DECIMAL ACTIVITHM: POWERS OF TWO PATTERNS

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 an alternating pattern, which I "stain"" as RED (2-multiple)-columns and NONRED (non-2-multiple)-columns.

Seeing, the child can be be guided to a Conclusion:The 2-ness (non-2-ness) pattern repeats every 10. Then the child can be guided to see that, equivalently, 2-ness (non-2-ness) is conserved by the transformation of adding 10.

This also means (as you know, and the child learns in Elementary Arithmetic Classes): "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, the child can be guided to see that the 4-ness forms another pattern which repeats every 100 (10-ness of 10-ness). And guided to see the 4-ness (or non-4-ness) pattern is conserved by the transformation of adding 100 to a number. This also means (something rarely, if ever taught!) 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 check the last 2 digits.

Next, you color (say) green the 8-multiples. The child can be guided to see that, from the coloring, that the 8-ness pattern repeats every 1000. Equivalently, 8-ness (non-8-ness) is conserved by the transformation of adding 1000.

And the child can be guided to realize (something rarely, if ever taught!) 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. The child can be guided to see that 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 (ever taught?) 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 n2-ness pattern (multiplying n 2's) repeats every 10n (multiplying n 10's). The child can be told that this INVOKES A 2-POWER CONSERVATION LAW: 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.

Why? BECAUSE WE HAVE IMPOSED THE PATTERN OF TENNESS AND MULTIPLES OF TENS UPON THE NUMBERS. Now, 2 IS A PRIMEFACTOR OF 10, SO IMPLICITLY WE HAVE IMPOSED THE PATTERN OF TWOS AND MULTIPLES OF TWOS. You can explain that this is AN INHERITANCE.