To find the slope \( m \) of the line that passes through the points \( (x_1, y_1) = (1, 1) \) and \( (x_2, y_2) = (9, 10) \), we use the formula for the slope:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the given points into the formula:
\[ m = \frac{10 - 1}{9 - 1} \]
Calculating the differences in the numerator and denominator:
\[ m = \frac{9}{8} \]
Thus, the slope of the line that passes through the points \( (1, 1) \) and \( (9, 10) \) is
\[ \boxed{\frac{9}{8}} \]