Lesson 4: Using Rates of Change to Build Tables and Graphs

Who do we use Rate of Change to calculate the change in the output given the change in the input?

F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima.

Lesson Description: In this lesson, students apply the concept of slope to build mathematical models from the ground up. Instead of just identifying a slope, they use it to "drive" the creation of tables and graphs. Key skills include: The "Starting Value" (b): Identifying the initial state of a function at x = 0.Iterative Table Building: Using the rate of change to calculate subsequent y-values (e.g., if the rate is +5, each new row in the table increases by 5).Graphing via Movement: Using a fixed starting point and a constant rate to plot a series of points that illustrate a trend. Non-Unit Rates: Handling rates of change that aren't whole numbers (e.g., a rate of 2/3) and understanding how that affects the "step" between points in a table versus a graph.
Purpose
The purpose of Section 9.4 is to demonstrate the predictive power of algebra. By using a rate of change to build a table or graph, students are essentially performing "forecasting." This is exactly how spreadsheets and financial models work: they take an initial value and apply a constant rule to see where the data will be in the future. This lesson reinforces the idea that a linear function is a repeatable pattern, giving students the confidence to project data points far beyond the provided information.
Depth of Knowledge (DOK) Level
DOK Level 2
Level 2 (Skill/Concept): This section requires students to apply a rule across multiple steps. They must demonstrate that they can take a rate (slope) and an initial value (intercept) and correctly generate a sequence of related values. This involves "mental movement"—knowing that a change in the x-column must be balanced by a specific, calculated change in the f(x)-column to maintain linearity.

A local carpet cleaning company charges $15 for each room plus a nonrefundable reservation fee of $25. Make a table of values for the total cost as the number of rooms to be cleaned increases.

If some students have difficulty with word problems because they cannot picture what the problem is trying to communicate, then sometimes it is easier for those students to graph or draw a picture of the given information. For Guided Practice 5, you may wish to have students do part b first by using the starting point and the rate of change to determine other points on the graph.

Use "you try" pg 216 and practice problems to assess students' understanding of the lesson concepts