Progression in calculations Years 5 and 6

Transcription

1 Progression in calculations Years 5 and 6

2 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 1 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.

3 Y5 and Y6 Addition & Subtraction Coun forwards or backwards in s eps of powers of 10 for any given number up o Sk p count ng forwards and backwards n steps of powers of 10 (.e. 10, 100, 1000, and ) should be ncorporated nto trans t on act v t es and pract sed regularly. In Year 5 pup ls work w th numbers up to as well as tenths, hundredths and thousandths. In Year 6 pup ls work w th numbers up to Using known fac s and unders anding of place value o derive CPA Support w th p ace va ue counters on a p ace va ue chart repeated y add ng the same counter and regroup ng as needed. Count ng st cks and number nes Pay part cu ar attent on to boundar es where regroup ng happens more than once and so more than one d g t changes. e.g = or = = = Us ng the follow ng language makes the log c expl c t: I know three ones plus four ones s equal to seven ones. Therefore, three ten thousands plus four ten thousands s equal to seven ten thousands. In Year 5 extend to mult ples of and as well as tenths, hundredths and thousandths. In Year 6 extend to mult ples of one m ll on. These der ved facts should be used to est mate and check answers to calculat ons = = = = = = = = =

4 Par i ioning one number and applying known fac s o add. CPA Part t on ng nto p ace va ue amounts canon ca part t on ng : = Pup ls can use th s strategy mentally or w th jott ngs as needed. Pup ls should be aware of the range of cho ces ava lable when dec d ng how to part t on the number that s to be added. They should be encouraged to count on from the number of greater value as th s w ll be more eff c ent. However, they should have an understand ng of the commutat ve law of add t on, that the parts can be added n any order. Pup ls have exper ence w th these strateg es w th smaller numbers from prev ous years and so the focus should be on develop ng flex b l ty and explor ng eff c ency. W th p ace va ue counters represent the arger number and then add each p ace va ue part of the other number. The mage above shows the thousands be ng added. Represent p ctor a y w th an empty number ne Part t on ng n d fferent ways non-canon ca part t on ng : Extend the Make ten strategy (see gu dance n Y1 or Y2) to count on to a mu t p e of = The strategy can be used w th dec ma numbers Make one = =

5 Deve op understand n same Sub rac ion by par i ioning and applying known fac s. Pup ls can use th s strategy mentally or w th jott ngs as needed. Pup ls should be aware of the range of cho ces ava lable when dec d ng how to part t on the number that s to be subtracted. Part t on ng nto p ac = 75 Pup ls have exper ence w th these strateg es w th smaller numbers from prev ous years and so the focus should be on develop ng flex b l ty and explor ng eff c ency. Represent p ctor a y eft Part t on ng n d ffere Extend the Make ten

6 Calcula e difference by coun ing back It s nterest ng to note that f nd ng the d fference s revers ble. For example, the d fference between 5 and 2 s the same as the d fference between 2 and 5. Th s s not the case for other subtract on concepts P ace the numbers e ther end o between them. Se ect eff c ent F nd ng the d fference s eff c e Calcula e difference by coun ing on Addition strategies can be used to find difference F nd ng the d fference s eff c e

7 Pup 541 Sub Pup 2.5 Round and adjus Ad Addi ion and sub rac ion using compensa ion 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 Near doubles Pup ls should be able to double numbers up to 100 and use th s to der ve doubles for mult ples of ten as well as dec mal numbers. These facts can be adjusted to calculate near doubles

8 Pup s sho Dec ma n Par i ion bo h numbers and combine he par s Pup ls should be secure w th th s method for numbers up to , us ng place value counters or D enes to show conceptual understand ng. If mult ple regroup ngs are requ red, then pup ls should cons der us ng the column method. Wri en column me hods for addi ion For th s In Year 5, pup ls are expected to be able to use formal wr tten methods to add whole numbers w th more than four d g ts as well as work ng w th numbers w th up to three dec mal places. Pup ls should th nk about whether th s s the most eff c ent method, cons der ng f mental methods would be more effect ve. Cont nue to use concrete man pulat ves alongs de the formal method. When add ng dec mal numbers w th a d fferent number of dec mal places, n order to avo d calculat on errors, pup ls should be encouraged to nsert zeros so that there s a d g t n every row. Th s s not necessary for calculat on and these zeros are not place holders as the value of the other d g ts s not changed by t be ng placed. 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 Comb ne 3 4

9 If you hav Wri en column me hods for sub rac ion In Year 5, pup ls are expected to be able to use formal wr tten methods to subtract whole numbers w th more than four d g ts as well as work ng w th numbers w th up to three dec mal places 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 and 4 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 th nk about f th s s the most eff c ent method, cons der ng whether mental strateg es (such as count ng on, us ng known number facts, compensat on etc.) may be l kel er to produce an accurate solut on. 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 The term subtract o You can r Or you ca

10 Y5 and Y6 Multiplication Mul iply and divide whole numbers and hose involving decimals by 10, 100 and 1000 Avoid saying that you add a zero when multiplying by ten and instead use the language of place holder Use place value counters and charts to visualise and then notice what happens to the digits When you mu t p y by ten ea become tens the tens becom When mu t p y ng who e num has a va ue that s ten t mes x 10 = When you d v de by ten each become tens and the tens be t a va ue that s ten t mes sm When d v d ng mu t p es of te each d g t has a va ue that s E.g = = 21.03

11 Using known fac s and place value o derive mul iplica ion fac s 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 s further extended and appl ed to f nd a range of related facts. Pup ls should work w th dec mals w th up to two dec mal places. These der ved facts should be used to est mate and check answers to calculat ons. These are the mu t p cat on facts pup s shou d be ab e to der ve from a known fact

12 Mu t p y by 4 b e.g = 32 Mu t p y by 8 b Mu t p y by 5 b e.g = 18 e.g = d Doubling and halving Pup ls should exper ence doubl ng and halv ng larger and smaller numbers as they expand the r understand ng of the number system. Doubl ng and halv ng can then be used n larger calculat ons. D v de by 4 by e.g = 5 e.g = 24 D v de by 8 by e.g = 5 D v de by 5 by

13 Jott ngs on Bead str ng Mul iply by par i ioning one number and mul iplying each par 8 x 14 = 8 Dis ribu ive law a x (b + c) = a x b + a x c Cu sena re Bu d on pup s understand ng of arrays of counters to represent mu t p cat on to see that area mode s can be a usefu representat on Using knowledge of fac ors In Year 5 pup ls are expected to be able to dent fy factor pa rs and th s knowledge can be used to calculate. Pup ls w ll be us ng the commutat ve and assoc at ve laws of mult pl cat on. Ca cu ate 6 Two and tw Commu a ive law a x b = b x a Associa ive law Three and e a x b x c = (a x b) x c = a x (b x c) They should explore and compare the d fferent opt ons and choose the most eff c ent order to complete calculat ons. Four and s

14 Formal wri en me hod of shor mul iplica ion 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. 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. Mul iplying by a 2-digi number Formal wri en me hod of long mul iplica ion In Year 6 pup ls are extended from mult pl cat on by a 1-d g t number to mult pl cat on by a 2-d g t number. Extend the place value chart model used n Year 4, us ng an add t onal row on the place value chart. Extend understand ng of the d strub t ve law to develop conceptual understand ng of the two rows of the formal wr tten method. D enes blocks can be used to construct area models to represent th s.

15 Y5 and Y6 Division Deriving fac s from known 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. Using knowledge of mul iples o divide Us ng an area model to part t on the whole nto mult ples of the d v sor (the number you are d v d ng by)

16 Using knowledge of fac ors o divide Pup ls explore th s strategy when us ng repeated halv ng. 2 x 2 = 4 and so f you d v de by 4 the same result can be ach eved by d v d ng by two and then by two aga n.

17 How hund How Shor division 8528 Dividing a 4-digi numbers by 1-digi numbers The thought process of the trad t onal algor thm s as follows: How many 4s n 8? 2 How many 4s n 5? 1 w th 1 rema n ng so regroup. How many 4s n 12? 3 How many 4s n 8? 2 Shar Warn ng: If you s mply apply place value knowledge to each step, the th nk ng goes wrong f you have to regroup. How many 4s n 8000? 2000 How many 4s n 500? 100 w th 1 rema n ng ( llog cal) The answer would be 125. Shar ng the d v dend bu lds conceptual understand ng however doesn t scaffold the th nk ng of the algor thm. Us ng place value counters and f nd ng groups of the d v sor for each power of ten w ll bu ld conceptual understand ng of the short d v s on algor thm. Area models are also useful representat ons, as seen w th other strateg es and exempl f ed for long d v s on. 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. 8 tho equa 12 Grou

18 Long division CPA Dividing a 4-digi number by a 2-digi number Follow the language structures of the short d v s on strategy. Instead of record ng the regrouped amounts as small d g ts the numbers are wr tten out below. Th s can be eas er to work w th when d v d ng by larger numbers. If d v d ng by a number outs de of the r known facts, pup ls should start by record ng some mult ples of that number to scaffold.