Back to index of science and math pages by Donald Sauter.

### Single-digit Multiplication - Stepping through from easy to hard

*** Introduction ***

(The first few paragraphs of introduction here are the same as in the companion page single-digit addition.)

If you've never noticed, almost everything you do in math is just some mixture of addition and multiplication (recognizing that subtraction is bound up with addition, and division with multiplication.) And since all addition and multiplication, no matter how big or complicated, is performed stepwise in the brain on two digits at a time, you'll see that the value of quick and sure answers to the single-digit additions and multiplications cannot be overstated.

There is simply no point in going any further in math without complete mastery of single-digit addition and multiplication.

But what's the problem? Isn't this kid's stuff? Didn't everybody learn addition in first grade and multiplication in third grade?

Not by a long shot. I've had much opportunity to observe the math skills of lots of people, from grade school to high school to college to intelligent, educated adults, and what I see is somewhere between heartbreaking and horrifying.

I have some definite notions of what the schools are doing wrong in math. For a start, calculators. Now, I can't swear no student has ever learned a bit of math in spite of calculators, but I assure you no student, anywhere, has ever learned math with a calculator. Grrrr... And there's a lunatic obsession with written explanations of "how you got your answer." Students are moved along on a conveyor belt regardless of how little has stuck. And, sorry about this, basic math is taught by teachers who were never taught basic math themselves. The insanity has been going on for several generations now.

But we're here to get down to multiplication and find a way to think about each basic fact. Obviously, the third grader himself won't be reading this, so the intended audience is anyone working with students in this critical area, or anyone wishing to fill in the gaps in his own basic math skills at any stage of life.

What we are gunning for is "quick answers" to all of the single-digit multiplications. It would be great if we could simply memorize them, but, being realistic about it, this is not an easy thing even for a "really smart" person. But "quick" is practically as good as "instant", and if a rapid-fire intermediate mental step or two is necessary, that is perfectly acceptable. But there must be no fumbling about. Just as in single-digit addition there is always a better way to get to the final answer than by counting up, in single-digit multiplication there is always a better way than by jumping up or down from a known "benchmark", such as getting 6x7 by adding another 7 to 5x7=35. Although that shows an excellent understanding of multiplication, it is too long and too dangerous.

And as with addition, NO FINGERS!!! It seems that doing the nines with the hands has become all the rage nowadays. Teachers, your crutch is crippling your students in math.

Here's the multiplication table:

```             0  1  2  3  4  5  6  7  8  9
-----------------------------
0x | 0  0  0  0  0  0  0  0  0  0
1x | 0  1  2  3  4  5  6  7  8  9
2x | 0  2  4  6  8 10 12 14 16 18
3x | 0  3  6  9 12 15 18 21 24 27
4x | 0  4  8 12 16 20 24 28 32 36
5x | 0  5 10 15 20 25 30 35 40 45
6x | 0  6 12 18 24 30 36 42 48 54
7x | 0  7 14 21 28 35 42 49 56 63
8x | 0  8 16 24 32 40 48 56 64 72
9x | 0  9 18 27 36 45 54 63 72 81

```

Some math curricula step through it line by line, from the "zeroes" to the "nines". This may look like a reasonable approach, but I believe we can find a more natural progression from easiest to hardest. As we step through, I try to suggest a way of thinking about each of the multiplication facts. For most people, there are just a few that cause trouble, so feel free to zero in on those.

Please don't miss the last section. It is crucial that one knows the multiplication table forwards and backwards.

*** The Zeroes (0x) ***

Multiplication by 0 always gives 0. "Zero times something" means you've got no "somethings", and that's nothing. Think of "times 0" as wiping out any number completely.

This is very simple, but, as mentioned in the addition page, there's the danger of mixing up what 0 does in the "addition mode" with what it does in the "multiplication mode". It does two completely different things. Always give +0 and x0, as trivial as they may be, a second thought.

(I'll devote a web page to it one day, but think of math as being divided into two "rooms" - the addition room and the multiplication room. Each room has its own set of tools, many of which correspond to, but are not identical with, tools in the other room. For example, each room has a "do nothing" number called the "identity element". In the addition room that number is 0; in the multiplication room it's 1. You must always be totally aware of which room you're operating in, addition or multiplication. No matter how complicated a math problem gets, your brain only ever operates in one room at one time. At any moment you are doing either addition or multiplication, nimbly stepping back and forth between the rooms as needed. And, to repeat myself, no matter how complicated the addition or multiplication gets, your brain only processes a pair of single digits at a time. In a nutshell: all of math is just a big mixture of single-digit addition and single-digit multiplication.)

But back to the zeroes:

0x0=0
0x1=0
0x2=0
0x3=0
0x4=0
0x5=0
0x7=0
0x8=0
0x9=0

Of course we don't really memorize these as ten separate multiplication facts; we just know that "zero times anything is zero!"

*** The Ones (1x) ***

One is the "do nothing" number when you're doing multiplication. One times something is the exact same something. No need to belabor it.

1x2=2
1x3=3
1x4=4
1x5=5
1x6=6
1x7=7
1x8=8
1x9=9

We leave off 1x0 because zero runs the show in that case. As with the zeroes table, we don't memorize these facts; we always know that 1 times anything is the same number.

Notice I leave 10 out of our basic multiplication table, the main excuse being that it's two digits. This isn't to say that multiplying by 10 isn't fundamental and important; it is very much so. But it comes from a different "place"; in fact, it's bound up with an understanding of place value. It's the easiest and most basic case of the "multiplying and dividing by powers of 10" ball of wax, and kids take right to slapping a zero on a number to multiply by 10. Sadly, I find that few students are proficient in the general case, adding and lopping zeroes and shifting the decimal point as needed. As with the basic addition and multiplication tables, there's no sense in going any further in math without a complete understanding of multiplying and dividing by powers of 10.

Moreover, I would never in a million years put 11 and 12 in the basic multiplication table. They are not basic; they are properly treated just like any other number when we start multiplying multi-digit numbers. And why on earth lumber kids with a 12x12, 144-element multiplication table when the basic 10x10, 100-element table already gives everybody enough trouble? Sure, multiplying 11 times a single digit is a cinch, and it's worth knowing the smaller multiples of 12 because of its (unfortunate) use in the British system of measure, but there is nothing fundamental about multiplying by 11 or 12.

*** The Twos (2x) ***

These are the even numbers up to 18. We nailed them all when we did the doubles in the addition table. I maintain that when we add a number to itself, we see it as a "doubling", which is really multiplication, so there's absolutely nothing new here.

2x1= 2
2x2= 4
2x3= 6
2x4= 8
2x5=10
2x6=12
2x7=14
2x8=16
2x9=18

Again, and from now on, I leave off x0 since zero runs the show in that case. But I leave in 2x1 even though we met 1x2 in the ones table because it belongs here naturally as well. Without it, we would be missing the natural starting point for "counting by twos" (the even numbers).

Another reason for leaving in 2x1 when we've already done 1x2 is because we don't make the same big deal over order as we did with addition. When adding, we usually load up the big number first in our brain because it serves as the jumping off point to get to the answer. In multiplication, one of the numbers will "run the show," but it's not as a jumping off point; it's more like a command code which tells us which procedure to use to get the answer. In this case, the command code 1, whether in the first or second spot, shouts out, "Do nothing!"

It's time for the same short breather we took at this point in the addition page. All we've done are the "zeroes" and the "ones", which are dead easy, and the "twos", which are dead easy and old hat. But together they constitute more than half of the single-digit multiplications, that is, 51 out of the 100 problems in the multiplication table. Dang, math is easy!

*** The Fives (5x) ***

After the "twos", the "fives" are the next easiest. Everyone learns the 5, 10, 15, 20... "counting by fives" sequence very easily, so it's just a matter of getting good at matching the right answer to the right problem. Here's another place where an almost unconscious recognition of evens versus odds is tremendously valuable: 5 times an even ends in "0"; 5 times an odd will end in "5".

5x1= 5
5x2=10
5x3=15
5x4=20
5x5=25
5x6=30
5x7=35
5x8=40
5x9=45

If there's any trouble with these, it will be with the bigger ones. The first two, 5x1 and 5x2, are old hat.

5x3: Easy, easy, easy. It's that number smack in between 10 and 20 and ending in "5", 15!

5x4: Also unbearably easy. Four is even, so it ends in "0". The first one after 10 is 20!

5x5: Probably the world's most popular multiplication problem. It just hollers 25!

5x6: Six is even, so it ends in "0". Obviously, it has to come right after 5x5=25; what else but 30? And remember from the addition page how 6 is one of our swirly 3, 6, 9 "dripping with threes" digits? Isn't it nice that the "6" multiplier gives rise to a "3" digit in the answer?

5x7: Crazy 7 is odd, so it has to end in "5"; has to be bigger than 25; can't be as big as the biggest one (45); what else but 35?

5x8: Eight is even, so it ends in "0"; a big "five" but less than 5x10=50; what else but 40? And isn't it nice that the "4" is a nice, neat, even half of the "8"?

5x9: Nine is odd so it has to end in "5"; biggest of them all (before 5x10=50); what else but 45?

*** The Nines (9x) ***

Nine is the biggest and clumsiest single-digit number, so why should the "nines" come next after the easy "fives"? Because 9 is right next to 10, and 10 is a pure joy to work with. Remember how we used 10 as a stepping stone when adding something to 9? We can do something similar when multiplying by 9.

Here's the nines table:

9x1= 9
9x2=18
9x3=27
9x4=36
9x5=45
9x6=54
9x7=63
9x8=72
9x9=81

We know 9x1 from the ones table, 9x2 from the twos table; and 9x5 from the fives table. Students latch on to the square 9x9=81 quickly and easily, but all the others can be troublesome.

First of all, notice that in every case, the sum of the digits in the answers is always 9. For 18, 1+8=9; for 27, 2+7=9, and so on. Burn that thought in: when you multiply a single-digit number by 9, the digits in the answer add up to 9. Got it?

You also know that 9 times something has to be less than 10 times the same something. And we can multiply by 10 blindfolded.

Putting that together, here's a simple mental procedure. Take 9x6 as an example. You know it certainly can't be as big as ten 6's, which is 60. So the answer has to "fall back" into the 50s. That gives the first digit, "5". Five plus what equals 9? Four! Tack it onto the "5", and there's your answer, 54.

I meet students who will subtract one 6 from 60 for this problem. While that shows a good math sense, I maintain a mental subtraction involving a borrow is generally more difficult and risky than the quick and easy procedure described above.

Here are the troublesome nines, from big to small, using this procedure.

9x8: Ten 8s are 80, so we fall back into the 70s. 7 plus 2 = 9, so the answer is 72.
9x7: Ten 7s are 70, so we fall back into the 60s. 6 plus 3 = 9, so the answer is 63.
9x6: Ten 6s are 60, so we fall back into the 50s. 5 plus 4 = 9, so the answer is 54.
9x4: Ten 4s are 40, so we fall back into the 30s. 3 plus 6 = 9, so the answer is 36.
9x3: Ten 3s are 30, so we fall back into the 20s. 2 plus 7 = 9, so the answer is 27.

I myself only use this procedure for the three big ones: 9x8, 9x7, and 9x6. I get 9x3 from the threes table (3x9), and 9x4 from the fours table (4x9). After we do those, you can decide what works best for you.

*** The Squares ***

A "square" is a number times itself. For some reason, it seems that the squares come a little more easily for students. Perhaps it's because there's only one multiplier to worry about, not two. Just like students can rattle off 9+9=18 when there are still weak spots with the smaller additions, students can blurt out 9x9=81 when there are many smaller multiplications giving trouble.

1x1= 1
2x2= 4
3x3= 9
4x4=16
5x5=25
6x6=36
7x7=49
8x8=64
9x9=81

1x1 and 2x2 are baby stuff.

3x3: No problem. This, of course, is the biggie in our 3, 6, 9 "counting by threes" sequence. Nine is three 3's. You can't get any threeier than that!

4x4: A couple of nice evens which give rise to a nice even in the mid-teens, 16.

I believe it's worth mentioning the "powers of 2" sequence frequently and early. It's fun, and these numbers composed of nothing but 2's come up over and over in math. A "4" is two 2's, so 4x4 is four 2's. Aha, 16 is composed of four 2's; and there it is in the fourth spot in the sequence: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024... (These numbers should look very familiar from computing. For instance, a kilobyte is 1024 bytes.)

5x5: Doubly easy since it's a square and one of our easy "fives". Everybody's favorite multiplication problem; none other than 25!

6x6: Six is one of our swirly 3, 6, 9 "dripping with threes" digits. Isn't it nice that the answer, 36, is made up of digits from the same family? And remember that 6x6 shares its answer with 4x9. When you come up with 36, you might confirm it with a quick thought, "Oh yeah, that's right because I know the answer is the same as for another problem (4x9)."

7x7: I'm not sure of the best way to think about this one. What goes on in my mind is that 7 is our "peskiest" single-digit number, and 49 is probably the most oddball answer in our multiplication table. If you didn't know better, you'd almost think it was prime. So 7x7 and 49 were just made for each other. :-) Remember that 7x7 is in the high 40's, and that it sits right beside another answer in the multiplication table, 48. So when 7x7 comes up, think, "Oh yeah, there's that 48,49 pair, and 7x7 goes with the big, oddball one, 49.

8x8: Students don't seem to have much trouble with this. Eight is big, so 8x8 has to be one of our biggest multiplications. The 60's is a good place for it. Eight is even, and it's kind of nice that the answer, 64, is made up of all even digits.

As with the 4x4 problem, it is worthwhile to play around with the "powers of two" sequence here. An "8" is three 2's, so 8x8=64 is six 2's. And there it is, the sixth in line in the sequence: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024... I assure you, it will come up over and over in math, and very shortly when you get to working with fractions, that 64 is six 2's. Use the first digit of 64 as a mnemonic. (For 32, use the sum of the digits to remind you that 32 is five 2's.)

9x9: As I've said, everyone seems to latch on to this one, 81, right off the bat. If you blank on it, invoke the "nines" procedure.

*** The Threes (3x) ***

What I wish I could say is, "The threes are easy!" After all, three is one of our smallest numbers, so what could be so hard? But the reality is that many, many students are very shaky on the "big threes", meaning 3x7, 3x8 and 3x9. What really breaks my heart is how many people stumble around for 3x6, even.

3x1= 3
3x2= 6
3x3= 9
3x4=12
3x5=15
3x6=18
3x7=21
3x8=24
3x9=27

3x1 and 3x2 are baby stuff, coming from the ones and twos tables, respectively. Likewise, 3x3 presents no problem; we discussed it in the squares.

3x4: This should be an easy one to nail by virtue of being a small multiplication, and yielding a nice "dozen".

3x5: We know this from the fives table.

3x6: This is apparently just on the border of being "too big" to be easy for everybody. Remember that it falls short of 20 and has to be even because one of the factors is even. It might help to remember that it shares an answer, 18, with another multiplication fact, 2x9. But I beg you, if 3x6=18 is not firmly memorized, please make an extra effort to do so.

3x7: Somehow, 3x7=21 always clicked for me. I know that 21, the number just around the "20" corner, is in the multiplication table, and 3x7 is it. If it doesn't click so naturally for you, get familiar with the family of "big threes" (21, 24, 27). Then, 3x7 has to be the smallest one.

3x8: Likewise, after burning in the "big threes" (21, 24, 27), you can quickly associate 3x8 with the middle one, 24. If you're still fumbling around a little, think: the "big threes" are spread out in the 20s, so 3x8, the middle one, has to be in the mid-20s. Eight is even so the answer has to be even.

Also remember that 3x8 shares an answer with another multiplication fact, 4x6. And since 4x6 is also a troublemaker for lots of people, when you find yourself hesitating on a multiplication problem that is not so big, try out 24! I'm assuming that when you do hit on the right answer, you will positively recognize it as such. There should be a feeling of, "Oh yeah, of course; I've seen that a million times."

3x9: You can get this from the nines procedure, but I think it's better to get it from the family of "big threes" (21, 24, 27). 3x9 has to be the biggest one, 27. A quick mental double check for 3x9=27 is, "Oh yeah, 27 is just 3 away from 30."

There, now I've hit you four times with the family of "big threes" (21, 24, 27). Don't forget them!

*** The Leftovers ***

Here are the single-digit multiplications that do not fall in any of the categories above. Since we've run the zeroes, ones, twos, threes, fives, and nines, the leftovers will be just 4, 6, 7, and 8 multiplied by each other. I also include 4x9 because I think it fits much better with the "fours".

4x6=24
4x7=28
4x8=32
4x9=36
6x7=42
6x8=48
7x8=56

Going down the list . . .

4x6: I mentioned 4x6 when we did 3x8 since they have the same answer, 24. Remember good, ol' 24 when you're stymied for a moment!

All you need is one even number as a multiplier to produce an even answer, and 4x6 has two. It cries out evenness, so doesn't 24, made up neatly of two small even digits, make a pretty little answer?

I hope this will never be necessary but, in a pinch, when you're stymied by 4 times something, you can get it by doubling and doubling again, since a 4 is two 2's. So, in this case, 6 doubled is 12, and 12 doubled is 24. But please don't come to rely on that.

4x7: Remember that this is in the upper 20s. Not only that, but it's just above another multiplication table answer. So think, 4x7 is the big one of 27 and 28. This makes sense, since the answer has to be even. And you can do a quick confirmation that 27 can't be the one because we know that goes with 3x9.

4x8 and 4x9: We already met 4x9 in the nines table, but I've always treated it as a fours table problem. Just as we got familiar with the "big threes" (21, 24, 27) in the 20s, there are two "big fours" in the 30s. They are 32 and 36. Think of these as the only two answers in the 30s you need to learn. (30 and 35 are "gimmes".) Burn them in: 32 and 36, both nice, even numbers. Once you've got them, you just match up 4x8 with the small one, and 4x9 with the big one.

A quick mental double check for 4x9=36 is, "Oh yeah, 36 is just 4 away from 40."

6x7 and 6x8: I also think of these as a related pair. We learned the "big threes" in the 20s (21, 24, 27) and the "big fours" in the 30s (32, 36). Think of these as the "big sixes" in the 40s, even though it doesn't include the biggest 6, 6x9=54. The "big sixes" are 42 and 48. Burn them in: 42 and 48, nice, even numbers near the bottom and top of the 40s. Once you've got them, just match up 6x7 with the small one, and 6x8 with the big one.

About 6x7, it always seemed weird to me that our cranky 7 and our "dripping with threes" 6 should produce such a prim and proper number as 42, constructed with two nice, little even digits. If that seems weird to you as well, remember it. Say to yourself, "Oh yeah, 6 times 7, that's the one with the surprisingly pretty answer in the low 40s."

And don't be ashamed to mentally recite this stupid little rhyme for 6x8: "Six times eight/Is for-ty-eight!"

7x8: This one was always instant for me. I hope that means there's something about it that makes it easier to memorize. Think: 7x8 is one of our bigger multiplication facts, and makes it up into the mid-50s. The even "8" means the answer has to be even. There are very few answers to choose from up there (see section below on the Multiplication Table Answers) and it's none other than 56.

A memory aid is that 7x8 "uses up" the four consecutive digits, 5, 6, 7, 8. Think 56=7x8; it's a straight flush!

*** Quick Recap ***

Here is the multiplication table:

```             0  1  2  3  4  5  6  7  8  9
-----------------------------
0x | 0  0  0  0  0  0  0  0  0  0
1x | 0  1  2  3  4  5  6  7  8  9
2x | 0  2  4  6  8 10 12 14 16 18
3x | 0  3  6  9 12 15 18 21 24 27
4x | 0  4  8 12 16 20 24 28 32 36
5x | 0  5 10 15 20 25 30 35 40 45
6x | 0  6 12 18 24 30 36 42 48 54
7x | 0  7 14 21 28 35 42 49 56 63
8x | 0  8 16 24 32 40 48 56 64 72
9x | 0  9 18 27 36 45 54 63 72 81

```

Rather than bashing through it row by row, we took a more comfortable approach, from easier to harder. Here it is again, in condensed form.

• The Zeroes (0x): zero times anything is zero!
• The Ones (1x): one times anything is the number itself!
• The Twos (2x): the even numbers up to 20.
• The Fives (5x): the familiar "five, ten, fifteen, twenty..." sequence.
• 5 x even number ends in "0".
• 5 x odd number ends in "5".
• The Nines (9x): just 9x8 9x7 9x6.
• Step 1: drop back from 10 times the number.
• Step 2: digits add up to 9.
• The Squares: just 3x3 4x4 6x6 7x7 8x8 9x9 - generally no problem.
• 7x7=49: "pesky" 7 produces "oddball" 49; think of 48,49 pair in the multiplication table.
• The Threes (3x): just 3x4 3x6 3x7 3x8 3x9.
• Burn in the "big threes" (21, 24, 27).
• The Leftovers: just 4, 6, 7, 8 times each other (with special guest 4x9).
• 4x6=24: Hits that juicy target in the mid-20s shared with 3x8.
• 4x7=28: Even number in high 20s; think of 27,28 pair in the multiplication table.
• 4x8=32: In the 30s; smallest of "big fours" (32 36).
• 4x9=36: In the 30s; biggest of "big fours" (32 36).
• 6x7=42: In the 40s; smallest of "big sixes" (42 48). Surprisingly pretty number for 6 times 7!
• 6x8=48: In the 40s, biggest of "big sixes" (42 48). "Six times eight/Is for-ty-eight!"
• 7x8=56: Big enough to reach mid-50s; uses up digits 5, 6, 7, 8!

*** IMPORTANT!!! A Look at the Multiplication Table Answers ***

Once again, here's the multiplication table we just learned.

```             0  1  2  3  4  5  6  7  8  9
-----------------------------
0x | 0  0  0  0  0  0  0  0  0  0
1x | 0  1  2  3  4  5  6  7  8  9
2x | 0  2  4  6  8 10 12 14 16 18
3x | 0  3  6  9 12 15 18 21 24 27
4x | 0  4  8 12 16 20 24 28 32 36
5x | 0  5 10 15 20 25 30 35 40 45
6x | 0  6 12 18 24 30 36 42 48 54
7x | 0  7 14 21 28 35 42 49 56 63
8x | 0  8 16 24 32 40 48 56 64 72
9x | 0  9 18 27 36 45 54 63 72 81

```

Now we have all the multiplication facts either memorized or else have a quick procedure to get the answer with no fumbling about. You would think we were done, and at this point in learning the addition table we were, but with multiplication there's something more we must do. We need complete familiarity with all those numbers we see in the multiplication table, the Multiplication Table Answers. For instance, when you see the number 48, you have to recognize it instantly as a Multiplication Table Answer (MTA). Then, if necessary, you would do a quick mental search to come up with the problem that matches the MTA, in this case 6x8. Remember that we've already used this matching technique for some of our multiplication facts, such as for the troublesome "big threes" (21, 24, 27).

When you do division, which is just multiplication going the other way, you will often find yourself trying out MTAs. For example, to divide 592 by 8, you know you have to pick off the first two digits, 59 (because 8 won't go into 5), and then examine MTAs near, and less than, 58. If you know your MTAs, you know that there are only two in the 50s, 54 and 56, and a quick mental examination will lead to, "Oh yeah, 56 is in the eights table, and, um, what was it? oh yeah, 7 times 8." (You say "oh yeah" a lot when cranking math in your brain!)

While we're on the subject, if no one has ever come out and fully admitted it to you, searching and trial and error are part and parcel of division. It's not a failing on your part; it's the nature of the beast.

If I've failed to make an absolutely convincing argument for a total familiarity with the Multiplication Table Answers, please take my word for it - it is CRUCIAL for a good facility with multiplication and division. And that is a BIG part of math! (The other part being addition and subtraction.)

If we look at the multiplication table above we see that it has 100 elements. But, lucky for us, there are really far fewer distinct answers in it. Here are the Multiplication Table Answers placed in their spots in the 100 chart. (A "100 chart" shows the numbers 1 to 100 laid out in natural rows of ten):

```           All of the Multiplication Table Answers

1s:   1   2   3   4   5   6   7   8   9  10
10s:   .  12   .  14  15  16   .  18   .  20
20s:  21   .   .  24  25   .  27  28   .  30
30s:   .  32   .   .  35  36   .   .   .  40
40s:   .  42   .   .  45   .   .  48  49   .
50s:   .   .   .  54   .  56   .   .   .   .
60s:   .   .  63  64   .   .   .   .   .   .
70s:   .  72   .   .   .   .   .   .   .   .
80s:  81   .   .   .   .   .   .   .   .   .
90s:   .   .   .   .   .   .   .   .   .   .

```

If you count them up, you'll see there are only 36 MTAs. Not bad, but it's even better than that. Let's put aside the multiplication facts that are dead easy. That would be the trivial "ones" and "twos", and those proud "fives" standing at attention in their two columns. Also, 3x4=12, which is a piece of cake. And promise me you'll remember 18, besides being 2x9, is also 3x6.

Once we weed the baby stuff out, we're left with:

```        The nontrivial Multiplication Table Answers

1s:   .   .   .   .   .   .   .   .   .   .
10s:   .   .   .   .   .   .   .   .   .   .
20s:  21   .   .  24   .   .  27  28   .   .
30s:   .  32   .   .   .  36   .   .   .   .
40s:   .  42   .   .   .   .   .  48  49   .
50s:   .   .   .  54   .  56   .   .   .   .
60s:   .   .  63  64   .   .   .   .   .   .
70s:   .  72   .   .   .   .   .   .   .   .
80s:  81   .   .   .   .   .   .   .   .   .
90s:   .   .   .   .   .   .   .   .   .   .

```

Now we're down to just 15 MTAs that might require some small effort to nail. That should amaze you. We've shown that . . .

All of the nontrivial multiplication and division in mathematics boils down to knowing just 15 numbers!

Has anyone ever told the schools this? Does the mathematics education establishment know it? Here's what the nontrivial MTAs look like simply strung out . . .

21   24   27   28   32   36   42   48   49   54   56   63   64   72   81

. . . but a striking way to view their sparseness is to work backwards through the 100 chart, line by line.

• There are NO answers in the 90s.
• There is only ONE answer in the 80s: 81
• There is only ONE answer in the 70s: 72
• There are only TWO answers in the 60s: 63 64
• There are only TWO answers in the 50s: 54 56
• There are only THREE answers in the 40s: 42 48 49
• There are only TWO answers in the 30s: 32 36
• There are only FOUR answers in the 20s: 21 24 27 28

Notice how the layout in the 100 chart makes clear something that has come up a few times earlier. Ignoring the trivial and dead easy multiplication facts, there are just three instances of consecutive numbers in the multiplication table: (27,28), (48,49) and (63,64). I've suggested that you might make use of these pairs. About the troublemaker 4x7, think, "Oh yeah, that's the bigger one of that pair in the upper 20s, what was it? oh yeah, 27 and 28!" About the peskiest square 7x7, think, "Oh yeah, that's the bigger one of that pair in the upper 40s, what was it? oh yeah, 48 and 49!" I don't use the (63,64) pair in such a way to get at the answer to either multiplication fact, 9x7 and 8x8, but, when doing division, it's important to remember that there are only two Multiplication Table Answers in the 60s, and that they are, in fact, side by side in the lower 60s - "Oh yeah, 63 and 64!"

Before we go, let's put all of the MTAs back in the chart:

```           All of the Multiplication Table Answers

1s:   1   2   3   4   5   6   7   8   9  10
10s:   .  12   .  14  15  16   .  18   .  20
20s:  21   .   .  24  25   .  27  28   .  30
30s:   .  32   .   .  35  36   .   .   .  40
40s:   .  42   .   .  45   .   .  48  49   .
50s:   .   .   .  54   .  56   .   .   .   .
60s:   .   .  63  64   .   .   .   .   .   .
70s:   .  72   .   .   .   .   .   .   .   .
80s:  81   .   .   .   .   .   .   .   .   .
90s:   .   .   .   .   .   .   .   .   .   .

```

And what do we have? We have a worksheet for all the basic division problems! Each number in the chart is a division problem (in some cases two.) There's no reason to frame basic division problems in the form 81÷9=__ or 35÷7=__ . If the student has gotten what's going on here, he'll tell you the division problem and the answer given just the number. If you say "81", he answers, "81 divided by 9 is 9". If you say "35", he answers "35 divided by 5 is 7", or, the other way around, "35 divided by 7 is 5." For a few of the two-digit MTAs, specifically 12, 16, 18, 24, and 36, there are two division problems associated with the number. Given 24, for example, the student comes back with, "24 divided by 4 is 6, and 24 divided by 3 is 8."

And, believe me, there's not much more to math than that. After recognizing that numbers are made up of factors we're equipped to work with fractions. Exponentiation is repeated multiplication, but we do not memorize exponentiation tables; we carry out all work with exponentials using just our basic addition and multiplication skills. Algebra is just arithmetic with letters introduced to stand for numbers. And then it's on to solving for the "unknown" using the self-evident logic, "whatever you do to one side of the equation, you do to the other side."

And that, my friends, is more than 99% of all the math that 99% of the population will ever need to know. Areas? Volumes? Slopes? Probabilities? Statistics? All applications of basic addition and multiplication. Even calculus is nothing more than a big, old bucketful of addition and multiplication.

Got it?

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