Triangle Quest: Building Bridges with Geometry

10th Grade

Project
9 weeks

Triangle Quest: Building Bridges with Geometry

Angela Torres

Purpose

The purpose of this project is to engage students in a hands-on exploration of mathematical concepts related to triangles and angles, emphasizing their practical application in real-world structures. By investigating the properties of similarity, right triangle geometry, and trigonometry, students will gain a deeper understanding of how these principles contribute to the design and stability of everyday objects and structures. Through creating scale models and participating in design challenges, students will apply mathematical reasoning and problem-solving skills to construct stable and aesthetically pleasing designs, thereby reinforcing their learning and fostering an appreciation for the role of mathematics in engineering and architecture.

Learning goals

In this project, students will explore the mathematical concepts of similarity, right triangle geometry, and basic trigonometry through hands-on activities and real-world applications. They will learn to identify and prove the congruence of angles and the similarity of triangles using various methods such as SSS, SAS, and AA. Students will gain a deep understanding of trigonometric ratios—sine, cosine, and tangent—and apply these to solve problems involving right triangles, including finding missing sides and angles. Additionally, students will investigate angles of elevation and depression to solve practical word problems. By the end of the project, students will be able to apply these mathematical principles to design and assess the stability of structures, such as bridges or buildings, and articulate their understanding through a scale model and a comprehensive report. This project emphasizes the importance of geometry and trigonometry in engineering and architecture, fostering critical thinking and problem-solving skills.

Standards

Common Core - CCSS.MATH.CONTENT.HSG.SRT.B.4: Prove theorems about triangles, including the criteria for triangle similarity based on angles and sides.
Common Core - CCSS.MATH.CONTENT.HSG.SRT.C.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Common Core - CCSS.MATH.CONTENT.HSG.CO.C.9: Prove theorems about lines and angles, including theorems involving parallel lines cut by a transversal.

Products

Throughout this 9-week project, students will engage in a series of hands-on activities and experiments to solidify their understanding of similarity, right triangle geometry, and trigonometry. They will begin by participating in a Design Challenge Kickoff, where they will construct simple structures using triangles to explore concepts of stability and aesthetics. As the project progresses, students will conduct experiments with right triangles to measure sides and angles, calculate trigonometric ratios, and apply these concepts to real-world scenarios. The culmination of the project will involve students creating a scale model of a bridge or building. This model will demonstrate their grasp of mathematical principles such as trigonometric ratios, proportions, and the Pythagorean theorem. Students will present their models along with a detailed written report that explains the mathematical reasoning behind their design choices. Additionally, students will produce a visual representation, like a mind map or infographic, to reflect on their learning journey, capturing key insights and emotional experiences related to the essential question: How do the properties of triangles and angles influence the design and stability of everyday objects and structures? This comprehensive approach ensures students not only meet the necessary standards and competencies but also develop a deeper understanding of the practical applications of mathematics in engineering and design.

Launch

To kick off the project, organize a "Triangle Exploration Day" where students participate in a series of interactive stations. Each station focuses on a different aspect of triangles and their properties. For example, one station could involve using a laser level and protractor to measure angles and verify congruency, while another station could have students use a dynamic geometry software to manipulate and explore the properties of similar triangles. Additionally, include a station where students experiment with building mini-structures using triangle-based frameworks to test stability and strength. This hands-on exploration will ignite curiosity and provide a practical foundation for the concepts they will delve into throughout the project.

Exhibition

To exhibit the final product of this project, organize a "Geometry in Design Expo" where students showcase their scale models of bridges or buildings. Invite classmates, teachers, parents, and community members to attend. Each student or group can present their model, explaining the mathematical principles of similarity, right triangle geometry, and trigonometry applied in their design. They should also display their visual representation, such as a mind map or infographic, to illustrate their learning journey. This event not only celebrates their achievements but also allows them to articulate their understanding and insights gained throughout the project.

Week 1	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Introduction to the Project - The teacher will present the project overview, objectives, and expectations. Students will discuss the purpose and significance of understanding triangles and angles in real-world structures.	Activity 2: Triangle Exploration Day - Students participate in interactive stations, each focusing on different aspects of triangles and their properties, such as measuring angles with a protractor, using dynamic geometry software, and building mini-structures.	Activity 3: Introduction to Similarity and Congruence - Students will learn about the criteria for triangle similarity and congruence (SSS, SAS, AA) through guided lessons and practice problems.	Activity 4: Hands-On Exploration of Triangle Stability - Students will construct simple triangular frameworks and test their stability. This will involve predicting outcomes, conducting experiments, and recording observations.	Activity 5: Reflection and Discussion - Students will reflect on their experiences from the Triangle Exploration Day and discuss their findings on triangle stability and congruence. This will be a guided class discussion to consolidate learning.
Deliverables	1. Deliverable 1: A short reflection piece on the Triangle Exploration Day, summarizing key learnings and insights. 2. Deliverable 2: A set of practice problems on triangle similarity and congruence, completed and submitted for review.
Preparation	1. Teacher Preparation Task 1: Prepare a presentation outlining the project overview, objectives, and expectations. 2. Teacher Preparation Task 2: Set up interactive stations for the Triangle Exploration Day, ensuring all materials and equipment are available and functioning. 3. Teacher Preparation Task 3: Prepare guided lessons and practice problems on the criteria for triangle similarity and congruence. 4. Teacher Preparation Task 4: Gather materials for constructing triangular frameworks, such as straws, connectors, and measuring tools. 5. Teacher Preparation Task 5: Develop a reflection and discussion framework to guide students in synthesizing their learning from the week's activities.

Week 2	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Deep Dive into Right Triangle Geometry - Students will explore the properties of right triangles, including the Pythagorean Theorem, through guided lessons and practice exercises.	Activity 2: Experiment with Right Triangle Measurements - Using measurement tools, students will measure sides and angles of right triangles, calculate trigonometric ratios, and verify the Pythagorean Theorem.	Activity 3: Introduction to Trigonometric Ratios - Students will learn about sine, cosine, and tangent, and how to apply these ratios to solve right triangle problems.	Activity 4: Hands-On Application of Trigonometry - Students will engage in activities that require them to use trigonometric ratios to find missing sides and angles in right triangles.	Activity 5: Group Reflection and Problem Solving Session - Students will work in groups to solve complex right triangle problems and reflect on their learning process, discussing strategies and solutions.
Deliverables	1. Deliverable 1: Completed practice exercises on right triangle geometry and the Pythagorean Theorem. 2. Deliverable 2: A set of solved problems using trigonometric ratios, demonstrating understanding of sine, cosine, and tangent applications.
Preparation	1. Teacher Preparation Task 1: Prepare lesson materials and handouts on right triangle geometry and the Pythagorean Theorem. 2. Teacher Preparation Task 2: Gather measurement tools and resources for the hands-on experiment with right triangles. 3. Teacher Preparation Task 3: Develop a comprehensive lesson plan for introducing trigonometric ratios with examples and applications. 4. Teacher Preparation Task 4: Create a set of practice problems that involve applying trigonometric ratios to solve right triangle problems. 5. Teacher Preparation Task 5: Organize materials for the group reflection and problem-solving session to facilitate discussion and collaboration.

Week 3	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Exploration of Angles of Elevation and Depression - Students will learn about angles of elevation and depression through guided lessons and real-world examples. They will practice identifying these angles in various scenarios.	Activity 2: Practical Application with Tools - Students will use tools like clinometers or smartphone apps to measure angles of elevation and depression in the classroom or schoolyard, recording their findings.	Activity 3: Problem-Solving with Angles - Students will apply their knowledge of angles of elevation and depression to solve word problems, working individually and in pairs to enhance understanding.	Activity 4: Design Challenge Preparation - Students will begin planning their scale model designs, incorporating concepts of angles of elevation and depression into their initial sketches and plans.	Activity 5: Group Discussion and Feedback - Students will present their initial design plans to the class, receiving feedback on their use of angles and discussing possible improvements.
Deliverables	1. Deliverable 1: Completed worksheet on angles of elevation and depression, including measured angles and solutions to related problems. 2. Deliverable 2: Initial design sketch and plan for the scale model, highlighting the use of angles of elevation and depression.
Preparation	1. Teacher Preparation Task 1: Prepare lesson materials and examples on angles of elevation and depression, ensuring clarity and relevance to real-world applications. 2. Teacher Preparation Task 2: Organize and provide tools for measuring angles, such as clinometers or smartphone apps, and ensure students understand their use. 3. Teacher Preparation Task 3: Develop a set of problem-solving exercises focusing on angles of elevation and depression, with varying levels of difficulty. 4. Teacher Preparation Task 4: Create guidelines and criteria for the design challenge preparation, outlining how students should integrate angles into their designs. 5. Teacher Preparation Task 5: Plan a structured feedback session for the group discussion, helping students to critique constructively and refine their ideas.

Week 4	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Introduction to Scale Modeling - Students will be introduced to the concept of scale modeling. The teacher will explain how scale models are used in engineering and architecture, focusing on the importance of accuracy and proportionality.	Activity 2: Scale Model Planning - Students will begin planning their scale models of a bridge or building. They will choose a real-world structure to model and determine the scale they will use, considering the mathematical principles of similarity and proportion.	Activity 3: Mathematical Analysis and Calculations - Students will perform mathematical calculations to determine the dimensions of their scale model. They will apply their knowledge of similarity, trigonometric ratios, and the Pythagorean theorem to ensure accuracy.	Activity 4: Construction of Scale Models - Students will start constructing their scale models using materials such as cardboard, wood, or plastic. They will focus on maintaining the structural integrity and aesthetic appeal of their models.	Activity 5: Peer Review and Feedback - Students will present their progress to their peers, receiving constructive feedback on their scale models. This session will help them identify areas for improvement and refine their models before completion.
Deliverables	1. Deliverable 1: A detailed plan and sketch of the chosen structure for the scale model, including scale calculations and initial design considerations. 2. Deliverable 2: Progress report on the construction of the scale model, highlighting challenges faced and solutions implemented.
Preparation	1. Teacher Preparation Task 1: Prepare a presentation on scale modeling, including examples from engineering and architecture, to illustrate the importance of accuracy and proportionality. 2. Teacher Preparation Task 2: Compile a list of potential structures that students can choose for their scale models, ensuring a variety of options that align with the project's learning goals. 3. Teacher Preparation Task 3: Develop guidelines and criteria for the scale model planning and construction, emphasizing the application of mathematical principles. 4. Teacher Preparation Task 4: Gather materials and resources needed for the construction of scale models, ensuring availability for all students. 5. Teacher Preparation Task 5: Design a structured peer review framework to facilitate constructive feedback and discussion among students, promoting collaboration and idea exchange.

Week 5	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Advanced Study of Trigonometric Applications - Students will delve deeper into the application of trigonometric ratios in real-world contexts, focusing on more complex scenarios and problem-solving techniques.	Activity 2: Case Study Analysis - Students will analyze case studies of engineering structures, such as bridges or buildings, to understand how trigonometry and geometric principles are applied in their design and construction.	Activity 3: Application to Design Projects - Students will apply the advanced trigonometric concepts learned to their ongoing design projects, refining their scale models and ensuring mathematical accuracy.	Activity 4: Collaborative Problem-Solving Workshop - In groups, students will tackle a set of challenging trigonometric problems, encouraging peer learning and collaborative strategies.	Activity 5: Mid-Project Review and Reflection - Students will participate in a mid-project review session, where they reflect on their progress, share insights, and set goals for the remainder of the project.
Deliverables	1. Deliverable 1: A detailed analysis report on a selected case study, highlighting the use of trigonometry in real-world engineering structures. 2. Deliverable 2: An updated version of the scale model design, incorporating advanced trigonometric applications with a rationale for any changes made.
Preparation	1. Teacher Preparation Task 1: Prepare materials and resources for teaching advanced trigonometric applications, including real-world examples and problem sets. 2. Teacher Preparation Task 2: Compile a selection of case studies for students to analyze, ensuring they are relevant and demonstrate the practical use of trigonometry. 3. Teacher Preparation Task 3: Develop a set of challenging trigonometric problems for the collaborative workshop, focusing on real-world applications and critical thinking. 4. Teacher Preparation Task 4: Organize the mid-project review session, creating a framework for student reflection and goal-setting, and prepare feedback forms. 5. Teacher Preparation Task 5: Ensure that students have access to any additional materials or tools needed for refining their scale models, such as measurement tools or software.

Week 6	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Introduction to Structural Design Principles - Students will learn about fundamental principles of structural design, focusing on stability, load distribution, and material selection through a guided lesson.	Activity 2: Material Experimentation - Students will experiment with different materials to understand their properties and how they affect the stability and strength of structures. This will involve testing materials such as wood, plastic, and metal for flexibility, strength, and weight.	Activity 3: Incorporating Design Principles into Models - Students will apply the principles learned to refine their scale models, ensuring that the chosen materials and design elements contribute to structural stability and aesthetic appeal.	Activity 4: Peer Collaboration and Critique - Students will collaborate in small groups to critique each other's models, providing constructive feedback on design choices and material usage. This activity will foster teamwork and critical thinking.	Activity 5: Reflection on Design Process - Students will write a reflective piece on the design process, considering the challenges faced and insights gained through the application of structural principles.
Deliverables	1. Deliverable 1: A comparative analysis report on the material experimentation, detailing the properties of each material tested and the implications for structural design. 2. Deliverable 2: A refined version of the scale model, incorporating insights from material experimentation and peer feedback, along with a rationale for design choices made.
Preparation	1. Teacher Preparation Task 1: Develop a lesson plan on structural design principles, including examples and case studies of successful structures. 2. Teacher Preparation Task 2: Gather a variety of materials for student experimentation, ensuring enough quantity for all students to test and analyze. 3. Teacher Preparation Task 3: Create guidelines for the material experimentation activity, providing instructions on how to test and record material properties. 4. Teacher Preparation Task 4: Design a peer collaboration and critique framework, encouraging constructive feedback and collaboration among students. 5. Teacher Preparation Task 5: Prepare a reflection framework to guide students in analyzing their design process and the application of structural principles.

Week 7	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Integration of Mathematical Concepts - Students will integrate various mathematical concepts learned throughout the project, such as similarity, trigonometric ratios, and structural principles, to enhance their scale models.	Activity 2: Advanced Design Adjustments - Students will make advanced adjustments to their scale models based on feedback received and integrate additional elements that improve structural integrity and aesthetic appeal.	Activity 3: Collaborative Design Review - In groups, students will conduct a design review session where they present their models and receive peer feedback. This session will focus on the practical application of mathematical principles.	Activity 4: Preparing for the Geometry in Design Expo - Students will begin preparing for the final exhibition by creating presentation materials, such as posters and infographics, that explain their design process and mathematical applications.	Activity 5: Rehearsal and Public Speaking Practice - Students will practice presenting their models and explaining their design choices, focusing on clear communication and articulation of mathematical principles.
Deliverables	1. Deliverable 1: A final draft of the scale model, incorporating all integrated mathematical concepts and design adjustments. 2. Deliverable 2: Presentation materials for the Geometry in Design Expo, including posters and infographics that detail the design process and mathematical applications.
Preparation	1. Teacher Preparation Task 1: Develop guidelines for integrating mathematical concepts into scale models, providing examples and resources for students. 2. Teacher Preparation Task 2: Organize materials and resources for the collaborative design review session, ensuring students have the tools needed for feedback and critique. 3. Teacher Preparation Task 3: Prepare a framework for the Geometry in Design Expo preparation, including instructions on creating presentation materials and practice sessions. 4. Teacher Preparation Task 4: Plan and facilitate rehearsal sessions, providing feedback on students' presentation skills and clarity of mathematical explanations. 5. Teacher Preparation Task 5: Ensure all necessary materials and equipment are available for students to finalize their scale models and presentation materials.

Week 8	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Final Adjustments and Finishing Touches - Students will make final adjustments to their scale models, ensuring all mathematical concepts are accurately represented and the models are structurally sound and aesthetically pleasing.	Activity 2: Detailed Report Writing - Students will begin drafting their comprehensive written reports, documenting the mathematical reasoning, design process, and the application of principles such as similarity, right triangle geometry, and trigonometry.	Activity 3: Visual Representation Creation - Students will create visual representations (mind maps or infographics) that capture their learning journey, highlighting key insights and the role of mathematics in their design.	Activity 4: Peer Review of Reports - In pairs or small groups, students will exchange their reports and provide constructive feedback, focusing on clarity, coherence, and the accuracy of mathematical explanations.	Activity 5: Final Presentation Preparation - Students will practice their presentations for the Geometry in Design Expo, focusing on clear articulation of their design process and mathematical applications.
Deliverables	1. Deliverable 1: A polished and finalized scale model, incorporating all necessary adjustments and enhancements. 2. Deliverable 2: A comprehensive written report detailing the design process, mathematical applications, and reasoning. 3. Deliverable 3: A visual representation (mind map or infographic) that reflects the learning journey and key insights.
Preparation	1. Teacher Preparation Task 1: Provide guidelines and examples for writing comprehensive reports, emphasizing the integration of mathematical reasoning. 2. Teacher Preparation Task 2: Organize materials and resources for creating visual representations, ensuring students have access to necessary tools and software. 3. Teacher Preparation Task 3: Develop a framework for peer review sessions, guiding students on how to provide constructive and useful feedback. 4. Teacher Preparation Task 4: Plan and facilitate practice presentation sessions, offering feedback on students' communication skills and clarity of mathematical explanations. 5. Teacher Preparation Task 5: Ensure all final materials and equipment are prepared for the Geometry in Design Expo, facilitating the smooth completion of student projects.

Week 9	Day 1	Day 2	Day 3	Day 4	Day 5
Activities	Activity 1: Final Rehearsal for Geometry in Design Expo - Students will conduct a final rehearsal of their presentations, focusing on refining their delivery and ensuring clarity in explaining the mathematical principles applied in their models.	Activity 2: Geometry in Design Expo - Students will exhibit their scale models at the expo, presenting their work to classmates, teachers, parents, and community members. They will explain the design process, the mathematical principles used, and answer questions from the audience.	Activity 3: Expo Debrief and Reflection - After the expo, students will participate in a debrief session to reflect on their experiences, receiving feedback from peers and teachers on their presentations and models.	Activity 4: Final Report Submission - Students will submit their comprehensive written reports, incorporating any final feedback received during peer reviews and presentations.	Activity 5: Project Reflection and Evaluation - Students will complete a reflection activity, evaluating their learning journey, the challenges faced, and the skills developed during the project. This will include self-assessment and goal-setting for future learning endeavors.
Deliverables	1. Deliverable 1: Participation in the Geometry in Design Expo, including the presentation of their scale model and explanation of the mathematical applications. 2. Deliverable 2: Final submission of the comprehensive written report, documenting the design process and mathematical reasoning. 3. Deliverable 3: A personal reflection piece evaluating their project experience, learning progress, and future goals.
Preparation	1. Teacher Preparation Task 1: Organize and finalize all logistics for the Geometry in Design Expo, including the setup of the venue, scheduling of presentations, and communication with attendees. 2. Teacher Preparation Task 2: Prepare feedback forms for expo attendees to provide constructive feedback on students' presentations and models. 3. Teacher Preparation Task 3: Develop a structured debrief session plan to guide students in reflecting on their expo experience and incorporating feedback. 4. Teacher Preparation Task 4: Ensure all students have access to any necessary resources or support for the final submission of their reports and reflection pieces. 5. Teacher Preparation Task 5: Plan a closing session to celebrate students' achievements, providing certificates or tokens of recognition for their hard work and dedication throughout the project.