Introduction to Slope

Explore the concept of slope. Drag points on a coordinate plane, calculate rise over run, understand positive, negative, zero, and undefined slopes, and build climbable stairs.

Interactive Sandbox

Rise & Run Grapher

Drag the two points A and B around the coordinate system. Snapping to coordinates is active by default to yield clean mathematical divisions. Watch the slope ratio calculate dynamically.

Slope Ratio Value

m = 1.00
Formula: m = Rise / Run
• Point A = (1, 2)
• Point B = (5, 6)

• Rise (Δy) = y₂ - y₁ = 4.0
• Run (Δx) = x₂ - x₁ = 4.0

m = 4 / 4 = 1.00
Concept HUD

Slope Classifications

Depending on the direction of rise and run, the slope falls into one of four distinct categories. The dashboard HUD lights up dynamically as you drag points A and B on the coordinate system above.

↗ Positive
Rise & Run same sign. Line climbs up left-to-right.
↘ Negative
Rise & Run opposite signs. Line falls left-to-right.
→ Zero Slope
Rise is 0. Horizontal line. y stays constant.
↑ Undefined
Run is 0. Vertical line. Division by zero!

Connecting to Linear Equations


Slope is the foundational element of linear equations. If you extend this line, it intercepts the vertical y-axis at y-coordinate b = 1.0.

y = 1.00x + 1.00
Staircase Simulation

Staircase Steepness Simulator

Architects design staircases using specific slope safety guidelines. Adjust step rise (height) and step run (depth) to build a flight of stairs, then test run the walker to see if the slope is safe for climbing!

Standard Slope: Safe and comfortable staircase.

Stair Step Ratio

Rise/Run = 0.50 / 0.80 = 0.63
Step Rise (Height) 0.50 m
Step Run (Depth) 0.80 m
According to building codes, comfortable staircases have a slope of around 0.5 to 0.7 (rise height ≈ 7 inches, run depth ≈ 11 inches). If the slope gets too high, it effectively behaves as a vertical ladder!
Self Evaluation

Conceptual Quiz

Review your mastery of slopes, calculation steps, and zero vs undefined conditions with this interactive quiz.

Question 1 of 5 Score: 0/0

Find the slope of a line passing through (2, 3) and (6, 11).

Correct Answer!

Explanation text goes here.

Curriculum Focus


  • Rise & Run sign variations
  • Horizontal / zero slope lines
  • Vertical / undefined slope lines
  • Conversion to linear equations