Key Proficiency Ideas for AusVELSLevels Five and SixVince Wright ()

The following table shows how the proficiency strands can be linked to specific content descriptors in AusVELs for Levels Fiveand Six. For assessment purposes teachers might select a small number of specific proficiencies that are relevant to a given unit with the aim of covering a range of proficiencies over time.

Content: Number and Algebra / Proficiencies
Number (Whole numbers, operations and integers)
Level Five
Identify and describe factors and multiples of whole numbers and use them to solve problems;
Use estimation and rounding to check the reasonableness of answers to calculations;
Solve problems involving multiplication of large numbers by one- or two-digit numbers using efficient mental, written strategies and appropriate digital technologies;
Solve problems involving division by a one digit number, including those that result in a remainder;
Use efficient mental and written strategies and apply appropriate digital technologies to solve problems.
Level Six
Identify and describe properties of prime, composite, square and triangular numbers;
Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers.
Investigateeverydaysituationsthatuse integers.Locateand representthesenumbers onanumberline / Understandings (Understand that…)
Numbers can be classified in many ways and a given number may belong to many different sets, e.g. a multiple of 5, a triangular number.
Factors of a number are those numbers that divide into it with no remainder. One is a factor of any whole number.
A number that has only two factors, one and itself, is called a prime number.
A number that has more than two factors is a composite or rectangular number because at least two rectangles can be formed with that number of objects.
Numbers of objects that form a triangular arrangement are called triangular numbers. So they are the sums of repeatedly adding consecutive counting numbers, 1, 1 + 2 = 3, 1 + 2 + 3 = 6,…
Multiples of a number are the set of numbers created by repeatedly adding the number to form a sequence.
The properties of whole numbers under multiplication (commutative, distributive and associative) are applied to make multiplication calculations easier.
Multiplication and division are inverse operations so one undoes the other, and the result of one operation can be found/checked using the other.
Negative integers are needed when a larger whole number is subtracted from a smaller whole number so represent a ‘made-up’ quantity less than zero (to the left of zero on a number line).
Fluency (With minimal mental effort…)
Represent whole numbers using appropriate materials to justify choices of calculation strategies, particularly the use of arrays for multi-digit multiplication and division.
Know multiplication facts to 10 x 10 = 100 and the related division facts for multiplication, e.g. 7 x 4 = 28 so 28 ÷ 4 = 7.
Use strategies like rounding and compensating, splitting factors additively (distributive) and multiplicatively (associative), and combining standard place values to multiply and divide whole numbers.
Use correct mathematical symbols to represent number relationships, including using equals to show sameness.
Represent addition and subtraction calculations using materials, equations, empty number lines, diagrams, and written algorithms.
Problem Solving (In authentic contexts…)
Formulate appropriate estimations and calculations to solve number problems in context, recognising which operation/s are involved and sequence the operations.
Represent calculations using materials, equations, empty number lines, diagrams, and written algorithms.
Reasoning (Think that…)
Situations can involve estimation, where approximate answers are sufficient, exact calculation is needed, or a combination of the two.
Calculations can be made mentally, in written form or using digital technology and it is important to choose appropriately from these methods to meet the demands of the situation.
Solutions involve both answer and method so strategies need to be justified, using materials if necessary, considering whether the method is valid (correct), efficient and generalizable (transfers to similar problems).
Sets of multiples can have patterns, e.g. multiples of 4are all even but not all even numbers are multiples of 4. Some strategies exist to test if a given number is a multiple of other numbers.
Content: Number and Algebra / Proficiencies
Number (Fractional numbers and decimals)
Level Five
Solve problems involving division by a one digit number, including those that result in a remainder.
Compare and order common unit fractions and locate and represent them on a number line.
Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator.
Recognise that the place value system can be extended beyond hundredths.
Compare, order and represent decimals.
Describe, continue and create patterns with fractions, decimals and whole numbers resulting from addition and subtraction.
Level Six
Compare fractions with related denominators and locate and represent them on a number line.
Solve problems involving addition and subtraction of fractions with the same or related denominators.
Find a simple fraction of a quantity where the result is a whole number, with and without digital technologies.
Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers
Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies.
Multiply and divide decimals by powers of 10.
Make connections between equivalent fractions, decimals and percentages.
Investigate and calculate percentage discounts of 10%, 25% and 50% on sale items, with and without digital technologies.
Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence. / Understandings (Understand that…)
Fractional numbers are needed when wholes (ones) are inadequate for a situation. These situations are related to division and involve equal sharing or measuring, e.g. How many 4-packs can I make with 23 cookies? The remainder can be expressed as a fraction, as a decimal or as a whole number depending on context.
A written fraction represents the connection between a count of the number of parts (numerator) and the number of those equal parts a whole (one) is made up of (denominator). So fractions can be greater than one when the numerator is greater than the denominator.
The same fraction has many an infinite number of equivalent names that come from partitioning or combining the number of equal parts that form one. So the numerators and denominators are changed by multiplication.
Fractions are numbers that, just like whole numbers, are related to one so can be ordered and represented on a number line. Whole numbers can also be written as fractional numbers.
The properties of operations that hold for whole numbers also hold for fractions, e.g. x 24 = 24 x . When a fraction is used as an operator, e.g. two-thirds of 24, this can be expressed as x 24 = 16 and the two-thirds behaves just like a whole number.
Decimal numbers are a special form of equivalent fractions because they involve denominators that are powers of ten, i.e. tenths, hundredths, thousandths, etc.
The effect of multiplying a decimal number by ten is for the digits to move one place to the left relative to the ones place and dividing by ten has the inverse opposite effect.
Fluency (With minimal mental effort…)
Represent fractional numbers, including decimals, using materials, diagrams (including number lines), and symbols.
Know the decimal and percentage equivalents to common fractions such as halves, quarters, eighths, tenths, thirds and fifths.
Add, subtract and multiply fractions using standard written forms and use material and diagrams to represent the quantities involved.
Find common fractions of quantities, including everyday percentages such as 10%, 25%,30%, 50%.
Problem Solving (In authentic contexts…)
Formulate number problems in context, recognising when fractional numbers are needed, carry out appropriate estimations and calculations, and present solutions using correct numbers and units.
Represent solutions using materials, equations, diagrams, and symbols.
Reasoning (Think that…)
Justify the choice of solution methods, particularly the choice to use fractions, decimals and percentages in context.
Recognise that fractions have different characteristics in different situations, e.g. as numbers, operators, answers to division, and apply the characteristic/s that are appropriate to the context.
Content: Number and Algebra
Content: Patterns and algebra
Level Five
Describe,continueandcreatepatternswithfractions, decimalsandwholenumbers resultingfromadditionand subtraction.
Useequivalentnumber sentencesinvolving multiplicationanddivisionto find unknownquantities.
Level Six
Continueandcreatesequencesinvolving wholenumbers,fractionsand decimals.
Describethe ruleusedtocreatethesequence.
Explorethe useofbracketsandorderof operationstowritenumbersentences. / Understanding (Understand that…)
Patterns can be repeating, e.g. 1, 3, 5, 1, 3, 5, .., or growing, e.g. 1, 3, 5, 7, …
Rules for patterns can be recursive (describe what happens from one term to the next), e.g. numbers increase by two, or functional (describe the relationship between term and number), e.g. the number is double the term less one.
Patterns can be investigated using numeric and figural (spatial) methods and relationships can be represented using tables, words, and symbols.
Conventions in the way expressions and equations are written are followed so that the symbols carry unambiguous meaning, e.g. 3 + 4 x 6 ≠ (3 + 4) x 6.
Fluency (With minimal mental effort…)
Continue and create patterns using materials, tables, symbols and words.
Write expressions and equations to represent sequences of operations on numbers.
Problem Solving (In authentic contexts…)
Recognise and describe elements of repeat in sequential patterns and rules for growing patterns to find unknown terms in the pattern.
Represent and solve unknowns in story problems by recording the operations using equations and solving them.
Reasoning (Think that…)
Rules for patterns can be found by looking at similarities and differences among several terms or noticing organisation in one term and checking to see if the structure applies to other terms.
Counting sequences of whole numbers, decimals and fractions have digital patterns that allow for prediction of the later terms and rules to determine if a given number belongs in a sequence, e.g. All whole numbers belong in the sequence , , …
Content: Statistics and Probability
Level Five
List outcomesofchanceexperiments involvingequallylikely outcomesandrepresentprobabilitiesofthose outcomesusingfractions.
Recognisethatprobabilitiesrangefrom0 to1.
Posequestionsandcollectcategoricalor numericaldatabyobservationorsurvey.
Constructdisplays,includingcolumn graphs, dotplotsandtables,appropriate for datatype,withandwithouttheuseof digitaltechnologies.
Describeand interpretdifferent datasets incontext
Level Six
Describeprobabilitiesusingfractions, decimalsandpercentages.
Conductchance experimentswithboth smallandlargenumbersoftrialsusing appropriatedigitaltechnologies.
Compareobservedfrequenciesacross experimentswithexpectedfrequencies.
Interpretandcomparea rangeofdata displays,includingside-by-sidecolumn graphsfor twocategoricalvariables.
Interpretsecondarydatapresentedin digitalmediaandelsewhere. / Understanding (Understand that…)
Chance, the likelihood of an outcome occurring, can be measured like other attributes by carry out trials of the event or creating a theoretical model. Probabilities can be expressed as numbers from 0 (impossible) to 1 (certain) so fractions, decimals and percentages are used as measures, e.g. 50% likelihood means a half chance of that outcome occurring.
The results of trial naturally vary, particularly with small samples, but as the sample size increases the results get more reliable. So as samples get larger the proportions of outcomes should more closely approximate the expected outcomes from a theoretical model.
Statisticians use an inquiry cycle to investigate questions, collect data or using existing data, and sort and represent the data in different ways to look for similarities and differences.
Category data comes from sorting things into groups based on an attribute and numeric data comes from counting and measuring. Category data can be displayed in different ways to highlight differences (e.g. picture graphs or column graphs) and proportions of the whole (pie charts). Numeric data can be displayed in non-grouped displays, e.g. dot or stem and leaf plots, or grouped displays.
Fluency (With minimal mental effort…)
Systematically record the results of experiments or data collection using tally charts.
Represent results of experiments or data collection (category and numeric data) in tables, column graphs, dot plots and make statements about patterns and differences in the graphs/tables. Use digital technology and hand-drawn methods to create graphs, including pie charts for category data.
Problem Solving (In authentic contexts…)
Answer questions using the Statistical Inquiry Cycle [Pose a problem (question), Plan, Gather data, Analyse (Sort) data, and Form conclusions].
Reasoning (Think that…)
Models of theoretical probability and the results of trial should agree, with some variation, if the sample size is big enough.
Similarities and differences can be found between and among groups that can tell us about the context being investigated, e.g. Are girls really taller than boys?
Choose representations strategically to find different patterns and differences in the data and relate the findings back to the context.
Content: Measurement / Proficiencies
Measurement
Level Five
Choose appropriate units of measurement for length, area, volume, capacity and mass;
Calculate the perimeter and area of rectangles using familiar metric units;
Compare 12- and 24-hour time systems and convert between them.
Level Six
Connect decimal representations to the metric system;
Convert between common metric units of length, mass and capacity;
Solve problems involving the comparison of lengths and areas using appropriate units;
Connect volume and capacity and their units of measurement;
Interpret and use timetables. / Understanding (Understand that…)
Measuring involves choosing an attribute and a unit to measure that attribute. Area (internal 2-D space) and perimeter (length around a perimeter) are different attributes of a shape so are measured with different units, e.g. square metres for area and metres for perimeter.
Smaller units of measure give more precision but the choice of units must be practical for meeting the demands of the situation.
A measure is given as a count (number) and a referent (unit), e.g. 145mm, so the same measure can be expressed in different ways, e.g. 167cm = 1.67m.
Capacity relates to the space occupied by a liquid or gas and volume relates to the internal space of an object, so units for capacity and volume connect to one another, e.g. 500mL = 500 cm3 (cubic centimetres).
Fluency (With minimal mental effort…)
Read metric scales to the required accuracy in whole and decimal numbers of units for length, area, volume, capacity and time and record the measures as numbers and units, e.g. 10.35 seconds.
Use multiplication to calculate areas and volumes.
Convert between whole number and decimal measures for length, e.g. 1.78m = 1780mm, and between common units of time, e.g. 1minutes = 90 seconds.
Read and interpret timetables in everyday use such as train or bus timetables and television schedules.
Problem Solving (In authentic contexts…)
Create and interpret questions to solve a problem (measurement always has a purpose). Choose appropriate units to solve a problem, planthen sequence actions,carry out the measurements and report the findings.
Reasoning (Think that…)
Seriation (ordering) requires consistent size relations so transitive reasoning applies, i.e. If A<B and B<C then A<C.
Recognise the attribute/s being measured in context and use units to measure to a sensible degree of precision or estimate/approximate if appropriate.
Use the proportional relationships between units of the same attribute, e.g. 100 times as many centimetres as metres in a given space, and relate equivalent measures of volume and capacity.
Content: Geometry / Proficiencies
Geometry
Level Five
Connect three-dimensional objects with their nets and other two-dimensional representations;
Use a grid reference system to describe locations;
Describe routes using landmarks and directional language;
Describe translations, reflections and rotations of two-dimensional shapes.
Identify line and rotational symmetries;
Apply the enlargement transformation to familiar two dimensional shapes and explore the properties of the resulting image compared with the original;
Estimate, measure and compare angles using degrees. Construct angles using a protractor.
Level Six
Construct simple prisms and pyramids;
Investigate combinations of translations, reflections and rotations, with and without the use of digital technologies;
Introduce the Cartesian coordinate system using all four quadrants;
Investigate, with and without digital technologies, angles on a straight line, angles at a point and vertically opposite angles. Use results to find unknown angles. / Understanding (Understand that…)
Three dimensional shapes are bounded by two dimensional shapes and surfaces (not flat). So nets for simple polyhedra are made from connected 2-D shapes by folding.
Three dimensional objects appear differently from different viewpoints and these viewpoints can be represented diagrammatically.
Locations can be described by assigning an origin and imposing a grid or co-ordinate system on a landscape. The map that results involves scale so representations on a map are proportional to length in reality.
Maps can be used to anticipate and plan journeys, including travel time and distances. Directions for movement can be given in distances and turns (angles) with reference to important landmarks.
Objects in the natural and people-made worldare often symmetrical which relates to their function, e.g. animals have reflective symmetry so they are balanced.
Angles are a measure of turn and are often static in the real world. Angles are measured in degrees which is an ancient system related to equal divisions of a year (360 divides evenly by many numbers).
Fluency (With minimal mental effort…)
Anticipate whether given nets will form as prism by imaging the folding.
Draw 3D models from different viewpoints using grid paper and build 3D models from 2D pictures of different viewpoints for the same model.
Draw lines of reflective symmetry for a given shape, describe the order of rotational symmetry connecting to the angle of rotation.
Measure angles to the nearest degree using a protractor and draw angles of a given number of degrees.
Problem Solving (In authentic contexts…)