making sense of algorithms for multidigit multiplication and division juli k. dixon, ph.d....
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Making Sense of Making Sense of Algorithms for Algorithms for
Multidigit Multidigit Multiplication and Multiplication and
DivisionDivision
Juli K. Dixon, Ph.D.Juli K. Dixon, Ph.D.
University of Central University of Central FloridaFlorida
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
Making Sense of Making Sense of MultiplicationMultiplication
Mutliply 4 x 27 using Mutliply 4 x 27 using strategies based on strategies based on multiplication rather than multiplication rather than division.division.
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
How might you solve this with mental computation?
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
Solve it with:Solve it with:
Invented StrategiesInvented Strategies
Making Sense of Making Sense of MultiplicationMultiplication
How might students invent How might students invent strategies for 12 x 29?strategies for 12 x 29?
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
Solve it with:Solve it with:
Invented StrategiesInvented Strategies
Base-ten BlocksBase-ten Blocks
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
Solve it with:Solve it with:
Invented StrategiesInvented Strategies
Base-ten BlocksBase-ten Blocks
Partial ProductsPartial Products
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
Solve it with:Solve it with:
Invented StrategiesInvented Strategies
Base-ten BlocksBase-ten Blocks
Partial ProductsPartial Products
How does it help us prepare for Algebra?
“Say, I think I see where we went off. Isn’t eight times seven fifty-six?”
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
Solve it with:Solve it with:
Invented StrategiesInvented Strategies
Base-ten BlocksBase-ten Blocks
Partial ProductsPartial Products
How does it help us prepare for Algebra?
Consider (x + 2)(x + 5)
Making Sense of Making Sense of MultiplicationMultiplication
Now Consider 12 x 15Now Consider 12 x 15
Solve it with:Solve it with:
Invented StrategiesInvented Strategies
Base-ten BlocksBase-ten Blocks
Partial ProductsPartial Products
Traditional AlgorithmTraditional Algorithm
Define, “demonstrating Define, “demonstrating understanding of the standard understanding of the standard algorithm.”algorithm.”
The context is important. The context is important. Consider a “sharing” problem.Consider a “sharing” problem.
Making Sense of Making Sense of DivisionDivision
Consider 532 ÷ 14. How Consider 532 ÷ 14. How can we make sense of can we make sense of this in a measurement this in a measurement context?context?
Making Sense of Making Sense of DivisionDivision
Consider 532 ÷ 14. How Consider 532 ÷ 14. How can we make sense of can we make sense of this in a measurement this in a measurement context?context?
Repeated subtraction takes too long.
Making Sense of Making Sense of DivisionDivision
Consider 532 ÷ 14. How Consider 532 ÷ 14. How can we make sense of can we make sense of this in a measurement this in a measurement context?context?
Repeated subtraction takes too long.Consider Partial Quotients.
Making Sense of Making Sense of DivisionDivision
Consider 532 ÷ 14. How Consider 532 ÷ 14. How can we make sense of can we make sense of this in a measurement this in a measurement context?context?
Repeated subtraction takes too long.Consider Partial Quotients.
But what happens when we get to decimals?