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Encyclopedia > Multiplication table

In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. A timetable is an organized list or schedule, usually set out in tabular form, providing information about a series of arranged events: in particular, the time at which it is planned these events will take place. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Before calculators were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying variables— to simplify and drastically speed up computation. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...


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. For other uses, see Decimal (disambiguation). ... Traditional mathematics is the term used for the style of mathematics instruction used for a period in the 20th century before the appearance of reform mathematics based on NCTM standards, so it is best defined by contrast with the alternatives. ...

Contents

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. Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ... In mathematics, multiplication is an elementary arithmetic operation. ... In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f. ...

× 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 and in many other places. Folder back The yellow Pee Chee Folder was a very common American school item in the second half of the 20th century. ...


Traditional use

The traditional rote learning of multiplication was based on memorisation of columns in the table, in a form like It has been suggested that Rote memory be merged into this article or section. ...

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 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. This picture illustrates how the hours on a clock form a group under modular addition. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... A Cayley table, after the 19th century British mathematician Arthur Cayley, describes the structure of a finite group by arranging all the possible products of all the groups elements in a square table reminiscent of an addition or multiplication table. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ...


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. The National Council of Teachers of Mathematics (NCTM) was founded in 1920. ... Investigations in Number, Data, and Space is a complete K-5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts. ... Skip counting is a mathematics technique taught in place of formal multiplication in standards-based mathematics textbooks such as TERC. Another similar method is coloring in squares in a 100s chart to show multiplication patterns. ... The National Council of Teachers of Mathematics (NCTM) was founded in 1920. ... Investigations in Number, Data, and Space is a complete K-5 mathematics curriculum, developed at TERC in Cambridge, Massachusetts. ... Mathematically Correct is a website created by educators, parents, citizens and mathematicians / scientists who are concerned about the direction of standards-based mathematics and education reform. ...


External links


  Results from FactBites:
 
Grade 3: Using a Multiplication Table to Divide: Developing the Concept (498 words)
In this lesson, students use a multiplication table to explore the relationship between missing factors in multiplication and quotients in division.
Have students follow these steps on their multiplication tables as you demonstrate on the overhead.
You can determine whether students understand how to use a multiplication table to multiply and divide by having them use their own words to explain the procedure they follow.
Multiplication table - Wikipedia, the free encyclopedia (258 words)
In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system.
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.
Multiplication tables can also define binary operations on groups, fields, rings, and other algebraic systems.
  More results at FactBites »

 
 

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