Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 32 Curriculum Development Division Ministry of Education Malaysia 1. Calculate 27 25 5 9 2. Calculate – 20 14 7 3 12 45 3. Calculate 4 11 2 4. Calculate 5 1 4 3 1 5. Simplify k m 3 6. Simplify (5 ) 2 m n 7. Simplify 14 3 6 1 1 x 8. Simplify (2 3 ) 2 a d n 9. Simplify x y 10 9 5 3 2 10. Simplify x x 1 20 4 TEST YOURSELF B1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 33 Curriculum Development Division Ministry of Education Malaysia 2.0 Division of Fractions Consider the following: 6 3 First, let’s assume this circle as 1 whole unit. Therefore, the above division can be represented visually as follows: This means that 6 units are being divided into a group of 3 units, or mathematically can be written as: 6 3 2 The above division can also be interpreted as ‘how many 3’s can fit into 6’. The answer is ‘2 groups of 3 units can fit into 6 units’. Consider now a division of a fraction by a fraction like this: . 8 1 2 1 LESSON NOTES How many 8 1 is in ? 2 1 6 units are being divided into a group of 3 units: 6 3 2
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 34 Curriculum Development Division Ministry of Education Malaysia This means ‘How many is in ? 8 1 2 1 The answer is 4: Consider now this division: . 4 1 4 3 This means ‘How many is in ? 4 1 4 3 The answer is 3: But, how do you calculate the answer? How many 4 1 is in ? 4 3
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 35 Curriculum Development Division Ministry of Education Malaysia Consider again 63 2. Actually, the above division can be written as follows: 3 1 6 3 6 6 3 Notice that we can write the division in the multiplication form. But here, we have to change the second number to its reciprocal. Therefore, if we have a division of fraction by a fraction, we can do the same, that is, we have to change the second fraction to its reciprocal and then multiply the fractions. Therefore, in our earlier examples, we can have: 4 2 8 1 8 2 1 8 1 2 1 (i) The reciprocal of a fraction is found by inverting the fraction. Change the second fraction to its reciprocal and change the sign to . The reciprocal of 8 1 is . 1 8 These operations are the same! The reciprocal of 3 is . 3 1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 36 Curriculum Development Division Ministry of Education Malaysia 3 1 4 4 3 4 1 4 3 (ii) The steps to divide fractions can therefore be summarized as follows: 2.1 Division of Simple Fractions Example: 7 3 5 2 = 3 7 5 2 = 15 14 Change the second fraction to its reciprocal and change the sign to . Multiply the two numerators together and the two denominators together. Steps to Divide Fractions: 1. Change the second fraction to its reciprocal and change the sign to . 2. Multiply the numerators together and multiply the denominators together. 3. Simplify the fraction (if needed). Tips: (+) (+) = + (+) (–) = – (–) (+) = – (–) (–) = + Change the second fraction to its reciprocal and change the sign to . The reciprocal of 4 1 is . 1 4
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 37 Curriculum Development Division Ministry of Education Malaysia 2.2 Division of Fractions With Common Factors Examples: 9 2 21 10 = 2 9 21 10 = 2 9 21 10 = 7 15 = 7 1 2 7 6 5 3 6 7 5 3 10 7 7 6 5 3 1 5 3 7 1 2 Express the fraction in division form. Change the second fraction to its reciprocal and change the sign to . Simplify by canceling out the common factors. Change the fraction back to a mixed number. Change the second fraction to its reciprocal and change the sign to . Then, simplify by canceling out the common factors. Multiply the two numerators together and the two denominators together. Remember: (+) (–) = (–) Multiply the two numerators together and the two denominators together.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 38 Curriculum Development Division Ministry of Education Malaysia 1. Find 6 25 12 35 Solution : 6 25 12 35 = 25 6 12 35 = 10 7 2. Simplify – 4 2 5x x Solution : – x 5x 2 4 = – 2 5 8 x 3. Simplify 2 x y Solution : 2 x y 2 1 x y x y 2 5 7 Change the second fraction to its reciprocal and change the sign to . Then, simplify by canceling out the common factors. Method I EXAMPLES Change the second fraction to its reciprocal and change the sign to . Multiply the two numerators together and the two denominators together. Express the fraction in division form. Change the second fraction to its reciprocal and change to . Multiply the two numerators together and the two denominators together. Remember: (+) (–) = (–) Multiply the two numerators together and the two denominators together. 2 1
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 39 Curriculum Development Division Ministry of Education Malaysia Multiply the numerator and the denominator of the given fraction with x 2 x y = 2 x y x x = x x x y 2 = x y 2 4. Simplify 5 (1 ) 1 r Solution: 5 (1 ) 1 r = 5 ) 1 (1 r r r = r r 5 1 The given fraction. r is the denominator of r 1 . Multiply the given fraction with r r . Note that: ) 1 1 (1 r r r Method II The numerator is also a fraction with denominator x Multiply the numerator and the denominator of the given fraction by x.
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 40 Curriculum Development Division Ministry of Education Malaysia 1. Calculate 2 21 7 3 2. Calculate 16 5 8 7 9 5 3. Simplify 3 8 4y y 4. Simplify k 2 16 5. Simplify 3 5 2 x 6. Simplify n m n m 3 4 2 2 7. Simplify 8 1 4 y 8. Simplify x x 1 1 TEST YOURSELF B2
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 41 Curriculum Development Division Ministry of Education Malaysia 9. Calculate 5 3(1 ) 4 1 10. Simplify y x 1 5 11. Simplify 3 2 9 4 x 1 12. Simplify 1 5 1 1 p
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 42 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF A: 1. 7 3 2. 2 1 3. 14 5 4. 4 1 5. 35 38 or 35 3 1 6. 14 3 7. 13 67 or 13 2 5 8. 45 73 or 45 28 1 9. s 3 10. w 6 11. 2a 5 12. 3 f 1 13. ab 2b 4a 14. pq q 5 p 15. m n 16. 2 3p 3 17. 10 16x 17y 18. x 2x 1 19. ( 1) 1 x x 20. 2 21. 2 8x y 22. 2 9 7 4 n n 23. r r 3 1 2 24. 2 2 2 6 p p 25. 2 2 10 7 4 6 n n n 26. m 1 m 27. n n 5 5 28. n n 3 3 29. 2 8 10 n n 30. m p 3 4 3 ANSWERS
Basic Essential Additional Mathematics Skills (BEAMS) Module UNIT 2: Fractions 43 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF B1: 1. 3 2 1 3 5 or 2. 8 1 1 8 9 or 3. 2 1 5 2 11 or 4. 5 2 1 5 7 or 5. k 3m 6. 2 5mn 7. 4 x 8. na nd 2 3 9. x y 5 3 3 10 10. 4 1 5x TEST YOURSELF B2: 1. 49 2 2. 9 5 1 9 14 or 3. 2 6 y 4. 8k 5. 5 x 6 6. m 6 7. 2( 1) 1 y 8. 1 2 x x 9. 20 9 10. xy 5x 1 11. 6 13x 12. 4 p 5
Unit 1: Negative Numbers UNIT 3 ALGEBRAIC EXPRESSIONS AND ALGEBRAIC FORMULAE 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 Curriculum Development Division Ministry of Education Malaysia
TABLE OF CONTENTS Module Overview 1 Part A: Performing Operations on Algebraic Expressions 2 Part B: Expansion of Algebraic Expressions 10 Part C: Factorisation of Algebraic Expressions and Quadratic Expressions 15 Part D: Changing the Subject of a Formula 23 Activities Crossword Puzzle 31 Riddles 33 Further Exploration 37 Answers 38
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 1 Curriculum Development Division Ministry of Education Malaysia MODULE OVERVIEW 1. The aim of this module is to reinforce pupils’ understanding of the concepts and skills in Algebraic Expressions, Quadratic Expressions and Algebraic Formulae. 2. The concepts and skills in Algebraic Expressions, Quadratic Expressions and Algebraic Formulae are required in almost every topic in Additional Mathematics, especially when dealing with solving simultaneous equations, simplifying expressions, factorising and changing the subject of a formula. 3. It is hoped that this module will provide a solid foundation for studies of Additional Mathematics topics such as: Functions Quadratic Equations and Quadratic Functions Simultaneous Equations Indices and Logarithms Progressions Differentiation Integration 4. This module consists of four parts and each part deals with specific skills. This format provides the teacher with the freedom to choose any parts that is relevant to the skills to be reinforced.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 2 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES Pupils who face problem in performing operations on algebraic expressions might have difficulties learning the following topics: Simultaneous Equations - Pupils need to be skilful in simplifying the algebraic expressions in order to solve two simultaneous equations. Functions - Simplifying algebraic expressions is essential in finding composite functions. Coordinate Geometry - When finding the equation of locus which involves distance formula, the techniques of simplifying algebraic expressions are required. Differentiation - While performing differentiation of polynomial functions, skills in simplifying algebraic expressions are needed. Strategy: 1. Teacher reinforces the related terminologies such as: unknowns, algebraic terms, like terms, unlike terms, algebraic expressions, etc. 2. Teacher explains and shows examples of algebraic expressions such as: 8k, 3p + 2, 4x – (2y + 3xy) 3. Referring to the “Lesson Notes” and “Examples” given, teacher explains how to perform addition, subtraction, multiplication and division on algebraic expressions. 4. Teacher emphasises on the rules of simplifying algebraic expressions. PART A: PERFORMING OPERATIONS ON ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Upon completion of Part A, pupils will be able to perform operations on algebraic expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 3 Curriculum Development Division Ministry of Education Malaysia PART A: PERFORMING BASIC ARITHMETIC OPERATIONS ON ALGEBRAIC EXPRESSIONS 1. An algebraic expression is a mathematical term or a sum or difference of mathematical terms that may use numbers, unknowns, or both. Examples of algebraic expressions: 2r, 3x + 2y, 6x 2 +7x + 10, 8c + 3a – n 2 , g 3 2. An unknown is a symbol that represents a number. We normally use letters such as n, t, or x for unknowns. 3. The basic unit of an algebraic expression is a term. In general, a term is either a number or a product of a number and one or more unknowns. The numerical part of the term, is known as the coefficient. Examples: Algebraic expression with one term: 2r, g 3 Algebraic expression with two terms: 3x + 2y, 6s – 7t Algebraic expression with three terms: 6x 2 +7x + 10, 8c + 3a – n 2 4. Like terms are terms with the same unknowns and the same powers. Examples: 3ab, –5ab are like terms. 3x 2 , 2 5 2 x are like terms. 5. Unlike terms are terms with different unknowns or different powers. Examples: 1.5m, 9k, 3xy, 2x 2 y are all unlike terms. LESSON NOTES 6 xy Coefficient Unknowns
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 4 Curriculum Development Division Ministry of Education Malaysia 6. An algebraic expression with like terms can be simplified by adding or subtracting the coefficients of the unknown in algebraic terms. 7. To simplify an algebraic expression with like terms and unlike terms, group the like terms first, and then simplify them. 8. An algebraic expression with unlike terms cannot be simplified. 9. Algebraic fractions are fractions involving algebraic terms or expressions. Examples: . 2 , 2 4 , 6 2 , 15 3 2 2 2 2 2 2 x xy y x y rg g r g h m 10. To simplify an algebraic fraction, identify the common factor of both the numerator and the denominator. Then, simplify it by elimination.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 5 Curriculum Development Division Ministry of Education Malaysia Simplify the following algebraic expressions and algebraic fractions: (a) 5x – (3x – 4x) 4 6 (e) s t (b) –3r –9s + 6r + 7s z x y 2 3 6 5 (f) (c) 2 2 2 4 rg g r g g f e (g) 2 p q 3 4 (d) (h) x x 3 2 1 3 Solutions: (a) 5x – (3x – 4x) = 5x – (– x) = 5x + x = 6x (b) –3r –9s + 6r + 7s = –3r + 6r –9s + 7s = 3r – 2s 2 2 2 4 (c) rg g r g r g r g r g r g 2 4 (2 ) 4 2 2 Perform the operation in the bracket. Arrange the algebraic terms according to the like terms. . Unlike terms cannot be simplified. Leave the answer in the simplest form as shown. Algebraic expression with like terms can be simplified by adding or subtracting the coefficients of the unknown. Simplify by canceling out the common factor and the same unknowns in both the numerator and the denominator. 1 1 EXAMPLES
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 6 Curriculum Development Division Ministry of Education Malaysia pq q p pq p pq q p q 3 4 3 4 3 4 (d) 12 3 2 6 2 2 4 3 3 4 6 (e) s t s t s t z xy z x y z x y 4 5 2 2 5 2 3 6 5 (f) f g e f g e g f e 2 2 1 (g) 2 x x x x x x x x x x 6 6 1 3 1 2 6 1 3 2 6 1 3 2 1 2 3 (2) 3 2 1 3 (h) The LCM of p and q is pq. The LCM of 4 and 6 is 12. Simplify by canceling out the common factor, then multiply the numerators together and followed by the denominators. Change division to multiplication of the reciprocal of 2g. Equate the denominator. 2 1
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 7 Curriculum Development Division Ministry of Education Malaysia ALTERNATIVE METHOD Simplify the following algebraic fractions: (a) x x 3 2 1 3 = x x 3 2 1 3 2 2 = 3 (2) (2) 2 1 3 (2) x x = x x 6 6 1 (b) 5 2 3 x = 5 2 3 x x x x x x x x x 5 3 2 5( ) ( ) 2( ) 3 x x x x x x x x x x 4 16 3 2(2 ) (2 ) 2 3 8(2 ) 2 2 2 2 3 8 2 2 3 8 (c) The denominator of 2x 3 is 2x. Therefore, multiply the algebraic fraction by x x 2 2 . Each of the terms in the numerator and denominator is multiplied by 2x. . The denominator of is 2 2 1 . Therefore, multiply the algebraic fraction by 2 2 . Each of the terms in the numerator and denominator of the algebraic fraction is multiplied by 2. The denominator of x 3 is x. Therefore, multiply the algebraic fraction by x x . Each of the terms in the numerator and denominator is multiplied by x.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 8 Curriculum Development Division Ministry of Education Malaysia x x x x x 36 21 8 28 21 (7) 4(7) 7 8 3(7) 7 7 4 7 8 3 4 7 8 3 (d) The denominator of 7 8 x is 7. Therefore, multiply the algebraic fraction by 7 7 . Each of the terms in the numerator and denominator is multiplied by 7. Simplify the denominator.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 9 Curriculum Development Division Ministry of Education Malaysia Simplify the following algebraic expressions: 1. 2a –3b + 7a – 2b 2. − 4m + 5n + 2m – 9n 3. 8k – ( 4k – 2k ) 4. 6p – ( 8p – 4p ) y 5x 3 1 5. 5 2 3 4 6. h k c a b 2 3 7 4 7. c d c d 3 8 2 4 8. yz z xy 9. w uv vw u 2 10. 6 5 2 11. x 5 4 2 4 12. x x TEST YOURSELF A
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 10 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES Pupils who face problem in expanding algebraic expressions might have difficulties in learning of the following topics: Simultaneous Equations – pupils need to be skilful in expanding the algebraic expressions in order to solve two simultaneous equations. Functions – Expanding algebraic expressions is essential when finding composite function. Coordinate Geometry – when finding the equation of locus which involves distance formula, the techniques of expansion are applied. Strategy: Pupils must revise the basic skills involving expanding algebraic expressions. PART B: EXPANSION OF ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVE Upon completion of Part B, pupils will be able to expand algebraic expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 11 Curriculum Development Division Ministry of Education Malaysia PART B: EXPANSION OF ALGEBRAIC EXPRESSIONS 1. Expansion is the result of multiplying an algebraic expression by a term or another algebraic expression. 2. An algebraic expression in a single bracket is expanded by multiplying each term in the bracket with another term outside the bracket. 3(2b – 6c – 3) = 6b – 18c – 9 3. Algebraic expressions involving two brackets can be expanded by multiplying each term of algebraic expression in the first bracket with every term in the second bracket. (2a + 3b)(6a – 5b) = 12a 2 – 10ab + 18ab – 15b 2 = 12a 2 + 8ab – 15b 2 4. Useful expansion tips: (i) (a + b) 2 = a 2 + 2ab + b 2 (ii) (a – b) 2 = a 2 – 2ab + b 2 (iii) (a – b)(a + b) = (a + b)(a – b) = a 2 – b 2 LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 12 Curriculum Development Division Ministry of Education Malaysia Expand each of the following algebraic expressions: (a) 2(x + 3y) (b) – 3a (6b + 5 – 4c) Solutions: (a) 2 (x + 3y) = 2x + 6y (b) –3a (6b + 5 – 4c) = –18ab – 15a + 12ac 9 12 3 2 (c) y = 12 3 2 9 3 2 y = 6y + 8 = (a + 3) (a + 3) = a 2 + 3a + 3a + 9 = a 2 + 6a + 9 When expanding two brackets, each term within the first bracket is multiplied by every term within the second bracket. 9 12 3 2 (c) y 2 (e) 3 2k 5 2 (d) (a 3) (f)( p 2)( p 5) 2 (d) (a 3) When expanding a bracket, each term within the bracket is multiplied by the term outside the bracket. When expanding a bracket, each term within the bracket is multiplied by the term outside the bracket. 1 3 1 4 EXAMPLES Simplify by canceling out the common factor, then multiply the numerators together and followed by the denominators.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 13 Curriculum Development Division Ministry of Education Malaysia (c) (4x – 3y)(6x – 5y) – 18 xy – 20 xy – 38 xy = 24x 2 – 38 xy + 15y 2 2 (e) 3 2k 5 = –3(2k + 5) (2k + 5) = –3(4k 2 + 20k + 25) = –12k 2 – 60k – 75 (f) ( p 2)(q 5) = pq – 5p + 2q – 10 ALTERNATIVE METHOD Expanding two brackets (a) (a + 3) (a + 3) = a 2 + 3a + 3a + 9 = a 2 + 6a + 9 (b) (2p + 3q) (6p – 5q) = 12p 2 – 10 pq + 18 pq – 15q 2 = 12p 2 + 8 pq – 15q 2 When expanding two brackets, each term within the first bracket is multiplied by every term within the second bracket. When expanding two brackets, write down the product of expansion and then, simplify the like terms. When expanding two brackets, each term within the first bracket is multiplied by every term within the second bracket.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 14 Curriculum Development Division Ministry of Education Malaysia Simplify the following expressions and give your answers in the simplest form. 4 3 1. 4 2n 6 1 2 1 2. q 3. 6x2x 3y 4. 2a b 2(a b) 5. 2( p 3) ( p 6) 3 2 6 3 1 6. y x y x 7. 1 2 1 2 e e 8. m n m2m n 2 9. f gf g g2 f g 10. h ih i 2ih 3i TEST YOURSELF B
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 15 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES Some pupils may face problem in factorising the algebraic expressions. For example, in the Differentiation topic which involves differentiation using the combination of Product Rule and Chain Rule or the combination of Quotient Rule and Chain Rule, pupils need to simplify the answers using factorisation. Examples: 2 2 2 2 3 3 2 3 3 3 4 2 3 4 (7 2 ) (3 ) (4 15) (7 2 ) (7 2 )[ 3(3 ) ] (3 ) ( 2) 7 2 (3 ) 2. 2 (7 5) (49 15) 2 [28(7 5) ] (7 5) (6 ) 1. 2 (7 5) x x x x x x x dx dy x x y x x x x x x x dx dy y x x Strategy 1. Pupils revise the techniques of factorisation. PART C: FACTORISATION OF ALGEBRAIC EXPRESSIONS AND QUADRATIC EXPRESSIONS LEARNING OBJECTIVE Upon completion of Part C, pupils will be able to factorise algebraic expressions and quadratic expressions.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 16 Curriculum Development Division Ministry of Education Malaysia PART C: FACTORISATION OF ALGEBRAIC EXPRESSIONS AND QUADRATIC EXPRESSIONS 1. Factorisation is the process of finding the factors of the terms in an algebraic expression. It is the reverse process of expansion. 2. Here are the methods used to factorise algebraic expressions: (i) Express an algebraic expression as a product of the Highest Common Factor (HCF) of its terms and another algebraic expression. ab – bc = b(a – c) (ii) Express an algebraic expression with three algebraic terms as a complete square of two algebraic terms. a 2 + 2ab + b 2 = (a + b) 2 a 2 – 2ab + b 2 = (a – b) 2 (iii) Express an algebraic expression with four algebraic terms as a product of two algebraic expressions. ab + ac + bd + cd = a(b + c) + d(b + c) = (a + d)(b + c) (iv) Express an algebraic expression in the form of difference of two squares as a product of two algebraic expressions. a 2 – b 2 = (a + b)(a – b) 3. Quadratic expressions are expressions which fulfill the following characteristics: (i) have only one unknown; and (ii) the highest power of the unknown is 2. 4. Quadratic expressions can be factorised using the methods in 2(i) and 2(ii). 5. The Cross Method can be used to factorise algebraic expression in the general form of ax 2 + bx + c, where a, b, c are constants and a ≠ 0, b ≠ 0, c ≠ 0. LESSON NOTES
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 17 Curriculum Development Division Ministry of Education Malaysia (a) Factorising the Common Factors i) mn + m = m (n +1) ii) 3mp + pq = p (3m + q) iii) 2mn – 6n = 2n (m – 3) (b) Factorising Algebraic Expressions with Four Terms i) vy + wy + vz + wz = y (v + w) + z (v + w) = (v + w)(y + z) ii) 21bm – 7bs + 6cm – 2cs = 7b(3m – s) + 2c(3m – s) = (3m – s)(7b + 2c) Factorise the first and the second terms with the common factor y, then factorise the third and fourth terms with the common factor z. . (v + w) is the common factor. Factorise the first and the second terms with common factor 7b, then factorise the third and fourth terms with common factor 2c. (3m – s) is the common factor. EXAMPLES Factorise the common factor m. . Factorise the common factor p. . Factorise the common factor 2n. .
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 18 Curriculum Development Division Ministry of Education Malaysia (c) Factorising the Algebraic Expressions by Using Difference of Two Squares i) x 2 – 16 = x 2 – 4 2 = (x + 4)(x – 4) ii) 4x 2 – 25 = (2x) 2 – 5 2 = (2x + 5)(2x – 5) (d) Factorising the Expressions by Using the Cross Method i) x 2 – 5x + 6 x x x x x 3 2 5 2 3 x 2 – 5x + 6 = (x – 3) (x – 2) ii) 3x 2 + 4x – 4 x x x x x 2 6 4 2 3 2 3x 2 + 4x – 4 = (3x – 2) (x + 2) The summation of the cross multiplication products should equal to the middle term of the quadratic expression in the general form. The summation of the cross multiplication products should equal to the middle term of the quadratic expression in the general form. a 2 – b 2 = (a + b)(a – b)
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 19 Curriculum Development Division Ministry of Education Malaysia ALTERNATIVE METHOD Factorise the following quadratic expressions: i) x 2 – 5x + 6 ac b + 6 – 5 –2 –3 (x – 2) (x – 3) 5 6 ( 2)( 3) 2 x x x x ii) x 2 – 5x – 6 ac b – 6 – 5 +1 – 6 (x + 1) (x– 6) 5 6 ( 1)( 6) 2 x x x x +1 (–6) = –6 +1 (–6) = –6 +1 – 6 = –5 a=+1 b= –5 c = –6 REMEMBER!!! An algebraic expression can be represented in the general form of ax 2 + bx + c, where a, b, c are constants and a ≠ 0, b ≠ 0, c ≠ 0. +1 (+ 6) = + 6 –2 (–3) = +6 –2 + (–3) = –5 a=+1 b= –5 c =+6
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 20 Curriculum Development Division Ministry of Education Malaysia TEST YOURSELF C (iii) 2x 2 – 11x + 5 ac b + 10 –11 –1 – 10 2 10 2 1 5 2 1 (2x – 1) (x – 5) 2 11 5 (2 1)( 5) 2 x x x x (iv) 3x 2 + 4x – 4 ac b – 12 + 4 – 2 +6 2 3 2 3 6 3 2 The coefficient of x 2 is 2, divide each number by 2. (+2) (+5) = +10 –1 (–10) = +10 –1 + (–10) = –11 –2 + 6 = 4 The coefficient of x 2 is 3, divide each number by 3. 3 (– 4) = –12 a=+2 b = –11 c =+5 a =+ 3 b=+ 4 c = –4 (3x – 2) (x + 2) The coefficient of x 2 is 2, multiply by 2: 2 1)( 5 2 5 5 2 1 2 1 x x x x x x The coefficient of x 2 is 3, multiply by 3: 3 2)( 2 3 2 2 3 2 3 2 x x x x x x 3 4 4 (3 2)( 2) 2 x x x x
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 21 Curriculum Development Division Ministry of Education Malaysia Factorise the following quadratic expressions completely. 1. 3p 2 – 15 2. 2x 2 – 6 3. x 2 – 4x 4. 5m 2 + 12m 5. pq – 2p 6. 7m + 14mn 7. k 2 –144 8. 4p 2 – 1 9. 2x 2 – 18 10. 9m 2 – 169 TEST YOURSELF C
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 22 Curriculum Development Division Ministry of Education Malaysia 11. 2x 2 + x – 10 12. 3x 2 + 2x – 8 13. 3p 2 – 5p – 12 14. 4p2 – 3p – 1 15. 2x 2 – 3x – 5 16. 4x 2 – 12x + 5 17. 5p 2 + p – 6 18. 2x 2 – 11x + 12 19. 3p + k + 9pr + 3kr 20. 4c 2 – 2ct – 6cw + 3tw
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 23 Curriculum Development Division Ministry of Education Malaysia TEACHING AND LEARNING STRATEGIES If pupils have difficulties in changing the subject of a formula, they probably face problems in the following topics: Functions – Changing the subject of the formula is essential in finding the inverse function. Circular Measure – Changing the subject of the formula is needed to find the r or from the formulae s = r or 2 2 1 A r . Simultaneous Equations – Changing the subject of the formula is the first step of solving simultaneous equations. Strategy: 1. Teacher gives examples of formulae and asks pupils to indicate the subject of each of the formula. Examples: y = x – 2 V r h A bh 2 2 1 y, A and V are the subjects of the formulae. PART D: CHANGING THE SUBJECT OF A FORMULA LEARNING OBJECTIVE Upon completion of this module, pupils will be able to change the subject of a formula.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 24 Curriculum Development Division Ministry of Education Malaysia PART D: CHANGING THE SUBJECT OF A FORMULA 1. An algebraic formula is an equation which connects a few unknowns with an equal sign. Examples: V r h A bh 2 2 1 2. The subject of a formula is a single unknown with a power of one and a coefficient of one, expressed in terms of other unknowns. Examples: A bh 2 1 a 2 = b 2 + c 2 T Tr h 2 2 1 3. A formula can be rearranged to change the subject of the formula. Here are the suggested steps that can be used to change the subject of the formula: (i) Fraction : Get rid of fraction by multiplying each term in the formula with the denominator of the fraction. (ii) Brackets : Expand the terms in the bracket. (iii) Group : Group all the like terms on the left or right side of the formula. (iv) Factorise : Factorise the terms with common factor. (v) Solve : Make the coefficient and the power of the subject equal to one. LESSON NOTES A is the subject of the formula because it is expressed in terms of other unknowns. a 2 is not the subject of the formula because the power ≠ 1 T is not the subject of the formula because it is found on both sides of the equation.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 25 Curriculum Development Division Ministry of Education Malaysia 1. Given that 2x + y = 2, express x in terms of y. Solution: 2x + y = 2 2x = 2 – y x = 2 2 y 2. Given that y x y 5 2 3 , express x in terms of y. Solution: y x y 5 2 3 3x + y = 10y 3x = 10y – y 3x = 9y x = 3 9y x = 3y No fraction and brackets. Group: Retain the x term on the left hand side of the equation by grouping all the y term to the right hand side of the equation. Fraction: Multiply both sides of the equation by 2. Group: Retain the x term on the left hand side of the equation by grouping all the y term to the right hand side of the equation. Solve: Divide both sides of the equation by 2 to make the coefficient of x equal to 1. Solve: Divide both sides of the equation by 3 to make the coefficient of x equal to 1. EXAMPLES Steps to Change the Subject of a Formula (i) Fraction (ii) Brackets (iii) Group (iv) Factorise (v) Solve
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 26 Curriculum Development Division Ministry of Education Malaysia 3. Given that x 2y , express x in terms of y. Solution: x 2 y x = (2y) 2 x = 4y 2 4. Given that p x 3 , express x in terms of p. Solution: p x 3 2 2 9 (3 ) 3 x p x p x p 5. Given that 3 x 2 x y , express x in terms of y. Solution: 2 2 2 2 2 2 2 3 2 3 2 y x y x x y x x y x x y Solve: Square both sides of the equation to make the power of x equal to 1. Fraction: Multiply both sides of the equation by 3. Solve: Square both sides of the equation to make the power of x equal to1. Group: Group the like terms Solve: Divide both sides of the equation by 2 to make the coefficient of x equal to 1. Solve: Square both sides of equation to make the power of x equal to 1. Simplify the terms.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 27 Curriculum Development Division Ministry of Education Malaysia 6. Given that 4 11x – 2(1 – y) = 2xp , express x in terms of y and p. Solution: 4 11x – 2 (1 – y) = 2xp 11x – 8(1 – y) = 8xp 11x – 8 + 8y = 8xp 11x – 8xp = 8 – 8y x(11 – 8p) = 8 – 8y x = p y 11 8 8 8 7. Given that n p x 5 2 3 = 1 – p , express p in terms of x and n. Solution: n p x 5 2 3 = 1 – p 2p – 3x = 5n – 5pn 2p + 5pn = 5n + 3x p(2 + 5n) = 5n + 3x p = n n x 2 5 5 3 Fraction: Multiply both sides of the equation by 4. Bracket: Expand the bracket. Group: Group the like terms. Factorise: Factorise the x term. Solve: Divide both sides by (11 – 8p) to make the coefficient of x equal to 1. Fraction: Multiply both sides of the equation by 5n. Solve: Divide both sides of the equation by (2 + 5n) to make the coefficient of p equal to 1. Group: Group the like p terms. Factorise: Factorise the p terms.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 28 Curriculum Development Division Ministry of Education Malaysia 1. Express x in terms of y. a) x y 2 0 b) 2x y 3 0 c) 2y x 1 d) 2 2 1 x y e) 3x y 5 f) 3y x 4 TEST YOURSELF D
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 29 Curriculum Development Division Ministry of Education Malaysia 2. Express x in terms of y. a) y x b) 2y x c) 3 2 x y d) y 1 3 x e) 3 x y x 1 f) x 1 y
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 30 Curriculum Development Division Ministry of Education Malaysia 3. Change the subject of the following formulae: a) Given that 2 x a x a , express x in terms of a . b) Given that x x y 1 1 , express x in terms of y . c) Given that f u v 1 1 1 , express u in terms of v and f . d) Given that 4 3 2 2 p q p q , express p in terms of q. e) Given that p 3m 2mn , express m in terms of n and p . f) Given that C C A B 1 , express C in terms of A and B . g) Given that y x y x 2 2 , express y in terms of x. h) Given that g l T 2 , express g in terms of T and l.
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 31 Curriculum Development Division Ministry of Education Malaysia CROSSWORD PUZZLE HORIZONTAL 1) – 4p, 10q and 7r are called algebraic . 3) An algebraic term is the of unknowns and numbers. 4) 4m and 8m are called terms. 5) V r h 2 , then V is the of the formula. 7) An can be represented by a letter. 10) 3 2 1 2 2 x x x x . ACTIVITIES
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 32 Curriculum Development Division Ministry of Education Malaysia VERTICAL 2) An algebraic consists of two or more algebraic terms combined by addition or subtraction or both. 6) 2 1 2 2 5 2 2 x x x x . 8) terms are terms with different unknowns. 9) The number attached in front of an unknown is called .
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 33 Curriculum Development Division Ministry of Education Malaysia RIDDLES RIDDLE 1 1. You are given 9 multiple-choice questions. 2. For each of the questions, choose the correct answer and fill the alphabet in the box below. 3. Rearrange the alphabets to form a word. 4. What is the word? 1 2 3 4 5 6 7 8 9 1. Calculate . 3 5 1 2 D) 5 1 O) 1 W) 3 11 N) 15 11 2. Simplify 3x 9y 6x 7y . F) 3x 2y W) 9x 16y E) 3x 2y X) 9x 2y 3. Simplify 3 2 p q . L) 6 2 p 3q A) 6 2 p 3q N) 6 3q 2 p R) 6 3p 2q
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 34 Curriculum Development Division Ministry of Education Malaysia 4. Expand 2(x 4) (x 7) . A) x 1 D) x 15 U) 3x 1 C) 3x 15 5. Expand 3a(2b 5c). S ) 6ab15ac C) 6ab15ac T) 6ab15ac R) 6ab15ac 6. Factorise 25 2 x . E) (x 5)(x 5) T) (x 5)(x 5) I) (x 5)(x 5) C) (x 25)(x 25) 7. Factorise pq 4q . D) pq(1 4q) E) q( p 4) T) p(q 4) S) q( p 4) 8. Factorise 8 12 2 x x . I ) (x 2)(x 6) W) (x 2)(x 6) F) (x 4)(x 3) C) (x 4)(x 3) 9. Given that 4 2 3 x x y , express x in terms of y. L) 5 y x C) 5 y x T) 11 y x N) 3 8 y x
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 35 Curriculum Development Division Ministry of Education Malaysia RIDDLE 2 1. You are given 9 multiple-choice questions. 2. For each of the questions, choose the correct answer and fill the alphabet in the box below. 3. Rearrange the alphabets to form a word. 4. What is the word? 1 2 3 4 5 6 7 8 9 1. Calculate . 3 1 5 x A) 3 5 x O) x x 3 5 I ) 5 3 x x N) 5 3 x 2. Simplify r p q 4 5 3 . F) q pr 4 15 R) pr q 15 4 W) r pq 20 3 B) r pq 5 3 3. Simplify z xy yz x 2 . N) 2 2 y D) 2 2 2z x L) 2 2z x I) 2 2 z x 4. Solve (3 ). 2 x y x x y E) x y xy 2 2 2 D) x y xy 2 2 2 I ) x y x xy 2 2 2 3 N) x y xy 2 2 2
Basic Essential Additional Mathematics Skills (BEAM) Module Unit 3: Algebraic Expressions and Algebraic Formulae 36 Curriculum Development Division Ministry of Education Malaysia 5. Expand 2 p 5 . I) 25 2 p N) 25 2 p D) 10 25 2 p p L) 10 25 2 p p 6. Factorise 2 7 15 2 y y . F) (2y 3)(y 5) D) (2y 3)(y 5) W) (2y 3)(y 5) L) (y 3)(2y 5) 7. Factorise 2 11 5 2 p p . R) (2 p 1)(p 5) B) (2 p 1)(p 5) F) ( p 1)(p 5) W) ( p 1)(2 p 5) 8. Given that C A C B ( 1) , express C in terms of A and B. L) B A B C R) B A C 1 C) B A AB C N) B A AB C 9. Given that 5 x y x 2 , express x in terms of y. O) 16 4 2 y x B) 24 4 2 y x I ) 2 2 1 y x U) 2 4 2 y x