Student Actions

Teacher Actions

• Develop the ability to communicate mathematically as they justify their interpretation to teachers and peers, and demonstrate their understanding of the pieces of the graph in the context of mathematical modeling problems

• Develop a deep and flexible conceptual understanding when exploring properties of linear piecewise functions and determining how and/or when to utilize these properties within mathematical models.

• Develop the ability to make conjectures, model and generalize when drawing conclusions about how range values apply to each piece of a piecewise function and extending the relationship of domain restrictions within a piecewise function to a contextual situation. 

• Use and connect mathematical representations of linear piecewise functions presented in a variety of ways including equations with restricted domain values, verbal interpretations, and graphed images for various mathematical modeling problems.

• Build procedural fluency from conceptual understanding for students as they learn to analyze only a segment of a linear graph and interpret and express the boundaries of each piece of the function as it relates to contextual situations.

• Pose purposeful questions to students about what they notice on the graph about the domain, range, or intervals of a linear piecewise function and how these properties relate to contextual situations. 

Key Understandings

Misconceptions 

• Piecewise linear function is a real-valued function defined on the real numbers or a segment thereof, whose graph is composed of line segment sections.

• Increasing intervals of a linear piecewise function contain a positive rate of change.

• Decreasing intervals of a linear piecewise function contain a negative rate of change.

• Constant intervals of a linear piecewise function contain no rate of change. 

• Students confuse or can not identify the part of the domain corresponding to a specific linear function.

• Students interchange the domain and range of each linear function portion.

• Students may not realize that two or more distinct lines are necessary to model contextual situations represented by a linear piecewise function.

• Students may not recognize that a horizontal portion of a linear piecewise function represents a constant interval where there is no change in the dependent values (a slope of zero). 

  Knowledge Connections

Prior Knowledge

Leads to 

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

• Identifying graphical properties/characteristics of linear functions (PA.A.2.3) 

• Graphing and analyzing piecewise functions with up to three branches of linear, quadratic and exponential segments (A2.F.1.8)