Number & Operations for Teachers 

Copyright David & Cynthia Thomas, 2009

Multiplication

Multiplication of whole numbers may be thought of as repeated addition. For example, the expression 3 x 2 = 6 is logically equivalent to the expression 2 + 2 + 2 = 6, both of which convey the idea “three groups of two.” Figure 3.1 shows four concept models for this expression: A number line model; a set model; an area model; and a tree model.

2 2 2

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0 1 2 3 4 5 6 7

Number Line Model

2 + 2 + 2

+ + =

Set Model

Three sets, each with two elements

Area Model

Three rows, each with two columns

Tree Model

Three main branches, each with two smaller branches

Figure 3.1: Concept models for 3x2 = 6

By contrast, each concept model in Figure 3.2 models the expression 2 x 3, or “two groups of three.” Note that commuting the order of the factors in these expressions does not change the value of their product.

3 3

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0 1 2 3 4 5 6 7

Number Line Model

3 + 3

+ =

Set Model

Two sets, each with three elements

Area Model

Two rows, each with three columns

Tree Model

Two main branches, each with three smaller branches

Figure 3.2: Concept models for 2 x 3 = 6

Example 3.1

Using an area model, represent 3 x 5. Represent 5 x 3.

Solution 3.1

3 x 5

5 x 3

Example 3.2

Give a written representation for the following concept model

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0 1 2 3 4 5 6 7 8

Solution 3.2

4 x 2

Example 3.3

Give a written representation for the following concept model

______________________________

0 1 2 3 4 5 6 7 8

Solution 3.3

2 x 4