Download teacher notes | sessions 10 - 15 - Sci

1
SECONDARY SCHOOL IMPROVEMENT
PROGRAMME (SSIP) 2016
GRADE 12
SUBJECT:
MATHEMATICS
TEACHER NOTES
(Page 1 of 15)
© Gauteng Department of Education
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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
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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
a0 a

20 2
 mOB
mOB 
mOA 
mOA
cos 2 x  8 is 1  8  9  3
5a
b2
a 5a

2 b2
 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  Â  28  at centre  2   at circumference
© Gauteng Department of Education
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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
x2
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
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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
x4
y
1
1
x4
© Gauteng Department of Education
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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
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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
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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
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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 41
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
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(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
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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
x20
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
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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
x2
1
1
1
The expression will be undefined if x  2 since


22
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
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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
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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
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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
/