Unit 3 — Modeling with Linear Functions, Linear Systems, & Exponential Functions
Description
This unit extends students' function work by formally introducing exponential functions alongside linear functions. Students construct functions from data pairs and verbal descriptions, interpreting rate of change and initial values. They deepen their understanding of systems of equations and inequalities, solving exactly and approximately, and interpreting solutions as viable or nonviable in context. Students develop proficiency with function notation and evaluate functions for inputs in their domains. The unit distinguishes between linear and exponential growth by examining how each grows over equal intervals. Students write both linear and exponential functions from graphs, tables, and descriptions, including arithmetic and geometric sequences. They interpret parameters in linear and exponential functions and use properties of exponents to produce equivalent forms. The unit includes analysis of function features such as intercepts, domains, ranges, and average rates of change, both for simple cases by hand and complex cases using technology.
Essential Questions
- How do we distinguish between linear and exponential functions based on how they grow?
- How can we write functions that model real-world relationships, and what do the parameters tell us?
- How do we solve and interpret systems of equations and inequalities graphically and algebraically?
- How do transformations affect the graphs of functions?
Learning Objectives
- Construct a function to model a linear relationship; interpret rate of change and initial value
- Describe qualitatively the functional relationship between quantities by analyzing a graph
- Solve systems of linear equations exactly and approximately
- Solve systems of linear inequalities and interpret solution regions
- Use function notation, evaluate functions, and interpret statements in context
- Distinguish between linear and exponential functions based on growth patterns
- Construct linear and exponential functions from graphs, tables, and descriptions
- Write linear and exponential functions given arithmetic and geometric sequences
- Interpret parameters in linear and exponential functions in terms of a context
- Analyze and compare properties of two functions in different representations
- Calculate and interpret average rate of change
- Graph functions including square root, cube root, and piecewise-defined functions
Supplemental Resources
- Printed graphs showing linear and exponential patterns for comparison
- Graphic organizers for organizing function information by representation
- Sticky notes for marking key features on graphs
- Index cards for writing and evaluating linear and exponential functions
- Sentence strips describing real-world linear and exponential situations
Expressions and Equations
Functions
Students engage in collaborative discussions about mathematical concepts, construct arguments to support mathematical claims using evidence, analyze and interpret information presented in diverse formats, and write informative explanations of mathematical processes and procedures.
Students apply mathematical reasoning to analyze scientific data, use quantitative relationships to describe phenomena, construct explanations based on evidence, and model real-world relationships in biological and physical systems.
Formative Assessments
- CPM checkups on systems of equations and exponential growth
- Quizzes on distinguishing linear from exponential and writing functions
- Observations of students solving system problems graphically
- Pair-and-share on comparing two functions in different forms
- Exit tickets on interpreting parameters in context
Summative Assessment
Unit 2 test on linear functions, systems of equations and inequalities, exponential functions, and function comparisons; performance assessment modeling real-world situations with linear and exponential functions
Benchmark Assessment
Benchmark covering major content (A.REI.C.6, A.CED.A.3, F.IF.A, F.LE.A, F.BF.A, 8.F.B)
Alternative Assessment
Students may demonstrate understanding through verbal explanation of function features, rate of change, and system solutions in place of written work. Number lines, function tables with filled-in values, and equation templates may be provided to support function construction and system solving.
IEP (Individualized Education Program)
Students may benefit from graphic organizers that visually distinguish linear from exponential growth patterns, helping them organize rate-of-change reasoning before writing or solving. Providing a reference sheet with function notation examples, exponent rules, and annotated graph templates supports processing during multi-step tasks involving systems or function construction. Output flexibility—such as allowing oral explanation of parameter interpretation or use of a calculator for computation-heavy steps—reduces barriers while keeping the focus on conceptual understanding. Breaking tasks like solving systems or analyzing two functions into clearly sequenced steps with checkpoints supports students in sustaining attention and monitoring their own progress.
Section 504
Extended time on quizzes and unit assessments supports students working through the multi-step reasoning required by systems of equations, function construction, and exponential modeling. Preferential seating and a low-distraction environment are particularly helpful during graphical analysis tasks that require sustained visual attention. Providing a clean printed copy of any graphs, tables, or function representations used in assessments—rather than requiring students to read them from a projected display—ensures consistent access to the information needed to respond accurately.
ELL / MLL
Previewing key vocabulary such as 'rate of change,' 'initial value,' 'exponential,' 'system,' and 'parameter' with visual examples and translated glossaries helps students connect mathematical language to meaning before engaging with tasks. Using graphs, tables, and visual models as entry points into function comparisons and contextual problems reduces reliance on dense text while keeping the mathematics accessible. Pairing simplified written directions with visual or worked examples when students are asked to write or interpret functions from descriptions supports both comprehension and productive engagement.
At Risk (RTI)
Connecting new content to students' prior work with linear equations and proportional reasoning provides a familiar entry point before introducing exponential growth and systems. Offering problems with reduced complexity first—such as interpreting a table before writing a function rule, or solving a system graphically before solving algebraically—builds confidence and reveals where gaps in understanding exist. Frequent brief check-ins during work time allow teachers to redirect misconceptions about growth patterns or function notation before they become entrenched, and structured graphic organizers can help students organize their thinking across the unit's range of function types.
Gifted & Talented
Students ready for greater depth can explore how linear and exponential models break down or need adjustment when applied to more complex real-world data, including an introduction to regression and residual analysis using technology. Investigating the behavior of piecewise or step functions in applied contexts—such as rate structures, tax brackets, or population modeling—extends the unit's function analysis work meaningfully. Encouraging students to analyze and compare multiple representations of the same situation, justify why one model type is more appropriate than another, and communicate their reasoning formally through written mathematical argument deepens both conceptual understanding and mathematical practice.