1 unpacking nsw mathematics 7-10 for the australian curriculum [email protected] this...
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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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Content descriptions
Sample of Year 7content
descriptions
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Content descriptions
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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The Australian Curriculum in NSW
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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The Australian Curriculum in NSW
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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Working Mathematically 2
Fluency is … Applying maths Using mathematics confidently, competently
and effectively ‘Develop skills in choosing appropriate
procedures … flexibly, accurately, efficiently’
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Working Mathematically 3
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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Working Mathematically 4
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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Working Mathematically 5
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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Stage 3 Number and Algebra
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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Stage 4 Number and Algebra
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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Stage 4 Number and Algebra
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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Stage 4 Number and Algebra
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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Stage 5.1 Number and Algebra
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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Stage 5.2 Number and Algebra
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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Stage 5.2 Number and Algebra
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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Stage 5.2 Number and Algebra
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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Stage 5.3 Number and Algebra
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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Stage 5.3 Number and Algebra
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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Stage 5.3 Number and Algebra
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
35Stage 3 Measurement and Geometry
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
36Stage 4 Measurement and Geometry
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)
37Stage 4 Measurement and Geometry
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)
38Stage 4 Measurement and Geometry
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
40Stage 5.2 Measurement and Geometry
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
41Stage 5.2 Measurement and Geometry
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)
42Stage 5.3 Measurement and Geometry
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 §
43Stage 5.3 Measurement and Geometry
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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Stage 4 Statistics and Probability
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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Stage 4 Statistics and Probability
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)
48Stage 5.2 Statistics and Probability
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
49Stage 5.3 Statistics and Probability
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
www.newcenturymaths.com.au