Unit 2 — Modeling with Linear Equations and Inequalities
Description
This unit develops students' ability to reason quantitatively with units and create equations modeling real-world situations. Students solve linear equations and inequalities in one variable, including literal equations with letter coefficients. They interpret expressions and understand solving as a process of reasoning. Students work with systems of linear equations in two variables, solving both algebraically and graphically, and interpret solutions in context. The unit introduces functions formally, establishing the definition and exploring how to determine if a relation is a function. Students compare functions represented in multiple forms and analyze linear functions using slope, intercepts, and tables of values. Throughout, students represent and solve equations graphically.
Essential Questions
- What does it mean to solve an equation, and how do we justify each step?
- How do we create equations from real-world descriptions and use them to solve problems?
- What is a function, and how are functions represented and compared?
- How do we interpret linear models in terms of the data they represent?
Learning Objectives
- Use units to understand and guide the solution of multi-step problems
- Solve linear equations and inequalities in one variable, including literal equations
- Explain each step in solving an equation using properties of equality
- Create linear equations and inequalities in one variable and in two variables
- Define and identify functions; use function notation
- Compare properties of functions represented in different ways
- Interpret the structure of linear expressions and equations
- Represent and solve equations graphically
- Create and interpret scatter plots; fit functions to data and interpret correlation coefficients
- Distinguish between correlation and causation
Supplemental Resources
- Printed word lists of linear function vocabulary and symbols
- Graphic organizers for comparing functions in multiple representations
- Sticky notes for annotating graphs and equations
- Grid paper for graphing linear equations and systems
- Index cards for practice comparing function properties
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 equation solving and function concepts
- Quizzes on linear systems and function properties
- Observation of students solving real-world problems with units
- Pair-and-share on comparing functions in different representations
- Exit tickets on interpreting slope and intercept in context
Summative Assessment
Unit 1 test covering linear equations, inequalities, systems, functions, and linear models; performance assessment using data to create and interpret linear models
Benchmark Assessment
— not configured —
Alternative Assessment
Students may demonstrate understanding through oral explanations of solving steps, with teacher scribing or recording responses, in place of written work. Number lines, equation mats with visual representations of variables and constants, and pre-made equation steps to sequence may be provided to support solving and interpreting linear equations and inequalities.
IEP (Individualized Education Program)
Students may benefit from graphic organizers that outline the steps for solving linear equations and inequalities, helping them track their reasoning through each property of equality. Providing reference sheets with key vocabulary, equation-solving steps, and function notation supports independence during multi-step problem work. Where written output is a barrier, allow students to demonstrate understanding of slope, intercepts, or function comparisons through oral explanation or annotated graphs. Extended time and chunked assignments help students process the layered demands of moving between algebraic, tabular, and graphical representations.
Section 504
Students should be given extended time on equation-solving tasks and assessments that require interpreting and creating linear models across multiple representations. Preferential seating and a low-distraction environment support sustained focus during graphing and systems work, where attention to detail is critical. Printed copies of any board work involving equations, graphs, or function comparisons reduce the burden of copying and allow students to stay engaged with the reasoning process.
ELL / MLL
Pre-teaching key terms such as function, slope, intercept, correlation, and inequality — using visual models, labeled graphs, and real-world contexts — helps students build the language needed to access this unit's content. Directions for multi-step equation and graphing tasks should be given in short, clear steps, and students should be encouraged to restate instructions in their own words before beginning. Bilingual glossaries and side-by-side representations (graph paired with equation paired with table) provide multiple access points for understanding how linear relationships are expressed in different forms.
At Risk (RTI)
Connecting equation-solving to familiar real-world situations — such as calculating costs, distances, or rates — gives students a concrete entry point before moving to abstract symbolic work. Reducing the number of problems on any given task while maintaining a focus on key concepts, such as setting up and solving one equation or identifying whether a relation is a function, helps students build confidence and mastery. Frequent check-ins during multi-step work allow teachers to catch and address misconceptions early, particularly around interpreting slope and intercept in context or reasoning through inequality solutions.
Gifted & Talented
Students can extend their understanding by exploring how systems of equations model real-world constraint problems, such as optimization scenarios, and by analyzing when and why unique, infinite, or no solutions arise in systems. Encouraging students to examine the relationship between correlation coefficients and the strength of linear models — including a critical look at what correlation does and does not imply about causation — deepens statistical reasoning beyond the unit's core expectations. Students may also investigate non-linear functions as a contrast to the linear models studied, using data sets to determine which type of function provides the better fit and justifying their reasoning mathematically.