A Calculation Policy for Church Road School

Understanding Mathematics Team July 2014

This Calculation Policy has been revised to take account of the new National Curriculum Programme of Study, published in 2013. Written calculations methods for the 4 rules of number need to be built on mental strategies. The principal focus of mathematics teaching in KS1 is to ensure that pupils develop confidence and mental fluency with whole numbers, counting and place value. This should involve working with numerals, words and the four operations, including with practical resources (for example: concrete objects and measuring tools). Written calculations should always be presented horizontally so that children use the most efficient method. If a calculation is presented vertically, children will automatically do a written calculation, which may not be the most efficient. Children should always be encouraged to ask :

Can I do it in my head?



Do I need jottings to help me use mental methods?



Would a written method be the most efficient for this calculation?

Estimation – “Is my answer sensible?” Children should be encouraged to estimate from the earliest stages. This policy is organised into steps and the suggested year group is written next to each step. This is a guide only as children will progress at different rates. Addition and subtraction Always build on mental strategies. Assess the children’s knowledge and understanding.

Can the children 

Count forward and backward in 1’s and 10’s?



+10, +1, -10, -1 to/from any two-digit number?



Partition and recombine numbers?



Use the mathematical language of addition and subtraction confidently?

Do the children 

Know number bonds to 10, 20 and 100?



Have a sense of the relative size of numbers?

Is their knowledge of place value and the relative value of digits secure? How good are their estimation skills?

Addition Step 1 (EYFS) No formal recording of addition. Lots of practical and oral work using the language and vocabulary of addition. Teacher models written addition – 3 cars + 4 cars = 7 cars Step 2 (EYFS) Continue with practical work. Lots of oral work, using the language and vocabulary of addition. Introduce the symbols + = Children draw informal pictures of 3 + 4 = 7

Children use a printed number line when they are ready. They draw their own number line to record steps in addition. The steps often bridge through a multiple of 10. (Introduced in Year 1)

8 + 7 = 15

Step 3 (Year 2)

More practical work. Use number squares and number lines to add two digit numbers. Draw own number lines to add 2 two digit numbers. 48 + 36 = 84 Children refine their number lines.

or

Draw own number lines to add a two digit number to a three digit number. 436 + 75 Draw own number lines to add a three digit number to a three digit number and refine their number lines.

456 + 373

Step 4 (Year 2) Partitioning 



The next stage is to record mental methods using partitioning. Add the tens and then the ones to form partial sums and then add these partial sums. 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.

48+36 = 48+30 +6 = 78 + 6 = 84 Add TU by partitioning. Add the most significant figure first – tens then ones. 48 + 36 40 + 30 = 70 8 + 6 = 14 70 + 14 = 84 Add HTU by partitioning. Add the most significant figure first – hundreds, then tens, then ones. 456 + 373 400+300=700 50+70=120 6+3=9 700+120+9=829 Step 5 (Year 2)

Partitioned numbers are then written under one another: Expanded method in columns 



Move on to a layout showing the addition of the tens to the tens and the ones to the ones separately. To find the partial sums the tens are added first and then the ones. As children gain confidence, ask them to start by adding the ones digits first always. The addition of the tens in the calculation 47 + 76 is described in the words 'forty plus seventy equals one hundred and ten', stressing the link to the related fact 'four plus seven equals eleven'.



When the children are confident and secure with the expanded method, they learn 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.

Adding the tens first (building on mental strategies):

Then adding the ones first:

Discuss how adding the ones first gives the same answer as adding the tens first. Refine over time to adding the ones digits first consistently.

Step 6 (Year 3) Column method 

 

In this method, recording is reduced further. Carry digits are recorded below the line, using the words 'carry ten' or 'carry one hundred', not 'carry one'. Later, extend to adding three two-digit numbers, two three-digit numbers and numbers with different numbers of digits. The words forty add seventy are used NOT four add seven.

Use column addition to add £, p. By the end of Year 4, children will be using the column method to add 4 digit numbers.

Step 7 (Introduced in Year 4) Use column addition to add decimals with up to two places.

4.3 +6.2 10.5

34.75 +23.6 58.35 1

58.69 35.73 +94.42 11 1

Step 8 (Years 5 and 6) Reinforce and use efficient method of addition.

Subtraction From Reception, subtraction is taught as “take away” or “what is the difference?” Both meanings and methods are taught side by side. The relative size of numbers is taught so that children can decide to count on or take away. Step 1 (EYFS) No formal recording of subtraction. Lots of practical and oral work using the language and vocabulary of subtraction. Teacher models written subtraction. 7cars – 3 cars = 4 cars

Step 2 (EYFS) Continue with practical work. Lots of oral work using the language and vocabulary of subtraction. Symbols - = are introduced for the children to write. Children draw informal pictures of 7 – 4 = 3

Children count back on a printed number line when ready. Step 3 (Year1) There is more practical work. Children use printed number lines and 100 squares to subtract U from TU by counting back. Then they subtract TU from TU by counting back. Children draw their own number lines to subtract U from TU, then TU from TU. They need to understand counting back methods before counting on.

Counting back 15 - 7 = 8

74 - 27 = 47

The steps may be recorded in a different order:

Or combined:

Counting on

Step 4 (Year 2)

Step 5 (Year2) Expanded layout (leading to column method) (No addition signs when partitioning)   

Children partition the numbers into tens and ones and write one under the other mirroring the column method, where ones are placed under ones and tens under tens. This does not link directly to mental methods of counting back or up but parallels the partitioning method for addition. It also relies on secure mental skills. 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 with partitioning.

Partitioned numbers are written under one another:

74-23= 74 -23

70 4 -20 3 50 1

74 – 27= 74 - 27

70 60 20

14 7

This should be explained in the following way: As it is not possible to subtract 7 from 4, the tens of the 74 need to be partitioned into 60 and 10. The 10 is moved across to the ones so the 4 becomes 14. We need to show this by crossing out the 70 and putting 60, then put a 1 (representing the 10) in front of the 4, making 14. (It is good practice to make the 1 as large as the 4 as this makes it clear the number is now 14). The 7 can now be subtracted from the 14.

563-241

563 -241

addition signs can be used to show partitioning.

563 − 271, adjustment from the hundreds to the tens, or partitioning the hundreds 563 -271 Begin by reading aloud the number from which we are subtracting: 'five hundred and sixty-three'. Then discuss the hundreds, tens and ones components of the number, and how 500 + 60 can be partitioned into 400 + 160. The subtraction of the tens becomes '160 minus 70', an application of subtraction of multiples of ten.

563 − 278, adjustment from the hundreds to the tens and the tens to the ones 563 -278 Here both the tens and the ones digits to be subtracted are bigger than both the tens and the ones digits you are subtracting from. Discuss how 60 + 3 is partitioned into 50 + 13, and then how 500 + 50 can be partitioned into 400 + 150, and how this helps when subtracting. 503 − 278, dealing with zeros when adjusting 503 -278 Here 0 acts as a place holder for the tens. The adjustment has to be done in two stages. First the 500 + 0 is partitioned into 400 + 100 and then the 100 + 3 is partitioned into 90 + 13. Step 6 (Year 3) Children use the formal column method without decomposition TU – TU 68 -32 36 HTU – TU 578 - 25 553

HTU - HTU 578 -325 253

Step 7 (Year 3) Children use the formal column method with decomposition. TU – TU

HTU - HTU

Step 8 (Year4) Children develop column subtraction to larger numbers and decimals.

Step 9 (Years 5 and 6) Children reinforce and use efficient method of subtraction.

Multiplication These notes show the steps in building up to using an efficient written method for multiplication. To multiply successfully, children need to be able to:       

recall all multiplication facts to 10 × 10 partition number into multiples of one hundred, ten and one work out products such as 70 × 5, 70 × 50, 700 × 5 or700 × 50 using the related fact 7 × 5 and their knowledge of place value add two or more single-digit numbers mentally add multiples of 10 (such as 60 + 70) or of 100 (such as 600 + 700) using the related addition fact, 6 + 7, and their knowledge of place value add combinations of whole numbers using the column method (see above). emphasise inverse operations from the earliest steps. 6 x 2 = 12 2 x 6 = 12 12 ÷ 6 = 2 12 ÷ 2 = 6

Note: It is important that children's mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for multiplication.

Multiplication Step 1 (Year 1) Group objects into 2s, 5s, and 10s. Teacher models the use of symbol x =

Step 2 (In no particular order – at the teacher’s discretion) Group counters, multilink etc. 3x4 ○○○ ○○○ ○○○ ○○○ Make and draw arrays ○○○ ○○○ ○○○ ○○○ Use repeated addition 3+3+3+3 Draw own number line for repeated addition.

Use the symbols x = Record number sentences

Step 3

3 x 4 = 12

3 x __ = 12

(Year 2)

Multiply one digit and two digit numbers by 10 by moving the digits one column to the left, using a place value grid. Multiply one digit and two digit numbers by 100 by moving the digits two columns to the left, using a place value grid.

Multiply TU x U by partitioning and recombining. 34 x 2 = 34

30

4

x2

60

8

= 68

Multiply HTU x U by partitioning and recombining. 342 x 4 = 342

Step 4

300

40

2

x4

1200

160

8

= 1368

(Year 3)

Multiply numbers up to 1000 by 10 then 100 by moving one/two columns to the left, using place value charts. Use the grid method to multiply TU x U, emphasising the link with partitioning.

x 40 3 5 200 15

Step 5

= 215

(Year 3)

Multiply whole numbers and decimals by 10, 100, 1000 by moving 1/2/3 columns to the left.

Use the grid method to multiply TU x TU.

x 20 7 50 1000 350 6 120 42

1350 162 1512 1

Use the grid method to multiply HTU x TU.

x 20 9 200 4000 1800 80 1600 720 6 120 54

5800 2320 174 8294 1

Step 6

(Year 4)

Introduce short multiplication TU x U and HTU x U 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 grid method of Step 4.

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.

Step 7 (Year 5)

Use the grid method to multiply decimals. Introduce long multiplication TU x TU showing expanded method first: 24 x 16 24 120 40 200 384

(6x 4) (6x 20) (10x 4) (10x20)

Then reduce it to standard written long multiplication:

24 x16 144 ² 240 384

Step 8 (Year 6) Introduce long multiplication HTU and ThHTU x TU Reinforce and use efficient method of multiplication including decimals and money.

Division These notes show the stages in building up to long division. To divide successfully in their heads, children need to be able to:   

 

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 partition two-digit and three-digit numbers into multiples of 100, 10 and 1 in different ways recall multiplication and division facts to 10 × 10, recognise multiples of onedigit numbers and divide multiples of 10 or 100 by a single-digit number using their knowledge of division facts and place value 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.

Note: It is important that children's mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for division .

Division Step 1 (EYFS) Lots of practical and oral work using the language and vocabulary of sharing. Groupings for play and and P.E. Practical activities with different numbers and types of objects to experience halving. Include activities involving measures; for example capacity, mass, money. Can you share these sweets between 2 people? Will there be any left over?

Step 2 (Y1) Share objects into equal groups. Children carry out practical tasks that involve sharing objects into equal groups to solve problems such as: ‘How many pencils are on each table if there are 4 tables and 12 pencils?’ They find combinations of groups of equal numbers of objects, such as working out the total number of blocks if there are three groups of five blocks, and they count in fives to check.

Step 3 (Y1) Share objects into equal groups

○○○ ○○○ ○○○ ○○○

Introduce and use the symbol ÷ Record number sentences e.g. 12 ÷ 3 = 4 Understand that division is repeated subtraction and calculate accordingly. 12 – 3 = 9 9–3=6 6–3=3 3–3=0

Step 4 (Y2) Chunk on a number line backwards and begin to relate division to multiplication using the correct mathematical language.

-3

-3

3 Record0as a written calculation: 12 ÷ 3 6= 4

-3

-3

9

12

Continue to discuss remainders as left or left over. Check the answer by going forward.

Step 5 (Y2) Continue to divide by chunking backwards on a number line. Know that division is the inverse of multiplication. Divide with remainders through chunking on a number line and recording.

-3

-2

0

2

-3

-3

8

5

11

11 ÷ 3 = 3 r2

Step 6 (Y3) Chunk backwards in larger groups, beginning with chunks of 10 then progressing to multiples of 10 and record as a division calculation.

-30

-30

10 x 3

10 x 3

-3

1x3

3 63  3 = 21

33

63

Step 7 (Y3) Division by chunking of TU ÷ U (informal written method) including remainders. 72  5 72 -50 (5 x 10) 22 -20 (5 x 4) 2 14

= 14 r 2

Step 8 (Y4) Short division of TU ÷ U eg: For 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.

The short division method is recorded like this :

This is then shortened to:

Step 9 (Y4/5) HTU ÷ U by chunking (expanded written method).

Step 10 (Y3/4) Short division of HTU ÷ U

For 291 ÷ 3, because 3 × 90 = 270 and 3 × 100 = 300, we use 270 and split the dividend of 291 into 270 + 21. Each part is then divided by 3.

The short division method is recorded like this:

This is then shortened to:

The carry digit '2' represents the 2 tens that have been exchanged for 20 ones. In the first recording above it is written in front of the 1 to show that a total of 21 ones are to be divided by 3. The 97 written above the line represents the answer: 90 + 7,or 9 tens and 7 ones.

Step 11(Y5/6) HTU ÷ TU

Long division (linking to chunking)

540 ÷ 24

24 × 20 = 480 and 24 × 30 = 720, so the answer lies between 20 and 30. So start by subtracting 480 from 560.

The recording above is the long division method. (Conventionally the digits of the answer are recorded above the line as shown below).

Reinforce and use efficient method.