| 1 |
Study Plan |
Study plan – Middle School – Year III |
| Objective: On completion of the course formative assessment a tailored study plan is created identifying the lessons requiring revision. |
| 2 |
Multiplication |
Multiples and factors of whole numbers |
| Objective: On completion of the lesson the student will be able to specify multiples and factors of whole numbers, and calculate the product of squared numbers. |
| 3 |
Surds |
Introducing surds |
| Objective: On completion of the lesson the student will be able to identify and know the properties of surds as irrational numbers and be able to distinguish them from rational numbers. |
| 4 |
Surds |
Some rules for the operations with surds |
| Objective: On completion of the lesson the student will know how to use the rules for division and multiplication of surds. |
| 5 |
Surds |
Simplifying surds |
| Objective: On completion of the lesson the student will know how to use the rules for simplifying surds using division and multiplication. |
| 6 |
Surds |
Creating entire surds |
| Objective: On completion of the lesson the student will be able to write numbers as entire surds and compare numbers by writing as entire surds |
| 7 |
Surds |
Adding and subtracting like surds |
| Objective: On completion of the lesson the student will be able to add and subtract surds and simplify expressions by collecting like surds. |
| 8 |
Surds |
Expanding surds |
| Objective: On completion of the lesson the student will be able to expand and then simplify binomial expressions involving surds. |
| 9 |
Surds |
Binomial expansions |
| Objective: On completion of the lesson the student will be able to expand and simplify the squares of binomial sums and differences involving surds. |
| 10 |
Surds |
Conjugate binomials with surds |
| Objective: On completion of the lesson the student will be able to expand and simplify conjugate binomial expressions involving surds. |
| 11 |
Surds |
Rationalising the denominator |
| Objective: On completion of the lesson the student will be able to rationalise denominators of fractions where the denominator involves surds. |
| 12 |
Algebraic expressions |
Algebraic expressions. |
| Objective: On completion of the lesson the student will understand some of the short cuts used in writing algebraic expressions, and the student will be able to write algebraic expressions down in a way that is easier to understand. |
| 13 |
Algebraic expressions |
Substitution into algebraic expressions. |
| Objective: On completion of the lesson the student will be able to replace pronumerals with numbers, and then perform the correct operations. |
| 14 |
Algebraic expressions |
Directed numbers: addition and subtraction. |
| Objective: On completion of the lesson the student will be able to add and subtract positive and negative numbers in any combination, and understand adding and subtracting positive and negative pronumerals. |
| 15 |
Algebraic expressions |
Directed numbers: multiplication and division. |
| Objective: On completion of the lesson the student will understand which combinations of signs produce a positive answer and which ones produce a negative answer. |
| 16 |
Algebraic expressions |
Simplifying algebraic expressions: adding like terms. |
| Objective: On completion of the lesson the student will be able to simplify and evaluate numerical expressions containing patterns, and be able to simplify algebraic expressions that contain like terms. |
| 17 |
Algebraic expressions |
Simplifying algebraic Expressions: subtracting like terms. |
| Objective: On completion of the lesson the student will be able to recognise the difference between like and unlike terms, and be able to simplify an expression using subtraction. |
| 18 |
Algebraic expressions |
Simplifying Algebraic expressions: combining addition and subtraction. |
| Objective: On completion of the lesson the student will understand how to approach algebraic expressions questions and avoid the most common mistakes. |
| 19 |
Algebraic expressions |
Simplifying algebraic expressions: multiplication |
| Objective: On completion of the lesson the student will be able to simplify expressions involving multiplication of pronumerals and write them in the simplest form. |
| 20 |
Algebraic expressions |
Simplifying algebraic expressions: division |
| Objective: On completion of the lesson the student will be able to use all the operations needed for simplifying algebraic expressions. |
| 21 |
Algebraic expressions |
Expanding algebraic expressions: multiplication |
| Objective: On completion of the lesson the student will be able mentally to multiply and remove parentheses from simple algebraic expressions in one step. |
| 22 |
Algebraic expressions |
Expanding algebraic expressions: negative multiplier |
| Objective: On completion of the lesson the student will be able to expand expressions using a negative multiplier. |
| 23 |
Algebraic expressions |
Expanding and simplifying algebraic expressions |
| Objective: On completion of the lesson the student will be familiar with expanding and simplifying algebraic expressions. |
| 24 |
Surds |
Rationalising binomial denominators |
| Objective: On completion of the lesson the student will be able to rationalise denominators of fractions where the denominator involves binomial expressions. |
| 25 |
Graphing binomials |
Binomial products. |
| Objective: On completion of the lesson the student will understand the term binomial product and be capable of expanding and simplifying an expression. |
| 26 |
Graphing binomials |
Binomial products with negative multiplier |
| Objective: On completion of the lesson the student will understand specific terms and be prepared to expand and simplify different monic binomial products. |
| 27 |
Graphing binomials |
Binomial products [non-monic]. |
| Objective: On completion of the lesson, the student will have examined more complex examples with binomial products. |
| 28 |
Squaring binomial |
Squaring a binomial. [monic] |
| Objective: On completion of the lesson the student should understand the simple one-step process of squaring a monic binomial. |
| 29 |
Squaring binomial |
Squaring a binomial [non-monic]. |
| Objective: On completion of the lesson the student will apply the same rule that is used with monic binomials. |
| 30 |
Factorising |
Expansions leading to the difference of two squares |
| Objective: On completion of the lesson the student will understand expansions leading to differences of 2 squares. |
| 31 |
Algebraic expressions-products |
Products in simplification of algebraic expressions |
| Objective: On completion of the lesson the student will understand simplification of algebraic expressions in step-by-step processing. |
| 32 |
Algebraic expressions-larger expansions |
Algebraic Expressions – Larger expansions. |
| Objective: On completion of the lesson the student will be capable of expanding larger algebraic expressions. |
| 33 |
Algebra-highest common factor |
Highest common factor. |
| Objective: On completion of the lesson the student will be capable of turning a simple algebraic expression into the product of a factor in parentheses and identifying the highest common factors of the whole expression. |
| 34 |
Factors by grouping |
Factors by grouping. |
| Objective: On completion of the lesson the student will be able to complete the process given just two factors for the whole expression. |
| 35 |
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. |
| 36 |
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. |
| 37 |
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. |
| 38 |
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. |
| 39 |
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. |
| 40 |
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. |
| 41 |
Factorising quads |
Factorisation of non-monic quadratic trinomials |
| Objective: On completion of the lesson the student will be capable of factorising any quadratic trinomial. |
| 42 |
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. |
| 43 |
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. |
| 44 |
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. |
| 45 |
Algebra-factorising |
Simplifying easy algebraic fractions. |
| Objective: On completion of the lesson the student will understand how to simplify algebraic fractions by factorising. |
| 46 |
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. |
| 47 |
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. |
| 48 |
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. |
| 49 |
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. |
| 50 |
Quadratic equations |
Introduction to quadratic equations. |
| Objective: On completion of the lesson the student will understand simple quadratic equations. |
| 51 |
Quadratic equations |
Quadratic equations with factorisation. |
| Objective: On completion of the lesson the student will be able to find both roots of a quadratic equation by factorising. |
| 52 |
Quadratic equations |
Solving quadratic equations. |
| Objective: On completion of the lesson the student will have gained more confidence in working with quadratic equations. |
| 53 |
Quadratic equations |
Completing the square |
| Objective: On completion of the lesson the student will understand the process of completing the square. |
| 54 |
Quadratic equations |
Solving quadratic equations by completing the square |
| Objective: On completion of the lesson the student will understand the reasoning behind completing the square. |
| 55 |
Quadratic equations |
The quadratic formula |
| Objective: On completion of the lesson the student will be familiar with the quadratic formula. |
| 56 |
Quadratic equations |
Problem solving with quadratic equations |
| Objective: On completion of the lesson the student will be able to express a problem as a quadratic equation and then solve it. |
| 57 |
Quadratic equations |
Solving simultaneous quadratic equations graphically |
| 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.. |
| 58 |
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. |
| 59 |
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. |
| 60 |
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. |
| 61 |
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. |
| 62 |
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. |
| 63 |
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. |
| 64 |
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. |
| 65 |
Statistics |
Frequency histograms and polygons |
| Objective: On completion of the lesson the student will be able to construct and interpret frequency histograms and polygons. |
| 66 |
Statistics |
Relative frequency |
| Objective: On completion of the lesson the student will be able to collect, display and make judgements about data. |
| 67 |
Statistics |
The range. |
| Objective: On completion of the lesson the student will be able to determine the range of data in either raw form or in a frequency distribution table. |
| 68 |
Statistic-probability |
The mode |
| Objective: On completion of the lesson the student will understand how to find the mode from raw data, a frequency distribution table and polygon. |
| 69 |
Statistic-probability |
The mean |
| Objective: On completion of the lesson the student will be able to calculate means from raw data and from a frequency table using an fx column. |
| 70 |
Statistic-probability |
The median |
| Objective: On completion of the lesson the student will be able to determine the median of a set of raw scores |
| 71 |
Statistic-probability |
Cumulative frequency |
| Objective: On completion of the lesson the student will be able to construct cumulative frequency columns, histograms and polygons. |
| 72 |
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. |
| 73 |
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. |
| 74 |
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. |
| 75 |
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. |
| 76 |
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. |
| 77 |
Pythagoras |
Find the hypotenuse |
| Objective: On completion of this lesson the student will be able to use Pythagoras’ Theorem to calculate the length of the hypotenuse. |
| 78 |
Pythagoras |
Pythagorean triples |
| Objective: On completion of the lesson the student will be able to use the 3-4-5 Pythagorean triple. |
| 79 |
Pythagoras |
Find the hypotenuse Part 2 |
| Objective: On completion of this lesson the student will be able to use Pythagoras’ Theorem to calculate the length of the hypotenuse using decimals and surds. |
| 80 |
Pythagoras |
Calculating a leg of a right-angled triangle |
| Objective: On completion of this lesson the student will be able to use Pythagoras’ Theorem to calculate the length of one of the shorter sides of a right triangle. |
| 81 |
Pythagoras |
Proofs of Pythagoras theorem |
| Objective: On completion of this lesson the student will have geometric proofs for Pythagoras’ Theorem |
| 82 |
Circle Geometry |
Theorem – Equal arcs on circles of equal radii subtend equal angles at the centre. Theorem – Equal angles at the centre of a circle on equal arcs. |
| Objective: On completion of the lesson the student will be able to prove that ‘Equal arcs on circles of equal radii, subtend equal angles at the centre’, and that ‘Equal angles at the centre of a circle stand on equal arcs.’ They should then be able to use these pro |
| 83 |
Circle Geometry |
Theorem – The perpendicular from the centre of a circle to a chord bisects the chord. Theorem – The line from the centre of a circle to the mid-point of the chord is perpendicular to the chord. |
| Objective: On completion of the lesson the student will be able to prove that ‘The perpendicular from the centre of a circle to a chord bisects the chord.’ and its converse theorem ‘The line from the centre of a circle to the mid-point of the chord is perpendicular’ |
| 84 |
Circle Geometry |
Theorem – Equal chords in equal circles are equidistant from the centres. Theorem – Chords in a circle which are equidistant from the centre are equal. |
| Objective: On completion of the lesson the student will be able to prove that equal chords in equal circles are equidistant from the centre. |
| 85 |
Circle Geometry |
Theorem – The angle at the centre of a circle is double the angle at the circumference standing on the same arc. |
| Objective: On completion of the lesson the student will be able to prove that the angle at the centre of a circle is double the angle at the circumference standing on the same arc. |
| 86 |
Circle Geometry |
Theorem – Angles in the same segment of a circle are equal. |
| Objective: On completion of the lesson the student will be able to prove that the angles in the same segment are equal. |
| 87 |
Circle Geometry |
Theorem – The angle of a semi-circle is a right angle. |
| Objective: On completion of the lesson the student will be able to prove that ‘The angle of a semi-circle is a right-angle.’ |
| 88 |
Circle Geometry |
Theorem – The opposite angles of a cyclic quadrilateral are supplementary. |
| Objective: On completion of the lesson the student will be able to prove that the opposite angles of a cyclic quadrilateral are supplementary. |
| 89 |
Circle Geometry |
Theorem – The exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite angle. |
| Objective: On completion of the lesson the student will be able to prove that the exterior angle at a vertex of a cyclic quadrilateral equals the interior opposite. |
| 90 |
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. |
| 91 |
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. |
| 92 |
Circle Geometry |
Theorem – The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. |
| Objective: On completion of the lesson the student will be able to prove that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. |
| 93 |
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 |
| 94 |
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. |
| 95 |
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. |
| 96 |
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. |
| 97 |
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. |
| 98 |
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. |
| 99 |
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. |
| 100 |
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. |
| 101 |
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. |
| 102 |
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. |
| 103 |
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. |
| 104 |
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. |
| 105 |
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. |
| 106 |
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. |
| 107 |
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. |
| 108 |
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. |
| 109 |
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. |
| 110 |
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. |
| 111 |
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. |
| 112 |
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. |
| 113 |
Exam |
Exam – Middle School – Year III |
| Objective: Exam |
