Brecknock Primary School

Calculation Policy

Introduction:

The CPA Approach:

This policy has been produced to encourage teaching calculation with understanding. The development towards efficient written methods of calculation has been structured

around the concrete – pictorial –abstract approach. This provides opportunities throughout for using mathematical vocabulary, developing mathematical thinking and using

multiple representations. Children should be encouraged to make links and spot patterns as they progress.

Concrete representation

The enactive stage - a student is first introduced to an idea or a skill by acting it out with real objects. In division, for example, this might be done by separating apples into groups of red ones and green ones or by sharing 12 biscuits amongst 6 children. This is a 'hands on' component using real objects and it is the foundation for conceptual understanding.

Pictorial representation

The iconic stage - a student has sufficiently understood the hands-on experiences performed and can now relate them to representations, such as a diagram or picture of the problem. In the case of a division exercise this could be the action of circling objects.

Abstract representation

The symbolic stage - a student is now capable of representing problems by using mathematical notation, for example: 12 ÷ 2 = 6.

Interim Methods:

Interim methods should be used to help children understand the internal logic of formal methods of recording calculations. Expanded methods should be seen as short-term;

they are stepping stones to formal written methods. Teachers should scaffold children’s thinking to guide them to the most eff icient methods, whilst at the same time valuing

their own ideas. Children should demonstrate they are secure with the current method they are using before moving on – this involves written evidence and a verbal

explanation using appropriate mathematical language of place value.

Misconceptions – ‘Good mistakes’

Difficult points need to be identified and anticipated when lessons are being designed and these need to be an explicit part of the teaching, rather than the teacher just

responding to children’s difficulties if they happen to arise in the lesson. Teachers should work together with other teachers, as well as use the “misconceptions in the key

objectives” support document, to help them consider various possible misconceptions and plan for them.

The teacher should be actively seeking to uncover possible difficulties because if one child has a difficulty it is likely that others will have a similar difficulty. Difficult points also

give an opportunity to reinforce that we learn most by working on and through ideas with which we are not fully secure or confident. Discussion about difficult points can be

stimulated by asking children to share thoughts about their own examples when these show errors arising from insufficient understanding. Mistakes are ‘good’ if we can learn

from them.

Notes for written methods:

Children should always be encouraged to consider if a mental calculation would be appropriate before using written methods.

Children should be taught when it is appropriate to do an approximation or estimate first and check with the inverse operation at the end.

Children who make persistent mistakes should return to the method that they can use accurately until they ready to move on.

Children should be encouraged to use the correct language and explain how they have answered a question (e.g. refer to the actual value of digits).

Teachers will discuss errors and diagnose problems then work through questions that caused difficulties step-by-step – not simply re-teach the method.

All new written methods should be presented alongside the previous method and children should be encouraged to explain ‘what’s the same’ and ‘what is different’.

Teachers will use meta-language to talk through new written methods e.g. ‘if you know this, then you know this…’

When revising methods or extending to more challenging numbers, teachers will refer back to expanded methods. This helps reinforce understanding and reminds children that they have an alternative to fall back on if they are having difficulties.

By upper Key Stage 2, children should be confident in choosing and using a strategy that they know will get them to the correct answer as efficiently as possible. At each stage of their development children must:

have a very clear grasp of place value in all year groups, in particular the models, images and representations used to secure mathematical understanding.

use the correct mathematical vocabulary, and be able to explain their mathematical reasoning when appropriate.

Demonstrate ‘arithmetical proficiency’ - being able to recall as many facts as possible (tables, number bonds and doubles) quickly and accurately, and then apply

these facts to calculations.

develop a repertoire of mental calculation strategies, alongside secure and efficient written methods.

understand when each method is needed using their sense of number to decide on the most appropriate way of tackling any calculation.

When are children ready to move on to written calculations?

Addition and subtraction

• Do they know addition and subtraction facts to 20? • Do they understand place value and can they partition numbers? • Can they add 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?

Multiplication and division

• Do they know the 2, 3, 4, 5 and 10 time table • Do they know the result of multiplying by 0 and 1? • Do they understand 0 as a place holder? • Can they multiply two and three digit numbers by 10 and 100? • Can they double and halve two digit numbers mentally? • Can they use multiplication facts they know to derive mentally other multiplication facts that they do not know? • Can they explain their mental strategies orally and record them using informal jottings?

Progression in Calculations

Addition

Objective and Strategies

Concrete Pictorial Abstract

Combining two parts to make a whole: part- whole model (aggregation)

= 4 + 2 4 + = 6 2 + + = 6

6

4

?

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.

Use the part-part

whole diagram as

shown above to move

onto more abstract

problems.

First use images within the bar (above). Then represent each object as part of a bar as a 1:1 representation before each quantity is represented approximately as a rectangular bar.

Starting at the bigger number and counting on (augmentation)

with no regrouping

Start with the larger number on the bead string and then count on to the smaller number 1 by 1 to find the answer.

Count on using number tracks / number lines / 100 grids

Start at the larger number on the number line and count on in ones or in one jump to find the answer. Move from a number track, to a marked number line to an empty number line. 7+ 4

0 1 2 3 4 5 6 7 8 9 10 11 12

When confident, encourage children to move on from counting in ones to more efficient jumps.

5 + 12 = 17 Place the larger number in your head and count on the smaller number to find your answer.

Regrouping to make 10. NOTE – this is a core skill that students must have a firm understanding in, to be successful with calculating more mentally and with more complex numbers. Ensure you are dedicating enough time to the concrete and pictorial phase. In KS2 and even UKS2, some children may need to go back to this skill if they are unable to bridge ten mentally. Using concrete and pictorial methods such as these are not limited to younger years.

6 + 5 = 11

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

Use pictures and images to practice regrouping: add the smaller number to make 10 (use the part-whole model when discussing partitioning) and then add what is left.

7 + 4= 11 If I am at seven, how many more do I need to make 10? How many more do I add on now? 28 + 63 = 91 If I am at sixty three, how many more do I need to make 70? How many more do I add on now? OR I am at sixty three; I add the tens to make 83. Then, how many more do I need to make 90? How many more do I have left to add on?

Use a number line to demonstrate counting on by partitioning.

Move on to larger numbers within 100. 63 + 28 = 91

28 is partitioned into 20 + 8. The tens are added first, then the ones are added by ‘bridging through a multiple of 10’

Adding three single digits NOTE – this is a great skill to teach in a series of addition lessons in KS1. This is not a core, fundamental skill that students must grasp in order to move into no regrouping addition.

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

Following on from making 10, make 10 with 2 of the digits (if possible) then add on the third digit.

Combine the two numbers that make 10 and then add on the remainder.

Add together three groups of objects. Draw a

picture to recombine the groups to make 10.

+ +

+

No regrouping addition in KS2

Use dienes or place value counters - add together the ones first then add the tens.

44 + 15 = 59 After practically using the base 10 blocks and place value counters, children can draw the counters to help them to solve additions.

The expanded method should be introduced alongside dienes or place value counters. ‘What’s the same, what’s different?’

Initially, children should be encouraged to partition the numbers. They should be adding from the right – ie starting with the ones, then the tens and finally the hundreds. Initially, equations for ones, tens and hundreds can be written in brackets alongside.

NB – This is an interim method to develop conceptual understanding. They are working towards the formal written method

The formal written method should be introduced alongside the expanded method. ‘What’s the same, what’s different?’

Mental method: Children should be shown the same calculation using an empty number line, partitioning the smaller number and counting on in ones and tens.

Compensation method: children should also be shown efficient ways of adding mentally, including adding 9 or 11 by adding 10 and adjusting by 1

35 + 9 = 44 +10

35 45 44

-1

Column method- regrouping TO + TO (up to 100)

Make both numbers on a place value grid using dienes or place value counters. 47 + 25 = 72

Add up the units (‘12 ones is 1 ten and 2 ones’) and regroup. Then add the tens (‘4 tens add 3 tens is 7 tens’).

As children move on to decimals, money and decimal place value counters can be used to support learning.

Expanded written method: Partitioning the numbers visually encourages children to make a link to the equations in brackets. This expanded method should be introduced alongside dienes or place value counters. ‘What’s the same, what’s different?’

If required, children can draw a pictoral representation of the columns and place value counters to further support their learning and understanding. Mental method: Children should be shown the same calculation using an empty number line, partitioning the smaller number and counting on in ones and tens. They should be encouraged to move on from bridging the 10, instead using their knowledge of number facts to 20 and place value to add the ones in one step. Explain this is more efficient as it requires fewer steps. Taught with Dienes or place value counters alongside.

This formal written method should be introduced alongside the expanded method. ‘What’s the same, what’s different?’

Column method- regrouping HTO + TO (through 100) and beyond NOTE – Even in KS2, your class may need concrete and pictorial representations in a lesson to help underpin the knowledge of the abstract columnar method. Differentiation across the class may need several groups of children working on different ways to add (concrete, pictorial, abstract) depending on their needs. Once a student understands conceptually why the method works, move them onto the abstract, formal written method. The aim of getting most of your class to the abstract stage by the end of the year. See below to cross-reference the standard of questions your Year Group should hit by the end of the year. Year 3: Add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction Year 4: Add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate

Year 5: Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)

Make both numbers on a place value grid using dienes or place value counters.

283 + 163 = 446

Model adding up the ones. Then add the tens (’14 tens is 1 hundred and 4 tens’) and regroup. Finally, add the hundreds (‘2 hundreds add 2 hundreds is 4 hundreds’).

Expanded written method:

Mental Method:

Formal written method (with regrouping and moving up to and beyond 4 digits):

Misconceptions to consider: Lining up digits Place holders Clear carrying and remembering to add it Encourage chn to quickly estimate answers by rounding and mentally adding – using their answer to check the ‘reasonableness’ of their written answer.

Subtraction

Objective and Strategies

Concrete Pictorial Abstract

Taking away ones

Use physical objects, counters, cubes, fingers etc. to show how objects can be taken away one at a time. 6 – 2 = 4

Cross out drawn objects to show what has been taken away.

Begin to introduce a number track alongside.

18 – 3 = ? ? = 8 - 2

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.

Use the bar model to make links with the inverse:

Ellen had 14 sweets. She gave 8 to Kallie. How many does

she have left?

Move to using numbers within the part-whole model. Make number families.

Encourage flexible thinking and create part-part whole problems with more than one answer

Extend:

10 ?

?

Counting back no regrouping

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.

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

Count back on a number tracks / number lines / 100 grids

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. The jumps are recorded above the representation, subtracting in tens before moving onto ones.

Children should be taught efficient mental strategies such as counting back and adding ones (compensation method).

This example shows counting back in 100s and then adding ones (to compensate). Other examples may be better suited to counting back in tens first then adding ones (to compensate).

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

Make 10 as first steps to regrouping / bridging

14 – 5 =

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?

Find the difference

Compare amounts and objects to find the difference.

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

Basic bar models can be used to help demonstrate finding the difference. Start by putting physical items inside the boxes (either the actual item or objects to represent them).

Count on to find the difference. 11 – 5 = 6

Increase efficiency using a number line. The first jump should be to the next multiple of ten followed by counting in multiples of ten before adding any remaining ones.

231 – 198 =

Hannah has 23 sandwiches, Helen has 15 sandwiches. Find the difference between the number of sandwiches. 23 – 15 = 8 (15 add what equals 23? Encourage using number bonds to get answer in one efficient step. Otherwise bridge ten and add the remainder)

Use other practical equipment such as bead strings. Begin to introduce the pictorial stage alongside.

Bar Model Draw bars to help demonstrate finding the difference between 2 numbers.

Counting on and taking away– Frogs and Robbers

With 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.

Play games in role deciding when it is best to be a frog, counting up (jumping) to subtract or when it is best to be a robber, taking away (stealing) to subtract.

Counting back efficiently when numbers are far apart AND when they do not require bridging 97 – 15 = 72

-5 -10

82 87 97 Counting up to find a small difference AND when bridging is required

42 – 39 = 3

42 39 40

+ 1 + 2

Based on the distance between numbers and whether bridging is required, decide which method is most efficient to perform mentally. Speed and accuracy are key.

Emphasise the inverse

‘Number families’ ‘Story of a number’ Get children to complete addition and subtraction sentences that use the same three numbers and explain how they are linked. Use as many images and models as possible. Use numbers up to 100 – don’t just focus on multiples of 10.

Column method without regrouping NOTE – Even in KS2, your class may need concrete and pictorial representations in a lesson to help underpin the knowledge of the abstract columnar method. Differentiation across the class may need several groups of children working on different ways to subtract (concrete, pictorial, abstract) depending on their needs. Once a student understands conceptually why the method works, move them onto the abstract, formal written method. The aim of getting most of your class to the abstract stage by the end of the year. NOTE 2 – always start teaching column method without regrouping before moving into regrouping. It is important students have an understanding of the steps/layout before attempting regrouping.

75 – 42 = 33

Use diennes and place value counters to make the bigger number first then take the smaller number away.

The pictorial and concrete representations demonstrate what is being taken away, with the answer left behind. The vertical, expanded equation alongside can explained verbally. Teachers should refer to the equation when modelling, in order to help them verbalise their understanding, encouraging the children to do the same.

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

Mental Calculation Efficient and flexible use of number line to:

- count back - count up - compensation method

Or, when no regrouping is required, partition and subtract

Formal written column subtraction:

Column method with regrouping NOTE – Even in KS2, your class may need concrete and pictorial representations in a lesson to help underpin the knowledge of the abstract columnar method. Differentiation across the class may need several groups of children working on different ways to subtract (concrete, pictorial, abstract) depending on their needs. Once a student understands conceptually why the method works, move them onto the abstract, formal written method. The aim of getting most of your class to the abstract stage by the end of the year. See below to cross-reference the standard of questions your Year Group should hit by the end of the year. Year 3: Add and subtract numbers with up to three digits, using formal written methods of columnar addition and subtraction Year 4: Add and subtract numbers with up to 4 digits using the formal written methods of columnar addition and subtraction where appropriate Year 5: Add and subtract whole numbers with more than 4 digits, including using formal written methods (columnar addition and subtraction)

Use diennes to start with before moving on to place value counters. Start with one exchange before moving onto subtractions with 2 exchanges. Where the ones cannot be subtracted, regrouping takes place.

In this example, seven tens is regrouped into six tens and ten ones. The equation becomes 12-7 and 60-40 respectively.

The three representations show what is happening with the equation recorded alongside at the end. Teachers should refer to the equation when modelling, in order to help them verbalise their understanding, encouraging the children to do the same.

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.

Written Method:

At this point, children are introduced to the formal written method without expanding for the first time. This should happen alongside the expanded method. Children should be asked ‘What’s the same, what’s different?’ Again, in this example, the ones cannot be subtracted so regrouping takes place. Three tens are regrouped into two tens and ten ones.

Mental Method: Efficient and flexible use of number line to:

- count back - count up - compensation method

Formal written column subtraction Teachers should chose numbers carefully, building up to multiple regrouping.

Children should be made aware of the method below but encouraged to use a number line when subtracting a decimal number from a whole number.

Multiplication

Objective and Strategies

Concrete Pictorial Abstract

Doubling Use practical activities to show how to double a number.

Draw pictures to show how to double a number. Make the link between doubling and halving:

Partition a number and then double each part before recombining it back together.

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. Write sequences with multiples of numbers. 2, 4, 6, 8, 10, ? 5, 10, 15, 20, 25 , 30, ? 3, 6, ? , 12, 15, ? 2, ?, ?, ?, 10

Repeated addition

Write addition sentences to describe objects and pictures.

Introduce the multiplication sign

2 + 2 + 2 + 2 + 2 = 10 5 x 2 = 10 Consider stem sentences to scaffold consideration of both the size of the group and the number of groups

Unitising In order to be able to reason multiplicatively, chn need to be able to unitise (treat a group as a single entity). Think about unitising in the context of the activities you engage in with a counting stick. Developing flexibility of thinking, the unit can have any value, it can be more than one or less than one. Each unit however has the same value

5 × 4 = five groups of 4? 4 x 5 = four groups of 5

There are 4 teams. There are 5 children in each team. How many children are taking part in this competition? 4 x 5 = 20 There are 5 cups. There are 4 children in each cup. How many children are there altogether in the cups? 5 x 4 = 20

Use different

objects to add

equal groups.

Arrays- showing commutative multiplication

When equal groups are arranged in equal rows, an array is formed.

Create arrays using counters/ cubes to show multiplication sentences. e.g. 6 x 4 = 6 groups of 4

Draw arrays in different rotations to find commutative multiplication sentences.

Use an array to write multiplication sentences and reinforce repeated addition.

5 + 5 + 5 = 15 3 + 3 + 3 + 3 + 3 = 15 5 x 3 = 15 3 x 5 = 15 Later, make links to the inverse

If I know…Then I know… All children: reason

mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language

Once children have learned some multiplication facts, it is important they learn to generalise these facts. Creating an If I know…Then I know…cloud can help with these generalisations. Although not a necessary step in formal methods, it allows children to not use formal methods to work out what should be known facts. i.e. 800 x 70 = A LKS2 cloud may look like: UKS2 cloud may look like:

Grid Method Year 3: Write and

calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one- digit numbers, using mental and progressing to formal written methods

Show the link with arrays to first introduce the grid method.

Children can draw the counters, using colours to show different amounts or just use circles in the different columns to show their thinking as shown below.

Start with multiplying by one digit numbers and showing the clear addition alongside the grid. Ensure children are secure with place value – especailly x 10 and x 100. Encourage them to verbalise the process to demonstrate understanding.

Children can record 56 for each answer, then adjust the answer to match how the question has been adjusted. e.g. 80 x 7 = 560 (the question has been multiplied by 10, so I must multiply my answer by 10. UKS2 children must do the multiple of ten before ‘taking back the decimal.’

e.g. 800 x 0.7 = 560.0 (the question has been multiplied by 100, so I must multiply my answer by 100. Then, the question was

divided by 10, so I must divide my answer by ten (essentially ‘taking back’ a 0)

Language can be adjusted to

the year group. LKS2 could

discuss how the question is

different, “there’s a zero.”

Older years can identify the

question has been

divided/multiplied by ten and

therefore, the answer must

be as well.

NOTE – use as a step to get LKS2 onto short multiplication.

The two digit number is partitioned horizontally with the tens digit coming first. The equation is then represented using counters (or an array).

Again, the two digit number is partitioned horizontally with the tens digit coming first. This time the equation is represented using place value counters or Dienes.

Bar modelling can support learners when identifying and solving problems with multiplication alongside the formal written methods. E.g. There are 3 bags of sweets. Each bag has 4 sweets inside. How many are there altogether?

E.g. There are 4 bags of sweets. Each bag has 30 sweets inside. How many are there altogether?

TO x O

HTO x O

HTO x TO

Ensure chn take care with presentation, neatly putting the total of each row alongside before adding them using column addition

Short column multiplication Year 3: Write and

calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one- digit numbers, using mental and progressing to formal written methods Year 4: Multiply two-

digit and three-digit numbers by a one-digit number using formal written layout Year 5: Multiply

numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two- digit numbers Year 6: Multiply multi-

digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication

The short multiplication method is introduced alongside the grid method and expanded form to aid understanding. Children should be told that these are interim methods – they are working towards becoming more efficient mathematicians. Teachers should ensure children can explain their understanding of each method verbally and written down before moving on. Children should be encouraged to discuss what is similar and what is different between the different strategies. Bar modelling can support learners when solving word problems with multiplication, alongside the formal written methods.

TO x O

HTO x O

Long column multiplication

Multiplying a two-digit number by a three-digit number should be introduced through the grid method before moving to long multiplication to aid understanding. When long multiplication is introduced, both equations should be presented so that the answers to the individual multiplication steps are on the same line. Children should be encouraged to discuss what is similar and what is different.

TO x TO

HTO x TO

Division

Objective and Strategies

Concrete Pictorial Abstract

Halving (can be done in conjuncture with doubling or as a separate unit).

Use practical activities to show how to half a number.

Draw pictures to show how to half a number. Make the link between doubling and halving:

Partition a number and then half each part before recombining it back together.

What can we

share evenly

between

two?

Dominoes

reinforce two

equal parts.

Sharing objects into groups (primarily KS1)

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

Children use pictures or shapes to share quantities.

9 ÷ 3 = ?

Share 9 buns between three people.

9 ÷ 3 = 3

Division as grouping (primarily LKS2) NOTE – Even in UKS2, your class may need concrete and pictorial representations in a lesson to help underpin the knowledge of the abstract methods. Differentiation across the class may need several groups of children working on different ways to subtract (concrete, pictorial, abstract) depending on their needs. Once a student understands conceptually why

Divide quantities into equal groups. Use cubes, counters, objects or place value counters to aid understanding.

Use a number line counting up from zero to show jumps in groups. The number of jumps equals the number of groups. This representation should be supported by grouping of concrete materials and other pictorial representations.

Note: NC times table requirements should be considered when choosing examples for the early written division methods (end of Y2: x2, x5 and x10. By the end of Y3: x3, x4 and x8. By the end of Y4: up to 12 x 12)

28 ÷ 7 = 4 Divide 28 into 7 groups. How many are in each group? Can you think of a number we could share equally between three people?

8 ÷ 2 = 4

the method works, move them onto the abstract, formal written method. The aim of getting most of your class to the abstract stage by the end of the year.

Use the bar model to assist. 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. Link to multiplication.

Division within arrays Year 3: Write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, including for two-digit numbers times one- digit numbers, using mental and progressing to formal written methods

Link division to multiplication by creating an array and thinking about the number sentences that can be created. Eg 15 ÷ 3 = 5 5 x 3 = 15 15 ÷ 5 = 3 3 x 5 = 15

Encourage chn to verbalise their understanding:

Draw an array and use lines to split the array into groups to make multiplication and division sentences.

Make links to part-part whole images used with addition and subtraction

Find the inverse of multiplication and division sentences by creating four linking number sentences. 7 x 4 = 28 4 x 7 = 28 28 ÷ 7 = 4 28 ÷ 4 = 7

Short division (When dividing by one digit) NOTE – Although the curriculum states that remainders are introduced in Year 5 only, some greater depth pupils may benefit from working with remainders in a formal method in Year 4. Remainders can be explored in concrete and pictorial methods in earlier years, especially linking to multiples/times table. i.e. Will 5 fit evenly into 26? No, because it is not in the 5x table. How close can we get and what’s left over? See next row for further examples of how to introduce remainders in younger years.

Year 3: write and calculate mathematical statements for multiplication and division using the multiplication tables that they know, using mental and progressing to formal written methods Year 4: Pupils practise to become fluent in the formal written method of short multiplication and short division with exact answers (no remainder). Year 5: Divide numbers up to 4 digits by a one- digit number using the formal written method of short division and interpret remainders appropriately for the context

Use place counters to divide. This will require secure knowledge of place value – children need to be able to explain, verbally or written, what is happening at each stage.

72 ÷ 3 = 24

Progress to more formal layout: TO ÷ O Chn make links to arrays. They can begin by physcially creating the arrays using counters. 96 ÷ 3 = 32

90 ÷ 3 = 30 (3 tens) 6 ÷ 3 = 2 HTO ÷ O

696 ÷ 3 = 232 600 ÷ 3 = 200 (2 hundreds) 90 ÷ 3 = 30 (3 tens) 6 ÷ 3 = 2 (2 ones)

Students can continue to use drawn diagrams with dots or circles to help them divide numbers into equal groups. Mental Calculation: Encourage chn to use a number line to count in multiples to divide more efficiently.

Formal Written:

Begin with divisions that divide equally with no remainder.

“4 goes into 9 2 times remainder

1. 4 goes into 16 4 times”

Move onto divisions with a remainder for Year 5 and 6 (see note on left for younger years and see row below).

Children need to be able to decide what to do after division and round up or down according to the context. Year 5 and 6: Finally move into decimal places to divide the total accurately.

Year 6 only: divisor a 2-digit number

Intro to division with a remainders Abstract starts in Year 5.

NOTE – Although calculating remainders formally begins in Year 5, it is good practice to introduce the concept through the concrete and pictorial stages in younger years.

Divide objects between groups and see how much is left over 14 ÷ 3 =

Talk with the child about splitting items in class and at home between different numbers of people. e.g. 20 ÷ 4 = 6 (Each child gets 6 equal pieces)

20 ÷ 3 = 6 remainder 2 (Each gets 6 pieces but

there are 2 pieces left over which can not be divided equally between 3 people)

Discuss what do could you do with various remainder (left-over) items? E.g. Cut food, put biscuits back in the tin, put money back in a purse, agree to share things like pens.

Jump forward in equal jumps on a number line then see how many more you need to jump to find a remainder. This should also be supported by grouping of concrete materials and other pictorial representations.

Draw dots and group them to divide an amount and clearly show a remainder. Use pictures to assist:

Complete written divisions and show the remainder using r. Use knowledge of times tables to assist.

Chunking (when dividing by two digits or more)

NOTE – This is an optional step to get children to use formal long division in Year 6. Formal long division is an excellent end goal, however, Year 6s can use short division for 2-digit divisor on SATs. Many children in UKS2 do not get on to this step and often the steps for chunking can be difficult for children with poor working memories. Consider carefully whether you introduce this step to your students or move straight to Long Division.

Year 5 and 6: divide numbers up to 4 digits by a two- digit whole number using a formal written method, and interpret remainders as whole number remainders, fractions, or by rounding, as appropriate for the context

HTO ÷ TO (no remainders): Chn should draw a RUB (really useful box) to help them to chunk efficiently. They should always start by x1, x2, x10, x5 in this order as it encourages them to make links (double x1 to get x2 answer and half x10 answer to get x5). Chn will need to make another box if the number they are dividing by is very large. Encourage chn to make links (like x20 by doubling their answer to x10 or x100 by timising their answer to x10 by 10)

Chn may benefit from seeing the chunks being taken away using a number line.

Mental Calculation The efficient grouping method is now used but with a two digit divisor. Again, the number of groups should be recorded above the jump. HTO ÷ TO (remainders interpreted as fractions or decimals)

Formal Long Division NOTE – This is a difficult step for children who struggle with working memory and have difficulty organising their work. Many children may not get onto this step by the end of KS2, however, short division can be used to calculate a 2-digit divisor.

This method is reliant on children knowing their times tables facts very well and also being able to make links between calculations. Without this, chunking should always be used.

Order of Operations Year 6: use their knowledge of the order of operations to carry out calculations involving the four operations.

BIDMAS = Brackets, Indices, Division, Multiplication, Addition and Subtraction (aka BODMAS or BEDMAS – orders, exponents respectively).

Start with Indices as children should be familiar with squared and cubed numbers. If children can calculate mentally/have automaticity, that is fine. Nearing the teaching of algebra, have them set up their books in proper notation to assist with the transition.

Teaching point – ensure students are not multiplying the root number by the indice. i.e. 62 is 6 x 6 (6 multiplied by itself two times) and not 6 x 2. This is a good mistake students can get stuck on.

Children working towards may need ‘jottings’ on the side of their book to help with a calculation.

NOTE – division and multiplication and addition and subtraction, are completed in the order in which they appear as they have equal ‘weighting’. So although it lists division before multiplication, if the question was 5 x 3 ÷ 2, you would do 5 x 3 first.

Students may find it helpful to write the acronym above the question and cross out operations that are irrelevant to the question to avoid completing the wrong order of operations.

“Layering” answers helps set up for algebra so although

it seems counter-intuitive to get children to lay out steps,

it can help for later notations. In Year 6, both the left and

right options are acceptable. The left helps children with

working memory and processing difficulties.

“Layering” can also help children sequence the order of

operations, preventing errors, by copying the information

in the next line that is not being calculated. For example,

students copy = 5 + and then solve the multiplication.

Language: Addition & Subtraction Years 3 and 4: add, addition, more, plus, increase, sum, total, altogether, double, near double, how many more to make…? how many more to make…? how many more is… than…? how much more is…? -, subtract, subtraction, take (away), minus, decrease, leave, how many are left/left over? how many fewer is… than…? how much less is…? difference between, half, halve, how many more/fewer is… than…? how much more/less is…? Is equal to, is the same as, tens boundary, hundreds boundary, inverse Years 5 and 6: add, addition, more, plus, increase, sum, total, altogether, double, near double, how many more to make…? subtract, subtraction, take (away), minus, decrease, leave, how many are left/left over? difference between, half, halve, how many more/fewer is… than…? how much more/less is…? Is equal to, sign, is the same as, tens boundary, hundreds boundary, units boundary, tenths boundary, inverse Multiplication & Division Year 3 and 4: lots of, groups of, times, multiply, multiplication, multiplied by, multiple of, product, once, twice, three times… ten times…times as (big, long, wide… and so on), repeated addition, array, row, column, double, halve, share, share equally, one each, two each, three each…group in pairs, threes… tens, equal groups of, divide, division, divided by, divided into, remainder, factor, quotient, divisible by, inverse Years 5 and 6: lots of, groups of, times, multiply, multiplication, multiplied by, multiple of, product once, twice, three times… ten times…times as (big, long, wide… and so on), repeated addition array, row, column, double, halve, share, share equally, one each, two each, three each…group in pairs, threes… tens, equal groups of, divide, division, divided by, divided into, dividend, divisor, remainder, factor, quotient, divisible by, inverse, fraction

Suggested Calculation Progression: Addition:

Step 1: I know when to add some more Step 2: I know to find the total Step 3: I add the right amount Step 4: I add the right amount and can count how many altogether Step 5: I can add numbers of objects to 10 Step 6: I can read number sentences: U + U = Step 7: I can arrange a number sentence: 5 blocks and 3 blocks is the same as 5 + 3 = Step 8: I can solve a number sentence U + U = Step 9: I can solve addition on a number line 3 + 4 = Step 10: I can add 1 to a number up to 20 16 + 1 = Step 11: I can add 2 or 3 to a number up to 20 16 + 3 = Step 12: I can add a U number to a number to 20 11 + 7 = Step 13: I can add a U to a TU number 28 + 1 = Step 14: I can add 10 to a TU tens number 40 + 10 =

Step 20: I can solve any TU + U 68 + 9 = Step 21: I can add any TU tens number to another one 80 + 90 = Step 22: I can add a TU tens number to a TU number 23 + 40 = Step 23: I can add any TU tens number to a TU number 23 + 90 = Step 24: I can add a TU number to a TU number 43 + 52 = Step 25: I can solve any TU + TU 43 + 88 = Step 26: I can solve HTU + TU 432 + 24 = Step 27: I can solve any HTU + TU 432 + 88 = Step 29: I can solve any HTU + HTU 741 + 353 = Step 30: I can solve HTU + HTU as money £385 + £467 = Step 31: I can solve any HTU + HTU as money £271 + £867 = Step 32: I can solve .t + .t 0.4 + 0.3 = Step 33: I can solve any .t + .t 0.8 + 0.9 = Step 34: I can solve U.t + U.t 3.4 + 2.5 = Step 35: I can solve any U.t + U.t 6.7 + 8.4 =

Step 15: I can add 10 to any TU number 38 + 10 = Step 16: I can add a U number to a TU tens number 30 + 4 = Step 17: I can solve TU + U 32 + 4 = Step 18: I can add a TU tens number to another one 20 + 60 = Step 19: I can solve any U + U in my head 8 + 9 =

Step 36: I can solve additions with U.th 4.37 + 4.62 = Step 37: I can solve any additions with U.th 3.85 + 8.67 = Step 38: I can solve additions with larger numbers 3819 + 9632 = Step 39: I can solve additions with several numbers 1202 + 45 + 367 = Step 40: I can solve U.th + U.t 3.33 + 2.5 = Step 41: I can solve any U.th + U.t 8.67 + 9.8 =

Subtraction: Step 1: I know when to take some away Step 2: I know to take some away, then count how many are left Step 3: I take away the right amount Step 4: I take away the right amount and count how many are left Step 5: I can take away numbers of objects to 10 Step 6: I can read a subtraction number sentence “6 - 4 =” Step 7: I can arrange a subtraction number sentence 6 blocks – 4 blocks Step 8: I can solve a subtraction number sentence 6 – 4 = Step 9: I can solve subtraction on a number line 6 – 4 = 2 Step 10: I can take 1 from a number to 20 16 – 1 = Step 11: I can take 2 or 3 from a number to 20 16 – 3 = Step 12: I can take a U from a number to 20 16 – 7 = Step 13: I can take 10 from a multiple of 10 80 – 10 = Step 14: I can take 10 from a TU number 43 – 10 = Step 15: I can take a multiple of 10 from a multiple of 10 80 – 30 = Step 16: I can take a U number from a multiple of 10 80 – 6 = Step 17: I can solve TU – U 48 – 5 = Step 18: I can solve any TU – U 43 – 7 = Step 19: I can solve any HTU – U 343 – 7 = Step 20: I can spot the next multiple of 10 73…80 Step 21: I can count to the next multiple of 10 73…80 = 7 Step 22: I know the gap to the next multiple of 10 80 – 73 = Step 23: I know the U gap from a multiple of 10 84 – 80 = Step 24: I know the total gap across a multiple of 10 84 – 73 = Step 25: I can take a multiple of 10 from any TU number 46 – 20 = Step 26: I can find the 2 gaps in a TU – TU question 46 – 17 = Step 27: I can solve any TU – TU 46 – 17 = Step 28: I can take any TU number from 100 100 – 35 =

Step 29: I can take 100 from any HTU number 682 – 100 = Step 30: I can solve any HTU – TU 682 – 35 = Step 31: I can solve ThHTU - TU 4628 – 35 = Step 32: I can solve HTU – HTU 628 – 235 = Step 33: I can solve HTU – HTU as money £6.28–£2.35 = Step 34: I can subtract numbers with .th 6.08 – 2.05 = Step 35: I can subtract numbers with .t 4.5 – 1.7 = Step 36: I can solve any whole number subtraction question 4603 – 176 = Step 37: I can subtract numbers with different decimal places 5.6 – 3.75 =

Multiplication: Step 1: I can set out groups of toys when I play sets out 3 lots of 4 cars Step 2: I can find the total amount of toys sets out 3 lots of 4 cars & finds the total Step 3: I can set out groups of blocks sets out 3 lots of 4 blocks Step 4: I can find the total amount of blocks sets out 3 lots of 4 blocks & finds total Step 5: I can draw groups of dots sets out 3 lots of 4 dots Step 6: I can find the total amount of dots sets out 3 lots of 4 dots & finds total Step 7: I can write out repeated addition 3 + 3 + 3 + 3 = Step 8: I can solve repeated addition 3 + 3 + 3 + 3 = 12 Step 9: I can solve U x U 4 x 2 = Step 10: I can solve a multiple of 10 x U 20 x 4 = Step 11: I can solve TU x U 23 x 4 =

X table facts to be known from memory (x2, x5, x10 x2, x3, x4, x5, x8) Step 12: I can solve any U x U 7 x 8 = Step 13: I can do any multiple of 10 x a multiple of 10 80 x 70 Step 14: I can solve any TU x U 86 x 7 = Step 15: I can solve HTU x U 725 x 6 = Step 16: I can solve TU x TU 38 x 69 = Step 17: I can solve HTU x TU 368 x 53 = Step 18: I can solve U x .t 0.3 x 4 = Step 19: I can solve U x .th 6 x 0.07 All x table facts to be known from memory, along with secure place value knowledge Step 20: I can solve any TU x U.th 26 x 5.24 =

Division: Step 1: I can give out objects fairly “Please share these sweets out fairly.” Step 2: I can count how many each person was given “How many does each person have?” Step 3: I can share an even number of objects between 2 people “Please share these sweets between us, and say how many we have each.” Step 4: I can halve an even number of objects “How many in that half?” Step 5: I can share 6,9,12 or 15 objects between 3 people 15 ÷ 3 = Step 6: I can share 6,9,12 or 15 objects into 3 15 ÷ 3 = Step 7: I can share 8, 12, 16 or 20 objects between 4 people 20 ÷ 4 = Step 8: I can share 8, 12, 16 or 20 objects into 4 20 ÷ 4 = Step 9: I can share equally to solve division problems (÷2,3,4) 28 ÷ 4 = Step 10: I can make groups of 2, 5 or 10 (“Please put these sweets into piles of 5.”) Step 11: I can find how many altogether by counting through each group (1,2,3,4,5 / 6,7,8,9,10 / 11,12,13,14,15) Step 12: I can find how many altogether by counting in 2’s, 5’s or 10’s Step 13: I can arrange a division number sentence 15 ÷ 3 = (15 blocks going into piles of 3) Step 14: I can solve a division number sentence with objects 5 piles of 3 Step 15: I can solve division using objects (with remainders) 17 ÷ 3 = 5 r2 (17 blocks going into piles of 3) Step 16: I can use a tables fact to find a division fact 15 ÷ 5 = Step 17: I can use a tables fact to find a division fact 17 ÷ 5 = (with remainders) Step 18: I can combine 2 or more tables facts to solve division 65 ÷ 5 = Step 19: I can combine 2 or more tables facts to solve division 69 ÷ 5 = (with remainders)

All x table facts to be known from memory, along with secure place value knowledge Step 20: I can use a tables fact to find a division fact 45 ÷ 9 = Step 21: I can use a tables fact to find a division fact 47 ÷ 9 = (with remainders) Step 22: I can combine 2 or more tables facts to solve division 117 ÷ 9 = Step 23: I can combine 2 or more tables facts to solve division 120 ÷ 9 = (with remainders) Step 28: I can use a coin fact to find a division fact 280 ÷ 14 = Step 29: I can use a coin fact to find a division fact 286 ÷ 14 = (with remainders) Step 30: I can combine 2 or more coin facts to solve division 286 ÷ 14 = Step 31: I can combine 2 or more coin facts to solve division 331 ÷ 14 = (with remainders) Step 32: I can use a tables fact to find a decimal division fact 2.4 ÷ 8 = Step 33: I can solve division involving decimal numbers

Fractions

Objective and Strategies

Concrete Pictorial Abstract

Identify parts and wholes A whole can be divided into many parts. Many parts can make one whole.

Your shape is the whole. Cut it into a maximum of 4 parts. Colour each part a different colour. Glue the parts together to return to the whole.

If China is the whole, then, Shanghai is a part

5 is the whole. 3 is a part. 2 is a part.

Introduce the concept of equal parts Check that children don’t think that just dividing a shape into any two pieces is halving but understand that they have to be equal

Show and talk about halves and quarters of objects using equal sharing and grouping, such as:

Sandwiches

Shapes

Continuous quantities – liquids, string etc

Practically make equal fractions and find objects that can be split into equal fractions (e.g. leaves)

Identify, write and shade fractions (begin with simple fractions like 1/2 and ¼) Compare fractions, making links between them (e.g. ¼ with 1/8 etc).

Show and talk about halves and quarters in real life practical contexts, such as:

• Half, quarter and three-quarter turn when telling the time (On a clock face, show half-past 7). • In PE, use everyday language to describe a movement – whole / half / quarter turns. • Use a floor robot to reach a particular place. • Show half and quarter of shapes in different ways.

Shade fractions of amounts

State the fraction that is shaded

Find simple fractions of amounts, for example, 1/2 of 6 = 3

Solve problems by sharing a group of items between two people and four people:

Biscuits

Bars of chocolate

Jar of sweets

String / ribbon

Four children share a pizza equally. Draw a diagram to show how much pizza each gets. What fraction of the pizza does each child get? Four children share 12 marbles equally. Draw a diagram to show how much pizza each gets. What fraction of the marbles does each child get?

Half of 8 = ? Link to doubling: ? + ? = 8 and ? x 2 = 8

Half of 12 is 2/4 of 12 is 1/4 of 20 = 3/4 of 20 =

Develop:

Count in fractions up to 10, starting from any numbers and using the 1/2 and 2/4 equivalence on the number line Count up and down in tenths (Year 3) moving on to fractions with different denominators

Use concrete and pictorial models of fractions to assist with counting. e.g. cut a sandwich up into 4. What fraction is each part? Then model adding one piece at a time, counting up in quarters until you get a whole again. Then take a part away. How much do you have now?

Count in halves up to 10, showing this on a number line and visually, e.g. as halves of a rectangular model.

On a number line labelled 0-1, mark 1/3, ¼, ½, ¾, 9/10 etc

Sequences: 1 ¼, 1 ½, 1 ¾, 2, ?

Make links to repeated addition

Recognise relationships between fractions, including equivalent fractions

e.g. 2/4 and 1 /2

Paper folding - What is the relationship between these fractions?

Compare two small Cuisenaire rods and say what fraction one rod is of the other another

Encourage the correct use of mathematical language when asking children to explain their

understanding

Begin to compare and order fractions and put them onto a number line

Use cubes to explain why 1/10 is smaller than 1/5 by comparing the lengths of the parts. When the denominator is a larger

number, the value of the fraction is smaller.

Move on to pictorial representations of fractions with the same numerator but

different denominators

Use fraction walls, and carefully chosen images to decide which fraction is bigger. Link fraction walls to the number line and

measurement scales and place fractions on the line to order them.

Order these fractions:

Formal Methods Addition, subtraction, multiplication and division of fractions

All formal methods should be taught alongside concrete resources and pictorial representations. Children should be encouraged to verbalise their understanding at every opportunity. It is important for teachers to model each method carefully, explaining what is going on at each stage. Questions should always be introduced in a context which is meaningful to the children.

Adding and Subtracting Fractions Year 3: Add and subtract fractions with the same denominator within one whole

Children should practice this with concrete materials (e.g. chocolate) and multiple representations before completing the abstract calculation. There is no requirement to go beyond a whole.

Introduce all concepts in context:

Year 4: Add and Subtract fractions where the answer may be an improper fraction

In Year 4, the expectation moves beyond a whole into improper fractions. There is no requirement to convert to mixed numbers.

Year 5: Add fractions with the same denominators and convert the answer from improper fractions to mixed numbers

The next step, in Year 5, now requires children to convert their answer from an improper fraction to a mixed number.

Year 5: Add and subtract fractions where one denominator is a multiple of the other

Children are converting to find the lowest common multiple for the first time – a secure understanding of equivalent fractions is therefore required. In the first instance, examples should be used that remain within a whole before practising with mixed numbers. Children draw upon their knowledge of multiples to find the lowest common multiple rather than multiplying the two denominators.

Year 6: Add and subtract fractions with different denominators

At this stage, both common denominators are converted to the lowest common multiple. Children will need to draw upon their times tables knowledge in identifying these. It may be useful to provide additional support through the inclusion of an arrow indicating what the numerator and denominator are being multiplied by when finding the equivalent fraction.

Year 6: Add and subtract a mixed number to a fraction where there are different denominators

The final stage requires children to again identify the lowest common multiple. A possible misconception here is that children may, in finding the equivalent fraction, multiply the whole number.

Multiplying Fractions Year 5: Multiply proper fractions and mixed numbers by whole numbers.

Proper Fractions: This should be introduced through repeated addition alongside a representation for the majority. With more confident children, you may want to go onto the formal method where they are required to convert the whole into an improper fraction (with 1 as the denominator).

4 x ¼ = 4/4 (one whole)

Mixed Numbers: The whole numbers should be multiplied first before multiplying the proper fraction through repeated addition (as covered in the proper fraction element of this objective).

Year 5: Multiply simple pairs of proper fractions writing the answer in its simplest form

The numerators of both fractions are multiplied together as are the denominators. This should be covered before moving onto the requirement to simplify

Dividing Fractions Year 6: Divide proper fractions by whole numbers (covering when the numerator is and is not a multiple of the whole number)

Children should explore the pictorial representation of dividing a fraction. For example, 1/2 ÷ 2 means children need to split one half into two equal pieces. In this example, two thirds is divided into three equal parts giving 6/9. 6/9 is then divided by 3 to give an answer of 2/9.

In the abstract method, the whole number is made into an improper fraction before it is changed to a reciprocal of the divisor. The final step is multiplying as explained previously. Children should be exposed to examples where the numerator is and is not a multiple of the whole number (for example 12/15 ÷ 6 and 4/5 ÷ 3).

Calculating Percentage

Year 5: solve problems which require knowing percentage and decimal equivalents of 1/2, 1/4, 1/5, 2/5, 4/5 and those fractions with a denominator of a multiple of 10 or 25 Year 6 - solve problems involving the calculation of percentages [for example, of measures, and such as 15% of 360] and the use of percentages for comparison

There are two effective ways of calculating percentage. Although there is a faster/shorter way, the longer method reinforces, conceptually, how students arrive at a given total. It is important children understand how they are working out these calculations for them to use as a building block for secondary school concepts. However, the longer method can sometimes be trickier for children with poor working memory. Choose the method that is appropriate for your students (and they may be displayed side by side to aid in understanding.) Children should be encouraged to discuss what is similar and what is different between the different strategies. To start percentage calculation with all students, begin with a multiplication cloud. Children can then see how some percenatges can be calculated just by looking at a number. It can also link to multiplication If I know…Then I know… This sets up the formal method below.

Although this is a much more efficient

method, it doesn’t have the conceptual

underpinning that the longer method

does. It is important children understand

how percentages are calculated before

moving to the shorter method.