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© Boardworks Ltd 2004 1 of 60 KS3 Mathematics A1 Algebraic expressions

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© Boardworks Ltd 2004 2 of 60 A1.1 Writing expressions Contents A1 Algebraic expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution

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© Boardworks Ltd 2004 3 of 60 Using symbols for unknowns + 9 = 17 Look at this problem: The symbolstands for an unknown number. We can work out the value of. = 8 because8 + 9 = 17

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© Boardworks Ltd 2004 4 of 60 Using symbols for unknowns Look at this problem: –= 5 The symbolsstand for unknown numbers.and In this example, and can have many values. For example,12 – 7 = 53.2 – –1.8 = 5or andare called variables because their value can vary.

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© Boardworks Ltd 2004 5 of 60 Using letter symbols for unknowns In algebra, we use letter symbols to stand for numbers. These letters are called unknowns or variables. Sometimes we can work out the value of the letters and sometimes we can’t. For example, We can write an unknown number with 3 added on to it as n + 3 This is an example of an algebraic expression.

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© Boardworks Ltd 2004 6 of 60 Writing an expression Suppose Jon has a packet of biscuits and he doesn’t know how many biscuits it contains. He can call the number of biscuits in the full packet, b. If he opens the packet and eats 4 biscuits, he can write an expression for the number of biscuits remaining in the packet as: b – 4

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© Boardworks Ltd 2004 7 of 60 Writing an equation Jon counts the number of biscuits in the packet after he has eaten 4 of them. There are 22. He can write this as an equation: b – 4 = 22 We can work out the value of the letter b. b = 26 That means that there were 26 biscuits in the full packet.

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© Boardworks Ltd 2004 8 of 60 Writing expressions When we write expressions in algebra we don’t usually use the multiplication symbol ×. For example, 5 × n or n × 5 is written as 5 n. The number must be written before the letter. When we multiply a letter symbol by 1, we don’t have to write the 1. For example, 1 × n or n × 1 is written as n.

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© Boardworks Ltd 2004 9 of 60 Writing expressions When we write expressions in algebra we don’t usually use the division symbol ÷. Instead we use a dividing line as in fraction notation. For example, When we multiply a letter symbol by itself, we use index notation. For example, n ÷ 3 is written as n 3 n × n is written as n 2. n squared

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© Boardworks Ltd 2004 10 of 60 Writing expressions Here are some examples of algebraic expressions: n + 7a number n plus 7 5 – n 5 minus a number n 2n2n 2 lots of the number n or 2 × n 6 n 6 divided by a number n 4 n + 54 lots of a number n plus 5 n3n3 a number n multiplied by itself twice or n × n × n 3 × ( n + 4) or 3( n + 4) a number n plus 4 and then times 3.

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© Boardworks Ltd 2004 11 of 60 Writing expressions Miss Green is holding n number of cubes in her hand: She takes 3 cubes away. n – 3 She doubles the number of cubes she is holding. 2 × n or 2n2n Write an expression for the number of cubes in her hand if:

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© Boardworks Ltd 2004 12 of 60 Equivalent expression match

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© Boardworks Ltd 2004 13 of 60 Identities When two expressions are equivalent we can link them with the sign. For example, x + x + x 3 x x + x + x is identically equal to 3 x This is called an identity. In an identity, the expressions on each side of the equation are equal for all values of the unknown. The expressions are said to be identically equal.

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© Boardworks Ltd 2004 14 of 60 A1.2 Collecting like terms Contents A1 Algebraic expressions A1.1 Writing expressions A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution A1.3 Multiplying terms

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© Boardworks Ltd 2004 15 of 60 Like terms An algebraic expression is made up of terms and operators such as +, –, ×, ÷ and ( ). A term is made up of numbers and letter symbols but not operators. For example, 3 a + 4 b – a + 5 is an expression. 3 a, 4 b, a and 5 are terms in the expression. 3 a and a are called like terms because they both contain a number and the letter symbol a.

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© Boardworks Ltd 2004 16 of 60 Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In algebra, a + a + a + a = 4 a The a’ s are like terms. We collect together like terms to simplify the expression. In arithmetic, 5 + 5 + 5 + 5 = 4 × 5

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© Boardworks Ltd 2004 17 of 60 Collecting together like terms 7 × b + 3 × b = 10 × b Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In algebra, 7 b + 3 b = 10 b 7 b, 3 b and 10 b are like terms. They all contain a number and the letter b. In arithmetic, (7 × 4) + (3 × 4) = 10 × 4 or

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© Boardworks Ltd 2004 18 of 60 Collecting together like terms Remember, in algebra letters stand for numbers, so we can use the same rules as we use for arithmetic. In algebra, x + 6 x – 3 x = 4 x x, 6 x, 3 x and 4 x are like terms. They all contain a number and the letter x. In arithmetic, 2 + (6 × 2) – (3 × 2) = 4 × 2

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© Boardworks Ltd 2004 19 of 60 Collecting together like terms When we add or subtract like terms in an expression we say we are simplifying an expression by collecting together like terms. An expression can contain different like terms. For example, 3 a + 2 b + 4 a + 6 b = 3 a + 4 a + 2 b + 6 b = 7 a + 8 b This expression cannot be simplified any further.

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© Boardworks Ltd 2004 20 of 60 Simplify these expressions by collecting together like terms. 1) a + a + a + a + a =5a5a 2) 5 b – 4 b = b 3) 4 c + 3 d + 3 – 2 c + 6 – d =4 c – 2 c + 3 d – d + 3 + 6 = 2 c + 2 d + 9 4) 4 n + n 2 – 3 n =4 n – 3 n + n 2 = 5) 4 r + 6 s – t Cannot be simplified Collecting together like terms n + n 2

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© Boardworks Ltd 2004 21 of 60 Algebraic perimeters Remember, to find the perimeter of a shape we add together the length of each of its sides. Write an algebraic expression for the perimeter of the following shapes: 2a2a 3b3b Perimeter = 2 a + 3 b + 2 a + 3 b = 4 a + 6 b 5x5x 4y4y 5x5x x Perimeter = 4 y + 5 x + x + 5 x = 4 y + 11 x

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© Boardworks Ltd 2004 22 of 60 Algebraic pyramids

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© Boardworks Ltd 2004 23 of 60 Algebraic magic square

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© Boardworks Ltd 2004 24 of 60 A1.3 Multiplying terms Contents A1 Algebraic expressions A1.1 Writing expressions A1.2 Collecting like terms A1.4 Dividing terms A1.5 Factorising expressions A1.6 Substitution

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© Boardworks Ltd 2004 25 of 60 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 = 4 a 1 × b = b We don’t need to write a 1 in front of the letter. b × 5 =5b5b We don’t write b 5. 3 × d × c = 3 cd 6 × e × e =6e26e2 We write letters in alphabetical order.

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© Boardworks Ltd 2004 26 of 60 Using index notation Simplify: x + x + x + x + x = 5 x Simplify: x × x × x × x × xx × x × x × x × x = x 5 x to the power of 5 This is called index notation. Similarly, x × xx × x = x 2 x × x × xx × x × x = x 3 x × x × x × xx × x × x × x = x 4

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© Boardworks Ltd 2004 27 of 60 We can use index notation to simplify expressions. For example, 3 p × 2 p =3 × p × 2 × p =6p26p2 q 2 × q 3 = q × q × q × q × q = q5q5 3 r × r 2 =3 × r × r × r =3r33r3 2 t × 2 t =(2 t ) 2 or4t24t2 Using index notation

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© Boardworks Ltd 2004 28 of 60 Grid method for multiplying numbers

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© Boardworks Ltd 2004 29 of 60 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 = 4 a + 4 b Brackets

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© Boardworks Ltd 2004 30 of 60 Using the grid method to expand brackets

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© Boardworks Ltd 2004 31 of 60 Expanding brackets then simplifying Sometimes we need to multiply out brackets and then simplify. For example, 3 x + 2(5 – x ) We need to multiply the bracket by 2 and collect together like terms. 3x3x + 10 – 2 x = 3 x – 2 x + 10 = x + 10

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© Boardworks Ltd 2004 32 of 60 Expanding brackets then simplifying Simplify 4 – (5 n – 3) We need to multiply the bracket by –1 and collect together like terms. 4 – 5 n + 3 = 4 + 3 – 5 n = 7 – 5 n

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© Boardworks Ltd 2004 33 of 60 Expanding brackets then simplifying Simplify 2(3 n – 4) + 3(3 n + 5) We need to multiply out both brackets and collect together like terms. 6n6n – 8 + 9 n + 15 = 6 n + 9 n – 8 + 15 = 15 n + 7

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© Boardworks Ltd 2004 34 of 60 Simplify 5(3 a + 2 b ) – 2(2 a + 5 b ) We need to multiply out both brackets and collect together like terms. 15 a + 10 b – 4 a –10 b = 15 a – 4 a + 10 b – 10 b = 11 a Expanding brackets then simplifying

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© Boardworks Ltd 2004 35 of 60 Algebraic multiplication square

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© Boardworks Ltd 2004 36 of 60 Pelmanism: Equivalent expressions

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© Boardworks Ltd 2004 37 of 60 Algebraic areas

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© Boardworks Ltd 2004 38 of 60 A1.4 Dividing terms Contents A1 Algebraic expressions A1.3 Multiplying terms A1.2 Collecting like terms A1.1 Writing expressions A1.5 Factorising expressions A1.6 Substitution

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© Boardworks Ltd 2004 39 of 60 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 is written as a + b c

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© Boardworks Ltd 2004 40 of 60 Like a fraction, we can often simplify expressions by cancelling. For example, n 3 ÷ n 2 = n3n3 n2n2 = n × n × nn × n × n n × nn × n 1 1 1 1 = n 6 p 2 ÷ 3 p = 6p26p2 3p3p = 6 × p × p 3 × p 2 1 1 1 = 2 p Dividing terms

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© Boardworks Ltd 2004 41 of 60 Algebraic areas

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© Boardworks Ltd 2004 42 of 60 Hexagon Puzzle

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© Boardworks Ltd 2004 43 of 60 A1.5 Factorizing expressions Contents A1 Algebraic expressions A1.1 Writing expressions A1.3 Multiplying terms A1.2 Collecting like terms A1.4 Dividing terms A1.6 Substitution

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© Boardworks Ltd 2004 44 of 60 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 5 x + 10 the terms 5 x and 10 have a common factor, 5. We can write the 5 outside of a set of brackets 5( x + 2) We can write the 5 outside of a set of brackets and mentally divide 5 x + 10 by 5. (5 x + 10) ÷ 5 = x + 2 This is written inside the bracket. 5( x + 2)

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© Boardworks Ltd 2004 45 of 60 Factorizing expressions Writing 5 x + 10 as 5( x + 2) is called factorizing the expression. Factorize 6 a + 8 6 a + 8 =2(3 a + 4) Factorize 12 – 9 n 12 – 9 n =3(4 – 3 n ) The highest common factor of 6a and 8 is 2.2. (6 a + 8) ÷ 2 =3 a + 4 The highest common factor of 12 and 9 n is 3.3. (12 – 9 n ) ÷ 3 =4 – 3 n

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© Boardworks Ltd 2004 46 of 60 Factorizing expressions Writing 5 x + 10 as 5( x + 2) is called factorizing the expression. 3 x + x 2 = x (3 + x ) 2 p + 6 p 2 – 4 p 3 = 2 p (1 + 3 p – 2 p 2 ) The highest common factor of 3 x and x 2 is x.x. (3 x + x 2 ) ÷ x =3 + x The highest common factor of 2 p, 6 p 2 and 4 p 3 is 2p.2p. (2 p + 6 p 2 – 4 p 3 ) ÷ 2 p = 1 + 3 p – 2 p 2 Factorize 3 x + x 2 Factorize 2 p + 6 p 2 – 4 p 3

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© Boardworks Ltd 2004 47 of 60 Algebraic multiplication square

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© Boardworks Ltd 2004 48 of 60 Pelmanism: Equivalent expressions

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© Boardworks Ltd 2004 49 of 60 A1.6 Substitution Contents A1 Algebraic expressions A1.1 Writing expressions A1.3 Multiplying terms A1.2 Collecting like terms A1.4 Dividing terms A1.5 Factorising expressions

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© Boardworks Ltd 2004 50 of 60 Work it out! 4 + 3 ×8 = 28 5 = 19 43 = 133 0.6 = 5.8 –7 = –17

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© Boardworks Ltd 2004 51 of 60 Work it out! 2 7 × 6 = 21 9 = 31.5 22 = 77 0.4 = 1.4 –3 = –10.5

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© Boardworks Ltd 2004 52 of 60 Work it out! 2 + 6 3 = 15 9 = 87 12 = 150 0.2 = 6.04 –4 = 22

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© Boardworks Ltd 2004 53 of 60 Work it out! 2( + 8)7 = 30 18 = 52 69 = 154 3.6 = 23.2 –13 = –10

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© Boardworks Ltd 2004 54 of 60 Substitution What does substitution mean? In algebra, when we replace letters in an expression or equation with numbers we call it substitution.

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© Boardworks Ltd 2004 55 of 60 How can be written as an algebraic expression? 4 + 3 × Using n for the variable we can write this as4 + 3 n We can evaluate the expression 4 + 3 n by substituting different values for n. When n = 54 + 3 n = 4 + 3 × 5 = 4 + 15 = 19 When n = 114 + 3 n = 4 + 3 × 11 = 4 + 33 = 37 Substitution

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© Boardworks Ltd 2004 56 of 60 can be written as 7n7n 2 We can evaluate the expression by substituting different values for n. 7n7n 2 When n = 4 7n7n 2 = 7 × 4 ÷ 2 = 28 ÷ 2 = 14 When n = 1.1 7n7n 2 = 7 × 1.1 ÷ 2 = 7.7 ÷ 2 = 3.85 7 × 2 Substitution

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© Boardworks Ltd 2004 57 of 60 can be written as n 2 + 6 We can evaluate the expression n 2 + 6 by substituting different values for n. When n = 4 n 2 + 6 = 4 2 + 6 = 16 + 6 = 22 When n = 0.6 n 2 + 7 = 0.6 2 + 6 = 0.36 + 6 = 6.36 2 + 6 Substitution

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© Boardworks Ltd 2004 58 of 60 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 2( + 8) Substitution

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© Boardworks Ltd 2004 59 of 60 Here are five expressions. 1) a + b + c 2) 3 a + 2 c 3) a ( b + c ) 4) abc 5) a b2 – cb2 – c Evaluate these expressions when a = 5, b = 2 and c = –1 = 5 + 2 + –1= 6 = 3 × 5 + 2 × –1= 15 + –2= 13 = 5 × (2 + –1)= 5 × 1= 5 = 5 × 2 × –1= 10 × –1= –10 = 5 ÷ 5= 1 5 22 – –122 – –1 = Substitution exercise

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© Boardworks Ltd 2004 60 of 60 Noughts and crosses - substitution

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