Etymology
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grandmother (n.)
early 15c., from grand- + mother (n.1), probably on analogy of French grand-mère. Replaced earlier grandame (c. 1200) and Old English ealdemodor.
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gran (1)
childish abbreviation of grandmother or granny, 1863.
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grandma (n.)
1793, shortening of grandmama (1749), childish or familiar form of grandmother (see grand- + mama).
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gammer (n.)
"old woman," 1570s, contraction of grandmother (corresponding to gaffer, but according to OED representing a different construction).
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granny (n.)
also grannie, 1660s, according to OED, most likely a diminutive and contraction of grannam, shortened form of grandame, rather than from grandmother. The sailor's granny knot (by 1803), originally granny's knot, readily jammed and insecure, is a reef or square knot with the second part crossed the wrong way, so called in contempt because "it is the natural knot tied by women or landsmen" [Smyth, "Sailor's Word-Book," 1867]. Granny Smith apples (1895) are said to have been named for Maria Ann Smith (d. 1870) of Australia, who originated them. Granny glasses attested from 1966.
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babushka (n.)
type of head covering for women, 1938, from Russian babushka "grandmother."
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nana 

child's word for "grandmother" or, sometimes, "nurse," 1844 (see nanny).

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ayah (n.)
"native nurse, children's governess," Anglo-Indian, 1782, from Portuguese aia, cognate with Spanish aya, Italian aja, etc., "nurse," from Latin avia "grandmother," fem. of avus "grandfather" (see uncle).
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