Comparing Rates of Change From Graphs, Tables, and Equations

🔑 Key Concepts

  • Rate of change tells how much one quantity changes compared with another quantity.
  • The rate of change of a linear relationship is the same as the slope of its graph.
  • A rate can be determined from a graph, equation, table, or verbal description.
  • To compare rates shown in different forms, rewrite each one as a unit rate.
  • When pace is measured in minutes per mile, the smallest number represents the fastest runner.

✏️ Worked Example — Comparing Three Runners’ Paces

🧠 Math Vocabulary

  • Rate of change: A comparison showing how much one quantity changes for each unit of another quantity.
  • Slope: The rate of change of a line, calculated as rise divided by run.
  • Unit rate: A rate that compares a quantity to exactly one unit of another quantity.
  • Pace: The amount of time required to travel one unit of distance, such as minutes per mile.
  • Constant rate: A rate that remains the same throughout a relationship.

💡 Main Idea

A linear rate of change can be represented by a graph, an equation, or a table. Although the representations look different, each one communicates the same mathematical relationship. To compare them, determine the unit rate from each representation and make sure the units are the same. In this problem, each rate is written in minutes per mile, so the smallest value represents the fastest pace.

📚 What You Should Already Know

Before this lesson, you should know how to read ordered pairs from a graph, identify the coefficient of a variable in an equation, and compare corresponding values in a table. You should also understand that slope can be calculated as rise over run and interpreted as a unit rate.

🚀 What Comes Next

Next, students will use rates of change to write linear equations, compare functions represented in different forms, and interpret slopes in real-world situations involving speed, cost, distance, temperature, and other measurable quantities.

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