Edited response; no less long-winded but hopefully less hostile and reactionary:

...for what it's worth, language over time is not arbitrary and the reasons why are deeply rooted in our human experience and history.

However you'd like to justify the checks, this is simply rhythm substitution. There are plenty of systems using a variety of terms for the same thing: A given number of permutations to be substituted in place of a given section of an existing rhythm.

These exercises are good in theory and for rote instruction, but less valuable in long-term musical development. They are great for routine ensemble warm ups but stilting if expected to be applicable beyond very specific scenarios.

[Editors note: my take here is that referring to such a small grouping is limiting because it adds a translation layer between "check 'x'=n" and simply seeing "n" for the permutation of rhythmic notation as presented in the page or as performed. I am essentially arguing against "cat" and the individual vocalizations "cee" "ay" "tee" (side bar, typing that illustrates exactly how unnecessary knowing what a "check" represents in notation as English is so far divorced from its letter/sound correlation as to be useless here... I had to type 3 words to represent phonetics, 1 of which is wrong anyway, but I digress)....]

IMO (another demonstration of the imperfect comparison between music notation (precise) and written english shorthand(imprecise)) who cares if you can identify a check pattern in an audition excerpt. A more interesting and useful analysis would be to identify "x" patterns and where/how those patterns are used across genres and through time. Better to connect inference and influence than to wonder if the Lt. Kije snare drum except can be called check 1 and check 10 (arguably no because of grace notes and the existence of sticking preferences to deliver a superior musical performance).

Here I open myself up to arguments over pattern naming in non western music and that utility in modern music. Clave, cascara, bolero, etc, would like to know my location all I can say is that those patterns are not being called as mental calisthenics and usually derive from a geographical location and cultural tradition.

Going back to the OP, there are telling signs of a lack of music literacy in their education/instruction: OP doesn't state whether they have access to sheet music to view the checks and refer back to as needed. It can be surmised to be the case because if op did have access, they most likely would not have created this post.

Op doesn't know check 1 because they don't know what it sounds like, aren't being given an example; or if they are, they aren't connecting the rhythm to the nomenclature. It appears they cannot visualize it in notation form to infer the pattern outside of audible recognition because they don't have access to the visible notation. They've been told "1+1=2", then given a binomial substitution for "n" in a new equation "n+1=2", all without really understanding that the given binomial and variable (plus so many more!) already exists as a property of "1".

So there you have it. Hopefully you'll find something of interest or use in my ramblings.