## Grade 9

## Mathematics Curriculum

**Year 9 Level Description**

The proficiency strands **understanding, fluency, problem-solving** and **reasoning** are an integral part of mathematics content across the three content strands: number and algebra, measurement and geometry, and statistics and probability. The proficiencies reinforce the significance of working mathematically within the content and describe how the content is explored or developed. They provide the language to build in the developmental aspects of the learning of mathematics. The achievement standards reflect the content and encompass the proficiencies.

At this year level:

**understanding**includes describing the relationship between graphs and equations, simplifying a range of algebraic expressions and explaining the use of relative frequencies to estimate probabilities and of the trigonometric ratios for right-angle triangles.**fluency**includes applying the index laws to expressions with integer indices, expressing numbers in scientific notation, listing outcomes for experiments, developing familiarity with calculations involving the Cartesian plane and calculating areas of shapes and surface areas of prisms.**problem-solving**includes formulating and modelling practical situations involving surface areas and volumes of right prisms, applying ratio and scale factors to similar figures, solving problems involving right-angle trigonometry and collecting data from secondary sources to investigate an issue.**reasoning**includes following mathematical arguments, evaluating media reports and using statistical knowledge to clarify situations, developing strategies in investigating similarity and sketching linear graphs.

## Number and Algebra

- Real Numbers
- Money and financial mathematics
- Patterns and Algebra
- Linear and non-linear relationships

Solve problems involving direct proportion. Explore the relationship between graphs and equations corresponding to simple rate problems (ACMNA208)

- identifying direct proportion in real-life contexts

Apply index laws to numerical expressions with integer indices (ACMNA209)

- simplifying and evaluating numerical expressions, using involving both positive and negative integer indices

Express numbers in scientific notation (ACMNA210)

- representing extremely large and small numbers in scientific notation, and numbers expressed in scientific notation as whole numbers or decimals.

Model and represent unit fractions including 1/2, 1/4, 1/3, 1/5 and their multiples to a complete whole (ACMNA058)

- partitioning areas, lengths and collections to create halves, thirds, quarters and fifths, such as folding the same sized sheets of paper to illustrate different unit fractions and comparing the number of parts with their sizes.
- locating unit fractions on a number line
- recognising that in English the term ‘one third’ is used (order: numerator, denominator) but that in other languages this concept may be expressed as ‘three parts, one of them’ (order: denominator, numerator) for example Japanese.

Extend and apply the index laws to variables, using positive integer indices and the zero index (ACMNA212)

- understanding that index laws apply to variables as well as numbers.

Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)

- understanding that the distributive law can be applied to algebraic expressions as well as numbers.
- understanding the relationship between expansion and factorization and identifying algebraic factors in algebraic expressions.

Find the distance between two points located on the Cartesian plane using a range of strategies, including graphing software (ACMNA214)

- investigating graphical and algebraic techniques for finding distance between two points
- using Pythagoras' theorem to calculate distance between two points.

Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (ACMNA294)

- investigating graphical and algebraic techniques for finding midpoint and gradient.
- recognising that the gradient of a line is the same as the gradient of any line segment on that line.

Sketch linear graphs using the coordinates of two points and solve linear equations (ACMNA215)

- determining linear rules from suitable diagrams, tables of values and graphs and describing them using both words and algebra.

Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations (ACMNA296)

graphing parabolas, and circles connecting x-intercepts of a graph to a related equation.

## Measurement & Geometry

- Using units of measurement
- Geometric reasoning
- Pythagoras and trigonometry

Calculate areas of composite shapes (ACMMG216)

- understanding that partitioning composite shapes into rectangles and triangles is a strategy for solving problems involving area.

Calculate the surface area and volume of cylinders and solve related problems (ACMMG217)

- analysing nets of cylinders to establish formulas for surface area.
- connecting the volume and capacity of a cylinder to solve authentic problems.

Solve problems involving the surface area and volume of right prisms (ACMMG218)

- solving practical problems involving surface area and volume of right prisms

Investigate very small and very large time scales and intervals (ACMMG219)

- investigating the usefulness of scientific notation in representing very large and very small numbers

Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar (ACMMG220)

- establishing the conditions for similarity of two triangles and comparing this to the conditions for congruence.
- using the properties of similarity and ratio, and correct mathematical notation and language, to solve problems involving enlargement (for example, scale diagrams).
- using the enlargement transformation to establish similarity, understanding that similarity and congruence help describe relationships between geometrical shapes and are important elements of reasoning and proof.

Solve problems using ratio and scale factors in similar figures (ACMMG221)

- establishing the relationship between areas of similar figures and the ratio of corresponding sides (scale factor)

Investigate Pythagoras’ Theorem and its application to solving simple problems involving right angled triangles (ACMMG222)

- understanding that Pythagoras' Theorem is a useful tool in determining unknown lengths in right-angled triangles and has widespread applications.
- recognising that right-angled triangle calculations may generate results that can be integers, fractions or irrational numbers.

Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles (ACMMG223)

- developing understanding of the relationship between the corresponding sides of similar right-angled triangles

Apply trigonometry to solve right-angled triangle problems (ACMMG224)

- understanding the terms adjacent and opposite sides in a right-angled triangle.
- selecting and accurately using the correct trigonometric ratio to find unknown sides (adjacent, opposite and hypotenuse) and angles in right-angled triangles.

## STATISTICS & PROBABILITY

- Chance
- Data representation and interpretation

List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays. Assign probabilities to outcomes and determine probabilities for events (ACMSP225)

- conducting two-step chance experiments.
- using systematic methods to list outcomes of experiments and to list outcomes favourable to an event.
- comparing experiments which differ only by being undertaken with replacement or without replacement.

Calculate relative frequencies from given or collected data to estimate probabilities of events involving 'and' or 'or' (ACMSP226)

- using Venn diagrams or two-way tables to calculate relative frequencies of events involving ‘and’, ‘or’ questions.
- using relative frequencies to find an estimate of probabilities of ‘and’, ‘or’ events.

Investigate reports of surveys in digital media and elsewhere for information on how data were obtained to estimate population means and medians (ACMSP227)

- investigating a range of data and its sources, for example the age of residents in Australia, Cambodia and Tonga; the number of subjects studied at school in a year by 14-year-old students in Australia, Japan and Timor-Leste.

Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly and from secondary sources (ACMSP228)

- comparing the annual rainfall in various parts of Australia, Pakistan, New Guinea and Malaysia.

Construct back-to-back stem-and-leaf plots and histograms and describe data, using terms including ‘skewed’, ‘symmetric’ and ‘bi modal’ (ACMSP282)

- using stem-and-leaf plots to compare two like sets of data such as the heights of girls and the heights of boys in a class.
- describing the shape of the distribution of data using terms such as ‘positive skew’, ‘negative skew’ and 'symmetric' and 'bi-modal'.

Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread (ACMSP283)

comparing means, medians and ranges of two sets of numerical data which have been displayed using histograms, dot plots, or stem and leaf plots.

## ACHIEVEMENT STANDARD

By the end of Year 9, students solve problems involving simple interest. They interpret ratio and scale factors in similar figures. Students explain similarity of triangles. They recognise the connections between similarity and the trigonometric ratios. Students compare techniques for collecting data from primary and secondary sources. They make sense of the position of the mean and median in skewed, symmetric and bi-modal displays to describe and interpret data.

Students apply the index laws to numbers and express numbers in scientific notation. They expand binomial expressions. They find the distance between two points on the Cartesian plane and the gradient and midpoint of a line segment. They sketch linear and non-linear relations. Students calculate areas of shapes and the volume and surface area of right prisms and cylinders. They use Pythagoras’ Theorem and trigonometry to find unknown sides of right-angled triangles. Students calculate relative frequencies to estimate probabilities, list outcomes for two-step experiments and assign probabilities for those outcomes. They construct histograms and back-to-back stem-and-leaf plots.