# Mathematics and statistics

- What is mathematics and statistics?
- Why study mathematics and statistics?
- Learning area structure
- Achievement objectives

## Achievement objectives

### Level 1

*In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:*

### Number and algebra

**Number strategies**

- Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.

**Number knowledge**

- Know the forward and backward counting sequences of whole numbers to 100.
- Know groupings with five, within ten, and with ten.

**Equations and expressions**

- Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures.

**Patterns and relationships**

- Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many.
- Create and continue sequential patterns.

### Geometry and measurement

**Measurement**

- Order and compare objects or events by length, area, volume and capacity, weight (mass), turn (angle), temperature, and time by direct comparison and/or counting whole numbers of units.

**Shape**

- Sort objects by their appearance.

**Position and orientation**

- Give and follow instructions for movement that involve distances, directions, and half or quarter turns.
- Describe their position relative to a person or object.

**Transformation**

- Communicate and record the results of translations, reflections, and rotations on plane shapes.

### Statistics

**Statistical investigation**

Conduct investigations using the statistical enquiry cycle:

- posing and answering questions
- gathering, sorting and counting, and displaying category data
- discussing the results.

**Statistical literacy**

- Interpret statements made by others from statistical investigations and probability activities.

**Probability**

- Investigate situations that involve elements of chance, acknowledging and anticipating possible outcomes.

### Level 2

*In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:*

### Number and algebra

**Number strategies**

- Use simple additive strategies with whole numbers and fractions.

**Number knowledge**

- Know forward and backward counting sequences with whole numbers to at least 1000.
- Know the basic addition and subtraction facts.
- Know how many ones, tens, and hundreds are in whole numbers to at least 1000.
- Know simple fractions in everyday use.

**Equations and expressions**

- Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.

**Patterns and relationships**

- Generalise that whole numbers can be partitioned in many ways.
- Find rules for the next member in a sequential pattern.

### Geometry and measurement

**Measurement**

- Create and use appropriate units and devices to measure length, area, volume and capacity, weight (mass), turn (angle), temperature, and time.
- Partition and/or combine like measures and communicate them, using numbers and units.

**Shape**

- Sort objects by their spatial features, with justification.
- Identify and describe the plane shapes found in objects.

**Position and orientation**

- Create and use simple maps to show position and direction.
- Describe different views and pathways from locations on a map.

**Transformation**

- Predict and communicate the results of translations, reflections, and rotations on plane shapes.

### Statistics

**Statistical investigation**

Conduct investigations using the statistical enquiry cycle:

- posing and answering questions
- gathering, sorting, and displaying category and whole-number data
- communicating findings based on the data.

**Statistical literacy**

- Compare statements with the features of simple data displays from statistical investigations or probability activities undertaken by others.

**Probability**

- Investigate simple situations that involve elements of chance, recognising equal and different likelihoods and acknowledging uncertainty.

### Level 3

*In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:*

### Number and algebra

**Number strategies**

- Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.

**Number knowledge**

- Know basic multiplication and division facts.
- Know counting sequences for whole numbers.
- Know how many tenths, tens, hundreds, and thousands are in whole numbers.
- Know fractions and percentages in everyday use.

**Equations and expressions**

- Record and interpret additive and simple multiplicative strategies, using, words, diagrams, and symbols, with an understanding of equality.

**Patterns and relationships**

- Generalise the properties of addition and subtraction with whole numbers.
- Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.

### Geometry and measurement

**Measurement**

- Use linear scales and whole numbers of metric units for length, area, volume and capacity, weight (mass), angle, temperature, and time.
- Find areas of rectangles and volumes of cuboids by applying multiplication.

**Shape**

- Classify plane shapes and prisms by their spatial features.
- Represent objects with drawings and models.

**Position and orientation**

- Use a co-ordinate system or the language of direction and distance to specify locations and describe paths.

**Transformation**

- Describe the transformations (reflection, rotation, translation, or enlargement) that have mapped one object onto another.

### Statistics

**Statistical investigation**

Conduct investigations using the statistical enquiry cycle:

- gathering, sorting, and displaying multivariate category and whole-number data and simple time-series data to answer questions
- identifying patterns and trends in context, within and between data sets
- communicating findings, using data displays.

**Statistical literacy**

- Evaluate the effectiveness of different displays in representing the findings of a statistical investigation or probability activity undertaken by others.

**Probability**

- Investigate simple situations that involve elements of chance by comparing experimental results with expectations from models of all the outcomes, acknowledging that samples vary.

### Level 4

### Number and algebra

**Number strategies and knowledge**

- Use a range of multiplicative strategies when operating on whole numbers.
- Understand addition and subtraction of fractions, decimals, and integers.
- Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
- Apply simple linear proportions, including ordering fractions.
- Know the equivalent decimal and percentage forms for everyday fractions.
- Know the relative size and place value structure of positive and negative integers and decimals to three places.

**Equations and expressions**

- Form and solve simple linear equations.

**Patterns and relationships**

- Generalise properties of multiplication and division with whole numbers.
- Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.

### Geometry and measurement

**Measurement**

- Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
- Convert between metric units, using whole numbers and commonly used decimals.
- Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
- Interpret and use scales, timetables, and charts.

**Shape**

- Identify classes of two- and three-dimensional shapes by their geometric properties.
- Relate three-dimensional models to two-dimensional representations, and vice versa.

**Position and orientation**

- Communicate and interpret locations and directions, using compass directions, distances, and grid references.

**Transformation**

- Use the invariant properties of figures and objects under transformations (reflection, rotation, translation, or enlargement).

### Statistics

**Statistical investigation**

Plan and conduct investigations using the statistical enquiry cycle:

- determining appropriate variables and data collection methods
- gathering, sorting, and displaying multivariate category, measurement, and time-series data to detect patterns, variations, relationships, and trends
- comparing distributions visually
- communicating findings, using appropriate displays.

**Statistical literacy**

- Evaluate statements made by others about the findings of statistical investigations and probability activities.

**Probability**

- Investigate situations that involve elements of chance by comparing experimental distributions with expectations from models of the possible outcomes, acknowledging variation and independence.
- Use simple fractions and percentages to describe probabilities.

### Level 5

*In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. **They will solve problems and model situations that require them to:*

### Number and algebra

**Number strategies and knowledge**

- Reason with linear proportions.
- Use prime numbers, common factors and multiples, and powers (including square roots).
- Understand operations on fractions, decimals, percentages, and integers.
- Use rates and ratios.
- Know commonly used fraction, decimal, and percentage conversions.
- Know and apply standard form, significant figures, rounding, and decimal place value.

**Equations and expressions**

- Form and solve linear and simple quadratic equations.

**Patterns and relationships**

- Generalise the properties of operations with fractional numbers and integers.
- Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.

### Geometry and measurement

**Measurement**

- Select and use appropriate metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time, with awareness that measurements are approximate.
- Convert between metric units, using decimals.
- Deduce and use formulae to find the perimeters and areas of polygons and the volumes of prisms.
- Find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders.

**Shape**

- Deduce the angle properties of intersecting and parallel lines and the angle properties of polygons and apply these properties.
- Create accurate nets for simple polyhedra and connect three-dimensional solids with different two-dimensional representations.

**Position and orientation**

- Construct and describe simple loci.
- Interpret points and lines on co-ordinate planes, including scales and bearings on maps.

**Transformation**

- Define and use transformations and describe the invariant properties of figures and objects under these transformations.
- Apply trigonometric ratios and Pythagoras’ theorem in two dimensions.

### Statistics

**Statistical investigation**

Plan and conduct surveys and experiments using the statistical enquiry cycle:

- determining appropriate variables and measures
- considering sources of variation
- gathering and cleaning data
- using multiple displays, and re-categorising data to find patterns, variations, relationships, and trends in multivariate data sets
- comparing sample distributions visually, using measures of centre, spread, and proportion
- presenting a report of findings.

**Statistical literacy**

- Evaluate statistical investigations or probability activities undertaken by others, including data collection methods, choice of measures, and validity of findings.

**Probability**

- Compare and describe the variation between theoretical and experimental distributions in situations that involve elements of chance.
- Calculate probabilities, using fractions, percentages, and ratios.

### Level 6

### Number and algebra

**Number strategies and knowledge**

- Apply direct and inverse relationships with linear proportions.
- Extend powers to include integers and fractions.
- Apply everyday compounding rates.
- Find optimal solutions, using numerical approaches.

**Equations and expressions**

- Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns.

**Patterns and relationships**

- Generalise the properties of operations with rational numbers, including the properties of exponents.
- Relate graphs, tables, and equations to linear, quadratic, and simple exponential relationships found in number and spatial patterns.
- Relate rate of change to the gradient of a graph.

### Geometry and measurement

**Measurement**

- Measure at a level of precision appropriate to the task.
- Apply the relationships between units in the metric system, including the units for measuring different attributes and derived measures.
- Calculate volumes, including prisms, pyramids, cones, and spheres, using formulae.

**Shape**

- Deduce and apply the angle properties related to circles.
- Recognise when shapes are similar and use proportional reasoning to find an unknown length.
- Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions.

**Position and orientation**

- Use a co-ordinate plane or map to show points in common and areas contained by two or more loci.

**Transformation**

- Compare and apply single and multiple transformations.
- Analyse symmetrical patterns by the transformations used to create them.

### Statistics

**Statistical investigation**

Plan and conduct investigations using the statistical enquiry cycle:

- justifying the variables and measures used
- managing sources of variation, including through the use of random sampling
- identifying and communicating features in context (trends, relationships between variables, and differences within and between distributions), using multiple displays
- making informal inferences about populations from sample data
- justifying findings, using displays and measures.

**Statistical literacy**

- Evaluate statistical reports in the media by relating the displays, statistics, processes, and probabilities used to the claims made.

**Probability**

Investigate situations that involve elements of chance:

- comparing discrete theoretical distributions and experimental distributions, appreciating the role of sample size
- calculating probabilities in discrete situations.

### Level 7

### Mathematics

**Patterns and relationships**

- Apply co-ordinate geometry techniques to points and lines.
- Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs.
- Use arithmetic and geometric sequences and series.
- Apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions.
- Choose appropriate networks to find optimal solutions.

**Equations and expressions**

- Manipulate rational, exponential, and logarithmic algebraic expressions.
- Form and use linear, quadratic, and simple trigonometric equations.
- Form and use pairs of simultaneous equations, one of which may be non-linear.

**Calculus**

- Sketch the graphs of functions and their gradient functions and describe the relationship between these graphs.
- Apply differentiation and anti-differentiation techniques to polynomials.

### Statistics

**Statistical investigation**

Carry out investigations of phenomena, using the statistical enquiry cycle:

- conducting surveys that require random sampling techniques, conducting experiments, and using existing data sets
- evaluating the choice of measures for variables and the sampling and data collection methods used
- using relevant contextual knowledge, exploratory data analysis, and statistical inference.

Make inferences from surveys and experiments:

- making informal predictions, interpolations, and extrapolations
- using sample statistics to make point estimates of population parameters
- recognising the effect of sample size on the variability of an estimate.

**Statistical literacy**

Evaluate statistically based reports:

- interpreting risk and relative risk
- identifying sampling and possible non-sampling errors in surveys, including polls.

**Probability**

Investigate situations that involve elements of chance:

- comparing theoretical continuous distributions, such as the normal distribution, with experimental distributions
- calculating probabilities, using such tools as two-way tables, tree diagrams, simulations, and technology.

### Level 8

### Mathematics

**Patterns and relationships**

- Apply the geometry of conic sections.
- Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions.
- Use permutations and combinations.
- Use curve fitting, log modelling, and linear programming techniques.
- Develop network diagrams to find optimal solutions, including critical paths.

**Equations and expressions**

- Manipulate trigonometric expressions.
- Form and use trigonometric, polynomial, and other non-linear equations.
- Form and use systems of simultaneous equations, including three linear equations and three variables, and interpret the solutions in context.
- Manipulate complex numbers and present them graphically.

**Calculus**

- Identify discontinuities and limits of functions.
- Choose and apply a variety of differentiation, integration, and anti-differentiation techniques to functions and relations, using both analytical and numerical methods.
- Form differential equations and interpret the solutions.

### Statistics

**Statistical investigation**

Carry out investigations of phenomena, using the statistical enquiry cycle:

- conducting experiments using experimental design principles, conducting surveys, and using existing data sets
- finding, using, and assessing appropriate models (including linear regression for bivariate data and additive models for time-series data), seeking explanations, and making predictions
- using informed contextual knowledge, exploratory data analysis, and statistical inference
- communicating findings and evaluating all stages of the cycle.

Make inferences from surveys and experiments:

- determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem
- using methods such as resampling or randomisation to assess the strength of evidence.

**Statistical literacy**

Evaluate a wide range of statistically based reports, including surveys and polls, experiments, and observational studies:

- critiquing causal-relationship claims
- interpreting margins of error.

**Probability**

Investigate situations that involve elements of chance:

- calculating probabilities of independent, combined, and conditional events
- calculating and interpreting expected values and standard deviations of discrete random variables
- applying distributions such as the Poisson, binomial, and normal.

Published on: 03 Apr 2014