SME
Grade 8 Math · Expressions & Equations
Slope and rate of change
Use this for “Find the slope of the line through (−2, −1) and (4, 2).”
Chart
From (−2, −1) to (4, 2)
m = rise ÷ run = 3 ÷ 6 = 1/2.
This is an example. Type your own values to start a new problem.
1
2
m
=
rise
run
=
−
(
)
−
(
)
Formulas
rise
=
y₂
−
y₁
(rise
=
second
y
−
first
y)
3 = 2 − (−1)
run
=
x₂
−
x₁
(run
=
second
x
−
first
x)
6 = 4 − (−2)
m
=
rise
÷
run
(slope
=
rise
÷
run)
1
2
=
3
÷
6
Assumptions
•
The slope is the rise divided by the run between any two points on the line.
•
Any two points give the same slope: the slope triangles are similar.
•
A line going down to the right has a negative rise, so a negative slope.
•
A run of 0 is a vertical line: it has no slope.
Step-by-step
WE KNOW
x₁ = −2 (first x), y₁ = −1 (first y), x₂ = 4 (second x), y₂ = 2 (second y)
FIND
rise (rise), run (run), slope (m)
Step 1 · Find rise (rise)
Use rise = y₂ − y₁ (rise = second y − first y)
The rise is how far up the second point is from the first.
rise = 2 − (−1)
rise = 3
Step 2 · Find run (run)
Use run = x₂ − x₁ (run = second x − first x)
The run is how far across the second point is from the first.
run = 4 − (−2)
run = 6
Step 3 · Find slope (m)
Use m = rise ÷ run (slope = rise ÷ run)
Divide the rise by the run.
m
=
3
6
m
=
1
2
Check
Put the numbers back in. Both sides should match.
3 = 2 − (−1) ✓
6 = 4 − (−2) ✓
1
2
=
3
÷
6
✓
Topic check
See what you know: 5 questions on this topic
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More problem types
Slope as a unit rate
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The signs of the slope and intercept
Use this for “Which line rises to the right and crosses the y-axis below 0?”