1

Objective and

Strategies

Concrete Pictorial Abstract

Halving in

practical

contents linked

to simple

problem solving

Start with links to real

life events. Cut a cake in

half, both halves being

exactly the same size.

Encourage children to use mark-making to support their

thinking about numbers and simple problems.

Sharing into two

groups linked to

simple problem

solving??

Share four cars between

two friends.

Counting beads, using

both hands to share the

beads in middle into two

ends.

How many different ways can you show half on the ten frame/bar/

grid/counting stick?

Part/part whole

Encourage children to use mark-making to support their

thinking about numbers and simple problems.

Halving by partitioning??

With decimals??

SIJS Division Policy

2

Objective and

Strategies

Concrete Pictorial Abstract

Sharing into

groups

I have ten cubes. Can

you share them equally

in 2 groups?

Share 9 buns between 3 people.

9÷3=3

9÷__=3

Recall division facts related to times tables.

3

Objective and

Strategies

Concrete Pictorial Abstract

Division as

grouping

Divide quantities into

equal groups. Use cubes,

counters, beads and

objects to aid

understanding.

Use a number line to show jumps in groups. The number of jumps

equals the number of groups.

Think of the bar as a whole. Split it into the number of groups you

are dividing by and work out how many would be within each group.

Use sections on a counting stick.

28÷7=4

Divide 28 into 7 groups. How many are in each group?

How many groups of 3 in 12?

How many jumps to get to 12?

Use x table grid

4

Objective and

Strategies

Concrete Pictorial Abstract

Dividing multi-

ples of 10, 100

and 1000 by 10,

100 and 1000

Pupils use the

strategy of shar-

ing into equal

groups of tens,

hundreds or

thousands to

reinforce their

understanding of

place value

Using beadstring

30÷10=3

Back of 100 square and Juli’s pictorial grid to show 1/10 and 1/100

6000÷200=30

I know there are five groups of 200 in 1000 and I have six

1000s and 5x6=30.

Place value chart

÷10 to the right one jump

÷100 to the right two jumps

Then zero the hero flies off

10x__ = 100x24

Odd ones out

0.7x100, 7x10, 0.07x1000

Division within

arrays

Link division to multipli-

cation by creating an

array and thinking about

the number sentences

that can be created.

E.g.

15÷3=5 5x3=15

15÷5=3 3x5=15

Draw an array and use lines to split the array into groups to make

multiplication and division sentences.

Find the inverse of multiplication and division sentences by

creating four linking number sentences.

7x4=28

4x7=28

28÷7=4

28÷4=7

5

Objective and

Strategies

Concrete Pictorial Abstract

Division with a

remainder

14÷3=

Divide objects between

groups and see how

much is left over.

Jump forward or back in equal jumps on a number line then see how

many more you need to jump to find a remainder.

Draw dots and group them to divide an amount and clearly show a

remainder.

14÷3=4r2

Complete written divisions and show the remainder using

r.

72 children are going on a trip with 4 teachers. A bus has

20 seats. How many buses should they book?

A 1m piece of ribbon is cut into equal pieces and a piece

measuring 4cm remains. What length could the equal piec-

es be? How many ways could the ribbon have been cut

into equal lengths?

6

Objective and

Strategies

Concrete Pictorial Abstract

Short division Use place value /

partitioning with Dienes.

448÷4=112

852÷4=213

8 hundreds shared into 4

equal groups

5 tens shared into 4

equal groups

(Regroup 1 ten for 10

ones)

12 ones shared into 4

equal groups

Students can continue to use drawn diagrams with dots or circles to

help them divide numbers into equal groups.

Encourage them to move towards counting in multiples to divide

more efficiently.

Begin with divisions that divide equally.

Move onto divisions with a remainder.

7

Objective and

Strategies

Concrete Pictorial Abstract

Long division The short division

method can be applied

for 11 and 12 using

times table knowledge.

Factors should be used

to break down the

calculation and apply

the short division

method.

1

Objective and

Strategies

Concrete Pictorial Abstract

Doubling Use practical activities to

show how to double a

number.

Double 3 is 6

Draw pictures to show how to double a number.

Double 3 is 6.

Multiply by 4 by doubling twice.

Multiply by 8 by doubling three times.

Partition a number and then double each part before re-

combining it back together.

What is the relationship between 3x2, 3x4 and 3x8?

I double a number and my answer is 64. What is the num-

ber?

SIJS Multiplication Policy

2

Objective and

Strategies

Concrete Pictorial Abstract

Repeated

addition

Use different objects to

add equal groups.

3+3+3

0.2 + 0.2 + 0.2 = 0.6

Write addition sentences to describe objects and pictures.

Write a story for … Include jottings.

Write addition statements as multiplication statements.

True or false? 7+7+7 is the same as 7x4?

3

Objective and

Strategies

Concrete Pictorial Abstract

Counting in

multiples

Count in multiples

supported by concrete

objects in equal groups.

X table grid

Use a number line with kangaroo character or pictures to continue

support in counting in multiples.

Use counting stick.

Count in multiples of a number aloud.

Write sequences with multiples of numbers.

2, 4, 6, 8, 10

5, 10, 15, 20, 25, 30

If I know 7x5, how does this help me with 8x5?

Missing number sequences including negative numbers

and decimals.

Count in multiples of 5 starting on a non-multiple of 5,

e.g. 3

4

Objective and

Strategies

Concrete Pictorial Abstract

Arrays - showing

commutative

multiplication

Create arrays using

counters/cubes to show

multiplication sentences.

Draw arrays in different rotations to find commutative multiplication

sentences.

Link arrays to area of rectangles.

Use an array to write multiplication sentences and

reinforce repeated addition.

5+5+5=15

3+3+3+3+3=15

5x3=15

How many different ways can you make 16 using

multiplication?

Sam is planting 12 onions. He wants them to be in equal

rows. How many different ways could he arrange them?

5

Objective and

Strategies

Concrete Pictorial Abstract

Times tables

using known

facts

E.g. double x2

table to find x4

table facts or

double 6x5 to

find 12x5

Use counters, beads and

cubes.

Multiplication grid

Using jottings, explain the relationship between 3x4 and

4x3.

If you know 3x4 is 12, what is the missing number?

__x3=12

6

Objective and

Strategies

Concrete Pictorial Abstract

Ten times bigger

Language of ten

times bigger

must be well

modelled and

understood

Dienes

Ruler mm cm

Possible misconception: move the decimal point

Encourage children to keep the decimal point stationary

and move the digits. Use Zero the Hero.

What is the relationship between the following:

2x3, 2x30, 20x3, 20x3x10

Missing numbers

__x10=15

5x__=15

7

Objective and

Strategies

Concrete Pictorial Abstract

Multiplying by

10,100 and 1000

5x1=5

5x10=50

3x1=3

3x100=300

Place value chart

Equivalent calculations

98x5=98x10÷2

Measures and money problems

Missing number questions

Multiplication of

2-digit numbers

with partitioning

(no regrouping)

13x3=39 13x3=39 3x12

10 and 2 make 12

3x2=6

3x10=30

30+6=36

Missing values on grid

8

Objective and

Strategies

Concrete Pictorial Abstract

Multiplication of

2-digit numbers

with partitioning

(regrouping)

3x25

20 and 5 make 25

3x5=15

3x20=60

AND

15=10 and 5

SO

60+10=70

70+5=75

Grid method Use Base 10

4 rows of 13

24x3=72 Start with multiplying by 1-digit numbers and showing

clear addition alongside the grid.

Multiply by a 2-digit number

Missing digits

9

Objective and

Strategies

Concrete Pictorial Abstract

Short

multiplication

It is important here that

children always multiply

the ones first and note

down their answer

followed by the tens

which they also note

down.

Bar modelling and umber lines can support learners when solving

problems with multiplication alongside the formal written methods.

Start with long multiplication, reminding the children

about lining up their numbers clearly in columns.

If it helps, children can write out what they are solving next

to their answer.

This moves to the more compact method.

1

SIJS Addition & Subtraction Policy Addition

Objective and Strategies

Concrete Pictorial Abstract

Combining two parts to make a whole: part-whole model

Use cubes to add two numbers together as a group or in a bar. Then recount all using

one-to-one correspondence. (The colours of the cubes are important for

correspondence)

Pupils could place ten on top of the whole as well as writing it down. The parts could

also be written alongside the concrete representation.

Use pictures to add two numbers together in a group or in a bar:

3 + 2 = 5 Use the part-part whole diagram as shown above to move into the abstract.

10 = 6 + 4 10 – 6 = 4 10 – 4 = 6 10 = 4 + 6 10 = ? + 6 10 - ? = 4

etc

Use starters such as: How many more to

make….?

Starting at the bigger number and counting on

Using counters then beads, start with the larger number and then count on the smaller number 1 by 1 to find the answer.

12 + 5 = 17 (on a marked number line)

Start at the larger number on the number line and count on in ones or in one jump to find the answer (blank number line with increments) Use spider to jump in 10s Use Frog to jump in 1s

5 + 12 = 17 Hold the larger number in your head and count on the smaller number to find your answer. 12, 13, 14, 15, 16, 17 Spider to jump on in 10s Frog to jump on in 1s 10s 1s

Make Ten strategy

6 + 5 = 11

Start with the bigger number and use the smaller number to make 10

Use pictures or a number line. Regroup or partition the smaller number to make 10. Using a ten frame:

7 + 4 =11 If I am at seven, how many more do I need to make 10? And how many more do I add on? (blank number line)

9

2

Regrouping (exchanging) to make 10

The colours of the beads on the bead string make it clear how many more need to be added to make ten.

Use multilink/counters on a ten frame:

9 + 5 = 14

This is a concrete/pictorial skill that will support the make ten strategy and column addition.

Adding multiples of 10

Using the vocabulary of 1 ten, 2 tens , 3 tens etc alongside 10, 20, 30 is important as pupils need to understand that it is a ten and not a one that is being added.

50 = 30 + 20

Using dienes equipment e.g. 34 + 20 = 54 INSERT PHOTO

Introduce the kangaroo who jumps in big jumps of multiples of 10, 100 etc on a number line:

3 tens + 5 tens = ____tens 30 + 50 = ______ Use a hundred square with spider counting up and down in 10s Using place value chart with place value counters to see how the tens column changes each time a multiple of 10 is added

50 + 20 = 70 Children count up in tens 50,60,70 or may recognise their number bonds 5 + 2 = 7 so 50 + 20 = 70

Using numbers up flip folders: What is 1 more than …? What is 12 more than…?

Adding near multiples of 10 by adjusting (+9, 19, 29; +1, +11, +29 etc)

Using dienes equipment as above changing the tens to ones to show the adjusting process 30 + 19 = 49 30 + 20 = 50 and adjust to give one back

Using a number line: The Wave (+ 9, 19, + 8 etc)

The Seagull ( + 1, 11, 21 etc)

Children to mentally add near multiples of 10 by adding the too much and giving one back e.g 23 + 9 45 + 19 44 + 29 or adding tens and adding one more: e.g – 44 + 11 32 + 21

3

Adding three single digit numbers

4 + 7 + 6 = 17 Put 4 and 6 together to make 10. Add on 7

The first bead string shows 4, 7 and 6. The colours of the bead string show that it makes more than ten. The second bead string shows 4,6 and then 7. The final bead string shows how they have now been put together to find the total.

Add together three groups of objects. Draw a picture to recombine the groups to make 10. Use 3 ten frames with counters to represent each amount and then combine:

Combine the two numbers that make 10 and then add on the remaining amount:

Include missing number questions:

4 + ? + 6 = 17

How many more to make…?

Place value addition (Partitioning)

Partition with cubes, dienes, beadstring, counters, place value counters.

Taking one number and partitioning the other. 22 + 17 = 39 “No work calculation”

22 + 17 = 39 Use informal jottings:

22 + 17

30 9

30 + 9 = 39

Using known facts (I know….so…)

Number bonds, Doubles, Near doubles…

Using dienes equipment: If 3 + 4 = 7 What is 0.3 + 0.4?

4

Column method – regrouping (carrying)

Using dienes equipment on a place value chart:

Make both numbers on a place value grid: 146 + 527

(carry row) Add up the units and exchange 10 ones for one 10

(carry row)

Add up the rest of the columns, exchanging the 10 counters from one column for the next place value column until every column has been added. As children move on to decimals, money and decimal place value counters can be used to support learning.

Children can draw a pictorial represntation of the columns and place value counters to further support their learning and understanding

The carry row should be above for consistency within written method

Expanded method:

Moving on to the compact method:

Need examples with carry row

5

Subtraction

Objective and Strategies

Concrete Pictorial Abstract

Taking away ones When this is first introduced, the concrete representation should be based upon the diagram. Real objects should be placed on top of the images as one-to-one correspondence progressing to representing the group of ten with tens rod and ones with ones cubes.

Use a variety of objects, counters, cubes etc to show how objects can be

taken away:

6 - 2 =4

Cross out drawn objects to show what has been taken away: 8 – 7 = 1 9 – 4 = 5 10 – 5 = 5

18 – 3 = 15

8 – 2 = 6

Counting back

Make the larger number in your subtraction. Use a variety of objects, cubes, counters etc and move them

away from the group as you take them away counting backwards as you go.

Use a bead string to move the beads along as you count backwards in ones.

13- 4 = 9

Count back on a number line or a number square: Frog counting back in ones (number line with increments and/or labelled) (number square) Start at the bigger number and count back the smaller numbers showing the jumps on the number line: (Kangaroo – count back in 10s; Frog – count back in 1’s) Using a number square:

Put 13 in your head, count back 4. What number are you at?

Record on a blank

number line

6

Find the difference

Compare amounts and objects to find the difference:

Use cubes to build towers or make bars to find the difference

Use basic bar models with items to find the difference

Draw bars to find the difference:

Count on to find the difference:

Hannah has 23 sandwiches, Helen has 15 sandwiches.

Find the difference between the number of sandwiches.

Part part whole model

Link to addition – use the part whole model to help explain the inverse between addition and subtraction.

If 10 is the whole and 6 is one of the

parts. What is the other part? 10 – 6 =

Use a pictorial representation of objects to show the part-part-whole model.

Move to using numbers within the part whole model:

Subtracting 10s

Use base 10 materials

Use spider – spider lives up in the ceiling and come down and goes back up in a straight line. Counting in 10s.

34 - 10 =

7

Making 10 Bridging

Make 14 on the ten frame. Take

away the four first to make 10 and then take away five more so you

have taken away 9. You are left

with the answer of 5.

14 – 8 = (14 – 4 – 4)

Start at 13. Take away 3 to reach 10. Then take away the remaining 4 so you have taken away 7 altogether. You have reached your

answer.

Moving on to using number lines with increments but no labelled.

16 – 8=

How many do we take off to reach the next 10? How many do we have left to take off?

Subtracting tens and adding extra ones Pupils must be taught to round the numbers that is being subtracted. Pupils will develop a sense of efficiency with this method, beginning to identify when this method is more efficient that

subtracting tens and then ones.

“The wave” 53 – 17 = 36

53 – 17 = 36 Round 17 to 20 53-20 = 33 20 – 17 = 3(number

bonds) 33+3 = 36 (we add because we took an extra 3 away when we subtracted 20 instead of 17)

8

Subtracting multiples of ten

Using the vocabulary of 1 ten, 2

tens, 3 tens etc alongside 10,20,30 is important as pupils need to

understand that it is a ten not a

one that Is being taken away.

Use beads aswell

5 tens – 2 tens = ___tens 50 – 20 = ____

Using a hundred square:

32 – 10 = 22 Look at the number

of tens in the largest number. Count back in tens to subtract the smaller number. 30,20. Add on the number of ones that we originally had. Counting back in 10s or 100s from any starting point 53,43,33…. 540, 440,340….

Counting back in multiples of ten and one hundred

Removing one group of 10 each

time

Kangaroo jumps back in 10s

Subtracting near multiples of ten and adjusting

The wave ( -9, -8 etc – take too much and give some back)

The seagull (-11, -12 etc – take 10 and take some more)

9

Adding using compensation rounding and adjustment

54,128 + 9987 54128 + 10, 000- 13 = 64,218 – 13 Highlight that the calculation can be done in another order:

Using near doubles: e.g

70 – 60 70 – 70 = 0 so add on 10 more is 10

Column method without regrouping

Use Base 10 to make the bigger

number then take the smaller number away.

Show how you partition numbers to subtract. Again make the larger

number first.

Draw the Base 10 or place value counters alongside the written calculation to help to show working.

Expanded method:

This will lead to a clear compact written method:

10s 1s

10

Column method with regrouping

Use Base 10 to start with before

moving on to place value counters. Start with one exchange before

moving onto subtractions with 2

exchanges. Make the larger number with the place value

counters

Make the larger number with the place value counters

Start with the ones, can I take away 8 from 4 easily? I need to

exchange one of my tens for ten

ones.

Now I can subtract my ones.

Now look at the tens, can I take

away 8 tens easily? I need to exchange one hundred for ten

Draw the counters onto a place value grid and show what you have taken away by crossing the counters out as well as clearly showing the exchanges you make.

When confident, children can find their own way to record the exchange/regrouping

Children can start their formal written method by

partitioning the number into clear place value columns. Expanded method:

Compact method:

This will lead to an understanding of subtracting any number including

decimals

11

tens.

Now I can take away eight tens

and complete my subtraction

Show children how the concrete method links to the written method

alongside your working. Cross out the numbers when exchanging and

show where we write our new amount.