A. Background Mathematics
Multiplication of a natural number and a fraction
In the multiplication of a natural number and a fraction, a distinction is made between (natural number) × (fraction) versus (fraction) × (natural number).
The expression 5 × 3/7 means "five 3/7s." It can be explained easily through the principle of repeating the same number (or fraction), represented as follows:
| 5 × | 3 | = | 3 | + | 3 | + | 3 | + | 3 | + | 3 | = | 3 + 3 + 3 + 3 + 3 | = | 5 × 3 | = | 15 | = 2 | 1 |
| 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | 7 |
On the other hand, 3/7 × 5 means "three-sevenths of the quantity 5." It can be explained using a number line as shown below:
The number line shows the quantity 5 divided into 7 portions, with 3/7 being three of those portions. That is,
| 3 | × 5 = 3 × (5 ÷ 7) = 3 × | 5 | = | 3 × 5 | = | 15 | = 2 | 1 |
| 7 | 7 | 7 | 7 | 7 |
Students will see that (5 × 3/7) is the same as (3/7 × 5), in other words, that the result is the same. When students have discovered that the result is the same if the order of multiplication is reversed, explain this principle as the "commutative law."