Ideas
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
2ab 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(3 x  2)  ( x  1)
 3 x 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)
Mathscape 9 and Mathscape 9
Extension School CD ROM
(Reasoning, Communicating)
2
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 3 Consumer arithmetic
3.1 Salaries and wages
Try this: Sue’s boutique
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
March 2004 web update
Substrand
Outcome
Consumer
Arithmetic
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
 calculating earnings for various
time periods from different
sources, including:
– 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
Working Mathematically
 read and interpret pay slips from
part-time jobs when questioning the
details of their own employment
(Questioning, Communicating)
 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 net earnings
considering deductions such as
taxation and superannuation
Mathscape 9 and Mathscape 9
Extension School CD ROM
 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)
3
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 4 Equations, inequations
and formulae
4.1 Inverse operations
4.2 One- and two-step equations
4.3 Equations with pronumerals on
both sides
4.4 Equations with grouping symbols
4.5 Equations with one fraction
4.6 Equations with more than one
fraction
4.7 Inequations
4.8 Solving worded problems
Try this: A prince and a king
4.9 Evaluating the subject of a
formula
Try this: Arm strength
4.10 Equations arising from
substitution
Try this: Floodlighting by formula
Focus on working mathematically:
Bushfires
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)
4
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: 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
x2  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)
5
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 5 Measurement
5.1 Length, mass and capacity
Try this: Bag of potatoes
5.2 Accuracy and precision
5.3 Time
Try this: Overseas call
5.4 Pythagoras’ theorem
Try this: Pythagorean proof by
Perigal
5.5 Solving problems with Pythagoras’
theorem
Try this: The box and the wall
5.6 Perimeter
5.7 Circumference
Try this: Command module
5.8 Converting units of area
5.9 Calculating area
Try this: The area of a circle
5.10 Area of a circle
5.11 Composite areas
Try this: Area
5.12 Problem involving area
Focus on working mathematically:
The Melbourne Cup
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Algebraic
Techniques
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
6
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: 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)
7
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
March 2004 web update
Text Reference
Substrand
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
Try this: Earthquakes
Focus on working mathematically:
World health
Language link with Macquarie
Chapter review
Data
Representati
on and
Analysis
Outcome
DS5.1.1
Key Ideas
Construct frequency tables
for grouped data
Find mean and modal class
for grouped data
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
Determine cumulative
frequency
 grouping data into class intervals
Find median using a
cumulative frequency table
or polygon
 constructing a histogram for grouped
data
 constructing a frequency table 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)
8
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: 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
March 2004 web update
Substrand
Probability
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
where n is the total number of
outcomes in the sample space
 using the formula to calculate
probabilities for simple events
 simulating probability experiments
using random number generators
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)
 recognise that probability estimates
become more stable as the number
of trials increases (Reasoning)
 recognise randomness in chance
situations (Communicating)
 apply the formula for calculating
probabilities to problems related to
card, dice and other games
(Applying Strategies)
9
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 8 Indices
8.1 Index notation
Try this: Power pulse graphs
8.2 Simplifying numerical
expressions using the index laws
8.3 The index law for
multiplication
8.4 The index law for division
8.5 The index law for further
powers
8.6 Miscellaneous questions on
the index laws
8.7 The zero index
Try this: Smallest to largest
8.8 The negative index
8.9 Products and quotients with
negative indices
Try this: Digit patterns
8.10 The fraction index
8.11 Scientific notation
8.12 Scientific notation on the
calculator
Focus on working mathematically:
Mathematics is the heart of
science
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
March 2004 web update
Substrand
Rational
Numbers
Outcome
NS5.1.1
Key Ideas
Knowledge and Skills
Working Mathematically
Define and use zero index
and negative integral indices
 describing numbers written in index
form using terms such as base,
power, index, exponent
 solve numerical problems involving
indices
(Applying Strategies)
Develop the index laws
arithmetically
 evaluating numbers expressed as
powers of positive whole numbers
Use index notation for
square and cube roots
 establishing the meaning of the zero
index and negative indices e.g. by
patterns
 explain the incorrect use of index
laws
e.g. why
32  34  96 (Communicating,
Reasoning)
Express numbers in
scientific notation (positive
and negative powers of 10)
32
31
30
3 1
9
3
1
1
3
 verify the index laws by using a
calculator
e.g. to compare the values of
32
1
9

1
32
 writing reciprocals of powers using
negative indices
1
1
e.g. 34  4 
81
3
 translating numbers to index form
(integral indices) and vice versa
 5
2
2
 1
,  5 2  and 5 (Reasoning)
 
 communicate and interpret technical
information using scientific notation
(Communicating)
 developing index laws arithmetically
by expressing each term in
expanded form e.g.
 explain the difference between
numerical expressions such as
2104 and 2 4 , particularly with
reference to calculator displays
(Communicating, Reasoning)
32  34  (3  3)  (3  3  3  3)  324  36
3 3 3 3 3
35  32 
 352  33
3 3
 solve problems involving scientific
notation
(Applying Strategies)
3   3  3 3  3 3  3 3  3  3
2 4
24
 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
10
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 8 Indices
8.1 Index notation
Try this: Power pulse graphs
8.2 Simplifying numerical expressions
using the index laws
8.3 The index law for multiplication
8.4 The index law for division
8.5 The index law for further powers
8.6 Miscellaneous questions on the
index laws
8.7 The zero index
Try this: Smallest to largest
8.8 The negative index
8.9 Products and quotients with
negative indices
Try this: Digit patterns
8.10 The fraction index
8.11 Scientific notation
8.12 Scientific notation on the
calculator
Focus on working mathematically:
Mathematics is the heart of science
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Rational
Numbers
Outcome
NS5.1.1
Key Ideas
Knowledge and Skills
Working Mathematically
 writing square roots and cube roots in
index form
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
11
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 8 Indices
8.1 Index notation
Try this: Power pulse graphs
8.2 Simplifying numerical
expressions using the index
laws
8.3 The index law for
multiplication
8.4 The index law for division
8.5 The index law for further
powers
8.6 Miscellaneous questions
on the index laws
8.7 The zero index
Try this: Smallest to largest
8.8 The negative index
8.9 Products and quotients
with negative indices
Try this: Digit patterns
8.10 The fraction index
8.11 Scientific notation
8.12 Scientific notation on the
calculator
Focus on working
mathematically:
Mathematics is the heart of
science
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Outcome
Algebraic
Techniques
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
 verify the index laws using a
calculator
e.g. use a calculator to
compare the values of (34 ) 2
e.g.
2 2  2 3  2 2 3  2 5
a m  a n  a mn
25  2 2  252  23
a m  a n  a mn
2   2
(a m ) n  a mn
2 3
6
 establishing that a0  1 using the index laws
a3  a3  a33  a0
e.g.
and
a3  a3  1

a0  1
 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
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
12
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
March 2004 web update
Text Reference
Substrand
Outcome
Chapter 8 Indices
8.1 Index notation
Try this: Power pulse graphs
8.2 Simplifying numerical
expressions using the index
laws
8.3 The index law for
multiplication
8.4 The index law for division
8.5 The index law for further
powers
8.6 Miscellaneous questions on
the index laws
8.7 The zero index
Try this: Smallest to largest
8.8 The negative index
8.9 Products and quotients with
negative indices
Try this: Digit patterns
8.10 The fraction index
8.11 Scientific notation
8.12 Scientific notation on the
calculator
Focus on working
mathematically:
Mathematics is the heart of
science
Language link with Macquarie
Chapter review
Algebraic
Techniques
PAS5.2.1
Key Ideas
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
 a  a
2
2
 a 12   a
 
 
and
Working Mathematically
 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
a  a2
then
 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
1
3 x 2 5 x 2
1
5x 0  1
9 x 5  3x 5  3x
a5  a7  a 2
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
Mathscape 9 and Mathscape
9 Extension School CD ROM
13
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
March 2004 web update
Text Reference
Substrand
Chapter 9 Geometry
9.1 Angles
9.2 Parallel lines
9.3 Triangles
Try this: The badge of the
Pythagoreans
9.4 Angle sum of a quadrilateral
9.5 Special quadrilaterals
Try this: Five shapes
9.6 Polygons
Try this: How many diagonals in a
polygon?
Focus on working mathematically:
A surprising finding
Language link with Macquarie
Chapter review
Properties of
Geometrical
Figures
Outcome
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
 find the size of the interior and
exterior angles of regular polygons
with 5,6,7,8, … sides
(Applying Strategies)
 establishing that the sum of the
exterior angles of any convex
polygon is 360
 applying angle sum results to find
unknown angles
 solve problems using angle sum of
polygon results (Applying Strategies)
Mathscape 9 and Mathscape 9
Extension School CD ROM
14
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
March 2004 web update
Text Reference
Substrand
Outcome
Chapter 10 The linear function
10.1 The number plane
10.2 Graphing straight lines
Try this: Size 8
10.3 Horizontal and vertical lines
10.4 Gradient of a line
Try this: Hanging around
10.5 The linear equation
Try this: Latitude and temperature
Mathscape 9 prelims page iv
Tuesday
10.6 The intersection of straight
lines
Focus on working mathematically:
Paper sizes in the printing industry
Language link with Macquarie
Chapter review
Coordinate
Geometry
PAS5.1.2
Key Ideas
Graph linear and simple nonlinear 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
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)
 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
 finding the gradient of a straight line from
the graph by drawing a right-angled
triangle after joining two points on the line
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: Working Mathematically
Text Reference
Chapter 10 The linear function
10.1 The number plane
10.2 Graphing straight lines
Try this: Size 8
10.3 Horizontal and vertical lines
10.4 Gradient of a line
Try this: Hanging around
10.5 The linear equation
Try this: Latitude and temperature
Mathscape 9 prelims page iv
Tuesday
10.6 The intersection of straight
lines
Focus on working mathematically:
Paper sizes in the printing
industry
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9
Extension School CD ROM
March 2004 web update
Substrand
Outcome
PAS5.1.2
Key Ideas
Graph linear and simple
non-linear relationships
from equations
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
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)
 determining whether a point lies on a
line by substituting into the equation of
the line
16
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
March 2004 web update
Text Reference
Substrand
Chapter 11 Trigonometry
11.1 Side ratios in right-angled
triangles
11.2 The trigonometric ratios
Try this: Height to base ratio
11.3 Trigonometric ratios using a
calculator
11.4 Finding the length of a side
11.5 Problems involving finding
sides
Try this: Make a hypsometer
11.6 Finding the size of an angle
11.7 Problems involving finding
angles
11.8 Angles of elevation and
depression
Try this: Pilot instructions
Focus on working mathematically:
Finding your latitude from the sun
Language link with Macquarie
Chapter review
Applies
trigonometry
to solve
problems
(diagrams
given)
including
those
involving
angles of
elevation and
depression
Mathscape 9 and Mathscape 9
Extension School CD ROM
Outcome
MS5.1.2
Key Ideas
Use trigonometry to
find sides and angles
in right-angled 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 right-angled 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
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)
 interpret diagrams in questions
involving angles of elevation and
depression (Communicating)
 relate the tangent ratio to gradient of
a line (Reflecting)
 using a calculator to find an angle correct
to the nearest degree, given one of the
trigonometric ratios of the angle
Trigonometry of Right-Angled Triangles
 selecting and using appropriate
trigonometric ratios in right-angled
triangles to find unknown sides, including
the hypotenuse
 selecting and using appropriate
trigonometric ratios in right-angled
triangles to find unknown angles correct
to the nearest degree
 identifying angles of elevation and
depression
 solving problems involving angles of
elevation and depression when given a
diagram
17
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 12 Co-ordinate geometry
12.1 The distance between two
points
12.2 The distance formula
12.3 The midpoint of an interval
12.4 The gradient formula
12.5 General form of the equation
of a line
Try this: Car hire
12.6 Parallel lines
Try this: Temperature rising
Focus on working mathematically:
Finding the gradient of a ski run
Language link with Macquarie
Chapter review
March 2004 web update
Substrand
Coordinate
Geometry
Outcome
PAS5.1.2
Key Ideas
Use a diagram to determine
midpoint, length and gradient
of an interval joining two
points on the number plane
Knowledge and Skills
Midpoint, Length and Gradient
 determining the midpoint of an
interval from a diagram
 graphing two points to form an
interval on the number plane and
forming a right-angled triangle by
drawing a vertical side from the
higher point and a horizontal side
from the lower point
 using the right-angled triangle drawn
between two points on the number
plane and Pythagoras’ theorem to
determine the length of the interval
joining the two points
Working Mathematically
 describe the meaning of the
midpoint of an interval and how
it can be found
(Communicating)
 describe how the length of an
interval joining two points can
be calculated using
Pythagoras’ theorem
(Communicating, Reasoning)
 relate the concept of gradient
to the tangent ratio in
trigonometry for lines with
positive gradients (Reflecting)
Mathscape 9 and Mathscape 9
Extension School CD ROM
18
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
Text Reference
Chapter 12 Co-ordinate geometry
12.1 The distance between two points
12.2 The distance formula
12.3 The midpoint of an interval
12.4 The gradient formula
12.5 General form of the equation of a line
Try this: Car hire
12.6 Parallel lines
Try this: Temperature rising
Focus on working mathematically:
Finding the gradient of a ski run
Language link with Macquarie
Chapter review
Mathscape 9 and Mathscape 9 Extension
School CD ROM
March 2004 web update
Substrand
Outcome
Coordinate
Geometry
PAS5.2.3
Key Ideas
Use midpoint,
distance and gradient
formulae
Knowledge and Skills
Midpoint, Distance and Gradient
Formulae
 using the average concept to
establish the formula for the midpoint,
M, of the interval joining two points
x1, y1  and x2 , y2  on the number
plane
 x  x y  y2 
M ( x, y )   1 2 , 1

2 
 2
 using the formula to find the midpoint
of the interval joining two points on
the number plane
 using Pythagoras’ theorem to
establish the formula for the distance,
d, between two points x1, y1  and
x2 , y2  on the number plane
Working Mathematically
 explain the meaning of each of
the pronumerals in the
formulae for midpoint, distance
and gradient (Communicating)
 use the appropriate formulae to
solve problems on the number
plane (Applying Strategies)
 use gradient and distance
formulae to determine the type
of triangle three points will form
or the type of quadrilateral four
points will form and justify the
answer (Applying Strategies,
Reasoning)
 explain why the following
formulae give the same
solutions as those in the lefthand column
d  ( x2  x1 ) 2  ( y2  y1 ) 2
d  ( x1  x2 ) 2  ( y1  y2 ) 2
 using the formula to find the distance
between two points on the number
plane
and
 using the relationship
m
rise
run
to establish the formula for the
gradient, m, of an interval joining two
points x1, y1  and x2 , y2  on the
gradient 
y1  y2
x1  x2
(Reasoning, Communicating)
number plane
m
y2  y1
x2  x1
 using the formula to find the gradient
of an interval joining two points on the
number plane
19
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.
Mathscape 9: Working Mathematically
March 2004 web update
Gradient/Intercept Form
 rearranging an equation in general
form
(ax + by + c = 0) to the
gradient/intercept form
 determining that two lines are parallel
if their gradients are equal
20
Copyright © Clive Meyers, Lloyd Dawe & Graham Barnsley 2004.
Published by Macmillan Education Australia.