Policy For Mathematics

Beaumont Primary School

Mathematics Policy Introduction

In accordance with the requirement of the National Curriculum at Key Stage 2 we promote the acquisition of mathematical skills and knowledge, and also the ability to investigate and solve mathematical problems. We believe that mathematics should be enjoyable and meaningful in order to develop a positive attitude to the subject. It is not intended to be taught to children by rote, we should aim to help children to learn that mathematics can be taught meaningfully with understanding. Purpose

Mathematics is a creative and highly interconnected discipline that has been developed over essential

to everyday life, critical to science, technology and engineering, and necessary for financial literacy and all forms of employment. A high quality education in maths therefore provides a foundation for understanding the world, the ability to reason mathematically, an appreciation of the beauty and power of mathematics, and a sense of enjoyment and curiosity about the subject.

Aims (embracing the new curriculum) The new National Curriculum for mathematics aims to ensure that all pupils:

become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately;

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

can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.

Mathematics is an interconnected subject in which pupils need to be able to move fluently between representations of mathematical ideas. The programmes of study are, by necessity, organised into distinct domains, but pupils should make rich connections across mathematical ideas to develop fluency, mathematical reasoning and competence in solving increasingly sophisticated problems. They should also apply their mathematical knowledge to science and other subjects. The expectation is that the majority of pupils will move through the programmes of study at broadly the same pace. However, decisions about when to progress should always be

stage. 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. Spoken Language The national curriculum for mathematics reflects the importance of spoken language in

cognitively, socially and linguistically. The quality and variety of language that pupils hear and speak are key factors in developing their mathematical vocabulary and presenting a mathematical justification, argument or proof. They must be assisted in making their thinking clear to themselves as well as others and teachers should ensure that pupils build secure foundations by using discussion to probe and remedy their misconceptions. Early Years

Work undertaken within the Foundation Stage is guided by the requirements and

children are given ample opportunity to develop their understanding of mathematics. Lessons in the Early Years aim to do this through varied activities that allow children to use, enjoy, explore, practise and talk confidently about mathematics. School Curriculum (KS1 and KS2) A new personalised curriculum has been devised for all pupils which includes all the age related programmes of study and incorporates the aims of the 2014 curriculum throughout. The new curriculum for each age phase is set under three unit headings per term with a range of domains covered within each. Planning, learning and teaching

The three aims of knowledge, reasoning and problem solving are to be incorporated into every lesson and to provide children with the independence to be challenged. Planning will allow for children to begin learning as soon as the lesson starts with open-ended thinking activities before encountering the review stage. Lessons will then include knowledge, reasoning and problem solving activities, all of which will have a purpose - a real life connection and a reason for learning including, where necessary, the outside environment. Mini plenaries will be planned for and

further and allow for the progression through activities to develop at the appropriate pace for each individual child. Final plenaries will be planned for and used effectively to consolidate, assess and take learning forward.

Questioning is the key to success in all our mathematics sessions and questions will be continuously adapted by the teacher and support staff based on assessment for learning.

Mental maths should be incorporated throughout all lessons and mental strategies for solving all mathematical concepts will be discussed and developed based on continuous assessment for learning.

Use of teaching assistant support is planned for in every part of the mathematics lessons to ensure they are used effectively in supporting, developing and assessing pupil progress throughout. We value the impact that T.A. support has on all our

with teachers. Regular training opportunities are given to keep them fully updated and develop their skills further. They are encouraged to share assessment observations made in mini-plenaries, final plenaries and through discussions with teaching staff to

Differentiation and Support

This is incorporated into all mathematics lessons and is done in various ways, such as:

setting challenging age-related knowledge, reasoning and problem solving tasks

understanding;

small, differentiated steps for all children to move through at a pace that suits that their needs;

understanding throughout lessons;

ensuring that marking and constructive feedback is personal, frequent and of a consistently high quality enabling pupils to understand how to improve and develop their work and planned in time for children to respond to feedback;

real life, practical links throughout all knowledge, reasoning and problem solving tasks, with whole class activities planned at the end of each unit;

range of practical, real life resources used to support all stages of learning within the class;

homework set, differentiated through the three aims - knowledge, reasoning and problem solving tasks;

intervention programmes/extra teacher support delivered where needed, both in class and through extra sessions planned outside the sessions i.e. pre-teaching;

visual stimulus/aids are provided for our visually impaired and English as an additional language pupils.

Assessment Assessment is regarded as an integral part of teaching and learning and is a continuous process. It is the responsibility of the class teacher to assess all pupils in their class. This is mainly achieved through mini-plenaries, questioning, marking, TA feedback and pupil self-assessment. Marking The main purpose of our marking policy is to ensure that as children progress through the school, they benefit from constructive guidance and next step questioning to challenge and consolidate their learning further. Display

All classrooms should have mathematics working walls which:

celebrate

ask questions to promoted mathematical thinking;

contain key vocabulary;

are an integral part of teaching. Contribution of mathematics to teaching in other curriculum areas

Wherever possible, skills and concepts learned within mathematics lessons are followed up and reinforced within all other curriculum areas. These links are made explicit to the children. English Mathematics contributes significantly to the teaching of English in our school, by actively promoting the skills of reading, writing, speaking and listening. For example, we encourage children to read and interpret problems in order to identify the mathematics involved. The children explain and present their work to others during plenary sessions. Younger children enjoy stories and rhyme that rely on counting and sequencing. Older children encounter mathematical vocabulary, graphs and charts when using non-fiction texts. Information Technology (IT) Children use and apply mathematics in a variety of ways when solving problems using IT. Younger children use IT to communicate results with appropriate mathematical symbols. Older children use it to produce graphs and tables when explaining their results or when creating repeating patterns, such as tessellations. When working on control, children use standard and non-standard measures for distance and angle. They use simulations to identify patterns and relationships. Personal, social and health educations (PSHE) and citizenship Mathematics contributes to the teaching of PSHE and citizenship. The work that children do outside their normal lessons encourages independent study and helps them to become increasingly responsible for their own learning. The planned activities that children do within

views. We present older children with real-life situations in their work on the spending of money. Spiritual, moral, social and cultural development The teaching of mathematics supports the social development of our children through the way we expect them to work with each other in lessons. Resources Teachers plan from Focus Maths scheme, which follows the format of the Revised Framework. Each teacher is equipped with a set of teaching resources to support an emphasis on resource-based teaching. All classes have a set of basic equipment to support the curriculum. Resource needs are audited annually and purchased in order of priority. A variety of representations of the number system should be accessible to children in every lesson e.g. numerals, 0-100 number squares and a variety of other grids, appropriate numbered and blank number lines and tracks, place value charts and grids.

Progression of calculation methods We have a policy for progression in calculation methods to ensure continuity and consistency throughout the school. (see Appendix 1) Problem Solving We have a separate policy for problem solving. (see Appendix 2) Policy of Equality Statement The Board of Governors and School is committed to a policy of equality and aims to ensure that no employee, job applicant, pupil or other member of the school community is treated less favourably on grounds of sex, race, colour, ethnic or national origin, marital status, age, sexual orientation, disability or religious belief. Date: Summer Term 2017 Review Date: Summer Term 2019

Appendix 1

How we teach calculati ons:

Calculation Policy for

Mathe mati cs

The fo llowing calculat ion policy has been devised to meet r equir ements of t he Nat ional Curr iculum 2014

for the t eaching and lear ning of mat hemat ics, and is also designed to give pupils a consist ent and

smooth pr ogr ession of lear ning in calculat ions acr oss t he school. Please not e t hat this calculat ion

policy is designed to build on pr ogr essively f r om the cont ent and methods est abl ished in the Ear ly

Year s Foundat ion St age.

Age st age expect at ions

The calculat ion policy is or ganised accor ding to age st age expect at ions as set out in t he Nat ional

Curr iculum 2014, however it is vit al that pupils ar e t aught accor ding to t he sta ge t hat t hey are

currently work ing at, being moved onto t he next level as soon as t hey ar e r eady, or wor king at a lower

st age unt il t hey ar e secur e enough to move on.

Providing a conte xt f or calculat ion: I t is impor t ant t hat any t ype of calculat ion is given a r eal lif e cont ext or p r oblem solving appr oach to

help build childr ens under st anding of the pur pose of calculat ion, and to help t hem

r ecognise when to use cer t ain operat ions and methods when f aced wit h pr oblems. This must be a

pr ior it y with in calculat ion lessons.

Choosing a calculat ion method:

Childr en need to be t aught and encour aged to use t he f ollowing pr ocesses in deciding what appr oach

th ey will t ake to a calculat ion, to ensur e t hey select the most appr opr iat e method f or the number s

involved:

Can I do it in my head using a

mental strategy?

Could I use some jottings to

help me?

Should I use a written method to

work it out?

Calculation Process:

Approximate,

Calculate,

Check it!

Year 1 Add w it h numbers up to 20

Use number ed number lines t o add, by count ing on in ones. Encourage child r en

t o star t with t he larger number and count on.

+1 +1 +1

Children should:

Have access t o a wide range of count ing equipment , ever yday obj ect s,

number t r acks and number lines, and be shown numbers in di f fe r ent con-

text s.

Read and wr ite t he addit ion (+) and equals (=) signs wit hin number sen-

te nces.

I nte r pr et additi on number sente nces and solve missing box pr oblems,

using concr et e obj ect s and number line additi on t o solve t hem: 8 + 3 =

15 + 4 = 5 + 3 + 1 = + = 6

This builds on f r om pr ior lear ning of adding by combining t wo set s of obj ect s

into one gr oup (5 cubes and 3 cubes) in Ear ly Year s.

8 + 5

Key vocabular y: add, mor e, plus, and, make, altogeth er, t ot al, equal t o, equals, double, most, count on, number line

Key skil ls f or addition at Y1:

Read and wr ite numbers t o 100 in numer als, incl. 1 20 in wor ds

Recall bonds t o 10 and 20, and addition fac t s wit hin 20

Count t o and acr oss 100

Count in mult iples of 1 2, 5 and 10

Solve simple 1-st ep pr oblems involving additio n, using obj ect s, number lines and pict or ial

r epr esent at ions.

Video c lips: Using a r ange of equipment and str at egies to r einfor ce addit ion sta te ment s / bonds t o 10

http://www.youtube.com/watch?v=OkW1Y11tGxw&list=PLQqF8sn28L9wjDm8uJEJcRCDDoY6raPE_

Year 2 Add with 2-digit numb ers Developing ment al fl uency with addi t ion and place value involving 2-digit number s, t hen est abl ish mor e f or mal method s.

Add 2 - digit nu mbers and te ns: Add 2- digit nu mbers and units:

Use empt y number lines,

concre t e equipment , hundred

squares e t c. t o build

confid ence and f luency in

mental addi t ion skills .

Add pairs of 2- digit numbers, moving t o t he par t itio ned c olumn method wh en

secure adding t ens and un it s:

23 + 34: STEP 1: Only provide

examples t hat do

NOT cross th e t ens

boundary unt il th ey

are secure wit h t he

meth od it self.

STEP 2: Once children can add a

mult iple of t en t o a 2-digit

number menta lly (e.g. 80+11), th ey

are re ady f or adding pairs of

2-digit numbers tha t DO cross

th e t ens boundary (e.g. 58 + 43).

58 + 43: STEP 3: Children who are

confid ent and accurat e wit h

th is stage should move ont o

th e expanded addit ion

meth ods wit h 2 and 3-digit

numbers (see Y3).

To suppor t under st anding, pupils may physically make and carr y out the calculat ion wit h

Dienes Base 10 apparat us or pl ace value count er s, th en compar e t heir pr act ical ver sion to

the wr it t en f or m, to help th em to build an under st anding of it .

Key vocabular y: add, mor e, plus, and, make, altogeth er, t ot al, equal t o, equals, double, most, count on, number line, sum, ten s, units, par t ition, addition, column, tens boundar y

Key skil ls f or addition at Y2: Add a 2-digit number and ones (e.g. 27 + 6)

Add a 2-digit number and t ens (e.g. 23 + 40) Add pair s of 2-digit number s (e.g. 35 + 47) Add th r ee single-digit number s (e.g. 5 + 9 + 7) Show t hat adding can be done in any or der (t he commut at ive law). Recall bonds to 20 and bonds of t ens to 100 (30 + 70 etc .) Count in st eps of 2, 3 and 5 and count in t ens f r om any number . Under st and t he place value of 2-digit number s (t ens and ones) Compar e and or der number s to 100 using < > and = signs. Read and wr it e number s to at least 100 in numer als and wor ds. Solve pr oblems with addit ion, using concr et e obj ect s, picto r ial r epr esent at ions, involving number s,

quant it ies and measur es, and applying ment al and wri tt en met hods.

Year 3 Add numbers with up t o 3-digits

I ntr oduce t he expanded column addit ion method:

Add t he unit s f ir st, in pr epar at ion

for the compact method.

In ord er t o car r y out t his method of addi t ion:

Child r en need t o r ecognise t he value of t he

hundr eds, te ns and unit s wit hout r ecor ding t he

par t itio ning.

Pupils need t o be able t o add in columns.

Move to t he compact column addit ion method, with arrying :

Add unit s f ir st.

arr numbers

under neath the

bottom line.

236

+ 73

309 1

Childr en who ar e ver y secur e and conf ident wit h 3-digit

expanded column addi t ion should be moved onto t he compact

column addit ion method, being introduced to carryin for

the f ir st t ime. Compar e t he expanded method to t he

compact column method t o develop an under st anding of the

pr ocess and t he r educed number of st eps involved.

Remind pupils the actual value is t hr ee t ens add seven

tens , not three add seven which equals t en t ens.

Key vocabular y: add, mor e, plus, and, make, alto get her , tot al, equal t o, equals, double, most , count on, number line, sum, t ens, unit s, par t it ion, plus, addit ion, column, t ens boundar y, hundre ds boundary, increase, ver tical, car expanded, compact

Key skil ls f or addition at Y3:

Read and wr it e number s to 1000 in numerals and wor ds. Add 2-digit number s ment ally, incl. th ose exceeding 100. Add a t hr ee- digit number and ones mentally (175 + 8) Add a t hr ee- digit number and t ens mentally (249 + 50)

Add a t hr ee- digit number and hundr eds mentally (381 + 400 )

Est imat e answer s to calculat ions, using inver se to check answer s. Solve pr oblems, including missing number pr oblems, using number f act s, place value, and mor e complex addit ion. Recognise place value of each digit in 3-digit number s (hundr eds, t ens, ones.) Cont inue to pr act ise a wide r ange of ment al addit ion st r at egies, ie. number bonds, adding t he near est

mult iple of 10, 100, 100 and adj ust ing, using near doubles, par t it ioning and r ecombining.

Video clip: D em on st rati on of exp an ded 3 -d igit co lu mn ad d iti o n

http://www.teachertube.com/viewVideo.php?video_id=24325

Year 4 Add nu mbers with up t o 4 digits

Move f r om expanded addition t o t he compact column met hod, adding units

f irst , and carrying numbers under neat h t he calculat ion. Also include money and

measur es context s.

e. g. 3517 + 396 = 3913

In t r oduce t he compact column addit ion method by

asking child r en to add t he t wo given number s t o-

get her using t he method t hat t hey ar e f amil iar

wit h (expanded column addit ion see Y3). Teacher

models t he compact met hod wit h carr ying, asking

childr en to discuss similar it ies and dif f er ences and

est abli sh how it is carr ied out .

Add unit s f ir st.

arr numbers

under neath the

bottom line.

Reinfo r ce cor r ect place value by r eminding

th em t he act ual value is 5 hundr eds add 3 hun-

dr eds, not 5 add 3, for example.

Use and apply t his method t o money and

measur ement values.

Key vocabulary: add, mor e, plus, and, make, alt oget her , tot al, equal t o, equals, double, most , count on, number line, sum, t ens, unit s, par t it ion, plus, addit ion, column, t ens boundar y, hundr eds boundar y,

increase, vertical, arry , expanded, compact , t housands, hundreds, digit s, inverse

Key ski lls f or addition at Y4: Select most appr opr iate method: ment al, j ott ings or wr it te n and explain why.

Recognise t he place value of each digit in a f our -digit number .

Round any number t o t he near est 10, 100 or 1000.

Esti mat e and use inver se oper at ions t o check answers.

Solve 2-st ep pr oblems in context , deciding which oper at ions and met hods t o use and why.

Find 1000 mor e or less t han a given number .

Cont inue t o pr act ise a wide r ange of ment al addition s t r ate gies, ie. number bonds, add t he

near est mult iple of 10, 100, 1000 and adjus t , use near doubles, par t it ioning and r ecombining.

Add numbers wit h up t o 4 digit s using t he f or mal wr it te n method of column additi on

Solve 2-st ep pr oblems in context s, deciding which oper at ions and met hods t o use and why.

Esti mat e and use inver se oper at ions t o check answers t o a calculat ion.

Year 5 Add nu mbers with more t han 4 digit s

including money, measur es and decimals wit h dif f er ent numbers of d ecimal

places.

The decimal point should be aligned in t he same

way as t he ot her place value columns, and must be in

t he same column in t he answer .

Numbers should exceed 4 digit s.

Pupils should be able t o add mor e t han t wo values,

car ef ully aligning place value columns.

Say tenths add 7 tenths

to r einfo r ce place value.

Empty decimal places can be

f illed with zer o to show t he

place value in each column.

Childr en sh ould:

Underst and the place value of te nt hs and hundr edt hs and use t his t o

align number s with dif fe r ent numbers of decimal places.

Key vocabulary: add, mor e, plus, and, make, alt oget her , tot al, equal t o, equals, double, most , count on, number line, sum, t ens, unit s, par t it ion, plus, addit ion, column, t ens boundar y, hundr eds boundar y, increase, car expanded, compact, ver t ical, t housands, hundr eds, digit s, inverse & decimal places, decimal point , t ent hs, hundr edt hs, t housandt hs

Key ski lls f or addition at Y5: Add number s ment ally wit h incr easingly lar ge number s, using and pr act ising a range of ment al st r at egies

ie. add t he near est mult iple of 10, 100, 100 and adj ust; use near doubles, inver se, par t it ioning and

r e-combining; using number bonds. Use r ounding to check answer s and accur acy. Solve mult i-st ep pr oblems in cont ext s, deciding which operat ions and metho ds to use and why. Read, wr ite , or der and compar e number s to at least 1 milli on and det ermine t he value of each digit . Round any number up to 1 000 000 to the nearest 10, 100, 1000, 10 000 and 100 000 . Add number s wit h mor e th an 4 digit s using fo rmal wr it t en method of columnar addi t ion.

Year 6 Add s everal num bers of incr easing complexity

Adding sever al number s wit h dif f er ent number s of

decimal places (including money and measur es):

Tent hs, hundr edt hs and t housandt hs should

be corr ectly aligned, wit h the decimal point

lined up ver t ically including in t he answer r ow.

Zer os could be added into any empty decimal

places, to show t her e is no value to add.

Empt y decimal places can be

fi lled wit h zero t o show t he

place value in each column.

Adding sever al number s wit h mor e t han 4 digit s.

Key vocabular y: add, mor e, plus, and, make, alto get her , tot al, equal t o, equals, double, most , count on, number line, sum, t ens, unit s, par t it ion, plus, addit ion, column, t ens boundar y, hundr eds boundar y, increase, car expanded, compact, ver t ical, t housands, hundr eds, digit s, invers e, decimal places, decimal point , t ent hs, hundr edt hs, t housandt hs

Key ski lls f or addition at Y6: Per f or m ment al calculat ions, including with mixed oper at ions and lar ge number s, using and

pr act ising a r ange of ment al st r ate gies.

Solve multi -st ep pr oblems in context , deciding which oper at ions and methods t o use and why.

Use esti mat ion t o check answer s t o calculat ions and dete r mine, in t he contex t of a pr oblem,

levels of a ccur acy.

Read, wr it e, or der and compar e numbers up t o 10 milli on and dete r mine t he value of each digit .

Round any whole number t o a r equir ed degr ee of accur acy.

Pupils under stand how t o add ment ally with lar ger number s and calculat ions of in cr easing

complex ity.

Year 1 Subt ract f r om numbers up t o 20

Childr en consolidat e under st anding of subt r act ion pract ically,

showing subt ract ion on bead st r ings, using cubes et c. and in

f amil iar cont ext s, and ar e int r oduced to mor e f or mal

r ecor ding using number lines as below:

Subt ract by ta king away

Read, wri t e and

int erpret number

sentences wit h

- and = signs.

Count back in ones on

a number ed number

line to t ake away, wit h

number s up to 20:

Find the distance betwe

-1 -1 -1 -1

Model subt ract ion using hundred

squares and numbered number

lines/t racks and pract ically.

This will be int r oduced

pr act ically with t he

language find the

distance between and

how many more ? in a

r ange of f amil iar cont ext s.

Seven is 3 more than four

am 2 years older than my

sister

Mental su btrac t ion

Child r en should st ar t r ecalling subt r act ion fa ct s up t o and wit hin 10 and 20, and

should be able t o subt ract zer o.

Key vocabulary: equal t o, t ake, t ake away, less, minus, subt ract , leaves, dist ance bet ween, how many mor e, how many f ewer / less t han, most, least, count back , how many left, how much less is_?

Key ski lls f or subt r action at Y1: Given a number , say one more or one less. Count to and over 100, fo r ward and back, f r om any number . Repr esent and use subtr act ion f acts to 20 and with in 20 . Subt ract wit h one- digit and tw o- digit number s to 20, including zer o. Solve one-st ep pr oblems t hat involve addi t ion and subt r act ion, using concr ete obj ects (ie bead st r ing,

obj ect s, cubes) and pict ur es, and missing number probl ems. Read and wr it e number s fr om 0 to 20 in numer als and wor ds.

Year 2 Subt ract w it h 2- digit n umbers Subt r act on a number line by count ing back, aiming t o develop

ment al subt r act ion skills.

This st rat egy will be used f or :

Use Dienes blocks

f or s ubt r act ion

calculat ions t oo.

2- digit numbers subt r act units (by t aking away / count ing back) e.g. 36 7

2- digit numbers subt r act t ens (by t aking away / count ing back) e.g. 48 30

Subtrac t ing pairs of 2- digit number s (see below:)

Subt ract ing pai rs of 2- digit numbers on a number line:

47 - 23 = 24 Par t it ion t he second number

and subt ract it in t ens and unit s, as below:

Move t owards mor e ef f icient

j umps back, as below:

Then subt ract

units .

Subt ra ct t ens

fi r st .

Combine met hods wit h use of a hundr ed

squar e t o r einf or ce under st anding of

number value and or der.

Teaching chil dr en to br idge thr ough te n

can help t hem to become mor e ef f icient, for

example 42 25:

Ment al st r ate gy - subt ract nu mber s close t oget her by count ing on:

St art wit h t he

smaller number

and count on t o

t he larg est .

Many ment al st r at egies are t aught . Children are t aught t o

recognise th at when numbers are close t oget her , it is more

effi cient t o count on t he diffe rence. They need t o be clear

about t he re lat ionship bet ween addit ion and subt r act ion.

Key vocabulary: equal t o, t ake, t ake away, less, minus, subt ract , leaves, dist ance bet ween, how many mor e, how many f ewer / less t han, most, least, count back , how many left, how much less is_? dif fe re nce, count on, st r ate gy, par t itio n, t ens, units

Key ski lls f or subt r action at Y2: Recognise t he place value of each digit in a t wo-digit number . Recall and use subt ract ion f act s to 20 flu ently, and der ive and use r elat ed f acts up to 100. Subt ract using concr ete obj ect s, pict or ial r epr esent at ions, 100 squar es and ment ally, including: a t wo-

digit number and ones, a t wo-digit number and t ens, and t wo t wo-digit number s. Show t hat subt ract ion of one number f r om anot her cannot be done in any order . Recognise and use inver se r elat ionship bet ween addit ion and subt ract ion, using this to check calcula-

t ions and missing number pr oblems. Solve simple addi t ion and subt r act ion pr oblems including measur es, using concr ete obj ect s, picto r ial

r epr esent at ion, and also applying t heir incr easing knowledge of ment al and wr it t en met hods. Read and wr it e number s to at least 100 in numer als and in wor ds.

Year 3 Subt racting wi t h 2 and 3-digit numbers. I nt r oduce partition ed column subt r action meth od.

STEP 1: int roduce

th is met hod wit h

examples where no

exchanging is

require d.

89 35 = 54

80 + 9 - 30 + 5

50 + 4

When learning to exchange , explore

partitioning in different ways so that pupils

understand th at when you exchange, th e VALUE

is th e same ie 72 = 70+2 = 60+12 = 50+22 et c.

Emphasise t hat th e value hasn changed, we

have j ust par t it ioned it in a dif f erent way.

STEP 2: int roduce

exchan through

pract ical subt ract ion. Make

th e larger number wit h Base

10, th en subt ract 47 f rom

it .

70 + 2 - 40 +

7 20 + 5 = 25

Before subtracting from the 72 blocks, they will need to exchange a row of 10 for ten units. Then subtract 7, and subtract 4 tens.

STEP 3: Once pupils are secure

wit h th e under st anding of

exchanging , they can use the

par t it ioned column meth od t o

subt ract any 2 and 3-digit numbers.

Subt r act ing money:

par t it ion int o e.g.

£1 + 30p + 8p

Counti ng on as a ment al st r at egy f or subt r actio n:

Cont inue t o re inf orce count ing on as a st rat egy f or close- to geth er numbers (e.g. 121 118),

and also for numbers that are nearl multiples of 10, 100, 1000 or £s, which make it easier

t o count on (e.g. 102-89, 131 79, or calculat ing change f r om £ 1 et c.).

Sta rt at th e smaller number and count on in t ens fir st , t hen count on in unit s t o f ind

th e re st of th e dif f erence:

Key vocabulary: equal t o, t ake, t ake away, less, minus, subt r act , leaves, dist ance bet ween, how many mor e, how many fe wer / less t han, most , least , count back , how many lef t , how much less is_? di f f er ence, count

on, st r at egy, par t it ion, t ens, unit s exchange, decr ease, hundreds, value, digit Key skil ls f or subt ract ion at Y3: Subt ract menta lly a: 3- digit number and ones, 3- digit number and t ens, 3- digit number and hundreds . Est imate answers and use inverse operat ions t o check.

Solve problems, including missing number problems.

Find 10 or 100 more or less tha n a given number.

Recognise th e place value of each digit in a 3-digit number . Count ing up di f fer ences as a mental st rat egy when numbers are close to get her or near mult i-

ples of 10 (see examples above) Read and wri t e numbers up t o 1000 in numerals and word s.

Appr oximat e,

Calculate ,

Check it!

Pract ise ment al subt ract ion st rat egies, such as subt ract ing near mult iples of 10 and adj ust ing (e.g. subt ract ing 19 or

21), and select most appropriat e met hods t o subt ract , explaining why.

Video clips: 1 Subt ra cti on t eaching childr en to consider t he most appr opri ate methods before calculat ing 2 I ntr oducing par t it ioned column subt ract ion met hod, from pra ct ical to writ t en

http://www.youtube.com/watch?v=RCCLseBLBSohttp://www.youtube.com/watch?v=dP8NlFLZzOg

Year 4 Subt ract w it h up t o 4- digit numb ers

Partitioned column subtraction with exchangin (decomposit ion):

As int roduced in Y3, but moving

to wards more complex numbers

and values. Use place value coun-

t ers to reinforce exchanging .

Compact column subt r action (see video)

Subt r act ing money: par t it ion

int o £1 + 30 + 5 for example.

To int r oduce the compact method, ask chil dr en to per fo r m a subt r act ion calculat ion with t he

f amil iar par t it ioned column subt ract ion t hen

display t he compact ver sion f or the calculat ion th ey have done. Ask pupils to consider how it r elat es to t he method t hey know, what is similar and what is dif f er ent, to develop an under st anding of it (shown on video).

Give plent y of oppor t unit ies t o

apply th is t o money and measures.

Ment al str ate gies

Always encourage children t o consider t he

best metho d f or t he numbers i nvolved

mental, count ing on, count ing back or wri t -

t en met hod (see video).

A var iet y of ment al st r at egies must be t aught and pr act ised, including count ing on to f ind

the dif f er ence wher e number s ar e closer t oget her , or wher e it is easier to

count on (see video below). Appr oximat e,

Calculate ,

Check it!

Key vocabular y: equal t o, t ake, t ake away, less, minus, subt r act , leaves, dist ance be- t ween, how many mor e, how many fe wer / less t han, most , least , count back , how many lef t , how much less is_? di f f er ence, count on, st r at egy, par t it ion, t ens, unit s exch ange, decr ease, hundr eds, value, digit , inverse

Key ski lls f or subt r action at Y4: Subt ract by count ing on wher e number s ar e close t oget her or th ey ar e near to mult iples of 10, 100 etc. Childr en select the most appr opr iat e and ef f icient methods f or given subt r act ion calculat ions. Est imat e and use inver se oper at ions to check answers. Solve addit ion and subt r act ion 2-st ep pr oblems, choosing which oper at ions and metho ds to use and why.

Solve simple measur e and money pr oblems involving f ract ions and decimals to t wo decimal places.

Find 1000 mor e or less t han a given number . Count backwar ds th r ough zer o, including negat ive number s. Recognise place value of each digit in a 4-digit number Round any number to the near est 10, 100 or 1000 Solve number and pr act ical pr oblems t hat involve t he above, wit h incr easingly lar ge posit ive numbers.

Videos: 1 Subtract ion t eaching childr en to consider t he most appr opri at e metho ds befo r e calculat ing 2 I ntr oducing par t it ioned column subt ract ion met hod, from pra ct ical to writ t en

3 Moving t o t he compact column method of subt r act ion (yout ube)

http://www.youtube.com/watch?v=RCCLseBLBSohttp://www.youtube.com/watch?v=dP8NlFLZzOghttp://www.youtube.com/watch?v=3ihxp2mqnhs

Year 5 Subtr act with at least 4- digit numbers

including money, measur es, decimals.

Compact column subtrac t ion

(with exchangin ).

Children who are st ill not

secure wit h number f act s

and place value will need t o

re main on th e par t it ioned

column met hod unt il r eady

f or t he compact met hod.

Subt ract ing wit h lar ger int eger s.

See moving to

the compact

method video.

Subt ract wit h decimal values, including mix t ur es

of int eger s and decimals, aligning t he decimal

point.

Creat e lot s of oppor t unit ies f or

subt ract ing and f inding di f f ere nces

wit h money and measures.

Add a zero in any empty decimal

places t o aid underst anding of

what t o subt ract in t hat column.

Appr oximat e,

Key vocabulary: equal t o, t ake, t ake away, less, minus, subt r act , leaves, dist ance bet ween, how many mor e, how many fe wer / less t han, most , least , count back , how

many lef t , how much less is_? di f f er ence, count on, st r at egy, par t it ion, t ens, unit s

Calculate ,

Check it!

exchange, decr ease, hundr eds, value, digit , invers e, t ent hs, hundr edt hs, decimal point , decimal

Key ski lls f or subt r action at Y5: Subt ract number s ment ally with incr easingly lar ge number s .

Use r ounding and est imat ion to check answer s to calculat ions and det er mine, in a r ange of cont ext s,

levels of accuracy . Solve addit ion and subt r act ion mult i-st ep pr oblems in cont ext, deciding which operat ions and methods

to use and why.

Read, wr ite , or der and compar e number s to at least 1 milli on and det ermine t he value of each digit . Count fo r war ds or b ackwar ds in st eps of power s of 10 f or any given number up to 1 millio n.

In t er pr et negat ive number s in cont ext, count ing f or wards and backwards wit h posit ive and negat ive in-

t eger s th r ough 0. Round any number up to 1 milli on to t he near est 10, 100, 1000 , 10 000 and 100 000 .

Video c lip:

Moving t o t he compact column method of subt r act ion (yout ube)

http://www.youtube.com/watch?v=3ihxp2mqnhs

Year 6 Subt r act ing wit h inc re asingly la rge and more c omplex numbers and decimal values.

Using the compact column

meth od t o subt r act more

complex intege r s

Using the compact column

meth od t o subt r act money

and measures, including

decimals with dif f er ent

number s of decimal places.

Empty decimal places can be

f illed with zero to show t he

place value in each column.

Pupils should be able t o apply t heir knowledge of a r ange of ment al st r ate gies,

ment al r ecall skills, and inf or mal and f or mal wr itte n met hods when select ing t he

most a ppropr iate met hod t o wor k out subt ract ion pr oblems.

Appr oximat e,

Key vocabular y: equal to , t ake, t ake away, less, minus, subt r act , leaves, dis- t ance bet ween, how many mor e, how many fe wer / less t han, most , least , count

Calculate ,

Check it!

back , how many lef t , how much less is_? di f f er ence, count on, st r at egy, par t i t ion, t ens, unit s exchange, decr ease, hundr eds, value, digit , invers e, t ent hs, hundr edt hs, decimal point , decimal

Key skil ls f or subt ract ion at Y6: Solve addit ion and subt r act ion mult i-st ep pr oblems in cont ext, deciding which operat ions and methods

to use and why.

Read, wr ite , or der and compar e number s up to 10 million and det er mine t he value of each digit Round any whole number to a r equir ed degr ee of accuracy Use negat ive number s in cont ext, and calculat e int er vals acr oss zer o.

Childr en need to ut ilise and consider a r ange of ment al subt r act ion st r at egies, j ott ings and wr it t en

methods befo r e choosing how to calculat e.

See pr evious videos for intr oducing t he compact column method.

Year 1 Multiply with c oncre te obj ects, arr ays and

pict orial rep r esenta t ions.

How many legs will 3 t eddies have?

Ther e ar e 3 sweet s in one bag.

How many sweet s ar e in 5 bags

altog et her?

2 + 2 + 2 = 6

3+3+3+3+3

= 15

Give children expe r ience of counti ng equal gr oup of obje cts in 2s,

5s and 10s.

Pr esent pr acti cal pr oblem solving activities involving counti ng equal

sets or gr oups, as above.

Key vocabular y: gr oups of, lots of, ti mes, arr ay, altogeth er , multi ply, count

Key skil ls f or mult iplicat ion at Y1:

Count in mult iples of 2, 5 and 10.

Solve one-st ep pr oblems involving mult iplic at ion, by calculat ing t he answer using concr et e obj ect s,

pict or ial r epr esent at ions and ar r ays wit h t he suppor t of t he te acher .

Make connect ions bet ween arr ays, number pat te r ns, and count ing in t wos, f ives and t ens.

Begin t o under sta nd doubling using concr et e obj ect s and pict or ial r epr esent at ions.

Year 2 Multiply using ar r ays and r epeat ed addit ion

(using at le ast 2s, 5s a nd 10s)

Use repe at ed ad diti on on a number line:

St ar t ing f r om zer o, make equal j umps up on

a number line to wor k out mult iplicat ion f acts and

wr it e mult ipl icat ion st at ement s using x and = signs.

#

Use array s:

5 x 3 = 3 + 3 + 3 + 3 = 15

3 x 5 = 5 + 5 + 5 = 15

Use ar r ays t o help te ach chil dr en t o under sta nd t he commut at ive law of

mult iplic at ion, and give examples such as 3 x = 6.

Use pr acti cal appar atus:

Use ment al r ecall:

Childr en should begin to r ecall mult iplica t ion f acts for 2, 5 and 10 t imes t ables

th r ough pr act ice in count ing and under st anding of t he oper at ion.

Key vocabulary: gr oups of, lot s of, ti mes, ar r ay, alto get her , mult iply, count , mult ipli ed by, r epeate d additi on, column, r ow, commut at ive, set s of, equal gr oups, t imes as big as, once, t wice, t hr ee t imes...

Key skil ls f or mult iplicat ion at Y2: Count in st eps of 2, 3 and 5 f r om zer o, and in 10s f r om any number .

Recall and use mult iplicat ion f act s f r om t he 2, 5 and 10 mult iplicat ion t ables, including r ecognising odds

and evens. Wr it e and calculat e number sta t ement s using t he x and = signs. Show t hat mult iplicat ion can be done in any or der (commut at ive).

Solve a r ange of pr oblems involving mult iplicat ion, using concr ete obj ect s, ar rays, r epeat ed addi t ion,

ment al metho ds, and mult iplicat ion f act s. Pupils use a var iet y of language to discuss and descr ibe mult iplicat ion.

Video c lips: Teaching for under st anding of mult ipl icat ion f acts (yout ube) Practical mult iplicat ion and t he commuta t ive law (yout ube)

http://www.youtube.com/watch?v=YPWmOVt8vgw&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiixhttp://www.youtube.com/watch?v=VGkjjVfnGYI&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiix&index=2

Year 3 Multiply 2- digits by a single digit number

I nt r oduce t he grid met hod for mult iplying 2- digit by single- digit s:

Link t he layout of t he gr id to an arr ay init ially:

160 + 24 = 1 84

In t r oduce t he gr id meth od wit h chil dr en physically making an arr ay to r epr esent t he

calculat ion (e.g. make 8 lots of 23 wit h 10s and 1s place value count er s), t hen t ranslat e

th is to gr id met hod f or mat (see video clip).

To do t his, c hildr en must be able t o:

Par t it ion number s into t ens and unit s

Mult iply mult iples of t en by a single digit (e.g. 20 x 4) using th eir knowledge of

mult iplicat ion f acts and place value

Recall and wor k out mult iplicat ion f acts in t he 2, 3, 4, 5, 8 and 10 t imes t ables.

Wor k out mult iplicat ion f acts not known by r epeat ed addi t ion or oth er t aught

ment al st rat egies (e.g. by commut at ive law, wor king out near mult iples and adj ust -

ing, using doubl ing etc .) St r at egies to suppor t this ar e r epeat ed addit ion using a

number line, bead bar s and arr ays:

Key vocabular y: gr oups of, lot s of, t imes, ar r ay, alto get her , mult iply, count, mult iplied by, r epeat ed ad- di t ion, column, r ow, commut at ive, set s of, equal gr oups, t imes, _t imes as big as, once, t wice, t hr ee t imes...,

par t itio n, grid met hod, mult iple, pr oduct , t ens, units, value Key skil ls f or mult iplicat ion: Recall and use mult iplicat ion f act s f or the 2, 3, 4, 5, 8 and 10 mult ipli cat ion t ables, and mult iply

mult iples of 10.

Wr it e and calculat e number sta t ement s using t he multi plicat ion t ables th ey know, including 2- digit x

single- digit , dr awing upon ment al metho ds, and pr ogressing to r eliable wr it t en method s. Solve mult iplicat ion pr oblems, including missing number pr oblems. Develop ment al st r at egies using commut at ivit y (e.g. 4 x 12 x 5 = 4 x 5 x 12 = 20 x 12 = 240 ) Solve simple pr oblems in cont ext s, deciding which operat ions and metho ds to use.

Develop ef f icient ment al methods to solve a r ange of pr oblems e.g using commut at ivit y (4 × 12 × 5 = 4 × 5 × 12 = 20 × 12 = 240 ) and f or missing number pr oblems x 5 = 20, 3 x = 18, x = 32

Video c lips: Teaching t he gr id method as an int er im st ep (par t it ioning and count er s to int r oduce gr id)

http://www.youtube.com/watch?v=qyTRtoqYi7Q&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiix

Year 4 Multiply 2 and 3- digits by a single digit, using all mult iplicat ion ta bles up to 12 x 12

Developing t he grid met hod:

500

150

+ 30

680

Encour age column

addi t ion to add

accur at ely.

Move onto short mult iplicat ion (see Y5) if and when childr en ar e conf ident

and accurate mult iplying 2 and 3-digit number s by a single digit th is way, and

are already confident in carryin for written addition.

Childr en sh ould be able t o:

Appr oximate before t hey calculat e, and make th is a r egular par t of t heir

calculat ing, going back to the appr oxi mat ion to check the r easonableness of t heir

answer . e.g:

346 x 9 is appr oxi mat ely 350 x 10 = 3500 Recor d an appr oximat ion t o check the f inal answer against.

Mult iply mult iples of t en and one hundr ed by a single-digit, using

th eir mult ipl icat ion t able knowledge.

Recall all t imes t ables up to 12 x 12

Appr oximat e,

Calculate ,

Check it!

Key vocabulary: gr oups of, lot s of, t imes, ar r ay, alto get her , mult iply, count, mult iplied by, r epeat ed addit ion, ar r ay, column, r ow, commut at ive, gr oups of, set s of, lot s of, equal gr oups, t imes, mult iply, t imes as big as, once, t wice, t hr ee t imes... par t i t ion, gr id met hod, tot al, mult iple, pr oduct , set s of, inverse

Key skil ls f or mult iplicat ion at Y4: Count in mult iples of 6, 7, 9, 25 and 1000 Recall mult iplicat ion f acts f or all multi plicat ion t ables up to 12 x 12. Recognise place value of digit s in up to 4-digit number s Use place value, known f acts and der ived f acts to mult iply ment ally, e.g. multi ply by 1, 10, 100, by 0, or to

mult iply 3 number s.

Use commut at ivity and ot her st r at egies ment ally 3 x 6 = 6 x 3 , 2 x 6 x 5 = 10 x 6 , 39x7 = 30 x 7 + 9 x 7. Solve pr oblems with incr easingly complex mult iplicat ion in a r ange of cont ext s. Count in mult iples of 6, 7, 9, 25 and 1000 Recognise t he place value of each digit in a f our -digit number (t housands, hundr eds, t ens, and ones)

x 300 20 7

4 1200 80 28

Year 5 Multip ly up t o 4- digits by 1 or 2 digit s.

I nt r oducing column multi plicat ion

In t r oduce by compar ing a gr id method calculat ion t o a shor t mult iplicat ion met h- od, to see how the st eps ar e r elat ed, but not ice how th er e ar e less st eps involved in t he column method (see video).

Childr en need to be t aught to appr oxi mat e fi r st, e.g. f or 72 x 38, th ey will use

r ounding: 72 x 38 is appr oxi mat ely 70 x 40 = 2800 , and use t he appr oximat ion

to check the r easonableness of t heir answer against .

Short multi plication f or m ulti plying by a single digit

Pupils could be asked t o work out a

given calculat ion using th e gr id, and

then compare it to yo column

meth od. What are t he similari t ies

and dif f erences? Unpick t he st eps

and show how it reduces th e st eps.

I nt r oduce long multi plication f or m ulti plying by 2 digits

18 x 3 on t he 1st row

(8 x 3 = 24, c arryi ng th e 2 for twenty, then 1 x 3).

Moving to wards m ore complex number s:

18 x 10 on th e 2nd row. Put

a zero in unit s fir st , th en

say 8 x 1, and 1 x 1.

Appr oximat e,

Calculate ,

(1234 x 6)

(1234 x 10)

Check it!

Key vocabulary gr oups of, lot s of, t imes, ar r ay, alto get her , mult iply, count , mult iplied by, r epeat ed ad- di t ion, column, r ow, commut at ive, set s of, equal gr oups, _t imes as big as, once, t wice, t hr ee t imes..., par t i- t ion, gr id met hod, tot al, mult iple, pr oduct , inver se, squar e, f act or, int eger , decimal, short /l ong mult i- plication, r ry

Key skil ls f or mult iplicat ion at Y5: I dent ify mult iples and f acto r s, using knowledge of mult iplicat ion t ables to 12x12. Solve pr oblems wher e lar ger number s ar e decomposed into t heir f acto r s Mult iply and divide int eger s and decimals by 10, 100 and 1000 Recognise and use squar e and cube number s and t heir notat ion Solve pr oblems involving combinat ions of operat ions, choosing and using calculat ions and methods appr opr iat ely.

Video c lips: Moving from grid method to a compact method Reinfor cing r apid t imes t able r ecall: Demonstra t ion of long multi plicat ion

http://www.youtube.com/watch?v=5ppOF53x_q0&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiixhttp://www.youtube.com/watch?v=5ppOF53x_q0&list=PLQqF8sn28L9yj34NpXK7Yffze7ZoXTiixhttp://www.youtube.com/watch?v=t_bnlB2KRL4

Year 6 Short a nd long mult iplica t ion as in Y5, and

mult iply decimals with up t o 2d.p by a single digit .

Line up th e decimal

point s in t he quest ion

and t he answer.

Remind children

tha t t he single

digit belongs in

th e unit s column.

This works well f or

mult iplying money (£ .p) and

oth er measures.

Childr en will be able t o:

Use r ounding and place value to make appr oximat ions befo r e calculat ing and use

th ese to check answer s against.

Use short mult iplica t ion (see Y5) to mult iply numbers wit h more th an 4- digits

by a single digit ; to mult iply money and measur es, and to mult iply decimals with

up to 2d. p. by a single digit .

Use long mult iplicat ion (see Y5) to mult iply number s wit h at least 4 digits by a

2- digit number. Appr oximat e,

Calculate ,

Check it!

Key vocabular y: gr oups of, lot s of, t imes, ar r ay, alto get her , mult iply, count, mult iplied by, r epeat ed ad- di t ion, ar r ay, column, r ow, commut at ive, set s of, equal gr oups, t imes as big as, once, t wice, t hr ee t imes... par t it ion, grid met hod, tot al, mult iple, pr oduct , invers e, squar e, f acto r , int eger , decimal, shor t / long mul- tiplication, arry , te nt hs, hundredt hs, decimal Key skil ls f or mult iplicat ion at Y6:

Recall mult iplic at ion fa ct s f or all t imes t ables up to 12 x 12 (as Y4 a nd Y5).

Mult iply multi -digit numbers, up t o 4-digit x 2-digit using long mult iplicat ion.

Per f or m ment al calculat ions wit h mix ed oper at ions and lar ge number s.

Solve multi -st ep pr oblems in a r ange of context s, choosing appr opr iat e combinat ions of oper a-

t ions and methods.

Esti mat e answer s using r ound and appr oximat ion and dete r mine levels of accur acy.

Round any int eger t o a r equir ed degr ee of accur acy.

Video clips:

Moving from grid method to a compact method (yout ube)

Reinfor cing r apid t imes t able r ecall: (yout ube)

Demonstra t ion of long multi plicat ion (SLEP)