Lesson 5: Interpreting Slope of a Linear Function

What do we need to know in order to determine if the function is linear?

What is the ratio to find rate of change?

What can a linear graph tell you about the relationship that it represents?

Rate of change
Slope
Input
Output
Units

F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Lesson Description This lesson moves away from calculation and toward interpretation and analysis. Students are given completed linear models and asked to "deconstruct" them. Key skills include: The Meaning of b: Identifying the y-intercept as the "starting value," "base fee," "initial height," or "original amount" at time zero (x = 0).The Meaning of m: Interpreting the slope as the "rate of change" or "unit rate" (e.g., "the temperature drops by 2 degrees for every hour that passes").Units of Measure: Correcting labeling the numerical values with their real-world units (e.g., distinguishing between a slope of "5" and a rate of "5 dollars per gallon").Predictive Analysis: Using the function to explain what will happen if the independent variable increases or decreases (e.g., "If x increases by 10, the total cost will increase by 50 dollars").
Purpose
The purpose of Section 10.5 is to ensure conceptual mastery. A student might be able to calculate a slope of -3.5, but the real-world value of that math lies in knowing that it represents a "depreciation of $3,500 per year" for a piece of equipment. This section prepares students for the data-driven decision-making required in Middlesex CC's professional programs, ensuring they can not only "do the math" but also explain the "why" and "how" to colleagues or clients.
Depth of Knowledge (DOK) Level
DOK Level 3
Level 3 (Strategic Thinking & Complex Reasoning): This is a high-level interpretive task. Students must analyze a symbolic model and produce a sophisticated verbal justification of its components. They are often asked to evaluate the validity of the model (e.g., "Does it make sense for this slope to be negative in this context?") and explain the implications of the mathematical relationship on a real-life outcome.

Students often misalign the corresponding output and input values, arriving at the wrong sign for rate of change and slope. Students can inspect the function to see if it is increasing or decreasing to reinforce the correct way to calculate rate of change

Use the context of the problem to highlight the meaning of the intercepts and slope.

Extension: The function A(m) = 200 - 1.25 m represents the balance in a bank account (in thousands of dollars) after m months. Identify the slope, vertical and horizontal intercepts and interpret their meanings

Use "you try" pg 242 and practice problems to assess students' understanding of the lesson concepts