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© Boardworks Ltd 2006 1 of 60 A1 Introduction to algebra S1–S4 Mathematics.

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1 © Boardworks Ltd of 60 A1 Introduction to algebra S1–S4 Mathematics

2 © Boardworks Ltd of 60 Contents A1 Introduction to algebra A A A A A A A1.1 Writing expressions A1.2 Collecting like terms A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution A1.3 Multiplying terms and expanding brackets

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

4 © Boardworks Ltd 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.

5 © Boardworks Ltd 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.

6 © Boardworks Ltd 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 a. If he opens the packet and eats 4 biscuits, he can write an expression for the number of biscuits remaining in the packet as: a – 4

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

8 © Boardworks Ltd 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.

9 © Boardworks Ltd 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

10 © Boardworks Ltd 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 and by itself again or n × n × n 3 × ( n + 4) or 3( n + 4) a number n plus 4 and then times 3.

11 © Boardworks Ltd 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:

12 © Boardworks Ltd of 60 Equivalent expression match

13 © Boardworks Ltd of 60 Contents A A A A A A A1.2 Collecting like terms A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution A1 Introduction to algebra A1.1 Writing expressions A1.3 Multiplying terms and expanding brackets

14 © Boardworks Ltd 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.

15 © Boardworks Ltd 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, = 4 × 5

16 © Boardworks Ltd 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

17 © Boardworks Ltd 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

18 © Boardworks Ltd 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.

19 © Boardworks Ltd 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 = 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

20 © Boardworks Ltd of 60 Algebraic perimeters Remember, to find the perimeter of a shape we add together the lengths of each of its sides. Write algebraic expressions for the perimeters 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

21 © Boardworks Ltd of 60 Algebraic pyramids

22 © Boardworks Ltd of 60 Algebraic magic square

23 © Boardworks Ltd of 60 Contents A A A A A A A1.3 Multiplying terms and expanding brackets A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution A1 Introduction to algebra A1.2 Collecting like terms A1.1 Writing expressions

24 © Boardworks Ltd 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.

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

26 © Boardworks Ltd 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

27 © Boardworks Ltd of 60 Grid method for multiplying numbers

28 © Boardworks Ltd of 60 Look at this algebraic expression: 4( a + b ) What do you think it means? Remember, in algebra we do not write the multiplication sign ×. So this expression 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

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

30 © Boardworks Ltd of 60 Look at this algebraic expression: Expanding expressions with brackets 3 y (4 – 2 y ) This means 3 y × (4 – 2 y ), but we do not usually write × in algebra. To expand or multiply out this expression we multiply every term inside the bracket by the term outside the bracket. 3 y (4 – 2 y ) =12 y – 6 y 2

31 © Boardworks Ltd of 60 Look at this algebraic expression: Expanding expressions with brackets – a (2 a 2 – 2 a + 3) When there is a negative term outside the bracket, the signs of the multiplied terms change. – a (2 a 2 – 3 a + 1) =–2 a a 2 – a In general,– x ( y + z ) =– xy – xz – x ( y – z ) =– xy + xz –( y + z ) =– y – z –( y – z ) =– y + z

32 © Boardworks Ltd 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

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

34 © Boardworks Ltd of 60 Expanding brackets and simplifying Expand and simplify:2(3 n – 4) + 3(3 n + 5) We need to multiply out both brackets and collect together like terms. 6n6n – n + 15 = 6 n + 9 n – = 15 n + 7 2(3 n – 4) + 3(3 n + 5) =

35 © Boardworks Ltd of 60 We need to multiply out both brackets and collect together like terms. 15 a + 10 b – 2 a – 5 ab = 15 a – 2 a + 10 b – 5 ab = 13 a + 10 b – 5 ab Expanding brackets and simplifying 5(3 a + 2 b ) – a (2 + 5 b ) = Expand and simplify:5(3 a + 2 b ) – a (2 + 5 b )

36 © Boardworks Ltd of 60 Algebraic multiplication square

37 © Boardworks Ltd of 60 Pelmanism: Equivalent expressions

38 © Boardworks Ltd of 60 Algebraic areas

39 © Boardworks Ltd of 60 Contents A A A A A A A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution A1 Introduction to algebra A1.2 Collecting like terms A1.1 Writing expressions A1.3 Multiplying terms and expanding brackets

40 © Boardworks Ltd 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

41 © Boardworks Ltd of 60 As with fractions, we can often simplify expressions by cancelling. For example, n 3 ÷ n 2 = n3n3 n2n2 = n × n × nn × n × n n × nn × n = n 6 p 2 ÷ 3 p = 6p26p2 3p3p = 6 × p × p 3 × p = 2 p Dividing terms

42 © Boardworks Ltd of 60 Hexagon puzzle

43 © Boardworks Ltd of 60 Contents A A A A A A A1.5 Factorizing expressions A1.6 Substitution A1 Introduction to algebra A1.4 Dividing terms A1.2 Collecting like terms A1.1 Writing expressions A1.3 Multiplying terms and expanding brackets

44 © Boardworks Ltd of 60 Factorizing expressions Factorizing an expression is the opposite of expanding it. a ( b + c ) ab + ac Expanding or multiplying out Factorizing Often: When we expand an expression we remove the brackets. When we factorize an expression we write it with brackets.

45 © Boardworks Ltd of 60 Factorizing expressions Expressions can be factorized by dividing each term by a common factor and writing this outside a pair of 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)

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

47 © Boardworks Ltd 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 = p – 2 p 2 Factorize 3 x + x 2 Factorize 2 p + 6 p 2 – 4 p 3

48 © Boardworks Ltd of 60 Factorization

49 © Boardworks Ltd of 60 Contents A A A A A A A1.6 Substitution A1 Introduction to algebra A1.5 Factorizing expressions A1.4 Dividing terms A1.2 Collecting like terms A1.1 Writing expressions A1.3 Multiplying terms and expanding brackets

50 © Boardworks Ltd of 60 Work it out! ×8 = 28 5 = = = 5.8 –7 = –17

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

52 © Boardworks Ltd of 60 Work it out! = 15 9 = = = 6.04 –4 = 22

53 © Boardworks Ltd of 60 Work it out! 2( + 8)7 = = = = 23.2 –13 = –10

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

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

56 © Boardworks Ltd 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 = × 2 Substitution

57 © Boardworks Ltd of 60 can be written as n We can evaluate the expression n by substituting different values for n. When n = 4 n = = = 22 When n = 0.6 n = = = Substitution

58 © Boardworks Ltd 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 = 42 2( + 8) Substitution

59 © Boardworks Ltd 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. = –1= 6 = 3 × × –1= 15 + –2= 13 = 5 × (2 + –1)= 5 × 1= 5 = 5 × 2 × –1= 10 × –1= –10 = 5 ÷ 5= – –122 – –1 = Substitution exercise

60 © Boardworks Ltd of 60 Noughts and crosses – substitution


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