Number & Operations for Teachers

Understanding the Standard Fraction Division Algorithm

According to the standard “invert and multiply” algorithm, the quotient a/b ¸ c/d is computed as

What is the logical basis for this algorithm? In terms of the following the operation 4/5 ¸ 1/3,

· The dividend 4/5 partitions the unit square/rectangle into 5 columns, four of which are shaded. In general, a dividend a/b is modeled by partitioning the unit square/rectangle into b columns then shading a of them.

· The divisor 1/3 partitions the unit square/rectangle into 3 rows, one of which is shaded. In general, a divisor c/d is modeled by partitioning the unit square/rectangle into d rows then shading c of them.

· As a result of this process, the unit square/rectangle in Example 5.1 is partitioned into fifteen cells. In general, the unit square/rectangle is modeled using bd cells.

· Into how many cells is the dividend partitioned? In Example 5.1, the dividend is modeled using 3x4 = 12 cells. In general, the dividend is modeled using ad cells, since the dividend is a columns wide and d rows deep.

· Into how many cells is the divisor partitioned? In Example 5.1, the divisor is modeled using five cells. In general, the divisor is modeled using bc cells, since the divisor is b columns wide and c rows deep.

· Consequently, the ratio of dividend to divisor, in terms of cells, is given by ad/bc. This is the same result obtained using the traditional “invert and multiply” algorithm.

A different argument based on algebraic manipulation proceeds as follows:

· If both the numerator and denominator of the complex fraction are multiplied by the same non-zero amount, the value of the expression will be unchanged.

· Multiply both numerator and denominator by d/c, the inverse of the denominator,

· Then