The Reversed Times Tables

Abstract

The product of 3x5 can be explained as five groups of three each. It can also be explained as three groups of five each. Because the result of 3x5 is the same as the result of 5x3, both explanations are correct. Which explanation is more acceptable in order to reach educational purposes?

The product of two numbers can be explained as the sum of numbers. For example, 3x5=5+5+5, the product of 3x5 is the sum of three fives. In algebra, the product of 3y is the sum of three y's. The definition of ny is the sum of n y's. In order to be consistent with the definition of products in algebra, products 3x5 should be explained as three groups of five each.

The product of 3x5 is explained as five groups of three each following from the traditional times tables.

1      2      3      4      5      6      7      8      9
1x1=1  2x1=2  3x1=3  4x1=4  5x1=5  6x1=6  7x1=7  8x1=8  9x1=9
1x2=2  2x2=4  3x2=6  4x2=8  5x2=10 6x2=12 7x2=14 8x2=16 9x2=18
1x3=3  2x3=6  3x3=9  4x3=12 5x3=15 6x3=18 7x3=21 8x3=24 9x3=27
1x4=4  2x4=8  3x4=12 4x4=16 5x4=20 6x4=24 7x4=28 8x4=32 9x4=36
1x5=5  2x5=10 3x5=15 4x5=20 5x5=25 6x5=30 7x5=35 8x5=40 9x5=45
1x6=6  2x6=12 3x6=18 4x6=24 5x6=30 6x6=36 7x6=42 8x6=48 9x6=54
1x7=7  2x7=14 3x7=21 4x7=28 5x7=35 6x7=42 7x7=47 8x7=56 9x7=63
1x8=8  2x8=16 3x8=24 4x8=32 5x8=40 6x8=48 7x8=56 8x8=64 9x8=92
1x9=9  2x9=18 3x9=27 4x9=36 5x9=45 6x9=54 7x9=63 8x9=72 9x9=81
                    

Each row of this table is constructed by placing the base in the front of "x" and placing the different numbers of 1 to 9 in the rear of "x". In order to explain products, the front number is treated as the amount in each group and the rear number is treated as the number of groups. This explanation is not only inconsistent with algebra but also inconsistent with our natural language. When the amount of an object is described, quantity is placed in the front and the unit is placed in the rear. For example, 3kg means three of one kilo-gram. In order to avoid this difficulty, the Traditional Times Tables can be rewritten as the following:

  1      2      3      4      5      6      7      8      9
1x1=1  1x2=2  1x3=3  1x4=4  1x5=5  1x6=6  1x7=7  1x8=8  1x9=9
2x1=2  2x2=4  2x3=6  2x4=8  2x5=10 2x6=12 2x7=14 2x8=16 2x9=18
3x1=3  3x2=6  3x3=9  3x4=12 3x5=15 3x6=18 3x7=21 3x8=24 3x9=27
4x1=4  4x2=8  4x3=12 4x4=16 4x5=20 4x6=24 4x7=28 4x8=32 4x9=36
5x1=5  5x2=10 5x3=15 5x4=20 5x5=25 5x6=30 5x7=35 5x8=40 5x9=45
6x1=6  6x2=12 6x3=18 6x4=24 6x5=30 6x6=36 6x7=42 6x8=48 6x9=54
7x1=7  7x2=14 7x3=21 7x4=28 7x5=35 7x6=42 7x7=49 7x8=56 7x9=63
8x1=8  8x2=16 8x3=24 8x4=32 8x5=40 8x6=48 8x7=56 8x8=64 8x9=72
9x1=9  9x2=18 9x3=27 9x4=36 9x5=45 9x6=54 9x7=63 9x8=72 9x9=81
                    

Each row of this new table is constructed by placing the base in the rear of "x" and placing the different numbers of 1 to 9 in the front of "x". For the explanation of the product, the rear is treated as the amount in each group and the front number is treated as the number of groups. This new times tables is called the Reversed Times Tables.

For educational purposes, the Traditional Times Tables are suggested to be replaced by the Reversed Times Tables.