Task# Decimal Expansions of Fractions

Illustrative MathematicsVaries

The goal of this task is to convert some fractions to decimals and then make conjectures about which fractions have terminating decimal expansions (as well as the length of those decimals). The teacher may wish to focus more on the conversion process and less on identifying which fractions have terminating or repeating decimal representations. In this case, only part (a) is needed and the key is that all remainders which appear in the long division algorithm are less than the divisor: this means that the decimal must either terminate (when there is a remainder of 0) or repeat (when we see the same non-zero remainder twice). The other parts of this task are looking forward to an additional level of structure, namely identifying which fractions have terminating decimals and which have repeating decimals.