| Progress
Map |
| Measurement
Model of Addition |
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- Linear
measure is additive for mixed
number measures.
- Models
addition as iteration of unit-measures,
e.g., iterates 2 + 3 by starting
at 2, then moving 3 for a result
of 5. Different expressions imply
differences in iteration.
- Invented
algorithms for addition, especially
decomposition strategies.
- Addition
is a joining of sets, including
direct modeling, counting, derived
facts.
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| Measurement
Model of Subtraction |
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- Subtraction
results in a measured difference
between mixed numbers.
- Subtraction
is viewed as resulting in a measured
difference between whole number
measures.
- Invented
algorithms for subtraction, especially
decomposition that employ place-value
strategies.
- Subtraction
is separation of sets, including
direct modeling, counting, derived
facts.
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| Measurement
Model of Multiplication |
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- Multiplication
by fraction (m/n) results in m
copies of n splits of a unit of
measure.
- Reversible
reasoning about total measured
quantity and unit-of-measure.
Unit of measure is 1/n times as
long as the total quantity. Total
measure is n times a unit-measure.
- Multiplication
by unit fractions (1/n) results
in 1 copy of n splits of a unit
measure. (1/4 is 1 copy of a unit-name
that is split into 4 congruent
parts)
- Multiplication
is viewed as n copies of units
of measure m. The result is a
measured quantity.
- Multiplication
is repeated addition.
- Multiplication
is computation. No other interpretation.
A way of getting results.
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| Measurement
Model of Division |
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- Division
of fractions is viewed as rescaling:
m/n / x/y results in the measure
of m/n in units of x/y.
- Equivalent
fractions are of equal measure.
- Division
is viewed as rescaling a measure,
so that m/n results in the measure
of m in units of n.
- Division
(m/n) is viewed as repeated subtraction
(m-n)
- Division
is viewed as a computation. No
other interpretation.
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