Multiplication table

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In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.

The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with our base-ten numbers. It is necessary to memorize the table up to 9 × 9, and often helpful up to 12 × 12 to be proficient in traditional mathematics. As noted below many schools in the United States adopted standards-based mathematics texts which completely omitted use or presentation of the multiplication table, though this practice is being increasingly abandoned in the face of protests that proficiency in elementary arithmetic is still important.

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[edit] In basic arithmetic

A multiplication table ("times table", as used to teach schoolchildren multiplication) is a grid where rows and columns are headed by the numbers to multiply, and the entry in each cell is the product of the column and row headings. Traditionally, the heading for the first row and first column contains the symbole multiplication operator.

× 1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 1 2 3 4 5 6 7 8 9 10 11 12 13 14
2 2 4 6 8 10 12 14 16 18 20 22 24 26 28
3 3 6 9 12 15 18 21 24 27 30 33 36 39 42
4 4 8 12 16 20 24 28 32 36 40 44 48 52 56
5 5 10 15 20 25 30 35 40 45 50 55 60 65 70
6 6 12 18 24 30 36 42 48 54 60 66 72 78 84
7 7 14 21 28 35 42 49 56 63 70 77 84 91 98
8 8 16 24 32 40 48 56 64 72 80 88 96 104 112
9 9 18 27 36 45 54 63 72 81 90 99 108 117 126
10 10 20 30 40 50 60 70 80 90 100 110 120 130 140
11 11 22 33 44 55 66 77 88 99 110 121 132 143 154
12 12 24 36 48 60 72 84 96 108 120 132 144 156 168
13 13 26 39 52 65 78 91 104 117 130 143 156 169 182
14 14 28 42 56 70 84 98 112 126 140 154 168 182 196

So, for example, 3×6=18 by looking up where 3 and 6 intersect.

This table does not give the zeros. That is because any real number times zero is zero.

Multiplication tables vary from country to country. They may have ranges from 1×1 to 10×10, from 2×1 to 9×9, or from 1×1 to 12×12 to quote a few examples. 10 x 10 is essential for use in long multiplication, but knowledge to 12 x 12 and higher can be used as shortcuts in other calculation methods. The most common example of such a table in the 1960s and 1970s was inside the reference section of the Pee Chee folder commonly used in United States schools and in many other places.

[edit] Traditional use

The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like

1 × 7 = 7
2 × 7 = 14
3 × 7 = 21
4 × 7 = 28
5 × 7 = 35
6 × 7 = 42
7 × 7 = 49
8 × 7 = 56
9 × 7 = 63
10 x 7 = 70
11 x 7 = 77
12 x 7 = 84
13 x 7 = 91
14 x 7 = 98
15 x 7 = 105

Learning the content of the (10x10) table is much less work than it superficially seems to be. (It should not be learnt as the table itself, but rather as connections between any two single-digit factors and the resulting product, until the connection becomes intuitive, much like vocabulary in a foreign language.) Because of the symmetry of the table 45 entries are in fact duplicates (55 entries left). The connection between 1 and any number as well as 10 and any number are trivial (36 entries left), the connections between 5 and any number can easily be derived from the multiplication by 10 and adding the occasional 5 for odd numbers(28 entries left). Multiplication by 2 is generally considered easy as well (21 entries left) and finally multiplication by 9 has an easily memorized pattern as well. Taking all those entries out of the table leaves all of 15 entries to be learnt by rote.

Bold text==Patterns in the tables==

For example, for multiplication by 6 a pattern emerges:

  2 × 6 = 12
  4 × 6 = 24
  6 × 6 = 36
  8 × 6 = 48
 10 × 6 = 60

 number × 6 = half_of_number_times_10  + number

The rule is convenient for even numbers, but also true for odd ones:

 1 × 6 = 05 +  1 =  6
 2 × 6 = 10 +  2 = 12
 3 × 6 = 15 +  3 = 18
 4 × 6 = 20 +  4 = 24
 5 × 6 = 25 +  5 = 30
 6 × 6 = 30 +  6 = 36
 7 × 6 = 35 +  7 = 42
 8 × 6 = 40 +  8 = 48
 9 × 6 = 45 +  9 = 54
10 × 6 = 50 + 10 = 60

[edit] In abstract algebra

Multiplication tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they can be called Cayley tables. For an example, see octonion.

[edit] Standards-based mathematics reform in the USA

In 1989, the NCTM developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which played down the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as TERC omit aids such as multiplication tables, instead guiding students to invent their own methods, including skip counting and coloring in multiples on 100s charts. It is thought by many that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. Standards organizations such as the NCTM had originally called for "de-emphasis" on basic skills in the late 1980s, but they have since refined their statements to explicitly include learning mathematics facts. Though later versions of texts such as TERC have been rewritten, the use of earlier versions of such texts has been heavily criticized by groups such as Where's the Math and Mathematically Correct as being inadequate for producing students proficient in elementary arithmetic.

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