Mathscape 9 Teaching Program
Page 1
Stage 5
MATHSCAPE 9
Term Chapter
1
2
4
Time
1. Rational numbers
2 weeks / 8 hrs
2. Algebra
2 weeks / 8 hrs
3. Consumer arithmetic
2 weeks / 8 hrs
4. Equations, inequations and formulae
2 weeks / 8 hrs
5. Measurement
3 weeks / 12 hrs
6. Data representation and analysis
2 weeks / 8 hrs
7. Probability
1 week / 4 hrs
8. Indices
3 weeks / 12 hrs
9. Geometry
2 weeks / 8 hrs
10. The linear function
2 weeks / 8 hrs
11. Trigonometry
3 weeks / 12 hrs
12. Co-ordinate geometry
2 weeks / 8 hrs
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 2
Chapter 1. Rational numbers
Text references
CD reference
Substrand
Mathscape 9
Chapter 1. Rational Numbers
(pages 1–25)
Significant figures
Recurring decimals
Rates
Rational numbers
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Round numbers to a specified number of significant figures.
Express recurring decimals as fractions.
Convert rates from one set of units to another.
NS5.2.1 (page 67): Rounds decimals to a specified number of significant
figures, expresses recurring decimals in fraction form and converts rates from
one set of units to another.
Working mathematically
Students learn to

recognise that calculators show approximations to recurring decimals e.g.






justify that 0.9  1 (Reasoning)
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)
2
3
displayed as 0.666667 (Communicating)
.
Knowledge and skills
Teaching, learning and assessment
Students learn about





identifying significant figures
rounding numbers to a specified number of significant figures
using the language of estimation appropriately, including:
 rounding
 approximate
 level of accuracy
using symbols for approximation e.g. 

TRY THIS
Fermi Problem (page 10): Estimation problem solving
Desert Walk (page 15): Problem solving
Passing Trains (page 20): Travel graph problem
FOCUS ON WORKING MATHEMATICALLY
Art, Magic Squares and Mathematics (page 20): If you would like to learn
how to make a magic square start with John Webb's article in the June
2000 journal of nrich, the mathematics enrichment page of the Millenium
Mathematics Project based at the University of Cambridge
http://nrich.maths.org/mathsf/journalf/jun00/art2/ There are many sites
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program


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
.

Page 3
. .
.
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


which will provide instructions but this is a good one to begin. From
January 2004 the nrich web home page can be found at
http://nrich.maths.org/public/viewer.php?obj_id=1376 and the home
page of the project at http://mmp.maths.org/index.html
The web page http://www-history.mcs.stand.ac.uk/history/Mathematicians/Durer.html will get you straight to
Albrecht Durer. You can scroll through the text to get a look at his
engraving Melancholia which is highlighted in blue. From here you can go
to the main index and look for "magic squares" under topics, or check
out Leonhard Euler under mathematicians.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 23)
CHAPTER REVIEW (page 24) a collection of problems to revise the
chapter.
Technology
Significant Figures: this spreadsheet in designed to round off a given number to a desired number of significant figures. To be used with the text.
Recurring Decimals: this spreadsheet converts recurring decimals to fractions.
Rates: this spreadsheet deals with rates and ratios in units.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 4
Chapter 2. Algebra
Text references
CD reference
Substrand
Mathscape 9
Chapter 2. Algebra
(pages 26–66)
Simplify (with fractional indices)
Expand
Railway tickets
Algebraic techniques
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Simplify, expand and factorise algebraic expressions including those involving
fractions or with negative and/or fractional indices.
PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions
involving fractions and negative and fractional indices.
Working mathematically
Students learn to


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)
Knowledge and skills
Teaching, learning and assessment
Students learn about



simplifying algebraic expressions involving fractions, such as
2 x 2 x 7 a 5a 2 y y 2ab 6




5
3 8 12 3 6 3
2b
expanding, by removing grouping symbols, and collecting like terms where
possible, algebraic expressions such as

TRY THIS
Flags (page 35): Algebraic problem solving
Overhanging the overhang (page 42): Practical
Railway Tickets (page 58): Complete a table and find a rule
FOCUS ON WORKING MATHEMATICALLY
Party Magic (page 59): Teachers may wish to down load the Party Magic
with Algebra worksheet in the technology folder for chapter 2 Algebra.
This worksheet explores the algebraic structure of the games using
technology.
The web link http://atschool.eduweb.co.uk/ufa10/tricks.htm at
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 5
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 2  6 x
14ab  12a 2
21xy  3x  9 x 2


Birmingham in England has great resources for students and teachers.
The web page http://www.umassmed.edu/bsrc/tricks.cfm has good links
and lots of activites to show that maths really can be fun.
For the addicted to fun and games check out Martin Gardner's books at
http://thinks.com/books/gardner.htm
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 61)
CHAPTER REVIEW (page 62) a collection of problems to revise the
chapter.
Technology
Simplify (with fractional indices): algebraic program that simplifies algebraic terms. To be used with the worksheets. Also to be used with the Focus on Working
mathematically section.
Expand: this program will expand a given algebraic expression.
Railway Tickets: worksheet to use with the “Try This” problem on page 58.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 6
Chapter 3. Consumer arithmetic
Text references
CD reference
Substrand
Mathscape 9
Chapter 3. Consumer Arithmetic
(pages 67–107)
Money
Consumer arithmetic
Duration
2 weeks / 8 hours
Key ideas
Outcomes
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.
Use compound interest formula.
Solve consumer arithmetic problems involving compound interest, depreciation and
successive discounts.
NS5.1.2 (page 70): Solves consumer arithmetic problems involving earning
and spending money.
NS5.2.2 (page 71): Solves Consumer arithmetic problems involving compound
interest, depreciation, and successive discounts.
Working mathematically
Students learn to





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)
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)
Knowledge and skills
Teaching, learning and assessment
Students learn about


calculating earnings for various time periods from different sources, including:
- wage
- salary
- commission
- piecework

TRY THIS
Sue’s Boutique (page 72): Problem Solving
Telephone Charges (page 92): Problem Solving
Progressive Discounting (page 98): Investigation
FOCUS ON WORKING MATHEMATICALLY
Sydney Market prices in 1831 (page 102): The purpose of the learning
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program





Page 7
- 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
calculating a ‘best buy’
calculating the result of successive discounts


activities is for students to think about the cost of living in Australia today
using market prices in 1831 as a starting point. As an extension students
are given opportunity to explore inflation and how the consumer price
index (CPI) is calculated. An invitation to a member of the Economics
staff to your class could be stimulating. Teachers should note that the
further apart the years being compared, the less valid it is to use the
relative prices of goods in those years to measure the standard of living.
This point is well made in the article by Nell Ingalls published on the web
site http://www.sls.lib.il.us/reference/por/features/98/money.html. This
is a useful source of information on the value of money.
A good summary of how the CPI is calculated in Australia can be found at
http://www.aph.gov.au/library/pubs/mesi/features/cpi.htm
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 105)
CHAPTER REVIEW (page 106) a collection of problems to revise the
chapter.
Technology
Money: series of worksheets to use with spreadsheets to explore Commission, Net Income, Piece Work, Salaries, Wages and a Weekly Budget.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 8
Chapter 4. Equations, inequations and formulae
Text references
CD reference
Substrand
Mathscape 9
Chapter 4: Equations, inequations and
formulae (pages 108–40)
Evaluating
Floodlighting
Algebraic techniques
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Solve linear and simple quadratic equations of the form ax  c
Solve linear inequalities
2
PAS5.2.2 (page 90): Solves linear and simple quadratic equations, solves
linear inequalities and solves simultaneous equations using graphical and
analytical methods.
Working mathematically
Students learn to







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)
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:
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 9
m
y2  y1
x2  x1
1 2
mv
2
4
V  r 3
3
SA  2r 2  2rh
E


(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)
Knowledge and skills
Teaching, learning and assessment
Students learn about

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 ax 2  c
solving equations arising from substitution into formulae



TRY THIS
A Prince and a King (page 129): Two Ancient Problems
Arm Strength (page 132): Formulae Investigation
Floodlighting by formula (page 136): Formulae Investigation
FOCUS ON WORKING MATHEMATICALLY
Bushfires (page 137): Teachers may wish to use a spreadsheet to evaluate
F given C and vice versa. Go to the Evaluating the subject.xls
spreadsheet in the technology folder for Chapter 4. There is also a useful
worksheet. Extension students could discuss whether F = 9C/5 + 32 is a
formula or an equation and what constitutes the difference -- see page 136
on Floodlighting for example. For newspaper reports of the fires try the
Sydney Morning herald web site http://www.smh.com.au/. If you type
'Sydney bushfires' into a search engine you will get a range of options.
http://www.gi.alaska.edu/ScienceForum/ASF13/1317.html will give
you a short account how the two men Daniel Fahrenheit and Anders
Celcius constructed their scales. This will be very useful link with the
study of science.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 138)
CHAPTER REVIEW (page 139) a collection of problems to revise the
chapter.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 10
Linear Inequalities
 solving inequalities such as
3x  1  9
2(a  4)  24
t4
 3
5
Technology
Evaluating: students analyse a spreadsheet and then design their own.
Floodlighting: activity to complement the “Try This” problem on page 136.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 11
Chapter 5. Measurement
Text references
CD reference
Substrand
Mathscape 9
Chapter 5:.Measurement
(pages 141–205)
Perigal
Measuring plane shapes
Circle measuring
Algebraic techniques
Duration
3 weeks / 12 hours
Key ideas
Outcomes
Develop formulae and use to find the area of rhombuses, trapeziums and kites.
Find the area and perimeter of simple composite figures consisting of two shapes
including quadrants and semicircles.
Find area and perimeter of more complex composite figures.
MS5.1.1 (page 126): Use formulae to calculate the area of quadrilaterals and
find areas and perimeters of simple composite figures.
MS5.2.1 (page 127): Find areas and perimeters of composite figures.
Working mathematically
Students learn to








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)
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)
Knowledge and skills
Teaching, learning and assessment
Students learn about


developing and using formulae to find the area of quadrilaterals:
- for a kite or rhombus, Area  12 xy where
x and y are the lengths of the
diagonals;
- for a trapezium, Area  12 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
TRY THIS
Bags of potatoes (page 147): Problem Solving
Overseas Call (page 154): Problem Solving
Pythagorean Proof by Perigal (page 160): Proof
The box and the wall (page 163): Problem Solving
Command Module (page 174): Investigation of Apollo 11
The area of a circle (page 185): Archimedes method
Area (page 195): Challenge Problem
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program



Page 12
including quadrants and semicircles
calculating the perimeter of simple composite figures consisting of two shapes
including quadrants and semicircles
calculating the area and perimeter of sectors
calculating the perimeter and area of composite figures by dissection into
triangles, special quadrilaterals, semicircles and sectors



FOCUS ON WORKING MATHEMATICALLY
The Melbourne Cup (page 198): These activities focus on units of
measurement linked to the Melbourne cup. The history of the cup provides
insight into the dramatic changes in the Australian way of life since the
race began in 1861. Students also explore the origin of the words used to
describe the units.
The web page http://www.unc.edu/~rowlett/units/ is a dictionary of
unusual units you will find fascinating.
If you type in 'Melbourne cup' into a search engine, you will have lots of
choice. Try the VRC web page
http://home.vicnet.net.au/~basiced3/cup/history.html
The web page http://www.equine-world.co.uk/about_horses/height.htm
will show you nice diagram on the way the heights of horses are
measured.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 199)
CHAPTER REVIEW (page 201) a collection of problems to revise the
chapter.
Technology
Perigal: Cabri Geometry interactive worksheet on the Pythagorean proof by Perigal.
Measuring Plane Shapes: this file contains hyperlinks to a number of interactive geometric diagrams.
Circle Measuring: a set of Cabri Geometry interactive worksheets that are used for students to explore the parts and use of circles.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 13
Chapter 6. Data representation and analysis
Text references
CD reference
Substrand
Mathscape 9
Chapter 6. Data Representation and
Analysis (pages 206–53)
Data analysis
Cumulative analysis
Data representation and analysis
Duration
2 weeks / 8 hours
Key ideas
Outcomes
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
DS5.1.1 (page 116): Groups data to aid analysis and constructs frequency and
cumulative frequency tables and graphs.
Working mathematically
Students learn to




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)
Knowledge and skills
Teaching, learning and assessment
Students learn about









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

TRY THIS
The English Language (page 232): Investigation
Earthquakes (page 246): Can we predict the number of Earthquakes there
will be in a year?
FOCUS ON WORKING MATHEMATICALLY
World Health (page 246): This investigation provides an opportunity for
students to analyse two indicators of world public health and to apply their
skills in Working mathematically. The objective is to show how statistical
evidence can play a role in arguing a case for the development of
programs to support global health. There is an excellent opportunity for
class discussion about the sort of data governments need in order to make
sensible policy decisions for global health.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 14


A good international web site is
http://www.globalhealth.gov/worldhealthstatistics.shtml
The frequently asked questions page at
http://www.globalhealth.gov/faq.shtml provides useful background
information for teachers
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 249)
CHAPTER REVIEW (page 250) a collection of problems to revise the
chapter.
Technology
Data Analysis: students Analyse data with the help of a spreadsheet.
Cumulative Analysis: students use the spreadsheet to calculate the median using the cumulative frequency
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 15
Chapter 7. Probability
Text reference
CD reference
Substrand
Mathscape 9
Chapter 8. Probability (pages 254–80)
Probability
Craps simulation
Weighted dice
Probability
Duration
1 week / 4 hours
Key ideas
Outcomes
Determine relative frequencies to estimate probabilities.
Determine theoretical probabilities.
NS5.1.3 (page 75): Determines relative frequencies and theoretical
probabilities.
Working mathematically
Students learn to






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)
Knowledge and skills
Teaching, learning and assessment
Students learn about





repeating an experiment a number of times to determine the relative frequency of
an event
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
number of favourable outcomes
P( A) =
n
where n is the total number of outcomes in the sample space
using the formula to calculate probabilities for simple events

TRY THIS
Two-Up (page 265): Experiment
The game of Craps (page 270): Simulation
Winning Chances (page 274): Problem Solving
FOCUS ON WORKING MATHEMATICALLY
Getting through traffic lights (page 275): This activity is designed to
introduce students to the idea of a simulation. It is designed for all students
to enjoy. Teachers should carry out the simulation first using the
technology they wish to use in class.
The Maths Online web site at
http://www.mathsonline.co.uk/nonmembers/resource/prob/ is a great
help to teachers looking for lesson plans to simulate real life probability
problems. Includes on line flash movies which will draw graphs directly
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program

Page 16
simulating probability experiments using random number generators


from your input.
For a good reference text with a CD ROM to simulate probability
problems using a graphics calculator try Winter MJ and Carlson RJ (2001)
Probability Simulations, Key Curriculum Press, Emeryville, California.
Barry Kissane's web page
http://wwwstaff.murdoch.edu.au/%7Ekissane/graphicscalcs.htm is
invaluable for CASIO users.
The Open University Centre for Mathematics Education at
http://mcs.open.ac.uk/cme/TIcourses/timain.html has some excellent
courses for teachers with little experience in using a TI graphics calculator.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 276)
CHAPTER REVIEW (page 278) a collection of problems to revise the
chapter.
Technology
Probability: the spreadsheet simulates the drawing of different coloured balls from a bag with replacement.
Craps Simulation: this spreadsheet explores the probabilities of winning and losing a game of craps.
Weighted Dice: dice simulation spreadsheet.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 17
Chapter 8. Indices
Text references
CD reference
Substrands
Mathscape 9
Chapter 8. Indices (pages 281–312)
Simplify (with fractional indices)
Expand
Rational numbers
Algebraic techniques
Duration
3 weeks / 12 hours
Key ideas
Outcomes
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)
Apply the index laws to simplify algebraic expressions (positive integral indices
only).
Simplify, expand and factorise algebraic expressions including those involving
fractions or with negative and/or fractional indices.
NS5.1.1 (page 65): Applies index laws to simplify and evaluate arithmetic
expressions and uses scientific notation to write large and small numbers.
PAS5.1.1 (page 87): Applies the index laws to simplify algebraic expressions.
PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions
involving fractions and negative and fractional indices.
Working mathematically
Students learn to

solve numerical problems involving indices (Applying Strategies)

explain the incorrect use of index laws e.g. why 32  34  96 (Communicating, Reasoning)

verify the index laws by using a calculator e.g. to compare the values of
 5
2

1
,  5 2  and 5 (Reasoning)
 
communicate and interpret technical information using scientific notation (Communicating)


explain the difference between numerical expressions such as 2  10 4 and 2 4 , particularly with reference to calculator displays (Communicating, Reasoning)
solve problems involving scientific notation (Applying Strategies)

verify the index laws using a calculator e.g. use a calculator to compare the values of (34 )2 and 38 (Reasoning)


explain why x 0  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)
2
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program



Page 18
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)
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?
5x0  1
9 x5  3x5  3x
a5  a7  a 2
2c  4 
1
2c 4
(Applying Strategies, Reasoning, Communicating)

explain the difference between particular pairs of algebraic expressions, such as x 2 and  2 x (Reasoning, Communicating)
Knowledge and skills
Teaching, learning and assessment
Students learn about




describing numbers written in index form using terms such as base, power, index,
exponent
evaluating numbers expressed as powers of positive whole numbers
establishing the meaning of the zero index and negative indices e.g. by patterns
32
9



31
3
30
1
3 1
1
3
3 2
1
1
9 32
1
1

4
81
3
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
32 4 3  3  3  3  3  3  3  3  324  38
writing reciprocals of powers using negative indices e.g. 3 4 
 

TRY THIS
Power Pulse Graphs (page 283): Investigation
Smallest to Largest (page 295): Problem Solving
Digit Patterns (page 300): Investigation
FOCUS ON WORKING MATHEMATICALLY
Mathematics is at the heart of Science (page 308): The Powers of 10 web
site http://www.powersof10.com/ should be explored before starting this
Working mathematically activity. There are excellent pictures and ideas
for creating absorbing lessons. The learning activities are suitable for
students working in pairs. Calculators are recommended. In particular try
the patterns section at
http://www.powersof10.com/powers/patterns/patterns.html
The ABC web site http://www.abc.net.au/science has a wealth of ideas to
enable students to see how mathematics lies at the heart of science. The Dr
Karl page has a live Q & A opportunity. The class could formulate a
question, send it in and listen to the answer on radio or online. There is
also a news page which provides great ideas for lesson starters. Teachers
are encouraged to liaise with science staff for further information and to
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program


Page 19
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
1





writing square roots and cube roots in index form 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
using the index laws previously established for numbers to develop the index laws
in algebraic form

e.g.
 

22  23  223  25
am  an  amn
25  22  25 2  23
am  an  amn
2   2
2 3



invite them to the lesson.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 309)
CHAPTER REVIEW (page 311) a collection of problems to revise the
chapter.
6
(a m ) n  a mn
establishing that a 0  1 using the index laws e.g. a 3  a 3  a 3 3  a 0 and
a3  a3  1  a 0  1
simplifying algebraic expressions that include index notation
e.g.
5x0  3  8
2 x 2  3x3  6 x5
12 a 6  3a 2  4a 4
2m3 (m 2  3)  2m5  6m3

applying the index laws to simplify expressions involving pronumerals
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
 a
2
Page 20

establishing that

using index laws to assist with the definition of the fractional index for square root
 a  a  a  a  a2  a
given
 a
2
a
2
and  a   a
 
1
2
1
then



a  a2
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
1
1
1
a 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  3x3
1
1
3 x 2 5 x 2
1
1
3
6 y 4 y 3
Technology
Simplify (with fractional indices): algebraic program that simplifies algebraic terms. To be used with the worksheet. Also to be used with the Focus on Working
mathematically section. Worksheet included.
Expand: this program will expand a given algebraic expression.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 21
Chapter 9. Geometry
Text references
CD reference
Substrand
Mathscape 9
Chapter 9. Geometry (pages 313–63)
N polygon
Exterior angle
Euler line
Properties of geometric figures
Duration
2 weeks / 8 hours
Key idea
Outcomes
Establish sum of exterior angles result and sum of interior angles result for polygons.
SGS5.2.1 (page 157): Develops and applies results related to the angle sum of
interior and exterior angles for any convex polygon.
Working mathematically
Students learn to



express in algebraic terms the interior angle sum of a polygon with n sides e.g. (n–2)  180 (Communicating)
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)
Knowledge and skills
Students learn about




Teaching, learning and assessment

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
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



TRY THIS
The badge of the Pythagoreans (page 337): Historical Problem
Five Shapes (page 348): Problem Solving
How many diagonals in a polygon? (page 353): Investigation
FOCUS ON WORKING MATHEMATICALLY
A surprising finding (page 354): In this activity we arrive at Pythagoras'
theorem from a cutting and pasting activity with hexagons of equal area. It
is designed as a fun activity for all students. However there is a deeper
mathematical idea which is really for teachers but could be used as an
extension. The 4 hexagons drawn on pages 355–356 each tessellate the
plane. See http://www.cut-the-knot.org/pythagoras/index.shtml for
details -- read proof 16 and 38 of Pythagoras' theorem.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 356)
CHAPTER REVIEW (page 357): A collection of problems to revise the
chapter.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 22
Technology
N Polygon: this geometry program draws regular polygons at speed and displays their diagonals. Explores a curious geometrical pattern that would be time
consuming if drawn by hand.
Exterior Angle: this learning activity makes use of the exterior angle property of a triangle. Students have the opportunity to apply the reasoning to solve a problem
in geometry.
Euler Line: the Euler line of a triangle is a line that passes through three special points of a triangle. Investigative exercise.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 23
Chapter 10. The linear function
Text references
CD reference
Substrand
Mathscape 9
Chapter 10. The Linear Function
(pages 364–99)
Line equation
Intersecting lines
Co-ordinate geometry
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Use a diagram to determine midpoint, length and gradient of an interval joining two
points on the number plane.
Graph linear and simple non-linear relationships from equations.
PAS5.1.2 (page 97): Determines the midpoint, length and gradient of an
interval joining two points on the number plane and graphs linear and simple
non-linear relationships from equations.
Working mathematically
Students learn to








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)
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)
Knowledge and skills
Teaching, learning and assessment
Students learn about

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

TRY THIS
Size 8 (page 374): Problem Solving
Hanging around (page 383): Problem Solving
Latitude and Temperature (page 389): Investigation
FOCUS ON WORKING MATHEMATICALLY
Paper Sizes in the printing industry (page 394): The web link
http://www.cl.cam.ac.uk/~mgk25/iso-paper.html is a good overview of
the length to breadth relationship of A4 to A3.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program


Page 24
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
Graphs of Relationships
 constructing tables of values and using coordinates to graph vertical and
horizontal lines such as
x  3, x  1


The web link http://www.twics.com/~eds/paper/papersize.html provides
you with more information about international paper sizes. Note that the B
series is about half way between two A sizes. It is intended as an
alternative to the A sizes, and used primarily for books, posters and wall
charts. Note that the ratio of length to breadth for the B series is also √2 : 1
and the ratio of their areas 2 : 1.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 395)
CHAPTER REVIEW (page 396) a collection of problems to revise the
chapter.
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
Technology
Line Equation: interactive program with accompanying worksheet.
Intersecting Lines: interactive program with accompanying worksheet.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 25
Chapter 11. Trigonometry
Text references
CD reference
Substrand
Mathscape 9
Chapter 11. Trigonometry
(pages 400–38)
Sine Cosine
SOHCAHTOA
Trigonometry
Duration
3 weeks / 12 hours
Key ideas
Outcomes
Use trigonometry to find sides and angles in right-angled triangles.
Solve problems involving angles of elevation and angles of depression from diagrams
MS5.1.2 (page 139): Applies trigonometry to solve problems (diagrams given)
including those involving angles of elevation and depression.
Working mathematically
Students learn to





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)
Knowledge and skills
Teaching, learning and assessment
Students learn about

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
 using a calculator to find an angle correct to the nearest degree, given one of the
trigonometric ratios of the angle

TRY THIS
Height to Base Ratio (page 408): Investigation
Make a Hypsometer (page 421): Practical
Pilot Instructions (page 432): Problem Solving
FOCUS ON WORKING MATHEMATICALLY: (page 433) Finding your
latitude from the sun
This is designed as a fun outdoor activity. Teachers need to prepare well in
advance because of the restricted days of the equinox. However the
activity could be carried out on a day close to the equinox if it happens to
be cloudy. An explanation of the difference can be found in Mathscape 9
Extension page 481. The geometry should be discussed carefully in class
before the outdoor lesson.
See what a sailor does to determine latitude using an astrolabe at
http://www.ruf.rice.edu/~feegi/measure.html
A great site to look at navigation in the 15th century is
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 26
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


http://www.ruf.rice.edu/~feegi/site_map.html
Read about advances in navigational technology from the Astrolabe to
today's Global Positioning System at
http://www.canadiangeographic.ca/Magazine/ND01/
findingourway.html
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING
(page 435)
CHAPTER REVIEW (page 436) a collection of problems to revise the
chapter.
Technology
Sine Cosine: explores the range of Trig graphs.
SOHCAHTOA: investigation of the tan ratio.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 27
Chapter 12. Co-ordinate geometry
Text references
CD reference
Substrand
Mathscape 9
Chapter 12. Co-ordinate Geometry (pages 439–70)
Intersecting lines
Crow flying
Co-ordinate geometry
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Use a diagram to determine midpoint, length and gradient of an
interval joining two points on the number plane.
Graph linear and simple non-linear relationships from
equations
Use midpoint, distance and gradient formulae.
Apply the gradient/intercept form to interpret and graph
straight lines.
PAS5.1.2 (page 97): Determines the midpoint, length and gradient of an interval joining two points
on the number plane and graphs linear and simple non-linear relationships from equations.
PAS5.2.3 (page 99): Uses formulae to find midpoint, distance and gradient and applies the
gradient/intercept form to interpret and graph straight lines.
Working mathematically
Students learn to







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)
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 left-hand column
d  ( x1  x2 )2  ( y1  y2 )2 and m 
y1  y2
(Reasoning, Communicating)
x1  x2
Knowledge and skills
Teaching, learning and assessment
Students learn about

Midpoint, Length and Gradient
TRY THIS
Car Hire (page 459): Problem Solving
Temperature Rising (page 462): Problem Solving
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program



Page 28
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
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



FOCUS ON WORKING MATHEMATICALLY
Finding the gradient of a ski run (page 463)
The resource book Kleeman, G. and Peters A. (2002) Skills in Australian Geography, Cambridge
University Press, Cambridge is your best guide for this activity. Try the Social Science
department for a copy or your school library.
For a good model of calculating gradient from contour maps go to
http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/map_sample_answer2.html.
However measurements are calculated in feet which are still used in the USA.
A good site written for scouts which looks at gradients, contours and features of ordinance survey
maps is http://www.scoutingresources.org.uk/mapping_contour.html
Note that the Sun moves from east to west through the northern sky in our (southern)
hemisphere. This means the sun will shine on the northern and eastern slopes during the day.
Hence the preference for these slopes. Just what we need to enjoy skiing.
EXTENSION ACTIVITIES, LET’S COMMUNICATE, REFLECTING (page 466)
CHAPTER REVIEW (page 468) a collection of problems to revise the chapter.
d  ( x2  x1 )2  ( y2  y1 )2



using the formula to find the distance between two points
on the number plane
using the relationship
rise
gradient 
run
to establish the formula for the gradient, m, of an interval
joining two points x1 , y1  and x2 , y2  on the number
plane
y  y1
m 2
x2  x1
using the formula to find the gradient of an interval joining
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Teaching Program
Page 29
two points on the number plane
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
Technology
Intersecting Lines: interactive program with accompanying worksheet.
Crow Flying: students use Pythagoras’ Theorem to investigate how much distance they would save if they could fly in a straight line (as the crow flies) across
city blocks. students create their own spreadsheet and investigate when the saving is greatest.
Midpoint: interactive geometry worksheet.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.