This sentence is an example of a run-on or comma splice. It has two independent clauses separated only by a comma. One way to fix this is to change the comma to a semicolon; if this were to be done, the corrected sentence would read, "Thomas had always struggled with trigonometry; he thought that circles should remain circles and let triangles and squiggles be their own shapes."

The clause, "she refused to speak with him for longer than necessary" is an independent clause and therefore must be separated from the sentence's initial independent clause ("Cal, with his good looks and smug tone, came off as arrogant to Sarah") with a semicolon, a comma followed by a conjunction, or a period, not just a comma.

The clause following "them" ("they were such huge fans of those shows") is an independent clause. It therefore needs to be joined to the first part of the sentence, the compound independent clause "I don't remember many Saturday morning cartoons, but May and John can name pretty much all of them") using a semicolon, a comma followed by a conjunction, or a period, not just a comma. (Note the correct use of a comma followed by the conjunction "but" in the sentence's initial compound independent clause.)

This sentence contains a comma splice. Two independent clauses, like the two in the example, cannot be separated by just a comma. Instead, they should be linked using a period, a semicolon, or a comma with a conjunction.

This sentence features a common error of punctuation known as a comma splice. Both clauses, separated here by a comma, are independent. Independent clauses can be separated by a semicolon, a period, or a comma with a conjunction after it. Any of those three options would fix the comma splice error in this sentence, and all three options are equally grammatically correct in this instance.

This is an example of comma splice. The two clauses (separated here by a comma) are both independent clauses (each is a grammatically complete clause that could stand alone as a full sentence). If they are linked with a comma and no conjunction, as in this example, it is a comma splice. They should be linked with a comma and a conjunction, a period, or a semicolon.

There are several acceptable ways to separate independent clauses, but simply using a single comma by itself is not one of them. To do so is to make an error known as the comma splice. The sentence above contains such a splice, and it should be corrected. The options to make this correction would be to add a coordinating conjunction after the comma, replace the comma with a period, or replace the comma with a semicolon.

There are several acceptable ways to separate independent clauses, but simply using a single comma by itself is not one of them. To do so creates an error known as the comma splice. The sentence above features a comma splice after "hands." The first clause, "the brilliant surgeon had steady hands," is a grammatically complete independent clause, as is the second clause, "she was an incredible pool player." The options for separating these two clauses are to place a period, a semicolon, or a coordinating conjunction and a comma between them. All three of these options are equally grammatically correct, whichever you choose is a matter of style.

There are several acceptable ways to separate independent clauses, but simply using a single comma by itself is not one of them. That is an error known as the comma splice. The three options available to correct such an error are a period, a conjunction with a comma, or a semicolon. These three options are all equally grammatically correct, which one you choose is a matter of style. The example sentence contains two independent clauses, so rather than a comma after "world" one of those three options needs to be inserted.

Both clauses in the example sentence are independent clauses (grammatically complete clauses that could stand on their own as sentences). When two independent clauses are connected into one sentence it is called a compound sentence. The correct ways to punctuate compound sentences are either a conjunction and a comma or a semicolon. The other option in this instance would be to simply replace the comma with a period.