Robert Hansen wrote , "While reviewing Shute's Elementary Algebra book I came across a reference to FOIL during the discussion of products and factoring of polynomials."
As William Betz had been writing math books for more than a decade by this time, it's likely 1929 was not the first time FOIL was written about."
I have renewed my search for the origins of Foil, and a little more about William Betz, which I will add to this as I do not have enough for much of a bio page for him.
In 2015 Thony Christie, a very knowledgeable math historian, wrote that he had never hear of the term Foil, and had to look it up. His historical experience, and the fact that he has lived and worked in both England and Germany for extended periods suggest that this was an American creation, not an import.
I also heard from Shemsi alhaddad, who grew up in Kuwait, is an associate professor of math. He writes, "not in Arabic (that I know of). Just taught as the distributive property.." in the 1980s. "It may be different now. I didn't hear of anything like FOIL or Sohcahtoa until I moved to the U.S"/
Pat Grew , a Canadian math teacher, was aware of its use; " My intro was in Ontario Curr in '84 as a teacher."
Here is an early bio in the Orient, the yearbook of East High School in Rochester, New York where he was head of the Math Dept, and eventually the Assistant Principal as well..
He is also listed in the Admin and Mathematics Dept in the faculty of the 1935 Orient. There is also a Martha Betz listed in the Modern Language Faculty, and a student named Betz on the baseball team picture,(pg 48) but it now seems they were not relations.
The 1940 census says he was married to a Elsie Betz, and had two children living in the house. since the oldest was only 12 in 1940. William lived until at least 1959 when he is listed, still at the same address, in the phone book.
If anyone has an earlier link to this use of FOIL, or about the life of William Betz, please advise.