Mathscape 9 Extension Teaching Program
Page 1
Stage 5
MATHSCAPE 9 EXTENSION
Term Chapter
1
2
3
4
Time
1. Rational numbers
2 weeks / 8 hrs
2. Algebra
3 weeks / 12 hrs
3. Consumer arithmetic
2 weeks / 8 hrs
4. Equations, inequations and formulae
2 weeks / 8 hrs
5. Measurement
2 weeks / 8 hrs
6. Data representation and analysis
2 weeks / 8 hrs
7. Probability
1 week / 4 hrs
8. Surds
2 weeks / 8 hrs
9. Indices
2 weeks / 8 hrs
10. Geometry
2 weeks / 8 hrs
11. The linear function
2 weeks / 8 hrs
12. Trigonometry
2 weeks / 8 hrs
13. Simultaneous equations
1 week / 4 hrs
14. Co-ordinate geometry
2 weeks / 8 hrs
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program
Page 2
Chapter 1. Rational numbers
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 1. Rational Numbers
(pages 1–24)
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

TRY THIS
Fermi Problem (page 10): Estimation problem solving
Desert Walk (page 15): Problem solving
Passing Trains (page 19): Travel graph problem
FOCUS ON WORKING MATHEMATICALLY
A number pattern from Galileo (page 20): This is an activity designed for
students to enjoy mathematics for its own sake. It illustrates the power of
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program
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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
.

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


inductive reasoning. Teachers are advised to be ready to discuss the
differences in inductive and deductive reasoning. It also provides an
opportunity for students to see how a famous scientist loved mathematics for
its own sake too. The web site at http://www-groups.dcs.st-and.ac.uk
/~history/Mathematicians/Galileo.html describes the major contributions
Galileo made to mathematics and science.
The NASA site http://www.jpl.nasa.gov/galileo/countdown/ has wonderful
information about the Galileo mission to Jupiter which ended in September
2003, including fly bys of the moons of Jupiter.
Other References:
Wood, L. and Perrett, G. (1997) Advanced Mathematical Discourse,
University of Technology Sydney. The chapter by Peter and Dubravka Petocz
on pattern and proof is very helpful for teachers.
Nelsen, R.B. (1993) Proof Without Words, The Mathematical Association of
America, Washington DC page 115 has a nice visual explanation for the result
explored in this activity on page 115.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 21)
CHAPTER REVIEW (page 23) 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: rates are converted in this interactive program.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program
Page 4
Chapter 2. Algebra
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 2. Algebra
(pages 25–77)
Simplify (with fractional indices)
Binomial Products
Perfect squares
Railway tickets
Algebraic techniques
Duration
3 weeks / 12 hours
Key ideas
Outcomes
Simplify, expand and factorise algebraic expressions including those involving
fractions or with negative and/or fractional indices.
Use algebraic techniques to simplify expressions, expand binomial products and
factorise quadratic expressions.
PAS5.2.1 (page 88): Simplifies, expands and factorises algebraic expressions
involving fractions and negative and fractional indices.
PAS5.3.1 (page 92): Uses algebraic techniques to simplify expressions, expand
binomial products and factorise quadratic expressions.
NS5.3.1 (page 68): Performs operations with surds and 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 xy2  8 xy(3x  2) (Reasoning, Communicating)



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develop facility with the algebraic symbol system in order to apply algebraic techniques to other strands and substrands (Applying Strategies, Communicating)
use factorising techniques to solve quadratic equations and draw graphs of parabolas (Applying Strategies, Communicating)
solve problems, such as: find a relationship that describes the number of diagonals in a polygon with n sides (Applying Strategies)
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)
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program
Page 5
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
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 2  6 x
14 ab  12 a 2
21xy  3x  9 x 2

simplifying algebraic expressions, including those involving fractions, such as


 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
expanding binomial products by finding the area of rectangles
x
e.g.
8
hence
x
x
2
8x
3
3x
24


TRY THIS
Flags (page 31): Algebraic problem solving
Overhanging the overhang (page 38): Practical
Railway Tickets (page 53): Complete a table and find a rule
Proof (page 66): Challenge
FOCUS ON WORKING MATHEMATICALLY
A number pattern from Blaise Pascal 1654 (page 71): In contrast with the
pure number pattern activity in chapter 1, this investigation provides a
challenge for students to apply their new knowledge to solve a problem. To
give them something to aim at I have given the specific answer for 19
flavours in question 1 and the general solution for n flavours in question 5.
The web site http://mathforum.org/library/drmath/view/59185.html
"Ask Dr Math" has a good explanation of the ice cream problem.
The intention is to encourage students to have a go, make a guess, think
intuitively, be Pascal like! Teachers are encouraged to use the Reflecting
exercise on page 73 as a way of deepening student understanding of the
individual preferences in which individual mathematicians like to work. A
useful book reference is Kline, Morris (1972) Mathematical Thought from
Ancient to Modern Times, Oxford University Press pages 295–7 for
general information on Pascal's love of intuition and the contributions he
made to mathematics.
Go to http://ptri1.tripod.com/ for a good overview of the number patterns
in Pascal's triangle. Note that this site is in the process of changing as the
book goes to press. You may wish to make a connection between Pascal's
triangle and the Sierpinski fractal, by colouring the odd numbers black and
leaving the even numbers white.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 72)
CHAPTER REVIEW (page 74) a collection of problems to revise the
chapter.
x  8x  3  x2  8x  3x  24
 x 2  11x  24
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program

Page 6
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
(a  b)(a  b)  a 2  b 2
(a  b) 2  a 2  2ab  b 2

factorising expressions:
- common factors
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.
Binomial Products: a worksheet using the executable “Expand”.
Perfect Squares: shows students how squaring expressions in Algebra can be represented by drawing. To be used with page 63.
Railway Tickets: worksheet to use with the “Try This” problem on page 58.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program
Page 7
Chapter 3. Consumer arithmetic
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 3. Consumer Arithmetic
(pages 78–116)
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

TRY THIS
Telephone Charges (page 101): Problem Solving
Progressive Discounting (page 107): Investigation
FOCUS ON WORKING MATHEMATICALLY (page 111) Sydney
Market prices in 1831
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program
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Page 8
- 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
calculating a ‘best buy’
calculating the result of successive discounts


The purpose of the learning activities is for students to think about the cost
of living in Australia today using market prices in 1831 as a starting point.
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, and to get help from
the web sites provided in the teacher support material.
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
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.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 112)
CHAPTER REVIEW (page 114) 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 Extension Teaching Program
Page 9
Chapter 4. Equations, inequations and formulae
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 4. Equations, inequations and
formulae (pages 117–53)
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
Solves a range of linear equations
2
PAS5.2.2 (page 90): Solves linear and simple quadratic equations, solves linear
inequalities and solves simultaneous equations using graphical and analytical
methods.
PAS5.3.2 (page 94): Solves linear, quadratic and simultaneous equations, solves
and graphs inequalities, and rearranges literal equations.
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:
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Mathscape 9 Extension Teaching Program
Page 10
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)
solve non-routine problems using algebraic techniques (Applying Strategies, Communicating)
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 (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

TRY THIS
A Prince and a King (page 1137): Two Ancient Problems
Floodlighting by formula (page 143): Formulae Investigation
FOCUS ON WORKING MATHEMATICALLY
Splitting the Atom (page 149): In this activity students learn about the
relationship between mass and energy, stated in the form of a famous
equation. Teachers are advised to work through the example first noting the
units for mass, speed and kinetic energy. It will be a good idea to complete
the Challenge questions before using them in class. The increase in mass is
small, just over a tenth of a milligram. This everyday implications of this
should be discussed.
In general the web links I have looked at are rather hard to read for students.
The link
http://www.phys.virginia.edu/classes/109N/lectures/mass_increase.html
contains good background information for teachers.
I recommend the following excellent book reference for students and
teachers wanting to read further. Guillen, M. (2000) Five Equations That
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.
Mathscape 9 Extension Teaching Program
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Page 11
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


Changed the World, Abacus Books, London. pages 215–266.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 150)
CHAPTER REVIEW (page 152) a collection of problems to revise the
chapter.
Linear Inequalities
 solving inequalities such as
3x  1  9
2(a  4)  24
t4
 3
5
Linear, Quadratic and Simultaneous Equations
 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
 solving problems involving linear equations
Technology
Evaluating: students analyse a spreadsheet and then design their own.
Floodlighting: activity to complement the “Try This” problem on page 143.
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Mathscape 9 Extension Teaching Program
Page 12
Chapter 5. Measurement
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 5. Measurement
(pages 155–210)
Perigal
Measuring plane shapes
Circle measuring
Algebraic techniques
Duration
2 weeks / 8 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

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





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;
TRY THIS
Pythagorean Proof by Perigal (page 170): Proof
Command Module (page 180): Investigation of Apollo 11
The area of a circle (page 191): Archimedes method
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Mathscape 9 Extension Teaching Program
Page 13
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
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
-







Area (page 200): Challenge Problem
FOCUS ON WORKING MATHEMATICALLY
The Solar System (page 203): For good pictures try http://www.the-solarsystem.net/
At http://www.nineplanets.org/ you can take a tour of the solar system
and find out lots about each planet. Highly recommended.
There are lots of interesting things to see at the NASA web site
http://www.nasa.gov/ including the recent exploration of the red planet
Mars. In August 2003 Mars was also at its closest point to Earth in 70 000
years. A drawing of the orbits of the two planets at their closest point will
help students grasp the periodic nature of this phenomenon.
On the web link http://www.exploratorium.edu/ronh/solar_system/ you
can build a model of the solar system on the spot. Try it out.
There is also a universe explorer called Celestia available for download
from http://www.shatters.net/celestia/. Worth a look. Very powerful.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 205)
CHAPTER REVIEW (page 206) a collection of problems to revise the
chapter.
Technology
Pythagoras Theorem: students use the worksheet and the program to discover how to use and prove Pythagoras’ theorem.
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 Extension Teaching Program
Page 14
Chapter 6. Data representation and analysis
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 6. Data representation and
analysis (pages 211–55)
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 236): Investigation
FOCUS ON WORKING MATHEMATICALLY
World Health (page 248): This investigation provides an opportunity for
students to analyse two indicators of world public health and to apply their
skills in Working mathematically. Teachers are encouraged to work
through the activities first.
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
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Mathscape 9 Extension Teaching Program
Page 15


governments need in order to make sensible policy decisions for global
health.
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
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 250)
CHAPTER REVIEW (page 252) 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
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Mathscape 9 Extension Teaching Program
Page 16
Chapter 7. Probability
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 7. Probability (pages 256–81)
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

TRY THIS
Two-Up (page 266): Experiment
The game of Craps (page 271): Simulation
Winning Chances (page 275): Problem Solving
FOCUS ON WORKING MATHEMATICALLY
A Party Game: Roll a six and eat the chocolate (page 276): This activity applies
probability concepts in a game which could actually be played in class with an
adventurous teacher! It is designed for students to enjoy.
Teachers should carry out the simulation in the Challenge questions first using the
technology they wish to use in class. A spreadsheet demonstration in a lab or the use of a
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Mathscape 9 Extension Teaching Program
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Page 17
using the formula to calculate probabilities for simple events
simulating probability experiments using random number generators


set of graphics calculators (see pages 266–8) would be appropriate.
Note that the activity is a binomial experiment. Choose questions appropriate for your
class. Some questions could be revisited at a later time in stage 5.
A simple introduction to the Chevalier's famous gambling problem can be found at http:
//www.ga.k12.pa.us/academics/us/Math/Geometry/stwk98/CASINO/History.htm
For a good example of a high school student's project on why you should not gamble go
to http://www.ga.k12.pa.us/academics/us/Math/Geometry/stwk98/CASINO
/index.htm Barry Kissane's web page http://wwwstaff.murdoch.edu.au/%7Ekissane
/graphicscalcs.htm is invaluable for CASIO users.
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 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.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 277)
CHAPTER REVIEW (page 279) 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.
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Mathscape 9 Extension Teaching Program
Page 18
Chapter 8. Surds
Text reference
CD reference
Substrand
Mathscape 9 Extension
Chapter 8. Surds (pages 282–310)
Simplify (with fractional indices)
Iterative square root
Real numbers
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Define the system of real numbers distinguishing between rational and
irrational numbers.
Perform operations with surds.
Use integers and fractions for index notation.
Convert between surd and index form
NS5.3.1 (page 68): Performs operations with surds and indices.
Working mathematically
Students learn to




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)
solve numerical problems involving surds and/or fractional indices (Applying Strategies)

explain why a particular sentence is incorrect e.g.
3  5  8 (Communicating, Reasoning)
Knowledge and skills
Teaching, learning and assessment
Students learn about


defining a rational number:
a
of two integers where b ≠ 0.
b
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
A rational number is the ratio



TRY THIS
Greater Number (page 291): Problem Solving
Imaginary Numbers (page 297): Introduction to i
Exact Values (page 304): Challenge
FOCUS ON WORKING MATHEMATICALLY
Fibonacci Numbers and the Golden Mean (page 305): This activity is the study of a
very important surd known as the golden mean. It appears in fictional writing such
as the Dan Brown's The Da Vinci Code for example, which could be used to show
that it not just mathematicians who find it interesting. The link with the work of
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Mathscape 9 Extension Teaching Program
Page 19

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
using the following results for x, y > 0:
 x
2


x
xy 
x. y
x

y
x
x2


Leonardo da Vinci is of great interest.
The web link
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html will give
you lots of information about Fibonacci Numbers and Nature.
The web link http://galaxy.cau.edu/tsmith/KW/golden.html is a good start to
explore the Golden Number .
The web link http://www-groups.dcs.st-andrews.ac.uk/~history
/Mathematicians/Fibonacci.html provides information on the life of Fibonacci
and his contribution to mathematics.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 307)
CHAPTER REVIEW (page 309) a collection of problems to revise the chapter.
y
using the four operations of addition, subtraction, multiplication and
division to simplify expressions involving surds
expanding expressions involving surds such as
 3  5  or 2  3 2  3 
2

rationalising the denominators of surds of the form
a b
c d
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.
Iterative Square Root: spreadsheet activity finds the square root by iteration.
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Mathscape 9 Extension Teaching Program
Page 20
Chapter 9. Indices
Text references
CD reference
Substrands
Mathscape 9 Extension
Chapter 9. Indices (pages 311–42)
Simplify (with fractional indices)
Rational numbers
Algebraic Techniques
Duration
2 weeks / 8 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
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Mathscape 9 Extension Teaching Program
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Page 21
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
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)
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
31
30
3 1
9
3
1
1
3

3 2
1
9
 312
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
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
 

TRY THIS
Smallest to Largest (page 323): Problem Solving
Digit Patterns (page 328): Investigation
FOCUS ON WORKING MATHEMATICALLY
Mathematics is at the heart of Science (page 338): 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 invite them to the lesson.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 339)
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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
 
CHAPTER REVIEW (page 340) a collection of problems to revise the
chapter.
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

e.g.
 

22  23  223  25
am  an  amn
25  22  25 2  23
am  an  amn
2   2
2 3
6
(a m ) n  a mn

using the index laws previously established for numbers to develop the index laws
in algebraic form

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
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Mathscape 9 Extension Teaching Program
 a
2
Page 23

establishing that

using index laws to assist with the definition of the fractional index for square root

2
1
 a and  a 2   a then a  a 2
 
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
given


 a
 a  a  a  a  a2  a
2
1
(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.
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Mathscape 9 Extension Teaching Program
Page 24
Chapter 10. Geometry
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 10. Geometry (pages 343–407)
nPolygon
Triangle angles
Fermat point
Exterior angle
Euler line
Properties of geometric figures
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Establish sum of exterior angles result and sum of interior angles result for polygons.
Identify similar triangles and describe their properties.
Apply tests for congruent triangles.
Use simple deductive reasoning in numerical and non-numerical problems.
Verify the properties of special quadrilaterals using congruent triangles.
SGS5.2.1 (page 157): Develops and applies results related to the angle sum of
interior and exterior angles for any convex polygon.
SGS5.2.2 (page 158): Develops and applies results for proving that triangles
are congruent or similar.
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)
apply the properties of congruent and similar triangles to solve problems, justifying the results (Applying Strategies, Reasoning)
apply simple deductive reasoning in solving numerical and non-numerical problems (Applying Strategies, Reasoning)
Knowledge and skills
Teaching, learning and assessment
Students learn about






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
determining what information is needed to show that two triangles are congruent
If three sides of one triangle are respectively equal to three sides of another

TRY THIS
The badge of the Pythagoreans (page 363): Historical Problem
Five Shapes (page 374): Problem Solving
How many diagonals in a polygon? (page 379): Investigation
An Investigation of Triangles (page 360): Investigation
Triangle Angles (page 392): Investigation
FOCUS ON WORKING MATHEMATICALLY
Does a triangle have a centre? (page 397): It is suggested that teachers do
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Mathscape 9 Extension Teaching Program
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Page 25
triangle, then the two triangles are congruent (SSS).
If two sides and the included angle of one triangle are respectively equal to two
sides and the included angle of another triangle, then the two triangles are
congruent (SAS).
If two angles and one side of one triangle are respectively equal to two angles and
the matching side of another triangle, then the two triangles are congruent (AAS).
If the hypotenuse and a second side of one right-angled triangle are respectively
equal to the hypotenuse and a second side of another right-angled triangle, then
the two triangles are congruent (RHS).
applying the congruency tests to justify that two triangles are congruent
applying the four triangle congruency tests in numerical exercises to find
unknown sides and angles


some lead up construction work with the circumcentre, orthocentre,
centroid and incentre of a triangle before starting this activity.
The site http://www.punahou.edu/acad/sanders/CenterTriangle.html
has some lovely applications for the centroid, circumcentre and incentre,
the teacher may wish to use one of these problems as a starter for the
lesson.
The Fermat point P is the solution to the optimisation problem which
requires the sum of the distances from P to each vertex to be a minimum.
A teaching suggestion is to find where to build a new electricity sub
station from three isolated towns so that the cost of laying cable (related
directly to the length required) to each town is a minimum. You could
work from a map of western NSW for example.
The site http://www2.evansville.edu/ck6/tcenters/ illustrates the classical
and more recent "centres" in some detail which will excite the
mathematical curiosity of students.
Higgins, P.M (2002) Mathematics for the Imagination, Oxford University
Press, pp 82-100 has excellent background material for this activity.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 400)
CHAPTER REVIEW (page 402) a collection of problems to revise the
chapter.
Technology
nPolygon: 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.
Triangle Angles: this program models the drawing of isosceles triangles.
Fermat Point: this program is designed to give students more experience in deductive reasoning in geometry.
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.
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Mathscape 9 Extension Teaching Program
Page 26
Chapter 11. The linear function
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 11. The Linear Function
(pages 408–39)
Line equation
Intersecting lines
Co-ordinate geometry
Duration
2 weeks / 8 hours
Key ideas
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.
Outcomes
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
Students learn about
Teaching, learning and assessment

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 417): Problem Solving
Hanging around (page 427): Problem Solving
Latitude and Temperature (page 433): Investigation
FOCUS ON WORKING MATHEMATICALLY
Television Advertising (page 433): A good reference text is Lowe, I.
(1991) Mathematics at work: Modelling your world volume 1, Australian
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Mathscape 9 Extension Teaching Program
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Page 27
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
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


Academy of Mathematics, pages 393–5.
Advanced students do not need to know the mathematical basis of
correlation to appreciate that it simply measures goodness of fit. The
correlation of x with y in the table is 0.83 which indicates strong linear
relationship. Such statistics are valuable in interpretation of the linear
model to the problem at hand.
A good web site which lists indexes used for financial forecasting is
http://www.neatideas.com/nasd.htm Teachers might like to discuss
forecasting of the money market as a very important business investment
strategy. For example TV news bulletins give stock indexes like the
NASDAQ and Dow Jones, money rates every day. Correlation coefficients
show how well trends in predicted values follow trends in actual values in
the past.
It is important for students to note that a strong correlation between two
variables does not necessarily imply a causal relationship. It is simply the
strength of the linear model which is indicated.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 435)
CHAPTER REVIEW (page 436) a collection of problems to revise the
chapter.
Technology
Line Equation: interactive program with accompanying worksheet.
Intersecting Lines: interactive program with accompanying worksheet.
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Mathscape 9 Extension Teaching Program
Page 28
Chapter 12. Trigonometry
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 12. Trigonometry
(pages 440–87)
Sine cosine
SOHCAHTOA
Trigonometry
Duration
2 weeks / 8 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.
Solve further trigonometry problems including those involving three-figure bearings.
MS5.1.2 (page 139): Applies trigonometry to solve problems (diagrams given)
including those involving angles of elevation and depression.
MS5.2.3 (page 140): Applies trigonometry to solve problems including those
involving bearings.
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)
solve simple problems involving three-figure bearings (Applying Strategies, Communicating)
interpret directions given as bearings (Communicating)
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

TRY THIS
Height to Base Ratio (page 448): Investigation
Make a Hypsometer (page 460): Practical
Pilot Instructions (page 470): Problem Solving
The Sine Rule (page 478): Investigation
FOCUS ON WORKING MATHEMATICALLY
Finding your latitude from the sun (page 479): This is designed as a fun
outdoor activity. Teachers need to prepare well in advance and study the
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Mathscape 9 Extension Teaching Program
Page 29


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
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
 using three-figure bearings (e.g. 035º, 225º) and compass bearings e.g. SSW
 drawing diagrams and using them to solve word problems which involve bearings
or angles of elevation and depression


diagrams carefully. The activity can be carried out on any sunny day and
an adjustment to the observed angle made for the number of days since the
last equinox. The geometry should be discussed carefully in class before
the outdoor measurement s are taken.
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
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
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 461)
CHAPTER REVIEW (page 484) a collection of problems to revise the
chapter.
Technology
Sine Cosine: explores the range of Trig graphs.
SOHCAHTOA: investigation of the tan ratio.
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Mathscape 9 Extension Teaching Program
Page 30
Chapter 13. Simultaneous equations
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 13. Simultaneous Equations
(pages 488–510)
Intersecting lines
Porous rocks
Algebraic techniques
Duration
1 week / 4 hours
Key ideas
Outcomes
Solve simultaneous equations using graphical and analytical methods for simple
examples.
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


use graphics calculators and spreadsheet software to plot pairs of lines and read off the point of intersection (Applying Strategies)
solve linear simultaneous equations resulting from problems and interpret the results (Applying Strategies, Communicating)
Knowledge and skills
Teaching, learning and assessment
Students learn about

Simultaneous Equations
 solving simultaneous equations using non-algebraic methods, such as ‘guess
and check’, setting up tables of values or looking for patterns
 solving linear simultaneous equations by finding the point of intersection of
their graphs
 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

TRY THIS
Find the Values (page 498): Problem Solving
A Pythagorean Problem (page 502): Problem Solving
FOCUS ON WORKING MATHEMATICALLY
Exploring for water, oil and gas—The density of air-filled porous rock (page 506):
This activity is an application of simultaneous equations in earth science. It has
been adapted for school use. Teachers will note that the equations to be solved are
not linear. They can be made so by setting x = 1/V in equations (1) and (2) on page
507. Students can then solve for D and x and finally D and V. The graphs are
straight lines and the solution easily verified.
The web link
http://www.earthsci.ucl.ac.uk/undergrad/geomaths/rev/simnb/sim9.htm sets
out the context of the porous rocks problem clearly but with of course different
symbolism. This example comes from a set of revision exercises in undergraduate
geomaths at the Dept of Earth Sciences at University College London. The
introduction to simultaneous equations is found at
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Mathscape 9 Extension Teaching Program
Page 31
http://www.earthsci.ucl.ac.uk/undergrad/geomaths/rev/simnb/MHsimnb.htm
The web link http://www.blackgold.ab.ca/leduc1/blkgld1.htm discusses the
particular case of oil in porous rocks.
On Mars today, low temperature and pressure limit the stability of liquid water.
The effects of pore sizes and atmospheric pressure on liquid water in rocks were
studied to examine the possibility of liquid water existing on Mars inside rocks and
in pore spaces in the soil. Read about it at
http://www.asgsb.org/programs/2000/24.html
 CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 507)
CHAPTER REVIEW (page 509) A collection of problems to revise the chapter.
Technology
Intersecting Lines: interactive program with accompanying worksheet.
Porous Rocks: worksheet to accompany the Focus on Working mathematically section.
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Mathscape 9 Extension Teaching Program
Page 32
Chapter 14. Co-ordinate geometry
Text references
CD reference
Substrand
Mathscape 9 Extension
Chapter 14. Co-ordinate Geometry
(pages 511–62)
Intersecting lines
Crow flying
Co-ordinate geometry
Duration
2 weeks / 8 hours
Key ideas
Outcomes
Use and apply various standard forms of the equation of a
straight line, and graph regions on the number plane.
PAS5.3.3 (page 102): Uses various standard forms of the equation of a straight line and graphs
regions on the number plane.
Working mathematically
Students learn to




recognise from a list of equations those that result in straight line graphs (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  2 x, y  2 x, y  2 x  1 and describe the transformation that would map one graph onto the other (Communicating)



describe the conditions for a line to have a negative gradient (Reasoning, 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)
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)
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)






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Mathscape 9 Extension Teaching Program


Page 33
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)
Knowledge and skills
Students learn about
Teaching, learning and assessment

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

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


TRY THIS
A line with no integer co-ordinates (page 525): Investigation
Car Hire (page 536): Problem Solving
Temperature Rising (page 540): Problem Solving
FOCUS ON WORKING MATHEMATICALLY
Finding the gradient of a ski run (page 554): 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 eastern and northern slopes during the day.
Hence the preference for these slopes. Just what we need to enjoy skiing.
CHALLENGE, LET’S COMMUNICATE, REFLECTING (page 557)
CHAPTER REVIEW (page 559) 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
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Mathscape 9 Extension Teaching Program

Page 34
using the relationship
rise
run
to establish the formula for the gradient, m, of an interval
joining two points x1 , y1  and x2 , y2  on the number plane
gradient 
m

y2  y1
x2  x1
using the formula to find the gradient of an interval joining
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.
Published by Macmillan Education Australia. © Macmillan Education Australia 2004.