Progression in Calculations

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x. -. ÷. +. Birchwood Primary School. Progression in Calculations. Introduction.

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x

-

÷

+

Birchwood Primary School

Progression

in Calculations

Introduction

Written methods of calculations are based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly.These skills lead on to more formal written methods of calculation.Strategies for calculation need to be supported by familiar models, images and practical activities to reinforce understanding. When teaching a new strategy it is important to start with numbers the child can easily manipulate so that they can understand the concept. The transition between levels should not be hurried. Children should not be moved onto the next method until they are secure in their current method and are able to explain it. Previous levels should be revisited to consolidate understanding when introducing a new strategy.A sound understanding of the number system is essential for children to carry out calculations efficiently and accurately.

Maths @ Birchwood
• A Typical Birchwood maths ‘unit’ will follow a series of lessons!!
• We use the wheel to direct our teaching….

40

8

• Mental Skills
• Recognise the size and position of numbers
• Count on in ones and tens
• Know number bonds to 10 and 20
• Add multiples of 10 to any number
• Partition and recombine numbers
• Bridge through 10
• Models, Images and apparatus
• Place value apparatus - diennes
• Arrow cards
• Number tracks
• Numbered number lines
• Marked but unnumbered number lines
• Empty number lines
• Hundred square
• Counting stick
• Models and Images charts
• ITPs – Number Facts, Ordering Numbers, Number Grid, Counting on and back in ones and tens
• Key Vocabulary
• plus
• and
• count on
• more
• sum
• total
• altogether
• increase

+

+

0

1

2

3

4

5

6

7

8

9

10

One more than three is four

1, 2, 3, 4, 5, 6 … there are 6 teddies

P8 or ELG’s

Recognise numbers 0 to 10

Level 1c

Count reliably up to 10 everyday objects

Level 1c

Find one more than a number

Level 1c

Begin to relate addition to combining two groups of objects

makes 5

and

Level 1b

Count in ones and tens

Level 1b

Count along a number line to add numbers together

3 + 2 = 5

Level 1b

6 + 4 = 10

Begin to use the + and = signs to record mental calculations in a number sentence

+

=

+

=

1

1

2

2

3

3

Level 1a

Know that addition can be done in any order

Level 1a

3 +5

Put the biggest number first and count on

Level 2c

Know doubles of numbers

+

5 5 = 10

Level 2b

8 + 7 = 15

Add two single-digit numbers that bridge 10

5

1

5

9

6

2

4

8

3

7

Level 2b

Know by heart all pairs of numbers with a total of 10

Level 2a

Begin to partition numbers in order to add

5

15

11

15

9

6

12

14

8

13

7

15

16

17

18

25

26

27

28

15

25

28

Level 2a

15 + 1 = 16

Know which digit changes when adding 1s or 10s to any number

15

16

15 + 10 = 25

25

15 + 20 = 35

15

25

35

Level 2a

Adding two two-digit numbers (without bridging)

Counting in tens and ones

Partitioning and recombining

15 + 13 = 28

Level 3c

Know by heart all pairs of numbers with a total of 20

6 and how many more make 20?

40

8

30

6

Level 3c

Add near multiples of 10 to a 2 digit number

25 + 21 = 46

Level 3c

Adding two two-digit numbers (bridging through tens boundary)

Using a number line

OR

Using place value cards and Dienes to partition numbers and recombine

48 + 36 = 84

40 + 30 + 8 + 6

40 + 30 = 70

8 + 6 = 14

70 + 14 = 84

Level 3b

T

U

Expanded method

It is important that the children have a good understanding of place value and partitioning using concrete resources and visual images to support calculations. The expanded method enables children to see what happens to numbers in the standard written method.

48 + 36

48

+36

T U

40 + 8

30 + 6

80 + 4

10

This method should also be used for adding HTU to HTU

Level 3a

4 8

+ 3 6

8 4

Standard written method

The previous stages reinforce what happens to the numbers when they are added together using more formal written methods.

1

This method should also be used for adding HTU to HTU

Level 4c

528 + 288

500 + 200 = 700

20 + 80 = 100

8 + 8 = 16

= 816

They must also add HTU mentally, starting wit the most significant digit first.

Level 4b +

£4.85

+ £3.38

£8.23

1 1

40

8

• Progression in Teaching Subtraction
• Mental Skills
• Recognise the size and position of numbers
• Count back in ones and tens
• Know number facts for all numbers to 20
• Subtract multiples of 10 from any number
• Partition and recombine numbers (only partition the number to be subtracted)
• Bridge through 10
• Models, Images and apparatus
• Place value apparatus - diennes
• Arrow cards
• Number tracks
• Numbered number lines
• Marked but unnumbered lines
• Hundred square
• Empty number lines.
• Counting stick
• Models and Images Charts
• ITPs – Number Facts, Counting on and back in ones and tens, Difference
• Key Vocabulary
• Subtract
• Take away
• Minus
• Count back
• Less
• Fewer
• Difference between

-

-

10

9

8

7

,

,

,

...

P8 or ELG’s

Ten green bottles hanging on the wall …

Five fat sausages frying in a pan …

Begin to count backwards in familiar contexts such as number rhymes or stories

Level 1c

Continue the count back in ones from any given number

Level 1c

Three teddies take away two teddies leaves one teddy

Begin to relate subtraction to ‘ taking away ’

Level 1c

Find one less than a number up to 10

Level 1b

Count back in tens

Level 1b

6 - 4 = 2

Begin to use the – and = signs to record a number sentence using numbers up to 10 whilst still using practical resources

The difference between 8 and 5 is 3

Level 1b

1

1

2

2

3

3

Compare 2 sets to find numerical difference

Level 1a

Maria had six sweets and she ate four. How many did she have left?

Understand the operation of subtraction and use related vocab. Subtract numbers when solving problems involving up to 10 objects in a range of contexts

6 - 4 = 2

Level 2c

13 – 1 =

21 – 10 =

Within the range 1- 30 say 1 or 10 less than any number

Level 2b

7 – 2 = 5

Know by heart subtraction facts for numbers up to 10

Level 2b

15 - 7 = 8

Subtract single digit numbers often bridging through 10

Level 2b

Subtract 1 from a two-digit number

45 - 1

Level 2a

Begin to find the difference by counting up from the smallest number

Level 2a

Subtract 10 from a two-digit number

20

30

43 – 23

25

45

20

23

33

43

43 –

43 – 20 = 23

23 – 3 = 20

20

3

- 3

- 10

- 10

Level 2a

Subtract multiples of 10 from any number

Level 2a

Begin to partition numbers in order to take away

Partition the number to be subtracted (no exchanging)

Level 3c

Decide whether to count on or count back

74 - 27 = 47

T

U

43 - 27 = 16

-

2

7

to subtract 7 units we need to exchange a ten for ten units

40 + 3

- 20 + 7

10 + 6

30

10 +

Level 3c

Subtract mentally a near multiple of 10 from a 2 digit number

56 – 21 = 35

Level 3c

Know by heart subtraction facts for numbers up to 10

Level 3b

Expanded method

It is important that the children have a good understanding of place value and partitioning using concrete resources and visual images to support calculations. The expanded method enables children to see what happens to numbers in the standard written method.

4 3

- 2 7

1 6

3

1

Level 3a

Standard written method

The previous stages reinforce what happens to numbers when they are subtracted using more formal written methods. It is important that the children have a good understanding of place value and partitioning.

Level 4b

Subtract decimals to 2 places

²3 .¹79ml

- 2. 86ml

______

0 .93ml

https://www.ncetm.org.uk/resources/40532

40

8

Progression in Teaching Multiplication

Mental Skills

Recognise the size and position of numbers

Count on in different steps 2s, 5s, 10s

Double numbers to 10

Quick recall of multiplication facts

Use known facts to derive associated facts

Multiplying by 10, 100, 1000 and understanding the effect

Multiplying by multiples of 10

Models, Images and apparatus

Place value apparatus

Arrays

100 squares

Number tracks

Numbered number lines

Marked but unnumbered lines

Empty number lines.

Multiplication squares

Counting stick

Models and Images charts

ITPs – Multiplication grid, Number Dials, Multiplication Facts

Vocabulary

Lots of

Groups of

Times

Multiply

Multiplication

Multiple

Product

Once, twice, three times

Array, row, column

Double

x

x

0

2

4

6

8

0

5

10

15

20

25

Level 1b

Count in tens from zero

0

20

30

40

50

Level 2c

Count in twos from zero

Level 2c

Count in fives from zero

Level 2b

Know doubles and corresponding halves to 20

2 2 2 2

+

+

+

2 4

x

4 2

x

Level 2a

2 + 2 + 2 + 2 = 8

4 x 2 = 10

2 multiplied by 4

4 lots of 2

Level 2a

Understand multiplication as an array

Level 2a

Understand how to represent arrays on a number line

x 5

Level 2a

2 x 5 = 10

6 x 5 = 30

Tables spider

3 x 5 = 15

4 x 5 = 20

10 x 5 = 50

8 x 5 = 40

Know by heart facts for the 2,5 & 10 multiplication tables

5 x 5 = 25

Level 3c

Use known facts to work out new ones

Use known facts and place value to carry out mentally simple x calculations

Level 3c

Know by heart facts for the 2,3,4,5 & 10 multiplication tables

10

10

3

3

4

4

40

12

10

3

40

12

4

10

10

3

4

40

40

12

20 ( 2 x 10 )

3

4

80

12

Level 3c

Use place value apparatus to support the multiplication of U x TU

4 x 13

Level 3c

Use place value apparatus to support the multiplication of U x TU alongside the grid method

4 x 13

40 + 12 = 52

Level 3b

Use place value apparatus to represent the multiplication of U x TU alongside the grid method

4 x 23

80 + 12 = 92

30

3

10

300

30

= 330 +

4

120

12

= 132

462

56 × 27 1120 (56 × 20) 392 (56 × 7)1512

1

Level 3a +

14 x 33

Multiplying TU x TU

300

120

30

+ 12

462

This may be as far as many children progress in their use of multiplication.

Level 4c

Standard written method

Level 4b

Use factors to multiply

Understand that …

24 x 20 = 24 x 2 x 10

24 x 50 = 24 x 5 x 10

https://www.ncetm.org.uk/resources/40530

40

8

Progression in Teaching Division

Mental Skills

Recognise the size and position of numbers

Count back in different steps 2s, 5s, 10s

Halve numbers to 20

Recognise division as repeated subtraction

Quick recall of division facts

Use known facts to derive associated facts

Divide by 10, 100, 1000 and understanding the effect

Divide by multiples of 10

Models, Images and apparatus

Counting apparatus

Arrays

100 squares

Number tracks

Numbered number lines

Marked but unnumbered lines

Empty number lines.

Multiplication squares

Models and Images charts

ITPs – Multiplication, remainders grid, Number Dials, Grouping,

Vocabulary

Lots of

Groups of

Share

Shared between

Group

Divide

Divide into

Division

Divided by

Remainder

Factor

Quotient

Divisible

÷

÷

0

10

20

30

?

2

4

6

8

Level 1c

Give every bear a sweet – how many sweets do we need?

Solve problems by sharing objects in a practical or role play context

Level 1b

Count back in tens

Give each bear 2 sweets – how many sweets do we need?

Level 1a

Solving practical problems by sharing into equal groups

Level 2c

Count back in twos

Level 2c

Count back in fives

0

5

10

15

Level 2b

Half of 6 is 3

½ of 6 = 3

Know halves to 20

Level 2a

Understand division as sharing

1 2 3 4 5

6 7 8 9 10 etc….

Level 2a

Understand division as grouping

1,2,3,4,5 6,7,8,9,10 etc

Level 2a

12 divided into groups of 3 gives 4 groups

12  3 = 4

12 divided into groups of 4 gives 3 groups

12  4 = 3

Reinforce division as grouping through the use of arrays

0

3

6

9

12

15

18

Level 3c

18 divided into groups of 3

18  3 = 6

Represent ‘groups’ for division on a number line using apparatus alongside the line to include remainders.

18  3 = 6

-3

-3

-3

-3

-3

-3

0

18

3

6

9

12

15

18  6 = 3

Level 3b

126 ÷ 21 = 6

-21

-21

-21

-21

-21

-21

84

105

126

0

21

42

63

180 ÷ 60 = 30

Level 3a

126 ÷ 21 = 6

(2 x21)

(2 x21)

(2 x21)

0

42

84

126

https://www.ncetm.org.uk/resources/43589

Level 4c

This must not be introduced to the children before Year 5

18

• 18 ÷3 = 6
• 1 8
• - 3 ( 1 x 3 )
• 1 5
• - 3 ( 1 x 3 )
• 1 2
• - 3 ( 1 x 3 )
• 9
• - 3 ( 1 x 3 )
• 6
• - 3 ( 1 x 3 )
• 3
• - 3 ( 1 x 3 )
• 0

(1 x 3)

15

Understand division as repeated subtraction using a vertical line and apparatus to make the links

(1 x 3)

12

(1 x 3)

9

(1 x 3)

6

(1 x 3)

3

(1 x 3)

0

Level 4b

What facts do I know about the 7 times-table?

Year 5 and 6 only once confident with previous method

Children need to see that as the numbers get larger, large chunk subtraction is the more efficient method. Multiples of the divisor (large chunks) are taken away. Multiplication facts are needed to see the size of the ‘chunk’.

Fact Box

1 x 7 = 7

2 x 7 = 14

5 x 7 = 35

10 x 7 = 70

20 X 7 = 140

50 x 7 = 350

100 x 7 = 700

518 ÷ 7 = 74

518

- 350 ( 50 x 7 )

168

- 140 ( 20 x 7 )

28

- 28 ( 4 x 7 )

0

100 ÷ 7 = 14 r 2

100

- 70 ( 10 x 7 )

30

- 28 ( 4 x 7 )

2

Level 4a

Year 6 Top set only

560 ÷ 24

2 3 r 8

2 4 5 6 0

4 8 0

8 0

7 2

8

Standard written method

Links directly to large chunk subtraction

How you can support your child
• Look for and talk about numbers in the environment
• Play games
• Shopping
• Counting on/back
• Number bonds
• Doubles/Halves
• Times tables
• Division facts
Websites
• http://nrich.maths.org/public/
• http://www.bbc.co.uk/schools/ks2bitesize/maths/number.shtml
• http://www.teachingtime.co.uk/
• http://www.teachingtables.co.uk