Multiplication Table Revisited

Every elementary school kid has a daunting math task to do: remembering the multiplication tables.
There are 9x9=81 (how do I know? I know the multiplication table!) entries and it sure is a daunting task for kids who have little attention span. But look, you MUST remember this table or you are stuck in your math education. You can't do division without this, you can't do fractions without this... Don't even think about Algebra, Geometry or Calculus.

Calculator you say? Yes it is helpful but kids still need to know this table.... because it is faster! Don't believe me? Does your calculator has a common-denominator button if you are to show steps to add two fractions? It is clumsy if you need a calculator every step of the way. I am sure you can probably find an app or two to do this (hey I wrote one). But it is still clumsy to use anything other than your head to do this fairly simple task.

Here we go: the original multiplication table.

×	1	2	3	4	5	6	7	8	9
1	1	2	3	4	5	6	7	8	9
2	2	4	6	8	10	12	14	16	18
3	3	6	9	12	15	18	21	24	27
4	4	8	12	16	20	24	28	32	36
5	5	10	15	20	25	30	35	40	45
6	6	12	18	24	30	36	42	48	54
7	7	14	21	28	35	42	49	56	63
8	8	16	24	32	40	48	56	64	72
9	9	18	27	36	45	54	63	72	81

I got good news for you. I know how to simplify it. Let me remind you of the 2 properties that you have heard of but thought it was completely useless of-course statements...
1. Multiplicative Identity Property: 1 × anything = anything.
2. Associative Property of Multiplication: a × b = b × a.

Look, the first property boom-boom trimmed one row and column involving the ones. You already know this part of the table. Look, the second property cut away a big triangular chunk. You DON'T need to remember what's 9 × 2, if you already remember 2 × 9. There is only ONE item to remember for the 9's, and that is 9 × 9=81. (Note: a × b is not always b × a. For regular numbers, yes. But this doesn't apply for other math entities such as the matrix.) You just boiled down the table from 81 entries to just 36 entries. Come on kids, you can remember this (by 2nd grade or so).

×	1	2	3	4	5	6	7	8	9
1
2		4
3		6	9
4		8	12	16
5		10	15	20	25
6		12	18	24	30	36
7		14	21	28	35	42	49
8		16	24	32	40	48	56	64
9		18	27	36	45	54	63	72	81

There are unbelievably many American 8 or 9 or even older kids who don't know this basic table.... who is doomed to fail in just about every math class in their probably-short school life.