Multiplication table
From Wikipedia, the free encyclopedia
In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.
The multiplication table was traditionally taught as an essential part of elementary arithmetics all around the world, as it lays the foundation for arithmetic operations. It is neccesary to memorize the table up to 9 x 9, and often helpful up to 12 x 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.
Contents |
[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 symbol for the multiplication operator.
| × | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| 2 | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 |
| 3 | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 |
| 4 | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 |
| 5 | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 |
| 6 | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 |
| 7 | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 |
| 8 | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 |
| 9 | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 |
| 10 | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 |
| 11 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 |
| 12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |
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.
The layout, however, follows the left-to-right, top-to-bottom movement of Western culture's reading. Since Descartes' promotion of the rectangular coordinate system, the presentation of the multiplication facts in the first quadrant makes pedagogical sense for the math learner. The XY Chart shown below has taken advantage of this.
The XY Chart multiplication table shows the facts in the first quadrant. Under this arrangement as multiplication learners navigate from the factors to products they follow the same steps as plotting points by following the two coordinates. Curiously, the layout of the XY Chart corresponds closely to the definition of Cartesian products.
[edit] Traditional use
The traditional rote learning of multiplication was based on memorisation 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
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 connetions between any two single-digit factors and the resulting product, until the connection becomes intuitiv, 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 occational 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.
[edit] 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 1989, the NCTM developed new standards which were based on the belief all students should learn higher order thinking skills, and de-emphasize teaching of traditional methods which 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 mutiples on 100s charts. It is thought by many that calculators have made it unneccesary or counterproductive to invest time in memorizing the multiplication table. A math textbook which lacks the multiplication table is probably one of a number of standards-based texts that were produced in the United States according to these beliefs, and adopted because of federal grant money to promote these methods. Standards organizations such as the NCTM had originally called for "de-emphasis" on basic skills in the late 80s, but they have since refined their statements to explicitly include learning mathematics facts. Though later versions of texts such as TERC have been rewritten, many schools in the United States continue to use widely adopted texts which do not include the multiplication table,[citation needed] and the usage 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 arithemetic.
