Lesson 3: 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
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 focuses on the contextual meaning of the slope $m$ in the linear function $f(x) = mx + b$. Students will learn to translate the numerical value of slope into a descriptive sentence. Key concepts include: Slope as Unit Rate: Understanding that slope represents the change in the dependent variable for every one-unit increase in the independent variable. Identifying Units: Determining the correct units for slope by using the "Units of y per Unit of x" formula (e.g., miles per hour, dollars per gallon, or degrees per minute).Interpreting Sign: Explaining what a positive slope (growth/increase) versus a negative slope (decay/decrease) means within a specific story. Comparing Steepness: Analyzing which of two slopes represents a "faster" rate of change in a practical scenario (e.g., which car is traveling faster or which bank account is growing more quickly).
Purpose
The purpose of Section 9.3 is to develop quantitative literacy. In fields like healthcare, business, and technology, "slope" is rarely called by its mathematical name; instead, it is referred to as "speed," "burn rate," "hourly wage," or "flow rate." By learning to interpret slope, students gain the ability to explain how a system is changing, which is a critical skill for data analysis and professional communication. This section transforms the slope from an abstract fraction into a meaningful measurement. Depth of Knowledge (DOK) Level
DOK Level 2 & 3
Level 2 (Skill/Concept): Students must identify the units of slope from a graph or table and perform basic interpretations (e.g., "The slope is 15, which means the cost increases by 15 for every additional item").
Level 3 (Strategic Thinking): Students must explain the significance of the slope in a complex scenario. They might be asked to predict what happens to the real-world outcome if the slope changes, or to compare two different linear models and explain why one rate of change is more desirable than the other in a given context.

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 road sign on a hill says 5% grade. The elevation of the road at that point is 1200 feet. Make a drawing of this situation. What would be the elevation of the road at an additional 2000 horizontal feet from the road sign?

Use "you try" pg 214 and practice problems to assess students' understanding of the lesson concepts