## Understanding fractions as representing multiplicative relationships

What happens when students do not grasp the idea that fractions represent multiplicative relationships?

Suppose a student is asked to explain which fraction is larger, 9/10 or 12/13. This could result in drawing two rectangles, one made up of ten equal parts and the other composed of thirteen equal parts. If the student forms the rectangles drawn to represent each whole by adding on small rectangular pieces, the equality of the wholes is lost, as is the multiplicative relationship.

*An example of an additive approach to representing larger denominators*

An additive interpretation of area typically results in counting units of area rather than subdividing a fixed unit. Adding units of area will result in the whole growing, as above.Adding three parts to ten parts results in an additive part-whole relationship (three plus ten is thirteen). Multiplicative part-whole relationships rely on subdividing a fixed unit (thirteen times the part makes the whole). The idea that fractional parts, as mathematical objects, must refer to same-sized wholes is central to the development of fractions as multiplicative quantities,the comparison of fractions and in particular equivalent fractions.