curriculum audit for new senior mathematics 11 and 12 (hsc)

New Senior Mathematics for Years 11 & 12 Second Edition

Curriculum audit

Curriculum reference Textbook reference—Preliminary

Textbook reference—HSC

1. Basic Arithmetic and Algebra

1.1 Review of arithmetical operations on rational numbers and quadratic surds.

Chapter 1 Arithmetic and surds1.1 Review of basics1.2 Repeating decimals1.3 Scientific notation (standard form)1.4 Significant figures and decimal

places1.5 Approximations1.6 Real numbers and surds1.7 Adding and subtracting surds1.8 The distributive law1.9 Rationalising denominators

Outcomes: P1, P2, P3

1.2 Inequalities and absolute values.

3.3 Simple linear inequalities7.3 Absolute value functions


1.3 Review of manipulation of and substitution in algebraic expressions, factorisation and operations on simple algebraic fractions.

Chapter 2 Algebra2.1 Simplifying algebraic expressions2.2 Substitution in formulae2.3 Basic polynomials2.4 Factorising by grouping in pairs2.5 Factorising using the difference of

two squares2.6 Sum and difference of two cubes2.7 Factorising quadratic trinomials2.8 Factorising non-monic trinomials2.9 Mixed factorisations2.10 Algebraic fractions2.11 Adding and subtracting algebraic

fractions2.12 Harder algebraic fractions


1.4 Linear equations and inequalities. Quadratic equations. Simultaneous equations.

Chapter 3 Equations and inequalities3.1 Linear equations in one variable3.2 Linear equations involving fractions3.3 Simple linear inequalities3.4 Quadratic equations3.5 Quadratic equations without a linear

term3.6 Quadratic equations without a

constant term3.7 General quadratic equations

… continued

3.8 Completing the square 3.9 Solving quadratic equations by

completing the square 3.10 Quadratic equations with non-rational

solutions3.11 Completing the square for non-monic

equations3.12 The quadratic formula3.13 Problems involving quadratic

equations3.14 Square roots and absolute value3.15 Simultaneous equations3.16 Problem solving with simultaneous

equations3.17 Solving simultaneous equations—linear

and second degree3.18 Solving simultaneous equations—linear

and second degree in the general form


2. Plane Geometry

2.1 Preliminaries on diagrams, notation, symbols and conventions.

Chapter 4 Plane geometry4.1 Angle review4.2 Parallel lines

Outcomes: P1, P2

2.2 Definitions of special plane figures.

4.3 Angle properties of triangles4.4 Quadrilaterals and polygons

Outcomes: P1, P2

2.3 Properties of angles at a point and of angles formed by transversals to parallel lines. Tests for parallel lines.Angle sums of triangles, quadrilaterals and general polygons.Exterior angle properties.Congruence of triangles. Tests for congruence.Properties of special triangles and quadrilaterals. Tests for special quadrilaterals.Properties of transversals to parallel lines.Similarity of triangles. Tests for similarity.Pythagoras’ theorem and its converse.Area formulae.

4.1 Angle review4.2 Parallel lines4.3 Angle properties of triangles4.4 Quadrilaterals and polygons4.5 Congruent triangles (Preliminary)4.6 Similar triangles (Preliminary)4.7 Intercept properties of parallel lines4.8 Pythagoras’ theorem4.9 Area formulae


2.4 Application of above properties to the solution of numerical exercises requiring one or more steps of reasoning.

Chapter 4 Plane geometry4.1–4.9


2.5 Application of above properties to simple theoretical problems requiring one or more steps of reasoning.

Chapter 11 Plane and coordinate geometrycoordinate geometry11.1 Congruent triangles (HSC)11.2 Similar triangles (HSC)

Outcomes: H1, H2, H5, H9

3. Probability

3.1 Random experiments, equally likely outcomes; probability of a given result.

Chapter 18 Probability18.1 Introduction to probability

3.2 Sum and product of results. 18.2 Venn diagrams18.3 Finite sample spaces

3.3 Experiments involving successive outcomes; tree diagrams.

18.4 Successive outcomes18.5 Independent events18.6 Dependent events


4. Real Functions of a Real Variable and their Geometrical Representation

4.1 Dependent and independent variables. Functional notation. Range and domain.

7 Functions and relations7.1 Functions and relations


4.2 The graph of a function. Simple examples.

7.2 Sketching basic functions7.3 Absolute value functions


4.3 Algebraic representation of geometrical relationships.

Locus problems.

8 Locus and regions8.1 Locus8.2 Circles8.3 Further locus


4.4 Region and inequality. Simple examples.

8.4 Non-linear inequalities


5. Trigonometric Ratios—Review and Some Preliminary Results

5.1 Review of the trigonometric ratios, using the unit circle.

Chapter 5 Trigonometric ratios and applications5.1 Angles of any magnitude


5.2 Trigonometric ratios of: –, 90° – , 180° ± , 360° ± .

5.1 Angles of any magnitude5.5 Trigonometric identities


5.3 The exact ratios. 5.3 Exact values of trigonometric ratios5.4 More trigonometric exact values


5.4 Bearings and angles of elevation.

5.6 Direction and bearings5.7 Angles of elevation and depression


5.5 Sine and cosine rules for a triangle. Area of a triangle, given two sides and the included angle.

5.8 The sine rule5.9 The cosine rule5.10 Area of a triangle


6. Linear Functions and Lines

6.1 The linear functiony = mx + b and its graph.

6 Coordinate geometry—straight lines6.1 Gradient of a straight line

Outcomes: P1, P4

6.2 The straight line: equation of a line passing through a given point with given slope; equation of a line passing through two given points; the general equation ax + by + c = 0; parallel lines; perpendicular lines.

6.1 Gradient of a straight line6.2 Equation of a straight line

Outcomes: P1, P4

6.3 Intersection of lines: intersection of two lines and the solution of two linear equations in two unknowns; the equation of a line passing through the point of intersection of two given lines.

6.3 Intersection of two lines

Outcomes: P1, P4

6.4 Regions determined by lines: linear inequalities.

6.4 Region and inequalities6.5 Simultaneous linear inequalities

Outcomes: P1, P4

6.5 Distance between two points and the (perpendicular) distance of a point from a line.

6.6 Midpoint of an interval and distance between two points

6.7 Perpendicular distance of a point from a line

Outcomes: P1, P4

6.7 The mid-point of an interval. 6.6 Midpoint of an interval and distance between two points

Outcomes: P1, P4

6.8 Coordinate methods in geometry.

Chapter 11 Plane and coordinate geometrycoordinate geometry11.3 Harder intercept properties of

parallel lines 11.4 Coordinate methods in

geometry


7. Series and Applications

7.1 Arithmetic series. Formulae for the nth term and sum of n terms.

Chapter 16 Series and applications16.1 Series and sigma notation (∑)16.2 Arithmetic series


7.2 Geometric series. Formulae for the nth term and sum of n terms.

16.3 Finite geometric series


7.3 Geometric series with a ratio between –1 and 1. The limit of xn, as n ∞ for |x|<1, and the concept of limiting sum for a geometric series.

16.4 Infinite geometric series


7.5 Applications of arithmetic series.Applications of geometric series: compound interest, simplified hire purchase and repayment problems.Applications to recurring decimals.

16.5 Compound interest applications

16.6 Further applications of series


8. The Tangent to a Curve and the Derivative of a Function

8.1 Informal discussion of continuity.

10 Differential calculus 10.1 Continuity and gradients of tangents

Outcomes: P1, P6, P7, P8

8.2 The notion of the limit of a function and the definition of continuity in terms of this notion. Continuity of f + g, f – g, fg in terms of continuity of f and g.

10.2 Limit and continuity


8.3 Gradient of a secant to the curve y = f(x).

10.1 Continuity and gradients of tangents


8.4 Tangent as the limiting position of a secant. The gradient of the tangent.

Equations of tangent and normal at a given point of the curve y = f(x).

10.1 Continuity and gradients of tangents10.4 Finding the derivative from first

principles


8.5 Formal definition of the gradient of y = f(x) at the point where x = c.

Notations f '(c), at x = c.

10.3 Gradient of a curve at a point—formal definition

10.4 Finding the derivative from first principles


8.6 The gradient or derivative as a function.

Notations f '(x), , (f(x)), y'

10.5 Conditions for differentiability


8.7 Differentiation of xn for positive integral n.

The tangent to y = xn.

10.6 More derivatives from first principles


8.8 Differentiation of and x–1 from first principles. For the two functions u and v, differentiation of Cu (C constant), u + v, u – v, uv. The composite function rule. Differentiation of u/v.

10.6 More derivatives from first principles10.7 The product rule10.8 The chain rule10.9 The quotient rule


8.9 Differentiation of: general polynomial, xn for n rational, and functions of the form {f(x)}n or f(x)/g(x), where f(x), g(x) are polynomials.

10.6 More derivatives from first principles10.7 The product rule10.8 The chain rule10.9 The quotient rule


9. The Quadratic Polynomial and the Parabola

9.1 The quadratic polynomialax2 + bx + c. Graph of a quadratic function. Roots of a quadratic equation. Quadratic inequalities.

9. Quadratic functions and the parabola9.1 Quadratic functions9.4 Relationship between roots and

coefficients9.5 Sign of a quadratic function


9.2 General theory of quadratic equations, relation between roots and coefficients. The discriminant.

9.2 Parabolas and discriminants 9.6 Further examples involving

discriminants9.8 Solution set of simultaneous

equations


9.3 Classification of quadratic expressions; identity of two quadratic expressions.

9.7 Identity of two quadratic expressions


9.4 Equations reducible to quadratics.

9.3 Equations reducible to quadratics


9.5 The parabola defined as a locus. The equation x2 = 4Ay. Use of change of origin when vertex is not at (0, 0).

9.9 The parabola as a locus


10. Geometrical Applications of Differentiation

10.1 Significance of the sign of the derivative.

12 Geometrical applications of differentiation

12.1 The sign of the derivative

Outcomes: H1, H5, H6, H7, H9

10.2 Stationary points on curves. 12.2 The first derivative and turning points


10.3 The second derivative. The

notations f "(x), , y".

12.3 The second derivative and concavity


10.4 Geometrical significance of the second derivative.

12.3 The second derivative and concavity

12.4 The second derivative and turning points


10.5 The sketching of simple curves.

12.5 Sketching rational algebraic functions


10.6 Problems on maxima and minima.

12.6 Problem solving with derivatives


10.7 Tangents and normals to curves.

12.7 Tangents and normals to a curve


10.8 The primitive function and its geometrical interpretation.

12.8 Primitive functions


11. Integration

11.1 The definite integral. 13 Integral calculus13.1 Area under a curve13.2 The definite integral and the

area under a curve


11.2 The relation between the integral and the primitive function.

13.3 The definite integral and the primitive function


11.3 Approximate methods: trapezoidal rule and Simpson’s rule.

13.5 Approximate methods of integration—trapezoidal rule

13.6 Approximate methods of integration—Simpson’s rule


11.4 Applications of integration: areas and volumes of solids of revolution.

13.4 More areas13.7 Area between two curves13.8 Area bounded by the

y-axis13.9 Volume of solids of revolution13.10 Average value of a function—

an application of integration13.11 Indefinite integrals


12. Logarithmic and Exponential Functions

12.1 Review of index laws, and definition of ar for a>0, wherer is rational.

Chapter 14 Exponential and logarithmic functions 14.1 Index laws with integers as

indices14.2 Index laws with fractional

indices14.3 Solving equations with

exponents

Outcomes: H1, H3, H5, H6, H7, H8, H9

12.2 Definition of logarithm to the base a. Algebraic properties of logarithms and exponents.

14.4 Logarithms14.5 Solving equations involving

logarithms


12.3 The functions y = ax andy = logax for a>0 and real x. Change of base.

14.6 Exponential functions


12.4 The derivatives of y = ax and y = logax. Natural logarithms and exponential function.

14.8 Applications of the exponential function

14.9 Natural logarithms


12.5 Differentiation and integration of simple composite functions involving exponentials and logarithms.

14.7 Integrating the exponential function

14.10 Integrals resulting in logarithmic functions

14.11 Applications of the logarithmic function


13. The Trigonometric Functions

13.1 Circular measure of angles. Angle, arc, sector.

Chapter 15 Trigonometric functions

15.1 Radian measure of an angle

15.2 Arc length and sector area of a circle

15.3 Angles of any magnitude

Outcomes:H1, H4, H5, H6, H7, H8, H9

13.2 The functions sin x, cos x, tan x, cosec x, sec x, cot x and their graphs.

5.2 Trigonometric graphs and equations


15.4 Graphs of trigonometric functions

15.5 Solution of trigonometric equations


13.3 Periodicity and other simple properties of the functions sin x, cos x and tan x.

15.6 Graphical solution of equations


13.4 Approximations to sin x,cos x, tan x, when x is small.

The result = 1.

15.7 Approximations when x is small


13.5 Differentiation of cos x,sin x, tan x.

15.8 Derivatives of trigonometric functions


13.6 Primitive functions of sin x, cos x, sec2x.

15.9 Primitives of trigonometric functions


13.7 Extension of 13.2–13.6to functions of the forma sin(bx + c), etc.

15.10 Applications of trigonometric functions


14. Applications of Calculus to the Physical World

14.1 Rates of change as derivatives with respect to time.

The notation , , etc.

Chapter 17 Applications of calculus to the physical world

17.1 Gradient as a rate of change

Outcomes: H1, H3, H4, H5, H6, H7, H8, H9

14.2 Exponential growth and decay; rate of change of population; the equation

= kN, where k is the population

growth constant.

17.2 Exponential growth and decay


14.3 Velocity and acceleration as time derivatives. Applications involving:

(i) the determination of the velocity and acceleration of a particle given its distance from a point as a function of time

(ii) the determination of the distance of a particle from a given point, given its acceleration or velocity as a function of time together with appropriate initial conditions.

17.3 Motion of a particle in a straight line

17.4 Other examples of motion


curriculum audit for new senior mathematics 11 and 12 (hsc)

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