A NY Times article contains this question:
Why is yawning is contagious?
Why are there two “is”’s in this sentence?
Similarly for these sentence fragments:
... as far as the freak was concerned was ... (source: University of Sheffield)
When I found out she was a cheapskate was ...
What's the purpose of the second verb in those sentences? (”As far as the freak was concerned was”, “When I found out she was a cheapskate was”, “Why is yawning is contagious”)
There is no "explanation". These are simply errors. You should not see these in any edited text, but this kind of stumbling over words is not uncommon in unrehearsed speech. The example in the lesson plan is a mistake. I suspect that part of the NY Times website is not as carefully checked as the main news articles and opinion pieces.
The sentence "Why is yawning contagious?" is a correct question. "why yawning is contagious" is a sentence fragment. It is a content clause that could be used like a noun in a sentence: "I'll tell you why yawning is contagious."
It is a mistake to include "is" twice. A native speaker would never do it intentionally. The same goes for all the other examples. They are mistakes.
The first example is more or less meaningless on its own. It can be parsed only when placed in a context.
[What was saleable] [as far as the freak was concerned] [was], of course, physical difference, in a form that was both marketable and palatable.
There is a content clause "what was saleable", which is modified by "as far as the freak was concerned", and this is the subject of the sentence. Then the main verb is "was". Compare this example
The apple that was on the table was red.
The first "was" is the verb in the relative clause "that was on the table". The second "was" is the main verb in the sentence.
The second is probably a mix of "When I found out what a cheapskate she was" and "When I found out she was a cheapskate" But note that neither of these are complete sentences. It could also be part of a longer sentence, with the first part being a content clause, but it's not well written:
[When I found out she was a cheapskate] [was] the day she took all the mini shampoos from the hotel.
The sentence with is repeated appears to be an error. The sentence with was twice is perfectly good, though a little confusing. It reads
What was saleable as far as the freak was concerned was, of course, physical difference, in a form that was both marketable and palatable.
In this sentence as far as the freak was concerned is a parenthetical clause. Without it, it reads
What was saleable ... was, of course, physical difference, in a form that was both marketable and palatable.
which is perfectly well-formed.
You may be confused because the parenthetical clause was not set off with commas - it could have been, but did not have to be - and the second was was followed by another parenthetical phase "of course", which was set off with commas.
Some of these sentences are simply errors, such as “*Why is yawning is contagious?” If I had to guess, perhaps the writer meant to type, “Why is yawning so contagious,” but accidentally typed si for so and then had it autocorrected to is.
Some of those sentences are correct, however. They have a clause that contains the verb is or was, and that clause is the subject of a sentence whose main verb is also is or was. These don’t make sense if you look only at those sentence fragments in isolation.
A simple example of this is the saying, “It is what it is.” The subject is “it,” the main verb is “is,” and the direct object is the noun clause “what it is.”
Let’s take a look at one of the fragments you provide a link to:
What was saleable as far as the freak was concerned was, of course, physical difference, in a form that was both marketable and palatable.
This is a complex sentence (in that it has several subordinate clauses with their own subjects and verbs). This is common in academic writing. To break this down:
In short, was can appear four times in the sentence because it appears in the main clause as well as in subordinate clauses. Each time, it has a different subject.
They're either mistakes, or else part of longer sentences where the grammar would be meaningful. Seems likely though that they're just typos. Newspapers have subeditors whose job is to pick out mistakes like this, but they're only human, sometimes errors get through. If you posted the entire sentences, rather than fragments, you'd get a better, more accurate opinion, but from what is here, they're simply incorrect.
If you could post the full sentences, a sentence either side might help as well, or even the entire paragraph. Sometimes people will break laws of grammar to make a joke or a point, or for stylistic reasons.
