eMath Module

eMath ModuleRambling Walk Around Singapore…

This activity is proudly organized, sponsored and supported by: • Kenneth Yuen 3O2(32),• John Phua 3O2(18),• Koh Tzehou 3O2(09), • Math Department of HCI• Singapore Tourism Board and

Committee• Singapore nature Society

Introduction

To make this walk much more exciting and

fun-filled, you will not only be exploring the nature side, parks as well as the cultural and historical landmarks of Singapore, but at each stop, you will be engaged in interesting and challenging mathematical activities.

In addition, the biggest takeaway would be to learn about Singapore’s heritage and history in j u s t a wa l k !

Instructions

The rambling walk would be from 08:00 to 15:00. All together you would be exploring 6 (excluding the

final destination) stops in Singapore. There are also prizes waiting for you to take home in

addition to the treasures you find! Here are the prizes:

Instructions

1st Prize: Attachment to Singapore Nature Society for June Holidays 2011

2nd Prize: 2 free tickets to the Singapore Math and Algebra Museum

3rd Prize: A Vanda Miss Joaquim plant To win the prizes, finish the activities at each stop and

reach the final stop first! Don’t miss this golden opportunity!

ATTRACTIVE PRIZES FOR JUST A RAMBLING WALK?? YES IT’s TRUE!!

Instructions

All diagrams are not drawn to scale. The tasks involve knowledge on trigonometry, such as

solving of 2D problems using sine rule, cosine rule, bearings, area of triangle and simple trigonometry.

The Geogebra applet that we have given you is to help you get from one stop to another. You may need to zoom in and out. We apologise for any inconveniences caused.

For answers which are not exact, leave it to 3 significant figures. For angles, leave it to 1 decimal place.

Geogebra Applet

There are 4 maps available in the applet. Please follow the instructions for the next 6 tasks and use the applet according to the maps given.

Instructions

Instructions

Items needed:Geogebra Applet (Laptop) Instructions + Activity Worksheet (This one) $10 (provided for lunch at Sixth Stop)A street directory book (in case you get lost)

So…where to start? Start You will be starting at Hwa Chong Institution, Bukit

Timah. 1st Stop Clue: Where is the founder of Singapore

standing at now?

Instructions

First Stop

First Stop

First Stop: Raffles Statue @ outside Victoria Concert Hall, Raffles Place

Geogebra Applet: Refer to Map 3 (Marina South)

1. Read the following extract: Sir Stamford Raffles was the first founder of

Singapore. He stepped on Singapore soil in 1819 and negotiated with the Temenggong then to build a trading port here.

Singapore was once a British colony till 1959, whereby Singapore was granted self-governance.

In 1965, when Singapore separated from the Federation of Malaya, Singapore became independent. This year marks the 46th birthday of the young nation, Singapore.

First Stop

2. Task 1A birthday cake for Singapore’s 46th Birthday was reported lost near the Raffles Statue. Using satellite imaging and technology, only the following information about the location of the cake could be known:

Distance between Raffles Statue and the cake = 57m. The bearing of the cake from Raffles Statue = x (less than

090°). The following diagram was drawn. (Refer to next slide)

First Stop

3. Diagram 1

Find x. From there, you should be able to find the cake and your clue for the next stop.

First Stop

4. Method and Answer for Task 1 Concept tested: Sine and Cosine Rule

First Stop

2 2 2

2 2 2

2

1

2( )( ) cos

57 50 2(50)(57)cos90

574975.822(5 . .)

sin sin 9050 75.822

sin 90sin ( 50)75.822

41.25741.3 (1 . .)

AC BC AB AB BC ABC

AC

ACAC s fx

x

xx d p

5. 2nd Stop Clues 2nd Stop Clue A

You are now at Marina South. (on Geogebra Applet Map 3)

The bearing of your next location should be 307-310° from Marina South.

Note: Any deviations from the clue may end up giving you a different location. Hence, clue B is given to you too as a reference.

2nd Stop Clue B: Find the flowers, the trees, and the nature all over here! The largest garden of Singapore!

First Stop

Second Stop

Second Stop: Singapore Botanical Gardens Geogebra Applet: Refer to Map 3 (Bukit Timah)

Second Stop

1. Read the following extract: Sir Stamford Raffles, the founder of Singapore and a

keen naturalist, established the first botanical and experimental garden on Government Hill (Fort Canning Hill) in 1822.

The Gardens is geared towards entrenching itself as a tropical botanical institution of international renown, a key tourist destination and a flagship park.

Second Stop

2. Task 2You are required to help Singapore Botanical Gardens to put up banners for an event. The event will be held at the Symphony Lake. There are three banners of different lengths. Only the lengths of 2 banners are known and the bearing of B from C is 065°. Find CAB. ∠From there, you should be able to put up all the banners in the way like the diagram and get your next clue from the staff.

Second Stop

3. Diagram 2

Symphony Lake 95m

65m

B

A

C

Second Stop

4. Method and Answer for Task 2: Concept tested: Sine Rule

Second Stop

sin sin 6565 95

sin 65sin 6595

38.3 (1 . .)

CAB

CAB

CAB d p

5. 3rd Stop Clues 3rd Stop Clue A

You are now at Marina South. (on Geogebra Applet Map 3)

The bearing of your next location should be 133-137° from the “Bukit Timah”.


3rd Stop Clue B: We are going to say goodbye to this place very soon.

Second Stop

Third Stop

Third Stop: KTM Station at Tanjong Pagar Geogebra Applet: Refer to Map 3 (Outram)

Third Stop

1. Read the following extract: The train service carries passengers from Singapore

to Kuala Lumper, and from Kuala Lumper back to Singapore.

Recently, it has been announced that the KTM station will be dismantled soon because it will be relocated to Johor or Woodlands.

Many people have since flocked to the station to take photos for memories, or even buy the train tickets to have a last train ride before it goes.

Third Stop

2. Task 3a:Watch a train arrive at Tanjong Pagar KTM Station from point X. After a train has arrived at the station, watch the train return to the train depot. The train should be travelling at an average speed of 21m/s and take about 20s to enter the train depot. Given that DX = 205m, XE = 315m, find angle DXE.

Third Stop

3. Diagram 3:

Third Stop

X

E: KTM Station

D: Train Depot

4. Task 3b:The KTM Station is intending to paint the area covered by DXE in black, but this proposal has been delayed because nobody know how to calculate the area. Also, it is too dangerous and inconvenient to measure the height of the triangle because trains frequently pass by DE. Give a try at calculating the area of DXE so that you get your next clue from the station control personnel.

Third Stop

5. Method and Answer for Task 3a: Concept tested: Cosine Rule

2 2 2

2 2 2

cos2( )( )

205 315 420cos2(205)(315)

105.8 (1 . .)

DX XE DEDXEDX XE

DXE

DXE d p

Third Stop

6. Method and Answer for Task 3b: Concept tested: Sine Rule and Area of Triangle Construct an additional line FX perpendicular to

the line DE.

Third Stop

X

E: KTM Station

D: Train Depot

F

6. Method and Answer for Task 3b:

Third Stop

𝐴𝑟𝑒𝑎=12×𝑏𝑎𝑠𝑒×h h𝑒𝑖𝑔 𝑡

¿

sin 𝑋𝐷𝐸 =

sin𝐸𝐷𝑋

sin 105.8420

= sin∠𝐹𝐸𝑋205

sin∠𝐹𝐸𝑋= sin 105.8420

×205

¿

Read from left to right.

7. 4th Stop Clues: 4th Stop Clue A:

You are now at Outram. (on Geogebra Applet Map 3) The bearing of your next location should be 075-078°

from the “Outram”. Note: Any deviations from the clue may end up giving you

a different location. Hence, clue B is given to you too as a reference.

4th Stop Clue B: Singapore’s most iconic and historical landmark, or more specifically Singapore’s mascot

Third Stop

Fourth Stop

Fourth Stop: Merlion Park @ One Fullerton Geogebra Applet: Refer to Map 2 or Map 3

(Marina South)

Fourth Stop

1. Read the following extract: The Merlion Park was originally located at the

Esplanade Bridge, but now when you come out from One Fullerton/Fullerton Hotel, you will notice this iconic landmark standing at the edge spitting out water.

The Merlion has become a famous landmark and represents Singapore’s historical development – from once a fishing village to a lion city, hence the half-lion and half-mermaid Merlion.

Fourth Stop

2. Task 4 The government wants to build a shelter for the poor

Merlion who has to withstand rain and hot sun day in and out. The shelter is to be triangular. However, the contractor is currently having troubles with finding out the 3 angles of the shelter. He only knows the following information:

The bearing of I from G is 099°. The bearing of H from I is 230°. The bearing of G from H is 015°. Find 3 angles of the shelter. When you are done, call the

contractor at 65778990 and he will tell you the clue for your next stop.

Fourth Stop

3. Diagram 4:

Fourth Stop

G

H

IMerlion

4. Method and Answer for Task 4: Concept tested: Bearings and Basic Angle

Geometry

Fourth Stop

KG

H

IMerlion

J

L 099°

230°015°

4. Method and Answer for Task 4:

The three angles are 35°, 49° and 96°.

Fourth Stop

360 230 130180 130 5050 15 35180 99 81360 230 81 49180 35 49 96

KIHLHIGHIGIKHIGHGI


You are now at Merlion Park. (on Geogebra Applet Map 2)

The bearing of your next location should be 072-078° from the Merlion Park.


5th Stop Clue B: The tallest Ferris Wheel on Earth. I’m not kidding!

Fourth Stop

Fifth Stop

Fifth Stop: The Singapore Flyer @ Marina Bay Geogebra Applet: Refer to Map 2 (Singapore

Flyer)

Fifth Stop

1. Read the following extract: Singapore Flyer was first conceived in early 2000s by

Patrick MacMahon of Melchers Project Management, a subsidiary of German company Melchers.

It is the tallest Ferris wheel in the world, taller than the London Eye and Star of Nanchang.

It has also become one of Singapore’s architectural landmarks, together with Esplanade, Marina Bay Sands and Gardens by the Bay.

Fifth Stop

2. Task 5: Take a ride up the Singapore Flyer. When you reach the top, you can look down at the Esplanade. Also, you get a panoramic view of the Marina Barrage and the Esplanade Bridge too. Your field of vision is illustrated in the diagram below:

Fifth Stop

3. Diagram 5:

The distance between the Suntec City and the Float @ Marina Bay is 100m. The bearing of M from N is 062° and the bearing of M from O is 312°. Points M, N, P and O are points on the ground and N, P, O lie on a straight line perpendicular to the North direction. Find the distance between Suntec City and Mandarin Oriental Hotel.

M: Singapore Flyer

N: Suntec City

O: The Float@ Marina Bay

P: Mandarin Oriental Hotel

062°

Fifth Stop

4. Method and Answer for Task 5: Concept tested: Bearings, Basic Trigonometry,

Algebraic Manipulation

Fifth Stop

90 62 28312 270 42

tan 28

tan 42

MNOMOP

MPNPMPPO

Continued on next slide

Fifth Stop

4. Method and Answer for Task 5:

tan 28

tan 42

100tan 28 tan 42(tan 42) (tan 28) 100(tan 28)(tan 42)

MPNP

MPPO

MP MP

MP MP

(tan 42 tan 28) 100(tan 28)(tan 42)100(tan 28)(tan 42)(tan 42 tan 28)33.423 (5 . )33.423tan 2862.872 (5 . .)62.9 (3 . .)

MP

MP

MP m s f

NP

NP m s fNP m s f

Read from left to right.


You are now at Marina East . (on Geogebra Applet Map 3)

The bearing of your next location should be 344-350° from the Marina East.


6th Stop Clue B: The most unique park in Singapore, with trees shaped like bottles.

Fifth Stop

Sixth Stop

Sixth Stop: Bottle Tree Park @ Yishun Geogebra Applet: Refer to Map 2 (Yishun)

Sixth Stop

1. Read the following extract: The most unique thing about this park is that it grows

bottle trees! The picture above tells it all. Besides that, residents can also fish at the big lake

there and have a seafood feast at a seafood restaurant nearby.

Not only so, if you don’t like fishing, prawning is an available alternative. This park definitely promises you tons of fun and that’s the power of nature!

Sixth Stop

2. Task 6a: First and foremost, have your lunch! We have provided

you $10 to have a good lunch at the Seafood Restaurant there.

The Bottle Tree Park Lake is circular in shape. Fishing nets have been placed inside the circle to form a square QRST. Another red fishing net is placed to separate the two types of fishes that residents can fish. U is a midpoint of RS. If the fishing net QR is 45m long, find the following: Angle The radius of the circular lake

Sixth Stop

3. Diagram 6:

Sixth Stop

4. Method and Answer to Task 6a: Concept tested: Basic Trigonometry, Pythagoras

Theorem

Sixth Stop

2 2

45tan22.5

63.4 (1 . .)

(45 45 )63.6 (3 . .)

d p

RadiusRadius m s f

5. Task 6b: You are now at Bottle Tree. (on Geogebra Applet Map

4) Your next destination is Khatib MRT, but first you need

to measure the bearing of Khatib MRT from Bottle Tree. A stationmaster would be present at your final destination and request for the bearing value.

Method and Answer for Task 6b: Use Geogebra Applet Map 4 to help you. Your bearing is

34-38°.

Sixth Stop

6. THE FINAL DESTINATION:Hwa Chong Institution. Take bus 852 back to the school from Khatib MRT

Station. HURRY!! The prizes are waiting for you to come

back fast!

Sixth Stop

Seventh Stop

Debrief Congratulations, you have finished the activity

Rambling Walk Around Singapore! Hope you have enjoyed the walk while solving challenging but fulfilling mathematical problems involving trigonometry.

P/S: As for the prizes, that will depend on whether you are the first, second, third or last.

Seventh Stop

THE ENDHave a nice day.

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eMath Module