Hexa-Home: Geometric Design for Modular Disaster Shelters
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Hexa-Home: Geometric Design for Modular Disaster Shelters

Grade 10Math20 days
In this project, 10th-grade students act as architectural engineers to design 'Hexa-Home,' a modular, interlocking disaster relief shelter system. Using geometric modeling, rigid transformations, and trigonometry, learners optimize structural stability and material efficiency while justifying the use of hexagonal designs over traditional shapes. The experience culminates in the creation of scaled 3D prototypes and a comprehensive technical portfolio that balances mathematical precision with real-world constraints like human livability and transport logistics.
Geometric ModelingArchitectural EngineeringModular DesignDisaster ReliefRigid TransformationsTrigonometryScale Prototyping
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we, as architectural engineers, use geometric modeling and transformations to design a modular, interlocking shelter system that balances structural stability, material efficiency, and livability for disaster relief efforts?

Essential Questions

Supporting questions that break down major concepts.
  • How do geometric transformations and rigid motions (translations, rotations, reflections) ensure that modular components can interlock and fit together without gaps?
  • How can we use similarity and scale factors to translate a miniature prototype into a full-scale, functional disaster relief shelter?
  • In what ways do the properties of triangles and trigonometric ratios determine the structural stability and weight distribution of the shelter's frame?
  • How does the choice of specific polygons (like hexagons) optimize floor space and material efficiency compared to traditional square designs?
  • How can we mathematically prove that our interlocking joints will remain secure under the stress of environmental factors?
  • How do we use geometric modeling to balance the constraints of portable shipping sizes with the human need for livable space?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Apply rigid transformations (translations, rotations, and reflections) to design interlocking modular components that fit together seamlessly.
  • Utilize scale factors and dilations to accurately transition a 3D geometric prototype into a full-scale architectural plan.
  • Apply properties of triangles and trigonometric ratios to determine the stability, load distribution, and precise angles of the shelter's frame.
  • Analyze and compare the efficiency of different polygons (specifically hexagons) in maximizing usable floor area while minimizing material usage.
  • Construct mathematical arguments and proofs to verify the structural integrity and geometric consistency of interlocking joints.
  • Use geometric modeling to solve design constraints involving portability, volume, and human livability.

Common Core State Standards for Mathematics

HSG-MG.A.3
Primary
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).Reason: This is the core of the project: using geometry to design a functional, modular disaster relief shelter within specific constraints.
HSG-CO.A.2
Primary
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).Reason: Students must use rigid motions (translations and rotations) to ensure that modular units interlock correctly and maintain their shape.
HSG-SRT.A.1
Primary
Verify experimentally the properties of dilations given by a center and a scale factor.Reason: Essential for the scaling component of the project where students move from a prototype to a full-scale model.
HSG-SRT.B.5
Primary
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.Reason: The project requires using similarity for scaling and congruence to ensure modular parts are identical and interchangeable.
HSG-SRT.A.2
Secondary
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.Reason: Supports the scaling process by ensuring the mathematical relationship between the model and the actual shelter is preserved.
HSG-CO.C.10
Secondary
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.Reason: Used when students need to prove why their triangular structural supports are stable or how angles within the hexagonal units interact.
HSG-SRT.C.8
Primary
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Reason: Recommended: Critical for determining heights, side lengths, and stability of the frame where right triangles are formed.
HSG-C.A.1
Supporting
Prove that all circles are similar.Reason: Supporting standard that may apply if students use circular joints or curved modular elements.
HSG-SRT.A.3
Secondary
Use properties of similarity transformations to establish the AA criterion for two triangles to be similar.Reason: Ensures students understand the geometric principles behind the similarity of their scaled structural components.

Entry Events

Events that will be used to introduce the project to students

Nature’s Architect: The Honeycomb Stress Test

The classroom is transformed into a 'Testing Lab' where students are given low-cost materials (straws, tape, cardstock) and asked to build a structure that can support 50 times its own weight. This hands-on inquiry leads to a discovery of why hexagons are nature's most efficient shape for strength and space-filling, setting the stage for their modular designs.
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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

The Hex-Cell Efficiency Audit

Before building, students must mathematically justify why the hexagon is the superior choice for modular disaster relief. In this activity, students compare the perimeter (representing material/wall costs) to the area (representing livable floor space) of various regular polygons. This audit sets the mathematical foundation for choosing a hexagonal footprint over traditional square or rectangular designs.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Calculate the area and perimeter of a square and a regular hexagon with the same fixed area (e.g., 200 square feet).
2. Determine the ratio of area to perimeter for each shape to identify which provides the most space per unit of wall material.
3. Decompose the hexagon into six equilateral triangles and use the properties of 30-60-90 triangles or the Pythagorean Theorem to find the apothem and height of the unit.
4. Draft a short proposal explaining why the hexagonal shape is the most efficient for rapid-deployment disaster relief.

Final Product

What students will submit as the final product of the activityA 'Geometric Feasibility Report' featuring calculations, a comparative table of shape efficiencies, and a written justification for the hexagonal design.

Alignment

How this activity aligns with the learning objectives & standardsHSG-MG.A.3: Students apply geometric methods to minimize material cost and maximize area. HSG-CO.C.10: Using triangle properties to analyze the interior geometry of a hexagon.
Activity 2

The Great Connection: Mapping the Interlock

A modular system is only effective if its parts fit together perfectly without gaps (tessellation). In this activity, students use coordinate geometry and rigid transformations to design the 'interlocking joint' of the Hexa-Home. They will demonstrate how a single modular unit can be translated or rotated to connect with another unit to create a scalable community layout.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Plot the vertices of a regular hexagon on a coordinate plane using its center as the origin.
2. Design a 'male' and 'female' interlocking joint (a notch or tab) on the sides of the hexagon.
3. Apply a translation transformation to the coordinates of the first hexagon to show exactly where the second hexagon must be placed to 'lock' into the first.
4. Perform a 60-degree rotation around a vertex to show how the units can branch out to create non-linear housing clusters.

Final Product

What students will submit as the final product of the activityA digital or hand-drawn 'Interlock Blueprint' showing at least three units connected, with transformation functions (e.g., T(x,y) = (x+h, y+k)) labeled for each connection point.

Alignment

How this activity aligns with the learning objectives & standardsHSG-CO.A.2: Students use translations and rotations to map one unit onto another. HSG-SRT.B.5: Using congruence to ensure every modular joint is identical and interchangeable.
Activity 3

Truss Trials: The Strength of Triangles

In this activity, students design the internal support frame for the Hexa-Home roof and walls. Since triangles are the most rigid geometric shape, students must 'triangulate' the hexagonal faces to ensure the structure doesn't collapse under environmental stress (like wind or snow). They will use trigonometry to calculate the exact lengths of the struts needed.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Divide the hexagonal ceiling or wall into a series of triangles to create a truss system.
2. Use Sine, Cosine, and Tangent (SOH CAH TOA) to calculate the lengths of the supporting beams based on a standard wall height.
3. Apply the Law of Sines or Cosines if the internal triangles are not right-angled.
4. Write a geometric proof or explanation showing why the chosen triangular arrangement (congruence) prevents the hexagonal frame from deforming.

Final Product

What students will submit as the final product of the activityA 'Structural Load Map'—a technical drawing of one wall or roof section with all internal triangular supports labeled with their lengths and interior angles.

Alignment

How this activity aligns with the learning objectives & standardsHSG-SRT.C.8: Using trigonometric ratios to solve for frame lengths. HSG-SRT.B.4: Proving properties of the internal triangular supports. HSG-CO.C.10: Applying triangle sum theorems to ensure structural stability.
Activity 4

From Model to Mountain: The Scale-Up Operation

Students move from their mathematical models to a physical prototype. In this activity, they will select a scale factor (e.g., 1:20) and use dilations to create a miniature model. They must prove that their prototype is 'similar' to the intended full-scale disaster relief unit, ensuring that all angles are preserved and all sides are proportional.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Choose a scale factor that allows the full-scale shelter (e.g., 10 feet wide) to fit on a desk (e.g., 6 inches wide).
2. Calculate the dilation of every line segment and joint from the original design to the prototype scale.
3. Construct the physical prototype based strictly on these dilated measurements.
4. Measure the angles of the physical prototype and compare them to the original mathematical design to verify that similarity transformations preserve angle measures.

Final Product

What students will submit as the final product of the activityA scaled 3D prototype (made of cardstock or 3D printed) accompanied by a 'Similarity Verification Sheet' comparing prototype dimensions to real-world dimensions.

Alignment

How this activity aligns with the learning objectives & standardsHSG-SRT.A.1: Verifying properties of dilations with a scale factor. HSG-SRT.A.2: Using similarity transformations to ensure the prototype is a faithful representation of the final build. HSG-SRT.A.3: Establishing similarity through AA or SSS criteria.
Activity 5

The Hexa-Home Master Spec

In the final phase, students address the 'human' side of the design problem. They must use geometric modeling to calculate the volume of the shelter (for air circulation) and the floor area (for beds and supplies). They will then create a 'Master Spec' that packages all their previous work into a final pitch for a disaster relief agency.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Calculate the total volume of the 3D hexagonal prism to ensure it meets minimum cubic-foot requirements for healthy air circulation.
2. Use geometric modeling to determine the most efficient layout for four standard-sized cots within the hexagonal floor plan, minimizing 'dead space'.
3. Analyze the 'packability' of the units by calculating how many folded modular units can fit into a standard shipping container (rectangular prism) using volume ratios.
4. Present the final design to a panel of 'Emergency Response Directors' (the class), defending the math behind the modularity and stability.

Final Product

What students will submit as the final product of the activityThe 'Hexa-Home Master Specification Portfolio' containing the efficiency audit, interlock blueprint, structural load map, scale verification, and a 3D digital walkthrough or physical model.

Alignment

How this activity aligns with the learning objectives & standardsHSG-MG.A.3: Final application of geometry to a design problem involving physical and human constraints. HSG-C.A.1: (Optional) Integrating circular ventilation or water storage components into the hexagonal grid.
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Hexa-Home: Modular Disaster Relief Design Rubric

Category 1

Design Optimization and Rationale

Evaluates the mathematical foundation for shape selection and space efficiency.
Criterion 1

Geometric Efficiency & Modeling (HSG-MG.A.3, HSG-CO.C.10)

Assessment of the student's ability to use geometric modeling to compare shapes, calculate area-to-perimeter ratios, and justify the efficiency of the hexagonal design for material conservation and space optimization.

Exemplary
4 Points

Mathematical justification is sophisticated and error-free. Demonstrates a deep understanding of how the apothem and height relate to area. Provides an innovative argument for material efficiency that exceeds basic requirements.

Proficient
3 Points

Accurately calculates area and perimeter for both shapes. Identifies the correct ratio and provides a clear, logical justification for the hexagonal design based on the mathematical evidence.

Developing
2 Points

Calculations contain minor errors. Justification is present but lacks a strong mathematical link between the area-to-perimeter ratio and the efficiency of the design.

Beginning
1 Points

Calculations are incomplete or significantly incorrect. Choice of shape is not supported by mathematical reasoning or the 'Geometric Feasibility Report' is missing key components.

Category 2

Transformational Geometry & Interlocking

Evaluates the application of transformations to ensure modular components fit together seamlessly.
Criterion 1

Rigid Motions and Modular Connectivity (HSG-CO.A.2, HSG-SRT.B.5)

Evaluation of how rigid transformations (translations and rotations) and coordinate geometry are used to design precise, interlocking 'male/female' joints for modularity.

Exemplary
4 Points

Transformations are mapped flawlessly using functional notation T(x,y). Joint design shows exceptional foresight into 3D interlocking. Rotation and translation proofs demonstrate perfect tessellation without gaps.

Proficient
3 Points

Correctly applies translations and rotations to hexagonal units on a coordinate plane. Interlocking joints are functional, and the 'Interlock Blueprint' clearly labels transformation functions.

Developing
2 Points

Transformations are attempted but contain errors in coordinate mapping. Joints may not align perfectly, or labels for transformations are missing/unclear.

Beginning
1 Points

Minimal evidence of coordinate geometry. Units do not interlock, or rigid transformations are not used to explain the connection points.

Category 3

Structural Engineering & Trig

Evaluates the use of triangles and trigonometry to ensure the shelter's physical stability.
Criterion 1

Trigonometry and Structural Integrity (HSG-SRT.C.8, HSG-SRT.B.4)

Assessment of the application of trigonometric ratios and triangle properties to design a stable, load-bearing internal support frame.

Exemplary
4 Points

Calculations for all struts and angles are precise and include advanced applications of Law of Sines/Cosines. Structural load map provides a professional-level technical drawing with a rigorous geometric proof of stability.

Proficient
3 Points

Correctly uses SOH CAH TOA and the Pythagorean Theorem to find beam lengths and angles. The structural load map is complete and demonstrates why the triangular arrangement ensures rigidity.

Developing
2 Points

Trigonometric ratios are applied but contain calculation errors. Structural map is partially labeled, or the explanation of triangular stability is weak.

Beginning
1 Points

Significant errors in trigonometric applications or no evidence of triangulation in the frame design. Structural map is missing or lacks mathematical labels.

Category 4

Scaling and Dilation Accuracy

Evaluates the transition from mathematical model to physical prototype via dilations.
Criterion 1

Similarity, Scaling, and Prototyping (HSG-SRT.A.1, HSG-SRT.A.2)

Evaluation of the student's ability to apply scale factors and dilations to create a physical prototype that is mathematically similar to the full-scale design.

Exemplary
4 Points

Dilation is perfectly executed across all dimensions. The similarity verification sheet provides an exhaustive comparison, proving that all angles were preserved and all side ratios are constant. Prototype quality is outstanding.

Proficient
3 Points

Scale factor is applied consistently to all measurements. The physical prototype is a faithful representation of the blueprint, and the verification sheet correctly identifies similarity criteria (AA or SSS).

Developing
2 Points

Scale factor is applied inconsistently, leading to some distortions in the prototype. Similarity verification is incomplete or contains errors in proportional reasoning.

Beginning
1 Points

Model does not match the dimensions of the blueprint. No evidence of a consistent scale factor or dilation process. Angle measures are not preserved.

Category 5

Synthesis and Final Spec Portfolio

Evaluates the integration of all project components into a final, livable solution.
Criterion 1

Constraints, Volume, and Professional Delivery (HSG-MG.A.3)

Assessment of the final portfolio's ability to address human constraints (volume for air, floor space for cots) and the professional delivery of the final pitch.

Exemplary
4 Points

The Master Spec is a comprehensive professional portfolio. Calculations for volume and packability are flawless. Presentation is highly persuasive, demonstrating deep metacognition and mastery of all project constraints.

Proficient
3 Points

Portfolio is complete and well-organized. Effectively models the layout for cots and calculates volume correctly. Presentation clearly communicates the design's benefits to the 'Emergency Directors.'

Developing
2 Points

Portfolio is missing key sections or contains calculation errors regarding volume and logistics. The 'human side' of the design is addressed only superficially.

Beginning
1 Points

Portfolio is disorganized or incomplete. Fails to account for human constraints like floor space or shipping volume. Presentation is ineffective or lacks mathematical support.

Reflection Prompts

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

How confident do you feel in your ability to use geometric transformations (translations and rotations) to ensure different parts of a complex structure fit together perfectly?

Scale
Required
Question 2

In designing the Hexa-Home, which geometric concept was most critical for ensuring that the modular units could be easily repeated and connected across a large-scale disaster relief site?

Multiple choice
Required
Options
Rigid Transformations (to ensure parts fit without gaps)
Dilations and Scale Factors (to move from model to reality)
Trigonometric Ratios (to determine stable angles and lengths)
Polygonal Efficiency (to maximize floor space vs. materials)
Question 3

Engineering often involves 'trade-offs.' Describe a specific moment where you had to balance a mathematical requirement (like maximizing area) with a human or physical constraint (like shipping size or livability). How did geometric modeling help you reach a decision?

Text
Required
Question 4

Reflect on your 'Similarity Verification Sheet.' When you moved from your mathematical design to your physical prototype, what was the most challenging part of maintaining accuracy through dilations, and how did you verify that your model remained 'similar' to the full-scale version?

Text
Required