S1–S4 Mathematics

1. S1–S4 Mathematics A1 Introduction to algebra

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

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

4. Using symbols for unknowns – = 5 The symbols and stand for unknown numbers. In this example, and can have many values. For example, 12 – 7 = 5 or 3.2 – –1.8 = 5 and are called variables because their value can vary. Look at this problem:

5. Using letter symbols for unknowns For example, We can write an unknown number with 3 added on to it as n + 3 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. This is an example of an algebraic expression.

6. 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. 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. 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 5n. 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. Writing expressions n n ÷ 3 is written as 3 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. n squared For example, n × n is written as n2.

10. Writing expressions 6 n Here are some examples of algebraic expressions: n + 7 a number n plus 7 5 –n 5 minus a number n 2n 2 lots of the number n or 2 ×n 6 divided by a number n 4n + 5 4 lots of a number n plus 5 a number n multiplied by itself and by itself again or n× n × n n3 3 × (n + 4) or 3(n + 4) a number n plus 4 and then times 3.

11. 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. or 2 ×n 2n

12. Equivalent expression match

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

14. 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, 3a + 4b – a + 5 is an expression. 3a, 4b, a and 5 are terms in the expression. 3a and a are called like terms because they both contain a number and the letter symbol a.

15. 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+a= 4a The a’s are like terms. We collect together like terms to simplify the expression.

16. 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 7b + 3b= 10b 7b, 3b and 10b are like terms. They all contain a number and the letter b.

17. 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 + 6x– 3x = 4x x, 6x, 3x and 4x are like terms. They all contain a number and the letter x.

18. 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, 3a + 2b + 4a + 6b = 3a + 4a + 2b + 6b = 7a + 8b This expression cannot be simplified any further.

19. Collecting together like terms Simplify these expressions by collecting together like terms. 1) a + a + a + a + a = 5a 2) 5b – 4b = b 3) 4c + 3d + 3 – 2c + 6 –d = 4c – 2c + 3d – d + 3 + 6 = 2c + 2d + 9 4) 4n + n2 – 3n = 4n– 3n+ n2= n+ n2 Cannot be simplified 5) 4r + 6s–t

20. Algebraic perimeters 2a 3b 5x 4y x 5x 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: Perimeter = 2a + 3b + 2a + 3b = 4a + 6b Perimeter = 4y + 5x + x + 5x = 4y + 11x

21. Algebraic pyramids

22. Algebraic magic square

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

24. 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 We don’t need to write a 1 in front of the letter. 1 ×b = b b× 5 = 5b We don’t write b5. We write letters in alphabetical order. 3 ×d×c = 3cd 6 ×e×e = 6e2

25. Using index notation Simplify: a + a + a + a + a = 5a a to the power of 5 Simplify: a×a×a×a×a = a5 This is called index notation. Similarly, a×a = a2 a×a×a = a3 a×a×a×a = a4

26. 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 or 4t2

27. Grid method for multiplying numbers

28. Brackets 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 = 4a + 4b

29. Using the grid method to expand brackets

30. Expanding expressions with brackets Look at this algebraic expression: 3y(4 – 2y) This means 3y × (4 – 2y), 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. 3y(4 – 2y) = 12y – 6y2

31. Expanding expressions with brackets In general, –x(y + z) = –xy – xz –x(y – z) = –xy + xz –(y + z) = –y – z –(y – z) = –y + z Look at this algebraic expression: –a(2a2 – 2a + 3) When there is a negative term outside the bracket, the signs of the multiplied terms change. –a(2a2 – 3a + 1) = –2a3 + 3a2 – a

32. 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

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

34. Expanding brackets and simplifying Expand and simplify: 2(3n– 4) + 3(3n + 5) We need to multiply out both brackets and collect together like terms. 2(3n– 4) + 3(3n + 5) = – 8 + 15 6n + 9n = 6n + 9n– 8 + 15 = 15n + 7

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

36. Algebraic multiplication square

37. Pelmanism: Equivalent expressions

38. Algebraic areas

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

40. Dividing terms a + b c 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, is written as (a + b) ÷ c

41. Dividing terms n3 6p2 n2 3p 6 ×p×p n×n×n 3 ×p n×n As with fractions, we can often simplify expressions by cancelling. For example, n3÷ n2 = 6p2÷ 3p = 1 1 2 1 = = 1 1 1 1 = n = 2p

42. Hexagon puzzle

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

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

45. Expressions can be factorized by dividing each term by a common factor and writing this outside a pair of brackets. Factorizing expressions 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 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) 5(x+ 2)

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

47. Writing 5x + 10 as 5(x + 2) is called factorizing the expression. Factorizing expressions Factorize 3x + x2 Factorize 2p + 6p2 – 4p3 The highest common factor of 3x and x2 is The highest common factor of 2p, 6p2 and 4p3 is x. 2p. (2p + 6p2 – 4p3) ÷ 2p = (3x + x2) ÷ x = 3 + x 1 + 3p– 2p2 3x + x2 = x(3 + x) 2p + 6p2 – 4p3 = 2p(1 + 3p– 2p2)

48. Factorization

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

50. Work it out! 4 + 3 × 0.6 43 –7 8 5 = 133 = –17 = 5.8 = 28 = 19