1. Revised Additional Mathematics syllabus
4/21/2017
Revised Additional Mathematics syllabus
Set Language and notation
Vectors(Relative Velocity)
Permutations and Combinations
Matrices
Calculus

2. Revised Additional Mathematics syllabus
4/21/2017
Revised Additional Mathematics syllabus
Syllabus Aims
To enable students to
consolidate and extend elementary mathematical skills;
further develop their knowledge of mathematical concepts and principles;
appreciate the interconnectedness of mathematical knowledge;
acquire a suitable foundation in mathematics for further study;
devise mathematical arguments and use and present them precisely and logically;
integrate information technology to enhance the mathematical experience;
conduct independent and/or cooperative enquiry and experiment.
develop the confidence to apply their mathematical skills in appropriate situations;
develop creativity and perseverance in the approach to problem solving;
derive enjoyment and satisfaction from engaging in mathematical pursuits, and gain an appreciation of the beauty, power and usefulness of mathematics.
appreciate the interconnectedness of mathematical knowledge;
acquire a suitable foundation in mathematics for further study;

3. Revised Additional Mathematics syllabus
4/21/2017
Revised Additional Mathematics syllabus
We aim to
Continue to build on the five inter-related aspects of the pentagon framework

4. Revised Additional Mathematics syllabus
4/21/2017
Revised Additional Mathematics syllabus
Appreciation
Interest
Confidence
Perseverance
Attitudes
Metacognition
Processes
Concepts
Skills
Monitoring one’s own thinking
Mathematical PROBLEM SOLVING
Estimation and Approximation
Mental Calculation
Communication
Use of mathematical tools
Arithmetic manipulation
Algebraic manipulation
Handling data
Thinking skills Heuristics
*The conceptualisation of the primary and secondary mathematics programme is based on the framework
5 inter-related aspects of the pentagon framework
Understanding of Concepts
Mastery of essential skills
Acquisition of mathematics processes
Cultivation of positive attitudes towards mathematics
Metacognitive insight about one’s thinking and learning
Numerical
Geometrical
Algebraic
Statistical

5. Revised Additional Mathematics syllabus
4/21/2017
Revised Additional Mathematics syllabus
To continue to underscore the importance of the affective aspects of mathematics learning
Meaningful
activities
confidence in applying mathematics,
enjoyment of mathematical pursuits,
appreciation of the power and beauty of mathematics
perseverance in problem solving
Meaningful activities to
- develop mathematical communication skills
- facilitate the development & explication of thinking skills & problem solving processes

6. Revised Additional Mathematics syllabus
4/21/2017
Revised Additional Mathematics syllabus
Set Language and notation
Permutations and Combinations
Matrices
Vectors (Relative Velocity)
Calculus
Rich Learning environment
Set Language and notation
A topic of discrete mathematics. Many primary activities use manipulatives materials to explore aspects of set relationships and operations. Set concepts have been investigated throughout the elementary grades and here, a formal vocabulary is now introduced.
Permutations and combinations
closely related to sets, again, elementary counting techniques have been introduced to pupils in primary activities. We are developing the strategies at a formal level at this stage.
Matrices
another discrete mathematics topic that can be a means of integrated whole rather than a set of isolated topic. Matrices are a powerful unifying concept, connecting ideas from storing data, solving linear systems, represent geometric figures to transformation etc.
Relative Velocity
has a rich learning environment with real life applications.
Using and applying mathematics
in practical tasks
real life problems
within mathematics itself

7. 3 mins 2 mins 1 min Start 200 m min-1 B A B A B A B A N
Example 1 on p1
4/21/2017
Relative Velocity
200 m min-1
3 mins A B
600 m
2 mins A B
400 m N
1 min A B
200 m
The figure shows a sequence of “aerial snapshots” of 2 boats. Boat A is travelling north and boat B is travelling north-east. At the start the 2 boats are together, but as time goes on they get farther and farther apart; however, it so happens (in this example) that B is travelling at just the right speed to keep it always due east of A.
A passenger in boat A; idly enjoying the trip, may be totally unaware that he is travelling north. As far as he is concerned, he is watching boat B moving away from him in an easterly direction, and their distance apart increasing at a rate of 200 m every minute. In these circumstances we say that the velocity of B relative to A is 200 m per minute east.
Start A B

8. Velocity vector of B relative to AP2
4/21/2017
Relative Velocity
Velocity vector of B relative to A
(displacement vector) = (velocity vector)  (time elapsed)
= (velocity vector of B) - (Velocity vector of A)
metres
200
600
400  A B
Start
Observation (1)
Thus, if an object is moving with constant velocity (i.e. its speed and direction of motion remain unchanged) then
(displacement vector) = (velocity vector)  (time elapsed)
Observation (2)
(velocity vector of B relative to A ) = (velocity vector of B) - (velocity vector of A )

9. South Bank N C D North Bank 60 m D C D C D C Relative Velocity
Example 2 on p2
4/21/2017
Relative Velocity
South Bank N
C D
North Bank
60 m D C D C D C
Two people are sitting in a boat on a river which is flowing due east. One of them, C, dives into the water and swims directly towards the south bank. The other, D, sits and watches her, allowing the boat to drift with the stream. If C has no landmark to aim for but simply 'points' herself due south as she swims, she will be carried downstream. However, the boat will also be carried downstream at the same rate so that D will continue to see C due south of him. If after one minute D sees C 60 m due south of him, he will reckon that C's velocity relative to the boat is 1 ms-1 (or 60 m min-1) due south.
Learning about Properties of vectors and Vector sums, Components of a vector -
River boat & The Plane and the wind - notes and animation -
Relative Motion (Frame of reference) -

10. (velocity of boat relative to water)
'true' velocity of the boat
= combination of the rower's effort
+ the effect of the current.
‘True’
Example 4 on p4
4/21/2017
Relative Velocity
(velocity of boat relative to water)
(velocity of water)
North Bank 3 2 
Effect of the current
Rower's effort
Final result:
travelling diagonally across the river N
Example 4
A man wishes to row from the north to the south bank of a straight stretch of river in which the current is flowing at 3 ms-1 due east. The river is 100 m wide and the man can row at 2 ms-1. If he rows steadily towards the opposite bank, in what direction and with what speed will he actually travel, and how far downstream from his starting-point will he arrive?
Solution
Velocity of the water = 3 ms-1 due east
Velocity of the boat relative to the water = 2 ms-1 due south
Using the formula for relative velocity: (velocity of boat relative to water) = (velocity of boat)  (velocity of water)
The triangle shows that it 'fits with common sense'; the velocity of 2 ms-1 southwards is due to the rower's efforts, the velocity of 3 ms-1 eastwards is due to the current and the sum vector (resultant) shows the final result - the man travels diagonally across the river.

11. How far downstream did he land?
Example 4 on p4
How far downstream did he land?
4/21/2017
Relative Velocity
The current makes no difference to his crossing time!
North Bank 3 2 
Effect of the current
Rower's effort
Final result:
travelling diagonally across the river
‘True’ N
To find how far downstream he lands, it is simplest to argue as follows:
If there were no current, the boat would go straight across at 2 ms-1, so that it would travel the 100m width of the river in 50s. Now the current makes no difference to his crossing time, since its only effect is to sweep the boat downstream, so that it still takes him 50s to cross even when the current is flowing. Since the current is flowing at 3 ms-1, then during these 50s it will sweep him downstream 150m, so that he will land 150m east of his starting-point.

12. Page 5
4/21/2017
Relative Velocity
Points of View
‘the velocity of A relative to B’
refers to velocity of A from the point view of a (sometimes imaginary) person who is moving with B.
The velocity of a swimmer relative to the water is her speed and direction of motion as seen by someone in a boat which is drifting with the current.
For example, the velocity of a swimmer relative to the water is her speed and direction of motion as seen by someone in a boat which is drifting with the current, and the velocity of a missile relative to an aircraft is its speed and direction of motion as seen by someone in the aircraft.

13. At what speed does it actually travel?
Page 5
4/21/2017
Points of View
At what speed does it actually travel?
Boat/bank = boat/water + water/bank
Probe/Jupiter = probe/Earth + Earth/Jupiter
(velocity of boat relative to bank) = (velocity of boat relative to water) + (velocity of water relative to bank)
From the point of view of someone on the ground or dry land, i.e. relative to the Earth.
(velocity of probe relative to Jupiter) = (velocity of probe relative to Earth) + (velocity of Earth relative to Jupiter)
Can we talk about the ‘actual’ velocity of an object at all? every velocity is relative to something
The phrase 'at what speed does it actually travel' carried the unwritten words 'from the point of view of someone on the ground' or 'from the point of view of someone on dry land'. We have made the unstated assumption that the ground, i.e. the Earth, is in some way fixed and that the 'actual' velocity of an object is its velocity relative to the Earth. This is all very well for motion taking place on or near the surface of the Earth, but we know that in fact the Earth is moving and that when scientists calculate the trajectory of a space probe to Jupiter, they have to take into account the velocity of Earth relative to Jupiter. We cannot really talk about the 'actual' velocity of an object at all; every velocity is relative to something, and it is often helpful to make this clear when we write down the solutions of problems. For example:
(velocity of boat relative to bank) = (velocity of boat relative to water) + (velocity of water relative to bank);
(velocity of missile relative to ground) = (velocity of missile relative to aircraft) + (velocity of aircraft relative to ground);
(velocity of probe relative to Jupiter) = (velocity of probe relative to Earth) + (velocity of Earth relative to Jupiter)

14. Q 1000m P ‘Aiming off’  Relative Velocity
Example 5 on p 6
4/21/2017
Relative Velocity
Q m P
‘Aiming off’
An aircraft wishes to travel from point P to point Q which is due west of P. If wind is blowing from the south-west, in which direction must the pilot head? How long will the journey take? 
Aircraft is not travelling westward
100
Example 5
An aircraft whose airspeed (i.e. its speed relative to the air) is 500 kmh-1 wishes to travel from point P to point Q which is 1000km due west of P. If the wind is blowing from the south-west at 100kmh-1, in what direction must the pilot head, and how long will the journey take?
Solution
Using the equation:
(Aircraft relative to ground) = (aircraft relative to air) + (air relative to ground).
If the pilot head west, i.e. he sets the velocity of the aircraft relative to the air due west, The wind is carrying him north of the direction he wants to go.
45
Wind
500
Velocity of aircraft relative to wind

15. Aircraft is 'aiming off' somewhat south Wind
Example 5 on p 6
4/21/2017
‘Aiming off’
100
500
45
Aircraft is not travelling westward
Velocity of aircraft relative to wind
Wind 
100
500
45 
Aircraft is 'aiming off' somewhat south
Wind
Aircraft is travelling westward
The wind is carrying him north of the direction he wants to go, so he must 'aim off' somewhat south.

16. Two canoeists A and B each paddle in still water at 5m/s
Two canoeists A and B each paddle in still water at 5m/s. They both leave at the same time from the same point on the rive bank. The river flows at 3m/s between straight parallel bank, 240m apart.
Canoeist A paddles in the direction that enables him to cross the river in the shortest distance. Canoeist B paddles in such a direction that he lands 240m downstream of the point where A lands.
Determine, with full working whether A or B lands first.