# Year 12 Methods (Units 3 and 4) Mathematics – Victoria

### VIC Year 12 Methods (Units 3 and 4) Mathematics

# | TOPIC | TITLE | |
---|---|---|---|

1 | Study Plan | Study plan – Year 12 | |

Objective: On completion of the course formative assessment a tailored study plan is created identifying the lessons requiring revision. | |||

2 | Simultaneous equns | Elimination method | |

Objective: On completion of the lesson the student will be able to solve 2 equations with 2 unknown variables by the elimination method. | |||

3 | Simultaneous equns | Elimination method part 2 | |

Objective: On completion of the lesson the student will be able to solve all types of simultaneous equations with 2 unknown variables by the elimination method. | |||

4 | Simultaneous equns | Applications of simultaneous equations | |

Objective: On completion of this lesson the student will be able to derive simultaneous equations from a given problem and then solve those simultaneous equations. | |||

5 | Difference of 2 squares | Difference of two squares | |

Objective: On completion of the lesson the student understand the difference of two squares and be capable of recognising the factors. | |||

6 | Common fact and diff | Common factor and the difference of two squares | |

Objective: On completion of the lesson the student will be aware of common factors and recognise the difference of two squares. | |||

7 | Quadratic trinomials | Quadratic trinomials [monic] – Case 1. | |

Objective: On completion of the lesson the student will understand the factorisation of quadratic trinomial equations with all terms positive. | |||

8 | Factorising quads | Factorising quadratic trinomials [monic] – Case 2. | |

Objective: On completion of the lesson the student will accurately identify the process if the middle term of a quadratic trinomial is negative. | |||

9 | Factorising quads | Factorising quadratic trinomials [monic] – Case 3. | |

Objective: On completion of the lesson the student will have an increased knowledge on factorising quadratic trinomials and will understand where the 2nd term is positive and the 3rd term is negative. | |||

10 | Factorising quads | Factorising quadratic trinomials [monic] – Case 4. | |

Objective: On completion of the lesson the student will understand how to factorise all of the possible types of monic quadratic trinomials and specifcally where the 2nd term and 3rd terms are negative. | |||

11 | Factorising quads | Factorisation of non-monic quadratic trinomials | |

Objective: On completion of the lesson the student will be capable of factorising any quadratic trinomial. | |||

12 | Factorising quads | Factorisation of non-monic quadratic trinomials – moon method | |

Objective: On completion of the lesson the student know two methods for factorisation of quadratic trinomials including the cross method. | |||

13 | Sum/diff 2 cubes | Sum and difference of two cubes. | |

Objective: On completion of the lesson the student will be cognisant of the sum and difference of 2 cubes and be capable of factorising them. | |||

14 | Algebraic fractions | Simplifying algebraic fractions. | |

Objective: On completion of the lesson the student should be familiar with all of the factorisation methods presented to this point. | |||

15 | Geometry-parabola | The parabola: to describe properties of a parabola from its equation | |

Objective: On completion of the lesson the student will be able to predict the general shape and important features of a parabola and then graph the parabola to check the predictions. | |||

16 | Functions and graphs | Quadratic polynomials of the form y = ax. + bx + c. | |

Objective: On completion of the lesson the student will be able to predict the general shape of a parabola and verify the predictions by sketching the parabola. The student will also be introduced to the discriminant and the axis. | |||

17 | Functions and graphs | Graphing perfect squares: y=(a-x) squared | |

Objective: On completion of the lesson the student will be able to analyse a curve and then check their work by graphing the curve. | |||

18 | Graphing roots | Graphing irrational roots | |

Objective: On completion of the lesson the student will be able to solve any polynomial which has real roots, whether they are rational or irrational. | |||

19 | Graphing-polynomials | Graphing complex polynomials: quadratics with no real roots | |

Objective: On completion of the lesson the student will be able to determine whether a quadratic has real or complex roots and then graph it. | |||

20 | Graphing-polynomials | General equation of a circle: determine and graph the equation | |

Objective: On completion of the lesson the student will be able to solve these types of problems. Working with circles will also help the student in the topic of circle geometry, which tests the student’s skills in logic and reasoning. | |||

21 | Graphing-cubic curves | Graphing cubic curves | |

Objective: On completion of this lesson the student will be able to graph a cubic given its equation or derive the equation of a cubic given its graph or other relevant information. | |||

22 | Absolute value equations | Absolute value equations | |

Objective: On completion of this lesson the student will be able to relate to graphs involving the absolute value function. The student will be capable of graphing the function given its equation and be able to solve for the intersection of an absolute value functio | |||

23 | Rect.hyperbola | The rectangular hyperbola. | |

Objective: On completion of the lesson the student will be able to analyse and graph a rectangular hyperbola and describe its important features. | |||

24 | Exponential function | The exponential function. | |

Objective: On completion of the lesson the student will be able to graph any equation in the form y equals a to the power x, where a is any positive real number apart from 1. | |||

25 | Log functions | Logarithmic functions. | |

Objective: On completion of this lesson the student will be able to define basic logarithmic functions and describe the relationship between logarithms and exponents including graph logarithmic functions. The student will understand the relationship between logarit | |||

26 | Functions | Notation and evaluations | |

Objective: On completion of the lesson the student will be understand different notations for functions. | |||

27 | Functions | More on domain and range | |

Objective: On completion of the lesson the student will be able to describe the domain and range using appropriate set notation. | |||

28 | Functions | Domain and range from graphical representations | |

Objective: On completion of the lesson the student will be able to describe the domain and range using appropriate set notation from graphical representations. | |||

29 | Functions | Evaluating and graphing piecewise functions | |

Objective: On completion of the lesson the student will be able to evaluate and graph piecewise functions. | |||

30 | Functions | Functions combinations | |

Objective: On completion of the lesson the student will be able to perform operations with functions while working with their domains. | |||

31 | Functions | Composition of functions | |

Objective: On completion of the lesson the student will understand composition of functions or a function of a function. | |||

32 | Functions | Inverse functions | |

Objective: On completion of the lesson the student will be able to find inverse functions, use the notation correctly and the horizontal line test will be used. | |||

33 | Functions | Rational functions Part 1 | |

Objective: On completion of the lesson the student will be able to work with the division of functions and to interpret this on the coordinate number plane showing vertical and horizontal asymptotes. | |||

34 | Functions | Rational functions Part 2 | |

Objective: On completion of the lesson the student will be able to use the degree of polynomials and polynomial division to assist in graphing rational functions on the coordinate number plane showing vertical, horizontal and slant asymptotes. | |||

35 | Functions | Parametric equations (Stage 2) | |

Objective: On completion of the lesson the student will be able to eliminate the parameter from a set of equations and identify appropriate restrictions on the domain and range. | |||

36 | Functions | Polynomial addition etc in combining and simplifying functions (Stage 2) | |

Objective: On completion of the lesson the student will have multiple techniques to understand and construct graphs using algebra. | |||

37 | Functions | Parametric functions (Stage 2) | |

Objective: On completion of the lesson the student will understand some standard parametric forms using trigonometric identities, appreciate the beauty of the the graphs that can be generated and an application to projectile motion. | |||

38 | Algebra-polynomials | Introduction to polynomials | |

Objective: On completion of the lesson the student will understand all the terminology associated with polynomials and be able to judge if any algebraic expression is a polynomial or not. | |||

39 | Algebra-polynomials | The sum, difference and product of two polynomials. | |

Objective: On completion of the lesson the student will be able to add subtract and multiply polynomials and find the degrees of the answers. | |||

40 | Algebra-polynomials | Polynomials and long division. | |

Objective: On completion of the lesson the student will understand the long division process with polynomials. | |||

41 | Remainder theorem | The remainder theorem. | |

Objective: On completion of the lesson the student will understand how the remainder theorem works and how it can be applied. | |||

42 | Remainder theorem | More on remainder theorem | |

Objective: On completion of the lesson the student will understand the remainder theorem and how it can be applied to solve some interesting questions on finding unknown coefficients of polynomials. | |||

43 | Factor theorem | The factor theorem | |

Objective: On completion of the lesson the student will be able to use the factor theorem and determine if a term in the form of x minus a is a factor of a given polynomial. | |||

44 | Factor theorem | More on the factor theorem | |

Objective: On completion of the lesson the student will fully understand the factor theorem and how it can be applied to solve some questions on finding unknown coefficients of polynomials. | |||

45 | Factor theorem | Complete factorisations using the factor theorem | |

Objective: On completion of the lesson the student will be able to factorise polynomials of a higher degree than 2 and to find their zeros. | |||

46 | Polynomial equations | Polynomial equations | |

Objective: On completion of the lesson the student will be capable of solving polynomial equations given in different forms. | |||

47 | Graphs, polynomials | Graphs of polynomials | |

Objective: On completion of the lesson the student will understand how to graph polynomials using the zeros of polynomials, the y intercepts and the direction of the curves. | |||

48 | Roots quad equations | Sum and product of roots of quadratic equations | |

Objective: On completion of the lesson the student will understand the formulas for the sum and product of roots of quadratic polynomials and how to use them. The student will understand how to form a quadratic equation given its roots. | |||

49 | Roots quad equations | Sum and product of roots of cubic and quartic equations | |

Objective: On completion of the lesson the student will be able to do problems on the sum and products of roots of cubic and quartic equations. | |||

50 | Approx roots | Methods of approximating roots | |

Objective: On completion of the lesson the student will be capable of finding approximate roots of polynomial equations using half the interval method. The student will be able to make a number of applications of this rule within the one question. | |||

51 | Newton’s approx | Newton’s method of approximation | |

Objective: On completion of the lesson the student will be able to use Newton’s method in finding approximate roots of polynomial equations and be capable of more than one application of this method. | |||

52 | Logarithms-Power of 2 | Powers of 2. | |

Objective: On completion of the lesson the student should be able to convert between logarithmic statements and index statements to the power of 2. | |||

53 | Logarithms-Equations and logs | Equations of type log x to the base 3 = 4. | |

Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the number from which the logarithm evolves. | |||

54 | Logarithms-Equations and logs | Equations of type log 32 to the base x = 5. | |

Objective: On completion of the lesson the student will have an enhanced understanding of the definition of a logarithm and how to use it to find an unknown variable which in this case is the base from which the number came. | |||

55 | Logarithms-Log laws | Laws of logarithms. | |

Objective: On completion of the lesson the student will be familiar with 5 logarithm laws. | |||

56 | Logarithms-Log laws expansion | Using the log laws to expand logarithmic expressions. | |

Objective: On completion of the lesson the student will be able to use the log laws to expand logarithmic expressions. | |||

57 | Logarithms-Log laws simplifying | Using the log laws to simplify expressions involving logarithms. | |

Objective: On completion of the lesson the student will be able to simplify logarithmic expressions using the log laws. | |||

58 | Logarithms-Log laws numbers | Using the log laws to find the logarithms of numbers. | |

Objective: On completion of the lesson the student will have an enhanced understanding of the use of the log laws and be able to do more applications with numerical examples. | |||

59 | Logarithms-Equations and logs | Equations involving logarithms. | |

Objective: On completion of the lesson the student will be able to solve equations with log terms. | |||

60 | Logarithms-Logs to solve equations | Using logarithms to solve equations. | |

Objective: On completion of the lesson the student will be able to use logarithms to solve index equations with the assistance of a calculator. | |||

61 | Logarithms-Change base formula | Change of base formula | |

Objective: On completion of the lesson the student will have seen the change of base formula for logarithms and be capable of using it to change the logarithm of one base to another base. | |||

62 | Logarithms-Graph-log curve | The graph of the logarithmic curve | |

Objective: On completion of the lesson the student will be able to draw a logarithmic curve to a given base and know the general properties of log curves. | |||

63 | Logarithms-Log curves | Working with log curves. | |

Objective: On completion of the lesson the student will be able to solve problems with log curves | |||

64 | Calculus | Limits | |

Objective: On completion of the lesson the student will be able to solve problems using limiting sum rule. | |||

65 | Calculus=1st prin | Differentiation from first principles. | |

Objective: On completion of the lesson the student will be able apply the first principles (calculus) formula to find the gradient of a tangent at any point on a continuous curve. | |||

66 | Calculus=1st prin | Differentiation of y = x to the power of n. | |

Objective: On completion of the Calculus lesson the student will be able to differentiate a number of expressions involving x raised to the power of n. | |||

67 | Calculus-differential, integ | Meaning of dy over dx – equations of tangents and normals. | |

Objective: On completion of the Calculus lesson the student will be able to apply differentiation and algebra skills to find the equation of the tangent and the normal to a point on a curve. | |||

68 | Calculus-differential, integ | Function of a function rule, product rule, quotient rule. | |

Objective: On completion of the Calculus lesson the student will understand how to use the chain rule, the product rule and the quotient rule. | |||

69 | Calculus-differential, integ | Increasing, decreasing and stationary functions. | |

Objective: On completion of the lesson the student will understand how to find the first derivative of various functions, and use it in various situations to identify increasing, decreasing and stationary functions. | |||

70 | Calculus | First Derivative – turning points and curve sketching | |

Objective: On completion of the Calculus lesson the student will be able to use the first derivative to find and identify the nature of stationary points on a curve. | |||

71 | Calculus-2nd derivative | The second derivative – concavity. | |

Objective: On completion of the Calculus lesson the student will be able to find a second derivative, and use it to find the domain over which a curve is concave up or concave down, as well as any points of inflexion. | |||

72 | Trig complementary angles | Complementary angle results. | |

Objective: On completion of the lesson the student will understand how to establish the complementary angle results for the sine and cosine ratios and then how to use these results to solve trig equations. | |||

73 | Trig identities | Trigonometric identities | |

Objective: On completion of the lesson the student will be able to simplify trigonometrical expressions and solve trigonometry equations using the knowledge of trig identities. | |||

74 | Trig larger angles | Angles of any magnitude | |

Objective: On completion of the lesson the student will be able to find the trigonometric values of angles of any magnitude by assigning angles to the four quadrants of the circle. | |||

75 | Trig larger angles | Trigonometric ratios of 0°, 90°, 180°, 270° and 360° | |

Objective: On completion of the lesson the student will learn how to find the Trigonometric Ratios of 0, 90, 180, 270 and 360 degrees. | |||

76 | Graph sine | Graphing the trigonometric ratios – I Sine curve. | |

Objective: On completion of the lesson the student will recognise and draw the sine curve exploring changes in amplitude and period. | |||

77 | Graph cosine | Graphing the trigonometric ratios – II Cosine curve. | |

Objective: On completion of the lesson the student will know how to recognise and draw the cosine curve exploring changes in amplitude and period. | |||

78 | Graphs tan curve | Graphing the trigonometric ratios – III Tangent curve. | |

Objective: On completion of the lesson the student will know how to recognise and draw the tan curve. | |||

79 | Graph reciprocals | Graphing the trigonometric ratios – IV Reciprocal ratios. | |

Objective: On completion of the lesson the student will know how to recognise and draw the curves of the reciprocal ratios: cosec, sec and cot. | |||

80 | Trig larger angles | Using one ratio to find another. | |

Objective: On completion of the lesson the student will find other trig ratios given one trig ratio and to work with angles of any magnitude. | |||

81 | Trig equations | Solving trigonometric equations – Type I. | |

Objective: On completion of the lesson the student will solve simple trig equations with restricted domains. | |||

82 | Trig equations | Solving trigonometric equations – Type II. | |

Objective: On completion of the lesson the student will solve trig equations with multiples of theta and restricted domains. | |||

83 | Trig equations | Solving trigonometric equations – Type III. | |

Objective: On completion of the lesson the student will solve trig equations with two trig ratios and restricted domains. | |||

84 | Absolute value or modulus | Simplifying absolute values | |

Objective: On completion of the lesson the student will be able to simplify expressions involving absolute values or the modulus of real numbers. | |||

85 | Absolute value or modulus | Solving for the variable | |

Objective: On completion of the lesson the student will be able to solve equations involving a single absolute value. | |||

86 | Absolute value or modulus | Solving and graphing inequalities | |

Objective: On completion of the lesson the student will be able to solve inequalities involving one absolute value. | |||

87 | Calculus – Curve sketching | Curve sketching | |

Objective: On completion of the Calculus lesson the student will be able to use the first and second derivatives to find turning points of a curve, identify maxima and minima, and concavity, then use this information to sketch a curve. | |||

88 | Calculus – Maxima minima | Practical applications of maxima and minima | |

Objective: On completion of the lesson the student will be able to apply calculus to a suite of simple maxima or minima problems. | |||

89 | Calculus – Integration | Integration – anti-differentiation, primitive function | |

Objective: On completion of the Calculus lesson the student will be able to use rules of integration to find primitives of some simple functions. | |||

90 | Calculus – Computation area | Computation of an area | |

Objective: On completion of the Calculus lesson the student will be able to select an appropriate formula to calculate an area, re-arrange an expression to suit the formula, and use correct limits in the formula to evaluate an area. | |||

91 | Calculus – Computation volumes | Computation of volumes of revolution | |

Objective: On completion of the Calculus lesson the student will know how to choose an appropriate volume formula, re-arrange an expression to suit the formula, and then calculate a result to a prescribed accuracy. | |||

92 | Calculus – Trapezoidal and Simpson’s rules | The Trapezium rule and Simpson’s rule | |

Objective: On completion of the Calculus lesson the student will know how to calculate sub-intervals, set up a table of values, then apply the Trapezoidal Rule, or Simpson’s Rule to approximate an area beneath a curve. | |||

93 | Statistics | Frequency distribution table | |

Objective: On completion of the lesson the student will be able to construct a frequency distribution table for raw data and interpret the table. | |||

94 | Statistics | Frequency histograms and polygons | |

Objective: On completion of the lesson the student will be able to construct and interpret frequency histograms and polygons. | |||

95 | Statistics | Relative frequency | |

Objective: On completion of the lesson the student will be able to collect, display and make judgements about data. | |||

96 | Statistic-probability | Calculating the median from a frequency distribution | |

Objective: On completion of the lesson the student will be able to determine the median from a cumulative frequency polygon. | |||

97 | Statistic-probability | Tree diagrams – not depending on previous outcomes | |

Objective: On completion of the lesson the student will be confident in drawing tree diagrams to list outcomes of a multi stage probability problem and then finding probabilities of certain events not depending on previous outcomes. | |||

98 | Statistic-probability | Tree diagrams – depending on previous outcomes | |

Objective: On completion of the lesson the student will be confident in drawing tree diagrams to list outcomes of other multi stage probability problems and then finding probabilities of certain events depending on previous outcomes. | |||

99 | Statistic-probability | The complementary result .. | |

Objective: On completion of the lesson the student will be capable of ascertaining the probability of certain results where the complementary event is involved. | |||

100 | Statistic-probability | P[A or B] When A and B are both mutually and NOT mutually exclusive | |

Objective: On completion of this lesson the student will be able to distinguish between mutually exclusive and non mutually exclusive events and be able to find the probabilities of both. | |||

101 | Statistic-probability | Binomial Theorem – Pascal’s Triangle | |

Objective: On completion of this lesson the student will use Pascal’s triangle and the binomial theorem to write the expansion of binomial expressions raised to integer powers. | |||

102 | Statistic-probability | Binomial probabilities using the Binomial Theorem | |

Objective: On completion of the lesson the student will be able to solve certain types of probability questions using the binomial theorem | |||

103 | Statistic-probability | Counting techniques and ordered selections – permutations | |

Objective: On completion of this lesson the student will be competent in using some new counting techniques used for solving probability. | |||

104 | Statistic-probability | Unordered selections – combinations | |

Objective: On completion of the lesson the student will be able to use the formula, n c r both with and without a calculator and be able to use it to solve probability problems where unordered selections happen. | |||

105 | Statistics – grouped data | Calculating mean, mode and median from grouped data | |

Objective: On completion of the lesson the student will be capable of identifying class centres, get frequency counts and determine the mean and mode values. | |||

106 | Statistics – Range and dispersion | Range as a measure of dispersion | |

Objective: On completion of the lesson the student will be able to determine the range and using it in decision making. | |||

107 | Statistics – Spread | Measures of spread | |

Objective: On completion of the lesson the student will be able to find the standard deviation, using a data set or a frequency distribution table and calculator. | |||

108 | Statistics – Standard deviation | Standard deviation applications | |

Objective: On completion of the lesson the student will be able to use standard deviation as a measure of deviation from a mean. | |||

109 | Statistics – Standard deviation | Normal distribution | |

Objective: On completion of the lesson the student will be able to use the standard deviation of a normal distribution to find the percentage of scores within ranges. | |||

110 | Statistics – Interquartile range | Measures of spread: the interquartile range | |

Objective: On completion of the lesson the student will be able to find the upper and lower quartiles and the interquartile range | |||

111 | Geometric transformations | Geometry transformations without matrices: reflection (Stage 2) | |

Objective: On completion of this lesson the student will use and understand the language used in geometric transformations and perform reflections in a number plane. | |||

112 | Geometric transformations | Geometry transformations without matrices: translation (Stage 2) | |

Objective: On completion of this lesson the student will perform translations in a number plane. | |||

113 | Geometric transformations | Geometry transformations without matrices: rotation (Stage 2) | |

Objective: On completion of this lesson the student will perform and construct rotations. | |||

114 | Geometric transformations | Geometry transformations without matrices: dilation or enlargement (Stage 2) | |

Objective: On completion of this lesson the student will perform the non-congruent transformation of dilation or emlargement and calculate scale factor. | |||

115 | Geometric transformations | The definition and concept of combined transformations resulting in an equivalent single transformation. | |

Objective: On completion of this lesson the student will combine reflections and glide transformations to produce single isometric transformations. | |||

116 | Logic | Inductive and deductive reasoning | |

Objective: On completion of this lesson the student will understand and use the terms hypothesis, conclusion, inductive and deductive. | |||

117 | Logic | Definition and use of counter examples | |

Objective: On completion of this lesson the student will be able to create counter examples to statements. | |||

118 | Logic | Indirect proofs | |

Objective: On completion of the lesson the student will be able to use indirect proofs by assuming the opposite of the statement being proved. | |||

119 | Logic | Mathematical induction | |

Objective: On completion of the lesson the student will be able to perform the process of mathematical induction for simple series. | |||

120 | Logic | Conditional statements (converse, inverse and contrapositive) (Stage 2) | |

Objective: On completion of the lesson the student will be able to form related conditional statements. | |||

121 | Exam | Exam – Year 12 – Mathematical Methods Units 3 and 4 | |

Objective: Exam |