Once the packet transitions from basic relations into linear equations, the cognitive load shifts to spatial geometry. Students are expected to calculate rate of change across uneven tables, graph linear functions directly from equations, and navigate fluidly between slope-intercept form ($y = mx + b$) and standard form ($Ax + By = C$).
The primary friction point here is the rate of change formula:
$$m = \frac{y2 - y1}{x2 - x1}$$
Students consistently mix up the coordinates, subtracting $x$ in the numerator and $y$ in the denominator. A second recurring trap occurs when negative coordinates enter the calculation. Faced with calculating the slope between $(-3, 4)$ and $(5, -2)$, students often write $-2 - 4$ correctly for the numerator, but reduce the denominator to $5 - 3$ instead of $5 - (-3)$, completely reversing the sign of the slope.
Graphing linear functions also highlights mechanical gaps. Wilson’s study guide purposely includes equations where the $y$-intercept is zero (e.g., $y = -\frac{2}{3}x$) or where the slope is an integer (e.g., $y = 4x - 5$). When $m$ is an integer, students forget that the horizontal "run" is 1, graphing a flat line instead of a steep incline.
Standard form conversions trigger algebraic sign errors. Converting $3x - 4y = 12$ into slope-intercept form requires two steps: subtracting $3x$ from both sides, then dividing every term by $-4$. Students routinely forget that dividing $-12$ by $-4$ produces a positive intercept, or they drop the negative sign attached to the $y$-coefficient altogether.
