Grade 11 advance mathematics book with solution

Mathematics units
Grade 11 advanced
Contents
11A.1 Number and algebra 139 11A.8 Algebra 3 199
11A.2 Geometry 1 149 11A.9 Trigonometry 2 209
11A.3 Algebra 1 157 11A.10 Probability 2 217
11A.4 Trigonometry 1 165 11A.11 Calculus 225
11A.5 Probability 1 175 11A.12 Vectors 237
11A.6 Algebra 2 185 11A.13 Geometry 2 245
11A.7 Measures 193 11A.14 Algebra 4 253

Mathematics units: Grade 11 advanced 135 teaching hours
UNIT 11A.4: Trigonometry 1
Sine and cosine rule
Solution of triangles in 2-D and 3-D
9 hours
UNIT 11A.10: Probability 2
Risk
Trends over time; moving
averages
Simulations using random
numbers
10 hours
15% 30%
1st semester
70 hours
2nd semester
65 hours
UNIT 11A.2: Geometry 1
Proof
Standard circle theorems
9 hours
UNIT 11A.7: Measures
Rates and compound
measures
3 hours
UNIT 11A.5: Probability 1
Empirical probability
Using mathematical models,
e.g. tree diagrams
Independent and dependent
events
10 hours
UNIT 11A.9: Trigonometry 2
Circular functions
Trigonometric equations and identities
7 hours
Reasoning
and
problem
solving
should
be
integrated
into
each
unit
UNIT 11A.0: Grade 10A revision
3 hours
UNIT 11A.1: Number and algebra
Sequences: finite and infinite
geometric sequences; sum of first
n squares and cubes; recurrence
relations
Binomial theorem
Permutations and combinations
12 hours
UNIT 11A.3: Algebra 1
Properties of graphs of functions,
including maxima and minima
Quadratic functions
12 hours
Cubic, reciprocal, sine and cosine
functions
Modulus and other non-standard
functions
Inverse functions
Composite functions
12 hours
UNIT 11A.11: Calculus
Limits
Introduction to calculus
Derivatives of standard functions
12 hours
Quadratic equations; real roots
Simultaneous equations (linear and
quadratic)
Inequalities, including solution sets
12 hours
Transformation of functions
Exponential function
Logarithms
11 hours
55%
UNIT 11A.13: Geometry 2
Transformations
Plans and elevations
7 hours
UNIT 11A.12: Vectors
Position vector; addition and subtraction
in 2-D and 3-D; vector diagrams
Scalar product; multiplication by scalar;
magnitude and direction; displacement
and velocity; unit vectors
6 hours

139 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.1 | Number and algebra © Education Institute 2005
GRADE 11A: Number and algebra
Series, combinatorics and the binomial theorem
About this unit
This is the only unit on number and algebra for
Grade 11 advanced. It brings together several
topics on algebra related to sequences and
combinatorics. The unit builds on work in number
and algebra in the Grade 10 advanced units.
The unit is designed to guide your planning and
teaching of mathematics lessons. It provides a
link between the standards for mathematics and
your lesson plans.
The teaching and learning activities should help
you to plan the content and pace of lessons.
Adapt the ideas to meet your students’ needs.
Supplement the activities where necessary with
appropriate tasks and exercises from textbooks
and other resources, including ICT.
For consolidation or extension activities, look at
the units for Grade 10 advanced or Grade 12
advanced.
Introduce the unit to students by summarising
what they will learn and how this builds on earlier
work. Review the unit at the end, drawing out the
main learning points, links to other work and
real-world applications.
Previous learning
To meet the expectations of this unit, students should already be able to
investigate the growth of simple patterns, and to generalise algebraic
relationships to model the behaviour of the patterns. They should be able to
identify and sum arithmetic and geometric series, and to convert any
recurring decimal to an exact fraction.
Expectations
By the end of the unit, students will work to expected degrees of
accuracy. They will recognise when to use ICT and do so efficiently. They
will use sigma notation, and will generate and sum simple recursive
sequences, including arithmetic and geometric series, to model the
behaviour of real-world situations. They will use formulae for the sum of the
squares and cubes of the first n positive integers. They will be familiar with
the patterns in Pascal’s triangle, find combinations and permutations, and
use the binomial theorem expansion of (1 + x)
n
, where n is a positive integer.
Students who progress further will differentiate between different kinds of
sequences and series with fluency, and manipulate formulae associated
with them with confidence. They will find more complex permutations and
combinations, and will extend their use of the binomial theorem.
Resources
The main resources needed for this unit are:
• overhead projector (OHP)
• Internet access, computer and data projector
• spreadsheet software such as Microsoft Excel
• computers with Internet access and spreadsheet software for students
• calculators for students
Key vocabulary and technical terms
Students should understand, use and spell correctly:
• recursive, sequence, series, convergent
• arithmetic, geometric, common difference, common ratio, sum to n terms,
sum to infinity
• permutation, arrangement, selection, combination, factorial
• binomial theorem, binomial coefficient, Pascal’s triangle
UNIT 11A.1
12 hours

Standards for the unit
12 hours
SUPPORTING STANDARDS
Grade 10A standards
CORE STANDARDS
Grade 11A standards
EXTENSION STANDARDS
Grade 12A standards
11A.1.13 Work to expected degrees of accuracy, and know when an
exact solution is appropriate.
11A.1.14 Recognise when to use ICT and when not to, and use it
efficiently.
12AQ.1.16
11A.4.1 Generate sequences from term-to-term definitions and from
position-to-term definitions, including recursive sequences, to
model the behaviour of real-world situations, for example
population growth.
Recognise when to use ICT and when not to,
and use it efficiently; use ICT to present findings
and conclusions.
10A.4.2 Generate sequences from term-to-
term and position-to-term
definitions; investigate the growth of
simple patterns, generalising
algebraic relationships to model the
behaviour of the patterns.
11A.4.2 Understand and use sigma notation for summing the terms
of a sequence.
10A.4.3 Identify and sum arithmetic
sequences, including the first n
consecutive positive integers, and
give a ‘geometric proof’ for the
formulae for these sums.
11A.4.3 Recognise an arithmetic progression (AP); sum an arithmetic
series and know the formula for the rth term of the series in
terms of the first term and the common difference between
terms.
10A.4.4 Identify and sum geometric
sequences and know the conditions
under which an infinite geometric
series can be summed.
11A.4.4 Recognise a geometric progression (GP); generate term-to-
term and position-to-term definitions for the terms of a GP in
terms of the common ratio between terms; sum a finite
geometric series.
10A.4.5 Convert any recurring decimal to an
exact fraction.
11A.4.5 Sum to infinity a convergent geometric series and know the
condition on the common ratio for an infinite geometric series
to be convergent.
11A.4.6 Understand and use formulae for the sum of:
• the squares of the first n positive integers;
• the cubes of the first n positive integers.
2 hours
Series notation
and formulae
2 hours
Arithmetic and
geometric series
8 hours
Binomial
theorem and
combinatorics
11A.4.7 Understand and use factorial notation and know that 0! = 1;
know the binomial theorem expansion of (1 + x)
n
for positive
integer n and that the term in x
r
has coefficient
n
Cr, where
n
Cr =
!
( )! !
n
n r r
−
; know how to use Pascal’s triangle to find
n
Cr; find permutations and combinations.
12AQ.4.1 Understand that
n
Cr is the number of
combinations of r different objects from n
different objects and that the number of
permutations of r different objects from n
different objects is r!
n
Cr, which is denoted by
n
Pr.
12AS.4.1 Find permutations and combinations.
Unit 11A.1

Activities
Objectives Possible teaching activities Notes School resources
2 hours
Series notation and
formulae
Understand and use sigma
notation for summing the
terms of a sequence.
Generate sequences from
term-to-term definitions and
from position-to-term
definitions, including
recursive sequences, to
model the behaviour of real-
world situations, for example
population growth.
Understand and use formulae
for the sum of:
• the squares of the first n
positive integers;
• the cubes of the first n
positive integers.
Series notation
Class discussion
Begin by explaining sigma notation as a compact way to present the sum of a set of numbers or
the terms of a sequence. Do this first by a number of simple examples, and then displayed on a
spreadsheet column, where the row() function can replace the index on sigma.
Conclude the presentation with a number of examples worked orally, turning series into sigma
notation with index and limits, and vice versa.
Examples
• Express in full:
5
1
( 1)
r
r r
=
+
∑
• Use sigma notation to express
7 6 5
2
5 4 3
+ + + .
A1=row(A1)*(row(A1)+1)
B1=2; B2=B1+A2.
A1 and B2 are replicated down the columns.
Extend discussion using the applet Sequences
(www.fi.uu.nl/wisweb/welcome_en.html).
This column is for
schools to note their
own resources, e.g.
textbooks, worksheets.
With more able pupils
mention of the
corresponding pi
notation for products
may be of interest.
Exercises
Get students to use the row() function in a spreadsheet to simulate the index in the sum as
shown above. Many of the textbook exercises designed for pencil and paper can be adapted to
this. Extend such exercises to incorporate the formulae for the sum of the first n natural
numbers, the sum of their squares, and the sum of their cubes.
• Express 5× 6 + 6×7 + 7× 8 + 8× 9 + … + 199× 200 using sigma notation, and hence, with
the aid of a spreadsheet, evaluate their sum.
• Use replication to generate the first 50 terms of the series 5, 5, 10, 15, 25, … Find its sum.
• Use replication to show the terms of
100
1 1
r
r
r
= +
∑ on a spreadsheet; hence calculate the sum.
• If 2 1
6
1
( 1)(2 1),
n
r
r n n n
=
= + +
∑ find a formula for
2
2
1
.
n
r n
r
= +
∑
• If un = n(n + 1)(n + 2), write down and simplify an expression for un+1 /un, and hence obtain a
recurrence relation for the sequence (un).
Exercises like these promote new ways of
looking at this sort of problem. They can lead to
applications where algebra models situations
such as accumulating bank deposits, growth,
appreciation and decay.
Use the applet Discrete dynamic models from
www.fi.uu.nl/wisweb/welcome_en.html to
explore and discuss some real-world situations.
Unit 11A.1

Standard series: arithmetic and geometric
Class discussion
Arithmetic and geometric series and the standard formulae for them were introduced in Grade
10 advanced. Restate them with sigma notation to revise their use.
Ask students to explain the term arithmetic sequence with one or more examples. Draw out the
notation usually used:
• a for the first term;
• d for the common difference;
• n for the number of terms.
Revise the formulae for the nth term and the sum of n terms.
Repeat the exercise for geometric series, including the notation r for the common difference.
Revise the formulae for the nth term and the sum of n terms.
In both cases, discuss the proof of the sum in as much detail as is necessary.
Turn to convergence. Ask for an examples of:
• an arithmetic series which converges;
• a geometric series which converges.
Students should be able to decide that no arithmetic series converges, and will probably recall
geometric series that converge. Use the discussion to re-establish the criterion ( < 1
r ) for
convergence.
On the web
The work on arithmetic series and geometric
series is featured on MathsNet at
www.mathsnet.net/asa2/2004/c2.html#1.
2 hours
Arithmetic and geometric
series
Recognise an arithmetic
progression (AP); sum an
arithmetic series and know
the formula for the rth term of
the series in terms of the first
term and the common
difference between terms.
Recognise a geometric
progression (GP); generate
term-to-term and position-to-
term definitions for the terms
of a GP in terms of the
common ratio between terms;
sum a finite geometric series.
Sum to infinity a convergent
geometric series and know
the condition on the common
ratio for an infinite geometric
series to be convergent. Exercises
Combine a mix of APs and GPs, ending with problem solving. Include
• routine exercises on the use of the formulae
(nth term, sum to n terms, sum to infinity for a GP);
• calculation of fractions equivalent to recurring decimals by considering them as convergent
geometric series;
• word problems that bring in applications.
There are two different models of growth involved here. Students need experience of a range of
problems so that they are clear about how these growth models compare.
If time permits at the end of this work, repeat the argument that establishes that all rational
numbers correspond to terminating or repeating decimals. This reinforces work on the existence
of irrationals characterised by non-repeating but non-terminating decimal expansions.
Alternatively, cover these points within the exercise once the routine examples have been
finished.
Include problems such as this one.
• The first, second and fourth terms of a
geometric progression are also in arithmetic
progression. Find the possible values of the
common ratio.

Arrangements (or permutations)
Class discussion
Discuss this problem.
• If towns A and B are connected by r distinct routes and towns B and C by s distinct routes,
how many distinct routes are there from A to C via B?
This simple problem establishes an important principle: that in such circumstances the answer
is r × s. (In the diagram the number of routes from A to C is 6 rather than 5.)
Now develop the argument for the number of permutations of n objects as n!.
Develop the number of permutations of r elements selected from n from this. The first choice
can be made in n ways, the second in n – 1, and so on, until the rth can be made in n – r + 1.
Hence by the r × s principle above, we have:
!
P ( 1)( 2)...( 1)
( )!
n
r
n
n n n n r
n r
= − − − + =
−
Take this example of use of factorial notation slowly, as students need to use it too. Consider
the special case of 0! = 1; do this by trying to get students to tell you what it should be so that
the notation remains consistent (e.g. in how many ways can 0 be arranged from n?).
The objective of this part of the work is to embed
in students’ minds several distinct problems and
their standardised solutions so that the same
situations can be recognised as elements of
more complex problems.
8 hours
Binomial theorem and
combinatorics
Work to expected degrees of
accuracy, and know when an
exact solution is appropriate.
Recognise when to use ICT
and when not to, and use it
efficiently.
Understand and use factorial
notation and know that 0! = 1;
know the binomial theorem
expansion of (1 + x)
n
for
positive integer n and that the
term in x
r
has coefficient
n
Cr,
where
n
Cr = !
!( )! ;
n
r n r
− know how
to use Pascal’s triangle to
find
n
Cr ; find permutations
and combinations.
Exercise
Include:
• basic drill on factorial notation;
• occurrence of 0! in settings which make its value of 1 clearly necessary for consistency;
• basic drill on permutations;
• problem solving where non-standard counting situations figure.
Examples
• How many ways can different groupings of four be taken for a photograph from a family
gathering of ten people? (Take the same four in a different order as a different grouping.)
If Bader and Ali are inseparable (and insist on being photographed together or not at all), in
how many ways can this be done?
• In how many ways can the digits in 98 765 be arranged if an even number must appear in the
first, third or fifth place?
There are many sets of examples on
permutations that will support this work.
Include ways of dealing with repetitions. For
example, how many ways can n objects be
arranged if three of them are identical? If we
take any one arrangement, regarding the three
identical items as distinguishable, they can be
rearranged in 3! ways; hence the total number of
arrangements is n!/ 3!.
Also include ways of dealing with restrictions.
For example, how many ways can n objects be
arranged if two of them must be separate? We
count the arrangements in which the two are
together, which is equivalent to the number of
arrangements of n – 1 items, except that the two
may be either way round, so it is 2 × (n – 1)!.
These are the arrangements to be excluded, so
the required figure is:
n! – 2 × (n – 1)! = (n – 2) × (n – 1)!

Selections (or combinations)
Class discussion
Establish the concept of selection (or combination) as distinct from permutation (or
arrangement). Have available a number of examples where the distinction is important.
The formula for
n
Pr can then be adapted to that for
n
Cr. Since any selection of r can be
rearranged in r! ways,
n
Pr = r! ×
n
Cr. Hence
!
C
( )! !
n
r
n
n r r
=
−
Use this result to discuss a number of examples, including at least one non-standard problem.
Do not expect too much at this stage, however, and focus clearly on repeatedly distinguishing
selections (or combinations) from arrangements (or permutations). The words do not
immediately connect with intuition, so students will easily remain confused.
!
P ( 1)...( 1)
( )!
n
r
n
n n n r
n r
= = − − +
−
where the number of factors is equal to the
number of elements of the permutation.
! ( 1)...( 1)
C
( )! ! 1 2 ...
n
r
n n n n r
n r r r
− − +
= =
− × × ×
where the number of factors in numerator and
denominator is equal.
On the web
The work in this section is featured on MathsNet
(www.mathsnet.net/asa2/2004/c2.html#1).
Exercise (and subsequent discussion)
There are many sets of exercises available on this topic. Include
• drill on the new notation
n
Cr;
• discovery that
n
Cr =
n
Cn–r, both from factorials and from context;
• distinguishing permutations from combinations;
• problem solving which focuses on counting.
Bring out in full discussion of these questions the different strategies which can lead to an
equivalent answer. More able pupils will benefit from seeing that there is indeed more than one
way of getting an answer and that sometimes one method is much more elegant or economical
than another.
Examples
• In how many ways can a committee of 5 be chosen from 10 people (a) so as to include both
the youngest and the oldest; (b) so as to exclude the youngest if it includes the oldest?
• Find n if (a)
n
C2 = 55, (b)
n
C2 =
n
C5.
• A committee of 6 is chosen from 10 men and 7 women so as to contain at least 3 men and 2
women. In how many ways can this be done if two particular women refuse to serve on the
same committee?
More able students will cope with the identity
n+1
Cr =
n
Cr +
n
Cr–1 by factorials. The exercise is
tricky but demanding, and makes a good
contrast with the simpler combinatorial argument
which can also justify it!
This result underlies Pascal’s triangle.
On the web
There are lots of websites containing interactive
versions of Pascal’s triangle which help students
to find the patterns, e.g. mathforum.org/
workshops/usi/pascal/mo.pascal.html.

The binomial theorem
Class discussion
Pascal’s triangle was used in Grade 10 to prepare for the binomial theorem. Review the Grade
10 work briefly.
Students will recognise that the expansion of (a + b)
n
can be done by using the nth row of the
triangle, and will remember (if hazily) how each line can be derived from the previous line.
Include in the review the long multiplication method of generating each new power from the last
to make clear why the rule of adding two terms from above to obtain the one below works.
Pose the problem of enumerating the nth line of the triangle without writing out all the lines that
precede it. Give students time to discuss this problem in small groups. This is an extension of
the use of formulae for nth terms or for the sum of n terms, which students have already met.
There are two ways of going about this. Each has its merits. Present both methods to students
but in different lessons.
The simpler argument is direct appeal to combinatorics. Ask students:
• How does the a5
b7
term arise in the expansion of (a + b)
12
?
The question should provoke thought. With some encouragement it should become clear that
the problem is like the routes between towns at the beginning of the unit. There are 2
12
routes
through the brackets since there is a choice of two (a or b) each time. The term in question
arises from the number of routes through the brackets which pass through exactly five of the
letters a, so that the rest of the letters are b; that is
12
C5.
Make the proposition based on the argument just considered that the rth term on the nth row is
n
Cr, remembering that there is a ‘row zero’ and a term numbered zero. The argument has in
effect proved that proposition. Get students to verify this on any of the early rows.
Postpone discussion of the second method (which uses a recursion relation to generate one
row from the next). Move on to the use of the new formula. Derive, for example, the first four
terms of (a + b)
12
to show that this can now be done without Pascal’s triangle.
The applet Pascal’s triangle is a useful visual aid
(see nlvm.usu.edu/en/nav/vlibrary.html).
This illustrates one possible ‘route’ through the
brackets to obtain a term a5
b7
.
Class exercise (with problem solving)
Drill on the binomial theorem is part of Grade 10; it is appropriate to revise it here but also to
press on to consider harder questions with larger values of the index n and to investigate some
of the properties of Pascal’s triangle in more sophisticated terms.
Include:
• elementary expansions of (a + b)
n
in full;
• examples of the type (1 + x + x2
)
4
where the first few terms are required;
• problems on the structure of Pascal’s triangle, such as:
– show that the coefficient of x in (1 + x)
n
is the nth triangular number;
– prove that
0
( 1) 0
n
r
r
n
r
=
⎛ ⎞
− =
⎜ ⎟
⎝ ⎠
∑ .

Harder problems with the binomial theorem
Class discussion
Ask questions about a possible trinomial theorem. For example:
• How would you expand (x + y + z)
n
?
Allow time for the suggestion of treating the problem as (x + [y + z])
n
to emerge, i.e. to use a
repeated application of the binomial theorem.
Work an example such as finding the first four terms of the expansion of (1 + x + 2x2
)
5
(see the
model solution on the right).
Make sure that students appreciate not only the strategy for solving that kind of problem but
also the need to organise their work so that accuracy does not become a problem.
Model solution
2 5
2 2 2
2 3
2 3
3 2 2
2 3 3
2 3
(1 2 )
5 5 4
1 ( 2 ) ( 2 )
1 1 2
5 4 3
( 2 ) ...
1 2 3
1 5 10 10( 4 ...)
3
10( 2 ...)...
1
1 5 10 10 40 10 ...
1 15 10 50 ...
x x
x x x x
x x
x x x x
x x x
x x x x x
x x x
+ +
⋅
= + + + +
⋅
⋅ ⋅
+ + +
⋅ ⋅
= + + + + +
+ + ⋅ ⋅ +
= + + + + + +
= + + + +
Work through some harder applications, such as this one.
• Find the exact values of the sum and the product of + 7
(2 3) and − 7
(2 3) .
Hence show that the integral part of + 7
(2 3) is 10083.
Some non-routine problems on the binomial theorem can be found by choosing the Maths
Finder option on the Nrich website at www.nrich.maths.org/public/index.php.
Model solution
7
7 6 5
4 3 2
2 2 3 3
7 5 3 2
3 6 4
2 2 3
7
7 7
7
(2 3)
2 7 2 3 21 2 3
35 2 3 3 35 2 3
21 2 3 3 7 2 3 3 3
2 21 2 3 35 2 3
7 2 3 7 2 3 35 2 3 3
21 2 3 3 3 3
5042 2911 3
Similarly (2 3) 5042 2911 3
Hence (2 3) (2 3) 10084
Now (2 3)
+
= + ⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅
= + ⋅ ⋅ + ⋅ ⋅
+ ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅
+ ⋅ ⋅ ⋅ + ⋅
= +
− = −
+ + − =
+ ×
2
7 2 7
7
7
7
(2 3) (2 3 ) 1
Then we have
1
0 (2 3) 1
(2 3)
So 0 10084 (2 3) 1
and the result follows.
− = − =
< − = <
+
< − − <

Assessment
Examples of assessment tasks and questions Notes School resources
In a TV quiz show a contestant can triple her winnings if she survives from one round to the
next. Write down a formula for her prize Pn+1 in the (n + 1)th round in terms of her prize Pn in the
nth round. Write an alternative formula for Pn+1 in terms of n.
The prize for winning in the first round is QR 1000. What is the minimum number of rounds that
will have to be contested to win at least QR 700 000?
A sequence is defined by un+2 = un+1 – un with u1 = 5 and u2 = –4. Write down the first eight
terms of the sequence.
A sequence is defined by un = n(n – 1) + 41. Write down the first twelve terms of this sequence.
What do you notice about these terms? Form a conjecture about this sequence and carry out
further tests to see if your conjecture is correct.
Assessment
Set up activities that allow
students to demonstrate what
they have learned in this unit.
The activities can be provided
informally or formally during
and at the end of the unit, or
for homework. They can be
selected from the teaching
activities or can be new
experiences. Choose tasks
and questions from the
examples on the right to
incorporate in the activities.
In a certain country, there is a net increase in population from one year to the next of 5%. Set
up a recurrence relation to describe the population in year n + 1 in terms of the population in
year n. Find the population in year n + 4 compared to the population in year n. Use your formula
to find the number of years it takes to double the population from year n.
A woman buys a car and pays in monthly instalments. The car costs QR60 500 and interest is
charged on any outstanding debt at a monthly rate of r%. The woman pays back a fixed
amount each month of QR M. Set up a recurrence relation connecting the amount owed, An+1,
after n + 1 months in terms of the amount owed, An, at the end of the nth month. How many
months will it take to repay the debt if M = 1200 and r = 1.2? How much will the woman have
then paid for the car? Investigate repayments for different values of M and r.
Find
10
1
( 1).
r +
∑
Find 2
r
∑ for integer values of r from 1 to 10.
Write out in full
4
1
( 1)
( 1)
r
r r
−
+
∑ and use partial fractions to evaluate this sum.
Rewrite these sums using sigma notation:
7
2
+ 8
2
+ 9
2
+ 10
2
1
⁄121 –
1
⁄144 +
1
⁄169 –
1
⁄196 +
1
⁄225
At the end of every year a car loses 30% of its value at the start of the year. Construct a
formula, in terms of the original purchase price, to give the value of the car n years after
purchase. After how many years will the car first be worth less than 90% of its original value?
Show that
( 1)( 2)...( 1)
C .
!
n
r
n n n n r
r
− − − +
=
Unit 11A.1

Use the binomial theorem to expand (x – 2y)
4
.
Fifteen European countries are about to set up three new regulatory bodies, the Avocado
Authority, the Broccoli Board and the Courgette Commission. Each of the new bodies to be set
up will have one member from each of 6 different countries. Some countries may be
represented on more than one body, and some may be represented on none.
a. In how many ways can the 6 countries represented on the Avocado Authority be chosen?
Find the number of ways in which the three bodies can be made up in each of the following
cases. Give your answers in scientific notation to three significant figures (e.g. 1.23 × 10
4
). Note
that the order in which countries are allocated to bodies does not matter.
b. The 6 countries for each body are chosen freely from the 15 original countries.
c. France insists on being represented on each body and Britain insists on not being
represented on any of them, but otherwise the countries are chosen freely.
d. No country may be on both the Avocado Authority and the Broccoli Board, but otherwise the
countries are chosen freely.
MEI
In this question, a circle consists of a sequence of sectors with angles a1, a2, a3, …,
as shown in the figure. All angles are measured in degrees. Four cases are considered.
a. In the first case, the angles a1, a2, a3, a4, a5, a6, … form a periodic sequence
1°, 2°, 3°, 1°, 2°, 3°, … How many sectors will fill the circle?
b. In the second case, a1 = 8.5° and the angles form an arithmetic progression with common
difference 1°. Verify that 20 sectors fill the circle exactly.
c. In the third case, the angles form an arithmetic progression with common difference 0.5°,
and 30 sectors fill the circle exactly. Find a1.
d. In the fourth case, the angles form a geometric progression with a1 = 90° and common ratio
3
⁄4. Find how many sectors have angle greater than 1°. Now show that no matter how many
sectors are used they will always fit into the circle.
MEI

149 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.2 | Geometry 1 © Supreme Education Council 2005
GRADE 11A: Geometry 1
Circles and proofs
About this unit
This is the first of two units on geometry for
Grade 11 advanced. It introduces the circle
theorems, building on the approach started in
Grade 10.
your lesson plans.
advanced.
Previous learning
prove that the perpendicular from the centre of a circle to a chord bisects the
chord; and that the two tangents from an external point to a circle are of
equal length.
Expectation
By the end of the unit, students will use geometry to solve theoretical
problems. They will prove standard circle theorems. They will use a dynamic
geometry system to conjecture results and to explore geometric proof.
Students who progress further will solve more complex geometrical
problems.
Resources
• dynamic geometry system (DGS) such as:
Geometer’s Sketchpad
(see www.keypress.com/sketchpad)
Cabri Geometrie
(see www.chartwellyorke.com/cabri.html)
• computers with Internet access and dynamic geometry software for
students
• graphics calculators for students
• proof, theorem, converse
• angle subtended by an arc at the centre and at the circumference
• major arc, minor arc, segment, sector, tangent, chord, cyclic quadrilateral,
concyclic, collinear
UNIT 11A.2
9 hours

9 hours
Grade 10A standards
CORE STANDARDS
Grade 11A standards
EXTENSION STANDARDS
Grade 12A standards
10A.1.6 Develop short chains of logical
reasoning, using correct
mathematical notation and terms.
11A.1.6 Develop chains of logical reasoning, using correct mathematical
notation and terms.
12AS.1.6
11A.1.7 Explain their reasoning, both orally and in writing.
10A.1.8 Generate simple mathematical proofs,
and identify exceptional cases.
11A.1.8 Generate mathematical proofs, and identify exceptional cases.
Develop chains of logical reasoning,
using correct terminology and
mathematical notation, including
symbols for logical implication.
10A.1.9 Generalise whenever possible. 11A.1.11 Conjecture alternative possibilities with ‘What if …?’ and ‘What if
not …?’ questions.
10A.1.14 Recognise when to use ICT and when
not to, and use it efficiently.
11A.1.14 Recognise when to use ICT and when not to, and use it
efficiently.
12AQ.1.16
11A.8.1 Use a dynamic geometry system to conjecture results and to
explore geometric proof.
Recognise when to use ICT and when
not to, and use it efficiently; use ICT to
present findings and conclusions.
2 hours
Circle theorems 1
2 hours
Circle theorems 2
3 hours
Circle theorems 3
2 hours
Proofs
10A.6.9 Prove the circle theorems:
• The perpendicular from the centre
of the circle to a chord bisects the
chord.
• The two tangents from an external
point to a circle are of equal length.
11A.8.9 Prove the circle theorems:
• The angle subtended by an arc at the centre of the circle is
twice the angle subtended by the arc at a point on the circle,
including, as a special case, the angle in a semicircle is a right
angle.
• Angles in the same segment subtended by a chord are equal.
• The angle subtended by a chord at the centre of a circle is
twice the angle between the chord and the tangent to the
circle at an end point of the chord.
• When two chords BC and DE in a circle intersect at A then
AB × AC = AD × DE.
• Opposite angles of a cyclic quadrilateral are supplementary.
Unit 11A.2

Activities
Investigating circle properties
Investigation
Begin by asking students to use a dynamic geometry system to investigate the relationship
between the angle at the centre of a circle subtended by an arc and an angle subtended by the
same arc at the circumference. Structure this as follows.
• Draw a circle and mark two points A and B on it that are not opposite ends of a diameter.
• Draw the two radii joining the points to the centre of the circle O.
• Measure angle AOB and label the angle with that measurement. Check that your
measurement changes dynamically as you move A or B on the circumference of the circle.
• What is the measured angle when A and B are at opposite ends of a diameter? Are there any
other special cases?
• With points A and B more or less in their original positions, mark a third point C, distinct from
both A and B, on the major arc AB.
• Measure and label angle ACB.
• What do you notice if you move A or B as before? Can you turn that observation into a
conjecture? Is it restricted in any way? Are there any special cases?
• Check that your conjecture is still valid if this time you move C. Is there any restriction on C?
• Can you prove your conjecture?
This figure was produced in Cabri Geometrie. It
shows the configuration used in the
investigation.
On the web
Waldo’s Interactive Maths pages have
sections on the circle theorems. See
www.waldomaths.com.
This column is for
own resources, e.g.
2 hours
Circle theorems 1
Develop chains of logical
mathematical notation and
terms.
Explain their reasoning, both
orally and in writing.
Conjecture alternative
possibilities with ‘What if …?’
and ‘What if not …?’
questions.
Recognise when to use ICT
and when not to, and use it
efficiently.
Use a dynamic geometry
system to conjecture results
and to explore geometric
proof.
Prove the circle theorems:
• The angle subtended by an
arc at the centre of the
circle is twice the angle
subtended by the arc at a
point on the circle,
including, as a special
case, the angle in
semicircle is a right angle.
• Angles in the same
segment subtended by a
chord are equal.
Class discussion
Help students to draw together their conclusions. Not all their conclusions will be the same or
expressed in the same way. Most students should realise that the angle at the circumference is
always half the angle at the centre (once they have set aside the way in which the result
changes when C passes over A or B, or vice versa). The result that opposite angles of a cyclic
quadrilateral add to 360° may emerge.
Once conclusions are settled, raise the question of proof. There are two issues:
• that the results need to be proved;
• how you do it.
Review the properties of an isosceles triangle, which underpin the proof (see the details on the
right).
Do not rush to the hierarchy of results, namely that for a given arc:
• the angle at the centre is twice the angle at the circumference;
• all angles at the circumference are equal.
Rather, allow students to see first that these two results are connected: one leads to the other.
This may help students to focus on the first result as the more fundamental and challenging.
To establish the result:
• draw the circle and points A, B, C and O, with
lines AO, BO, AC and BC;
• invite suggestions on how to proceed, then
draw OC;
• establish that the two triangles are isosceles
(because of equal radii);
• invite other suggestions on how to proceed;
• extend CO and establish the 2x and 2y;
• show that ∠AOB = 2(x + y) = 2 ∠ACB.
Unit 11A.2

Exercises
Follow this with examples of calculations based on these results. A variety of simple
configurations are needed which help students to see the basic configuration in different
orientations.
Examples
Find the angles marked a, b and c in the figures on the right.
2 hours
Circle theorems 2
terms.
Generate mathematical
proofs, and identify
exceptional cases.
questions.
Prove the circle theorem:
• Opposite angles of a cyclic
quadrilateral are
supplementary.
More circle theorems
Class discussion
Establish the further results about cyclic quadrilaterals, namely:
• opposite angles of a cyclic quadrilateral are supplementary;
• the external angle of a cyclic quadrilateral is equal to the interior opposite angle.
Do this by posing the problem on the right, with 110°, a and b marked on the diagram. Ask
students to suggest ways of calculating the other two angles of the quadrilateral. Some students
should suggest connecting the diagram with what has gone before by drawing in radii. It is
important to do this so that students can ask ‘What if …?’ questions for themselves, and see
that a certain degree of lateral thinking is essential.
Once one or two examples have been studied, allow students time to make the conjectures (the
theorems just quoted) and to write out or contribute the steps of a general proof.
For reference these results can be referred to by:
• opposite ∠s of a cyclic quadrilateral;
• external ∠ of a cyclic quadrilateral.
These can be abbreviated if appropriate.
Steps in the calculation:
• do not draw the dotted radii in first – just mark
110°, a and b;
• invite suggestions on how to proceed, then
draw the dotted radii;
• use the angle properties to calculate 220°,
2a (= 360° – 220°), then a, then b (= 110°).

Exercises
These should extend the work in the last set of exercises. Stick to calculations for the most part.
Include the interesting result (see the diagram on the right) that produces two parallel lines AB
and DC. Ask students:
• If ∠BAD = 80°, what is ∠ADC?
• What can you say about AB and DC in this case?
• Does this result generalise?
On the web
The Geometry section of MathsNet has a
section on circle theorems. See
www.mathsnet.net/geometry.
3 hours
Circle theorems 3
terms.
proofs, and identify
exceptional cases.
questions.
Prove the circle theorems:
• The angle subtended by a
chord at the centre of a
circle is twice the angle
between the chord and the
tangent to the circle at an
end point of the chord.
• When two chords BC and
DE in a circle intersect at A
then AB × AC = AD × DE.
The alternate segment
Class discussion
Establish the tangent results. Use class discussion to establish the proof or – if appropriate for
the group – use a structured investigation using a dynamic geometry system to obtain the
tangent result. There are two associated results:
• the alternate segment property;
• the equivalent relationship specified in the standards.
The full argument is shown on the right.
In the diagram, 2x3 = 2x4 since both subtend the
same chord.
x1 + y1 = 90° (∠ between tangent and radius)
y1 = y2 (isosceles U)
so 2x3 = 180° – y1 – y2
= 2x1 (∠ at the centre)
and x1 = x4 (alternate segment result)

Exercises
Set questions (mainly calculations) on the new properties.
As this is the third extension of the work, students by now should be quick to absorb the new
properties and find it much easier to make progress. More able students will benefit from trying
some proof questions once they have mastered the basic drills.
Examples
• In the figure on the right, calculate all the angles marked by a letter when:
– e = 72°, u = 64°;
– x = 58°, f = 53°.
• T is a point outside a circle, and TA and TC are tangents which meet the circle at A and C
respectively. B is a point on the minor arc AC and D is a point on the major arc AC. Sketch
the figure and work out as many angles as you can when:
– ∠TAB = 41°, ∠BAC = 28°, ∠CAD = 38°;
– ∠ABC = 152°, ∠TCB = 10°, ∠DAC = 31°.
Intersecting chords
Class discussion
Establish the intersecting chords property.
To prove the intersecting chords theorem, return to the diagram used for showing that two
angles on the same arc are equal.
Refer students to the diagram, and ask them:
• Which triangles are similar and why?
• What does this say about the relationship between x, y, z and u?
• Deduce that x × y = u × z.
There is an equivalent proof when the point where the two chords intersect lies outside the
circle. Students may be able to establish the first result by themselves using the hints above; if
instead you do this as class discussion, get them to do the second case for themselves.
Exercises
Set calculations on the new properties. This section will make a refreshing change since all the
previous work has focused on angles. Use the opportunity to revise work on similarity which has
been implicated in the proofs.
Examples
• With respect to the figure used above right:
– if x = 6 cm, y = 4 cm, u = 2cm, calculate z;
– if x + y = 10 cm, y = 7 cm, u = 3cm, calculate u + z.
When P lies outside the circle, the
cyclic quadrilateral results establish
that UPDE and UPAB are similar. It
follows that PE × PA = PD × PB. If the
limit is taken as B and D approach C,
then both products are also equal to PC
2
.

Mathematical logic
Class discussion
Students have rehearsed thoroughly in stages the various angle properties associated with a
circle. In addition, in the intersecting chords theorem they have revisited work on similar
triangles. The work has been based on the properties of an isosceles triangle.
Take the opportunity to review the angles work by asking students to rehearse once again the
arguments that establish the angle properties associated with a circle. List the various results as
the discussion develops. Do one or more examples of proofs if appropriate.
Exercises
These should now focus on proof. Revise if necessary the way a proof should be presented,
and insist that this procedure is followed. This work will seem much more difficult so allow time
for it to be mastered.
Examples
• XLY is a tangent to a circle of which LM is chord. The bisectors of ∠XLM and ∠YLM cut the
circle at A and B. Prove that AB is a diameter. What sort of figure is ALBM?
• Two circles touch internally at V. A line through V cuts the circles again at U and W. Prove
that the tangents at U and W are parallel. Does a similar result follow if the circles touch
externally at V?
• A and C are two distinct points on a circle, not at opposite ends of a
diameter. B is any point on the major arc AC. Line BD is drawn so
that it bisects angle ABC and meets the circle at D. Show that, no
matter where B is, point D is always the same.
• Two chords of a circle of length 10 cm and 12cm intersect so that
the longer bisects the shorter. Find in what proportions the longer is
divided by the shorter, giving your answer to two significant figures.
2 hours
Proofs
terms.
questions.
proofs.
Class discussion
To round off this unit, consider some ‘What if …?’ questions. For example, reverse the cyclic
quadrilateral result. Ask:
• If non-collinear points P, Q, R and S are such that ∠PQS = ∠PRS, do the points lie on a
circle?
This is similar to the converse of Pythagoras’ theorem in that it is one of the few cases of a
theorem and its converse where the two are not trivially related.
Make sure that all students realise that this is a serious question that needs logical argument
and proof.
On the web
There is a page devoted to the intersecting
chords theorem at www.cut-the-knot.org/
proofs/IntersectingChordsTheorem.shtml.
Model solution of problem in final class
discussion
We start with x1 and x2 equal.
We assume that R lies outside the unique circle
passing through P, Q and S.
Draw in PR′ to get a further equal angle x3.
Then PR and PR′ have corresponding angles x2
and x3 equal so must be parallel. But they
intersect at P, which is a contradiction.
The same follows if we assume that R lies inside
the circle. So R must lie on the circle and the
four points are indeed concyclic.

Assessment
In triangle ABC, the altitudes BN and CM of the triangle ABC intersect at S.
∠MSB is 40° and ∠SBC is 20°.
Prove that triangle ABC is an isosceles triangle.
TIMSS Grade 12
Each side of the regular hexagon ABCDEF is 10 cm long.
Find the length of the diagonal AC.
TIMSS Grade 12
Assessment
Two circles with centres at A and B have radii of 7cm and 10 cm as shown in the
diagram.
The length of the common chord PQ is 8 cm.
Calculate the length of AB.
TIMSS Grade 12
A straight line intersects a circle at points A and B. The circle in turn intersects
another circle at D and C. AD is produced until it intersects the second circle at E,
and BC similarly at F. Prove that AB is parallel to EF.
The diagram shows two circles, centres A and B. OPX and OQY are tangents to
the circles.
a. State the mathematical name for the quadrilateral ABYQ. Give your reasons.
b. Angle AOQ is 20°. The line OAB cuts the larger circle at Z, as shown.
Calculate the size of angle BZY.
Show your working.
MEI
Unit 11A.2

157 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.3 | Algebra 1 © Education Institute 2005
GRADE 11A: Algebra 1
Quadratics and graphs
About this unit
This is the first of four units on algebra for Grade
11 advanced. In it students will connect their
algebraic and graphical knowledge as they see
how a quadratic form relates to its graph. The
unit builds on the work on algebra in the Grade
10 advanced units.
your lesson plans.
advanced.
Previous learning
solve quadratic equations exactly, by factorisation, by completing the square
and by using the quadratic formula.
Expectations
By the end of the unit, students will use their knowledge of
interconnections in mathematics and a range of strategies to solve
problems. They will find the axis of symmetry of the graph of a quadratic
function, and the coordinates of its turning point. They will know what the
discriminant tells them about the real roots of a quadratic equation, and they
will find approximate solutions of quadratic equations using graphical
methods. Through their study of functions and their graphs, and the solution
of associated equations, students will appreciate a range of numerical and
algebraic applications in the real world.
Students who progress further will solve more complex problems and will
know what the discriminant tells them about the complex roots of a quadratic
equation.
Resources
• spreadsheet software such as Microsoft Excel
• graph plotting software such as:
Autograph
(see www.autograph-math.com)
Graphmatica
(free from www8.pair.com/ksoft)
• computers with Internet access, spreadsheet and graph plotting software
for students
• quadratic, linear, root, repeated
• symmetry, axis, turning point, maximum, minimum
• discriminant, non-negative
UNIT 11A.3
12 hours

12 hours
Grade 10A standards
CORE STANDARDS
Grade 11A standards
EXTENSION STANDARDS
Grade 12A standards
11A.1.3 Identify and use interconnections between mathematical topics.
11A.1.5 Use a range of strategies to solve problems, including working
the problem backwards and then redirecting the logic forwards;
set up and solve relevant equations and perform appropriate
calculations and manipulations; change the viewpoint or
mathematical representation, and introduce numerical,
algebraic, graphical, geometrical or statistical reasoning as
necessary.
11A.5.2 Model a range of situations with appropriate quadratic functions.
11A.5.3 Find the axis of symmetry of the graph of a quadratic function,
and the coordinates of its turning point by algebraic
manipulation; understand the effect of varying the coefficients a,
b and c in the expression ax2
+ bx + c.
10A.4.11 Solve quadratic equations exactly,
by factorisation, by completing the
square and by using the quadratic
formula.
11A.5.4 Given a quadratic equation of the form ax2
+ bx + c = 0, know
that:
• the discriminant Δ = b2
– 4ac must be non-negative for the
exact solution set in to exist;
• there are two distinct roots if Δ is positive and one repeated
root if Δ is zero.
12AQ.6.1 Given a quadratic equation of the form
ax2
+ bx + c = 0, know that if the
discriminant Δ = b2
– 4ac is negative,
there are two complex roots, which are
conjugate to each other.
3 hours
Plotting and
using quadratic
graphs
3 hours
Completing the
square
3 hours
Constructing
quadratic
graphs
3 hours
Modelling and
problem
solving with
quadratics
11A.5.5 Find approximate solutions of the quadratic equation
ax2
+ bx + c = 0 by reading from the graph of y = ax2
+ bx + c the
x-coordinate(s) of the intersection point(s) of the graph of this
function and the x-axis.
Unit 11A.3

Activities
3 hours
Plotting and using
quadratic graphs
Find approximate solutions
of the quadratic equation
ax2
+ bx + c = 0 by reading
from the graph of
y = ax2
+ bx + c the
x-coordinate(s) of the
intersection point(s) of
the graph of this function
and the x-axis.
Use a range of strategies to
solve problems, including
working the problem
backwards and then
redirecting the logic forwards;
set up and solve relevant
equations and perform
appropriate calculations and
manipulations; change the
viewpoint or mathematical
representation, and introduce
numerical, algebraic,
graphical, geometrical or
statistical reasoning as
necessary.
Plotting quadratic graphs
Class discussion
Grade 10 advanced work has familiarised students with quadratic graphs using ICT. The
purpose of the first section of this unit is for students to become familiar with pencil and paper
techniques for graphs so that they better appreciate the advantages of using ICT.
Begin by discussing the technique for plotting curves as opposed to straight lines. Stress:
• constructing a table of values;
• choosing scales;
• plotting points;
• drawing a smooth curve.
The table of values
Usually the problem will dictate the domain for x (the range of the x-scale). Where this is not the
case, students must decide by experiment what range to use. Make the (usually integral) values
in the domain the first row of the table, and thereafter give each term (with its sign) another row.
The range for y and the scales on the axes
Aim to use scales that are a compromise between having a square-shaped graph and
manageability. Avoid, for example, 3 units per 2 cm, which would be difficult both to plot and to
read; and avoid tall thin or short fat graphs. Avoid also large scales that result in widely
separated points; the result is an unreliable graph between the points. This advice is best
offered after students have experimented a little for themselves and have put together the
resultant graphs for general criticism.
Plotting points
Use dots for points, so that a good curve conceals them effectively. In graphs from scientific
experiments, students may have used crosses to plot points. In a mathematical graph the points
are only means to an end and not in themselves of significance; nevertheless, the resulting
curve must pass through all of them. In a science experiment, the points are scientific
measurements that are individually important; often the aim is to get a straight line and not all
the measured points will necessarily lie on it.
Joining the points
Take time to allow students to practise this skill. Get students to turn the paper round to use the
natural facility of the wrist to best advantage. The curve should be lightly drawn with a sharp but
soft pencil first, and then firmed up as its precise trace becomes clear. Once drawn, it should
not be possible to discern where the original points were by bumps of sudden curvature. Once
the plots have been made, if there are wide gaps between any pairs of adjacent points, the
points should be supplemented by extra ones.
Table of values for the graph y = x2
– 3x + 1 on
the interval –1 ≤ x ≤ 4:
x –1 0 1 2 3 4
x2
1 0 1 4 9 16
–3x 3 0 –3 –6 –9 –12
+1 1 1 1 1 1 1
y 5 1 –1 –1 1 5
Note the distinction of the first row from the rest
by heavy underlining! A frequent mistake is to
add this row into the sum of each column.
This example suggests a scale of 2cm per unit
on the x-axis (so a width of 10 cm) and (in this
case) the same for the y-axis (so a height of
about 12cm); the result 10cm × 12 cm is
roughly square.
Use Autograph in support of this section to show
the correct answers for the graphs. Microsoft
Excel can also be used to display tables such as
that above quickly for the checking of answers.
This column is for
own resources, e.g.
Unit 11A.3

Exercises
Set questions such as those on the right. Some new items of vocabulary are introduced here. At
the end of this section review them for future reference.
Give students enough practice to ensure that they master:
• the graph plotting skills;
• the correspondence between graphs and algebraic problems such as solving a quadratic.
Make sure that the questions set allow discussion of potential pitfalls, such as the distinction
between 2x2
and (2x)
2
and between (–x)
2
and –x2
.
Include at least one case of a graph that does not cut the x-axis so that its corresponding
quadratic equation is insoluble.
Examples
• Plot points and draw a smooth curve for the
function y = x2
+ 3x – 2 for values of x which
satisfy –5 ≤ x ≤ 2. For what values of x is
y = 0? What are the coordinates of the turning
point? Is it a maximum or a minimum?
• Draw the graph of y = 2x2
– 5x – 2 for values
of x which satisfy –1 ≤ x ≤ 6. Find:
– the values of x for which y = 0;
– the coordinates of the turning point;
– the equation of the axis of symmetry.
How would you use the graph you have
drawn to solve also the equation
0 = 2x2
– 5x – 4?
3 hours
Completing the square
Identify and use
interconnections between
mathematical topics.
Find the axis of symmetry of
the graph of a quadratic
function, and the coordinates
of its turning point by
algebraic manipulation;
understand the effect of
varying the coefficients a, b
and c in the expression
ax2
+ bx + c.
Given a quadratic equation of
the form ax2
+ bx + c = 0,
know that:
• the discriminant
Δ = b2
– 4ac must be non-
negative for the exact
solution set in to exist;
• there are two distinct roots
if Δ is positive and one
repeated root if Δ is zero.
Completing the square
Class discussion
The last section introduced the idea of finding maximum and minimum values of a function.
Move on to relating those values to the parameters of ax2
+ bx + c, and the technique of
completing the square.
Consider this question.
• What is the minimum value of (say) x2
+ 3x – 2?
Some students may suggest drawing a graph. Since that is time-consuming, what follows is an
alternative approach.
Remind students about work in Grade 10 on solving quadratics by completing the square.
Introduce the completion of the square technique by writing:
x2
+ 3x – 2 = (x +
3
⁄2)
2
– (
3
⁄2)
2
– 2
= (x +
3
⁄2)
2
– 4
1
⁄4
Once the details of the technique are clear, ask students to volunteer:
• the coordinates of the minimum (or maximum) point of the corresponding curve;
• the equation of the axis of symmetry.
Extend the technique to cope with cases where the coefficient of x is either or both:
• non-zero;
• negative.
If students display little confidence in the algebra
featured here, use Autograph to show that the
original and the completed square form have the
same graphs.
On the web
Quadratics, functions and equations are
extensively covered on MathsNet at
www.mathsnet.net/algebra.
Mathworld has a lengthy article beginning at
high-school level and going beyond at
mathworld.wolfram.com/QuadraticEquation.html.
St Andrews’ Mathematics has a history article at
www-groups.dcs.st-and.ac.uk/~history/
HistTopics/Quadratic_etc_equations.html.

Exercises
Set questions to practise this new technique which tie this work to the previous section on
graphs. If students have retained their graphs, for example, they can use algebraic techniques
to verify their answers or to make them more precise.
Examples
• Find a and b such that x2
– 4x + 1 = (x – a)
2
+ b.
• Find a, b and c such that 2x2
– 4x + 1 = a(x – b)
2
+ c.
• Find the coordinates of the turning point and the axis of symmetry of the curve
y = 5 + 4x – 2x2
.
• Does the equation 1 + x + x2
= 0 have any solutions? Explain your answer carefully.
Using completing the square
Class discussion
Conclude this section of work by considering the general equation ax2
+ bx + c = 0 (a ≠ 0) and
derive the quadratic formula. Emphasise that:
• the formula method is based on completion of the square;
• it explains how to decide which cases have roots and which do not (corresponding to curves
which do or do not cut the x-axis) according to the values of Δ = b2
– 4ac.
The full derivation of this result is included in work for Grade 10 advanced.
The three cases are
• Δ > 0: two distinct roots;
• Δ = 0: two coincident roots (so just one);
• Δ < 0: no roots.
Exercises
Set questions that require calculation of the discriminant in different contexts.
Examples
• Show that the equation x2
– x + 1 = 0 has no roots.
• Show that the line y = 2x – 1 does not intersect the curve y = x2
+ x + 1.
• Find the range of values of m for which x2
– mx + 4 = 0 has roots.

Linking algebra and graphs
Class discussion
Students’ experience now encompasses a wide spectrum of techniques in relation to quadratic
functions and equations.
Summarise in as much detail as needed the technique of completing the square and how the
parameters that figure in the derivation relate to the representation of a function on a graph. The
particular points to mention are that:
• a quadratic graph is always a parabola;
• the intercept on the y-axis is at (0, c);
• the value of a determines the orientation as well as the ‘thinness’ of the parabola;
• the axis of symmetry is
2
b
x
a
= − ;
• the minimum/maximum point is
2
4
,
2 4
b ac b
a a
⎛ ⎞
−
−
⎜ ⎟
⎝ ⎠
;
• the corresponding quadratic equation will or will not have roots depending on the value of
b2
– 4ac.
More able students may appreciate a brief
digression into curves such as x = y2
.
3 hours
Constructing quadratic
graphs
working the problem
backwards and then
necessary.
Find the axis of symmetry of
the graph of a quadratic
function, and the coordinates
of its turning point by
algebraic manipulation;
understand the effect of
varying the coefficients
a, b and c in the expression
ax2
+ bx + c.
Exercises
Set questions to be explored with graphics calculators (or Autograph) so that the
correspondence between the algebra and graphs is exercised.
As a simple alternative to Autograph or graphics calculators, use the applet Grapher from
nlvm.usu.edu/en/nav/vlibrary.html. The trace function is a particularly good feature of this applet
as it allows students to see the values of x and y at particular points of the graph.
Examples
• Find an equation that models the graph on the right.
• Find an equation which models the graph obtained by reflecting the one shown about the line
y = –2.
• Find a quadratic function with minimum value 2 and which cuts the y-axis at y = 2.
• Consider the graph of y = x2
. By drawing a suitable straight line, find an approximate solution
to the equation 10x2
– 2x + 25 = 0.

3 hours
Modelling and problem
solving with quadratics
working the problem
backwards and then
necessary.
Model a range of situations
with appropriate quadratic
functions.
Modelling and problem solving
Class discussion
Introduce a problem that is sufficiently complex to preclude guesswork and that requires a
quadratic equation for solution. For this purpose, review the approach, namely:
• specifying the unknown carefully, either by defining it in words or by drawing an appropriate
diagram;
• formulating an equation;
• solving the equation;
• interpreting the solution by giving a clear answer to the original question.
Emphasise the elements of modelling implicit in this process:
• attempting to represent the problem as a piece of mathematics;
• solving the mathematical problem in its own terms;
• relating the solution critically to the original situation.
This is a good opportunity to revise some Grade 10 work, such as factorising and solving
quadratics by formula.
Exercises
Set problems such as the examples below which:
• require modelling techniques;
• revisit work covered in Grade 10.
Examples, with model solutions on the right
1 The sum of the squares from n + 1 to n + 4 is 294. What is n?
(Hint: use the formula for the sum of squares.)
2 A herdsman took a certain number of goats to market and sold them for QR198 in total. If he
had sold them for QR2 less each he would have needed to sell four more to have the same
takings. How many goats did he sell?
Further problem solving
Some problems on quadratic functions can be found by choosing the Maths Finder option on
the Nrich website (www.nrich.maths.org/public/index.php). Try the problems Parabolic patterns
and More parabolic patterns.
Model solutions
Problem 1
1
6
1
6
2
2
3 2 2
3 2
2
2
( 4)( 5)(2 8 1)
( 1)(2 1) 294
( 9 20)(2 9)
(2 3 1) 1764
2 18 40 9 81
180 2 3 1764
24 120 1584 0
5 66 0
( 11)( 6) 0
6 or 11
n n n
n n n
n n n
n n n
n n n n n
n n n
n n
n n
n n
n
+ + + +
− + + =
+ + +
− + + =
+ + + +
+ − − − =
+ − =
+ − =
+ − =
= −
So, given that n must be positive, it is 6.
Problem 2
Number of
goats
Price per
goat
n 198/n
n + 4 198/(n + 4)
2
2
2
198 198
2
4
198( 4) 198 2 ( 4)
792 2 8
2 8 792 0
4 396 0
( 22)( 18) 0
18 or 22
n n
n n n n
n n
n n
n n
n n
n
− =
+
⇒ + − = +
⇒ = +
⇒ + − =
⇒ + − =
⇒ + − =
⇒ = −
So the number of goats was 18.
On the web
Quadratic equations applied to problem solving
are covered in a page from Purplemath at
www.purplemath.com/modules/quadprob.htm.

Assessment
Curve A is the reflection in the x-axis of y = x2
.
What is the equation of curve A?
A fountain at ground level sprays out jets of water. Each jet is a parabola. The jet that
sprays the farthest has equation y = –x2
+ 8x – 15. Factorise this expression.
Hence find (a) where the fountain jet is positioned in this xy-coordinate system and
(b) how far from the fountain jet the water hits the ground.
Calculate the greatest height that the water reaches.
Assessment
incorporate in the activities. Huda throws a ball to Mariam who is standing 20 m away.
The ball is thrown and caught at a height of 2.0 m above the ground.
The ball follows the curve with equation y = 6 + c(10 – x)
2
, where c is a constant.
Calculate the value of c by substituting x = 0, y = 2 into the equation.
An n-sided polygon has
1
⁄2 n(n – 3) diagonals.
A polygon has 104 diagonals. How many sides does it have?
y = (x – 3)
2
+ 5 is a quadratic function of x. What is the minimum value of this function and for
what value of x does it occur? What is the maximum range of the function? Give the equation of
the axis of symmetry of the function. Write an alternative form for the equation defining the
function. Sketch the graph of this function.
The diagram shows a rectangular field. Yasir walks round the field along two straight paths from
A to B and from B to C. He walks a total of 160 metres from A to C. Lara walks in a straight line
diagonally across the field, from A to C. She walks a total of 120 metres.
Let the distance AB be x metres. Write down an equation for x and show that it simplifies to
x2
– 160x + 5600 = 0
Solve this equation to find the distance AB, given that it is shorter than the distance BC.
MEI
Unit 11A.3

165 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.4 | Trigonometry 1 © Education Institute 2005
GRADE 11A: Trigonometry 1
Using sine, cosine and area rules
About this unit
This is the first of two units on trigonometry for
Grade 11 advanced. It builds on the work on
trigonometry in Grade 10 and extends problems
to the case of triangles which are not right-
angled.
your lesson plans.
advanced.
Previous learning
To meet the expectations of this unit, students should already know and use
the standard trigonometric ratios to find the remaining sides of a right-angled
triangle given one side and one angle, or to find the angles given two sides.
Expectations
By the end of the unit, students will use trigonometry to solve practical
and theoretical problems, breaking them down into smaller tasks. They will
know and use the sine and cosine rules, and will calculate the area of a
triangle using ½ absin C.
Students who progress further will solve more complex problems.
Resources
• graph plotting software such as:
Autograph
(see www.autograph-math.com)
Graphmatica
(free from www8.pair.com/ksoft)
• dynamic geometry system (DGS) such as:
Geometer’s Sketchpad
(see www.keypress.com/sketchpad)
Cabri Geometrie
(see www.chartwellyorke.com/cabri.html)
• computers with Internet access, graph plotting and dynamic geometry
software for students
• sine (sin), cosine (cos), arc sine (arcsin), arc cosine (arccos)
UNIT 11A.4
9 hours

9 hours
Grade 10A standards
CORE STANDARDS
Grade 11A standards
EXTENSION STANDARDS
Grade 12A standards
11A.1.4 Break down complex problems into smaller tasks.
10A.6.5 Know the standard trigonometric ratios and
their standard abbreviations, e.g. for sine of
θ, given an angle θ in a right-angled
triangle, and use these ratios to find the
remaining sides of a right-angled triangle
given one side and one angle or to find the
angles given two sides.
11A.8.2 Know and use the sine rule and the cosine rule to solve
triangles
11A.8.3 Solve triangle problems in two and three dimensions.
2 hours
Calculating areas
of triangles
2 hours
The sine rule
2 hours
The cosine rule
3 hours
Problem solving
11A.8.4 Calculate the area of a triangle using ½ absin C.
Unit 11A.4

Activities
2 hours
Calculating areas of
triangles
Calculate the area of a
triangle using ½ absin C.
Symmetries of sine and cosine graphs: the area rule
Practical activity
Either with Autograph or with a graphics calculator (in degree mode), get students to produce a
display of the graphs of y = sinx and y = cosx over the domain 0° ≤ x ≤ 180°.
Point out the symmetries. Ask students to speculate on what happens if the domain is extended
further; they can of course verify any conjecture with a further display.
Demonstration of symmetries in Autograph
This column is for
own resources, e.g.
Class discussion
From the graphs, establish that:
sinx° = sin (180 – x)° and cos x° = –cos (180 – x)°
Ask students to confirm these results using random values on a calculator (for example, by
comparing 40° and 140°). Consider extension to sines and cosines of angles in the range zero
to four right angles if appropriate for more able students.
Sketch a case of an acute-angled triangle with two sides and an included angle given. Ask the
class how this information can be used to calculate the area of the triangle. Remind students of
the conventions for naming angles and sides of triangles (small letters for sides and
corresponding capital letters for opposite angles).
Do this by providing a number of sketched triangles with measurements marked and getting
students to verbalise the data in the correct format.
Establish the area rule, ½ absin C, using the conventional notation.
Show that the relationship sin x° = sin (180 – x)° enables this to work even when angle C is
bigger than a right angle, and that the case where angle C is a right angle reduces to the
standard formula ½bh.
The naming convention for sides: corresponding
sides and opposite angles use the same letter.
Exercises
Give practice on this rule in simple cases.
Examples
• Find the area of UPQR in which PQ = 4cm, QR = 5cm and ∠PQR = 40°.
• UXYZ has area 49 m
2
and two sides XY = 10.7 m and YZ = 13.5 m. Find the two possible
sizes of ∠XYZ, and sketch the two triangles.
On the web
The BBC's Bitesize site covers the work of the
trigonometrical formulae at
www.bbc.co.uk/schools/gcsebitesize/
maths/shapeh/areaofatrianglerev1.shtml.
Unit 11A.4

2 hours
The sine rule
Know and use the sine rule
and the cosine rule to solve
triangles.
Investigation
Provide two examples of triangles in which two angles and one side are specified, making
essential use of the naming convention. (For example, in UABC, a = 5cm, ∠B = 40°, ∠C = 55°;
in UXYZ, ∠X = 20°, ∠Y = 140°, x = 17cm.)
Get students to:
• sketch these, with the given information clearly shown;
• investigate what further information can be determined from these starting points.
Students should be able to see that they can calculate:
• the third angle in each case;
• the area;
• at least one of the unknown sides.
Class discussion
Draw UABC in which ∠B = 55° and ∠C = 40°, and BC = 5 cm. Do this both as a sketch on the
board or OHT, with an indication of how it would be done exactly as a construction, and also in
a dynamic geometry system (DGS), where the unknowns can be measured by using the
software’s capabilities. Challenge students to emulate the software, i.e. to do it by calculation.
The diagram was produced in DGS by creating line segment BC and choosing the two
unnamed points so that the angles have the right magnitudes (essentially as a process of
construction would). The software then measures the lengths of the segments AC and AB.
Ask for proposals of how to calculate the remaining sides; if the answer is not forthcoming, ask
which altitude may be calculated from the given information. (In the example of UABC above, it
is possible to calculate an altitude BH, where H lies on AC.) Discuss this calculation, to see how
it can lead to the calculation of b. Ask students:
• Is it also possible to calculate c, and to calculate the area?
Avoid giving practice in this as a routine, since it
would reinforce what is ultimately an inefficient
procedure. Nevertheless, students should
discover the way that the sine rule is proved,
namely:
sin sin
sin sin
sin
for the purpose
sin
sin of calculating
to establish
sin sin sin the sine rule
h
A h c A
c
h
C h a C
a
h
a
C
c A
C a
a c b
A C B
= ⇒ =
= ⇒ =
⇒ =
⎧
= ⎨
⎩
⎧
⎛ ⎞
⇒ = = ⎨
⎜ ⎟
⎝ ⎠ ⎩

Rework the calculation with a, B, C instead of the numbers given, so that the sine rule emerges
as
sin sin
b c
B C
= .
Invite students to suggest what would result if the same procedure were used to calculate a
(see the notes on the right).
Demonstrate another case to show the routine of using the sine rule in such a situation.
Establish the procedures:
• write down the sine rule;
• tick the letters which represent known measures (to establish what may be found);
• do the calculations.
More able students should be encouraged to derive
sin
sin
b A
a
B
= before substituting.
The sine rule can also be established using the
circle geometry results of Unit 10A.2. Consider
UABC and its circumcircle centre O.
An isosceles triangle is formed by the two radii
OA and OB (= r) and the side AB = c of UABC.
Hence the two angles marked x are each equal
to ∠C (∠s on same arc). Then, in UAOH,
1
⁄2 c = HB = rsinC, i.e. 2
sin
c
r
C
=
This argument can be repeated for each pair
(b, B) and (a, A), so the sine rule follows.
Exercises
Get students to practise this technique on several examples. Use the opportunity to reinforce
clear presentation:
• draw a diagram first with letters and information marked;
• set out calculations clearly;
• consider appropriate accuracy for the final answers.
Class discussion
Demonstrate the use of the sine rule to find an unknown angle (for example, in UABC,
a = 5cm, b = 7cm, ∠B = 50°; find ∠A), taking care to avoid the ambiguous case (by
specifying cases where the angle to be determined is not opposite the largest side).
Exercises
Set questions which use this technique on several similar cases.
More able students may show interest in the
ambiguous case and how the calculation ties in
with the process of construction, i.e. how two
different triangles can be realised from the same
data in each case.

2 hours
The cosine rule
Know and use the sine rule
and the cosine rule to solve
triangles.
Working up to and using the cosine rule
Investigation
Ask students to sketch triangles in each of these cases:
• two sides and an included angle are given;
(for example in UABC, b = 7cm, c = 6cm, ∠A = 65°);
• three sides are given
(for example in UXYZ, x = 6cm, y = 5cm, z = 4cm).
Ask students to consider what further information can be readily calculated (the area in the first
case and nothing in the second).
Class discussion
Take the first case of triangle ABC where two sides and an included angle are specified. Ask:
• What construction was helpful when considering the sine rule?
If no answer is forthcoming, introduce one of the two altitudes (BH or CK). Derive the cosine
rule by applying Pythagoras’ theorem to the two resultant right-angled triangles.
Spend some moments considering the structure (symmetry and arrangements of letters) of this
formula.
More able students should be able to write down equivalent forms for b2
and c2
, although less
confident students at this stage may prefer to have all such equivalent forms explicitly available
to them to make a choice.
Round this discussion off by showing that the cosine rule reduces to Pythagoras’ theorem in the
case of an angle that is 90°.
Note that care is needed here to avoid straying
into difficulties. The method to use is:
2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2
( )
2
2
2
2 cos
b x h
c a x h
a ax x h
c b a ax
c a b ax
a b ab C
= +
= − +
= − + +
⇒ − = −
⇒ = + −
= + −
Stress that at the next-to-last line the whole
right-hand side is known. It may not be obvious
to students that x too is known; give them the
chance to volunteer how to calculate it by
elementary trigonometry.
Exercises
Set questions which require use of the cosine rule to calculate unknown sides. More able
students should be able to manage √(b2
+ c2
– 2bccosA) with no intermediate steps.
In later questions, extend the work into:
• two-stage calculations, such as those that require calculation of the area;
• word problems.
Although this appears to be an exercise similar
to that on the sine rule, students will have
difficulty at first in obtaining the correct answer
from a calculator; the common mistake (when
calculating a2
) is to enter b2
+ c2
– 2bc and
erroneously carry out a calculation; check early
that that this pitfall has been avoided.

Class discussion
Starting from a presentation of the sine and cosine rules, ask students to consider the case of a
triangle with three sides given. Ask:
• Which formula can be used to yield results?
This is an opportunity to help students to appreciate which formula is of value in any given
context. Less confident students will only make progress if they realise that the sine rule is of no
value (in a particular case) before trying the cosine rule. Demonstrate:
• how the cosine rule can be rearranged (see the notes on the right);
• the three equivalent forms;
• how to use the appropriate version to find an unknown angle when three sides are given.
Students should be familiar with the formula
2 2 2
cos
2
b c a
A
bc
+ −
=
and its two equivalents.
Exercises
Set exercises in two stages:
• cases of three sides given;
• miscellaneous examples where the solution depends only on a single use of the sine or
cosine rule.
3 hours
Problem solving
Break down complex
problems into smaller tasks.
Solve triangle problems in
two and three dimensions
Using the sine and cosine rule to solve problems
Class discussion
Review the three formulae (area rule, sine rule, cosine rule) and the naming conventions.
Present a problem that involves all three as an illustration of strategy. Indicate to students the
importance of carrying forward an accurate (rather than truncated) answer in successive parts
of the question to make the final answer(s) sufficiently accurate.
Example
In UABC, a = 4cm, b = 6cm, c = 7cm; find one of the angles, and hence find the area.
As a general rule, where it is not clear how to
proceed, write down the sine rule to see if it can
help. If not, try the cosine rule.
Solution
The sine rule cannot be used here.
2 2 2
2 2 2
arccos
2
6 7 4
arccos
2 6 7
34.771...
34.8 to 1 d.p.
b c a
A
bc
+ −
=
+ −
=
× ×
=
=
The interim value should be stored in one of the calculator’s memories. For the area we have:
1
2 sin 0.5 6 7 sin34.771...
11.97...
12.0 to 3 s.f.
bc A = × × ×
=
=
So the area is 12.0cm
2
.
On the web
Mathcentre has worked examples of the
rules at www.mathcentre.ac.uk/students.php/
all_subjects/trigonometry/sine_cosine/
resources/52.

Exercises
Set miscellaneous problems of increasing complexity on the area, sine and cosine rules.
Include:
• miscellaneous one-stage calculations where the main emphasis is choice and use of the
correct rule;
• two-stage calculations where the first step is obvious or indicated;
• problems where strategy is important;
• problems posed in words to develop problem solving skills.
Less confident students should be able to
progress beyond one-stage problems here.
Problems in three dimensions
Class discussion
Discuss a problem in three dimensions.
Example
A solid with four sides – a tetrahedron – has vertices A, B, C and D. AB = 4cm, AC = 5cm,
AD = 6cm, BD = 7cm, ∠ABC = 80°, ∠DAC = 40°. Sketch the figure and mark the distances
and angles given. Find:
• ∠ACB;
• ∠ABD;
• BC;
• DC;
• ∠DBC;
• the total surface area of the figure.
Stress the main points of good technique:
• a good diagram;
• the repeated drawing of plane figures which represent each triangle worked on;
• taking a strategic view of the problem.
Steps of the solution
• Draw UABC and use the sine rule to calculate ∠ACB – it must be acute.
• Draw UABD and use the cosine rule to calculate ∠ABD.
• Use UABC again to calculate ∠BAC.
• Use UABC again and the cosine rule to calculate BC.
• Draw UADC and use the cosine rule to calculate DC.
• Use the cosine rule to calculate ∠DBC.
• Calculate the area of each triangular side and add those values together.

Exercises
Set problems with three-dimensional situations. Start with one-stage calculations to begin with,
moving gradually to full-scale problem solving as in the worked example above or the example
below.
Example:
The Great Pyramid of Cheops in Egypt is built on a square base with side 230 metres.
Each face of the pyramid is at 52° to the horizontal.
Calculate the height of the pyramid.
Calculate the inclination of an edge of the pyramid to the horizontal.
The Great Pyramid of Cheops at Giza
(source: www.kingtutshop.com)

Assessment
Show that Pythagoras’ theorem is a special case of the cosine rule.
A triangle has its three angles in the ratio 2 : 3: 4.
Find to two significant figures the ratio of the lengths of its sides.
A helicopter at airfield A received a distress call from a boat. The position of the boat, B, was
given as 147km from the airfield, on a bearing of 072°.
A man on the boat is flown to hospital.
Calculate the distance the helicopter travelled from the boat to the hospital at H.
Assessment
On another occasion the helicopter travelled from the airfield on a bearing of 218° to fly a
pregnant woman at W to the hospital.
The helicopter then flew on a bearing of 081° to the hospital, H.
Calculate the distance the helicopter travelled from where it picked up the woman to the
hospital.
The two sides of a canal are straight, parallel and the same height above the water level. Jana
and Sharifa want to find the width of the canal. They measure 100m on the canal bank and
stand facing each other at points J and S. Jana measures the angle she turns through to look at
a post, P, as 25°. Sharifa measures the angle she turns through to look at the post as 15°.
Calculate the width of the canal.
The diagram shows two concentric circles. OP = 3.0cm and OQ = 5.0cm.
a. Calculate the shaded area.
b. Calculate the length of PQ when the angle POQ is 130°.
Give your answer to a suitable degree of accuracy.
R and S lie on the same concentric circles as P and Q.
c. The area of triangle ROS is 5cm
2
. Find the possible sizes of angle ROS.
MEI
Unit 11A.4

175 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.5 | Probability 1 © Education Institute 2005
GRADE 11A: Probability 1
Independent and dependent events
About this unit
This unit is the first of two on probability for
Grade 11 advanced. In it students meet
elementary ideas of probability and how to
formalise them into mathematical notation,
building on the work on elementary probability in
Grades 8 and 9.
your lesson plans.
advanced.
Previous learning
To meet the expectations of this unit, students should already be able to use
mathematics to model and predict the outcomes of real-world applications.
They should be able to select statistics and a range of charts, graphs and
tables to present findings.
Expectations
By the end of the unit, students will use a simple mathematical model to
calculate, for a particular set of events, the theoretical probability of obtaining
a particular outcome for a random variable associated with those events.
They will calculate probabilities of single and combined events. They will use
tree diagrams to represent and calculate the probabilities of compound
events when the events are independent and when one event is conditional
on another.
Students who progress further will readily relate the definitions of
independence and exclusive events to practical situations in order to apply
their knowledge.
Resources
• computers with Internet access for students
• scientific calculators for students
• spinners, dice, coins, drawing pins for simple probability demonstrations,
or simulations of these
• event, exclusive, independent, exhaustive
• experiment, random variable, outcome
• empirical probability, theoretical probability, conditional probability, relative
frequency, expected frequency, symmetry
UNIT 11A.5
10 hours

10 hours
Grade 8 and 9 standards
CORE STANDARDS
Grade 11A standards
EXTENSION STANDARDS
Grade 12A standards
9.8.5 Use relative frequency as an
estimate of probability and use this
to compare outcomes of
experiments.
11A.1.2 Use mathematics to model and predict the outcomes of
real-world applications; compare and contrast two or
more given models of a particular situation.
12AS.1.2 Use mathematics to model and predict the outcomes of
substantial real-world applications, and to compare and
contrast two or more given models of a particular
situation.
12AQ.1.2 Use statistical techniques to model and predict the
outcomes of statistical situations, including real-world
applications; work to definitions and perform appropriate
tests.
11A.1.10 Approach a problem systematically, recognising when it
is important to enumerate all outcomes.
12AQ.1.10 Approach complex problems systematically, recognising
when and how it is important to enumerate all outcomes.
11A.12.1 Know that all probability values lie between 0 and 1, and
that the extreme values correspond respectively to
impossibility and certainty of occurrence.
12AQ.11.1 Know that all probability values lie between 0 and 1, and
that the extreme values correspond respectively to
impossibility and certainty of occurrence; calculate
probabilities.
12AQ.11.2 Know that all possible outcomes for an experiment form
the sample space for that experiment; use the sample
space to calculate probabilities for each outcome.
11A.12.2 Understand that a random variable has a range of
values that cannot be predicted with certainty, and
investigate common examples of random variables;
measure the empirical probability (relative frequency) of
obtaining a particular value of a random variable.
12AQ.11.3 Understand that a random variable has a range of
values that cannot be predicted with certainty;
investigate common examples of random variables;
measure the empirical probability (relative frequency) of
obtaining a particular value of a random variable.
8.8.9 Use problem conditions to calculate
theoretical probabilities for possible
outcomes.
3 hours
Empirical
probability
2 hours
Mutually
exclusive and
exhaustive
events
2 hours
Independent
events
3 hours
Probability
trees
9.8.7 Compare experimental and
theoretical probability in different
contexts.
11A.12.3 Use a simple mathematical model to calculate, for a
particular set of events, the theoretical probability of
obtaining a particular outcome for a random variable
associated with those events.
12AQ.11.4 Know that a probability distribution for a random variable
assigns the probabilities of all the possible values of the
variable and that these values total to 1; use a simple
mathematical probability distribution to calculate, for a
particular set of events, the theoretical probability of
obtaining a particular outcome for a random variable
associated with those events.
11A.12.5 Understand when two events are mutually exclusive,
and when a set of events is exhaustive; know that the
sum of probabilities for all outcomes of a set of mutually
exclusive and exhaustive events is 1, and use this in
probability calculations.
12AQ.11.8 Understand when two events are mutually exclusive,
and when a set of events is exhaustive; know that the
sum of probabilities for all outcomes of a set of mutually
exclusive and exhaustive events is 1, and use this in
probability calculations.
Unit 11A.5

10 hours
Grade 8 and 9 standards
CORE STANDARDS
Grade 11A standards
EXTENSION STANDARDS
Grade 12A standards
11A.12.6 Know that when two events A and B are mutually
exclusive the probability of A or B, denoted by P(A ∪ B),
is P(A) + P(B), where P(A) is the probability of event A
alone and P(B) is the probability of event B alone.
9.8.6 Know that if A and B are mutually
exclusive, the probability of A or B
is the sum of the probabilities of A
and of B.
11A.12.7 Know that two events A and B are independent if the
probability of A and B occurring together, denoted by
P(A ∩ B), is the product P(A) × P(B).
12AQ.11.9 Know that:
• when two events A and B are mutually exclusive
the probability of A or B, denoted by P(A ∪ B), is
P(A) + P(B), where P(A) is the probability of event A
alone and P(B) is the probability of event B alone;
• two events A and B are independent if the probability
of A and B occurring together, denoted by P(A ∩ B), is
the product P(A) × P(B);
• when two events A and B are not mutually exclusive
the probability of A or B, denoted by P(A ∪ B), is
P(A ∪ B) = P(A) + P(B) – P(A ∩ B), where P(A) is
the probability of event A alone, P(B) is the probability
of event B alone and P(A ∩ B) is the probability of
both A and B occurring together.
9.8.10 List systematically all the possible
outcomes of an experiment.
11A.12.8 Use tree diagrams to represent and calculate the
probabilities of compound events when the events are
independent and when one event is conditional on
another.
12AQ.11.10 Use tree diagrams to represent and calculate the
probabilities of compound events when the events are
independent and when one event is conditional on
another.
11A.12.9 Know that in general if event B is dependent on event A,
then the probability of A and B both occurring is
P(A ∩ B) = P(A) × P(B|Α), where P(B |Α) is the
conditional probability of B given that A has occurred.

Grade 11 advance mathematics book with solution

Grade 11 advance mathematics book with solution

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