Students investigate how circle relationships, rotations, scale factor, and volume can solve a real measurement problem by designing and revising a circular product prototype such as a roundabout model, fountain base, or organizer. Across design jams, measurement labs, and rotation stations, they use sketches, tools, photo evidence, and calculations to connect arc length, sector area, radians, and geometric constructions to a useful structure or product. The work builds toward explaining how parts of a circle reveal distance, area, and spatial relationships in everyday contexts while meeting shared decisions, critique, and revision goals. By presenting their model to classmates, families, and a community engineering connection, students show both mathematical reasoning and how they handled challenges, feedback, and collaboration.

Students will measure and compare circles, arcs, sectors, and radians, then use those relationships to solve a real measurement problem in a scaled prototype. They will construct and analyze inscribed regular polygons, explain rotations and dilations in a design, and use circle theorems and proportional reasoning to justify their choices. Students will apply formulas and informal arguments for circumference, area, and volume, including cylinders and related solids, to revise and improve a product such as a roundabout, fountain base, or organizer. They will collaborate through project logs, critique rounds, and presentations to communicate mathematical thinking, document revisions with evidence, and reflect on challenges, decisions, and growth.

Students will create a project log that includes dated photos, sketches, radius and diameter measurements, arc length and sector area calculations, radian conversions, and notes from each revision. In teams, they will build and revise a scaled circular prototype such as a tabletop organizer, fountain base, or traffic roundabout model that uses inscribed regular polygons, rotations, dilations, and volume reasoning to solve a real measurement problem. They will also produce display-ready artifacts for the Rotation Station Expo and Circle Snapshots Gallery Walk, including a labeled scale drawing, calculation board or QR-linked digital explanation, and brief reflection summary. By the end, each team will present a final model and performance-task explanation showing how circle relationships, scale factor, and volume informed their design decisions.

Open with a Design Jam and Rotation Relay: teams rotate through short stations using string, rulers, protractors, and grid paper to find centers of rotation, test dilations, and measure arc length and sector area on real objects like lids, wheels, and roundabout photos. Then introduce a civil-engineering challenge from a city transportation example and ask teams to sketch a quick prototype for a circular product or structure that solves a measurement problem. Students record first ideas, photos, and rough calculations in a project log, then share one connection between the station work and their design choice. Close with a brief exit circle where each student names one circle concept they can already use in a real product and one question they want to investigate.

Host a Rotation Station Expo where teams display their circular product prototype, scale drawing, and project log with QR codes linking to photo evidence, sketches, calculations, and revision notes. Add a Circle Snapshots Gallery Walk so classmates, families, teachers, and the civil engineer can leave sticky-note questions and feedback about how each design used arc length, sector area, radians, scale factor, and volume to solve a real measurement problem. Include a short presentation at each station in which students explain one design revision, justify a key circle relationship or construction, and reflect on a challenge, a shared team decision, and how they improved the model.