Definition: Slope is a number that measures how steep a line is, found by dividing the vertical change by the horizontal change between two points.
What is slope?
Slope tells how much y changes when x increases by 1. It is rise over run: m = (y2 – y1) / (x2 – x1).
How do you find it?
Use two points, say (2, 5) and (6, 17). The rise is 17 – 5 = 12 and the run is 6 – 2 = 4. So m = 12 / 4 = 3.
What does the sign mean?
- Positive: the line rises left to right.
- Negative: it falls.
- Zero: it is horizontal.
- Undefined: it is vertical.
What does it mean in real life?
Mathbot’s drone climbs from 40 feet at 2 seconds to 100 feet at 7 seconds. The slope is 60 / 5 = 12 feet per second, its climbing rate.
How does it connect to proportional relationships?
For y = kx, the slope equals k. Any two points on the line give the same slope.
What is a common mistake?
Mixing the order of subtraction. If you start with y2 on top, you must start with x2 on the bottom. Also, run over rise is upside down.
Example: A line through (1, 9) and (5, 1) has slope -2.
Where to learn it
- Slope and Proportional Relationships on Graphs – lesson
- Modeling Real Situations With Linear Functions – lesson