# 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 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.

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
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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:
• 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.
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Resources:
At level 2 look for evidence that students can ...

• add and subtract 2-digit whole numbers (involving simple recombining) using addition basic facts and simple place value knowledge [early] or start to use part-whole strategies (such as place value partitioning):
• Subtraction:

Multiplication/Division (Combine or Partitioning)

• use symmetry to partition small sets of objects:  or grouping diagrams [early]
• use simple basic facts knowledge to solve multiplication and division problems
• use additive strategies (such as repeated addition/subtraction, deriving from known basic facts: How many are there?
• Multiplication:
• Division

Fractions of sets, shapes and quantities

• start to partition ("chunk") to find simple unit fractions (halves, thirds, quarters, fifths, and tenths).
• Partitioning sets and shapes
• involve finding half of small quantities (under 20) without using sets [early] or use basic facts knowledge and simple additive strategies (such as repeated addition) to find fractions of small quantities (e.g., finding 1/4 of 12 can involve recognising that 3 + 3 + 3 + 3 = 12).
• Fractions as operators

*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.
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Resources:
At level 3 look for eivdence that students can ...

• add and subtract 2- and 3-digit numbers that involve recombining and use part-whole strategies such as place value partitioning, tidy numbers, rounding and compensation [early];
• use a range of different and more sophisticated part-whole strategies and larger 3-digit numbers that involve multiple recombining;
• utilise the inverse relationship of addition and subtraction (reversibility or complementary addition);
• Subtraction:

Multiplication/Division (Combine or Partitioning)

• use multiplicative strategies such as doubling and halving* [early], combining place value partitioning with basic facts, rounding and adjustment (such as compensation), and derived basic facts:
• Multiplication:
• Divsion:

*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

• partition sets and shapes into common fraction parts. Partitioning sets and shapes:
• identify the fraction part: Part-whole fractions
• use additive (such as repeated addition) and simple multiplicative strategies (repeated halving) to find common fractions of quantities (e.g., 1/4 of 60, 1/2 of 60 is 30, and 1/2 of 30 is 15) [early] or start to use multiplication and division facts, and simple multiplicative strategies to find common fractions of quantities (e.g., for 1/4 of 24, involve recognising that 4 lots of 6 make 24, so 1/4 of 24 is 6;
• 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.
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Resources:
At level 4 look for evidence that students can ...

Multiplication/Division (Combine or Partitioning)

• a broad range of additive and multiplicative strategies (such as partitioning both factors, and multiplicative reversibility) to solve maths problems involving:

• whole numbers: Estimation involving multiplication:
• ratio:  Show how many
• fractions
Students can work out the simplest form of fractions (equivalent fractions):
• Finding fractions of quantities:
• percentages: finding a range of percentages of quantities:
• Multiplication with decimals:
• Multiplication
• Division

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.
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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
• create and continue 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
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Resources:
Sequential, Spatial and number patterns
At level 2 look for evidence that students can ...

• continue a simple repeating pattern and can identify the repeated part [early] or continue the pattern and describe how simple patterns with two variable (features such as shading and shape) repeats:
• continue a simple spatial or number pattern that is increasing by a simple addition of subtraction (sequential) rule [early]: or continue patterns with more complex sequential rules (more complex addition or subtraction) for spatial and number patterns:

* 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.
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Resources:
Sequential, spatial and number patterns
At level 3 look for evidence that students can ...

• continue repeating patterns, identify the repeating part, and predict the shapes for given ordinal positions (e.g., what the 35th shape would look like):
• continue patterns with more complex sequential rules (more complex addition or subtraction), and describe the sequential rule for the pattern:
• describe spatial and number patterns giving a sequential/recursive rule:
spatial patterns:  number patterns:

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.
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Resources:
Spatial and number patterns
At level 4 look for evidence that students can ...

• continue patterns and identify functional/direct rules for given spatial and number patterns [early]:
• represent patterns in tables or on appropriate graphs:
• use a functional/direct rule to find any given members of the pattern: functional rules:
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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Resources:

Weight

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.
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Resources:
Measurement
At level 2 look for evidence that students can ...

Length

• measure a length with a scale to the nearest whole number:

Area

• measure area of shapes using whole unit squares:

Volume/capacity

• compare the volume/capacity of objects to a given standard unit measure:
• measure the volume of objects using whole cubed units;

Temperature

Weight

• compare the weight of objects to a given standard unit measure:

Time

• measure (analogue and digital) time to the nearest half/quarter hour:
• add and subtract time problems to the nearest half hour:
• add and subtract time problems involving days, weeks and months:

* 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
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Resources:
Measurement
At level 3 look for evidence that students can ...

Length

• measure length using a given unit of measurement or :
• measure length to the nearest half unit of measurement [early] or nearest tenth of a unit of measurement:

Area

• measure area of shapes using an array (multiplication of lengths) or unit squares and half measures:
Note: solving array problems may also indicate evidence for early multiplicative strategies;

Volume/capacity

• measure volume of objects using cubed units and half measures:
• identify capacities of everday containers:

Temperature

• measure temperature on a scale to the nearest tenth of a unit:

Weight

• measure weight on a scale to the nearest tenth of a unit:

Time

Units of measurement

• select appropriate units of measurement for time and the attributes of objects [early]:.

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

• calculate length and perimeter of shapes/objects:
• Note: solving perimeter problems may also indicate evidence for additive strategies (Number and Algebra, Level 2/3/4)

Area

• calculate area of shapes/objects:
Note: solving area problems may also indicate evidence for early multiplicative strategies (Number and Algebra, Level 3)

Volume

• measure volume of objects with whole number [early] or decimal measures:
Note: solving volume problems may also indicate evidence for advanced multiplicative strategies (early Number and Algebra, Level 4)

Temperature

Weight

• measure weight on a scale to the nearest tenth of a unit:

Angle

• measure the angle of shapes or object in degrees:

Time

• measure (analogue and digital) time (including tenth of a second);
• add and subtract analogue and digital time problems:

Units of measurement

• convert between different whole number [early]:
or different decimal 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
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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.
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Resources:
At level 2 look for evidence that students can ...
Shapes

• can sort shapes based on two features:

Nets

• create a net for a cubes:
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
Resources:
At level 3 look for evidence that students can...
Shapes

Nets

• identify and create nets for rectangular prisms:

Views

• draw different views of 3-dimensional shapes/objects:
• make a 3-dimensional shape from given views:
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
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Resources:
Rotation
Transformations Level 2
[early]
• represent reflections, translations, and rotations by creating and describing patterns.

• represent and describe the symmetries of a shape.
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Resources:
Transformations
At level 2 look for evidence that students can ...

• reflect, translate or rotate shapes to make a pattern and/or describe how a pattern was made:
• show, identify and describe aspects of symmetry of a shape:
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.
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Resources:
Transformations
At level 3 look for evidence that students can ...

• show and describe the result of transformations upon given shapes and patterns:
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.
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Resources:
Transformations
At level 4 look for evidence that students can ...

• describe the transformations that have changed a shape or pattern:
• identify how an enlargement transformations has changed a shape or pattern:
• identify shapes or patterns and how they change or remain under transformation:
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.
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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
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Resources:

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

• give personal locations and directions (including left, right, forward and backwards, and half and quarter turns) [early]:
• describe locations of places or objects on simple maps:

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.
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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 ...

• use simple scale [early]:
• use more detailed compass points (including NE, SE, NW, SW):
• use more complex scales;
• use co-ordinates;
*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.
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Resources:
Table construction:
Graph interpretation:

Probability:
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.
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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.
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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.
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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.
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Resources:
At level 2 look for evidence that students can ...
Probability

• identify the most likely outcomes for probability situations involving counting small numbers of objects [early] or identify the most likely outcomes for probability situations involving frequency or area models of probability, explaining their reasoning or acknowledging uncertainty:  Spinner probabilities - Spin a surprise;
• describe the probabilities for simple events using everyday language [early] or describe the probabilities for events using everyday language including area models: Possible events - Different spinners
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.
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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.
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Resources:
At level 4 look for evidence that students can ...
Probability

• order the probability of given outcomes [early] or with  with justification:  Spinner for a board
• express as fractions the likelihoods of outcomes for situations involving chance:
• perform experiments to test probability predictions:

Combinations

• identify all combinations using models of all outcomes, including tree diagrams of two [early] or three independent outcomes:
• identify all combinations using tree diagrams