Lesson Designs | Jaylee Collier 1

Project Based Instruction (PBI)

Project-Based Instruction/Learning (PBI/L) is a student-centered approach to learning that engages students in exploring real-world problems, questions, and challenges through projects. In a high school mathematics classroom, PBI helps students see the relevance of mathematics beyond textbooks by connecting concepts to authentic situations they may encounter in their daily lives, future careers, and communities. This approach allows teachers to meet students where they are by providing multiple entry points for learning, encouraging collaboration, creativity, and critical thinking. Through projects, all students have opportunities to contribute their unique strengths, perspectives, and ideas, ensuring that every voice is valued in the learning process. PBI also promotes deeper understanding, increased engagement, student ownership of learning, and the development of important skills such as communication, problem-solving, and teamwork, helping students become confident and capable learners both inside and outside the mathematics classroom.

Below, you can view two PBI units. The first is a short five-day unit, while the second is a six-week unit you can explore the materials below.

How can we design affordable identical mini homes and shared neighborhood spaces so that, even with the same layout and furniture, every home feels unique?

This lesson addresses Kentucky standards KY.HS.G.4 and KY.HS.G.31 by engaging students in understanding and applying rigid transformations including rotations, reflections, and translations, to determine and justify congruence of geometric figures. Students use geometric reasoning to predict the effects of transformations, create and analyze figures under given constraints, and apply these methods to solve design problems. Through these learning experiences, students demonstrate their ability to apply transformation rules, evaluate and justify congruence using rigid motion criteria, compare sequences of transformations, and design congruent figures that meet specific requirements.

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Day 1: Congruent Parts

The central focus is for students to discover how geometric transformations (translations, reflections, and rotations) can be used to create designs that look and feel different while still maintaining congruence. By using the client brief, students will gain experience with a real-world context of identical floor plans being transformed to create variety in a neighborhood which in the end they will have full creative freedom with. As students are exploring what makes homes feel the same or different and considering essential features of a home, students will begin to show how mathematical ideas of transformation and congruence apply directly to design.

Day 1 Lesson Plan

Day 2: Congruent Parts

The central focus of this lesson is for students to be able to explain and use corresponding parts and congruence by applying them in architectural design. Students will move from informally noticing similarities and differences across various blueprints to formally understanding how congruence can be established. Using contractor feedback, revising blueprints, and starting design of their neighborhoods with congruence notations, students will be determining which transformations stitch as rotations, translations, and reflections are required to achieve specific outcomes. By the end of the lesson, students will be able to explain and use congruence criteria in multiple contexts, showing how corresponding parts not only prove mathematical relationships but also improve the clarity, consistency, and functionality of home designs.

Day 2 Lesson Plan

Day 3: Costs and Congruence

The central focus of this lesson is to connect geometric reasoning about rigid motions and triangle-congruence with the design and budget task: students will use rotations, reflections, and translations (and SSS/SAS/ASA for triangles) to justify when two homes or roof trusses are truly congruent, they will record corresponding parts and transformation sequences, and design congruent figures that meet specific constraints. Those mathematical proofs then have real consequences in the neighborhood budgeting activity proving congruence allows for bulk-cost savings so students must communicate precise geometric reasoning, compare different valid transformation paths, and make collaborative design decisions grounded in both math and practical constraints.

Day 3 Lesson Plan

Day 4: Proofs

The central focus of this lesson is to deepen students’ understanding of congruence and transformations using formal geometric reasoning to then apply to their neighborhoods. Building on their prior neighborhood design work, students will construct, analyze, and present formal proofs, during their presentations, that justify the congruence of their homes and roof trusses. Through guided and independent proof writing, they will learn how to organize logical arguments, apply transformation rules, and use precise mathematical language to communicate their reasoning. The lesson focuses on the real-world relevance of proofs by showing mathematical justification and architectural design decisions, cost efficiency, and professional communication in the context of preparing their final neighborhood presentations.

Day 4 Lesson Plan

Day 5: Presentations and Justifications

The central focus of this lesson is to have students apply their understanding of geometric transformations and congruence in a real-world context by finalizing and presenting their neighborhood designs. Students will demonstrate how they used rotations, reflections, and translations, along with SSS, SAS, and ASA criteria, to justify congruence between homes and roof trusses. The lesson emphasizes both mathematical reasoning and practical application, requiring students to communicate their logic clearly, analyze multiple transformation pathways, and make design decisions that meet specific constraints. Through this process, students connect geometric concepts to design challenges, reflecting on how congruence and transformations influence both mathematical proofs and architectural outcomes.

Day 5 Lesson Plan

Calendar of Activities

Assessment Plan

Throughout this unit, I used a variety of informal assessments during discussions and work time to gauge student understanding in real time. At the end of each class period, students completed an Architectural Journal entry, which served as the primary formative assessment tool. I reviewed these journals before the next day’s lesson to identify misconceptions, track progress toward each objective, and adjust instruction as needed. My full assessment plan, including all rubrics, scoring criteria, and descriptions of every journal prompt, can be viewed using the link to the right. Additional assessment materials, including copies of each Architectural Journal handout, are available in the handouts tab.

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If true equality means every community receives identical resources and opportunities, how can we design multiple thriving communities that each feel distinctly meaningful while proving that sameness can be the foundation for infinite diversity across all aspects of human life?

This six-week project-based unit addresses Kentucky standards KY.HS.G.4, KY.HS.G.5, KY.HS.G.30, and KY.HS.G.31 by engaging students as architectural firms designing mathematically congruent yet experientially distinct housing developments. Students master rigid transformations (rotations, reflections, translations) and congruence criteria (SSS, SAS, ASA) while solving an authentic design challenge: creating six identical 800-square-foot homes on a 2-acre site within a $900,000 budget that feel uniquely different to residents. Through sustained mathematical reasoning, students analyze congruence statements to determine required transformations, compare multiple transformation sequences that achieve the same outcomes, construct formal geometric proofs, and generate creative solutions to design constraints while critically evaluating whether mathematical sameness truly creates equitable community outcomes. Students demonstrate mastery by presenting professional proposals that balance rigorous mathematical argumentation with compelling client communication, culminating in a synthesis that connects geometric reasoning to questions about housing equity and community development.

Day 1: Project Launch and Community Identity

Day 1 Lesson Plan

Day 8: ASA

The central focus of this lesson is for students to deepen their understanding of triangle congruence, specifically ASA, and to see how congruence criteria emerge from both geometric construction and real-world architectural design. Through hands-on construction, scenario analysis, proof critique, and application to their community blueprints, students will explore why certain combinations of angles and sides guarantee congruence and how these mathematical ideas support accurate, efficient, and consistent building practices. As students compare ASA to SSS and SAS and connect each criterion to authentic design decisions, they will build a clearer sense of how rigid motions and corresponding parts justify congruent structures in real-world settings. This foundation will prepare them to use precise mathematical reasoning throughout the broader community design project.

Day 8 Lesson Plan

Day 12: Congruence Proofs

The central focus of this lesson is for students to apply geometric transformations and congruence criteria to write clear, formal mathematical proofs grounded in their own community design blueprints. By identifying congruent structures within their designs, selecting appropriate criteria such as SSS, SAS, or ASA, and constructing formal proofs using statements and reasons, students will deepen their understanding of how rigid motions justify triangle congruence and why this reasoning is essential in real-world architectural contexts. Through peer review, revision, and client-friendly communication, students will connect mathematical precision to professional decision-making, recognizing how formal proof supports safety, cost efficiency, structural integrity, and client confidence in architectural planning.

Day 12 Lesson Plan

Day 16: New Constraints

Day 16 Lesson Plan

Day 35: Assessment

The central focus of this lesson is for students to transfer their understanding of geometric transformations, triangle congruence, and design-based reasoning into a completely new context by applying these ideas to NASA’s modular Mars habitat challenge. Students will use rigid motions, congruence criteria, geometric measurements, and spatial reasoning to design identical habitat shells, create varied interiors, place modules across Martian terrain, and justify their decisions through formal mathematical proofs. This task asks students to take everything they learned about congruence, transformations, and design constraints in the community unit and apply it flexibly to a novel scenario where identical structures must support fairness, safety, efficiency, and diverse human needs.

Day 35 Lesson Plan

Calendar of Activities

Week 1: Students investigate how congruent structures can feel different through transformations while learning how identity, values, and infrastructure shape a thriving community.

Week 2: Students develop mastery of the SSS, SAS, and ASA criteria and analyze transformation sequences to prepare mathematically rigorous foundations for their community designs.

Week 3: Students create and refine their full neighborhood layout by positioning congruent homes, constructing formal proofs, crafting interior differences, and optimizing budget decisions.

Week 4: Students revise their community designs in response to zoning or environmental challenges, compile mathematical evidence, and prepare polished presentations for the client panel.

Week 5: Students present their completed community designs to a client panel, evaluate peers’ mathematical reasoning, and analyze what makes a design mathematically and practically successful.

Week 6: Students demonstrate individual mastery of transformations and congruence, analyze the relationship between mathematical sameness and equity, and reflect on their growth across the entire project.

View all 30 days here.

Calendar of Activities