Tuesday, October 20, 2009

Algebra band descriptors

The following are amended descriptors for NAPLAN band Algebra
a) Level 8 Band 10
i) factorise quadratic expressions including the difference of two squares, e.g.
ii) manipulate algebraic formulae, equations and expressions, finding common factors and multiplying two linear expressions
iii) derive and use more complex formulae and change the subject of a formula
iv) evaluate algebraic formulae, substituting fractions, decimals and negative numbers
v) solve inequalities in two variables and find the solution set
vi) sketch, interpret and identify graphs of linear, quadratic, cubic and reciprocal functions, and graphs that model real situations
vii) understand the effect on a graph of addition of (or multiplication by) a constant
viii) identifies the pair of values that satisfies an algebraic expression.
ix) Solves a multistep algebra problem
b) Level 7 Band 9
i) square a linear expression, and expand and simplify the product of two linear expressions of the form (x ± n) and simplify the corresponding quadratic expression
ii) use algebraic and graphical methods to solve simultaneous linear equations in two variables
iii) solve inequalities in one variable and represent the solution set on a number line
iv) use formulae from mathematics and other subjects; substitute numbers into expressions and formulae; derive a formula and, in simple cases, change its subject
v) find the next term and nth term of quadratic sequences and functions and explore their properties
vi) plot graphs of simple quadratic and cubic functions, e,g, y= x2, y= 3x2 + 4,
vii) Solve algebraic equations with one variable and expressions involving multiple operations with negative values
viii) Determines an algebraic expression to model a relationship
c) Level 6 Band 8
i) use systematic trial and improvement methods and ICT tools to find approximate solutions to equations such as x3 +x= 20
ii) construct and solve linear equations with integer coefficients, using an appropriate method
iii) generate terms of a sequence using term-to-term and position-to-term definitions of the sequence, on paper and using ICT; write an expression to describe the nth term of an arithmetic sequence
iv) plot the graphs of linear functions, where y is given explicitly in terms of x; recognise that equations of the form y =. mx + c correspond to straight-line graphs
v) construct functions arising from real-life problems and plot their corresponding graphs; interpret graphs arising from real situations
vi) Determines equivalent algebraic expressions
d) Level 5 Band 7
i) construct, express in symbolic form, and use simple formulae involving one or two operations
ii) use and interpret coordinates in all four quadrants
iii) Solves equivalent algebraic expressions and expressions involving multiple operations
e) Level 4 Band 6
i) begin to use simple formulae expressed in words
ii) use and interpret coordinates in the first quadrant
f) Level 3 Band 5
i) recognise a wider range of sequences
ii) begin to understand the role of '=' (the 'equals' sign)
iii) Solves a simple algebraic expression by substituting a given number
g) Level 2 Band 4
i) Recognise sequences of numbers, including odd and even numbers

Calculating band descriptors

The following are amended descriptors for NAPLAN band Calculating
a) Level 8 Band 10
i) use fractions or percentages to solve problems involving repeated proportional changes or the calculation of the original quantity given the result of a proportional change
ii) solve problems involving calculating with powers, roots and numbers expressed in standard form, checking for correct order of magnitude and using a calculator as appropriate
b) Level 7 Band 9
i) calculate the result of any proportional change using multiplicative methods
ii) understand the effects of multiplying and dividing by numbers between 0 and 1
iii) add, subtract, multiply and divide fractions
iv) make and justify estimates and approximations of calculations; estimate calculations by rounding numbers to one significant figure and multiplying and dividing mentally
v) use a calculator efficiently and appropriately to perform complex calculations with numbers of any size, knowing not to round during intermediate steps of a calculation
c) Level 6 Band 8
i) calculate percentages and find the outcome of a given percentage increase or decrease
ii) divide a quantity into two or more parts in a given ratio and solve problems involving ratio and direct proportion
iii) use proportional reasoning to solve a problem, choosing the correct numbers to take as 100%, or as a whole
iv) add and subtract fractions by writing them with a common denominator, calculate fractions of quantities (fraction answers), multiply and divide an integer by a fraction
v) calculates duration between digital times

d) Level 5 Band 7
i) use known facts, place value, knowledge of operations and brackets to calculate including using all four operations with decimals to two places
ii) use a calculator where appropriate to calculate fractions/percentages of quantities/measurements
iii) understand and use an appropriate non-calculator method for solving problems that involve multiplying and dividing any three digit number by any two digit number
iv) solve simple problems involving ordering, adding, subtracting negative numbers in context
v) solve simple problems involving ratio and direct proportion
vi) apply inverse operations and approximate to check answers to problems are of the correct magnitude
vii) Calculates and compares the area of similar shapes
e) Level 4 Band 6
i) use a range of mental methods of computation with all operations
ii) recall multiplication facts up to 10 x 10 and quickly derive corresponding division facts
iii) use efficient written methods of addition and subtraction and of short multiplication and division
iv) multiply a simple decimal by a single digit
v) solve problems with or without a calculator
vi) check the reasonableness of results with reference to the context or size of numbers
f) Level 3 Band 5
i) derive associated division facts from known multiplication facts
ii) add and subtract two digit numbers mentally
iii) add and subtract three digit numbers using written method
iv) multiply and divide two digit numbers by 2, 3, 4 or 5 as well as 10 with whole number answers and remainders
v) use mental recall of addition and subtraction facts to 20 in solving problems involving larger numbers
vi) solve whole number problems including those involving multiplication or division that may give rise to remainders
g) Level 2 Band 4
i) use the knowledge that subtraction is the inverse of addition and understand halving as a way of 'undoing' doubling and vice versa
ii) use mental recall of addition and subtraction facts to 10
iii) use mental calculation strategies to solve number problems including those involving money and measures
iv) record their work in writing
v) choose the appropriate operation when solving addition and subtraction problems

Data band descriptors

The following are amended descriptors for NAPLAN band Data
a) Level 8 Band 10
i) estimate and find the median, quartiles and interquartile range for large data sets, including using a cumulative frequency diagram
ii) compare two or more distributions and make inferences, using the shape of the distributions and measures of average and spread including median and quartiles
iii) know when to add or multiply two probabilities
iv) use tree diagrams to calculate probabilities of combinations of independent events
b) Level 7 Band 9
i) suggest a problem to explore using statistical methods, frame questions and raise conjectures; identify possible sources of bias and plan how to minimise it
ii) select, construct and modify, on paper and using ICT suitable graphical representation to progress an enquiry including frequency polygons and lines of best fit on scatter graphs
iii) estimate the mean, median and range of a set of grouped data and determine the modal class, selecting the statistic most appropriate to the line of enquiry
iv) compare two or more distributions and make inferences, using the shape of the distributions and measures of average and range
v) understand relative frequency as an estimate of probability and use this to compare outcomes of an experiment
vi) examine critically the results of a statistical enquiry, and justify the choice of statistical representation in written presentation
vii) Interprets data to determine the relationship between mean, median and mode.
c) Level 6 Band 8
i) designs survey or experiment to capture the necessary data from one or more sources; design, trial and, if necessary, refine data collection sheets; construct tables for large discrete and continuous sets of raw data, choosing suitable class intervals; design and use two-way tables
ii) select, construct and modify, on paper and using ICT:
(1) pie charts for categorical data
(2) bar charts and frequency diagrams for discrete and continuous data
(3) simple time graphs for time series
(4) scatter graphs
iii) and identify which are most useful in the context of the problem
iv) find and record all possible mutually exclusive outcomes for single events and two successive events in a systematic way
v) know that the sum of probabilities of all mutually exclusive outcomes is 1 and use this when solving problems
vi) communicate interpretations and results of a statistical survey using selected tables, graphs and diagrams in support
vii) Interprets stem and leaf plots
viii) Describes probability using decimals
d) Level 5 Band 7
i) ask questions, plan how to answer them and collect the data required
ii) in probability, select methods based on equally likely outcomes and experimental evidence, as appropriate
iii) understand and use the probability scale from 0 to 1
iv) understand and use the mean of discrete data and compare two simple distributions, using the range and one of mode, median or mean
v) understand that different outcomes may result from repeating an experiment
vi) interpret graphs and diagrams, including pie charts, and draw conclusions
vii) create and interpret line graphs where the intermediate values have meaning
viii) Matches data in a table to points on a graph
e) Level 4 Band 6
i) collect and record discrete data
ii) group data, where appropriate, in equal class intervals
iii) continue to use Venn and Carroll diagrams to record their sorting and classifying of information
iv) construct and interpret frequency diagrams and simple line graphs
v) understand and use the mode and range to describe sets of data
vi) Calculates the average using data in a table
f) Level 3 Band 5
i) gather information
ii) construct bar charts and pictograms, where the symbol represents a group of units
iii) use Venn and Carroll diagrams to record their sorting and classifying of information
iv) extract and interpret information presented in simple tables, lists, bar charts and pictograms
g) Level 2
i) sort objects and classify them using more than one criterion
ii) understand vocabulary relating to handling data
iii) collect and sort data to test a simple hypothesis
iv) record results in simple lists, tables, pictograms and block graphs
v) communicate their findings, using the simple lists, tables, pictograms and block graphs they have recorded

Number band descriptors

The following are amended descriptors for NAPLAN band Numbers
a) Level 8 Band 10
i) understand the equivalence between recurring decimals and fractions
ii) Solves complex multistep problems involving percentages and decimals.
b) Level 7 Band 9
i) understand and use proportionality
c) Level 6 Band 8
i) use the equivalence of fractions, decimals and percentages to compare proportions
ii) Solves problems involving ratio and percentage increase.
iii) Solves a multi step rates problem involving spend and mixed numbers,
iv) Describes probability using decimals.
d) Level 5 Band 7
i) use understanding of place value to multiply and divide whole numbers and decimals by 10, 100 and 1000 and explain the effect
ii) round decimals to the nearest decimal place and order negative numbers in context
iii) recognise and use number patterns and relationships
iv) use equivalence between fractions and order fractions and decimals
v) reduce a fraction to its simplest form by canceling common factors
vi) understand simple ratio
vii) Calculates with percentages and converts mixed numbers to improper fractions
e) Level 4 Band 6
i) recognise and describe number patterns
ii) recognise and describe number relationships including multiple, factor and square
iii) use place value to multiply and divide whole numbers by 10 or 100
iv) recognise approximate proportions of a whole and use simple fractions and percentages to describe these
v) order decimals to three decimal places
vi) begin to understand simple ratio
vii) Solves problems involving addition and subtraction of decimals
viii) Continues number patterns in a table of values
f) Level 3 Band 5
i) understand place value in numbers to 1000
ii) use place value to make approximations
iii) recognise negative numbers in contexts such as temperature
iv) use simple fractions that are several parts of a whole and recognise when two simple fractions are equivalent
v) begin to use decimal notation in contexts such as money
g) Level 2 Band 4
i) count sets of objects reliably
ii) begin to understand the place value of each digit; :se this to order numbers up to 100
iii) begin to use halves and quarters and relate the concept of half of a small quantity to the concept of half of a shape

Shape Band Descriptors

The following are amended descriptors for NAPLAN band Shape
a) Level 8 Band 10
i) understand and use congruence and mathematical similarity
ii) understand and use trigonometrical relationships in right-angled triangles, and use these to solve problems, including those involving bearings
iii) understand the difference between formulae for perimeter, area and volume in simple contexts by considering dimensions
iv) Interprets a map and uses multiple operations with decimals to calculate a distance
v) Calculates the surface area of a triangular prism
b) Level 7 Band 9
i) understand and apply Pythagoras' theorem when solving problems in 2-D
ii) calculate lengths, areas and volumes in plane shapes and right prisms
iii) enlarge 2-D shapes, given a centre of enlargement and a fractional scale factor, on paper and using ICT; recognise the similarity of the resulting shapes
iv) find the locus of a point that moves according to a given rule, both by reasoning and using ICT
v) recognise that measurements given to the nearest whole unit may be inaccurate by up to one half of the unit in either direction
vi) understand and use measures of speed (and other compound measures such as density or pressure) to solve problems
vii) Applies knowledge of angle properties of polygons to calculate angle size.
c) Level 6 Band 8
i) classify quadrilaterals by their geometric properties
ii) solve geometrical problems using properties of angles, of parallel and intersecting lines, and of triangles and other polygons
iii) identify alternate and corresponding angles; understand a proof that the sum of the angles of a triangle is 1800 and of a quadrilateral is 360
iv) devise instructions for a computer to generate and transform shapes and paths
v) visualise and use 2-D representations of 3-D objects
vi) enlarge 2-D shapes, given a centre of enlargement and a positive whole-number scale factor
vii) know that translations, rotations and reflections preserve length and angle and map objects onto congruent images
viii) use straight edge and compasses to do standard constructions
ix) deduce and use formulae for the area of a triangle and parallelogram, and the volume of a cuboid; calculate volumes and surface areas of cuboids
x) know and use the formulae for the circumference and are, of a circle
xi) Determines direction using compass points and angle size of turn
d) Level 5 Band 7
i) use a wider range of properties of 2-D and 3-D shapes and identify all the symmetries of 2-D shapes
ii) use language associated with angle and know and use the angle sum of a triangle and that of angles at a point
iii) reason about position and movement and transform shapes
iv) measure and draw angles to the nearest degree, when constructing models and drawing or using shapes
v) read and interpret scales on a range of measuring instruments, explaining what each labeled division represents
vi) solve problems involving the conversion of units and make sensible estimates of a range of measures in relation to everyday situations
vii) understand and use the formula for the area of a rectangle and distinguish area from perimeter
viii) visualize the net of a cube with distinctive faces.
e) Level 4 Band 6
i) use the properties of 2-D and 3-D shapes
ii) make 3-D models by linking given faces or edges and draw common 2-0 shapes in different orientations on grids
iii) reflect simple shapes in a mirror line, translate shapes horizontally or vertically and begin to rotate a simple shape or object about its centre or a vertex
iv) choose and use appropriate units and instruments
v) interpret, with appropriate accuracy, numbers on a range of measuring instruments
vi) find perimeters of simple shapes and find areas by counting squares
vii) identifies scale and locates position on a plan using coordinates and a key
viii) interprets information in a graph by comparing two points
f) Level 3 Band 5
i) classify 3-D and 2-D shapes in various ways using mathematical properties such as reflective symmetry for 2-D shapes
ii) begin to recognise nets of familiar 3-D shapes, e.g. cube, cuboid, triangular prism, square-based pyramid
iii) recognise shapes in different orientations and reflect shapes, presented on a grid, in a vertical or horizontal mirror line
iv) describe position and movement
v) use a wider range of measures including non-standard units and standard metric units of length, capacity and mass in a range of contexts
vi) use standard units of time
vii) Identifies symmetry in irregular 2-D shapes.
g) Level 2 Band 4
i) use mathematical names for common 3-D and 2-D shapes
ii) describe their properties, including numbers of sides and corners
iii) describe the position of objects
iv) distinguish between straight and turning movements, recognise right angles in turns and understand angle as a measurement of turn
v) begin to use a wider range of measures including to use everyday non-standard and standard units to measure length and mass
vi) begin to understand that numbers can be used not only to count discrete objects but also to describe continuous measures

Using and Applying Mathematics Band Descriptors

The following are amended descriptors for NAPLAN band Using and Applying Mathematics
a) Level 8 Band 10
i) Develop and follow alternative methods and approaches
ii) reflect on lines of enquiry when exploring mathematical tasks
iii) Select and combine known facts and problem solving strategies to solve problems of increased complexity
iv) Convey mathematical meaning through precise and consistent use of symbols
v) Examine generalisations or solutions reached in an activity, commenting constructively on the reasoning and logic or the process employed, or the results obtained
vi) Distinguish between practical demonstration and proof; know underlying assumptions, recognizing their importance and limitations, and effect of varying them
vii) Applies ratio to calculate a proportion
b) Level 7 Band 9
i) Solve increasingly demanding problems and evaluate solutions; explore connections in mathematics across a range of contexts: number, algebra, shape, space and measures, and handling data; refine or extend the mathematics used to generate fuller solutions
ii) Give reasons for choice of presentation, explaining selected features and showing insight into the problem structure.
iii) Justify generalizations, arguments or solutions.
iv) Appreciate the difference between mathematical explanation and experimental evidence
v) Applies a proportion to calculate distance
c) Level 6 Band 8
i) Solve problems and carry through substantial tasks by breaking them into smaller, more manageable tasks, using a range of efficient techniques, methods and resources, including ICT; give solutions to an appropriate degree of accuracy.
ii) Interpret, discuss and synthesize information in a variety of mathematical forms
iii) Present a concise, reasoned argument, using symbols, diagrams, graphs and related explanatory texts.
iv) Use logical argument to establish the truth of a statement
v) Applies knowledge of intersecting lines to calculate angle size.
d) Level 5 Band 7
i) Identify and obtain necessary information to carry through a task and solve mathematical problems
ii) Check results, considering whether these are reasonable
iii) Solve word problems and investigations from a range of contexts
iv) show understanding of situations by describing them mathematically using symbols, words and diagrams
v) draw simple conclusions of their own and give an explanation of their reasoning
e) Level 4 Band 6
i) develop own strategies for solving problems
ii) use their own strategies within mathematics and in applying mathematics to practical contexts
iii) present information and results in a clear and organised way
iv) search for a solution by trying out ideas of their own
f) Level 3 Band 5
i) select the mathematics they use in a wider range of classroom activities
ii) try different approaches and find ways of overcoming difficulties that arise when they are solving problems
iii) begin to organise their work and check results
iv) use and interpret mathematical symbols and diagrams
v) understand a general statement by finding particular examples that match it
vi) review their work and reasoning
g) level 2 Band 4
i) select the mathematics they use in some classroom activities
ii) discuss their work using mathematical language
iii) begin to represent their work using symbols and simple diagrams
iv) predict what comes next in a simple number, shape or spatial pattern or sequence and give
v) reasons for their opinions
vi) explain why an answer is correct

Band Descriptors for NAPLAN

there aren't any right now, because NAPLAN does not work as advertised, and because the Australian Government has not achieved a national curriculum. So NAPLAN cannot admit what it actually is doing with NAPLAN tests and its bands. So, following will be some work towards amending UK's Key Stage 3 band descriptors for NAPLAN.

Using and Applying Mathematics band descriptors

Shape band descriptors

Number band descriptors

Data band descriptors

Calculating band descriptors

Algebra band descriptors

Limitations
The band descriptors are basically guess work, and may have discrepancies which will be ironed out over time, or ignored.
The UK Key Stage 3 Descriptors have been copied and NAPLAN descriptors have been inserted where they seem to apply.
Level 2 Band 4 is not based on any examination of Band 4 descriptors.