Student Actions

Teacher Actions

• Develop accurate and appropriate procedural fluency by creating tables, writing equations, and graphing functions to determine linear and nonlinear relationships.

• Develop a deep and flexible conceptual understanding by exploring linear and nonlinear functions to understand the relationship between the independent and dependent variables.

• Develop mathematical reasoning by exploring functions to determine whether they are linear or nonlinear and justifying their reasoning mathematically to their teacher and peers. 

• Use and connect mathematical representations to engage students in connecting the concepts of linear and nonlinear functions to their equations, graphs, tables, and charts.

• Pose purposeful questions and examples to connect students to real-world situations by using examples of several different  functions as they explore the similarities and differences of those functions.

• Implement tasks that promote reasoning and problem-solving by having students interpret the mathematical relationship between tables, graphs, and equations of  linear or nonlinear functions.

• Encourage productive struggle as students work through real-world situations to determine whether a function is linear or nonlinear. 

Key Understandings

Misconceptions 

• A function is linear if it can take the form of f(x)=mx+b.

• Distinguish between linear or nonlinear functions by using tables, graphs, or equations.

• Determine a parent function or its graph as linear or nonlinear.

• Linear functions grow by equal intervals (consistent addition patterns) while exponential functions grow by equal factors (consistent multiplication patterns) over equal intervals.

• A function is considered nonlinear when:

• there are non-constant differences among output values

• the graph is not a straight line

• the equation contains a variable with any power besides 1

• the equation contains a variable in the denominator of a rational term 

• Students may mix up the input and the output and incorrectly identify a relationship as linear when it is nonlinear.

• If a function is not in slope-intercept form, the students may not recognize it as a linear function.

• Students may confuse arithmetic (equal intervals) and geometric intervals (equal factors over equal intervals) without understanding that arithmetic growth is modeled with addition and subtraction and that geometric growth is modeled with multiplication and division. 

• When using tables to identify functions, students may not consider the rate of changebetween all provided data points and only consider individual data points instead.

• Ex: Given this table, students may see a non-linear pattern because they only consider the numbers in the table and do not calculate the rate of change, when in fact the change is an equal interval 

  Knowledge Connections

Prior Knowledge

Leads to 

• Identifying functions as linear if it’s in slope intercept form or has a graph of a nonvertical line (PA.A.1.3)

• Representing linear functions with multiple representations (PA.A.2.1)

• Identifying, describing and analyzing linear relationships (PA.A.2.2)

• Identifying slope as the constant rate of change for a linear function (PA.A.2.3) 

• Recognize arithmetic sequences are linear using multiple representations and finding the next term in a sequence (A2.A.3.1)

• Recognize geometric sequences are exponential using multiple representations and finding the next term in a sequence (A2.A.3.2)