Fewer know that the physical Torah is called a “Sefer Torah” — a “book” or “scroll” of Torah. “Torah” itself refers to the content, which appears as the first five books in Jewish scriptures (popularly known as “Genesis,” “Exodus,” “Leviticus,” “Numbers,” and “Deuteronomy”).
In fact, these are the first five books in every biblical tradition. While various Christian traditions differ regarding which books are included and in which order, all agree with each other and with Judaism about the first five books.
In Judaism, for learning purposes, Torah gradually became divided into weekly readings. Each one is termed a “parshah” or “sidrah.” A “sidrah” is an order, arrangement, row, etc; the noun being derived from the Hebrew verb sadar/סדר –– to arrange, to put in order.
Chapter/verse numbers were added in the Middle Ages in Europe by Christian scholars. Later adopted by Jewish scholars, it’s still the preferred Jewish practice to quote the words of a source and indicate chapter/verse numbers in a footnote. The weekly divisions are standardized (with some variations) in the Jewish world at this time, although this was not always so.
Each “parshah” or “sidrah” has its own title, usually based on a word found in the first sentence. These divisions aren’t seen in the written text of Torah, but appear in printed books called a “chumash.” A chumash is used for private reading and study, and to follow the public reading in synagogue services.
Torah is typically read publicly on Mondays, Thursdays and Saturdays. On Saturday/Shabbat, Torah is read in the morning service; later, a selection from the next following sidrah is read in the afternoon service, which is then re-read in the Monday and Thursday morning services.
Public reading of a Sefer Torah in synagogue is called the “Kriyat ha-Torah.” “Kriyah” comes from the Hebrew verb “ka’ra,” which means “to recite” or “to read aloud.” Elsewhere, it can mean “to call.” For example, “G-d called [Va’yik’ra] to Mosheh…” . In either case, it clearly refers to something audible.
Public reading of Scripture is a practice that began with Judaism and extended into Christianity, Islam and elsewhere. In other traditions, knowledge of scripture was confined to priests and others specifically trained to read and understand it. Protestant tradition veered radically in the opposite direction, holding that everyone should read scripture on their own.
In the mid-80’s, for example, I visited a group in Brooklyn. I remember them as a Sikh group named “3HO,” but when I looked up “3HO” online, it’s a kundalini yoga group, so my memory might be off here. In any case, they recited their scriptures in an annual, unbroken 3-day period. Even if no one is in the room but the reader, the scriptures are recited, day and night. As requested of me, I removed my shoes, covered my head, sat on the floor and listened intently. I don’t remember clearly whether the readings were in English or another language; English, I think. I also don’t remember anything surprising musically; just simple, practical chant. While I was there, one reader replaced another — without a pause in the reading. It was quite fascinating. This kind of public reading has its ultimate source in Jewish practice.
Similar to the Hebrew “ka’ra/kriyah,” “Qur’an” comes from an Arabic verb referring to reading aloud:
“The word qur’ān appears about 70 times in the Quran itself, assuming various meanings. It is a verbal noun (masdar) of the Arabic verb ‘qara’a’…[same as Hebrew “ka’ra”] meaning ‘he read’ or ‘he recited’. The Syriac equivalent is…’qeryānā’, which refers to [a] ‘scripture reading’ or ‘lesson’. While some Western scholars consider the word to be derived from the Syriac, the majority of Muslim authorities hold the origin of the word is ‘qara’a’ itself [note the lack of reference to a Hebrew source based on Jewish practice]. Regardless, it had become an Arabic term by Muhammad’s lifetime. An important meaning of the word is the ‘act of reciting’, as reflected in an early Quranic passage: ‘It is for Us to collect it and to recite it (qur’ānahu)’.” 
In a scene in the film “Lawrence of Arabia,” Prince Faisal (Alec Guinness) is having the Qur’an read to him. There’s some discussion in Islam as to whether the “mitzvah” of reading or learning Qur’an needs to be done aloud, or if it can be done by silent reading:
“…secrecy is further away from ostentation [i.e. in public reading]. Therefore it is more virtuous with respect to the one who fears that. But if he does not fear ostentation, then to raise the voice is more virtuous, on condition that he does not does not disturb others who are praying, sleeping… The evidence that raising the voice is more virtuous is that there are more actions involved therein. Because it conveys the benefit of it to other people, it stirs the heart of the reciter and gathers his attention towards contemplation, and turns his hearing towards it. And because it repels sleep, increases vigor, and arouses others who are sleeping or are heedless and invigorates them. So whenever one of these intentions is present then reciting loudly is more virtuous.” 
The same source further states a preference that the Qur’an be read on similar days to those on which Torah is read:
“The preferred days (for recitation) are: Friday, Monday, Thursday…” 
Despite the latitude given for silent reading of the Qur’an, great care is given in public recitation:
“There are two types of recitation: Murattal is at a slower pace, used for study and practice. Mujawwad refers to a slow recitation that deploys heightened technical artistry and melodic modulation, as in public performances by trained experts. It is directed to and dependent upon an audience, for the Mujawwad reciter seeks to involve the listeners.” 
It’s possible, therefore, to suggest that the choice of the word “Qur’an” is related to, or derived from, the Hebrew word for the public recitation of Torah. It should be added, though, that it can also imply that the surahs were “recited” to Mohamed by G-d, and/or that the surahs were learned by heart and recited by the early Muslims (as is, in fact, the case), before an authoritative edition was produced (a process which itself was not immediate).
The Qur’an is divided into sections, each of which is called a “surah”:
“Each chapter or portion of the Qur’an is called a ‘Sura,’ which means a ‘Degree’ or ‘Step,’ by which we mount up.” 
Like a ‘sidrah,’ each “surah” has its own title — usually based on a word in one of the first sentences:
“SURAH: lit. ‘A row or series.’ A term used exclusively for the chapters [‘sections,’ actually] of the Qur’an…These chapters are called after some word which occurs in the text…The ancient Jews divided the whole law of Moses into fifty-four ‘siderim’, or ‘sections’ which were named after the same manner as the ‘Surahs’ of the Qur’an.” 
There is even some scholarly conjecture that the word ‘surah’ is based on “sidrah.” For example:
“The very word ‘surah’ owes its existence, as I believe, to a misreading of ‘sidrah’.” 
A Scripture called a “recitation,” divided into sections; the similarity of “surah/sidrah;” the titles of each section taken from words within that section itself — all can suggest the Muslim borrowing of a form originally Jewish; specifically rabbinic.
This in no way denies the inspired quality of the Qur’an. Rembrandt didn’t invent oil painting; Shakespeare didn’t invent playwriting; one might even say that Einstein didn’t invent Physics. In each case, inspiration poured itself, as it were, into a form that was already in existence — investing that form with a unique radiance.
We can see from the above that Torah has had an amazing impact far beyond the boundaries of the Jewish world alone.
We can be immensely proud of our tradition.
That could inspire us to learn more of it.
 Va’yik’ra/Lev. 1:1
 ibid., p. 8
[note the substitution of Friday for Saturday, in keeping with the Muslim designation of Friday as a “sabbath.” In Jewish tradition, “Monday” and “Thursday” were designated, traditionally by Ezra, because these were days in which many people would come into town to market their goods and buy others’, thereby being preferable for educational purposes as well.]
 Ali, Abdullah Yusuf; The Holy Qur’an; Text, Translation and Commentary; Tahrike Tarsile Qur’an, Inc., © 1988; p. 13 [note: The Qur’an was later also divided into chapters and verses, while retaining the surah-divisions]
 Hughes, Thomas Patrick; A Dictionary of Islam; W.H. Allen, 1885; p. 623
[note: The author of this quote overlooks that the Jewish division of scripture pre-dates the Islamic one, thereby missing the opportunity to see the historical connection between the two.]
 Abrahams, Israel in the Jewish Quarterly Review, Oct. 1907; vol. XX, no. 77, p. 877
[I have not found any empirical proof of this by any writer.]