Progression in Maths - calculation methods

1. x - ÷ + Progression in Maths - calculation methods Abbey Primary School – December 2012

2. Introduction • Written methods of calculations are reliable and efficient procedures which can be used in many different contexts. However, they are of no use to someone who applies them inaccurately or who cannot judge whether the answer is reasonable. • The progression towards a written method is crucial, since it is based on steps which are done mentally and which need to be secured first. • A structured approach to developing written methods relies on: • Apparatus • Apparatus + picture • Apparatus + picture + recording • Picture + recording • Recording • Developing a common whole school approach enhances pupils’ confidence and aids continuity across the school. • It can be hard for a pupil to hold all the steps of a calculation in their heads, so informal jottings should be encouraged at all times. • 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.

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 Number tracks numicon 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 greater more Most sum total altogether increase

4. 0 1 2 3 4 5 6 7 8 9 10 1, 2, 3, 4, 5, 6 … there are 6 teddies One more than three is four Recognise numbers 0 to 10 Cardinality: Knowing that 6 objects is the number 6. Count reliably up to 10 everyday objects Find one more than a number Count in ones and tens Count all and makes 5 Count on (keeping number in your head / hiding the first pattern) 3 + 2 = 5 Begin to use the + and = signs to record mental calculations in a number sentence 6 + 4 = 10 Know doubles of numbers and their corresponding halves 5 5 = 10 +

5. 5 1 5 9 6 2 4 8 3 7 + = + = 1 1 2 2 3 3 Know by heart all pairs of numbers with a total of 20 Know that addition can be done in any order + 3 3 +5 Put the biggest number first and count on 5 8 Begin to partition and recombine numbers in order to add

6. 15 15 16 17 18 25 26 27 28 15 25 28 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 Adding two two-digit numbers (without bridging) Counting in tens and ones 15 + 13 = 28 15 + 10 + 3 = 28 Adding two two-digit numbers (bridging through tens boundary) Using a number line 48 + 30 + 6 = 84 48 + 36 = 84

7. T U 40 8 30 6 Adding two two-digit numbers (bridging through tens boundary) Using place value cards and place value apparatus to partition numbers and recombine 48 + 36 14 (8 + 6) 70 (40 + 30) 84 48 + 36 = 84 Remember to add the least significant digits (e.g. units in this case) first 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 = 84 T U 40 + 8 + 30 + 6 80 + 4 10

8. 4 8 + 3 6 8 4 1 Standard written method The previous stages reinforce what happens to the numbers when they are added together using more formal written methods. 225 + 156 = 381 200 + 20 + 5 + 100 + 50 + 6 300 + 80 + 1 10 2 2 5 +1 5 6 3 8 1 1

9. 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 and Images • Counting apparatus • Place value apparatus • Place value 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 (specific use) • minus • count back • less • fewer • difference between -

10. 10 9 8 7 , , , ... 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 Continue to count back in ones from any given number Three teddies take away two teddies leaves one teddy Take away Find one less than a number Count back in tens If I take away four shells there are six left Count backwards along a number line to ‘ take away

11. 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 sentence 6 - 4 = 2 Know by heart subtraction facts for numbers up to 10 and 20 Begin to find the difference 40 – 27 = 13 Find the difference using the number line + 10 + 1 + 1 + 1 27 28 29 30 40

12. Find the difference using the number line 40 – 27 = 13 + 10 + 3 27 30 40 Find the difference using the number line 74 - 27 = 47 Extend to 3 digit numbers and decimals

13. 43 - 27 = 16 to subtract 7 units we need to exchange a ten for ten units 40 + 3 - 20 + 7 10 + 6 30 10 + 4 3 - 2 7 1 6 3 1 column subtraction (without exchanging) 40 + 3 - 30 + 2 10 + 1 • 43 – 32 = 11 • 4 3 • 3 2 • 1 1 Extend to 3 digit numbers without exchanging. 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. column subtraction (with exchanging) 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.

14. 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 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 Vocabulary lots of groups of times multiply multiplication multiple product once, twice, three times array, row, column double repeated addition x

15. 0 10 20 30 40 50 0 2 4 6 8 10 Count in tens from zero Count in twos from zero Count in fives from zero 5 0 10 15 20 25 30 Know doubles and corresponding halves Know multiplication tables to 10 x 10 x 5 2 x 5 = 10 6 x 5 = 30 3 x 5 = 15 8 x 5 = 40 Use known facts to work out new ones

16. 2 2 2 2 + + + 2 4 x 4 2 x Understand multiplication as repeated addition 2 + 2 + 2 + 2 = 8 4 x 2 = 8 2 multiplied by 4 4 lots of 2 Understand multiplication as an array Understand how to represent arrays on a number line

17. 10 10 3 3 4 4 40 12 10 3 40 12 4 10 10 3 4 20 3 4 80 12 Use place value apparatus to support the multiplication of U x TU 13 x 4 = 52 Use place value apparatus to support the multiplication of U x TU alongside the grid method 13 x 4 = 52 40 + 12 = 52 Use place value apparatus to represent the multiplication of U x TU alongside the grid method 23 x 4 = 92 80 + 12 = 92

18. 30 3 10 300 30 4 120 12 300 120 30 + 12 462 Multiplying TU x TU 33 x 14 = 462 Use same method to multiply HTU X U

19. 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 Vocabulary lots of groups of share group halve half divide division divided by remainder factor quotient divisible ÷

20. 0 10 20 30 40 50 0 0 5 2 4 10 6 15 20 8 10 25 Count on in tens Count on in twos Count on in fives Know halves Half of 6 is 3 ½ of 6 = 3 If 2 x 10 = 20 then 20  10 = 2 20 2 = 10 Use known multiplication facts to work out corresponding division facts

21. Understand division as sharing Understand division as grouping 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

22. 18  4 = 4 r 2 0 3 6 9 12 15 18 18 divided into groups of 3 18  3 = 6 Represent ‘groups’ for division on a number line using apparatus alongside the line 1 x 3 1 x 3 1 x 3 1 x 3 1 x 3 1 x 3 3 0 6 9 12 15 18 Remember to ask the question – how many groups of 3 in 18? Introduce remainders 1 x 4 r 2 1 x 4 1 x 4 1 x 4 0 4 8 12 16 18 How many groups of 4s in 18? There are 4 groups of 4 with 2 left over.

23. Children need to see that as the numbers get larger, chunking groups is the more efficient method. 65  5 = 13 50 15 10 x 5 3 x 5 0 50 65 How many groups of 5 in 65? There are 13 groups of 5 in 65?