KS 3 Mathematics A 1 Algebraic expressions 1

KS 3 Mathematics A 1 Algebraic expressions 1 of 60 © Boardworks Ltd 2004

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

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

Using symbols for unknowns Look at this problem: – The symbols In this example, For example, and 4 of 60 and 12 – 7 = 5 =5 stand for unknown numbers. can have many values. or 3. 2 – – 1. 8 = 5 are called variables because their value can vary. © Boardworks Ltd 2004

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. 5 of 60 © Boardworks Ltd 2004

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 6 of 60 © Boardworks Ltd 2004

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. 7 of 60 © Boardworks Ltd 2004

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. 8 of 60 © Boardworks Ltd 2004

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, n n ÷ 3 is written as 3 When we multiply a letter symbol by itself, we use index notation. n squared For example, n × n is written as n 2. 9 of 60 © Boardworks Ltd 2004

Writing expressions Here are some examples of algebraic expressions: n+7 a number n plus 7 5–n 5 minus a number n 2 n 2 lots of the number n or 2 × n 6 divided by a number n 4 n + 5 4 lots of a number n plus 5 n 3 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. 10 of 60 © Boardworks Ltd 2004

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

Equivalent expression match 12 of 60 © Boardworks Ltd 2004

Identities When two expressions are equivalent we can link them with the sign. x + x is identically For example, equal to 3 x x + x 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. 13 of 60 © Boardworks Ltd 2004

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

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. 15 of 60 © Boardworks Ltd 2004

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

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

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

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. 19 of 60 © Boardworks Ltd 2004

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

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: 2 a 3 b Perimeter = 2 a + 3 b + 2 a + 3 b = 4 a + 6 b 5 x 4 y x 5 x 21 of 60 Perimeter = 4 y + 5 x = 4 y + 11 x © Boardworks Ltd 2004

Algebraic pyramids 22 of 60 © Boardworks Ltd 2004

Algebraic magic square 23 of 60 © Boardworks Ltd 2004

Contents A 1 Algebraic expressions A 1. 1 Writing expressions A 1. 2 Collecting like terms A 1. 3 Multiplying terms A 1. 4 Dividing terms A 1. 5 Factorising expressions A 1. 6 Substitution 24 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 = 4 a 1×b=b We don’t need to write a 1 in front of the letter. b × 5 = 5 b We don’t write b 5. 3 × d × c = 3 cd We write letters in alphabetical order. 6 × e = 6 e 2 25 of 60 © Boardworks Ltd 2004

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

Using index notation We can use index notation to simplify expressions. For example, 3 p × 2 p = 3 × p × 2 × p = 6 p 2 q 2 × q 3 = q × q × q = q 5 3 r × r 2 = 3 × r × r = 3 r 3 2 t × 2 t = (2 t)2 27 of 60 or 4 t 2 © Boardworks Ltd 2004

Grid method for multiplying numbers 28 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 = 4 a + 4 b 29 of 60 © Boardworks Ltd 2004

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

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. 3 x + 10 – 2 x = 3 x – 2 x + 10 = x + 10 31 of 60 © Boardworks Ltd 2004

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 32 of 60 © Boardworks Ltd 2004

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. 6 n – 8 + 9 n + 15 = 6 n + 9 n – 8 + 15 = 15 n + 7 33 of 60 © Boardworks Ltd 2004

Expanding brackets then simplifying 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 34 of 60 © Boardworks Ltd 2004

Algebraic multiplication square 35 of 60 © Boardworks Ltd 2004

Pelmanism: Equivalent expressions 36 of 60 © Boardworks Ltd 2004

Algebraic areas 37 of 60 © Boardworks Ltd 2004

Contents A 1 Algebraic expressions A 1. 1 Writing expressions A 1. 2 Collecting like terms A 1. 3 Multiplying terms A 1. 4 Dividing terms A 1. 5 Factorising expressions A 1. 6 Substitution 38 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 39 of 60 is written as a+b c © Boardworks Ltd 2004

$Dividing terms Like a fraction, we can often simplify expressions by cancelling. For example,$

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

Algebraic areas 41 of 60 © Boardworks Ltd 2004

Hexagon Puzzle 42 of 60 © Boardworks Ltd 2004

Contents A 1 Algebraic expressions A 1. 1 Writing expressions A 1. 2 Collecting like terms A 1. 3 Multiplying terms A 1. 4 Dividing terms A 1. 5 Factorizing expressions A 1. 6 Substitution 43 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 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 and mentally divide 5 x + 10 by 5. (5 x + 10) ÷ 5 = x + 2 This is written inside the bracket. 5(x + 2) 44 of 60 © Boardworks Ltd 2004

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

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

Algebraic multiplication square 47 of 60 © Boardworks Ltd 2004

Pelmanism: Equivalent expressions 48 of 60 © Boardworks Ltd 2004

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

Work it out! 7 × 0. 4 22 – 3 6 9 2 ====–

Work it out! 0. 2 12 – 4 3 9 2 +6 ===6. 04

Work it out! 2( – 13 3. 6 18 69 7 + 8) ===23.

Substitution What does substitution mean? In algebra, when we replace letters in an expression or equation with numbers we call it substitution. 54 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 + 3 n We can evaluate the expression 4 + 3 n by substituting different values for n. When n = 5 4 + 3 n = 4 + 3 × 5 = 4 + 15 = 19 When n = 11 4 + 3 n = 4 + 3 × 11 = 4 + 33 = 37 55 of 60 © Boardworks Ltd 2004

Substitution 7× can be written as 7 n 2 2 7 n We can evaluate the expression by substituting different 2 values for n. When n = 4 7 n 2 = 7× 4÷ 2 = 28 ÷ 2 = 14 When n = 1. 1 7 n 2 = 7 × 1. 1 ÷ 2 = 7. 7 ÷ 2 = 3. 85 56 of 60 © Boardworks Ltd 2004

Substitution 2 +6 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 = 42 + 6 = 16 + 6 = 22 When n = 0. 6 n 2 + 7 = 0. 62 + 6 = 0. 36 + 6 = 6. 36 57 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 58 of 60 © Boardworks Ltd 2004

Substitution exercise Here are five expressions. 1) a + b + c = 5 + 2 + – 1 = 6 2) 3 a + 2 c = 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 b 2 – c 5) = =5÷ 5=1 a 5 Evaluate these expressions when a = 5, b = 2 and c = – 1 59 of 60 © Boardworks Ltd 2004