Rubric for Additional Mathematics Project Work 2012

Name : ………………………………………….. Form 5 : ………..

I. C. No. : ………………………………… Index Number : .……………………

A. Report Presentation Range of

marksMark

awardedSubtotal

A1. General Aspect(6%)

Appropriate title 1Content page 1Systematic presentation 1 – 2Creativity 1 – 2

A2. Introduction (10%)

Comprehensive introduction – may include history, values and other information relevant to the task including pictures and captions.

5 – 7

Satisfactory introduction. 1 – 4Relating given task to real-life situations. 1 – 3

B. Task SpecificationB1. Specifying the Task (10%)

Identify and state all given information and required results in effective mathematical statements.

8 – 10

Identify and state given information and required results. 4 – 7Identify and state given information and required results incompletely.

1 – 3

C. Problem solvingC1. Procedure (40%)

In-depth understanding of the questions. Use very efficient problem-solving strategies.Answer all questions correctly.

38 – 40

Understand the questions.Use strategies that lead to solutions of the problems.Answer all questions correctly.

20 – 37

Partially understand the questions. Use strategies that are partially useful. Do not answer all questions correctly.

10 – 19

C2. Communication & Reasoning (20%)

Communicate precisely and efficiently using mathematical representations such as symbols / graphs/ tables / diagrams.Refined and complex reasoning

18 - 20

Communicate effectively using mathematical representations such as symbols / graphs / tables /diagrams.Use effective mathematical reasoning

10 – 17

Communicate ineffectively. Limited use of mathematical representations such as symbols / graphs / tables / diagrams. Some evidence of mathematical reasoning

5 - 9

C3. making and proving conjecture (10%)

Evidence of making and proving conjectures. 10

D. Conclusion / GeneralisationD1. Conclusion / Generalisation

Draw conclusion / generalization 2 - 4

Total score (A1 + A2 + B1 + C1 + C2 + C3 + D1)

CONTENTS

Titles pagesAppreciation 1

A Brief History Function 2-3

Part 1 4

Part 2

Part 3

Further Exploration

Reflection

OBJECTIVE

ADDITIONAL MATHEMATICS PROJECT WORK 2012

Part 1

A family is an important institution in a society. Everyone’s hope is to have a happy family. A closely knitted family will ensure the well-being of the community and hence the nation. Below is an example of Zulkefli and Aminah’s family tree. Answer the questions below based on the family tree given.

Diagram 1

a) Based on Diagram 1, list at least 5 different sets (Example : set of sons).b) Based on Diagram 1, draw at least three arrow diagrams to show the relation

of the family. Hence, i) determine the domain, codomain, objects, images and range.ii) recognise the type of relation.iii) represent each relation above using other methods.

c) Based on (b), which relation is a function? State your reason.

Part 2a) You are required to come out with an attractive and creative card. Your

card must have the following information.i) a family photoii) a family tree

b) Write a short description about your family in not more than 150 words.

Part 3

Based on Diagram 3, answer the following questions.

Diagram 3

Find

Zulkefli & Aminah

Daughter A & Son-in-law

Grand daughter E& grandson-in-law

Grand son J & grand daughter-

in-law

Great grand daughter P

Grand son K

Son C & daughter-in-law

Grand daughter F

Grand son G & grand daughter-in-law

Grand son H

Grand daughter L

Great grand daughter M

Great grandson N

Daughter B & son-in-law

Son D & daughter-in-

law

−1

0

2

2

3

5

4

9

25

f g

a) function and function .b) the inverse function and .

c) composite function , , and .

d) function and . Is = ?e) What can you deduce from the following composite function?

and

Hence, deduce the value of .

Further Exploration

a) The function g is defined by . Find

i) , , , , and .

ii) Hence, state the function and , where n is a positive integer.

b) Given the function is defined by . Find . Therefore

deduce the inverse functions for the followings. Check your answers using another method.

i)

ii)

iii)

Reflection

While you were carrying out this project, what did you learn? What moral values did you put into practice? Represent your opinion or feelings creatively by using symbols, illustrations, paintings or songs.

END OF PROJECT PAPER

First and foremost, I would like to thank to my Additional Mathematics teacher, Mrs.Cheeh Oi Muei as she gives us important guidance and commitment during this project work. She is supportive figure throughout the whole project.

I also would like to give thanks to all my friends for helping me and always supporting me to help complete this project work. They have sharing information with other people including me. Without them, this project would never be finished.

For their strong support, I would like to express my gratitude to my beloved parents and also helping me to complete the project and they are always be my side and I hope they will still be there in the future.

Last but not least, I would also like to thank to all my teachers and my friends for helping me collect the much needed data for this. Not forgetting to all the people who were involve directly or indirectly towards making this project a reality.

Appreciation

A function ƒ takes an input x, and returns an output ƒ(x). One metaphor describes the function as a "machine" or "black box" that for each input returns a corresponding output.

In mathematics, a function is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output. An example of such a relation is defined by the rule f(x) = x2, which relates an input x to its square, which are both real numbers. The output of the function f corresponding to an input x is denoted by f(x) (read "f of x"). If the input is –3, then the output is 9, and we may write f(–3) = 9.

The input to a function is often called the argument and the output is often called the value. Inputs and outputs need not be numbers – they can be elements of any set, for instance geometric figures. For example, a function could associate a triangle with the number 3, a square with the number 4, and so on.

There are many ways to describe or represent a function. Some functions may be described by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function. In science, many functions are given by a table that gives the outputs for selected inputs. A function can be described through its relationship with other functions, for example as an inverse function or as a solution of a differential equation. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, and division of functions. Another important operation defined on functions is function composition, where the output from one function becomes the input to another function.

The input and output are often expressed as an ordered pair. In the example above, we have the ordered pair <–3, 9>. This ordered pair can be viewed as the Cartesian coordinates of a point on the graph of the function. But no picture can exactly define every point in an infinite set. In modern mathematics, a function is defined by its set of inputs, called the domain, a set containing the outputs, called its

A Brief History Of Function

codomain, and the set of all paired input and outputs, called the graph. For example, we could define a function using the rule f(x) = x2 by saying that the domain and codomain are the real numbers, and that the ordered pairs are all pairs of real numbers <x, x2>. Collections of functions with the same domain and the same codomain are called function spaces, the properties of which are studied in such mathematical disciplines as real analysis and complex analysis

a) Set of sons = { son C, son D }Set of daughter = { daughter A, daughter B}Set of grand daughter = { grand daughter E, grand daughter F, grand daughter L }Set of grand son = { grand son G, grand son H, grand son J, grand son K}Set of great grand = { great grand daughter M, great grandson N, great grand daughter P }

b) Daughter of

Daughter A

Daughter B

Zulkefli

Part 1

i) Domain = { Daughter A, Daughter B }Codomain = { Zulkefli }Objects = Daughter A, Daughter BImages = ZulkefliRange = { Zulkefli }

ii) Many-to-One relation

iii) { (Daughter A, Zulkefli) , (Daughter B, Zulkefli) }

Father of

i) Domain = { Zulkefli }Codomaim = { Daunghter A, Daughter B, Son C, Son D }Objects = ZulkefliImages = Daughter A, Daughter B, Son C, Son DRange = { Daughter A, Daughter B, Son C, Son D }

Husband of

Zulkefli

Daughter A

Daughter B

Son C

Son D

Son C

Son D

Daughter in law

Daughter in law

i)

Domain = { Son C, Son D }Codomain = { Daughter in law }Objects = Son C, Son DImages = Daughter in lawRange = { Daughter in law }

ii)

One-to-one relationiii)

{ ( Son C, Daughter in law) , ( Son D, Daughter in law) }

c)

Is this a function or not?

We need to ask ourselves, does every first element (or input) correspond with EXACTLY ONE second element (or output)?  In this case, the answer is yes. Zulkefli is related with his daughter A, daughter B, Son C and Son D.

Note that a relation can still be a function if an output value associates with more than one input value as shown in this example.  But again, it would be a no the other way around, where an input value corresponds to two or more output values.

So, this relation would be an example of a function.  

ii)

Hj.Nasir &

Abdul Halim and Jamilah Hamzan and

ZawiyahHalizah and Ismail

Hasimah and Abdullah

Nor JalilahNor SyazanaMuhammad

Nor Mirza NadiaNazyraWira Asyraf

Nur AtikahHisyamuddinMuhammad Arif

Family Tree

Part 2

b)

Family is an important unit of society. It holds great importance in social life. It is the strongest unit of society. A society is made up of families.

A family is the first school in which a child receives the basic values of life. He learns good manners in the family. The morals and values learnt in family become our guiding force. They make our character. They lay the foundation of our thinking. I feel fortunate to be born in a family where values are inculcated in early childhood.

I belong to a middle class family. There are six members in my family. They are my parents, I and my brother and two sisters. My father is the head of the family. He enjoys a commanding position. His decision is final in family matters. Nobody can dare to go against him. Everybody respects him. He is the guardian of the family. He is a cool and considerate man. He is just and fair. His decision is never influenced by any other. He is a technician. He always helping us in our study.

My mother is a worker in Courts Sdn. Bhd. She is soft-natured and caring. She takes great care of us. She does not care for her comfort for us. She looks after our grandparents. She helps the poor and the needy. She is religious and God-fearing.

Our family has been known for discipline and values. We give great importance to values and morals in life. Since our early childhood we are taught to respect the elders and love the children. We learnt the lesson of punctuality and honesty from our grandfather. It is due to the good education of our grandparents that we could excel both in sports and education. Since our childhood we have been put into the habit of rising early in the morning. This has a natural effect on our health and physical fitness.

A Description About My Family

Our family is like a heaven. There is peace, prosperity, love and care. The younger’s have regard and respect for the elders while the elders shower them with their love and affection. The instructions of the elders are followed with great respect. If a member has some problem the whole family is beside him. I am proud of my family.

TAMPAL KAD

a) Function f f(x) = x + 3

Function g g(x) = x2

Part 3

b) Inverse function f(x) = x + 3let y = x + 3 x = y – 3f -1 (x) = x – 3

Inverse function g(x) = x2

let y = x2

x =

g -1 (x) =

c) Composite functionfg ( x)= f(g( x)) = x + 3 = (x2 ) + 3 = x2 + 3

gf ( x ) = g(f ( x )) = x2

= (x + 3 ) 2

= x2 + 6x + 9

f 2 ( x ) = ff ( x ) = x + 3 = (x + 3 ) + 3 = x + 6

g 2 ( x ) = gg ( x ) = x2

= (x2) 2

= x 4

d) =

=

( fg ) -1 ( x ) → let y = x2 + 3 x2 = y – 3

x =

( fg ) -1 ( x ) =

Conclusion : = ( fg ) -1 ( x )

e) ff -1 ( x ) = f(f -1 ( x )) = x + 3

= ( x – 3 ) + 3 = x

gg -1 ( x + 1) = g(g -1 ( x + 1))

= ( ) 2

= x + 1

So, ff -1 ( x ) ≠ gg -1 ( x + 1)

ff -1 ( k 2 + 2 ) = k 2 + 2

a)

i)

Further Exploration

g 2 : x = gg: x =

= ÷

=

= -

g 3 : x = g2 g : x =

= - ( )

g 4 : x = g 2 g 2 : x =

= x

g 5 : x = g 4 g : x =

g 6 : x = g 5 g : x =

So g 30 : x =

ii) g 2n : x =

g 2n + 1 : x =

b)

Let y =

y ( cx + d ) = ax + b cxy + dy = ax + b cxy – ax = b – dy x ( cy – a ) = b – dy

x =

f -1: x =

ii)

f -1: x =

iii)

f -1: x =

iv)

f -1: x =

Back when I was a child

Before life removed all the innocence

My father would lift me high

And dance with my mother and me and then

Spend me around till I fell asleep

Then up the stairs he would carry me

And I knew for sure I was loved

I f I could get another chance

Another walk, another dance with him

I’d play a song that would never ever end

How I’d love to dance with my father again

When I and my mother would disagree

To get my way I would run from her to him

He’d make me laugh just to comfort me

Then finally make me do just what my momma said

Later that night, when I was asleep

He left a dollar under my sheet

Never dreamed that he would be gone from me

If I could steal

One final glance

One final step

Reflection

One final dance with him

I’d play a song that would never ever end

Cause I’d love to dace my father again

Sometimes, I’d listen outside her door

I’d hear how my mother cried for him

I’d pray for even more than me

I know I’m praying for much to much

But could you send back the only man she loved

I know you don’t do it usually

But dear Lord she’s dying to dance with my father again

Every night I fall asleep

And this is all I ever dream