0.5

At 0.5, the work of a student progressing towards the standard at Level 1 demonstrates, for example:

• use of descriptive terms such as longer, taller and heavier to compare length and mass of pairs of familiar objects
• use of yesterday, today and tomorrow and the ability to name the corresponding days
• awareness that a clock changes in response to time in a daily cycle
• use of a die or similar device to determine the range of outcomes in a game
• interpretation of pictographs, and collection and sorting of items or data in preparation for the creation of a pictograph

• Awareness of time

1.0 Standard

At Level 1, students compare length, area, capacity and mass of familiar objects using descriptive terms such as longer, taller, larger, holds more and heavier. They make measurements using informal units such as paces for length, handprints for area, glasses for capacity, and bricks for weight.

They recognise the continuity of time and the natural cycles such as day/night and the seasons. They correctly sequence days of the week. They use informal units such as heartbeats and hand claps at regular intervals to measure and describe the passage of time.

They recognise and respond to unpredictability and variability in events, such as getting or not getting a certain number on the roll of a die in a game or the outcome of a coin toss. They collect and display data related to their own activities using simple pictographs.

• Comparison of length

1.25

At 1.25, the work of a student progressing towards the standard at Level 2 demonstrates, for example:

• informal measurement of length by making, describing and comparing personal units
• use of a clock to determine the hour
• ordering of days, weeks, months and years
• understanding of distinction between cold, cool, warm, hot and boiling
• awareness that some events are equally likely to occur; for example, a head or a tail showing when a coin is tossed

• First experiences with chance 
• Reading the hour on a clock

1.5

At 1.5, the work of a student progressing towards the standard at Level 2 demonstrates, for example:

• use of uniform units for length; for example, cm as a unit for measuring length
• informal measurement of area and mass by making, describing and comparing personal units
• knowledge of the relationship between analogue and digital clocks
• knowledge of the outcomes of chance events such as rolling a die
• interpretation of pictographs, bar and column graphs

1.75

At 1.75, the work of a student progressing towards the standard at Level 2 demonstrates, for example:

• informal measurement of capacity by making, describing and comparing personal units
• construction of a time line for daily activity and use of a diary for recording daily events
• drawing of an analogue clock to match a given digital time and of reading an analogue clock to the nearest half hour
• ordering of familiar events in terms of their probability between impossible and certain
• collection and recording of categorical and numerical data

• Pictographs and bar graphs

2.0 Standard

At Level 2, students make, describe and compare measurements of length, area, volume, mass and time using informal units. They recognise the differences between non-uniform measures, such as hand-spans, to measure length, and uniform measures, such as icy-pole sticks. They judge relative capacity of familiar objects and containers by eye and make informal comparisons of weight by hefting. They describe temperature using qualitative terms (for example, cold, warm, hot). Students use formal units such as hour and minute for time, litre for capacity and the standard units of metres, kilograms and seconds.

Students recognise the key elements of the calendar and place in sequence days, weeks and months. They describe common and familiar time patterns and such as the time, duration and day of regular sport training and tell the time at hours and half-hours using an analogue clock, and to hours and minutes using a digital clock.

Students predict the outcome of chance events, such as the rolling of a die, using qualitative terms such as certain, likely, unlikely and impossible. They collect simple categorical and numerical data (count of frequency) and present this data using pictographs and simple bar graphs.

• The idea of a unit

2.25

At 2.25, the work of a student progressing towards the standard at Level 3 demonstrates, for example:

• use of formal units of measurement; for example, metres to measure length, and hour, minute and second for time
• application of estimations using personal units, such as pace length and arm span, and comparison with measures using formal units, such as metres and centimetres
• use of ruler and tape measure (linear scale) and trundle wheel (circular scale) to validate estimates of length
• setting of temperature in Celsius on a circular scale; for example, on an oven, and estimation of temperature in degrees Celsius
• displays of data as a column or bar graph

2.5

At 2.5, the work of a student progressing towards the standard at Level 3 demonstrates, for example:

• estimation and measurement of mass, volume and capacity of common objects; for example, kilogram of flour, litre of soft drink
• reading of analogue clocks to the nearest quarter of an hour
• construction and interpretation of a daily timetable
• identification of events which are equally likely
• construction of an appropriately labelled bar graph

• Reading clocks to quarter hours

2.75

At 2.75, the work of a student progressing towards the standard at Level 3 demonstrates, for example:

• calculation of area through multiplication of the length of a rectangle by its width
• estimation of angle in terms of quarter turns and half turns
• investigation of the fairness of events such as gambling and games through experimentation
• comparison of the likelihood of everyday events and linking of events with statements about how likely they are to occur
• understanding of the distinction between discrete and continuous scales

• Measuring area 
  

At Level 3, students estimate and measure length, area, volume, capacity, mass and time using appropriate instruments. They recognise and use different units of measurement including informal (for example, paces), formal (for example, centimetres) and standard metric measures (for example, metre) in appropriate contexts. They read linear scales (for example, tape measures) and circular scales (for example, bathroom scales) in measurement contexts. They read digital time displays and analogue clock times at five-minute intervals. They interpret timetables and calendars in relation to familiar events. They compare the likelihood of everyday events (for example, the chances of rain and snow). They describe the fairness of events in qualitative terms. They plan and conduct chance experiments (for example, using colours on a spinner) and display the results of these experiments. They recognise different types of data: non-numerical (categories), separate numbers (discrete), or points on an unbroken number line (continuous).They use a column or bar graph to display the results of an experiment (for example, the frequencies of possible categories).

• Fairness relates to having an equal chance of winning

3.25

At 3.25, the work of a student progressing towards the standard at Level 4 demonstrates, for example:

• estimation and measurement of perimeter of polygons
• conversion between metric measurements for length; for example, 0.27m = 27cm
• estimation and measurement of angles in degrees to the nearest 10°
• use of fractions to assign probability values between 0 and 1 to probabilities based on symmetry; for example, Pr(six on a die) = ¹/₆
• identification of mode and range for a set of data

3.5

At 3.5, the work of a student progressing towards the standard at Level 4 demonstrates, for example:

• estimation and measurement of surface area; for example, use of square metres, and area of land; for example, use of hectares
• awareness of the accuracy of measurement required and the appropriate tools and units
• subdivision of a circle into two sectors according to a given proportion for arc length
• design of questionnaires to obtain data from a sample of the population
• sorting of data using technology

3.75

At 3.75, the work of a student progressing towards the standard at Level 4 demonstrates, for example:

• conversion between metric units; for example, L to mL, and understanding of the significance of thousands and thousandths in the metric system
• simulation of simple random events
• calculation and analysis of the stability of a sequence of long run frequencies where the number of trials increases, say from 5 to 10 to 20 to 100
• interpretation of pie charts and histograms
• identification of the median for a set of data

• Choosing Appropriate Graphical Displays
• Median as another central measure
  

4.0 Standard

At Level 4, students use metric units to estimate and measure length, perimeter, area, surface area, mass, volume, capacity time and temperature. They measure angles in degrees. They measure as accurately as needed for the purpose of the activity. They convert between metric units of length, capacity and time (for example, L–mL, sec–min).

Students describe and calculate probabilities using words, and fractions and decimals between 0 and 1. They calculate probabilities for chance outcomes (for example, using spinners) and use the symmetry properties of equally likely outcomes. They simulate chance events (for example, the chance that a family has three girls in a row) and understand that experimental estimates of probabilities converge to the theoretical probability in the long run.

Students recognise and give consideration to different data types in forming questionnaires and sampling. They distinguish between categorical and numerical data and classify numerical data as discrete (from counting) or continuous (from measurement). They present data in appropriate displays (for example, a pie chart for eye colour data and a histogram for grouped data of student heights). They calculate and interpret measures of centrality (mean, median, and mode) and data spread (range).

• Perimeter and area are not the same
• Converting between measurement units
• Time intervals

4.25

At 4.25, the work of a student progressing towards the standard at Level 5 demonstrates, for example:

• development and use of formulas for the area and perimeter of triangles and parallelograms
• determination of the internal and external angle sums for a polygon and confirmation by measurement
• estimation of the likely maximum and minimum error associated with a measurement
• appropriate use of zero to indicate accuracy of measurement; for example, a piece of timber 2.100m long is accurate to the nearest mm
• recognition of the mean value of a set of measurements as the best estimate, and that the range could represent the associated error

4.5

At 4.5, the work of a student progressing towards the standard at Level 5 demonstrates, for example:

• use of appropriate units and measurement of length, perimeter, area, surface area, mass, volume, capacity, angle, time and temperature, in context
• calculation of total surface area of prisms, including cylinders, by considering their nets
• contrast between the stability of long run relative frequency and the variation of observations based on small samples
• construction of dot plots, and stem and leaf plots to represent data sets

• Dot plots and stem-and-leaf plots

4.75

At 4.75, the work of a student progressing towards the standard at Level 5 demonstrates, for example:

• understanding of the distinction between error and percentage error
• use of random numbers to assist in probability simulations and the arithmetic manipulation of random numbers to achieve the desired set of outcomes
• calculation of theoretical probability using ratio of number of ‘successful’ outcomes to total number of outcomes
• use of tree diagrams to explore the outcomes from multiple event trials
• display and interpretation of dot plots, and stem and leaf plots, including reference to mean, median and mode as measures of centre

• Area of a circle
• A critical approach to summary statistics and graphs
  

5.0 Standard

At Level 5, students measure length, perimeter, area, surface area, mass, volume, capacity, angle, time and temperature using suitable units for these measurements in context. They interpret and use measurement formulas for the area and perimeter of circles, triangles and parallelograms and simple composite shapes. They calculate the surface area and volume of prisms and cylinders.

Students estimate the accuracy of measurements and give suitable lower and upper bounds for measurement values. They calculate absolute percentage error of estimated values.

Students use appropriate technology to generate random numbers in the conduct of simple simulations.

Students identify empirical probability as long-run relative frequency. They calculate theoretical probabilities by dividing the number of possible successful outcomes by the total number of possible outcomes. They use tree diagrams to investigate the probability of outcomes in simple multiple event trials.

Students organise, tabulate and display discrete and continuous data (grouped and ungrouped) using technology for larger data sets. They represent uni-variate data in appropriate graphical forms including dot plots, stem and leaf plots, column graphs, bar charts and histograms. They calculate summary statistics for measures of centre (mean, median, mode) and spread (range, and mean absolute difference), and make simple inferences based on this data.

• Short run variation and long-run stability

5.25

At 5.25, the work of a student progressing towards the standard at Level 6 demonstrates, for example:

• conversion between units and between derived units
• use of pythagoras theorem to calculate the length of a hypotenuse
• use of symmetry and scale to calculate side lengths in triangles
• representation of compound events involving two categories and the logical connectives and, or and not using lists, grids (lattice diagrams), tree diagrams, venn diagrams and karnaugh maps (two-way tables) and the calculation of associated probabilities
• representation of statistical data using technology

• Converting between derived units

5.5

At 5.5, the work of a student progressing towards the standard at Level 6 demonstrates, for example:

• calculation and application of ratio, proportion and rate of change such as concentration, density and the rate of filling a container
• use of pythagoras theorem to calculate the length of a side other than a hypotenuse
• use of trigonometric ratios to calculate unknown sides in a right-angled triangle
• display of data as a box plot including calculation of quartiles and inter-quartile range and the identification of outliers
• qualitative judgment of positive or negative correlation and strength of relationship and, if appropriate, application of gradient to find a line of good fit by eye

• Calculations involving rates

5.75

At 5.75, the work of a student progressing towards the standard at Level 6 demonstrates, for example:

• conversion between degrees and radians, and use of radians when calculating arc length and area of sectors
• use of Pythagoras' theorem in three-dimensional applications
• calculation of unknown angle in a right-angled triangle using trigonometric ratios
• use of surveys as a means of obtaining information about a population, including awareness that sample results will not always provide a reasonable estimate of population parameters
• placement of a line of best fit on a scatter plot using technology and, where appropriate, use of a line of best fit to make predictions

• Deeper understanding of Pythagoras' theorem

6.0 Standard

At Level 6, students estimate and measure length, area, surface area, mass, volume, capacity and angle. They select and use appropriate units, converting between units as required. They calculate constant rates such as the density of substances (that is, mass in relation to volume), concentration of fluids, average speed and pollution levels in the atmosphere. Students decide on acceptable or tolerable levels of error in a given situation. They interpret and use mensuration formulas for calculating the perimeter, surface area and volume of familiar two- and three-dimensional shapes and simple composites of these shapes. Students use Pythagoras’ theorem and trigonometric ratios (sine, cosine and tangent) to obtain lengths of sides, angles and the area of right-angled triangles.

They use degrees and radians as units of measurement for angles and convert between units of measurement as appropriate.

Students estimate probabilities based on data (experiments, surveys, samples, simulations) and assign and justify subjective probabilities in familiar situations. They list event spaces (for combinations of up to three events) by lists, grids, tree diagrams, Venn diagrams and karnaugh maps (two-way tables). They calculate probabilities for complementary, mutually exclusive, and compound events (defined using and, or and not). They classify events as dependent or independent.

Students comprehend the difference between a population and a sample. They generate data using surveys, experiments and sampling procedures. They calculate summary statistics for centrality (mode, median and mean), spread (box plot, inter-quartile range, outliers) and association (by-eye estimation of the line of best fit from a scatter plot). They distinguish informally between association and causal relationship in bi-variate data, and make predictions based on an estimated line of best fit for scatter-plot data with strong association between two variables.