Key Features Exploration

A1.PAFR.2.1 Transform linear, quadratic, exponential, and linear absolute value functions to equivalent forms to identify slope and y-intercept for linear, vertex, and roots (if any) for quadratic and linear absolute value, and y-intercept for exponential.
A1.PAFR.2.3 Solve and graph linear, quadratic, exponential, and linear absolute value equations given in tabular, symbolic, and/or verbal forms using intercepts, domain and range, intervals of increasing and decreasing, vertex (maximum and minimum), end-behavior, and symmetry, and interpret these in terms of mathematical and real-world situations. A1.PAFR.3.3 Translate among graphical, tabular, verbal, and symbolic representations in function notation, to identify intercepts, intervals where the function is increasing, decreasing, constant, maximums and minimums, and symmetries and explain their meanings in real-world and mathematical situations. Focus Process Standard(s) MPS.PS.1: Problem Solving

Learning Target(s) I can identify key features of a quadratic given a graph, a table, or an equation. I can graph a parabola given an equation or a table.

Overview

This 90-minute lesson engages Algebra 1 students in a deep exploration of quadratic functions, aligned with the International Baccalaureate (IB) Mathematics: Applications and Interpretation framework. The lesson targets the identification and interpretation of key quadratic features—vertex, roots, intercepts, symmetry, intervals of increase/decrease, domain and range, and end behavior—using multiple representations: graphs, tables, equations, and verbal descriptions.

This inquiry-driven approach promotes problem solving (MPS.PS.1), critical thinking, and conceptual understanding through collaboration and technology integration, meeting the specified South Carolina A1.PAFR standards while exemplifying IB learner profile attributes such as inquirers, communicators, and reflective thinkers.

Learning Objectives

By the end of this lesson, students will be able to:

LO1: Identify key features of quadratic functions (vertex, roots, intercepts, axis of symmetry, intervals of increase/decrease, domain, range, and end behavior) from graphs, tables, equations, and verbal descriptions.
(IB MYP Math Criterion A: Knowledge & Understanding; S1 & S3)
LO2: Transform quadratic functions among standard, vertex, and factored forms to facilitate identification of key features.
(IB MYP Math Criterion B: Investigating Patterns)
LO3: Graph a parabola accurately given an equation or table by applying transformations and feature analysis.
(IB MYP Math Criterion C: Communicating)
LO4: Translate between graphical, tabular, symbolic, and verbal function representations using function notation, interpreting features in mathematical and real-world contexts.
(IB MYP Math Criterion D: Applying Mathematics in Real-Life Contexts)
LO5: Collaborate effectively through Think-Pair-Share and compare-and-connect routines to deepen conceptual understanding and communication skills.
(IB Approaches to Learning: Communication & Collaboration)

Materials & Resources

Graphing calculators or laptops/tablets with Desmos (or similar graphing tech)
Whiteboards and markers (for pair/group work)
Printed handouts with quadratic functions in different forms (standard, vertex, factored) and their corresponding tables and graphs
Projector or interactive whiteboard for teacher models
Student notebooks and pencils

Lesson Breakdown

Time	Activity	Description & IB-Aligned Pedagogical Focus
0–10 min	Engagement & Activation:
Think-Pair-Share: Forms of Quadratic Functions	- Present the quadratic function ( f(x) = x^2 - 4x + 3 ) in three forms (standard, factored, vertex).

Students spend 1 min thinking individually about how each form highlights different features (roots, vertex, y-intercept).
In pairs, students discuss insights, then share with class.
Teacher highlights links to algebraic manipulation and geometric interpretation (IB Learner profile: Open-Minded, Reflective).
Connects to LO2 and LO4. | | 10–25 min | Exploration:
Identifying Features from Representations | - Provide student pairs with a set of quadratic functions in various forms (graph, table, equation).
Students identify features: vertex, roots, intercepts, symmetry, intervals of increasing/decreasing, domain, range, end behavior.
Encourage function notation accuracy.
Teacher circulates to scaffold, ask probing questions.
Emphasize IB criterion A knowledge & understanding and MPS.PS.1 problem solving through representation translation (LO1, LO3, LO4). | | 25–35 min | Activity Synthesis & Discussion | - Select pairs to present findings on vertex and roots for different representations.
Class discussion to consolidate understanding of how forms reveal features differently.
Emphasize that y-intercept = ( f(0) ), how vertex form shows max/min clearly, factored form shows roots.
Support metacognition and math discourse (IB ATL Communication). | | 35–50 min | Comparative Analysis:
Compare and Connect Quadratic Graphs | - Present two quadratic graphs (e.g., one with two roots facing downward, one with no roots facing upward).
Students (pairs) complete guided worksheet with prompts:
- Circle negative y-values on one graph and positive on another.
- Identify axis of symmetry.
- Describe end behavior.
- Write domain and range in interval notation.
Use Think-Pair-Share and whole class share-out to discuss observations.
Focus on IB criterion A & D: apply knowledge and interpret in real-world contexts related to increasing/decreasing intervals, max/min values, and symmetry (LO1, LO5). | | 50–60 min | Technology Integration:
Desmos Exploration | - Students use graphing technology to input quadratic equations from earlier activities.
Manipulate coefficients to observe changes in vertex, roots, and shape.
Explore transformations and confirm graphical findings through digital experimentation (IB Approaches to Learning: Information Literacy).
Teacher facilitates reflection on the link between algebraic and graphical insight (LO2, LO4). | | 60–75 min | Guided Practice:
Graphing Parabolas from Equations and Tables | - Individually or in pairs, students receive a quadratic equation or table of values.
Students complete graph on graph paper or digitally.
Prompt: Identify and plot vertex, intercepts, axis of symmetry; determine domain and range; note intervals where function increases/decreases.
Teacher circulates providing support and formative feedback (MYP Criterion C). (LO3). | | 75–85 min | Real-World Application & Interpretation | - Present a verbal scenario modeled by a quadratic function (e.g., projectile motion, profit maximization).
Students interpret key features in context: What does vertex represent? What do roots signify?
Discuss how understanding these helps solve real problems (IB Criterion D).
Facilitated via pair discussion and class sharing. | | 85–90 min | Cool Down & Reflection | - Quick-write: "Today I can…" statements addressing LOs.
Exit-ticket question: Identify vertex and roots from given function form or graph.
Teacher collects to inform future instruction and reflect on lesson effectiveness (IB ATL: Reflection). |

Differentiation & Inclusion

English Language Learners: Use graphic organizers with vocabulary scaffold (vertex, intercept, symmetry) and sentence starters during discussions. Visual supports throughout.
Students with Disabilities: Partner with peers for verbal explanation; provide partially completed tables or graphs; audio resources as needed.
Advanced Learners: Challenge to write quadratic functions in different forms from graphs or data, explaining transformations. Explore connections to quadratic inequalities or domain restrictions.

Assessment & Feedback

Formative Assessment:
- Observation of discussions, pair work insights, technology exploration.
- Quick checks during guided practice.
- Exit ticket for immediate gauge of understanding.
Summative Connections:
- Use data from activities and exit ticket to design follow-up assessments aligned with A1.PAFR.2.1 and 2.3 standards focusing on transformation and interpretation of quadratics in diverse forms.

Reflection and Extensions

Encourage students to maintain a mathematical journal with function transformation notes and real-world examples.
Extension project: Students create short presentations or posters relating quadratic functions to physics, economics, or biology.
Link to next units on exponential functions, deepening their representational fluency and application skills aligned with IB interdisciplinary learning.

Alignment with IB Curriculum and Standards

IB MYP Criterion	Description	Supported Learning Objectives & Activities
Criterion A: Knowledge & Understanding	Demonstrate knowledge and understanding of mathematical concepts and techniques.	Identification of key features from equations, tables, graphs; interpreting domain, range, and behavior (LO1).
Criterion B: Investigating Patterns	Apply mathematical concepts to develop conjectures and generalizations.	Transforming forms of quadratics to reveal vertex, roots (LO2).
Criterion C: Communicating	Organize and express mathematical ideas coherently.	Graphing practice, verbal and written explanations, descriptive discussion (LO3).
Criterion D: Applying Mathematics in Real Life Contexts	Apply mathematics to interpret real-world scenarios.	Application to verbal models and context interpretation (LO4).
ATL Skills: Communication, Collaboration, Reflection	Use strategies and tools to construct understanding collaboratively.	Think-Pair-Share, Compare-Connect, technology exploration, reflective quick-write (LO5).

This lesson plan crucially builds mathematical fluency and conceptual understanding by interweaving multiple representations, explicit vocabulary focus, and collaborative strategies. The integration of technology and real-world contexts exemplify IB’s global-minded approach to mathematics education, encouraging students to become confident, thoughtful problem solvers ready for higher-level inquiry.