Graphs · Slope · 4.1

Slope = rise ÷ run.

Slope measures steepness: how much the line rises (change in y) for each unit it runs (change in x). Slope = rise ÷ run. From (0, 0) to (3, 2), rise 2 over run 3 → slope 2/3.

01 / THE IDEA

See it work.

Watch the method

Slope = rise ÷ run

Rise 2, run 3 → slope 2/3.

1 / 2
-6-666
1

Rise over run

Rise = change in y = 2; run = change in x = 3.

23\frac{2}{3}
2

Simplify

The slope is rise ÷ run.

m=2/3m = 2/3
02 / YOUR TURN

Practice until it feels automatic.

Your turn

Work it out.

Type a number, e.g. 5.

streak 0
best 0
-6-666
Your answer:
— / WHY IT WORKS

The idea underneath.

Rise over run

Pick two points; the change in y over the change in x is the slope. Steeper lines have bigger slopes.

Why it works

Up 2, over 3 → 2/3.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Find the slope of the line through (1,-3) and (-6,-5).

1

Rise over run

Rise = change in y = -2; run = change in x = -7. -2/-7

2

Simplify

The slope is rise ÷ run. m = 2/7

EX 2medium

Find the slope of the line through (-5,3) and (5,5).

1

Rise over run

Rise = change in y = 2; run = change in x = 10. 2/10

2

Simplify

The slope is rise ÷ run. m = 1/5

EX 3hard

Find the slope of the line through (4,-1) and (-1,3).

1

Rise over run

Rise = change in y = 4; run = change in x = -5. 4/-5

2

Simplify

The slope is rise ÷ run. m = -4/5

04 / COMMON TRAPS

Where students slip.

Trap

Using run ÷ rise.

Fix

Slope is rise ÷ run (y over x).

Trap

Counting the run backwards.

Fix

Go from the same start point for both.

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Find the slope of the line through (-5,4) and (-2,6).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
C2

Find the slope of the line through (4,-4) and (-2,2).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
C3

Find the slope of the line through (-1,-3) and (-4,6).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
C4

Find the slope of the line through (6,-2) and (-2,-4).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
Apply

Apply (4)

A1

Find the slope of the line through (4,-3) and (6,1).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
A2

Find the slope of the line through (5,5) and (-3,3).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
A3

Find the slope of the line through (-4,1) and (-5,-1).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
A4

Find the slope of the line through (3,2) and (1,-6).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
Extension

Extension (4)

E1

Find the slope of the line through (6,-2) and (-6,6).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
E2

Find the slope of the line through (3,3) and (-1,4).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
E3

Find the slope of the line through (4,-6) and (-3,4).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).
E4

Find the slope of the line through (-1,-2) and (-3,2).

Need a hint?
  • Find the rise (change in y).
  • Divide by the run (change in x).