This mathtivity motivates mathematics appreciation, articulating math as patterns. First I tell children two true (but ignored) stories.

An archaeologist traveled, as airplane passenger, over terrain he knew at ground level. From this perspective the archaeologist saw, outlined on the ground below, the buried fragments of walls and buildings. Later, diggers discovered ruins dating from the Roman invasion of Britain (around 120 AD), during Emperor Hadrian's reign. People had lived upon these ruins for centuries without being aware of them.

I explain this to children: "Those people were too close to see buried patterns. Same way, you're too close to arithmetic to see the number patterns!"

The second story teaches the children "how to rise above the number-blob".

In 1839 German biologist, Theodore Schwann, discovered (in the kind of pattern-searching I describe in the associated file on the aCTIVITHM) that the basic plant and animal unit (gnomon!) is the cell, which reproduces (subdivides) during the process called "mitosis" to exhibit subpatterns. STAINING the cell highlights the subpatterns.

These subpatterns became labeled by two Greek words: "chromo" for "colored"; "soma" for "body" -- hence, produced the word "chromosomes". (Later chromosomes were identified as hereditary carriers, with genes of "DNA".)

I exhort the children: "STAIN the ARITHMETIC BLOB -- to highlight its number subpatterns!"

To do so, you create 10 x 10 grids as representation of decimal numeration. Then you label COLUMNS with the numbers, "0-9"; and you label ROWS with the numbers "0-9, 10-19, 20-29", etc.

BEHOLD! decimal numeration imposes patterns of tens; patterns of tens of tens (hundreds); patterns of tens of tens of tens (thousands); etc.

BEBEHOLD! 10 = 2 x 5, which invokes the 2-ness and 5-ness subpatterns.

To see this, you color (say) red the 2-multiples -- 0, 2, 4, 6, ..., 98. This yields alternate red (2-multiple) and white (non-2-multiple) columns. Conclusion:The 2-ness (or non-2-ness) pattern repeats every 10. Equivalently,2-ness (or non-2-ness) is conserved by the transformation of adding 10. This also means (as you know): "A number is a 2-multiple if, and only if, its last digit is even."

Next, you color (say) blue the 4-multiples (that is, 2-ness of 2-ness). From the coloring, you see that the 4-ness pattern repeats every 100 (10-ness of 10-ness). And the 4-ness (or non-4-ness) pattern is conserved by the transformation of adding 100 to a number. This also means that a number is a 4-multiple if, and only if, the last 2 digits form a 4-multiple.) Thus, 35716 is a 4-multiple because its last two digits, 16, form a 4-mlutiple. That's all you need to check to detect 4-multipleness: -- just the last 2 digits.

Next, you color (say) green the 8-multiples. The coloring shows that the 8-ness pattern repeats every 1000. Equivalently, 8-ness (non-8-ness) is conserved by the transformation of adding 1000. And this means that A number is an 8-multiple if, and only if, its last 3 digits form an 8-mlutiple. Thus, 357016 is an 8-multiple because its last 3 digits, 016, form an 8-mlutiple.

Next, color (say) yellow the16-multiples. Now, 16-ness (2 x 2 x 2 x 2) repeats every 10,000 (10x 10 x 10 x 10). Equivalently, 16-ness (non-16-ness) pattern is conserved by adding 10,000. And this means that a number is a 16-multiple if, and only if, its last 4 digits forms a 16-multiple. Thus, 3573264 is a 16-multiple because its last 4 digits forms a 16-multiple.

In general, the n-2-ness pattern (multiplying n 2's) repeats every n-10-ness (multiplying n 10's), invoking an n-10-ness conservation law. This means that a number is a multiple of an n-product of 2's if, and only if, its last n digits forms a multiple of an n-product of 2's.

By coloring multiples of 5 or multiples of 25 (5 x 5) or multiples of 125 (5 x 5 x 5), etc., the children find similarly repeating patterns upon this replacement of 2's by 5's. Thus, conservation laws follow. And, in general, a number written in DECIMAL NUMERALS is a multiple of an n-fives-product if, and only, if its last n digits forms such a product

By colored patterns, the children discover that decimal patterning also exhibits interesting subpatterns for 9 and 11, which are the immediate neighbors of the base or gnomon of ten.

You color (say) brown the 9-multiples: 9, 18, 27, 36, 45, ... , 81, 90, 99. Result: a diagonal pattern, going from upper grid-right to lower left. Why? Because 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. 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 MOTHER OF ALL PATTERNS: 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. (This is also yields the algorithm known as "The Bookeeper's Check".)

Starting over, you color (say) purple the 11-multiples: 0, 11, 22, 33, 44, 55, 66, 77, 88, 99. Result: a counterdiagonal pattern, running upper left to lower right -- or counterdiagonally to 9-pattern. Why? Well, 11 = 10 + 1. Adding 10 drops one row; adding 1 shifts one column right. Note: The "alternating sum" (alternately adding and subtracting) of digits of eleven-multiples equals zero. Hence, the algorithm of "casting out elevens". Consider 176: 1 - 7 + 6 = 0. 176 is an 11-multiple: 176 = 11 x 16. (I created the label,"codigital root", for the 11-pattern.)

The codigital root of a "decimal" number is conserved by the transformations of alternate-sum of digits; the transformations of casting out 11's; the transformationsof dividing by 11.

In 6 class-hours, I taught 24 third-grade children to determine if a 10-digit number has factor 144 = 16 x 9, using the 16-rule and "casting out nines". Children pasted walls with colored patterns. If a 10-digit number's last 4 digits had a colored pattern, the number has factor 16; all 9s casting out indicates factor 9; if both cases, 144. Try this on (a) 1729306348; (b) 1525906448; (c) 1727906448. Solution: (a) 9, not 16; (b) 16, not 9; (16) both, hence 144. Each child had a different 3-problem test. Each answered correctly.

Children also experiment with eight-grids ("octal" numeration in computer science), wherein sevenness assumes the diagonal pattern of nines in ten-grids; nineness assumes the counterdiagonal pattern of elevens in ten-grids -- because seven and nine are immediate neighbors of eight (as are nine and eleven for ten). Thus, as articulated in THE MOTHER OF ALL PATTERNS: THE ACTIVITHM, the subpatterns depend upon the ACTION of imposing a particular pattern!

This mathtivity teaches useful arithmetical tricks (extending to basis "x" in numerical algebra!).

It also prepares for Conservation Laws in physics: Conservation of Energy, Conservation of ngular Momentum, etc.. And it echoes the religious faith of a cherished old hymn, "Oh, Thou Unchanging!".