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© Boardworks Ltd 2004 1 of 63 N9 Mental methods KS3 Mathematics

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© Boardworks Ltd 2004 2 of 63 N9.1 Order of operations Contents N9 Mental methods N9.2 Addition and subtraction N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

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© Boardworks Ltd 2004 3 of 63 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 = 2 NOT 7 – 3 – 2 = 7 – 1 = 6

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© Boardworks Ltd 2004 4 of 63 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 = 16 NOT 8 + 2 × 4 = 10 × 4 = 40

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© Boardworks Ltd 2004 5 of 63 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

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© Boardworks Ltd 2004 6 of 63 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

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© Boardworks Ltd 2004 7 of 63 Using a division line What is ? 13 + 8 7 When we use a horizontal line for division the dividing line acts as a bracket. 13 + 8 7 = (13 + 8) ÷ 7 = 21 ÷ 7 = 3

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© Boardworks Ltd 2004 8 of 63 Using a division line What is ? 24 + 8 24 – 8 Again, the dividing line acts as a bracket. = 32 ÷ 16 = 2 = (24 + 8) ÷ (24 – 8) 24 + 8 24 – 8

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© Boardworks Ltd 2004 9 of 63 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).

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© Boardworks Ltd 2004 10 of 63 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 × 7 2 then Indices, = 100 – 2 × 49 then Division and Multiplication, = 100 – 98 and then Addition and Subtraction = 2

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© Boardworks Ltd 2004 11 of 63 BIDMAS Remember BIDMAS: RACKETS NDICES (OR POWERS) IVISION ULTIPLICATION DDITION UBTRACTION B B I I D D M M A A S S

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© Boardworks Ltd 2004 12 of 63 Using BIDMAS What is (3.4 + 4.6) 2 (6 + 5 × 2) – 8 × 0.5 ? Brackets first, then Indices, then Division and Multiplication, 8282 16 – 8 × 0.5 = 64 4 – 8 × 0.5 = = 16 – 4 = 12 and then Addition and Subtraction

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© Boardworks Ltd 2004 13 of 63 Using a calculator We can use a calculator to evaluate more difficult calculations. For example, (5 2 + 7 2 ) 7 - 5 This can be entered as: (5 x2x2 + 7 x2x2 )÷ (7-5) = 4.3 (to 1 d.p.) Always use an approximation to check answers given by a calculator.

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© Boardworks Ltd 2004 14 of 63 Positioning brackets

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© Boardworks Ltd 2004 15 of 63 Target numbers

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© Boardworks Ltd 2004 16 of 63 N9.2 Addition and subtraction Contents N9 Mental methods N9.1 Order of operations N9.3 Multiplication and division N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

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© Boardworks Ltd 2004 17 of 63 Complements match

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© Boardworks Ltd 2004 18 of 63 Counting on and back

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© Boardworks Ltd 2004 19 of 63 Using partitioning to add What is 276 + 68? 276 + 68 =200 + 70 + 6 + 60 + 8 What is 63.8 + 4.7? 63.8 + 4.7= 60 + 3 + 0.8 + 4 + 0.7 = 6 + 8 + 70 + 60 + 200 = 14 + 130 + 200 = 344 = 0.8 + 0.7 + 3 + 4 + 60 = 1.5 + 7 + 60 = 68.5

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© Boardworks Ltd 2004 20 of 63 Adding by counting up What is 276 + 68? 276 + 60 336 + 8 344 What is 63.8 + 4.7? 63.867.8 + 0.7 68.5 + 4 276+ 60+ 8= 344 63.8+ 4+ 0.7= 68.5

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© Boardworks Ltd 2004 21 of 63 Using compensation to add What is 276 + 68? 276+ 70– 2 276344 – 2 346 = 344 What is 63.8 + 4.7? 63.8+ 5– 0.3 63.868.5 – 0.3 68.8 = 68.5 + 70 + 5

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© Boardworks Ltd 2004 22 of 63 Using partitioning to subtract What is 564 – 437? 564– 400– 30 127 – 30 164 – 400 564 – 7 What is 22.5 – 6.4? 22.5– 2– 4 16.1 – 4 16.5 – 2 22.5 = 16.1 = 127 – 0.4 20.5134 – 0.4 – 7

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© Boardworks Ltd 2004 23 of 63 Subtracting by counting up 437537 + 3 5406.416.4 + 6 22.5 What is 564 – 437? What is 22.5 – 6.4? + 100 + 10 + 20+ 4 560564 100+ 3+ 20+ 4= 127 + 0.1 22.4 10+ 6+ 0.1= 16.1

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© Boardworks Ltd 2004 24 of 63 Using compensation to subtract What is 564 – 437? 564– 500+ 63 64564 = 127 What is 22.5 – 6.4? 22.5– 6.5+ 0.1 1616.122.5 = 16.1 127 + 0.1 + 63– 500 – 6.5

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© Boardworks Ltd 2004 25 of 63 Addition pyramid

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© Boardworks Ltd 2004 26 of 63 N9.3 Multiplication and division Contents N9 Mental methods N9.1 Order of operations N9.2 Addition and subtraction N9.4 Numbers between 0 and 1 N9.5 Problems and puzzles

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© Boardworks Ltd 2004 27 of 63 Using partitioning to multiply whole numbers We can work out 7 × 43 mentally using partitioning. 43 = 40 + 3 So, 7 × 43 = (7 × 40) + (7 × 3) = 280 + 21 = 301 What is 7 × 43?

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© Boardworks Ltd 2004 28 of 63 Using partitioning to multiply decimals 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 What is 3.2 × 40?

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© Boardworks Ltd 2004 29 of 63 Using the distributive law to multiply 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 What is 0.6 × 29?

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© Boardworks Ltd 2004 30 of 63 Using a grid to multiply

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© Boardworks Ltd 2004 31 of 63 Using a grid to multiply

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© Boardworks Ltd 2004 32 of 63 Using factors to multiply whole numbers 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: 26 × 2 × 2 × 3 = 52 × 2 × 3 What is 26 × 12? = 104 × 3 = 312

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© Boardworks Ltd 2004 33 of 63 Using factors to multiply decimals 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 = = 6.3 × 2 What is 0.7 × 18? = 12.6 = 0.7 × 9 × 2

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© Boardworks Ltd 2004 34 of 63 Using doubling and halving 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 What is 7.5 × 8?

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© Boardworks Ltd 2004 35 of 63 Using factors to divide whole numbers 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 = = 34 ÷ 10 What is 68 ÷ 20? = 3.4 68 ÷ 2 ÷ 10

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© Boardworks Ltd 2004 36 of 63 Using factors to divide decimals 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 = = 6.2 ÷ 2 ÷ 2 = 3.1 ÷ 2 = 1.55 What is 12.4 ÷ 8? 31 ÷ 2 = 15.5 so 3.1 ÷ 2 =1.55 12.4 ÷ 2 ÷ 2 ÷ 2

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© Boardworks Ltd 2004 37 of 63 Using partitioning to divide We can work out 486 ÷ 6 by partitioning 486. 486 = 480 + 6 So, 486 ÷ 6 = (480 ÷ 6) + (6 ÷ 6) = 80 + 1 = 81 What is 486 ÷ 6?

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© Boardworks Ltd 2004 38 of 63 Using fractions to divide whole numbers We can simplify 420 ÷ 40 by writing the division as a fraction and then cancelling. 420 ÷ 40 = 420 40 420 40 21 2 = 2 = 10 1 / 2 What is 420 ÷ 40? = 10.5

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© Boardworks Ltd 2004 39 of 63 Using fractions to divide decimals We can simplify 2.6 ÷ 0.8 by writing the division as a fraction. 2.6 ÷ 0.8 = 2.6 0.8 13 4 = 4 = 3 1 / 4 What is 2.6 ÷ 0.8? = × 10 26 8 8 = 3.25

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© Boardworks Ltd 2004 40 of 63 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? 7 × 600 =7 × 6 × 100 = 42 × 100 = 4200 What is 2.3 × 4000? 2.3 × 4000 =2.3 × 4 × 1000 = 9.2 × 1000 = 9200

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© Boardworks Ltd 2004 41 of 63 Dividing by multiples of 10, 100 and 1000 What is 24 ÷ 80? 24 ÷ 80 =24 ÷ 8 ÷ 10 = 3 ÷ 10 = 0.3 What is 4.5 ÷ 500? 4.5 ÷ 500 =4.5 ÷ 5 ÷ 100 = 0.9 ÷ 100 = 0.009 We can use our knowledge of place value to divide by multiples of 10, 100 and 1000.

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© Boardworks Ltd 2004 42 of 63 Noughts and crosses 1

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© Boardworks Ltd 2004 43 of 63 N9.4 Numbers between 0 and 1 Contents N9.3 Multiplication and division N9 Mental methods N9.1 Order of operations N9.2 Addition and subtraction N9.5 Problems and puzzles

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© Boardworks Ltd 2004 44 of 63 Multiplying by multiples of 0.1 and 0.01 Multiplying by 0.1Dividing by 10is the same asMultiplying by 0.01Dividing by 100is the same as What is 4 × 0.8? 4 × 0.8 =4 × 8 × 0.1 = 32 × 0.1 = 32 ÷ 10 = 3.2 What is 15 × 0.03? 15 × 0.03 =15 × 3 × 0.01 = 45 × 0.01 = 45 ÷ 100 = 0.45

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© Boardworks Ltd 2004 45 of 63 Dividing by multiples of 0.1 and 0.01 Dividing by 0.1Multiplying by 10is the same asDividing by 0.01Multiplying by 100is the same as What is 36 ÷ 0.4? 36 ÷ 0.4 =36 ÷ 4 ÷ 0.1 = 9 ÷ 0.1 = 9 × 10 = 90 3 ÷ 0.02 =3 ÷ 2 ÷ 0.01 = 1.5 ÷ 0.01 = 1.5 × 100 = 150 What is 3 ÷ 0.02?

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© Boardworks Ltd 2004 46 of 63 Multiplying by small multiples of 0.1

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© Boardworks Ltd 2004 47 of 63 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.

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© Boardworks Ltd 2004 48 of 63 Noughts and crosses 2

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© Boardworks Ltd 2004 49 of 63 N9.5 Problems and puzzles Contents N9.4 Numbers between 0 and 1 N9.3 Multiplication and division N9 Mental methods N9.1 Order of operations N9.2 Addition and subtraction

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© Boardworks Ltd 2004 50 of 63 Chequered sums

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© Boardworks Ltd 2004 51 of 63 Arithmagons – whole numbers

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© Boardworks Ltd 2004 52 of 63 Arithmagons - decimals

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© Boardworks Ltd 2004 53 of 63 Arithmagons –integers

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© Boardworks Ltd 2004 54 of 63 Arithmagons – two significant figures

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© Boardworks Ltd 2004 55 of 63 Circle sums – whole numbers

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© Boardworks Ltd 2004 56 of 63 Circle sums - integers

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© Boardworks Ltd 2004 57 of 63 Circle sums – one decimal place

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© Boardworks Ltd 2004 58 of 63 Circle sums –two decimal places

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© Boardworks Ltd 2004 59 of 63 Productagons – using times tables

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© Boardworks Ltd 2004 60 of 63 Productagons – using factors

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© Boardworks Ltd 2004 61 of 63 Productagons – using partitioning

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© Boardworks Ltd 2004 62 of 63 Productagons – using place value

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© Boardworks Ltd 2004 63 of 63 Product triangle

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