From the Ground Up
Dimensions of Affordable Housing
One of the biggest issues in San Diego today is a lack of affordable housing. Housing shortages have driven up home and rental prices, making it difficult for many residents to find housing or afford to remain in the city. San Diego’s Office of the Independent Budget Analyst reports that as of 2022, the current pace of development is far behind what’s needed to meet goals, and that “an extreme amount of housing production is needed” to impact affordability.
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In this interdisciplinary Math-Humanities project, students used geometry to analyze floor plans and develop a floor plan for a property that would address the city's need for more affordable housing. In their Humanities classes, students learned about historical and contemporary barriers to housing access, and they researched affordable housing solutions in a foreign country of their choice. Once they had developed their floor plans, students used dilation and proportion to create a 3D model of their property from cardboard and foam board. They also created websites to showcase their learning and link to local organizations that are working to create more affordable housing in their community.
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Essential Questions:
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How does geometry help us understand housing development, construction, and affordability?​
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How does housing affordability and availability impact different groups of people? What are some potential solutions to address these issues?
Dilini Perera, professional architect and alumnus of NewSchool of Architecture and Design, gives a presentation on constructing architectural models.
Top-down view of model shows the floor plan.
Student-made 3D model of a 12-unit affordable housing complex intended for a vacant parcel in La Jolla.
Student measures and cuts panels for the exterior walls of her model.
Student presents his group's floor plan for peer review.
PBL Design Notes
Funky Floor Plans
Student hand-out
This was my first time designing and implementing an interdisciplinary project, and while it presented unique challenges, it also offered students a unique depth of learning in both subjects. My Humanities teaching partner Zak and I worked closely to develop our curricula, test products, and refine our lesson sequencing so the content fit together and supported students in their project-based learning. Our final products were designed to complement each other; students used what they learned in their research essays to create their floor plans, and a QR code on their 3D model linked to their websites, where readers could learn more about the design of the model, read students' essays and reflections, and follow links to partner organizations in the community.
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I particularly enjoyed writing lessons that invited students to explore geometry through architecture and design. Some highlight math lessons included "FAR from Ohio Street," where students found the floor area ratio (FAR) of several buildings on the same city block in North Park. During this lesson, students visualized different floor area ratios, created a mathematical definition for housing density, and the discussed the significance of density for housing supply and affordability. Another standout lesson was "Funky Floor Plans," in which students analyzed rooms of different regular polygonal shapes with the same perimeter. Students found that rooms with more walls enclosed more floor area, hypothesized that a circle maximized area for a given perimeter/circumference, discussed why almost all homes are rectangular with rectangular rooms, and learned about some non-rectangular housing developments around the world. Some students carried this analysis into their floor plan drafts; for example, two groups designed hexagonal prism buildings, and other group included split hexagons in their design.
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I also learned a lot about finding and working with community partners. During this project we collaborated with two main partners: Casa Familiar, a nonprofit advocacy and community development organization in San Ysidro, and NewSchool of Architecture and Design, a local university with degree programs in Architecture, Design, and Construction Management. My teaching partner and I organized two field trips and two guest speakers throughout the 8 week project. With Casa Familiar, students explored existing affordable housing developments in San Ysidro as well as two lots intended for development under a Community Land Trust. At NewSchool, students learned about the school and principles of design, saw undergraduate student designs for the San Ysidro Land Trust parcels, and explored the architecture studio and workshop spaces to learn more about model making. These experiences added an extra layer of authenticity and purpose to the project; in their reflections, many students wrote about the field trips and guest speakers as highlights of the project experience.
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Overall, I'm excited to continue planning interdisciplinary projects for students to see math alive in the world around them and central to solving problems that matter.
Content Standards
HSG-SRT.A.1
Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through
the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
HSG-GMD.A.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
HSG-GMD.A.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
HSG-GMD-A.5
Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k², and k³, respectively; determine length, area and volume measures using scale factors.
HSG-SRT.A.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
HSG-MG.A.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
HSG-MG.A.3
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).
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