Unit 4 — Quadratic Equations, Functions & Polynomials
Description
This unit develops students' understanding of polynomials and quadratic functions. Students add, subtract, and multiply polynomials, recognizing that polynomials form a system analogous to the integers. They factor expressions, particularly identifying differences of squares. The unit focuses on solving quadratic equations using multiple methods including inspection, square roots, factoring, completing the square, and the quadratic formula. Students derive the quadratic formula from completing the square and recognize when solutions are complex, expressing them in the form a ± bi. Students create quadratic equations from real-world situations. The unit interprets key features of quadratic functions including intercepts, vertices, intervals of increase and decrease, and domain. Students sketch graphs from verbal descriptions and use equivalent forms to reveal specific function properties. They perform and analyze transformations of quadratic functions, identify zeros and construct rough graphs of polynomials, and compare properties of quadratic functions in different representations. The unit concludes with properties of rational and irrational numbers.
Essential Questions
- How are polynomial operations related to integer operations?
- What are the advantages of different forms of quadratic expressions and equations?
- How do we determine which method is most efficient for solving a quadratic equation?
- How do transformations affect the graph of a quadratic function?
Learning Objectives
- Add, subtract, and multiply polynomials
- Recognize and factor differences of squares
- Solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula
- Derive the quadratic formula from completing the square
- Write complex solutions in a ± bi form
- Create quadratic equations from real-world problems
- Interpret key features of quadratic functions from graphs and tables
- Sketch graphs of quadratic functions given verbal descriptions
- Factor and complete the square to reveal zeros and extreme values
- Graph quadratic functions showing intercepts and extreme values
- Compare properties of two quadratic functions in different forms
- Identify effects of transformations on quadratic functions
- Find approximate solutions when a linear function intersects a quadratic function
- Identify zeros of cubic functions and use them to sketch graphs
- Explain conclusions about sums and products of rational and irrational numbers
Supplemental Resources
- Graphic organizers for organizing quadratic solution methods
- Printed polynomial expressions for factoring practice
- Rulers and grid paper for graphing quadratic functions
- Index cards for identifying key features of parabolas
- Sentence strips describing real-world quadratic situations
Expressions and Equations
Functions
The Number System
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 polynomial operations and factoring
- Quizzes on quadratic solution methods and function features
- Observations of students selecting efficient solution methods
- Pair-and-share on interpreting quadratic functions in different forms
- Exit tickets on describing transformations and their effects
Summative Assessment
Unit 3 test on polynomials, quadratic equations, quadratic functions, and rational and irrational numbers; performance assessment modeling real-world situations with quadratic functions
Benchmark Assessment
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Alternative Assessment
Students may demonstrate understanding of polynomial operations and quadratic solving methods through a combination of oral explanation, step-by-step guided worksheets with worked examples, and use of manipulatives or visual models to show factoring and completing the square. For solving quadratic equations, students may use a reference sheet listing the methods and may respond to problems with reduced scope (e.g., focusing on one solution method rather than multiple methods in a single task).
IEP (Individualized Education Program)
Students may benefit from graphic organizers that outline the steps for each quadratic solution method, helping them track multi-step algebraic processes such as completing the square or applying the quadratic formula. Providing a reference sheet with polynomial operation rules, factoring patterns like differences of squares, and the quadratic formula supports processing without reducing the rigor of the content. Where written output is a barrier, allow students to demonstrate understanding of function features—such as vertices, intercepts, and transformations—through oral explanation or annotated graphs. Break longer tasks involving multiple solution methods into clearly sequenced steps, and offer frequent check-ins to catch and address procedural misconceptions early.
Section 504
Extended time should be made available for assessments covering multi-step processes such as solving quadratic equations and graphing functions, as these tasks place significant demand on working memory and processing speed. Preferential seating and a low-distraction environment support sustained focus during lessons that require students to track multiple algebraic representations simultaneously. Providing a printed copy of any board work involving graphs, equations, or step-by-step procedures ensures students are not disadvantaged by difficulties with copying.
ELL / MLL
Key vocabulary for this unit—including terms such as polynomial, quadratic, vertex, intercept, zero, transformation, and rational and irrational numbers—should be introduced with visual supports such as labeled diagrams and graphic organizers before and during instruction. Pairing symbolic representations with verbal descriptions and graphs helps students connect mathematical language to meaning, particularly when interpreting quadratic function features or describing transformations. Simplified, step-by-step written directions for multi-stage procedures support comprehension, and allowing students to initially discuss reasoning with a partner in their home language before sharing in English builds confidence and accessibility.
At Risk (RTI)
Connect new content to students' prior knowledge of linear functions and integer operations, using these familiar structures as a bridge to polynomial operations and quadratic behavior. Reducing the number of problems on practice tasks to focus on one solution method at a time—rather than all methods simultaneously—allows students to build mastery incrementally before being asked to select among approaches. Visual representations of quadratic graphs, including clearly labeled vertices and intercepts, provide concrete entry points for students who struggle with purely symbolic work. Positive framing of incremental progress, particularly as students navigate the more abstract methods like completing the square, supports motivation and persistence.
Gifted & Talented
Students who demonstrate early mastery of standard quadratic solution methods can be challenged to investigate the relationships among the different methods—for example, exploring why completing the square always works and how the discriminant determines the nature of solutions algebraically and graphically. Extending into complex solutions and their geometric interpretation, or exploring polynomial behavior beyond quadratics including connections to cubic functions and their zeros, offers meaningful depth. Students might also investigate real-world modeling scenarios that require selecting and justifying the most efficient solution method, or examine the properties of rational and irrational numbers through a more formal proof-based lens.