33065561 Additional Mathematics Repaired)

PROJECT WORK FORADDITIONAL MATHEMATHICS

-2010-

PROJECT WORK 1

Name Narendar a/l Samuthiram

Class 5 Elit

I/C 931013-10-5371

Teacher Puan Munirah Bt Ibrahim

School Sekolah Menengah Dato’ Mohd Said Nilai

Content

CONTENTS PAGES

OBJECTIVE 2

INTRODUCTION 3-4

PART 1 (QUESTION) 5-8

PART 2 (FURTHER EXPLORATION) 9-16

REFLECTION 17

RUBRIC 18-19

Objectivei. To apply and adapt a variety of problem-solving strategies to solve

problems;

ii. To improve thinking skills;

iii. To promote effective mathematical communication;

iv. To develop mathematical knowledge through problem solving in a way

that increases students’ interest and confidence;

v. To use the language of mathematics to express mathematical ideas

precisely;

vi. To provide learning environment that stimulates and enhances effective

learning;

vii. To develop positive attitude towards mathematics.

Introduction

The diagram below shows the gate of an art gallery. A concrete structure is built at the upper part of the gate and the words ‘ART GALLERY’ is written on it. The top of the concrete structure is flat whereas at the bottom is parabolic in shape. The concrete structure is supported by two vertical pillars at both ends.

The distance between the two pillars is 4 metres and the height of the pillar is 5 metres. The height of the concrete structure is 1 metre. The shortest distance from point A of the concrete structure to point B, that is the highest point on the parabolic shape, is 0.5 metres.

Question:

(a) The parabolic shape of the concrete structure can be represented by various functions depending on the point of reference. Based on different points of reference, obtain at least three different functions which can be used to represent the curve of this concrete structure.

(b) The front surface of this concrete structure will be painted before the words ‘ART GALLERY’ is written on it. Find the area to be painted.

Solution:(a)Function 1

Maximum point (0,4.5) and pass through point (2,4)y=a¿

b=0 , c=4.5

y=a¿

y=a x2+4.5−−−(1)

Substitute (2,4 ) into(1)

4=a¿

4 a=−0.5

a=−0.125

∴ y=−0.125 x2+4.5

Function 2

Maximum point (0, 0.5) and pass through point (2, 0)y=a¿

b=0 , c=0.5

y=a¿

y=a x2+0.5−−−(2)

Substitute (2 ,0 )into (2)

0=a¿

4 a=−0.5

a=−0.125

∴ y=−0.125 x2+0.5

Function 3

Maximum point (2, 4.5) and pass through point (0, 4)y=a¿

b=2 , c=4.5

y=a¿

Substitute (0 ,4 )into (3)

4=a¿

4 a=−0.5

a=−0.125

∴ y=−0.125¿

(b)

Area to be painted= Area of rectangle - Area under the curve¿4×1−2∫

0

2

(−0.125 x2+0.5 )dx

¿4−2 [−0.125 x3

3+0.5 x ]

0

2

¿4−2( 23−0)

¿223m2

Further Exploration(a) You are given four different shapes of concrete structures as shown in

the diagrams below. All the structures have the same thickness of 40cm and are symmetrical

(i) Given that the cost to construct 1 cubic metre of concrete is RM840.00, determine which structure will cost the minimum to construct.

(ii) As the president of Arts Club, you are given the opportunity to decide on the shape of the gate to be constructed. Which shape would you choose? Explain and elaborate on your reasons for choosing the shape.

Solution:

(i) Structure 1

Area=223m2

Volume=Area×Thickness

¿223m2×0.4m

¿ 1615m3

Cost=1615m3× RM 840

¿ RM 896

Structure 2

Area=Area of Rectangle−Area of Triangle

¿1m×4m−12×4m×0.5m

¿4m2−1m2

¿3m2

Volume=Area×Thickness

¿3m2×0.4m

¿1.2m3

Cost=1.2m3×RM 840

¿ RM 1008

Structure 3

Area=Area of Rectangle−Area of Trapezium

¿1m×4m−(4m+1m)

2×0.5m

¿4m2−54m2

¿2.75m2

Volume=Area×Thickness

¿2.75m2×0.4m

¿1.1m3

Cost=1.1m3×RM 840

¿ RM 924

Structure 4

Area=Area of Rectangle−Area of Trapezium

¿1m×4m−(2m+4m)

2×0.5m

¿4m2−1.5m2

¿2.5m2

Volume=Area×Thickness

¿2.5m2×0.4m

¿1m3

Cost=1m3×RM 840

¿ RM 840

∴Structure4will cost theminimum¿construct , that is RM 840.

(ii) As the president of the Arts Club, I will decide Structure 4 as the shape of the gate to be constructed. It is because Structure 4 will cost the minimum and it is easier to be constructed compared to Structure 1 which is a curve.

(b) The following questions refer to the concrete structure in the diagram below.

If the value of k increases with a common difference of 0.25 m;

(i) Complete Table 1 by finding the values of k and the corresponding areas of the concrete structure to be painted.

(ii) Observe the values of the area to be painted from Table 1. Do you see any pattern? Discuss.

k (m) Area to be painted(m2)0.00 4×1−0+4

2×0.5=3

0.25 4×1−0.25+42

×0.5=2.9375

0.50 4×1−0.5+42

×0.5=2.875

0.75 4×1−0.75+42

×0.5=2.8125

1.00 4×1−1+42×0.5=2.75

1.25 4×1−1.25+42

×0.5=2.6875

1.50 4×1−1.5+42

×0.5=2.625

1.75 4×1−1.75+42

×0.5=2.5625

2.00 4×1−2+42×0.5=2.5

(ii) There is a pattern in the area to be painted.

The area to be painted decreases as the k increases 0.25m and form a series of numbers:

3, 2.9375, 2.875, 2.8125, 2.75, 2.6875, 2.625, 2.5625, 2.5

We can see that the difference between each term and the next term is the same.

2.9375−3=−0.06252.875−2.9375=−0.06252.8125−2.875=−0.06252.75−2.8125=−0.06252.6875−2.75=−0.06252.625−2.6875=−0.06252.5625−2.625=−0.06252.5−2.5625=−0.0625

∴ We can deduce that this series of numbers is an Arithmetic Progression (AP), with a common difference, d=−0.0625

In conclusion, when k increases 0.25m, the area to be painted decreases by -0.0625m2

(c) Express the area of the concrete structure to be painted in terms of k. Find the area when k approaches the value of 4 and predict the shape of the concrete structure.

Theareaof the concrete structure¿be painted

¿4×1−(k+4 )

2×0.5

¿4− k4+1

¿3− k4

∴ k→4

k4→1

Areaof concrete structure ¿be painted→3−1

→2m2

The shape of the concrete structure will be a rectangle with length 4m and breadth 0.5m, which may look like this:

Reflection

While I conducting the project, I had learned some moralvalues that I practice. This project had taught me toresponsible on the works that are given to me to becompleted. This project also had make me felt moreconfident to do works and not to give up easily when wecould not find the solution for the question. I also learned tobe more discipline on time, which I was given about amonth to complete this project and pass up to my teacherjust in time. I also enjoy doing this project during my schoolholiday as I spend my time with friends to complete thisproject and it had tightened our friendship.

RUBRIC FOR ADDITIONAL MATHEMATICS PROJECT WORK 1/2010Name: Narendar a/l Samuthiram Form: 5 Elit

A. Report Presentation

1. General

Aspect (6%)

Appropriate title. 1

Contents page. 1

Systematic presentation. 1 – 2

Creativity. 1 – 2

2. Introduction

(7%)

Comprehensive introduction – may include history/moral/aesthetical values and others.

5 – 7

Satisfactory introduction. 3 – 4

Incomplete introduction. 1 – 2

B. Task Specification

1. Specify the

Task (5%)

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

4 – 5

Identify and state given information and required results incompletely.

1 – 3

C. Problem-Solving

1. Procedure

(42 %)

Evidence of making and proving conjectures. 3 – 5

In-depth understanding of the problem.

Use very efficient problem –solving strategies.

Answer all questions correctly.

20 – 25

Understand the problem.

Use strategies that lead to solutions of the problems.

Answer all questions correctly.

10 – 19

Partially understand the problem.

Use strategies that are partially useful.

Do not answer all questions correctly.

1 – 9

Concise and efficient communication using symbol/tables/diagrams when necessary.

8 – 12

Not concise and inefficient communication. 1 – 7

2. Findings

(20%)

Answer all questions correctly. 8 – 12

Partially answer questions correctly. 1 – 7

Detail discussion of findings. 5 – 8

Satisfactory discussion of findings. 1 – 4

3. Exploration

(10%)

Explore the task concisely and efficiently. 6 – 10

Explore the task. 1 – 5

D. Conclusion

1. Conclusion

(5 %)Draw relevant conclusions/generalisation. 2 – 5

E. Reflection

1. Reflection

(5 %)Comprehensive reflection. 1 – 5

Total score (A1 + A2 + B1 + C1 + C2 + C3 + D1 + E1)100