South Molton C of E Primary School Progression in Calculations

1. x - ÷ + South Molton C of E Primary School Progression in Calculations

2. Introduction The purpose of this policy is to outline and exemplify the strategies and methods that we will teach children to enable them to become highly proficient and skilled in mental and written calculations for the four operations of addition (+), subtraction (-), multiplication (x) and division (÷). It will also show the progression in skills and strategies through the maths curriculum. Written methods of calculations are based on mental strategies and therefore children’s ability to calculate mentally and use a range of known facts is given a high priority. It is vital that children develop a strong and secure conceptual understanding of the four operations. Therefore we use apparatus such as NUMICON, Base 10, Cuisenaire and other relevant resources to support the development of this understanding. We also use the connective model of maths so that children understand the language, the images and models, the strategies and how to apply them for each operation. It is important that children are able to apply the most efficient and suitable method for a given calculation and that they think like mathematicians, choosing and applying the best strategies and estimating prior to calculating. They should also check their calculations using the inverse. Symbols Language Mathematical image/picture Context Figure1.1 The connective model of learning mathematics (adapted by the Devon Primary Maths Team)

3. The Importance of Mental Maths Methods • It is vitally important the all children develop their ability to calculate mentally using all four operations and are able to draw upon their bank of known facts to support them when calculating with larger numbers and decimal numbers. • Learning addition and subtraction facts as well as been able to count on and back in 1s, 10s, 100s, 1000s as well as decimal steps from any start number are crucial in supporting all mental calculations for addition and subtraction. They also support the use of more formal written methods. • Learning times table facts up to 12 x 12 and the associated division facts provides children with the key skills to enable them to multiply and divide both mentally and using written methods. Understanding what happens when a number is multiplied or divided by 10, 100 and 1000, supports their understanding of formal written methods. • Known facts need to be regularly practised so the children can recall them at speed and with accuracy. There is also a strong emphasis on teaching them to utilise these facts to support other calculations.

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 and Images • NUMICON • Cuisenaire • Base 10 • Place value apparatus • Place value cards • Number tracks • Numbered number lines • Marked but blank 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 1, 2, 3, 4, 5, 6 … there are 6 teddies One more than three is four Recognise numbers 0 to 10 Count reliably up to 10 everyday objects Find one more than a number Count in ones and tens Begin to relate addition to combining two groups of objects and makes 5 Count along a number line to add numbers together 3 + 2 = 5 Begin to use the + and = signs to record mental calculations in a number sentence 6 + 4 = 10 Know doubles of numbers 5 5 = 10 +

6. 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 10 and 20 Know that addition can be done in any order 3 +5 Put the biggest number first and count on 4 + 6 = 1 + 9 Understand ‘equals’ sign = as ‘same as’ 4 + 6 = 1 + 9 Begin to partition numbers in order to add

7. 15 +3 +10 15 25 28 40 8 30 6 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 Partitioning and recombining 15 + 13 = 28 Using place value cards and Base 10 to partition numbers and recombine 48 + 36 = 84 40 + 8 30 + 6 70 + 14 = 84

8. Using a number line This is also a mental method which can be done mentally or written down. It doesn’t matter how smaller number is added once it has been partitioned. I.e. It can be added hundreds first or ones. 376 + 258 = 634 + 50 + 200 + 8 376 576 626 634 Standard written method – Column Addition The previous stages reinforce what happens to the numbers when they are added together using more formal written methods. This method can be used for any numbers and decimals. 48 + 36 48 + 36 84 258 + 376 634 1 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 NUMICON Base 10 Cuisenaire 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 Minus Count back Less Fewer Difference between Decrease -

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 the count back in ones from any given number Three teddies take away two teddies leaves one teddy Begin to relate subtraction to ‘ taking 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 Count below the number line when counting back 4 5 6 7 8 9 10

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 45 – 1 = Subtract 1 from a two-digit number -1 43 44 45 Subtract 10 from a two-digit number -10 45 – 10 = 45 35 45 – 30 = Subtract multiples of 10 from any number -10 -10 -10 15 25 35 45

12. 4 3 - 2 7 1 6 4 1 ¹3 - 2 7 1 8 6 3 3 1 Begin to find the difference by counting up from the smallest number. This method is absolutely crucial in developing a good understanding of subtraction. It is also the most efficient mental method. It should be used when numbers are close together. Base 10 apparatus & HTU grids can be used as additional support if necessary 74 - 27 = 47 Now where’s the answer? 403 - 227 = 176 +3 +3 +70 +100 400 403 227 230 300 Standard written method It is important that the children have a good understanding of place value and partitioning before using this method. You don’t “borrow” or “exchange”, you TAKE a ten or hundred. 10

13. 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 and Images NUMICON Cuisenaire 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 40 8 x

14. 2 2 2 2 + + + 0 2 4 6 8 0 5 10 15 20 25 Count in tens from zero 0 10 20 30 40 Count in twos from zero Count in fives from zero Know doubles and corresponding halves Understand multiplication as repeated addition 2 + 2 + 2 + 2 = 8 4 x 2 = 8 2 multiplied by 4 4 lots of 2

15. 2 4 x 4 2 x Know multiplication tables to 12X12 by end of Year 4. x5 Use known facts to work out new ones 2x5=10 3x5=15 6x5=30 8x5=40 10x5=50 Understand that … 24 x 20 = 24 x 2 x 10 24 x 50 = 24 x 5 x 10 Use factors to multiply Understand multiplication as an array Understand how to represent arrays on a number line 4 4 2 hops of 4 2 2 2 2 4 hops of 2

16. 3 30 30 10 300 = 330 + 4 120 12 = 132 462 13 Use place value apparatus to support multiplication 4 4 x 13 = Multiplying TU x TU partitioning the numbers Multiplying U x TU 4 x 13 = X 10 3 4 40 12 = 52 Multiplying TU x TU 14 x 33 Most children will use this method. With this method it is vital that children make an estimate of the answer and can multiply by 10 and 100.

17. Formal Written Methods for Multiplication:Short Multiplication and Long Multiplication

18. 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 NUMICON 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 Group Divide Division Divided by Remainder Factor Quotient Divisible ÷

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

20. 0 3 6 9 12 15 18 Understand division as sharing USING APPARATUS. Understand division as grouping USING APPARATUS. 18 divided into groups of 3 18  3 = 6 Represent ‘groups’ for division on a number line using apparatus alongside the line 18  3 = 6 18  6 = 3 8  2 = 4 4 jumps of 2 Understand division as repeated addition using a horizontal line and apparatus to make the links +2 +2 +2 +2 0 2 4 6 8

21. What facts do I know about the 7 times-table? Children need to see that as the numbers get larger, large chunk addition is the more efficient method. Multiples of the divisor (large chunks) are added. 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 Estimate answer first 40x7 20x7 10x7 4x7 0 280 420 490 518 519  7= 74 r 1 40x7 20x7 10x7 4x7 r 1 0 280 420 490 518 519

22. Formal Written Methods for DivisionShort Division and Long Division