NSW Mathematics Extension 1 Stage 6 Syllabus 2017 PDF

NSW Syllabus for the Australian Curriculum Mathematics Extension 1 Stage 6 Syllabus  Year 11 Mathematics Extension 1 Year 11...

NSW Syllabus for the Australian Curriculum Mathematics Extension 1 Stage 6 Syllabus  Year 11 Mathematics Extension 1 Year 11 Course Content Year 11 Course Structure and Requirements The course is organised in topics, with the topics divided into subtopics. Mathematics Extension Topics Subtopics Functions ME-F1 Further Work with Functions Year 11 course ME-F2 Polynomials (60 hours) Trigonometric Functions ME-T1 Inverse Trigonometric Functions ME-T2 Further Trigonometric Identities Calculus ME-C1 Rates of Change Combinatorics ME-A1 Working with Combinatorics For the Year 11 course: The Mathematics Advanced Year 11 course should be taught prior to or concurrently with this course. Students should experience content in the course in familiar and routine situations as well as unfamiliar situations. Students should be provided with regular opportunities involving the integration of technology to enrich the learning experience. Mathematics Extension 1 Stage 6 Syllabus (2017) 29  Year 11 Topic: Functions Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › manipulates algebraic expressions and graphical functions to solve problems ME11-2 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Topic Focus The topic Functions involves the use of both algebraic and graphical conventions and terminology to describe, interpret and model relationships of and between changing quantities. This topic provides the means to more fully understand the behaviour of functions, extending to include inequalities, absolute values and inverse functions. A knowledge of functions enables students to discover connections between algebraic and graphical representations, to determine solutions of equations and to model theoretical or real-life situations involving algebra. The study of functions is important in developing students’ ability to find, recognise and use connections, to communicate concisely and precisely, to use algebraic techniques and manipulations to describe and solve problems, and to predict future outcomes in areas such as finance, economics and weather. Subtopics ME-F1 Further Work with Functions ME-F2 Polynomials Mathematics Extension 1 Stage 6 Syllabus (2017) 30  Year 11 Functions ME-F1 Further Work with Functions Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › manipulates algebraic expressions and graphical functions to solve problems ME11-2 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Subtopic Focus The principal focus of this subtopic is to further explore functions in a variety of contexts including: reciprocal and inverse functions, manipulating graphs of functions, and parametric representation of functions. The study of inequalities is an application of functions and enables students to express domains and ranges as inequalities. Students develop proficiency in methods to identify solutions to equations both algebraically and graphically. The study of inverse functions is important in higher Mathematics and the calculus of these is studied later in the course. The study of parameters sets foundations for later work on projectiles. Content F1.1: Graphical relationships Students: 1 examine the relationship between the graph of 𝑦 = 𝑓(𝑥) and the graph of 𝑦 = and hence 𝑓(𝑥) sketch the graphs (ACMSM099) examine the relationship between the graph of 𝑦 = 𝑓(𝑥) and the graphs of 𝑦 2 = 𝑓(𝑥) and 𝑦 = √𝑓(𝑥) and hence sketch the graphs examine the relationship between the graph of 𝑦 = 𝑓(𝑥) and the graphs of 𝑦 = |𝑓(𝑥)| and 𝑦 = 𝑓(|𝑥|) and hence sketch the graphs (ACMSM099) examine the relationship between the graphs of 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) and the graphs of 𝑦 = 𝑓(𝑥) + 𝑔(𝑥) and 𝑦 = 𝑓(𝑥)𝑔(𝑥) and hence sketch the graphs apply knowledge of graphical relationships to solve problems in practical and abstract contexts AAM F1.2: Inequalities Students: solve quadratic inequalities using both algebraic and graphical techniques solve inequalities involving rational expressions, including those with the unknown in the denominator solve absolute value inequalities of the form |𝑎𝑥 + 𝑏| ≥ 𝑘, |𝑎𝑥 + 𝑏| ≤ 𝑘, |𝑎𝑥 + 𝑏| < 𝑘 and |𝑎𝑥 + 𝑏| > 𝑘 Mathematics Extension 1 Stage 6 Syllabus (2017) 31  Year 11 F1.3: Inverse functions Students: define the inverse relation of a function 𝑦 = 𝑓(𝑥) to be the relation obtained by reversing all the ordered pairs of the function examine and use the reflection property of the graph of a function and the graph of its inverse (ACMSM096) – understand why the graph of the inverse relation is obtained by reflecting the graph of the function in the line 𝑦 = 𝑥 – using the fact that this reflection exchanges horizontal and vertical lines, recognise that the horizontal line test can be used to determine whether the inverse relation of a function is again a function write the rule or rules for the inverse relation by exchanging 𝑥 and 𝑦 in the function rules, including any restrictions, and solve for 𝑦, if possible when the inverse relation is a function, use the notation 𝑓 −1 (𝑥) and identify the relationships between the domains and ranges of 𝑓(𝑥) and 𝑓 −1 (𝑥) when the inverse relation is not a function, restrict the domain to obtain new functions that are one-to-one, and compare the effectiveness of different restrictions solve problems based on the relationship between a function and its inverse function using algebraic or graphical techniques AAM F1.4: Parametric form of a function or relation Students: understand the concept of parametric representation and examine lines, parabolas and circles expressed in parametric form – understand that linear and quadratic functions, and circles can be expressed in either parametric form or Cartesian form – convert linear and quadratic functions, and circles from parametric form to Cartesian form and vice versa – sketch linear and quadratic functions, and circles expressed in parametric form Mathematics Extension 1 Stage 6 Syllabus (2017) 32  Year 11 Functions ME-F2 Polynomials Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › manipulates algebraic expressions and graphical functions to solve problems ME11-2 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Subtopic Focus The principal focus of this subtopic is to explore the behaviour of polynomials algebraically, including the remainder and factor theorems, and sums and products of roots. Students develop knowledge, skills and understanding to manipulate, analyse and solve polynomial equations. Polynomials are of fundamental importance in algebra and have many applications in higher mathematics. They are also significant in many other fields of study, including the sciences, engineering, finance and economics. Content F2.1: Remainder and factor theorems Students: define a general polynomial in one variable, 𝑥, of degree 𝑛 with real coefficients to be the expression: 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 , where 𝑎𝑛 ≠ 0 – understand and use terminology relating to polynomials including degree, leading term, leading coefficient and constant term use division of polynomials to express 𝑃(𝑥) in the form 𝑃(𝑥) = 𝐴(𝑥). 𝑄(𝑥) + 𝑅(𝑥) where deg 𝑅(𝑥) < deg 𝐴(𝑥) and 𝐴(𝑥) is a linear or quadratic divisor, 𝑄(𝑥) the quotient and 𝑅(𝑥) the remainder – review the process of division with remainders for integers – describe the process of division using the terms: dividend, divisor, quotient, remainder prove and apply the factor theorem and the remainder theorem for polynomials and hence solve simple polynomial equations (ACMSM089, ACMSM091) Mathematics Extension 1 Stage 6 Syllabus (2017) 33  Year 11 F2.2: Sums and products of roots of polynomials Students: solve problems using the relationships between the roots and coefficients of quadratic, cubic and quartic equations AAM – consider quadratic, cubic and quartic equations, and derive formulae as appropriate for the sums and products of roots in terms of the coefficients determine the multiplicity of a root of a polynomial equation – prove that if a polynomial equation of the form 𝑃(𝑥) = 0 has a root of multiplicity 𝑟 > 1, then 𝑃′ (𝑥) = 0 has a root of multiplicity 𝑟 − 1 graph a variety of polynomials and investigate the link between the root of a polynomial equation and the zero on the graph of the related polynomial function – examine the sign change of the function and shape of the graph either side of roots of varying multiplicity Mathematics Extension 1 Stage 6 Syllabus (2017) 34  Year 11 Topic: Trigonometric Functions Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › applies concepts and techniques of inverse trigonometric functions and simplifying expressions involving compound angles in the solution of problems ME11-3 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Topic Focus The topic Trigonometric Functions involves the study of periodic functions in geometric, algebraic, numerical and graphical representations. It extends to exploration and understanding of inverse trigonometric functions over restricted domains and their behaviour in both algebraic and graphical form. A knowledge of trigonometric functions enables the solving of problems involving inverse trigonometric functions, and the modelling of the behaviour of naturally occurring periodic phenomena such as waves and signals to solve problems and to predict future outcomes. The study of the graphs of trigonometric functions is important in developing students’ understanding of the connections between algebraic and graphical representations and how this can be applied to solve problems from theoretical or real-life scenarios and situations. Subtopics ME-T1 Inverse Trigonometric Functions ME-T2 Further Trigonometric Identities Mathematics Extension 1 Stage 6 Syllabus (2017) 35  Year 11 Trigonometric Functions ME-T1 Inverse Trigonometric Functions Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › applies concepts and techniques of inverse trigonometric functions and simplifying expressions involving compound angles in the solution of problems ME11-3 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Subtopic Focus The principal focus of this subtopic is for students to determine and to work with the inverse trigonometric functions. Students explore inverse trigonometric functions which are important examples of inverse functions. They sketch the graphs of these functions and apply a range of properties to extend their knowledge and understanding of the connections between algebraic and geometrical representations of functions. This enables a deeper understanding of the nature of periodic functions, which are used as powerful modelling tools for any quantity that varies in a cyclical way. Content Students: define and use the inverse trigonometric functions (ACMSM119) – understand and use the notation arcsin 𝑥 and sin−1 𝑥 for the inverse function of sin 𝑥 when 𝜋 𝜋 − ≤ 𝑥 ≤ (and similarly for cos 𝑥 and tan 𝑥) and understand when each notation might be 2 2 appropriate to avoid confusion with the reciprocal functions 𝜋 𝜋 – use the convention of restricting the domain of sin 𝑥 to − ≤ 𝑥 ≤ , so the inverse function 2 2 exists. The inverse of this restricted sine function is defined by: 𝑦 = sin−1 𝑥, −1 ≤ 𝑥 ≤ 1 and 𝜋 𝜋 − ≤𝑦≤ 2 2 – use the convention of restricting the domain of cos 𝑥 to 0 ≤ 𝑥 ≤ 𝜋 , so the inverse function exists. The inverse of this restricted cosine function is defined by: 𝑦 = cos −1 𝑥, −1 ≤ 𝑥 ≤ 1 and 0 ≤ 𝑦 ≤ 𝜋 𝜋 𝜋 – use the convention of restricting the domain of tan 𝑥 to − < 𝑥 < , so the inverse function 2 2 exists. The inverse of this restricted tangent function is defined by: 𝑦 = tan−1 𝑥, 𝑥 is a real π π number and − < 𝑦 < 2 2 – classify inverse trigonometric functions as odd, even or neither odd nor even Mathematics Extension 1 Stage 6 Syllabus (2017) 36  Year 11 sketch graphs of the inverse trigonometric functions use the relationships sin (sin−1 𝑥) = 𝑥 and sin−1 (sin 𝑥) = 𝑥, cos (cos −1 𝑥) = 𝑥 and cos −1 (cos 𝑥) = 𝑥, and tan (tan−1 𝑥) = 𝑥 and tan−1 (tan 𝑥) = 𝑥 where appropriate, and state the values of 𝑥 for which these relationships are valid prove and use the properties: sin−1 (−𝑥) = − sin−1 𝑥, cos −1 (−𝑥) = 𝜋 − cos −1 𝑥, 𝜋 tan−1 (−𝑥) = − tan−1 𝑥 and cos −1 𝑥 + sin−1 𝑥 = 2 solve problems involving inverse trigonometric functions in a variety of abstract and practical situations AAM Mathematics Extension 1 Stage 6 Syllabus (2017) 37  Year 11 Trigonometric Functions ME-T2 Further Trigonometric Identities Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › applies concepts and techniques of inverse trigonometric functions and simplifying expressions involving compound angles in the solution of problems ME11-3 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Subtopic Focus The principal focus of this subtopic is for students to define and work with trigonometric identities to both prove results and manipulate expressions. Students develop knowledge of how to manipulate trigonometric expressions to solve equations and to prove results. Trigonometric expressions and equations provide a powerful tool for modelling quantities that vary in a cyclical way such as tides, seasons, demand for resources, and alternating current. The solution of trigonometric equations may require the use of trigonometric identities. Content Students: derive and use the sum and difference expansions for the trigonometric functions sin (𝐴 ± 𝐵), cos (𝐴 ± 𝐵) and tan (𝐴 ± 𝐵) (ACMSM044) – sin (𝐴 ± 𝐵) = sin 𝐴 cos 𝐵 ± cos 𝐴 sin 𝐵 – cos (𝐴 ± 𝐵) = cos 𝐴 cos 𝐵 ∓ sin 𝐴 sin 𝐵 tan 𝐴 ± tan 𝐵 – tan (𝐴 ± 𝐵) = 1 ∓ tan 𝐴 tan 𝐵 derive and use the double angle formulae for sin 2𝐴 , cos 2𝐴 and tan 2𝐴 (ACMSM044) – sin 2𝐴 = 2 sin 𝐴 cos 𝐴 – cos 2𝐴 = cos 2 𝐴 − sin2 𝐴 = 2 cos 2 𝐴 − 1 = 1 − 2 sin2 𝐴 2 tan 𝐴 – tan 2𝐴 = 1 − tan2 𝐴 𝐴 derive and use expressions for sin 𝐴, cos 𝐴 and tan 𝐴 in terms of 𝑡 where 𝑡 = tan (the 𝑡-formulae) 2 2𝑡 – sin 𝐴 = 1+𝑡 2 1−𝑡 2 – cos 𝐴 = 1+𝑡 2 2𝑡 – tan 𝐴 = 1−𝑡 2 Mathematics Extension 1 Stage 6 Syllabus (2017) 38  Year 11 derive and use the formulae for trigonometric products as sums and differences for cos 𝐴 cos 𝐵, sin 𝐴 sin 𝐵, sin 𝐴 cos 𝐵 and cos 𝐴 sin 𝐵 (ACMSM047) – cos 𝐴 cos 𝐵 = 12[cos(𝐴 − 𝐵) + cos(𝐴 + 𝐵)] – sin 𝐴 sin 𝐵 = 12[cos(𝐴 − 𝐵) − cos(𝐴 + 𝐵)] – sin 𝐴 cos 𝐵 = 12[sin(𝐴 + 𝐵) + sin(𝐴 − 𝐵)] – cos 𝐴 sin 𝐵 = 12[sin(𝐴 + 𝐵) − sin(𝐴 − 𝐵)] Mathematics Extension 1 Stage 6 Syllabus (2017) 39  Year 11 Topic: Calculus Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › applies understanding of the concept of a derivative in the solution of problems, including rates of change, exponential growth and decay and related rates of change ME11-4 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Topic Focus The topic Calculus involves the study of how things change and provides a framework for developing quantitative models of change and deducing their consequences. It involves the development of the connections between rates of change and related rates of change, the derivatives of functions and the manipulative skills necessary for the effective use of differential calculus. The study of calculus is important in developing students’ knowledge and understanding of related rates of change and developing the capacity to operate with and model situations involving change, using algebraic and graphical techniques to describe and solve problems and to predict outcomes with relevance to, for example the physical, natural and medical sciences, commerce and the construction industry. Subtopics ME-C1 Rates of Change Mathematics Extension 1 Stage 6 Syllabus (2017) 40  Year 11 Calculus ME-C1 Rates of Change Outcomes A student: › uses algebraic and graphical concepts in the modelling and solving of problems involving functions and their inverses ME11-1 › applies understanding of the concept of a derivative in the solution of problems, including rates of change, exponential growth and decay and related rates of change ME11-4 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Subtopic Focus The principal focus of this subtopic is for students to solve problems involving the chain rule and differentiation of the exponential function, and understand how these concepts can be applied to the physical and natural sciences. Students develop the ability to study motion problems in an abstract situation, which may in later studies be applied to large and small mechanical systems, from aeroplanes and satellites to miniature robotics. Students also study the mathematics of exponential growth and decay, two fundamental processes in the natural environment. Content C1.1: Rates of change with respect to time Students: describe the rate of change of a physical quantity with respect to time as a derivative – investigate examples where the rate of change of some aspect of a given object with respect to time can be modelled using derivatives AAM – use appropriate language to describe rates of change, for example ‘at rest’, ‘initially’, ‘change of direction’ and ‘increasing at an increasing rate’ 𝑑𝑄 find and interpret the derivative , given a function in the form 𝑄 = 𝑓(𝑡), for the amount of a 𝑑𝑡 physical quantity present at time 𝑡 describe the rate of change with respect to time of the displacement of a particle moving along the 𝑑𝑥 𝑥-axis as a derivative or 𝑥̇ 𝑑𝑡 describe the rate of change with respect to time of the velocity of a particle moving along the 𝑑2 𝑥 𝑥-axis as a derivative or 𝑥̈ 𝑑𝑡 2 Mathematics Extension 1 Stage 6 Syllabus (2017) 41  Year 11 C1.2: Exponential growth and decay Students: construct, analyse and manipulate an exponential model of the form 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 to solve a practical growth or decay problem in various contexts (for example population growth, radioactive decay or depreciation) AAM 𝑑𝑁 – establish the simple growth model, = 𝑘𝑁, where 𝑁 is the size of the physical quantity, 𝑁 = 𝑑𝑡 𝑁(𝑡) at time 𝑡 and 𝑘 is the growth constant 𝑑𝑁 – verify (by substitution) that the function 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 satisfies the relationship = 𝑘𝑁, with 𝐴 𝑑𝑡 being the initial value of 𝑁 – sketch the curve 𝑁(𝑡) = 𝐴𝑒 𝑘𝑡 for positive and negative values of 𝑘 – recognise that this model states that the rate of change of a quantity varies directly with the size of the quantity at any instant 𝑑𝑁 establish the modified exponential model, = 𝑘(𝑁 − 𝑃), for dealing with problems such as 𝑑𝑡 ‘Newton’s Law of Cooling’ or an ecosystem with a natural ‘carrying capacity’ AAM 𝑑𝑁 – verify (by substitution) that a solution to the differential equation = 𝑘(𝑁 − 𝑃) is 𝑑𝑡 𝑁(𝑡) = 𝑃 + 𝐴𝑒 𝑘𝑡 , for an arbitrary constant 𝐴, and 𝑃 a fixed quantity, and that the solution is 𝑁 = 𝑃 in the case when 𝐴 = 0 – sketch the curve 𝑁(𝑡) = 𝑃 + 𝐴𝑒 𝑘𝑡 for positive and negative values of 𝑘 – note that whenever 𝑘 < 0, the quantity 𝑁 tends to the limit 𝑃 as 𝑡 → ∞, irrespective of the initial conditions – recognise that this model states that the rate of change of a quantity varies directly with the difference in the size of the quantity and a fixed quantity at any instant solve problems involving situations that can be modelled using the exponential model or the modified exponential model and sketch graphs appropriate to such problems AAM C1.3: Related rates of change Students: solve problems involving related rates of change as instances of the chain rule (ACMSM129) AAM develop models of contexts where a rate of change of a function can be expressed as a rate of change of a composition of two functions, and to which the chain rule can be applied Mathematics Extension 1 Stage 6 Syllabus (2017) 42  Year 11 Topic: Combinatorics Outcomes A student: › uses concepts of permutations and combinations to solve problems involving counting or ordering ME11-5 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Topic Focus The topic Combinatorics involves counting and ordering as well as exploring arrangements, patterns, symmetry and other methods to generalise and predict outcomes. The consideration of the expansion of (𝑥 + 𝑦)𝑛 , where 𝑛 is a positive integer, draws together aspects of number theory and probability theory. A knowledge of combinatorics is useful when considering situations and solving problems involving counting, sorting and arranging. Efficient counting methods have many applications and are used in the study of probability. The study of combinatorics is important in developing students’ ability to generalise situations, to explore patterns and to ensure the consideration of all outcomes in situations such as the placement of people or objects, setting-up of surveys, jury or committee selection and design. Subtopics ME-A1 Working with Combinatorics Mathematics Extension 1 Stage 6 Syllabus (2017) 43  Year 11 Combinatorics ME-A1 Working with Combinatorics Outcomes A student: › uses concepts of permutations and combinations to solve problems involving counting or ordering ME11-5 › uses appropriate technology to investigate, organise and interpret information to solve problems in a range of contexts ME11-6 › communicates making comprehensive use of mathematical language, notation, diagrams and graphs ME11-7 Subtopic Focus The principal focus of this subtopic is to develop students’ understanding and proficiency with permutations and combinations and their relevance to the binomial coefficients. Students develop proficiency in ordering and counting techniques in both restricted and unrestricted situations. The binomial expansion is introduced, Pascal’s triangle is constructed and related identities are proved. The material studied provides the basis for more advanced work, where the binomial expansion is extended to cases for rational values of 𝑛, and applications in calculus are explored. Content A1.1: Permutations and combinations Students: list and count the number of ways an event can occur use the fundamental counting principle (also known as the multiplication principle) use factorial notation to describe and determine the number of ways 𝑛 different items can be arranged in a line or a circle – solve problems involving cases where some items are not distinct (excluding arrangements in a circle) solve simple problems and prove results using the pigeonhole principle (ACMSM006) – understand that if there are 𝑛 pigeonholes and 𝑛 + 1 pigeons to go into them, then at least one pigeonhole must hold 2 or more pigeons – generalise to: If 𝑛 pigeons are sitting in 𝑘 pigeonholes, where 𝑛 > 𝑘, then there is at least one 𝑛 pigeonhole with at least pigeons in it 𝑘 – prove the pigeonhole principle understand and use permutations to solve problems (ACMSM001) 𝑛! – understand and use the notation 𝑛 𝑃𝑟 and the formula 𝑛 𝑃𝑟 = (𝑛−𝑟)! solve problems involving permutations and restrictions with or without repeated objects (ACMSM004) understand and use combinations to solve problems (ACMSM007) 𝑛 𝑛! – understand and use the notations ( ) and 𝑛 𝐶𝑟 and the formula 𝑛 𝐶𝑟 = 𝑟 𝑟!(𝑛−𝑟)! (ACMMM045, ACMSM008) solve practical problems involving permutations and combinations, including those involving simple probability situations AAM Mathematics Extension 1 Stage 6 Syllabus (2017) 44  Year 11 A1.2: The binomial expansion and Pascal’s triangle Students: expand (𝑥 + 𝑦)𝑛 for small positive integers 𝑛 (ACMMM046) – note the pattern formed by the coefficients of 𝑥 in the expansion of (1 + 𝑥)𝑛 and recognise links to Pascal’s triangle 𝑛 – recognise the numbers ( ) (also denoted 𝑛 𝐶𝑟 ) as binomial coefficients (ACMMM047) 𝑟 derive and use simple identities associated with Pascal’s triangle (ACMSM009) – establish combinatorial proofs of the Pascal’s triangle relations 𝑛 𝐶0 = 1, 𝑛 𝐶𝑛 = 1; 𝑛 𝐶𝑟 = 𝑛−1 𝐶𝑟−1 + 𝑛−1 𝐶𝑟 for 1 ≤ 𝑟 ≤ 𝑛 − 1; and 𝑛 𝐶𝑟 = 𝑛 𝐶𝑛−𝑟 Mathematics Extension 1 Stage 6 Syllabus (2017) 45  Year 12 Mathematics Extension 1 Year 12 Course Content Year 12 Course Structure and Requirements The course is organised in topics, with the topics divided into subtopics. Mathematics Extension 1 Topics Subtopics Proof ME-P1 Proof by Mathematical Induction Year 12 course Vectors ME-V1 Introduction to Vectors (60 hours) Trigonometric Functions ME-T3 Trigonometric Equations Calculus ME-C2 Further Calculus Skills ME-C3 Applications of Calculus Statistical Analysis ME-S1 The Binomial Distribution For the Year 12 course: The Mathematics Advanced Year 12 course should be taught prior to or concurrently with this course. The Mathematics Advanced Year 11 course is a prerequisite. Students should experience content in the course in familiar and routine situations as well as unfamiliar situations. Students should be provided with regular opportunities involving the integration of technology. Mathematics Extension 1 Stage 6 Syllabus (2017) 46  Year 12 Topic: Proof Outcomes A student: › applies techniques involving proof or calculus to model and solve problems ME12-1 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Topic Focus The topic Proof involves the communication and justification of an argument for a mathematical statement in a clear, concise and precise manner. A knowledge of proof enables a level of reasoning, justification and communication that is accurate, concise and precise. The study of proof is important in developing students’ ability to reason, justify, communicate and critique mathematical arguments and statements necessary for problem solving and generalising patterns. Subtopics ME-P1 Proof by Mathematical Induction Mathematics Extension 1 Stage 6 Syllabus (2017) 47  Year 12 Proof ME-P1 Proof by Mathematical Induction Outcomes A student: › applies techniques involving proof or calculus to model and solve problems ME12-1 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Subtopic Focus The principal focus of this subtopic is to explore and to develop the use of the technique of proof by mathematical induction to prove results. Students are introduced to mathematical induction for a limited range of applications so that they have time to develop confidence in its use. Students develop the use of formal mathematical language and argument to prove the validity of given situations using inductive reasoning. The logical sequence of steps in the proof technique needs to be understood and carefully justified, thus encouraging clear and concise communication which is useful both in further study of mathematics and in life. Content Students: understand the nature of inductive proof, including the ‘initial statement’ and the inductive step (ACMSM064) prove results using mathematical induction 𝑛(𝑛+1)(2𝑛+1) – prove results for sums, for example 1 + 4 + 9 + ⋯ + 𝑛2 = for any positive integer 𝑛 6 (ACMSM065) – prove divisibility results, for example 32𝑛 − 1 is divisible by 8 for any positive integer 𝑛 (ACMSM066) identify errors in false ‘proofs by induction’, such as cases where only one of the required two steps of a proof by induction is true, and understand that this means that the statement has not been proved recognise situations where proof by mathematical induction is not appropriate Mathematics Extension 1 Stage 6 Syllabus (2017) 48  Year 12 Topic: Vectors Outcomes A student: › applies concepts and techniques involving vectors and projectiles to solve problems ME12-2 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Topic Focus The topic Vectors involves mathematical representation of a quantity with magnitude and direction and its geometrical depiction. This topic provides a modern language and approach to explore and explain a range of object behaviours in a variety of contexts from theoretical or real-life scenarios. A knowledge of vectors enables the understanding of the behaviour of objects in two dimensions and ways in which this behaviour can be expressed, including the consideration of position, displacement and movement. The study of vectors is important in developing students’ understanding of an object’s representation and behaviour in two dimensions using a variety of notations, and how to use these notations effectively to explore the geometry of a situation. Vectors are used in many fields of study, including engineering, structural analysis and navigation. Subtopics ME-V1 Introduction to Vectors Mathematics Extension 1 Stage 6 Syllabus (2017) 49  Year 12 Vectors ME-V1 Introduction to Vectors Outcomes A student: › applies concepts and techniques involving vectors and projectiles to solve problems ME12-2 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Subtopic Focus The principal focus of this subtopic is to introduce the concept of vectors in two dimensions, use them to represent quantities with magnitude and direction, and understand that this representation can allow for the exploration of situations such as geometrical proofs. Students develop an understanding of vector notations and how to manipulate vectors to allow geometrical situations to be explored further. The example of projectile motion as an application of vectors is then introduced. These concepts are explored further in the Mathematics Extension 2 course. Content V1.1: Introduction to vectors Students: define a vector as a quantity having both magnitude and direction, and examine examples of vectors, including displacement and velocity (ACMSM010) – explain the distinction between a position vector and a displacement (relative) vector  define and use a variety of notations and representations for vectors in two dimensions (ACMSM014) – use standard notations for vectors, for example: a~ , ⃗⃗⃗⃗⃗ 𝐴𝐵 and 𝐚 – represent vectors graphically in two dimensions as directed line segments – define unit vectors as vectors of magnitude 1, and the standard two-dimensional perpendicular unit vectors ~i and j ~ – express and use vectors in two dimensions in a variety of forms, including component form, ordered pairs and column vector notation perform addition and subtraction of vectors and multiplication of a vector by a scalar algebraically and geometrically, and interpret these operations in geometric terms AAM – graphically represent a scalar multiple of a vector (ACMSM012) – use the triangle law and the parallelogram law to find the sum and difference of two vectors – define and use addition and subtraction of vectors in component form (ACMSM017) – define and use multiplication by a scalar of a vector in component form (ACMSM018) Mathematics Extension 1 Stage 6 Syllabus (2017) 50  Year 12 V1.2: Further operations with vectors Students: define, calculate and use the magnitude of a vector in two dimensions and use the notation u ~ for the magnitude of a vector u~  x ~i  y j ~ – prove that the magnitude of a vector, u~  x ~i  y j , can be found using: u~  x ~i  y j  x 2  y 2 ~ ~ – identify the magnitude of a displacement vector ⃗⃗⃗⃗⃗ 𝐴𝐵 as being the distance between the points 𝐴 and 𝐵 u~ – convert a non-zero vector u~ into a unit vector û~ by dividing by its length: u~ˆ  u~ define and use the direction of a vector in two dimensions define, calculate and use the scalar (dot) product of two vectors u~  x1 ~i  y1 j and v~  x2 ~i  y 2 j ~ ~ AAM – apply the scalar product, u ~ ~ v , to vectors expressed in component form, where u~  v~  x1 x2  y1 y2 – use the expression for the scalar (dot) product, u~  v~  u~ v~ cos where 𝜃 is the angle between vectors u~ and v~ to solve problems – demonstrate the equivalence, u~  v~  u~ v~ cos  x1 x2  y1 y2 and use this relationship to solve problems 2 – establish and use the formula v~ v~  v~ – calculate the angle between two vectors using the scalar (dot) product of two vectors in two dimensions examine properties of parallel and perpendicular vectors and determine if two vectors are parallel or perpendicular (ACMSM021) define and use the projection of one vector onto another (ACMSM022) solve problems involving displacement, force and velocity involving vector concepts in two dimensions (ACMSM023) AAM prove geometric results and construct proofs involving vectors in two dimensions including to proving that: AAM – the diagonals of a parallelogram meet at right angles if and only if it is a rhombus (ACMSM039) – the midpoints of the sides of a quadrilateral join to form a parallelogram (ACMSM040) – the sum of the squares of the lengths of the diagonals of a parallelogram is equal to the sum of the squares of the lengths of the sides (ACMSM041) Mathematics Extension 1 Stage 6 Syllabus (2017) 51  Year 12 V1.3: Projectile motion Students: understand the concept of projectile motion, and model and analyse a projectile’s path assuming that: – the projectile is a point – the force due to air resistance is negligible – the only force acting on the projectile is the constant force due to gravity, assuming that the projectile is moving close to the Earth’s surface model the motion of a projectile as a particle moving with constant acceleration due to gravity and derive the equations of motion of a projectile AAM – represent the motion of a projectile using vectors – recognise that the horizontal and vertical components of the motion of a projectile can be represented by horizontal and vertical vectors – derive the horizontal and vertical equations of motion of a projectile – understand and explain the limitations of this projectile model use equations for horizontal and vertical components of velocity and displacement to solve problems on projectiles apply calculus to the equations of motion to solve problems involving projectiles (ACMSM115) AAM Mathematics Extension 1 Stage 6 Syllabus (2017) 52  Year 12 Topic: Trigonometric Functions Outcomes A student: › applies advanced concepts and techniques in simplifying expressions involving compound angles and solving trigonometric equations ME12-3 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Topic Focus The topic Trigonometric Functions involves the study of periodic functions in geometric, algebraic, numerical and graphical representations. It extends to include the exploration of both algebraic and geometric methods to solve trigonometric problems. A knowledge of trigonometric functions enables students to manipulate trigonometric expressions to prove identities and solve equations. The study of trigonometric functions is important in developing students’ understanding of the connections between algebraic and graphical representations and how this can be applied to solve problems from theoretical or real-life scenarios, for example involving waves and signals. Subtopics ME-T3 Trigonometric Equations Mathematics Extension 1 Stage 6 Syllabus (2017) 53  Year 12 Trigonometric Functions ME-T3 Trigonometric Equations Outcomes A student: › applies advanced concepts and techniques in simplifying expressions involving compound angles and solving trigonometric equations ME12-3 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Subtopic Focus The principal focus of this subtopic is to consolidate and extend students’ knowledge in relation to solving trigonometric equations and to apply this knowledge to practical situations. Students develop complex algebraic manipulative skills and fluency in applying trigonometric knowledge to a variety of situations. Trigonometric expressions and equations provide a powerful tool for modelling quantities that vary in a cyclical way such as tides, seasons, demand for resources, and alternating current. Content Students: convert expressions of the form 𝑎 cos 𝑥 + 𝑏 sin 𝑥 to 𝑅 cos(𝑥 ± 𝛼) or 𝑅 sin(𝑥 ± 𝛼) and apply these to solve equations of the form 𝑎 cos 𝑥 + 𝑏 sin 𝑥 = 𝑐, sketch graphs and solve related problems (ACMSM048) solve trigonometric equations requiring factorising and/or the application of compound angle, double angle formulae or the 𝑡-formulae prove and apply other trigonometric identities, for example cos 3𝑥 = 4cos 3 𝑥 − 3 cos 𝑥 (ACMSM049) solve trigonometric equations and interpret solutions in context using technology or otherwise Mathematics Extension 1 Stage 6 Syllabus (2017) 54  Year 12 Topic: Calculus Outcomes A student: › applies techniques involving proof or calculus to model and solve problems ME12-1 › uses calculus in the solution of applied problems, including differential equations and volumes of solids of revolution ME12-4 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Topic Focus The topic Calculus involves the study of how things change and provides a framework for developing quantitative models of change and deducing their consequences. It involves the development of analytic and numeric integration techniques and the use of these techniques in solving problems. The study of calculus is important in developing students’ knowledge, understanding and capacity to operate with and model situations involving change, and to use algebraic and graphical techniques to describe and solve problems and to predict future outcomes with relevance to, for example science, engineering, finance, economics and the construction industry. Subtopics ME-C2 Further Calculus Skills ME-C3 Applications of Calculus Mathematics Extension 1 Stage 6 Syllabus (2017) 55  Year 12 Calculus ME-C2 Further Calculus Skills Outcomes A student: › applies techniques involving proof or calculus to model and solve problems ME12-1 › uses calculus in the solution of applied problems, including differential equations and volumes of solids of revolution ME12-4 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Subtopic Focus The principal focus of this subtopic is to further develop students’ knowledge, skills and understanding relating to differentiation and integration techniques. Students develop an awareness and understanding of the interconnectedness of topics across the syllabus, and the fluency that can be obtained in the use of calculus techniques. Later studies in mathematics place prime importance on familiarity and confidence in a variety of calculus techniques as these are used in many different fields. Content Students: find and evaluate indefinite and definite integrals using the method of integration by substitution, using a given substitution – change an integrand into an appropriate form using algebra prove and use the identities sin2 𝑛𝑥 = 12(1 − cos 2𝑛𝑥) and cos 2 𝑛𝑥 = 12(1 + cos 2𝑛𝑥) to solve problems solve problems involving ∫ sin 2 𝑛𝑥 𝑑𝑥 and ∫ cos 2 𝑛𝑥 𝑑𝑥 𝑑𝑦 1 find derivatives of inverse functions by using the relationship = 𝑑𝑥 𝑑𝑥 𝑑𝑦 solve problems involving the derivatives of inverse trigonometric functions 1 𝑎 integrate expressions of the form or (ACMSM121) √𝑎2 −𝑥 2 𝑎2 +𝑥 2 Mathematics Extension 1 Stage 6 Syllabus (2017) 56  Year 12 Calculus ME-C3 Applications of Calculus Outcomes A student: › applies techniques involving proof or calculus to model and solve problems ME12-1 › uses calculus in the solution of applied problems, including differential equations and volumes of solids of revolution ME12-4 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Subtopic Focus The principal focus of this subtopic is to develop an understanding of applications of calculus in a practical context, including the more accessible kinds of differential equations and volumes of solids of revolution, to solve problems. Students develop an awareness and understanding of the use of differential equations which arise when the rate of change in one quantity with respect to another can be expressed in mathematical form. The study of differential equations has important applications in science, engineering, finance, economics and broader applications in mathematics. Content C3.1: Further area and volumes of solids of revolution Students: calculate area of regions between curves determined by functions (ACMSM124) sketch, with and without the use of technology, the graph of a solid of revolution whose boundary is formed by rotating an arc of a function about the 𝑥-axis or 𝑦-axis AAM calculate the volume of a solid of revolution formed by rotating a region in the plane about the 𝑥-axis or 𝑦-axis, with and without the use of technology (ACMSM125) AAM determine the volumes of solids of revolution that are formed by rotating the region between two curves about either the 𝑥-axis or 𝑦-axis in both real-life and abstract contexts AAM C3.2: Differential equations Students: recognise that an equation involving a derivative is called a differential equation recognise that solutions to differential equations are functions and that these solutions may not be unique sketch the graph of a particular solution given a direction field and initial conditions – form a direction field (slope field) from simple first-order differential equations – recognise the shape of a direction field from several alternatives given the form of a differential equation, and vice versa – sketch several possible solution curves on a given direction field Mathematics Extension 1 Stage 6 Syllabus (2017) 57  Year 12 solve simple first-order differential equations (ACMSM130) 𝑑𝑦 – solve differential equations of the form = 𝑓(𝑥) 𝑑𝑥 𝑑𝑦 – solve differential equations of the form = 𝑔(𝑦) 𝑑𝑥 𝑑𝑦 – solve differential equations of the form = 𝑓(𝑥)𝑔(𝑦) using separation of variables 𝑑𝑥 recognise the features of a first-order linear differential equation and that exponential growth and decay models are first-order linear differential equations, with known solutions model and solve differential equations including to the logistic equation that will arise in situations where rates are involved, for example in chemistry, biology and economics (ACMSM132) AAM Mathematics Extension 1 Stage 6 Syllabus (2017) 58  Year 12 Topic: Statistical Analysis Outcomes A student: › applies appropriate statistical processes to present, analyse and interpret data ME12-5 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Topic Focus The topic Statistical Analysis involves the exploration, display and interpretation of data via modelling to identify and communicate key information. A knowledge of statistical analysis enables careful interpretation of situations and an awareness of the contributing factors when presented with information by third parties, including its possible misrepresentation. The study of statistical analysis is important in developing students’ ability to consider the level of reliability that can be applied to the analysis of current situations and to predict future outcomes. It supports the development of understanding of how conclusions drawn from data can be used to inform decisions made by groups such as scientific investigators, business people and policy-makers. Subtopics ME-S1 The Binomial Distribution Mathematics Extension 1 Stage 6 Syllabus (2017) 59  Year 12 Statistical Analysis ME-S1 The Binomial Distribution Outcomes A student: › applies appropriate statistical processes to present, analyse and interpret data ME12-5 › chooses and uses appropriate technology to solve problems in a range of contexts ME12-6 › evaluates and justifies conclusions, communicating a position clearly in appropriate mathematical forms ME12-7 Subtopic Focus The principal focus of this subtopic is to develop an understanding of binomial random variables and their uses in modelling random processes involving chance and variation. Students develop an understanding of binomial distributions and associated statistical analysis methods and their use in modelling binomial events. Binomial probabilities and the binomial distribution are used to model situations where only two outcomes are possible. The use of the binomial distribution and binomial probability has many applications, including medicine and genetics. Content S1.1: Bernoulli and binomial distributions Students: use a Bernoulli random variable as a model for two-outcome situations (ACMMM143) – identify contexts suitable for modelling by Bernoulli random variables (ACMMM144) use Bernoulli random variables and their associated probabilities to solve practical problems (ACMMM146) AAM – understand and apply the formulae for the mean, 𝐸(𝑋) = 𝑥̅ = 𝑝, and variance, Var(𝑋) = 𝑝(1 − 𝑝), of the Bernoulli distribution with parameter 𝑝, and 𝑋 defined as the number of successes (ACMMM145) understand the concepts of Bernoulli trials and the concept of a binomial random variable as the number of ‘successes’ in 𝑛 independent Bernoulli trials, with the same probability of success 𝑝 in each trial (ACMMM147) – calculate the expected frequencies of the various possible outcomes from a series of Bernoulli trials use binomial distributions and their associated probabilities to solve practical problems (ACMMM150) AAM – identify contexts suitable for modelling by binomial random variables (ACMMM148) – identify the binomial parameter 𝑝 as the probability of success – understand and use the notation 𝑋~Bin(𝑛, 𝑝) to indicate that the random variable 𝑋 is distributed binomially with parameters 𝑛 and 𝑝 – apply the formulae for probabilities 𝑃(𝑋 = 𝑟) = 𝑛 𝐶𝑟 𝑝𝑟 (1 − 𝑝)𝑛−𝑟 associated with the binomial distribution with parameters 𝑛 and 𝑝 and understand the meaning of 𝑛 𝐶𝑟 as the number of ways in which an outcome with 𝑟 successes can occur – understand and apply the formulae for the mean, 𝐸(𝑋) = 𝑥̅ = 𝑛𝑝, and the variance, Var(𝑋) = 𝑛𝑝(1 − 𝑝), of a binomial distribution with parameters 𝑛 and 𝑝 Mathematics Extension 1 Stage 6 Syllabus (2017) 60  Year 12 S1.2: Normal approximation for the sample proportion Students: use appropriate graphs to explore the behaviour of the sample proportion on collected or supplied data AAM – understand the concept of the sample proportion 𝑝̂ as a random variable whose value varies between samples (ACMMM174) explore the behaviour of the sample proportion using simulated data AAM – examine the approximate normality of the distribution of 𝑝̂ for large samples (ACMMM175) understand and use the normal approximation to the distribution of the sample proportion and its limitations AAM Mathematics Extension 1 Stage 6 Syllabus (2017) 61 Glossary Glossary term Elaboration Aboriginal and Aboriginal Peoples are the first peoples of Australia and are represented by Torres Strait over 250 language groups each associated with a particular Country or Islander Peoples territory. Torres Strait Islander Peoples whose island territories to the north east of Australia were annexed by Queensland in 1879 are also Indigenous Australians and are represented by five cultural groups. An Aboriginal and/or Torres Strait Islander person is someone who:  is of Aboriginal and/or Torres Strait Islander descent  identifies as an Aboriginal person and/or Torres Strait Islander person, and  is accepted as such by the Aboriginal and/or Torres Strait Islander community in which they live. Bernoulli The Bernoulli distribution is the probability distribution of a random variable distribution which takes the value 1 with ‘success’ probability 𝑝, and the value 0 with ‘failure’ probability 𝑞 = 1 − 𝑝. The Bernoulli distribution is a special case of the binomial distribution, where 𝑛 = 1. Bernoulli random A Bernoulli random variable has two possible values, namely 0 representing variable failure and 1 representing success. The parameter associated with such a random variable is the probability 𝑝 of obtaining a 1. Bernoulli trial A Bernoulli trial is an experiment with only two possible outcomes, labelled ‘success’ and ‘failure’. binomial coefficient The coefficient of the term 𝑥 𝑛−𝑟 𝑦 𝑟 in the expansion of (𝑥 + 𝑦)𝑛 is called a 𝑛 binomial coefficient. It is written as 𝑛 𝐶𝑟 or ( ) where 𝑟 = 0, 1,... , 𝑛 and is 𝑟 𝑛! given by: 𝑟!(𝑛−𝑟)! binomial The binomial distribution with parameters 𝑛 and 𝑝 is the discrete probability distribution distribution of the number of successes in a sequence of 𝑛 independent Bernoulli trials, each of which yields success with probability 𝑝. binomial expansion A binomial expansion describes the algebraic expansion of powers of a binomial expression. binomial random A binomial random variable 𝑋 represents the number of successes in 𝑛 variable independent Bernoulli trials. In each Bernoulli trial, the probability of success is 𝑝 and the probability of failure is: 𝑞 = 1 − 𝑝 column vector A vector, a~ , in two dimensions can be represented in column vector notation notation. For example the ordered pair a~  4, 5 can be represented in  4 column vector notation as: a~     5 Mathematics Extension 1 Stage 6 Syllabus (2017) 62 Glossary term Elaboration combination A combination is a selection of 𝑟 distinct objects from 𝑛 distinct objects, where order is not important. The number of such combinations is denoted 𝑛 𝑛! by 𝑛 𝐶𝑟 or ( ), and is given by: 𝑟 𝑟!(𝑛−𝑟)! component form of The component form of a vector, v~ , expresses the vector in terms of unit a vector vectors ~i , a unit vector in the 𝑥-direction, and j , a unit vector in the ~ 𝑦-direction. For example the ordered vector pair v~  4, 3 can be represented as: v~  4 ~i  3 j ~ differential equation A differential equation is any equation containing the derivative of an unknown function. direction field A direction field (or slope field) is a graphical representation of the tangent lines to the solutions of a first-order differential equation. displacement vector A displacement vector represents the displacement from one point to another. factor theorem The factor theorem states that a polynomial 𝑃(𝑥) has a factor (𝑥 − 𝑘) if and only if 𝑃(𝑘) = 0, ie 𝑘 is a root of the equation 𝑃(𝑥) = 0. The factor theorem links the factors and zeros of a polynomial. factorial The product of the first 𝑛 positive integers is called the factorial of 𝑛 and is denoted by 𝑛!. 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)(𝑛 − 3) × … × 3 × 2 × 1 By definition: 0! = 1 fundamental The fundamental counting principle states that if one event has 𝑚 possible counting principle outcomes and a second independent event has 𝑛 possible outcomes, then there are a total of 𝑚 × 𝑛 possible outcomes for the two combined events. integrand An integrand is a function that is to be integrated. logistic equation 𝑑𝑁 The logistic equation is the differential equation = 𝑘𝑁(𝑃 − 𝑁) where 𝑘, 𝑃 𝑑𝑡 𝑑𝑁 are constants. Thus: if 𝑁 = 0 or 𝑁 = 𝑃, =0 𝑑𝑡 mathematical Mathematical induction is a method of mathematical proof used to prove induction statements involving the natural numbers. Also known as proof by induction or inductive proof. The principle of induction is an axiom and so cannot itself be proven. multiplicity of a root Given a polynomial 𝑃(𝑥), if 𝑃(𝑥) = (𝑥 − 𝑎)𝑟 𝑄(𝑥) , 𝑄(𝑎) ≠ 0 and 𝑟 is a positive integer, then the root 𝑥 = 𝑎 has multiplicity 𝑟. Mathematics Extension 1 Stage 6 Syllabus (2017) 63 Glossary term Elaboration parameter 1. A parameter is a quantity that defines certain characteristics of a function or system. For example 𝜃 is a parameter in 𝑦 = 𝑥 cos 𝜃 2. A parameter can be a characteristic value of a situation. For example the time taken for a machine to produce a certain product. permutation A permutation is an arrangement of 𝑟 distinct objects taken from 𝑛 distinct objects where order is important. The number of such permutations is denoted by 𝑛 𝑃𝑟 and is equal to: 𝑛 𝑛! 𝑃𝑟 = 𝑛(𝑛 − 1)... (𝑛 − 𝑟 + 1) = (𝑛−𝑟)! The number of permutations of 𝑛 objects is 𝑛!. position vector The position vector of a point 𝑃 in the plane is the vector joining the origin to 𝑃. remainder theorem The remainder theorem states that if a polynomial 𝑃(𝑥) is divided by (𝑥 − 𝑘), the remainder is equal to 𝑃(𝑘). sample proportion The sample proportion (𝑝̂ ) is the fraction of samples out of 𝑛 Bernoulli trials 𝑥 which were successes (𝑥), that is: 𝑝̂ = 𝑛 For large 𝑛, 𝑝̂ has an approximately normal distribution. scalar A scalar is a quantity with magnitude but no direction. statement A statement is an assertion that can be true or false but not both. Mathematics Extension 1 Stage 6 Syllabus (2017) 64 NSW Syllabus for the Australian curriculum Physics Stage 6 Syllabus  Year 11 Physics Year 11 Course Content Year 11 Course Structure and Requirements Modules Indicative hours Depth studies Module 1 Kinematics 60 Year 11 Module 2 Working course Dynamics Scientifically *15 hours Skills Module 3 in Modules 1–4 (120 hours) Waves and Thermodynamics 60 Module 4 Electricity and Magnetism *15 hours must be allocated to depth studies within the 120 indicative course hours. Requirements for Practical Investigations Scientific investigations include both practical investigations and secondary-sourced investigations. Practical investigations are an essential part of the Year 11 course and must occupy a minimum of 35 hours of course time, including time allocated to practical investigations in depth studies. Practical investigations include: undertaking laboratory experiments, including the use of appropriate digital technologies fieldwork. Secondary-sourced investigations include: locating and accessing a wide range of secondary data and/or information using and reorganising secondary data and/or information. Physics Stage 6 Syllabus 32  Year 11 Working Scientifically Skills It is expected that the content of each skill will be addressed by the end of the Stage 6 course. Questioning and Predicting Outcomes A student: › develops and evaluates questions and hypotheses for scientific investigation PH11/12-1 Content Students: develop and evaluate inquiry questions and hypotheses to identify a concept that can be investigated scientifically, involving primary and secondary data (ACSPH001, ACSPH061, ACSPH096) modify questions and hypotheses to reflect new evidence Planning Investigations Outcomes A student: › designs and evaluates investigations in order to obtain primary and secondary data and information PH11/12-2 Content Students: assess risks, consider ethical issues and select appropriate materials and technologies when designing and planning an investigation (ACSPH031, ACSPH097) justify and evaluate the use of variables and experimental controls to ensure that a valid procedure is developed that allows for the reliable collection of data (ACSPH002) evaluate and modify an investigation in response to new evidence Physics Stage 6 Syllabus 33  Year 11 Conducting Investigations Outcomes A student: › conducts investigations to collect valid and reliable primary and secondary data and information PH11/12-3 Content Students: employ and evaluate safe work practices and manage risks (ACSPH031) use appropriate technologies to ensure and evaluate accuracy select and extract information from a wide range of reliable secondary sources and acknowledge them using an accepted referencing style Processing Data and Information Outcomes A student: › selects and processes appropriate qualitative and quantitative data and information using a range of appropriate media PH11/12-4 Content Students: select qualitative and quantitative data and information and represent them using a range of formats, digital technologies and appropriate media (ACSPH004, ACSPH007, ACSPH064, ACSPH101) apply quantitative processes where appropriate evaluate and improve the quality of data Analysing Data and Information Outcomes A student: › analyses and evaluates primary and secondary data and information PH11/12-5 Content Students: derive trends, patterns and relationships in data and information assess error, uncertainty and limitations in data (ACSPH004, ACSPH005, ACSPH033, ACSPH099) assess the relevance, accuracy, validity and reliability of primary and secondary data and suggest improvements to investigations (ACSPH005) Physics Stage 6 Syllabus 34  Year 11 Problem Solving Outcomes A student: › solves scientific problems using primary and secondary data, critical thinking skills and scientific processes PH11/12-6 Content Students: use modelling (including mathematical examples) to explain phenomena, make predictions and solve problems using evidence from primary and secondary sources (ACSPH006, ACSPH010) use scientific evidence and critical thinking skills to solve problems Communicating Outcomes A student: › communicates scientific understanding using suitable language and terminology for a specific audience or purpose PH11/12-7 Content Students: select and use suitable forms of digital, visual, written and/or oral forms of communication select and apply appropriate scientific notations, nomenclature and scientific language to communicate in a variety of contexts (ACSPH008, ACSPH036, ACSPH067, ACSPH102) construct evidence-based arguments and engage in peer feedback to evaluate an argument or conclusion (ACSPH034, ACSPH036) Physics Stage 6 Syllabus 35  Year 11 Module 1: Kinematics Outcomes A student: › designs and evaluates investigations in order to obtain primary and secondary data and information PH11/12- 2 › conducts investigations to collect valid and reliable primary and secondary data and information PH11/12-3 › selects and processes appropriate qualitative and quantitative data and information using a range of appropriate media PH11/12-4 › analyses and evaluates primary and secondary data and information PH11/12-5 › solves scientific problems using primary and secondary data, critical thinking skills and scientific processes PH11/12-6 › describes and analyses motion in terms of scalar and vector quantities in two dimensions and makes quantitative measurements and calculations for distance, displacement, speed, velocity and acceleration PH11-8 Content Focus Motion is a fundamental observable phenomenon. The study of kinematics involves describing, measuring and analysing motion without considering the forces and masses involved in that motion. Uniformly accelerated motion is described in terms of relationships between measurable scalar and vector quantities, including displacement, speed, velocity, acceleration and time. Representations – including graphs and vectors, and equations of motion – can be used qualitatively and quantitatively to describe and predict linear motion. By studying this module, students come to understand that scientific knowledge enables scientists to offer valid explanations and make reliable predictions, particularly in regard to the motion of an object. Working Scientifically In this module, students focus on designing, evaluating and conducting investigations to examine trends in data and solve problems related to kinematics. Students should be provided with opportunities to engage with all the Working Scientifically skills throughout the course. Content Motion in a Straight Line Inquiry question: How is the motion of an object moving in a straight line described and predicted? Students: describe uniform straight-line (rectilinear) motion and uniformly accelerated motion through: – qualitative descriptions – the use of scalar and vector quantities (ACSPH060) conduct a practical investigation to gather data to facilitate the analysis of instantaneous and average velocity through: – quantitative, first-hand measurements – the graphical representation and interpretation of data (ACSPH061) Physics Stage 6 Syllabus 36  Year 11 calculate the relative velocity of two objects moving along the same line using vector analysis conduct practical investigations, selecting from a range of technologies, to record and analyse the motion of objects in a variety of situations in one dimension in order to measure or calculate: – time – distance – displacement – speed – velocity – acceleration use mathematical modelling and graphs, selected from a range of technologies, to analyse and derive relationships between time, distance, displacement, speed, velocity and acceleration in rectilinear motion, including: 1 – 𝑠 = 𝑢𝑡 + 2 𝑎𝑡 2 – 𝑣 = 𝑢 + 𝑎𝑡 – 𝑣 2 = 𝑢2 + 2𝑎𝑠 (ACSPH061) Motion on a Plane Inquiry question: How is the motion of an object that changes its direction of movement on a plane described? Students: analyse vectors in one and two dimensions to: – resolve a vector into two perpendicular components – add two perpendicular vector components to obtain a single vector (ACSPH061) represent the distance and displacement of objects moving on a horizontal plane using: – vector addition – resolution of components of vectors (ACSPH060) describe and analyse algebraically, graphically and with vector diagrams, the ways in which the motion of objects changes, including: – velocity – displacement (ACSPH060, ACSPH061) describe and analyse, using vector analysis, the relative positions and motions of one object relative to another object on a plane (ACSPH061) analyse the relative motion of objects in two dimensions in a variety of situations, for example: – a boat on a flowing river relative to the bank – two moving cars – an aeroplane in a crosswind relative to the ground (ACSPH060, ACSPH132) Physics Stage 6 Syllabus 37  Year 11 Module 2: Dynamics Outcomes A student: › designs and evaluates investigations in order to obtain primary and secondary data and information PH11/12-2 › selects and processes appropriate qualitative and quantitative data and information using a range of appropriate media PH11/12-4 › solves scientific problems using primary and secondary data, critical thinking skills and scientific processes PH11/12-6 › describes and explains events in terms of Newton’s Laws of Motion, the law of conservation of momentum and the law of conservation of energy PH11-9 Content Focus The relationship between the motion of objects and the forces that act on them is often complex. However, Newton’s Laws of Motion can be used to describe the effect of forces on the motion of single objects and simple systems. This module develops the key concept that forces are always produced in pairs that act on different objects and add to zero. By applying Newton’s laws directly to simple systems, and, where appropriate, the law of conservation of momentum and law of conservation of mechanical energy, students examine the effects of forces. They also examine the interactions and relationships that can occur between objects by modelling and representing these using vectors and equations. In many situations, within and beyond the discipline of physics, knowing the rates of change of quantities provides deeper insight into various phenomena. In this module, the rates of change of displacement, velocity and energy are of particular significance and students develop an understanding of the usefulness and limitations of modelling. Working Scientifically In this module, students focus on designing, evaluating and conducting investigations and interpreting trends in data to solve problems related to dynamics. Students should be provided with opportunities to engage with all the Working Scientifically skills throughout the course. Content Forces Inquiry question: How are forces produced between objects and what effects do forces produce? Students: using Newton’s Laws of Motion, describe static and dynamic interactions between two or more objects and the changes that result from: – a contact force – a force mediated by fields Physics Stage 6 Syllabus 38  Year 11 explore the concept of net force and equilibrium in one-dimensional and simple two-dimensional contexts using: (ACSPH050) – algebraic addition – vector addition – vector addition by resolution into components solve problems or make quantitative predictions about resultant and component forces by applying the following relationships: – 𝐹⃗ AB = −𝐹⃗BA – 𝐹𝑥 = 𝐹cos𝜃, 𝐹𝑦 = 𝐹sin𝜃 conduct a practical investigation to explain and predict the motion of objects on inclined planes (ACSPH098) Forces, Acceleration and Energy Inquiry question: How can the motion of objects be explained and analysed? Students: apply Newton’s first two laws of motion to a variety of everyday situations, including both static and dynamic examples, and include the role played by friction 𝑓⃗friction = 𝜇𝐹⃗𝑁 (ACSPH063) investigate, describe and analyse the acceleration of a single object subjected to a constant net force and relate the motion of the object to Newton’s Second Law of Motion through the use of: (ACSPH062, ACSPH063) – qualitative descriptions – graphs and vectors – deriving relationships from graphical representations including 𝐹⃗net = 𝑚𝑎⃗ and relationships of uniformly accelerated motion apply the special case of conservation of mechanical energy to the quantitative analysis of motion involving: – work done and change in the kinetic energy of an object undergoing accelerated rectilinear motion in one dimension 𝑊 = 𝐹∥ 𝑠 = 𝐹𝑠cos𝜃 – changes in gravitational potential energy of an object in a uniform field Δ𝑈 = 𝑚𝑔Δℎ conduct investigations over a range of mechanical processes to analyse qualitatively and ∆𝐸 quantitatively the concept of average power 𝑃= ,𝑃 = 𝐹∥ 𝑣 = 𝐹𝑣cos𝜃 including but not ∆𝑡 limited to: – uniformly accelerated rectilinear motion – objects raised against the force of gravity – work done against air resistance, rolling resistance and friction Physics Stage 6 Syllabus 39  Year 11 Momentum, Energy and Simple Systems Inquiry question: How is the motion of objects in a simple system dependent on the interaction between the objects? Students: conduct an investigation to describe and analyse one-dimensional (collinear) and two-dimensional interactions of objects in simple closed systems (ACSPH064) analyse quantitatively and predict, using the law of conservation of momentum ∑ 𝑚 𝑣⃗before = ∑ 𝑚 𝑣⃗after and, where appropriate, conservation of kinetic energy 1 2 21 ∑ 𝑚𝑣before = ∑ 2 𝑚𝑣after , the results of interactions in elastic collisions (ACSPH066) 2 investigate the relationship and analyse information obtained from graphical representations of force as a function of time evaluate the effects of forces involved in collisions and other interactions, and analyse quantitatively the interactions using the concept of impulse Δ𝑝⃗ = 𝐹⃗net Δ𝑡 analyse and compare the momentum and kinetic energy of elastic and inelastic collisions (ACSPH066) Physics Stage 6 Syllabus 40  Year 11 Module 3: Waves and Thermodynamics Outcomes A student: › conducts investigations to collect valid and reliable primary and secondary data and information PH11/12-3 › selects and processes appropriate qualitative and quantitative data and information using a range of appropriate media PH11/12-4 › solves scientific problems using primary and secondary data, critical thinking skills and scientific processes PH11/12-6 › communicates scientific understanding using suitable language and terminology for a specific audience or purpose PH11/12-7 › explains and analyses waves and the transfer of energy by sound, light and thermodynamic principles PH11-10 Content Focus Wave motion involves the transfer of energy without the transfer of matter. By exploring the behaviour of wave motion and examining the characteristics of wavelength, frequency, period, velocity and amplitude, students further their understanding of the properties of waves. They are then able to demonstrate how waves can be reflected, refracted, diffracted and superposed (interfered) and to develop an understanding that not all waves require a medium for their propagation. Students examine mechanical waves and electromagnetic waves, including their similarities and differences. Students also examine energy and its transfer, in the form of heat, from one place to another. Thermodynamics is the study of the relationship between energy, work, temperature and matter. Understanding this relationship allows students to appreciate particle motion within objects. Students have the opportunity to examine how hot objects lose energy in three ways: first, by conduction, and, second, by convection – which both involve the motion of particles; and, third, the emission of electromagnetic radiation. An understanding of thermodynamics is a pathway to understanding related concepts in many fields involving Science, Technology, Engineering and Mathematics (STEM). Working Scientifically In this module, students focus on conducting investigations, collecting and processing data and information, interpreting trends in data and communicating scientific ideas about waves and thermodynamics. Students should be provided with opportunities to engage with all the Working Scientifically skills throughout the course. Physics Stage 6 Syllabus 41  Year 11 Content Wave Properties Inquiry question: What are the properties of all waves and wave motion? Students: conduct a practical investigation involving the creation of mechanical waves in a variety of situations in order to explain: – the role of the medium in the propagation of mechanical waves – the transfer of energy involved in the propagation of mechanical waves (ACSPH067, ACSPH070) conduct practical investigations to explain and analyse the differences between: – transverse and longitudinal waves (ACSPH068) – mechanical and electromagnetic waves (ACSPH070, ACSPH074) construct and/or interpret graphs of displacement as a function of time and as a function of position of transverse and longitudinal waves, and relate the features of those graphs to the following wave characteristics: – velocity – frequency – period – wavelength – displacement and amplitude (ACSPH069) solve problems and/or make predictions by modelling and applying the following relationships to a variety of situations: – 𝑣 = 𝑓𝜆 1 – 𝑓=𝑇 Wave Behaviour Inquiry question: How do waves behave? Students: explain the behaviour of waves in a variety of situations by investigating the phenomena of: – reflection – refraction – diffraction – wave superposition (ACSPH071, ACSPH072) conduct an investigation to distinguish between progressive and standing waves (ACSPH072) conduct an investigation to explore resonance in mechanical systems and the relationships between: – driving frequency – natural frequency of the oscillating system – amplitude of motion – transfer/transformation of energy within the system (ACSPH073) Physics Stage 6 Syllabus 42  Year 11 Sound Waves Inquiry question: What evidence suggests that sound is a mechanical wave? Students: conduct a practical investigation to relate the pitch and loudness of a sound to its wave characteristics model the behaviour of sound in air as a longitudinal wave relate the displacement of air molecules to variations in pressure (ACSPH070) investigate quantitatively the relationship between distance and intensity of sound conduct investigations to analyse the reflection, diffraction, resonance and superposition of sound waves (ACSPH071) investigate and model the behaviour of standing waves on strings and/or in pipes to relate quantitatively the fundamental and harmonic frequencies of the waves that are produced to the physical characteristics (eg length, mass, tension, wave velocity) of the medium (ACSPH072) analyse qualitatively and quantitatively the relationships of the wave nature of sound to explain: – beats 𝑓beat = |𝑓2 − 𝑓1 | (𝑣wave +𝑣observer ) – the Doppler effect 𝑓 ′ =𝑓 (𝑣wave −𝑣source ) Ray Model of Light Inquiry question: What properties can be demonstrated when using the ray model of light? Students: conduct a practical investigation to analyse the formation of images in mirrors and lenses via reflection and refraction using the ray model of light (ACSPH075) conduct investigations to examine qualitatively and quantitatively the refraction and total internal reflection of light (ACSPH075, ACSPH076) predict quantitatively, using Snell’s Law, the refraction and total internal reflection of light in a variety of situations conduct a practical investigation to demonstrate and explain the phenomenon of the dispersion of light conduct an investigation to demonstrate the relationship between inverse square law, the intensity of light and the transfer of energy (ACSPH077) solve problems or make quantitative predictions in a variety of situations by applying the following relationships to: 𝑐 – 𝑛𝑥 = 𝑣 – for the refractive index of medium 𝑥, 𝑣𝑥 is the speed of light in the medium 𝑥 – 𝑛1 sin 𝜃1 = 𝑛2 sin 𝜃2 (Snell’s Law) 𝑛 – sin 𝜃c = 2 𝑛1 – 𝐼1 𝑟12 = 𝐼2 𝑟22 – to compare the intensity of light at two points, 𝑟1 and 𝑟2 Physics Stage 6 Syllabus 43  Year 11 Thermodynamics Inquiry question: How are temperature, thermal energy and particle motion related? Students: explain the relationship between the temperature of an object and the kinetic energy of the particles within it (ACSPH018) explain the concept of thermal equilibrium (ACSPH022) analyse the relationship between the change in temperature of an object and its specific heat capacity through the equation 𝑄 = 𝑚𝑐Δ𝑇 (ACSPH020) investigate energy transfer by the process of: – conduction – convection – radiation (ACSPH016) conduct an investigation to analyse qualitatively and quantitatively the latent heat involved in a change of state model and predict quantitatively energy transfer from hot objects by the process of thermal conductivity apply the following relationships to solve problems and make quantitative predictions in a variety of situations: – 𝑄 = 𝑚𝑐Δ𝑇, where c is the specific heat capacity of a substance 𝑄 𝑘𝐴∆𝑇 – = where 𝑘 is the thermal conductivity of a material 𝑡 𝑑 Physics Stage 6 Syllabus 44  Year 11 Module 4: Electricity and Magnetism Outcomes A student: › develops and evaluates questions and hypotheses for scientific investigation PH11/12-1 › analyses and evaluates primary and secondary data and information PH11/12-5 › communicates scientific understanding using suitable language and terminology for a specific audience or purpose PH11/12-7 › explains and quantitatively analyses electric fields, circuitry and magnetism PH11-11 Content Focus Atomic theory and the laws of conservation of energy and electric charge are unifying concepts in understanding the electrical and magnetic properties and behaviour of matter. Interactions resulting from these properties and behaviour can be understood and analysed in terms of electric fields represented by lines. Students use these representations and mathematical models to make predictions about the behaviour of objects, and explore the limitations of the models. Students also examine how the analysis of electrical circuits’ behaviour and the transfer and conversion of energy in electrical circuits has led to a variety of technological applications. Working Scientifically In this module, students focus on developing questions and hypotheses, processing and analysing trends and patterns in data, and communicating ideas about electricity and magnetism. Students should be provided with opportunities to engage with all the Working Scientifically skills throughout the course. Content Electrostatics Inquiry question: How do charged objects interact with other charged objects and with neutral objects? Students: conduct investigations to describe and analyse qualitatively and quantitatively: – processes by which objects become electrically charged (ACSPH002) – the forces produced by other objects as a result of their interactions with charged objects (ACSPH103) – variables that affect electrostatic forces between those objects (ACSPH103) using t