Dividing fractions inquiry

The prompt

Mathematical inquiry processes: Reason; extend to other cases; generalise. Conceptual field of inquiry: Division of fractions; reciprocals.

The prompt introduces students to dividing a fraction by another fraction. The statement can be seen as answering two questions:

The use of a mixed number in the first part of the prompt helps students understand the division of fractions more clearly, particularly when the teacher gives them a diagram like the one below through which they can visualise the statements. Using the improper fraction has led to confusion in which students attempt to spot a superficial pattern between threes and fours.

Once students have a conceptual understanding, the teacher could introduce the equivalence of one and a third and four-thirds, which, in turn, could lead to a discussion of reciprocals.

The inquiry can be extended into other statements, such as 

Students have been enthusiastic to extend their understanding beyond unit fractions by changing the dividend and divisor. Initially they might continue to consider thirds and quarters (see example below) before using bars split, for example, into 15 parts for thirds and fifths.