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Slope as Rate of Change

A free Pre-Algebra lesson from the “Coordinate Plane, Functions, and Slope” unit, with a worked example and practice problems including step-by-step solutions.

Slope is the constant rate of change of a linear relationship: rise over run = (y2 - y1) / (x2 - x1). In context, slope answers 'how much does y change for each 1-unit change in x?' A positive slope rises, a negative slope falls, and a slope of 0 is a horizontal line. In Coordinate Plane, Functions, and Slope, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.

What you'll learn

Why it matters: Miles per hour, dollars per item, degrees per minute, and any 'per' rate is the slope of the relationship between the two quantities.

Worked example

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

  1. Rise = 11 - 3 = 8.
  2. Run = 6 - 2 = 4.
  3. Slope = 8 / 4 = 2.
  4. Connect the calculation back to Slope as Rate of Change so the method, not just the arithmetic, is clear.

Answer: 2

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response). Counts include repeated prompts and variants.

Practice problems

1. Find the slope through (0, 0) and (5, 15).

Show solution
  1. Rise 15, run 5.
  2. Slope = 15/5 = 3.

Answer: 3

2. Find the slope through (1, 2) and (4, 8).

Show solution
  1. Rise 6, run 3.
  2. Slope = 6/3 = 2.

Answer: 2

3. Find the slope through (-3, -1) and (1, 7).

Show solution
  1. Rise 7 - (-1) = 8, run 1 - (-3) = 4.
  2. Slope = 8/4 = 2.

Answer: 2

4. A pool fills 200 gallons in 5 minutes. What is the rate in gallons per minute?

Show solution
  1. Divide gallons by minutes.
  2. 200 / 5 = 40 gallons/min.

Answer: 40

5. A car travels 240 miles in 4 hours. What is the rate in miles per hour?

Show solution
  1. Divide miles by hours.
  2. 240 / 4 = 60 mph.

Answer: 60

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