1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 121 132 144

It's very important to memorize these up to at least ten times ten.

In the interest of promoting number base diversity...


Octal
|---|---|---|---|---|---|---|---|---|---|
|01 |02 |03 |04 |05 |06 |07 |10 |11 |12 |
|---|---|---|---|---|---|---|---|---|---|
|02 |04 |06 |10 |12 |14 |16 |20 |22 |24 |
|---|---|---|---|---|---|---|---|---|---|
|03 |06 |11 |14 |17 |22 |25 |30 |33 |36 |
|---|---|---|---|---|---|---|---|---|---|
|04 |10 |14 |20 |24 |30 |34 |40 |44 |50 |
|---|---|---|---|---|---|---|---|---|---|
|05 |12 |17 |24 |31 |36 |43 |50 |55 |62 |
|---|---|---|---|---|---|---|---|---|---|
|06 |14 |22 |30 |36 |44 |52 |60 |66 |74 |
|---|---|---|---|---|---|---|---|---|---|
|07 |16 |25 |34 |43 |52 |61 |70 |77 |106|
|---|---|---|---|---|---|---|---|---|---|
|10 |20 |30 |40 |50 |60 |70 |100|110|120|
|---|---|---|---|---|---|---|---|---|---|
|11 |22 |33 |44 |55 |66 |77 |110|121|132|
|---|---|---|---|---|---|---|---|---|---|
|12 |24 |36 |50 |62 |74 |106|120|132|144|
|---|---|---|---|---|---|---|---|---|---|

Hexadecimal
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|01 |02 |03 |04 |05 |06 |07 |08 |09 |0A |0B |0C |0D |0E |0F |10 |11 |12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|02 |04 |06 |08 |0A |0C |0E |10 |12 |14 |16 |18 |1A |1C |1E |20 |22 |24 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|03 |06 |09 |0C |0F |12 |15 |18 |1B |1E |21 |24 |27 |2A |2D |30 |33 |36 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|04 |08 |0C |10 |14 |18 |1C |20 |24 |28 |2C |30 |34 |38 |3C |40 |44 |48 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|05 |0A |0F |14 |19 |1E |23 |28 |2D |32 |37 |3C |41 |46 |4B |50 |55 |5A |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|06 |0C |12 |18 |1E |24 |2A |30 |36 |3C |42 |48 |4E |54 |5A |60 |66 |6C |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|07 |0E |15 |1C |23 |2A |31 |38 |3F |46 |4D |54 |5B |62 |69 |70 |77 |7E |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|08 |10 |18 |20 |28 |30 |38 |40 |48 |50 |58 |60 |68 |70 |78 |80 |88 |90 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|09 |12 |1B |24 |2D |36 |3F |48 |51 |5A |63 |6C |75 |7E |87 |90 |99 |A2 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|0A |14 |1E |28 |32 |3C |46 |50 |5A |64 |6E |78 |82 |8C |96 |A0 |AA |B4 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|0B |16 |21 |2C |37 |42 |4D |58 |63 |6E |79 |84 |8F |9A |A5 |B0 |BB |C6 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|0C |18 |24 |30 |3C |48 |54 |60 |6C |78 |84 |90 |9C |A8 |B4 |C0 |CC |D8 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|0D |1A |27 |34 |41 |4E |5B |68 |75 |82 |8F |9C |A9 |B6 |C3 |D0 |DD |EA |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|0E |1C |2A |38 |46 |54 |62 |70 |7E |8C |9A |A8 |B6 |C4 |D2 |E0 |EE |FC |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|0F |1E |2D |3C |4B |5A |69 |78 |87 |96 |A5 |B4 |C3 |D2 |E1 |F0 |FF |10E|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|10 |20 |30 |40 |50 |60 |70 |80 |90 |A0 |B0 |C0 |D0 |E0 |F0 |100|110|120|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|11 |22 |33 |44 |55 |66 |77 |88 |99 |AA |BB |CC |DD |EE |FF |110|121|132|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|12 |24 |36 |48 |5A |6C |7E |90 |A2 |B4 |C6 |D8 |EA |FC |10E|120|132|144|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Tips for Remembering Specific Multiplication Facts

Back in my day

When I was in second grade, my family moved and I started at a new school. The first week I was there, I came home and told my mom that the teacher said we had to write the multiplication tables. Mom tried to tell me that I probably only had to write the zeros, or the ones, and that most likely my class would be working on multiplication for weeks, but I wouldn’t listen. I was new to the school, I had not understood the assignment, and being my little type A eight-year-old self, I stayed up half the night writing all the multiplication tables:


0 X 0 = 0     1 X 0 = 0     2 X 0 = 0     3 X 0 = 0     4 X 0 = 0 	5 X 0 = 0
0 X 1 = 0     1 X 1 = 1     2 X 1 = 2     3 X 1 = 3     4 X 1 = 4 	5 X 1 = 5
0 X 2 = 0     1 X 2 = 2     2 X 2 = 4     3 X 2 = 6     4 X 2 = 8 	5 X 2 = 10
0 X 3 = 0     1 X 3 = 3     2 X 3 = 6     3 X 3 = 9     4 X 3 = 12	5 X 3 = 15
0 X 4 = 0     1 X 4 = 4     2 X 4 = 8     3 X 4 = 12    4 X 4 = 16       5 X 4 = 20
0 X 5 = 0     1 X 5 = 5     2 X 5 = 10    3 X 5 = 15    4 X 5 = 20	5 X 5 = 25

all the way up to 

0 X 12 = 0    1 X 12 = 12   2 X 12 = 24   3 X 12 = 36   4 X 12 = 58     5 X 12 = 60


Of course, the above illustration only shows the zeros through fives. I wrote the sixes, sevens, eights, nines, tens, elevens, and twelves, too. And when I got to school it turned out my mom had been right (again), and I had done about six weeks worth of work. Sigh.

Newfangled advancements

These days, I teach multiplication to students with learning disabilities. Thank goodness they (we) no longer teach it the way they did when I was in school. These days, students are asked to write just the answers in order, not the whole problem, and they only have to go up to 10 X 10.

As mentioned in previous writeups, the lower numbers are not so hard to memorize. Zero times anything is zero; that takes care of ten whole facts right there (we usually start with 0 X 1, not 0 X 0). The ones are easy; it’s just counting up: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Little kids often learn the twos and the fives early on, so they’re not usually a problem, even for special populations. Even if you’re a kid who has trouble memorizing, you can count up by threes or fours fairly quickly, especially if you’re allowed to use your fingers.1 Multiplying three times four, for instance, could be accomplished as follows:

Three, four, five, six, seven, eight, nine, ten, eleven, twelve; the answer is twelve.

Or, alternately, counting by fours:

Four, five, six, seven, eight, nine, ten, eleven, twelve; the answer is twelve.

The cold, hard facts

The sixes, sevens, and eights are harder, but only the upper half of each, because the lower halves have already been learned. Six times zero, one, two, three, four, and five have already been memorized (or a technique for finding the answer found) when working on the zero, one, two, three, four, and five tables. So really, when you get right down to it, there are only a handful of hard-to-remember facts:

 
6 X 6 = 36           7 X 6 = 42            8 X 6 = 48    
6 X 7 = 42           7 X 7 = 49            8 X 7 = 56
6 X 8 = 48           7 X 8 = 56            8 X 8 = 64

And of those facts, three are repeated ( 6 X 7 = 42 and 7 X 6 = 42, etc.) So there are only six hard facts to memorize. That’s when we start to get creative.

The key is to come up with something that is memorable. It can be stupid, or silly, or gross (gross is particularly good, when working with little boys), as long as it sticks in the kids’ heads. There are books published that contain pictures and stories for each math fact; unfortunately, I don’t have any titles to offer. But I know the books exist, and have two examples that came from them. 2

The first is for the fact 4 X 4, which isn’t one of the really hard ones. The picture in the book shows a 4 X 4 truck, and makes the point that in order to be old enough to drive the 4 X 4, you have to be 16. (Get it? 4 X 4 = 16). The next example is for 6 X 6, and shows two sixes, dressed as legionnaires, making their way across the desert. It’s hot, they’re thirsty. They’re thirsty sixes. Thirty-six. 6 X 6 = 36.

The other way to remember 6 X 6 = 36 is the phrase, or prompt, “half again”, as in half of six, and then six again.

The next fact involves a little story.

A few years ago there was an article in the news about two first graders who had ‘gotten married’ on the playground, but then, after a while, they were just not getting along. The magic was gone. Of course, they got divorced. That should have been the end of it, but it was one of those messy divorces. The children’s parents got involved, and there were restraining orders to keep the kids away from each other, and eventually the whole fiasco made the news.

Tell the students this story, and then explain that the prompt for 6 X 7 is “six year old and seven year old, say ‘I do’…”

which rhymes with the answer, 42.

6 X 8 Is much less involved. It’s just the rhyme, “six times eight, fork and plate…”

which rhymes with the answer, 48.

7 X 7 Is a sports fact: “touchdown, 7 points” leads to the San Francisco 49ers football team,

and hence the answer, 49.3

7 X 8 Works better if you write it down; “the answer and the question are all in a line

56 = 7 X 8 ; (5, 6, 7, 8).

8 X 8 Is the best, for the gross-out factor: “I ate and I ate until I got sick on the floor..."

(ate = 8, sick = sixty, floor = four; the answer is 64).

You may have noticed that I have left out the nines times tables.

Tricks for learning the nines

Write the numbers zero through 9 vertically, and then write the numbers 9-0 next to them:4

0   9

1   8

2   7

3   6

4   5

5   4

6   3

7   2

8   1

9   0

That works for most people, but there is another way, which doesn't involve any writing. To multiply n times 9, hold up both hands, palms facing away. ( YOU'RE GOING TO HAVE TO DO THIS TO MAKE IT WORK. GO ON, HOLD UP YOUR HANDS.) Count from the left pinkie over n fingers (for instance, if you are figuring out 4 X 9, count over 4 fingers, to your left index finger) and fold that finger down. Everything to the left of the folded finger is tens (in this example, three fingers = thirty) and the fingers to the right are ones (in this example, six fingers = 6), so the answer is 36.

In this lovely ascii example of 6 X 9, the sixth finger from the left (the thumb on the right hand) is folded down. Everything to the left of that thumb becomes a ten, and everything to the right, a one:

 
    | | |            | | |  
  | | | |            | | | |    
  | | | |  /         | | | |      
          /      ( )
  1 2 3 4 5          1 2 3 4
   tens               ones
    5                  4

A common mistake with this method is that fingers on the left hand are always 10's, and fingers on the right 1's. This is simply not the case. Try 8 X 9 ; fold down your right middle finger. You should have seven fingers to the left of the folded finger, and two to the right, signifying the answer 72.

Cool, huh?

_____________________

1 I'm a strong believer in counting on one's fingers. If you've got 'em, use 'em. Eventually, either you'll get to the point that you don't need to count that way anymore, or you'll get really, really fast (and more subtle) at it. Either way, it's an acceptable modification. 2 If you happen to know the title of this (or similar) books, please let me know. 3My mom used to say “close to fifty” or “miner 49er”, which both link to the answer, but not so much to the question. But what ever works, works. 4Something similar works for the eights. The first column has the numbers 0-8, with four repeated, and the second column counts down by twos, starting with eight. It's a bit more involved, but it works for some kids:

0   8

1   6

2   4

3   2

4   0

4   8

5   6

6   4

7   2

8   0

For more math fun, see TouchMath.

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