Blueprint Builders Slides Blueprint Builders
ALGEBRAIC DESIGN & SCALE MODELING
The Project Challenge
You have been commissioned to design a Tiny Home that is both functional and affordable.
"Design is not just what it looks like and feels like. Design is how it works... and how the math adds up."
Deliverables:
1 A set of algebraic blueprints
2 A 3D physical scale model
3 A budget & constraint presentation
Design Constraints
Budget
Your total material cost \(C\) must follow this inequality:
\(C \le \$1,500\)
Includes flooring, walls, and fixtures.
Floor Area
The total living area \(A\) must satisfy:
\(200 < A < 450\)
Square footage measured from the interior walls.
Efficiency
The ratio of window area to wall area \(R\) must be:
\(R \ge 0.15\)
Ensures natural light and energy efficiency.
The Scale Factor
Your model must be built to a scale of 1:24.
The Conversion Equation:
\(m = \frac{1}{24} a\)
Where \(m\) is model size and \(a\) is actual size.
If your actual wall is 12 feet long, how many inches will it be on your model?
MODEL
Actual Home
1" = 2' scale
The Final Presentation
A
Concept Pitch
Who is your client? What was your design inspiration?
B
The Math Proof
Show the inequalities used to prove you stayed within budget and area limits.
C
Model Showcase
Walk us through your scale model. Point out architectural features.
Rubric Highlights
Accuracy of Scaling Equations 25%
Inequality Application (Budget/Area) 25%
Model Craftsmanship & Scale 30%
Presentation Clarity 20%
Blueprint Builders Project Guide Blueprint Builders
Project Specifications & Guide
PROJECT ID: AL-2026-X
REVISION: 01-A
STATUS: ACTIVE
Lead Architect:
Date Issued:
I. Project Mission
The urban development board has tasked you with designing a prototype for a functional, affordable "Tiny Home" to be used in high-density urban areas. Your design must demonstrate a mastery of mathematical modeling, spatial efficiency, and resource management.
II. Mathematical Constraints
Requirement Algebraic Inequality Variables Budget Limit \(C \le 1500\) \(C = \) Total Cost (\$) Floor Area Range \(200 < A < 450\) \(A = \) Area (sq ft) Height Limit \(h \le 14\) \(h = \) Max Height (ft) Window Ratio \(R \ge 0.15\) \(R = \) Windows/Wall Area
III. Resource Pricing (Actual Scale)
Wall Paneling: $8 / linear foot
Flooring (Oak): $12 / sq foot
Standard Window: $250 / unit
Entry Door: $600 / unit
Kitchenette Unit: $450 / unit
Roofing Material: $10 / sq foot
Standard Scaling Rule
All models must be built to a 1:24 Scale. This means 1 inch on your model represents 24 inches (2 feet) in actual space.
MODEL SIZE = \(\frac{1}{24} \times\) ACTUAL SIZE
OFFICIAL
IV. Final Deliverables
1
Architectural Calculation Log
A detailed math packet showing all equations used to calculate area, total costs, and scaling conversions. Must include proofs for all inequalities.
2
Scale Model Prototype
A physical model built precisely at 1:24 scale. Use materials like foam board, balsa wood, or recycled cardboard. Interior and exterior walls must be present.
3
Design Presentation
A 5-minute presentation pitching your Tiny Home. Explain how you balanced budget constraints with living requirements.
V. Project Rubric
Criteria Exemplary (Full Pts) Pts Math Accuracy All algebraic equations for area and cost are correctly formulated and solved. Scaling is perfect throughout. 30 Constraint Adherence Proof provided for all 4 inequalities. All design choices strictly fall within the required numerical ranges. 20 Scale Craftsmanship Model is durable, visually professional, and measurements are exact within 1/16th of an inch. 30 Communication Presentation is persuasive, clear, and uses appropriate mathematical and architectural terminology. 20
Milestone Checklist:
Phase 1: Initial Floor Plan & Dimension Calculations
Phase 2: Budget Audit & Inequality Proofs
Phase 3: Scaling dimensions for Model (1:24)
Phase 4: Physical Construction & Finishing Touches
Blueprint Builders Calculation Log Calculation Log
Project: Blueprint Builders
Architect:
Date:
1
Floor Area Calculation
Show the equation for your total floor area \(A\). Remember: \(200 < A < 450\) sq ft.
Inequality Proof:
200 < __________ < 450
2
Window Ratio Analysis
Constraint: \(R = \frac{\text{Total Window Area}}{\text{Total Wall Area}} \ge 0.15\)
Total Wall Area (sq ft):
Total Window Area (sq ft):
Is \(R \ge 0.15\)?
YES
NO
3
Total Cost Equation (\(C\))
Formulate your multi-variable cost equation. Reference the Pricing Guide.
Example: \(C = 8w + 12f + 250n + \dots\)
4
Final Budget Audit
Component Unit Cost Quantity Subtotal Walls (Lin. Ft) $8.00 Flooring (Sq Ft) $12.00 Roofing (Sq Ft) $10.00 Windows (Each) $250.00 Fixtures/Other Varies Total Calculated Cost (\(C\)): $
Inequality Check: \(C \le 1500\)
Pass
Fail
5
Model Scaling Table (1:24)
Convert your actual dimensions (ft) to model dimensions (in). Use: \(m = \frac{1}{24} \times (a \times 12)\) or simply \(m = \frac{a}{2}\).
Feature Name Actual Length (ft) Calculation Model Length (in) Main Wall A / 2 Main Wall B / 2 Interior Wall 1 / 2 Door Width 3 ft 3 / 2 1.5" Floor Height / 2 Window Width / 2
*Conversion Tip: Because 1:24 scale means 1 inch = 2 feet, simply divide your actual feet by 2 to get model inches!
Blueprint Builders Teacher Guide Teacher Guide
Blueprint Builders PBL
Topic Focus
Algebra & Inequalities
Pacing Strategy (10 Days)
DAYS 1-2
Project Launch & Algebra Review. Introduce the "Tiny Home" brief. Practice setting up multi-variable inequalities and area equations.
DAYS 3-4
Mathematical Planning. Students complete the Calculation Log. Teacher audits budget equations and scaling factors before construction starts.
DAYS 5-8
Scale Construction. Building the 1:24 model. Students must prove measurements on their model match their calculations exactly.
DAYS 9-10
Presentations. Architects pitch their designs. Peers evaluate designs based on the math proof and aesthetic appeal.
Algebraic Pitfalls to Watch For
Linear vs. Area Scaling: Remind students that if length scales by 1/24, area scales by \((1/24)^2\). However, for this project, they calculate actual area first, then just build linear segments at 1:24.
Variable Definition: Ensure students clearly define what their variables represent (e.g., \(w = \) number of windows, \(f = \) square feet of flooring).
Inequality Direction: Students often flip \(<\) and \(>\) when modeling "no more than" vs. "at least".
Construction Tips
Materials for Success:
Foam board (best for walls)
Low-temp glue guns
Precision rulers (32nds/16ths)
Graph paper for templates
Modeling Walls:
Remind students to account for wall thickness in their models! If they want 10ft interior space, the exterior cut must be slightly longer.
Example "Gold Standard" Proofs
Area Modeling (Rectangle + Alcove)
A student designs a main living space of 20x15 and a bathroom of 8x10.
Living Area \(= 20 \times 15 = 300\) sq ft
Bath Area \(= 8 \times 10 = 80\) sq ft
Total \(A = 380\) sq ft
Proof: \(200 < 380 < 450\) (TRUE)
Budget Modeling (Multi-Variable Inequality)
Example variable set: \(w\) (perimeter lin ft), \(f\) (floor sq ft), \(n\) (windows).
\(C = 8w + 12f + 250n + 600d + 450k\)
Values: \(w=110, f=380, n=2, d=1, k=1\)
\(C = 8(110) + 12(380) + 250(2) + 600(1) + 450(1)\)
\(C = 880 + 4560 + 500 + 600 + 450\)
Wait! This exceeds $1,500. This is where student iteration happens!
Correction Required: Student must switch to cheaper flooring or reduce area.
Presentation Questioning Prompts:
