Presentation on theme: " - + x ? Progression in Calculations: A guide for parents"— Presentation transcript:
1 - + x ? Progression in Calculations: A guide for parents South Farnham School
2 IntroductionWritten methods of calculations are based on mental strategies. Each of the four operations builds on mental skills which provide the foundation for jottings and informal written methods of recording. Skills need to be taught, practised and reviewed constantly. These skills lead on to more formal written methods of calculation. Strategies for calculation need to be supported by familiar models and images to reinforce understanding. When teaching a new strategy it is important to start with numbers that the child can easily manipulate so that they can understand the concept. The transition between stages should not be hurried as not all children will be ready to move on to the next stage at the same time, therefore the progression in this document is outlined in stages. An indication as to which stage each year group begins their focus is also given; however previous stages may need to be revisited to consolidate understanding when introducing a new strategy. A sound understanding of the number system is essential for children to carry out calculations efficiently and accurately.
3 Progression in Teaching Addition Mental SkillsRecognise the size and position of numbersCount on in ones and tensKnow number bonds to 10 and 20Add multiples of 10 to any numberPartition and recombine numbersBridge through 10Key Vocabularyaddadditionplusandcount onmoresumtotalaltogetherincrease+408
4 Recognise numbers 0 to 10123456789101, 2, 3, 4, 5, 6 … there are 6 teddiesCount reliably up to 10 everyday objectsFind one more than a numberOne more than three is fourCount in ones and tensBegin to relate addition to combining two groups of objectsandmakes 5Count along a number line to add numbers together3 + 2 = 5Begin to use the + and = signs to record mental calculations in a number sentence6 + 4 = 10Know doubles of numbers= 10+
5 Know by heart all pairs of numbers with a total of 10 and 20 92837465+2=311+2=3Know that addition can be done in any order+ 3Put the biggest number first and count on3 + 5588 + 7 = 15Add two single-digit numbers that bridge 10Begin to partition numbers in order to add
6 = 16Know which digit changes when adding 1s or 10s to any number1516= 251525= 35152535Adding two two-digit numbers (without bridging)Counting in tens and onesPartitioning and recombining1516171825262728152528= 28Year 3Adding two two-digit numbers (bridging through tens boundary)Using a number lineORUsing place value cards and place value apparatus to partition numbers and recombine408306= 84= 14= 70= 84
7 Standard written method Expanded methodIt is important that the children have a good understanding of place value and partitioning using concrete resources and visual images to support calculations. The expanded method enables children to see what happens to numbers in the standard written method.TU48+ 36T U4 8(U)(T)8 4Year 4Year 4/54 8+ 3 68 4Standard written methodThe previous stages reinforce what happens to the numbers when they are added together using more formal written methods.Year 6 consolidates this method by using larger numbers and adding decimals.1
8 Progression in Teaching Subtraction Mental SkillsRecognise the size and position of numbersCount back in ones and tensKnow number facts for all numbers to 20Subtract multiples of 10 from any numberPartition and recombine numbers (only partition the number to be subtracted)Bridge through 10Key Vocabularysubtracttake awayminuscount backlessfewerdifference between-408
9 7 8 9 10 , ... Ten green bottles hanging on the wall … Five fat sausages frying in a pan …Begin to count backwards in familiar contexts such as number rhymes or stories78910,...Continue the count back in ones from any given numberThree teddies take away two teddies leaves one teddyBegin to relate subtraction to ‘ taking away ’Find one less than a numberCount back in tensIf I take away four shells there are six leftCount backwards along a number line to ‘ take away
10 Maria had six sweets and she ate four. How many did she have left? Begin to use the – and = signs to record mental calculations in a number sentence6 - 4 = 2Know by heart subtraction facts for numbers up to 10 and 20= 8+3+5‘Counting on’Subtract single digit numbers often bridging through 1071015Begin to find the difference by counting up from the smallest numberBegin to partition numbers in order to take away
11 -1Subtract 1 from a two-digit number45 - 14445Year 3Subtract 10 from a two-digit number-103545Subtract multiples of 10 from any number453525-1043 –43 – 20 = 2323 – 3 = 20203+ 3+ 10+ 743 – 2343403023Partition the number to be subtracted (no exchanging)Year 4Decide whether to count on or count back= 47Now what’s the answer?
12 to subtract 7 units we need to exchange a ten for ten units Partitioning number to be subtracted – with exchanging (links to counting back on number line)2074343 –43 – 20 = 2 323 – 7 = 1 6207= 16to subtract 7 units we need to exchange a ten for ten units= 16Expanded methodIt is important that the children have a good understanding of place value and partitioning using concrete resources and visual images to support calculations. The expanded method enables children to see what happens to numbers in the standard written method.27-TU10 +30Year 4/5Standard written methodThe previous stages reinforce what happens to numbers when they are subtracted using more formal written methods. It is important that the children have a good understanding of place value and partitioning.4 31 613Year 6 consolidate this method with larger numbers
13 Progression in Teaching Multiplication Mental SkillsRecognise the size and position of numbersCount on in different steps 2s, 5s, 10sDouble numbers up to 10Recognise multiplication as repeated additionQuick recall of multiplication factsUse known facts to derive associated factsMultiplying by 10, 100, 1000 and understanding the effectMultiplying by multiples of 10Vocabularylots ofgroups oftimesmultiplymultiplicationmultipleproductonce, twice, three timesarray, row, columndoublerepeated additionx408
14 Use known facts to work out new ones Count in tens from zero10203040Count in twos from zero8642Count in fives from zero510152025Know doubles and corresponding halvesKnow multiplication tables to 10 x 10x 52 x 5 = 106 x 5 = 303 x 5 = 158 x 5 = 40Use known facts to work out new ones
15 +Year 3= 84 x 2 = 82 multiplied by 44 lots of 2Understand multiplication as repeated additionx4 22 4xUnderstand multiplication as an arrayUnderstand how to represent arrays on a number lineUnderstand that …24 x = x x 1024 x = x x 10Use factors to multiply
16 Use place value apparatus to support the multiplication of U x TU 4103Use place value apparatus to support the multiplication of U x TU4 x 1341034012Year 4Use place value apparatus to support the multiplication of U x TU alongside the grid method410340124 x 13= 524103Use place value apparatus to represent the multiplication of U x TU alongside the grid method124010344 x 23128020 ( 2 x 10 )34= 92
17 Standard written method Multiplying TU x TU14 x 3310430330012012== 13246230012030+ 12462Year 5/656 × (56 × 7) (56 × 20) 15121Standard written method
18 Progression in Teaching Division Mental SkillsRecognise the size and position of numbersCount back in different steps 2s, 5s, 10sHalve numbers to 20Recognise division as repeated subtractionQuick recall of division factsUse known facts to derive associated factsDivide by 10, 100, 1000 and understanding the effectDivide by multiples of 10Vocabularylots ofgroups ofsharegrouphalvehalfdividedivisiondivided byremainderfactorquotientdivisible÷408
19 If 2 x 10 = 20 then 20 10 = 2 20 2 = 10 Count back in tens 30 20 Count back in twos10864?Count back in fives51015Know halvesHalf of 6 is 3½ of 6 = 3If 2 x 10 = 20then20 10 = 220 2 = 10Use known multiplication facts to work out corresponding division facts
20 Year 3 Year 4 12 divided into groups of 3 gives 4 groups 12 3 = 4 Understand division as sharingUnderstand division as groupingYear 412 divided into groups of 3 gives 4 groups12 3 = 412 divided into groups of 4 gives 3 groups12 4 = 3Reinforce division as grouping through the use of arrays
21 Year 5 look at repeated subtraction of larger chunks 18 divided into groups of 318 3 = 6Represent ‘groups’ for division on a number line using apparatus alongside the line36912151818 3 = 61818 6 = 31818 ÷3 = 61 8( 1 x 3 )1 51 29( 1 x 3 )63( 1 x 3 )Year 4/5- 315- 312Understand division as repeated subtraction using a vertical line and apparatus to make the links- 39- 3Year 5 look at repeated subtraction of larger chunks6- 33- 3
22 What facts do I know about the 7 times-table? Children need to see that as the numbers get larger, large chunk subtraction is the more efficient method. Multiples of the divisor (large chunks) are taken away. Multiplication facts are needed to see the size of the ‘chunk’.Year 5/6Fact Box1 x 7 = 72 x 7 = 145 x 7 = 3510 x 7 = 7020 x 7 = 14050 x 7 = 350100 x 7 = 700518 ÷ 7 = 74518( 50 x 7 )168( 20 x 7 )28( 4 x 7 )100 ÷ 7 = 14 r 2100( 10 x 7 )30( 4 x 7 )2Year 6560 ÷ 242 3 r 8(20x24)8 0(3x24)8Standard written methodLinks directly to large chunk subtraction
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