teaching program year 9

New Century Maths 9 Stages 5.1/5.2 teaching program (p. 1)

Teaching program

New Century Maths 9 Stages 5.1/5.2 for the Australian Curriculum

Year 9 topics

Week SEMESTER 1

Week SEMESTER 2 Term 1

1 1. Pythagoras theorem

(Measurement and Geometry)

Term 3

1 7. Equations

(Number and Algebra)

2

2

3

3

4

2. Working with numbers

(Number and Algebra) 4

8. Earning money


5

5

6

6

9. Investigating data

(Statistics and Probability)

7

3. Algebra


7

8

8

9 9 Lost time

10

Lost time

10 10. Surface area

and volume

Term 2 1

4. Trigonometry


Term 4 1


2

2

3

3

11. Coordinate geometry

and graphs

4

5. Indices


4


5

5

6

6

12. Probability


7

6. Geometry


7

8

8

13. Congruent and

similar figures

9

Lost time

9


10

10

CURRICULUM STRANDS

Number and Algebra Measurement and Geometry Statistics and Probability


Year 10 topics

Week SEMESTER 1

Week SEMESTER 2 Term 1

1 1. Interest and depreciation


Term 3 1

8. Trigonometry


2

2

3

2. Coordinate geometry


3

4

4

9. Simultaneous equations


5

3. Surface area and volume


5

6

6

10. Probability


7

4. Algebra


7

8

8

9

Lost time

9

11. Geometry


10

10

Term 2

1 5. Investigating data


Term 4

1

2

2

3

3

OPTION TOPICS

4

6. Graphs


4

Stage 5.3 chapters

12. Surds

5

5

13. Products and factors

14. Quadratic equations

6

6

and the parabola

7

7. Equations and inequalities


7

8

8

9

9

10

Lost time 10

Lost time

CURRICULUM STRANDS

Number and Algebra Measurement and Geometry Statistics and Probability


1. PYTHAGORAS THEOREM Time: 3 weeks (Term 1, Week 1) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 1

NSW and Australian Curriculum references: Measurement and Geometry

Right-angled triangles (Pythagoras) / Real numbers

Investigate the concept of irrational numbers, including (8NA186)

Right-angled triangles (Pythagoras) / Pythagoras and trigonometry

Investigate Pythagoras theorem and its application to solving simple problems involving right-angled triangles (NSW Stage 4 / 9MG222)

NSW Stage 4 outcomes

A student:

MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

MA4-2 WM applies appropriate mathematical techniques to solve problems

MA4-16 MG applies Pythagoras theorem to calculate side lengths in right-angled triangles and solves related problems

INTRODUCTION

This is actually the revision of a Stage 4 topic (NSW syllabus) introduced in Year 8, and is not technically Stage 5 work. Note,

however, that in the national Australian curriculum Pythagoras theorem is introduced in Year 9. Emphasis should be placed

upon understanding the theorem and using it to solve problems involving the sides of right-angled triangles. Students should

gain an understanding of Pythagoras theorem, rather than just being able to recite the formula. Answers for unknown sides

should be given as exact surds or decimal approximations.

CONTENT

1 Squares, square roots and surds 8NA186 U C

investigate the concept of irrational numbers

2 Pythagoras theorem 9MG222 U F R C

identify the hypotenuse as the longest side in any right-angled triangle and also as the side opposite the right angle

establish the relationship between the lengths of the sides of a right-angled triangle in practical ways, including using

digital technologies

3 Finding the hypotenuse 9MG222 U F

4 Finding a shorter side 9MG222 U F

solve practical problems involving Pythagoras theorem, approximating the answer as a decimal and giving an exact

answer as a surd

5 Mixed problems 9MG222 F

6 Testing for right-angled triangles 9MG222 U F R C

use the converse of Pythagoras theorem to establish whether a triangle has a right angle

7 Pythagorean triads 9MG222 U F C

identify a Pythagorean triad as a set of three numbers such that the sum of the squares of the first two equals the square

of the third

8 Pythagoras theorem problems 9MG222 F PS C

9 Revision and mixed problems

RELATED TOPICS

Year 8: Pythagoras theorem, Geometry, Area and volume, Congruent figures Year 9: Trigonometry, Coordinate geometry

Year 10: Coordinate geometry, Trigonometry

PROFICIENCY STRANDS / WORKING MATHEMATICALLY

U = Understanding (knowing and relating maths): Understanding how the sides of a right-angled triangle are related by

Pythagoras theorem

F = Fluency (applying maths): Selecting appropriate techniques to find unknown sides and test right-angled triangles

PS = Problem solving (modelling and investigating with maths): Using Pythagoras theorem to solve measurement

problems

R = Reasoning (generalising and proving with maths): Proving that a triangle is right-angled given the lengths of its


sides

C = Communicating (describing and representing maths): Describing and explaining Pythagoras theorem in words and

as a formula

EXTENSION IDEAS

How did mathematicians find square roots before calculators and computers? Investigate Newtons method.

Perigals dissection and other formal proofs of Pythagoras theorem

Pythagoras and the Pythagoreans, history of Pythagoras theorem

Harder problems: two-stage or in three dimensions, for example, longest diagonal in a rectangular prism

History of Pythagorean triads, properties of Pythagorean triads

Length of an interval on the number plane (also in the Coordinate geometry topic)

Irrational numbers, graphing surds on a number line, simplifying surds

The real number system, proof that 2 is irrational

TEACHING NOTES AND IDEAS

Pythagoras theorem was actually discovered by others, centuries before Pythagoras was born around 580 BCE.

Use knotted rope to show how ancient Egyptians builders made a 3-4-5 triangle to create a right angle.

State Pythagoras theorem in words and as a formula. Stress that it works for right-angled triangles only. Emphasise correct setting-out of solutions. Check answers. Obviously its wrong if the hypotenuse is shorter than one of the other sides.

For students who have difficulty solving equations, it may be easier to teach x2 = p2 q2, where x is the unknown side. Students then remember to add if x is the hypotenuse and subtract if x is one of the shorter sides.

Demonstrate that the length 2 can be constructed using a right-angled isosceles triangle.

There are different formulas for creating Pythagorean triads, such as (p2 q2, 2pq, p2 + q2), (n,2

12 n,

2

12 n) for odd n, (2n

+ 1, 2n2 + 2n, 2n2 + 2n + 1). Multiplying or dividing a triad by a constant gives another triad: we can use this to create

decimal triads such as (2.8, 9.6, 10).

Pythagorean triads (useful for triangle problems): (3, 4, 5) (5, 12, 13) (6, 8, 10) (7, 24, 25) (8, 15, 17) (9, 12, 15) (9, 40, 41) (10, 24, 26) (11, 60, 61) (12, 16, 20) (12, 35, 37) (13, 84, 85) (14, 48, 50) (15, 20, 25) (15, 36, 39) (16, 30, 34) (16, 63, 65)

(18, 24, 30) (18, 80, 82) (20, 21, 29) (20, 48, 52) (20, 99, 101) (21, 28, 35) (21, 72, 75) (24, 32, 40) (24, 45, 51) (24, 70, 74)

(25, 60, 65) (27, 36, 45) (28, 45, 53) (28, 96, 100) (30, 40, 50) (30, 72, 78) (32, 60, 68) (33, 44, 55) (33, 56, 65) (35, 84 , 91)

(36, 48, 60) (36, 77, 85) (39, 52, 65) (39, 80, 89) (40, 42, 58) (40, 75, 85) (40, 96, 104) (42, 56, 70) (45, 60, 75) (48, 55, 73)

(48, 64, 80) (48, 90, 102) (51, 68, 85) (54, 72, 90) (56, 90, 106) (57, 76, 95) (60, 63, 87) (60, 80, 100) (60, 91, 109) (63, 84,

105) (65, 72, 97) (66, 88, 110) (69, 92, 115) (72, 96, 120) (80, 84, 116).

ASSESSMENT IDEAS

Open-ended problems, for example, the length of the hypotenuse is 10 (or 10 ). What are the possible lengths of the other two sides?

Research assignment on Pythagoras and Pythagoras theorem.

Matching activities: Pythagoras theorem to diagrams.

Writing activity explaining Pythagoras theorem.

TECHNOLOGY

Spreadsheets can be used to find unknown sides or generate Pythagorean triads. Use the Internet to research the history of

Pythagoras and irrational numbers. Use dynamic geometry software to explore and prove Pythagoras theorem.

LANGUAGE

Hypotenuse is an ancient Greek word: hypo means under while teinousa means stretching because the hypotenuse stretches under a right angle.

Explain and reinforce the logic behind the converse of Pythagoras theorem.

From the NSW syllabus: The meaning of exact answer will need to be taught explicitly. Students may find some of the terminology/vocabulary encountered in word problems involving Pythagoras theorem difficult to interpret, for example, foot of a ladder, inclined, guy wire.


2. WORKING WITH NUMBERS Time: 3 weeks (Term 1, Week 4) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 2

NSW and Australian Curriculum references: Number and Algebra

Computation with Integers / Number and place value

Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (8NA183)

Fractions, Decimals and Percentages / Real numbers

Investigate terminating and recurring decimals (8NA184)

Solve problems involving the use of percentages, including percentage increases and decreases, with and without digital technologies (8NA187)

Ratios and Rates / Real numbers

Solve a range of problems involving rates and ratios, with and without digital technologies (8NA188)

Solve problems involving direct proportion; explore the relationship between graphs and equations corresponding to simple rate problems (9NA208)

Financial Mathematics / Money and financial mathematics

Solve problems involving profit and loss, with and without digital technologies (8NA189)

Solve problems involving simple interest (9NA211)

Time / Using units of measurement

Solve problems involving duration, including using 12- and 24-hour time within a single time zone (8MG199)


A student:

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts

MA5.1-2 WM selects and uses appropriate strategies to solve problems

MA5.1-4 NA solves financial problems involving earning, spending and investing money

MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions

MA5.2-2 WM interprets mathematical or real-life situations, systematically applying appropriate strategies to solve problems

INTRODUCTION

This topic reinforces mostly Stage 4 Number skills necessary for Year 9 and 10, with the only new concepts being simple

interest and converting rates. This is a short refresher topic that revises mental, pen-and-paper and calculator skills so dont

dwell too long on particulars. Keep it simple and make the lessons appropriate to the ability of your class. You may even like to

set part of this topic as a revision assignment rather than re-teach it all. Ensure that estimating and checking of answers are

reinforced during lessons. Also emphasise the importance of mental computation skills, such as in increasing $140 by 20%.

CONTENT

1 Integers 8NA183 U F

2 Decimals 8NA183 U F

carry out the four operations with rational numbers and integers, using efficient mental and written strategies and

appropriate digital technologies

3 Terminating and recurring decimals 8NA184 U F R C

investigate terminating and recurring decimals

4 Fractions 8NA183 U F

5 Percentages 8NA183 U F

6 Operations with percentages 8NA187 F PS C solve problems involving the use of percentages, including percentage increases and decreases

7 Percentages and money 8NA189 F PS C solve problems involving profit, loss, discounts and GST

8 Simple interest 9NA211 F PS C solve problems involving simple interest

9 Ratios and rates 8NA188 U F PS C solve a range of problems involving ratios and ratios, with and without digital technologies


10 Stage 5.2: Converting rates 9NA208 U F R C

convert between units for rates, for example, kilometres per hour to metres per second

11 Time differences 8MG199 U F PS C solve problems involving duration, including using 12- and 24-hour time within a single time zone


RELATED TOPICS

Year 8: Working with numbers, Fractions and percentages, Ratios, rates and time

Year 9: Indices

Year 10: Interest and depreciation


U = Understanding (knowing and relating maths): Understanding operations with integers, decimals, percentages, ratios

and rates, and the relationships between them

F = Fluency (applying maths): Using appropriate strategies for evaluating expressions: mental, pen-and-paper, calculator

PS = Problem solving (modelling and investigating with maths): Solving problems using integers, decimals,

percentages, ratios and rates

R = Reasoning (generalising and proving with maths): Finding patterns in terminating and recurring decimals, reasoning

to convert rates

C = Communicating (describing and representing maths): Using correct notation for integers, decimals, percentages,

ratios and rates

EXTENSION IDEAS

Rational vs irrational numbers

Investigate the history of calculation methods, for example, Italian multiplication

Investigate the value of 0.9 . Is it really equal to 1? Convert recurring decimals to fractions (Stage 5.3)

Investigate unfamiliar calculator keys


Encourage students to develop a number sense rather than rely on the calculator too often. Check that answers make sense. Estimate first.

Include open-ended questions such as finding two fractions that have a sum of 5

11 or three fractions between

1

8 and

1

5.

Investigate patterns in the recurring decimals of the fraction families of the sixths, sevenths and ninths.

Some decimals are neither terminating nor recurring. Their digits run endlessly, but without repeating, for example, 2 = 1.4142135 and = 3.1415926

Investigate the percentage forms of fraction families such as the eighths and the sixths. What are 162

3% and 37.5% as

fractions?

Students should learn calculator shortcuts for percentage calculations, such as multiplying by 1.15 to increase a number by 15%. Also investigate the [%] key if appropriate.

Dont round if an exact answer (fraction or surd) is required.

For what type of problems is it appropriate to round? What is the difference between 7 cm and 7.0 cm?

The unitary method is quite powerful and can be applied to percentages, fractions, decimals, ratios and rates.

ASSESSMENT IDEAS

Non-calculator test.

Revision assignment.

TECHNOLOGY

Use calculators to evaluate mixed expressions, including the use of the parentheses and ANS keys, but beware of cheap

calculators that do not follow order of operations rules. Students can use the spreadsheet to round or order decimals, or convert fractions to terminating and recurring decimals.


LANGUAGE

Reinforce the language of approximation: approximate, write correct to, round to, n decimal places, nearest tenth. Note that the NSW syllabus now prefers the term rounding to rounding off.

Terminating means stopping; recurring means repeating.

When expressing quantities as percentages or fractions, reinforce the importance of the quantity that follows of in the question, such as What percentage of the class are boys? This quantity appears in the denominator of the calculation. Also differentiate between cost price and selling price.


3. ALGEBRA Time: 3 weeks (Term 1, Week 7) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 3


Algebraic Techniques 1 and 2 / Patterns and algebra

Factorise algebraic expressions by identifying numerical factors (8NA191)

Factorise algebraic expressions by identifying algebraic factors (NSW Stage 4)

Simplify algebraic expressions involving the four operations (8NA192)

Algebraic Techniques / Patterns and algebra

Apply the four operations to simple algebraic fractions with numerical denominators (10NA232)

Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (9NA213)


A student:


MA5.2-3 WM constructs arguments to prove and justify results

MA5.2-6 NA simplifies algebraic fractions

INTRODUCTION

This topic reinforces mostly Stage 4 Algebra skills, with the only new concepts being algebraic fractions and expanding

binomial products for Stage 5.2. In Year 8, students learned to simplify algebraic expressions, including the processes of

expanding and factorising. This topic is fairly technical and abstract so each skill should be revised with care and precision

appropriate to the level of the class. Students should practise and master each skill before moving onto the next one.

CONTENT

1 From words to algebraic expressions 7NA177 U F PS R C move fluently between algebraic and word representations as descriptions of the same situation

2 Substitution 7NA176 U F PS create algebraic expressions and evaluate them by substituting a given value for each variable

3 Adding and subtracting terms 8NA192 U F R C

4 Multiplying and dividing terms 8NA192 U F R C

simplify algebraic expressions involving the four operations

5 Stage 5.2: Adding and subtracting algebraic fractions 10NA232 U F R C

6 Stage 5.2: Multiplying and dividing algebraic fractions 10NA232 U F R C

apply the four operations to simple algebraic fractions

7 Expanding expressions 9NA213 U F R C

apply the distributive law to the expansion of algebraic expressions and collect like terms where appropriate

8 Factorising expressions 8NA191 U F R C

factorise algebraic expressions

9 Stage 5.2: Expanding binomial products 9NA213 U F R C

apply the distributive law to the expansion of binomials


RELATED TOPICS

Year 8: Algebra, Equations

Year 9: Indices, Equations

Year 10: Algebra, Equations and inequalities


U = Understanding (knowing and relating maths): Knowing how variables and algebraic expressions work to describe

and generalise number patterns and rules


F = Fluency (applying maths): Interpreting and writing algebra fluently and selecting the right strategy to simplify,

expand and factorise expressions

PS = Problem solving (modelling and investigating with maths): Using expressions and formulas to represent and solve

problems

R = Reasoning (generalising and proving with maths): Using algebra to represent, generalise and simplify number

patterns and rules

C = Communicating (describing and representing maths): Describing number patterns and rules algebraically

EXTENSION IDEAS

More challenging problems involving substitution and translating worded statements into algebraic expressions

Special binomial products (Stage 5.3), for example, (x + 5)(x 5), (x + 2)2

Factorising quadratic expressions (Year 10, x2 + bx + c only)

Factorising by grouping in pairs (Stage 5.3)


Investigate pattern in problems such as handshakes, page numbering, angle sum of a polygon, checkerboard squares, creases in paper-folding, Pascals triangle.

From the NSW syllabus: To gain an understanding of algebra, students must be introduced to the concepts of pronumerals, expressions, unknowns, equations, patterns, relationships and graphs in a wide variety of contexts. For each successive

context, these ideas need to be redeveloped. Students need gradual exposure to abstract ideas as they begin to relate to

algebraic terms to real situations.

Demonstrate adding algebraic terms in constructing perimeter formulas and multiplying terms in constructing area formulas.

Some students believe 4a + 2b a = [4a +] [2b ] a = 5a 2b. Encourage them to group each term with the sign before it: 4a [+ 2b] [ a] = 3a + 2b.

Common mistakes: 2a a = 2, 3b2 = 3b 3b. Explain that the index 2 belongs to the b only.

For simplifying algebraic terms, include mixed exercises so that students experience all four operations and identify which rule to use. Include terms that are constants or which have powers.

Demonstrate operations with numerical fractions before moving onto algebraic fractions. Adding and subtracting problems may be restricted to fractions with numerical denominators and monomial (one-term) numerators only.

NSW syllabus: Check expansions and factorisations by performing the reverse process. Include examples involving negative terms.

Describe the process involved when expanding a binomial product. There are many approaches: distributive law, long multiplication, areas of rectangles. Encourage students to look for patterns in their expanded results.

Demonstrate the equivalence of expansions and factorisations, for example (x + 2)(x 2) = x2 4 by substituting a value for x in both sides of the identity. Use a spreadsheet or graphics calculator.

Evaluate 982 by expanding (100 92)2. Evaluate 19 x 21 by expanding (20 1)(20 + 1). Investigate the mental calculation trick for squaring a 2-digit number ending in 5, found in Mental Skills 2A in Chapter 2

ASSESSMENT IDEAS

Writing activity on the use of variables and simplifying algebraic expressions

Research assignment or poster on the algebraic rules or the history/meaning of algebra

Vocabulary test

TECHNOLOGY

Note that spreadsheet formulas are written differently to algebraic formulas. CAS (Computer Algebra Systems) can be used to

simplify, expand or evaluate algebraic expressions.

LANGUAGE

Reinforce the meanings of variable, term, expression, simplify, evaluate, substitute, expand and factorise.

An algebraic term consists of a number and/or a variable, for example, 4p2. An algebraic expression is a phrase containing terms and one or more arithmetic operation, for example, 5x + 6. An equation is a sentence containing an expression, an = sign and an answer, for example, 5x + 6 = 26.

The word expand comes from writing out an expression the long way without brackets. Draw a diagram using rectangles and an array of dots to show equivalences such as 3(n + 2) = 3n + 6.

Emphasise the difference between expand and factorise, as students will often do the opposite of what is requested.

binomial = algebraic expression with two terms, for example 2ab b2 or x + 5, from the Latin bi nomen, two names.


4. TRIGONOMETRY Time: 3 weeks (Term 2, Week 1) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 4


Right-angled triangles (Pythagoras) / Pythagoras and trigonometry

Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled

triangles (9MG223)

Apply trigonometry to solve right-angled triangle problems (9MG224)


A student:



MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context

MA5.1-10 MG applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression



MA5.2-13 MG applies trigonometry to solve problems, including problems involving bearings

INTRODUCTION

This Stage 5 Measurement topic is entirely new to students, but they have met related areas such as geometry, scale drawings,

Pythagoras theorem, ratios and equations at Stage 4. Do not rush through this topicspend some time investigating right-

angled triangles and the sine, cosine and tangent ratios before applying them to solve problems. Stage 5.1 students work with

angles in degrees only, while Stage 5.2 students work in degrees and minutes. Ensure that students receive plenty of practice in

setting out their work correctly.

CONTENT

1 The sides of a right-angled triangle 9MG223 U C

2 The trigonometric ratios 9MG223 U C

3 Similar right-angled triangles 9MG223 U R C

use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles

4 Trigonometry on a calculator 9MG223 U F

5 Finding an unknown side 9MG224 U F PS

select and use appropriate trigonometric ratios in right-angled triangles to find unknown sides, where the given angle is

measured in degrees

(STAGE 5.2) find the lengths of unknown sides in right-angled triangles where the given angle is measured in degrees

and minutes

apply trigonometry to solve right-angled triangle problems

6 Finding more unknown sides 9MG224 U F PS

select and use appropriate trigonometric ratios in right-angled triangles to find the hypotenuse

7 Finding an unknown angle 9MG224 U F PS

select and use appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest

degree

(STAGE 5.2) find the size in degrees and minutes of unknown angles in right-angled triangles


RELATED TOPICS

Year 8: Pythagoras theorem Year 9: Pythagoras theorem, Geometry, Congruent and similar figures Year 10: Trigonometry



U = Understanding (knowing and relating maths): Learning basic trigonometry concepts and using them to find

unknown sides and angles in right-angled triangles

F = Fluency (applying maths): Applying appropriate methods to find unknown sides and angles

PS = Problem solving (modelling and investigating with maths): Apply trigonometric methods to real-life problems

R = Reasoning (generalising and proving with maths): Using similar right-angled triangles as a basis for introducing

trigonometry

C = Communicating (describing and representing maths): Using the terminology of trigonometry to describe the sides

and ratios of lengths in right-angled triangles

EXTENSION IDEAS

Angles of elevation and depression, bearings (Year 10).

The exact ratios, complementary relations such as cos 25 = sin 65, trigonometry of obtuse angles (Year 10 Stage 5.3)

The sine, cosine and tangent graphs (Year 10 Stage 5.3)


Note that you may need to give a brief explanation of similar triangles when introducing trigonometry, now that similar figures are no longer taught in Stage 4.

Make a clinometer. Calculate the heights of trees, flagpoles and buildings using trigonometry.

Investigate the history of the Babylonian base 60 system used in measuring angle size (and time). Students have already used the degrees-minutes-seconds button on the calculator for time calculations in Stage 4.

The trigonometric ratios are constant for a particular angle size, no matter how large the (similar) right-angled triangle. Compare measured values with calculator values. See NelsonNet worksheet Investigating the tangent ratio.

Students could verify their answers to trigonometric problems using scale drawings.

Students should set out their solutions properly and use correct trigonometric terminology. Encourage them to check the reasonableness of answers to trigonometric problems by making a rough scale drawing. Students need practice in drawing

diagrams for a given problem. Have students devise a problem for a given diagram and swap problems.

ASSESSMENT IDEAS

Practical test involving clinometers

Research project on the history or applications of trigonometry

TECHNOLOGY

Make sure that students have set their calculators in degrees mode. Display an old book of trigonometric tables to show what

students used before calculators became widely available. Use a spreadsheet to compare the ratios of the sides of similar right-

angled triangles. The trigonometric ratios can be calculated on a spreadsheet but the angle sizes must be converted from degrees

to radians first.

LANGUAGE

From the NSW syllabus: The word trigonometry is derived from two Greek words meaning triangle and measurement.

Angles of elevation and depression, and bearings, will be introduced in Year 10.

Stress that the hypotenuse is a fixed side in a right-angled triangle, while the opposite and adjacent sides depend upon the angle quoted. Students already know the hypotenuse from Pythagoras theorem.

From the NSW syllabus: Emphasis should be placed on correct pronunciation of sin as sine.

Encourage students to devise mnemonics for the trigonometric ratios.

The word minute comes from the Latin pars minuta prima, meaning the first (prima) division of a degree or hour. The word second comes from pars minuta secunda, meaning the second (secunda) division of a degree or hour.


5. INDICES Time: 3 weeks (Term 2, Week 4) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 5

NSW and Australian Curriculum references: Number and Algebra / Measurement and Geometry

Indices / Patterns and algebra

Extend and apply the index laws to variables, using positive-integer indices and the zero index (9NA212)

Simplify algebraic products and quotients using index laws (9NA231)

Indices / Real numbers

Apply index laws to numerical expressions with integer indices (9NA209)

Apply index laws to algebraic expressions involving integer indices (NSW Stage 5.2)

Numbers of any magnitude / Real numbers

Express numbers in scientific notation (9NA210) NSW Stage 5 outcomes

A student:



MA5.1-5 NA operates with algebraic expressions involving positive-integer and zero indices, and establishes the meaning of negative indices for numerical bases

MA5.1-9 MG interprets very small and very large units of measurement, uses scientific notation, and rounds to significant figures


MA5.2-7 NA applies index laws to operate with algebraic expressions involving integer indices

INTRODUCTION

In this topic, students are introduced to the index laws and negative indices. It examines indices both numerically and

algebraically, applying them so that students wont make mistakes such as 52 56 = 108. More time should be spent on

examining the number patterns generated by repeated multiplication so that the different types of powers are more readily

understood, especially the zero and negative indices. Scientific notation is also introduced for writing large and small numbers

using powers of ten.

CONTENT

1 Multiplying and dividing terms with the same base 9NA212 U F R C

2 Power of a power 9NA212 U F R C extend and apply the index laws to variables, using positive-integer indices

3 Powers of products and quotients 9NA231 U F R C simplify algebraic products and quotients using index laws

4 The zero index 9NA212 U F R C extend and apply the index laws to variables, using the zero index

5 Negative indices 9NA209 U F R C apply index laws to numerical expressions with integer indices

(NSW, STAGE 5.2) apply index laws to algebraic expressions involving integer indices

6 Summary of the index laws 9NA209 U F R C

7 Significant figures NSW U F R C identify significant figures and round numbers to a specified number of significant figures

8 Scientific notation 9NA210 U F R C express numbers in scientific notation

order numbers expressed in scientific notation

9 Scientific notation on a calculator 9NA210 U F PS R C enter and read scientific notation on a calculator

solve problems involving scientific notation


RELATED TOPICS

Year 8: Working with numbers, Algebra

Year 9: Algebra, Surface area and volume

Year 10: Algebra



U = Understanding (knowing and relating maths): Relating the index laws, zero and negative indices and scientific

notation

F = Fluency (applying maths): Using the correct method for simplifying algebraic expressions involving indices, reading

and writing scientific notation

PS = Problem solving (modelling and investigating with maths): Solving problems involving scientific notation

R = Reasoning (generalising and proving with maths): Using index laws to generalise rules about operating on terms

with indices

C = Communicating (describing and representing maths): Writing the index laws, significant figures and scientific

notation competently

EXTENSION IDEAS

Fractional indices, negative powers of fractions (Stage 5.3)

Engineering notation, a form of scientific notation


Begin with a numerical approach to the index laws. Demonstrate zero and negative powers by extending the repeated multiplication pattern backwards.

Open-ended question: find two terms that can be divided to give 27.

Verify the index laws by using a calculator. Explain why a particular algebraic sentence, for example, a3 a2 = a6, is incorrect.

Common student errors: 5x0 = 1, 9x5 3x5 = 3x, 2c-4 = 4

1

2c, 2a2 = 4a2, (3b)2 = 3b2.

Use scientific notation to express and rank astronomical distances (such as planets), populations and areas of countries.

ASSESSMENT IDEAS

Assignment: Research the names of the big numbers or metric prefixes.

TECHNOLOGY

Investigate the following calculator keys: [xy], 3

, y

x . Examine their counterparts on a graphics calculator or spreadsheet.

LANGUAGE

For 24, 2 is called the base and 4 is called the power, index or exponent.

From the NSW syllabus: Teachers should use fuller expressions before shortening them, for example, 24 should be expressed as 2 raised to the power of 4, before 2 to the power of 4 and finally 2 to the 4.


6. GEOMETRY Time: 2 weeks (Term 2, Week 7) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 6


Angle relationships / Geometric reasoning

Identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal, and the

relationships between them (7MG163)

Investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning (7MG164)

Properties of Geometrical Figures 1 / Geometric reasoning

Classify triangles according to their side and angle properties and describe quadrilaterals (7MG165)

Demonstrate that the angle sum of a triangle is 180 and use this to find the angle sum of a quadrilateral (7MG166)

Properties of Geometrical Figures

Apply the result for the interior angle sum of a triangle to find, by dissection, the interior angle sum of polygons with more than three sides (NSW Stage 5.2)

Establish that the sum of the exterior angles of any convex polygon is 360 (NSW Stage 5.2)


A student:



MA5.2-14 MG calculates the angle sum of any polygon and uses minimum conditions to prove triangles are congruent or similar

INTRODUCTION

This short topic revises Stage 4 geometry concepts with angles, triangles and quadrilaterals before turning to the interior and

exterior angle sums of polygons for Stage 5.2 students. Although Year 9 marks the start of more formal geometry, the emphasis

is still upon discovering properties informally through construction and measurement rather than by deductive proofs using

congruent triangles. Promote the language of geometry and the correct use of reasoning, with attention given to drawing clear

diagrams and setting out proofs and solutions carefully.

CONTENT

1 Angle geometry 7MG163, 164 U F R C

identify corresponding, alternate and co-interior angles when two straight lines are crossed by a transversal, and the

relationships between them

investigate conditions for two lines to be parallel and solve simple numerical problems using reasoning

2 Triangle geometry 7MG165, 166 U F R C classify triangles according to their side and angle properties and solve related numerical problems using reasoning

apply the angle sum of a triangle and that any exterior angle of a triangle equals the sum of the two interior opposite angles

3 Quadrilateral geometry 7MG165, 166 U F R C classify quadrilaterals according to their side and angle properties and solve related numerical problems using reasoning

apply the angle sum of a quadrilateral

4 Stage 5.2: Angle sum of a polygon NSW U F R C apply the result for the interior angle sum of a triangle to find, by dissection, the interior angle sum of polygons with

more than three sides

5 Stage 5.2: Exterior angle sum of a convex polygon NSW U F R C establish that the sum of the exterior angles of any convex polygon is 360


RELATED TOPICS

Year 8: Geometry, Area and volume

Year 9: Surface area and volume, Congruent and similar figures

Year 10: Surface area and volume, Geometry



U = Understanding (knowing and relating maths): Learning geometrical concepts, definitions, terminology and notation

F = Fluency (applying maths): Applying correct procedures, rules and notation to solve find unknown angles

R = Reasoning (generalising and proving with maths): Using logic and reasoning to find unknown angles

C = Communicating (describing and representing maths): Classifying triangles, quadrilaterals and polygons and using

correct geometrical terminology

EXTENSION IDEAS

Investigate the history of geometry and Euclid.

From NSW syllabus: Students who recognise class inclusivity and minimum requirements for definitions may address this Stage 4 content concurrently with content in Stage 5 Properties of Geometrical Figures where properties of triangles and

quadrilaterals are deduced from formal definitions. For example, is a parallelogram a trapezium?

Formal proofs in deductive geometry.


Resources: geometrical instruments, paper and scissors, charts and posters, geometry and drawing software.

From NSW syllabus: Students should give reasons when finding unknown angles. For some students, formal setting-out

could be introduced. For example, PQR = 70 (corresponding angles, PQ || SR).

Properties of triangles and quadrilaterals should be demonstrated informally (by symmetry, paper-folding, protractor and ruler measurement), rather than by congruent triangle proofs.

Students should have experience in classifying triangles and quadrilaterals using their properties and minimal conditions, for example, which quadrilaterals diagonals bisect each other?

From NSW syllabus: A range of examples of the various triangles and quadrilaterals should be given, including quadrilaterals containing a reflex angle and figures presented in different orientations.

The properties of special quadrilaterals allow us to develop formulas for finding their areas in the topic Surface area and volume, for example, the diagonal properties of the kite and rhombus.

In how many different ways can you demonstrate the angle sum of a triangle (or quadrilateral)?

Proving properties of quadrilaterals by congruent triangles will be covered in the topic Congruent and similar figures.

The exterior angle sum of a convex polygon is 360: if you walk around the perimeter of a closed figure, the total of your turns should be a revolution.

ASSESSMENT IDEAS

Writing activity or poster summary on the properties of triangles, quadrilaterals and polygons

Vocabulary test

What shape am I? puzzles

Research/investigation assignment on properties of triangles, quadrilaterals or polygons

Assignment on setting out a geometry proof

TECHNOLOGY

There is much scope in this topic to use dynamic geometry software such as GeoGebra. The Internet is full of dynamic

geometry animations and applets that demonstrate the properties of triangles, quadrilaterals and polygons shown in this topic.

LANGUAGE

Students need practice in interpreting geometrical descriptions. Work in pairs, with one student describing a figure while the other tries to draw it.

Avoid using the term base angles for isosceles triangles because it may be misleading, depending upon the orientation of the triangle. Instead, use the angles opposite the equal sides or the two angles next to the uneven side.

From the NSW syllabus: The diagonals of a convex quadrilateral lie inside the figure.


7. EQUATIONS Time: 3 weeks (Term 3, Week 1) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 7


Equations / Linear and non-linear relationships

Solve linear equations using algebraic and graphical techniques, and verify solutions by substitution (8NA194)

Solve linear equations (9NA215)

Substitute values into formulas to determine an unknown (10NA234)

Solve problems involving linear equations, including those derived from formulas (10NA235)

Solve linear equations involving simple algebraic fractions (10NA240)

Solve simple quadratic equations using a range of strategies (10NA241)


A student:



MA5.2-8 NA solves linear and simple quadratic equations

INTRODUCTION

This topic revises and builds upon the Stage 4 concept of equations and the formal methods for solving them. Like many

algebra skills, the process of equation-solving is detailed and technical, requiring careful and precise understanding and

practice, so dont rush through this topic. The second half of this topic introduces more complex equations for Stage 5.2

students, namely equations with algebraic fractions, simple quadratic equations, and solving equations after substitution into

formulas.

CONTENT

1 Two-step equations 8NA194 U F R

2 Equations with variables on both sides 8NA194 U F R

solve linear equations using algebraic techniques

3 Equations with brackets 8NA194, 9NA215 U F R

solve linear equations involving grouping symbols

4 Equation problems 8NA194 U F PS R C

solve real-life problems by using pronumerals to represent unknowns

5 Stage 5.2: Equations with algebraic fractions 10NA240 U F R

solve linear equations involving simple algebraic fractions

6 Stage 5.2: Simple quadratic equations ax2 = c 10NA241 U F R C

solve simple quadratic equations of the form ax2 = c

7 Stage 5.2: Equations and formulas 10NA234, 235 U F PS R C

substitute values into formulas to determine an unknown

solve problems involving linear equations, including those derived from formulas


RELATED TOPICS

Year 8: Algebra, Equations, Graphing linear equations

Year 9: Algebra, Coordinate geometry and graphs

Year 10: Coordinate geometry, Algebra, Equations and inequalities, Simultaneous equations


U = Understanding (knowing and relating maths): Understanding the steps for solving equations

F = Fluency (applying maths): Selecting correct techniques for solving equations

PS = Problem solving (modelling and investigating with maths): Solving real-life problems using equations and

formulas

R = Reasoning (generalising and proving with maths): Using algebraic operations to solve equations

C = Communicating (describing and representing maths): Describing the solution to real-life problems in words after


solving an equation

EXTENSION IDEAS

Harder formulas and word problems, constructing formulas

Equations with the unknown in the denominator

Inequalities (Year 10)

Simple cubic equations ax3 = c (Stage 5.3)

Simultaneous equations


Stress that the goal of solving an equation is to have the variable on its own on the left side of the equation and the value on the right side.

Examples of Stage 5.2 equations with algebraic fractions from NSW syllabus:

(denominators should be numerical).

When solving a word problem, identify the unknown quantity and call it x, say. After solving, check that its solution sounds reasonable.

Examples of formulas: perimeter and area, circle formulas, speed, metric conversions (for example, Celsisus to Fahrenheit), Pythagoras theorem, angle sum of a polygon, E = mc2.

ASSESSMENT IDEAS

Writing activity comparing and evaluating the different methods of solving an equation.

TECHNOLOGY

CAS calculators and the WolframAlpha website can be used to solve equations.

LANGUAGE

An algebraic expression refers to a phrase containing terms and arithmetic operations, such as 2a + 5, while an algebraic

equation refers to a sentence involving an expression and an equals sign, such as 2a + 5 = 13.

Encourage students to set out their solutions to equations neatly with equals signs aligned in the same column.

quadratic = algebraic expression in which the highest power of x is 2, eg 5x2 3x + 4.


8. EARNING MONEY Time: 2 weeks (Term 3, Week 4) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 8

NSW Curriculum reference: Number and Algebra

Financial mathematics

Solve problems involving earning money (NSW Stage 5.1)


A student:




MA5.1-4 NA solves financial problems involving earning, spending and investing money

INTRODUCTION

In this short practical topic, students apply their Number skills to situations involving earning money and paying income tax.

This topic is actually unique to the NSW syllabus and does not appear in the national Australian curriculum, but it has been

retained so that Year 9 students can be more financially literate with their income and tax calculations. Attention should be

given towards making examples as realistic as possible, with current wage and tax rates being found on the Internet.

CONTENT

1 Wages and salaries NSW U F PS C

solve problems involving earning money

calculate weekly, fortnightly, monthly and yearly earnings

2 Overtime pay NSW U F PS R C

3 Commission, piecework and leave loading NSW U F PS C

calculate earnings from wages, overtime, commission and piecework

calculate annual leave loading

4 Income tax NSW U F C

determine annual taxable income using current tax rates

5 PAYG and net pay NSW U F C

use published tables or online calculators to determine the weekly, fortnightly or monthly tax to be deducted from a

workers pay under the Australian pay-as-you-go (PAYG) taxation system

calculate net earnings after deductions and taxation are taken into account


RELATED TOPICS

Year 8: Fractions and percentages, Ratios, rates and time

Year 9: Working with numbers

Year 10: Interest and depreciation


U = Understanding (knowing and relating maths): Learning about the different types of income and their associated

calculations

F = Fluency (applying maths): Applying appropriate calculations for income and taxation problems

PS = Problem solving (modelling and investigating with maths): Solving a variety of real-life problems involving

earning an income

R = Reasoning (generalising and proving with maths): Understanding the logic and reasoning behind calculating

overtime pay

C = Communicating (describing and representing maths): Interpreting and using the terminology of earning and

taxation


EXTENSION IDEAS

Back-to-front problems, for example, given the final pay after annual leave loading or overtime pay was added, find the original pay

Calculating tax refunds or debts


Resources: job advertisements in newspapers and on websites, tax tables, payslips.

Use employment sections of newspapers to compare current wages and salaries of occupations.

Liaise with the HSIE faculty or the schools careers adviser for resources.

Discuss types of jobs where overtime, commission and piecework occur. Investigate the advantages and disadvantages of each type of income.

ASSESSMENT IDEAS

Practical or problem-solving test/assignment

Collage/poster/case study on the different ways of earning money.

TECHNOLOGY

Use spreadsheets to calculate pays, net incomes and income tax.

LANGUAGE

The abbreviation K comes from the Greek word khilioi meaning thousand. It is used in many job advertisements, for example, a salary of $80K.


9. INVESTIGATING DATA Time: 3 weeks (Term 3, Week 6) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 9

NSW and Australian Curriculum references: Statistics and Probability

Single Variable Data Analysis / Data representation and interpretation

Calculate mean, median, mode and range for sets of data, and interpret these statistics in the context of data (7SP171)

Investigate the effect of individual data values, including outliers, on the mean and median (8SP207)

Investigate reports of surveys in digital media and elsewhere for information on how data was obtained to estimate population means and medians (9SP227)

Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly from secondary sources (9SP228)

Construct back-to-back stem-and-leaf plots and histograms and describe data using terms, using terms including skewed, symmetric and bi-modal (9SP282)

Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread (SP283)

Data Collection and Representation / Data representation and interpretation

Investigate techniques for collecting data, including census, sampling and observation (8SP284)


A student:




MA5.1-12 SP uses statistical displays to compare sets of data, and evaluates statistical claims made in the media


INTRODUCTION

In this Statistics topic, students begin to look at data sets as a whole, analysing the shape of a distribution and comparing the

statistical measures for two data sets. This unit builds upon concepts learned in Stage 4 such as histograms, stem-and-leaf plots,

types of data and samples vs census. Stage 5.2 students also examine bias in sampling, surveys and questionnaires.

CONTENT

1 The mean, median, mode and range 7SP171, 8SP207 U F PS R C

calculate mean, median, mode and range for sets of data, and interpret these statistics in the context of data

investigate the effect of individual data values, including outliers, on the mean and median

2 Histograms and stem-and-leaf plots 9SP282, 228 U F PS R C

construct back-to-back stem-and-leaf plots and histograms

identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly from secondary sources

3 The shape of a distribution 9SP282 U F PS R C describe data using terms, including skewed, symmetric and bi-modal

4 Comparing data sets 9SP283 F PS R C compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location

(centre) and spread

5 Sampling and types of data 8SP284, 9SP228 U F R C investigate techniques for collecting data, including census, sampling and observation

6 Stage 5.2: Bias and questionnaires 9SP227 F PS R C investigate reports of surveys in digital media and elsewhere for information on how data was obtained to estimate

population means and medians


RELATED TOPICS

Year 8: Investigating data

Year 10: Investigating data



U = Understanding (knowing and relating maths): Knowing the various types of data displays and statistical measures

F = Fluency (applying maths): Interpreting data sets through their graphs and statistical measures

PS = Problem solving (modelling and investigating with maths): Analysing data to solve problems and draw conclusions

R = Reasoning (generalising and proving with maths): Making generalisations and drawing conclusions from statistical

displays and measures

C = Communicating (describing and representing maths): Classifying, representing and interpreting data in different

forms and using correct statistical terminology

EXTENSION IDEAS

Interquartile range, box-and-whisker plots (Year 10)

Grouped data, class intervals, median class (no longer part of syllabus)

Replicate or implement a major statistical investigation.


This topic lends itself to investigation projects, The class may be surveyed on a number of characteristics: height, arm span, shoe size, heartbeat rate, reaction time, number of children in family, number of people living at home, hours slept last

night, number of letters in first name, number of cars or mobile phones owned at home, reaction time.

Examples of surveys: TV/radio ratings, opinion polls, phone polls, CD sales, quality control. Survey the number of left-handed or blue-eyed students in the class or Year group and use this to estimate the number with the same feature in the

school or whole of Australia.

Survey the number of left-handed or blue-eyed students in the class or Year 9 and use this to estimate the number with the same feature in the school or whole of Australia.

Question when it is more appropriate to use the mode or median, rather than the mean, when analysing data. Which is higher, the mean or median price of Australian homes?

Sometimes, a sample is biased because it is too small or does not represent the population accurately, for example, men only, adults only.

ASSESSMENT IDEAS

Include open-ended questions: The range of a set of eight scores is 10 and the mode is 3. What might the scores be?

Plan, implement and report on a statistical investigation.

Vocabulary test.

Investigate the use and abuse of statistics and statistical graphs in the media.

Research the role of the Australian Bureau of Statistics.

TECHNOLOGY

Explore the statistical and graphing features of a spreadsheet, GeoGebra, Fx-Stat, graphics/CAS calculators or software. Use a

spreadsheet to examine the effects of altering data, for example, outliers. Visit the CensusAtSchool website

www.abs.gov.au/censusatschool.

LANGUAGE

This topic contains much statistical jargon, so a student-created glossary may be useful.

Reinforce the terminology measures of location and measures of spread.

Population may refer to a collection of items as well as people.

Spend considerable time explaining the difference between discrete and continuous data.

Strictly speaking, the term bi-modal does not mean two modes. A bi-modal distribution actually has two peaks, with the higher one being the mode. However, in this context, mode has the same meaning as peak.


10. SURFACE AREA AND VOLUME Time: 3 weeks (Term 3, Week 10) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 10


Numbers of any magnitude / Using units of measurement

Investigate very small and very large time scales and intervals (9MG219)

Area and surface area, Volume / Using units of measurement

Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites (8MG196)

Develop the formulas for volumes of rectangular and triangular prisms and prisms in general; use formulas to solve problems involving volume (8MG198)

Calculate the areas of composite shapes (9MG216)

Calculate the surface area and volume of cylinders and solve related problems (9MG217)

Solve problems involving the surface area and volume of right prisms (9MG218)


A student:



MA5.1-8 MG calculate the areas of composite shapes, and the surface areas of rectangular and triangular prisms

MA5.1-9 MG interprets very small and very large units of measurement, uses scientific notation, and rounds to significant figures


MA5.2-2 WM interpret mathematical or real-life situations, systematically applying appropriate strategies to solve problems

MA5.2-11 MG calculates the surface areas of right prisms, cylinders and related composite solids

MA5.2-12 MG applies formulas to calculate the volumes of composite solids composed of right prisms and cylinders

INTRODUCTION

This Measurement topic builds upon and extends concepts and skills learned in Stage 4, particularly in area and volume, before

introducing surface area. Rather than learn a set of facts and formulas, the emphasis is upon understanding each idea met in this

topic. This is achieved by applying the skills to a variety of real problems. Practice in estimating, the correct setting-out of

solutions and the rounding of answers should feature prominently in the teaching of this topic.

CONTENT

1 The metric system 9MG219 U F R C

interpret the meanings of prefixes for very small and very large units of measurement, such as nano, micro, mega,

giga and tera

convert between units of measurement of digital information, for example, gigabytes to terabytes, megabytes to

kilobytes

investigate very small and very large time scales and intervals

2 Limits of accuracy of measuring instruments NSW U R C

describe the limits of accuracy of measuring instruments (0.5 unit of measurement)

3 Perimeters and areas of composite shapes 9MG216 U F R

calculate the perimeters and areas of composite shapes

4 Areas of quadrilaterals 8MG196 U F PS R

find areas of parallelograms, trapeziums, rhombuses and kites

5 Circumferences and areas of circular shapes 9MG216 U F PS R

calculate the areas of composite figures by dissection into quadrants, semi-circles and sectors

6 Surface area of a prism 9MG218 U F PS R

solve problems involving the surface areas of right prisms

7 Stage 5.2: Surface area of a cylinder 9MG217 U F PS R

calculate the surface areas of cylinders and solve related problems

8 Volumes of prisms and cylinders 8MG198 U F PS R

solve problems involving volume and capacity of right prisms and cylinders



RELATED TOPICS

Year 8: Pythagoras theorem, Geometry, Area and volume Year 9: Pythagoras theorem, Geometry, Congruent and similar figures Year 10: Surface area and volume


U = Understanding (knowing and relating maths): Knowing the concepts of area, surface area and volume, and their

formulas

F = Fluency (applying maths): Selecting correct strategies to convert between metric units and calculate areas and

volumes

PS = Problem solving (modelling and investigating with maths): Solving problems involving area, surface area and

volume

R = Reasoning (generalising and proving with maths): Using and adapting formulas for calculating perimeters, areas and

volumes

C = Communicating (describing and representing maths): Understanding the metric prefixes and the terminology of

measurement error

EXTENSION IDEAS

Percentage error.

Investigate unusual units of measurement such as nautical mile, Richter scale, decibel, light year.

Herons formula for the area of a triangle with sides of length a, b and c.

Areas of irregular figures: traverse surveys, Simpsons rule.

Volume of a pyramid, cone or sphere (Stage 5.3).

Circumference of the Earth, latitude and longitude (small and great circles) on the Earths surface.


Resources: measuring instruments such as stopwatches, nets of solid shapes, paper, scissors.

There should be some discussion on the accuracy of measuring instruments. A good starting point is the electronic timing of track and swimming events.

Investigate the measurement of very small objects and very large objects. How thin is a sheet of paper?

Include perimeter and area problems where extra information is given or Pythagoras theorem must be used. Investigate maximum area problems.

The area of a rhombus or kite can be cut up and rearranged into two congruent triangles or one rectangle. This method actually works for any quadrilateral with diagonals that are perpendicular.

The area of a trapezium can be cut up and rearranged into two triangles or one rectangle.

Emphasise how area involves multiplying two dimensions or powers of 2 while volume involves three dimensions or powers of 3. Compare the area formula for a circle to that of a square: both involve powers of 2.

With composite area problems, encourage students to look for opportunities for combining two semi-circles.

ASSESSMENT IDEAS

Investigate paper and envelope sizes, the legal size of an envelope, history of , areas of countries or Australian states, the

Imperial system of measurement, digital memory sizes.

Practical activity/assignment/test on area, surface area and volume.

Open-ended and back-to-front questions: A triangular prism has a volume of 36 cm3. What could its dimensions be?

TECHNOLOGY

Drawing and animation software may be used to demonstrate area and volumes of geometrical figures. Also search for

animations and applets from the Internet.

LANGUAGE

See the NSW syllabus for the Latin and Greek meanings of the metric prefixes.

From NSW syllabus: Students are expected to be able to determine whether the prisms and cylinders referred to in practical problems are closed or open (one end only or both ends).

From NSW syllabus: The abbreviation m2 is read as 'square metre(s)' and not 'metre(s) squared' or 'metre(s) square'.


11. COORDINATE GEOMETRY AND GRAPHS Time: 3 weeks (Term 4, Week 3) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 11


Linear relationships / Linear and non-linear relationships

Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (9NA214)

Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (9NA294)

Sketch linear graphs using the coordinates of two points (9NA215)

Interpret and graph linear relationships using the gradient-intercept form of the equation of a straight line (NSW Stage 5.2)

Solve linear equations using graphical techniques (8NA194)

Ratios and rates / Real numbers

Solve problems involving direct proportion; explore the relationship between graphs and equations corresponding to simple rate problems (9NA208)

Non-linear relationships / Linear and non-linear relationships

Graph simple non-linear relations, with and without the use of digital technologies (9NA296)

Explore the connection between algebraic and graphical representations of relations such as simple quadratics, circles and exponentials using digital technology as appropriate (10NA239)


A student:



MA5.1-6 NA determines the midpoint, gradient and length of an interval, and graphs linear relationships




MA5.2-5 NA recognises direct and direct proportion, and solves problems involving direct proportion

MA5.2-9 NA uses the gradient-intercept form to interpret and graph linear relationships

MA5.2-10 NA connects algebraic and graphical representations of simple non-linear relationships

INTRODUCTION

This topic marks the start of formal coordinate geometry. Students have already graphed linear equations in Year 8 but this

Stage 5 topic extends their knowledge to the methods of finding the length, midpoint and gradient of an interval. Stage 5.2

students also examine the gradient-intercept equation of a line and are introduced to the concept of direct proportion. Finally,

students graph parabolas and circles. There is much scope for using graphing software such as GeoGebra in this topic.

CONTENT

1 The length of an interval 9NA214 U F R C find the distance between two points located on the Cartesian plane using a range of strategies, including graphing

software

2 The midpoint of an interval 9NA294 U R C

3 The gradient of a line 9NA294 U F R C find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including

graphing software

4 Graphing linear equations 9NA215 U F R C sketch linear graphs using the coordinates of two points

determine whether a point lies on a line by substitution

5 Stage 5.2: The gradient-intercept formula y = mx + b NSW U F R C interpret and graph linear relationships using the gradient-intercept form of the equation of a straight line

6 Stage 5.2: Finding the equation of a line y = mx + b NSW U F R C find the gradient and y-intercept of a straight line from its graph and use these to determine the equation of the line

7 Solving linear equations graphically 8NA194 U F R C solve linear equations using graphical techniques


8 Stage 5.2: Direct proportion 9NA208 U F PS R C solve problems involving direct proportion and explore the relationship between graphs and equations corresponding to

simple rate problems

9 Graphing quadratic equations 9NA296, 10NA239 U F R C graph simple non-linear relations, with and without the use of digital technologies

graph parabolic relationships of the form y = ax2 and y = ax2 + c

10 Graphing circles 9NA296, 10NA239 U F R C sketch circles of the form x2 + y2 = r2

explore the connection between algebraic and graphical representations of relations such as simple quadratics and circles using digital technology as appropriate


RELATED TOPICS

Year 8: Graphing linear equations

Year 10: Coordinate geometry, Graphs, Simultaneous equations


U = Understanding (knowing and relating maths): Relating linear and non-linear equations to their graphs

F = Fluency (applying maths): Using appropriate techniques to graph equations and to find the length, midpoint and

gradient of an interval

PS = Problem solving (modelling and investigating with maths): Solving problems involving direct proportion

R = Reasoning (generalising and proving with maths): Generalising how the variables in an equation affect its graphs

shape and other features

C = Communicating (describing and representing maths): Describing and interpreting relationships using equations and

graphs.

EXTENSION IDEAS

The formulas for distance, midpoint and gradient of an interval (Stage 5.3)

Gradients of parallel and perpendicular lines (Year 10)

Inverse proportion, graphing hyperbolas and exponential curves (Year 10)


Resources: number plane grid paper, graphics calculator, graphics software.

Investigate large, small, positive, negative, zero and fractional gradients. Demonstrate how a negative gradient has a

negative run. Show that the gradient ratio is constant for a straight line.

Gradient is also used to describe land, roads and hills in construction and hiking.

The general form of the linear equation ax + by + c = 0 will be introduced in the Year 10 topic, Coordinate geometry.

All points that lie on the line have coordinates that satisfy the linear equation. Points that dont lie on the line do not satisfy the equation.

When graphing, remind students to label the axes and graph, and to show the scale on both axes.

The parabola is a conic section formed by the intersection of a cone by a plane that cuts it at a steeper angle to its base than its axis. The path of a projectile (object thrown) is a parabola, as is the shape of a satellite dish, concave lens or car

headlight.

ASSESSMENT IDEAS

Practical graphing test using pen-and-paper or technology.

Open-ended questions, for example, find two points that are 2 units apart, if the midpoint of an interval is (1, 4), what could the endpoints of the interval be?

TECHNOLOGY

Use a graphics calculator, graphing software or spreadsheets to complete tables of values and graph linear and non-linear

equations.


LANGUAGE

Develop the idea of the midpoint as an average. Remind students that the midpoint is a point, so the answer should be a pair of coordinates.

Why does the gradient-intercept equation have that name?

The Cartesian plane is another name for the number plane, named after the French philosopher and mathematician Ren Descartes.


12. PROBABILITY Time: 2 weeks (Term 4, Week 6) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 12

NSW and Australian Curriculum references: Statistics and Probability

Probability 1 / Chance

Identify complementary events and use the sum of probabilities to solve problems (8SP204)

Probability / Chance

List all outcomes for two-step chance experiments, with and without replacement, using tree diagrams or arrays; assign probabilities to outcomes and determine probabilities for events (9SP225)

calculate relative frequencies from given or collected data to estimate probabilities of events involving and or or (9SP226)


A student:




MA5.1-13 SP calculates relative frequencies to estimate probabilities of simple and compound events




MA5.2-17 SP describes and calculates probabilities in multi-step chance experiments

INTRODUCTION

This short topic revises and extends probability concepts introduced in Year 8, especially Venn diagrams and two-way tables.

The focus is upon interpreting descriptions of events using the words and, or, at least and not, so there are many

opportunities for class discussion and language activities. Tree diagrams to represent the sample space of two-step experiments

are introduced for Stage 5.2 students, so spend considerable time teaching and practising drawing these as students often have

difficulty understanding them.

CONTENT

1 Probability 8SP204 U F PS R C identify complementary events and use the sum of probabilities to solve problems

2 Relative frequency 9SP226 U F PS R C calculate relative frequencies from given or collected data to estimate probabilities of events involving and or or

3 Venn diagrams 9SP226 U F PS R C represent events in Venn diagrams and solve related problems

describe events using language of at least, exclusive or (A or B but not both), inclusive or (A or B or both) and and

calculate probabilities of events from data contained in Venn diagrams

4 Two-way tables 9SP226 U F PS R C represent events in two-way tables and solve related problems

calculate probabilities of events from data contained in two-way tables

5 Stage 5.2: Two-step experiments 9SP225 U F PS R C list all outcomes for two-step chance experiments, with and without replacement, using tree diagrams or arrays, and

assign probabilities to outcomes and determine probabilities for events


RELATED TOPICS

Year 8: Probability

Year 10: Probability



U = Understanding (knowing and relating maths): Relating the ways of listing sample spaces to calculating probabilities

of events, including two-step events

F = Fluency (applying maths): Interpreting and drawing Venn diagrams, two-way tables, lists and tree diagrams

competently

PS = Problem solving (modelling and investigating with maths): Solving problems involving overlapping categories and

two-stage experiments

R = Reasoning (generalising and proving with maths): Interpreting situations involving multiple categories and using

logic to interpret statements involving and, or, at least and not

C = Communicating (describing and representing maths): Representing events using Venn diagrams, two-way tables,

lists and tree diagrams

EXTENSION IDEAS

Three-step experiments (Year 10)

Probability tree diagrams that have probability values listed on branches, addition and product rules

Probability simulations using technology

More complex Venn diagrams, set notation (union vs intersection)

Investigate probability expressed as odds (ratio), for example, 10 to 1

Counting techniques, the birthday problem

Investigating the probability of winning games of chance and gambling

Investigate the use of probability in insurance, for example, life expectancy


Resources: Dice, coins, counters, spinners, playing cards, probability simulation software.

Students were introduced to Venn diagrams and two-way tables in Year 8.

Do not assume that all students have had experience with the properties of playing cards: suits, colours, deck of 52. Be sensitive to religious and cultural differences in attitudes towards gambling.

Graph the results of a probability experiment on a dot plot or histogram.

What happens to relative frequencies as the number of experimental trials increases?

If a coin is tossed seven times and comes up heads each time, what is the probability that the next toss is also a head?

ASSESSMENT IDEAS

Writing and comprehension activities on describing events involving mutually exclusive and overlapping activities

Experimental probability investigation or simulation

Research project on the applications or history of probability, for example, insurance premiums, planning for roads and new

communities

TECHNOLOGY

Random numbers can be generated on the calculator, graphics calculator and spreadsheet. Spreadsheets and other software may

be used to simulate a chance situation. The Internet is also a rich source for probability simulations.

LANGUAGE

Students should know the difference between an outcome and an event: an event contains one or more outcomes of an experiment.

Inclusive or = A or B or both, exclusive or = A or B but not both, mutually exclusive means A and B are not overlapping and cannot both happen

What is the difference between at least 4 and 4 or more? Students (even in Year 12) often think that the two phrases mean the same thing.

Note that in the new syllabus the term two-step experiment replaces two-stage experiment. Clearly explain the difference between with replacement and without replacement.


13. CONGRUENT AND SIMILAR FIGURES Time: 3 weeks (Term 4, Week 8) Text: New Century Maths 9 Stages 5.1/5.2, Chapter 13


Properties of Geometrical Figures 2 / Geometric reasoning

Define congruence of plane shapes using transformations (8MG200)

Develop the conditions for congruence of triangles (8MG201)

Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (8MG202)

Properties of Geometrical Figures / Geometric reasoning

Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (9MG220)

Solve problems using ratio and scale factors in similar figures (9MG221)

Establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems using reasoning (8MG202)


A student:




MA5.1-11 MG describes and applied the properties of similar figures and scale drawings




INTRODUCTION

This Geometry topic revises the concept of congruence met in Year 8 and contrasts it with similarity introduced here. Geometrical

properties are meant to be discovered through construction and measurement (including the use of technology) rather than formal

deductive reasoning, which is often beyond the grasp of Year 9 students. The tests for congruent and similar triangles are covered,

but not formal proofs for them as this is done in Year 10 for congruent triangles or at Stage 5.3 for similar triangles. There is much

scope in this topic for practical activities, reasoning tasks and class discussions.

CONTENT

1 Congruent figures 8MG200 U F C define congruence of plane shapes using transformations

2 Tests for congruent triangles 8MG201 U F PS R C develop the conditions for congruence of triangles

3 Using congruence to prove geometrical properties 8MG202 U F PS R C establish properties of quadrilaterals using congruent triangles and angle properties, and solve related numerical problems

using reasoning

4 Similar figures 9MG220 U F R C use the enlargement transformation to explain similarity

5 Properties of similar figures 9MG220, 221 U F R C

6 Scale diagrams 9MG221 U F PS C solve problems using ratio and scale factors in similar figures

7 Stage 5.2: Tests for similar triangles 9MG220 U F PS R C investigate the minimum conditions needed, and establish the four tests, for two triangles to be similar


RELATED TOPICS

Year 8: Geometry, Congruent figures

Year 9: Trigonometry, Geometry

Year 10: Trigonometry, Geometry



U = Understanding (knowing and relating maths): Connecting together the properties of congruent and similar figures

F = Fluency (applying maths): Selecting appropriate tests and strategies for identifying similar triangles and applying similar

figures

PS = Problem solving (modelling and investigating with maths): Using similar figures and scale factors to interpret scale

diagrams

R = Reasoning (generalising and proving with maths): Using congruence to prove properties of triangles and quadrilaterals

C = Communicating (describing and representing maths): Describing and identifying the tests for congruent and similar

triangles, interpreting the scale on a diagram

EXTENSION IDEAS

Formal proofs of congruent a

teaching program year 9

Documents

algebra number

geometry number

algebra measurement

geometry measurement

algebra term

geometry term

geometry statistics

pythagoras theorem measurement