Maths Calculations: Addition and Subtraction Skills and Methods

1. Appleton Wiske Primary School Parents Information Evening Tuesday 14th October 2008 7pm

2. Reasons for recording: • To communicate to others, “to talk maths.” • To record findings • To present information or data for self or others to interpret or use • To describe or explain methods • To record answers • To help work out a calculation by recording interim steps • To help solve a problem by sketching a diagram • To help organise information so that it is easy to interpret • To help remember key facts and ideas

3. MATHS CALCULATIONS • Development and progression of skills • How we teach these skills • Real life contexts • Assessment for Learning – asking questions • Talk maths – Is that bigger? Why?

4. ADDITION

5. To add successfully, children need to be able to: • Recall all addition pairs to 9+9 • Recall bonds to 10 • Add mentally a series of single digit numbers e.g. 5+8+4 • Add multiples of 10 e.g. 70+60 (using their knowledge of related addition facts (7+6) and knowledge of place value) • Add multiples of 100 e.g. 700+600 • Partition two and three digit numbers into multiples of one hundred, ten and one in different ways. • *It is important that mental calculations are practised and secured alongside written methods.* *It is important that mental calculations are practised and secured alongside written methods.*

6. Addition Vocabulary • And • Make • More • Altogether • Add • Plus • Sum • Total • Addition

7. Early Recordings At this stage children begin to record their work using pictures, symbols and numerals • Children can record in a variety of ways and generally the recording is very informal using non-standard methods. • Example:  Makes 4

8. . Early Recordings . 5 4

9. The Full Number Line • The children can start to count on using a numbered line. 0 1 2 3 4 5 6 7 8 9 10 Start on2. Count on 4 more. Makes 6.

10. Horizontal Method • Children begin to use and understand the + and = signs to record mental calculations in a number sentence. 5+2=7

11. The Empty Number Line • The mental methods that lead to column addition generally involve partitioning, e.g. adding the tens and ones separately, often starting with the tens. • Children need to be able to partition numbers in ways other than in to tens and ones • This helps them make multiples of ten by adding in steps. • The empty number line helps to record the steps on the way to calculating the total. • 8+7=15+2 +5 8 10 15  • Steps in addition can be recorded on a number line. The steps often bridge through a multiple of ten. 

12. The Empty Number Line • 48+36=84  • +30 +2 +4  • 48 78 80 84

13. Partitioning • The next stage is to record mental methods using partitioning. • Adding the most significant digit first or adding the least significant digit first. • Examples:Record steps in addition using partitioning. • 76+47= • 70+40= 110 *Add the tens digits • 6+7=13 *Add the units digits • 110+13=123 *Add the two totals • 76+7=83 *Add one number and the unit of the other • 83+40=123 *Add the tens that are left to the new total

14. Expanded Method in Columns -an Interim Step in Understanding Addition • Move on to a layout showing separately the addition of the tens to the tens and the units to the units. • Write the numbers in columns: • Adding the tens first Adding the units first. • 76 76 • + 47+47 • 110 13 • 13110 • 123 123 • Adding the tens first gives the same answer as adding the units first.

15. Expanded Method in Columns • The addition of the tens in the calculation 76+47 is described in the words “seventy plus forty equals one hundred and ten.” • The expanded method leads children to the more compact standard method so that they understand the structure and efficiency of the standard method. • The amount of time children spend on this method depends on how secure they are in their recall of number facts and their understanding of place value.

16. The Standard Method • In the standard method, recording is reduced further. Carry digits are recorded below the answer line, using the words “carry ten” or “carry one hundred”, not “carry one”. • Extend to adding three two- digit numbers, two three- digit numbers and numbers with different numbers of digits. 76 458 +47+366 123824 11 11 • The standard method of column addition is efficient when used with larger whole numbers and decimals. • Once learned and understood the method is quick and reliable.

17. Decimals • £4.21 + £3.87 72.5km + 54.6km   4.21 72.5 +3.87+54.6 8.08127.1 1 1 1 £8.08 127.1km

18. Grading of Difficulty for + Carrying units to tens 47 +25 237 +516 Carrying tens to hundreds 371 +485 541 +293 No carrying 23 +42 624 +315 Extra digit in answer 94 +73 561 +718

19.  More than two numbers to be added 35 62 +24 237 148 +516 Different numbers of digits 24 375 + 48 Carrying units to tens and tens to hundreds 376 +485 547 +295

20. SUBTRACTION

21. Subtraction To subtract successfully, children need to be able to: • Recall all addition and subtraction facts to 20; • Subtract multiples of 10 e.g. 160-70=90 using a subtraction fact • 16-7=9 and their knowledge of place value; • Partition two digit and three digit numbers in to multiples of one • hundred,ten and one in different ways e.g. partition 74 into 70+4 NB: It is important that children’s mental methods of calculation are practised and secured alongside their learning and use of an efficient written method for subtraction.

22. Stage 1: Early RecordingsStage 1: Early Recording At this stage children begin to record their work using pictures, symbols and numerals. Children can record in a variety of ways and generally the recording is very informal using non-standard methods. Example: Take away 1 car. How many have you got left?

23. Early Recordings Example Counting on (finding the difference)

24. Stage 2 The Number Line The children can subtract by counting on using a numbered line 1 2 3 4 5 6 7 8

25. Stage 3: Horizontal MethodChildren begin to use and understand the – and = signs to record mental calculations in a number sentence. When subtracting within 10 or 20 children may record using simple subtraction sentences e.g.5-4=1 10-7=3 17-4=3

26. Stage 4: The Empty Number LineThe empty number line helps to record or explain the steps in mental subtraction.Example: Counting on(finding the difference)84-56= 28+4 +20 +4____________________________56 60 80 84Answer= 28

27. Stage 4 Cont.Counting back84-56=28 -50 -6 84 34 28 Counting on783-356=427 +4 +40 +300 +83________________________________________356 360 400 700 783

28. Stage 5: Partitioning • Subtraction can be recorded using partitioning to write equivalent calculations that can be carried out mentally. • Example • 84-56= • 84-50=34 • 34-6=28

29. Stage 6: Standard Column Subtraction (starting with 2 digits) without decomposition • Partitioning the numbers into tens and units and writing one under the other with the answers being placed under each column you are working on. • The language of subtraction is critical to aide children’s understanding. • “8 units take away 6 units equals 2 units. • 8 tens take 5 tens equals 3 tens” Example 88 - 56 32 657 - 526 1 3 1

30. Stage 7: Standard Column Subtraction with Decomposition • “Three take six we can’t do it. • Exchange one ten from eight tens leaves me with seven tens. • Thirteen take six equals seven. • Seven tens or seventy take away five tens or fifty leaves me with two tens or twenty. • Seven hundred take away three hundred leaves me with four hundred. • Answer equals four hundred and twenty seven.” Example 7 1 783 - 356 427

31. Stage 3-6: Decimals Example £8.95 - £4.38 72.5m – 4.6m

32. Grading of Difficulty for –This is an analysis of the difficulty of the calculation and is independent of the method used. Similar calculations should be done mentally.

33. Continued

34. Minus SubtractCounting on/back FewerLess than Take awayDifference between Vocabulary

35. 40 8 • Progression in Teaching Subtraction • Mental Skills • Recognise the size and position of numbers • Count back in ones and tens • Know number bonds to 10, 20, 100, 1 • Subtract multiples of 10/100 from any number • Partition and recombine numbers (only partition the number to be subtracted) • Bridge through 10 • Calculate halves of numbers • Models and Images • Counting and sorting equipment • Deans • Place value cards • 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 -

36. 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 or hundreds If I take away four shells there are six left Count backwards along a number line to ‘ take away

37. 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 15 - 7 = 8 Subtract single digit numbers often bridging through 10 Begin to find the difference by counting up from the smallest number Begin to partition numbers in order to take away

38. 43 – 23 -10 -10 20 23 33 43 25 35 45 43 – 43 – 20 = 23 23 – 3 = 20 20 3 - 3 - 10 - 10 -1 Subtract 1 from a two-digit number 45 - 1 44 45 Subtract 10 from a two-digit number -10 45 - 10 35 45 Subtract multiples of 10 from any number 45 - 20 Partition the number to be subtracted (no exchanging) Decide whether to count on or count back 74 - 27 = 47 Now what’s the answer?

39. 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 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. 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.

41. MULTIPLICATION

42. To multiply successfully, children need to be able to: • Recall all multiplication facts to 10x10: • Partition numbers into multiples of one hundred, ten and one: • Work out products such as 70x5, 70x50, 700x5 or 700x50 using the related fact 7x5 and their knowledge of place value: • Add two or more single digit numbers mentally: • Add multiples of ten e.g. 60 + 70 or of 100 e.g. 600 + 700 using the related addition fact 6+7 and their knowledge of place value: • Add combinations of whole numbers using the column method.

43. Early recording – Introduced in Yr 2 Three sets of two Three groups of two Three lots of two Three times two

44. Horizontal Method • Understand 4x2=2+2+2+2 • Use sets, drawing ** ** ** ** • Represent repeated addition and arrays as multiplication

45. Mental Multiplication Using Partitioning • Informal recording from Year Four might be • 38x7=(30x7)+(8x7) • = 210+56 • =266

46. Grid layout –expanded method • X 30 8 • 7 210 56 =266 • Constantly using knowledge of multiplying and dividing by 10 • Knowledge of times tables vital – feeds into so many other areas of maths

47. Extended grid layout • 56x27 • X 50 6 • 20 1000 120 1120 • 7 350 42 +392 • = 1512 • This method of multiplication is accepted in comprehensive schools up to Yr 9 and usually is a child’s preferred method.

48. Vertical Format Expanded Working 38 X 7 56 (8x7) + 210 (30x7) 266 • 56 • X 27 • 42 (7x6) • 350 (7x50) • 120 (20x6) • + 1000 (20x50) • 1512

49. Decimals • Pupils progress to calculating with decimals e.g. 4.9 x6 and then 4.29x6 • Use either grid method or vertical format

50. DIVISION