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

# TOPIC TITLE
1 Study Plan Study plan – High School – Year I – Science and Engineering
Objective: On completion of the course formative assessment a tailored study plan is created identifying the lessons requiring revision.
2 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.
3 Coordinate Geometry-midpoint, slope Mid-point formula
Objective: On completion of the lesson the student will be able to understand the mid point formula and use it practically.
4 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.
5 Geometry-constructions Circumcentre and incentre (Stage 2)
Objective: On completion of the lesson the student will be able geometrically construct the circumcentre and incentre for a triangle and to use Pythagoras’ Theorem to calculate values.
6 Geometry-constructions Orthocentre and centroids (Stage 2)
Objective: On completion of the lesson the student will be able geometrically construct the orthocentre and centroid for a triangle and to use algebra to calculate values.
Objective: On completion of the lesson the student will be able to use coordinate geometry to apply the midsegment properties of a triangle.
Objective: On completion of this lesson the student will understand the properties that classify quadrilaterals.
9 Geometry-quadrilaterals Using the Properties of a Parallelogram
Objective: On completion of this lesson the student will be able to use and prove the properties of a parallelogram.
10 Geometry-quadrilaterals Proving a Shape is a Parallelogram
Objective: On completion of this lesson the student will be able to use properties to prove a given quadrilateral is a parallelogram.
11 Geometry-quadrilaterals Properties of the Rectangle, Square and Rhombus
Objective: On completion of this lesson students will be able to use the properties of the rectangle, square and rhombus for formal proofs and to find values.
12 Geometry-quadrilaterals Properties of the Trapezium and Kite
Objective: On completion of this lesson students will be able to use the properties of the trapezium and kite for formal proofs and to find values.
Objective: On completion of this lesson the student will know the relationships between quadrilaterals and use coordinate methods to prove some of the properties.
14 Geometry-locus Constructions and loci – single condition
Objective: On completion of the lesson the student will understand the term locus and describe several using a single condition.
15 Geometry-locus Constructions and loci – multiple conditions
Objective: On completion of the lesson the student will describe a locus that satisfies multiple conditions on a number plane.
Objective: On completion of the lesson the student will be able to calculate the gradient of a line given its inclination, or angle to the positive direction of the x-axis; or its rise and run.
Objective: On completion of the lesson the student will be able to calculate the gradient of a line given any two points on the line and also be capable of checking whether 3 or more points lie on the same line and what an unknown point will make to parallel lines.
18 Trigonometry-ratios Using the calculator to find an angle given a trigonometric ratio.
Objective: On completion of the lesson the student will be capable of using a calculator to find the value of an unknown angle when given a trigonometric ratio.
19 Trigonometry- ratios Using the trigonometric ratios to find an angle in a right-angled triangle.
Objective: On completion of the lesson the student will be able to find the value of an unknown angle in a right angle triangle given the lengths of 2 of the sides.
20 Coordinate Geometry-straight line The straight line.
Objective: On completion of the lesson the student will be able to draw a line which is parallel to either axis and comment on its gradient, where that gradient exists.
21 Coordinate Geometry-slope, etc. Lines through the origin.
Objective: On completion of the lesson the student will be able to draw a line which passes through the origin of the form y=mx and comment on its gradient compared to the gradients of other lines through the origin and use the information to solve problems.
22 Coordinate Geometry-equation of line General form of a line and the x and y Intercepts.
Objective: On completion of the lesson the student will be able to change the equation of a straight line from the form, written as y=mx+c, into the general form and vice versa.
23 Coordinate Geometry-intercept Slope intercept form of a line.
Objective: On completion of the lesson the student will be able to find the slope and intercept given the equation and given the slope and intercept, derive the equation.
24 Coordinate Geometry-point slope Point slope form of a line
Objective: On completion of the lesson the student will understand how to derive the equation of a straight line given the gradient and a point on the line.
25 Co-ordinate Geometry-Two point formula Two point formula: equation of a line which joins a pair of points.
Objective: On completion of the lesson the student will be able to calculate the equation of a line given any two named points on the line.
26 Co-ordinate Geometry-Intercept form Intercept form of a straight line: find the equation when given x and y
Objective: On completion of the lesson the student will have an effective and efficient method for calculating the equation of a straight line.
27 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.
28 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.
29 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.
30 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.
31 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.
32 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.
33 Translations Transformations – reflections
Objective: On completion of the lesson the student will be able to take a pre-image and using the appropriate techniques, accurately show its image after reflection.
34 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.
35 Geometric transformations Geometry transformations without matrices: translation (Stage 2)
Objective: On completion of this lesson the student will perform translations in a number plane.
36 Geometric transformations Geometry transformations without matrices: rotation (Stage 2)
Objective: On completion of this lesson the student will perform and construct rotations.
37 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.
38 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.
39 Geometry-circles The equation of a circle: to find radii of circles
Objective: On completion of the lesson the student will be able to describe a circle mathematically given its equation or its graph. Additionally, the student will be able to work out the equation of a circle given its centre and radius.
40 Geometry-circles The semicircle: to select the equation given the semi circle and vice versa
Objective: On completion of the lesson the student will be able to sketch a semicircle given its equation and derive the equation of a given semicircle.
41 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.
42 Conic sections Circles
Objective: On completion of the lesson the student will identify the radius of a circle given in standard form.
43 Conic sections Ellipses
Objective: On completion of the lesson the student will identify focus, vertices and axes of an ellipse.
44 Circle Geometry Theorem – The tangent to a circle is perpendicular to the radius drawn to it at the point of contact.
Objective: On completion of the lesson the student will be able to prove that the tangent and the radius of a circle are perpendicular at the point of contact.
45 Circle Geometry Theorem – Tangents to a circle from an external point are equal.
Objective: On completion of the lesson the student will be able to prove that tangents to a circle from an external point are equal.
46 Circle Geometry-chords Theorem – The products of the intercepts of two intersecting chords are equal.
Objective: On completion of the lesson the student will be able to prove that ‘The product of the intercepts of two intersecting chords are equal.’, and use this result to complete questions that require this knowledge.
47 Circle Geometry-tangents Theorem – The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point. [Including Alternate Proof]
Objective: On completion of the lesson the student will be able to prove and apply ‘The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point ‘, and use this result to complete q
48 Circle Geometry Theorem – When circles touch, the line of the centres passes through the point of contact.
Objective: On completion of the lesson the student will be able to prove that ‘ When two circles touch, the line of the centres passes through the point of contact’, and use this result to complete questions that require it.
49 Circle Geometry-non-collinear Theorem – Any three non-collinear points lie on a unique circle whose centre is the point of concurrency of the perpendicular bisectors of the intervals joining these points.
Objective: On completion of the lesson the student will be able to prove that ‘ Any three non-collinear points lie on a unique circle whose centre is the point of concurrency of the perpendicular bisectors of the intervals joining these points’, and use this knowled
50 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.
51 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.
52 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.
53 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.
54 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.
55 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.
56 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.
57 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.
58 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
59 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.
60 Algebra-inequalities Solving Inequalities.
Objective: On completion of the lesson the student will understand the ‘greater than’ and ‘less than’ signs, and be able to perform simple inequalities.
61 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.
62 Functions Notation and evaluations
Objective: On completion of the lesson the student will be understand different notations for functions.
63 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.
64 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.
65 Functions Evaluating and graphing piecewise functions
Objective: On completion of the lesson the student will be able to evaluate and graph piecewise functions.
66 Functions Functions combinations
Objective: On completion of the lesson the student will be able to perform operations with functions while working with their domains.
67 Functions Composition of functions
Objective: On completion of the lesson the student will understand composition of functions or a function of a function.
68 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.
69 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.
70 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.
71 Trigonometry-ratios Trigonometric ratios.
Objective: On completion of the lesson the student will be able to identify the hypotenuse, adjacent and opposite sides for a given angle in a right angle triangle. The student will be able to label the side lengths in relation to a given angle e.g. the side c is op
72 Trigonometry-ratios Using the calculator.
Objective: On completion of the lesson the student will be able to use the calculator to find values for the sine, cosine and tangent ratios of acute angles.
73 Trigonometry-ratios Using the trigonometric ratios to find unknown length. [Case 1 Sine].
Objective: On completion of the lesson the student will be able to use the sine ratio to calculate lengths and distances.
74 Trigonometry-ratios Using the trigonometric ratios to find unknown length. [Case 2 Cosine].
Objective: On completion of the lesson the student will be able to use the cosine ratio to find the length of the adjacent side of a right angle triangle.
75 Trigonometry-ratios Using the trigonometric ratios to find unknown length. [Case 3 Tangent Ratio].
Objective: On completion of the lesson the student will be able to use the tangent ratio to calculate the length of the opposite side in a right angle triangle.
76 Trigonometry-ratios Unknown in the denominator. [Case 4].
Objective: On completion of the lesson the student will understand how to use the trig ratios to calculate lengths and distances when the denominator is unknown.
77 Trigonometry-compass Bearings – the compass.
Objective: On completion of the lesson the student will be able to identify compass bearings, compass bearings with acute angles and 3 figure bearings from true north.
78 Trigonometry-elevation Angles of elevation and depression.
Objective: On completion of the lesson the student will be able to identify angles of depression and angles of elevation, and the relationship between them.
79 Trigonometry-practical Trigonometric ratios in practical situations.
Objective: On completion of the lesson the student will be able to use trigonometric ratios to solve problems involving compass bearings and angles of depression and elevation.
80 Trigonometry-exact ratios Trigonometric ratios of 30., 45. and 60. – exact ratios.
Objective: On completion of the lesson the student will be able to find the exact sine, cosine and tangent ratios for the angles 30., 45.and 60.
81 Trigonometry-cosine rule The cosine rule to find an unknown side. [Case 1 SAS].
Objective: On completion of the lesson the student will be able to use the cosine rule to find the length of an unknown side of a triangle knowing 2 sides and the included angle.
82 Trigonometry-cosine rule The cosine rule to find an unknown angle. [Case 2 SSS].
Objective: On completion of the lesson the student will be able to find the size of an unknown angle of a triangle using the cosine rule given the lengths of the 3 sides.
83 Trigonometry-sine rule The sine rule to find an unknown side. Case 1.
Objective: On completion of the lesson the student will be able to use the Sine rule to find the length of a particular side when the student is given the sizes of 2 of the angles and one of the sides.
84 Trigonometry-sine rule The sine rule to find an unknown angle. Case 2.
Objective: On completion of the lesson the student will be able to use the sine rule to find an unknown angle when given 2 sides and a non-included angle.
85 Trigonometry-areas The area formula
Objective: On completion of the lesson the student will be able to use the sine formula for finding the area of a triangle given 2 sides and the included angle.
Objective: On completion of the lesson the student will understand simple quadratic equations.
Objective: On completion of the lesson the student will be able to find both roots of a quadratic equation by factorising.
Objective: On completion of the lesson the student will have gained more confidence in working with quadratic equations.
89 Quadratic equations Completing the square
Objective: On completion of the lesson the student will understand the process of completing the square.
Objective: On completion of the lesson the student will understand the reasoning behind completing the square.
Objective: On completion of the lesson the student will be familiar with the quadratic formula.
Objective: On completion of the lesson the student will be able to express a problem as a quadratic equation and then solve it.
Objective: On completion of the lesson the student will better understand why quadratic equations have two solutions and will be capable of solving quadratic equations and problems graphically..
94 Trig-reciprocal ratios Reciprocal ratios.
Objective: On completion of the lesson the student will be able to identify and use the reciprocal trigonometric ratios of sine, cosine and tan, that is, the cosecant, secant and cotangent ratios.
95 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.
96 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.
97 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.
98 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.
99 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.
100 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.
101 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.
102 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.
103 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.
104 Trig equations Solving trigonometric equations – Type I.
Objective: On completion of the lesson the student will solve simple trig equations with restricted domains.
105 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.
106 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.
107 Exam Exam – High School – Year I – Science and Engineering
Objective: Exam