How to make your own multiplication table without really knowing how to add.

This weekend, I have been helping my six year old son Dagon with his multiplication.

Learning his multiplication table is not so natural for Dagon. I don't believe this to be unusual. Many children have trouble learning to multiply. I began to reflect on why this is. Memorizing the multiplication table means learning the answers to 45 multiplication problems. Each answer is a two digit number. Patterns relating these two digit numbers do not become easily visible to a person who is not fluent in two digit addition.

The goal of this hub is to point out that the problem is considerably simplified if we break it into two parts. Namely find the second digit for each answer and only after having done this, find the first digits. The table of second digits has considerably more symmetry than the full multiplication table.

Here's what we need to know:

A. We must be able to compare single digit numbers and determine which is bigger.

B. We must be able to count. For the purposes of this hub, that corresponds to memorizing the sequence of numbers: 1 2 3 4 5 6 7 8 9.

C. We must also memorize the number sequence: 2 4 6 8 0. This is essentially counting by twos.

D. We must memorize the long number sequence: 3 6 9 2 5 8 1 4 7. (For Dagon this is the hardest part.) The sequence is of course the last digit of counting by threes. The sequence can be easily recovered by reading the numbers on a numeric keypad down the columns right-to-left.

E. We must memorize the number sequence  4 8 2 6 0. (This is adding by fours. Notice the close relationship to the sequence in C.)

E. We must memorize the number sequence: 5 0

F. We must be able to invert the order of sequences.

That's it.

In the first column of the last digit table place the sequence in B.

In the second column place the sequence in C, repeating it when the sequence runs out

In the third column place the sequence in D.

In the fourth column place the sequence in E, repeating when the sequence runs out.

In the fifth column place the sequence in F, repeating when the sequence runs out.

So far we have constructed what is usually considered the easy part of the table. We have the results of multiplication by numbers up to 5, or at least their final digits.

To get the rest of the last digit table we do the following.

To get the sixth column, we invert the order of the fourth column. To get the seventh column, we invert the order of the third column. To get the eighth column, we invert the order of the second column. To get the ninth column, we invert the first column.

Many people know that the multiplication table is symmetric across the left to right diagonal because multiplication is commutative. We are exploiting a less well known symmetry of the last digit table. It is also symmetric across the right to left diagonal. This follows from the identity

(10-x)(10 -y) = 100 - 10 x - 10 y + xy

The right hand side is the same as xy except for multiples of 10.

Having found the last digits in the table, how do we find the first digits? We employ a simple carrying algorithm. The first digit of every number on the first row is 0. We go down each column separately. Each time, we go from one row to the next, we leave the first digit the same if the last digit has increased, and we increase the first digit by 1 if the last digit has decreased.

Here's an example: Let's say we want to know all the multiples of 7. We remember the last digits of all the multiples of 3. (This was the sequence in D.)

3    6    9    2   5   8   1   4   7

To get the last digits of the multiples of 7, we just invert this list

7   4    1   8   5    2    9   6   3

The first number has 0 as its first digit

07   4   1   8    5    2    9   6    3

4 is smaller than 7 so the second first digit should be increased by 1 from 0.

07  14  1  8   5   2   9   6   3

1 is smaller than 4 so we increase from 1 to  2

07  14   21     8    5   2    9   6   3

8 is larger than 1 so we leave the first digit at 2

07  14  21   28   5   2   9    6  3

5 is smaller than 8 so we increase.

07  14  21  28  35  2   9  6   3

2 is smaller than 5 so we increase

07  14  21  28  35  42  9   6   3

9 is larger than 2 so we hold at 4

07  14  21  28   35  42  49  6  3

6 is smaller than 9 so we increase to 5.

07  14  21   28  35   42  49   56  3

3 is smaller than 6 so we increase from 5 to 6

07  14  21 28 35 42 49 56  63

Voila! The multiples of 7.

What is the use of our snazzy new algorithm for generating the multiplication table? Since it generates the same multiplication table as the conventional algorithm, it must in some sense be equivalent. However, our algorithm seems to generate a little less waste. If you like making tables by adding and you want to show all the work in your additions (carrying the 1 etc.) you need to do scratchwork on a separate page and include only the answers in your finished multiplication table.Here however you can do each step on a single sheet of paper, with no step more complicated than adding 1. At the end, you have a finished multiplication table. I claim that for a student, this is a less tedious way of writing out a table.

Does this have any pedagogical uses? Can it help one to memorize the table? I don't really know. But I claim that the separation between finding the second and first digits is very natural. These steps involve different kinds of mathematics and conflating them may be what leads to some of the difficulties in memorizing the table. If we build the table this way, we see more of the hidden grammar of single digit multiplication.

How would I use this to teach Dagon? I would have him write out the multiplication table a lot of times. I would hope this would get him more accustomed to the shape of the table. Eventually when someone asked him what goes in a particular place in that table, it would be a location he was familiar with from having generated it so many times.

For the moment, the main thing impeding me from asking Dagon to do that is that he has some physical difficulty writing. When that is overcome, this may become a very practical approach.

If asked to explain to a professional mathematician what is good about this approach, I would respond that any approach to the multiplication table which increases the emphasis on the ring of integers modulo 10 cannot help but be clearer than the original presentation.