PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images

Recording Examples

1. Say the number names backwards in order starting from ten / twenty.

0 - 10 progressing to 20

May assign number symbol to number spoken but not necessarily.

N/A

2. Say the number names backwards in order starting from any given number.

0 - 10 progressing to 20

Beginning to assign number symbol to number spoken e.g. can pick out number 6 on a number track when asked.

N/A

‘Find me number 6’

3. Use some language of quantity such as ‘less than’, ‘fewer’ or ‘least’.

N/A Real life opportunities e.g. There are fewer pencils in one pot than the other

N/A I have fewer/ less raisins than you do!

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images

Recording Examples

4. Recognise that a group of things changes in quantity when something is taken away.

0 - 10 Real life opportunities e.g.

There are fewer bricks in this tower because I took some / 4 away.

N/A There are 5 cars in the car park. 2 drive away. How many are left?

5. Find the number that is one less than a given number with practical objects.

0 - 10 1 less than 4 is 3.

N/A

One less than five is four. Ordering numicon and finding ‘one less’.

6. Physically take away a given number of practical objects from a set and say how many are left This should go alongside stage 7.

0 - 5, progressing to 10 This should go alongside counting back within 10

Concrete objects 5 - 2 =

N/A

7 - 3 =

5 - 3 = Encourage children to move and touch as they count.

Take one peg away.

What shape fits over the

remaining pegs?

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images

Recording Examples

7. Use number stories and rhymes to reinforce physical takeaway. This should go alongside stage 6.

0 - 10 10 cheeky monkeys jumping on the bed, Frogs on a log etc…

N/A Use physical objects to represent the monkeys/ frogs etc and remove them as the song is sung: ‘9 little speckled frogs, sat on a speckled log, etc…’

1 2 3 4 5 6 7 8 9

8. Placing concrete objects onto a structured number track remove items from the track to take away. .

1 - 10 Progressing to 20 (may count beyond this range).

Link to maths outdoors- jumping back on a number

track Jumping back- to

consolidate ‘one less’ etc…

Practical

9 – 6 = 3 Introduce board games at this point to reinforce number order and link to number tracks- eggs snakes and ladders, ‘counting back to fall down the snake’.

Link to ‘role play’ e.g. place 10 chairs out to make a bus; 10 children sit in, then one gets off the bus. Children must see the number track ‘in context’ and should not be moved on to ‘abstract’ models / images in isolation.

1 2 3 4 5 6 7 8 9

9, 8, 7 ….

E.g. Remove cubes etc… from the track to show 9 – 6 = 3 Then, reinforce by counting back from 9 in ones (8, 7, and 6). Reinforce 9 as the starting point, not part of the jump back.

Begin to count backwards in familiar contexts such as number

rhymes or stories.

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images

Recording Examples

9. Subtraction as ‘hiding’ an amount. How many left? Begin to show link to addition E.g. 4 – 3 = 1 1 + 3 = 4

1 - 10 Progressing to 20 (may count beyond this range).

4 – 1 = 3

4 pencils are in the pot.

‘Eyes closed’. How many now?

How many did I take away/ hide?

Practical

.

10. Subtraction as ‘physical removal of an object’, linking to ‘striking out’ drawn images on a page.

1 - 5/10 Pictiorial moving to iconic to aid efficieny.

Draws pictures, then ‘dots’ instead of teddies.

Draws 8 shells. Cross out 4. Leaves 4. . leading to…

8 – 4 = 4.

Arrangement of objects progresses from

‘random’ to linear.

Linking subtraction to addition /difference/

‘How many more?’

Make the link between the 2 operations using

‘hide and reveal strategies’.

I had 10 pegs, how many did I take away?

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images

Recording Examples

11. Place the concrete objects onto a structured

number line. Remove amount/ cross out amount to find how many left. ‘Take away’ by ‘covering up’ a numicon piece with another.

Within 20

‘Touches and counts back on structured number line. May draw linear picture / dots and cross out.

5 – 2 = 3 Children need to become confident at the ‘breakable chain’ and ‘bi- directional chain’ levels of counting (Fuson 1988), so that they are able to start the counting at any point and count forwards and backwards.

5 – 3 = 2

10 – 3 = 7

9 – 5 = 4

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images

Recording Examples

12. Understanding subtraction as difference by counting up, where 2 numbers are close together on a number line.

Within 10/ 20 initially.

The difference between 6 and 8 is 2 8 – 6 = 2

13. Use a

structured number line / bead string to count back.

Within 20 and beyond. Add on single digit number, then teens number.

13 – 5 = 8- link to bead string

Records as + calculation 13 – 5 = 8 and can explain how it was done.

14 – 5 = 9

4 less than 10 is 6.

14. Use unstructured number line to subtract ones from a teens- number.

Within 20 Jumping back in ones. Counting back MUST be secure within 20.

Blu tac structured number line to board, show jotting for example 10 – 4 = 6. Remove structured number line to leave an ‘unstructured number line’.

Model method within 10 to show the process of moving from a structured number line to an unstructured number line eg 10 count back 4 = 6. Draw the jottings above the number line, then remove it to leave the unstructured number line. Write numbers under the jottings.

6 10

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number range

Possible Models / Images

Recording / Examples

15. Use an unstructured number line, starting at the biggest number and counting back in tens and

ones. Children begin to use the structured number line as a tool to support mental calculation.

Within 50, then progress to 100. Use the structured number line to support early place value knowledge i.e. subtraction of multiples of 10 from a 2-digit number.

What could the numbers be? Link to real life application e.g. subtracting 20p from 37p to leave 17p.

Then progress to counting back more than one multiple of 10 as shown below:

16. Bridging through 10 when appropriate i.e. partition the ones digit when crossing the tens boundary.

Within 100.

56 - 27 Subtract 20. Subtract 7 by partitioning 7 into 6 and 1 to allow bridging through the multiple of 10.

Use numicon on a number track to show how to bridge through 10 by partitioning and recombining the ones. Example 1: 23 – 5 = 18 Example 2: 75 – 8 = 67 (ref p92 Toolkit 2 Guide)

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images Recording / Examples

17. Partition the ones digit when crossing the tens or hundreds boundary.

TO - TO HTO - TO Numbers to 1000.

Dienes

Children need to know the value of each digit in a number, and should be able to partition a two digit number confidently. Partition the number into tens and ones. Subtract groups of ten and then partition the ones digit to cross the tens boundary.

18. Count on from the smallest number to the largest number to ‘find the difference’.

TO – TO Numbers to 100.

Dienes

When children understand the concept of difference, through practical activity and can confidently subtract by counting backwards on a number line, they are ready to begin to use ‘counting on’ to find the difference. They need to be able to decide whether it is more efficient to count on or back, depending on the numbers involved. E.g. If the numbers are close together then ‘count on’ e.g. difference between 28 and 35.

Children using this method should be able to say or write the corresponding subtraction calculation in response to a given addition calculation e.g. 35 – 28 = 7, 28 + 7 = 35. They should be able to check their calculation using the inverse operation.

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

SEE FOLLOWING PROGRESSION

Criteria for readiness for formal written methods of addition and subtraction:

Do the children know addition and subtraction facts to 20?

Do they understand place value and can they partition numbers into hundreds, tens and ones?

Do they use and apply the commutative and associative laws of addition?

Can they add at least three single-digit numbers mentally?

Can they add and subtract any pair of two-digit numbers mentally?

Can they explain their mental strategies orally and record them using informal jottings?

This checklist is merely a guide and, in deciding whether children are ready to move on, teachers need to be sure that they are not making

judgements that will adversely affect the children’s progress. Some may not be making the progress expected, but may make progress by

being moved on, e.g. children with specific learning difficulties whose mathematical understanding is sound but who have difficulty in learning

addition and subtraction facts to 20.

Overview

1. Establish mental methods, based on a good understanding of place value in numbers

2. Present calculations in a horizontal format.

3. Show children how to set out written calculations vertically, initially using expanded layouts that record their mental methods. Use of the

expanded format helps children to keep place value in mind and significantly reduces the number of errors.

4. As children become more confident, refine the written record into a more compact method.

5. Extend to larger numbers and to decimals. (Children may initially need to return to a more expanded layout.)

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images Recording / Examples

19.

Use expanded column method to subtract integers (without exchange). Use informal jottings to partition numbers and recombine.

HTO - HTO HTO – TO Numbers to 1000.

Dienes H T O

100 + 20 + 3

Children should be able to work out this calculation mentally. This method should be taught as an introduction to the column method only. Children should be taught to arrange their numbers so that the digits are in aligning columns, ensuring correct place value. Partition numbers using place value labelling. Largest number on the top. Subtract the hundreds. Subtract the tens. Subtract the ones. Recombine the number. H T O 200 + 40 + 8 - 100 + 20 + 5

100 + 20 + 3 = 123 Also, model subtracting the ones numbers first and moving towards the highest integers. When children are confident with this method, they can be introduced to ‘exchanging’.

20. Use expanded column method to subtract integers (with exchange).

HTO – HTO HTO – TO Numbers to 1000.

Dienes as above. Children should be confident with partitioning numbers in different ways to understand the ideas of ‘exchanging’. Partition the two numbers into hundreds, tens and ones. It is at this stage that the children must know to subtract the least significant numbers first.

PROGRESSION IN SUBTRACTION DOCUMENT FOR LYMINGTON INFANT AND LYMINGTON JUNIOR SCHOOL: SUBTRACTION -

Copyright Lymington Infant and Junior Schoo1 2015

Step Skill Number Range

Possible Models / Images Recording / Examples

21. Use the compact column method for subtraction (decomposition)

3 and 4 digit numbers. Could include 5 digit numbers and decimal numbers when place value is secure.

Dienes This method is for those children that have a secure understanding of place value. Use place value labelling above the calculation to reinforce (Th H T O). It is at this stage that the children must know to subtract the least significant numbers first. Children must learn to exchange tens for ones or

H T 0 hundreds for tens.

5 34 18 It is important to ensure that the numbers are referred

- 1 2 9 to using correct place value e.g. the 2 in the tens

4 1 9 column has a value of 20. When subtracting decimal numbers the decimal points must line up, one on top of the other. When teaching this use 1/10 (t), 1/100 (h) and 1/1000 (th) signs to link to fractions. T O t h 9 8 . 3 9 - 4 2 . 8 1 5 5 . 5 8

22. Use the compact column method to subtract integers and decimal numbers (mixed).

Numbers with up to 3 decimal places and integers with up to 5 digits. Numbers in the context of money and measure.

Dienes When subtracting decimal numbers the decimal points must line up, one on top of the other. Children must recognise that an integer has an ‘invisible’ decimal point at the end of the number. When teaching this use 1/10 (t), 1/100 (h) and 1/1000 (th) signs to link to fractions. T O t h 4 9 . 6 7 8 9 . 9 6 5 9 . 2 7 5 - 5 . 4 6 - 7 . 7 - 3 . 1 4 4 4 . 2 1 8 2 . 2 6 5 6 . 1 3 5