Bryan Paul Sullo

Multiplication without Memorization

In Uncategorized on 29 May 2015 at 10:43 am


Round about fourth grade, we were expected to learn the multiplication table—the big ten-by-ten grid of all the single-digit integers and their resulting products. Well, I never got the hang of it. Sure, I grasped the concept of multiplication, probably better than most of my classmates, but I couldn’t for the life of me remember that 7 x 8 = 56, or that 4 x 6 = 24, or any of the other, seemingly random facts.

To this day, I can’t remember most of the multiplication table. My brain doesn’t work that way. I have a hard time memorizing lists of meaningless things. What I can remember though are rules. Rules have meaning. They can be understood and reasoned out and then applied to a problem. I’d like to share my multiplication rules with the world. This may be completely useless in the computer age, but if I can help some struggling fourth-grader somewhere (or even a struggling adult) to overcome a problem that seems to be so simple for most, it will be worth it.

Like I said above, the traditional memorization approach is a ten-by-ten grid of the integers from zero to nine. To be fair, their aren’t one hundred facts to memorize: Zero and one are taken care of with rules (as I’ll explain below). Of the remaining sixty-four facts, twenty-eight are just duplicates, which leaves thirty-six facts to be memorized—still a pretty hefty quantity of random number groups.

So, that’s thirty-six facts and two rules—thirty-eight items to memorize in all. My method consists of a single fact and five rules—six items in all. Which one sounds easier to you?

Fair warning, if you’re one of the majority of people who had no problem memorizing the multiplication table, this is going to sound a lot more complex than just remembering a bunch of numbers. If you (or your child) are like me, however, this is truly easier. Trust me.

Let’s start with the two rules that are common to both methods:


The first rule (for zero) is called “Always Zero”. Zero times anything else is always zero. Pretty easy to comprehend and to remember.


The second common rule is called “Always X”. It applies to anything multiplied by one. One times X is always X. Again, there’s no big mystery there.


In fact, I don’t know that anyone would teach multiplication by asking their students to memorize each of the nineteen combinations with one and zero. They’re going to teach the two rules I mentioned above, and then move on. Why, then, do they teach the rest of the multiplication table as discrete facts, when there are rules that can take care of most of them?

Here is where my method starts to diverge from the traditional memorization model. I’ll start each paragraph with the number to which each rule applies.


My next rule is called the “Rule of X + X”. Anything times two is just that number plus itself. Yes, that’s how we’re taught the concept, but we’re encouraged to memorize that 4×2=8. Why? We’ve already memorized, when we learned addition, that 4+4=8, so why memorize a different fact to represent the same thing. (Even for people who had trouble memorizing addition, it’s simple enough to add by counting and come up with the right answer.)


“Wait!” you say. “What happened to three and four?” I’ll get to that later. The “Rule of Fives” is simply to count by fives: five, ten, fifteen, twenty… Want to get five times six? Just count by fives, six times. Use your fingers to keep track if it helps. It’s actually very fast.

Another way to do the “Rule of Fives” is to say that any even number times five is just the even number divided by two with a zero tacked on the end (2×5=10, 4×5=20, etc.) and any odd number times five is just the lower even number times five, with a five tacked on the end. (It’s a neat trick, but it’s actually making two rules for fives, and that’s too complicated for me.)


“The Rule of Nines” is a fun one. To get the product of any single-digit integer and nine, take the original integer and subtract one. Take that result and figure out whatever you need to add to it to get nine. Put the two numbers side by side. For example, 7×9=? Subtract one from 7 and get 6. You need 3 more than 6 to make 9, so the answer is 63. (7-1=6, 9-6=3, 7×9=63)


That’s it. That’s all you need to remember. Oh, almost. I left out three, four, six, seven, and eight. Seems like a lot, but they’re not if you understand the system. There’s nothing to memorize about them, and many of the possible combinations are already covered by the rules above.


For threes, you can just use the rule for two (the “Rule of X + X”) and add X again, because X times 3 is just X more than X times 2. 8×3=(8×2)+8

There are only 5 cases where you need to use this method.


For fours, you can just use the rule for five (the “Rule of Fives”) and subtract X. 8×4=(8×5)-8. If you want something even simpler, Use the “Rule of X + X” twice: 8×4=8×2×2.

There are only 4 cases where this method is necessary.


For Sixes, its “Rule of Fives” plus X.

Only three cases where this method is needed.


Multiplying by eight is just the “Rule of Nines” minus X.

This is only needed in 2 cases: 8×7 and 8×8.


That leaves sevens. Seven is a special case. Every combination of anything times seven has already been covered by another rule, except one: Seven times seven. That’s forty-nine. Nothing to do but memorize that one fact.

  1. This is a fantastic aid!

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