Mathematics ARBs and the National Standards progressions

Many of the mathematics ARBs can support teachers with decisions they need to make around the 2009 National Standards progressions.  They can provide evidence of learning that can contribute to establishing their overall teacher judgements (OTJ).

Mathematics ARB items have been developed around curriculum levels and are generally trialled with groups of students. The data from the trials provides the difficulty level, i.e., how hard it was for the trial students to get the correct answer. However, it is the strategies that students use to complete tasks that indicate where they are on the National Standards progressions, rather than the ARB item or the difficulty level. An  ARB item at one curriculum level could be used to identify a range of  curriculum levels related to the standards. 
For example students might answer questions on a level 3 item about addition. The strategies that students use can indicate which step on the National Standard progression they are working at:

  • advanced counting strategies suggest curriculum level 1 (specifically Year 2)
  • early additive strategies suggest curriculum level 2
  • advanced additive/early multiplicative strategies suggest curriculum level 3
  • advanced multiplicative /early proportional strategies suggest curriculum level 4.

The term [early] is used to indicate the earlier year level of the standard (Years 3, 5 and 7) . For example an assessment statement at level 3 that has [early] following it indicates that the statement is appropriate for a Year 5 level.

These mathematics assessment resources from the ARBs have been mapped to the key foci from the 2009 National Standards Mathematics progressions. The statements provided in the National Standards selector are an interpretation of the standards with a link to ARB items that can be used to provide evidence relating to the standard at that level.

Click on the Curriculum area and Curriculum level to view the description of the National Standards step and related ARB resources. 

  Level 1 Level 2   Level 3   Level 4
Number & Algebra Number & Algebra
(Level 1: Year 1&2)
Number & Algebra
(Level 2: Year 3&4)
Number & Algebra (Level 3: Year 5&6) Number & Algebra
(Level 4: Year 7&8)
Geometry & Measurement Geometry & Measurement
(Level 1: Year 1&2)
Geometry & Measurement
(Level 2: Year 3&4)

Geometry & Measurement
(Level 3: Year 5&6)

Geometry & Measurement
(Level 4: Year 7&8)
Statistics Statistics
(Level 1: Year 1 &2)
Statistics
(Level 2: Year 3 &4)
Statistics
(Level 3: Year 5&6)
Statistics
(Level 4: Year 7&8)
Component: 
Number Level 1
[early]
  • apply counting-all strategies
 
  • apply counting-on, counting-back, skip-counting, and simple grouping strategies to combine or partition whole numbers
  • use equal sharing and symmetry to find fractions of sets, shapes, and quantities
Click to View Resources
Resources: 
 
 
At level 1 look for evidence that students can ...
 
Counting
  • count all item in a set [early]  
  • use counting on strategies or skip counting strategies to solve simple addition and subtraction problems:  
Addition/subtraction
 
Equal sharing
Number Level 2

[early] Apply basic addition facts and knowledge of place value and symmetry to:

  • combine or partition whole numbers;
  • find fractions of sets, shapes, and quantities.

           Apply basic addition and subtraction facts, simple multiplication facts, and knowledge of place value and symmetry to:

  • combine or partition whole numbers;
  • find fractions of sets, shapes, and quantities.
Click to View Resources
Resources: 
At level 2 look for evidence that students can ...
Addition/Subtraction (Combine or Partitioning)

Multiplication/Division (Combine or Partitioning)

Fractions of sets, shapes and quantities

*The resource Fractions and sets could be used without the sets provided as evidence of students ability to find fractions of quantities (fractions as operators).

 

Number Level 3
[early] Apply additive and simple multiplicative strategies and knowledge of symmetry to:
  • combine or partition whole numbers;
  • find fractions of sets, shapes, and quantities.
          Apply additive and simple multiplicative strategies flexibly to:
  • combine or partition whole numbers, including performing mixed operations and using addition and subtraction as inverse operations;
  • find fractions of sets, shapes, and quantities.
Click to View Resources
Resources: 
At level 3 look for eivdence that students can ...
Addition/Subtraction (Combine or Partitioning)

Multiplication/Division (Combine or Partitioning)

*The resources Using doubling and halving and Using doubling and halving II explore the use of the doubling and halving strategy.

Fractions of sets shapes and quantities

Number Level 4
[early] Apply additive and multiplicative strategies flexibly to whole numbers, ratios, and equivalent fractions (including percentages);
  • apply additive strategies to decimals;
  • balance positive and negative amounts.
          Apply multiplicative strategies flexibly to whole numbers, ratios, and equivalent fractions (including decimals and percentages);
  • use multiplication and division as inverse operations on whole numbers;
  • apply additive strategies flexibly to decimals and integers.
Click to View Resources
Resources: 
At level 4 look for evidence that students can ...
 
Addition/Subtraction (Combine or Partitioning)

Multiplication/Division (Combine or Partitioning)

Note: using multiplication as the inverse of division and vice versa is evidence of multiplication at level 4.

Algebra Level 1
[early]
  • continue sequential patterns and number patterns based on ones.
 
  • create and continue sequential patterns by identifying the unit of repeat
  • continue number patterns based on ones, twos, fives, and tens.
Click to View Resources
Resources: 
At level 1 look for evidence that students can ...
 
Repeated patterns
  • continue repeated patterns [early] or create and continue repeated patterns and identify the unit of repeat  
Number patterns
Algebra Level 2
[early]
  • create and continue sequential patterns with one or two variables by identifying the unit of repeat;
  • continue spatial patterns and number patterns based on simple addition or subtraction.
 
  • create and continue, and give the rule for sequential patterns with two variables;
  • continue spatial patterns and number patterns based on repeated addition or subtraction
Click to View Resources
Resources: 
Sequential, Spatial and number patterns
At level 2 look for evidence that students can ...

* The resource Missing shapes has both types of repeating patterns.
+ some resources also ask students to identify/describe rules (early level 3) but these are very simple sequential rules (early/level 2).

Algebra Level 3
[early]
  • create, continue, and predict further members of sequential patterns with two variables;
  • describe spatial and number patterns, using rules that involve spatial features, repeated addition or subtraction, and simple multiplication.
 
  • determine members of sequential patterns, given their ordinal positions;
  • describe spatial and number patterns, using:

    • tables and graphs;
    • rules that involve spatial features, repeated addition or subtraction, and simple multiplication.
Click to View Resources
Resources: 
Sequential, spatial and number patterns
At level 3 look for evidence that students can ...

Note: if a student describes a pattern using a functional (direct) rule this may be evidence that they are operating at Algebra, early Level 4 for Patterns and relationships (e.g., Repeating bead patterns)

Algebra Level 4
[early]
  • find and represent relationships in spatial and number patterns, using:

    • tables and graphs;
    • general rules for linear relationships.

 

  • find and represent relationships in spatial and number patterns, using:

    • tables and graphs;
    • equations for linear relationships;
    • recursive rules for non-linear relationships;
  • apply inverse operations to simple linear relationships.
Click to View Resources
Resources: 
Measurement Level 1
[early]
  • compare the lengths, areas, volumes or capacities, and weights of objects directly
 
  • compare the lengths, areas, volumes or capacities, and weights of objects and the durations of events, using self-chosen units of measurement
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Measurement Level 2
[early]
  • measure the lengths, areas, volumes or capacities, and weights of objects and the duration of events, using linear whole-number scales and applying basic addition facts to standard units.
 
  • measure the lengths, areas, volumes or capacities, weights, and temperatures of objects and the duration of events, reading scales to the nearest whole number and applying addition, subtraction, and simple multiplication facts to standard units.
Click to View Resources
Resources: 
Measurement
At level 2 look for evidence that students can ...
 
Length

Area

Volume/capacity

Temperature

Weight

Time

* Looking at length, weight, and capacity can be used as evidence for length, capacity and weight.

Measurement Level 3
[early]
  • measure time and the attributes of objects choosing appropriate standar units and working with them to the nearest tenth;
 
 
  • mesure time and the attributes of objects, choosing appropriate standard units;
  • use arrays to find the areas of rectangles and the volumes of cuboids, given whole-number dimensions
Click to View Resources
Resources: 
Measurement
At level 3 look for evidence that students can ...
 
Length

Area

Volume/capacity

Temperature

Weight

Time

Units of measurement

 

Measurement Level 4
[early]
  • measure time and the attributes of objects, using metric and other standard measures;
  • make simple conversions between units, using whole numbers;
  • use side or edge lengths to find the perimeters and areas of rectangles and parallelograms and the volumes of cuboids, given whole-number dimensions.
 
  • use metric and other standard measures;
  • make simple conversions between units, using decimals;
  • use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
Click to View Resources
Resources: 
Measurement
At level 4 look for evidence that students can ...
Length/perimeter

Area

Volume

Temperature

Weight

Angle

Time

Units of measurement

*The resource The Danish flag can provide evidence of students' ability to calculate perimeter and area of a complex shape (Level 4).
+The resource Estimating the room can provide evidence of students' ability to measure (estimate) width, height and length as well as volume at level 4.

Shape Level 1
[early]
  • sort objects and shapes by a single feature and describe the feature, using everyday language
 
  • sort objects and shapes by different features and describe the features, using mathematical language
Click to View Resources
Resources: 
Shape properties: 
Shape Level 2

[early]

  • sort objects and two- and three-dimensional shapes by their features, identifying categories within categories.
 
  • sort objects and two- and three-dimensional shapes by two features simultaneously;
  • create nets for cubes.
Click to View Resources
Resources: 
Shape Level 3
[early]
  • sort two- and three-dimensional shapes, considering the presence and/or absence of features simultaneously and justifying the decisions made;
  • create nets for rectangular prisms;
  • draw plan, front, and side views of objects.

 

  • sort two- and three-dimensional shapes (including prisms), considering given properties simultaneously and justifying the decisions made;
  • identify nets for rectangular prisms;
  • draw or make objects, given their plan, front, and side views.
Click to View Resources
Shape Level 4
[early]
  • sort two- and three-dimensional shapes into classes, defining properties and justifying the decisions made;
  • create or identify nets for rectangular prisms and other simple solids;
  • draw plan, front, side, and perspective views of objects.

 

  • sort two- and three-dimensional shapes into classes, considering the relationships between the classes and justifying the decisions made;
  • create or identify nets for rectangular prisms and other simple solids, given particular requirements;
  • draw or make objects, given their plan, front, and side views or their perspective views.
Click to View Resources
Resources: 
At level 4 look for evidence that students can ...
Shapes
 
Nets

Views

Transformation Level 1
[early]
  • represent reflections and translations by creating patterns
 
  • represent reflections and translations by creating and describing patterns
Click to View Resources
Resources: 
Transformations Level 2
[early]
  • represent reflections, translations, and rotations by creating and describing patterns.
 
  • represent and describe the symmetries of a shape.
Click to View Resources
Resources: 
Transformations
At level 2 look for evidence that students can ...

Transformations Level 3
[early]
 
  • represent and describe the results of reflection, rotation, and translation on shapes.
 
  • represent and describe the results of reflection, rotation, and translation on shapes or patterns.
Click to View Resources
Transformations Level 4
[early]
 
  • Identify and describe the transformations that have produced given shapes or patterns.
 
  • Identify and describe the features of shapes or patterns that change or do not change under transformation.
Click to View Resources
Position Level 1
[early]
  • describe personal locations and give directions, using everyday language.
 
  • describe personal locations and give directions, using steps and half- or quarter-turns.
Click to View Resources
Position Level 2

[early]

  • describe personal locations and give directions, using whole-number measures and half- or quarter-turns.

 

  • describe personal locations and give directions, using simple maps
Click to View Resources
Resources: 

Position and orientation
At level 2 look for evidence that students can ...

 
Position Level 3
[early]
 
  • describe locations and give directions, using grid references and points of the compass.
 
  • describe locations and give directions, using grid references, turns, and points of the compass.
Click to View Resources
Resources: 

Position and orientation
At early level 3 look for evidence that students can ...

  • use grid references or compass points (North, South, East, West) for directions and a standard measures for distance: 
Position Level 4
[early]
  • describe locations and give directions, using grid references, simple scales, turns, and points of the compass.
 
  • describe locations and give directions, using scales, bearings, and co-ordinates.
Click to View Resources
Resources: 

Position and orientation
At level 4 look for evidence that students can ...

*Distance and bearing scaffolds the use of compass bearings.
Statistics Level 1
[early]
  • investigate questions by using the statistical enquiry cycle (with support), gathering, displaying, and/or counting category data.
 
  • investigate questions by using the statistical enquiry cycle (with support), gathering, displaying, and/or identifying similarities and differences in category data
  • describe the likelihoods of outcomes for a simple situation involving chance, using everyday language.
Click to View Resources
Resources: 
Table construction:
Graph interpretation:
 
 
Statistics Level 2
[early]
investigate questions by using the statistical enquiry cycle (with support):
  • gather and display category and simple whole-number data;
  • interpret displays in context.
 
investigate questions by using the statistical enquiry cycle independently:
  • gather and display category and simple whole-number data;
  • interpret results in context.
Click to View Resources
Resources: 
At level 2 look for evidence that students can ...
Statistical enquiry cycle

Gather and display data

Analyse and interpret data

Statistics Level 3
[early]
 
investigate summary and comparison questions by using the statistical enquiry cycle:
  • gather, display, and identify patterns in category and whole-number data;
  • interpret results in context.
 
investigate summary and comparison questions by using the statistical enquiry cycle:
  • gather or access multivariate category and whole-number data;
  • sort data into categories or intervals, display it in different ways, and identify patterns;
  • interpret results in context, accepting that samples vary.
Click to View Resources
Resources: 
At level 3 look for evidence that students can ...
Statistical enquiry cycle

  • pose a simple question to investigate, plan who to sample, and how to collect and display the data: Sport on TV;

Gather and display data

  • construct graphs or tables of category or whole number data, including dot plots with category data, or scaffolded stem-and-leaf graphs [early]: Marine fish - Lunch food;
  • construct graphs or tables of category or whole number data, including multivariate graphs, dot plots with whole number data, or stem-and-leaf graphs: Science test results - Ice-cream flavours - Waiting time;

Analyse and interpret data

Statistics Level 4

[early]

investigate summary, comparison, and relationship questions by using the statistical enquiry cycle:

  • gather or access multivariate category and measurement data;
  • sort data and display it in multiple ways, identifying patterns and variations;
  • interpret results in context, accepting that samples vary and have no effect on one another.

 

investigate summary, comparison, and relationship questions by using the statistical enquiry cycle:

  • gather or access multivariate category, measurement, and time-series data;
  • sort data and display it in multiple ways, identifying patterns, variations, relationships, and trends and using ideas about middle and spread where appropriate;
  • interpret results in context, identifying factors that produce uncertainty.
Click to View Resources
Resources: 
At level 4 look for evidence that students can...
Statistical enquiry cycle

Gather and display data

Analyse and interpret data

Probability Level 1
  • describe the likelihoods of outcomes for a simple situation involving chance, using everyday language.
Click to View Resources
Resources: 
At level 1 look for evidence that students can ...
Probability

identify the likelihood of simple outcomes 
Probability Level 2
[early]
 
  • compare and explain the likelihoods of outcomes for a simple situation involving chance.
 
  • compare and explain the likelihoods of outcomes for a simple situation involving chance, acknowledging uncertainty.
Click to View Resources
Resources: 
At level 2 look for evidence that students can ...
Probability

Probability Level 3
[early]
  • order the likelihoods of outcomes for simple situations involving chance, experimenting or listing all possible outcomes
 
  • order the likelihoods of outcomes for situations involving chance, considering experimental results and models of all possible outcomes.
Click to View Resources
Resources: 
At level 3 look for evidence that students can ...
Probability

Combinations
At level 3 look for evidence of:

Probability Level 4
[early]
  • order the likelihoods of outcomes for situations involving chance, checking for consistency between experimental results and models of all possible outcomes.
 
  • express as fractions the likelihoods of outcomes for situations involving chance, checking for consistency between experimental results and models of all possible outcomes.
Click to View Resources
Resources: 
At level 4 look for evidence that students can ...
Probability

Combinations