## Level 5: Multiplication and division as operations

Essentially this means that two composite units are coordinated in the operation of multiplication. When multiplication and division are treated as operations, it means that both the number in each unit and the number of units are treated as composites. Instead of re-creating the construction of each row as a composite unit, students can take the items in a column as indicators or markers of row composites. Functioning as symbolic pointers, these markers indicate row composites without requiring re-presentation of the items within those rows. It is, in fact, the use of this indicating function that implements the distribution of one composite over the elements of the other as required in a multiplicative scheme (Steffe, 1992).

10 by 6 array – composites not coordinated

For example, “3 groups of 4 makes 12” or “3 fours are 12”. In solving a partially screened array task such as shown in the following diagram both the ‘four’ and the ‘five’ are treated as composites. The student does not need to use repeated addition but knows that four fives are twenty.

At this level the student can immediately recall, or easily derive, a wide range of multiplication and division facts. In doing this the student uses multiplication and division as inverses of each other in a wide range of contexts and problems.

Linking multiplication and division

A student is able to demonstrate an understanding of coordinating groups beyond simple recall of number facts. For example, a student able to correctly recall 4 x 7 = 28, may not necessarily be able to determining how many groups of 7 are needed to make up 28.

Further, at this level the student is able to work with remainders in division problems and understands the relationship between x and ÷, as well as the structure of the array of multiples and groups. This understanding of the structure of multiplication is important for understanding area and volume.