When Chuck is dressing for his practice exam, the first shot of him in front of the mirror he ties his tie up to the collar. In the next shot he is fixing his hair and you can see that the tie is not tied to the top of the collar.
When the family is having breakfast and the mom announces her wedding plans, we see the sister Meg fill a bowl with cereal and milk. Moments later when they argue, she throws her cereal at Gloria. But Gloria is shown as perfectly dry, when she should be covered in cereal and milk.
The engraved plate on McLoed's clock is supposed to include the first few lines of Vergil's Aeneid in Latin; the words, however, are completely jumbled, beginning with "Cano, arma que (sic!) virum qui, profugus fata" etc. It ought to be "Arma virumque cano, Troiae qui primus ab oris..." and so on.
Meg's braces are wrong for the time period (summer 1968) when the movie was supposed to take place. She should be wearing braces the were steel bands that wrapped around each tooth with a wire bracket on the front of the band. Meg has braces that are used today, which are a bracket glued to the tooth to hold the wire that straighten the teeth. The newer method didn't start to be used until around 1980.
When McLeod is teaching Chuck geometry, he uses the bay window across from Chuck as a dry erase board. That part of the movie takes place in 1968 and dry erase markers were not invented until 1975.
When Gibson is giving Stahl his mock exam results from his truck, you can see a boom mic reflected in the cars windscreen.
When giving the method for finding the center of a circle, McLeod says and draws that the line perpendicular to the midpoint of AB goes through C, which is extraneous. A more correct geometric construction would not have named C until AB had been bisected and the line from the midpoint at right angles to AB had been drawn to the circle, making the point at which the two meet C.
When McLeod explains to Norstadt what he does for a living he hands off several magazines and says "Here's a few you might have seen" - it is unlikely that such a teacher would make the mistake of using the singular "here is" as opposed to the plural "here are" when talking about several magazines...