Between the Dots

A line does not jump from one grid point to the next. It passes through every point in between. On y = 1/2x + 1, the value x = 1 gives y = 3/2. The point (1, 3/2) is a point of the line, half a unit above 1. A rule with a fraction gives points between the grid points, and the line passes through them too.

The line y = 1/2x + 1 rises half a unit for every unit to the right

For y = (1/2)x + 1, choosing x = 0, 2, or 4 gives integer y-coordinates 1, 2, and 3. These points are convenient to plot. For a coefficient of 1/3, choosing multiples of 3 similarly avoids fractional products.