Students investigate how volume, surface area, and unit-rate comparisons help determine the most efficient 3D solid for a real storage or floor-plan need. Through a fast build-and-test launch, iterative modeling, and critique, they apply geometry formulas and informal reasoning about volume to make and defend design choices. The week culminates in a public gallery walk where students present a scaled prototype or digital model to classmates and a local architecture partner, explaining how their design fits the space and minimizes wasted material.

Students will calculate and compare volume and surface area for cylinders, pyramids, cones, spheres, prisms, and composite solids to determine which design best fits a real storage or floor-plan need. They will use unit-rate comparisons and informal arguments, including Cavalieri’s principle where appropriate, to justify why one solid is more space-efficient than another. Students will collaborate to build, test, revise, and present a scaled prototype or digital model that communicates mathematical reasoning clearly to peers and a local architecture audience. They will reflect on how their math decisions and teamwork shaped the final design choice.

Students create quick comparison charts, annotated sketches, and first-draft physical or digital solid models as they test prisms, cylinders, pyramids, cones, spheres, and composite solids for a real storage or floor-plan challenge. Midway through the week, each team produces a revised prototype after peer critique, with labeled calculations for volume, surface area, and unit-rate efficiency, plus a short informal argument connecting their choices to Cavalieri’s principle when relevant. By the end, teams present a scaled prototype or digital model of the most space-efficient design, accompanied by a design brief that explains fit, space use, material tradeoffs, and why their solid best matches the purpose. These final products are displayed in a gallery walk for classmates and the architecture partner, who review and respond to each team’s solution.

Open with a fast “Build-and-Battle” challenge: teams examine mystery containers and solid models, then predict which shape holds the most volume for the least surface area in the same space. Students test their ideas by filling, measuring, stacking, or comparing models, then record first impressions about which solid seems most efficient for a real storage or floor-plan need. Introduce a local architecture firm’s design brief and floor-plan or storage challenge so teams can connect the math to an authentic client need. Close with a quick share-out where teams defend one early claim about the best solid and name one question they need to investigate next.

Host a Shape Showdown Gallery Walk where teams display their scaled prototypes or digital models alongside clear labels showing volume, surface area, and unit-rate comparisons for their chosen solid. Invite a local architecture firm to review each design against the storage or floor-plan challenge, ask questions about fit and efficiency, and give brief feedback on how well the model uses space with minimal surface area. Classmates and guests can leave feedback notes naming one strong math decision and one possible revision, and each team gives a short presentation explaining why their solid was the best choice for the purpose. End with a brief celebration that highlights strong reasoning, collaboration, and clear communication across the designs.