Discovering a simple fraction pattern during my lesson planning

One of the joys of being a K-12 mathematics teacher is revisiting old mathematics and seeking to see concepts in a new light. Teaching requires this type of stuff.

For instance, could you consider the table above? I think of obtaining the successive decimal fractions by dividing the previous decimal by 2. But division, at least to me, is often more cumbersome than multiplication--irrespective of the tight bond between the two.

Here is the pattern that pops out at me, as I am here in the early A.M. planning my lesson: after we obtain 1/2 = 0.5 and 1/4 = 0.25 as benchmarks, we simply do the following--

Think of 1/2^n, for integer n >1, as always ending in 25. We gain the other digits between the decimal point and the 25 as follows.

We need to know that 1/8 = 0.125. From there, we can almost mentally halve the number or do a simple whole-number multiplication with a pencil and paper.

Knowing that 1/8 = 0.125, we focus on the digit between 0 and 25, which is 1. We take the number D formed by this digit, multiply D by 5, and add 1. We have 5D + 1, which gives us the digits I will represent as [5D + 1].

So, 1/2^n for n > 3 has the form 1/([0...0][5D + 1]25), where the denominator isn't a multiplication expression but rather an expression of digits. Only the expression 5D is representative of multiplication.

So, now, here we have it:

1/16 is derived from 1/8, and 1/32 is derived from 1/16, and 1/64 is derived from 1/32, in the algorithm outlined above.

Given 1/8 = 0.125, we have that 1/16 = 0.[0][5*1 + 1]25 = 0.[0][6]25 = 0.0625;

given 1/32 = 1/[0][5*6 + 1]25 = 0.0[31]25 = 0.03125;

given 1/32 = 0.03125, we have that 1/64 = 0.0[5*31 + 1]25 = 0.015625, etc.

We need to keep track of the number of digits between the decimal point and the 5 to ensure that each successive decimal in its simplest form has one more digit than the previous one.

Patterns abound in school mathematics! They are immune to curriculum changes!