Teaching Models

Multiplication and Division Basic Facts

Multiplication can be defined as repeated addition. For example, 3 × 6 can be viewed as 6 + 6 + 6. More generally, for any positive integer n, n × b can be represented as n × b = b + b + … + b where the sum on the right consists of n addends.

A rectangular array provides a visual model for multiplication. For example, the product 3 × 6 can be represented as shown. Besides representing 3 × 6 as an array of 18 unit squares, this model also shows that a rectangle with a height of 3 units and a base of 6 units has an area of 3 × 6 square units, or 18 square units.

Diamond array

By displaying 18 diamonds as 3 rows with 6 diamonds in each row, this array provides a visual representation of 3 × 6 as 6 + 6 + 6.

An equivalent area model can be made in which the diamonds in the array are replaced by unit squares.

Besides representing 3 × 6 as an array of 18 unit squares, this model also shows that a rectangle with a height of 3 units and a base of 6 units has an area of 3 × 6 square units, or 18 square units.

unit squares

Multiplication is a binary operation in that it operates on a pair of numbers to produce another number. Given a pair of numbers a and b called factors, multiplication assigns them a value a × b = c, called the product.

The use of doubling is based on the Associative Property of Multiplication, which states that for any numbers, a, b, and c,

a × (b × c) = (a × b) × c.

As a result, 4 × 6 = (2 × 2) × 6 = 2 × (2 × 6) = 2 × 12, or 12 + 12 = 24

As shown above, mathematical properties are often used to simplify computation. Below are four multiplication properties stated in words, shown with both a numeric example and an algebraic example.

Associative Property of Multiplication
When numbers or variables are multiplied, the factors can be grouped in different ways without changing the result.

(2 × 3) × 4 = 2 × (3 × 4)
(a × b) × c = a × (b × c)

Commutative Property of Multiplication
When numbers or variables are multiplied, the order of the factors can be changed without changing the result.

2 × 3 = 3 × 2
a × b = b × a

Zero Property of Multiplication
When a number or variable is multiplied by 0, the result is 0.

2 × 0 = 0
a × 0 = 0

Property of One for Multiplication
When a number or variable is multiplied by 1, the result is the same number or variable.

2 × 1 = 2
a × 1 = a

The Property of One for Multiplication is also known as the Identity Property of Multiplication.

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

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

These products of 9 all have two digits whose sum is 9. Furthermore, the tens digit is 1 less than the number multiplying 9. This pattern can be explained using the base-ten positional system. If a multiple of 9 has this property and its unit digit is positive, then 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 operations leave the sum of the digits unchanged. Since 2 × 9 has the above property, so do the multiples of 9 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 pattern breaks down.

Relating Multiplication and Division
Division is the inverse of multiplication. Given that b is not equal to zero (0), the statement

a ÷ b = c

is true if and only if a = b × c. The relationship provides a basis for checking the correctness of answers to division problems. The fact that 68 ÷ 4 = 17 can be confirmed by the observation that 68 = 4 × 17. The statement a ÷ b = c is read as “a divided by b is equal to c,” where a is the dividend, b is the divisor, and c is the quotient.

The fact that the multiplication problem 3 × 5 = 15 can be represented by a 3 × 5 array of objects provides a basis for using arrays to represent the division problems 15 ÷ 3 = 5 and 15 ÷ 5 = 3.

Grouped arrays

In the first case, a group of 15 objects is separated into three groups of 5, while in the second case the group of 15 objects is separated into five groups of 3. Such pictures can help students use known multiplication facts to solve certain division problems. For example, to solve 20 ÷ 4 = ?, the students should learn to think 4 × ? = 20. Another way to show the relationship between multiplication and division is by using a number line. On a number line, the basic fact 5 × 3 = 15 can be represented in terms of skip counting as five steps of 3 each.

Count up number line

Because division is the inverse operation of multiplication, division facts can be represented in terms of repeated subtraction. The division fact 15 ÷ 3 = 5 corresponds to repeated subtraction of 3 from 15, which can be shown as five "backward" steps of 3 each on the number line.

Count down number line

While arrays and skip counting provide useful representations of multiplication and division, it is important for students to be able to arrive at 5 × 3 = 15 and 15 ÷ 3 = 5 without making use of such strategies. In the case of multiplication, students are expected to commit the basic one-digit multiplication facts to memory.

One way of helping students understand and use the relationship between multiplication and division is through the use of fact families like the following.

3 × 4 = 12
4 × 3 = 12
12 ÷ 3 = 4
12 ÷ 4 = 3
  2 × 9 = 18
9 × 2 = 18
18 ÷ 2 = 9
12 ÷ 9 = 2
  6 × 7 = 42
7 × 6 = 42
42 ÷ 6 = 7
42 ÷ 7 = 6

In the general case, where a, b, and c are nonzero numbers and a × b = c, the corresponding four-member fact family is of the form:

a × b = c
b × a = c
c ÷ a = b
c ÷ b = a

Teaching Model 4.2: Relate Multiplication and Division Algebra


Houghton Mifflin Math Grade 4