Lesson 5: Is the Function Linear
Duration of Days: 2
Lesson Objective
Determine if slope of a data set has constant, or changing slope.
Understand that linear functions are lines which have constant slope and that positive and negative slope indicate if the functions is increasing or decreasing
What do we need to know in order to determine if a function is linear? The slope must be?
What is the ratio to find rate of change and slope?
What can a linear graph tell you about the relationship that it represents?
Slope
Slope-intercept form
Line
Linear
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 In this lesson, students develop a set of criteria to test for linearity across multiple representations. They will learn that a function is only linear if it possesses a constant rate of change. The lesson covers: The Table Test: Checking if the ratio of the change in output to the change in input (Delta y\Delta x) remains consistent throughout the entire data set. The Equation Test: Identifying if the variables are raised only to the first power and are not in the denominator or under a radical. The Graph Test: Visually confirming if the relationship forms a single, straight line with a consistent "steepness."Distinguishing Non-Linearity: Recognizing common non-linear patterns, such as curves (quadratic/exponential) or abrupt changes in direction (absolute value).
Purpose
The purpose of Section 9.5 is to develop critical evaluation skills. In data science and statistics, assuming a relationship is linear when it is actually curved can lead to massive errors in prediction. By teaching students to ask "Is this linear?", we are training them to verify their assumptions before applying a formula. This section ensures that students don't just "plug in numbers," but instead analyze the underlying nature of the data they are working with.
Depth of Knowledge (DOK) Level
DOK Level 2 & 3
Level 2 (Skill/Concept): Using the slope formula to check multiple pairs of points in a table to see if the slope is constant. Comparing an equation to the standard f(x) = mx + b form to determine its type.
Level 3 (Strategic Thinking): Given a set of real-world data, students must justify why a linear model may or may not be appropriate. They must provide evidence (mathematical or contextual) to support their conclusion, such as pointing out that a rate of growth is increasing over time rather than staying constant.
Calculating slope as change in x over change in y, or calculating slope as the change between the coordinates of one point over change between the coordinates of a second point
If students automatically assume that the left-most point has to be (x1, y1) and the point farther right is (x2, y2), then explain that the designation of (x1, y1) and (x2, y2) is arbitrary. Write pairs of points on index cards. Give one card to each student. Have them find the slope both ways. Then ask which way made the subtraction easier.
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 218 and practice problems to assess students' understanding of the lesson concepts