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MathsWorld 9—VELS links by chapter
Chapter 1
Real numbers
Level
Standard/Progression point
MathsWorld 9
At Level 5, students identify complete factor sets
for natural numbers and express these natural
numbers as products of powers of primes (for
example, 36 000 = 25 × 32 × 53).
They know the decimal equivalents for the unit
fractions ½, 1/3, ¼, 1/5, 1/8, 1/9 and find equivalent
representations of fractions as decimals, ratios and
percentages (for example, a subset: set ratio of 4:9
can be expressed equivalently as
4/9 = 0.4 ≈ 44.44%).
Students use knowledge of perfect squares when
calculating and estimating squares and square roots
of numbers
(for example, 202 = 400 and
302 = 900 so √700 is between 20 and 30).
They evaluate natural numbers and simple fractions
given in base-exponent form (for example, 54 = 625
and (2/3)2 = 4/9).
*Chapter pre-test Q 9, 10, 11
1.3: Factors and prime factors
Examples 1, 2, 3
Ex. 1.3 Q 1 – 3
*Chapter pre-test Q 3, 13
Ex. 1.1 Q 2, 3, 4, 5, 6
Number
5.0
5.0
5.0
5.0
5.0
5.0
5.0
5.0
5.0
5.0
They calculate squares and square roots of rational
numbers that are perfect squares (for example,
√0.81 = 0.9 and √(9/16) = ¾).
They calculate cubes and cube roots of perfect
cubes (for example, 3√64 = 4).
Using technology they find square and cube roots of
rational numbers to a specified degree of accuracy
(for example, 3√200 = 5.848 to three decimal
places).
Students use a range of strategies for approximating
the results of computations, such as front-end
estimation and rounding
(for example, 925 ÷ 34 ≈ 900 ÷ 30 = 30).
Students use efficient mental and/or written
methods for arithmetic computation involving
rational numbers, including division of integers by
two-digit divisors.
They use technology for arithmetic computations
involving several operations on rational numbers of
any size.
*Chapter pre-test Q 1e,f, 6g-j, 13-16
1.4: Irrational numbers
Example 1
Ex. 1.4 Q 3
*Chapter pre-test Q 1e,f, 6g,h, 14b,c
1.2: Integer powers of rational
numbers
Examples 1 – 5
Ex. 1.2 Q 5
*Chapter pre-test Q 6h-j, 12e,f
*Chapter pre-test Q 6g, 16h,i
*Chapter pre-test Q 16d-j
1.4: Irrational numbers
Example 1
Try this! p 30
Ex. 1.4 Q 3
MathsWorld 9 Practice and
Enrichment Workbook (and CD)
Technology toolkit
TI 83/84 1.1, 1.2 p 161
TI 89 1.1, 1.2 p 195
*Chapter pre-test Q 7, 16
Most questions require students to round
answers to a specified or to a sensible
number of decimal places.
*Chapter pre-test Q 1 – 13
*Chapter pre-test Q 16
Also Chapters 2, 5, 7, 9, 10
See MathsWorld 9 Practice and
Enrichment Workbook (and CD)
Technology toolkit
TI 83/84 1.1, 1.2 p 161
TI 89 1.1, 1.2 p 195
MathsWorld 9 Teacher edition
Copyright  Macmillan Education Australia. Unauthorised copying prohibited.
1
MathsWorld 9—VELS links by chapter
Chapter 1
Real numbers
Level
Standard/Progression point
MathsWorld 9
5.25
•
5.25
•
5.5
•
1.1: Rational numbers
p5
1.4: Irrational numbers
Try this! p 27
1.3: Factors and prime factors
Examples 1, 2, 3
(See note in Teacher edition p 18 and
Year 9 Cumulative test)
1.4: Irrational numbers
Examples 2, 3
Ex. Q 6, 7, 9, 10, 15
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 1.1 p 161
TI 89 1.1 p 195
5.5
•
5.5
•
5.5
•
5.5
•
5.75
•
relationships between real, rational, irrational
integers and natural numbers on a Venn
diagram
determination of lowest common multiple
through investigation of prime factors
simplification of surds, for example, 12  2 3
solution of proportion problems using real
numbers
calculation of approximate values for  , the
golden ratio, using measurement, definition,
and successive ratios of the Fibonacci
sequence
computation involving natural numbers,
integers, finite decimals and surds, without the
aid of technology, giving exact answers as
applicable.
calculation of the remainder after division by
using multiplication (as needed for Euclid's
method)
3
knowledge of the equivalence of  1  and 103
 
 10 
Analysis task 1: The golden ratio
Chapter pre-test Q 1 – 13
1.1: Rational numbers
Examples 1, 2, 3, 4
Ex. 1.1 Q 1 – 11
1.2: Integer powers of rational
numbers
Examples 1, 3, 4, 5
Ex. 1.2 Q 1 – 11
1.3: Factors and prime factors
Examples 1, 2, 3, 4, 5
Ex. 1.3 Q 1 – 11
1.4: Irrational numbers
Examples 1, 2, 3, 4
Ex. 1.4 Q 9 – 17
1.5: Adding and subtracting surds
Examples 1, 2
Ex. 1.5 Q 1, 3 – 14
1.6: Multiplying and dividing surds
Examples 1, 2, 3
Ex. 1.5 Q 1 – 10
1.3: Factors and prime factors
Example 4
Ex. 1.3 Q 4, 5
1.2: Integer powers of rational
numbers
Try this! p 14
Example 4
Ex. 1.4 Q 6, 7, 8
MathsWorld 9 Teacher edition
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2
MathsWorld 9—VELS links by chapter
Chapter 1
Real numbers
Level
Standard/Progression point
MathsWorld 9
5.75
•
1.7: Rationalising the denominator
Examples 1, 2
Ex. 1.7 Q 1 – 7
Try this! p 40
rationalisation of expressions where division by
a square root is involved, for example,
5
15

3
3
6.0
At Level 6, students comprehend the set of real
numbers containing natural, integer, rational and
irrational numbers.
6.0
They represent rational numbers in both fractional
and decimal (terminating and infinite recurring)
forms (for example, 14/ 25 = 1.16, = 47/ 99 ).
6.0
They comprehend that irrational numbers have an
infinite non-terminating decimal form.
They specify decimal rational approximations for
square roots of primes, rational numbers that are not
perfect squares, the golden ratio φ, and simple
fractions of π correct to a required decimal place
accuracy.
Students use the Euclidean division algorithm to
find the greatest common divisor (highest common
factor) of two natural numbers 9 (for example, the
greatest common divisor of 1071 and 1029 is 21
since 1071 = 1029 × 1 + 42, 1029 = 42 × 24 + 21
and 42 = 21 × 2 + 0).
Students carry out arithmetic computations
involving natural numbers, integers and finite
decimals using mental and/or written algorithms
(one- or two-digit divisors in the case of division).
They carry out exact arithmetic computations
involving fractions and irrational numbers such as
square roots
(for example, √18 = 3√2, √( 3/2 ) = (√6)/ 2) and
multiples and fractions of π (for example
π + π/ 4 = 5 / 4).
6.0
6.0
6.0
6.0
Chapter Warm-up Try this! p 4
1.1: Rational numbers
Ex. 1.1 Q 1
1.4: Irrational numbers
p 27 Try this!
*Chapter pre-test Q 3, 4
Chapter Warm-up Try this! p 4
1.1: Rational numbers
Examples 1, 2, 3, 4
Ex. 1.1 Q 2 – 11
1.4: Irrational numbers
Try this! p 30
1.4: Irrational numbers
Example 1
Ex. 1.4 Q 3, 6
Analysis task 1: The golden ratio
1.3: Factors and prime factors
Examples 4, 5
Ex. 1.3 Q 4 – 8
*Chapter pre-test Q 1 – 13
(See Teacher edition for comment, and
Year 9 Cumulative Revision Test)
1.4: Irrational numbers
Examples 2, 3
Ex. 1.4 Q 9 – 13
1.5: Adding and subtracting surds
Examples 1, 2
Ex. 1.5 Q 1 – 14
1.6: Multiplying and dividing surds
Examples 1, 2, 3
Ex. 1.6 Q 1 – 10
1.7: Rationalising the denominator
Examples 1, 2
Ex. 1.7 Q 1 – 7
Structure
5.0
At Level 5 students identify collections of numbers
as subsets of natural numbers, integers, rational
numbers and real numbers.
5.0
They test the validity of statements formed by the
use of the connectives and, or, not, and the
quantifiers none, some and all, (for example, ‘some
natural numbers can be expressed as the sum of two
squares’).
Chapter Warm-up Try this!
p4
1.1: Real numbers
Ex. 1.1 Q 1
1.5: Adding and subtracting surds
Ex. 1.5 Q 6
MathsWorld 9 Teacher edition
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3
MathsWorld 9—VELS links by chapter
Chapter 1
Real numbers
Level
Standard/Progression point
5.0
Students apply the commutative, associative, and
distributive properties in mental and written
computation (for example, 24 × 60 can be
calculated as 20 × 60 + 4 × 60 or as 12 × 12 × 10).
They use exponent laws for multiplication and
division of power terms (for example 23 × 25 = 28,
20 = 1, 23 ÷ 25 = 2−2, (52)3 = 56 and (3 × 4)2 = 32 ×
42).
Students generalise from perfect square and
difference of two square number patterns
(for example, 252 = (20 + 5)2 = 400 + 2 × (100) +
25 = 625. And 35 × 25 = (30 + 5) (30 - 5) = 900 −
25 = 875)
They identify the identity element and inverse of
rational numbers for the operations of addition and
multiplication
(for example, ½ + − ½ = 0 and 2/3 × 3/2 = 1).
5.0
5.0
5.0
5.75
•
6.0
At Level 6, students classify and describe the
properties of the real number system and the subsets
of rational and irrational numbers.
They use irrational numbers such as π ,  and
common surds in calculations in both exact and
approximate form.
6.0
expression of irrational numbers in both exact
and approximate form
MathsWorld 9
1.2: Integer powers of rational
numbers
Examples 3, 4, 5
Ex. 1.2 Q 1 – 10
1.4: Irrational numbers
Example 1
1.5: Adding and subtracting surds
Ex. 1.5 Q 11
Analysis task 1: The golden ratio
Try this! p 27
1.4: Irrational numbers
Ex. 1.4 Q 1, 2
1.4: Irrational numbers
Examples 1, 2, 3
Ex. 1.4 Q 12, 13
1.5: Adding and subtracting surds
Examples 1, 2
Ex. 1.5 Q 3 – 12
1.6: Multiplying and dividing surds
Examples 1, 2, 3
Ex. 1.6 Q 1 – 10
Analysis task 1: The golden ratio
Analysis task 2: Federation Square tiles
Working
mathematically
5.0
At Level 5, students formulate conjectures and
follow simple mathematical deductions (for
example, if the side length of a cube is doubled,
then the surface area increases by a factor of four,
and the volume increases by a factor of eight).
1.6: Multiplying and dividing surds
Try this! p 40
MathsWorld 9 Teacher edition
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4
MathsWorld 9—VELS links by chapter
Chapter 1
Real numbers
Level
Standard/Progression point
MathsWorld 9
6.0
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
6.0
They use geometry software or graphics calculators
to create geometric objects and transform them,
taking into account invariance under
transformation.
1.4: Irrational numbers
Ex. 1.4 Q 5, 6, 7
1.5: Adding and subtracting surds
Ex. 1.5 Q 2
1.6: Multiplying and dividing surds
Try this! p 40
1.7: Rationalising the denominator
Try this! p 45
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 1.1 p 161
TI 89 1.1 p 195
Analysis task 1: The golden ratio
MathsWorld 9 Teacher edition
Copyright  Macmillan Education Australia. Unauthorised copying prohibited.
5
MathsWorld 9—VELS links by chapter
Chapter 2
Length, area and volume
Level
Standard/Progression point
MathsWorld 9
5.0
Using technology they find square and cube roots of
rational numbers to a specified degree of accuracy
(for example, 3√200 = 5.848 to three decimal
places).
2.3: Pythagoras’ theorem
Ex. 2.3 Q 1 – 4, 9
MathsWorld 9 Practice and
Enrichment Workbook (and CD)
Technology toolkit
TI 83/84 1.1, 1.2 p 161
TI 89 1.1, 1.2 p 195
5.0
Students use a range of strategies for approximating
the results of computations, such as front-end
estimation and rounding
(for example, 925 ÷ 34 ≈ 900 ÷ 30 = 30).
They use approximations to π in related
measurement calculations
(for example, π × 52 = 25π = 78.54 correct to two
decimal places).
Number
5.0
5.25
•
5.75
•
5.75
•
representation and recognition of large and
small numbers in scientific notation
division and multiplication of numbers in index
form, including application to scientific
notation.
3
knowledge of the equivalence of  1  and 103
 
 10 
5.75
•
application of scientific notation and recalled
approximations to squares and square roots to
approximate values for expressions.
*Chapter pre-test Q 6
2.4: Calculating perimeter
Examples 2, 3
Ex. 2.4 Q 4 – 8
2.5: Area
Examples 1d, 2b, 3, 4
Ex. 2.5 Q 1i – l, 2e – i ,
5e – j, 7
2.6: Surface area
Examples 6, 7
Ex. 2.6 Q 3 – 5, 8
2.7: Volume
Examples 3, 5
Ex. 2.7 Q 5 – 8, 13, 16, 17
Analysis task 1: Chemical storage tanks
Analysis task 2: Melbourne Central cone
2.2: Scientific notation
Examples 1 – 7
Try this! p 69
Ex. 2.2 Q 1 – 9
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 1.1 p 161
TI 89 1.1 p 195
2.2: Scientific notation
Examples 6, 7
Ex. 2.2 Q 5 – 8
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 1.1 p 161
TI 89 1.1 p 195
2.2: Scientific notation
Try this! p 69
Example 4
See Teacher edition p 71 for additional
example and questions
MathsWorld 9 Teacher edition
Copyright  Macmillan Education Australia. Unauthorised copying prohibited.
6
MathsWorld 9—VELS links by chapter
Chapter 2
Length, area and volume
Level
Standard/Progression point
MathsWorld 9
6.0
They specify decimal rational approximations for
square roots of primes, rational numbers that are not
perfect squares, the golden ratio φ, and simple
fractions of π correct to a required decimal place
accuracy.
They perform computations involving very large or
very small numbers in scientific notation (for
example, 0.0045 × 0.000028 = 4.5 × 10 −3 × 2.8 ×
10−5 = 1.26 × 10−7).
They use appropriate estimates to evaluate the
reasonableness of the results of calculations
involving rational and irrational numbers, and the
decimal approximations for them.
Analysis task 3: Short shoelaces
6.0
6.0
6.0
They carry out computations to a required accuracy
in terms of decimal places and/or significant
figures.
2.2: Scientific notation
Examples 3, 5, 6, 7
Ex. 2.2 Q 5 – 9
2.1: Significant figures and
measurement errors
Try this! p 61
Ex. 2.3 – 2.7
Students should be encouraged in all
exercises to estimate answers and check
for reasonableness
2.1: Significant figures and
measurement errors
Examples 1, 2
Ex. 2.1 Q 1 – 4
All exercises in sections 2.4 to 2.7
include questions that require students to
round answers to a given or sensible
number of significant figures.
Space
5.0
They use two-dimensional nets to construct a
simple three-dimensional object such as a prism or
a platonic solid.
They recognise and describe boundaries, surfaces
and interiors of common plane and threedimensional shapes, including cylinders, spheres,
cones, prisms and polyhedra.
They recognise the features of circles (centre,
radius, diameter, chord, arc, semi-circle,
circumference, segment, sector and tangent) and use
associated angle properties.
2.6: Surface area
Example 3
Ex. 2.6 Q 2
2.4: Calculating perimeter
2.5: Area
2.6: Surface area
2.7: Volume
2.4: Calculating perimeter
p 87
Example 3
2.5: Area
p 93
Example 4
5.0
At Level 5, students measure length, perimeter,
area, surface area, mass, volume, capacity, angle,
time and temperature using suitable units for these
measurements in context.
5.0
They interpret and use measurement formulas for
the area and perimeter of circles, triangles and
parallelograms and simple composite shapes.
*Chapter pre-test
Q 4 – 10
2.4: Calculating perimeter
2.5: Area
2.6: Surface area
2.7: Volume
*Chapter pre-test Q 4–8
2.4: Calculating perimeter
2.5: Area
6.0
6.0
Measurement,
Chance and Data
MathsWorld 9 Teacher edition
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7
MathsWorld 9—VELS links by chapter
Chapter 2
Length, area and volume
Level
Standard/Progression point
MathsWorld 9
5.0
They calculate the surface area and volume of
prisms and cylinders.
5.0
Students estimate the accuracy of measurements
and give suitable lower and upper bounds for
measurement values.
5.0
They calculate absolute percentage error of
estimated values.
5.25
•
5.25
•
5.5
•
6.0
At Level 6, students estimate and measure length,
area, surface area, mass, volume, capacity and
angle.
6.0
They select and use appropriate units, converting
between units as required.
*Chapter pre-test Q 9, 10
2.6: Surface area
Examples 1, 2, 3, 6, 7
Ex. 2.6 Q 1 – 6, 8
2.7: Volume
Examples 1, 2, 3
Ex. 2.7 Q 1 – 11
Analysis task 1: Chemical storage tanks
*Chapter pre-test Q 3
2.1: Significant figures and
measurement errors
Ex. 2.1 Q 6 – 8
2.1 Significant figures and
measurement errors
Try this! p 63
Example 2
Ex. 2.1 Q 9, 11
*Chapter pre-test Q 2e, f
2.6: Surface area
Example 6
2.7: Volume
Example 3
Ex. 2.7 Q 4, 5, 7, 8, 11, 15, 16
2.3: Pythagoras' theorem
Examples 1, 2, 3, 5
Ex. 2.3 Q 1, 2, 3, 4, 5, 6, 12, 14
Analysis task 3: Short shoelaces!
2.3: Pythagoras' theorem
Example 6
Ex. 2.3 Q 8, 9, 10, 11, 13
*Chapter pre-test
Q 4 – 10
2.4: Calculating perimeter
Examples 1, 2, 3
Ex. 2.4 Q 1 – 8
2.5: Area
Examples 1, 2, 3, 4
Ex. 2.5 Q 1 – 10
2.6: Surface area
Examples 1 – 7
Ex. 2.6 Q 1 – 8
2.7: Volume
Examples 1, 2, 3, 4
Ex. 2.7 Q 1 – 17
Analysis task 1: Chemical storage tanks
Analysis task 2: Melbourne Central cone
*Chapter pre-test Q 1, 2
2.5: Area
Ex. 2.5 Q 4
2.6: Surface area
Example 6
2.7: Volume
Example 3
Ex. 2.7 Q 8, 14, 15, 16
conversion between units and between derived
units
use of Pythagoras' theorem to calculate the
length of the hypotenuse
use of Pythagoras' theorem to calculate the
length of a side other than the hypotenuse
MathsWorld 9 Teacher edition
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8
MathsWorld 9—VELS links by chapter
Chapter 2
Length, area and volume
Level
Standard/Progression point
MathsWorld 9
6.0
Students decide on acceptable or tolerable levels of
error in a given situation.
6.0
They interpret and use mensuration formulas for
calculating the perimeter, surface area and volume
of familiar two- and three-dimensional shapes and
simple composites of these shapes.
Students use Pythagoras’ theorem and trigonometric
ratios (sine, cosine and tangent) to obtain lengths of
sides, angles and the area of right-angled triangles.
2.1 Significant figures and
measurement errors
Ex. 2.1 Q 6 – 11
2.4: Calculating perimeter
2.5: Area
2.6: Surface area
2.7: Volume
2.3: Pythagoras’ theorem
Examples 1, 2, 3, 4, 5, 6
Ex. 2.3 Q 1 – 18
2.4: Calculating perimeter
Try this! p 85
Ex. 2.4 Q 3
2.6: Surface area
Example 5
Ex. 2.6 Q 6
Analysis task 3: Short shoe laces
6.0
Structure
5.75
•
6.0
Students form and test mathematical conjectures;
for example, ‘What relationship holds between the
lengths of the three sides of a triangle?’
They use irrational numbers such as π ,  and
common surds in calculations in both exact and
approximate form.
2.7: Volume
Try this! p 113 – 115
5.0
Students use variables in general mathematical
statements.
Formulae used in all sections
5.25
•
5.25
•
2.3: Pythagoras’ theorem
Ex. 2.3 Q 19
See also MathsWorld 8
Chapter 5 Algebra toolbox
Analysis task 2: Odds and evens
All sections
5.5
•
5.5
•
6.0
expression of irrational numbers in both exact
and approximate form
Questions involving Pythagoras' theorem
and 
Questions involving Pythagoras' theorem
and 
Working
mathematically
6.0
presentation of algebraic arguments using
appropriate mathematical symbols and
conventions
evaluation of the appropriateness of the results
of their own calculations
justification or proof of generalisations made
from specific cases
selection and use of technology to explore
geometric and algebraic relationships and data
trends
They follow formal mathematical arguments for the
truth of propositions.
2.3: Pythagoras’ theorem
Try this! p 74
Analysis task 1: Chemical storage tanks
Analysis task 3: Short shoelaces!
2.3: Pythagoras’ theorem
Try this! p 74
MathsWorld 9 Teacher edition
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9
MathsWorld 9—VELS links by chapter
Chapter 2
Length, area and volume
Level
Standard/Progression point
MathsWorld 9
6.0
Students choose, use and develop mathematical
models and procedures to investigate and solve
problems set in a wide range of practical, theoretical
and historical contexts (for example, exact and
approximate measurement formulas for the volumes
of various three dimensional objects such as
truncated pyramids).
They judge the reasonableness of their results based
on the context under consideration.
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
Analysis task 1: Chemical storage tanks
Analysis task 3: Short shoelaces!
6.0
6.0
Sections 2.3 – 2.7
All exercises
Analysis task 1: Chemical storage tanks
MathsWorld 9 Teacher edition
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10
MathsWorld 9—VELS links by chapter
Chapter 3
Mathematical thinking
Level
Standard/Progression point
MathsWorld 9
They use appropriate estimates to evaluate the
reasonableness of the results of calculations
involving rational and irrational numbers, and the
decimal approximations for them.
Chapter 3: Mathematical thinking
3.2: Extended modelling tasks with
technology
Practice problem 2
Try this! p 144
Try this! p 149
5.25
•
6.0
Students use Pythagoras’ theorem and trigonometric
ratios (sine, cosine and tangent) to obtain lengths of
sides, angles and the area of right-angled triangles.
Chapter 3: Mathematical thinking
3.1: Mathematical modelling
Example problem 1 p 134 - 136
3.2: Extended modelling tasks with
technology
Try this! p 151, 159
Problem set 3.2 Q 1, 2
Chapter 3: Mathematical thinking
3.1: Mathematical modelling
Example problem 1Try this! p 136
3.2: Extended modelling tasks with
technology
Extended example problem 1 Try this! p
151
Problem set 3.2 Q1, 2
Number
6.0
Measurement,
Chance and Data
use of Pythagoras' theorem to calculate the
length of the hypotenuse
Working
mathematically
5.5
•
5.75
•
6.0
6.0
selection and use of technology to explore
geometric and algebraic relationships and data
trends
recognition of functionality of technology and
its limitations, such as image resolution,
discontinuities in graphs and systematic error
in computation through rounding
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
Students choose, use and develop mathematical
models and procedures to investigate and solve
problems set in a wide range of practical, theoretical
and historical contexts (for example, exact and
approximate measurement formulas for the volumes
of various three dimensional objects such as
truncated pyramids).
3.2: Extended modelling tasks with
technology
Try this! p 154
3.2: Extended modelling tasks with
technology
Extended example problems 1, 2
3.2: Extended modelling tasks with
technology
Extended example problems 1, 2
Chapter Warm-up Try this! p 132
3.1 Mathematical modelling
Example problem 1Try this! p 136
Practice problems 1, 2
Example problems 2, 3
Problem set 3.1
3.2: Extended modelling with
technology
Extended example problem 1 Try this!
pp 149, 150, 151, 159, 161, 162, 163
Problem set 3.2
MathsWorld 9 Teacher edition
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11
MathsWorld 9—VELS links by chapter
Chapter 3
Mathematical thinking
Level
Standard/Progression point
MathsWorld 9
6.0
They use geometry software or graphics calculators
to create geometric objects and transform them,
taking into account invariance under
transformation.
3.2: Extended modelling tasks with
technology
Extended example problem 1
Try this! p 149
Problem set 3.2 Q 1, 2
MathsWorld 9 Teacher edition
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12
MathsWorld 9—VELS links by chapter
Chapter 4
Algebra toolbox 1
Level
Standard/Progression point
MathsWorld 9
Students apply the commutative, associative, and
distributive properties in mental and written
computation (for example, 24 × 60 can be
calculated as 20 × 60 + 4 × 60 or as 12 × 12 × 10).
Number examples regularly used before
generalising, particularly in conjunction
with use of a geometric model (e.g., p
203). See further note and examples in
Teacher edition
4.2: What does solving mean?
Example 7
Ex. 4.2 Q 27
Structure
5.0
5.0
5.0
Students use inverses to rearrange simple
mensuration formulas, and to find equivalent
algebraic expressions
(for example, if P = 2L + 2W, then W = P/2 − L. If
A = πr2 then r = √A/π).
They solve simple equations (for example, 5x+ 7 =
23, 1.4x − 1.6 = 8.3, and 4x2 − 3 = 13) using tables,
graphs and inverse operations.
5.0
They recognise and use inequality symbols.
5.0
They solve simple inequalities such as y ≤ 2x+ 4
and decide whether inequalities such as x2 > 2y are
satisfied or not for specific values of x and y.
5.0
Students generalise from perfect square and
difference of two square number patterns
(for example, 252 = (20 + 5)2 = 400 + 2 × (100) +
25 = 625. And 35 × 25 = (30 + 5) (30 – 5) = 900 −
25 = 875)
5.25
•
5.5
•
factorisation of algebraic expressions by
extracting a common factor
expansion of products of algebraic factors, for
example,  2 x  1 x  5  2 x2  9 x  5
*Chapter pre-test Q 10
4.2: What does solving mean?
Examples 1, 2, 3, 4, 5
Ex. 4.2 Q 1 – 17
4.2: What does solving mean?
Example 6
Ex. 4.2 Q 20, 21, 24, 25, 26
4.2: What does solving mean?
Example 6
Ex. 4.2 Q 20, 21, 24-26
(See note and further examples in
Teacher edition p 184)
Number examples regularly used before
generalising, particularly in conjunction
with use of a geometric model. See
further note and examples in Teacher
edition p 214
4.4: Factorising algebraic expressions
Try this! p 197, 198, 199
Examples 1, 2
Ex. 4.4 Q 1 – 12
4.3: Expanding algebraic expressions
Try this! p 192, 193
Examples 1, 2, 3
Ex. 4.3 Q 1 - 6
4.5: Expanding binomials
Try this! p 204, 205
Examples 1, 2
Ex. 4.5 Q 1 – 13
4.7: Perfect squares and difference of
squares
Try this! p 214, 216, 218
Examples 1, 2, 3, 4, 5
Ex. 4.7 Q 1 – 14
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13
MathsWorld 9—VELS links by chapter
Chapter 4
Algebra toolbox 1
Level
Standard/Progression point
MathsWorld 9
5.5
•
5.5
•
5.75
•
4.3: Expanding algebraic expressions
Try this! p 192, 193
Examples 1, 2, 3
Ex. 4.3 Q 1 – 6
4.4: Factorising algebraic expressions
Try this! p 197
Examples 1, 2
Ex. 4.4 Q 1 – 3, 5 – 12
4.5: Expanding binomials
Try this! p 204, 205
Examples 1, 2
Ex. 4.5 Q 1 – 13
4.6: Factorising quadratic trinomials
Try this! p 210, 211
Examples 2, 3
Ex. 4.6 Q 2 – 9
4.7: Perfect squares and differences of
squares
Try this! p 214, 218
Examples 1, 2, 3, 4, 5
Ex. 4.7 Q 1 – 14
4.8: Index form with pronumerals
Examples 3 – 9
Ex. 4.8 Q 3 – 13
Analysis task 1: Pascal's triangle and
binomial expansions
Analysis task 2: Completing the square
4.2: What does solving mean?
Try this! p 185
Example 7
Ex. 4.2 Q 22a - r
4.6 Factorising quadratic trinomials
Examples 2, 3
Ex. 4.6 Q 2 – 9
4.7: Perfect squares and difference of
squares
Examples 2, 3, 5
Ex. 4.7 Q 4, 6, 9, 10, 11, 13
equivalence between algebraic forms; for
example, polynomial, factorised and turning
point form of quadratics
use of inverse operations to re-arrange formulas
to change the subject of a formula
factorisation of simple quadratic expressions
and use of the null factor law for solution of
equations
MathsWorld 9 Teacher edition
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14
MathsWorld 9—VELS links by chapter
Chapter 4
Algebra toolbox 1
Level
Standard/Progression point
MathsWorld 9
6.0
Students apply the algebraic properties (closure,
associative, commutative, identity, inverse and
distributive) to computation with number, to
rearrange formulas, rearrange and simplify
algebraic expressions involving real variables.
*Chapter pre-test Q 5 – 9
4.3: Expanding algebraic expressions
Try this! p 192, 193
Examples 1, 2, 3
Ex. 4.3 Q 1 - 6
4.4: Factorising algebraic expressions
Try this! p 197, 198
Examples 1, 2
Ex. 4.4 Q 1 - 12
4.5: Expanding binomials
Try this! p 204, 205
Examples 1, 2
Ex. 4.5 Q 1 – 13
4.6: Factorising quadratic trinomials
Try this! p 210, 212
Examples 1, 2
Ex. 4.6 Q 1 - 9
4.7: Perfect squares and difference of
squares
Try this! p 214, 216, 218
Examples 1, 2, 3, 4, 5
Ex. 4.7 Q 1 – 14
At Level 5, students formulate conjectures and
follow simple mathematical deductions (for
example, if the side length of a cube is doubled,
then the surface area increases by a factor of four,
and the volume increases by a factor of eight).
They substitute numbers for variables (for example,
in equations, inequalities, identities and formulas).
4.6: Factorising quadratic trinomials
Try this! p 210
4.7: Perfect squares and difference of
squares
Try this! p 214
*Chapter pre-test Q 2 – 4
Chapter Warm-up Try this! p 169
4.1: Formulas and substitution
Examples 1, 2, 3, 4 , 5
Try this! p 172
Ex. 4.1 Q 1 – 10
4.2: What does solving mean?
Example 3
Ex. 4.2 Q 12, 13, 15 – 19
4.3: Expanding algebraic expressions
Try this! p 192
4.4: Factorising algebraic expressions
Try this! p 199
4.6: Factorising quadratic trinomials
Try this! p 210
4.7: Perfect squares and differences of
squares
Try this! p 214, 218
Working
mathematically
5.0
5.0
5.75
•
representation and manipulation of symbolic
expressions using technology
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MathsWorld 9—VELS links by chapter
Chapter 4
Algebra toolbox 1
Level
Standard/Progression point
MathsWorld 9
6.0
At Level 6, students formulate and test conjectures,
generalisations and arguments in natural language
and symbolic form (for example, ‘if m2 is even then
m is even, and if m2 is odd then m is odd’).
6.0
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
4.5: Expanding binomials
Try this! p 204, 205
4.7: Perfect squares and difference of
squares
Try this! p 214
Analysis task 1: Pascal’s triangle and
binomial expansions
Analysis task 2: Completing the square
4.1: Formulas and substitution
Try this! p 172
4.2: What does solving mean?
Try this! p 179
Example 3
Ex. 4.2 Q 12 – 19
4.3: Expanding algebraic expressions
Try this! p 192
4.4: Factorising algebraic expressions
Try this! p 199
4.6: Factorising quadratic trinomials
Try this! p 210
4.7: Perfect squares and difference
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16
MathsWorld 9—VELS links by chapter
Chapter 5
Ratios and rates
Level
Standard/Progression point
MathsWorld 9
They know the decimal equivalents for the unit
fractions ½, 1/3, ¼, 1/5, 1/8, 1/9 and find equivalent
representations of fractions as decimals, ratios and
percentages (for example, a subset: set ratio of 4:9
can be expressed equivalently as
4/9 = 0.4 ≈ 44.44%).
Students understand ratio as both set: set
comparison (for example, number of boys : number
of girls) and subset: set comparison (for example,
number of girls : number of students), and find
integer proportions of these, including percentages
(for example, the ratio number of girls: the number
of boys is 2 : 3 = 4 : 6 = 40% : 60%).
They use ratios of number pairs to understand
constant rate of change.
5.3 Percentages
Ex. 5.3 Q 1, 2
Number
5.0
5.0
5.0
5.0
They use number lines, graphs, numerical or
algebraic means to solve proportion problems and
percentage problems as proportion relative to 100.
5.25
•
5.25
•
5.5
•
5.5
•
6.0
solution of problems involving ratio and
proportion
calculation and use of percentage change in
practical situations, for example, discounts.
calculation of the whole given the size of a
percentage; for example, if a 20% discount is
$7, what was the original value?
solution of proportion problems using real
numbers
They represent rational numbers in both fractional
and decimal (terminating and infinite recurring)
forms (for example, 14/ 25 = 1.16, = 47/ 99 ).
*Chapter pre-test Q 1, 2, 3, 4, 5
Try this! p 244
5.1: Ratio and proportion
Example 1
Ex. 5.1 Q 1, 3, 5, 11
*Chapter pre-test Q 7, 8, 10
5.5: Constant and variable rates
Try this! p. 286
Example 1
Ex. 5.5 Q 1
5.1: Ratio and proportion
Examples 5, 6, 7
Ex. 5.1 Q 3 – 7, 8 – 14
5.3: Percentages
Examples 1, 4, 5, 6, 7
Ex. 5.3 Q 3 – 10
*Chapter pre-test Q 2 – 9
5.1: Ratio and proportion
Examples 5, 6, 7
Ex. 5.1 Q 6 – 18
5.3: Percentages
Examples 4 – 12
Ex. 5.3 Q 6 – 19
5.3: Percentages
Examples 5, 7
Ex. 5.3 Q 7, 8
5.1: Ratio and proportion
Examples 3, 4
Ex. 5.1 Q 1, 2
Space
6.0
They determine the effect of changing the scale of
one characteristic of two- and three-dimensional
shapes (for example, side length, area, volume and
angle measure) on related characteristics.
5.1: Ratio and proportion
Ex. 5.1 Q 15, 16, 17
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17
MathsWorld 9—VELS links by chapter
Chapter 5
Ratios and rates
Level
Standard/Progression point
MathsWorld 9
5.5
•
6.0
They calculate constant rates such as the density of
substances (that is, mass in relation to volume),
concentration of fluids, average speed and pollution
levels in the atmosphere.
*Chapter pre-test Q 7, 8, 9
5.2: Rates
Examples 1 – 9
Ex. 5.2 Q 1 – 16
5.5 Constant and variable rates
Try this! p 286, 287-288
Example 1
Ex. 5.5 Q 1 – 12
Analysis task 1: How much water do we
use?
Analysis task 2: Grand Prix
Analysis task 3: Compound interest
5.2: Rates
Examples 1 – 9
Ex. 5.2 Q 1 – 16
Measurement,
Chance and Data
calculation and application of ratio, proportion
and rate of change such as concentration,
density, and the rate of filling a container
Working
mathematically
5.0
Students use variables in general mathematical
statements.
5.5
•
6.0
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
selection and use of technology to explore
geometric and algebraic relationships and data
trends
5.3: Percentages
Examples 9, 10, 11, 12
Ex. 5.3 Q 12 – 16
Analysis task 1: How much water do we
use?
Analysis task 2: Grand Prix
Analysis task 3: Compound interest
Analysis task 1: How much water do we
use?
Analysis task 2: Grand Prix
Analysis task 3: Compound interest
MathsWorld 9 Teacher edition
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18
MathsWorld 9—VELS links by chapter
Chapter 6
Two-dimensional space
Level
Standard/Progression point
MathsWorld 9
5.0
At Level 5, students construct two-dimensional and
simple three-dimensional shapes according to
specifications of length, angle and adjacency.
5.0
They use the properties of parallel lines and
transversals of these lines to calculate angles that
are supplementary, corresponding, allied (cointerior) and alternate.
5.0
They describe and apply the angle properties of
regular and irregular polygons, in particular,
triangles and quadrilaterals.
5.0
They recognise congruence of shapes and solids.
5.0
They make tessellations from simple shapes.
5.0
They use lines, grids, contours, isobars, scales and
bearings to specify location and direction on plans
and maps.
5.25
•
5.5
•
*Chapter pre-test Q 4, 5
6.4: Angles in a circle
Example 5
Ex. 6.4 Q 4, 5
*Chapter pre-test Q 1, 2d, f
6.1: Angles, parallel lines and triangles
Example 4
Ex. 6.1 Q 4a – i, 6d, f
6.2: Quadrilateral properties
Example 2b
Ex. 6.2 2a, c, d, e, f, g, i
*Chapter pre-test Q 2, 5, 6b, 8, 9, 10
Q 2, 3
6.1: Angles, parallel lines and triangles
Examples 5, 8
Ex. 6.1 Q 3b – i, 4a – h, 6a, b, c, 7 – 11,
19 – 21
6.2: Quadrilateral properties
Example 2
Ex. 6.2 Q 2 – 8
6.3: Polygons
Examples 1, 2
Ex. 6.3 Q 1 – 7
Analysis task 1: Pascal’s angle trisector
6.1: Angles, parallel lines and triangles
Example 1
Ex. 6.1 Q 7
6.3: Polygons
Ex. 6.3 Q 6
6.1: Angles, parallel lines and triangles
Example 9
Ex. 6.1 16, 17, 19, 20
*Chapter pre-test Q 3
6.1: Angles, parallel lines and triangles
Examples 1, 2
Ex. 6.1 Q 7
6.4: Angles in a circle
p 335
Examples 1, 2, 3, 4
Ex. 6.4 Q 1, 2, 3
5.5
•
6.0
They recognise the features of circles (centre,
radius, diameter, chord, arc, semi-circle,
circumference, segment, sector and tangent) and use
associated angle properties.
Space
knowledge of sets of conditions for pairs of
triangles to be congruent
recognition of features of circles (centre,
radius, diameter, chord, arc, semi-circle,
segment, sector and tangent) and the associated
angle properties
investigation of angle properties of circles and
tangents
6.4: Angles in a circle
Try this! p 335-336
Analysis task 2: Road accident analysis
Analysis task 3: Cyclic quadrilaterals
6.4: Angles in a circle
Try this! p 335-6
Examples 1, 2, 3, 4, 5
Ex. 6.4 Q 1 – 5
Analysis task 2: Road accident analysis
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19
MathsWorld 9—VELS links by chapter
Chapter 6
Two-dimensional space
Level
Standard/Progression point
MathsWorld 9
Students use Pythagoras’ theorem and trigonometric
ratios (sine, cosine and tangent) to obtain lengths of
sides, angles and the area of right-angled triangles.
Analysis task 2: Road accident analysis
Students form and test mathematical conjectures;
for example, ‘What relationship holds between the
lengths of the three sides of a triangle?’
6.4: Angles in a circle
Try this! p 336
Analysis task 3: Cyclic quadrilaterals
5.0
Students explain geometric propositions (for
example, by varying the location of key points
and/or lines in a construction).
5.5
•
5.5
•
6.1: Angles, parallel lines and triangles
Try this! p 310
Ex. 6.1 Q 9
6.2: Quadrilateral properties
Example 1
Ex. 6.2 Q 1, 4
6.4: Angles in a circle
Try this! p 336, 338
6.1: Angles, parallel lines and triangles
Try this! p 307
Ex. 6.1 Q 1, 2, 9, 19
6.2: Quadrilateral properties
Try this! p 310
Example 1
Ex. 6.2 Q 1, 6, 7
6.4: Angles in a circle
Try this! pp 335-336
Analysis task 3: Cyclic quadrilaterals
Measurement,
Chance and Data
6.0
Structure
6.0
Working
mathematically
6.0
justification or proof of generalisations made
from specific cases
selection and use of technology to explore
geometric and algebraic relationships and data
trends
At Level 6, students formulate and test conjectures,
generalisations and arguments in natural language
and symbolic form (for example, ‘if m2 is even then
m is even, and if m2 is odd then m is odd’).
6.0
They follow formal mathematical arguments for the
truth of propositions.
6.0
They generalise from one situation to another, and
investigate it further by changing the initial
constraints or other boundary conditions.
6.1: Angles, parallel lines and triangles
Try this! p 307, 310
Example 2
Ex. 6.1 Q 1, 2, 8, 9, 19
6.2: Quadrilateral properties
Example 1
Ex. 6.2 Q 1, 3, 4, 6, 7
6.3: Polygons
Try this! p 331
6.4: Angles in a circle
Try this! pp 335-336
Analysis task 3: Cyclic quadrilaterals
6.1: Angles , parallel lines and
triangles
Try this! p 307, 310
Example 2
6.2: Quadrilateral properties
Example 1
6.4: Polygons
Try this! p 331
MathsWorld 9 Teacher edition
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20
MathsWorld 9—VELS links by chapter
Chapter 6
Two-dimensional space
Level
Standard/Progression point
MathsWorld 9
6.0
They use geometry software or graphics calculators
to create geometric objects and transform them,
taking into account invariance under
transformation.
6.4: Angles in a circle
Try this! p 335-336,
Analysis task 3: Cyclic quadrilaterals
MathsWorld 9 Teacher edition
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21
MathsWorld 9—VELS links by chapter
Chapter 7
Similarity and trigonometry
Level
Standard/Progression point
MathsWorld 9
5.0
Students use a range of strategies for approximating
the results of computations, such as front-end
estimation and rounding
(for example, 925 ÷ 34 ≈ 900 ÷ 30 = 30).
Most questions require students to round
answers to a specified or to a sensible
number of decimal places.
5.25
•
*Chapter pre-test Q 2, 3, 7, 10
7.2: Similar triangles
Example 2
Ex. 7.2 Q 2, 4, 5, 8, 11, 12, 13
Number
solution of problems involving ratio and
proportion
Space
5.0
They use the properties of parallel lines and
transversals of these lines to calculate angles that
are supplementary, corresponding, allied (cointerior) and alternate.
They relate similarity to enlargement from a
common fixed point.
They use lines, grids, contours, isobars, scales and
bearings to specify location and direction on plans
and maps.
5.0
5.0
5.25
•
6.0
At Level 6, students represent two- and threedimensional shapes using lines, curves, polygons
and circles.
6.0
Students use the conditions for shapes to be
congruent or similar.
6.0
They apply isometric and similarity transformations
of geometric shapes in the plane.
6.0
They determine the effect of changing the scale of
one characteristic of two- and three-dimensional
shapes (for example, side length, area, volume and
angle measure) on related characteristics.
application of the angle properties of parallel
lines and transversals to other geometrical
problems
7.5: Applying trigonometry
Try this! p 398
Try this! p 360
Ex. 7.1 Q 6
7.5: Applying trigonometry
Example 3
Ex. 7.5 Q 9, 10, 12
7.5: Applying trigonometry
Try this! p 398
Example 3
Ex. 7.5Q 5
7.5: Applications of trigonometry
Examples 1, 2, 3
Ex. 7.5 All questions
7.1: Similarity and scale
Examples 1, 2
Ex. 7.1 Q 1, 2
7.2: Similar triangles
Examples 1, 2
Ex. 7.2 Q 1, 3, 5, 6, 7, 9, 10, 11
7.1: Similarity and scale
Try this! p 360
Ex. 7.1 Q 6, 9
Chapter Warm-up p 358
7.1: Similarity and scale
Try this! p 361
Ex. 7.1 Q 3, 4, 5, 7, 8, 10, 11, 12
Measurement,
Chance and Data
5.25
•
use of similarity and scale to calculate side
lengths in triangles
7.2: Similar triangles
Example 2
Ex. 7.2 Q 2, 4, 5, 7, 8, 11, 12, 13
7.3: Trigonometric ratios
Try this! p 374, 375
MathsWorld 9 Teacher edition
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22
MathsWorld 9—VELS links by chapter
Chapter 7
Similarity and trigonometry
Level
Standard/Progression point
MathsWorld 9
5.5
•
5.75
•
7.3: Trigonometric ratios
Try this! pp 374, 375
Examples 1 – 3
Ex. 7.3 Q 5 – 17
Analysis task 2: Boom angles
7.4: Calculating angles
Examples 1, 2
Ex. 7.4 Q 3 – 9
6.0
Students decide on acceptable or tolerable levels of
error in a given situation.
Students use Pythagoras’ theorem and trigonometric
ratios (sine, cosine and tangent) to obtain lengths of
sides, angles and the area of right-angled triangles.
Analysis task 3: Angle errors
5.25
•
All sections
5.5
•
6.0
use of trigonometric ratios to calculate
unknown sides in a right-angled triangle
calculation of unknown angle in a right-angled
triangle using trigonometric ratios
7.3: Trigonometric ratios
Ex. 7.3 Q 15, 16
Analysis task 2: Boom angles part b
Working
mathematically
6.0
6.0
6.0
6.0
evaluation of the appropriateness of the results
of their own calculations
selection and use of technology to explore
geometric and algebraic relationships and data
trends
At Level 6, students formulate and test conjectures,
generalisations and arguments in natural language
and symbolic form (for example, ‘if m2 is even then
m is even, and if m2 is odd then m is odd’).
They judge the reasonableness of their results based
on the context under consideration.
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
They use geometry software or graphics calculators
to create geometric objects and transform them,
taking into account invariance under
transformation.
Calculators are used in sections 7.3 to 7.5
for angles and trigonometric calculations.
7.2: Similar triangles
Ex. 7.2 Q 6, 9, 11
Analysis task 1: Quadrilateral midpoints
Sections 7.2 – 7.5
Analysis task 2: Boom angles
Analysis task 3: Angle errors
Analysis task 1: Quadrilateral midpoints
MathsWorld 9 Teacher edition
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23
MathsWorld 9—VELS links by chapter
Chapter 8
Functions and modelling
Level
Standard/Progression point
MathsWorld 9
5.0
Students use coordinates to identify position in the
plane.
*Chapter pre-test Q 1
8.2: Formulating functions
Example 1b
Ex. 8.2 Q 1 – 5
5.75
•
8.3: Linear functions
Ex. 8.3 Q 9
Space
application of geometrical transformations to
graphs
Measurement,
Chance and Data
5.5
•
5.75
•
6.0
They distinguish informally between association
and causal relationship in bi-variate data, and make
predictions based on an estimated line of best fit for
scatter-plot data with strong association between
two variables.
qualitative judgement of positive or negative
correlation and strength of relationship and, if
appropriate, application of gradient to find a
line of good fit by eye.
placement of a line of best fit on a scatter plot
using technology and, where appropriate, use
of a line of best fit to make predictions.
8.7: Mathematical models
Try this! p 487, 489
Ex. 8.7 Q 6, 7, 14
8.7: Mathematical modelling
Try this! p 485
Example 1
Try this! p 489
Ex. 8.7 Q 6, 7, 14
Try this! p 489
8.7: Mathematical models
Example 1
Ex. 8.7 Q 5, 6, 7, 8, 12, 14
Structure
5.0
They recognise and use inequality symbols.
5.0
They solve simple inequalities such as y ≤ 2x+ 4
and decide whether inequalities such as x2 > 2y are
satisfied or not for specific values of x and y.
Students identify a function as a one-to-one
correspondence or a many-to-one correspondence
between two sets.
5.0
5.0
They represent a function by a table of values, a
graph, and by a rule.
5.0
They describe and specify the independent variable
of a function and its domain, and the dependent
variable and its range.
8.1: What is a function?
Examples 3, 4
Ex. 8.1 Q 9, 10
*Chapter pre-test Q 4, 5
8.1: What is a function?
Try this! p 424
Ex. 8.1 Q 7
*Chapter pre-test Q 1, 2, 3, 6, 8
8.1: What is a function?
Try this! p 427
Examples 1, 2
Ex. 8.1 Q 5 – 8
*Chapter pre-test Q 8
8.1: What is a function?
Examples 2, 3
Ex. 8.1 Q 7, 8
8.2: Formulating functions
Example 1
Ex. 8.2 Q 2, 3, 4, 5
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24
MathsWorld 9—VELS links by chapter
Chapter 8
Functions and modelling
Level
Standard/Progression point
5.0
They construct tables of values and graphs for
linear functions.
5.0
They use linear and other functions such as f(x) = 2x
− 4,
xy = 24, y = 2x and y = x2 − 3 to model various
situations.
5.25
•
5.25
•
5.25
•
5.5
•
solution of equations by graphical methods
identification of linear, quadratic and
exponential functions by table, rule and graph
in the first quadrant
knowledge of the quantities represented by the
constants m and c in the equation
y = mx + c
representation of numbers in a geometric
sequence (constant multiple, constant
percentage change) as an exponential function
MathsWorld 9
8.2: Formulating functions
Example 1
Ex. 8.2 Q 1, 3, 5
8.3: Linear functions
Try this! p 443
Examples 3, 4, 5, 6, 7, 8
4, 5, 6
Chapter Warm-up Try this!
8.2: Formulating functions
Example 1
Ex. 8.2 Q 1 – 5
8.3: Linear functions
Ex. 8.3 Q 17 – 21
8.4: Reciprocal functions
Try this! p 463
Example 1
Ex. 8.4 Q 1 – 5
8.5: Exponential functions
Try this! p 469
Example 1
Ex. 8.5 Q 1, 3, 4, 5
8.6: Quadratic functions
Try this! p 476
Examples 1, 2
Ex. 8.6 Q 1, 2, 3
8.3: Linear functions
Example 9
Ex. 8.3 Q 22
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 2.2 p 167; 2.4 p 169
TI 89 2.2 p 204; 2.4 p 206
8.3: Linear functions
Examples 1 – 9
Ex. 8.3 Q 1 – 25
8.5: Exponential functions
Example 1
Ex. 8.5 Q 1 – 6
8.6: Quadratic functions
Examples 1, 2
Ex. 8.6 Q 1 – 4
*Chapter pre-test
Q 2, 3, 7, 8
8.3: Linear functions
Try this! p 443
Examples 1, 2, 3
Ex. 8.3 Q 1, 6, 7, 12, 17, 18, 19, 20
8.2: Formulating functions
Ex. 8.2 Q 2
8.5: Exponential functions
Try this! p 470
Ex. 8.5 Q 1, 2, 3
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MathsWorld 9—VELS links by chapter
Chapter 8
Functions and modelling
Level
Standard/Progression point
MathsWorld 9
5.5
•
8.3: Linear functions
Ex. 8.3 Q 9, 10
5.75
•
5.75
•
6.0
Students identify and represent linear, quadratic and
exponential functions by table, rule and graph (all
four quadrants of the Cartesian coordinate system)
with consideration of independent and dependent
variables, domain and range.
knowledge of the relationship between
geometrical and algebraic forms for
transformations
testing of sequences by calculating first
difference, second difference or ratio between
consecutive terms to determine existence of
linear, quadratic and exponential functions
representation of algebraic models for sets of
data using technology
8.3: Linear functions
p 445
Example 3
Ex. 8.3 Q 2
8.5: Exponential functions
Try this! p 470
p 472
Ex. 8.5 Q 1
8.7: Mathematical models
Example 2
Ex. 8.7 Q 1 – 3
Analysis tasks 1: Water hyacinth
Analysis task 2: Video and DVD sales
8.5: Exponential functions
Try this! p 469
Example1
Ex. 8.5 Q 3 – 5
8.6: Quadratic functions
Try this! p 476, 479
Ex. 8.6 Q 3
8.7: Mathematical models
Try this! p 485, 489
Example 2
Ex. 8.7 Q 4, 6, 8, 11, 12, 13, 14, 15
Analysis task 1: Water hyacinth
Analysis task 3: Water tank costs
*Chapter pre-test
Q1–8
8.3: Linear functions
Try this! p 443
Examples 1 – 9
Ex. 8.3 Q 1 – 25
8.5: Exponential functions
Try this! p 470
Example 1
Ex. 8.5 Q 1 – 6
8.6: Quadratic functions
Try this! p 476
Examples 1, 2
Ex. 8.6 Q 1 - 4
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MathsWorld 9—VELS links by chapter
Chapter 8
Functions and modelling
Level
Standard/Progression point
MathsWorld 9
6.0
They distinguish between these types of functions
by testing for constant first difference, constant
second difference or constant ratio between
consecutive terms (for example, to distinguish
between the functions described by the sets of
ordered pairs
{(1, 2), (2, 4), (3, 6), (4, 8) …}; {(1, 2), (2, 4), (3,
8), (4, 14) …}; and {(1, 2), (2, 4), (3, 8), (4, 16)
…}).
8.3: Linear functions
Examples 3, 4
Ex. 8.3 Q 2
8.5: Exponential functions
Try this! p 470
Example 1
Ex. 8.5 Q 1 – 6
8.7: Functions and modelling
Example 2
Ex. 8.7 Q 1, 3, 13
Analysis task 1: Water hyacinth
Students use variables in general mathematical
statements.
Students develop simple mathematical models for
real situations (for example, using constant rates of
change for linear models).
They develop generalisations by abstracting the
features from situations and expressing these in
words and symbols.
They predict using interpolation (working with
what is already known) and extrapolation (working
beyond what is already known).
Variables used in all sections
Working
mathematically
5.0
5.0
5.0
5.0
5.25
•
5.5
•
5.75
•
6.0
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
development of alternative algebraic models for
a set of data and evaluation of their relative
merits
selection and use of technology to explore
geometric and algebraic relationships and data
trends
recognition of functionality of technology and
its limitations, such as image resolution,
discontinuities in graphs and systematic error
in computation through rounding
8.2: Formulating functions
Example 1
Ex. 8.2 Q 1 – 5
Chapter Warm-up Try this! p 420
8.7: Mathematical models
Try this! p 485
Examples 1, 2
Ex. 8.7 Q 5, 6, 7, 8, 9
8.7: Mathematical models
Ex. 8.7 Q 15
8.5: Exponential functions
Ex. 8.5 Q 3, 4
8.7: Mathematical models
Example 2
Ex. 8.7 Q 4, 6, 8, 14, 15
Analysis task 1: Water hyacinths
Analysis task 2: Video and DVD sales
Analysis task 3: Water tank costs
Tip: What viewing window is that? p 454
8.6: Quadratic functions
Try this! p 476
8.7: Mathematical models
Ex. 8.7 Q 4, 11
8.3: Linear functions
Example 9
Ex. 8.3 Q 3, 9, 10, 11, 25
8.6: Quadratic functions
Try this! p 476
8.7: Mathematical models
Example 2
Ex. 8.7 Q 8, 13, 14, 15
Analysis task 1: Water hyacinths
Analysis task 2: Video and DVD sales
Analysis task 3: Water tanks
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27
MathsWorld 9—VELS links by chapter
Chapter 9
Chance
Level
Standard/Progression point
MathsWorld 9
They know the decimal equivalents for the unit
fractions ½, 1/3, ¼, 1/5, 1/8, 1/9 and find equivalent
representations of fractions as decimals, ratios and
percentages (for example, a subset: set ratio of 4:9
can be expressed equivalently as
4/9 = 0.4 ≈ 44.44%).
They represent rational numbers in both fractional
and decimal (terminating and infinite recurring)
forms (for example, 14/ 25 = 1.16, = 47/ 99 ).
*Chapter pre-test Q 1, 3
Examples 1, 2, 3
5.0
Students identify empirical probability as long-run
relative frequency.
5.0
They calculate theoretical probabilities by dividing
the number of possible successful outcomes by the
total number of possible outcomes.
5.0
They use tree diagrams to investigate the
probability of outcomes in simple multiple event
trials.
5.25
•
*Chapter pre-test Q 6
9.3: Probability
Try this! p 536
Example 3
Ex. 9.3 Q 8 – 12
*Chapter pre-test Q 4, 5
9.3: Probability
Try this! p 536
Example 3
Ex. 9.3 Q 8 – 12
9.4: Diagrams and tables
Examples 1, 2
Ex. 9.4 Q 1, 2, 6
9.4: Diagrams and tables
Examples 1, 2, 3, 4, 5, 6
Ex. 9.4 Q 1 – 13
Analysis task 1: At the fair
Number
5.0
6.0
*Chapter pre-test Q 6
9.3: Probability
Ex 9.3 Q 10
Measurement,
Chance and Data
6.0
6.0
6.0
representation of compound events involving
two categories and the logical connectives and,
or and not using lists, grids (lattice diagrams),
tree diagrams, Venn diagrams and Karnaugh
maps (two-way tables) and the calculation of
associated probabilities
Students estimate probabilities based on data
(experiments, surveys, samples, simulations) and
assign and justify subjective probabilities in
familiar situations.
They list event spaces (for combinations of up to
three events) by lists, grids, tree diagrams, Venn
diagrams and Karnaugh maps (two-way tables).
They calculate probabilities for complementary,
mutually exclusive, and compound events (defined
using and, or and not).
9.3: Probability
Try this! p 536
Example 3
Ex. 9.3 Q 8 – 12
9.4: Diagrams and tables
Examples 1, 2, 6
Ex. 9.4 Q 1, 2, 6, 7, 8, 12
9.4: Diagrams and tables
Example 3
Ex. 9.4 Q 3, 5, 69, 10, 11, 13
Structure
5.0
They use Venn diagrams and tree diagrams to show
the relationships of intersection, union, inclusion
(subset) and complement between the sets.
9.2: Venn diagrams
Examples 1, 2
Ex. 9.2: Q 1 – 4
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MathsWorld 9—VELS links by chapter
Chapter 9
Chance
Level
Standard/Progression point
MathsWorld 9
5.0
They list the elements of the set of all subsets
(power set) of a given finite set and comprehend the
partial-order relationship between these subsets
with respect to inclusion (for example, given the set
{a, b, c} the corresponding power set is {Ø, {a},
{b}, {c}, {a, b}, {b, c}, {a, c}, {a, b, c}}.)
They test the validity of statements formed by the
use of the connectives and, or, not, and the
quantifiers none, some and all, (for example, ‘some
natural numbers can be expressed as the sum of two
squares’).
They apply these to the specification of sets defined
in terms of one or two attributes, and to searches in
data-bases.
9.1: The language of sets
Examples 4, 5
Ex. 9.1 Q 7, 8
5.0
5.0
5.25
•
5.5
•
relationships between two sets using a Venn
diagram, tree diagram and Karnaugh map
expression of the relationship between sets
using membership,  , complement, ′,
intersection,  , union,  , and subset,  ,
for up to two sets.
Student express relations between sets using
membership,  , complement, ′, intersection,  ,
union,  , and subset,  , for up to three sets.
They represent a universal set as the disjoint union
of intersections of up to three sets and their
complements, and illustrate this using a tree
diagram, Venn diagram or Karnaugh map.
6.0
6.0
9.2: Venn diagrams
Try this! p 522, 524
Examples 1, 2, 3, 4
Ex. 9.2 Q 1 – 17
9.2: Venn diagrams
Try this! p 522, 524
Examples 1, 2, 3, 4
Ex. 9.2 Q 1 – 17
9.2: Venn diagrams
Example 4
Ex. 9.2 Q 1g – 15
9.4: Diagrams and tables
Examples 3, 4, 5
Ex. 9.4 Q 3, 4, 5, 10, 11
9.1: The language of sets
Examples 1 – 9
Ex. 9.1 Q 1 – 17
9.1: The language of sets
Examples 1, 4, 6, 7, 8, 9
Ex. 9.1 Q 1 – 15
9.2: Venn diagrams
Examples 1, 2
Ex. 9.2
9.4: Diagrams and tables
Examples 1, 2, 3, 4, 5
Ex. 9.4
Working
mathematically
5.75
6.0
•
simulation of events using technology
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
Analysis task 3: A day at the races
Analysis task 3: A day at the races
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29
MathsWorld 9—VELS links by chapter
Chapter 10
Analysing data
Level
Standard/Progression point
MathsWorld 9
5.0
Students use appropriate technology to generate
random numbers in the conduct of simple
simulations.
5.0
Students organise, tabulate and display discrete and
continuous data (grouped and ungrouped) using
technology for larger data sets.
5.0
They represent uni-variate data in appropriate
graphical forms including dot plots, stem and leaf
plots, column graphs, bar charts and histograms.
5.0
They calculate summary statistics for measures of
centre (mean, median, mode) and spread (range,
and mean absolute difference), and make simple
inferences based on this data.
5.25
•
5.5
•
5.75
•
10.1: Sampling and questionnaires
Examples 1, 2
Try this! p 565
Ex. 10.1 Q 4
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 1.7 p 165
TI 89 1.8 p 199
*Chapter pre-test
Q 1, 2, 3, 7, 8, 9, 10
10.3: Representing data
Examples 1, 2, 3, 4
Ex. 10.3 Q 1 – 11
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 6.1– 6.6 p 183–190
TI 89 6.1– 6.5 p 218–224
*Chapter pre-test Q 1, 2, 3, 9, 10
10.3: Representing data
Examples 1, 2, 3, 4
Ex.10.3 Q 1 – 11
*Chapter pre-test Q 3, 4, 5, 6
10.4: Summarising data
Examples 1, 2, 3, 4
Ex. 10.4 Q 1 – 15
10.3: Representing data
Ex. 10.3 Q 6 – 9
10.4: Summarising data
p. 604
Ex. 10.4 Q 1 – 14
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 6.1– 6.6 p 183–190
TI 89 6.1– 6.5 p 218–224
10.4: Summarising data
Example 4
Ex. 10.4 Q 5 – 8, 12
10.5: Boxplots
Examples 1, 2
Ex. 10.5 Q 1 – 10
10.1: Sampling and questionnaires
Example 4
Ex. 10.1 Q 1, 2, 3, 7, 8, 9, 10, 11, 12
Measurement,
Chance and Data
6.0
representation of statistical data using
technology
display of data as a box plot including
calculation of quartiles and inter-quartile range
and the identification of outliers
use of surveys as a means of obtaining
information about a population, including
awareness that sample results will not always
provide a reasonable estimate of population
parameters.
Students comprehend the difference between a
population and a sample.
10.1: Sampling and questionnaires
Analysis task 3: Investigating sample size
MathsWorld 9 Teacher edition
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30
MathsWorld 9—VELS links by chapter
Chapter 10
Analysing data
Level
Standard/Progression point
6.0
They generate data using surveys, experiments and
sampling procedures.
They calculate summary statistics for centrality
(mode, median and mean), spread (box plot, interquartile range, outliers) and association (by-eye
estimation of the line of best fit from a scatter plot).
6.0
MathsWorld 9
*Chapter pre-test
Q3–6
10.4: Summarising data
Examples 1 – 4
Ex. 10.4 Q 1 – 15
10.5: Boxplots
Examples 1, 2
Ex. 10.5 Q 1 – 10
Working
mathematically
5.0
Students use variables in general mathematical
statements.
5.5
•
6.0
6.0
selection and use of technology to explore
geometric and algebraic relationships and data
trends
They generalise from one situation to another, and
investigate it further by changing the initial
constraints or other boundary conditions.
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
10.4: Summarising data
Examples 1, 2
10.4: Summarising data
Exercise 10.4 Q 1 - 11
Analysis task 3: Investigating sample size
10.1: Sampling and questionnaires
Examples 1, 2
Ex. 10.1 Q 4
10.2: Interpreting statistical
information
Example 2
Ex. 10.2 Q 2
10.4: Summarising data
Ex. 10.4 All questions
10.5: Boxplots
Tip p 611
Ex. 10.5 Q 7, 8, 9
Analysis task 1: Investigating languages
Analysis task 2: Who can spell?
Analysis task 3: Investigating sample size
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31
MathsWorld 9—VELS links by chapter
Chapter 11
Algebra toolbox 2
Level
Standard/Progression point
MathsWorld 9
•
Analysis task 1: A family of parabolas
Space
5.75
application of geometrical transformations to
graphs
Structure
5.0
They recognise and use inequality symbols.
5.25
•
5.5
•
5.75
•
5.75
•
solution of equations by graphical methods
knowledge of the relationship between
geometrical and algebraic forms for
transformations
factorisation of simple quadratic expressions
and use of the null factor law for solution of
equations
formulation of pairs of simultaneous equations
and their graphical solution
*Chapter pre-test
Q5
11.2: Other techniques for solving
equations
Examples 1, 2
Try this! p 662, 665
Ex. 11.2 Q 1 – 12
Analysis task 2: Simba's SMS costs
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 2.2 p 167; 2.4 p 169
TI 89 2.2 p 204; 2.4 p 206
11.1: Solving quadratic equations
Try this! p 649, 651
Ex. 11.1 Q 5, 6, 11
11.1: Solving quadratic equations
Examples 2, 3
Ex. 11.1 Q 4, 7, 14
11.2: Other techniques for solving
equations
Ex. 11.2 Q 2, 3, 5, 7
Analysis task 2: Simba’s SMS costs
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 2.2 p 167; 2.4 p 169
TI 89 2.2 p 204; 2.4 p 206
Working
mathematically
5.5
•
selection and use of technology to explore
geometric and algebraic relationships and data
trends
Chapter Warm-up Try this! p 641
Analysis task 2: Simba’s SMS costs
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 2.1– 2.4 p 166–169
TI 89 2.1– 2.4 p 204– 206
MathsWorld 9 Teacher edition
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32
MathsWorld 9—VELS links by chapter
Chapter 11
Algebra toolbox 2
Level
Standard/Progression point
MathsWorld 9
5.75
•
6.0
They generalise from one situation to another, and
investigate it further by changing the initial
constraints or other boundary conditions.
They select and use technology in various
combinations to assist in mathematical inquiry, to
manipulate and represent data, to analyse functions
and carry out symbolic manipulation.
11.2: Other techniques for solving
equations
Examples 1, 2, 3
Try this! p 662
Ex. 11.2 Q 12, 13, 14
Analysis task 3: Scale issues
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 3.1, 3.2 p 171, 172
TI 89 3.1, 3.2 p 208, 209
Analysis task 1: A family of parabolas
6.0
recognition of functionality of technology and
its limitations, such as image resolution,
discontinuities in graphs and systematic error
in computation through rounding
Chapter Warm-up Try this! p 641
Analysis task 1: A family of parabolas
MathsWorld 9 Practice and
Enrichment Workbook (and CD):
Technology toolkit
TI 83/84 1.7– 1.8 p 164–165; 2.1– 2.4 p
166–169; 6.1 – 6.7 p 183–192
TI 89 1.7– 1.9 p 198–200; 2.1– 2.4 p
202– 206; 6.1 – 6.7 p 219–227
MathsWorld 9 Teacher edition
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33