Focus UASA KSSM (F1,2 & 3) 2024 MATHEMATICS

CODE QR ► Full Solutions for UASA Practices ► Form 1.2.3 UPSA & UASA Model Papers with Answers 6 sets PELANGI BESTSELLER DIGITAL BUDDIES FREE INFO VIDEO INFOGRAPHIC • Ng Seng How* • Ooi Soo Huat* • Yong Kuan Yeoh* • Moy Wah Goon • Dr. Chiang K. W. • Samantha Neo *Textbook Author MATHEMATICS Dual Language Programme UASA 1∙2∙3 KSSM FORM

viii CONTENTS FORM 1 Learning Area Number and Operations Rational Numbers 1 1 Chapter 1.1 Integers 2 1.2 Basic Arithmetic Operations involving Integers 4 1.3 Positive and Negative Fractions 8 1.4 Positive and Negative Decimals 12 1.5 Rational Numbers 14 UASA Practice 1 16 Factors and Multiples 18 2 Chapter 2.1 Factors, Prime Factors and Highest Common Factor (HCF) 19 2.2 Multiples, Common Multiples and Lowest Common Multiple (LCM) 23 UASA Practice 2 26 Squares, Square Roots, Cubes 3 and Cube Roots 28 Chapter 3.1 Squares and Square Roots 29 3.2 Cubes and Cube Roots 36 UASA Practice 3 44 Learning Area Relationship and Algebra Ratios, Rates and Proportions 46 4 Chapter 4.1 Ratios 47 4.2 Rates 51 4.3 Proportions 53 4.4 Ratios, Rates and Proportions 54 4.5 Relationship between Ratios, Rates and Proportions with Percentages, Fractions and Decimals 59 UASA Practice 4 63 Algebraic Expressions 65 5 Chapter 5.1 Variables and Algebraic Expressions 66 5.2 Algebraic Expressions involving Basic Arithmetic Operations 71 UASA Practice 5 76 Linear Equations 78 6 Chapter 6.1 Linear Equations in One Variable 79 6.2 Linear Equations in Two Variables 82 6.3 Simultaneous Linear Equations in Two Variables 85 UASA Practice 6 88 Linear Inequalities 90 7 Chapter 7.1 Inequalities 91 7.2 Linear Inequalities in One Variable 94 UASA Practice 7 99 Learning Area Measurement and Geometry Lines and Angles 100 8 Chapter 8.1 Lines and Angles 101 8.2 Angles related to Intersecting Lines 116 8.3 Angles related to Parallel Lines and Transversals 118 UASA Practice 8 124

ix Basic Polygons 127 9 Chapter 9.1 Polygons 128 9.2 Properties of Triangles and the Interior and Exterior Angles of Triangles 129 9.3 Properties of Quadrilaterals and the Interior and Exterior Angles of Quadrilaterals 133 UASA Practice 9 137 Perimeter and Area 140 10 Chapter 10.1 Perimeter 141 10.2 Area of Triangles, Parallelograms, Kites and Trapeziums 146 10.3 Relationship between Perimeter and Area 154 UASA Practice 10 157 Learning Area Discrete Mathematics Introduction to Set 161 11 Chapter 11.1 Set 162 11.2 Venn Diagrams, Universal Sets, Complement of a Set and Subsets 166 UASA Practice 11 173 Learning Area Statistics and Probability Data Handling 175 12 Chapter 12.1 Data Collection, Organization and Representation Process, and Interpretation of Data Representation 176 UASA Practice 12 192 Learning Area Measurement and Geometry The Pythagoras’ Theorem 195 13 Chapter 13.1 The Pythagoras’ Theorem 196 13.2 The Converse of Pythagoras’ Theorem 199 UASA Practice 13 202 FORM 2 Learning Area Number and Operations Patterns and Sequences 204 1 Chapter 1.1 Patterns 205 1.2 Sequences 206 1.3 Patterns and Sequences 209 UASA Practice 1 212 Learning Area Relationship and Algebra Factorisation and Algebraic 2 Fractions 215 Chapter 2.1 Expansion 216 2.2 Factorisation 220 2.3 Algebraic Expressions and Laws of Basic Arithmetic Operations 224 UASA Practice 2 226 Algebraic Formulae 228 3 Chapter 3.1 Algebraic Formulae 229 UASA Practice 3 234

x Learning Area Measurement and Geometry Polygon 236 4 Chapter 4.1 Regular Polygon 237 4.2 Interior Angles and Exterior Angles of Polygons 239 UASA Practice 4 245 Circles 248 5 Chapter 5.1 Properties of Circles 249 5.2 Symmetry and Chords 251 5.3 Circumference and Area of a Circle 253 UASA Practice 5 259 Three-Dimensional Geometrical 6 Shapes 262 Chapter 6.1 Geometric Properties of ThreeDimensional Shapes 263 6.2 Nets of Three-Dimensional Shapes 265 6.3 Surface Area of ThreeDimensional Shapes 267 6.4 Volume of Three-Dimensional Shapes 274 UASA Practice 6 280 Learning Area Relationship and Algebra Coordinates 283 7 Chapter 7.1 Distance in a Cartesian Coordinate System 284 7.2 Midpoint in the Cartesian Coordinate System 287 7.3 The Cartesian Coordinate System 289 UASA Practice 7 291 Graphs of Functions 293 8 Chapter 8.1 Functions 294 8.2 Graphs of Functions 297 UASA Practice 8 304 Speed and Acceleration 306 9 Chapter 9.1 Speed 307 9.2 Acceleration 312 UASA Practice 9 314 Gradient of a Straight Line 316 10 Chapter 10.1 Gradient 317 UASA Practice 10 323 Learning Area Measurement and Geometry Isometric Transformations 326 11 Chapter 11.1 Transformations 327 11.2 Translation 328 11.3 Reflection 335 11.4 Rotation 340 11.5 Translation, Reflection and Rotation as an Isometry 346 11.6 Rotational Symmetry 348 UASA Practice 11 350 Learning Area Statistics and Probability Measures of Central 12 Tendencies 352 Chapter 12.1 Measures of Central Tendencies 353 UASA Practice 12 363 Simple Probability 366 13 Chapter 13.1 Experimental Probability 367 13.2 The Probability Theory Involving Equally Likely Outcomes 369 13.3 Complement of an Event Probability 372 13.4 Simple Probability 374 UASA Practice 13 376

xi FORM 3 Learning Area Number and Operations Indices 378 1 Chapter 1.1 Index Notation 379 1.2 Laws of Indices 380 UASA Practice 1 386 Standard Form 387 2 Chapter 2.1 Significant Figures 388 2.2 Standard Form 391 UASA Practice 2 395 Consumer Mathematics: Savings and Investments, 3 Credit and Debt 397 Chapter 3.1 Savings and Investments 398 3.2 Credit and Debt Management 405 UASA Practice 3 412 Learning Area Measurement and Geometry Scale Drawings 414 4 Chapter 4.1 Scale Drawings 415 UASA Practice 4 423 Trigonometric Ratios 425 5 Chapter 5.1 Sine, Cosine and Tangent of Acute Angles in Right-angled Triangles 426 UASA Practice 5 434 Angles and Tangents of Circles 436 6 Chapter 6.1 Angle at the Circumference and Central Angle Subtended by an Arc 437 6.2 Cyclic Quadrilaterals 440 6.3 Tangents to Circles 443 6.4 Angles and Tangents of Circles 449 UASA Practice 6 451 Plans and Elevations 453 7 Chapter 7.1 Orthogonal Projections 454 7.2 Plans and Elevations 457 UASA Practice 7 468 Loci in Two Dimensions 472 8 Chapter 8.1 Loci 473 8.2 Loci in Two Dimensions 474 UASA Practice 8 480 Learning Area Relationship and Algebra Straight Lines 483 9 Chapter 9.1 Straight Lines 484 UASA Practice 9 494 Answers 497 SCAN ● Kertas Model UPSA Tingkatan 1 ● Kertas Model UASA Tingkatan 1 ● Kertas Model UPSA Tingkatan 2 ● Kertas Model UASA Tingkatan 2 ● Kertas Model UPSA Tingkatan 3 ● Kertas Model UASA Tingkatan 3 ● Jawapan • Form 1 UPSA Model Paper • Form 1 UASA Model Paper • Form 2 UPSA Model Paper • Form 2 UASA Model Paper • Form 3 UPSA Model Paper • Form 3 UASA Model Paper • Answers https://plus.pelangibooks.com/Resources/ FocusKSSM2024/MathematicsF1-F3/KertasModel.html

1 – 4° C Do you know, the temperature in some deserts is very high during the day and very low at night? The air in the desert is very dry and caused the heat to drop very fast at night. The average temperature in most deserts reaches 38°C during the day. Whereas in some deserts, the temperature decreases to –4°C at night. This temperature varies depending on the location of the deserts. In your opinion, does the difference of both temperature is high? Learning Area: Number and Operations Form 1 1 Chapter Rational Numbers Identity Law Distributive Law Associative Law Commutative Law Integer Rational number Fraction Decimal Zero KEYWORDS INFOGRAPHIC Access to

Mathematics Form 1 Chapter 1 Rational Numbers 2 1 Form 1.1 Integers 1.1.1 Recognise positive and negative numbers based on real-life situations 1. We usually use positive and negative numbers in our daily life. (a) A positive number is a number that is greater than 0. A positive number is written with or without the (+) sign. For example, +1, +3.5, + 1 2 or 1, 3.5, 1 2 are positive numbers. (b) A negative number is a number that is less than 0. A negative number is written with (–) sign. For example, –1, –3.5, – 1 2 are negative numbers. MATHS MATHS INFO INFO • 1 is an integer. • 3.5 is a decimal. • 1 2 is a fraction. 2. The situations that can be represented by positive and negative numbers: (a) The movement to the north is represented by a positive number and the movement to the south is represented by a negative number. (b) The upward movement is represented by a positive number and the downward movement is represented by a negative number. (c) The temperature that is higher than 0°C is represented by a positive number and the temperature that is lower than 0°C is represented by a negative number. (d) Profit is represented by a positive number and loss is represented by a negative number. (e) The increase in the price is represented by a positive number and the decrease in the price is represented by a negative number. Example 1 Represent each of the following by using a positive number or a negative number. (a) A loss of RM750. (b) The price of a packet of nasi lemak has increased by 50 cents. (c) Price of share market of Company P has increased by RM2.10 within 1 month. Solution (a) –RM750 (b) +50 cents or 50 cents (c) +RM2.10 Try questions 1 – 2 in Formative Practice 1.1 Example 2 Complete the following sentences. (a) 400 m to the right is written as +400. Thus, 250 m to the left is written as . (b) 56 km to the west is written as –56. Thus, 65 km to the east is written as . (c) 35 m above the sea level is written as +35. Thus, 12 m below the sea level is written as . Solution (a) –250 (b) +65 or 65 (c) –12 Try question 3 in Formative Practice 1.1 1.1.2 Recognise and describe integers 1. In Section 1.1 & 1.2, only positive integers and negative integers are discussed. (a) A positive integer is an integer that is greater than 0. For example, +2, +5, +17 or 2, 5, 17 are positive integers. (b) A negative integer is an integer that is less than 0. For example, –2, –5, –17 are negative integers.

Mathematics Form 1 Chapter 1 Rational Numbers 3 Form 1 Tips • Integers are …, –2, –1, 0, 1, 2, … • The positive integers are 1, 2, 3, 4, … • The negative integers are …, –4, –3, –2, –1 Determine the positive integer and the negative integer. –3, +4.5, 8, +16, –2.4, –4 1 2 Solution Positive integer: 8, +16 Negative integer: –3 Example 3 Try question 4 in Formative Practice 1.1 1.1.3 Represent integers on number lines and make connections between the values and positions of the integers with respect to other integers on the number line 1. Integers can be represented using a horizontal number line or a vertical number line. (a) Horizontal number line –4 –3 –2 Negative integers The value is decreasing Positive integers The value is increasing –1 0 1 2 3 4 Zero Note that the positive integers are located on the right hand side of zero whereas the negative integers are located on the left hand side of zero. MATHS MATHS INFO INFO Zero is neither a positive integer nor a negative integer. Zero is the separator for positive units and negative units on a number line. (b) Vertical number line –3 –2 Negative integers The value is decreasing Positive integers Zero The value is increasing –1 0 1 2 3 Note that the positive integers are located above zero whereas the negative integers are located below zero. Example 4 Represent the following integers on number line. (a) –3, –2, 2, 3 (b) –20, –16, –8, 4, 8 Solution (a) –3 –2 –1 0 1 2 3 (b) –20 –16 –8 –12 –4 0 4 8 Try question 5 in Formative Practice 1.1 1.1.4 Compare and arrange integers in order 1. On a number line, the integers to the right are greater than the integers to the left or the integers to the left are less than the integers to the right. 2. Based on the position of integers on a number line, the integers can either be arranged in ascending order or in descending order.

Mathematics Form 1 Chapter 1 Rational Numbers 4 1 Form Determine the greater integer. (a) –2 or 4 (b) –1 or –7 Solution (a) 4 is greater than –2. A positive number is always greater than a negative number. (b) –7 –6 –5 –4 –3 –2 –1 0 –1 is located on the right hand side of –7. Thus, –1 is greater than –7. Example 5 Try questions 6 – 7 in Formative Practice 1.1 (a) Arrange –3, –4, 0, –1 and 2 in ascending order. (b) Arrange –4, –1, 3, 1 and –3 in descending order. Solution (a) –4 –3 –2 –1 0 1 2 Ascending order: –4, –3, –1, 0, 2 (b) –4 –3 –2 –1 0 1 2 3 Descending order: 3, 1, –1, –3, –4 Example 6 Try questions 8 – 9 in Formative Practice 1.1 Formative Practice 1.1 1. Determine the positive numbers and the negative numbers. –10, 2 5 , 7.3, 5, –67, – 3 4 , –8.9, 0, +3 2. Represent each of the following by using a positive number or a negative number. (a) An elevator goes down 7 levels. (b) 500 m above sea level. (c) A loss of RM450. (d) The price of share market of Maju Company increased by RM0.50. (e) 150 m below sea level. (f) The decrease of water level by 3 cm. 3. Complete the following sentences. (a) The movement of a car 85 km to the north is written as +85. Thus, the movement of a van 100 km to the south is written as . (b) 8°C below the freezing point is written as –8. Thus, 38°C above freezing point is written as . (c) An eagle flies 80 m above sea level. The position of the eagle is written as +80. A turtle is located 45 m below sea level. The position of the turtle is written as . 4. Determine the positive integer and the negative integer. –8, 4, 5.1, –12.8, 7.06, –3, 1 7 5. Represent the following integers on number lines. (a) –2, 1, –3, 2, 4 (b) –3, –9, 6, 3, –15 6. Determine the greater integer. (a) 8 or –1 (b) –15 or –19 7. Determine the largest integer and the smallest integer. (a) –8, 6, –10, 4, 1, –3 (b) 9, –13, 14, –16, 8, 12 8. Arrange each of the following sets of integers in ascending order. (a) –6, –4, 1, –2, 3, –5 (b) 9, –8, –6, 7, –10, 3 (c) –3, 0, 11, –14, –16, 18 (d) –13, –3, 4, –19, 19, –4 9. Arrange each of the following sets of integers in descending order. (a) –1, –6, 5, 0, 3, –4 (b) 5, –2, 7, –1, 3, –6 (c) 4, 1, –3, –7, 2, –1 (d) –12, –5, –8, –3, –1, –10 Basic Arithmetic Operations involving Integers 1.2 1.2.1 Addition and subtraction of integers 1. Addition of two or more integers is the process of finding the sum of the integers.

Mathematics Form 1 Chapter 1 Rational Numbers 5 Form 1 2. Subtraction between two integers is the process of finding the difference between the two integers. 3. Addition or subtraction of two integers can be represented by using a number line. Example 7 Calculate. (a) –1 + 4 (b) 2 + (–4) (c) 3 – 5 (d) –6 – (–3) Solution (a) –1 + 4 –1 0 1 4 units to the right 2 3 = 3 (b) 2 + (–4) –2 –1 0 4 units to the left 1 2 = 2 – 4 = –2 (c) 3 – 5 –2 –1 0 5 units to the left 1 2 3 = –2 (d) –6 – (–3) –6 –5 –4 3 units to the right –3 = –6 + 3 = –3 Try question 1 in Formative Practice 1.2 1.2.2 Multiplication and division of integers 1. The sign of the product for multiplication of integers is shown below. • (+) × (+) = (+) • (+) × (–) = (–) • (–) × (+) = (–) • (–) × (–) = (+) 2. The sign of the quotient for division of integers is shown below. • (+) ÷ (+) = (+) • (+) ÷ (–) = (–) • (–) ÷ (+) = (–) • (–) ÷ (–) = (+) Example 8 Calculate. (a) –4 × 2 (b) 3 × (–4) (c) –7 × (–6) Solution (a) –4 × 2 = –(4 × 2) (–) × (+) = (–) = –8 (b) 3 × (–4) = –(3 × 4) (+) × (–) = (–) = –12 (c) –7 × (–6) = +(7 × 6) (–) × (–) = (+) = 42 Try question 2 in Formative Practice 1.2 Example 9 Calculate. (a) 35 ÷ (–5) (b) –12 ÷ 2 (c) –24 –8 Solution (a) 35 ÷ (–5) = –(35 ÷ 5) (+) ÷ (–) = (–) = –7 (b) –12 ÷ 2 = –(12 ÷ 2) (–) ÷ (+) = (–) = –6 (c) –24 –8 = 24 8 (–) ÷ (–) = (+) = 3 Try question 3 in Formative Practice 1.2

Mathematics Form 1 Chapter 1 Rational Numbers 6 1 Form 1.2.3 Perform computations involving combined basic arithmetic operations of integers 1. The order of operations involving addition, subtraction, multiplication, division and brackets is as shown below. Brackets ↓ × or ÷ from left to right ↓ + or – from left to right Example 10 Solve each of the following. (a) –8 + (–3) – (–5) (b) 2 × (–8) ÷ 4 Solution (a) –8 + (–3) – (–5) = –8 – 3 + 5 + / – from left to right = –11 + 5 = –6 (b) 2 × (–8) ÷ 4 = –(2 × 8) ÷ 4 × / ÷ from left to right = –16 ÷ 4 = –(16 ÷ 4) = –4 Try question 4 in Formative Practice 1.2 Example 11 Calculate each of the following. (a) 3 + (–5) × 2 – 7 (b) –6 + (7 – 10) ÷ (–3) (c) –7 + (–5) 1 – (–2) Solution (a) 3 + (–5) × 2 – 7 = 3 + –(5 × 2) – 7 = 3 + (–10) – 7 = –7 – 7 = –14 × / ÷ from left to right + / – from left to right (b) –6 + (7 – 10) ÷ (–3) = –6 + (–3) ÷ (–3) = –6 + [(–3) ÷ (–3)] = –6 + 1 = –5 (c) –7 + (–5) 1 – (–2) = –7 – 5 1 + 2 = –12 3 = –4 Simplified Try question 5 in Formative Practice 1.2 1.2.4 Describe the laws of arithmetic operations 1. The law of arithmetic operations: (a) Commutative Law a + b = b + a a × b = b × a (b) Associative Law (a + b) + c = a + (b + c) (a × b) × c = a × (b × c) (c) Distributive Law a × (b + c) = a × b + a × c a × (b – c) = a × b – a × c (d) Identity Law a + 0 = a a + (–a) = 0 a × 0 = 0 a × 1 = a a × 1 a = 1 Brackets ( ) × / ÷ from left to right + / – from left to right Perform the basic arithmetic operations for numerator and denominator separately.

Mathematics Form 1 Chapter 1 Rational Numbers 7 Form 1 1.2.5 Perform efficient computations by using the laws of basic arithmetic operations Example 12 Calculate each of the following using efficient computation. (a) (105 + 47) + 13 (b) (4 × 9) × 25 (c) 3 × 102 (d) 39 × 5 + 12 × 5 Solution (a) (105 + 47) + 13 = 105 + (47 + 13) Associative Law = 105 + 60 = 165 (b) (4 × 9) × 25 = (9 × 4) × 25 = 9 × (4 × 25) = 9 × 100 = 900 (c) 3 × 102 = 3 × (100 + 2) = 3 × 100 + 3 × 2 = 300 + 6 = 306 (d) 39 × 5 + 12 × 5 = (39 + 12) × 5 Distributive Law = 51 × 5 = 255 Try question 6 in Formative Practice 1.2 1.2.6 Solve problems involving integers Example 13 The initial temperature of an ice cube in a beaker is 0°C. When a pinch of salt was added, the temperature of the ice cube dropped by 2°C. Then, the ice cube was heated until the temperature increased by 35°C. Find the current temperature, in °C, of the melted ice cube. Commutative Law Associative Law Distributive Law Solution Current temperature of the melted ice cube = 0°C – 2°C + 35°C = –2°C + 35°C = 33°C Try question 7 – 8 in Formative Practice 1.2 Example 14 Daily Application Company A & D has 3 branches, namely, P, Q and R. In 2016, branch P had a loss of RM14 230, branch Q gained a profit of RM32 000 and branch R had a loss of two times of branch P. Explain whether Company A & D gained profit. Solution Branch P had a loss of RM14 230. ➞ –RM14 230 Branch Q had a profit of RM32 000. ➞ +RM32 000 Branch R had two times the loss of branch P. ➞ 2 × (–RM14 230) –RM14 230 + RM32 000 + 2 × (–RM14 230) = –RM14 230 + RM32 000 + (–RM28 460) = –RM14 230 + RM32 000 – RM28 460 = –RM10 690 - sign shows loss. Thus, Company A & D did not gain profit. Company A & D had a loss of RM10 690. Try question 9 in Formative Practice 1.2 Common mistake 1 INFO Formative Practice 1.2 1. Calculate each of the following. (a) 3 + (+2) (b) –3 + 6 (c) 7 + (–8) (d) –5 + (–9) (e) 4 – 7 (f) 2 – (–9) (g) –6 – 4 (h) –7 – (–3) 2. Evaluate each of the following. (a) 8 × (–2) (b) –4 × 5 (c) (–6) × (–7) (d) (–3) × (–12)

Mathematics Form 1 Chapter 1 Rational Numbers 8 1 Form 3. Evaluate each of the following. (a) 45 ÷ (–3) (b) –28 ÷ 4 (c) (–49) ÷ (–7) (d) –36 –3 4. Calculate. (a) 5 – (–2) + (–7) (b) –4 + 3 – (–2) (c) –21 ÷ 7 × (–4) (d) 81 ÷ (–9) × (–3) 5. Evaluate. (a) –7 × (–2) + (–4) (b) 8 ÷ (–4) – (–2) (c) 42 – (–16) ÷ (–8) (d) 7 + (–5) × (–2) (e) 10 – 20 ÷ (–2) + (–5) (f) –6 × [12 + (–7)] – (–2) (g) 16 – (–2) –5 + 2 (h) –23 + 7 1 – (–3) 6. Solve each of the following using efficient calculation. (a) 401 + 82 + 8 (b) 500 + 47 – 17 (c) 49 × 50 × 2 (d) 3 × 81 ÷ 9 (e) 701 × 4 (f) 15 × 3 + 25 × 3 7. An eagle flew 20 m above sea level. A fish was 35 m vertically below the eagle. Find the position of the fish. 8. The water level in a reservoir decreases by 2 m every day. Find the total decrease of water level, in m, after 3 days. 9. In a Mathematics quiz, for each correct answer will be awarded 3 marks, for each wrong answer will be deducted 2 marks and for each unanswered question will be deducted 1 mark. The quiz has 16 questions and the contestants who scored 35 and above will be given a prize. (a) Leong obtained 13 correct answers and 2 wrong answers. Determine whether Leong will be awarded a prize. Give your reason. (b) John obtained 13 correct answers. Is it possible for John to receive the prize? Give your reason. HOTS Analysing HOTS Evaluating Positive and Negative Fractions 1.3 1.3.1 Represent positive and negative fractions on number lines 1. Representing the positive and the negative fractions on a number line is the same as integers. 4 –– 3 3 –– 3 2 –– 3 1 –– 3 1 – 3 1 –– 3 1 –– 3 1 –– 3 1 –– 3 1 +– 3 1 +– 3 1 +– 3 1 +– 3 2 – 3 3 – 3 4 – 3 0 Negative fractions Zero Positive fractions The value is decreasing The value is increasing 2. Note that the positive fractions are located on the right hand side of zero whereas the negative fractions are located on the left hand side of zero. MATHS MATHS INFO INFO The fractions can also be represented by using vertical number line. Example 15 Represent the following fractions on number lines. (a) 1 5 , – 2 5 , – 4 5 , 3 5 (b) – 5 6 , 1 3 , – 2 3 , – 1 6 Solution (a) 4 –– 5 3 –– 5 2 –– 5 1 –– 5 1 – 5 2 – 5 3 – 5 0 (b) 5 –– 6 –– = – – 2 –– 3 1 –– 6 3 –– 6 2 –– 6 1 – 6 1 – 3 0 4 6 2 3 – = – 2 6 1 3 Try question 1 in Formative Practice 1.3

Mathematics Form 1 Chapter 1 Rational Numbers 9 Form 1 1.3.2 Compare and arrange fractions in order 1. On a number line, the fractions to the right is larger than the fractions to the left. 2. Based on the position of fractions on a number line, the fractions can either be arranged in ascending order or in descending order. Example 16 Determine the largest fraction. (a) – 1 2 or 1 3 (b) –3 7 8 or –1 3 4 (c) –1 2 3 or –1 5 6 Solution (a) 1 3 is larger than – 1 2 . A positive fraction is always larger than a negative fraction. (b) –3 7 8 or –1 3 4 Compare the whole number of the two mixed numbers: –1 is larger than –3. Thus, –1 3 4 is larger than –3 7 8 . (c) –1 2 3 or –1 5 6 ➞ –1 4 6 or –1 5 6 ➞ –10 6 or –11 6 Compare the values of numerators for the two negative fractions with the same denominator: –10 is larger than –11. – 10 6 is larger than – 11 6 . Thus, –1 2 3 is larger than –1 5 6 . Try question 2 in Formative Practice 1.3 Example 17 (a) Arrange – 1 5 , 1 10, – 1 2 , 1 5 and – 3 10 in ascending order. (b) Arrange 1 4 , – 1 8 , – 1 4 , 3 8 and – 1 2 in descending order. Solution (a) – 1 5 , 1 10 , – 1 2 , 1 5 , – 3 10 ↓ ↓ ↓ ↓ ↓ – 2 10 , 1 10 , – 5 10, 2 10 , – 3 10 Draw a number line to represent the given fractions. 5 –– 10 3 –– 10 2 –– 10 Original fractions 1 –– 2 1 –– 5 1 – 10 1 – 5 3 –– 10 2 – 10 1 – 10 Thus, the fractions arranged in ascending order is – 1 2 , – 3 10, – 1 5 , 1 10, 1 5 . (List the fractions on a number line from left to right.) Alternative Method – 1 5 , 1 10 , – 1 2 , 1 5 , – 3 10 ↓ ↓ ↓ ↓ ↓ – 2 10 , 1 10 , – 5 10 , 2 10 , – 3 10 Compare the numerators with the same denominators. Arrange the fractions based on the values of numerators, from the smallest value to the largest value. –5 10, –3 10, –2 10, 1 10, 2 10 = – 1 2 , – 3 10, – 1 5 , 1 10, 1 5 Equalize the denominator Equalize the denominator

Mathematics Form 1 Chapter 1 Rational Numbers 10 1 Form Brackets ↓ × or ÷ from left to right ↓ + or – from left to right Example 18 Solve each of the following. (a) –1 1 8 + 1 2 × 3 4 (b) 1 1 8 ÷ 1–1 1 2 – 3 4 2 (c) –1 1 3 ÷ 4 9 + 1 4 × 2 1 2 Solution (a) –1 1 8 + 1 2 × 3 4 = – 9 8 + 1 2 × 3 4 = – 9 8 + 1 1 2 × 3 4 2 = – 9 8 + 3 8 = – 6 8 = – 3 4 The lowest term (b) 1 1 8 ÷ 1–1 1 2 – 3 4 2 = 9 8 ÷ 1– 3 2 – 3 4 2 = 9 8 ÷ 1– 6 4 – 3 4 2 = 9 8 ÷ 1– 9 4 2 = 9 8 × 1– 4 9 2 Change ÷ into × and inverse the fraction – 9 4 . = – 1 2 (c) –1 1 3 ÷ 4 9 + 1 4 × 2 1 2 = – 4 3 ÷ 4 9 + 1 4 × 5 2 Change the mixed number to improper fraction. × / ÷ from left to right Change to improper fraction. Equalize the denominators for the calculation in the bracket Do the operations inside the brackets first Change to improper fraction. (b) 1 4 , – 1 8 , – 1 4 , 3 8 , – 1 2 ↓ ↓ ↓ ↓ ↓ 2 8 , – 1 8 , – 2 8 , 3 8 , – 4 8 Draw a number line to represent the given fractions. 4 –– 8 2 –– 8 1 –– 8 2 – 8 3 – 8 Original fractions 1 –– 2 1 –– 8 1 – 4 3 – 8 1 –– 4 Thus, the fractions arranged in descending order are 3 8 , 1 4 , – 1 8 , – 1 4 , – 1 2 . (List the fractions on a number line from right to left.) Alternative Method 1 4 , – 1 8 , – 1 4 , 3 8 , – 1 2 ↓ ↓ ↓ ↓ ↓ 2 8 , – 1 8 , – 2 8 , 3 8 , – 4 8 Compare the numerators that have the same denominators. Arrange the fractions based on the values of numerators, from the largest value to the smallest value. 3 8 , 2 8 , –1 8 , –2 8 , –4 8 = 3 8 , 1 4 , – 1 8 , – 1 4 , – 1 2 Equalize the denominators Try question 3 in Formative Practice 1.3 1.3.3 Perform computations involving combined basic arithmetic operations of positive and negative fractions 1. The order of operations involving addition, subtraction, multiplication, division and brackets for fractions is the same as the order of operations involving integers. Equalize the denominators

Mathematics Form 1 Chapter 1 Rational Numbers 11 Form 1 = 1– 4 3 ÷ 4 9 2 + 1 1 4 × 5 2 2 × / ÷ from left to right = 1– 4 3 × 9 4 2 + 1 1 4 × 5 2 2 = –3 + 5 8 = –2 3 8 Try question 4 in Formative Practice 1.3 1.3.4 Solve problems involving fractions Example 19 Daily Application The table below shows the profit and loss of a company in three consecutive years. Year Profit / Loss 2014 Profit of RM1 1 3 million 2015 Loss of RM2 1 4 million 2016 2 times of the profit in the year 2014 Calculate the profit or loss, in million, of the company in the period of three years. Give your answer in fraction form. Solution 1 1 3 + 1–2 1 4 2 + 2 × 1 1 3 Negative sign shows loss. = 1 1 3 – 2 1 4 + 12 × 4 3 2 = 4 3 – 9 4 + 12 × 4 3 2 = 4 3 – 9 4 + 8 3 = 16 12 – 27 12 + 32 12 = 21 12 = 7 4 = 1 3 4 Thus, the company gained a profit of RM1 3 4 million in the period of three years. Try questions 5 – 6 in Formative Practice 1.3 HOTS Challenge Fill in the blanks with ‘+’, ‘‒’, ‘×’ and ‘÷’ without repetition to get the largest value. (‒8)   (‒10) 8 (‒4) 1 Solution: 4 (–) × (–) = (+) (–8) × (–10) + 8 – (–4) ÷ 1 1442443 14243 1 Get the 2 Get the larger value smaller value 3 (+) + (+) = (+) Formative Practice 1.3 1. Represent each of the following on number lines. (a) 1 7 , –   2 7 , 3 7 , –   4 7 (b) 1 3 , –   2 3 , 0, –1 1 3 (c) 1 6 , –   1 2 , –   1 3 , 1 3 (d) –   1 4 , 1 8 , 1 2 , –   3 8 2. Determine the largest fraction. (a) –   2 3 , 4 5 (b) –5 1 2 , –1 1 2 (c) –2 3 5 , –2 7 10 (d) –3 7 8 , –2 1 9 , –2 1 6 3. Arrange each of the following in (i) ascending order, (ii) descending order. (a) 1 2 , –   3 8 , –   1 2 , 5 8 , –   1 8 (b) –   2 5 , 1 10, 1 2 , 3 5 , –   7 10 (c) –1 1 3 , –   5 6 , 4 9 , –1 1 6 , 7 9 (d) 1 1 4 , –   5 8 , –1 1 2 , –1 7 8 , 3 2 4. Calculate each of the following. (a) 1 2 + 1–2 1 4 2 – 1–   3 8 2 Common mistake 2 INFO

Mathematics Form 1 Chapter 1 Rational Numbers 12 1 Form (b) –2 1 2 × 4 5 ÷ 1–   2 3 2 (c) 1 1 9 × 1–   1 5 2 – 5 6 (d) –   7 8 + 3 10 × 5 12 (e) –   3 4 ÷ 11 1 2 – 5 8 2 (f) 1–3 3 4 + 5 122 × 1–   8 152 (g) –1 1 4 – 4 9 ÷ 2 2 3 × 1 2 (h) –2 1 2 ÷ 5 8 + 3 4 × 1–   1 6 2 (i) –   2 7 × 1 2 5 – 1–   5 6 2 ÷ 1 1 4 5. A submarine was submerged as deep as 3 5 of 250 m per minute. Find the new position, in m, of the submarine after 1 4 hour. HOTS Applying 6. Normala buys 6 packets of sugar with mass of 3 10 kg each. She needs 2 1 2 kg of sugar to bake a cake. Is Normala’s sugar enough for her to bake the cake? Give your reason. HOTS Applying HOTS Analysing Positive and Negative Decimals 1.4 1.4.1 Represent positive and negative decimals on number lines 1. Representing the decimals on a number line is the same as integers. –0.1 –0.1 –0.1 –0.1 +0.1 +0.1 +0.1 +0.1 0 Negative decimals Positive decimals Zero The value is decreasing The value is increasing –0.4 –0.3 –0.2 –0.1 0.1 0.2 0.3 0.4 2. Note that the positive decimals are located on the right hand side of zero whereas the negative decimals are located on the left hand side of zero. MATHS MATHS INFO INFO Decimals can also be represented by using vertical number line. Example 20 Represent the following decimals on number lines. (a) 1.0, –1.5, –0.5, 1.5, –2.0 (b) –1.2, 1.2, –4.8, 3.6, –2.4 Solution (a) –2.0 –0.5 –1.0 0 –1.5 0.5 1.0 1.5 (b) –4.8 –2.4 –1.2 –3.6 0 1.2 2.4 3.6 Try question 1 in Formative Practice 1.4 1.4.2 Compare and arrange decimals in order 1. On a number line, the decimals to the right is larger than the decimals to the left. 2. Based on the position of decimals on the number line, the decimals can either be arranged in ascending order or in descending order. Example 21 Determine the largest decimal. (a) 1.8 or –4.5 (b) –5.35 or –8.6 (c) –3.62, –4.82 or –3.67 Solution (a) 1.8 is larger than –4.5. A positive decimal is always larger than a negative decimal.

Mathematics Form 1 Chapter 1 Rational Numbers 13 Form 1 (b) –5.35 –5 is larger than –8. –8.6 –5.35 is larger than –8.6. (c) –3.62, –4.82 or –3.67 Compare the value of first digit, –3 is larger than –4. Thus, –4.82 is the smallest. For –3.62 and –3.67, compare the value of the last digit because the values of the first digit and the second digit are the same. Since, –0.02 is larger than –0.07, –3.62 is larger than –3.67. Therefore, –3.62 is the largest. Try question 2 in Formative Practice 1.4 Example 22 (a) Arrange –1.6, 1.4, –3.8, –2.5 and 2.35 in ascending order. (b) Arrange 3.28, –4.1, –1.03, 2.2 and –2.3 in descending order. Solution (a) Positive numbers: 1.4 and 2.35 Compare the value of the first digit, 1 and 2. 2 is larger than 1. Thus, 2.35 is larger than 1.4. Arrange in ascending order: 1.4, 2.35 Negative numbers: –1.6, –3.8 and –2.5 Compare the value of the first digit and arrange the decimals in ascending order based on the value of the first digit, that is –3.8, –2.5, –1.6. Thus, the decimals arranged in ascending order are –3.8, –2.5, –1.6, 1.4, 2.35. (b) Positive numbers: 3.28, 2.2 Compare the value of the first digit, 3 and 2. 3 is larger than 2. Thus, 3.28 is larger than 2.2. Arrange in descending order: 3.28, 2.2 Negative numbers: –4.1, –1.03 and –2.3 Compare the value of the first digit and arrange the decimals in descending order based on the value of the first digit, that is –1.03, –2.3, –4.1. Thus, the decimals arranged in descending order are 3.28, 2.2, –1.03, –2.3, –4.1. Try question 3 in Formative Practice 1.4 1.4.3 Perform computations involving combined basic arithmetic operations of positive and negative decimals 1. The order of operation involving addition, subtraction, multiplication, division and brackets for decimals is the same as integers. Brackets ↓ × or ÷ from left to right ↓ + or – from left to right Example 23 Solve each of the following. (a) 4.2 + (–1.25) × 8.2 (b) 8.91 ÷ (–0.02 – 1.6) (c) (–5.2 + 1.48) – 3.12 × 2.5 Solution (a) 4.2 + (–1.25) × 8.2 × / ÷ from left to right = 4.2 + (–10.25) = 4.2 – 10.25 = –6.05 (b) 8.91 ÷ (–0.02 – 1.6) = 8.91 ÷ (–1.62) = –(8.91 ÷ 1.62) = –5.5 (c) (–5.2 + 1.48) – 3.12 × 2.5 = –3.72 – 3.12 × 2.5 = –3.72 – 7.8 = –11.52 Try question 4 in Formative Practice 1.4 Perform computation in the bracket first. Perform computation in the bracket first. × / ÷ from left to right

Mathematics Form 1 Chapter 1 Rational Numbers 14 1 Form 1.4.4 Solve problems involving decimals Example 24 Daily Application A turtle was 10.28 m below the sea level. The vertical distance between a fish and the turtle is 3.85 m. State the possible positions, in m, of the fish. Solution There are two possible positions. Either the fish is located above the turtle or the fish is located below the turtle. If the fish is located above the turtle, then the position of the fish = –10.28 + 3.85 = –6.43 m If the fish is located below the turtle, then the position of the fish = –10.28 – 3.85 = –14.13 m Thus, the possible position of the fish is either 6.43 m below the sea level or 14.13 m below the sea level. Try questions 5 – 6 in Formative Practice 1.4 Formative Practice 1.4 1. Represent the following decimals on number lines. (a) 0.2, –0.1, 0.3, –0.4, –0.2 (b) –1.3, –3.9, 2.6, –5.2, 3.9 2. Determine the largest decimal. (a) –2.3 or 2.9 (b) –5.6 or –7.3 (c) –1.12, –3.52 or –1.11 3. Arrange each of the following in (i) ascending order, (ii) descending order. (a) 3.87, –1.4, –6.7, 4.5, –3.31 (b) –0.4, 0.9, –5.2, 1.4, –3.0 (c) 2.33, –3.22, –4.11, 1.55, –1.44 (d) –5.42, –5.44, 0.3, –2.9, 0.03 4. Calculate each of the following. (a) 0.7 + (–2.4) – (–0.04) (b) –1.5 × 3.2 ÷ 4 (c) (–1.4) × 3.25 – 4.8 (d) 0.245 + (–2.3) × (–3.8) (e) –5.6 ÷ (–1.5 – 1.3) (f) (–1.75 + 4.85) × (–2.15) (g) –2.25 – 1.125 × 3.8 ÷ 0.25 (h) –0.5 ÷ 0.8 + 1.55 × (–3.24) (i) –4.8 × 1.4 – (–2.28) ÷ 0.625 5. In an experiment, the initial temperature of a metal rod was 35.3°C. The temperature of the metal rod dropped by 45.5°C and then increased by 6.2°C. Find the final temperature, in °C, of the metal rod. HOTS Applying 6. Danny bought 2 500 units and 3 000 units of shares of a company at the price of RM1.76 per unit and RM1.68 per unit respectively. At the end, he sold all the shares at the price of RM1.75 per unit. Determine whether Danny gained profit. Give your reason. HOTS Applying HOTS Analysing 1.5 Rational Numbers 1.5.1 Recognise and describe rational numbers 1. A rational number is a number that can be written in the form of a b , where a and b are integers and b ≠ 0. Example 25 Determine whether the following numbers are rational numbers. Explain your answer. (a) 1 2 3 (b) –0.24 (c) 4 Solution (a) 1 2 3 = 5 3 Express the number in improper fraction. Thus, 1 2 3 is a rational number. (b) –0.24 = – 24 100 Express in the fraction of hundredths. = – 6 25 In the lowest term. Thus, –0.24 is a rational number.

Mathematics Form 1 Chapter 1 Rational Numbers 15 Form 1 Tips –0.24 can be expressed in the fraction form by using a scientific calculator. Press (–) 0 . 2 4 = SHIFT d /c (c) 4 = 4 1 Thus, 4 is a rational number. Try question 1 in Formative Practice 1.5 Example 26 Calculate each of the following. (a) 4.7 – 1–1 5 8 2 × 1 5 , give the answer as a decimal. (b) 1–0.75 + 1 8 ÷ 1 2 2 × 0.8, give the answer as a fraction. Solution (a) 4.7 – 1–1 5 8 2 × 1 5 = 4.7 – (–1.625) × 0.2 = 4.7 – (–0.325) = 4.7 + 0.325 = 5.025 (b) 1–0.75 + 1 8 ÷ 1 2 2 × 0.8 = 1– 75 100 + 1 8 ÷ 1 2 2 × 8 10 Change into fraction. = 1– 3 4 + 1 8 × 2 2 × 4 5 = 1– 3 4 + 1 4 2 × 4 5 = – 2 4 × 4 5 = – 2 5 Try questions 2 – 3 in Formative Practice 1.5 Change into decimals. Example 27 Daily Application A submarine was located 650 m below the sea level. The submarine rose 20.5 m per minute for 20 minutes and then descended 135 2 5 m. Find the final position, in m, of the submarine. Solution Final position of the submarine = –650 + 20 × 20.5 – 135 2 5 = –650 + 410 – 135.4 = –240 – 135.4 = –375.4 Thus, the final position of the submarine was 375.4 m below the sea level. Try questions 4 – 6 in Formative Practice 1.5 Fill in the blanks with the symbol ‘+’ or ‘‒’ to get the largest value. 10 1–   1 2 2 2    (–7.5) TIMSS Formative Practice 1.5 1. Determine whether the numbers –8, 3.15, 1 1 5 and – 2 3 are rational numbers. Give your reason. 2. Calculate each of the following. Give your answer in decimal form. (a) –5.2 + 2 1 2 × 1–   4 5 2 (b) [6 + (–2.13)] × 2 3 (c) –2 1 2 ÷ [5.2 – (–2.8)] (d) –2.53 + 1–   1 8 – 3 4 ÷ 0.62

Mathematics Form 1 Chapter 1 Rational Numbers 16 1 Form 3. Calculate each of the following. Give your answer in fraction form. (a) 2.25 + 1–1 1 2 2 × 8 (b) –   1 8 ÷ 0.1 + 1–   1 4 2 (c) 3.65 + 1–2 1 5 – 22 × 2 1 4 (d) 1–1.625 + 1 1 8 ÷ 1 1 2 2 × 0.4 4. Puan Hasnah had 4.5 kg of sugar. She used 1 4 of the sugar to bake a cake. Calculate the mass, in kg, of the remaining sugar. Give the answer in fraction form. HOTS Applying 5. The initial depth of a pond was 2.52 m. When the surrounding temperature increased, the water level of the pond descended 1 3 of the initial level. Then, water was pumped into the pond until the levels increased by 12.5 cm. Calculate the current height, in m, of the water level of the pond. Give the answer in decimal form. HOTS Applying 6. In a Mathematics Quiz, each contestant is given 10 questions. Each correct answer will be awarded 3 marks, each wrong answer will be deducted 1.5 marks and each unanswered question will be deducted 1 2 marks. The table shows the number of questions answered by 3 contestants. Contestant Number of questions answered correctly Number of questions answered wrongly Emilia 5 4 Ker Er 5 5 Tharishini 4 0 Among the three contestants, who scored the highest marks? Explain your answer. HOTS Analysing HOTS Evaluating Section A 1. Which of the following is an integer? A 3 4 C √5 B 6.7 D 0 2. Which of the following shows ascending order? A 2, 7 2 , 0.5, –2, –1 B –12, –14, –16, –18, –20 C –0.25, 1 4 , 3 5 , 4.2, 14 3 D 0.9, 0.6, 0.3, 0, –0.3 3. 2 7 + 2 7 + 2 7 + 3 7 = × 2 7 + 5 7 The value in is A 2 C 4 B 3 D 5 4. The diagram below shows a number line. P and Q are decimal numbers. –1.7 P Q –0.8 –0.5 Determine the value of P and of Q. A P = –1.4, Q = –1.3 B P = –1.1, Q = –1.4 C P = –1.3, Q = –1.4 D P = –1.4, Q = –1.1 5. Which of the following is the correct calculation for 6(–2 + 8) × 4.8 ÷ 3 5 ? A –36 × 8 B 36 × 4.8 × 5 3 C 36 × 4.8 × 3 5 D 6(–6) × 8 UASA Practice 1 Full solution

Mathematics Form 1 Chapter 1 Rational Numbers 17 Form 1 Section B 1. (a) Arrange the following numbers in descending order. [1 mark] 7, –6 4 5 , 3.8, 3 1 5 , –4.8, –10 (b) Match. [3 marks] 7 + 3 = 3 + 7 (4 + 7) + 3 = 4 + (7 + 3) Associative law Distributive law Commutative law 3(3 + 5) = 3 × 3 + 3 × 5 2. (a) Fill in the blanks with ‘+’, ‘-’, ‘×’, ‘÷’. [2 marks] (i) −9 (−7) = −2 × [6 (−2)] (ii) 6 + (−8) 4 = −4 (−1) (b) Fill in the blanks with ‘+’ or ‘−‘ to obtain the smallest value. [2 marks] (i) 1 4 (−6) 5.3 (ii) −3.5 7 5 (−8) 3. (a) The diagram below shows four number cards. –10 11 –13 5 Fill in the blanks with the suitable number from the diagram to form an arrangement of integers in descending order. [2 marks] 8 , , –8 , , –12 (b) Mark (3) for the rational number and (7) for the irrational number. [2 marks] (i) π (ii) −3.5 Section C 1. (a) Cindy has 2 480 pieces of stamps. The number of Ahmad’s stamps is 1 8 of Cindy’s stamps, while Muthu has four times of the number of Ahmad’s stamps. All the stamps are collected and arranged into 10 albums equally. Calculate the number of stamps in each album. [3 marks] (b) (i) State the biggest and the smallest fractions based on the fraction numbers below. [2 marks] – 1 9 , 1 6 , 1 4 , – 1 5 , 1 8 (ii) The diagram below shows a number line. M –0.4 0.4 N 2.0 State the value of M and of N. [2 marks] (c) The diagram below shows a number line. S –2 1.5 T – 1 4 Determine the value of S – T. [3 marks] HOTS Analysing 2. (a) (i) Sam wanted to simplify 20 9 . The following are Sam’s working steps. 20 9 = 20 4 + 5 Step 1 = 20 4 + 20 5 Step 2 = 5 + 4 Step 3 = 9 Step 4 Sam’s friend, Daniel told him that his working step was incorrect. Which step of Sam’s working is incorrect? What is the actual answer? [2 marks] HOTS Applying (ii) Given 7.5 ÷ 2 5 − 3.51 = x × 0.5. Find the value x. [2 marks] (b) Given that 3 4 of Alvin’s money is equal to 2 3 of Daniel’s money. If Daniel has RM360, calculate the total amount of their money. [3 marks] (c) Mr. Foo bought 2 000 unit shares of Company A & B at the price of RM2.48 per unit. On the next day, the shares decreased by 5 cents per unit and Mr. Foo bought another 3 000 unit shares. If Mr. Foo sold all his shares at the price of RM2.46 per unit, determine whether Mr. Foo gained profit or loss. Hence, give your opinion on why Mr. Foo bought the shares when the price of shares decreased. [3 marks] HOTS Analysing HOTS Evaluating

Mathematics Form 2 Chapter 1 Patterns and Sequences 204 Did you know that beehives have a matching pattern and often hexagonal? If the beehive is circular, there will be empty spaces between the shapes when arrange next to each other and there will be waste of space. In your opinion, why triangle and square are not chosen? Learning Area: Number and Operations Form 2 1 Chapter Patterns and Sequences Pattern Sequence Term Even numbers Odd numbers Pascal’s triangle Fibonacci numbers Algebraic expression Complete the sequence Extend the sequence KEYWORDS INFOGRAPHIC Access to

Mathematics Form 2 Chapter 1 Patterns and Sequences 205 Form 2 1.1 Patterns 1.1.1 Recognise and describe patterns 1. Pattern is an arrangement of shapes, colours, numbers, letters and many more that enables us to identify and describe the behaviours of the next object or event. For instance, (a) 2, 4, 6, 8, … is a set of even numbers. +2 +2 +2 +2 2, 4, 6, 8, … Set of even numbers starts with 2 and the pattern is adding 2 to obtain each subsequent number. (b) 1, 3, 5, 7, … is a set of odd numbers. +2 +2 +2 +2 1, 3, 5, 7, … Set of odd numbers starts with 1 and the pattern is adding 2 to obtain each subsequent number. (c) 1, 1, 2, 3, 5, 8, 13, 21, … is a set of Fibonacci numbers. 1, 1, 2, 3, 5, 8, 13, 21, … + + + + + + In a set of Fibonacci numbers, the first two numbers are 1. The pattern of each subsequent number is the sum of the two numbers before it. (d) The set of numbers in the diagram below is known as the Pascal’s triangle. 1 7 21 35 35 21 7 1 6 15 20 15 6 1 5 10 10 5 1 4 6 4 1 3 3 1 2 1 1 1 1 1 1 1 1 1 128 64 32 16 8 4 2 1 1 + 7 + ... + 7 + 1 1 + 6 + ... + 6 + 1 1 + 5 + ... + 5 + 1 1 + 4 + 6 + 4 + 1 1 + 3 + 3 + 1 1 + 2 + 1 1 + 1 1 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (i) Each number in the Pascal’s triangle is the direct sum of the two numbers on the top where each row starts and ends with 1. (ii) There are many patterns of numbers in rows and diagonals in this triangle. For instance, • sum of numbers in each row will form a list of numbers 1, 2, 4, 8, 16, …. • list of numbers 1, 3, 6, 10, 15, 21, … is the triangular numbers. (e) The pattern for the arrangement of tables and chairs in the following diagram is adding two chairs for one addition of table. 2. When we are describing a pattern, we must state the beginning of the patterns and each change that happens in the pattern. Example 1 Identify and describe the following set of numbers. (a) 22, 18, 14, 10, … (b) 3, 5, 8, 13, 21, … Solution (a) –4 –4 –4 –4 22,   18,   14,   10,   … The set of numbers starts with 22 and subtract 4 to obtain each subsequent number. (b) 3, 5, 8, 13, 21, … + + + The set of numbers starts with 3 and 5. Each subsequent number is the sum of the two numbers before it. Try question 1 in Formative Practice 1.1

Mathematics Form 2 Chapter 1 Patterns and Sequences 206 Form 2 Example 2 The diagram above shows a tiled wall in a kitchen. Identify and describe the pattern of the tiled wall. Solution The pattern of the tiled wall is the combination of triangles which consist of three trapeziums. Try questions 2 – 6 in Formative Practice 1.1 Formative Practice 1.1 1. Identify and describe the pattern for the following sets of numbers. (a) 4, 20, 100, 500, … (b) 1, 3, 4, 7, 11, … (c) 2, 8, 16, 64, 128, … (d) 3, 4, 6, 9, 13, … 2. Identify and describe the pattern for the following sets of shapes or letters. (a) (b) ABCXABCXXABCXXX 3. The diagram below shows the growth of shoot of a tree. Identify and describe the pattern for addition of leaves along the growth of the shoot. 4. The diagram below shows a pattern on an earthen jar. Identify and describe the pattern on the earthen jar. 5. Identify and describe the pattern for the scale of thermometer in the diagram below. 20ºC 10ºC 0ºC 6. The diagram below shows the arrangements of trapezium tables surrounded by chairs. Identify and describe the pattern for the arrangements of tables and chairs. Table Chair 1.2 Sequences 1.2.1 Recognise sequences 1. A list of succession of numbers created from a set of geometric shapes, numbers or objects is called a sequence. For instance, The set of geometric shapes above is a sequence because the geometric shape starts with 3-sided polygon and the pattern is adding one side for each subsequent shape.

Mathematics Form 2 Chapter 1 Patterns and Sequences 207 Form 2 Example 3 Explain whether each of the following is a sequence. (a) 15, 12, 10, 5, –3, … (b) First Second Third Fourth Fifth Solution (a) –3 –2 –5 –8 15,   12,   10,   5,   –3,   … It is not a sequence because the list of numbers does not follow a particular pattern. (b) It is a sequence because the set of geometrical shapes follows a particular pattern. The pattern starts with a hexagon, followed by two triangles, and then repeats. Try question 1 in Formative Practice 1.2 1.2.2 Identify and describe the pattern of a sequence, complete and extending the sequence Example 4 Identify and describe the pattern for each of the following sequences. Then, complete the sequence. (a) 9, 13, , 21, 25, , . (b) , 18, 6, , 2 3 , 2 9 , . Solution (a) The pattern of the sequence is add 4 to obtain each subsequent number. 9, 13, 17, 21, 25, 29 , 33 +4 +4 +4 +4 +4 +4 (b) The pattern of the sequence is divide by 3 to obtain each subsequent number. 54, 18, 6, 2, 2 3 , 2 9 , 2 27 ÷3 ÷3 ÷3 ÷3 ÷3 ÷3 Tips The reverse of division is multiplication. In Example 4(b), the first term can be obtained by multiplying 18 by 3. 54, 18, … ×3 Try questions 2 – 3 in Formative Practice 1.2 Example 5 Identify and describe the pattern for each of the following sequences. Hence, extend three subsequent numbers in the sequence. (a) 64, 59, 54, 49, … (b) –1, 2, –4, 8, … Solution (a) The sequence starts with 64 and subtract 5 to obtain each subsequent number. 64, 59, 54, 49, 44, 39, 34 –5 –5 –5 –5 –5 –5 (b) The sequence starts with –1 and multiply by –2 to obtain each subsequent number. –1, 2, –4, 8, –16, 32, –64 ×(–2) ×(–2) ×(–2) ×(–2) ×(–2) ×(–2) Tips A sequence of numbers can be obtained by adding, subtracting, multiplying or dividing. Try questions 4 – 5 in Formative Practice 1.2 Example 6 Identify and describe the pattern for the following sequence. Then, extend two diagrams for the sequence. First Second Third Fourth Fifth

Mathematics Form 2 Chapter 1 Patterns and Sequences 208 Form 2 Solution The pattern starts with triangle, rhombus and followed by trapezium, and then repeats. Thus, the two subsequent diagrams are as follow. Sixth Seventh Try questions 6 – 8 in Formative Practice 1.2 What is the number needed to fill in the ‘empty’ triangle? 3 4 8 4 6 3 5 13 2 7 9 5 4 14 6 Solution: 3 4 8 4 fi ff 6 3 5 13 fi ff 8 + 4 = 3 × 4 5 + 13 = 6 × 3 2 7 9 5 fi ff 4 14 6 5 fi ff 9 + 5 = 2 × 7 14 + 6 = 4 × 5 HOTS Challenge Formative Practice 1.2 1. Explain whether each of the following is a sequence. (a) 5, 10, 40, 240, 1 920, … (b) 7, 11, 18, 23, 24, … (c) 2. Identify and describe the pattern for each of the following sequences. Hence, complete the sequence. (a) 8, 13, , 23, 28, (b) 76, 72, 68, , , 56 (c) 2 000, , 500, 250, , 125 2 (d) , 27, 81, , 729, 2 187 3. Complete the empty spaces in the following Pascal’s triangle. 1 1 1 1 2 1 1 3 1 4 6 1 1 5 10 5 1 4. Identify and describe the pattern for each of the following sequences. Hence, extend three subsequent numbers in the sequence. (a) 5, –5, 5, –5, , , (b) 9, 20, 31, 42, , , (c) 55, 49, 43, 37, , , (d) 488, –244, 122, , , 5. Identify and describe the pattern for the following sequence. Hence, extend two subsequent numbers of the sequence. 1 9 = 0.111 …, 2 9 = 0.222 …, 3 9 = 0.333 … 6. Identify and describe the pattern for the following sequence. Hence, extend two subsequent diagrams in the sequence. (i) (ii) (iii)

Mathematics Form 2 Chapter 1 Patterns and Sequences 209 Form 2 7. Identify and describe the pattern for the following sequence. Then, complete diagrams (i), (ii) and (v) of the sequence. (i) (ii) (iii) (iv) (v) (vi) 8. Mahad arranges matchsticks to form a few patterns as shown in the diagram above. (a) Identify and describe the pattern for the sequence. (b) Draw two subsequent patterns that will be formed by Mahad. 1.3 Patterns and Sequences 1.3.1 Make generalisation about the pattern of a sequence Example 7 First row Second row Third row The diagram above shows an arrangement of canned drinks in a supermarket. The first row has 3 cans. The second row has 4 cans, the third row has 5 cans and so on. Make a generalisation for the pattern of the arrangement of canned drinks using numbers, words and algebraic expressions. Solution Using numbers: 3, 4, 5, 6, … Using words: The first row has 3 cans and each row below it has one can more than the row before it. Using algebraic expressions: Row Sequence 1 1 + 2 = 3 2 2 + 2 = 4 3 3 + 2 = 5 4 4 + 2 = 6 5 5 + 2 = 7 … … n n + 2 n + 2, where n represents the number of canned drinks in the n row (from the top). Try questions 1 and 2 in Formative Practice 1.3 1.3.2 Determine a term 1. Each element in a sequence is known as a term. For instance, in the sequence 5, 10, 15, 20, 25, … the first term is 5, the second term is 10, the third term is 15 and so on. Example 8 Determine the 8th term in the following sequence. 4, 7, 10, 13, 16, … Solution Extend the sequence by adding 3 until the 8th term. 4, 7, 10, 13, 16, 19, 22, 25 +3+3+3+3+3+3+3 The 8th term in the sequence is 25.

Mathematics Form 2 Chapter 1 Patterns and Sequences 210 Form 2 Alternative Method This method is suitable to determine the bigger term in a sequence. Position of term Sequence 1 3(1) + 1 = 4 2 3(2) + 1 = 7 3 3(3) + 1 = 10 4 3(4) + 1 = 13 … … n 3(n) + 1 The nth term = 3n + 1 The 8th term = 3(8) + 1 = 25 or ×3 n = 1, 2, 3, 4, 5, …, n 3n = 3, 6, 9, 12, 15, …, 3n +1 3n + 1 = 4, 7, 10, 13, 16, …, 3n + 1 Substitute n = 8 Position Term Further explanation of Alternative Method Example 8 INFO Tips 3 × 1 + 1 = 4 Term difference Number Position of term × + or – = Term Example 9 Determine the 12th term in the following sequence. 18, 14, 10, 6, 2, … Solution Extend the sequence by subtracting 4 until the 12th term. 18, 14, 10, 6, 2, –2, –6, –10 –14, –18, –22, –26 –4–4–4–4–4–4–4 –4 –4 –4 –4 The 12th term of the sequence is –26. Alternative Method Position of term Sequence 1 22 – 4(1) = 18 2 22 – 4(2) = 14 3 22 – 4(3) = 10 4 22 – 4(4) = 6 … … n 22 – 4n The nth term = 22 – 4n The 12th term = 12 – 4(12) = –26 Substitute n = 12 Further explanation of Alternative Method Example 9 INFO Tips To determine the nth term of a sequence where the order of terms is decreasing, we subtract a multiple of n from a fixed number. For instance, the nth term of 15, 13, 11, 9, … is 17 – 2n. Try question 3 in Formative Practice 1.3 In the diagram above, 9 matchsticks are used to form 4 triangles in a row. How many triangles can be formed in this way if 55 matchsticks are used? TIMSS Common mistake INFO

Mathematics Form 2 Chapter 1 Patterns and Sequences 211 Form 2 1.3.3 Solve problems Example 10 Daily Application During an annual school sports, a school band has performed various triangular formations at the school field. The triangular formations of 2, 3, 4 and 5 bandsmen at the triangular bases are as shown in the diagram. 2 3 4 5 Determine the total number of bandsmen needed to form a triangular formation of 8 bandsmen at the triangular base. Solution Number of bandsmen at the triangular base 2 3 4 5 6 7 8 Total number of bandsmen 3 6 10 15 21 28 36 +3 +4 +5 +6 +7 +8 The total number of bandsmen needed to form a triangular formation of 8 bandsmen at the triangular base is 36. Try questions 4 – 8 in Formative Practice 1.3 Formative Practice 1.3 1. Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 The diagram above shows the pattern of straight lines drawn passing through a number of points. Make a generalisation of pattern of the number of points passed through by each of the straight lines using numbers, words and algebraic expressions. 2. Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat Koko Oat A grocery store exhibits cereal boxes in an attractive arrangement to attract customers. The diagram above shows the arrangement for the first six top rows. Make a generalisation of pattern of the number of cereal boxes by using numbers, words and algebraic expressions. 3. Determine the 12th term in each of the following sequences. (a) 1, 5, 9, 13, … (b) 6, 11, 16, 21, … (c) 35, 32, 29, 26, … (d) 40, 35, 30, 25, … (e) 190, 180, 160, 150, … 4. The diagram below shows part of a calendar showing the dates for the month of January year 2001. Determine the date of the last Monday for the month of January year 2001. January 2001 Monday 1 2 3 4 5 6 7 8 9 10 11 Tuesday Wednesday Thursday Friday Saturday Sunday 5. A diver can dive to a depth of –3 m after one minute, –7 m after two minutes and –11 m after three minutes. Assume that the diver keeps diving following such pattern, determine the position of the diver after 8 minutes. 6. 1 layer 2 layers 3 layers The diagram above shows the first three layers of a well built from unit cubes. How many layers of the well are built by using 256 unit cubes?

Mathematics Form 2 Chapter 1 Patterns and Sequences 212 Form 2 UASA Practice 1 7. The Form 2 students of SMK Taman Bahagia are planning to attend a Mathematics workshop. The cost of transportation is RM15 per student and the cost of participation is RM25 per student. (a) Complete the following table for the first 5 students. Number of students Total cost (RM) 1 40 2 80 (b) Identify and describe the pattern that relates the number of students and the total cost. Hence, write an algebraic expression for the total cost in terms of proper variables. (c) If the total cost is RM1 760, how many students will be attending the Mathematics workshop? HOTS Analysing 8. 0 1 2 3 4 5 10 15 20 t n n The graph above shows the terms in a sequence where t n represents the nth term. (a) Based on the graph, make a generalisation of pattern of the terms in the sequence by using numbers, words and algebraic expressions. (b) Hence, determine t 10 of the sequence. HOTS Analysing Section A 1. Which of the following pattern is a sequence? A 4, 19, 34, 48, … B 145, 129, 113, 99, … C 1 4 , 1 8 , 1 16, 1 32, … D –0.4, –0.8, –0.16, –0.32, … 2. 70, 64, 58, 52, … Describe the pattern for the set of numbers above. A Add 4 to the previous number. B Subtract 6 from the previous number. C Divide the previous number by 3. D Multiply the previous number by 4. 3. Which of the following is a set of Fibonacci numbers? A 1, 2, 4, 8, … B 1, 2, 6, 24, … C 1, 5, 6, 11, … D 10, 20, 30, 40, … 4. 4 5 6 7 8 9 10 11 12 13 14 15 16 The diagram above shows a number line. What are the next two numbers in the sequence of the number line? A 18, 23 B 18, 24 C 19, 24 D 19, 25 5. Which of the following matches is incorrect? Starts with 100 and divide by 2. Starts with 100 and subtract 50. Starts with 1 and add 7. Starts with 1 and multiply by 3. 1, 4, 7, 10, 13, 16, … 1, 3, 9, 27, 81, … 100, 50, 25, 12.5, … 100, 50, 0, –50, –100, … A B C D Full solution

Mathematics Form 2 Chapter 1 Patterns and Sequences 213 Form 2 Section B 1. (a) Mark (✓) for the correct statement and (7) for the incorrect statement describing the pattern of the set of numbers 8, 5, 9, 6, 10, 7, … [2 marks] (i) The set of numbers begins with 8. Subtract 3 and then add 4 alternatively to obtain each of subsequent number. (ii) The set of numbers begins with 8 and 5. Add 1 simultaneously to both of the numbers to obtain each of two subsequent numbers. (b) 41 5 ? 13 9 ? 21 17 33 29 1 25 The diagram above shows a sequence of unorganised numbers with two unknown numbers. Find the two unknown numbers and write the number sequence in ascending order. [2 marks] 2. The diagram below represents patterned beads. Mark (✓) for the pattern of letters that is suitable to represent the arrangement of the beads and (✗) for the pattern of letters that is not suitable to represent the arrangement of the beads. [4 marks] A-B-A-B-B-C-C-B-D A-B-C-B-B-C-C-A-D D-C-D-C-C-A-A-C-B D-C-C-B-B-A-A-C-D Section C 1. (a) First Second Third Fourth sequence sequence sequence sequence The diagram above shows a pattern formed by arranging triangular cards. (i) State the number of triangle used in the fifth order. [1 mark] (ii) Farid has 42 pieces of triangle cards. Find the maximum sequence that can be formed using the cards. [2 marks] (iii) Hence, state the number of unused cards. [1 mark] (b) Triangular numbers are a list of numbers which can be represented by arrangements of triangular dots. The diagram below shows the first four terms in the triangular numbers. Determine the 10th term. [2 marks] 1 3 6 10 HOTS Applying (c) The first two numbers in a set of numbers are 3 and 5. (i) Write two different sets of number patterns starting with the two numbers. [2 marks] (ii) Determine the 8th term of each of the set. [2 marks] 2. (a) The diagram above shows an experiment to measure the distance travelled in each second of a trolley moving down a slope. The results of the experiment are shown in the following table. Time (s) The distance travelled (cm) 1 2 3 4 5 3 5 7 9 11 Complete the table below based on the result of the experiment. HOTS Applying [4 marks] Write a generalisation of pattern for the distance travelled by the trolley using numbers words algebraic expression

Mathematics Form 2 Chapter 1 Patterns and Sequences 214 Form 2 (b) A B C (i) Identify and describe the pattern for the sequence in the diagram above. [2 marks] (ii) Hence, extend two more diagrams in the sequence. [2 marks] (iii) Show that the sequence has more than one pattern. [2 marks] HOTS Analysing 3. Farid pulls apart blocks from a structure of brick game following a certain pattern. (a) Write the pattern as a number sequence. [2 marks] (b) How many blocks are pulled apart by Farid for each structure? [4 marks] (c) How many blocks are left when Farid pulls apart the blocks one more time? [4 marks] 4. (a) A primary school in a rural area is facing a decrease in registration of new students due to the population migration to the city. It is found that there is a uniform decrease of 110 students for every subsequent year. On the first year, the registration of new students is 2 100 students. (i) Write an algebraic expression of the number of students for a particular year. [2 marks] (ii) In which year the number of registration of new students will be decreasing until the number is less than 800 students for the first time? [2 marks] HOTS Analysing (b) January 2017 Sunday 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Monday Tuesday Wednesday Thursday Friday Saturday The diagram above shows a calendar. Add the three horizontal numbers and the three vertical numbers in the shaded rectangles. (i) What can you observe from the sum of both sets of the three numbers? [2 marks] (ii) How does the sum of three numbers relate to the number at the centre? [2 marks] (iii) Repeat (i) and (ii) with two other sets of three numbers. Describe the pattern of any two sets of three numbers that exist in the same way on the calendar. [2 marks] HOTS Analysing

378 A DNA molecule can split in half to form a copy of itself. Each of the copies will become a molecule that is identical to its original. These molecules will keep splitting and each splitting causes the number of DNA molecules doubled and this situation can be written as 2 × 2 × 2 × 2 × 2 × ... . How can we represent this multiplication in the simplest form? Learning Area: Number and Operations Form 3 1 Chapter Indices Index Base Repeated multiplication Index notation Law of indices Fractional index Exponent Power KEYWORDS INFOGRAPHIC Access to

Mathematics Form 3 Chapter 1 Indices 379 Form 3 1.1 Index Notation 1.1.1 Represent repeated multiplication in index form 1. The repeated multiplication of a number a can be written in index form by using the index notation, an, as shown in the following. a × a × a × ... × a = an Repeated multiplication of a by n times Index Base In index form, an, a is known as the base and n is power that is known as index. n is also known as exponent. MATHS MATHS INFO INFO • Index notation an means the value of a is multiplied repeatedly by n times. • an is read as a raised to the nth power. Example 1 Express each of the following repeated multiplication in index form. (a) 6 × 6 × 6 × 6 (b) 0.8 × 0.8 × 0.8 × 0.8 × 0.8 × 0.8 (c) 1– 2 7 2 × 1– 2 7 2 × 1– 2 7 2 Solution (a) 6 × 6 × 6 × 6 = 64 (b) 0.8 × 0.8 × 0.8 × 0.8 × 0.8 × 0.8 = 0.86 (c) 1– 2 7 2 × 1– 2 7 2 × 1– 2 7 2 = 1– 2 7 2 3 Explanation for the solutions INFO Try question 1 in Formative Practice 1.1 1.1.2 Rewrite a number in index form and vice versa 1. We can express a number in index form by the following steps: Write the number in the form of repeated multiplication. Write the product using base and index. For example, 32 = 2 × 2 × 2 × 2 × 2 = 25 2 as the base that is multiplied by 5 times. 2. We can find the value of a number in index form by the following steps. Write the number in the form of repeated multiplication. Find the product. For example, 74 = 7 × 7 × 7 × 7 = 2 401 7 is multiplied by 4 times Tips Using scientific calculator, press 7 ^ 4 = Example 2 Express each of the following numbers in index form. (a) 81 (b) –100 000 (c) 64 125 Solution (a) 81 = 9 × 9 or 81 = 3 × 3 × 3 × 3 = 92 = 34 (b) –100 000 = (–10) × (–10) × (–10) × (–10) × (–10) = (–10)5 (c) 64 125 = 4 5 × 4 5 × 4 5 = 1 4 5 2 3 Try question 2 in Formative Practice 1.1

Mathematics Form 3 Chapter 1 Indices 380 Form 3 Example 3 Find the value of each of the following. (a) 26 (b) (–8)4 (c) 1– 3 7 2 3 Solution (a) 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64 (b) (–8)4 = (–8) × (–8) × (–8) × (–8) = 4 096 (c) 1– 3 7 2 3 = 1– 3 7 2 × 1– 3 7 2 × 1– 3 7 2 = – 27 343 Tips • (–a)n is a negative value if n is an odd number. • (–a)n is a positive value if n is an even number. Try questions 3 – 6 in Formative Practice 1.1 Formative Practice 1.1 1. Express each of the following repeated multiplication in index form. (a) 5 × 5 × 5 × 5 × 5 × 5 × 5 (b) (–3) × (–3) × (–3) × (–3) (c) (–1.2) × (–1.2) × (–1.2) × (–1.2) × (–1.2) (d) 4 9 × 4 9 × 4 9 2. Express each of the following numbers in index form. (a) 729 (b) –32 (c) 1 296 10 000 (d) 0.0625 3. Find the value of each of the following. (a) 35 (b) (–4)6 (c) 1 5 6 2 3 (d) (–0.2)7 4. Given that a × a × a × a = 9n , where a and n are positive integers. State the value of a and of n. 5. Given k7 = –128, find the value of k. 6. The product of the repeated multiplication of a number by n times is 7  776. Determine the value of n. 1.2 Laws of Indices 1.2.1 Relate the multiplication of numbers in index form with the same base 1. Multiplication of numbers in index form with the same base can be related by repeated multiplication as follows. 93 × 95 = (9 × 9 × 9) × (9 × 9 × 9 × 9 × 9) = 98 Repeated multiplication of 9 by 5 times. Repeated multiplication of 9 by 3 times. Total number of repeated multiplications of 9 by 8 times (which is 3 + 5 = 8 times). 2. The product of two numbers in index form with the same base can be obtained by adding the indices of both numbers. am × an = am + n Example 4 Simplify each of the following. (a) 45 × 43 (b) 136 × 139 × 13 (c) 2p7 × 5p4 (d) xy 2 × x 3 y 5 Solution (a) 45 × 43 = 45 + 3 Retain the bases and add the indices. = 48 (b) 136 × 139 × 13 = 136 + 9 + 1 13 = 131 = 1316 (c) 2p7 × 5p4 = 2 × 5 × p7 × p4 = 10 × p7 + 4 = 10p11 (d) xy 2 × x 3 y 5 = x × x3 × y 2 × y 5 = x 1 + 3 × y 2 + 5 = x 4 y 7 Try questions 1 and 2 in Formative Practice 1.2 Simplify: 7a5 × 4a3 TIMSS

Mathematics Form 3 Chapter 1 Indices 381 Form 3 1.2.2 Relate the division of numbers in index form with the same base 1. Division of numbers in index form with the same base can be related by repeated multiplication as follows. 58 ÷ 56 = 58 56 58 ÷ 56 = 5 × 5 × 5 × 5 × 5 × 5 × 5 × 5 5 × 5 × 5 × 5 × 5 × 5 58 ÷ 56 = 52 58 ÷ 56 = 58 – 6 2. The quotient of two numbers in index form with the same base can be obtained by subtracting the indices of both numbers. am ÷ an = am – n where m and n are positive integers, a ≠ 0 Example 5 Simplify each of the following. (a) 1211 ÷ 125 (b) 308 302 (c) 48r 21 ÷ 6r14 (d) a7 b8 a4 b Solution (a) 1211 ÷ 125 = 1211 – 5 = 126 (b) 308 302 = 308 – 2 = 306 (c) 48r 21 ÷ 6r 14 = 48 6 r 21 – 14 = 8r 7 (d) a7 b8 a4 b = a7 – 4 × b8 – 1 = a3 b7 Try questions 3 and 4 in Formative Practice 1.2 Indices are subtracted, 8 – 6 = 2 Retain the bases and subtract the indices. 1.2.3 Relate the numbers in index form raised to a power 1. Numbers in index form raised to a power can be related by repeated multiplication as follows. Repeated multiplication of 42 by 3 times. (42 )3 = 42 × 42 × 42 (42 )3 = (4 × 4) × (4 × 4) × (4 × 4) (42 )3 = 4 × 4 × 4 × 4 × 4 × 4 (42 )3 = 46 (42 )3 = 42 × 3 2. Numbers in index form raised to a power can be simplified by multiplying its indices. (am)n = am × n where m and n are positive integers. Example 6 Simplify each of the following. (a) (67 )4 (b) (p3 )5 (c) (3x2 )4 (d) 1 m8 n5 2 2 Solution (a) (67 )4 = 67 × 4 = 628 (b) (p3 )5 = p3 × 5 = p15 (c) (3x2 )4 = 34 × x 2 × 4 = 81x8 (d) 1 m8 n5 2 2 = m8 × 2 n5 × 2 = m16 n10 Tips • (ap × bq )n = apn × bqn • (ap ÷ bq ) n = 1 ap bq 2 n = apn bqn Try questions 5 and 6 in Formative Practice 1.2 Indices are multiplied, 2 × 3 = 6 Common mistake INFO

Mathematics Form 3 Chapter 1 Indices 382 Form 3 1.2.4 Verify that a0 = 1 and a–n = 1 an 1. We can simplify an ÷ an as follows: an ÷ an = an – n or an ÷ an = an an = a0 = 1 Therefore, it is verified that a0 = 1 where a ≠ 0 For example, 90 = 1, 1 7 8 2 0 = 1, (–0.6)0 = 1 2. For a0 ÷ an, we can simplify as follows: a0 ÷ an = a0 – n or a0 ÷ an = a0 an = a–n = 1 a n Therefore, it is verified that a–n = 1 an where a ≠ 0 For example, 3–1 = 1 3 , z –5 = 1 z 5 Example 7 State each of the following in 1 an . (a) 5–3 (b) 0.7–6 Solution (a) 5–3 = 1 53 (b) 0.7–6 = 1 0.76 Try question 7 in Formative Practice 1.2 Example 8 State each of the following in a–n. (a) 1 69 (b) 1 r 3 1 1 a0 = 1 Solution (a) 1 69 = 6–9 (b) 1 r 3 = r –3 Tips • 1 a–n = an • 1 a b 2 –n = 1 b a 2 n Try question 8 in Formative Practice 1.2 Example 9 Express (a) 1 2 9 2 –1 in the form of positive index. (b) 1 x y 2 4 in the form of negative index. Solution (a) 1 2 9 2 –1 = 1 2 9 a–n = 1 an = 1 ÷ 2 9 = 9 2 (b) 1 x y 2 4 = x 4 y 4 = 1 ÷ y 4 x 4 = 1 ÷ 1 y x 2 4 = 1 1 y x 2 4 = 1 y x 2 –4 1 an = a–n Try question 9 in Formative Practice 1.2

Mathematics Form 3 Chapter 1 Indices 383 Form 3 1.2.5 State the relationship between fractional indices and roots and powers 1. The relationship between the fractional indices and the roots can be written as: a 1 n = n√—a For example, 25 1 2 = √ —25 and 128 1 7 = 7 √ —128 Conversely, √ —25 = 25 1 2 and 7 √ —128 = 128 1 7 2. For the number in fractional index a m n : •  a m n = a m × 1 n = (am) 1 n = n√ —am •  a m n = a 1 n × m = (a 1 n )m = (n√ —a)m Example 10 State each of the following in the form of n√ —a. (a) 8 1 3 (b) 256 1 8 Solution (a) 8 1 3 = 3 √ —8 (b) 256 1 8 = 8 √ —256 Try question 10 in Formative Practice 1.2 Example 11 Write each of the following in the form of a 1 n . (a) √ —121 (b) 6 √ —729 Solution (a) √ —121 = 121 1 2 (b) 6 √ —729 = 729 1 6 Try question 11 in Formative Practice 1.2 Example 12 State each of the following in the form of n√ —am or (n√ —a)m. (a) 27 8 3 (b) 100 3 2 Solution (a) 27 8 3 = 3 √ —278 or 27 8 3 = (3 √ —27)8 (b) 100 3 2 = √ —1003 or 100 3 2 = (√ —100)3 Try question 12 in Formative Practice 1.2 Example 13 Express each of the following in the form of a m n . (a) 5 √ —2434 (b) (√ —0.04)5 Solution (a) 5 √ —2434 = 243 4 5 (b) (√ —0.04)5 = 0.04 5 2 Try question 13 in Formative Practice 1.2 Example 14 Without using calculator, find the value of (a) 32 7 5 (b) 625 3 4 Solution (a) 32 7 5 = (5 √ —32)7 = 27 =128 (b) 625 3 4 = (4 √ —625)3 = 53 = 125 Tips Check answer by using scientific calculator. (a) Press 3 2 ^ ( 7 ab/c 5 ) = (b) Press 6 2 5 ^ ( 3 ab/c 4 ) = Try question 14 in Formative Practice 1.2

Mathematics Form 3 Chapter 1 Indices 384 Form 3 1.2.6 Perform operations involving laws of indices Example 15 Simplify each of the following. (a) 35 × 38 ÷ 37 (b) 74 × (72 )6 73 × 78 Solution (a) 35 × 38 ÷ 37 = 35 + 8 – 7 = 36 (b) 74 × (72 )6 73 × 78 = 74 × 72 × 6 73 × 78 = 74 + 12 – 3 – 8 = 75 Try question 15 in Formative Practice 1.2 Example 16 Find the value of each of the following. (a) √ —65 ÷ 36 × 6 7 2 (b) (4 9 2 × 54 ) ÷ (25 × 5 3 2 )2 Solution (a) √ —65 ÷ 36 × 6 7 2 = 6 5 2 ÷ 62 × 6 7 2 = 6 5 2 – 2 + 7 2 = 64 = 1 296 (b) (4 9 2 × 54 ) ÷ (25 × 5 3 2 )2 = (22 × 9 2 × 54 ) ÷ (25 × 2 × 5 3 2 × 2 ) Use (ap × bq ) n = apn × bqn = (29 × 54 ) ÷ (210 × 53 ) = 29 – 10 × 54 – 3 Use am ÷ an = am – n = 2–1 × 5 = 5 2 Try question 16 in Formative Practice 1.2 Example 17 Simplify each of the following. (a) 54y 3 6y × (3 √ —y 2 )6 (b) (m2 n 4 3 )9 m7 n10 Solution (a) 54y 3 6y × (3 √ —y 2 )6 = 54y 3 6y × (y 2 3 )6 = 54y 3 6y × y 4 = 54 6 × y 3 – 1 – 4 = 9y –2 = 9 y2 (b) (m2 n 4 3 )9 m7 n10 = m2 × 9n 4 3 × 9 m7 n10 Use (ap × bq ) n = apn × bqn = m18n12 m7 n10 = m18 – 7n12 – 10 Use am ÷ an = am – n = m11n2 Try question 17 in Formative Practice 1.2 1.2.7 Solve problems Example 18 Given 8x – 1 = 32, find the value of x. Solution 8x – 1 = 32 (23 )x – 1 = 25 23x – 3 = 25 3x – 3 = 5 x = 8 3 Example 19 Daily Application A Science quiz consists of two parts. There are 210 ways to answer 10 true-false questions in part A and 28 ways to answer 8 true-false questions in part B. How many ways to answer all 18 questions in the quiz? Solution Number of ways to answer all questions = 210 × 28 = 210 + 8 = 218 Try questions 18 – 20 in Formative Practice 1.2

Mathematics Form 3 Chapter 1 Indices 385 Form 3 Formative Practice 1.2 1. Simplify each of the following. (a) 37 × 34 (b) 185 × 182 (c) 56 × 5 × 58 (d) k 9 × k10 × k3 2. Simplify each of the following. (a) 6x2 × 2x4 (b) m3 n × m7 n5 (c) 4ab3 × 7a9 b2 (d) p2 q6 × p5 r 4 × q9 r 10 3. Simplify each of the following. (a) 76 ÷ 74 (b) 2913 ÷ 295 (c) 128 123 (d) y16 ÷ y7 4. Simplify each of the following. (a) 18h10 ÷ 3h4 (b) a4 b17 ÷ ab3 (c) k9 h5 k2 h3 (d) 128x20y7 32x12y 5. Simplify each of the following. (a) (93 ) 6 (b) (0.48 )5 (c) (d2 )7 (d) (y 6 )2 6. Simplify each of the following. (a) (4x5 )3 (b) (p2 r 9 )4 (c) 1 34 y 7 2 2 (d) 1 pq2 s3 2 7 7. Write each of the following in the form of 1 an . (a) 4–5 (b) 50–2 (c) p–1 (d) x–6 8. State each of the following in the form of a–n . (a) 1 94 (b) 1 148 (c) 1 q12 (d) 1 (2y)7 9. Express (a) in the form of positive index (i) 1 5–1 (ii) 1 10–2 (iii) 1 p q 2 –3 (b) in the form of negative index (i) 63 (ii) 3 8 (iii) 1 x y 2 6 10. State each of the following in the form of n √ — a. (a) 4 1 2 (b) 16 1 4 (c) h 1 15 11. Write each of the following in the form of a 1 n . (a) √ — 25 (b) 6 √ — 64 (c) 38√ — k 12. State each of the following in the form of n √ — am or ( n √ — a)m. (a) 81 5 4 (b) 125 2 3 (c) x 3 7 (d) y 9 2 13. State each of the following in the form of a m n . (a) 3 √ — 272 (b) √ — 0.045 (c) (5 √ — p)3 (d) (√ — q)7 14. Without using calculator, calculate the value of (a) 64 2 3 (b) 81 5 4 (c) 25 3 2 (d) 8 7 3 15. Simplify each of the following. (a) 65 × 67 ÷ 68 (b) k10 × k4 ÷ k9 (c) 235 × (233 )4 2310 × 233 (d) (x7 )3 x5 × x2 16. Find the value of each of the following. (a) 27 2 3 × 9 5 2 ÷ 34 (b) √ — 83 ÷ 16 × 2 1 2 (c) (25 × 27 2 3 )2 ÷ (54 × 9 5 2 ) (d) 4 2 3 × 2 5 3 32 2 5 17. Simplify each of the following. (a) p3 q–4 × (2p 2 3 q)6 (b) 28t 5 7t × ( 4 √ — t 3)8 (c) (a3 b 1 2 )4 a5 b6 (d) xy 4 × (x 3 y)5 (√ — xy 6)8 18. Given that (2a4 )3 × 2x 2a6 = 32ay , where x and y are integers. Find the value of x and of y. 19. Water in a dam is channelled to a water treatment plant at the rate of 243 m3 per minute. Find the time taken, in hours and minutes, to channel 813 m3 of water to the water treatment plant. 20. A type of bacteria reproduces by continuous cell division as shown in the diagram below. In a certain condition, a number of 29 parent cells have been reproduced at the same time. Find the number of daughter cells if all the parent cells undergone 8 times of cell division process. HOTS Applying

Mathematics Form 3 Chapter 1 Indices 386 Form 3 Section A 1. Simplify. 53 × (52 )7 A 512 B 517 C 521 D 527 2. Given n √32m = 4. Find the value of m and of n. A m = 2, n = 5 B m = 5, n = 2 C m = 3, n = 4 D m = 4, n = 3 3. Which of the following values is not equal to 256? A 28 B 44 C 83 D 162 4. Given 6x × 6x + 1 = 7 776. Find the value of x. A 5 B 4 C 3 D 2 Section B 1. (a) 1 h6 6 √ — h h6 A h–6 h4 ×h2 h 1 6 B The diagram above shows indices. Complete the equivalent pairs below based on the given example. [2 marks] A B Example 6 √ — h = h 1 6 1 h6 = h6 = (b) a2 × a3 a8 ÷ a2 a4 a–2 a12 a2 (a2 )3 The diagram above shows the operations involving indices. Based on the diagram, choose and write the operations that give the answer as shown in the following circle map. i-THINK [2 marks] a a 6 4 a–2 2. (a) Fill in the blanks with the correct numbers. (i) 5 √ — p2 = p [1 mark] (ii) (–8) = 1 [1 mark] (b) Given 5 √ — 8 × 5 √ — 8 × 5 √ — 8 × 5 √ — 8 = 2y , find the value of y. [2 marks] Section C 1. (a) Simplify. (i) (p3 )5 ÷ p7 × 1 p2 [2 marks] (ii) x 1 2 × x 3 2 x –5 [2 marks] (b) Given that 3n × 27 (3n )2 = 81, find the value of n. [3 marks] (c) Given 1 x y 2 n – 1 = 1 y x 2 n – 5 , find the value of n. [3 marks] HOTS Analysing 2. (a) Simplify √ — m4 n × (mn3 ) 2 mn 5 2 . [3 marks] (b) Simplify 18n + 1 × 31 – n 2n – 1 . [3 marks] (c) Given 16(25x ) = 125(32y ), find the value of x and of y. [4 marks] HOTS Analysing UASA Practice 1 Full solution

497 6. (a) 491 (b) 530 (c) 4 900 (d) 27 (e) 2 804 (f) 120 7. 15 m below sea level 8. Water level decreases by 6 m. 9. (a) Leong will not receive a prize because he answered 13 questions correctly, 2 questions incorrectly and 1 question not answered; The total marks that he obtained is 34. (b) John will likely receive a prize. If he answered 13 questions correctly and 3 questions not answered, so the total marks obtained is 36 marks. If he answered 13 questions correctly, 1 question incorrectly and 2 questions not answered, then the total marks obtained is 35 marks. Formative Practice 1.3 1. (a) 4 7 – – 2 7 – – 1 7 – 3 7 – (b) 1 3 –1– 2 3 – – 0 1 3 – (c) 1 2 – – 1 3 – – 1 6 – 1 3 – (d) 3 8 – – 1 4 – – 1 8 – 1 2 – 2. (a) 4 5 (b) –1 1 2 (c) –2 3 5 (d) –2 1 9 3. (a) (i) – 1 2 , – 3 8 , – 1 8 , 1 2 , 5 8 (ii) 5 8 , 1 2 , – 1 8 , – 3 8 , – 1 2 (b) (i) – 7 10, – 2 5 , 1 10, 1 2 , 3 5 (ii) 3 5 , 1 2 , 1 10, – 2 5 , – 7 10 (c) (i) –1 1 3 , –1 1 6 , – 5 6 , 4 9 , 7 9 (ii) 7 9 , 4 9 , – 5 6 , –1 1 6 , –1 1 3 (d) (i) –1 7 8 , –1 1 2 , – 5 8 , 1 1 4 , 3 2 (ii) 3 2 , 1 1 4 , – 5 8 , –1 1 2 , –1 7 8 4. (a) –1 3 8 (b) 3 (c) –1 1 18 (d) – 3 4 (e) – 6 7 (f) 1 7 9 (g) –1 1 3 (h) –4 1 8 (i) 4 15 5. 2 250 m below sea level 6. Not enough. Still short of 7 10 kg of sugar. Formative Practice 1.4 1. (a) –0.4 –0.2 –0.1 0.2 0.3 (b) –5.2 –3.9 –1.3 2.6 3.9 2. (a) 2.9 (b) –5.6 (c) –1.11 3. (a) (i) –6.7, –3.31, –1.4, 3.87, 4.5 (ii) 4.5, 3.87, –1.4, –3.31, –6.7 (b) (i) –5.2, –3.0, –0.4, 0.9, 1.4 (ii) 1.4, 0.9, –0.4, –3.0, –5.2 (c) (i) –4.11, –3.22, –1.44, 1.55, 2.33 (ii) 2.33, 1.55, –1.44, –3.22, –4.11 (d) (i) –5.44, –5.42, –2.9, 0.03, 0.3 (ii) 0.3, 0.03, –2.9, –5.42, –5.44 4. (a) –1.66 (b) –1.2 (c) –9.35 (d) 8.985 (e) 2 (f) –6.665 (g) –19.35 (h) –5.647 (i) –3.072 5. –4°C 6. Danny gained a profit of RM185. TIMSS ‒; +; ‒ Formative Practice 1.5 1. All the numbers given are rational numbers. –8 = –8 1 ; 3.15 = 63 20 ; 1 1 5 = 6 5 2. (a) –7.2 (b) 2.58 (c) –0.3125 (d) –3.905 3. (a) –9 3 4 (b) –1 1 2 FORM 1 1 Rational Numbers Chapter Formative Practice 1.1 1. Positive numbers: 2 5 , 7.3, 5, +3 Negative numbers: –10, –67, – 3 4 , –8.9 (Zero is not a positive number and not a negative number) 2. (a) –7 levels (b) +500 m (c) –RM450 (d) +RM0.50 (e) –150 m (f) –3 cm 3. (a) –100 (b) +38 or 38 (c) –45 4. Positive integer: 4; Negative integer: –8, –3 5. (a) –3 –2 1 2 4 (b) –15 –9 –3 3 6 6. (a) 8 (b) –15 7. (a) Largest integer: 6; Smallest integer: –10 (b) Largest integer: 14; Smallest integer: –16 8. (a) –6, –5, –4, –2, 1, 3 (b) –10, –8, –6, 3, 7, 9 (c) –16, –14, –3, 0, 11, 18 (d) –19, –13, –4, –3, 4, 19 9. (a) 5, 3, 0, –1, –4, –6 (b) 7, 5, 3, –1, –2, –6 (c) 4, 2, 1, –1, –3, –7 (d) –1, –3, –5, –8, –10, –12 Formative Practice 1.2 1. (a) 5 (b) 3 (c) –1 (d) –14 (e) –3 (f) 11 (g) –10 (h) –4 2. (a) –16 (b) –20 (c) 42 (d) 36 3. (a) –15 (b) –7 (c) 7 (d) 12 4. (a) 0 (b) 1 (c) 12 (d) 27 5. (a) 10 (b) 0 (c) 40 (d) 17 (e) 15 (f) –28 (g) –6 (h) –4 ANSWERS

Purchase eBook here! MATHEMATICS Dual Language Programme UASA 1∙2∙3 KSSM FORM PELANGI ISBN: 978-629-470-420-6 W.M: RM37.95 / E.M: RM38.95 CC037234 • Matematik • Sains • Sejarah • Mathematics • Science • Geografi TITLES IN THIS SERIES • Comprehensive Notes • Example and Solution • Tips • Common Mistakes REVISION • Formative Practices • UASA Practices • Form 1.2.3 UPSA & UASA Model Papers • Answers REINFORCEMENT & ASSESSMENT CODE QR • Maths Info • HOTS Challenge • Daily Application • TIMSS Challenge • Digital Resources EXTRA FEATURES CODE QR FOCUS KSSM Form 1.2.3 – a complete and precise series of reference books with special features to enhance students’ learning as a whole. This series covers the latest Kurikulum Standard Sekolah Menengah (KSSM) and integrates Ujian Akhir Sesi Akademik (UASA) requirements. A great resource for every student indeed!