How to Remember Multiplication Tables

wordpress.com has a “stats” feature that allows you, among other things, to see what search terms people have used to find your blog. It turns out that most hits that I get from search engines are by people wanting to know how to memorize their multiplication tables (they find the post Multiplication Tables, which doesn’t actually give any advice on memorizing them). I’ve started to feel so guilty about this that I decided I should write a post giving tips on memorizing multiplication tables (or times tables) hoping that it would be helpful to some of those folks searching for help. Then maybe they will read some of the other posts and get hooked on math

So here goes:

We start with something that sounds totally unrelated (but stick with me): James A. Michener’s Novel Texas. In that book there is a character named Miss Barlow, who is a school teacher. Michener has her: “Standing before a massive map of Texas that showed all the counties in outline only, she said softly ‘Your Texas has two-hundred and fifty four counties.’” Miss Barlow goes on to challenge the students to be able to name, from memory, every county when pointed out on the map. One big obstacle to achieving this feat (besides the shear immensity of the number of counties) is that many of the counties in West Texas and the Panhandle are nearly-identical little rectangles. But never fear! This teacher has a method to cut this memorization down to size. First you select any five counties that you wish. The only criterion for the selection of your counties that must be met is that each must be in a different region of Texas. In other words, you can’t group them all together. You have to pick counties scattered around Texas. Once you have those five counties nailed down and committed to memory you “could build upon them the relationships required in Texas history,” and continue to learn the names of the surrounding counties, until you have all 254 memorized.

Now let’s look at the multiplication table and see what it is like. Here it is, the 0 – 12 multiplication table in all its glory:

The only thing that I’ve left off are the numbers! But don’t worry about those! We’ll get to them soon enough. (By the way, why do we ever ask kids to memorize the 0 – 12 multiplication table? We use a base ten system. Memorizing 0 – 9 should be perfectly sufficient). And here is Texas:

Almost eerie how similar they are isn’t it? Look at all of those identical little squares in the multiplication table and those counties stacked one on top of another in Texas. Let’s try to approach the multiplication table from Miss Barlow’s view point. Actually the multiplication table is much easier than the map of Texas. Texas has 254 counties, but our multiplication table has entries to remember.

Here’s how I remember them.

Also, let’s face the fact that there are some really easy ones: The zero-times-blank-row and the blank-times-zero-column are a piece of cake, so is the one-times-blank-row and blank-times-one-column, and let’s face the fact that the ten-times-blank-row and blank-times-ten-column are just as easy. So already we have knocked out a big chunk of the table:

Of course everyone knows that 3 times 7 is the same thing as 7 times 3. We can get rid of almost half of our squares:

Looking more like Texas all the time. Now 5′s are easy, at least if you are multiplying by an even number, just chop it in half and throw on a zero. There’s also a handy trick for 9′s. For instance 9 times 7 is 63. The 6 is one less than 7 and the 3 is 9-6. It works with all of the single digit numbers, for instance 9 times 3 is 27.

Multiplying a single digit number by 11 is also easy, just write it twice, for instance 8 times 11 is 88. Now maybe it’s not exactly easy, but I think that everybody should be able to memorize the double of every number from 2 to 12:

Once you know those, it is pretty easy to multiply 12 by any number less than 5. Just write the number and then its double:

and

We don’t have much left to memorize! Finally, I think that it is worth while to memorize all of the squares. Here are the ones that we haven’t dealt with so far:

But maybe you don’t want to do that much work. So how about you just memorize and . Then if you memorize one more fact: , and look at the table, you might notice something:

Every square left blank is directly adjacent to (above, below, to the right or the left) a filled in square. Now build upon these the relationships required in arithmetic: For instance, let’s say that you want to know , well you know , so . You want to know ? Well, you know , so . Voila! Multiplication is a snap. Of course it’s going to take a lot of practice, but good luck! I hope it helps.

Synchronicity

I first heard the word synchronicity when I was sitting in an airport, waiting for a plane, working on this math problem (for fun) and over-hearing a man’s cell phone conversation. I thought that it was a good word for this phenomenon. Here it is:

Have you noticed that 3 times 7 is 21 and if you expand 7 base 3 it is 21. That is:

This seems like an interesting coincidence. Could it be that this happens for any other numbers? Maybe it happens infinitely often, maybe never. Asking such questions is where the adventure begins. Of course we are implicitly working base 10 here. That is, in the first line the string of numerals “21″ is interpreted as a base 10 number. In the second one it is (explicitly) interpreted base 3. You could allow 10 to be replace by other bases also.

If we restrict our attention to base 10 (in the first line), I claim that, by exhaustive computer search, there are only five other instances of this phenomenon. They are:

and

and

and

and

and

Multiplication tables

After reading Lockhart’s Lament (see yesterday’s post) you may wonder what my experience with math in school was and how I got to be a mathematician. Here are a few, rather disjointed vignetes :

Once (upon a time) in 1st grade my teacher was teaching us about multiplication. She said that she had a little secret that she knew: when you multiply a number by one you get the same number that you started out with! So there you go, we should have no trouble multiplying by one now. I thought that it was great that she had discovered this fact and I wanted to get in on the act also. I had noticed a pattern myself: when you multiply by two you get double the number that you started out with! Now there was a fact that my teacher hadn’t yet discovered! So I went to her and told her this little secret that I had learned. I thought that surely she would gather the class and tell them that: hey look, here is another little trick! She never did. I wasn’t crushed, but I was puzzled. After all, at the time at least, my secret seemed just as good as hers. I mention this for two reasons:
1) Because of the incident I know that I had a natural, intuitive understanding of what “doubling” meant before learning about multiplying by 2, and, at the time that I was first learning multiplication, these two things seemed like distinct and separate operations. (If I didn’t remember this incident, I wouldn’t remember how I thought about doubling and multiplying by 2 at that age).
2) Also at that age I thought of math as a process of discovery that everyone was allowd to participate in (which it is!). The system hadn’t yet drummed that idea out of me. I guess that the system never really did drum it out of me completely.

My second memory comes from first grade also, I’m pretty sure, but it could have been second grade. We were learning how to add multi-digit numbers and we were supposed to be working on a sheet where we calculated dozens of such sums. I thought it was cool that the technique we were using would let you add two numbers no matter how big they were! So I flipped my sheet over, wrote down two really big numbers, and started adding them. My teacher caught me before I had gotten very far and made me turn my sheet back over and work on what I was supposed to be doing.
Retrospective comments:
1) Why did she do that? Wouldn’t I have gotten just as much practice from working out my own sum? Yes! And I would have been having more fun. Again, I was participating in discovery for myself. But, of course, it would have been harder for her to check my work for accuracy…
2) I was having some fun with the playground in my mind! These big numbers had nothing to do with any real-life application. Kids just like big numbers! In fact kids love big numbers. My niece really likes the number 1,082. It was a kid who made up the words googol and googolplex. Why don’t we let kids play with numbers in a math class?

The next scene is from fourth grade. My teacher called two students to the front of the room. (I think that she had arranged this with them ahead of time). On the board she had a problem for each of them. One was to calculate $\latex 7+7+7$. The other was to calculate . They set about calculating. Well, I should say that the first student set about calculating . The second student simply wrote down . When they had both finished, our teacher said “See, multiplication is just a fast way of adding.” I was appalled. I thought about what would have happened had it been me up there. I didn’t have memorized, so I would have had to calculate . It’s the exact same problem! There is no easier way (that I was aware of at the time) to calculate other than calculating . Of course, if you already have the answer memorized, then of course you are going to be able to do it quickly. You just write it down! I don’t know if it was this demonstration, or something else (or just the fact that I am bad at memorizing facts) that turned me off to memorizing my times tables. In any case, I remember staying after school (or maybe it was staying in from recess) in order to learn my times tables with the other slow kids.
My comments here are:
1) Looking back I realize that the teacher could have had a much better demonstration if she had given something like to the first kid and to the second, or, even better, for the first and for the second, then we would have seen the power of this cool little algorithm for multiplying. You know, the one that you learned–just do it a digit at a time.
2) I don’t think (and I could be wrong on this) that multiplying is just a quick way to add. It certainly can be a quick way to add, but I think that going from adding to multiplying is a pretty big leap. Multiplying is a whole other animal. It is related to adding, and it interacts with adding in interesting ways, but I think that it is more than just fast adding. (Like I said I could be wrong. This is a philosophical issue, as much as a mathematical one.)

That’s enough for now, but look for more such vignettes in later posts.