Progression in Calculations

1. x - ÷ + Birchwood Primary School Progression in Calculations

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 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, images and practical activities to reinforce understanding. When teaching a new strategy it is important to start with numbers the child can easily manipulate so that they can understand the concept. The transition between levels should not be hurried. Children should not be moved onto the next method until they are secure in their current method and are able to explain it. Previous levels should 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. Maths @ Birchwood • A Typical Birchwood maths ‘unit’ will follow a series of lessons!! • We use the wheel to direct our teaching…. • STAGE 1 – Establish what we already know/address any misconceptions

4. 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, Images and apparatus • Place value apparatus - diennes • Arrow cards • Number tracks • Numbered number lines • Marked but unnumbered number lines • Empty number lines • Hundred square • Counting stick • Bead string • Models and Images charts • ITPs – Number Facts, Ordering Numbers, Number Grid, Counting on and back in ones and tens • Key Vocabulary • add • addition • plus • and • count on • more • sum • total • altogether • increase + +

5. 0 1 2 3 4 5 6 7 8 9 10 One more than three is four 1, 2, 3, 4, 5, 6 … there are 6 teddies P8 or ELG’s Recognise numbers 0 to 10 Level 1c Count reliably up to 10 everyday objects Level 1c Find one more than a number Level 1c Begin to relate addition to combining two groups of objects makes 5 and Level 1b Count in ones and tens Level 1b Count along a number line to add numbers together 3 + 2 = 5 Level 1b 6 + 4 = 10 Begin to use the + and = signs to record mental calculations in a number sentence

6. + = + = 1 1 2 2 3 3 Level 1a Know that addition can be done in any order Level 1a 3 +5 Put the biggest number first and count on Level 2c Know doubles of numbers + 5 5 = 10 Level 2b 8 + 7 = 15 Add two single-digit numbers that bridge 10

7. 5 1 5 9 6 2 4 8 3 7 Level 2b Know by heart all pairs of numbers with a total of 10 Level 2a Begin to partition numbers in order to add

8. 5 15 11 15 9 6 12 14 8 13 7 15 16 17 18 25 26 27 28 15 25 28 Level 2a 15 + 1 = 16 Know which digit changes when adding 1s or 10s to any number 15 16 15 + 10 = 25 25 15 + 20 = 35 15 25 35 Level 2a Adding two two-digit numbers (without bridging) Counting in tens and ones Partitioning and recombining 15 + 13 = 28 Level 3c Know by heart all pairs of numbers with a total of 20 6 and how many more make 20?

9. 40 8 30 6 Level 3c Add near multiples of 10 to a 2 digit number 25 + 21 = 46 Level 3c Adding two two-digit numbers (bridging through tens boundary) Using a number line OR Using place value cards and Dienes to partition numbers and recombine 48 + 36 = 84 40 + 30 + 8 + 6 40 + 30 = 70 8 + 6 = 14 70 + 14 = 84

10. Level 3b 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 children to see what happens to numbers in the standard written method. 48 + 36 48 +36 T U 40 + 8 30 + 6 80 + 4 10 This method should also be used for adding HTU to HTU Level 3a 4 8 + 3 6 8 4 Standard written method The previous stages reinforce what happens to the numbers when they are added together using more formal written methods. 1 This method should also be used for adding HTU to HTU

11. Level 4c 528 + 288 500 + 200 = 700 20 + 80 = 100 8 + 8 = 16 = 816 They must also add HTU mentally, starting wit the most significant digit first. Level 4b + £4.85 + £3.38 £8.23 1 1 Add must add decimals to 2 decimal places

12. 40 8 • Progression in Teaching Subtraction • Mental Skills • Recognise the size and position of numbers • Count back in ones and tens • Know number facts for all numbers to 20 • Subtract multiples of 10 from any number • Partition and recombine numbers (only partition the number to be subtracted) • Bridge through 10 • Models, Images and apparatus • Place value apparatus - diennes • Arrow cards • Number tracks • Numbered number lines • Marked but unnumbered lines • Hundred square • Empty number lines. • Counting stick • Bead strings • Models and Images Charts • ITPs – Number Facts, Counting on and back in ones and tens, Difference • Key Vocabulary • Subtract • Take away • Minus • Count back • Less • Fewer • Difference between - -

13. 10 9 8 7 , , , ... P8 or ELG’s 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 stories Level 1c Continue the count back in ones from any given number Level 1c Three teddies take away two teddies leaves one teddy Begin to relate subtraction to ‘ taking away ’ Level 1c Find one less than a number up to 10 Level 1b Count back in tens Level 1b 6 - 4 = 2 Begin to use the – and = signs to record a number sentence using numbers up to 10 whilst still using practical resources

14. The difference between 8 and 5 is 3 Level 1b 1 1 2 2 3 3 Compare 2 sets to find numerical difference Level 1a Maria had six sweets and she ate four. How many did she have left? Understand the operation of subtraction and use related vocab. Subtract numbers when solving problems involving up to 10 objects in a range of contexts 6 - 4 = 2 Level 2c 13 – 1 = 21 – 10 = Within the range 1- 30 say 1 or 10 less than any number Level 2b 7 – 2 = 5 Know by heart subtraction facts for numbers up to 10

15. Level 2b 15 - 7 = 8 Subtract single digit numbers often bridging through 10 Level 2b Subtract 1 from a two-digit number 45 - 1 Level 2a Begin to find the difference by counting up from the smallest number Level 2a Subtract 10 from a two-digit number 20 30

16. 43 – 23 25 45 20 23 33 43 43 – 43 – 20 = 23 23 – 3 = 20 20 3 - 3 - 10 - 10 Level 2a Subtract multiples of 10 from any number Level 2a Begin to partition numbers in order to take away Partition the number to be subtracted (no exchanging) Level 3c Decide whether to count on or count back 74 - 27 = 47 Now where’s the answer?

17. T U 43 - 27 = 16 - 2 7 to subtract 7 units we need to exchange a ten for ten units 40 + 3 - 20 + 7 10 + 6 30 10 + Level 3c Subtract mentally a near multiple of 10 from a 2 digit number 56 – 21 = 35 Level 3c Know by heart subtraction facts for numbers up to 10 Level 3b 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.

18. 4 3 - 2 7 1 6 3 1 Level 3a 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. Level 4b Subtract decimals to 2 places ²3 .¹79ml - 2. 86ml ______ 0 .93ml https://www.ncetm.org.uk/resources/40532

19. 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 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, Images and apparatus Place value apparatus Arrays 100 squares Number tracks Numbered number lines Marked but unnumbered lines Empty number lines. Multiplication squares Counting stick Models and Images charts ITPs – Multiplication grid, Number Dials, Multiplication Facts Vocabulary Lots of Groups of Times Multiply Multiplication Multiple Product Once, twice, three times Array, row, column Double Repeated addition x x

20. 0 2 4 6 8 0 5 10 15 20 25 Level 1b Count in tens from zero 0 20 30 40 50 Level 2c Count in twos from zero Level 2c Count in fives from zero Level 2b Know doubles and corresponding halves to 20

21. 2 2 2 2 + + + 2 4 x 4 2 x Level 2a 2 + 2 + 2 + 2 = 8 4 x 2 = 10 2 multiplied by 4 4 lots of 2 Understand multiplication as repeated addition Level 2a Understand multiplication as an array Level 2a Understand how to represent arrays on a number line x 5 Level 2a 2 x 5 = 10 6 x 5 = 30 Tables spider 3 x 5 = 15 4 x 5 = 20 10 x 5 = 50 8 x 5 = 40 Know by heart facts for the 2,5 & 10 multiplication tables 5 x 5 = 25

22. Level 3c Use known facts to work out new ones Use known facts and place value to carry out mentally simple x calculations Level 3c Know by heart facts for the 2,3,4,5 & 10 multiplication tables

23. 10 10 3 3 4 4 40 12 10 3 40 12 4 10 10 3 4 40 40 12 20 ( 2 x 10 ) 3 4 80 12 Level 3c Use place value apparatus to support the multiplication of U x TU 4 x 13 Level 3c Use place value apparatus to support the multiplication of U x TU alongside the grid method 4 x 13 40 + 12 = 52 Level 3b Use place value apparatus to represent the multiplication of U x TU alongside the grid method 4 x 23 80 + 12 = 92

24. 30 3 10 300 30 = 330 + 4 120 12 = 132 462 56 × 27 1120 (56 × 20) 392 (56 × 7)1512 1 Level 3a + 14 x 33 Multiplying TU x TU 300 120 30 + 12 462 This may be as far as many children progress in their use of multiplication. Level 4c Standard written method Level 4b Use factors to multiply Understand that … 24 x 20 = 24 x 2 x 10 24 x 50 = 24 x 5 x 10 https://www.ncetm.org.uk/resources/40530

25. 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, Images and apparatus 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, remainders grid, Number Dials, Grouping, Vocabulary Lots of Groups of Share Shared between Group Divide Divide into Division Divided by Remainder Factor Quotient Divisible ÷ ÷

26. 0 10 20 30 ? 2 4 6 8 Level 1c Give every bear a sweet – how many sweets do we need? Solve problems by sharing objects in a practical or role play context Level 1b Count back in tens Give each bear 2 sweets – how many sweets do we need? Level 1a Solving practical problems by sharing into equal groups Level 2c Count back in twos Level 2c Count back in fives 0 5 10 15

27. Level 2b Half of 6 is 3 ½ of 6 = 3 Know halves to 20 Level 2a Understand division as sharing 1 2 3 4 5 6 7 8 9 10 etc…. Level 2a Understand division as grouping 1,2,3,4,5 6,7,8,9,10 etc Level 2a 12 divided into groups of 3 gives 4 groups 12  3 = 4 12 divided into groups of 4 gives 3 groups 12  4 = 3 Reinforce division as grouping through the use of arrays

28. 0 3 6 9 12 15 18 Level 3c 18 divided into groups of 3 18  3 = 6 Represent ‘groups’ for division on a number line using apparatus alongside the line to include remainders. 18  3 = 6 -3 -3 -3 -3 -3 -3 0 18 3 6 9 12 15 18  6 = 3 Level 3b 126 ÷ 21 = 6 -21 -21 -21 -21 -21 -21 84 105 126 0 21 42 63 180 ÷ 60 = 30 Level 3a 126 ÷ 21 = 6 (2 x21) (2 x21) (2 x21) 0 42 84 126 https://www.ncetm.org.uk/resources/43589

29. Level 4c This must not be introduced to the children before Year 5 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 (1 x 3) 15 Understand division as repeated subtraction using a vertical line and apparatus to make the links (1 x 3) 12 (1 x 3) 9 (1 x 3) 6 (1 x 3) 3 (1 x 3) 0

30. Level 4b What facts do I know about the 7 times-table? Year 5 and 6 only once confident with previous method 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’. 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 ( 50 x 7 ) 168 - 140 ( 20 x 7 ) 28 - 28 ( 4 x 7 ) 0 100 ÷ 7 = 14 r 2 100 - 70 ( 10 x 7 ) 30 - 28 ( 4 x 7 ) 2 Level 4a Year 6 Top set only 560 ÷ 24 2 3 r 8 2 4 5 6 0 4 8 0 8 0 7 2 8 Standard written method Links directly to large chunk subtraction

31. How you can support your child • Look for and talk about numbers in the environment • Play games • Shopping • Counting on/back • Number bonds • Doubles/Halves • Times tables • Division facts

32. Websites • http://nrich.maths.org/public/ • http://www.bbc.co.uk/schools/ks2bitesize/maths/number.shtml • http://www.teachingtime.co.uk/ • http://www.teachingtables.co.uk • http://www.bbc.co.uk/schools/laac/menu.shtml • http://www.woodlands-junior.kent.sch.uk/interactive/literacy/index.htm • http://www.coxhoe.durham.sch.uk/