Standard: 8.F.A3 – Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s^2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
Grade level: Grade 8
Subject: Mathematics
Domain: Functions
Teacher Overview
This standard focuses on helping students understand the concept of linear functions and how they can be represented graphically by the equation y = mx + b. It also emphasizes the importance of distinguishing between linear and nonlinear functions, which is a foundational skill in algebra and higher-level mathematics. Before tackling this standard, students need to be comfortable with basic algebraic concepts and graphing techniques. They should be able to plot points on a coordinate plane and understand simple linear equations.
Mastering this standard will prepare students for more advanced topics in algebra, such as function transformations and the analysis of different types of functions in various contexts.
Common Misconception 1
A common misconception is that all functions are linear. This is incorrect because there are many types of functions, such as quadratic, exponential, and logarithmic functions, that are not linear.
Intervention 1
One effective intervention is to use graphing calculators or software to visually demonstrate the differences between linear and nonlinear functions. By plotting various functions, students can see firsthand how their graphs differ.
Common Misconception 2
Another misconception is that if a function passes through the origin (0,0), it must be linear. This is not true because many nonlinear functions, such as y = x^2, also pass through the origin.
Intervention 2
To address this misconception, provide students with multiple examples of functions that pass through the origin, both linear and nonlinear. Discuss the characteristics that differentiate them.
Prerequisite Knowledge
Students should have a basic understanding of algebraic expressions and equations, including how to plot points on a coordinate plane and interpret simple graphs.
Subsequent Knowledge
Students will develop the ability to analyze more complex functions and understand the concept of function transformations, as well as apply linear and nonlinear functions to solve real-world problems.
Instructional Activities
- Graphing various linear and nonlinear functions using graph paper or graphing software.
- Creating real-world scenarios where students must identify and classify functions as linear or nonlinear.
- Group discussions and presentations on the characteristics of linear and nonlinear functions.
- Interactive online simulations that allow students to manipulate variables in functions and observe the effects on their graphs.
