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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

Practice problems

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the slope through (0, 0) and (5, 15).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Rise 15, run 5.
  4. Slope = 15/5 = 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the slope through (1, 2) and (4, 8).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Rise 6, run 3.
  4. Slope = 6/3 = 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the slope through (-3, -1) and (1, 7).
  2. For slope or rate of change, compare vertical change to horizontal change and keep the sign attached to the direction of the change.
  3. Rise 7 - (-1) = 8, run 1 - (-3) = 4.
  4. Slope = 8/4 = 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: A pool fills 200 gallons in 5 minutes. What is the rate in gallons per minute?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Divide gallons by minutes.
  4. 200 / 5 = 40 gallons/min.
  5. Check the result by substituting or estimating: the response should match 40 and make sense in the original problem.

Answer: 40

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: A car travels 240 miles in 4 hours. What is the rate in miles per hour?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Divide miles by hours.
  4. 240 / 4 = 60 mph.
  5. Check the result by substituting or estimating: the response should match 60 and make sense in the original problem.

Answer: 60

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