Commutativity

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Commutativity

a property of the addition and multiplication of numbers expressed by the identities a + b = b + a and ab = ba. In a more general sense, the operation a * b is termed commutative if a * b = b *a. Addition and multiplication of polynomials, for example, have the property of commutativity; vector multiplication (see VECTOR PRODUCT) is not commutative since [a,b] = — [b,a].

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This is a very powerful representation of the commutative property without even having to rotate the array

That is 13 x 6 can be considered as (10 x 6) + (3 x 6) or in combination with the commutative property, 6 x 13 can be considered as (6 x 10) + 3 x 10).

The multiplicative situation

The children had already learned the commutative property of multiplication (e.

This approach to learning mathematics led children to a deeper understanding of the meaning of multiplication and its connection to other mathematical ideas, such as the commutative property and its inverse operation.

Meaning, memory, and multiplication: integrating patterns and properties with basic facts

Depending on the student's understanding of basic number concepts when remediation begins, typical development highlights the importance of integrating a foundational focus on number concepts within any approach to intervention, including number-after knowledge, the commutative property of addition, the additive identity property of zero, the notion that a whole can be decomposed into parts in different ways, and the relationship between subtraction and addition.

Second (and more to the heart of the conceptual decomposition strategies condition), tutors conducted conceptual lessons using the number line and blocks to focus on number-after knowledge, the commutative property of addition, the additive identity property of zero, the notion that a whole can be decomposed into parts in different ways, and the relationship between subtraction and addition.

A framework for remediating number combination deficits

Figure 2 shows the result of clicking on 3 after having clicked on 2, illustrating the commutative property of multiplication: 2 x 3 = 6 and 3 x 2 = 6.

Enhancing children's conceptual understanding of mathematics through Chartworld software

Finally, explanations that were based on the commutative property of multiplication were considered MB explanations.

First and second graders use of mathematically-based and practically-based explanations for multiplication with zero

Twelve of the students, having an informal concept of the commutative property, recognised common factors and reduced before multiplying.

Algebra students' difficulty with fractions: an error analysis

students establish equivalence relationships between mathematical expressions, using properties such as the commutative property of multiplication.

Exploring open-ended tasks as teacher learning

She had memorised most facts to 6x10, but seemed unaware of the commutative property of multiplication.

From here to there: the path to computational fluency with multi-digit multiplication: Janette Bobis highlights the critical links between number sense and a child's fluency with mental and written computation when learning how to perform multi-digit multiplication

There were a number of students that demonstrated an understanding of the commutative property of numbers (7 x 8 and 8 x 7, 30 + 26 and 26 + 30).

If it is broken, how can you fix it? Garry Clark describes the use of some simple software as a device to challenge children's calculation strategies