additional mathematics project work - utamajpnperak.moe.gov.my/jpn/attachments/article/5011/3.4...

PERAK STATE

ADDITIONAL

MATHEMATICS

PROJECT WORK

2017

PERAK STATE EDUCATION DEPARTMENT

Additional Mathematics Project Work YEAR 2017

PAGE | 1


Worksheet 1 : Let’s coordinate!

PART 1.1

A simple equation like y = f(x) can be expressed beautifully as a graph on the Cartesian plane. This

awesome representation was made possible by a French mathematician.

Write an anecdote on the contributions and achievements of this famous mathematician.

PART 1.2

Use simple materials to design a game based on coordinates.

Your design should include the following:

1. A suitable name for the game.

2. The mathematical concepts involved.

3. Rules of the game.

Games can be powerful tools that significantly boost personal development, learning

achievement and pupils’ success if the games are specially designed

to develop specific skills or concepts.


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Worksheet 2 : Let’s get into shape!

Quartz crystal has a beautiful geometrical shape. Follow the instructions below to build a model of

this beautiful crystal.

(a) Table 1 shows the coordinates of the vertices of two sets of triangles. Observe the pattern

shown and complete the table.

SET 1 SET 2

Triangle Coordinates of vertices Triangle Coordinates of vertices

A ( 0 , 1 ) ( 2 , 0 ) ( 2 , 2 ) P ( 5 , 0 ) ( 5 , 2 ) ( 7 , 1 )

B ( 0 , 3 ) ( 2 , 2 ) ( 2 , 4 ) Q ( 5 , 2 ) ( 5 , 4 ) ( 7 , 3 )

C ( 0 , 5 ) ( 2 , 4 ) ( 2 , 6 ) R ( 5 , 4 ) ( 5 , 6 ) ( 7 , 5 )

D S

E T

F U

TABLE 1

(b) Six rectangles, drawn on a Cartesian plane, are shown on page 3. On the Cartesian plane,

draw all the twelve triangles shown in Table 1 to complete the layout for the model of a

quartz crystal.

(c) Use the exact layout creatively to build the model.

[ The model built must be handed over to your teacher at the end of this project work. ]

(d) Calculate the total surface area, in cm2, of the model that you have built.


PAGE | 3


y

x 0

1 2 3 4 5 6 7

1

2

3

4

5

6

7

8

9

10

11

12


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Worksheet 3 : Enjoy the journey!

A straight line joins point P ( x1 , y1 ) and point Q ( x2 , y2 ).

Derive the formula for the length of PQ.

Diagram 1 shows the only roads that connect Town A, Town B and Town C.

Diagram 1

Ali drives from Town A to visit his friends who stay in Town B and Town C. After the visit, he returns

to Town A.

1. Calculate the total distance of the journey.

2. A new straight road is to be built connecting Town C to either Town A or Town B.

Which new road should Ali prefer for the above journey? Explain.

x ( km )

( 11 , – 8 )

Town B

( 5 , 8 )

y ( km )

J( 2 , 4 )

O

( – 7 , – 8 )

Town C Town A

Derive The Formula

Use The Formula


PAGE | 5


Worksheet 4 : Coconut delight!

A straight line joins point P ( x1 , y1 ) and point Q ( x2 , y2 ). Point X divides internally the line PQ in

the ratio m : n.

Derive the formula for the coordinates of X.

Hence, state the formula for the coordinates of the midpoint of PQ.

Diagram 2 shows three coconut trees planted on a piece of land drawn on a Cartesian plane.

Diagram 2

1. Salleh, the owner of the land, would like to plant another coconut tree in such a way the four

coconut trees are at the vertices of a parallelogram.

Where should Salleh plant the fourth coconut tree? Show your calculation.

2. Puteri, Salleh’s wife, would like to plant two more coconut trees.

(a) One tree is to be planted between the coconut trees at locations B and C such that

the three coconut trees form a straight line. Further, the location of the tree is such

that its distance from C is three times its distance from B.

Where should Puteri plant the coconut tree?

(b) The other tree is to be planted 10 m away from the coconut tree at location A such

that this coconut tree and the two coconut trees at locations A and B form a straight

line.

Where should Puteri plant the coconut tree?

Derive The Formula

Use The Formula

B(5 , 30)

A(– 10 , 10)

C(35 , 40)

y (m)

x (m) o


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Worksheet 5 : Share and share alike!

A( x1 , y1 ) . B( x2 , y2 ) , C( x3 , y3 ) and D( x4 , y4 ) are the vertices of quadrilateral ABCD. Derive the formula for the area of triangle ABC. Hence, deduce the formula for the area of quadrilateral ABCD and state the condition required.

Diagram 3 shows a piece of land drawn on a Cartesian plane. This piece of land is owned by four

brothers.

The four brothers agreed to share this piece of land equally amongst themselves. They decide to

divide it into four triangular plots of equal areas as shown in Diagram 3.

1. Calculate the area, in km2, of each plot of triangular land.

2. Determine the coordinates of point X.

Derive The Formula

Use The Formula

O

y (km)

x (km)

X

( 8 , 6 )

( 10 , 0 )

( 0 , 2 )

( 2 , 10 )

Diagram 3


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Worksheet 6 : Close encounter!

1. A straight line passes through point A( x1 , y1 ) and point B( x2 , y2 ).

State the formula for the gradient of the straight line.

2. The gradients of two straight lines are m1 and m2 respectively.

State the condition for the two straight lines to be

(a) parallel,

(b) perpendicular.

Diagram 4 shows the map of a town drawn on a Cartesian plane.

Diagram 4

Rita begins from location ( 12 , 18 ) and drives along a straight road which allows her to be always

equidistant from the police station and the fire station at any time.

By considering gradients only, determine the location on the road when Rita is nearest the petrol

station.

If Rita drives with an average speed of 80 km h – 1, calculate the time, in minutes, she takes to reach

this nearest location.

State The Formula

Use The Formula

( 12 , 18 )

( 4 , – 1 )

( – 3 , 2 )

( – 4 , 5 )

y (km)

x (km) O

Police Station

Petrol Station

Fire Station


PAGE | 8


Worksheet 7 : Nursery fun!

State the formula for finding the equation of a straight line given the following:

1. Gradient and the y-intercept.

2. Gradient and one point on the line.

3. Two points on the line.

4. The x-intercept and the y-intercept.

Diagram 5 shows a nursery bed in the shape of an irregular pentagon drawn on a Cartesian plane.

The gradients of boundaries AB and BC are 2

1 and – 1 respectively.

1. Find the equations of all the five boundaries of the nursery bed.

2. Calculate the area of the nursery bed by using two methods, one of which must be

integration.

State The Formula

Use The Formula

O x (m)

y (m)

A( 0 , 5 )

B

C( 7 , 4 )

D (6 , 2 )

E( 3 , 0 )

Diagram 5


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Worksheet 8 : Money-minded!

Diagram 6 is a map showing village P, village Q and a highway drawn on a Cartesian plane.

1. A road junction is to be built along the highway connecting Village P and Village Q to the

highway. The equation of the highway is y = 2x + 1.

Determine the location of the road junction so as to minimize the construction cost.

Use two methods. [Drawing is not allowed but the use of EXCEL is recommended.]

2. A school is to be built as far away from the highway as possible and it must be 5 km from

each of the two villages P and Q.

Determine the location of the school by solving simultaneously

(a) two non-linear equations,

(b) one non-linear equation and one linear equation.

x (km) O

y (km)

( 8 , 7 )

( 4 , – 1 )

Diagram 6

Village P

Village Q


PAGE | 10


Worksheet 9.1 : Hello! Nice meeting you!

Point P ( x , y ) lies on a Cartesian plane.

Define the position vector of point P relative to the origin.

Three students, Johan, Kassim and Latif are jogging in a recreational park. Diagram 7 shows their

initial positions on a Cartesian plane.

The position vectors of Johan, Kassim and Latif relative to the origin O are 𝑂𝐽⃗⃗⃗⃗ , 𝑂𝐾⃗⃗⃗⃗⃗⃗ and 𝑂𝐿⃗⃗⃗⃗ ⃗ respectively. It

is given that 𝑂𝐽⃗⃗⃗⃗ =

t36

t4, 𝑂𝐾⃗⃗⃗⃗⃗⃗ =

t4

t29 and 𝑂𝐿⃗⃗⃗⃗ ⃗ =

12t4

t42 where t is the time in seconds.

(a) Identify the names of student A, student B and student C.

(b) Find the equation of the path taken by each of the three students.

(c) Determine whether Johan, Kassim and Latif will meet one another at a common location by using a

vector method.

If they meet, determine that location by using two methods, one of which must a vector method.

[ Drawing is not allowed.]

(d) If Vx and Vy are the horizontal and vertical velocities of a student respectively, express, for each

student, Vx and Vy in terms of i and j by using differentiation.

Hence, find the resultant velocity V of each student in the form x i + y j to determine who jogs the

fastest.

O x (m)

y (m)

Diagram 7

Student C

Student B

Student A

Define Concept

Guna Konsep


PAGE | 11


Worksheet 9.2 : The grand line-up!

Three men, Sam, Paul and Martin, dressed in Spiderman outfit, are at a shopping complex to promote the

movie ‘ The Return of Spiderman III ’. Diagram 8 is a Cartesian plane showing their positions at a particular

time.

The position vectors of Sam, Paul and Martin relative to the origin O are 𝑂𝑆⃗⃗⃗⃗ ⃗, 𝑂𝑃⃗⃗⃗⃗ ⃗ and 𝑂𝑀⃗⃗ ⃗⃗ ⃗⃗ respectively. It

is given that 𝑂𝑆⃗⃗⃗⃗ ⃗ =

9t6

2t2, 𝑂𝑃⃗⃗⃗⃗ ⃗ =

2t

5t and 𝑂𝑀⃗⃗ ⃗⃗ ⃗⃗ =

5t32

t27 where t is the time in minutes.

(a) Find the coordinates of Paul and Martin when they are at the locations shown in Diagram 8.

(b) On the same axes, draw the loci of the three men.

(c) At a certain time, the positions of Sam, Paul and Martin lie on a straight line.

(i) Determine the time this situation happens.

Use two methods.

(ii) Find the equation of the straight line.

(iii) Find the ratio of the distance between Sam and Paul to the distance between Sam and

Martin at that time.

x

y

O

Sam

Paul

Martin

3

Diagram 8


PAGE | 12


Worksheet 10 : Loving it, HOTS!

Solve the following problems. You can do it!

1. Diagram 9 shows a scale drawing of two parallelograms, A and B.

(a) Copy the two parallelograms.

(b) On each parallelogram, draw suitable coordinate axes that will enable you to state the

coordinates of each vertex of the parallelogram easily.

(c) Hence, by considering integer coordinates only, calculate a possible area for each

parallelogram.

2. (a) Diagram 10 shows two straight lines, PQ and RS, intersecting at the origin O, and the line x =

2 drawn on a Cartesian plane.

The gradients of PQ and RS are m1 and m2 respectively.

Show that m1 m2 = – 1.

(b) Show that it is possible for the two lines 2x – ( m + 1 )y = 5 and 3my = x – 1 to be parallel

but impossible to be perpendicular.

Diagram 9

A B

5 units 13 units

5 units

13 units

O x

y

S

R

Q

P

Diagram 10

x = 2


PAGE | 13


3. Find the perpendicular distance of the point ( 8 , 3 ) from the line y = 2x – 3.

4. A ( – 1 , 7 ) and B ( 7 , 13 ) are two points on the Cartesian plane. Point P moves such that the product

of the gradients of PA and PB is always – 1.

(a) Find the equation of locus P.

(b) The x-coordinate of point Q is – 2 and it lies on the locus of P.

Calculate the area of triangle AQB in two ways.

5. Point P moves on a Cartesian plane such that it is always at a constant distance from point

C ( 0 , 12 ). Diagram 11 shows part of the locus of point P which passes through point A ( – 8 , 8 ) and

point B ( 4 , 4 ).

(a) Find the equation of locus P.

(b) Calculate the area of the shaded region.

6. During the cold season, the thickness of a sheet of ice in a lake is 1.5 m. When the weather gets

warmer, the sheet of ice melts at a constant rate. After 2 weeks, the thickness of the sheet of ice is

1.2 m.

(a) If y is the thickness, in m, of the sheet of ice after x weeks, express y in terms of x.

(b) Sketch the graph of y against x.

(c) How many weeks are needed for the sheet of ice to melt completely?

C( 0 , 12 )

A( – 8 , 8 )

O

y

x

Diagram 11

Locus of P

B( 4 , 4 )


PAGE | 14


Worksheet 11 : Reflection

There are many more topics in the Additional Mathematics syllabus that involve the use of coordinates.

Coordinates are just amazingly useful in many branches of mathematics! In this project work, you have the

experience of using coordinates to solve just a few problems. But as John Dewey, the renowned educationist,

said:

So, reflect creatively on this awesome aspect of coordinate geometry. Poems, songs, doodle art ………….. are

just some suggestions of creativities for you to consider.

We do not learn from experience……. We learn from reflecting on experience.

additional mathematics project work - utamajpnperak.moe.gov.my/jpn/attachments/article/5011/3.4...

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