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February 2013 Coaches Meeting

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Fluently add and subtract within ____ using strategies and algorithms (3 rd and 4 th grade) based on place value, properties of operations, and/or relationship between addition and subtraction.

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Solve 7 + 52 +186 without paper and pencil. Private think time Share with your shoulder partner how you arrived at your answer.

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AlgorithmsSome algorithms taught at home No algorithms 17 students19 students20 students 12% correct26 % correct45% correct Range of answers 9308-29 Range of answers 989-114 Range of answers 617-139 Research from Constance Kamii. Young Children Reinvent Arithmetic: Implications of Piaget’s Theory, Columbia University, 2000

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Flexible methods of computation involve taking apart and combining numbers in a variety of ways. Invented strategies are flexible methods of computing that vary with the numbers and the situation. Flexible methods for computation require a good understanding of the operations and the properties of operations. The traditional algorithms are clever strategies for computing that have been developed over time. They work for all numbers but are often far from the most efficient or useful method of computing.

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Counts by ones Use of base-ten models Direct Modeling Supported by written recordings Mental methods Invented Strategies Usually requires guided development Traditional Algorithms

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Students make fewer errors. Less re-teaching is required. Students develop number sense. Invented strategies are the basis for mental computation and estimation. Flexible methods are often faster than the traditional algorithms. Algorithms invention is itself a significantly important process of “doing mathematics.”

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Invented strategies are number oriented rather than digit oriented. 45 + 32 think of 40 + 30 rather than 4 +3 Invented strategies are left-handed rather than right handed. Invented strategies are flexible rather than rigid. Try each mentally 465 + 230 and 526 + 98 Did you use the same method?

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Students will not invent the traditional methods because they are right-handed methods and are simply not natural. This means that the teacher must introduce and explain each algorithm. Students will sense that they are the “right way” or “real way” to compute and will abandon any flexible method they have been developing. They will happen.

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Add Tens, Add Ones and Combine 46 + 38 40 + 30 is 70 6 and 8 is 14 70 and 14 is 84 Move Some to Make Tens 46 + 38 Take 2 from the 46 and put it with the 38 to make a 40. Now you have 44 and 40 more is 84.

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Take Tens from Tens, Then Subtract Ones 73-46 70 minus 40 is 30 Take away 6 more is 24 Now add in the 3 ones 27 Take Away Tens, Then Ones 73 – 46 73 minus 40 is 33 Then take way 6 3 makes 30 and 3 more is 27

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Yesterday there were 57 penguins sitting on the iceberg. Later 34 penguins joined them. How many penguins are now on the iceberg? There were 72 penguins sitting on the iceberg. 49 penguins jumped into the icy water. How many penguins are still on the iceberg ?

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