Dr. Cotter on What 6 × 3 Means

Sometimes 6 × 3 is thought of as “6 groups of 3.” However, consistency with the other arithmetic operations requires a second look. When adding 6 + 3, we start with 6 and transform it by adding 3. When subtracting 6 – 3, we start with 6 and transform it by removing 3. When dividing 6 ÷ 3, we start with 6 and transform it by dividing it into 3 groups or into groups of 3s. Likewise, 6 × 3 means we start with 6 and transform it by duplicating it a total of 3 times.

In the array (an arrangement of quantities in rows and columns) model, 6 × 3, 6 represents the horizontal quantity and 3 the vertical quantity. This is also consistent with the coordinate system; in (6, 3), the first number, 6, indicates the horizontal number and 3, the vertical number.

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