© boardworks ltd 2004 1 of 63 n9 mental methods ks3 mathematics

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N9 Mental methods

KS3 Mathematics

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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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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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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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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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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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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 + 87

= (13 + 8) ÷ 7

= 21 ÷ 7

= 3

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Using a division line

What is ?24 + 824 – 8

Again, the dividing line acts as a bracket.

= 32 ÷ 16

= 2

= (24 + 8) ÷ (24 – 8)24 + 824 – 8

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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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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

then Division and Multiplication,= 100 – 98

and then Addition and Subtraction= 2

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BIDMAS

Remember BIDMAS: RACKETS

NDICES (OR POWERS)

IVISION

ULTIPLICATION

DDITION

UBTRACTION

BBIIDDMMAASS

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Using BIDMAS

What is(3.4 + 4.6)2

(6 + 5 × 2)– 8 × 0.5 ?

Brackets first,

then Indices,

then Division and Multiplication,

82

16– 8 × 0.5=

64

4– 8 × 0.5=

= 16 – 4

= 12and then Addition and Subtraction

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Using a calculator

We can use a calculator to evaluate more difficult calculations.

For example, (52 + 72)

7 - 5

This can be entered as:

( 5 x2 + 7 x2 ) ÷ ( 7 - 5 )

= 4.3 (to 1 d.p.)

Always use an approximation to check answers given by a calculator.

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Positioning brackets

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Target numbers

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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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Complements match

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Counting on and back

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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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Adding by counting up

What is 276 + 68?

276

+ 60

336

+ 8

344

What is 63.8 + 4.7?

63.8 67.8

+ 0.7

68.5

+ 4

276 + 60 + 8 = 344

63.8 + 4 + 0.7 = 68.5

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Using compensation to add

What is 276 + 68?

276 + 70 – 2

276 344

– 2

346

= 344

What is 63.8 + 4.7?

63.8 + 5 – 0.3

63.8 68.5

– 0.3

68.8

= 68.5

+ 70

+ 5

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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

20.5

134

– 0.4

– 7

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Subtracting by counting up

437 537

+ 3

540

6.4 16.4

+ 6

22.5

What is 564 – 437?

What is 22.5 – 6.4?

+ 100

+ 10

+ 20 + 4

560 564

100 + 3 + 20 + 4 = 127

+ 0.1

22.4

10 + 6 + 0.1 = 16.1

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Using compensation to subtract

What is 564 – 437?

564 – 500 + 63

64 564

= 127

What is 22.5 – 6.4?

22.5 – 6.5 + 0.1

16 16.1 22.5

= 16.1

127

+ 0.1

+ 63 – 500

– 6.5

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Addition pyramid

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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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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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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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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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Using a grid to multiply

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Using a grid to multiply

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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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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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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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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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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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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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Using fractions to divide whole numbers

We can simplify 420 ÷ 40 by writing the division as a fraction and then cancelling.

420 ÷ 40 = 42040

42040

21

2= 21

2

= 101/2

What is 420 ÷ 40?

= 10.5

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Using fractions to divide decimals

We can simplify 2.6 ÷ 0.8 by writing the division as a fraction.

2.6 ÷ 0.8 = 2.60.8

13

4= 13

4

= 31/4

What is 2.6 ÷ 0.8?

=

× 10

× 10

268

268

= 3.25

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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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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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Noughts and crosses 1

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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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Multiplying by multiples of 0.1 and 0.01

Multiplying by 0.1 Dividing by 10is the same as

Multiplying by 0.01 Dividing 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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Dividing by multiples of 0.1 and 0.01

Dividing by 0.1 Multiplying by 10is the same as

Dividing by 0.01 Multiplying 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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Multiplying by small multiples of 0.1

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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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Noughts and crosses 2

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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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Chequered sums

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Arithmagons – whole numbers

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Arithmagons - decimals

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Arithmagons –integers

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Arithmagons – two significant figures

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Circle sums – whole numbers

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Circle sums - integers

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Circle sums – one decimal place

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Circle sums –two decimal places

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Productagons – using times tables

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Productagons – using factors

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Productagons – using partitioning

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Productagons – using place value

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Product triangle