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.

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.

Traditional use

Circa 493 A.D., Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144" (Maher & Makowski 2001, p.383)

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.

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

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.

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.

See also

• Vedic square

External links

• Underwater multiplication tables - interactive game
• Learn Multipication Tables through real Hip Hop Music with a certified Math Teacher
• Times Tables Games
• Free Multiplication Game for Windows
• Learning the multiplication while playing domino in a browser