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Subtraction structures Partitioning and Reduction: Same number line form: different language and meaning. ? 0 25 Partitioning: A quantity/group is split up (partitioned) in some way. -6 There are 25 sweets in a tin. Bob takes 6 sweets. How many are left for Julie? Reduction: The opposite of augmentation. Reduce 25 by 6. Start at 25 and count back 6. Comparison: Using subtraction (or complementary addition) to compare two groups of objects or quantities. Key language: What is the difference? Tank 1 How many more fish are in Tank 1? Tank 2 Subtraction of the smaller number from the larger number will find the difference. However, when comparing (for example when using number lines) pupils may use complementary addition to ‘count up’ to the higher number in order to find the difference. Subtraction as the inverse of addition: 23 -4 19 23 +4 19 1 Subtraction: Number lines Subtraction/Taking away Additional notes and guidance. This policy covers both subtraction/taking away and complementary addition. See the additional guidance in the appendix. Counting back in ones Link to a counting stick, multiples of a given number etc. Include arrows at either end and zero whenever possible to support conversations and a deeper understanding of number. Regular increments/spaces on the number line. More efficient jumps: Bridging through multiples of 5, 10, 100/partitioning etc. Pupils should be encouraged to make informal jottings during number line work. Horizontal jottings/simple recordings/written methods will emerge from number line work. From example: 14 – 4 – 5 – 2 = 3 The move to blank number lines; away from regular increments. Link to partitioning. 2 Complementary addition: KEY WORD = DIFFERENCE What is the difference between 12 and 5 (or 5 and 12)? Counting on in ones Additional notes and guidance. This policy covers both subtraction/taking away and complementary addition. See the additional guidance in the appendix. More efficient jumps: Bridging through multiples of 5, 10, 100/partitioning etc. Finding the difference The move to blank number lines; away from regular increments. Link to partitioning. 64 – 27 = 37 3 Informal written methods. Subtraction/Taking away As pupils move beyond simple number line work – informal methods may emerge. Additional notes and guidance. Subtractive informal, vertical written approaches are best illustrated through vertical number lines. Link to partitioning. (Informal written methods may emerge for complementary addition. Care needs to be taken to emphasise the relationship between these additive approaches and the concept of subtraction/taking away.) 4 Partitioning and partial (subtraction) sums. Additional notes and guidance. Partitioning one number: Progression from horizontal to vertical written methods Moving partitioning and partial subtraction sums into a vertical form: Care needs to be taken to make clear the relationship between the additive elements of these methods (e.g. the use of the ‘+’ sign indicating partitioning) and the concept of subtraction/taking away. Note: No decomposition/exchanging is needed in this example. This partitioning approach can be extended to HTU – TU or HTU – HTU etc. Working RIGHT to LEFT 5 Partitioning with decomposition/exchanging. Additional notes and guidance. Notes: a) Establish that the subtraction can be carried out. i.e. the smaller number can be subtracted from the larger number. b) Ask: “How can [the larger number] be written differently?” This discussion is supported by the language of place value. Working RIGHT to LEFT Note: The word decompose means “to separate into smaller and simpler parts or elements”. The word exchange can be used in both directions. “One ten has been exchanged for ten ones” “Ten units can be exchanged for one ten” 6 Progression to compact vertical subtraction: Formal decomposition Notes: a) Establish that the subtraction can be carried out. i.e. the smaller number can be subtracted from the larger number. b) Ask: “How can [the larger number] be written differently?” This discussion is supported by the language of place value. Additional notes and guidance. Illustrate partitioned and compact methods side-by-side. See demonstration PowerPoint. Working RIGHT to LEFT Note: The word decompose means “to separate into smaller and simpler parts or elements”. The word exchange can be used in both directions. “One ten has been exchanged for ten ones” “Ten units can be exchanged for one ten” Note: We are working RIGHT to LEFT. 7 Similar principles to those above can be applied to money and decimals. For example: Subtracting/taking away: Finding the difference using complementary addition: Compact formal subtraction with decomposition: The difference 8 Appendix ‘Taking away’ vs. ‘Complementary addition’ Note 1: Pupils may not immediately associate “finding the difference” by using complementary addition with “subtracting” or “taking away”. Note 2: Pupils should experience both ‘subtractive’ ‘taking away’ and ‘additive’ ‘complementary addition’ approaches. It would be inefficient for a pupil to use ‘taking away’ on a number line to calculate 91 – 85 and likewise it would be inefficient for a pupil to use complementary addition to calculate 91 – 7, say. (Clearly mental approaches may be preferable in both cases) The concept of ‘difference’ can be the link between the two methods. Context and question format The setting/context for a question and how a question is asked (format/structure) will influence how pupils visualise the problem. This may then influence the informal or formal methods they use. For example: consider the various formats below and how they may influence a pupils’ visualisation and subsequent working. What is 45 take away 6? What is the difference between 45 and 6? What is the difference between 6 and 45? Reduce 45 by 6. A pile of 45 sweet goes down by 6. How many are left? Jane takes 6 sweets from a jar of 45. How many are left for Daniel? Jane and Daniel have a jar of 45 sweets. How many are left for Daniel if Jane eats 6? Jane needs 45 sweets. She has 6 in a packet. How many more sweets does she need? What must be added to 6 to get 45? How many bigger is 45 compared to 6? How many less is 6 compared to 45? All these questions could be written as ‘45 – 6’. As with all calculations finding the numerical answer is only the first step. Understanding and progression is supported when the numerical answer is interpreted correctly within the given context/question. Opportunities should always be taken to make links between the different operations: Example: “The difference between 19 and 7 is 12. What is 7 add 12?” etc. 9