How to explain it
The anchor students hold onto: To find k: pick any point (x, y) on the line — not the origin — and compute k = y ÷ x. To compare two relationships, the larger k means the steeper graph and the greater rate.
Students extend proportional graphs to slope-intercept form y = mx + b in Slope-Intercept Form, then graph and compare equations of all forms in Graphing Linear Equations.
Worked examples
Example 1
Find k from an equation
Find k for y = 6x.
Step 1y = kx form: the coefficient of x is k.
Step 2Read off: k = 6.
Step 3Unit rate: 6 units of y per 1 unit of x.
Answerk = 6; equation is y = 6x.
Example 2
Compare two relationships
Compare y = 4x and y = x.
Step 1Read each k: first equation k = 4; second k = 1.
Step 2Compare: 4 > 1.
Step 3The larger k means a steeper line and greater rate.
Answery = 4x has the greater unit rate (k = 4 vs k = 1).
Common mistakes
What students write
Dividing x by y instead of y by x — computing k = x / y returns the reciprocal, not k.
The fix
k = y / x: the output variable (y) is always the numerator. Divide y by x, not x by y.
Try this
A student claims y = 3x + 2 is a proportional relationship because the graph is a straight line and has x in the equation. Identify the student's error. Explain how to check whether a relationship is proportional.
What students write
Calling y = 3x + 2 proportional because it has y = kx structure with a coefficient visible.
The fix
Substitute x = 0: y = 2, not 0. The graph misses the origin — proportional requires y = 0 when x = 0.
Teacher tip
Head off the two predictable errors before they happen. First: k = y / x: the output variable (y) is always the numerator. Divide y by x, not x by y. Second: Substitute x = 0: y = 2, not 0. The graph misses the origin — proportional requires y = 0 when x = 0.