MULTIPLICATION TABLES
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Multiplication Tables: x2 and x3
Constructed by John Eshleman, Ed.D.
November 11, 1998; April 9, 2000
Dr. Ogden Lindsley has repeatedly asserted (e.g., at the ABA conventions) that we "do not learn multiplication." He often has repeated the observation that behavior "lives in the multiply world." The Standard Celeration Chart uses a "multiply-divide" scale up the left (the y-axis). Celerations have multiply or divide values, and carry multiplication or division signs. Multiplication forms a key element of standard celeration charting.
Most of us are very familiar with the traditional multiplication tables. Therefore, I created the following multiplication tables to illustrate both "times twoing," and "times threeing." Both tables show multiplication as a series. In both cases, there is a starting digit, running from 0 to 9. Obviously, multiplying anything by 0 results in a product of 0. Lindsley thought that, nevertheless, I should include that "times zero" line, so I did.
X2 TABLE
Directions: Use the successive multiply table by going across rows. Every number in a row on the table is X2 (two times) the number to its left, and /2 (divide by two) the number to its right. Note that multiplying 0 by 2 does not change the value. Note: For multiplying, read each series across.
Times Two Table Start: | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
2 |
4 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
2 |
4 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
2048 |
3 |
6 |
12 |
24 |
48 |
96 |
192 |
384 |
768 |
1536 |
3072 |
4 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
2048 |
4096 |
5 |
10 |
20 |
40 |
80 |
160 |
320 |
640 |
1280 |
2560 |
5120 |
6 |
12 |
24 |
48 |
96 |
192 |
384 |
768 |
1536 |
3072 |
6144 |
7 |
14 |
28 |
56 |
112 |
224 |
448 |
896 |
1792 |
3584 |
7168 |
8 |
16 |
32 |
64 |
128 |
256 |
512 |
1024 |
2048 |
4096 |
8192 |
9 |
18 |
36 |
72 |
144 |
288 |
576 |
1152 |
2304 |
4608 |
9216 |
Times Two Table -- NOTES:
The series starting at 1, 2, 4, 8 are the same, having only differing starting points.
The same goes for the series starting at 3, and 6.
The series starting at 5, curiously, becomes the same as that starting at 1,
except each product has a 0 at the end of the number.
This makes the series starting at 5 easily predictable.
The series starting at 7, if you look at the last digit, then the last two digits,
and then later the last three digits, bears a relationship to the series that starts at 3.
Also, it has 4, 8, 6, 2,..., repeating last digit.
In the series that starts at 9, if you add up the digits of each product,
they will add to 9 or 18 (which also adds to 9). E.g., 1152: 1 + 1 + 5 + 2 = 9.
************************************************************
Times Two on the Chart
We can show "times two" on the Standard Celeration Chart. (Keep in mind that the chart section shown here represents a "likeness" of a portion of the actual Standard Celeration Chart. While the "times two" slope is very close to that of the actual "times two" slope, it may differ slightly. That's why, for real decision-making purposes, and publication purposes, one should always use actual charts. The figure shown here is for illustrative purposes only.)
Note that the series starts at 1, just as with the Table above. On the Daily Behavior Chart, each celeration period runs 1 week (7 days). Each thick "up and down" line stands for Sunday. The first Sunday after the starting point shows a frequency of 2 per minute. By the next Sunday, the frequency has sped up to 4 per minute. By the tenth week, it has reached 1024 per minute.
If you were to draw a "times two" celeration line from any of the starting points from 1 to 9, you would end up with (a) a celeration line parallel to the one drawn above, and (b) values at each Sunday corresponding to those in the Table above. Hence, the chart itself could be used as a geometric "multiplication table," as opposed to the numeric tables.
Note that the zero row of the Table above has no representation. That's because multiply-divide scales, known as "logarithmic" scales to some, have no zero value. Furthermore, as Lindsley tends to say, you cannot multiply out of zero.
X3 TABLE
Directions: Use the successive multiply table by going across rows. Every number in a row on the table is X3 (three times) the number to its left, and /3 (divide by three) the number to its right. Note that multiplying 0 by 3 does not change the value. Note: For multiplying, read each series across.
Times Three Table Start | 1st | 2nd | 3rd | 4th | 5th | 6th | 7th | 8th | 9th | 10th |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
0 |
1 |
3 |
9 |
27 |
81 |
243 |
729 |
2187 |
6561 |
19683 |
59049 |
2 |
6 |
18 |
54 |
162 |
486 |
1458 |
4374 |
13122 |
39366 |
118098 |
3 |
9 |
27 |
81 |
243 |
729 |
2187 |
6561 |
19683 |
59049 |
177147 |
4 |
12 |
36 |
108 |
324 |
972 |
2916 |
8748 |
26244 |
78732 |
236196 |
5 |
15 |
45 |
135 |
405 |
1215 |
3645 |
10935 |
32805 |
98415 |
295245 |
6 |
18 |
54 |
162 |
486 |
1458 |
4374 |
13122 |
39366 |
118098 |
354294 |
7 |
21 |
63 |
189 |
567 |
1701 |
5103 |
15309 |
45927 |
137781 |
413343 |
8 |
24 |
72 |
216 |
648 |
1944 |
5832 |
17496 |
52488 |
157464 |
472392 |
9 |
27 |
81 |
243 |
729 |
2187 |
6561 |
19683 |
59049 |
177147 |
531441 |
Times Three Table -- NOTES:
In each series, the digits add up to 3, 6, or 9 for every product.
E.g., 729: 7 + 2 + 9 = 18: 1 + 8 = 9. And 531441: 5 + 3 + 1 + 4 + 4 + 1 = 18: 1 + 8 = 9.
The series starting at 1, 3, and 9 are the same, having only different starting points.
The same goes for the series starting at 2 and 6.
The series starting at 5 repeats the same last two digits in sequence: 05, 15, 45, 35, ....
The series starting at 7 repeats the same last digit in sequence: 1, 3, 9, 7, ....
This is the same with the series that start at 1, 3, 9, and 7!
The series starting at 2, 4, 6, and 8 all repeat the same last digit in sequence: 2, 6, 8, 4, ....
Lindsley email comments regarding multiplication tables:
"Use the successive multiply table by going across rows. Every number in a row of the
x2 table is x2 (twice) the number to its left, and /2 (one half) the number to its right."
"There should be a top row starting at zero with all zeros in its row."
"Also, The numbers in the top line to the right of S have no value. They only confuse.
I didn't put them in my tables. I just had "Start with" there."
"Traditional muliplication tables are successive add as you go across the rows.
But you get a multiplication product of a row number and and a column number where they intersect."
"I think you need three types of tables."
"1. Traditional successive add series tables for showing addition and for finding
multiplication products from row and column intersections.."
"2. Successive multiply tables to show how series double, triple, and quadriple."
"3. A times self table ( 2 power) which shows the result of multiplying a number in the
left row by itself the number of times at the top of the column at tyue intersection.
This table gets very big very fast so you onoy need a few rows and columns."
Text File Versions of the Multiplication Tables
Text Version of the X2 Table:
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048
3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072
4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096
5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120
6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144
7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168
8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192
9, 18, 36, 72, 144, 288, 576, 1152, 2304, 4608, 9216
Text Version of the X3 Table:
1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049
2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098
3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147
4, 12, 36, 108, 324, 972, 2916, 8748, 26244, 78732, 236196
5, 15, 45, 135, 405, 1215, 3645, 10935, 32805, 98415, 295245
6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294
7, 21, 63, 189, 567, 1701, 5103, 15309, 45927, 137781, 413343
8, 24, 72, 216, 648, 1944, 5832, 17496, 52488, 157464, 472392
9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441
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Copyright 2000 by John W. Eshleman, Ed.D.
Revised -- July 3, 2000.
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