preface It is hoped that this book will help students to gain confidence in the subject and be better equipped to face the examinations.

Transcription

1 preface Secondary 1 Mathematics Tutorial 1A and 1B are designed to prepare Secondary 1 students in their understanding and application of mathematical concepts, skills and processes. What s covered in this book? Written in accordance with the latest syllabus, each chapter includes Objectives, Key Concepts and Formulae and Worked Examples to supplement and complement the lessons taught in school. Practice questions are structured as core, consolidation and challenging to ensure steady improvement and quick mastery of concepts. Books 1A and 1B cover all the topics for the entire school year. Additional feature that provides important notes, tips, examples and common student errors. Important concepts are highlighted to enhance understanding and retention. Tests and Assessments There are 4 assessment papers: Mid Year Examination Paper 1 and (found in Book 1A) and End of Year Examination Paper 1 and (found in Book 1B). Answer Key Fully worked solutions are provided for students to understand better how each problem is solved. These also serve as a tool for self study and assessment. It is hoped that this book will help students to gain confidence in the subject and be better equipped to face the examinations. The Editorial Team

2 Contents Chapter Objectives Page 1 Prime Numbers, Factors and Multiples Recognise prime numbers Perform prime factorisation on a composite number Find the HCF and LCM of a group of numbers using prime factorisation Find the square root and cube root of a number using prime factorisation Real Numbers and Approximation Identify rational and irrational numbers Perform the four operations on real numbers Approximate using decimal places and significant figures Estimate the results of computations 1 Introduction to Algebra Evaluate algebraic expressions and formulae Express real-world problems in algebraic terms 44 4 Algebraic Manipulation Evaluate and simplify algebraic expressions Factorise algebraic expressions by extracting common factors Express real world problems in algebraic terms 61 Solving Linear Equations and Inequalities Solve linear equations Formulate linear equations to solve word problems Number Patterns Recognise number patterns and find the terms of a sequence Find the general term of a sequence Solve problems involving number patterns and sequences Ratio, Rate and Speed Find ratios involving two or more quantities Calculate average speed Convert speed from one unit to another Solve problems involving ratio, rate and speed Mid Year Examination Paper 1 16 Mid Year Examination Paper 16 Fully worked solutions S1 S8 Complete the course with Secondary 1 Mathematics Tutorial 1B (Chapters 8 14)

3 Chapter 1 Prime Numbers, Factors and Multiples Objectives Recognise prime numbers Perform prime factorisation on a composite number Find the HCF and LCM of a group of numbers using prime factorisation Find the square root and cube root of a number using prime factorisation Key Concepts and Formulae 1 Whole numbers E.g. 0, 1,,, 4,,... Neither Prime nor Composite (0 and 1 only) Prime Numbers has only factors, 1 and itself E.g.,,, 7, 11,... Composite Prime Numbers has more than factors E.g. 4, 6, 8, 9, 10 when expressed in index notation is. Prime Factorisation is the process of expressing a composite number as a product of its prime factors. E.g. 04 = 7 4 Square and Square roots: Square of 6 = 6 = 6 6 = 6 A perfect square is a number whose square root is a whole number. E.g. 1, 4, 9, 16,, 6 are perfect squares. Cube and Cube roots: Cube of = = 1 1 = A perfect cube is a number whose cube root is a whole number. E.g. 1, 8, 7, 64, 1 are perfect cubes. 6 Highest Common Factor (HCF) and Lowest Common Multiple (LCM) can be found using Prime Factorisation.

4 WORKeD example 1 Find the prime factorisation of 10 in index notation. Solution: Method 1: Using the factor tree: 10 7 Hence, the prime factorisation of 10 is. Method : Using successive division: Only use prime numbers when finding the factors Hence, the prime factorisation of 10 is.

5 WORKeD example Find the HCF and LCM of the numbers 40, 60 and 100. Solution: Method 1: Using prime factorisation: 40 = 60 = In Method 1, you will need to find the prime factorisation of each number first. 100 = HCF = Extract common factors with lowest index. LCM = Extract common factors with highest index and all remaining factors. HCF = = 0 LCM = = 600 Method : Using successive division: Common prime factors 40, 60, 100 0, 0, 0 10, 1,,, Stop dividing when there are no common factors between any two numbers Use only prime numbers when performing prime factorisation. HCF = = 0 LCM = = 600

6 WORKeD example Buses on different routes start their journey from the interchange at regular intervals. Buses for route A leave the interchange every 18 minutes, buses for route B leave every 0 minutes and buses for route C leave every 4 minutes. Given that all the first buses for all routes leave at 7 a.m., what time will they next leave together? Solution: Successive division is a more efficient way to find LCM/HCF. Find the LCM of 18, 0 and 4. 18, 0, 4 9, 10, 1 9,, 6 9,,,, 1 1,, 1 1, 1, 1 LCM = = minutes = hours The buses will next leave together at 10 a.m. WORKeD example 4 Using prime factorisation, find the (a) square root of 784, (b) cube root of 16. Solution: a a = a b b b = b (a) Using prime factorisation, 784 = = ( 7) ( 7) = 7 = 8 (b) Using prime factorisation, 16 = 16 = ( ) ( ) ( ) = = 6 4

7 Prime Factorisation core Practice 1 Express the following numbers in index notation. index = Base To find the prime factorisation of a number: Method 1: Factor tree Method : Successive division (a) 7 (b) (c) (d) 7 (e) (f) (g) (h) (i) (j) 19 19

8 Find the prime factorisation of the following numbers. A number is divisible by if the sum of its digits is divisible by. A number is divisible by if its last digit is 0 or. (a) 00 (b) 94 (c) 7 (d) 1000 (e) 10 (f) 17 6