Progression In Calculation – EYFS to Year 6. Aims To have an overview of the skills children need to calculate. To understand how to support your child.

Progression In Calculation – EYFS to Year 6

Aims

• To have an overview of the skills children need to calculate.

• To understand how to support your child with maths.

• To be more aware of the models, images and resources used to support the teaching and learning of maths.

• Think about the progression from mental towards written methods.

Session 1

Addition

Beginning to add

Practical, counting objects and relating addition to combining two groups of objects

Beginning to use a number track

Use of the number track- hopping and recording.(a) 2 and 3 makes 5 0 1 2 3 4 5 6 7 8 9 10

Mental Strategies for Addition

Secure mental addition requires the ability to: •recall key number facts instantly (number pairs to 10, 20 & 100, doubles etc) and to apply these to similar calculations•recognise that addition can be done in any order and use this to add mentally different combinations of one and two digit numbers• partition two-digit numbers in different ways, including adding the tens and units separately before recombining•understand the language of addition including more than, sum, plus, greater than, total, altogether etc)

Written methods for Addition

Stage 1: The empty number lineThe empty number line helps to record the steps on the way to calculating the total. The steps often bridge through a multiple of 10.

8 + 7 = 15

48 + 36 = 84 or:

Over to you!

Use a number line to find answers to these sums.

53 + 24

86 + 17

149 + 38


Stage 2: PartitioningThe next stage is to record mental methods using partitioning. Partitioning both numbers into tens and ones mirrors the column method where ones are placed under ones and tens under tens. This also links to mental methods. Eg: 47 + 76 = 47 + 70 + 6 = 117 + 6 = 123 or 47 + 76 = 40 + 70 + 7 + 6 = 110 + 13 = 123 Partitioned numbers are then written under one another:

Over to you!

Use partitioning to find answers to these sums.

65 + 38

71 + 26

94 + 45


Stage 3: Expanded method in columns Children can now move on to a layout showing the addition of the tens to the tens and the ones to the ones separately. Children should start by adding the ones digits first. NB The addition of the tens in the calculation 47 + 76 is described as 40 + 70 = 110 as opposed to 4 + 7 = 11.


Stage 3: Expanded method in columns The expanded method leads children to the more compact method so that they understand its structure and efficiency. The amount of time that should be spent teaching and practising the expanded method will depend on how secure the children are in their recall of number facts and in their understanding of place value.

Over to you!

Use the expanded column method to find answers to these sums.

65 + 38

123 + 59

315 + 172


Stage 4: Column methodIn this method, recording is reduced further. Carry digits are recorded below the line, using the words 'carry 10' or 'carry 100', not 'carry 1'. Later, extend to adding three two-digit numbers, two three-digit numbers, numbers with different numbers of digits and decimals.

Session2Subtraction and take away

Introducing ‘take away’Begins with practical demonstrations of subtraction relating to ‘take away’. Use of number tracks, pictures and songs (10 green bottles, 5 little speckled frogs).

Beginning to take away

• Number tracks leading to number lines introduced for recording ‘jumps’ back.

• 8-3=51 2 3 4 5 6 7 8

Mental Strategies for Subtraction

Secure mental subtraction requires the ability to: • recall key subtraction facts instantly (inverse of number pairs to 10, 20 & 100, halves etc) and to apply these to similar calculations• mentally subtract combinations of one and two digit numbers• understand that subtraction is the inverse of addition and recognise that subtraction can’t be done in any order (it has to start with the larger number) • understand the language of subtraction including less, minus, take away, difference between etc)

The problem with subtractionTypical Questions

• Sam has saved 57p. Her sister has saved 83pHow much more money does Sam have than his sister?• Samir is running a 50 metre potato race. He drops out after

18 metresHow much further does he have to go?• Nisha and Charlie weigh fruit. Nisha’s weighs 38g. Charlies

weighs 50g.How much heavier is Charlies fruit than Nishas?• One sunflower is now 38cm high. Another is 83cm high.What is the difference between the heights of the sunflowers?

Progression In Subtraction - Difference

Written methods for Subtraction

Stage 1: The empty number lineThe empty number line helps to record the steps in mental subtraction.

• Counting Up - the steps can also be recorded by counting up from the smaller number to find the difference

or


Stage 1: The empty number lineWith practice, children will need to record less information and decide whether to count back or forward. It is useful to ask children whether counting up or back is the more efficient for calculations such as 57 - 12, 86 - 77 or 43 - 28.

With three-digit numbers the number of steps can again be reduced, provided that children are able to work out answers to calculations such as 178 + ? = 200 and 200 + ? = 326 mentally.

or

Over to you!

When would you use a numberline?

59 - 1186 – 68142 – 3592-9


Expanded column method It can also be applied to three and four digit numbers. Example: 741 - 367


Expanded column method Depending on the numbers it can get quite complicated and this stage may need a lot of time and perseverance!

Over to you!

Use the expanded column method to find answers to these sums.

73 - 39

123 - 58

315 - 177


Stage 4: Column methodThe expanded method is eventually reduced to:

Over to you!

Use the compact column method to find answers to these sums.

83 - 58

166 - 47

402 - 175

Session 3

Multiplication

Beginning to multiplyWhen we begin to multiply we start by counting in steps of 2, 3 across a numberline.

Mental Strategies for Multiplication

To multiply successfully, children need to be able to:• recall all multiplication facts to 10 × 10• apply times tables facts to similar calculations such as 7 x 5 70 × 5, 70 × 50, 700 × 5 or 700 × 50 using their knowledge of place value;• partition numbers into multiples of Hundreds, Tens and Units• add two or more single-digit numbers, multiples of 10 and 100 and combinations of whole numbers using the column method.• understand the language of multiplication including lots of, groups of, times, multiply, product

Written methods for Multiplication

Initially multiplication is introduced as ‘repeated addition’ using vocabulary such as ‘lots of’ or ‘groups of’ and real objects or pictures.

3 lots of 3 = 9 leading to 3 x 3 = 9

Solving Multiplication Calculations – before written methods

Use of arrays to solve simple problems • • • • • • • • 4 x 2 = 8 2 x 4 = 8

5x6=30 0 5 10 15 20 25 30

Use a numberline to multiply


In KS2 the aim is that children develop rapid recall of all times tables to 12 x 12 and can use an efficient written method for • two-digit by one-digit multiplication by the end of Year 4 (TU x U)•two-digit by two-digit multiplication by the end of Year 5 (TU x TU)•three-digit by two-digit multiplication by the end of Year 6 (HTU x TU)


Stage 1: Mental multiplication using partitioningThis allows the tens and ones to be multiplied separately to form partial products. These are then added to find the total product. Either the tens or the ones can be multiplied first but it is more common to start with the tens. This can look like......

14 3 (10 4) 3(10 3) (4 3) 30 12 42


Stage 2: The Grid MethodThis links directly to the mental method. It is an alternative way of recording the

same steps. It is better to place the number with the most digits in the left-hand column of the grid so that it is easier to add the partial products. For TU x TU, the partial products in each row are added, and then the two sums at the end of each row are added to find the total product


The next step is to move the number being multiplied (38 in the example shown) to an extra row at the top. Presenting the grid this way helps children to set out the addition of the partial products in preparation for the standard method.

Over to you!

Have a go at solving these multiplications using the grid method.

65 x 8

74 x 45

92 x 53


Stage 3: Expanded short multiplicationThe next step is to represent the method in a column format, but showing the working. Attention should be drawn to the links with the grid method above. Children should describe what they do by referring to the actual values of the digits in the columns. For example, the first step in 38 × 7 is ‘thirty multiplied by seven’, not ‘three times seven’, although the relationship 3 × 7 should be stressed. Most children should be able to use this expanded method for TU × U by the end of Year 4.

30 8 7210 30 7 210

56 8 7 56266

38 7210

56266


Stage 3: Expanded short multiplicationThe same steps can be used when introducing TU x TU.

1

56 27

1000 50 20 1000120 6 20 120350 50 7 350

42 6 7 421512

1

56 27

1120 56 20 392 56 7

1512

Over to you!

Have a go at solving these multiplications using the expanded short method.

32 x 8

56 x 15

78 x 37


Stage 4: Short multiplicationThe expanded method is eventually reduced to the standard method for short multiplication. The recording is reduced further, with carry digits recorded below the line. If, after practice, children cannot use the compact method without making errors, they should return to the expanded format of stage 3.

5

38 7266

The step here involves adding 210 and 50 mentally with only the 5 in the 50 recorded. This highlights the need for children to be able to add a multiple of 10 to a two-digit or three-digit number mentally before they reach this stage


Stage 5: Long multiplicationThis is applied to TU x TU as follows.

The carry digits in the partial products of 56 × 20 = 120 and 56 × 7 = 392 are usually carried mentally.The aim is for most children to use this long multiplication method for TU × TU by the end of Year 5.

1

56 27

1120 56 20 392 56 7

1512


In Year 6, children apply the same steps to multiply HTU x TU

1

286 29

4000 200 20 40001600 80 20 1600

120 6 20 1201800 200 9 1800

720 80 9 720 54 6 9 54

8294

1

286 29

5720 286 20 2574 286 9

8294

Start with the grid method, asking the children to estimate their answer first.

This expanded method is cumbersome, so there is plenty of incentive to move on to a more efficient method.

Children who are already secure with multiplication for TU × U and TU × TU should have little difficulty in using the same method for HTU × TU.

Session 4

Division

Last but not least……

• Many children can partition and multiply with confidence. But this is not the case for division. One reason for this may be that mental methods of division, stressing the correspondence to mental methods of multiplication, have not in the past been given enough attention.

• The aim is that children use mental methods when appropriate, but for calculations that they cannot do in their heads they use an efficient written method accurately and with confidence.

• The stages building up to long division span Years 4 to 6 - first introducing TU ÷ U, then extending to HTU ÷ U, and finally HTU ÷ TU.

Mental Strategies for Division

To divide successfully, children need to be able to: • partition two-digit and three-digit numbers into multiples of 100, 10 and 1 • recall multiplication and division facts to 10 × 10 and recognise multiples of one-digit numbers• know how to find a remainder working mentally - for example, find the remainder when 48 is divided by 5; • understand and use multiplication and division as inverse operations.• understand and use the vocabulary of division - for example in 18 ÷ 3 = 6,the 18 is the dividend, the 3 is the divisor and the 6 is the quotient;

Written methods for Division

Initially division is introduced as ‘sharing’ using real objects or pictures.

Share 10 apples equally between 2 children which eventually becomes 10 ÷ 2 = 5

Beginning to divide

Solve problems using repeated addition along a number line. +5 +5 +5 +5 +5 +5 30 ÷ 5 = 6 0 5 10 15 20 25 30

Finding remainders after simple division Know when to round the remainder up or down, depending on context of problem +4 +4 +4 rem 2 14 ÷ 4 = 3 r 2 0 4 8 12 14


To carry out written methods of division successful, children also need to be able to:

• understand division as repeated subtraction; • estimate how many times one number divides into another - for example, how many 6s there are in 47, or how many 23s there are in 92; • multiply a two-digit number by a single-digit number mentally; • subtract numbers using the column method.


Stage 3: 'Expanded' method for TU ÷ U and HTU ÷ UThis method, often referred to as 'chunking', is based on subtracting multiples of the divisor, or 'chunks'. It is useful for reminding children of the link between division and repeated subtraction. However, children need to recognise that chunking is inefficient if too many subtractions have to be carried out.


Refining the 'Expanded' method for HTU ÷ UInitially children subtract several chunks, but with practice they should look for the biggest multiples that they can find to subtract, to reduce the number of steps.Once they understand and can apply the expanded method, children should try the standard method for short division. For most children this will be at the end of Year 5 or the beginning of Year 6

Over to you!

Have a go at solving these divisions using the short method for TU ÷ U.

76 ÷ 8

91 ÷ 7

92 ÷ 4


Stage 2: Short division of TU ÷ UFor 81 ÷ 3, the dividend of 81 is split into 60, the highest multiple of 3 that is also a multiple 10 and less than 81, to give 60 + 21.Each number is then divided by 3.

leading to


Stage 4: Long division for HTU ÷ TUThe next step is to tackle HTU ÷ TU, which for most children will be in Year 6. The layout on the right, which links to chunking, is in essence the 'long division' method. Conventionally the 20, or 2 tens, and the 3 ones forming the answer are recorded above the line, as in the second recording.

Over to you!

Have a go at solving these divisions using ‘chunking’.

86 ÷ 7

156 ÷ 5

178 ÷ 8

Progression In Calculation – EYFS to Year 6. Aims To have an overview of the skills children need to calculate. To understand how to support your child.

Documents