Slope & rate of change
The idea
Slope puts a number on steepness: between two points on a line, it is the vertical change divided by the horizontal change — rise over run. In a real context, that very same number is a rate of change: dollars saved per week, kilometers per hour, points per game. You already plot points on the coordinate plane, so slope is the next question to ask of any two of them: for each step across, how far does the line climb or fall?
On a true line, the slope is the line's personality — pick any two points and the ratio comes out identical, which is exactly what makes prediction possible. Watch for two confusions. First, height is not steepness: a line drawn high on the grid can still be flat, while a low line can be steep; slope measures tilt, not position. Second, a negative slope is not an error — it simply reports a quantity falling as you move right, like money draining from an account.
Worked example
Maya saves at a steady rate. In week 2 her account holds $35, and in week 6 it holds $95. How much does she save per week, and how much will she have in week 7?
- Treat the facts as points (2, 35) and (6, 95), with weeks across and dollars up.
- Compute rise over run: the rise is 95 − 35 = 60 dollars while the run is 6 − 2 = 4 weeks, so the slope is 60 ÷ 4 = 15 dollars per week.
- Interpret the slope before using it: steady saving means every single week adds exactly $15, between any two weeks you pick.
- Predict week 7 by extending one week past week 6: 95 + 15 = 110 dollars, and as a check, week 2 plus five weeks of saving gives 35 + 5 × 15 = 110 as well.
Answer. Maya saves $15 per week, so her account will hold $110 in week 7.
Check your understanding
- Why does any pair of points on the same line give the same slope, and what would differing answers tell you?
- What does a negative slope mean in a savings story, and what would a slope of zero mean?
- How is rise over run connected to a unit rate like dollars per week?
- Why does subtracting the coordinates in a consistent order matter when computing slope?
Build the foundations first
Slope & rate of change builds on these concepts. If any feel shaky, start there.