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Basic Essential Additional Mathematics Skills

Curriculum Development Division

Ministry of Education Malaysia

Putrajaya

2010


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First published 2010

Curriculum Development Division,


Aras 4-8, Blok E9

Pusat Pentadbiran Kerajaan Persekutuan

62604 Putrajaya

Tel.: 03-88842000 Fax.: 03-88889917

Website: http://www.moe.gov.my/bpk

Copyright reserved. Except for use in a review, the reproduction or utilization of this

work in any form or by any electronic, mechanical, or other means, now known or

hereafter invented, including photocopying, and recording is forbidden without prior

written permission from the Director of the Curriculum Development Division, Ministry

of Education Malaysia.


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TABLE OF CONTENTS

Preface i

Acknowledgement ii

Introduction iii

Objective iii

Module Layout iii

BEAMS Module:

Unit 1: Negative Numbers

Unit 2: Fractions

Unit 3: Algebraic Expressions and Algebraic Formulae

Unit 4: Linear Equations

Unit 5: Indices

Unit 6: Coordinates and Graphs of Functions

Unit 7: Linear Inequalities

Unit 8: Trigonometry

Panel of Contributors


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ACKNOWLEDGEMENT

The Curriculum Development Division,

Ministry of Education wishes to express our

deepest gratitude and appreciation to all

panel of contributors for their expert

views and opinions, dedication,

and continuous support in

the development of

this module.

ii


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Additional Mathematics is an elective subject taught at the upper secondary level. This

subject demands a higher level of mathematical thinking and skills compared to that required

by the more general Mathematics KBSM. A sound foundation in mathematics is deemed

crucial for pupils not only to be able to grasp important concepts taught in AdditionalMathematics classes, but also in preparing them for tertiary education and life in general.

This Basic Essential Additional Mathematics Skills (BEAMS) Module is one of the

continuous efforts initiated by the Curriculum Development Division, Ministry of Education,

to ensure optimal development of mathematical skills amongst pupils at large. By the

acronym BEAMS itself, it is hoped that this module will serve as a concrete essential

support that will fruitfully diminish mathematics anxiety amongst pupils. Having gone

through the BEAMS Module, it is hoped that fears induced by inadequate basic

mathematical skills will vanish, and pupils will learn mathematics with the due excitement

and enjoyment.

INTRODUCTION

OBJECTIVE

The main objective of this module is to help pupils develop a solid essential mathematics

foundation and hence, be able to apply confidently their mathematical skills, specifically

in school and more significantly in real-life situations.

iii

MODULE LAYOUT

This module encompasses all mathematical skills and knowledge

taught in the lower secondary level and is divided into eight units as

follows:


Unit 2: Fractions

Unit 3: Algebraic Expressions and Algebraic Formulae

Unit 4: Linear Equations

Unit 5: Indices

Unit 6: Coordinates and Graphs of Functions

Unit 7: Linear Inequalities

Unit 8: Trigonometry


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Each unit stands alone and can be used as a comprehensive revision of a particular topic.

Most of the units follow as much as possible the following layout:

Module Overview

Objectives

Teaching and Learning StrategiesLesson Notes

Examples

Test Yourself

Answers

The Lesson Notes, Examples and Test Yourself in each unit can be used as

supplementary or reinforcement handouts to help pupils recall and understand the basic

concepts and skills needed in each topic.

Teachers are advised to study the whole unit prior to classroom teaching so as to familiarize

with its content. By completely examining the unit, teachers should be able to select any part

in the unit that best fit the needs of their pupils. It is reminded that each unit in this module is

by no means a complete lesson, rather as a supporting material that should be ingeniously

integrated into the Additional Mathematics teaching and learning processes.

At the outset, this module is aimed at furnishing pupils with the basic mathematics

foundation prior to the learning of Additional Mathematics, however the usage could be

broadened. This module can also be benefited by all pupils, especially those who arepreparing for the Penilaian Menengah Rendah (PMR) Examination.

iv


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Advisors:

Haji Ali bin Ab. Ghani AMN

DirectorCurriculum Development Division

Dr. Lee Boon HuaDeputy Director (Humanities)


Mohd. Zanal bin Dirin

Deputy Director (Science and Technology)Curriculum Development Division

Editorial Advisor:

Aziz bin SaadPrincipal Assistant Director

(Head of Science and Mathematics Sector)Curriculum Development Division

Editors:

Dr. Rusilawati binti OthmanAssistant Director

(Head of Secondary Mathematics Unit)Curriculum Development Division

Aszunarni binti Ayob

Assistant DirectorCurriculum Development Division

Rosita binti Mat ZainAssistant Director



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Abdul Rahim bin BujangSM Tun Fatimah, Johor

Ali Akbar bin AsriSM Sains, Labuan

Amrah bin BahariSMK Dato Sheikh Ahmad, Arau, Perlis

Aziyah binti Paimin

SMK Kompleks KLIA, , Negeri Sembilan

Bashirah binti SelemanSMK Sultan Abdul Halim, Jitra, Kedah

Bibi Kismete binti Kabul KhanSMK Jelapang Jaya, Ipoh, Perak

Che Rokiah binti Md. IsaSMK Dato Wan Mohd. Saman, Kedah

Cheong Nyok TaiSMK Perempuan, Kota Kinabalu, Sabah

Ding Hong EngSM Sains Alam Shah, Kuala Lumpur

Esah binti DaudSMK Seri Budiman, Kuala Terengganu

Haspiah binti BasiranSMK Tun Perak, Jasin, Melaka

Hon May WanSMK Tasek Damai, Ipoh, Perak

Horsiah binti AhmadSMK Tun Perak, Jasin, Melaka

Kalaimathi a/p RajagopalSMK Sungai Layar, Sungai Petani, Kedah

Kho Choong Quan

SMK Ulu Kinta, Ipoh, Perak

Lau Choi FongSMK Hulu Klang, Selangor

Loh Peh ChooSMK Bandar Baru Sungai Buloh, Selangor

Mohd. Misbah bin Ramli

SMK Tunku Sulong, Gurun, KedahNoor Aida binti Mohd. ZinSMK Tinggi Kajang, Kajang, Selangor

Noor Ishak bin Mohd. SallehSMK Laksamana, Kota Tinggi, Johor

Noorliah binti AhmatSM Teknik, Kuala Lumpur

Nor Aidah binti JohariSMK Teknik Setapak, Selangor

Writers:


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Layout and Illustration:

Aszunarni binti Ayob Mohd. Lufti bin Mahpudz

Assistant Director Assistant Director Curriculum Development Division Curriculum Development Division

Writers:

Nor Dalina binti IdrisSMK Syed Alwi, Kangar, Perlis

Norizatun binti Abdul SamidSMK Sultan Badlishah, Kulim, Kedah

Pahimi bin Wan SallehMaktab Sultan Ismail, Kelantan

Rauziah binti Mohd. AyobSMK Bandar Baru Salak Tinggi, Selangor

Rohaya binti ShaariSMK Tinggi Bukit Merajam, Pulau Pinang

Roziah binti Hj. ZakariaSMK Taman Inderawasih, Pulau Pinang

Shakiroh binti AwangSM Teknik Tuanku Jaafar, Negeri Sembilan

Sharina binti Mohd. Zulkifli

SMK Agama, Arau, Perlis

Sim Kwang YawSMK Petra, Kuching, Sarawak

Suhaimi bin Mohd. TabieeSMK Datuk Haji Abdul Kadir, Pulau Pinang

Suraiya binti Abdul HalimSMK Pokok Sena, Pulau Pinang

Tan Lee FangSMK Perlis, Perlis

Tempawan binti Abdul AzizSMK Mahsuri, Langkawi, Kedah

Turasima binti MarjukiSMKA Simpang Lima, Selangor

Wan Azlilah binti Wan NawiSMK Putrajaya Presint 9(1), WP Putrajaya

Zainah binti KebiSMK Pandan, Kuantan, Pahang

Zaleha binti Tomijan

SMK Ayer Puteh Dalam, Pendang, Kedah

Zariah binti HassanSMK Dato Onn, Butterworth, Pulau Pinang


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UNIT 1

NEGATIVE NUMBERS

B a s i c E s s e n t i a l

A d d i t i o n a l M a t h e m a t i c s S k i l l s




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TABLE OF CONTENTS

Module Overview 1

Part A: Addition and Subtraction of Integers Using Number Lines 2

1.0 Representing Integers on a Number Line 3

2.0 Addition and Subtraction of Positive Integers 3

3.0 Addition and Subtraction of Negative Integers 8

Part B: Addition and Subtraction of Integers Using the Sign Model 15

Part C: Further Practice on Addition and Subtraction of Integers 19

Part D: Addition and Subtraction of Integers Including the Use of Brackets 25

Part E: Multiplication of Integers 33

Part F: Multiplication of Integers Using the Accept-Reject Model 37

Part G: Division of Integers 40

Part H: Division of Integers Using the Accept-Reject Model 44

Part I: Combined Operations Involving Integers 49

Answers 52


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Basic Essential Additional Mathematics Skills (BEAMS) Module


1



MODULE OVERVIEW

1. Negative Numbers is the very basic topic which must be mastered by everypupil.

2. The concept of negative numbers is widely used in many AdditionalMathematics topics, for example:

(a) Functions (b) Quadratic Equations

(c) Quadratic Functions (d) Coordinate Geometry

(e) Differentiation (f) Trigonometry

Thus, pupils must master negative numbers in order to cope with topics inAdditional Mathematics.

3. The aim of this module is to reinforce pupils understanding on the concept ofnegative numbers.

4. This module is designed to enhance the pupils skills in using the concept of number line; using the arithmetic operations involving negative numbers; solving problems involving addition, subtraction, multiplication and

division of negative numbers; and applying the order of operations to solve problems.

5. It is hoped that this module will enhance pupils understanding on negativenumbers using the Sign Model and the Accept-Reject Model.

6. This module consists of nine parts and each part consists of learning objectiveswhich can be taught separately. Teachers may use any parts of the module as

and when it is required.


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2



TEACHING AND LEARNING STRATEGIES

The concept of negative numbers can be confusing and difficult for pupils to

grasp. Pupils face difficulty when dealing with operations involving positive and

negative integers.

Strategy:

Teacher should ensure that pupils understand the concept of positive and negative

integers using number lines. Pupils are also expected to be able to performcomputations involving addition and subtraction of integers with the use of the

number line.

PART A:

ADDITION AND SUBTRACTION

OF INTEGERS USING

NUMBER LINES

LEARNING OBJECTIVE

Upon completion of Part A, pupils will be able to perform computationsinvolving combined operations of addition and subtraction of integers using a

number lines.


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3



PART A:

ADDITION AND SUBTRACTION OF INTEGERS

USING NUMBER LINES

1.0 Representing Integers on a Number Line

Positive whole numbers, negative numbers and zero are all integers. Integers can be represented on a number line.

Note: i) 3 is the opposite of +3

ii) (2) becomes the opposite of negative 2, that is, positive 2.

2.0 Addition and Subtraction of Positive Integers

3 2 1 0 1 2 3 4

LESSON NOTES

Rules for Adding and Subtracting Positive Integers

When adding a positive integer, you move to the right on anumber line.

When subtracting a positive integer, you move to the lefton a number line.

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

Positive integers

may have a plus sign

in front of them,

like +3, or no sign in

front, like 3.


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4



(i) 2 + 3

Alternative Method:

EXAMPLES

Adding a positive integer:

Start by drawing an arrow from 0 to 2, and then,

draw an arrow of 3 units to the right:

2 + 3 = 5

5 4 3 2 1 0 1 2 3 4 5 6

Start

with 2

Add a

positive 3


Start at 2 and move 3 units to the right:

2 + 3 = 5

Make sure you start from

the position of the first

integer.

5 4 3 2 1 0 1 2 3 4 5 6


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5



(ii) 2 + 5

Alternative Method:


Start by drawing an arrow from 0 to2, and then,draw an arrow of 5 units to the right:

2 + 5 = 3

5 4 3 2 1 0 1 2 3 4 5 6

Add a

positive 5


the position of the firstinteger.

5 4 3 2 1 0 1 2 3 4 5 6


Start at2 and move 5 units to the right:

2 + 5 = 3


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6



(iii) 25 =3

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6

Subtracting a positive integer:

Start by drawing an arrow from 0 to 2, and then,

draw an arrow of 5 units to the left:

25 =3

Subtract a

positive 5


Start at 2 and move 5 units to the left:

25 =3

5 4 3 2 1 0 1 2 3 4 5 6



integer.


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7



(iv) 32 =5

Alternative Method:


Start by drawing an arrow from 0 to3, and

then, draw an arrow of 2 units to the left:

32 =5

5 4 3 2 1 0 1 2 3 4 5 6

Subtract a

positive 2

5 4 3 2 1 0 1 2 3 4 5 6


Start at3 and move 2 units to the left:

32 =5


the position of the firstinteger.


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8



3.0 Addition and Subtraction of Negative Integers

Consider the following operations:

41 = 3

42 = 2

43 = 1

44 = 0

45 =1

46 =2

Note that subtracting an integer gives the same result as adding its opposite. Adding orsubtracting a negative integer goes in the opposite direction to adding or subtracting a positive

integer.

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4

4 + (5) =1

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 44 + (6) =2

4 + (1) = 3

4 + (2) = 2

4 + (3) = 1

4 + (4) = 0


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9



Rules for Adding and Subtracting Negative Integers

When adding a negative integer, you move to the left on anumber line.

When subtracting a negative integer, you move to the righton a number line.

3 2 1 0 1 2 3 4

3 2 1 0 1 2 3 4


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10



(i) 2 + (1) =3

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6

Adding a negative integer:

Start at2 and move 1 unit to the left:

2 + (1) =3

EXAMPLES

5 4 3 2 1 0 1 2 3 4 5 6



then, draw an arrow of 1 unit to the left:

2 + (1) =3

Add a

negative 1



integer.

This operation of

2 + (1) =3

is the same as

21 =3.


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11



(ii) 1 + (3) =2

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6


Start at 1 and move 3 units to the left:

1 + (3) =2

Add a

negative 3

5 4 3 2 1 0 1 2 3 4 5 6


Start by drawing an arrow from 0 to 1, then, draw an arrow of

3 units to the left:

1 + (3) =2



integer.

This operation of

1 + (3) =2

is the same as

13 =2


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12



(iii) 3(3) = 6

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6

Subtracting a negative integer:

Start at 3 and move 3 units to the right:

3(3) = 6

5 4 3 2 1 0 1 2 3 4 5 6


Start by drawing an arrow from 0 to 3, and

then, draw an arrow of 3 units to the right:

3(3) = 6

Subtract a

negative 3

This operation of

3(3) = 6

is the same as

3 + 3 = 6



integer.


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13



(iv) 5(8) = 3

Alternative Method:

5 4 3 2 1 0 1 2 3 4 5 6


Start at5 and move 8 units to the right:

5(8) = 3

5 4 3 2 1 0 1 2 3 4 5 6

Subtract a

negative 8

This operation of

5(8) = 3

is the same as

5 + 8 = 3



then, draw an arrow of 8 units to the right:

5(8) = 3


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14



Solve the following.

1. 2 + 4

2. 3 + (6)

3. 2(4)

4. 35 + (2)

5. 5 + 8 + (5)

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

5 4 3 2 1 0 1 2 3 4 5 6

TEST YOURSELF A


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15




This part emphasises the first alternative method which include activities and

mathematical games that can help pupils understand further and master the

operations of positive and negative integers.

Strategy:

Teacher should ensure that pupils are able to perform computations involving

addition and subtraction of integers using the Sign Model.

PART B:


OF INTEGERS USING

THE SIGN MODEL

LEARNING OBJECTIVE

Upon completion of Part B, pupils will be able to perform computations

involving combined operations of addition and subtraction of integers usingthe Sign Model.


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16



PART B:


USING THE SIGN MODEL

In order to help pupils have a better understanding of positive and negative integers, we have

designed the Sign Model.

Example 1

What is the value of 35?

NUMBER SIGN

3 + + +

5

WORKINGS

i. Pair up the opposite signs.

ii. The number of the unpaired signs is

the answer.

Answer 2

+ + +

LESSON NOTES

EXAMPLES

The Sign Model

This model uses the + and signs. A positive number is represented by + sign. A negative number is represented by sign.


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17



Example 2

What is the value of 53 ?

NUMBER SIGN

3 _ _ _

5

WORKINGS

There is no opposite sign to pair up, sojust count the number of signs.

_ _ _ _ _ _ _ _

Answer 8

Example 3

What is the value of 53 ?

NUMBER SIGN

3

+5 + + + + +

WORKINGS

i. Pair up the opposite signs.

ii. The number ofunpaired signs is the

answer.

Answer 2

_

+ + +

_

+

_

+


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18




1. 4 + 8 2. 84 3. 127

4. 55 5. 574 6. 7 + 43

7. 4 + 37 8. 62 + 8 9. 3 + 4 + 6

TEST YOURSELF B


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19



PART C:

FURTHER PRACTICE ON


OF INTEGERS


This part emphasises addition and subtraction of large positive and negative integers.

Strategy:

Teacher should ensure the pupils are able to perform computation involving addition

and subtraction of large integers.

LEARNING OBJECTIVE

Upon completion of Part C, pupils will be able to perform computationsinvolving addition and subtraction of large integers.


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20



PART C:

FURTHER PRACTICE ON ADDITION AND SUBTRACTION OF INTEGERS

In Part A and Part B, the method of counting off the answer on a number line and the Sign

Model were used to perform computations involving addition and subtraction ofsmallintegers.

However, these methods are not suitable if we are dealing with large integers. We can use the

following Table Model in order to perform computations involving addition and subtraction

of large integers.

LESSON NOTES

Steps for Adding and Subtracting

Integers

1. Draw a table that has a column for + and a columnfor.

2. Write down all the numbers accordingly in thecolumn.

3. If the operation involves numbers with the samesigns, simply add the numbers and then put the

respective sign in the answer. (Note that we

normally do not put positive sign in front of a

positive number)

4. If the operation involves numbers with differentsigns, always subtract the smaller number from

the larger number and then put the sign of the

larger number in the answer.


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21



Examples:

i) 34 + 37 =+

34

37

+71

ii) 6520 =+

65 20

+45

iii)

73 + 22 =

+

22 73

51

iv) 228338 =+

228 338

110

Subtract the smaller number from

the larger number and put the sign

of the larger number in the

answer.

We can just write the answer as

45 instead of +45.




answer.




answer.

Add the numbers and then put the

positive sign in the answer.

We can just write the answer as

71 instead of +71.


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22



v) 428316 =+

428316

744

vi) 863 127 + 225 =+

225 863

127

225 990

765

vii) 234 675 567 =+

234 675

567

234 1242

1008


negative sign in the answer.

Add the two numbers in the

column and bring down the number

in the + column.


the larger number in the third row

and put the sign of the larger

number in the answer.



in the + column.






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23



viii) 482 + 236 718 =+

236 482

718

236 1200

964

ix) 765 984 + 432 =

+

432 765

984

432 1749

1317

x) 1782 + 436 + 652 =+

436

652

1782

10881782

694



in the + column.







in the + column.





Add the two numbers in the +

column and bring down the numberin the column.






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24




1. 4789 2. 5448 3. 33125

4. 352556 5. 345437456 6. 237 + 564318

7. 431 + 366778 8. 652517 + 887 9. 233 + 408689

TEST YOURSELF C


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25




This part emphasises the second alternative method which include activities to

enhance pupils understanding and mastery of the addition and subtraction of

integers, including the use of brackets.

Strategy:

Teacher should ensure that pupils understand the concept of addition and subtraction

of integers, including the use of brackets, using the Accept-Reject Model.

PART D:


OF INTEGERS INCLUDING THE

USE OF BRACKETS

LEARNING OBJECTIVE

Upon completion of Part D, pupils will be able to perform computations

involving combined operations of addition and subtraction of integers, includingthe use of brackets, using the Accept-Reject Model.


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26



PART D:


INCLUDING THE USE OF BRACKETS

To Accept or To Reject? Answer

+ ( 5 ) Accept +5 +5

( 2 ) Reject +2 2

+ (4) Accept 4 4

(8) Reject 8 +8

LESSON NOTES

The Accept - Reject Model

+ sign means to accept.

sign means to reject.


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27



i) 5 + (1) =

Number To Accept or To Reject? Answer

5+ (1)

Accept 5Accept 1

+51

+ + + + +

5 + (1) = 4

We can also solve this question by using the Table Model as follows:

5 + (1) = 51

+

5 1

+4

EXAMPLES

This operation of

5 + (1) = 4

is the same as

51 = 4

Subtract the smaller number fromthe larger number and put the sign


answer.

We can just write the answer as 4

instead of +4.


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ii) 6 + (3) =


6+ (3)

Reject 6Accept3

63

6 + (3) = 9


6 + (3) =63 =

+

6

3

9

This operation of

6 + (3) =9

is the same as

63 =9


negative sign in the answer.


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iii) 7(4) =


7(4)

Reject 7Reject4

7+4

+ + + +

7(4) = 3


7(4) =7 + 4 =

+

4 7

3

This operation of

7(4) =3

is the same as

7 + 4 =3




answer.

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