An important part of becoming a CFI is learning how to find the answers, from the official sources, of such questions. As a CFI, never teach 'tribal knowledge'. That is, things that you 'know' but have no idea where it comes from or why it is done. Students will ask questions, and you'll have answers that 'you've been told' or 'the way you've done it', but you don't know why. When you are asked such questions, find the source documents and show your student how the answer is found so that they will be able to find answers when they are studying for their rating and, later, when they are out flying on their own.
For your question, the first place to look is 14 CFR 61 - CERTIFICATION: PILOTS, FLIGHT INSTRUCTORS, AND GROUND INSTRUCTORS. Easy to find by searching "far 61".
Look through the index and you'll find a Subpart H - Flight Instructors Other than Flight Instructors With a Sport Pilot Rating.
The second item in Subpart H is 61.183 - Eligibility requirements. (I actually looked first in 61.185 - Aeronautical knowledge but that didn't have the information on knowledge tests)
61.183 (e) & (f) say,
(e) Pass a knowledge test on the areas listed in § 61.185(a)(1) of this part, unless the applicant:
(1) Holds a flight instructor certificate or ground instructor certificate issued under this part;
(2) Holds a teacher's certificate issued by a State, county, city, or municipality that authorizes the person to teach at an educational level of the 7th grade or higher; or
(3) Is employed as a teacher at an accredited college or university.
(f) Pass a knowledge test on the aeronautical knowledge areas listed in § 61.185(a)(2) and (a)(3) of this part that are appropriate to the flight instructor rating sought;
So (e) requires the FOI knowlege test unless the applicant meets (e)(2) or (e)(3).
(f) referrs us back to 61.185(a)(2) and (a)(3) which say,
(2) The aeronautical knowledge areas for a recreational, private, and commercial pilot certificate applicable to the aircraft category for which flight instructor privileges are sought; and
(3) The aeronautical knowledge areas for the instrument rating applicable to the category for which instrument flight instructor privileges are sought.
(f)(2) requires the FIA for those seeking flight instructor privileges in category 'airplane'. (f)(3) requires the FII for those seeking instrument instructor privileges.
Please excuse the long reply. I wanted to demonstrate the process which I am recommending to adopt for both studying to become a CFI and for when you are working as a CFI.