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### High School Year III – Science and Engineering Mathematics

# TOPIC TITLE
1 Study Plan Study plan – High School – Year III – Science and Engineering
Objective: On completion of the course formative assessment a tailored study plan is created identifying the lessons requiring revision.
2 Algebraic equations Equations involving grouping symbols.
Objective: On completion of the lesson the student will be able to solve equations using grouping symbols
3 Algebraic equations Equations involving fractions.
Objective: On completion of the lesson the student will know how to solve equations using fractions.
4 Algebra- formulae Equations resulting from substitution into formulae.
Objective: On completion of the lesson the student will be able to substitute into formulae and then solve the resulting equations.
5 Algebra-factorising Simplifying easy algebraic fractions.
Objective: On completion of the lesson the student will understand how to simplify algebraic fractions by factorising.
6 Algebraic fractions Simplifying algebraic fractions using the index laws.
Objective: On completion of the lesson the student will be able to simplify most algebraic fractions using different methodologies.
7 Algebra-negative indices Algebraic fractions resulting in negative indices.
Objective: On completion of the lesson the student will be able to understand how to simplify an algebraic fractional expression with a negative index, and also how to write such an expression without a negative index.
8 Factorisation Factorisation of algebraic fractions including binomials.
Objective: On completion of the lesson the student should be able to simplify more complex algebraic fractions using a variety of methods.
9 Algebraic fractions-binomial Cancelling binomial factors in algebraic fractions.
Objective: On completion of the lesson the student should be able to factorise binomials to simplify fractions.
10 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.
11 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.
12 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.
13 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.
14 Coordinate geometry Solve by graphing
Objective: On completion of the lesson students will use the slope intercept form of a line to create graphs and find points of intersection.
15 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.
16 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.
17 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.
18 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
19 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.
20 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.
21 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
22 Conic sections Introduction to conic sections and their general equation
Objective: On completion of the lesson the student will identify the conic section from the coefficients of the equation.
23 Conic sections The parabola x. = 4ay
Objective: On completion of the lesson the student will identify the focus and directrix for a parabola given in standard form.
24 Conic sections Circles
Objective: On completion of the lesson the student will identify the radius of a circle given in standard form.
25 Conic sections Ellipses
Objective: On completion of the lesson the student will identify focus, vertices and axes of an ellipse.
26 Conic sections Hyperbola
Objective: On completion of the lesson the student will identify focus, vertices, axes and asymptotes of a hyperbola.
27 Functions Definition, domain and range
Objective: On completion of this lesson the student will be able to select functions from relations by referring to the domain and range.
28 Functions Notation and evaluations
Objective: On completion of the lesson the student will be understand different notations for functions.
29 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.
30 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.
31 Functions Evaluating and graphing piecewise functions
Objective: On completion of the lesson the student will be able to evaluate and graph piecewise functions.
32 Functions Functions combinations
Objective: On completion of the lesson the student will be able to perform operations with functions while working with their domains.
33 Functions Composition of functions
Objective: On completion of the lesson the student will understand composition of functions or a function of a function.
34 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.
35 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.
36 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.
37 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.
38 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.
39 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.
40 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.
41 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.
42 Algebra-polynomials Polynomials and long division.
Objective: On completion of the lesson the student will understand the long division process with polynomials.
43 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.
44 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.
45 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.
46 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.
47 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.
48 Polynomial equations Polynomial equations
Objective: On completion of the lesson the student will be capable of solving polynomial equations given in different forms.
49 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.
50 Calculus Limits
Objective: On completion of the lesson the student will be able to solve problems using limiting sum rule.
51 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.
52 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.
53 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.
54 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.
55 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.
56 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.
57 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.
58 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.
59 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.
60 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.
61 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.
62 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.
63 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.
64 Similar triangles Similar triangles
Objective: On completion of the lesson the student will be able to identify which test to use to show two triangles are similar.
65 Similar triangles Using similar triangles to calculate lengths
Objective: On completion of the lesson the student will be able to calculate lengths using similar triangles.
66 Overlapping triangles Examples involving overlapping triangles
Objective: On completion of the lesson the student will be able to calculate unknown sides in overlapping or adjacent similar triangles.
67 Geometry – triangles Triangle inequality theorem
Objective: On completion of the lesson the student will understand and use the triangle inequality theorem.
68 Geometry To identify collinear points, coplanar lines and points in 2 and 3 dimensions
Objective: On completion of the lesson the student will use correct terms to describe points, lines, intervals and rays.
69 Coordinate Geometry-the plane Distance formula.
Objective: On completion of the lesson the student will be able to calculate the distance between any two points on the number plane and interpret the results.
70 Co-ordinate Geometry-Parallel lines equations Parallel lines: identify equation of a line parallel to another
Objective: On completion of the lesson the student will be able to decide if two or more lines are parallel or not and to solve problems involving parallel lines.
71 Co-ordinate Geometry-Perpendicular lines Perpendicular lines.
Objective: On completion of the lesson the student will be able to derive the equation of a line, given that it is perpendicular to another stated line.
72 Co-ordinate Geometry-Inequalities Inequalities on the number plane.
Objective: On completion of the lesson the student will be able to derive the expression for an inequality given its graph. The student will also be able to solve some problems using inequalities.
73 Co-ordinate Geometry-Theorems Perpendicular distance
Objective: On completion of the lesson the student will be able to derive the formula to calculate the distance between a given point and a given line. The student will also be able to calculate the distance between parallel lines.
74 Co-ordinate Geometry-Theorems Line through intersection of two given lines
Objective: On completion of the lesson the student will be able to calculate the equation of a line which goes through the intersection of two given lines and also through another named point or satisfies some other specified condition.
75 Co-ordinate Geometry-Theorems Angles between two lines
Objective: On completion of the lesson the student will be able to calculate the angle between given lines and derive the equation of a line given its angle to another line.
76 Co-ordinate Geometry-Theorems Internal and external division of an interval
Objective: On completion of the lesson the student will be able to divide an interval according to a given ratio and to calculate what point divides an interval in a given ratio for both internal and external divisions.
77 Vectors Vectors
Objective: On completion of the lesson the student will be able to represent a vector in matrix and diagrammatic form, as well as add two vectors using matrices and/or a diagram.
78 Simultaneous equations Number of solutions (Stage 2)
Objective: On completion of the lesson of the lesson the student will identify simultaneous equations that are consistent, inconsistent or the same.
79 Vectors 2 vector addition in 2 and 3D (stage 2)
Objective: On completion of the lesson the student will understand and use component forms for vector resolution.
80 Matrices Basic concepts – Matrices
Objective: On completion of the lesson the student will have had an introduction to matrices
81 Matrices Addition and subtraction of matrices
Objective: On completion of this lesson the student will be able to recognise when addition and subtraction of matrices is possible, and perform these operations.
82 Matrices Scalar matrix multiplication
Objective: On completion of this lesson the student will be able to perform scalar multiplication of a matrix.
83 Matrices Multiplication of one matrix by another matrix
Objective: On completion of the lesson the student will be able to state whether matrix by matrix multiplication is possible, predict the order of the answer matrix, and then perform matrix by matrix multiplication.
84 Matrices Translation in the number plane
Objective: On completion of the lesson the student will be able to place ordered pairs into a matrix, then perform translation by addition using a transformation matrix, then extract ordered pairs from an answer matrix.
85 Matrices Translation by matrix multiplication
Objective: On completion of the lesson the student will be able to convert ordered pairs to elements of a matrix, multiply matrices together, where possible, and interpret the answer matrix on a number plane.
86 Transformations Special transformations – reflections, rotations and enlargements.
Objective: On completion of the lesson the student will be able to perform transformations: to rotate, reflect and change the size of various shapes and or points where applicable.
87 Polar coordinates Plotting polar coordinates and converting polar to rectangular
Objective: On completion of the lesson the student will understand the polar coordinate system and relate this to the rectangular coordinate system.
88 Polar coordinates Converting rectangular coordinates to polar form
Objective: On completion of the lesson the student will understand the polar coordinate system and report these from rectangular coordinates.
89 Polar coordinates Write and graph points in polar form with negative vectors (Stage 2)
Objective: On completion of the lesson the student will be using negative angles and negative vector lengths.
90 Exam Exam – High School – Year III – Science and Engineering
Objective: Exam