1 UNPACKING NSW MATHEMATICS 7-10 FOR THE AUSTRALIAN CURRICULUM [email protected] This PowerPoint presentation can be downloaded from

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UNPACKING NSW MATHEMATICS 7-10 FOR THE AUSTRALIAN

CURRICULUM

[email protected] PowerPoint presentation can be downloaded from

www.newcenturymaths.com.au

© April 2013

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The Australian Curriculum: Maths

In the beginning (Dec 2010), there was ACARA’s Australian curriculum V1.0

‘Complete version’ V3.0 published January 2012

Three strands: Number and AlgebraMeasurement and GeometryStatistics and Probability

More statistics and probability: met every year from primary school

Stronger use of Information and Communication Technology (ICT): in calculation, statistics, graphing and geometry, opportunities for graphics/CAS calculators, spreadsheets, dynamic geometry

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Content descriptions

ACARA developed content descriptions: the mandatory content to be learned

The Australian curriculum lists content by Year level

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Sample of Year 7content

descriptions

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An example of a Year 7 content description ACMNA174 = Australian Curriculum

Mathematics Number and Algebra Content Description 174

Icons = Literacy, ICT capability, Critical and creative thinking, Numeracy, Ethical understanding

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NSW doesn’t like this curriculum: not enough detail here; needs more

precise statementsAll 33 Year 7

content descriptions

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The Australian Curriculum in NSW

The Board of Studies published its own NSW syllabus based on the Australian curriculum (AC) but with clearer detail and additional content

After two drafts, published in September 2012 Same strands and content descriptions as AC

but new names for substrands Stages 1 to 5 retained Stage 5 continuum retained: Stages 5.1, 5.2,

5.3 Basically, it’s the current NSW syllabus

adapted to the AC’s mandatory content descriptions

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Stage 5.1 + recommended Stage 5.2 content () leads to Stage 6 Mathematics General course

Stage 5.2 leads to Stage 6 Mathematics General or with recommended Stage 5.3 content (§) leads to Stage 6 Mathematics (‘2U’) course

Stage 5.3 leads to Stage 6 Mathematics Extension 1

Four Stage 5.3 optional topics (#) for Mathematics Extension 1 pathway: Polynomials, Logarithms, Functions, Circle geometry

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Additional topics include Pythagoras’ theorem in Stage 4, Earning money in Stage 5

So the NSW syllabus is significantly different from the Australian curriculum: it’s the AC + more!

Make sure the textbook you use is based on the NSW syllabus, not a ‘national’ AC book

Syllabus implementation begins next year (2014) with Years 7 and 9

Years K-6, 8 and 10 in the following year (2015)

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Strands and substrands

Numberand Algebra

Measurementand Geometry

Statisticsand Probability

Computation with integers Length Data collection and representation

Financial mathematics Area and surface area

Ratios and rates Volumes Single variabledata analysis

Fractions, decimals and percentages

Numbers of any magnitude

Algebraic techniques Time Bivariatedata analysis

Indices Properties ofgeometrical figures

Probability

Equations Angle relationships

Linear relationships Right-angled triangles

Non-linear relationships Trigonometry and Pythagoras’ theorem

Logarithms / Polynomials / Functions

Circle geometry

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Content strands

Three strands = ‘nouns’ of maths curriculumNumber and AlgebraMeasurement and GeometryStatistics and Probability

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Working Mathematically

Working Mathematically = ‘verbs’ of maths curriculum

Has five interrelated components:

Understanding

Fluency

Problem solving

Reasoning

Communicating

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Working Mathematically 1

Understanding is … More than just learning facts Knowing and relating maths: ‘deep

understanding’ Forming connections, knowing ‘why’ as well

as ‘how’ ‘Students build a strong foundation that

enables them to adapt and transfer mathematical concepts’

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Fluency is … Applying maths Using mathematics confidently, competently

and effectively ‘Develop skills in choosing appropriate

procedures … flexibly, accurately, efficiently’

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Problem solving is … Modelling and investigating with maths More than just applying maths to simple

problems Interpreting a rich, elaborate problem,

selecting an appropriate model/strategy, solving the problem and evaluating and justifying the solution

‘Formulate … and investigate problem situations, and communicate solutions effectively’

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Reasoning is … Generalising and proving with maths Higher-order thinking to connect specific facts

to general principles Using algebra, logic, formal proof and

justification ‘Students develop an increasingly

sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising’

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Communicating is … Describing and explaining maths Representing mathematical theory and

solutions in written, oral and graphical form Using words, algebraic symbols, special

notations, diagrams, graphs and tables ‘Describe, represent and explain

mathematical situations, concepts, methods and solutions to problems’

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Learning across the curriculum

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AdditionalNSW

content description

NSW outcome

s

Australian curriculum

content description

NSW elaborations of

Australian curriculum

contentWorking mathematically + Learning across the curriculum components tagged to content:• Problem solving• Critical and creative thinking

Strand + substrand

Background information

and language

notes

Sample page from the NSW syllabus

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• By strands Number and Algebra Measurement and Geometry Statistics and Probability

Unpacking Stages 3 to 5 content

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Stage 3 Number and Algebra

o More content in Stage 3o Place value up to ten milliono Factors and multiples, HCF, LCM (NEW to Stage

3)o Removed from syllabus: Roman numeralso Introduction to integers (not operations)o Prime and composite numberso Square and triangular numbers (NEW to Stage

3)o Simple budgeting (NEW to Stage 3)o Multiplication by 2-digit number (‘Long

multiplication’)o Division by 1-digit number (‘Short division’)o Rounding and estimating answerso Order of operations (NEW to Stage 3)

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o Adding and subtracting fractions (including mixed numerals) with related denominators (eg 3 and 6)

o Fraction of a numbero Rounding, adding and subtracting decimalso Multiplying decimals by whole numberso Dividing decimals by one-digit numberso Multiplying and dividing decimals by 10, 100,

1000o Percentageso Discounts of 10%, 25%, 50% (NEW to Stage 3)o Patterns and algebra: number patterns,

describing rules, tables of values, 5 x = 4 x 10, x 5 = 125

o The number plane, four quadrants (NEW to Stage 3)

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o Divisibility tests, factor treeso Division by 2-digit number (‘Long division’)

(NSW)o Mental computation: Associative, commutative

and distributive laws (NEW)o Removed from syllabus: History of number,

special numbers (eg Pascal’s triangle, Fibonacci)o Decimalso Integers, including order of operationso Fractions and percentageso GST (NEW, NSW)o Ratios and rateso Best buys (NEW)

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o Powers and roots, including order of operationso Irrational numbers (NEW)o Algebra concepts introduced more gradually,

without ‘matchstick patterns’ (moved to Stage 3)

o Variables, substitution, from words to algebrao Operations with algebraic termso Expanding and factorising expressionso 2- and 3-step equations: brackets and x on both

sideso Moved to Stage 5.2: Algebraic fractions, solving

equations after substituting into a formula, solving linear inequalities

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o Index laws using numbers, power of zero (NEW)o Simple quadratic equations x2 = c (NEW, NSW)o Moved to Stage 3: The number planeo Transformations on the number plane:

coordinates of P’ after translation, reflection, rotation (multiple of 90) of P (NEW)

o Graphing linear equationso Solving linear equations graphically, eg 3x – 7 =

5 by graphing y = 3x – 7 and reading x when y = 5 (NEW)

o Graphing simple non-linear relationships using technology (NEW, NSW)

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Stage 5.1 Number and Algebra

o Overall, the Stage 5.1 course is harder than before but students should still complete Stage 4 also

o Earning money topic: wages, salaries, overtime, commission, piecework, annual leave loading (NSW)

o Income tax and PAYG tax (NEW, NSW)o Simple interesto Term payments (NSW)o Removed from syllabus: Bonuses, credit card,

lay-by, deferred payment, loanso Compound interest (not formula)

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o Index lawso Zero and negative indiceso Removed from Stages 5.1/5.2: Powers of ½

and 1/3

o Coordinate geometry: Length, midpoint and gradient of an interval (no formulas)

o Graphing linear equationso Gradients of parallel lines (NEW)o Graphing simple quadratics, circles and

exponentials (y = 3x) using technology (NEW)

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o Compound interest formula = recommended for Maths General 2

o Depreciation (NSW) o Removed from syllabus: Successive discounts,

reducible interest loanso Converting between rateso Direct proportion and graph (NEW)o Moved from Stage 4: Conversion graphs (NSW)o Inverse proportion and graph (NEW, NSW)

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o More negative indices, including (a/b)-1

o Operations and equations with algebraic fractions

o Expanding binomial products (a + b)(c + d) (NEW to Stage 5.2)

o Factorising monic quadratics (x2 – 3x – 40), Solving monic quadratic equations (x2 – 3x – 40 = 0) (NEW to Stage 5.2)

o Solving equations after substituting into a formula

o Simultaneous linear equationso Solving linear inequalities

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o Moved to Stage 5.3: Length, midpoint and gradient formulas, powers of ½ and 1/3

o Graphing y = mx + b, gradient and y-intercept o Converting from general form to y = mx + bo Finding the equation of a line o Equations of parallel lineso Equations of perpendicular lines (NEW to Stage

5.2)o Graphing simple quadratics (y = ax2 + c), circles (x2 + y2 = r2) and exponentials (y = ±ax

+ c) (NEW to Stage 5.2) = recommended for Maths General 2

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o Stage 5.3 continues to be challenging and dense

o Converting recurring decimals to fractions (NSW) § = recommended for 2U Maths

o Fractional indices: am/n §o Simple cubic equations ax3 = c (NEW, NSW) §o Changing the subject of a formula (NSW) §o Simultaneous equations involving linear and non-

linear equations (NSW) §o Distance, midpoint, gradient formulas §o y – y1 = m(x – x1), ax + by + c = 0, line through

two points §o Equations of parallel and perpendicular lines §o Coordinate methods in geometry §

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o Graphing rates of change (eg filling different-shaped bottles), travel graphs with variable speed (NSW)

o Operations and equations involving algebraic fractions with binomial numerators § = rec for 2U Maths

o Special binomials: (a + b)2, (a + b)(a – b) §o Factorisation strategies, including grouping in

pairs §o Factorising all types of quadratics (3x2 + x – 4)

and solving all types of quadratic equations §o Surds, including binomial expansions and

rationalising the denominator §

2 5 4 1 2 3 1eg ,

6 3 4 3 2

x x y y

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o Graphing parabolas: axis of symmetry, vertex, intercepts § = recommended for 2U Maths

o Graphing hyperbolas and circles not centred at (0, 0) §

o Point of intersection of a line and parabola, hyperbola, circle (NSW) §

o Graphing cubic and higher-power functions (NSW) §o Transformations of graphs §o Polynomials topic # = option topic for Maths Ext 1

o Logarithms topic: logarithm laws, exponential and logarithmic graphs and equations: log4x = -2 #

o Functions and graphs topic (NSW) #o Removed from syllabus: Regions in the number

plane (graphing linear inequalities)

34Stage 3 Measurement and Geometry

o Length and perimetero Square kilometre and hectareo Area of square, rectangle, triangleo Volume and capacityo Volume of a rectangular prismo Mass: measuring, tonnes, gross vs neto 24-hour time, Australian time zoneso Measuring duration, timetables, timelineso Prisms and pyramids, nets of solids


o Types of triangles, quadrilaterals, polygonso Transformations: translation, reflection, rotation

(NEW)o Diagonals of plane shapeso Line and rotational symmetryo Enlargementso Parts of a circle: semi-circle, quadrant, sectoro Measuring angle sizeo Angle geometry: angles in a right angle, on a

straight line, at a point, vertically opposite angles (NEW to Stage 3)

o Map reading: grid references (eg H3), compass directions, giving travel directions


o More work on area, especially circular shapes, while surface area has been moved to Stage 5

o Moved to Stage 3: Estimating and measuring lengthso Moved to Stage 5.1: Limits of accuracy of measuring

instruments o Perimeter of shapeso Perimeter and area of a circle, semi-circle, quadranto Arc length, perimeter and area of a sector (NEW to

Stage 4)o Giving exact answers in terms of (NEW, NSW)o Converting between units of areao Area of rectangle, triangle, parallelogramo Area of trapezium, rhombus, kite (NEW to Stage 4)


o Moved to Stage 5.1: Nets and surface areas of prisms

o Different views of prismso Converting between units of volume, and

capacityo Volume of rectangular prism and cylindero Time differences and time calculationso International time zones (NSW)o Angle geometryo Corresponding, alternate and cointerior angleso Pythagoras’ theorem topic, including

Pythagorean triads (NSW Stage 4, AC Year 9)


o Triangles and quadrilaterals: types, properties, angle sum

o Line and rotational symmetryo Exterior angle of triangleo Congruent figures (not similar figures)o Tests for congruent triangles (not proofs) (NEW

to Stage 4)o Moved to Stage 5.1: Similar figureso Removed from syllabus: Solids topic

(polyhedra), geometrical constructions

39Stage 5.1 Measurement and Geometry

o Areas of composite shapeso Surface areas of rectangular and triangular prismso Significant figures (NEW to Stage 5.1, NSW)o Metric prefixes: nano, micro, mega, giga, tera

(NEW)o Digital memory: kB, MB, GB, TB (NEW, NSW)o Moved from Stage 4: Limits of accuracy of

measuring instrumentso Scientific notationo Trigonometry (degrees only, not minutes)o Angles of elevation and depressiono Moved from Stage 4: Similar figures, scale factor,

scale diagrams


o Surface area of a cylindero Surface area and volume of composite solidso Moved to Stage 5.3: Volume of pyramids,

cones, spheres

o Trigonometry (degrees and minutes) =

recommended for Maths General 2

o Three-figure bearings o Compass bearings (16 points of the compass),

eg WSW


o Simple deductive reasoning, more formal proofs

o Congruent triangle proofso Using congruent triangles to prove properties

of triangles and quadrilateralso Tests for similar triangles (not formal proofs)o Angle sum of a convex polygon, exterior angle

sum (NSW)


o Surface area and volume of pyramids, cones, spheres

o More trigonometryo Pythagoras’ theorem and trigonometry

problems in 3D (NEW) § = recommended for 2U Maths

o Trigonometric ratios up to 360: unit circle (NEW) §

o Trigonometric functions: graphs, period (NEW) §o Trigonometric relations, exact ratios (NSW) §o Simple trigonometric equations (NEW) §o Sine and cosine rules, area of triangle ½ ab sin

C §


o Formal and abstract geometry proofs §o Similar triangle proofs §o Surface areas and volumes of similar solids

(NSW) §o Formal definitions and tests for quadrilaterals §o Circle geometry # = option topic for Maths Ext 1

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Stage 3 Statistics and Probability

o Generally, more advanced work as Statistics and Probability now learned throughout primary school

o Column and line graphs, tableso Collecting data, dot plots, side-by-side (clustered)

column graphs (NEW to Stage 3)o Moved to Stage 4: Sector graphs, divided bar

graphs, the meano Investigating statistical claims in the media: bias,

misleading graphs (NEW to Stage 3)o The language of chance, the range of probabilityo Sample spaces, probability as a fraction, decimal,

percentage (NEW to Stage 3)o Comparing expected frequency with observed

frequency over repeated trials (NEW to Stage 3)

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o Generally, more advanced worko Collecting data: census, sampling, observationo Random sampling, surveys and biaso Categorical vs numerical datao Statistical graphs and tables: divided bar

graphs, sector graphs, line graphso Frequency tables, histograms and polygonso Stem-and-leaf plots, dot plotso Removed from syllabus: Step graphso Moved to Stage 5.2: Conversion graphs, scatter

plots

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o The mean, median, mode and range: from set of scores and statistical graphs

o Clusters, gaps and outliers: effects on mean and median (NEW)

o Variation of means and proportions in random samples drawn from the same population (NEW)

o Sample space and probabilityo The range of probabilityo Complementary eventso Venn diagrams and two-way tables (NEW)o Mutually exclusive events (NEW, NSW) o Meanings of ‘and’, ‘or’, ‘at least’, ‘at most’,

‘not’ (NEW)

47Stage 5.1 Statistics and Probability

o Back-to-back stem-and-leaf plotso Shape of distributions: skewed, symmetric, bi-

modal (NEW)o Comparing two data sets using mean, median

and rangeo Statistics in the media: misleading claims and

graphso Relative frequency, experimental probabilityo Venn diagrams and two-way tables: probability

of events involving ‘and’ or ‘or’ (NEW)


o Interquartile range, five-number summaries and box-and-whisker plots = recommended for Maths General 2

o Comparing data sets o Evaluating statistical reports in the media, bias o Bivariate data and scatter plots (NEW)o Two- and three-step experiments, with or without

replacement: tree diagrams (NEW to Stage 5.2)o Independent and dependent events (NEW to Stage

5.2)o Conditional probability (NEW)o Moved to Stage 5.3: Standard deviationo Removed from syllabus: Grouped data (class

intervals), Cumulative frequency graphs


o Investigating a report of a study (NEW)o Comparing means and standard deviations of

data setso Line of best fit (NEW)

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• December 2012: Final draft published by ACARA

• 4 courses: Essential Maths, General Maths,

Mathematical Methods, Specialist Maths

• Specialist Maths must be studied with Mathematical Methods

• 2016? 2018? 2020?: Implementation in NSW• Board of Studies needs to develop syllabuses

first, possibly using current course model, including a ‘Maths Extension 1’ course

The Australian curriculumfor senior maths

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The end• Any questions?

• Download sample chapters, teaching programs and a copy of this PowerPoint from www.newcenturymaths.com.au

• Select NCM for the Australian curriculum

[email protected]

www.newcenturymaths.com.au

Documents

1 UNPACKING NSW MATHEMATICS 7-10 FOR THE AUSTRALIAN CURRICULUM [email protected] This PowerPoint presentation can be downloaded from