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Developmental Stages in Calculation

Developmental Stages in Calculation. Colehill First School 1st March 2013. Addition and Subtraction. Part 1. Concrete Stage: Addition. Concrete stage: with real objects putting together two sets of objects, with a number sentence: 3 + 2 = 5. Concrete Stage: Subtraction.

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Developmental Stages in Calculation

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  1. Developmental Stages in Calculation Colehill First School 1st March 2013

  2. Addition and Subtraction Part 1

  3. Concrete Stage: Addition • Concrete stage: with real objects putting together two sets of objects, with a number sentence: 3 + 2 = 5

  4. Concrete Stage: Subtraction • Concrete stage: with real objects • taking a set of objects away from a larger set, with number sentence 4 – 2 = 2

  5. Counting Forwards (Addition) and Backwards (Subtraction) • Counting forwards along labelled number tracks or lines, with number sentences: 3 + 4 = 7

  6. Counting Forwards (Addition) and Backwards (Subtraction) • Counting backwards along labelled number tracks or lines, with number sentences: 5 – 2 = 3

  7. Place Value Addition and Subtraction • Place value addition using tens and units: 20 + 3 = 23

  8. Place Value Addition and Subtraction • Place value subtraction using tens and units: 15 – 5 = 10

  9. Addition and Subtraction using the Hundred Square:

  10. Addition By Counting On Using A Blank Number Line: • 47 + 36 = 83 47 50 80 83 47 77 83

  11. Subtraction By Counting Back Using A Blank Number Line: • 37 – 24 = 13 13 20 30 37 13 17 37

  12. Expanded Vertical Layout For addition: For subtraction: 47 + 327 – 116 76 300 20 7 - 13 (7+6) 100 10 6 110 (40 + 70) 200 10 1 123= 211

  13. Subtraction Using The Expanded Vertical Layout With Decomposition: 53 – 28 = 50 3 - 40 13 20 820 8 20 5 = 25

  14. The Compact Written Method • For addition: 47+ 76 123 1

  15. The Compact Written Method • For subtraction: 53 – 28: 4 513 – 2 82 5

  16. Multiplication and Division Part 2

  17. Multiplication • Concrete stage • putting together equal sets, with counting: 1 2 3 4 / 5 6 7 8 / 9 10 11 12

  18. Multiplication • Drawing stage • representing the concrete stage in pictures, with repeated addition: 5 + 5 = 10

  19. Multiplication • Counting forwards in jumps of greater than 1, both mentally and along labelled number tracks or lines, or on the hundred square, with number sentences written as repeated addition: 3 + 3 + 3 = 9

  20. Multiplication • Introducing the multiplication symbol as a shorthand form of recording: 2 + 2 + 2 = 6 or 3 x 2 = 6 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35 or 7 x 5 = 35

  21. Multiplication • Recognising equivalent multiplication, e.g. using arrays: 2 x 4 = 8 * * * * * * * * 4 x 2 = 8 * * * * * * * *

  22. Multiplication • Place Value: Multiplying by 10 and multiples of 10, using 0 as a place holder: • 5 x 10 = 50 • 5 x 100 = 500 • 5 x 1000 = 5000

  23. Multiplication • Multiplication using partitioning: e.g. 38 x 7 = (30 x 7) + (8 x 7) 124 x 6 = (100 x 6) + (20 x 6) + (4 x 6)

  24. Multiplication • Grid Layout (expanded method): 124 x 6 X 100 20 4 6 600 + 120 + 24 =744

  25. Multiplication • Vertical method for multiplication: 38 x 124 x 7 6 210 (30 x 7) 600 (100 x 6) 56 (8 x 7) 120 (20 x 6) 24 ( 4 x 6) 266 744

  26. Multiplication • Multiplication using the compact written method: 38 X 124 X 7 6 266 744 5 1 2

  27. Division • Concrete stage : sharing sharing a set of objects between a group of people: 4 shared between 2 people gives 2 each.

  28. Division • Drawing stage : sharing • representing the concrete stage in pictures, recording using the division symbol. e.g Sharing 6 sweets between 3 friends: Joe Lucy Lee * * * * * * 6 : 3 = 2

  29. Division • Concrete stage: grouping making groups or sets of a certain number: e.g. 15 grouped into 5s gives 3 groups: # # # # # # # # # # # # # # #

  30. Division Division as repeated subtraction: • Counting backwards in jumps of greater than 1 along labelled number tracks or lines, or on hundred squares, with number sentences: 8 -:- 4 = 2

  31. Division • Division as repeated subtraction without the number line, continuing until 0 (or a remainder) is reached, e.g. for 18 -:- 6: 18 – 6 = 12 – 6 = 6 – 6 = 0 I took 6 away three times, and so 18 -:- 6 = 3 And for 20 -:- 6: 20 – 6 = 14 – 6 = 8 – 6 = 2 I took 6 away 3 times, leaving 2 at the end, so: 20 -:- 6 = 3 remainder 2

  32. Division • Division using ‘chunking’ • Chunking means subtracting larger groups, or chunks, of the divisor number. This saves time and reduces the length of the repeated subtraction: e.g. for 72 -:- 6 72 – 60 (10 X 6) 12 – 12 (2 X 6) 0 So, 72 -:- 6 = 12

  33. Division Short formal written method for division: 12 6 72 and with remainders: 20 r 4 or 20.5 8 164

  34. Division • Long formal written method for division: • 78 remainder 3 7 549 49 59 56 3

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