Shabbos, Chapter 17, Mishnah 5. The Mishnah lays down a general rule: "Kol ha-kelim ha-nitalin b'Shabbos - shivreihen nitalin b'Shabbos" - any utensil that may be carried on Shabbos, its broken pieces may also be carried once it has broken. The permission, however, comes with a condition: "u'vilvad she'yehu osin me'ein melachah" - the fragments must still be fit to serve some function.
The examples the Mishnah brings:
"Shivrei areivah" - a kneading trough used for dough that broke, whose pieces are fit "l'chasos bahen es pi he'chavis" - to cover the mouth of a barrel with them. This is not the original function, but it is a real function, and therefore the pieces are not considered muktzeh.
"Shivrei zechuchis" - fragments of a glass vessel that are fit "l'chasos bahen es pi ha'pach" - to cover the mouth of a flask, an opening smaller than the first, and they too may be carried.
The principle in both examples is the same: the intact vessel served as a container, while the fragment serves as a cover, and nevertheless it is enough that it fills some function in order to permit carrying it.
The view of Rabbi Yehudah:
Rabbi Yehudah disagrees and rules more strictly: "u'vilvad she'yehu osin me'ein melachtan" - some function is not enough; the fragment must continue to serve something of its original work and purpose:
Fragments of a kneading trough - they must be fit "litzok l'sochan mikpah" - to pour porridge into them, so that they continue to serve as a container.
Fragments of glass - they must be fit "litzok l'sochan shemen" - to pour oil into them, and this use is sufficient.
In summary: in this Mishnah the Tanna Kamma and Rabbi Yehudah disagree about the scope of the permission to carry broken vessel pieces on Shabbos. According to the Tanna Kamma any use at all is enough, even though it is not the original function: fragments of a kneading trough to cover the mouth of a barrel, and fragments of glass to cover the mouth of a flask. According to Rabbi Yehudah the use must resemble the original work: the fragments must be fit for pouring porridge or oil into them, that is, they must continue to serve as a container.