Multiplication and Division with Remainder by
Counting with Factor-Product Tables
PATTERNS MAKE MULTIPLICATION EASY
There are 3 table shapes x 6 counting methods x 1 direction = 18 combinations. You could also claim there are more,
considering:
2 VARIATIONS like Factor A & B,
4 zig-zags of standard, left, right or upside-down TURNS or
4 COUNTING EXCEPTIONS of the tens place with single or double exceptions to the rule of single or double counting.
If you counted all the variations, there would be 3 table shapes x 7 counting methods x 4 directions x 2 decimal places
(tens & ones) = 168 possible combinations. Yet, you only need to remember and use 18 combination of rules to get the
products. Unlike these factor-product tables, there are no easy-to-recognize patterns or rules that can substituted for
memorizing the 121 separate products from a standard multiplication table of factors 0-10. Patterns and rules show a
relationship between separate facts that make them easier to remember.
For example, remembering a person’s “appearance” by combining gender, age group and hair color is easier to
remember because the elements are related within categories. Consider the following categories, not as definite to spark
controversy, but as an example how combinations are recognized and remembered. The combination of 2 genders (male,
female), 6 age groups (infant, child, adolescent, young, middle-aged, senior) and 8 hair colors (bald, blonde, red, brown,
black, gray, white, or dyed an unnatural color) yield 96 possible combinations, but that number is irrelevant to remember
the combined elements of appearance, especially considering some combinations are more often found (a gray-haired
senior man) while others are rare or non-existent (a male or female infant with unnaturally dyed hair).
The visual patterns in the GUIDE combining table shape, counting method and direction make them recognizable and
memorable as much as the patterns of gender, age group and hair color make a person easy to recognize and remember
despite the many potential combinations. (fade-in picture over GUIDE with words: gender ― male, age group ― senior,
hair color ― white).
TABLE SHAPE: The factors matching table shapes are easy to remember because the first 3 factors (0, 1, 2), the middle
factor (5) and the last factor (10) belong to the rectangle tables. Or, you can remember the rectangle factors are some
form of 0 & 1 (0, 1, 10) and a number that when flipped vertically, looks like the other (2 & 5). The remaining factors are
split into odd numbers (3, 7, 9) for tic-tac-toe grid table and even (4, 6, 8) for double 4-square table.
COUNTING METHOD: Both tens and ones place counting of the products for 0x table are Factor A counting, while 1x are
both Factor B counting. 10x has Factor A counting for ones and Factor B for tens. 5x table has Alternating 0 & 5 counting
for ones place. The remaining tables have an increasing pattern of tens counting: 2, 3 & 4x have row counting, 5, 6, 7
have double counting, and 8 & 9 have single counting. Ones place is even counting for even factors 2, 4, 6 & 8. Ones
place for remaining odd factors 3, 7 & 9 is single counting. Ones place for remaining even factors 4, 6 & 8 are even
counting.
COUNTING DIRECTION: The standard zig-zag is the counting direction for all tens places for all factors and ones place of the rectangle table factors 0, 1, 2, 5 and 10. It accounts for almost 73% (16/22) of all counting directions of tens and ones places, while the zig-zags turned left, right or upside-down, each count for 9% (2/22). There are other methods for finding products without memorizing the standard multiplication facts, but they are for just a few factors, not easy to recognize and use, and lack an approach to integrate all factors. They fail to be as simple and integrated as the factor-product tables because they try to find patterns by comparing the whole products with each other instead of counting the tens and ones place separately to reveal the patterns before combining both into products. MISSION STATEMENT: We provide innovative educational content, strategies, programs and media proven better than standard teaching methods. Our strategies are engaging, build on the familiar and include many senses and formats. The components are effective because of sound teaching methodology, testing and supported by research. “Mr. Brant teaches what other methods can’t”TM
Remembering combinations is easier when you see they are grouped by number, scale, shape or decimal place. Row
and double counting are only used in the tens place and alternate 0 & 5 of 5 and even counting of 2, 4, 6, 8 are only in
the ones place. The ones place also groups counting methods by number. First, factor A or B counting contains a 0, 1 or
both (10). Then, there is alternate 0 & 5 counting of 5. The factors left (2, 4, 6, 8) are even counted while odds 3, 7, 9 are
single counted. The tens place counting exceptions factor order 6, 7, 8, 9 increases as the distance from first cell for
exceptions decreases: last two cells, middle column and first two cells. Zig-zag counting directions groups rectangle
shape factors 0, 1, 2, 5 & 10 by standard zig-zag and the remaining factors by number order and clockwise zig-zag
turn order: left 3 & 4, right 6 & 7 and upside-down 8 & 9.