## Multiplication Facts and Patterns

A multiplication table for the integers from 0 to 10 contains 121 entries. These entries correspond to the basic multiplication facts that need to be mastered in the elementary grades. To ease this task, the fundamental properties of multiplication can be used to reduce the number of facts that require actual memorization.

For example, because 6 × 7 = 42, there is no need to memorize 7 × 6 = 42, since the Commutative Property of Multiplication states that 6 × 7 = 7 × 6. Thus, mastering of the 66 shaded entries (on or below the diagonal) is the same as mastering all 121 entries.

The Zero Property of Multiplication states that any number multiplied by zero is zero. That is, for any number a, a × 0 = 0 × a = 0. This fact can be used to reduce the number of facts requiring memorization by 11, leaving only 55 facts.

The Identity Property of Multiplication states that for all numbers a, a × 1 = 1 × a = a. Use of this property leaves 45 facts to be memorized.

When a number is multiplied by 10, the product is found by writing a zero after the number. This leaves 36 basic facts that must be memorized.

Prior knowledge of arithmetic can be used for help with the remaining 36 multiplication facts. For instance, facts involving doubling can be understood as addition facts for two identical addends. Doubling can then be used to recall multiplication facts involving 4 or 8. For example, once 2 × 6 = 12 is known, then 4 × 6 = 2 × 6 + 2 × 6 = 12 + 12 = 24. Similarly, 8 × 6 = 4 × 6 + 4 × 6 = 24 + 24 = 48.

The Associative Property of Multiplication, which states that for any numbers a, b, and c, a × (b × c) = (a × b) × c can be used to find doubles.

Example: 4 × 6 = (2 × 2) × 6 = 2 × (2 × 6) = 2 × 12

The Distributive Property can also be used to find doubles. This property states that for any numbers a, b, and c, a × (b + c) = (a × b) + (a × c) and (a + b) × c = (a × c) + (b × c).

Examples:

8 × 6 = (4 + 4) × 6 = (4 × 6) + (4 × 6) = 2 × (4 × 6)
7 × 8 = (2 + 5) × 8 = (2 × 8) + (5 × 8) = 16 + 40 = 56

The pattern of digits arising in multiples of 9 is easily remembered.

2 × 9 = 18
3 × 9 = 27
4 × 9 = 36
5 × 9 = 45
6 × 9 = 54
7 × 9 = 63
8 × 9 = 72
9 × 9 = 81

These products of 9 all have two digits whose sum is 9. Furthermore, the tens digit is 1 less than the factor with which 9 is multiplied. This pattern can be explained by the fact that the next multiple of 9 must also have this property. This is because the next multiple of 9 can be obtained by adding 10 (increasing the tens digit by 1) and subtracting 1 (decreasing the ones digit by 1). Together, these two actions leave the sum of the digits unchanged. Since 2 × 9 has the above property, so do the multiples by 3, 4, 5, 6, 7, 8, 9, and 10. It is only after the ones digit is reduced to 0, as in 10 × 9 = 90, that the argument breaks down.

Teaching Model 9.2: Multiply with 3