Progression in Calculations

1. x - ÷ + Progression in Calculations Cheam Park Farm Junior School

2. 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 Cheam Park Farm Junior 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. Year 3 Year 4 Year 5 Year 6 M.Stafford June 2007 These guidelines have been produced using suggestions from 4S. Thank-you

3. 40 8 Progression in teaching addition + 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 Words linked to + add, addition, and, count on, plus, sum, more, altogether, increase

4. 40 8 30 6 Year 3 Add two single-digit numbers that bridge 10 Year 4 8 + 7 = 15 BIGGEST number first 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 3 Year 4 Year 5 48+36=84 40 + 30 + 8 + 6 40 + 30 = 70 8 + 6 = 14 70 + 14 = 84

5. T U T U 40 + 8 30 + 6 80 + 4 10 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 3 Summer term Year 4 Year 5 48 + 36 48 +36 4 8 + 3 6 8 4 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 4 More able Extend this method to include thousands and decimals depending on needs of child Year 5 Year 6

6. 40 8 30 6 Year 3 Add two single-digit numbers that bridge 10 Year 4 8 + 7 = 15 BIGGEST number first 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 3 Year 4 Year 5 48+36=84 40 + 30 + 8 + 6 40 + 30 = 70 8 + 6 = 14 70 + 14 = 84

7. - 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 Interactive teaching Programmes 40 8 Words linked to - take away, subtract, subtraction, count back, minus, less, decrease, difference between

8. Subtract single digit numbers often bridging through 10 15 - 7 = 8 Year 3 Year 4 Biggest number FIRST 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

9. -10 43 – 23 -10 35 -10 45 20 23 33 43 25 35 45 43 – 43 – 20 = 23 23 – 3 = 20 20 3 - 3 - 10 - 10 Subtract 10 from a two-digit number 45 - 10 Year 3 Subtract multiples of 10 from any number Year 3 45 - 20 Year 4 Partition the number to be subtracted (no exchanging) Year 3 Year 4 Decide whether to count on or count back 74 - 27 = 47 Year 3 Year 4 Year 5

10. 4 3 20 7 T U 43 - 27 = 16 - 2 7 to subtract 7 units we need to exchange a ten for ten units 43 – 43 – 20= 2 3 23 – 7 = 1 6 40 + 3 - 20 + 7 10 + 6 30 10 + 20 7 4 3 - 2 7 1 6 3 1 Partitioning number to be subtracted – with exchanging (links to counting back on number line) 43 - 27 = 16 Year 3 Year 4 Year 5 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. Year 4 Year 5 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. Extend this method to include thousands and decimals depending on needs of child Year 4 Year 5 Year 6

11. 40 8 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 x 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 Words linked to x multiply, multiplication, multiple, double, array, times, lots of

12. 0 10 20 30 40 50 0 2 4 6 8 10 Year 3 Count in tens from zero Year 4 Count in twos from zero Year 3 Year 4 Count in fives from zero (all multiples) Year 3 Year 4 0 5 10 15 20 25 30 Year 3 Year 4 Double 35 Halve 98 Know doubles and corresponding halves Year 5 Year 6 Know multiplication tables to 10 x 10 6 x 5 = 30 2 x 5 = 10 x 5 Year 3 3 x 5 = 15 Year 4 Year 5 Use known facts to work out new ones 8 x 5 = 40 Year 6

13. 2 2 2 2 + + + 2 4 x 4 2 x 40 8 Understand multiplication as repeated addition 2 + 2 + 2 + 2 = 8 4 x 2 = 8 2 multiplied by 4 4 lots of 2 Year 3 Year 4 Understand multiplication as an array Year 3 Year 4 3x48= Use place value to support the multiplication of U x TU Year 3 X3=120 40 Year 4 X3=24 8 3x48=(3x40)+(3x8) Year 5 Understand that … 24 x 20 = 24 x 2 x 10 24 x 50 = 24 x 5 x 10 Use factors to multiply Year 4 Year 5 Year 6

14. 10 3 4 30 3 10 300 30 40 12 = 330 + 4 120 12 = 132 300 120 30 + 12 462 462 10 3 40 12 4 56× 27 1120 (56 × 20) 392 (56 × 7)1512 1 Use place value apparatus to support the multiplication of U x TU alongside the grid method 4 x 13 Year 4 Year 5 40 + 12 = 52 Multiplying TU x TU, HTU, ThHTU and decimals Year 4 Year 5 Year 6 14 x 33 Year 5 Year 6 Standard written method

15. 40 8 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 ÷ Words linked to ÷ group, divide, division, divided by, divisible, factor, share, half, halve, remainder, quotient

16. 0 10 20 30 ? 4 6 8 10 Count back in tens Year 3 Count back in twos Year 3 Count back in fives Year 3 Year 4 All multiples 0 5 10 15 Know halves Year 3 Year 4 Half of 6 is 3 ½ of 6 = 3 Half of 130 Year 5 Year 6 Use known multiplication facts to work out corresponding division facts If 2 x 10 = 20 then 20  10 = 2 20 2 = 10 Year 3 Year 4 Year 5 Year 6

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

18. 0 3 6 9 12 15 18 Represent ‘groups’ for division on a number line using apparatus alongside the line 18 divided into groups of 3 18  3 = 6 Year 3 Year 4 18  3 = 6 0 18 18  6 = 3 18 • 18 ÷3 = 6 • 1 8 • - 3 ( 1 x 3 ) • 1 5 • - 3 ( 1 x 3 ) • 1 2 • - 3 ( 1 x 3 ) • 9 • - 3 ( 1 x 3 ) • 6 • - 3 ( 1 x 3 ) • 3 • - 3 ( 1 x 3 ) • 0 Understand division as repeated subtraction using a vertical line and apparatus to make the links - 3 15 - 3 12 Year 4 - 3 Year 5 9 - 3 6 - 3 3 - 3 0

19. 560 ÷ 24 2 3 r 8 2 4 5 6 0 - 4 8 0 8 0 - 7 2 8 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 4 Year 5 Year 6 Fact Box 1 x 7 = 7 2 x 7 = 14 5 x 7 = 35 10 x 7 = 70 20 x 7 = 140 50 x 7 = 350 100 x 7 = 700 518 ÷ 7 = 74 518 - 350 ( 50x 7 ) 168 - 140 ( 20x 7 ) 28 - 28 ( 4x 7 ) 0 100 ÷7 = 14 r 2 100 - 70 ( 10x 7 ) 30 - 28 ( 4x 7 ) 2 Standard written method Links directly to large chunk subtraction Year 5 Year 6