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KS3 Mathematics

KS3 Mathematics. A1 Algebraic expressions. A1 Algebraic expressions. Contents. A1.2 Collecting like terms. A1.3 Multiplying terms. A1.1 Writing expressions. A1.4 Dividing terms. A1.5 Factorizing expressions. A1.6 Substitution. Using symbols for unknowns. + 9 = 17. The symbol.

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KS3 Mathematics

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  1. KS3 Mathematics A1 Algebraic expressions

  2. A1 Algebraic expressions Contents A1.2 Collecting like terms A1.3 Multiplying terms A1.1 Writing expressions A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution

  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, 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

  7. 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.

  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 twice 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. Identities When two expressions are equivalent we can link them with the  sign. x+ x+ xis identically equal to 3x For example, x+ x+ x 3x 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.

  14. A1 Algebraic expressions Contents A1.1 Writing expressions A1.3 Multiplying terms A1.2 Collecting like terms A1.4 Dividing terms A1.5 Factorizing expressions A1.6 Substitution

  15. 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.

  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, 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.

  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, (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.

  18. 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.

  19. 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.

  20. 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

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

  22. Algebraic pyramids

  23. Algebraic magic square

  24. A1 Algebraic expressions Contents A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.4 Dividing terms A1.5 Factorising expressions A1.6 Substitution

  25. 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

  26. Using index notation Simplify: x + x + x + x + x = 5x x to the power of 5 Simplify: x×x×x×x×x = x5 This is called index notation. Similarly, x×x = x2 x×x×x = x3 x×x×x×x = x4

  27. 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

  28. Grid method for multiplying numbers

  29. 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

  30. Using the grid method to expand brackets

  31. 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. – 2x 3x + 10 = 3x – 2x + 10 = x+ 10

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

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

  34. 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

  35. Algebraic multiplication square

  36. Pelmanism: Equivalent expressions

  37. Algebraic areas

  38. A1 Algebraic expressions Contents A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.4 Dividing terms A1.5 Factorising expressions A1.6 Substitution

  39. 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

  40. Dividing terms n3 6p2 n2 3p 6 ×p×p n×n×n 3 ×p n×n Like a fraction, we can often simplify expressions by cancelling. For example, n3÷ n2 = 6p2÷ 3p = 1 1 2 1 = = 1 1 1 1 = n = 2p

  41. Algebraic areas

  42. Hexagon Puzzle

  43. A1 Algebraic expressions Contents A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.5 Factorizing expressions A1.4 Dividing terms A1.6 Substitution

  44. Some expressions can be simplified by dividing each term by a common factor and writing the expression using 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)

  45. 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)

  46. 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)

  47. Algebraic multiplication square

  48. Pelmanism: Equivalent expressions

  49. A1 Algebraic expressions Contents A1.1 Writing expressions A1.2 Collecting like terms A1.3 Multiplying terms A1.6 Substitution A1.4 Dividing terms A1.5 Factorising expressions

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

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