Toward the end of the movie the Hebrew School teacher announces that "The Weather Service has issued a Tornado Warning". The National Weather Service was known as the Weather Bureau until late in 1970.
The school buses are painted yellow and black. During that time and into the 1970's Minnesota was one of two states (Nebraska was the other) that used orange and black for all school buses. It was only much later that they joined the more universal yellow and black color scheme.
In Minnesota, deer hunting season with rifles occurs in the autumn, while "tornado season" occurs from late spring to late summer. Two different dates in the film imply that the time of the year is in April and May, and the trees in the film have not changed color as they start to do in late August/early September. However, Larry Gopnik's neighbors return from deer hunting, well before the tornado scene at the end of the film.
When the son is studying for his Bar Mitzvah using an LP, the cover of a famous LP of Cantor Joseph Rosenblatt is shown. This record was in actuality a re-issue of Rosenblatt's famous cantorial recordings on the Victor label in the early 20th century, and not what the boy is listening to.
Larry Gopnik is writing an equation on the board in class. At one point he writes delta p equals the square root of ^2 - ^2 which would be zero, but the correct equation has the squared inside the bracket in the first term under the square root: ^2-^2. This is an equation for the root mean square deviation of momentum in Heisenberg's uncertainty principle of quantum mechanics. Later in the scene, after the students leave and Sy Ableman appears, the equation is in the correct form.
The mezuzah (a Scriptural passage usually enclosed in a small container, attached to the door of a Jewish home) affixed to the doorpost of the Sansky house, Larry Gopnik's neighbor, is on the left side of the entrance and tilted towards the outside. Mezuzahs are traditionally affixed to the right side of the entrance and tilt towards the house.
Larry tells the feds that they are sitting shiva over Sy Ableman. "Sitting Shiva" is done only by the seven relatives which are: father, mother, brother, sister, spouse, son or daughter. And since they are not one of the seven relatives they aren't supposed to be sitting shiva.
Yarmulkas (skull caps) are not worn in Hebrew class by students some of the time and then they are other times. The lettering on the school bus is in Hebrew which implies the school is not a reform movement school. They would always wear yarmulkas in class.
Larry's first name is spelled differently throughly the course of the movie. The sign on his office door reads "Prof. Lawrence Gopnik", but the letter from Meshbesher is addressed to "Laurence Gopnik" .
The contents of principal Turchik's desk drawer changes as Danny and his reefer buddy search for Danny's radio. In the brief instant the drawer closes, several new items can be seen that were not there when the boys were ransacking the drawer moments earlier, including a set of Groucho Marx glasses, a toy ray gun, a yo-yo, a garden hose nozzle and a Playboy magazine. Gone from the drawer are a gyroscope, Pez dispenser in the shape of a pumpkin, and a set of finger cuffs.
F Troop was an ABC show, which would have been channel 9 in the Twin Cities at that time. Channel 4, that they kept trying to find, was CBS (WCCO). There was nothing on channel 7, a station they complained came in poorly.
The blessing that is recited from the bima before the bar mitzvah reads his Torah portion should be "...asher bachar banu..." but in the movie the blessing is said, "...asher natan lanu...," which is the blessing to be recited after a Torah portion is read, and both were supposed to be recited by whoever was called up to the Torah, which in this case was Dani the Bar Mitzva boy.
(at around 1h 10 mins) Larry is standing in front of the blackboard. The equation behind his right shoulder, "hbar = h/2pi = 6.6 x 10^-34 in [mks] units" is wrong. hbar = 1.05 x 10^-34 Js in mks units.
Danny is called up as Maftir, the last reader for that weekly Torah portion. However, what he actually reads is the beginning of Parashat Behar, which is read by the first reader, not the last. The first reader would typically be a "cohen", a descendant of Aaron, the High Priest and brother of Moses. In light of their family name, it is reasonable to assume that the Coen brothers are themselves "cohanim".
Danny's transistor radio behaves like a 1980s Walkman. When it is confiscated it has rewound to (or been paused at) the beginning of the song. Marshak mysteriously knows what Danny was listening to when the device was stopped.