For example, a student who can recognise that 3/7 is equivalent to 6/14, which is less than one-half (7/14), will be in a strong position to both create an appropriate proper fraction
and successfully challenge opponent plays.
Simply that proper fractions
with N as denominator, where N has factors other than 2 or 5, will produce repeating sequences of length (N - 1) or less.
In Lessons 1 through 15, the focus was on proper fractions
and improper fractions equal to 1.
Locating proper fractions
on number lines: Effect of length and equivalence.
Initially the student should start working with dividing whole numbers by proper fractions
to develop the notion that division does not always yield a quotient that is smaller than the dividend.
Teachers will need to decide whether or not to specify that only proper fractions
--or perhaps proper fractions
plus improper fractions equivalent to 1--"count" for some of these activities.