One thing that I like about math is that it is so orderly! If you stick with it, things compute. They stay constant and objective. At the same time, math is a marvelous mystery. There is always something new and thrilling to discover. The patterns can be awe inspiring and even beautiful. In so many ways, I can see the hand of a Master Designer (our Gracious Creator) in math. Who else could invent all of the intricacies of geometry or the sense of justice found in algebraic equations?

I have had a load of fun computing the sums of digits in multiplication facts. I know this is a bit complicated, but stick with me for a minute. Take a piece of paper, and write the numbers 1 to 20 down the side. In the next column, write the sums of the digits of each number.

For example, the digit sum of 11 is 2, the digit sum of 12 is 3, etc. If you get a sum over 9, add the two digits of that number. For example, the intermediate digit sum of 59 is 14, and adding 1 and 4, the final sum is 5.

The pattern for multiples of 1 thus goes like this: 1, 2, 3, 4, 5, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9... until infinity.

Now, in another column, write the multiples of 2 up to 40, and the corresponding digit sums. The new pattern will be 2, 4, 6, 8, 3, 5, 7, 9, 2, 4, 6, 8, 3, 5, 7, 9.... (Even numbers in ascending order, then odd numbers in ascending order.)

Now do the multiples and digit sums for other numbers.

The digit sums for multiples of 3 will be 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9.... (Every multiple of 3 has a sum of digits as a multiple of 3 also. That’s a quick way to see if a number is divisible by 3.)

The digit sums of multiples of 4 will be 4, 8, 3, 7, 2, 6, 1, 5, 9, 4, 8, 3, 7, 2, 6.... (Every other number is decreased by one. Add 4, subtract 5, add 4, subtract 5, etc.)

The digit sums of multiples of 5 are 5, 1, 6, 2, 7, 3, 8, 4, 9.... (Every other number is increased by one. Subtract 4, add 5, subtract 4, add 5. Compare this with the 4s.)

The digit sums of multiples of 6 are 6, 3, 9, 6, 3, 9.... (Compare this with the digit sums for 3.)

For seven, the digit sums are 7, 5, 3, 1, 8, 6, 4, 2.... (Odd numbers is descending order, even numbers is descending order.)

For eight, the digit sums are 8, 7, 6, 5, 4, 3, 2, 1, 9... (Numbers in descending order.)

For nine, all digit sums are 9!

If you summarize the results, you find pairs of corresponding number patterns.

1: numbers in ascending order

8: numbers in descending order

2: even numbers is ascending order, odd numbers is ascending order

7: odd numbers is descending order, even numbers is descending order

3: multiples of 3 in ascending order

6: multiples of 3 in descending order

4: +4, -5, +4, -5...

5: +5, -4, +5, -4...

9 is unique, in that all digit sums are equal to 9. (Another quick way to verify your multiplication answers!)

Now, to go a bit further: If you do multiples of 10, 11, etc. you will find that the same patterns repeat. The digit sums for multiples of 10 are the same as for 1, 11 is the same as 2, 12 is the same as 3, etc. This happens in intervals of 9, so the digit sums for multiples of 5 are the same as for multiples of 14, 23, 32, etc.

Are you thoroughly confused yet? Try it again. Middle school students might be able to understand this. Dr. Ruth Beechick mentions it in her book

**! (This is an excellent book.)**

*You Can Teach Your Child Successfully, Grades 4-8*I wrote this article about 10 years ago in my Hope Chest e-magazine. I've had fun tonight browsing through my archives of old issues on the computer!

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