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Year 9(5.2) 2014 Mathematics Program

Term 1

1 2 3 4 5 6 7 8 9 10

1. Working with Numbers

(30min per week NAPLAN prep)

2. Pythagoras’ theorem


3. Algebra

(30min per week NAPLAN prep)Task 1 –(20%)

Term 2

1 2 3 4 5 6 7 8 9 10

4. Trigonometry


5. Indices 6. Earning Money

Task 2 – Half Yearly . (30%)

Term 3

1 2 3 4 5 6 7 8 9 10

7. Equations 8. Geometry 9. Investigating DataSurface Area and Volume

Task 3 – Assignment (10%)

Term 4

1 2 3 4 5 6 7 8 9 10 11

10. Surface Area and Volume

11. Coordinate Geometry and Graphs 12. Probability 13. Congruent and Similiar Figures

Task 4 – Yearly exam (40%)

WORKING WITH NUMBERSUnit Overview: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 don’t 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%.

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

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 MA5.2-1 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

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games and spelling

tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au 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.Content (sign and date) Quality Teaching Ideas Resources

o Integers ordering, adding, subtracting, multiplying and dividing integers

NCM Stages 5.1/5.2 pg 42 -89

o Decimals carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies

o Terminating and recurring decimals define each and state whether a fraction/decimal is

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

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terminating or recurring. (E) Some decimals are neither terminating nor recurring. Their digits run

endlessly, but without repeating, for example, = 1.4142135 … and π = 3.1415926 …

o Fractions : improper and mixed numbers, ordering fractions, simplifying fractions, adding, subtracting, multiplying and dividing fractions.

(X)Include open-ended questions such as finding two fractions that have

a sum of or three fractions between and .

o Percentages :converting between percentages, fractions and decimals

(M)Investigate the percentage forms of ‘fraction families’ such as the

eighths and the sixths. What are 16 % and 37.5% as fractions? (M)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.

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

o Operations with percentages solve problems involving the use of percentages, including percentage increases and decreases

o Percentages and money solve problems involving profit, loss, discounts and GST

o Simple interest solve problems involving simple interest

o Ratios and rates solve a range of problems involving ratios and ratios, with and without digital technologies

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

o Time differences solve problems involving duration, including using 12- and 24-hour time within a single time zone

Assessment

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Evaluation

PYTHAGORAS’ THEOREMUnit OverviewThis 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.

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

MA4-1 WM communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols MA4-2 WM applies appropriate mathematical techniques to solve problems

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory

games and spelling tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au 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”’.

Content Quality Teaching Ideas Resourceso Squares, square roots and surds

investigate the concept of irrational numbers

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

NCM 9 5.1/5.2Pg 2 -39

o Pythagoras’ theorem 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

o Finding the hypotenuse using Pythagoras theorem.

(M)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 it’s wrong if the hypotenuse is shorter than one of the other sides.

(X)Demonstrate that the length can be constructed using a right-angled isosceles triangle.

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o Finding a shorter side solve practical problems involving Pythagoras’ theorem, approximating the answer as a decimal and giving an exact answer as a surd

(E) scaffold the steps to solve a problem using pythagorus i.e. Step 1.....

o Mixed problems students identify the and use theorem find hyp or shorter side

o Testing for right-angled triangles use the converse of Pythagoras’ theorem to establish whether a triangle has a right angle

o Pythagorean triads 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

(X)There are different formulas for creating Pythagorean triads, such as (p2 – q2,

2pq, p2 + q2), (n,n2−12 ,

n2+12 ) 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).

o Pythagoras’ theorem problems mixed problems

(E)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.

Assessment

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Evaluation

ALGEBRA

Unit OverviewThis 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.

Outcomes MA5.2-6 NA simplifies algebraic fractions

MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-3 WM constructs arguments to prove and justify results


games and spelling tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au

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’.

Content Quality Teaching Ideas Resources From words to algebraic move fluently

between algebraic and word representations as descriptions of the same situation

(E) Investigate pattern in problems such as handshakes, page numbering, angle sum of a polygon, checkerboard squares, creases in paper-folding, Pascal’s triangle.

Substitution create algebraic expressions and evaluate them by substituting a given value for each variable

Adding and subtracting terms (M) Demonstrate adding algebraic terms in constructing perimeter formulas and multiplying terms in constructing area formulas.

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(E) 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.

(M) Common mistakes: 2a – a = 2, 3b2 = 3b × 3b. Explain that the index 2 belongs to the b only.

Multiplying and dividing terms

Adding and subtracting algebraic fractions

(M) 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.

Multiplying and dividing algebraic fractions

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

Factorising expressions factorise algebraic expressions

(M)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.

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

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

(X) Factorising quadratic expressions (Year 10, x2 + bx + c only)(X) Factorising by grouping in pairs (Stage 5.3)

Expanding binomial products apply the distributive law to the expansion of binomials

(X)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.

Assessment Strategies: 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

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TRIGONOMETRY

Unit OverviewThis 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 topic—spend 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.

Outcomes MA5.2-13 MG applies trigonometry to solve problems, including problems involving bearings MA5.1-10 MG applies trigonometry, given diagrams, to solve problems, including problems involving angles of elevation and depression

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

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise,

memory games and spelling tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au 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.

Content Quality Teaching Ideas Resourceso The sides of a right-angled triangle

labelling 3 sides of a right angle triangle(hypotenuse, opposite,adjacent)

NCM95.1/5.2Pg 132- 169

o The trigonometric ratios : sin,cos,tan- learn a mnemonic.

o (X)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.

o Similar right-angled triangles use



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

o Trigonometry on a calculator degrees, minutes

o Finding an unknown side select and use appropriate trigonometric ratios in right-angled triangles to find unknown sides, where the given angle is measured in degrees and minutes

o (X)Students could verify their answers to trigonometric problems using scale drawings.

o (M)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.

o Finding more unknown sides select and use appropriate trigonometric ratios in right-angled triangles to find the hypotenuse

o Finding an unknown angle select and use appropriate trigonometric ratios in right-angled triangles to find unknown angles correct to the nearest degree and minute

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

Assessment

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INDICES

Unit OverviewIn this topic, students are introduced to the index laws and negative indices. It examines indices both numerically and algebraically, applying them so that students

won’t 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.

Outcomes 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

MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions


games and spelling tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au

Content Quality Teaching Ideas Resourceso Multiplying and dividing terms with the

same base (M) Begin with a numerical approach to the index laws. Demonstrate zero

and negative powers by extending the repeated multiplication pattern backwards.

o Power of a power extend and apply the index laws to variables, using positive-integer indices

o Powers of products and quotients simplify algebraic products and quotients using index laws

(M) Open-ended question: find two terms that can be divided to give 27. (E) Verify the index laws by using a calculator. Explain why a particular

algebraic sentence, for example, a3 × a2 = a6, is incorrect.

o The zero index extend and apply the index laws to variables, using the zero index

o Negative indices apply index laws to numerical expressions with integer indices (NSW, STAGE 5.2) apply index laws to algebraic expressions involving integer

(E) Common student errors:

5x0 = 1, 9x5 ÷ 3x5 = 3x, 2c-4 = , 2a2 = 4a2, (3b)2 = 3b2.



indices

o Summary of the index laws

o Significant figures NSW identify significant figures and round numbers to a specified number of significant figures

o Scientific notation express numbers in scientific notation and order numbers expressed in scientific notation

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

o Scientific notation on a calculator enter and read scientific notation on a calculator and solve problems involving scientific notation

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

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EARNING MONEY

Unit OverviewIn 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.

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

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


tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au The abbreviation K comes from the Greek word khilioi meaning thousand. It is used in many job advertisements, for example, a salary of $80K.’

Content Quality Teaching Ideas Resourceso Wages and salaries

solve problems involving earning moneycalculate weekly, fortnightly, monthly and yearly earnings

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

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

o Overtime payo Commission, piecework and leave loading

calculate earnings from wages, overtime, commission and pieceworkcalculate annual leave loading

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

o Income taxdetermine annual taxable income using current tax rates

o PAYG and net payuse published tables or online calculators to determine the weekly, fortnightly or monthly tax to be deducted from a worker’s pay under the Australian ‘pay-as-you-go’ (PAYG) taxation system, calculate net earnings after deductions

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



and taxation are taken into account

Assessment

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GEOMETRYUnit OverviewThis 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.

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

MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and solutions MA5.2-3 WM constructs arguments to prove and justify results


tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au

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’.

Content Quality Teaching Ideas Resourceso Angle geometry identifies 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

(M) 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)’.

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

o Triangle geometry 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

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

(M) Students should have experience in classifying triangles and quadrilaterals using their properties and minimal conditions, for example, which quadrilateral’s diagonals bisect each other?

o Quadrilateral geometry classify quadrilaterals according to their side and angle properties and solve related numerical problems using reasoning. apply the angle sum of a quadrilateral.

(M) 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’.

(M)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. (E) In how many different ways can you demonstrate the angle sum of a triangle

(or quadrilateral)?

o Stage 5.2 : Angle sum of a polygon 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

(X)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.

o Stage 5.2 : Exterior angle sum of a convex polygon establish that the sum of the exterior angles of any convex polygon is 360°

Assessment 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 proofLanguage

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Topic7. EQUATIONSUnit OverviewThis 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 don’t 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.

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

Content Quality Teaching and Learning Strategies

o Two-step equationso Equations with variables on both sides solve

linear equations using algebraic techniqueso Equations with brackets solve linear equations

involving grouping symbolso Equation problems solve real-life problems by

using pronumerals to represent unknownso Stage 5.2 : Equations with algebraic fractions

solve linear equations involving simple algebraic fractions

o Stage 5.2 : Simple quadratic equations ax2 = c solve simple quadratic equations of

the form ax2 = co Stage 5.2 : Equations and formulas

substitute values into formulas to determine an unknown. Solve problems involving linear equations, including those derived from formulas

o Revision and mixed problems

(M)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.

(M)Examples of Stage 5.2 equations with algebraic

fractions from NSW syllabus:

(denominators should be numerical).

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

(M)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.

(X)Harder formulas and word problems, constructing formulas

(X)Equations with the unknown in the denominator (X)Inequalities (Year 10) (X)Simple cubic equations ax3 = c (Stage 5.3) (M)Simultaneous equations

Assessment

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.

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Topic8. EARNING MONEYUnit Overview

Ou

Content

o NSW U F PS R Co Revision and mixed problems

Quality Teaching and Learning Strategies

Liaise with the HSIE faculty or the school’s careers adviser for resources.

Assessment

Practical or problem-solving test/assignment Collage/poster/case study on the different ways of earning money.

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

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Topic9. INVESTIGATING DATAUnit OverviewIn 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.

Outcomes 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 MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context MA5.1-12 SP uses statistical displays to compare sets of data, and evaluates statistical claims made in

the media MA5.2-3 WM constructs arguments to prove and justify resultsContent Quality Teaching and Learning Strategies

o The mean, median, mode and range calculate mean, median, mode and range for sets of data, and interpret these statistics in the context of data. investigate the effect of

(M)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

individual data values, including outliers, on the mean and median

o Histograms and stem-and-leaf 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

o The shape of a distribution describe data using terms, including ‘skewed’, ‘symmetric’ and ‘bi-modal’

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

o Sampling and types of data investigate techniques for collecting data, including census, sampling and observation

o Stage 5.2 : Bias and questionnaires investigate reports of surveys in digital media and elsewhere for information on how data was obtained to estimate population means and medians


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.

(M)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.

(E)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.

(X)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?

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

Assessment 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’.

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Topic10. SURFACE AREA AND VOLUMEUnit OverviewThis 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.

Outcomes MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.2-1 WM selects and uses appropriate strategies to solve problems 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 MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and

solutions 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


o The metric system 9MG219 U F R Cinterpret 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 kilobytesinvestigate very small and very large time scales and intervals

o Limits of accuracy of measuring instruments NSW U R Cdescribe the limits of accuracy of measuring instruments (±0.5 unit of measurement)

o Perimeters and areas of composite shapes 9MG216 U F Rcalculate the perimeters and areas of composite shapes

o Areas of quadrilaterals 8MG196 U F PS Rfind areas of parallelograms, trapeziums, rhombuses and kites

o Circumferences and areas of circular shapes 9MG216 U F PS Rcalculate the areas of composite figures by dissection into quadrants, semi-circles and sectors

o Surface area of a prism 9MG218 U F PS Rsolve problems involving the surface areas of right prisms

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

o calculate the surface areas of cylinders and solve related problems

o Volumes of prisms and cylinders 8MG198 U

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

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

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

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

(M)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.

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

(M)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.

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

F PS Ro solve problems involving volume and capacity

of right prisms and cylinderso Revision and mixed problems

Assessment Investigate paper and envelope sizes, the legal size of an envelope, history of π, 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?’

TechnologyDrawing 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'.


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Topic11. COORDINATE GEOMETRY AND GRAPHSUnit OverviewThis 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.

Outcomes MA5.1-1 WM uses appropriate terminology, diagrams and symbols in mathematical contexts MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context MA5.1-6 NA determines the midpoint, gradient and length of an interval, and graphs linear relationships MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and


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


o The length of an interval find the distance between two points located on the Cartesian plane using a range of strategies, including

(M)Resources: number plane grid paper, graphics calculator, graphics software.

(M)Investigate large, small, positive, negative, zero and fractional gradients. Demonstrate how a

graphing softwareo The midpoint of an intervalo The gradient of a line find the midpoint and

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

o Graphing linear equations sketch linear graphs using the coordinates of two points determine whether a point lies on a line by substitution

o Stage 5.2 : The gradient-intercept formula y = mx + b interpret and graph linear relationships using the gradient-intercept form of the equation of a straight line

o Stage 5.2 : Finding the equation of a line y = mx + b find the gradient and y-intercept of a straight line from its graph and use these to determine the equation of the line

o Solving linear equations graphically solve linear equations using graphical techniques

o Stage 5.2 : Direct proportion solve problems involving direct proportion and explore the relationship between graphs and equations corresponding to simple rate problems

o Graphing quadratic equations 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

o Graphing circles sketch circles of the form x2 + y2 = r2

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


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

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

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

(M)All points that lie on the line have coordinates that satisfy the linear equation. Points that don’t lie on the line do not satisfy the equation.

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

(X)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.

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

(X)Gradients of parallel and perpendicular lines (Year 10)

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

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

Open-ended questions, for example, find two points that are units apart, if the midpoint of an interval is

(1, 4), what could the endpoints of the interval be?TechnologyUse 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 René Descartes.Registration

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Topic:12. PROBABILITYUnit OverviewThis 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.

Outcomes 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 MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context MA5.1-13 SP calculates relative frequencies to estimate probabilities of simple and compound events MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and


to solve problems MA5.2-3 WM constructs arguments to prove and justify results MA5.2-17 SP describes and calculates probabilities in multi-step chance experiments


o Probabilityidentify complementary events and use the sum of probabilities to solve problems

o 2 Relative frequencycalculate relative frequencies from given or

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

Do not assume that all students have had experience with the properties of playing cards: suits, colours, deck of 52. Be sensitive to religious

collected data to estimate probabilities of events involving ‘and’ or ‘or’

o 3 Venn diagramsrepresent events in Venn diagrams and solve related problemsdescribe 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

o 4 Two-way tablesrepresent events in two-way tables and solve related problemscalculate probabilities of events from data contained in two-way tables

o 5 Stage 5.2: Two-step experimentslist 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

o 6 Revision and mixed problems

and cultural differences in attitudes towards gambling.

(M)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?

(E)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 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

TechnologyRandom 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’.


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Topic:13. CONGRUENT AND SIMILAR FIGURESUnit OverviewThis 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.

Outcomes 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 MA5.1-3 WM provides reasoning to support conclusions that are appropriate to the context MA5.1-11 MG describes and applied the properties of similar figures and scale drawings MA5.2-1 WM selects appropriate notations and conventions to communicate mathematical ideas and


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


o Congruent figures define congruence of plane shapes using transformations

o Tests for congruent triangles develop the conditions for congruence of triangles

o Using congruence to prove geometrical properties

o establish properties of quadrilaterals using

(M)Resources: paper, scissors, dynamic geometry software such as GeoGebra, scale diagrams, maps and plans, summary charts.

(M)Investigate congruence in cultural and religious design patterns. From NSW syllabus: ‘Congruent figures are embedded in a variety of designs, for example, tapa cloth, Aboriginal designs, Indonesian ikat designs, Islamic designs, designs used in ancient

congruent triangles and angle properties, and solve related numerical problems using reasoning

o Similar figures use the enlargement transformation to explain similarity

o Properties of similar figureso Scale diagrams solve problems using ratio and

scale factors in similar figureso Stage 5.2 : Tests for similar triangles investigate

the minimum conditions needed, and establish the four tests, for two triangles to be similar


Egypt and Persia, window lattice, woven mats and baskets’.

(E)Students should be encouraged to prove results orally before writing them up. Introduce scaffolds of proofs where students fill in the blanks.

(M)Enlarge diagrams, logos, floor plans and sports fields such as basketball courts. Investigate the scale used in cameras, photocopiers, projectors, telescopes, microscopes, mirrors.

(E)Are all equilateral triangles similar? Are all rectangles similar? Are all isosceles triangles similar?

(X)When forming a proportion equation involving similar triangles, make x appear in the numerator.

(E)Trigonometry is based on similar right-angled triangles.

(M)Formal proofs of congruent and similar triangles, formal proofs of properties of triangles and quadrilaterals using congruence

(X)The golden ratio, the A series of paper sizes (X)Tessellations, including semi-regular tessellations

Assessment Writing activities, especially in identifying congruent and similar triangles or in writing a proof Practical test, including interpreting scale diagrams and identifying similar figures.

TechnologyUse dynamic geometry to investigate the properties of congruent and similar figures. The Math Open Reference website www.mathopenref.com contains animations demonstrating the tests for congruent and similar triangles.

Language Use matching angles rather than corresponding to avoid confusion with corresponding angles found when a

transversal crosses two lines. From the NSW syllabus: ‘This syllabus has used “matching” to describe angles and sides in the same position: however, the use of the word “corresponding” is not incorrect.’

Encourage students to set out their geometrical answers logically, step-by-step and giving reasons. The mathematical symbol ‘≡’ means ‘is identical to’ in algebra and ‘is congruent to’ in geometry. In geometry, the word similar has a different meaning to its everyday one. Remember to name the vertices of congruent and similar figures in matching order.


Be wary that in NSW, there is a continual debate on whether the tests for similar triangles can be abbreviated by initials in the same way as the tests for congruent triangles. The Australian curriculum lists these abbreviations in its glossary (using AAA for ‘equiangular’), but the NSW syllabus does not formally acknowledge them.

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Topic

Unit Overview

Outcomes

Literacy Students learn word bank using a variety of different teaching strategies for example close passage, definition matching exercise, memory games

and spelling tests. Word bank can be found at NCM 8 (pg 3) located at www.nelsonnet.com.au

Content Quality Teaching Ideas Resources



Assessment

Language

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