It seems your student didn’t understand fractions, nor that the expression could be interpreted as division, and especially had trouble thinking in general terms. You asked for “a good approach to teaching this.” You got an inductive approach to generalize, answers focused on the meaning of fractions and division, and one on powers.
Here is an approach to generalization based on meaning, not inductive reasoning. It is from Davydov’s implementation of Vygotsky’s theory, references below. The approach is to start with unspecified quantities that are concrete and general, such as two line segments or two strips of wood not of equal length. Use the shorter segment as a basic unit to measure the longer segment. How many of the shorter unit fits into the longer segment? Say that the longer one is exactly 3 times the shorter. Then the answer is 3. Label the shorter line as “U” for the basic unit, and the longer as “Q” for the quantity measured. The action of measuring can be expressed as Q/U, which is measurement division: how many U’s in Q? Then for these segments we can write Q/U=3. (First graders actually get this, although it is developed in many steps.)
Then make Q the same length as U. How many U’s in Q--clearly only one. Now Q/U=1. Since Q=U in this case, we don’t need two different letters. We can write it as U/U=1. How many units in one unit? It is one unit. Now go from the general to the specific to enrich and clarify the meaning. If U is 5 then 5/5=1. There is one 5 in 5, and so on with other numbers.
I used a length of 3 at first because it is easier to understand the meaning of the measurement if the units are not the same. After establishing the measurement and division expression, then move to equal lengths. I wouldn’t think that a high school student would have a problem understanding this.
Vygotsky’s approach is to start with the general and “ascend to the specific.” The emphasis is on understanding relationships, then doing computation. For example, the additive relationships of A + B = C, C – B = A, and C – A = B are thoroughly examined in concrete situations, then symbols, before problems are given. The initial tasks are to understand the relationships. Then problems are created by making one of the three elements the unknown.
Moxhay, P. (2008, September). “Assessing the scientific concept of number in primary school children.” Paper presented at the International Society for Cultural and Activity Research conference, San Diego.
Schmittau, J. and Morris, A. (2004). “The Development of Algebra in the Elementary Mathematics Curriculum of V.V. Davydov” in The Mathematics Educator, Vol.8, No.1, 60 – 87.