Quadratic Facades: Designing with Mathematical Precision
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Quadratic Facades: Designing with Mathematical Precision

Grade 9Math10 days
In this project, ninth-grade students explore the application of quadratic equations and their graphs in architectural design by creating unique building facades. Using hands-on activities, students learn to determine the domain and range of quadratic functions, graph them, and understand the effects of changes in parameters on the graphs. They solve quadratic equations using various methods to calculate dimensions and transform their designs aesthetically and structurally. The project connects mathematical inquiry with real-world architectural solutions, fostering creativity and a deeper understanding of geometry and algebra.
Quadratic FunctionsGraphingArchitectural DesignMathematical InquiryEquation SolvingParameter TransformationBuilding Facades
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can quadratic equations and their graphical representations be utilized to design an architecturally unique building facade that stands out?

Essential Questions

Supporting questions that break down major concepts.
  • What role do quadratic equations play in architectural design, specifically in creating building facades?
  • How can the graph of a quadratic function represent real-world structures, such as building facades?
  • How do alterations to the quadratic function f(x) = x², like changes in parameters a, b, c, and d, affect its graph and thus the design of a building facade?
  • In what ways can understanding the domain and range of a quadratic function assist in architectural planning and design?
  • What mathematical strategies can be employed to solve quadratic equations, and how are these strategies useful in real-life architectural design?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Students will be able to identify and apply quadratic functions to design aesthetically pleasing and structurally sound building facades.
  • Students will graph quadratic functions and understand their key attributes, such as the vertex, axis of symmetry, and intercepts.
  • Students will explore how different parameters in a quadratic function affect the shape and position of its graph, aiding in architectural design.
  • Students will solve quadratic equations using various methods to provide solutions for architectural challenges related to building facade design.

Texas Essential Knowledge and Skills (TEKS)

A.6(A)
Primary
Determine the domain and range of quadratic functions and represent the domain and range using inequalitiesReason: Understanding the domain and range is crucial for designing a building facade that fits within a specific space or size constraints.
A.7(A)
Primary
Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetryReason: Key attributes of the graph, like vertex and symmetry, are essential in creating aesthetically pleasing and structurally sound building facades.
A.7(C)
Primary
Determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x – c), f(bx) for specific values of a, b, c, and dReason: Understanding how parameter changes affect the quadratic graph helps students manipulate designs for aesthetic or structural purposes.
A.8(A)
Primary
Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formulaReason: Solving quadratic equations is foundational for determining dimensions and other critical measurements in architectural designs.

Entry Events

Events that will be used to introduce the project to students

Mystery of the Curved Skyline

Kick-off the project by presenting students with a stunning, digitally-rendered city skyline composed of unique, quadratic-inspired building facades. Challenge them to decode the mystery of these designs by identifying the quadratic equations and transformations that could create such architectural features. This concept connects architectural beauty with mathematical inquiry, prompting students to unravel and recreate the quadratics behind real-world facades.
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Portfolio Activities

Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.
Activity 1

Quadratic Domain Explorers

Students will explore the concept of domain and range in quadratic functions, laying the groundwork for designing facades that fit specific spatial constraints in architecture.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Introduce students to the concept of domain and range using simple quadratic equations.
2. Demonstrate how to determine domain and range in quadratic functions through inequalities.
3. Assign exercises where students identify domain and range of different quadratic equations.
4. Guide students to represent these using inequalities, applying their learning to potential facade designs.

Final Product

What students will submit as the final product of the activityA worksheet with quadratic equations and their domain and range expressed as inequalities.

Alignment

How this activity aligns with the learning objectives & standardsAligns with TEKS A.6(A) as it focuses on determining domain and range of quadratic functions and representing them with inequalities.
Activity 2

Graphing Gurus

In this activity, students will learn to graph quadratic functions, identify key features, and understand how these attributes influence architectural aesthetics.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Review fundamental graphing techniques and how they apply to quadratic functions.
2. Teach students to identify a graph's vertex, axis of symmetry, and intercepts.
3. Assign graphing exercises where students plot quadratic functions and label key attributes.
4. Challenge students to envision how these features can be translated into building facades.

Final Product

What students will submit as the final product of the activityA completed set of graph plots showcasing different quadratic functions with identified key attributes.

Alignment

How this activity aligns with the learning objectives & standardsAddresses TEKS A.7(A), focusing on graphing quadratic functions and identifying elements like vertex, axis of symmetry, and intercepts.
Activity 3

Transformation Architects

Students will delve into how altering parameters in a quadratic equation impacts the graph, showcasing their effect on architectural designs.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Introduce the parent function f(x) = x² and its basic graph.
2. Experiment with changes to the parameters a, b, c, and d in the function to observe graph transformations.
3. Engage students in predicting the effects of each parameter change.
4. Encourage students to explore creating variations of a facade by applying these transformations.

Final Product

What students will submit as the final product of the activityA portfolio of transformed quadratic function graphs illustrating potential variations for building facades.

Alignment

How this activity aligns with the learning objectives & standardsCorrelates with TEKS A.7(C) by determining effects on quadratic graphs when parameters a, b, c, and d are modified.
Activity 4

Quadratic Equation Solvers

This activity empowers students with strategies to solve quadratic equations, providing critical skills for architectural calculations.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Teach four methods for solving quadratic equations: factoring, square roots, completing the square, and quadratic formula.
2. Provide practice exercises for each method.
3. Discuss how solving quadratic equations helps in calculating dimensions in architectural contexts.
4. Conduct a mini-project where students solve real-world geometry problems related to building designs.

Final Product

What students will submit as the final product of the activityA compilation of solved quadratic equations, demonstrating mastery in various solving methods and their real-world applications.

Alignment

How this activity aligns with the learning objectives & standardsSupports TEKS A.8(A), focusing on solving quadratic equations using distinct methods.
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Quadratic Architecture Design Rubric

Category 1

Domain and Range Understanding

Assessment of students' proficiency in determining and representing the domain and range of quadratic functions in the context of architectural design.
Criterion 1

Identification of Domain and Range

Ability to correctly determine the domain and range of given quadratic functions.

Exemplary
4 Points

Identifies domain and range accurately and consistently across all functions, with insightful representations using inequalities.

Proficient
3 Points

Correctly identifies domain and range for most functions, representing them effectively using inequalities.

Developing
2 Points

Shows partial understanding by occasionally determining domain and range correctly, with basic representations.

Beginning
1 Points

Struggles to determine domain and range, often producing incorrect or incomplete representations.

Criterion 2

Application to Design Constraints

The extent to which students apply their understanding of domain and range to architectural design elements.

Exemplary
4 Points

Applies domain and range knowledge creatively to design constraints, producing precise and innovative architectural solutions.

Proficient
3 Points

Effectively integrates domain and range considerations into design constraints, producing functional architectural solutions.

Developing
2 Points

Attempts to incorporate domain and range into design, with limited success in addressing constraints.

Beginning
1 Points

Minimal consideration of domain and range in design constraints, resulting in impractical architectural solutions.

Category 2

Graphing and Key Features Identification

Evaluation of students' abilities to graph quadratic functions and identify key features essential for architectural design.
Criterion 1

Accuracy of Graphing

The precision with which students graph quadratic functions and identify key features such as vertex, intercepts, and axis of symmetry.

Exemplary
4 Points

Graphs all functions with high accuracy, consistently identifying all key features accurately.

Proficient
3 Points

Accurately graphs most functions and identifies most key features correctly.

Developing
2 Points

Graphs functions with some accuracy and partially identifies key features.

Beginning
1 Points

Produces inaccurate graphs with minimal identification of key features.

Criterion 2

Integration of Graph Features into Design

How well students translate graph features into creative architectural design elements.

Exemplary
4 Points

Innovatively uses graph features to create visually striking and structurally sound architectural designs.

Proficient
3 Points

Effectively incorporates graph features into architectural designs, contributing to both appearance and structure.

Developing
2 Points

Attempts to use graph features in designs, with varying levels of success in creativity and practicality.

Beginning
1 Points

Shows minimal integration of graph features, resulting in designs that lack coherence and structural integrity.

Category 3

Transformations and Parameter Effects

Assessment of students' understanding of parameter changes on quadratic graphs and their application in architectural contexts.
Criterion 1

Predictive Understanding of Transformations

The ability to predict the effects of parameter changes on quadratic graphs and apply them to design.

Exemplary
4 Points

Demonstrates a deep understanding of parameter transformations, predicting and applying effects with high precision in designs.

Proficient
3 Points

Shows good predictive understanding of transformations, effectively applying changes in designs.

Developing
2 Points

Recognizes some transformation effects, applying basic changes with limited accuracy.

Beginning
1 Points

Struggles to predict transformation effects, resulting in inaccurate application to designs.

Category 4

Solving Quadratic Equations

Evaluation of students' abilities to solve quadratic equations using multiple methods and their application in architectural design solutions.
Criterion 1

Method Proficiency

Skillfulness in using various methods to solve quadratic equations (factoring, square roots, completing the square, quadratic formula).

Exemplary
4 Points

Exhibits mastery in all four solving methods, applying them with precision to solve equations accurately.

Proficient
3 Points

Proficient in using most methods accurately to solve equations.

Developing
2 Points

Shows emerging skill in using solving methods with some accuracy and completeness.

Beginning
1 Points

Struggles with solving methods, leading to frequent errors in equation solutions.

Criterion 2

Application of Solutions to Design

Extent to which students use solutions from quadratic equations in practical architectural design scenarios.

Exemplary
4 Points

Seamlessly integrates equation solutions into design projects, demonstrating advanced problem-solving and application skills.

Proficient
3 Points

Effectively applies solutions to inform designs, with sound problem-solving abilities.

Developing
2 Points

Attempts to apply solutions to designs with limited success and practical insight.

Beginning
1 Points

Minimal application of solutions in design context, with little demonstration of practical problem-solving.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning
Question 1

Reflect on how understanding quadratic functions and their graphical representations can influence architectural design, particularly in creating unique building facades.

Text
Required
Question 2

On a scale of 1 to 5, how confident do you feel in using quadratic functions to design building facades?

Scale
Required
Question 3

What was the most challenging aspect of using quadratic equations in architectural design?

Text
Required
Question 4

Which method of solving quadratic equations did you find most effective for architectural planning and why?

Multiple choice
Optional
Options
Factoring
Square Roots
Completing the Square
Quadratic Formula
Question 5

How did changes in the parameters a, b, c, and d affect your understanding of designing building facades with quadratics?

Text
Required