Imposing base ten singles out numbers nine and eleven as special since they are "the immediate neighbors" of the base ten.
When a child colors nines in a tens pattern, the child sees a DIAGONAL PATTERN, from UPPER RIGHT TO LOWER LEFT.Why? Because, IN decimal notation, 9 = 10 - 1. Adding 10 drops you one row; subtracting 1 shifts one column left -- diagonally. Furthermore, digits of 9-multiples sum to 9: 18, 1 + 8 = 9; 27, 2 + 7 = 9; 36, 3 + 6 = 9; 45, 4 + 5 = 9; ... 99, 9 + 9 = 18, 1 + 8 = 9. Etc.
A number's digital root is the repeated sum of its digits. Thus, 4 is the digital root of 12478 since 1 + 2 + 4 + 7 + 8 = 22 and 2 + 2 = 4. Equivalently, "casting out the nines" in 12478 -- outing 1 and 8, 2 and 7 -- yields digital root 4. Equivalently, permuting (rearranging, as in anagrams of words) the digits of 12478 -- say, 12478->87421 or 12478->48217 or 12478->87421->48217 -- conserves its digital root, 4.
Gradually, the child can be guided to see that THE DIGITAL ROOT IS AN INVARIANT UNDER THE DIGIT-PERMUTATION TRANSFORMATION! Also, this means that 12478, 87421, 48217, and all its other permutations, yield remainder 4 when divided by 9.
This is another instance of THE ACTIVITHM. For, all the properties I have so far described arise from the ACTION of imposing the decimal pattern on collections of ones, that is, collecting ones in tens and tens of tens, etc.. Now, in the nines-pattern, we find that the digital root of a "decimal" number is conserved by the transformations of summing its digits; by the transformations of casting out 9's; by the transformations of permuting its digits; by the transformations of dividing by 9.
Thus, starting in the 10x10 grid with the number 7, we find that 7 + 9 = 16, 1 + 6 = 7. 16 + 9 = 25, 2 + 5 = 7. 25 + 9 = 34, 3 + 4 = 7. Etc., as can be seen.
This is also yields the algorithm known as "The Bookeeper's Check".
When a child colors elevens in a tens pattern, the child sees (in my jargon) a CODIAGONAL PATTERN, from UPPER LEFT TO LOWER RIGHT, the MIRROR IMAGE OF THE DIAGONAL NINES PATTERNJ.Why? Because, IN decimal notation, 11 = 10 + 1. Adding 10 drops you one row; adding 1 shifts one column right -- codiagonally.
We saw that THE DIGITS IN A NINES MULTIPLE ALWAYS SUM TO NINE, DIRECLTY OR BY REPEATED SUM. Similarly, we find that THE DIGITS IN AN ELEVENS ALWAYS DIFFERENCE TO ZERO. Thus, 11 results in 1 -1 = 0; 22 results in 2 - 2 = 0; 33 results in 3 - 3 = 0; etc. For more than two digit numerals, the rule becomes "alternating sum", described below. And we'll find that this suggests elevens algorithms comparabled to the nines algorithms.
For nines, And we found a digital root as the repeated sum of its digits. For elevens, we have a rule which I call the codigital root -- the repeated ALTERNATING SUM of its digits. By "alternating sum", mathematicians mean ALTERNATELY ADDING AND SUBTRACTING. Consider the number, 121 = 11 x 11. Starting form the right, we add 1, subtract 2, add 1, or, 1 - 1 + 1 = 0. Ending in zero, we know the number is a MULTIPLE OF ELEVEN.
This method is known as "casting out the elevens".
The child can gradually be guided to see this in an elevens in tens-grid.