1
SECONDARY SCHOOL IMPROVEMENT
PROGRAMME (SSIP) 2015
GRADE 12
SUBJECT:
MATHEMATICS
TEACHER SUGGESTIONS
(PAGE 1 OF 14)
© Gauteng Department of Education
2
SESSION NO:
15
TOPIC: EUCLIDEAN GEOMETRY (REVISION OF GR 11 CIRCLE GEOMETRY)
Teacher note:
Before tackling the typical examination questions, it is of utmost importance to first
discuss the content notes which revise each of the circle theorems. Make sure that
the learners understand each theorem. Also revise the proofs of the theorems that
can be tested in the matric final examination. Ensure that the learners understand
how the theorem is proved rather than the rote learning of concepts.
Once you are satisfied that the learners know the theorems, move on to the typical
examination questions. The first three questions ask basic questions using the
theorems. Work through these questions thoroughly. Questions 4 and 5 are more
traditional but are basic examination questions to get the learners going with
geometry. The homework questions take the learners further into numerical types as
well as non-numerical riders.
In Consolidation Session 19, learners are required to deal with more advanced types
of questions. In this session, the emphasis is on basic revision and understanding of
geometry.
Make sure that the learners use proper layout and correct reasons for statements
made. Knowing how to prove that a quadrilateral is cyclic or that a line is a tangent to
a circle at a point is extremely important.
When answering geometry questions, it is important for learners to fill in useful
information on the diagram before trying to do a formal answer. Brainstorming the
diagram is an important first step in the process of doing geometry. Let the learners
use different colours to mark off equal angles.
Don’t forget to revise the “plus” statements which include:
 the sum of the interior angles of a triangle
 the exterior angle of a triangle equals the sum of the interior opposite angles
 the opposite angles of a cyclic quadrilateral add up to one hundred and eighty
degrees.
SESSION NO:
16
TOPIC: EUCLIDEAN GEOMETRY (GRADE 12 TRIANGLE GEOMETRY)
Teacher note:
Before tackling the typical examination questions, it is of utmost importance to first
discuss the content notes which revise each of the triangle theorems. Make sure that
© Gauteng Department of Education
3
the learners understand each theorem. Also revise the proofs of the theorems that
can be tested in the matric final examination. Ensure that the learners understand
how the theorem is proved rather than the rote learning of concepts.
Once you are satisfied that the learners know the theorems, move on to the typical
examination questions and discuss each question in detail. The homework questions
will reinforce the concepts.
In Consolidation Session 19, learners are required to deal with more advanced types
of questions. In this session, the emphasis is on basic revision and understanding of
triangle geometry.
Make sure that the learners use proper layout and correct reasons for statements
made.
Before teaching Grade 12 geometry, make sure that learners master the following
important concepts required for triangle geometry.
The technique of cross multiplication is extremely useful when working with ratios
in triangle geometry.
Here is a quick revision of the technique of cross multiplication:
Consider the lengths AB, DE, AC and DF:
AB AC
If
then:

DE DF
DE DF
AB DE
(a)
(b)


AB AC
AC DF
DF AC
(c)
(d)
AB  DF  AC  DE

DE AB
AB  DF
AC  DE
(e)
(f)
 AC
AB 
DE
DF
Make sure that your learners know the meaning of ratios. For example, the ratio
AB 2
 does not necessarily mean that the length of AB is 2 and the length of BC is
BC 3
2 (see content notes).
AB 8cm

it is clear that the lengths of the line segments are given. The
BC 12cm
AB 8cm 2(4)cm 2



ratio can be simplified as follows:
BC 12cm 3(4)cm 3
On the diagrams, it is necessary to write the lengths as 2p and 3p if the ratio 2:3 is
In the ratio
given and the actual lengths are unknown.
© Gauteng Department of Education
4
When working with similar triangles, always make sure that the pairs of equal angles
correspond. For example:
If there are two triangles that must be proved similar, match the equal angles:
Â1
 2
B̂1
B̂2
Ĉ
D̂
When proving that the two triangles are similar, make sure that the equal angles
correspond: ABC|||DBA and not ABC|||ABD .
In some of the typical examination and homework questions, learners will be required
to locate triangles. The following strategy will help you in these types of examples.
The strategy outlined below is taken from the DoE approved Grade 12 Mind Action
Series Mathematics textbook (Allcopy Publishers). Permission to use this strategy
has been obtained from the authors.
1.
Try taking letters from the top and bottom and see if you can locate similar
triangles.
BC AB

AC AD
Refer to the diagram and see which triangles appear:
Is ABC a triangle?
Is ABD a triangle?
If there are two triangles, match the angles:
Â1
 2
B̂1
B̂2
Ĉ
D̂
When proving that the two triangles are similar, make sure that the equal
angles correspond:
ABC|||DBA
Use the theorems proved in previous grades to prove that two pairs of
corresponding angles are equal. The third pair will always be equal due to the
sum of the angles of a triangle.
2.
Try taking letters from the left and right and see if you can locate similar
triangles.
BC
AB

AC
AD
© Gauteng Department of Education
5
Refer to the diagram and see which triangles appear:
Is ABC a triangle?
Is ACD a triangle?
SESSION NO:
17
TOPIC: STATISTICS
Teacher note:
In this session, the calculator programmes for CASIO and SHARP calculators have
been provided.
Before tackling the typical examination questions and homework questions, it is
essential to revise the concepts discussed in the content notes section. Here are a
few comments regarding these concepts.
Ensure that learners know the meaning of mean, median, mode, lower quartile,
upper quartile, range and inter-quartile range.
Five number summaries and box and whisker plots are important and learners
can score well in this section. Outliers need to be revised in the context of box and
whisker plots as well as in scatter plot diagrams. Tukey’s method of determining
outliers is part of the CAPS syllabus. An outlier is any number in a data set, which
falls outside the interval:
3
3


Q1  2  IQR ; Q3  2  IQR  where Q1 and Q3 represent the lower and upper
quartiles respectively. IQR represents the inter-quartile range ( IQR  Q3  Q1 )
Learners often struggle to draw cumulative frequency curves. A suggestion is to
write graph points. For each point, the first coordinate represents the upper boundary
of the class interval. The second co-ordinate represents the cumulative frequency of
the ages. This provides a structure for learners. Reading the value of quartiles or
percentiles from the graph is important. Remember that the readings are estimates
and not actual values.
Variance and standard deviation can be calculated using a table or a calculator.
The table method and the use of a calculator is discussed in the session. It is
extremely important for learners to know how to use a calculator to do these
calculations.
The distribution of data in terms of positive and negative skew is important. It is
important to note that a set of data is
© Gauteng Department of Education
6



positively skewed or skewed to the right if the mean  median
negatively skewed or skewed to the left if the mean  median
normally distributed if the mean  median
It is important for learners to understand the concept of a scatterplot diagram. It is
used to determine the relationship between two variables. If data represented on a
scatterplot has a linear trend, then a line of best fit (regression line) can be
determined by means of a calculator. Remember that the line of best fit passes
through the mean point ( x ; y ) .
Outliers need to be excluded from the data when determining the equation of the
line of best fit since they tend to move the line of best fit out of its accurate position.
Make sure that your learners know how to interpolate and extrapolate using the
equation of the line of best fit. Interpolation is estimating within the given range of
data values whereas extrapolation is estimating outside the given range of data
values.
Using a calculator to calculate the correlation coefficient is an important skill for
learners to master. This value determines the strength of the linear correlation.
SESSION NO:
18
TOPIC: PROBABILITY GRADE 12
Teacher note:
Probability in Grade 12 is quite complicated and requires a whole session. Revision
of Grade 10 and 11 Probability will take place in session 21.
One of the major pitfalls of teaching Probability is the rote learning of the rules
without a proper understanding of them. It is absolutely essential to use as many
concrete examples as possible to explain how the rules work.
The fundamental counting principle (rule 1) states that:
If one operation can be done in m ways and a second operation can be done in n
ways then the total possible number of different ways in which both operations can
be done is m  n.
The best way to explain this principle is to use concrete examples to illustrate what is
happening.
This principle is critical to all of the other concepts in this session so it is imperative
that learners understand this rule.
© Gauteng Department of Education
7
When dealing with words containing letters that are all different, there are two types
of word arrangements that can be made. The first is where letters may be repeated.
In this case we use exponential notation. For example, there are a total of
4  4  4  4  44 word arrangements for the word EXAM. Each letter may be
repeated so there are 4 possible letters in each position.
However, when letters may not be repeated, we use factorial notation. Ensure that
learners understand factorial notation before dealing with this concept. For example,
there are a total of 4  3  2 1  4! word arrangements for the word EXAM if the letters
are not to be repeated.
This concept of not repeating the letters leads to the second rule, which states that
the number of arrangements of n different things taken in n different ways is given by
n!
The third rule states that the number of arrangements of n different things taken r at
a time is given by
n!
.
(n  r )!
This rule can be quite confusing for the weaker learner. The following approach is
suggested. For example, suppose that we must determine in how many ways 10
people can occupy 7 different positions.
10 people can occupy the first position. However, only 9 people can occupy the
second position since one person has already occupied the first position. In the third
position we have a possible 8 people who can occupy this position. In the fourth
position, there are 7 possible people. In the fifth, there are 6. In the sixth, there are 5
and in the seventh there are 4. We write this as follows: 10  9  8  7  6  5  4
The third rule states that this is the same as: 10  9  8  7  6  5  4 
10!
.
(10  7)!
It is the opinion of the author that first writing this as 10  9  8  7  6  5  4 provides for
greater understanding of the concept.
The alternative
10!
can be discussed but many learners tend to struggle with
(10  7)!
this notation.
Regarding arrangements of objects in a row, avoid the rote learning of rules. Rather
let the learners draw pictures and use concrete examples to understand the
processes involved.
For example, consider the situation where 3 boys and 2 girls sit in a row with one boy
and his girlfriend having to sit together.
© Gauteng Department of Education
8
The best way to teach this is to treat the couple as one “object”. Use the idea that
when two people marry they become one! The 5 people are now treated as 4
“objects”: 1 couple (1 boy and 1 girl) and 3 other people (2 boys, 1 girl).
couple
boy
boy
girl
In the first position, there are 4 choices from the 4 objects. Once the first position has
been occupied, there are 3 choices for the second position. For the third position
there are 2 choices and then 1 choice for the fourth position.
From the counting principle, this can be written as 4  3  2 1  4!
However, the couple can sit together in 2 ways: BG or GB.
This can be written as 2!.
Therefore we can write the answer as: 2!  4!  48
There is an alternative method available.
There are 4 possible seating arrangements:
couple ___ ___ ___
___ couple ___ ___
___ ___ couple ___
___ ___ ___ couple
There are 2! ways for the couple to sit together and there are 3! ways that the others
can then sit. This can be written as 4  2!  3!  48
When dealing with letter arrangements where letters are repeated in a word, the
following principles apply:
When repeated letters are treated as different objects, the second rule applies. For
example, the number of word arrangements that can be made from the word DAD
are 3!
The situation changes when repeated letters are treated as identical objects.
With the word DAD, the possible word arrangements are
3!
2!
The numerator represents the total number of letters in the word DAD, which is 3.
© Gauteng Department of Education
9
The denominator represents the fact that the letter D is written twice in the word.
Linking all of this to probability is important.
SESSION NO:
19
TOPIC: CONSOLIDATION OF EUCLIDEAN GEOMETRY (GRADE 11 AND 12)
Teacher note:
In this session the emphasis will be on revision and extension of all Grade 11 and 12
geometry theorems. Before learners attempt the typical examination questions, make
sure that they revise all circle and triangle theorems. This is provided in the content
notes section.
The typical examination questions show learners the layout of geometry in the final
examination. A theorem must be proved followed by riders linking to the theorem.
The questions in this exercise are more advanced than those in sessions 15 and 16.
This is a preparation for geometry in the final examination (Paper 2).
Remember that geometry is 50 marks out of 150 which is a third of the paper. It is
therefore imperative that learners work through this session thoroughly. The content
section contains all of the theorems and sessions 15 and 16 contain the proofs of
theorems required for the exam.
Make sure that learners use different colours to brainstorm the diagram first before
attempting to write up a final solution. When the final solution is given, learners must
provide correct reasons for their statements.
The homework questions provide further revision and extension for the learners. The
standard of these questions is in well in line with what will probably be asked in this
year’s final examination. It is the opinion of the author that the geometry section in
the final exam will probably be less demanding than the questions provided in this
session. By working through the questions in this session, learners will be wellprepared for geometry in the final exam.
Please go over the teacher notes for session 15 and 16 as well. The methods
suggested have been used successfully by the author over the years.
© Gauteng Department of Education
10
SESSION NO:
20
TOPIC: CONSOLIDATION OF STATISTICS (GRADE 11 AND 12)
Teacher note:
This session is a consolidation session which revises all Grade 10, 11 and 12
Statistics. Ensure that your learners understand all of the theory in the content notes
section. The use of a calculator is extremely important in Statistics. Learners must
ensure that they know how to calculate the mean, standard deviation, line of best fit
and correlation coefficient with ease.
The typical examination questions are presented in the same way as to be expected
in the final examination (Paper 2).
Once again, please ensure that learners know the meaning of mean, median, mode,
lower quartile, upper quartile, range and inter-quartile range.
Five number summaries and box and whisker plots are important and learners
can score well in this section. Outliers need to be revised in the context of box and
whisker plots as well as in scatter plot diagrams. Tukey’s method of determining
outliers is part of the CAPS syllabus. An outlier is any number in a data set which
falls outside the interval:
3
3


Q1  2  IQR ; Q3  2  IQR  where Q1 and Q3 represent the lower and upper
quartiles respectively. IQR represents the inter-quartile range ( IQR  Q3  Q1 )
Learners often struggle to draw cumulative frequency curves. A suggestion is to
write graph points. For each point, the first coordinate represents the upper boundary
of the class interval. The second co-ordinate represents the cumulative frequency of
the ages. This provides a structure for learners. Reading the value of quartiles or
percentiles from the graph is important. Remember that the readings are estimates
and not actual values.
Variance and standard deviation can be calculated using a table or a calculator.
The table method and the use of a calculator is discussed in the session. It is
extremely important for learners to know how to use a calculator to do these
calculations.
The distribution of data in terms of positive and negative skew is important. It is
important to note that a set of data is

positively skewed or skewed to the right if the mean  median

negatively skewed or skewed to the left if the mean  median

normally distributed if the mean  median
© Gauteng Department of Education
11
It is important for learners to understand the concept of a scatterplot diagram. It is
used to determine the relationship between two variables. If data represented on a
scatterplot has a linear trend, then a line of best fit (regression line) can be
determined by means of a calculator. Remember that the line of best fit passes
through the mean point ( x ; y ) .
Outliers need to be excluded from the data when determining the equation of the
line of best fit since they tend to move the line of best fit out of its accurate position.
Make sure that your learners know how to interpolate and extrapolate using the
equation of the line of best fit. Interpolation is estimating within the given range of
data values whereas extrapolation is estimating outside the given range of data
values.
Using a calculator to calculate the correlation coefficient is an important skill for
learners to master. This value determines the strength of the linear correlation.
SESSION NO:
21
TOPIC: CONSOLIDATION OF PROBABILITY (GRADE 10, 11 AND 12)
Teacher note:
Probability is a difficult topic for learners to understand and it is so important to make
sure that they understand all the terminology and concepts. This session will serve to
revise all Grade 10, 11 and 12 concepts and to provide learners with typical question
layout as to be expected in the final examination (paper 2).
Here are some important concepts from Grade 10, 11 and 12. Please ensure that
learners understand them fully. The content notes discuss them in more detail.
Consider the probability rule P(A or B)  P(A)  P(B)  P(A and B) .
If P(A and B)  0 there is an intersection and the events are inclusive.
If P(A and B)  0 there is no intersection and the events are mutually exclusive.
If P(A or B)  1, then the events are exhaustive (contain all elements of the sample
space between them)
If P(A or B)  1, then the events are not exhaustive (don’t contain all elements of the
sample space between them)
If P(A or B)  1 and P(A and B)  0 , then the events are complementary since they
are both exhaustive and mutually exclusive.
Tree diagrams are extremely important for representing the outcomes involving
replacement and non-replacement. Make sure that your learners understand how
to represent probabilities on a tree diagram.
© Gauteng Department of Education
12
Understanding independent events in the context of contingency tables is
important.
Venn diagrams are useful for representing inclusive and mutually exclusive events.
Make sure that your learners master all of the examples on venn diagrams,
particularly venn diagrams involving more than two circles. Learners should always
start from the intersection and work from there.
The fundamental counting principle (rule 1) states that:
If one operation can be done in m ways and a second operation can be done in n
ways then the total possible number of different ways in which both operations can
be done is m  n.
The best way to explain this principle is to use concrete examples to illustrate what is
happening.
This principle is critical to all of the other concepts in this session so it is imperative
that learners understand this rule.
When dealing with words containing letters that are all different, there are two types
of word arrangements that can be made. The first is where letters may be repeated.
In this case we use exponential notation. For example, there are a total of
4  4  4  4  44 word arrangements for the word EXAM. Each letter may be
repeated so there are 4 possible letters in each position.
However, when letters may not be repeated, we use factorial notation. Ensure that
learners understand factorial notation before dealing with this concept. For example,
there are a total of 4  3  2 1  4! word arrangements for the word EXAM if the letters
are not to be repeated.
This concept of not repeating the letters leads to the second rule, which states that
the number of arrangements of n different things taken in n different ways is given by
n!
The third rule states that the number of arrangements of n different things taken r at
a time is given by
n!
.
(n  r )!
This rule can be quite confusing for the weaker learner. The following approach is
suggested. For example, suppose that we must determine in how many ways 10
people can occupy 7 different positions.
10 people can occupy the first position. However, only 9 people can occupy the
second position since one person has already occupied the first position. In the third
position we have a possible 8 people who can occupy this position. In the fourth
© Gauteng Department of Education
13
position, there are 7 possible people. In the fifth, there are 6. In the sixth, there are 5
and in the seventh there are 4. We write this as follows: 10  9  8  7  6  5  4
The third rule states that this is the same as: 10  9  8  7  6  5  4 
10!
.
(10  7)!
It is the opinion of the author that first writing this as 10  9  8  7  6  5  4 provides for
a greater understanding of the concept.
The alternative
10!
can be discussed but many learners tend to struggle with
(10  7)!
this notation.
Regarding arrangements of objects in a row, avoid the rote learning of rules. Rather
let the learners draw pictures and use concrete examples to understand the
processes involved.
For example, consider the situation where 3 boys and 2 girls sit in a row with one boy
and his girlfriend having to sit together.
The best way to teach this is to treat the couple as one “object”. Use the idea that
when two people marry they become one! The 5 people are now treated as 4
“objects”: 1 couple (1 boy and 1 girl) and 3 other people (2 boys, 1 girl).
couple
boy
boy
girl
In the first position, there are 4 choices from the 4 objects. Once the first position has
been occupied, there are 3 choices for the second position. For the third position
there are 2 choices and then 1 choice for the fourth position.
From the counting principle, this can be written as 4  3  2 1  4!
However, the couple can sit together in 2 ways: BG or GB.
This can be written as 2!.
Therefore we can write the answer as: 2!  4!  48
There is an alternative method available.
There are 4 possible seating arrangements:
couple ___ ___ ___
___ couple ___ ___
___ ___ couple ___
___ ___ ___ couple
There are 2! ways for the couple to sit together and there are 3! ways that the others
can then sit. This can be written as 4  2!  3!  48
When dealing with letter arrangements where letters are repeated in a word, the
following principles apply:
© Gauteng Department of Education
14
When repeated letters are treated as different objects, the second rule applies. For
example, the number of word arrangements that can be made from the word DAD
are 3!
The situation changes when repeated letters are treated as identical objects.
With the word DAD, the possible word arrangements are
3!
2!
The numerator represents the total number of letters in the word DAD, which is 3.
The denominator represents the fact that the letter D is written twice in the word.
Linking all of this to probability is important.
© Gauteng Department of Education