Coordinating units at different levels
Coordinating units is an important activity in multiplication and in dealing with fractions. Initially students can only deal with one level of units. They may identify the number of parts in a group or the number of groups but they cannot simultaneously coordinate both units. When students have units at only one level these units are in effect parts rather than parts related multiplicatively to the whole.
Students who can multiplicatively coordinate units at two levels with fractions recognise how to iterate or repeatedly create unit fractions, such as one-third, to form a whole. In anticipating that they can do this they are effectively simultaneously coordinating units at the level of the part (one-third) and at the level of the whole (three-thirds).
Coordinating three levels of units is involved in understanding the construction of improper fractions. A student who conceives of five-quarters as an improper fraction appreciates the quantity is made up of five units, any one of which can repeatedly be produced. Moreover, he or she recognises that four quarters make one whole and the fifth quarter goes beyond one whole. Students who have only two levels of units may operate with fractions greater than one, but they don’t produce improper fractions. Typically, if they produce five quarters they will re-label them as five fifths.
One from one-and-a-half