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1 SECONDARY SCHOOL IMPROVEMENT PROGRAMME (SSIP) 2016 GRADE 12 SUBJECT: MATHEMATICS TEACHER NOTES (Page 1 of 15) © Gauteng Department of Education 2 PRE TEST AND POST TEST: The SSIP sessions will commence with a Pre test, which covers the subject content topics in Session 10 to 15. The tests must be handed to the learners and answers must be written on a sheet of paper with their name. The SSIP programme will conclude with a Post Test, which will again cover the subject content in Sessions 10 to 15. PLEASE ensure that learners adhere strictly to the time limit of 25 minutes. Mark Sheet: You will be required to provide a mark sheet for all the learners in your SSIP class, that indicates their pre test and post test marks. This mark sheet must be submitted to the SSIP Co-Ordinator at your venue, at the end of the SSIP holiday programme. PRE AND POST TEST MEMORANDUM : QUESTION 1 Discussion: C sin 50 sin 2(25) 2sin 25 cos 25 2sin 25 cos 25 cos 25 cos 25 QUESTION 2 Discussion: C cos 2 20 sin 20 cos 70 cos 2 20 sin 20 sin 20 cos 2 20 sin 2 20 cos 2(20) cos 40 QUESTION 3 Discussion: D (sin x cos x)2 sin 2 x 2sin x cos x cos 2 x (sin x cos x)2 1 2sin x cos x sin x cos x 1 2k QUESTION 4 Discussion: B 2cos 2 x 7 2cos 2 x 1 8 cos 2 x 8 © Gauteng Department of Education 3 The maximum value of cos2x is 1 since the range of y cos 2 x is [1;1] Therefore the maximum value of QUESTION 5 Discussion: D a0 a 20 2 mOB mOB mOA mOA cos 2 x 8 is 1 8 9 3 5a b2 a 5a 2 b2 ab 2a 10 2a . ab 10 10 b a QUESTION 6 Discussion: C f (0) a (0) b 2 b b 2 f ( x) ax 2 f (1) a(1) 2 5 a 2 a 3 f ( x) 3x 2 f (2) 3(2) 2 8 QUESTION 7 Discussion: Draw radius OD. 56 28 34 B ˆ 90 tangent radius ODE ˆ 56 int s of DOE x  28 at centre 2 at circumference © Gauteng Department of Education 4 QUESTION 8 Discussion: A ( 2 1)2 (3 8) (2 2 2 1) 2 (3 8) (3 2 2)(3 2 2) 9 4(2) 1 QUESTION 9 Discussion: C ( 2 1)2 (3 8) 4x 2 y 3z 9 x 22 x 2 y 2x y 3z 32 x z 2x y 2 z 2 (2 x)2 (2 x)2 4 x2 4 x2 8x2 QUESTION 10 C Discussion: 1 The graph is given by the equation y 2 QUESTION 11 C Discussion: The graph of the derivative of f ( x) 2 x3 6 x 4 is x2 1 f ( x) 6 x 2 6 which is the graph of a parabola. QUESTION 12 Discussion: D 3x 2 3x 3x f ( x) x 1 f ( x) f / ( x) 1 QUESTION 13 Discussion: D sin(40 10) sin 40 cos 20 cos 40 sin 20 © Gauteng Department of Education 5 It is also true that sin(40 10) sin 30 cos 60 QUESTION 14 Discussion: B cos 2 35 sin 2 (35) cos 2 35 sin(35) 2 cos 2 35 sin 35 2 cos 2 35 sin 2 35 cos 2(35) cos 70 QUESTION 15 B Discussion: The y-intercept of the line we want is 2 since this line passes through (0 ; 2) The line we want is parallel to y 3 x 4 . Therefore the gradient of the line we want is also 3 . Therefore the equation we want is y 3 x 2 QUESTION 16 D Discussion: The radius of the circle is r 48 16 3 4 3 QUESTION 17 D Discussion: The vertical asymptote is x 4 which means that x 4 is written in the denominator and the horizontal asymptote is y 1 . The equation can therefore be written in the form: y 1 1 or x4 y 1 1 x4 © Gauteng Department of Education 6 QUESTION 18 Discussion: D x 2 36 x 2 36 0 ( x 6)( x 6) 0 6 x 6 QUESTION 19 Discussion: A 2 x 0 x 2 x 2 QUESTION 20 D Discussion: The limit represents both the gradient of f at x 4 as well as the gradient of the tangent to f at x 4 (20 x 2) (40) SESSION NO: 10 (CONSOLIDATION) TOPIC: REVISION OF TRIGONOMETRY Trigonometry is a topic where learners can score a lot of marks. Unfortunately, many learners struggle with this topic and perform poorly in examinations. It is therefore so important to spend a lot of time on the basics. Make sure that your learners revise the basic principles of Trigonometry. Make sure that your learners master the concepts from Grade 10 which include: (a) Trigonometric ratios in right-angles triangles (determining lengths of sides and the size of angles) (b) The quadrants in which the trigonometric ratios are positive and negative. (c) Evaluating expressions using a calculator (rounding off is important) (d) Solving basic trigonometric equations using a calculator. Important concepts to master in Grade 11 include: (a) Problems involving Pythagoras. (b) Reduction formulae. Make sure that your learners understand that: © Gauteng Department of Education 7 sin(180 ) sin sin(180 ) sin cos(90 ) sin (1) (2) (3) but sin 2 (180 ) ( sin )2 sin but sin180 0 but cos(90 ) sin (4) cos2 225 cos2 45 2 cos 225 cos(180 45) 2 2 2 2 1 ( cos 45) 4 2 2 2 (c) Negative angles. Make sure that your learners understand that: sin() sin cos() cos tan() tan (a) (b) sin( 90) sin (90 ) sin(90 ) cos (d) Angles greater than 360 . Make sure that your learners understand that whenever the angle is greater than 360 , keep subtracting 360 from the angle until you get an angle in the interval 0 ;360 . For example, tan 765 tan(765 2(360)) tan 45 1 (e) General solutions of trigonometric equations. You can use either the reference angle approach (see Section B) or the following method: If cos a and 1 a 1 If sin a and 1 a 1 then sin 1 (a) k.360 (k Z) or 180 sin 1 (a) k.360 If tan a and a R (k Z) then tan 1 (a) k.180 (f) then cos 1 ( a) k.360 ( k Z) (k Z) When teaching the sine, cosine and area rules, ensure that your learners know when to use the rules. Focus on basic examples before doing the more complicated types involving more than one triangle. Emphasise the following: The sine rule is used when you are given: (1) Two sides and a non-included angle (2) More than one angle and a side The cosine rule is used when you are given: (1) Two sides and the included angle (2) Three sides The area rule is used when you are given: (1) Two sides and the included angle © Gauteng Department of Education 8 It is so important for learners to be able to integrate compound and double angles into the Grade 11 concepts, which include Pythagoras problems, identities, trigonometric equations, trigonometric graphs and the sine, cosine and area rules. Once you are confident that your learners have mastered the basics, then discuss the typical examination questions and then let them do the homework questions. SESSION NO: 11 (CONSOLIDATION) TOPIC: REVISION OF CALCULUS Please ensure that your learners understand the following well: First principles and differentiation (a) When determining the gradient from first principles, learners tend to make the following mistakes: f ( x) x 2 ( x h) 2 x 2 h 0 h [ f ( x ) is omitted in the second line and x2 should be ( x 2 ) ] f ( x) lim Many learners expand the expression ( x h) 2 incorrectly by writing: ( x 2 h 2 ) or x2 h2 or x2 2xh h2 Sometimes the limit symbol is ignored or written incorrectly: ( x 2 2 xh h 2 ) x 2 h 0 h ( x 2 2 xh h 2 ) x 2 f ( x) h 2 x 2 xh h 2 x 2 f ( x) h 2 2 xh h f ( x) h h(2 x h) f ( x) h f ( x) (2 x h) f ( x) lim (b) ( x 2 2 xh h 2 ) x 2 h 0 h 2 2 ( x 2 xh h ) x 2 lim h 0 h 2 x 2 xh h 2 x 2 lim h 0 h 2 2 xh h lim h h(2 x h) lim h (2 x h) f ( x) lim Learners often make mistakes with the different derivative notations: The symbol f ( x ) is either not used or introduced too early. © Gauteng Department of Education 9 3 2 x4 3 f ( x ) x 4 2 f ( x ) 6 x 5 f ( x) 3 2 x4 3 f ( x) x 4 2 f ( x) 6 x 5 f ( x) dy Dy is sometimes written as which is incorrect and learners Dx dx may also make the following mistakes: The symbol 3 4 x 2 3 y 4 x 41 2 y 3 2x dy 3 3 x 1 x 2 dx 2 2 y The symbol D x is sometimes handled incorrectly as follows: 3 Dx 4 2x 3 D x x 4 2 D x 6 x 5 (c) 3 Dx 4 2x 3 x 4 2 6 x 5 Learners often do not understand the meaning of f (a) and f (a) . Emphasise the difference: f (a) is the gradient of the function at x a whereas f (a) is the value of y corresponding to the value of x for the function. Cubic functions and tangents to functions at given points (a) (b) (c) (d) Make sure that learners know how to factorise cubic equations before sketching the graphs of cubic functions. Examiners often require learners to write the intercepts with the axes, stationary points and points of inflection in coordinate form ( a ; b) . Make sure that the learners are aware of this. Emphasise the relationship between the graph of a function and the graph of its derivative is important in that it explains to the learners why the second derivative is zero at a point of inflection. The point of inflection is determined by equating the second derivative to zero and solving for x. An alternative method is to add up the x-coordinates of the turning points and divide by 2. © Gauteng Department of Education 10 (e) A graph is concave up (happy) if f (a) 0 and concave down (sad) if f (a) 0 Problems involving maximum and minimum values Expressions involving area or volume can be seen as graphs of quadratic or cubic functions. This will assist learners to identify which particular value of x yields a maximum or minimum value. Concepts including derivative graphs, problems involving speed and displacement are also discussed in this lesson. SESSION NO: 12 (CONSOLIDATION) TOPIC: REVISION OF ANALYTICAL GEOMETRY Analytical Geometry is an important topic that carries a lot of marks in the matric final exam. Make sure that learners know the basic formulae and then practise lots of examples involving applications of these formulae. The properties of quadrilaterals are extremely important in Analytical Geometry. Make sure learners know how to prove that a quadrilateral is a parallelogram, rectangle, square, rhombus or trapezium by knowing the properties of these quadrilaterals. Completing the square is an important technique for determining the centre of a circle and its radius. Ensure that learners know how to do this. Determining the coordinates of the intercepts of a circle with the axes is important and learners often struggle with this concept. Re-writing a linear equation of the form ax by c in the form y ax q is essential when finding the gradient of the line. Make sure that your learners know how to do this. There are two methods of determining the equation of a straight line joining two points. The first method is to use the general equation y mx c in which m represents the gradient and the value of c can be determined by substituting one of the points on the line into this equation. The second method is to use the formula y y1 m( x x1 ) . All you have to do is now substitute m and a point ( x1 ; y1 ) on the line into this formula and the equation of the line is easily obtained. Some very advanced level 4 questions are included in this lesson. Question 3 in the typical exam questions involves many concepts. This question will challenge the top learners. Question 5 in the homework section is also quite challenging. © Gauteng Department of Education 11 SESSION NO: 13 (CONSOLIDATION) TOPIC: REVISION OF ALGEBRA It is important for learners to revise Grade 11 Algebra as it is tested extensively in Paper 1. When learners are required to solve a quadratic equation, always get the equation into its standard form ax2 bx c 0 . Then factorise if it is easy to do so or use the formula. Make sure that learners read the question to find out whether the answers must be left in surd form or in round off decimal form. b b 2 4ac x where a 0 The formula is: 2a Emphasise to the learners that the quadratic formula can be used to determine the solutions of any quadratic equation of the form ax2 bx c 0 where a 0 . For example, the quadratic equation x2 2 x 8 0 can be solved in two ways: Method 1 (Factorisation) Method 2 (Quadratic formula) x2 2 x 8 0 ( x 4)( x 2) 0 x 4 0 or x20 x 4 or x 2 (2) ( 2) 2 4(1)( 8) x 2(1) 2 36 2 6 2 2 x 4 or x 2 x Ensure that learners know how to round off decimal answers if the solutions of a quadratic equation are irrational. The use of a parabola is recommended when solving quadratic inequalities. Make sure that the coefficient of the term in x 2 is positive. Don’t forget to change the sign of the inequality when multiplying or dividing throughout by a negative. The method of elimination is far quicker with simultaneous linear equations than the method of substitution. With simultaneous equations involving linear and non-linear equations, always work with the linear equation first and choose a variable with a coefficient of 1. Make that variable the subject of the formula. This will make the substitution into the non-linear equation a much more efficient process. © Gauteng Department of Education 12 Make sure that learners do not confuse the difference between the concept of an undefined number and a non-real number. Undefined numbers occur when division by zero happens. The square root of a negative number is non-real. 1 For example, consider the expression x2 1 1 1 The expression will be undefined if x 2 since 22 0 0 The expression will be non-real if x 2 0 , i.e. x 2 Emphasise to learners that the nature of the roots of a quadratic equation is determined by the expression b2 4ac in the quadratic formula. Ensure that they know how to determine the nature of the roots of a quadratic equation without solving the equation. Learners must also be able to determine the value(s) of a variable for which the nature of the roots is given. The level 4 questions are quite challenging and in line with the new types of exam questions being asked. SESSION NO: 14 (CONSOLIDATION) TOPIC: REVISION OF FUNCTIONS Grade 11 functions needs to be revised early in the year. Before dealing with the Grade 11 functions, it is extremely important to ensure that your learners revise the concept of a function. Learners must understand the meaning of the notation f (2) 3 , which is so important in this chapter as well as in Calculus. In the new curriculum, the emphasis is on investigating the effect of the parameters a, b, p and q for the following functions: a y a( x p)2 q y q y ab x p q y ax q x p The mother graph for quadratic functions is y ax 2 . The graph of y a( x p)2 q is obtained by shifting the mother graph p units horizontally and then q units vertically. This method is highly effective since it makes the sketching of these functions meaningful. For example, the graph of y x 2 1 is the graph of y x 2 (mother graph) shifted 1 unit downwards. Immediately the learners can see that the graph has no x-intercepts without having to do an algebraic manipulation to get non-real solutions for the equation 0 x2 1 . © Gauteng Department of Education 13 Also, the graph of y x 2 1 is the graph of y x 2 (mother graph) shifted 1 unit downwards. Learners will clearly see that the graph has x-intercepts which can be determined algebraically. The graph of y 2( x 1) 2 8 is the graph of y 2 x 2 (mother graph) shifted 1 unit left and 8 units downwards. The concept of a “mother graph” can be applied to the other functions as well. With hyperbolic and exponential functions, the table method is used to shift points on the “mother graph”. The coordinates of the newly formed graph can then be indicated on the graph together with the asymptotes. Make sure that your learners know how to determine the equations of given graphs, understand the concepts of domain, range, increasing and decreasing. The graphical meaning of solving simultaneous equations is important. SESSION NO: 15 (CONSOLIDATION) TOPIC: REVISION OF GRADE 11 EUCLIDEAN GEOMETRY Grade 11 Geometry is very important as it impacts on Grade 12 Geometry. The nine circle geometry theorems must be understood and mastered by the learners in order for them to achieve success in solving riders. The first thing to do with the learners in this session is to revise the theorems thoroughly. The proofs of the four examinable theorems are present in the content notes (Section B) and it advisable to revise these in detail. Questions 1-3 in Section A will help learners to master the theorems using basic numerical types. In Question 4-5, learners will revise more complicated riders involving proving that quadrilaterals are cyclic and that lines are tangents to circles. Section C contains mixed numerical types as well as riders. It is important for the learners to work on all of these questions in preparation for exams. © Gauteng Department of Education 14 SSIP 2016: JUNE/JULY HOLIDAY PROGRAMME PRE AND POST TEST LEARNER RESULTS SHEET SSIP TEACHER NAME: __________________________________________ SUBJECT: _____________________________________________________ VENUE: ______________________________ DATE: __________________ LEARNER INITIALS AND SURNAME PRE TEST MARK / 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 28 30 © Gauteng Department of Education POST TEST IMPROVED MARK YES / NO / 15 LEARNER INITIALS AND SURNAME PRE TEST MARK / 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 Average: © Gauteng Department of Education POST TEST IMPROVED MARK YES / NO /