Year 9
Mathematics
ASMATY9
Year 9 Mathematics Achievement Standard
20 statements aligned to content descriptors and elaborations
Achievement standard (as ACARA publishes it)
Achievement standard
Click any sentence to jump to its statement and linked content descriptors.
By the end of Year 9, students recognise and use rational and irrational numbers to solve problems. They extend and apply the exponent laws with positive integers to variables. Students expand binomial products, and factorise monic quadratic expressions. They find the distance between 2 points on the Cartesian plane, and the gradient and midpoint of a line segment. Students use mathematical modelling to solve problems involving change in financial and other applied contexts, choosing to use linear and quadratic functions. They graph quadratic functions and solve monic quadratic equations with integer roots algebraically. Students describe the effects of variation of parameters on functions and relations, using digital tools, and make connections between their graphical and algebraic representations.
They apply formulas to solve problems involving the surface area and volume of right prisms and cylinders. Students solve problems involving ratio, similarity and scale in two-dimensional situations. They determine percentage errors in measurements. Students apply Pythagoras’ theorem and use trigonometric ratios to solve problems involving right-angled triangles. They use mathematical modelling to solve practical problems involving direct proportion, ratio and scale, evaluating the model and communicating their methods and findings. Students express small and large numbers in scientific notation. They apply the enlargement transformation to images of shapes and objects, and interpret results. Students design, use and test algorithms based on geometric constructions or theorems.
They compare and analyse the distributions of multiple numerical data sets, choose representations, describe features of these data sets using summary statistics and the shape of distributions, and consider the effect of outliers. Students explain how sampling techniques and representation can be used to support or question conclusions or to promote a point of view. They determine sets of outcomes for compound events and represent these in various ways. Students assign probabilities to the outcomes of compound events. They design and conduct experiments or simulations for combined events using digital tools.
The 20 achievement standard statements
Open a statement to see the content descriptors ACARA links to it, then open a descriptor's elaborations.
By the end of Year 9, students recognise and use rational and irrational numbers to solve problems.
Content descriptorAC9M9N01 Number
recognise that the real number system includes the rational numbers and the irrational numbers, and solve problems involving real numbers using digital tools
Elaborations (5)
- Elaboration investigating the real number system by representing the relationships between irrationals, rationals, integers and natural numbers and discussing the difference between exact representations and approximate decimal representations of irrational numbers
- Elaboration using a real number line to indicate the solution interval for inequalities of the form ax+b<c ; for example, 2x+7<0, or of the form ax+b>c; for example, 1.2x-5.4>10.8
- Elaboration using positive and negative rational numbers to solve problems; for example, for financial planning such as budgeting
- Elaboration solving problems involving the substitution of real numbers into formulas, understanding that solutions can be represented in exact form or as a decimal approximation, such as calculating the area of a circle using the formula A=\mathrm{π}r^2 and specifying the answer in terms of π as an exact real number; for example, the circumference of a circle with diameter 5 units is 5\mathrm\pi units, and the exact area is \mathrm\pi(5/2)^2=25/4\mathrm\pi square units which rounds to 19.63 square units, correct to 2 decimal places
- Elaboration investigating the position of rational and irrational numbers on the real number line, using geometric constructions to locate rational numbers and square roots on a number line; for example, \sqrt2 is located at the intersection of an arc and the number line, where the radius of the arc is the length of the diagonal of a one-unit square
They extend and apply the exponent laws with positive integers to variables.
Content descriptorAC9M9A01 Algebra
apply the exponent laws to numerical expressions with integer exponents and extend to variables
Elaborations (7)
- Elaboration representing decimals in exponential form; for example, 0.475 can be represented as 0.475\;=\;4/10+7/100+5/1000\;=\;4\times10^{-1}+7\times10^{-2}+5\times10^{-3} and 0.00023 as 23\times10^{-5}
- Elaboration simplifying and evaluating numerical expressions, involving both positive and negative integer exponents, explaining why; for example, 5^{-3}=1/5^3=(1/5)^3=1/125 and connecting terms of the sequence 125, 25, 5, 1, 1/5, 1/25, 1/125… to terms of the sequence 5^3, 5^2, 5^1, 5^0,5^{-1},5^{-2},5^{-3}...
- Elaboration relating the computation of numerical expressions involving exponents to the exponent laws and the definition of an exponent; for example, 2^3\div2^5\;=\;2^{-2}\;=\;1/2^2=1/4 and (3\times5)^2\;=\;3^2\times5^2\;=\;9\times25\;=\;225
- Elaboration recognising exponents in algebraic expressions and applying the relevant exponent laws and corresponding conventions; for example, for any non-zero natural number a, a^0\;=\;1, x^1\;=\;x, r^2\;=\;r\times r, h^3\;=\;h\times h\times h, y^4\;=\;y\times y\times y\times y, and 1/w \times 1/w=1/w^2 = w^{-2}
- Elaboration relating simplification of expressions from first principles and counting to the use of exponent laws; for example, (a^2)^3\;=\;(a\times a)\;\times\;(a\times a)\;\times\;(a\times a)\;=\;a\times a\times a\times a\times a\times a\;=\;a^6; b^2\times b^3\;=\;(b\times b)\times(b\times b\times b)\;=\;b\times b\times b\times b\times b\;=\;b^5; y^4/y^2\;=\;y\times y\times y\times y/y\times y\;=\;y^2/1\;=\;y^2 and (5a)^2\;=\;(5\times a)\times(5\times a)\;=\;5\times5\times a\times a\;=\;25\times a^2\;=\;25a^2
- Elaboration applying the exponent laws to simplifying expressions involving products, quotients, and powers of constants and variables; for example, (2xy)^3/xy^4\;=\;8x^3y^3/xy^4\;=\;8x^2y^{-1}
- Elaboration relating the prefixes for SI units from pico- (trillionth) to tera- (trillion) to the corresponding powers of 10; for example, one pico-gram = 10^{-12} gram and one terabyte = 10^{12} bytes
Students expand binomial products, and factorise monic quadratic expressions.
Content descriptorAC9M9A02 Algebra
simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions
Elaborations (3)
- Elaboration expanding combinations of binomials such as (x+7)(x+8),\;(x+7)(x-8),\;(x-7)(x+8),\;(x-7)(x-8) to identify expansion and factorisation patterns related to (x+a)(x+b)\;=\;x^2+(a+b)x+ab, where a and b are integers
- Elaboration using manipulatives such as algebra tiles or area models to expand or factorise algebraic expressions with readily identifiable binomial factors; for example, (x+1)(x+3)\;=\;x^2+4x+3,\;(x-5)^2\;=\;x^2-10x+25 or (x-3)^2+4\;=\;x^2-6x+9+4\;=\;x^2-6x+13
- Elaboration recognising the relationship between expansion and factorisation, and using digital tools to systematically explore the factorisation of x^2+mx+n where m and n are integers
They find the distance between 2 points on the Cartesian plane, and the gradient and midpoint of a line segment.
Content descriptorAC9M9A03 Algebra
find the gradient of a line segment, the midpoint of the line interval and the distance between 2 distinct points on the Cartesian plane
Elaborations (6)
- Elaboration recognising that the gradient of a line is calculated using the gradient of a line segment on that line and is independent of which 2 distinct points on the line are used for this calculation
- Elaboration using digital tools and transformations to illustrate that parallel lines in the Cartesian plane have the same gradient and that the relationship between the gradients of pairs of perpendicular lines is that their product is (-1)
- Elaboration using Pythagoras’ theorem to establish the distance between 2 points in the Cartesian plane and applying this using horizontal and vertical distances and coordinates
- Elaboration investigating graphical and algebraic techniques for finding the midpoint and gradient of the line segment between 2 points
- Elaboration using dynamic graphing software and superimposed images; for example, playground equipment, ramps and escalators, to investigate gradients in context and their relationship to rule of a linear function, and interpret gradient as a constant rate of change in linear modelling contexts
- Elaboration investigating how coordinate geometry and aspects of linear algebra play a fundamental role in machine learning and predictive algorithms; for example, object detection and navigation by autonomous vehicles
Students use mathematical modelling to solve problems involving change in financial and other applied contexts, choosing to use linear and quadratic functions.
Content descriptorAC9M9A02 Algebra
simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions
Elaborations (3)
- Elaboration expanding combinations of binomials such as (x+7)(x+8),\;(x+7)(x-8),\;(x-7)(x+8),\;(x-7)(x-8) to identify expansion and factorisation patterns related to (x+a)(x+b)\;=\;x^2+(a+b)x+ab, where a and b are integers
- Elaboration using manipulatives such as algebra tiles or area models to expand or factorise algebraic expressions with readily identifiable binomial factors; for example, (x+1)(x+3)\;=\;x^2+4x+3,\;(x-5)^2\;=\;x^2-10x+25 or (x-3)^2+4\;=\;x^2-6x+9+4\;=\;x^2-6x+13
- Elaboration recognising the relationship between expansion and factorisation, and using digital tools to systematically explore the factorisation of x^2+mx+n where m and n are integers
Content descriptorAC9M9A04 Algebra
identify and graph quadratic functions, solve quadratic equations graphically and numerically, and solve monic quadratic equations with integer roots algebraically, using graphing software and digital tools as appropriate
Elaborations (7)
- Elaboration recognising that in a table of values, if the second difference between consecutive values of the dependent variable is constant, then it is a quadratic
- Elaboration graphing quadratic functions using digital tools and comparing what is the same and what is different between these different functions and their respective graphs; interpreting features of the graphs such as symmetry, turning point, maximum and minimum values, and determining when values of the quadratic function lie within a given range
- Elaboration solving quadratic equations algebraically and comparing these to graphical solutions
- Elaboration using graphs to determine the solutions of quadratic equations; recognising that the roots of a quadratic function correspond to the x-intercepts of its graph and that if the graph has no x-intercepts, then the corresponding equation has no real solutions
- Elaboration relating horizontal axis intercepts of the graph of a quadratic function to the factorised form of its rule using the null factor law; for example, the graph of the function y=x^2-5x+6 can be represented as y=(x-2)(x-3) with x-axis intercepts where either (x-2)=0 or (x-3)=0
- Elaboration recognising that the equation x^2=a, where a>0, has 2 solutions, x=\sqrt a and x=-\sqrt a; for example, if x^2=39 then x=\sqrt{39}=6.245 correct to 3 decimal places, or x=-\sqrt{39}=-6.245 correct to 3 decimal places, and representing these graphically
- Elaboration graphing percentages of illumination of moon phases in relation to First Nations Australians’ understandings that describe the different phases of the moon
Content descriptorAC9M9A05 Algebra
use mathematical modelling to solve applied problems involving change including financial contexts; formulate problems, choosing to use either linear or quadratic functions; interpret solutions in terms of the situation; evaluate the model and report methods and findings
Elaborations (6)
- Elaboration modelling practical contexts using linear functions such as cooking times that include resting or cooling times, or water leakage from water tanks, using tables and graphs or digital tools and algebraically
- Elaboration modelling measurement situations and determining the perimeter and areas of rectangles where the length, l, of the rectangle is a linear function of its width, w; for example, l=w,\;l=w+5,\;l=3w,\;l=2w+7
- Elaboration modelling practical contexts using simple quadratic functions, tables and graphs (hand drawn or using digital tools) and algebraically, interpreting features of the graphs such as the turning point and intercepts in context; for example, area, paths of projectiles, parabolic mirrors, satellite dishes
- Elaboration modelling and solving problems involving financial contexts using linear functions; for example, combinations of purchases of different items when they have a set amount of money to spend, profit/loss situations and trade quotes involving call out fees
- Elaboration modelling situations involving change; for example, change in daily temperature during the ski season, fluctuation of speed above and below the speed limit, acceleration and deceleration of a car coming to and moving off from a set of traffic lights
- Elaboration modelling the hunting techniques of First Nations Australians using quadratic functions and exploring the effect of increasing the number of hunters to catch more prey
They graph quadratic functions and solve monic quadratic equations with integer roots algebraically.
Content descriptorAC9M9A02 Algebra
simplify algebraic expressions, expand binomial products and factorise monic quadratic expressions
Elaborations (3)
- Elaboration expanding combinations of binomials such as (x+7)(x+8),\;(x+7)(x-8),\;(x-7)(x+8),\;(x-7)(x-8) to identify expansion and factorisation patterns related to (x+a)(x+b)\;=\;x^2+(a+b)x+ab, where a and b are integers
- Elaboration using manipulatives such as algebra tiles or area models to expand or factorise algebraic expressions with readily identifiable binomial factors; for example, (x+1)(x+3)\;=\;x^2+4x+3,\;(x-5)^2\;=\;x^2-10x+25 or (x-3)^2+4\;=\;x^2-6x+9+4\;=\;x^2-6x+13
- Elaboration recognising the relationship between expansion and factorisation, and using digital tools to systematically explore the factorisation of x^2+mx+n where m and n are integers
Content descriptorAC9M9A04 Algebra
identify and graph quadratic functions, solve quadratic equations graphically and numerically, and solve monic quadratic equations with integer roots algebraically, using graphing software and digital tools as appropriate
Elaborations (7)
- Elaboration recognising that in a table of values, if the second difference between consecutive values of the dependent variable is constant, then it is a quadratic
- Elaboration graphing quadratic functions using digital tools and comparing what is the same and what is different between these different functions and their respective graphs; interpreting features of the graphs such as symmetry, turning point, maximum and minimum values, and determining when values of the quadratic function lie within a given range
- Elaboration solving quadratic equations algebraically and comparing these to graphical solutions
- Elaboration using graphs to determine the solutions of quadratic equations; recognising that the roots of a quadratic function correspond to the x-intercepts of its graph and that if the graph has no x-intercepts, then the corresponding equation has no real solutions
- Elaboration relating horizontal axis intercepts of the graph of a quadratic function to the factorised form of its rule using the null factor law; for example, the graph of the function y=x^2-5x+6 can be represented as y=(x-2)(x-3) with x-axis intercepts where either (x-2)=0 or (x-3)=0
- Elaboration recognising that the equation x^2=a, where a>0, has 2 solutions, x=\sqrt a and x=-\sqrt a; for example, if x^2=39 then x=\sqrt{39}=6.245 correct to 3 decimal places, or x=-\sqrt{39}=-6.245 correct to 3 decimal places, and representing these graphically
- Elaboration graphing percentages of illumination of moon phases in relation to First Nations Australians’ understandings that describe the different phases of the moon
Students describe the effects of variation of parameters on functions and relations, using digital tools, and make connections between their graphical and algebraic representations.
Content descriptorAC9M9A06 Algebra
experiment with the effects of the variation of parameters on graphs of related functions, using digital tools, making connections between graphical and algebraic representations, and generalising emerging patterns
Elaborations (4)
- Elaboration investigating transformations of the graph of y=x to the graph of y=ax+b by systematic variation of a and b and interpretating the effects of these transformations using digital tools; for example, y=x\rightarrow y=2x (vertical enlargement as a>1) \rightarrow y=2x-1 (vertical translation) and y=x\rightarrow y=1/2x (vertical compression as a<1) \rightarrow y=-1/2x (reflection in the horizontal axis) \rightarrow y=-1/2x+3 (vertical translation)
- Elaboration investigating transformations of the parabola y=x^2 in the Cartesian plane using digital tools to determine the relationship between graphical and algebraic representations of quadratic functions, including the completed square form; for example, y=x^2\rightarrow y=1/3x^2 (vertical compression as a<1) \rightarrow y=1/3(x-5)^2 (horizontal translation) \rightarrow y=1/3(x-5)^2+7 (vertical translation) or y=x^2\rightarrow y=2x^2 (vertical enlargement as a>1) \rightarrow y=-2x^2 (reflection in the horizontal axis) \rightarrow y=-2(x+6)^2 (horizontal translation) \rightarrow y=-2(x+6)^2+10 (vertical translation)
- Elaboration experimenting with digital tools by applying transformations to the graphs of functions, such as reciprocal y=\frac1x, square root y=\sqrt x, cube y=x^3 and exponential functions, y=2^x, y=(1/2)^x, identifying patterns
- Elaboration investigating how experimenting with the effects of the variation of parameters of related functions can provide artificial intelligence researchers insights into the predictive behaviour of artificial intelligence models
They apply formulas to solve problems involving the surface area and volume of right prisms and cylinders.
Content descriptorAC9M9M01 Measurement
solve problems involving the volume and surface area of right prisms and cylinders using appropriate units
Elaborations (4)
- Elaboration analysing nets of objects to generate short cuts and establish formulas for surface area
- Elaboration determining the amount of material needed to make can-coolers for a class fundraising project and working out the most cost-efficient way to cut out the pieces
- Elaboration finding different prisms that have the same volume but different surface areas, making conjectures as to what type of prism would have the smallest or largest surface area
- Elaboration investigating objects and technologies of First Nations Australians, analysing and connecting surface area and volume, and exploring their relationship to their capacity
Students solve problems involving ratio, similarity and scale in two-dimensional situations.
Content descriptorAC9M9M03 Measurement
solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles
Elaborations (7)
- Elaboration investigating the applications of Pythagoras’ theorem in authentic problems, including applying Pythagoras’ theorem and trigonometry to problems in surveying and design
- Elaboration applying the formula for calculation of distances between points on the Cartesian plane from their coordinates, emphasising the connection to vertical and horizontal displacements between the points
- Elaboration understanding the relationship between the corresponding sides of similar right-angled triangles and establishing the relationship between areas of similar figures and the ratio of corresponding sides, the scale factor
- Elaboration using images of proportional relationships to estimate actual measurements; for example, taking a photograph of a person standing in front of a tree and using the image and scale to estimate the height of the tree, discussing the findings and ways to improve the estimates
- Elaboration investigating theorems and conjectures involving triangles; for example, the triangle inequality, and generalising links between the Pythagorean rule for right-angled triangles, and related inequalities for acute and obtuse triangles; determining the minimal sets of information for a triangle from which other measures can all be determined
- Elaboration using knowledge of similar triangles, Pythagoras’ theorem, rates and algebra to design and construct a Biltmore stick used to measure the diameter and height of a tree, and calculating the density and dry mass to predict how much paper could be manufactured from the tree
- Elaboration investigating how autonomous vehicles solve spatial problems using algorithms based on geometric properties relating to angles, distances and scale
They determine percentage errors in measurements.
Content descriptorAC9M9M04 Measurement
calculate and interpret absolute, relative and percentage errors in measurements, recognising that all measurements are estimates
Elaborations (5)
- Elaboration investigating error as a percentage of the exact value; for example, comparing an estimation of the number of people expected to come to an event by subtracting the actual number that turned up to give an error, then converting this into a percentage error
- Elaboration using absolute value in a percentage error formula; considering when they would use the absolute value and when they would not, depending upon the context
- Elaboration calculating the percentage errors in expected budgets to actual expenditure
- Elaboration estimating the accuracy of measurements in practical contexts and giving suitable lower and upper bounds for measurement values
- Elaboration investigating how calculating and interpreting absolute, relative and percentage errors in measurements relates to artificial intelligence systems such as regression models, estimating uncertainty and recommendation systems
Students apply Pythagoras’ theorem and use trigonometric ratios to solve problems involving right-angled triangles.
Content descriptorAC9M9M03 Measurement
solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles
Elaborations (7)
- Elaboration investigating the applications of Pythagoras’ theorem in authentic problems, including applying Pythagoras’ theorem and trigonometry to problems in surveying and design
- Elaboration applying the formula for calculation of distances between points on the Cartesian plane from their coordinates, emphasising the connection to vertical and horizontal displacements between the points
- Elaboration understanding the relationship between the corresponding sides of similar right-angled triangles and establishing the relationship between areas of similar figures and the ratio of corresponding sides, the scale factor
- Elaboration using images of proportional relationships to estimate actual measurements; for example, taking a photograph of a person standing in front of a tree and using the image and scale to estimate the height of the tree, discussing the findings and ways to improve the estimates
- Elaboration investigating theorems and conjectures involving triangles; for example, the triangle inequality, and generalising links between the Pythagorean rule for right-angled triangles, and related inequalities for acute and obtuse triangles; determining the minimal sets of information for a triangle from which other measures can all be determined
- Elaboration using knowledge of similar triangles, Pythagoras’ theorem, rates and algebra to design and construct a Biltmore stick used to measure the diameter and height of a tree, and calculating the density and dry mass to predict how much paper could be manufactured from the tree
- Elaboration investigating how autonomous vehicles solve spatial problems using algorithms based on geometric properties relating to angles, distances and scale
Content descriptorAC9M9SP01 Space
recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity
Elaborations (4)
- Elaboration understanding the terms “base”, “altitude”, “hypotenuse”, and “adjacent” and “opposite” sides to an angle, in a right-angled triangle, and identifying these for a given right-angled triangle
- Elaboration investigating patterns to reason about nested similar triangles that are aligned on a coordinate plane, connecting ideas of parallel sides and identifying the constancy of ratios of corresponding sides for a given angle
- Elaboration establishing an understanding that the sine of an angle can be considered as the length of the altitude of a right-angled triangle with a hypotenuse of length one unit and similarly the cosine as the length of the base of the same triangle, and relating this to enlargement and similar triangles
- Elaboration relating the tangent of an angle to the altitude and base of nested similar right-angled triangles, and connecting the tangent of the angle at which the graph of a straight line meets the positive direction of the horizontal coordinate axis to the gradient of the straight line
They use mathematical modelling to solve practical problems involving direct proportion, ratio and scale, evaluating the model and communicating their methods and findings.
Content descriptorAC9M9M03 Measurement
solve spatial problems, applying angle properties, scale, similarity, Pythagoras’ theorem and trigonometry in right-angled triangles
Elaborations (7)
- Elaboration investigating the applications of Pythagoras’ theorem in authentic problems, including applying Pythagoras’ theorem and trigonometry to problems in surveying and design
- Elaboration applying the formula for calculation of distances between points on the Cartesian plane from their coordinates, emphasising the connection to vertical and horizontal displacements between the points
- Elaboration understanding the relationship between the corresponding sides of similar right-angled triangles and establishing the relationship between areas of similar figures and the ratio of corresponding sides, the scale factor
- Elaboration using images of proportional relationships to estimate actual measurements; for example, taking a photograph of a person standing in front of a tree and using the image and scale to estimate the height of the tree, discussing the findings and ways to improve the estimates
- Elaboration investigating theorems and conjectures involving triangles; for example, the triangle inequality, and generalising links between the Pythagorean rule for right-angled triangles, and related inequalities for acute and obtuse triangles; determining the minimal sets of information for a triangle from which other measures can all be determined
- Elaboration using knowledge of similar triangles, Pythagoras’ theorem, rates and algebra to design and construct a Biltmore stick used to measure the diameter and height of a tree, and calculating the density and dry mass to predict how much paper could be manufactured from the tree
- Elaboration investigating how autonomous vehicles solve spatial problems using algorithms based on geometric properties relating to angles, distances and scale
Content descriptorAC9M9M05 Measurement
use mathematical modelling to solve practical problems involving direct proportion, rates, ratio and scale, including financial contexts; formulate the problems and interpret solutions in terms of the situation; evaluate the model and report methods and findings
Elaborations (5)
- Elaboration modelling situations involving direct proportion such as pro rata pay rates, exchange rates, multiple quotes for a job, conversion between scales or other appropriate science contexts; for example, Hooke’s law and other science contexts involving wave lengths and frequencies
- Elaboration modelling situations that impact on image editing used in social media and how proportion may not be maintained and can result in distorted images
- Elaboration modelling situations involving compliance with building and construction standards in design and construction, such as the rise and tread of staircases, and vertical and horizontal components of escalators
- Elaboration modelling situations involving the application of rates in practical contexts; for example, density, birth, flow or heartbeats
- Elaboration exploring fire techniques in land management practices used by First Nations Australians that use proportion relationships, including the rate of fire spread in different fuel types to wind speed, temperature and relative humidity
Content descriptorAC9M9SP01 Space
recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity
Elaborations (4)
- Elaboration understanding the terms “base”, “altitude”, “hypotenuse”, and “adjacent” and “opposite” sides to an angle, in a right-angled triangle, and identifying these for a given right-angled triangle
- Elaboration investigating patterns to reason about nested similar triangles that are aligned on a coordinate plane, connecting ideas of parallel sides and identifying the constancy of ratios of corresponding sides for a given angle
- Elaboration establishing an understanding that the sine of an angle can be considered as the length of the altitude of a right-angled triangle with a hypotenuse of length one unit and similarly the cosine as the length of the base of the same triangle, and relating this to enlargement and similar triangles
- Elaboration relating the tangent of an angle to the altitude and base of nested similar right-angled triangles, and connecting the tangent of the angle at which the graph of a straight line meets the positive direction of the horizontal coordinate axis to the gradient of the straight line
Content descriptorAC9M9SP02 Space
apply the enlargement transformation to shapes and objects using dynamic geometry software as appropriate; identify and explain aspects that remain the same and those that change
Elaborations (3)
- Elaboration comparing the ratio of lengths of corresponding sides of similar triangles and angles
- Elaboration using the properties of similarity to solve problems involving enlargement
- Elaboration investigating and generalising patterns in length, angle, area and volume when side lengths of shapes and objects are enlarged or dilated by whole and rational numbers; for example, comparing an enlargement of a square and a cube of side length 2 units by a factor of 3 increases the area of the square, 2^2, to (3\times2)^2 = 9\times 2^2=9 times the original area and the volume of the cube, 2^3, to (3\times2)^3=27\times 2^3=27 times the volume
Students express small and large numbers in scientific notation.
Content descriptorAC9M9M02 Measurement
solve problems involving very small and very large measurements, time scales and intervals expressed in scientific notation
Elaborations (3)
- Elaboration representing very large and small real numbers in scientific notation, converting real numbers expressed in scientific notation into decimal form; for example, the approximate geological age of the earth is 4.6\times 10^9 years, the mass of a sugar molecule is 5.68\times 10^{-21} grams
- Elaboration using knowledge of place value and applying exponent laws to operate with numbers expressed in scientific notation in applied contexts; for example, performing calculations involving extremely small numbers in scientific and other contexts
- Elaboration examining the degree of accuracy that different measurement instruments provide in a science laboratory, such as a measuring cylinder compared with a pipette, and recording data values to the correct degree of accuracy using appropriate scientific notation
They apply the enlargement transformation to images of shapes and objects, and interpret results.
Content descriptorAC9M9SP02 Space
apply the enlargement transformation to shapes and objects using dynamic geometry software as appropriate; identify and explain aspects that remain the same and those that change
Elaborations (3)
- Elaboration comparing the ratio of lengths of corresponding sides of similar triangles and angles
- Elaboration using the properties of similarity to solve problems involving enlargement
- Elaboration investigating and generalising patterns in length, angle, area and volume when side lengths of shapes and objects are enlarged or dilated by whole and rational numbers; for example, comparing an enlargement of a square and a cube of side length 2 units by a factor of 3 increases the area of the square, 2^2, to (3\times2)^2 = 9\times 2^2=9 times the original area and the volume of the cube, 2^3, to (3\times2)^3=27\times 2^3=27 times the volume
Students design, use and test algorithms based on geometric constructions or theorems.
Content descriptorAC9M9SP03 Space
design, test and refine algorithms involving a sequence of steps and decisions based on geometric constructions and theorems; discuss and evaluate refinements
Elaborations (3)
- Elaboration creating an algorithm using pseudocode or flow charts to apply the triangle inequality, or an algorithm to generate Pythagorean triples
- Elaboration creating and testing algorithms designed to construct or bisect angles, using pseudocode or flow charts
- Elaboration developing an algorithm for an animation of a geometric construction, or a visual proof, evaluating the algorithm using test cases
They compare and analyse the distributions of multiple numerical data sets, choose representations, describe features of these data sets using summary statistics and the shape of distributions, and consider the effect of outliers.
Content descriptorAC9M9ST03 Statistics
represent the distribution of multiple data sets for numerical variables using comparative representations; compare data distributions with consideration of centre, spread and shape, and the effect of outliers on these measures
Elaborations (4)
- Elaboration describing the shape of the distribution of data using terms such as “positive skew”, “negative skew” and “symmetric” and “bi-modal”
- Elaboration using stem-and-leaf plots to compare 2 like sets of data such as the heights of girls and the heights of boys in a class
- Elaboration constructing grouped histograms that show trends in health issues such as lung cancer, leukemia, stroke and diabetes, and using the graph to justify, verify or invalidate claims
- Elaboration exploring comparative data presented in reports by National Indigenous Australians Agency in regard to “Closing the Gap”, discussing the comparative distributions within the context of the data; for example, comparative data presented in the “Closing the Gap – Prime Minister’s Report”
Content descriptorAC9M9ST04 Statistics
choose appropriate forms of display or visualisation for a given type of data; justify selections and interpret displays for a given context
Elaborations (5)
- Elaboration comparing data displays using mean, median and range to describe and interpret numerical data sets in terms of centre and spread using histograms, dot plots, or stem-and-leaf plots
- Elaboration choosing the type of representations based on the data type: categorical (nominal or ordinal) or numerical (discrete or continuous)
- Elaboration using different visualisations of data, including non-standard representations such as infographics, and discussing their purpose, intended audience; evaluating how well they communicate responses to statistical questions of interest
- Elaboration comparing and interpreting stacked bar charts, area charts and line graphs, discussing how they represent larger categories that can be subdivided into smaller categories and how information that can be obtained from these displays can be used for comparison
- Elaboration using digital tools, including generative artificial intelligence, to generate different data displays and visualisations using existing data sets, and discussing which form is more appropriate for the given context
Content descriptorAC9M9ST05 Statistics
plan and conduct statistical investigations involving the collection and analysis of different kinds of data; report findings and discuss the strength of evidence to support any conclusions
Elaborations (4)
- Elaboration planning and conducting an investigation using questions together with analysis of secondary data set collected from online data bases such as the Australian Bureau of Statistics
- Elaboration planning and conducting an investigation relating to consumer spending habits; modelling market research on what teenagers are prepared to spend on technology compared to clothing, with consideration of sample techniques and potential sources of bias
- Elaboration investigating where would be the best location for a tropical fruit plantation by conducting a statistical investigation comparing different variables such as the annual rainfall in various parts of Australia, Indonesia, New Guinea and Malaysia, land prices and associated farming costs
- Elaboration posing statistical questions, collecting, representing and interpreting data from different sources in relation to reconciliation, considering the relationships between variables
Students explain how sampling techniques and representation can be used to support or question conclusions or to promote a point of view.
Content descriptorAC9M9ST01 Statistics
analyse reports of surveys in digital media and elsewhere for information on how data was obtained to estimate population means and medians
Elaborations (5)
- Elaboration investigating and evaluating statistical reports in the media and other places by linking claims to displays, statistics and representative data
- Elaboration investigating the use of statistics in reports regarding the growth of Australia's trade with other countries of the Asia region
- Elaboration exploring how natural language processing (NLP) tools can be used to help analyse large volumes of survey reports in digital media, expediting various processes, and discussing the benefits or limitations of using these tools
- Elaboration 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, Singapore, Japan, South Korea and Timor-Leste
- Elaboration analysing reports of public opinion surveys on environmental issues, such as land clearing, wind farms or single use plastics; discussing methods of data collection and the reasonableness of any inferences made
Content descriptorAC9M9ST02 Statistics
analyse how different sampling methods can affect the results of surveys and how choice of representation can be used to support a particular point of view
Elaborations (3)
- Elaboration investigating and analysing different visualisations of data such as infographics found in the media and commenting on the strengths, weaknesses and possible biases of particular examples
- Elaboration discussing the impact of decreased landline usage or an increased aversion to answering calls from unknown numbers on survey data
- Elaboration exploring potential cultural bias relating to First Nations Australians by critically analysing sampling techniques in statistical reports
They determine sets of outcomes for compound events and represent these in various ways.
Content descriptorAC9M9P01 Probability
list all outcomes for compound events both with and without replacement, using lists, tree diagrams, tables or arrays; assign probabilities to outcomes
Elaborations (4)
- Elaboration discussing two-step chance experiments, such as the game of Heads and Tails, describing the different outcomes and their related probabilities
- Elaboration using systematic methods such as lists or arrays to record outcomes and assign probabilities, such as drawing the names of students from a bag to appoint 2 team leaders
- Elaboration using a tree diagram to represent a three-stage event and assigning probabilities to these events; for example, selecting 3 cards from a deck, assigning the probability of drawing an ace, then a king, then a queen of the same suit, with and without replacing the cards after every draw
- Elaboration assigning probabilities to compound events involving the random selection of people from a given population; for example, selecting 2 names at random from all of the students at a high school and assigning the probability that they are both in Year 9
Students assign probabilities to the outcomes of compound events.
Content descriptorAC9M9P01 Probability
list all outcomes for compound events both with and without replacement, using lists, tree diagrams, tables or arrays; assign probabilities to outcomes
Elaborations (4)
- Elaboration discussing two-step chance experiments, such as the game of Heads and Tails, describing the different outcomes and their related probabilities
- Elaboration using systematic methods such as lists or arrays to record outcomes and assign probabilities, such as drawing the names of students from a bag to appoint 2 team leaders
- Elaboration using a tree diagram to represent a three-stage event and assigning probabilities to these events; for example, selecting 3 cards from a deck, assigning the probability of drawing an ace, then a king, then a queen of the same suit, with and without replacing the cards after every draw
- Elaboration assigning probabilities to compound events involving the random selection of people from a given population; for example, selecting 2 names at random from all of the students at a high school and assigning the probability that they are both in Year 9
Content descriptorAC9M9P02 Probability
calculate relative frequencies from given or collected data to estimate probabilities of events involving “and”, inclusive “or” and exclusive “or”
Elaborations (3)
- Elaboration understanding that relative frequencies from large data sets or long-run experiments can provide reliable measures of probability and can be used to make predictions of decisions
- Elaboration using Venn diagrams or two-way tables to estimate frequencies of events involving “and”, “or” questions
- Elaboration designing, testing and refining an algorithm used to determine relative frequencies from a generated data set, to estimate different probabilities
They design and conduct experiments or simulations for combined events using digital tools.
Content descriptorAC9M9P01 Probability
list all outcomes for compound events both with and without replacement, using lists, tree diagrams, tables or arrays; assign probabilities to outcomes
Elaborations (4)
- Elaboration discussing two-step chance experiments, such as the game of Heads and Tails, describing the different outcomes and their related probabilities
- Elaboration using systematic methods such as lists or arrays to record outcomes and assign probabilities, such as drawing the names of students from a bag to appoint 2 team leaders
- Elaboration using a tree diagram to represent a three-stage event and assigning probabilities to these events; for example, selecting 3 cards from a deck, assigning the probability of drawing an ace, then a king, then a queen of the same suit, with and without replacing the cards after every draw
- Elaboration assigning probabilities to compound events involving the random selection of people from a given population; for example, selecting 2 names at random from all of the students at a high school and assigning the probability that they are both in Year 9
Content descriptorAC9M9P02 Probability
calculate relative frequencies from given or collected data to estimate probabilities of events involving “and”, inclusive “or” and exclusive “or”
Elaborations (3)
- Elaboration understanding that relative frequencies from large data sets or long-run experiments can provide reliable measures of probability and can be used to make predictions of decisions
- Elaboration using Venn diagrams or two-way tables to estimate frequencies of events involving “and”, “or” questions
- Elaboration designing, testing and refining an algorithm used to determine relative frequencies from a generated data set, to estimate different probabilities
Content descriptorAC9M9P03 Probability
design and conduct repeated chance experiments and simulations, using digital tools to compare probabilities of simple events to related compound events, and describe results
Elaborations (4)
- Elaboration using digital tools to conduct probability simulations that demonstrate the relationship between the probability of compound events and the individual probabilities
- Elaboration comparing experiments which differ only by being undertaken with replacement or without replacement
- Elaboration conducting two-step chance experiments using systematic methods to list outcomes of experiments and to list outcomes favourable to an event
- Elaboration using repeated trials of First Nations Australian children’s instructive games; for example, Gorri from all parts of Australia, to calculate the probabilities of winning and not winning
This achievement standard describes the expected learning for Year 9 Mathematics. Use it to plan teaching and assessment with the Australian Curriculum v9.
At a Glance
| Strand | Substrand | CDs | Elaborations |
|---|---|---|---|
| Algebra | — | 6 | 33 |
| Measurement | — | 5 | 24 |
| Number | — | 1 | 5 |
| Probability | — | 3 | 11 |
| Space | — | 3 | 10 |
| Statistics | — | 5 | 21 |
| Total | 23 | 104 | |
Frequently Asked Questions
What should students know by the end of Year 9 Mathematics?
By the end of Year 9, students recognise and use rational and irrational numbers to solve problems. They extend and apply the exponent laws with positive integers to variables. Students expand binomial products, and factorise monic quadratic expressions. Read the full standard
How many Content Descriptors support this standard?
23 Content Descriptors support this Achievement Standard (Algebra: 6, Measurement: 5, Number: 1, Probability: 3, Space: 3, Statistics: 5).
How does this compare to Year 8?
The Year 8 Mathematics standard (ASMATY8) covers the preceding curriculum level. Standards build progressively, with Year 9 expectations extending what was introduced in Year 8.
Is this from the latest Australian Curriculum?
Yes, this Achievement Standard is from the Australian Curriculum version 9.0 (AC v9), the most current version published by ACARA.