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Let us examine each of the four parashiyot, the verses in the Torah that apply to them, and what our Sages have said in their regard.
The first parasha on our calendar is Parashat Shekalim, which deals with the half-shekel coin. The Torah (Parashat Ki Tissa, Exodus 30:11-16) states that this served two purposes. First, it was to count the Children of Israel in a census, as heads were not to be counted so that the evil eye would not plague them (Rashi ad loc). Instead, they were counted with coins valued at half a shekel. The verse explains that these coins were then to be used for a kappara – an atonement. Rashi (ad loc.) explains that some of that money was used for the communal sacrifices to be offered on the altar throughout the year.
The first Mishna in J.T. Shekalim (1:1) states that “on the first day [Rosh Chodesh] of Adar [the Beit Din] would announce the shekalim contribution…” The Gemara asks, “Why on the first day of Adar?” The Gemara answers, “So that they will bring their shekalim in the proper time.” The Riv’van (R. Yehuda b. Binyamin HaRofei) explains in his commentary (ad loc.) that the time referred to is Rosh Chodesh Nisan, as the Gemara (B.T. Megilla 29b) explains concerning the verse (Numbers 28:14), “Zot olat chodesh bechodsho – This is the olah (burnt offering) sacrifice of each month in its month,” meaning the first of the month. At this time, “chaddesh – renew” from a new Terumah (collection), the Tamid and Mussaf sacrifices which were brought on Rosh Chodesh Nisan. They were acquired with the new shekalim coins [collected at that time].”
Riv’van then compares this announcement to the laws of Pesach; the Gemara (Pesachim 6a) states that we are to query and expound on the laws of Pesach 30 days before Pesach begins, which would be the 15th of Adar. Thus, all announcements are always made 30 days ahead, which for shekalim would be Rosh Chodesh Adar. Therefore, Parashat Shekalim is read on, or immediately prior to, Rosh Chodesh Adar.
Now, since we are bereft of the Holy Temple and we have no korbanot, we read Parashat Shekalim to commemorate them.
Second on the calendar is Parashat Zachor, based on the verses in Parashat Ki Tetze dealing with Amalek (Deuteronomy 25:17-19), “Zachor et asher asah lecha Amalek baderech betzet’chem mimitzrayim – Remember what Amalek did to you on the way as you were leaving Egypt.” The verse then explains, “Asher karcha baderech va’yezanev becha kol ha’nechshalim acharecha ve’ata ayef ve’yage’a, velo yarei Elokim – How he met you on the way, and he struck those of you who were hindmost, all the weak ones at your rear, when you were faint and exhausted, and he did not fear Hashem.”
The verse then instructs, “Vehaya behani’ach Hashem Elokecha lecha mikol oy’vecha misaviv ba’aretz asher Hashem Elokecha noten lecha nachala lerishtah, timcheh et zecher Amalek mitachat hashamayim, lo tishkach – It shall be when Hashem your G-d has given you rest from all your enemies all around, in the land that Hashem your G-d gives you for an inheritance to possess it, you shall blot out the memory of Amalek from under heaven, you shall not forget.”
The Mishna (Megilla 29a) explains that if Rosh Chodesh Adar falls on a Sabbath, we read Parashat Shekalim on that Sabbath. However, if Rosh Chodesh falls in the middle of the week, we read Parashat Shekalim on the Sabbath preceding Rosh Chodesh; we then interrupt the reading of the parashiyot, and resume with Parashat Zachor on the Sabbath after that.
Rashi s.v. “Umafsikin leshabbat haba’ah” explains that we endeavor to read Parashat Zachor on the Sabbath just before Purim in order to connect the eradication of Amalek with the downfall of Haman. This is another reason for reading Parashat Zachor during this time of year.
The third of the four parashiyot on our calendar is Parashat Parah, the section found at the beginning of Parashat Chukkat (Numbers 19:1-22), which discusses the red heifer without a blemish that Moses was commanded to hand to Eleazar the priest, to be sacrificed. The verses detail the entire procedure, which the Torah refers to as a chok, a law for which we do not know the reason.
Rashi (Megilla 29a) s.v. “para aduma” explains that the red heifer was brought to warn the Jews to purify themselves of any ritual defilement in order to offer the korban pesach in a ritually pure state.
Thus we see that this parasha is timely to the weeks before Pesach, which is why we read it at this time. (Rashi ad loc. s.v. “Ba’revi’it – hachodesh hazeh lachem” quotes the Jerusalem Talmud, stating that in actuality this should be the fourth parasha, pursuant to the sequence of events in the Torah).
Finally, the last of the Four Parashiyot is Parashat HaChodesh (Exodus 12:1-20), which contains the concept of Rosh Chodesh, the first commandment given to the Children of Israel, upon which our calendar is based, including the festivals, the first of which is Pesach. This section also contains the commandment of the paschal sacrifice and its laws.
This parasha is read on the Sabbath preceding Rosh Chodesh Nisan, except if Rosh Chodesh occurs on a Sabbath, in which case it is read on that Sabbath. Rashi (Megilla 29a) notes that since this section contains the laws of Pesach, the Mishna instructs us to read it at this time.
Thus we see that all four parashiyot as delineated in the Mishna (Megilla 29a) are designated to be read in a timely manner on these four specific Shabbatot.