Comparing Repayment Plans Using Rate of Change

🔑 Key Concepts

• Rate of change measures how quickly a quantity increases or decreases.
• Slope represents the rate of change in a linear relationship.
• The initial value is the amount present when the input is \(0\).
• A negative slope means the amount owed decreases over time.
• Rate of change can be calculated from a table or a graph.
• In \(y=mx+b\), \(m\) is the weekly change and \(b\) is the initial amount owed.

✏️ Worked Example — Comparing Two Repayment Plans

🧠 Math Vocabulary

• Rate of change: How much one quantity changes for each unit of another quantity.
• Slope: The rate of change of a linear relationship, calculated as change in \(y\) divided by change in \(x\).
• Initial value: The starting amount when the input is \(0\).
• Y-intercept: The initial value \(b\) in the equation \(y=mx+b\).
• Negative rate of change: A rate showing that a quantity decreases as the input increases.

💡 Main Idea

Tables and graphs can represent the same type of linear relationship. Calculate the slope to determine how much the amount owed changes each week, identify the initial amount as the y-intercept, and use \(y=mx+b\) to write an equation for each repayment plan.

📚 What You Should Already Know

Students should know how to read tables and coordinate graphs, calculate change between values, interpret ordered pairs, and convert months to weeks using four weeks per month.

🚀 What Comes Next

Students will compare linear relationships represented by tables, graphs, and equations and apply rate of change to situations involving cost, distance, speed, savings, and other real-world quantities.

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