Root extraction, part II: cube roots

As you might guess, this post builds on “Root extraction, part I“, which gave a way to visualize the traditional square root algorithm geometrically, an approach that has the advantage that each step appears natural and easily motivated.

Our goal herein is to do much the same for cube roots. The point is to find a geometric construction, ideally one well-suited to physical manipulatives, in which the steps in building the successive digits of the cube root of a number are transparent.  As with the post on square roots, I make no claims to originality in what follows.

Example:  Find . 

Root extraction, part I: square roots

I recently discussed the traditional algorithms for computing square and cube roots in my History of Math class.   Our reading, on mathematics in ancient China, gave both algorithms as a set of rules for manipulating number rods.  For me, it was fascinating to see past the text: the rules as given would transfer directly to an abacus/soroban calculation, and were essentially the same as the rules that prior generations of American schoolchildren would have been drilled on in school.

My students (mostly high school math teachers) found the book’s explanation of the method obscure;  the key is to view the process geometrically, rather than as a mechanical set of rules for manipulating digits.

I make no claim of originality in what follows; I offer it here in part because I can’t find any lucent discussions along these lines on the web.