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Algebraic pyramids 1 of 60 © Boardworks Ltd 2004 Algebraic magic square 2 of 60 © Boardworks Ltd 2004 Contents A1 Algebraic expressions A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.4 Dividing terms A1.5 Factorising expressions A1.6 Substitution 3 of 60 © Boardworks Ltd 2004 Multiplying terms together In algebra we usually leave out the multiplication sign ×. Any numbers must be written at the front and all letters should be written in alphabetical order. For example, 4 × a = 4a 1×b=b We don’t need to write a 1 in front of the letter. b × 5 = 5b We don’t write b5. 3 × d × c = 3cd We write letters in alphabetical order. 6 × e × e = 6e2 4 of 60 © Boardworks Ltd 2004 Using index notation Simplify: x + x + x + x + x = 5x Simplify: x × x × x × x × x = x5 x to the power of 5 This is called index notation. Similarly, x × x = x2 x × x × x = x3 x × x × x × x = x4 5 of 60 © Boardworks Ltd 2004 Using index notation We can use index notation to simplify expressions. For example, 3p × 2p = 3 × p × 2 × p = 6p2 q2 × q3 = q × q × q × q × q = q5 3r × r2 = 3 × r × r × r = 3r3 2t × 2t = (2t)2 6 of 60 or 4t2 © Boardworks Ltd 2004 Grid method for multiplying numbers 7 of 60 © Boardworks Ltd 2004 Brackets Look at this algebraic expression: 4(a + b) What do do think it means? Remember, in algebra we do not write the multiplication sign, ×. This expression actually means: 4 × (a + b) or (a + b) + (a + b) + (a + b) + (a + b) =a+b+a+b+a+b+a+b = 4a + 4b 8 of 60 © Boardworks Ltd 2004 Using the grid method to expand brackets 9 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Sometimes we need to multiply out brackets and then simplify. For example, 3x + 2(5 – x) We need to multiply the bracket by 2 and collect together like terms. 3x + 10 – 2x = 3x – 2x + 10 = x + 10 10 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Simplify 4 – (5n – 3) We need to multiply the bracket by –1 and collect together like terms. 4 – 5n + 3 = 4 + 3 – 5n = 7 – 5n 11 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Simplify 2(3n – 4) + 3(3n + 5) We need to multiply out both brackets and collect together like terms. 6n – 8 + 9n + 15 = 6n + 9n – 8 + 15 = 15n + 7 12 of 60 © Boardworks Ltd 2004 Expanding brackets then simplifying Simplify 5(3a + 2b) – 2(2a + 5b) We need to multiply out both brackets and collect together like terms. 15a + 10b – 4a –10b = 15a – 4a + 10b – 10b = 11a 13 of 60 © Boardworks Ltd 2004 Algebraic multiplication square 14 of 60 © Boardworks Ltd 2004 Pelmanism: Equivalent expressions 15 of 60 © Boardworks Ltd 2004 Algebraic areas 16 of 60 © Boardworks Ltd 2004 Contents A1 Algebraic expressions A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.4 Dividing terms A1.5 Factorising expressions A1.6 Substitution 17 of 60 © Boardworks Ltd 2004 Dividing terms Remember, in algebra we do not usually use the division sign, ÷. Instead we write the number or term we are dividing by underneath like a fraction. For example, (a + b) ÷ c 18 of 60 is written as a+b c © Boardworks Ltd 2004 Dividing terms Like a fraction, we can often simplify expressions by cancelling. For example, 3 n n3 ÷ n2 = 2 n 2 6p 6p2 ÷ 3p = 3p 1 1 1 n×n×n = n×n 6×p×p = 3×p =n = 2p 1 19 of 60 2 1 1 1 © Boardworks Ltd 2004 Algebraic areas 20 of 60 © Boardworks Ltd 2004 Hexagon Puzzle 21 of 60 © Boardworks Ltd 2004 Contents A1 Algebraic expressions A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution 22 of 60 © Boardworks Ltd 2004 Factorizing expressions Some expressions can be simplified by dividing each term by a common factor and writing the expression using brackets. For example, in the expression 5x + 10 the terms 5x and 10 have a common factor, 5. We can write the 5 outside of a set of brackets and mentally divide 5x + 10 by 5. (5x + 10) ÷ 5 = x + 2 This is written inside the bracket. 5(x + 2) 23 of 60 © Boardworks Ltd 2004 Factorizing expressions Writing 5x + 10 as 5(x + 2) is called factorizing the expression. Factorize 6a + 8 Factorize 12 – 9n The highest common factor of 6a and 8 is 2. The highest common factor of 12 and 9n is 3. (6a + 8) ÷ 2 = 3a + 4 (12 – 9n) ÷ 3 = 4 – 3n 6a + 8 = 2(3a + 4) 12 – 9n = 3(4 – 3n) 24 of 60 © Boardworks Ltd 2004 Factorizing expressions Writing 5x + 10 as 5(x + 2) is called factorizing the expression. Factorize 3x + x2 The highest common factor of 3x and x2 is x. (3x + x 2) ÷x=3+x Factorize 2p + 6p2 – 4p3 The highest common factor of 2p, 6p2 and 4p3 is 2p. (2p + 6p2 – 4p3) ÷ 2p = 1 + 3p – 2p2 3x + x2 = x(3 + x) 2p + 6p2 – 4p3 = 2p(1 + 3p – 2p2) 25 of 60 © Boardworks Ltd 2004 Algebraic multiplication square 26 of 60 © Boardworks Ltd 2004 Pelmanism: Equivalent expressions 27 of 60 © Boardworks Ltd 2004 Contents A1 Algebraic expressions A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.4 Dividing terms A1.5 Factorising expressions A1.6 Substitution 28 of 60 © Boardworks Ltd 2004 Work it out! 4 + 3 × 0.6 43 –7 8 5 ===–17 133 5.8 28 19 29 of 60 © Boardworks Ltd 2004 Work it out! 7 × 0.4 22 –3 6 9 2 ====–10.5 31.5 1.4 21 77 30 of 60 © Boardworks Ltd 2004 Work it out! 0.2 12 –4 3 9 2 +6 ===6.04 150 22 15 87 31 of 60 © Boardworks Ltd 2004 Work it out! 2( –13 3.6 18 69 7 + 8) ===23.2 –10 154 30 52 32 of 60 © Boardworks Ltd 2004 Substitution What does substitution mean? In algebra, when we replace letters in an expression or equation with numbers we call it substitution. 33 of 60 © Boardworks Ltd 2004 Substitution How can 4 + 3 × be written as an algebraic expression? Using n for the variable we can write this as 4 + 3n We can evaluate the expression 4 + 3n by substituting different values for n. When n = 5 4 + 3n = 4 + 3 × 5 = 4 + 15 = 19 When n = 11 4 + 3n = 4 + 3 × 11 = 4 + 33 = 37 34 of 60 © Boardworks Ltd 2004 Substitution 7× can be written as 7n 2 2 7n We can evaluate the expression by substituting different 2 values for n. When n = 4 7n 2 = 7×4÷2 = 28 ÷ 2 = 14 When n = 1.1 7n 2 = 7 × 1.1 ÷ 2 = 7.7 ÷ 2 = 3.85 35 of 60 © Boardworks Ltd 2004 Substitution 2 +6 can be written as n2 + 6 We can evaluate the expression n2 + 6 by substituting different values for n. When n = 4 n2 + 6 = 42 + 6 = 16 + 6 = 22 When n = 0.6 n2 + 7 = 0.62 + 6 = 0.36 + 6 = 6.36 36 of 60 © Boardworks Ltd 2004 Substitution 2( + 8) can be written as 2(n + 8) We can evaluate the expression 2(n + 8) by substituting different values for n. When n = 6 2(n + 8) = 2 × (6 + 8) = 2 × 14 = 28 When n = 13 2(n + 8) = 2 × (13 + 8) = 2 × 21 = 41 37 of 60 © Boardworks Ltd 2004 Substitution exercise Here are five expressions. 1) a + b + c = 5 + 2 + –1 = 6 2) 3a + 2c = 3 × 5 + 2 × –1 = 15 + –2 = 13 3) a(b + c) = 5 × (2 + –1) = 5 × 1 = 5 4) abc = 5 × 2 × –1= 10 × –1 = –10 22 – –1 b2 – c 5) = =5÷5=1 a 5 Evaluate these expressions when a = 5, b = 2 and c = –1 38 of 60 © Boardworks Ltd 2004 Noughts and crosses - substitution 39 of 60 © Boardworks Ltd 2004