Today, we take our pencils for granted. We don't worry too much if we lose or break one. We constantly sharpen them, and when they grow short and stubby we just throw them away. There was a time when pencils were considered much more valuable items, and more costly - relatively speaking - than they are today. In order to get the maximum use from pencils, right down to the nub, pencil lengtheners, or extenders, were used.
A pencil lengthener is a hollow-ended shaft into which a short stub of a pencil can be inserted. The added length means added life to an otherwise used-up pencil. A pencil stub less than an inch in length could be inserted into its new 'handle', and another mile or two of pencil mark could be extracted from it.
The photo below shows an Eberhard Faber 'Van Dyke' #331 pencil lengthener. The hollow black end has internal threads which grip a standard hexagonal pencil shaft, sans eraser, very tightly, adding about five inches to its length. My unauthoritative best-guess as to its vintage is circa 1940s.