Multiplication table

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In mathematics, a multiplication table (sometimes, less formally, a times 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. Many educators believe it is necessary to memorize the table up to 9 × 9.

In his 1820 book The Philosophy of Arithmetic,[1] mathematician John Leslie published a multiplication table up to 99 × 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 25 × 25.

Contents

Traditional use

In 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 × 10 = 10
  2 × 10 = 20
  3 × 10 = 30
  4 × 10 = 40
  5 × 10 = 50
  6 × 10 = 60
  7 × 10 = 70
  8 × 10 = 80
  9 × 10 = 90
 10 x 10 = 100
 11 x 10 = 110
 12 x 10 = 120
 13 x 10 = 130
 14 x 10 = 140
 15 x 10 = 150
 16 x 10 = 160
 17 x 10 = 170
 18 x 10 = 180
 19 x 10 = 190
 100 x 10 = 1000

This form of writing the multiplication table in columns with complete number sentences is still used in some countries instead of the modern grid above.

[edit] Patterns in the tables

There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:

    →                 →
  1 2 3             2   4
↑ 4 5 6 ↓         ↑       ↓
  7 8 9             6   8
    ←                 ←
    0                 0
 Fig. 1             Fig. 2

For example, to memorize all the multiples of 7:

  1. Look at the 7 in the first picture and follow the arrow.
  2. The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
  3. The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
  4. After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
  5. Proceed in the same way until the last number, 3, which corresponds to 63.
  6. Next, use the 0 at the bottom. It corresponds to 70.
  7. Then, start again with the 7. This time it will correspond to 77.
  8. Continue like this.

Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 1 to 9, except 5.

[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 National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. It is thought by many[who?] that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method.

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