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Problem: If I planted 4 rows of strawberry plants and put 18 plants in each row how many plants would I have altogether? Doubling and Halving Double 4 is 8 and half of 18 is 9 so 8 x 9 = 72 Trebling and Thirding Triple 4 is 12 and a third of 18 is 6 so 12 x 6 = 72 Problem: If I planted 4 rows of strawberry plants and put 18 plants in each row how many plants would I have altogether? Rounding and Compensating 20 x 4 = 80. However that’s 2 plants too many in each row so I need to take away 4 x 2 which is 8, so 80 – 8 = 72 Problem: If I planted 4 rows of strawberry plants and put 18 plants in each row how many plants would I have altogether? Place Value Partitioning 18 x 4 is the same as 10 x 4 = 40 + 8 x 4 = 32 so 40 + 32 = 72 Problem: If I wanted to buy 72 cans of soft drink how many 6 packs would I need to buy? Reversibility 12 x 6 is 72 therefore 72 ÷ 6 = 12 Problem: If I wanted to buy 72 cans of soft drink how many 6 packs would I need to buy? Proportional Adjustment Half of 72 is 36 and half of 6 is 3 so 36 ÷ 3 = 12. A third of 72 is 24 and a third of 6 is 2. 24 ÷ 2 = 12. Problem: If I wanted to buy 72 cans of soft drink how many 6 packs would I need to buy? Divisibility I know 60 is a multiple of 6 and so is 12 so 10 + 2 = 12. Problem: I cut 24 metres of material into lengths of 0.75m. How many pieces do I have? Conversion and Commutativity 24 ÷ ¾ = so ¾ x = 24? If 24 is ¾ then ¼ must be 8. The answer is 8 x 4 = 32 pieces. Problem: I cut 24 metres of material into lengths of 0.75m. How many pieces do I have? Doubling and Halving with Place Value 24 ÷ 0.75 is the same as 48 ÷ 1.5 which is the same as 96 ÷ 3 = 32. Commutativity Commutative operations in everyday life: Putting on shoes resembles a commutative operation, since which shoe is put on first is unimportant. Either way, the end result (having both shoes on), is the same. The commutativity of addition is observed when paying for an item with cash. Regardless of the order in which the notes or coins are given, they always give the same total. Addition: 7 + 4 is the same as 4 + 7 Multiplication: 8 x 3 is the same as 3 x 8 Reversibility Opposite operations are reversible: Division ( ÷ ) is the opposite of multiplication ( × ). Subtraction ( − ) is the opposite of addition ( + ). 15 ÷ 3 = 5 can be reversed to 5 × 3 = 15 12 + 7 = 19 can be reversed to 19 − 7 = 12