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# Mathematics and statistics

## Achievement objectives  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:

• 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. 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:

• 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. 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. 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

• 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. 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. 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

• 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.

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:

### 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.

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:

### 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