Progression in Calculations ÷ Cranmere Primary School

Introduction Written 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 needs to be taught, practised and reviewed constantly. These lead on to more formal written methods of calculations. This guidance is to support the teaching of calculations at Cranmere Primary School. 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. Previous stages may need to be revisited to consolidate understanding when introducing a new strategy while other children may need to be moved on to the next step. Suggestions for year groups have been included but this rather depends on the needs of the class. N.Wood Sept Year 3Year 4Year 5Year 6Year 2Year 1

408 Progression in teaching addition Words linked to + add, addition, and, count on, plus, sum, more, altogether, increase Mental skills Recognise the size and position of numbers Count on in ones and tens Know number bonds to 10 and 20 Add multiples of 10 to any number Partition and recombine numbers Bridge through 10 Models and Images Counting apparatus Place value apparatus Place value cards Numbered number lines Marked but unnumbered number lines Empty number lines Hundred square Counting stick Bead strings Interactive teaching Programmes Use of numicon

Counting Year 1 Number and place value (statutory requirements) -count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number -count, read and write numbers to 100 in numerals, count in different multiples including ones, twos, fives and tens. Year 2 Number and place value (statutory requirements) -count in steps of 2,3 and 5 from 0, and count in tens from any number, forward or backward. Year 3 Number, place value and rounding (statutory requirements) -count from 0 in multiples of 4, 8, 50 and 100; finding 10 or 100 more or less than a given number -count up and down in tenths; recognise that tenths arise from dividing an object into 10 equal parts and in dividing one-digit numbers or quantities by 10 Year 4 Number, place value and rounding (statutory requirements) -count in multiples of 6, 7, 9, 25 and count backwards through zero to include negative numbers -count up and down in hundredths; recognise that hundredths arise when dividing an object by a hundred and dividing tenths by ten Year 5 Number, place value, approximation and estimation (statutory requirements) -count forwards or backwards in steps of powers of 10 for any given number up to

Reception Recognise numerals 0 to 10 and understand the meaning of each number by recognising and knowing their number clusters Count reliably up to 10 everyday objects 1,2,3,4,5,6…ther e are 6 teddies altogether. Find one more than a number One more than three is four The number after 3 is 4 Count in ones as they become more familiar with using numbers on a number line and 100 square Then recognise numerals up to 20 and understand the meaning of each number by recognising and knowing their clusters

Begin to relate addition to combining two groups of objects and Begin to use the + and =signs to record mental calculations in a number sentence = 10 Children begin to solve problems using doubling, halving, grouping and sharing =10 Children show mental recall of number bonds to 10 and use these for problem solving. Some children show mental recall of number bonds to 20. Year 1 Counting choir Count, read and write numbers to 100 in numerals, count in different multiples including ones, twos, fives and tens and recognise patterns with help of a 100 square. Continue with the use of cluster cards in order for children to develop the skill of decomposing and recomposing numbers mentally. Know by heart all pairs of numbers with a total of 10 and 20.

Progression in difficulty when adding on a number line (see page ) 1. 1 digit + 1 digit Begin to partition numbers in order to add two 2 digit numbers Year 2 Counting choir Count in steps of 2,3,and 5 from 0 and from any of its multiples. Count in tens and ones from any number, forward or backward. Alternate the above counting within one session to help with calculating on a number line. Continue with the use of cluster cards in order for children to develop the skill of decomposing and recomposing numbers mentally.

Year 2 Continue to practice the mental recall of all pairs of number bonds with a total of 10 and 20. Know which digit changes when adding 1s or 10s to any number = = 25 Continue with partitioning two digit numbers into their Tens and Units and extend to partitioning three digit numbers into their H, T, U. In Year 2. children learn to add using blank number line. Progression in difficulty when adding on a number line can be seen on the following pages.

Add two single-digit numbers that bridge = 15 BIGGEST number first = = = 84 Adding two two-digit numbers (bridging through the tens boundary) Using a number line OR Using place value cards and place value apparatus to partition numbers and recombine Year =84 Year 4 Year 3 Year 5 Year 4 Year 2 Year 1 Year 2

From year 1. children continue to add using a blank number line. Progression in difficultly when adding on a number line.

TU T U Expanded method It 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 the children to see what happens to numbers in the standard written method Year 4 Year 5 Year 3 Column Method standard written method The previous stages reinforces what happens to the numbers when they are added together using more formal methods 1 Year 5Year 6 Extend this method to include thousands and decimals depending on needs of child Year 3 Year 4

Progression in teaching subtraction Mental skills Recognise the size and position of numbers Count on in ones and tens Know number facts for all numbers to 20 Subtract multiples of 10 from any number Partition and recombine numbers Bridge through 10 Models and Images Counting apparatus Place value apparatus Place value cards Numbered number lines Marked but unnumbered number lines Empty number lines Hundred square Counting stick Bead strings Models and Images, charts Interactive teaching Programmes Words linked to - take away, subtract, subtraction, count back, count on, minus, less, decrease, fewer, difference between 408

Reception Begin to count backwards in familiar contexts such as number rhymes or stories Ten green bottles hanging on the wall… 10, 9, 8, 7… Continue the count back in ones from any given number Begin to relate subtraction to ‘taking away’ Find one less than any number 1,2,3,4,5,6,7,8,9.10 What is the number before 4? Count back in ones Count backwards along a number line to take away Some children show mental recall of number bonds and use these for problem solving

Year 1 Begin to use the – and = signs to record mental calculations in a number sentence Know by heart subtraction facts for numbers up to 10 and 20 Continue to use cluster cards in order for children to develop the skill of decomposing and recomposing numbers mentally and to help with calculating on a number line

= 8 Biggest number FIRST Subtract single digit numbers often bridging through 10 Year 3 Begin to find the difference by counting up from the smallest number Year 3 Year 4 Partition numbers in order to take away Year 3 Year 4 Year 2 Year 1 Year 2 Year 1 Year 2

Subtract 10 from a two-digit number Year 3 Subtract multiples of 10 from any number Partition the number to be subtracted (no exchanging) – – 43 – 20 = – 3 = 20 3 Decide whether to count on or count back = 47 Year 3 Year 4 Year 5 Year 4 Year 2 20

Depending on the calculation children may wish to use the counting up method or the counting back method

Partitioning number to be subtracted – with exchanging (links to counting back on number line) = TU 43 – 43 – 20 = – 7 = Expanded method It 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 to subtract 7 units we need to exchange a ten for ten units = 16 Standard written method The 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. Year 3Year 4 Year 5Year 6 Extend this method to include thousands and decimals depending on needs of child Year 4 Year 5 Year 4 Year 3

Compact column method of subtraction

Progression in Teaching Multiplication Mental Skills Recognise the size and position of numbers Count on in different steps 2s, 5s, 10s Double numbers up to 10 Recognise multiplication as repeated addition Quick recall of multiplication facts Use known facts to derive associated facts Multiplying by 10, 100, 1000 and understanding the effect Multiplying by multiples of 10 Models and Images Counting apparatus Place value apparatus Arrays 100 squares Number tracks Numbered number lines Marked but unnumbered lines Empty number lines. Multiplication squares Counting stick Bead strings Models and Images charts ITPs – Multiplication grid, Number Dials, Multiplication Facts 408 Words linked to x multiply, multiplication, multiple, double, array, times, lots of, repeated addition, groups of, product

Count in tens from zero Count in twos from zero Count in fives from zero (all multiples) Know doubles and corresponding halves Know multiplication tables to 12x 12 x 5 6 x 5 = 30 Use known facts to work out new ones 2 x 5 = 10 8 x 5 = 40 3 x 5 = Year 1 Year 2Year 4 Year 3 Year 5 Year 6 Year 2 Double 35 Halve 98 Year 3 Year 5

Understand multiplication as repeated addition = 8 4 x 2 = 8 2 multiplied by 4 4 lots of 2 Understand multiplication as an array Year x48= 40 8 X3=120 X3=24 Use place value to support the multiplication of U x TU 3x48=(3x40)+(3x8) Year 3 Year 4 Understand that … 24 x 20 = 24 x 2 x x 50 = 24 x 5 x 10 Use factors to multiply Year 4 Year 5 Year 6 Year 4 Year 5 Year 6 Year 2

Use place value apparatus to support the multiplication of U x TU alongside the grid method = x 13 Year 4 Year 5 Multiplying TU x TU, HTU, ThHTU and decimals x 33 = = Year 5 Year 6 Year 4

Short and Long Multiplication Begin with numbers where carrying is not involved. Then move onto multiplying 3 digit numbers by a single digit without carrying. Always start multiplying by the unit number.

When knowledge is secure, higher numbers are used to introduce carrying Children need to be taught that the 0 in the second row is written as a placeholder because we are now multiplying the tens with each digit.

Progression in Teaching Division Mental Skills Recognise the size and position of numbers Count back in different steps 2s, 5s, 10s Halve numbers to 20 Recognise division as repeated subtraction Quick recall of division facts Use known facts to derive associated facts Divide by 10, 100, 1000 and understanding the effect Divide by multiples of 10 Models and Images Counting apparatus Arrays 100 squares Number tracks Numbered number lines Marked but unnumbered lines Empty number lines. Multiplication squares Models and Images charts ITPs – Multiplication grid, Number Dials, Grouping, Remainders 40 8 Words linked to ÷ group, divide, division, divided by, divisible, factor, share, half, halve, remainder, quotient ÷

Year 1 Counting choir Count in steps of 2s, 5s and 10s forward and back from 0 and from any multiple of 10 using the 100 square and taking the opportunity to discuss patterns that are recognised. Children learn to share objects practically 15 eggs shared between 5 baskets = 3 each Children use grouping to solve problems involving division. With the help of laminated sheets children place the given number of objects into groups using the correct mathematical vocabulary. 10 cookies into groups of 2. How many groups? Use bead strings to help children solve division problems. The children find out how many jumps of 5 they can make between 0 and into groups of 5 = 4 groups 20 into jumps of 5 = 4 jumps

Count back in tens Count back in twos Count back in fives Know halves Use known multiplication facts to work out corresponding division facts Half of 6 is 3 ½ of 6 = ? If 2 x 10 = 20 then 20  10 = 2 20  2 = 10 Year 2 Year 3 Year 4 Year 6 Year 5 Half of 130 Year 5Year 6 Year 3 Year 1 Year 3 Year 4 Year 2

Understand division as sharing Understand division as grouping Reinforce division as grouping through the use of arrays 12 divided into groups of 3 gives 4 groups 12  3 = 4 12 divided into groups of 4 gives 3 groups 12  4 = 3 Year 2 Year 3 Year 4 Year 2

Represent ‘groups’ for division on a number line using apparatus alongside the line  6 = 3 18  3 = 6 18 divided into groups of 3 18  3 = 6 Understand division as repeated subtraction using a vertical line and apparatus to make the links ÷3 = ( 1 x 3 ) ( 1 x 3 ) ( 1 x 3 ) ( 1 x 3 ) ( 1 x 3 ) ( 1 x 3 ) Year 3 Year 4 Year 5 Year 3 Year 2

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’. 100 ÷ 7 = 14 r ( 10 x 7 ) ( 4 x 7 ) 2 Fact Box 1 x 7 = 7 2 x 7 = 14 5 x 7 = x 7 = x 7 = x 7 = x 7 = 700 What facts do I know about the 7 times-table? 518 ÷ 7 = ( 50 x 7 ) ( 20 x 7 ) ( 4 x 7 ) 0 Standard written method Links directly to large chunk subtraction 560 ÷ r Year 5 Year 6 Year 5 Year 4