math puzzle with random letters that decode into numbers?

Question

1. A A , B
   A A A , C
   A A A A , D
   A A A A A , E
   A A A A A A , F
   A A A A A A A , G
   A A A A A A A A , H
   A A A A A A A A A , I

2. A J A K B
   A J A J A K C
   A J A J A J A K D
   B J A K C
   B J B K D 
   C J D K G

3. C L A K B
   D L A K C
   G L B K E

4. A J M K A
   D J M K D
   C L C K M

5. E J E K AM
   E J F K AA
   AM J AM K BM
   CM J B K CB

6. B N C K F
   B N E K AM
   E N AB K FM
   AM N AM K AMM
 
7. H O B K D
   AM O E K B
   AMM O BM K E

8. A O AM K PA
   A O AMM K PMA
   A O B K PE
   A O D K PBE

9. C N C K C Q K I
   D N D K D Q K AF
   AM Q K AMM

10. R AMM K AM
    R BE K E
    R G Q K G

Source: https://education.cherryarbordesign.com/docs/mathunplugged/mathematicalMessages.pdf

Answer 1

From top to bottom:

Case 1

establishes A .. I as the counting system. That is, it tells us they are the digits 1 .. 9.

Case 2

introduces the symbols J and K. All of the lines are satisfied if we interpret J as addition and K as the equivalence predicate (==)

Case 3

introduces L. Expectations from case 2 tell us that L should be subtraction.

Case 4

introduces M. We observe that it's the 0 digit.

Case 5

introduces juxtaposition. It tells us that digits stuck to each other should be interpreted as decimal.

Case 6

introduces N. We would expect to learn multiplication somewhere around this time, and it is indeed the case.

Case 7

introduces O. Previous cases help us recognize the division operator.

Case 8

introduces P. Matching with the division results tell us that P represents the decimal point, perhaps with a zero next to it.

Case 9

introduces Q. It represents the sQuare operator, and it's postfix.

Case 10

introduces R, for square Root, a prefix unary operator. Nice timing, possibly deliberate.