Progression in calculations Years 3 and 4

Progression in calculations Years 3 and 4

Introduction At the centre of the mastery approach to the teaching of mathematics is the belief that all pupils have the potential to succeed. They should have access to the same curriculum content and, rather than being extended with new learning, they should deepen their conceptual understanding by tackling challenging and varied problems. Similarly, with calculation strategies, pupils must not simply rote learn procedures but demonstrate their understanding of these procedures through the use of concrete materials and pictorial representations. This document outlines the different calculation strategies that should be taught and used in Years 3 to 6, in line with the requirements of the 2014 Primary National Curriculum. Presentation of calculations You will see that throughout this document, calculations are presented in a variety of ways. It is important for pupils mathematical understanding to experience and work with calculations and missing numbers in different positions relative to the = symbol. Examples used in classwork and independent work should reflect this.

Progression in calculations Year 3

Y3 Addition & Subtraction S ra egy & guidance Add and sub rac numbers men ally, including: a hree-digi number and ones; a hree-digi number and ens; a hree-digi number and hundreds Pup ls learn that th s s an appropr ate strategy when they are able to use known and der ved number facts or other mental strateg es to complete mental calculat ons w th accuracy. To beg n w th, some pup ls w ll prefer to use th s strategy only when there s no need to regroup, us ng number facts w th n 10 and der vat ons. More conf dent pup ls m ght choose from a range of mental strateg es that avo d wr tten algor thms, nclud ng (but not exhaust vely): known number facts w th n 20, It s mporta and pup s s w th numbe d g t numbe der ve answ No regroup 345 + 30 1128 + 300 326 + 342 der ved number facts, Make ten, round and adjust See Year 2 gu dance for exempl f cat on of these the use of concrete man pulat ves other than D enes blocks s mportant n re nforc ng the use of these strateg es. It s mportant that pup ls are g ven plenty of (scaffolded) pract ce at choos ng the r own strateg es to complete calculat ons eff c ently and accurately. Expl c t l nks need to be made between fam l ar number facts and the calculat ons that they can be useful for and pup ls need to be encouraged to a m for eff c ency. W th some 416 + 25 383 + 130 611 + 194 1482 + 900

Regrou 204 1 S ra egy & guidance Wri en column me hod for calcula ions ha require regrouping wi h up o 4-digi s D enes blocks should be used alongs de the p ctor al representat ons dur ng d rect teach ng and can be used by pup ls both for support and challenge. Place value counters can also be ntroduced at th s stage. Th s work rev ses and re nforces deas from Key Stage 1, nclud ng the focus on place value see Year 2 exempl f cat on. As for t wr tten concret Aga n on cou D rect teach ng of the columnar method should requ re at least one element of regroup ng, so that pup ls are clear about when t s most useful to use t. Ask ng them Can you th nk of a more eff c ent method? w ll challenge them to apply the r number sense / number facts to use eff c ent mental methods where poss ble. As n Year 2, pup ls should be g ven plenty of pract ce w th calculat ons that requ re mult ple separate nstances of regroup ng. In Year 3 they become more fam l ar w th calculat ons that requ re regroup ng to regroup. Understand ng must be secured through the cons dered use of man pulat ves and mages, comb ned w th careful use of language. Pup ls should be challenged as to whether th s s the most eff c ent method, cons der ng whether mental methods (such as count ng on, us ng known number facts, round and adjust etc.) may be l kel er to produce an accurate solut on. Pup ls requ r ng support m ght develop the r conf dence n the wr tten method us ng numbers that requ re no regroup ng. See Un t mater als for extra gu dance on th s strategy. 5 + 6 = Regrou 672 + 1 468 + 6 275 + 3 1035 -

S ra egy & guidance Find 10, 100 more or less han a given number 142 + 100 = 242 As pup ls become fam l ar w th numbers up to 1000, place value should be emphas sed and compar sons drawn between add ng tens, hundreds (and, n the last un t of the Summer term, thousands), nclud ng use of concrete man pulat ves and appropr ate mages. After n t al teach ng, th s should be ncorporated nto trans t on act v t es and pract sed regularly.

Y3 Multiplication Strategy & guidance Doubling to derive new multiplication facts CPA 3 x 3 = 9 3 x 6 = double 9 = 18 Pupils continue to make use of the idea that facts from easier times tables can be used to derive facts from related times tables using doubling as a strategy. This builds on the doubling strategy from Year 2.

S ra egy & guidance Skip coun ing in mul iples of 2, 3, 4, 5, 6, 8 and 10 Rehearsal of prev ously learnt tables as well as new content for Year 3 should be ncorporated nto trans t on act v t es and pract sed regularly. Use of par -par -whole model wi h arrays and bar models o es ablish commu a ivi y and inverse rela ionship be ween mul iplica ion and division In these contexts pup ls are able to dent fy all the equat ons n a fact fam ly. Ten imes grea er Pup ls s work on th s must be f rmly based on concrete representat ons the language of ten t mes greater must be well modelled and understood to prevent the numer cal m sconcept on of add ng a zero.

S ra egy & guidance Mul iplying by 10 and 100 Bu ld ng on the ten t mes greater work, pup ls use appropr ate D enes blocks and place value counters to mult ply 2, 3, 4, 5 and 10 by 10, 100 and 1000. Using known fac s for mul iplying by mul iples of 10 and 100 Pup ls grow ng understand ng of place value, allows them to make use of known facts to der ve mult pl cat ons us ng powers of 10. It s mportant to use tables w th wh ch they are already fam l ar (.e. not 7 or 9 tables n Year 3)

S ra egy & guidance Mul iplica ion of 2-digi numbers wi h par i ioning (no regrouping) Ch ldren should always cons der whether part t on ng s the best strategy f t s poss ble to use strateg es such as doubl ng (some may use doubl ng tw ce for 4), they need to choose the most eff c ent strategy. Ch ldren may w sh to make jott ngs, nclud ng a full gr d as exempl f ed here but gr d method s not a formal method and ts only purpose s to record mental calculat ons. Th s supports the development of the necessary mental calculat ng sk lls but does not h nder the ntroduct on of formal wr tten methods n Year 4. Concrete man pulat ves are essent al to develop understand ng.

S ra egy & guidance Mul iplica ion of 2-digi numbers wi h par i ioning (regrouping) Us ng concrete man pulat ves and later mov ng to us ng mages that represent them, supports pup ls early understand ng, lead ng towards formal wr tten methods n Year 4. Once aga n, th s s a mental strategy, wh ch they may choose to support w th nformal jott ngs, nclud ng a full gr d, as exempl f ed here. Pup ls must be encouraged to make use of the r known mult pl cat on facts and the r knowledge of place value to calculate, rather than count ng man pulat ves.

My who e s 500 an parts s 100. How m S ra egy & Guidance Y3 Division Dividing mul iples of 10, 100 and 1000 by 10, 100 and 1000 using scaling down Pup ls use the strategy of scal ng down, represent ng numbers w th concrete man pulat ves and mak ng the value ten t mes smaller. Dividing mul iples of 10, 100 and 1000 by 10, 100 and 1000 using grouping Pup ls d v de by 10, 100 and 1000 by mak ng groups of the d v sor. 500 100 =

Progression in calculations Year 4

Pay part cu ar a than one d g t c E.g. 990 + 10 o S ra egies & Guidance Y4 Addition & Subtraction Coun forwards and backwards in s eps of 10, 100 and 1000 for any number up o 10 000. Pup ls should count on and back n steps of ten, one hundred and one thousand from d fferent start ng po nts. These should be pract sed regularly, ensur ng that boundar es where more than one d g t changes are ncluded. Coun forwards and backwards in en hs and hundred hs Using known fac s and knowledge of place value o derive fac s. Add and sub rac mul iples of 10, 100 and 1000 men ally Pup ls extend th s knowledge to mentally add ng and subtract ng mult ples of 10, 100 and 1000. Count ng n d fferent mult ples of 10, 100 and 1000 should be ncorporated nto trans t on act v t es and pract sed regularly. Adding and sub rac ing by par i ioning one number and applying known fac s. By Year 4 pup ls are conf dent n the r place value knowledge and are calculat ng mentally both w th calculat ons that do not requ re regroup ng and w th those that do. See Y3 gu danc man pu at ves a understand ng

3527 + 29 4523 39 4523 39 S ra egies & Guidance Round and adjus 3527 + 29 Pup ls should recogn se that th s strategy s useful when add ng and subtract ng near mult ples of ten. They should apply the r knowledge of round ng. It s very easy to be confused about how to adjust and so v sual representat ons and log cal reason ng are essent al to success w th th s strategy. Bu ld flex b l ty by complet ng the same calculat on n a d fferent order. Comp et Comp et Near doubles 1600 + 15 Pup ls should be able to double numbers up to 100 and use th s to der ve doubles for mult ples of ten. These facts can be adjusted to calculate near doubles.

Wri S ra egies & Guidance en column me hods for addi ion Place value counters are a useful man pulat ve for represent ng the steps of the formal wr tten method. These should be used alongs de the wr tten layout to ensure conceptual understand ng and as a tool for expla n ng. Th s method and the language to use are best understood through the tutor al v deos found the toolk t. on Wri en column me hods for sub rac ion Place value counters are a useful man pulat ve for represent ng the steps of the formal wr tten method. These should be used alongs de the wr tten layout to ensure conceptual understand ng and as a tool for expla n ng. Th s method and the language to use are best understood through the tutor al v deos on the toolk t.

S ra egies & Guidance Calcula ing wi h decimal numbers Ass gn d fferent values to D enes equ pment. If a D enes 100 block has the value of 1, then a tens rod has a value of 0.1 and a ones cube has a value of 0.01. These can then be used to bu ld a conceptual understand ng of the relat onsh p between these. Place value counters are another useful man pulat ve for represent ng dec mal numbers. 24.2 + 13.4 = All of the calculat on strateg es for ntegers (whole numbers) can be used to calculate w th dec mal numbers.

Year 4 Multiplication S ra egies & Guidance Mul iplying by 10 and 100 When you mult ply by ten, each part s ten t mes greater. The ones become tens, the tens become hundreds, etc. When mult ply ng whole numbers, a zero holds a place so that each d g t has a value that s ten t mes greater. Repeated mult pl cat on by ten w ll bu ld an understand ng of mult ply ng by 100 and 1000 Using known fac s and place value for men al mul iplica ion involving mul iples of 10 and 100 Pup ls use the r grow ng knowledge of mult pl cat on facts, place value and der ved facts to mult ply mentally. Emphas s s placed on understand ng the relat onsh p (10 t mes or 100 t mes greater) between a known number fact and one to be der ved, allow ng far larger fact fam l es to be der ved from a s ngle known number fact. Knowledge of commutat v ty (that mult pl cat on can be completed n any order) s used to f nd a range of related facts.

34 6 30 6 (4 x 2 S ra egies & Guidance Mul iplying by par i ioning one number and mul iplying each par 14 x 6 Pup ls bu ld on mental mult pl cat on strateg es and develop an expl c t understand ng of d str but ve law, wh ch allows them to explore new strateg es to make more eff c ent calculat ons. As well as part t on ng nto tens and ones (a fam l ar strategy), they beg n to explore compensat ng strateg es and factor sat on to f nd the most eff c ent solut on to a calculat on. Dis ribu ive law a x (b + c) = a x b + a x c Men al mul iplica ion of hree 1-digi numbers, using he associa ive law Pup ls f rst learn that mult pl cat on can be performed n any order, before apply ng th s to choose the most eff c ent order to complete calculat ons, based on the r ncreas ngly soph st cated number facts and place value knowledge. Four p How m

S ra egies & Guidance Shor mul iplica ion of 3-digi number by 1-digi number To beg n w th pup ls are presented w th calculat ons that requ re no regroup ng or only regroup ng from the ones to the tens. The r conceptual understand ng s supported by the use of place value counters, both dur ng teacher demonstrat ons and dur ng the r own pract ce. W th pract ce pup ls w ll be able to regroup n any column, nclud ng from the hundreds to the thousands, nclud ng be ng able to mult ply numbers conta n ng zero and regroup ng through mult ple columns n a s ngle calculat on. Th s method and the language to use are best understood through the tutor al v deos found on the toolk t.

210 210 210 Y4 Division S ra egies & Guidance Dividing by 10 and 100 When you d v de by ten, each part s ten t mes smaller. The hundreds become tens and the tens become ones. Each d g t s n a place that g ves t a value that s ten t mes smaller. When d v d ng mult ples of ten, a place holder s no longer needed so that each d g t has a value that s ten t mes smaller. E.g. 210 10 = 21 Derived fac s Pup ls use the r grow ng knowledge of mult pl cat on facts, place value and der ved facts to mult ply mentally. Understand ng of the nverse relat onsh p between mult pl cat on and d v s on allows correspond ng d v s on facts to be der ved.

24 100 S ra egies & Guidance Shor division of 4-digi numbers by 1-digi numbers D v s on Pup ls start w th d v d ng 4-d g t numbers by 2, 3 and 4, where no regroup ng s requ red. Place value counters are used s multaneously n a place value chart, to develop conceptual understand ng. They progress to calculat ons that requ re regroup ng n the hundreds or tens columns. Pup ls bu ld on the r conceptual knowledge of d v s on to become conf dent w th d v d ng numbers where the tens d g t s smaller than the d v sor, extend ng th s to any d g t be ng smaller than the d v sor. Exempl f cat on of th s method and the language to use are best understood through v ew ng the tutor al v deos found on the toolk t. D v s on Division of a one- or wo-digi number by 10 and 100, iden ifying he value of he digi s in he answer as ones, en hs and hundred hs 24 10 = When you d v de by ten, each part s ten t mes smaller. The tens become ones and the ones become tenths. Each d g t s n a place that g ves t a value that s ten t mes smaller.