Quadratic Functions Exploration

Overview

This 60-minute lesson is designed for 11th-grade students in a general math classroom. It focuses on deepening students' understanding of quadratic functions, their graphs, and key characteristics. The lesson aligns with Common Core State Standards (CCSS) for Algebra II and incorporates differentiated instruction to meet diverse learners' needs. The activities engage students through visual, collaborative, and technology-enhanced learning modalities.

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Standards and Learning Targets

Common Core State Standards:

• HSA-REI.B.4: Solve quadratic equations in one variable.
• HSF-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand and using technology.
• HSF-IF.C.8b: Use the process of factoring and completing the square in a quadratic function to show zeros, maximum or minimum values, and axis of symmetry.

Learning Targets:

• Students will identify the key features of quadratic functions including vertex, axis of symmetry, zeros, and direction of opening.
• Students will interpret and graph quadratic functions from various forms (standard, vertex, and factored).
• Students will apply technology to explore transformations of quadratic graphs.
• Students will communicate mathematical reasoning in written and verbal form.

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Materials Needed

• Graphing calculators or laptops/tablets with graphing software
• Whiteboards and markers
• Quadratic function activity handout (includes practice problems and graph templates)
• Projector or smartboard for demonstrations
• Student math notebooks
• Colored pencils or markers

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Lesson Procedure

1. Introduction & Review (10 minutes)

• Engage: Begin by posing a real-world problem involving projectile motion (e.g., the path of a basketball shot).
• Quickly review the forms of quadratic functions (standard, vertex, and factored) and key vocabulary (vertex, axis of symmetry, roots/zeros).
• Use the projector to visually display a parabola and point out these features.

Differentiation: Use a graphic organizer chart to support ELL and learners who benefit from visual representations.

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2. Guided Instruction & Modeling (15 minutes)

• Model how to convert between different forms of quadratic functions and how to identify key features from each form.
• Demonstrate graphing a quadratic function both by hand and using a graphing calculator or graphing software.
• Highlight how changes to parameters in the equation affect the graph (stretch/compression, shifts, reflecting).

Inclusion: Use think-aloud strategies—explain your reasoning steps slowly and clearly for auditory learners and students with processing differences.

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3. Hands-On Exploration (20 minutes)

• Activity: Students work in pairs to complete the provided activity handout.

• They will:

  • Identify key features of quadratic functions in different forms.
  • Graph the quadratics on grid paper by hand.
  • Use graphing calculators/software to confirm accuracy.
  • Make predictions about how changing coefficients affects the graph, then test those predictions with technology.

• Circulate to provide scaffolded support and prompt higher-level thinking questions:

  • “What happens to the vertex when you change the ‘h’ value in vertex form?”
  • “How can you find the zeros without graphing?”

Differentiation: Assign more complex quadratics with transformations to advanced learners and provide step-by-step prompts for students needing additional support.

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4. Collaborative Reflection and Discussion (10 minutes)

• Regroup for a class discussion.
• Invite pairs to share insights about transformations of quadratic functions and how technology aided their understanding.
• Use a whiteboard to create a “master list” of transformations and their effects on the graph.

Inclusion: Encourage all students to contribute by offering multiple modes of participation (verbal, written, drawing).

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5. Closing and Formative Assessment (5 minutes)

• Conduct a quick exit ticket: Students write down one key feature of quadratic functions and one question they still have.
• Review a few responses aloud to assess understanding and inform future lessons.

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Assessment and Feedback

• Formative assessment through observation during group work, class discussions, and exit tickets.
• Use student work on handouts and graphs for informal assessment.
• Provide timely, specific feedback highlighting conceptual understanding and areas for improvement.

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Extensions and Homework

• Extension: Challenge students to write and graph a quadratic function modeling a real-world problem of their choice (e.g., skateboard ramp shape or satellite dish curve).
• Homework: Complete additional quadratic function problems emphasizing identifying features and graphing.

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Teacher Reflection Prompts

• Did the use of technology deepen students’ conceptual understanding?
• Were students able to articulate how changes in parameters affect the quadratic graph?
• Which students struggled most and how can differentiated supports be improved?
• How effective was the collaborative exploration in promoting engagement?

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This lesson harnesses visualization, technology, and collaborative reasoning to engage 11th graders in mastering quadratic functions while meeting rigorous US standards and inclusive practices.