Calculation policy

“A vision”

The teaching of Mathematics at Cogenhoe Primary is underpinned by two factors: the School’s ethos Inspiring a life-long commitment to learning and the school’s values Curiosity, Compassion, Courage, confidence and pride. Cogenhoe Primary School has embarked on a journey towards a mastery approach

of teaching Mathematics where the transition will be a gradual process over a few years. The rationale behind this is to provide a more challenging curriculum for all children and as well as the national curriculum which states:

The expectation is that most pupils will move through the programmes of study at broadly the same pace. Pupils who grasp concepts rapidly should be challenged through being offered rich and sophisticated problems before any acceleration through new content. Those who are not sufficiently fluent with

earlier material should consolidate their understanding, including through additional practice, before moving on.

To support this approach, School have invested in joining the Enigma Maths Hub and the subject leader has started a 2 year affiliation with the hub in order to implement the mastery approach.

Mastery in Mathematics will follow the ‘Big 5 ideas’ (fig.1) Our teaching for mastery is underpinned by the NCETM’s (National Centre for Excellence in the Teaching of

Mathematics) 5 Big Ideas. Opportunities for Mathematical Thinking allow children to make chains of reasoning connected with the other areas of their mathematics. A focus

on Representation and Structure ensures concepts are explored using concrete, pictorial and abstract (CPA approach) representations. Teachers use both procedural and

conceptual Variation within their lessons and there remains an emphasis on Fluency with a focus on number and times table facts. Running through all of these facets of Mathematics is Coherence which is achieved through the planning of very small

connected steps to link each skill and lesson within a topic.

Fig. 1

What is a mastery lesson?

There are many elements to a mastery lesson, below are some of the core principles that Cogenhoe Primary School teachers follow:

1. Teacher input usually lasts around 20/30 minutes giving ample time for independent practice whilst the teacher or Teaching Assistant delivers rapid intervention should somebody require it. Independent practice includes reasoning, problem solving and higher-order thinking activities.

2. Lessons are sharply focused with one new objective introduced at a time.

3. Difficult points and potential misconceptions are identified in advance and strategies to address them planned. Key questions are planned, to challenge thinking and develop learning for all pupils.

4. The use of high quality materials and tasks (White Rose Maths Hub) support learning in lessons.

5. There is regular interchange between concrete/contextual ideas and their abstract/symbolic representation.

6. Quicker Achievers are challenged through problem solving and reasoning activities such as True or False, What is the same/different?, Always, sometimes, never, missing digits, spot the mistake etc

7. Teacher-led discussion is interspersed with short tasks involving pupil to pupil discussion and completion of short activities. Formative assessment is carried out throughout the lesson; the teacher regularly checks pupils’ knowledge and understanding and adjusts the lesson accordingly. This forms part of the mastery learning instructional process.

8. All in one sheet, where children have no ceiling to their learning and for each lesson have a chance to attempt reasoning and problem solving activities.

What is the CPA (concrete, pictorial and abstract) approach?

Concrete, Pictorial, Abstract (CPA) is a highly effective approach to teaching that develops a deep and sustainable understanding of Maths in pupils. It is an essential technique within the Singapore method of teaching for mastery.

Concrete step of CPA

Concrete is the “doing” stage. During this stage, children use concrete objects to model problems. Unlike traditional Maths teaching methods where teachers demonstrate how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical (concrete)

objects. With the CPA framework, every abstract concept is first introduced using physical, interactive concrete materials.

Pictorial step of CPA

Pictorial is the “seeing” stage. Here, visual representations of concrete objects are used to model problems. This stage encourages children to make a mental connection between the physical object they just handled and the abstract pictures, diagrams or models that represent the objects from the

problem.

Building or drawing a model makes it easier for children to grasp difficult abstract concepts (for example, fractions). Simply put, it helps students visualise abstract problems and make them more accessible.

Abstract step of CPA

Abstract is the “symbolic” stage, where children use abstract symbols to model problems. Students will not progress to this stage until they have demonstrated that they have a solid understanding of the concrete and pictorial stages of the problem. The abstract stage involves the teacher introducing

abstract concepts (for example, mathematical symbols). Children are introduced to the concept at a symbolic level, using only numbers, notation, and mathematical symbols (for example, +, –, x, /) to indicate addition, multiplication or division.

CPA within Cogenhoe

Concrete objects will differ between the year groups and could include: counters, base ten, ten frames, multi-link cubes or beads. A teacher will move back and forth between the stages in order to reinforce skills and concepts. Children, throughout the school, will be encouraged to reason or problem solve

through pictures and diagrams on occasions.

Pictorial representation could include bar modelling, drawing of arrays or number bond diagrams. Again, children are encouraged to reasoning through pictures and diagrams to support explanations and justify their answers.

What is bar modelling?

Bar modelling is an essential Maths mastery strategy. A Singapore-style of Maths model, bar modelling allows pupils to draw and visualize mathematical concepts to solve problems.

Bar modelling and the CPA approach

The bar model method draws on the Concrete, Pictorial, Abstract (CPA) approach. The process begins with pupils exploring problems via concrete objects. Pupils then progress to drawing pictorial diagrams, and then to abstract notations. The example below explains how bar modelling moves from concrete

Maths models to pictorial representations.

As shown, the bar method is primarily pictorial. Pupils will naturally develop from handling concrete objects, to drawing pictorial representations, to creating abstract rectangles to

illustrate a problem. With time and practice, pupils will no longer need to draw individual boxes/units. Instead, they will label one long rectangle/bar with a number. At this stage, the bars will be somewhat proportional. So, in the example above, the purple bar representing 12 cookies is longer than the orange

The lasting advantages of bar modelling

The Singapore Maths model method — bar modelling — provides pupils with a powerful tool for solving word problems. However, the lasting power of bar modelling is that once pupils master the approach, they can easily use bar models year after year across many Maths topics. For example, bar modelling is

an excellent technique for tackling ratio problems, volume problems, fractions, and more.

Importantly, bar modelling leads students down the path towards mathematical fluency and number sense. Maths models using concrete or pictorial rectangles allow pupils to understand complex formulas (for example, algebra). Instead of simply following the steps of any given formula, students will

possess a strong understanding of what is actually happening when applying or working with formulas.

Progression in calculations

Year 1 Year 2 Year 3 Year 4 Year 5 Year 6Number - addition and subtraction

Understanding addition and subtraction

Choose an appropriate strategy to solve a calculation based upon the numbers involved (recall a known fact, calculate mentally, use a jotting)

Choose an appropriate strategy to solve a calculation based upon the numbers involved (recall a known fact, calculate mentally, use a jotting, written method)

Choose an appropriate strategy to solve a calculation based upon the numbers involved (recall a known fact, calculate mentally, use a jotting, written method)

Choose an appropriate strategy to solve a calculation based upon the numbers involved (recall a known fact, calculate mentally, use a jotting, written method)

Choose an appropriate strategy to solve a calculation based upon the numbers involved (recall a known fact, calculate mentally, use a jotting, written method)

Read, write and interpret mathematical statements involving addition (+), subtraction (-) and equals (=) signs

Show that addition of two numbers can be done in any order (commutative) and subtraction of one number from another cannotUnderstand subtraction as take away and difference (how many more, how many less/fewer)

Understand and use take away and difference for subtraction, deciding on the most efficient method for the numbers involved, irrespective of context

Addition and subtraction facts

Represent and use number bonds and related subtraction facts within 20

Recall and use addition and subtraction facts to 20 fluently, and derive and use related facts up to 100Recall and use number bonds for multiples of 5 totalling 60 (to support telling time to nearest 5 minutes)

Recall and use addition and subtraction facts for 100 (multiples of 5 and 10)Derive and use addition and subtraction facts for 100Derive and use addition and subtraction facts for multiples of 100 totalling 1000

Recall and use addition and subtraction facts for 100Recall and use addition and subtraction facts for multiples of 100 totalling 1000Derive and use addition and subtraction facts for 1 and 10 (with decimal numbers to

Recall and use addition and subtraction facts for 1 and 10 (with decimal numbers to one decimal place)Derive and use addition and subtraction facts for 1 (with decimal numbers to two decimal places)

Recall and use addition and subtraction facts for 1 (with decimal numbers to two decimal places)

one decimal place)

Mental methods

Select a mental strategy appropriate for the numbers

Select a mental strategy appropriate for the numbers

Select a mental strategy appropriate for the numbers

Select a mental strategy appropriate for the numbers

Select a mental strategy appropriate for the numbers

Add and subtract one-digit and two-digit numbers to 20, including zero (using concrete objects and pictorial representations)

Add and subtract numbers using concrete objects, pictorial representations, and mentally, including:- a two-digit number and ones- a two-digit number and tens- two two-digit numbers- adding three one-digit numbers

Add and subtract numbers mentally, including:- a three-digit number and ones- a three-digit number and tens- a three-digit number and hundreds

Add and subtract mentally combinations of two and three digit numbers and decimals to one decimal place

Add and subtract numbers mentally with increasingly large numbers and decimals to two decimal places

Perform mental calculations, including with mixed operations and large numbers and decimals

Written methods

*Written methods are informal at this stage – see mental methods for expectation of calculations

*Written methods are informal at this stage – see mental methods for expectation of calculations

Add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction

Add and subtract numbers with up to 4 digits and decimals with one decimal place using the formal written methods of columnar addition and subtraction where appropriate

Add and subtract whole numbers with more than 4 digits and decimals with two decimal places, including using formal written methods (columnar addition and subtraction)

Add and subtract whole numbers and decimals using formal written methods (columnar addition and subtraction)

Estimating and checking calculations

Recognise and use the inverse relationship between addition and subtraction and use this to check calculations and solve missing number problems

Estimate the answer to a calculation and use inverse operations to check answers

Estimate and use inverse operations to check answers to a calculation

Use rounding to check answers to calculations and determine, in the context of a problem, levels of accuracy

Use estimation to check answers to calculations and determine, in the context of a problem, an appropriate degree of accuracy

Order of operations

Use their knowledge of the order of operations to carry out calculations involving the four operations

Solving addition and subtraction problems including those with missing numbers

Solve one-step problems that involve addition and subtraction, using concrete objects and pictorial representations, and missing number problems such as7 = - 9

Solve problems with addition and subtraction including those with missing numbers:- using concrete objects and pictorial representations, including those involving numbers, quantities and measures- applying their increasing knowledge of mental and written methods

Solve problems, including missing number problems, using number facts, place value, and more complex addition and subtraction

Solve addition and subtraction two-step problems in contexts, deciding which operations and methods to use and whySolve addition and subtraction problems involving missing numbers

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and whySolve addition and subtraction problems involving missing numbers

Solve addition and subtraction multi-step problems in contexts, deciding which operations and methods to use and whySolve problems involving addition, subtraction, multiplication and division, including those with missing numbers

Addition

Objective Concrete Pictorial Abstract Greater Depth/challenge examples

Combining two parts to

make a whole (use other

resources too e.g. eggs,

shells, teddy bears, cars).

Part-part- whole

Or part-part-part-whole

The parts add together to make the

whole.

Use cubes to add two numbers together as a

group or in a bar.

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

4 + 3 = 7Four is a part, 3 is a

part and the whole is seven.

4 + 3 = 710= 6 + 4

Use the part-part whole diagram as shown above to

move into the abstract.

7

= ?? + 3

Counting on using number lines

using cubes or Numicon.

4 + 2 = 6 A bar model which encourages the children to count on, rather than count all.

Or

The abstract number line:

What is 2 more than 4?

What is the sum of 2 and 4?

What is the total of 4 and 2?

4 + 2 = ??

I have added to numbers and made 12. One

number is 4 what is the other?

12 + ?? = 1818 = ?? + 12

14 is the answer, what was the addition

question?

What do you notice?

12 + 4 = 1612 + 5 = 1712 + 6 = 18

14 + 5 = 1913 + 6 = 1912 + 7 = 19

Regrouping to make 10; using ten frames and

counters/cubes or using Numicon.

This is an essential skill that will

support column addition later on.Pupils should be

encouraged to start at the bigger

number and use the smaller number to

make ten.Also, the empty

spaces on the ten frame make it clear how many more are

needed to make ten.

The colours of the beads on the bead string make it clear

how many more need to be added to

make ten.

6 + 5 = 11A ten frame allows a visual regrouping of 10 and what

is left

Numicon can show how 5 can partition into 4 and 1 so the 4 is added to the 6 to regroup as 10 and then

there is one left over.

Or, use toys/household objects to place into a

group of 10 before adding on the final one item.

9 + 4 = 13

Use counters or cubes on the frame

Or, a visual

representation of the toys/objects used from the concrete stage.

6 + 5 = 115 partitions into 4 and 1

So, 6 + 4 = 1010 + 1 = 1

Children to develop an understanding of

equality e.g.6 + □ = 11

6 + 5 = 5 + □6 + 5 = □ + 4

Reinforce the = sign means ‘balanced with’ or ‘is equal

to’ rather than ‘answer’.

5 partitions into 4 and 1 so I can add 6 and 4 to make 10

then add the 1 left over, how can this help with 16 + 5?

Exploring the relationship (bonds) to 10. What can

single digits be partitioned into to enable the children to make 10

first e.g.

Is partitioning 5 into 3 and 2 going to help us with 6 +

5? Why not?

Further questions:I’m thinking of a number. I’ve

subtracted 5 and the answer is 7. What number was

I thinking of? Explain how you know.

I’m thinking of a number. I’ve added 8 and the answer is 19.

What number was I thinking of? Explain how you know.

I know that 7 and 3 is 10. How can I find 8 + 3? How could you

work it out?Shopping list under 20p. I have

20p to spend. If I spend 20p exactly, which two items could I

buy?And another two, and another

two.If I bought one of the items how

much change would I have? And another one,and another one.

Adding three single digit numbers (make

ten first)Pupils may need to try different combinations

before they find the two numbers that

make 10.

4 + 6 + 7 = 17

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.

4 + 6 + 7 = 17

??4 6 7

10 7

4

?? 6

7

Children identify bonds to 10 to simplify the question. If no bond is

present, children partition one number to create a bond e.g.

6 + 5 + 3 =5 partitions into 4 and 1

4 + 6 + 7 = 17

4 + ?? + 7 = 17

Children identify missing numbers

So, 6 + 4 = 10 then add 3 and 1 = 14Adding multiples of

tenUsing 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.It also emphasises the link to known number

facts. E.g. ‘2 + 3 is equal to 5. So 2 tens +

3 tens is equal to 5 tens.

30 + 20 = 50

MLC, Dienes, Base 10 are used to represent that 10

is added not 1.

Here, 3 tens add 4 tens is 7 tens. So 30 add 40 is 70. Plus the ones = 76

36+ 40

__________ 76

61+??

_________91

Partitioning to add (no regrouping)

Place value grids and practical aids should be used as shown in the diagram before

moving onto the pictorial

representations.

When not regrouping, partitioning is a

mental strategy and does not need formal recording in columns. This representation prepares them for

using column addition with formal recording.

41 + 8

Partition into 40 and 1.Add the ones 1 + 8 = 9

Add the 40 = 49

41 + 8 =

Bar Model

??41 8

Lines can be represented for 10s and dots for ones.24 + 13 = 37

41 + 8

1 + 8 = 940 + 9 = 49

Missing digits poses a problem to children.

26 + ? = 29

?4 + 5 = 59

Give children the answer and see if there is more

than one question possible.

Use pictorial representations of the practical work. Place value columns

prepare children for more formal column addition.

Bar Model

??24 13

Partitioning one number, then adding

tens and onesPupils can choose for themselves which of

the numbers they wish to partition.

22 + 17

Here 17 is partitioned into 10 and 7. The 10 is added

to the 22 then the 7 is added.

This process works with Dienes, Base 10 and other practical representations.

22 + 17 22 + 17 =

22 + 10 = 3232 + 7 = 39

68 + 2 + 26 = 96

What question could this

partitioned question

represent? = 68 +

28 = ??Counting on and back in tens and hundredsEfficient children will identify columns that

are not affected i.e when adding 10 the

ones are not affected.

2 tens add 1 ten is 3 tens

3 tens add 1 ten is 4 tens

32, 42, 52, 62

134, 234, 334, 434

172, 162, 152, 142

782, 682, 582,

Missing numbers in a sequence e.g. 27, ??, ??,

57

Make ten strategyHow pupils choose to apply this strategy is up to them; however,

the focus should always be on

efficiency.

38 + 15

Children identify the bond needed to make the next

10 e.g. 38 + 2 = 40. 40 + 13

38 + 15

= 53Introducing column

method for addition, regrouping only

Practical resources and place value grids

should be used as shown in the

diagrams.

Place value counters, cubes, Dienes can all help children with regrouping

(exchanging).

36 +25The 10 ones can be placed together and exchanged

into the next column i.e. 6 + 5 = 11, so 10 of them are exchanged leaving 1 in the ones column. Now there

are 6 tens.This can work for all practical resources.

24 + 17

Bar Model??

24 + 17

Note here when exchanging that the ‘exchange rate’ is 10 for 1 so only 1 is marked in

the next column.

Challenge children with missing digits in the

question.

24 17HTO + TO, HTO + HTO. When there are 10 ones in the

1s column- we exchange for 1 ten, when there are 10

tens in the 10s column- we

exchange for 1 hundred.

243 + 368 =Using place value counter

or other practical resources, children

demonstrate exchanging 10 or 1 into the next

column.

243 + 368

Pictorial representation of the practical practise.

243+368611

1 1

When exchanging 10 for 1, the 1 can be placed ‘on the step’ or below the answer

box.

?47+3?8625

Column addition including decimals

Children use previously learnt skills and apply to decimal numbers and numbers with

different number of digits. Exchanging 10

for 1 remains the same as previously

taught.

Use of place value grids populated with place value

counters or cubes etc to represent the number

Children choose their own representation e.g. dots to represent 1

value in each column

Sally is cycling in a race. She has cycled 3.145 km so far and

has 4.1 km left to go. What is the total

distance of the race?

0.?68+0.2??= 0.5?3

Do any of the ?’s have more than one possibility?

Conceptual differences in

addition examples21 + 34

Word problems:In year 3, there are 21 children and in

year 4, there are 34 children.How many children in total?

21 + 34 = 55. Prove it

Missing digit problems:

SubtractionTaking away

onesWhen this is firstintroduced, the

concrete representation

should be basedup on the diagram.Real objects should be placed on top of the images as one –

to – one correspondence sothat pupils can take

them away, progressing to

representing the group of ten with a tens rod and ones with ones cubes

Use physical objects, counters, cubes etc to

show how objects can be taken away.

6 – 2 = 4

4 – 2 = 2

Cross out

drawn objects

to show what has been taken away.

8 – 1 = 7

5 – 1 = 4

6 - ? = 2? = 9 – 3?? -2 = 5

Counting backSubtracting 1,

2, or 3 by counting back.

Pupils should be encouraged to rely on number

bonds knowledge as time goes on,

rather thanusing counting

back as their main strategy

Read, write and interpret

mathematical statements involving

addition (+), subtraction (–) and

equals (=) signs.

Represent and use number bonds and related subtraction

facts within 20.

Add and subtract one-digit and two-digit

numbers to 20, including zero.

Solve one-step

Make the larger number in your subtraction. Move

the beads along your bead string as you count backwards in ones.

13 – 4

Use counters and move them away from the group

as you take them away counting backwards as you

go.

Count back on a number line or number track

Start at the bigger number and count back the smaller number showing the

jumps on the number line.

This can progress all the way to counting back using two 2 digit

numbers.

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

at?

?? – 4 = 1217-?? = 15

problems that involve addition and

subtraction, using concrete objects and

pictorial representations, and

missing number problems such as 7 =

– 9.

Find the difference

Solve problems with addition and subtraction:i. using concrete objects

and pictorial representations,

including those involving numbers, quantities and

measuresii. Applying their

increasing knowledge of mental and written

methods.

Recall and use addition and subtraction facts to 20 fluently, and derive

and use related facts up to 100.

Add and subtract numbers using concrete

objects, pictorial representations, and mentally, including:

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

Count on to find the difference.Draw

bars to find the

difference between 2 numbers.

Hannah has 23 sandwiches, Helen has 15 sandwiches.

Find the difference between the number of sandwiches.

The difference between the two coins I have is 10p. One coin is 20p.

What is the other coin?

i. a two-digit number and ones

ii. a two-digit number and tens

iii. two two-digit numbers

iv. adding three one-digit numbers.

Show that addition of two numbers can be

done in any order (commutative) and subtraction of one

number from another cannot.

Recognise and use the inverse relationship

between addition and subtraction and use this to check calculations and

solve missing number problems.

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

Use a pictorial representation of objects to show the part part whole

model.

Ensure that children are exposed to part-part-whole where the whole and

parts switch places i.e the whole to the right/top/below of the parts

Move to using numbers within the part whole model.

The whole is 15. What could the parts be? Can you think of more than

one example?

Use of three or more parts

10

5

one of the parts. What is the other part?

10 - 6 =

6?? 2

Make 10As with addition,

children see that it is more efficient to subtract to get to

ten first then subtract again from ten. Knowledge of number bonds to and from ten and twenty are vital.

14 – 9 =

Make 14 on the ten frame. Take away the four first to

make 10 and then takeaway one more so you

have taken away 5. You are left with the answer of

9.

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.

16 – 8=

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

How many do we have left to take off?

Which of these is the odd one out? Why?

13 - 614 - 715 – 9

Is there a pattern you can spot?

Column method without

regroupingSubtract numbers with up to three

digits, using formal written

methods of

Use Base 10 to make the bigger number then take

the smaller number away.75 - 42

Draw the Base 10 or place value counters alongside the written

calculation to help to show working.

Start with expanded method to subtract

This will lead to a clear

82 - ?? = 51

columnar subtraction.

Add and subtract numbers with up to

three digits, using formal written methods of

columnar addition and subtraction.

Show how you partition numbers to subtract. Again

make the larger number first.

written column subtraction.

Column method with regrouping

Add and subtract numbers with up to 4 digits using the formal

written methods of columnar addition and

subtraction where

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.

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

Children can start their formal written method by

partitioning the number into clear place value columns.

appropriate.

Estimate and use inverse operations to check

answers to a calculation.

Solve addition and subtraction two-step problems in contexts,

deciding which operations and methods

to use and why.

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 tens.

Now I can take away eight tens and complete my

subtraction

Show children how the concrete method links to the

written method alongside

confident, children can find their own way to record the

exchange/regrouping.

Just writing the numbers as shown here shows that the child understands

the method and knows when to exchange/regroup.

your working. Cross out the numbers when exchanging

and show where we write our new amount.

Column method

Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and

subtraction).

Add and subtract numbers mentally with

increasingly large numbers.

Use rounding to check answers to calculations and determine, in the context of a problem,

levels of accuracy

Solve addition and subtraction multi-step problems in contexts,

deciding which operations and methods

to use and why

Use their knowledge of the order of operations to carry out calculations

involving the four operations.

Solve addition and

Use concrete materials to represent columnar

subtraction with decimal numbers. Decimal

numbers can also be represented with base

equipment.

Take away the digit furthest right first.

Exchanging required here.

One ten becomes 10 ones.

As above, a pictorial representation of the place value columns including

decimals.

263 – 26.5Extra 0 can be added so that

place value columns line up/match

subtraction multi-step problems in contexts,

deciding which operations and methods

to use and why. Solve problems involving

addition, subtraction, multiplication and

division.

Use estimation to check answers to calculations and determine, in the

context of a problem, an appropriate degree of

accuracy.

Then take away the tenths.

You can’t take 7 away from 5 so more exchanging is needed – one ten for 10 ones. So now it is 15 - 7

This leave 1 ten take away 1 ten.

MultiplicationDoubling Use practical activities to

show how to double a number.

Draw pictures to show how to double a number.

Partition a number and then double each part

before recombining it back together.

Double what is 10?Double what is 36?

Counting in multiples

Count in multiples supported by concrete

objects in equal groups.Use a

number line or pictures to continue support in counting in multiples.

Count in multiples of a number aloud.

Skip counting.

Write sequences with multiples of numbers.

2, 4, 6, 8, 105, 10, 15, 20, 25 , 30

Is 15 in the 5 times table? Prove it

Repeated addition

There are ____ in each group. There

are ____ groups. We have to add ____

_____ times.

Use pictorial representation including number lines to solve problems

Write addition sentences to describe objects and

pictures.2 x 5

What does this show?

Abstract number line showing 3 groups of 4

4 + 4 + 4 = 12

Using concrete resources, children create equal groups in order to be

added

3 x 4 = 12

Arrays- showing

commutative multiplication____ lots of ____ is the same as ____

lots of ____.

Create arrays using counters/ cubes to show multiplication sentences.

Draw arrays in different rotations

to find

commutative (the idea that order of digits doesn’t affect the answer)

multiplication sentences.Link arrays to areas of rectangles.

Use an array to write multiplication sentences and reinforce repeated

addition.

3 x 2 = 62 x 5 = 10

Using the inverse (to be

taught alongside division)

____ lots of ____ is ____ so ____

divided by ____ is ____.

As above, use arrays to test the inverse

Partition to multiply____ can be

partitioned into ____ and _____.____ lots of ___

ones is ____.____ lots of ____

tens is ____.____ ones add ____

tens is ____.

24 x 3

Using practical equipment, children make 24 3 times then add it all together.

Note, once 10 is reached in any column, exchanging

must happen (10:1)

15 x 4

Children to be encouraged to show the steps they have

taken.

When I partition to

multiply, after multiplying

the tens digit, the answer will always

end in a place holder (0).

True or false?

Column multiplication

(Grid multiplication)We always need to start at the ones.

____ ones times ____ ones is ___

ones.____ ones times ____ tens is ____ tens. Because we are multiplying by

ten, we need to add in a zero as a place

value holder.

Children to be supported by base 10 with smaller numbers before representing

numbers using base 10.

Bar modelling and number 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.

???X 5 10350500

= 860

We cannot have more than one digit in any place value

column, so we need to exchange ___ ones as ____ ten

(and etc as needed)

It is important at this stage that they always multiply the ones first and note down their

answer followed by the tens which they note

below.

This moves to the more compact method.

Note where the exchanging is placed within the calculation

Divisionshare, group, divide, divided by, half, divisor, dividend, quotient, remainder, exchange

Sharing objects into

groups____ shared equally

between ____ is _____

I have 10 cubes, can you share them equally in 2

groups?

Children use pictures or shapes to share quantities.

Share 9 buns between three people.

9 ÷ 3 = 3

8 ÷ 2 = 4

Repeated subtraction

We need to divide ____ into groups of ____, so we need to take away ____ each time. We have ____

groups of ____.

6 ÷ 2 =

Division as grouping

____ split into ___ groups means there

would be ____ in each group.

Divide quantities into equal groups.

Use cubes, counters, objects or place value

counters to aid understanding.

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

12 ÷ 3

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.

28 ÷ 7 = 4

Divide 28 into 7 groups. How many are in each

group?

Division within arrays

Link division to multiplication by

creating an array and thinking about the

number sentences that can be created.

Eg 15 ÷ 3 = 55 x 3 = 1515 ÷ 5 = 33 x 5 = 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.

7 x 4 = 284 x 7 = 2828 ÷ 7 = 428 ÷ 4 = 7

Division with a remainder

A remainder is what is left over after

splitting into equal groups.

____ divided by ___ gives ____ equal

groups, with _____

14 ÷ 3 =Divide objects between

groups and see how much is left over

Jump forward in equal jumps on a number line then see how many more you need to jump to find a remainder.

Complete written divisions and show the remainder

using r.

29 ÷ 8 = 3 r 5

remaining.

Cuisenaire over a ruler can also be used.

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

Remainder could be expressed as a fraction.

Remainder is the numerator and denominator is the

dividing number.

Short division (no exchange)In division, we start

from the largest place value column. We start from the

right.

___ is __ tens and ___ ones. ____ tens divided by ____ is ____. _____ ones

divided by _____ is _____. ____ add

____ is _____.e.g. 36 is 3 tens and

6 ones. 3 tens divided by 3 is one ten. 6 ones divided

Should first be shown using base 10 and shared

into groups, to understand the place value.

Use place value counters to divide using the bus stop method alongside

Digits under the bus stop could also be circled in groups of the divisor.

36 ÷ 3 36 ÷ 3 = 12

by 3 is 2 ones. One ten add 2 ones is 12.

Short division (with

exchange)e.g. 42 is 4 tens and

2 ones. We can share 3 tens equally with one ten in each

group but there is one ten left over.

We need to exchange this ten for ten ones. Now we have twelves ones. 12 shared

between 3 is 4 ones. In each group there

is one ten and 4 ones. 10 add 4 is 14.

42 ÷ 3=

Start with the biggest place value, we are sharing 40

into three groups. We can put 1 ten in each group and we have 1 ten left

over.

We exchange this ten for ten ones and then share

the ones equally among the groups.

We look how much is in 1 group so the answer is 14.

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.

Start with division without remainders

Move onto divisions with a remainder.

Finally move into decimal places to divide the total

accurately.

Long divisionSystematically go

through every small step using a

deliberate structure that is easy to

understand which numbers are being referred to e.g. 7

hundreds, not just ‘the 7’.

Use base 10 and place value counters to secue

understanding

Use partitioning to support understanding of division, e.g. 364 ÷ 14

Look for numbers which are clear multiples of the divisor,

e.g 364 = 140 + 140 + 70 + 14

Divide each multiple by the divisor140 ÷ 14 = 10140 ÷ 14 = 10

70 ÷ 14 = 514 ÷ 14 = 1

Combine the answers together to find the total.

364 ÷ 14 = 10 + 10 + 5 + 1 = 26

Write out the divisors times table and use these facts to

help.

If there are digits ‘left over’ after the final step in the

above example, these become the remainder. Can the fraction remainder be

simplified?

1 x 45 = 452 x 45 = 903 x 45 = 1354 x 45 = 1805 x 45 = 2256 x 45 = 2707 x 45 = 3158 x 45= 3609 x 45 = 40510 x 45 =450