teaching program year 10 (advanced)

New Century Maths Advanced 10+10A Stages 5.2/5.3 teaching program (p. 1)

Teaching program

New Century Maths Advanced 10+10A Stages 5.2/5.3 for the Australian Curriculum

(Syllabus extracts Board of Studies NSW 2012)

Year 10 topics Week SEMESTER 1

Week SEMESTER 2 Term 1

1 1. Surds

(Number and Algebra)

Term 3 1

8. Graphs (continued)

2

2

9. Trigonometry

(Measurement and Geometry)

3

2. Interest and depreciation


3

4

4

5

3. Coordinate geometry


5

6

6

10. Simultaneous equations


7

7

11. Quadratic equations

and the parabola

8

4. Surface area and volume


8


9

9

10

5. Products and factors


10

12. Probability

(Statistics and Probability)

Term 2

1 Term 4

1

2

2

13. Geometry


3

6. Investigating data


3

4

4

OPTION TOPICS

(recommended for Stage 6

5

5

Mathematics Extension 1)

6

7. Equations and

logarithms#

6

14. Polynomials#

15. Circle geometry#

7

(Number and Algebra) 7

16. Functions#

8

8

9

8. Graphs


9

Lost time

10

10

= Stage 5.3 content recommended for students progressing to Stage 6 Mathematics

# = Stage 5.3 content recommended for students progressing to Stage 6 Mathematics Extension 1

CURRICULUM STRANDS

Number and Algebra Measurement and Geometry Statistics and Probability


Year 9 topics

Week SEMESTER 1

Week SEMESTER 2 Term 1

1 1. Pythagoras theorem

and surds

Term 3 1

7. Equations


2

(Measurement and Geometry) 2

3

3

4

2. Working with numbers

(Number and Algebra) 4

8. Earning money


5

5

6

3. Products and factors


6

9. Investigating data


7

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


1. SURDS Recommended for Stage 5.2 students intending to study the Stage 6 Mathematics course

Time: 2 weeks (Term 1, Week 1)

Text: New Century Maths Advanced 10+10A Stages 5.2/5.3, Chapter 1, p.1.

NSW and Australian Curriculum references: Number and Algebra

Surds and indices / Real numbers

Define rational and irrational numbers and perform operations with surds and fractional indices (10ANA264)

NSW Stage 5 outcomes

A student:

MA5.3-1 WM uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

MA5.3-2 WM generalises mathematical ideas and techniques to analyse and solve problems efficiently

MA5.3 NA performs operations with surds and indices

INTRODUCTION

This Stage 5.3 topic covers operations with surds, including binomial products and rationalising the denominator. Students have

already met surds when solving problems involving Pythagoras theorem, lengths on the number plane and simple quadratic equations. Binomial expansions involving surds build upon skills from the Year 9 topic Products and factors. This is quite a

technical topic, so make sure students spend considerable time developing their knowledge and practising their manipulation

skills.

CONTENT

1 Stage 5.3: Surds and irrational numbers 10ANA264 U F R C define rational and irrational numbers

2 Stage 5.3: Simplifying surds 10ANA264 U F R perform operations with surds

3 Stage 5.3: Adding and subtracting surds 10ANA264 U F R

4 Stage 5.3: Multiplying and dividing surds 10ANA264 U F R

5 Stage 5.3: Binomial products involving surds 10ANA264 U F R C

expand expressions involving surds, for example, 2 3 2 3 6 Stage 5.3: Rationalising the denominator 10ANA264 U F C

rationalise the denominators of surds of the form a b

c d

7 Revision and mixed problems

RELATED TOPICS

Year 9: Pythagoras theorem and surds, Products and factors Year 10: Coordinate geometry, Products and factors, Trigonometry, Quadratic equations and the parabola

PROFICIENCY STRANDS / WORKING MATHEMATICALLY

U = Understanding (knowing and relating maths): Knowing what a surd is and how to perform operations on them

F = Fluency (applying maths): Using the correct strategy to simplify expressions involving surds, including rationalising

the denominator

R = Reasoning (generalising and proving with maths): Use the properties of surds to simplify expressions, including

rationalising the denominator

C = Communicating (describing and representing maths): Understanding the concepts of surds and irrational numbers

EXTENSION IDEAS

The real number system and classifying types of numbers

Newtons method for calculating square roots. The cube root formula.

Rationalising binomial denominators such as73

2

.

The golden section.


The graph of y = x (top half of a sideways parabola).

TEACHING NOTES AND IDEAS

When simplifying surds, encourage students to find a factor that is a square number. List the first 10 square numbers on the board for easy reference: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

Demonstrate how surd arithmetic follows the rules of algebra (as it should), for example, collecting like terms.

Demonstrate that the length 2 can be constructed using a right-angled isosceles triangle. By construction, graph surds on the number line. See Worksheet Surds on the number line.

Ancient Greek mathematicians believed that all numbers were rational and the world could be explained by rational numbers. Pythagoras proved this was false when calculating the diagonal of a square with sides 1 unit long.

Investigate how square roots were found before calculators were invented: Newtons method.

For the A series of paper sizes, investigate the ratio of length to width ( 2 : 1) or the lengths of the diagonals.

ASSESSMENT IDEAS

Research assignment on Pythagoras and the discovery of surds, or the golden section.

TECHNOLOGY

Spreadsheets can be used to approximate surds using the =SQRT formula or evaluate a square root using Newtons formula. Use the Internet to research the history of Pythagoras and surds. Use CAS (computer algebra system) software to manipulate

surds.

LANGUAGE

A rational number is a number than can be expressed in the ratio b

a where a and b are integers and b 0. An irrational

number cannot be expressed in this form. As decimals, they do not terminate, but they are not recurring either.

A surd is a root of a number that is not a square or any other power. All surds are irrational, but not all irrational numbers are surds. For example, is irrational but is not a surd.

The Latin word surdus means muffled or indistinct.

Note that 2a = a if and only if a is positive. More generally,

2 .a a

What is meant by rationalising the denominator?


2. INTEREST AND DEPRECIATION Time: 2 weeks (Term 1, Week 3)

Text: New Century Maths Advanced 10+10A Stages 5.2/5.3, Chapter 2, p.???


Financial mathematics / Money and financial mathematics

Solve problems involving earning money (NSW Stage 5.1)

Solve problems involving simple interest (9NA211)

Connect the compound interest formula to repeated applications of simple interest using appropriate digital technologies (10NA229)


A student:

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

MA5.2-4 NA solves financial problems involving compound interest

INTRODUCTION

In this short Financial Mathematics topic, students revise the mathematics of earning an income and paying income tax from

Year 9 before being introduced to the concept of compound interest, the compound interest formula and depreciation. Half of

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 Stage 5 students can be more financially literate with the mathematics of earning, saving and borrowing.

Classroom examples should be as realistic as possible, with current rates being found on the Internet.

CONTENT

1 Earning an income NSW U F PS C

solve problems involving earning money

calculate weekly, fortnightly, monthly and yearly earnings

calculate earnings from wages, overtime, commission and piecework

calculate annual leave loading

2 Income tax NSW U F C

determine annual taxable income using current tax rates

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

3 Simple interest 9NA211 U F PS C apply the simple interest formula I = PRN to solve problems related to investing money at simple interest rates

solve problems involving simple interest

4 Compound interest 10NA229 U F PS C

calculate compound interest for two or three years using repetition of the formula for simple interest

5 Compound interest formula 10NA229 U F PS R C

establish and use the formula A = P(1 + R)n to find compound interest

solve problems involving compound interest

6 Term payments NSW U F PS C calculate the cost of buying expensive items by paying an initial deposit and making regular repayments that include

simple interest

7 Depreciation NSW U F PS R C

use the compound interest formula to calculate depreciation


RELATED TOPICS

Year 9: Earning money



U = Understanding (knowing and relating maths): Learning about the different ways of calculating income, income tax, interest, term payments and depreciation

F = Fluency (applying maths): Having the financial literacy to make appropriate calculations for income, income tax, simple and compound interest

PS = Problem solving (modelling and investigating with maths): Finding unknown amounts in problems involving earning an income, simple interest, compound interest and depreciation

R = Reasoning (generalising and proving with maths): Understanding the logic and reasoning behind the compound interest formula and depreciation

C = Communicating (describing and representing maths): Using the terminology of financial mathematics appropriately

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

Compound interest tables and graphs, the exponential graph

Credit card calculations and charges, debit cards, hidden costs

Term payments charges, deferred payments


Resources: job advertisements and interest rates from newspapers and websites, tax tables, payslips, savings and loans brochures from banks and credit unions, depreciation tables from tax guides, spreadsheets.

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.

Investigate the compound interest formula from first principles by examining repeated percentage increase, for example, increasing by 8% = 108% = 1.08.

Students should learn the skill of expressing the interest rate, R, as a decimal. They should not round the value of R when calculating interest compounded monthly.

From the NSW syllabus, Stage 5.2: Calculate and compare investments for different compounding periods.

Discuss whether it makes sense to round up or down to the nearest cent when calculating interest.

Compare simple interest with compound interest. Which one earns more?

Examine the different types of savings and investment accounts available.

With depreciation, will the value of the item ever be zero?

Make problems as realistic as possible. Some students may be starting part-time jobs now and earning incomes.

Collect examples of term payments and interest rates from store catalogues such as Harvey Norman and The Good Guys. Compare total paid with cash price.

ASSESSMENT IDEAS

Practical or problem-solving test/assignment

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

Compound interest assignment comparing different interest rates, principals or compounding periods.

Spreadsheet or graphics calculator test.

TECHNOLOGY

Use spreadsheets or graphics calculators to calculate incomes, tax, interest and depreciation. Graph the progress of an

investment under compound interest. From the NSW syllabus, Stage 5.2: Internet sites may be used to find commercial rates for home loans and to find home loan calculators.

LANGUAGE

Some students have difficulty differentiating between interest and interest rate when answering questions.

Note that flat interest = simple interest.


3. COORDINATE GEOMETRY Recommended for Stage 5.2 students intending to study the Stage 6 Mathematics course




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 problems using various standard forms of the equation of a straight line (NSW Stage 5.3)

Solve problems involving parallel and perpendicular lines (10NA238)


A student:

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


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

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



MA5.3-3 WM uses deductive reasoning in presenting arguments and formal proofs

MA5.3-8 NA uses formulas to find midpoint, gradient and distance on the Cartesian plane, and applies standard forms of the equation of a straight line

INTRODUCTION

This algebra topic revises and extends coordinate geometry concepts and skills introduced in the Year 9 topic, Coordinate

geometry and graphs. It examines intervals and lines on the number plane as well as various forms of the equation of a straight

line. The general form of a linear equation and the point-gradient form are met for the first time, as well as the equations of

parallel and perpendicular lines. There is much scope for using graphing software such as GeoGebra in this topic. Note that the

formulas for the length, midpoint and gradient of an interval are now Stage 5.3 concepts only, and students will meet non-linear

graphs in the Graphs topic.

CONTENT

1 Length, midpoint and gradient of an interval 9NA214, 9NA294 U F R C find the distance between two points located on the Cartesian plane using a range of strategies, including graphing

software

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

(STAGE 5.3) use the formulas to find the length, midpoint and gradient of the interval joining two points on the

Cartesian plane

(STAGE 5.3) find the angle of inclination of a line with gradient m using m = tan

2 Parallel and perpendicular lines 10NA238 U F R C determine that parallel lines have equal gradients

determine that straight lines are perpendicular if the product of their gradients is -1

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

(STAGE 5.3) sketch the graph of a line by using its equation to find the x- and y-intercepts

determine whether a point lies on a line by substitution

4 The gradient-intercept equation 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

5 The general form of a linear equation ax + by + c = 0 NSW U F R C rearrange an equation of a straight line in the form ax + by + c = 0 ('general form') to gradient-intercept form to

determine the gradient and the y-intercept of the line


6 Stage 5.3: The point-gradient form of a linear equation NSW U F R C find the equation of a line passing through a point (x1, y1) with a given gradient m using the point-gradient form y y1 =

m(x x1) and gradient-intercept form y = mx + b

find the equation of a line passing through two points

7 Finding the equation of a line 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

8 Equations of parallel and perpendicular lines 10NA238 U F R C find the equation of a straight line parallel or perpendicular to another given line

9 Stage 5.3: Coordinate geometry problems NSW F PS R C solve a variety of problems by applying coordinate geometry formulas


RELATED TOPICS

Year 9: Pythagoras theorem and surds, Products and factors, Equations, Coordinate geometry and graphs Year 10: Products and factors, Graphs, Simultaneous equations


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

F = Fluency (applying maths): Using appropriate techniques to graph equations and to find the equation of a line

PS = Problem solving (modelling and investigating with maths): Proving geometrical properties of triangles and

quadrilaterals using coordinate geometry methods

R = Reasoning (generalising and proving with maths): Using the properties of parallel and perpendicular lines to find

their equations algebraically

C = Communicating (describing and representing maths): Writing the equation of a line in different ways and

describing its properties using correct terminology

EXTENSION IDEAS

Back-to-front problems involving distance, midpoint and gradient of an interval

The two-point and intercept forms of the linear equation: 1 2 1

1 2 1

y y y y

x x x x

and 1

x y

a b .

Open-ended problems: (a) Find two points that are 2 units apart; (b) If the midpoint of an interval is (1, 4), what could the endpoints of the interval be?

3D coordinate geometry, polar coordinates, latitude and longitude.


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

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

Describing gradient as the ratio of difference in y over difference in x will lead smoothly to the meaning of derivative in the Stage 6 Mathematics (calculus) course.

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

Identify the x- and y-intercepts of a line.

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.

Explain why the x-axis has equation y = 0. Explain why the y-axis has equation x = 0.

There are many forms of the linear equation. However, y = mx + b can be used to solve most problems involving finding the equation of a line.

Students need practice in converting between y = mx + b and ax + by + c = 0.

A handy formula: The gradient of the line ax + by + c = 0 isb

a .

Use coordinate geometry formulas to determine or prove the properties of triangles and quadrilaterals.

ASSESSMENT IDEAS

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


Graphing test or graphics calculator test.

Test/assignment on proving geometrical properties using coordinate geometry methods.

TECHNOLOGY

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

LANGUAGE

The Cartesian (number) plane is named after 17th century French philosopher and mathematician Ren Descartes (pronounced day-cart), who was one of the first to develop analytical geometry on the number plane.

Why do the gradient-intercept and point-gradient forms of a linear equation have those names?

The y-intercept is a value, b, not a point (0, b). So y = 2x 6 has a y-intercept of -6, not (0, -6).


4. SURFACE AREA AND VOLUME Time: 2 weeks (Term 1, Week 8)


NSW and Australian Curriculum references: Measurement and Geometry

Area and surface area, Volume / Using units of measurement

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)

Solve problems involving surface area and volume for a range of prisms, cylinders and composite solids (10MG242)

Solve problems involving surface area and volume of right pyramids, right cones, spheres and related composite solids

(10AMG271)

Properties of Geometrical Figures / Geometric reasoning

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


A student:

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


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




MA5.3-13 MG applies formulas to find the surface areas of right pyramids, right cones, spheres and related composite solids

MA5.3-14 MG applies formulas to find the volumes of right pyramids, right cones, spheres and related composite solids

INTRODUCTION

This topic extends surface area and volume concepts to more advanced solid shapes. Students will examine the processes and

formulas involved in calculating the surface areas and volumes of pyramids, cones and spheres, sometimes using Pythagoras

theorem first. They also examine the area and volume relationships between pairs of similar 2D and 3D figures. As this is a

Measurement topic, there are opportunities for investigation, practical work and open-ended problem-solving. 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 Surface area of a prism 9MG218, 10MG242 U F PS R C

solve problems involving the surface areas of right prisms

2 Surface area of a cylinder 9MG217, 10MG242 U F PS R

calculate the surface areas of cylinders and solve related problems

3 Stage 5.3: Surface area of a pyramid 10AMG271 U F PS R C

apply Pythagoras theorem to find the slant heights, base lengths and perpendicular heights of right pyramids and cones

find the surface area of right pyramids

4 Stage 5.3: Surface areas of cones and spheres 10AMG271 U F PS R C

find the surface areas of right cones and spheres

5 Surface areas of composite solids 10MG242, 10AMG271 U F PS R C

solve a variety of practical problems related to surface areas of prisms, cylinders and related composite solids

(STAGE 5.3) solve a variety of practical problems related to surface areas of pyramids, cones, spheres and related

composite solids

6 Volumes of prisms and cylinders 10MG242 U F PS R C


solve problems involving volume and capacity of right prisms and cylinders

find the volumes of solids that have uniform cross-sections that are sectors, including semi-circles and quadrants

7 Stage 5.3: Volumes of pyramids, cones and spheres 10AMG271 U F PS R

solve problems involving the volumes of right pyramids, right cones and spheres

8 Stage 5.3: Volumes of composite solids 10AMG271 U F PS R

find the volumes of composite solids that include right pyramids, right cones and hemispheres

9 Stage 5.3: Areas of similar figures 9MG221 U F PS R C establish and apply for two similar figures with similarity ratio 1 : k that matching areas are in the ratio 1 : k2

10 Stage 5.3: Surface areas and volumes of similar solids NSW U F PS R C establish and apply for two similar solids with similarity ratio 1 : k that matching surface areas are in the ratio 1 : k2

and matching volumes are in the ratio 1 : k3


RELATED TOPICS

Year 9: Pythagoras theorem and surds, Surface area and volume, Congruent and similar figures.


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

F = Fluency (applying maths): Selecting correct strategies to calculate surface areas and volumes

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

R = Reasoning (generalising and proving with maths): Understanding the reasoning behind the formula for the surface

area and volume of a cylinder

C = Communicating (describing and representing maths): Identifying the parts of a pyramid, cone and sphere when

calculating their surface areas and volumes

EXTENSION IDEAS

Proofs of the formulas for the volumes of pyramids and cones (see syllabus for ideas).

Right vs oblique pyramids and cones.

Proofs of the formulas for the surface area and volume of a sphere (see syllabus for ideas).

Link to Health and PE: why do babies dehydrate so quickly and why do mice eat so much?

Find the surface area of the best can. Design a carton to hold a litre of milk, considering ease of packing, storage, stacking,

minimum surface area, convenience of use, attractive design.


Resources: chart of surface area and volume formulas, nets or models of solid shapes, paper, scissors, measuring containers for capacity.

Demonstrate that the formulas for surface area have two dimensions while the formulas for volume have three dimensions.

The formulas for the volumes of right prisms, cylinders, pyramids and cones also work for oblique prisms, cylinders, pyramids and cones, provided that the perpendicular height is used.

Use sand and cardboard models to demonstrate the 1 : 3 relationship between the volumes of pyramids and prisms (and cones and cylinders).

From the NSW syllabus, Stage 5.3: A more systematic development of the volume formulas for pyramids, cones and

spheres can be given after integration is developed in Stage 6 (where the factor 1

3 emerges essentially because the primitive

of x2 is 31

3x ). Note that for the sphere, SA = 4r2 is the derivative of V =

34

3r .

What is the formula for the surface area of a hemisphere?

Include problems where extra information is given, or composite solids are involved.

For problems involving surface area and/or composite solids, encourage students to leave partial answers unrounded otherwise the final result will be inaccurate.

Include back-to-front problems where the surface area or volume is given.

Find applications of surface area and volume in building and construction, e.g. backyard pool, packing material.

Link to algebra: show that the formula for the surface area of a cylinder SA = 2r2 + 2rh may be factorised to SA = 2r(r + h) for easier calculation. Similarly with the formula for the surface area of a cone SA = rl + r2.


Investigate the A series of paper sizes. See Worksheet Investigating paper sizes. Why is a scale factor of 71% and 141% used on photocopiers to respectively halve or double the area of a sheet?

ASSESSMENT IDEAS

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

Open-ended and back-to-front questions: The volume of a triangular prism is 540 cm3. What might its dimensions be?

Research project.

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

From the 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), depending on the context.

From the NSW syllabus, Stage 5.3: The difference between the perpendicular heights and the slant heights of pyramids and cones should be made explicit to students.

A right prism has side faces that are rectangular and perpendicular to its cross-section. An oblique prism has side faces that are parallelograms and that are not perpendicular to its cross-section. Similarly, a right cylinder has its axis (of rotation)

perpendicular to its cross-section. An oblique cylinders axis is not perpendicular to its cross-section. The definitions are similar for right and oblique pyramids and cones.

A pyramid or cone with its top sliced off is called a frustum.


5. PRODUCTS AND FACTORS Recommended for Stage 5.2 students intending to study the Stage 6 Mathematics course




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)

Define rational and irrational numbers and perform operations with surds and fractional indices (10ANA264)

Indices / Patterns and algebra

Simplify algebraic products and quotients using index laws (10NA231)

Algebraic Techniques / Patterns and algebra

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

Factorise algebraic expressions by taking out a common algebraic factor (10NA230)

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

Apply the four operations to algebraic fractions with pronumerals in the denominator (NSW Stage 5.2)

Add and subtract algebraic fractions with numerical denominators, including those with binomial numerators (NSW Stage 5.3)

Expand binomial products and factorise monic quadratic expressions using a variety of strategies (10NA233) Algebraic Techniques / Linear and non-linear relationships

Expand binomial products and factorise algebraic expressions using a variety of techniques (10ANA269)


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



MA5.2-6 NA simplifies algebraic fractions, and expands and factorises quadratic expressions

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


MA5.3-5 NA selects and applies appropriate algebraic techniques to operate with algebraic expressions

INTRODUCTION

This revision topic reinforces algebra skills from the Year 9 topics Products and factors and Indices, especially the index laws,

expanding binomial products and factorising quadratic expressions. This topic is fairly technical and abstract so each skill

should be revised with care and precision appropriate to the level of the class. Spend considerable time in class examining the

patterns found in expansions and practising the abstract algebraic manipulations. The aim is to develop a systematic approach

to expansion and factorisation.

CONTENT

1 The index laws 9NA209, 10NA231 U F R C simplify algebraic products and quotients using index laws

2 Stage 5.3: Fractional indices 10ANA264 U F R C apply index laws to demonstrate the appropriateness of the definitions for fractional indices

3 Adding and subtracting algebraic fractions 10NA232 U F R C

apply the four operations to simple algebraic fractions

(STAGE 5.3) add and subtract algebraic fractions with binomial numerators

4 Multiplying and dividing algebraic fractions 10NA232 U F R C

5 Expanding and factorising expressions 9NA213, 10NA230 U F R C

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

factorise algebraic expressions by taking out a common algebraic factor

6 Expanding binomial products 9NA213, 10NA233 U F R C


apply the distributive law to the expansion of binomials

7 Stage 5.3: Factorising special binomial products 10ANA269 U F R C

factorise algebraic expressions involving grouping in pairs and a difference of two squares

8 Factorising quadratic expressions 10NA233 U F R C

factorise monic quadratic expressions x2 + bx + c using a variety of strategies

9 Stage 5.3: Factorising quadratic expressions of the form ax2 + bx + c 10ANA269 U F R C

factorise non-monic quadratic expressions ax2 + bx + c using a variety of strategies

10 Stage 5.3: Mixed factorisations 10ANA269 F R C

11 Stage 5.3: Factorising algebraic fractions NSW U F R C

factorise and simplify complex algebraic expressions involving algebraic fractions


RELATED TOPICS

Year 9: Products and factors, Indices

Year 10: Surds, Equations and logarithms, Quadratic equations and the parabola, Polynomials (option topic)


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

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

R = Reasoning (generalising and proving with maths): Using algebra to represent and generalise the index laws and special products

C = Communicating (describing and representing maths): Using the language of expanding and factorising correctly

EXTENSION IDEAS

More challenging problems involving the index laws and negative and fractional indices

More challenging problems involving expanding and factorising


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

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

1

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

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 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 of New Century Maths 9

Include open-ended questions such as (x )(x ) = x2 5x ... or what two terms could be multiplied to give 4a2 + 8a?

Students will need these factorising strategies when they solve the quadratic equations in the topic Quadratic equations and the parabola. The quadratic formula will also be introduced then.

Factorising ax2 + bx + c by grouping in pairs is a powerful method because it can be applied to problems where a is negative.

With the many types of factorisation, students need to use a systematic approach to decide which method to use. Have them design a poster on this.

Encourage students to check that an expression is fully factorised. Include quadratic trinomials where a simple numerical factor can be taken out first, eg 2x2 10x + 12 = 2(x2 5x + 6).

ASSESSMENT IDEAS

Writing activity on the use of variables and simplifying algebraic expressions, or on the processes of expanding and factorising and the patterns found in the special products.

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

Vocabulary test


TECHNOLOGY

CAS (Computer Algebra Systems) and websites such as Wolfram Alpha can be used to simplify, expand or evaluate algebraic

expressions. Use spreadsheets to evaluate and verify equivalent expressions.

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.

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.

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

From the NSW syllabus, Stage 5.3: When factorising (or expanding) algebraic expressions, students should be encouraged to describe the given expression (or expansion) using the appropriate terminology (for example, difference of two squares, monic quadratic trinomial) to assist them in learning the concepts and identifying the appropriate process.

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

trinomial = algebraic expression with three terms, for example x2 x + 4.

monomial = algebraic expression with one term, for example 5b3.

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


6. INVESTIGATING DATA Time: 3 weeks (Term 2, Week 3)


NSW and Australian Curriculum references: Statistics and Probability

Single Variable Data Analysis / Data representation and interpretation

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

Determine quartiles and interquartile range (10SP248)

Construct and interpret box plots and use them to compare data sets (10SP249)

Compare shapes of box plots to corresponding histograms and dot plots (10SP250)

Evaluate statistical reports in the media and other places by linking claims to displays, statistics and representative data (10SP253)

Investigate reports of studies in digital media and elsewhere for information on their planning and implementation (10ASP277)

Calculate and interpret the mean and standard deviation of data and use these to compare data sets (10ASP278)

Bivariate Data Analysis / Data representation and interpretation

Use scatter plots to investigate and comment on relationships between two numerical variables (10SP251)

Investigate and describe bivariate numerical data where the independent variable is time (10SP252)

Use information technologies to investigate bivariate numerical data sets. Where appropriate use a straight line to describe the relationship allowing for variation (10ASP279)


A student:

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



MA5.2-15 SP uses quartiles and box plots to compare sets of data, and evaluates sources of data

MA5.2-16 SP investigates relationships between two statistical variables, including their relationship over time




MA5.3-18 SP uses standard deviation to analyse data

MA5.3-19 SP investigates the relationship between numerical variables using lines of best fit, and explores how data is used to inform decision-making processes

INTRODUCTION

In this Statistics topic, students consolidate their statistical skills by meeting interquartile range, standard deviation, box plots,

bivariate data, scatter plots and lines of best fit. The shape of a frequency distribution is revised first, but the rest of the topic

will be new to students. The objective of this topic is to compare statistical measures for different sets of data. Aim to include

analysis of data from class surveys and students own experiences. Because this is an interpretation and investigation topic, there is much scope for writing and literacy activities.

CONTENT

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

2 Quartiles and interquartile range 10SP248 U F PS R C determine quartiles and interquartile range

3 Stage 5.3: Standard deviation 10ASP278 U F PS C investigate the meaning and calculation of standard deviation using a small set of data

find the standard deviation of a set of data using digital technologies

compare the relative merits of the range, interquartile range and standard deviation as measures of spread

4 Stage 5.3: Comparing means and standard deviations 10ASP278 F PS R C use the mean and standard deviation to compare two sets of data

5 Box plots 10SP249 U F PS R C

construct and interpret box plots

6 Parallel box plots 10SP249 U F PS R C

compare two or more sets of data using parallel box plots drawn on the same scale


7 Comparing data sets 10SP250 F PS R C compare shapes of box plots to corresponding histograms and dot plots

8 Scatter plots 10SP251 U F R C describe, informally, the strength and direction of the relationship between two variables displayed in a scatter plot, for

example, strong positive relationship, weak negative relationship, no association

9 Stage 5.3: Line of best fit 10ASP279 U F PS R C use digital technologies, such as a spreadsheet, to construct a line of best fit for bivariate numerical data

use lines of best fit to determine what might happen between known data values (interpolation) and what might happen beyond known data values (extrapolation)

10 Bivariate data involving time 10SP252 U F R C recognise the difference between an independent variable and its dependent variable

investigate and describe bivariate numerical data where the independent variable is time

11 Statistics in the media 10SP253 U F PS R C evaluate statistical reports in the media and other places by linking claims to displays, statistics and representative data

12 Stage 5.3: Investigating statistical studies 10ASP277 PS R C investigate reports of studies in digital media and elsewhere for information on their planning and implementation


RELATED TOPICS

Year 9: Investigating data


U = Understanding (knowing and relating maths): Knowing the concepts of quartiles, interquartile range, standard

deviation, box plots, bivariate data, scatter plots and lines of best fit

F = Fluency (applying maths): Constructing a five-number summary and box-and-whisker plot from a set of data

PS = Problem solving (modelling and investigating with maths): Analysing data presented in different forms to solve

problems and draw conclusions

R = Reasoning (generalising and proving with maths): Drawing conclusions about a set of data from box plots and

scatter plots

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

forms and using correct statistical terminology

EXTENSION IDEAS

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

Replicate or implement a major statistical investigation.

Calculating standard deviation the long way.

The normal curve and the 68%, 95%, 99% confidence intervals, z-scores.

Investigate what happens to the range, mean and standard deviation of a data set if a constant is added to each scores or if the values are multiplied by a constant. Use a spreadsheet where appropriate.


Resources: Graphics calculator, statistical and graphing software, spreadsheets, databases, newspapers and magazines, Australian Bureau of Statistics (www.abs.gov.au), Bureau of Meteorology (www.bom.gov.au).

This topic lends itself to investigation projects. The class may be surveyed on a number of characteristics and the data analysed: height, arm span, shoe size, heartbeat rate, reaction time, health and PE data, number of children in family,

number of people living at home, hours slept last night, number of letters in first name, number of vehicles/TV sets/mobile

phones owned at home.

Students should be able to calculate quartiles, interquartile range and standard deviation from data displayed in different forms: list of scores, frequency table, dot plot, stem-and-leaf plot.

Examine the statistics from the sports page of a newspaper or website.

The standard deviation is an average of the deviations of the scores from the mean of a data set.

Students are not expected to calculate standard deviation the long way using a formula, only interpret its value (population

standard deviation only n or xn) using the calculators statistics mode. Students are not expected to analyse the relative positions of the mean, mode and median in skewed distributions.

Why is the interquartile range a better measure of spread than the range sometimes? Why is the standard deviation often the best measure of spread?


From the NSW syllabus, Stage 5.2: Bivariate data analysis explores relationships between variables, including through the use of scatter plots and lines of best fit, and is generally used for explanatory purposes. A researcher investigating the

proportion of eligible voters who actually vote in an election might consider a single variable, such as age. If wanting to use

a bivariate approach, the researcher might compare age and gender, or age and income, or age and education, etc.

Find newspaper articles in which statistics have been misinterpreted. Students could analyse the statistics used in media claims or use statistics to justify an argument themselves.

ASSESSMENT IDEAS

Plan, implement and report on a statistical investigation.

Statistical survey.

Vocabulary test, Statistical graphs and displays test.

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

Research the role of the Australian Bureau of Statistics or the Australian Census.

TECHNOLOGY

From the NSW syllabus, Stage 5.2: Graphics calculators and other statistical software will display box plots for entered data, but students should be aware that results may not always be the same. This is because the technologies use varying methods for

creating the plots, eg some software packages use the mean and standard deviation by default to create a box plot. 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, such adding outliers or doubling every score. Visit the CensusAtSchool

website www.abs.gov.au/censusatschool.

LANGUAGE

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

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.

Reinforce the terminology measures of location and measures of spread.

Name the five measures found in a five-number summary.


7. EQUATIONS AND LOGARITHMS# Recommended for Stage 5.2 students intending to study the Stage 6 Mathematics course #Recommended for students intending to study the Stage 6 Mathematics Extension 1 course




Equations / Linear and non-linear relationships

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

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

Solve linear inequalities and graph their solutions on a number line (10NA236)

Solve linear equations involving simple algebraic fractions (10NA240)

Solve complex linear equations involving algebraic fractions (NSW Stage 5.3)

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

Solve simple cubic equations (NSW Stage 5.3)

Rearrange literal equations (NSW Stage 5.3)

Logarithms / Real numbers

Use the definition of a logarithm to establish and apply the laws of logarithms (10ANA265)

Solve simple exponential equations (10ANA270)


A student:




MA5.2-8 NA solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques




MA5.3-7 NA solves complex linear, quadratic, simple cubic and simultaneous equations, and rearranges literal equations

MA5.3-11 NA uses the definition of a logarithm to establish and apply the laws of logarithms

INTRODUCTION

This topic revises equation-solving skills from the Year 9 topic, Equations before introducing students to (monic) quadratic

equations of the form x2 + bx + c = 0, linear inequalities, logarithms and exponential and logarithmic equations. According to

the NSW syllabus, the logarithms content of this topic is recommended for students who intend to study the Stage 6

Mathematics Extension 1 course next year. Simultaneous equations will be covered in a separate topic, while harder quadratic

equations will be met in the topic Quadratic equations and the parabola. The processes behind solving equations and

inequalities and manipulating logarithmic expressions are detailed and technical, requiring careful and precise understanding

and practice, so dont rush through this topic.

CONTENT

1 Equations with algebraic fractions 10NA240 U F R

solve linear equations involving simple algebraic fractions

(NSW STAGE 5.3) solve complex linear equations involving algebraic fractions

2 Quadratic equations x2 + bx + c = 0 10NA241 U F R C solve simple quadratic equations of the form ax2 = c, leaving answers in exact form and as decimal approximations

solve quadratic equations of the form x2 + bx + c = 0, by factorisation

3 Stage 5.3: Simple cubic equations ax3 = c NSW U F R C

solve simple cubic equations of the form ax3 = c

4 Equation problems 10NA235 U F PS R C

solve real-life problems by using pronumerals to represent unknowns


5 Equations and formulas 10NA234, 10NA235 U F PS R C

substitute values into formulas to determine an unknown

solve problems involving linear equations, including those derived from formulas

6 Stage 5.3: Changing the subject of a formula NSW U F R C

rearrange literal equations (change the subject of formulas)

7 Graphing inequalities on a number line 10NA236 U F C

represent simple inequalities on the number line

8 Solving inequalities 10NA236 U F R

solve linear inequalities and graph their solutions on a number line

9 Stage 5.3: Logarithms# 10ANA265 U F R C define logarithms and convert between statements written in logarithmic and index form

10 Stage 5.3: Logarithm laws# 10ANA265 U F R C deduce and apply logarithm laws and properties

11 Stage 5.3: Exponential and logarithmic equations# 10ANA270 U F R C solve simple equations involving exponents and logarithms


RELATED TOPICS

Year 9: Products and factors, Equations

Year 10: Coordinate geometry, Products and factors, Graphs, Simultaneous equations, Quadratic equations and the parabola,

Functions (option topic)


U = Understanding (knowing and relating maths): Knowing the different types of equations and inequalities

F = Fluency (applying maths): Using the definition and properties of logarithms to evaluate and simplify expressions and

solve exponential and logarithmic equations

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

formulas

R = Reasoning (generalising and proving with maths): Understanding the logic behind the steps in solving equations and

inequalities

C = Communicating (describing and representing maths): Writing and simplifying expressions written in logarithmic

form

EXTENSION IDEAS

Harder formulas and word problems, constructing formulas

Equations with the unknown in the denominator, equations involving powers and roots

Logarithms were invented by the Scottish mathematician John Napier. He also devised a calculation system for multiplying and dividing called Napiers rods.

The history and applications of logarithms. From the NSW syllabus, Stage 5.3: Relate logarithms to practical scales, eg Richter, decibel and pH scales.

Evaluate logarithms on the calculator using the change of base law

Investigate how logarithm tables were used in calculations before calculators.


Encourage students to check solutions to equations and inequalities by substituting back.

With equations involving fractions, denominators should be numerical (no variables).

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.

Why cant you find the logarithm of a negative number or zero? Why is log 1 = 0?

The logarithms laws are derived from the index laws.

Logarithms convert a difficult multiplication and division pen-and-paper calculation into a simpler addition and subtraction one. Show the class a logarithm table from pre-calculator days. See Worksheet Logarithm tables.

The graphs of the exponential and logarithmic functions are examined in the Stage 5.3 option topic Functions.


ASSESSMENT IDEAS

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

Writing activity describing the process of solving an inequality.

Test, assignment or project on calculating using logarithm tables.

TECHNOLOGY

CAS calculators and the Wolfram Alpha website can be used to solve equations. Investigate logarithms on a calculator, graphics

calculator or spreadsheet.

LANGUAGE

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.

From the NSW syllabus: The square root sign signifies a positive number (or zero). Thus 9 = 3 (only). However, the

two numbers whose square is 9 are 9 or - 9 , i.e. 3 or -3.

Some students believe that x < 5 and x 4 mean the same thing. Explain the difference.

What is the origin of the terms exponent and logarithm?

From the NSW syllabus, Stage 5.3: Teachers need to emphasise the correct language used in connection with logarithms, eg logaax = x is 'log to the base a, of a to the power of x, equals x'.


8. GRAPHS Recommended for Stage 5.2 students intending to study the Stage 6 Mathematics course




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)

Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (10ANA267)

Describe, interpret and sketch cubics, other curves and their transformations (NSW Stage 5.3)


A student:

MA5.1-7 NA graphs simple non-linear relationships




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

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



MA5.3-4 NA draws, interprets and analyses graphs of physical phenomena

MA5.3-9 NA sketches and interprets a variety of non-linear relationships

INTRODUCTION

This algebra topic revises and extends concepts in proportion and non-linear graphs from the Year 9 topic Coordinate geometry

and graphs. Last year, students met the idea of direct proportion and the graphs of simple parabolas and circles, but here they

are introduced to inverse proportion, conversion graphs, distance-time graphs, more parabolas and circles, the cubic curve,

graphs of higher powers, the hyperbola and the exponential curve. There is much scope for using graphing software such as

GeoGebra in this topic. In many parts of this topic, the NSW syllabus goes beyond the Australian curriculum (see the sections

labelled NSW in red in CONTENT below).

CONTENT

1 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

2 Inverse proportion NSW U F PS R C identify and describe everyday examples of inverse (indirect) proportion

3 Conversion graphs NSW U F PS R C interpret and use conversion graphs to convert from one unit to another

4 Stage 5.3: Distance-time graphs NSW U F PS R C interpret distance-time graphs when the speed is variable

5 Stage 5.3: Graphs of change NSW U F PS R C interpret graphs where the rate of change is variable and describe the rate of change at different points on the graph

sketch a graph from a simple description, given a variable rate of change

6 The parabola y = ax2 + c 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

determine the x-coordinate of a point on a parabola, given the y-coordinate of the point


7 Stage 5.3: The parabola y = a(x r)2 NSW U F R C graph parabolas of the form y = a(x r)2 from the graph of y = ax2

8 Stage 5.3: The cubic curve y = ax3 + c NSW U F R C graph cubic curves of the form y = ax3 + c

9 Stage 5.3: The power curves y = axn + c NSW U F R C graph curves of the form y = axn + c and y = a(x r)n from the graph of y = axn

10 Stage 5.3: The hyperbola y = k

x

10ANA267 U F R C

graph a variety of hyperbolic curves, including y = k

x+ c and y =

k

x r

11 The exponential curve y = ax 9NA296, 10NA239 U F R C sketch, compare and describe simple exponential curves of the form y = ax + c

12 Stage 5.3: The circle (x h)2 + (y k)2 = r2 9NA296, 10NA239 U F R C sketch circles of the form x2 + y2 = r2

(STAGE 5.3) sketch circles of the form (x h)2 + (y k)2 = r2

13 Identifying graphs 10NA239 F R C match graphs of straight lines, parabolas, circles and exponentials to the appropriate equations

(STAGE 5.3) identify and name different types of graphs from their equations


RELATED TOPICS

Year 9: Coordinate geometry and graphs

Year 10: Coordinate geometry, Equations and logarithms, Simultaneous equations, Quadratic equations and the parabola.

Polynomials (option topic), Functions (option topic)


U = Understanding (knowing and relating maths): Understanding the concepts of direct and inverse proportion

F = Fluency (applying maths): Using appropriate techniques to graph parabolas, cubic curves, higher-power curves,

hyperbolas, exponential curves and circles

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

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

Direct and inverse proportion problems involving the square, cube or square root of a variable

The conic sections.

The parabola as a locus of all points equidistant from a fixed point and line

The square root graph y = x , the semi-circle y = 2 2r x


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

Students should construct practical graphs based on their personal experiences and display them in class, for example, noise level in a classroom, mood swings of a crowd during a football match. Teachers could develop a collection of interesting

graphs for use in the classroom.

When graphing, encourage students to label axes, use a suitable scale and label the graph.

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 hyperbola is a conic section formed by the intersection of a cone by a plane parallel to 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. The path of some comets is a parabola.

The graph of y = ax2 + bx + c will be covered in the topic Quadratic equations and the parabola. The graph of cubic curves of the form y = a(x r)(x s)(x t) will be covered in the Stage 5.3 option topic Polynomials.


All points that lie on the graph have coordinates that satisfy its equation. Points that dont lie on the graph do not satisfy the equation.

Compound interest and population growth can be modelled by exponential equations and graphs.

From the NSW syllabus, Stage 5.3: Determine whether a particular point is inside, on, or outside a given circle.

ASSESSMENT IDEAS

Writing and literacy activities involving practical graphs: tell me a story.

Given a worded description of a journey or situation, draw the graph.

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

Matching situations/equations to their graphs.

TECHNOLOGY

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

equations.

LANGUAGE

Why is it called direct proportion? Why is it called inverse proportion?

y = ax is called an exponential equation where a is the base and x is the power, index or exponent.

The words hyperbola and hyperbole (meaning exaggeration) both come from the Greek hyperbole, meaning to throw excessively. The hyperbola is like an exaggerated or excessive parabola (para means alongside).

The word asymptote comes from the Greek asumptotos meaning not together falling (a-sum-ptotos).


9. TRIGONOMETRY Recommended for Stage 5.2 students intending to study the Stage 6 Mathematics course



NSW and Australian Curriculum references: Measurement and Geometry

Right-angled triangles (Pythagoras) / Pythagoras and trigonometry

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

Solve right-angled triangle problems, including those involving direction and angles of elevation and depression (10MG245)

Establish the sine, cosine and area rules for any triangle and solve related problems (10AMG273)

Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies (10AMG274)

Solve simple trigonometric equations (10AMG275)

Apply Pythagoras' theorem and trigonometry to solving three-dimensional problems in right-angled triangles (10AMG276)


A student:

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




MA5.3-15 MG applies Pythagoras theorem, trigonometric relationships, the sine rule, the cosine rule and the area rule to solve problems, including problems involving three dimensions.

INTRODUCTION

This is the second trigonometry topic for Stage 5.3 students, a sequel to the right-angled trigonometry topic learned in Year 9.

Here, students extend their knowledge of trigonometry beyond right angles and right-angled triangles, examining the

trigonometric ratios of obtuse and reflex angles, some trigonometric relations, the graphs of the trigonometric functions, the

sine and cosine rules, and the area of a triangle sine formula. This topic is also recommended for Stage 5.2 students intending to

study the Stage 6 Mathematics course next year, where similar content will be met.

CONTENT

1 Right-angled trigonometry 9MG224, 10MG245 U F PS R C

find the lengths of unknown sides in right-angled triangles where the given angle is measured in degrees and minutes

find the size in degrees and minutes of unknown angles in right-angled triangles

solve a variety of practical problems involving angles of elevation and depression, including problems for which a diagram is not provided

2 Bearings 10MG245 U F PS R C

solve a variety of practical problems involving bearings, including problems for which a diagram is not provided

3 Stage 5.3: Pythagoras theorem and trigonometry in 3D 10AMG276 F PS R C

solve a variety of practical problems involving right-angled triangles in three dimensions, including problems for which a diagram is not provided

4 Stage 5.3: Trigonometric relations NSW U F R C

prove and use the relationships between the sine and cosine ratios of complementary angles in right-angled triangles:

sin A = cos(90 A), cos A = sin(90 A)

determine and use the exact sine, cosine and tangent ratios for angles of 30, 45 and 60


5 Stage 5.3: The trigonometric functions 10AMG274 U F R C

use the unit circle to define trigonometric functions for angles between 0 and 360, and graph them

prove than tan = sin

cos

use the unit circle or graphs of trigonometric functions to establish and use the following relationships for obtuse angles:

sin A = sin(180 A), cos A = -cos(180 A), tan A = -tan(180 A)

6 Stage 5.3: Trigonometric equations 10AMG275 U F R C

determine the possible acute and/or obtuse angle(s), given a trigonometric ratio

7 Stage 5.3: The sine rule 10AMG273 U F PS R C

prove the sine rule and use it to find an unknown side in a triangle

8 Stage 5.3: The sine rule for angles 10AMG273 U F PS R C

use the sine rule to find an unknown angle in a triangle, including in problems where there are two possible solutions

9 Stage 5.3: The cosine rule 10AMG273 U F PS R C

prove the cosine rule and use it to find an unknown side in a triangle

10 Stage 5.3: The cosine rule for angles 10AMG273 U F PS R C

use the cosine rule to find an unknown angle in a triangle

11 Stage 5.3: The area of a triangle 10AMG273 U F PS R C

prove and use the area rule to find the area of a triangle

12 Stage 5.3: Problems involving the sine and cosine rules 10AMG273 F PS R C

select and use the appropriate rule to find unknowns in non-right-angled triangles

solve a variety of practical problems that involve non-right-angled triangles, including problems where a diagram is not provided


RELATED TOPICS

Year 9: Pythagoras theorem and surds, Trigonometry Year 10: Surds


U = Understanding (knowing and relating maths): Extending the trigonometric ratios to angles beyond 90

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): Apply geometric reasoning in trigonometry problems involving

angles beyond 90 and triangles that are not right-angled

C = Communicating (describing and representing maths): Expressing solutions to equations and problems involving

trigonometry

EXTENSION IDEAS

Harder problems involving angles of elevation/depression, bearings, overlapping triangles.

Surveying and navigation, polar coordinates.

Radian and grad measure for angles.

Examining negative angles and angles beyond 360

Trigonometric identities, the tangent rule.

Herons formula for the area of a triangle.


Resources: clinometers, geometrical instruments, magnetic compass, surveying and navigation charts, graphing software.


Take the class outside and use the school fields for surveying and orienteering activities. Calculate the heights of trees, flagpoles and buildings using trigonometry.

Construct triangles or scale diagrams and compare measured results with calculated results using trigonometry.

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.

From the NSW syllabus, Stage 5.3: The graphs of the trigonometric functions mark the transition from understanding trigonometry as the study of lengths and angles in triangles to the study of waves, as will be developed in the Stage 6 calculus courses.

The m = tan relationship between the gradient and angle of inclination of a line is learned in the Coordinate geometry topic.

Students are not required to memorise the proofs of the sine, cosine and area rules.

The sine rule demonstrates that the longest side of a triangle is opposite the largest angle. The sine rule is quoted as a triple but in its application only a pair is used.

The cosine rule is an extension of Pythagoras theorem.

Emphasise that the sine, cosine and area rules work for all triangles. When is it appropriate to use each rule? What happens to the rules when one of the angles is 90?

Measure triangles and calculate their areas, using as many different methods as possible.

Prove that the area of an equilateral triangle with side length x is 3

4

x.

ASSESSMENT IDEAS

Practical test involving clinometers and magnetic compasses.

Research project on the history or applications of trigonometry.

Surveying, navigation or orienteering project.

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.

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

With compass bearings, stress the terminology: the bearing of P from O. See syllabus Language notes (Stage 5.2) for more details.

Ambiguous means to have more than one meaning, and the ambiguous case when using the sine rule occurs when there are two possible answers for an unknown angle: one acute, one obtuse. That means that there are two possible triangles

that can be drawn with the given lengths and angle sizes.


10. SIMULTANEOUS EQUATIONS Time: 1 week (Term 3, Week 6)




Solve linear simultaneous equations, using algebraic and graphical techniques, including with the use of digital technologies (10NA237)


A student:


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

MA5.2-8 NA solves linear and simple quadratic equations, linear inequalities and linear simultaneous equations, using analytical and graphical techniques

INTRODUCTION

In this short Stage 5.2 algebra topic, students are introduced to linear simultaneous equations and three different methods for

solving them: graphical method, elimination method and substitution method, building upon previous work on algebra and

graphing linear equations. As mentioned previously, the process of equation-solving is detailed and technical, and even more

so for simultaneous equations, requiring careful and precise understanding and practice, so dont rush through this topic.

CONTENT

1 Solving simultaneous equations graphically 10NA237 U F R C solve linear simultaneous equations by finding the point of intersection of their graphs

2 The elimination method 10NA237 U F R C solve linear simultaneous equations using appropriate algebraic techniques

3 The substitution method 10NA237 U F R C

4 Simultaneous equation problems 10NA237 U F PS C generate and solve linear simultaneous equations from word problems and interpret the results


RELATED TOPICS

Year 9: Equations, Coordinate geometry and graphs

Year 10: Coordinate geometry, Equations and logarithms, Quadratic equations and the parabola


U = Understanding (knowing and relating maths): Understanding the idea of simultaneous equations

F = Fluency (applying maths): Selecting an appropriate method for solving a pair of simultaneous equations

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

R = Reasoning (generalising and proving with maths): Understanding the logic behind the steps in solving simultaneous

equations

C = Communicating (describing and representing maths): Expressing solutions to simultaneous equations algebraically

and solutions to problems in words

EXTENSION IDEAS

Simultaneous equations involving linear and non-linear equations (covered in the topic Quadratic equations and the

parabola)

Using technology and websites to solve simultaneous equations


Encourage students to check solutions by substituting back. For word problems, check that the solution sounds reasonable.

When solving a word problem, students need practice in identifying the unknown quantity and calling it x, say.

When solving simultaneous equations, students often forget to give the solution for both variables x and y.

Open-ended question: if x + 2y = 9, what are some possible values of x and y?


ASSESSMENT IDEAS

Writing activity comparing and evaluating the different methods of simultaneous equations.

Simultaneous equations test/assignment comparing different methods of solution.

TECHNOLOGY

Investigate the use of CAS (computer algebra system) calculators/software, websites such as Wolfram Alpha, spreadsheets and

graphics calculators to solve simultaneous equations. Spreadsheets are good for guess-and-check strategies. From the NSW syllabus, Stage 5.2: Graphing software and graphics calculators allow students to graph two linear equations and to display the coordinates of the point of intersection of their graphs.

LANGUAGE

Why are they called simultaneous equations? Why do the three methods of solving simultaneous equations have those names?


11. QUADRATIC EQUATIONS AND THE PARABOLA Recommended for Stage 5.2 students intending to study the Stage 6 Mathematics course





Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts (10ANA269)

Non-linear relationships / Linear and non-linear relationships

Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (10ANA267)


A student:



MA5.3-7 NA solves complex linear, quadratic, simple cubic and simultane

teaching program year 10 (advanced)

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