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KS3 Mathematics

KS3 Mathematics. N9 Mental methods. N9 Mental methods. Contents. N9.2 Addition and subtraction. N9.1 Order of operations. N9.3 Multiplication and division. N9.4 Numbers between 0 and 1. N9.5 Problems and puzzles. Using the correct order of operations. What is 7 – 3 – 2?.

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KS3 Mathematics

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  1. KS3 Mathematics N9 Mental methods

  2. N9 Mental methods Contents N9.2 Addition and subtraction N9.1 Order of operations N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

  3. Using the correct order of operations What is 7 – 3 – 2? When a calculation contains more than one operation it is important that we use the correct order of operations. The first rule is we work from left to right so, 7 – 3 – 2 = 4 – 2 NOT 7 – 3 – 2 = 7 – 1 = 2 = 6

  4. Using the correct order of operations What is 8 + 2 × 4? The second rule is that we multiply or divide before we add or subtract. 8 + 2 × 4 = 8 + 8 NOT 8 + 2 × 4 = 10 × 4 = 16 = 40

  5. Brackets What is (15 – 9) ÷ 3? When a calculation contains brackets we always work out the contents of any brackets first. (15 – 9) ÷ 3 = 6 ÷ 3 = 2

  6. Nested brackets Sometimes we have to use brackets within brackets. For example, 10 ÷ {5 – (6 – 3)} These are called nested brackets. We evaluate the innermost brackets first and then work outwards. 10 ÷ {5 – (6 – 3)} = 10 ÷ {5 – 3} = 10 ÷ 2 = 5

  7. Using a division line 13 + 8 7 13 + 8 7 = (13 + 8) ÷ 7 What is ? When we use a horizontal line for division the dividing line acts as a bracket. = 21 ÷ 7 = 3

  8. Using a division line 24 + 8 24 + 8 24 – 8 24 – 8 = (24 + 8) ÷ (24 – 8) What is ? Again, the dividing line acts as a bracket. = 32 ÷ 16 = 2

  9. Multiplying by a bracket When we multiply by a bracket it is not always necessary to use the symbol for multiplication, ×. For example, 8 + 3(7 – 3) is equivalent to 8 + 3 × (7 – 3) = 8 + 3 × 4 = 8 + 12 = 20 Compare this to the use of brackets in algebraic expressions such as 3(a + 2).

  10. Indices What is 100 – 2(3 + 4)2 When indices appear in a calculation, these are worked out after brackets, but before multiplication and division. 100 – 2(3 + 4)2 Brackets first, = 100 – 2 × 72 then Indices, = 100 – 2 × 49 thenDivision and Multiplication, = 100 – 98 and thenAddition and Subtraction = 2

  11. BIDMAS B Remember BIDMAS: RACKETS I NDICES (OR POWERS) D IVISION M ULTIPLICATION A DDITION S UBTRACTION

  12. Using BIDMAS 82 64 = – 8 × 0.5 – 8 × 0.5 = 16 4 (3.4 + 4.6)2 What is – 8 × 0.5 ? (6 + 5 × 2) Brackets first, then Indices, thenDivision and Multiplication, = 16 – 4 and thenAddition and Subtraction = 12

  13. Using a calculator (52 + 72) 7 - 5  ( 5 5 7 7 ( ) x2 + x2 ) ÷ - We can use a calculator to evaluate more difficult calculations. For example, This can be entered as: = 4.3 (to 1 d.p.) Always use an approximation to check answers given by a calculator.

  14. Positioning brackets

  15. Target numbers

  16. N9 Mental methods Contents N9.1 Order of operations N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

  17. Complements match

  18. Counting on and back

  19. Using partitioning to add What is 276 + 68? 276 + 68 = 200 + 70 + 6 + 60 + 8 = 6 + 8 + 70 + 60 + 200 = 14 + 130 + 200 = 344 What is 63.8 + 4.7? 63.8 + 4.7 = 60 + 3 + 0.8 + 4 + 0.7 = 0.8 + 0.7 + 3 + 4 +60 = 1.5 + 7 + 60 = 68.5

  20. Adding by counting up + 60 + 8 276 336 344 + 4 + 0.7 63.8 67.8 68.5 What is 276 + 68? 276 + 60 + 8 = 344 What is 63.8 + 4.7? 63.8 + 4 + 0.7 = 68.5

  21. Using compensation to add + 70 – 2 276 344 346 + 5 – 0.3 63.8 68.5 68.8 What is 276 + 68? 276 + 70 – 2 = 344 What is 63.8 + 4.7? 63.8 + 5 – 0.3 = 68.5

  22. Using partitioning to subtract – 7 – 30 – 400 127 134 164 564 – 0.4 – 4 – 2 16.1 16.5 20.5 22.5 What is 564 – 437? 564 – 400 – 30 – 7 = 127 What is 22.5 – 6.4? 22.5 – 2 – 4 – 0.4 = 16.1

  23. Subtracting by counting up + 100 + 4 + 20 + 3 437 537 540 564 560 + 10 + 6 + 0.1 6.4 16.4 22.5 22.4 What is 564 – 437? 100 + 3 + 20 + 4 = 127 What is 22.5 – 6.4? 10 + 6 + 0.1 = 16.1

  24. Using compensation to subtract + 63 – 500 64 127 564 + 0.1 – 6.5 16 16.1 22.5 What is 564 – 437? 564 – 500 + 63 = 127 What is 22.5 – 6.4? 22.5 – 6.5 + 0.1 = 16.1

  25. Addition pyramid

  26. N9 Mental methods Contents N9.1 Order of operations N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

  27. Using partitioning to multiply whole numbers What is 7 × 43? We can work out 7 × 43 mentally using partitioning. 43 = 40 + 3 So, 7 × 43 = (7 × 40) + (7 × 3) = 280 + 21 = 301

  28. Using partitioning to multiply decimals What is 3.2 × 40? We can work out 3.2 × 40 by partitioning 3.2 3.2 = 3 + 0.2 So, 3.2 × 40 = (3 × 40) + (0.2 × 40) = 120 + 8 = 128

  29. Using the distributive law to multiply What is 0.6 × 29? We can work out 0.6 × 29 using the distributive law. 29 = 30 – 1 So, 0.6 × 29 = (0.6 × 30) – (0.6 × 1) = 18 – 0.6 = 17.4

  30. Using a grid to multiply

  31. Using a grid to multiply

  32. Using factors to multiply whole numbers What is 26 × 12? We can work out 26 × 12 by dividing 12 into factors. 12 = 4 × 3 = 2 × 2 × 3 So we can multiply 26 by 2, by 2 again and then by 3: 52 × 2 × 3 26 × 2 × 2 × 3 = = 104 × 3 = 312

  33. Using factors to multiply decimals What is 0.7 × 18? We can work out 0.7 × 18 by dividing 18 into factors. 18 = 9 × 2 So we can multiply 0.7 by 9 and then by 2: 0.7 × 18 = = 0.7 × 9 × 2 = 6.3 × 2 = 12.6

  34. Using doubling and halving What is 7.5 × 8? Two numbers can be multiplied together mentally by doubling one number and halving the other. We can repeat this until the numbers are easy to work out mentally. 7.5 × 8 = 15 × 4 = 30 × 2 = 60

  35. Using factors to divide whole numbers What is 68 ÷ 20? We can work out 68 ÷ 20 by dividing 20 into factors. 20 = 2 × 10 So we can divide 68 by 2 and then by 10: 68 ÷ 20 = 68 ÷ 2 ÷ 10 = 34 ÷ 10 = 3.4

  36. Using factors to divide decimals What is 12.4 ÷ 8? We can work out 12.4 ÷ 8 by dividing 8 into factors. 8 = 2 × 2 × 2 So we can divide 12.4 by 2, by 2 again and then by 2 a third time: 12.4 ÷ 8 = 12.4 ÷ 2 ÷ 2 ÷ 2 31 ÷ 2 = 15.5 so 3.1 ÷ 2 =1.55 = 6.2 ÷ 2 ÷ 2 = 3.1 ÷ 2 = 1.55

  37. Using partitioning to divide What is 486 ÷ 6? We can work out 486 ÷ 6 by partitioning 486. 486 = 480 + 6 So, 486 ÷ 6 = (480 ÷ 6) + (6 ÷ 6) = 80 + 1 = 81

  38. Using fractions to divide whole numbers 420 420 40 40 21 2 What is 420 ÷ 40? We can simplify 420 ÷ 40 by writing the division as a fraction and then cancelling. 420 ÷ 40 = 21 = 2 = 101/2 = 10.5

  39. Using fractions to divide decimals × 10 2.6 26 26 13 0.8 8 8 4 × 10 What is 2.6 ÷ 0.8? We can simplify 2.6 ÷ 0.8 by writing the division as a fraction. 2.6 ÷ 0.8 = = 13 = 4 = 31/4 = 3.25

  40. Multiplying by multiples of 10, 100 and 1000 We can use our knowledge of place value to multiply by multiples of 10, 100 and 1000. What is 7 × 600? What is 2.3 × 4000? 2.3 × 4000 = 2.3 × 4 × 1000 7 × 600 = 7 × 6 × 100 = 42 × 100 = 9.2 × 1000 = 4200 = 9200

  41. Dividing by multiples of 10, 100 and 1000 We can use our knowledge of place value to divide by multiples of 10, 100 and 1000. What is 24 ÷ 80? What is 4.5 ÷ 500? 24 ÷ 80 = 24 ÷ 8 ÷ 10 4.5 ÷ 500 = 4.5 ÷ 5 ÷ 100 = 3 ÷ 10 = 0.9 ÷ 100 = 0.3 = 0.009

  42. Noughts and crosses 1

  43. N9 Mental methods Contents N9.1 Order of operations N9.2 Addition and subtraction N9.4 Numbers between 0 and 1 N9.3 Multiplication and division N9.5 Problems and puzzles

  44. Multiplying by multiples of 0.1 and 0.01 Multiplying by 0.1 is the same as Dividing by 10 Multiplying by 0.01 is the same as Dividing by 100 What is 4 × 0.8? What is 15 × 0.03? 4 × 0.8 = 4 × 8 × 0.1 15 × 0.03 = 15 × 3 × 0.01 = 32 × 0.1 = 45 × 0.01 = 32 ÷ 10 = 45 ÷ 100 = 3.2 = 0.45

  45. Dividing by multiples of 0.1 and 0.01 Dividing by 0.1 is the same as Multiplying by 10 Dividing by 0.01 is the same as Multiplying by 100 What is 36 ÷ 0.4? What is 3 ÷ 0.02? 36 ÷ 0.4 = 36 ÷ 4 ÷ 0.1 3 ÷ 0.02 = 3 ÷ 2 ÷ 0.01 = 9 ÷ 0.1 = 1.5 ÷ 0.01 = 9 × 10 = 1.5 × 100 = 90 = 150

  46. Multiplying by small multiples of 0.1

  47. Multiplying by decimals between 1 and 0 When we multiply a number n by a number greater than 1 the answer will be bigger than n. When we multiply a number n by a number between 0 and 1 the answer will be smaller than n. When we divide a number n by a number greater than 1 the answer will be smaller than n. When we divide a number n by a number between 0 and 1 the answer will be bigger than n.

  48. Noughts and crosses 2

  49. N9 Mental methods Contents N9.1 Order of operations N9.2 Addition and subtraction N9.5 Problems and puzzles N9.3 Multiplication and division N9.4 Numbers between 0 and 1

  50. Chequered sums

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