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Zipline Geometry: Trig Ratios in Design

Grade 10Math4 days

Students design a safe and exhilarating zipline course by applying trigonometric ratios to calculate cable lengths, angles, and select appropriate materials. They build scale models and analyze potential risks and construction challenges. This project-based learning experience culminates in a formal design proposal incorporating calculations, diagrams, and risk mitigation strategies. Through this project, students connect mathematical concepts to real-world engineering design.

TrigonometryGeometryZipline DesignAnglesRatiosReal-World ApplicationProblem-Solving

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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we design a safe and exhilarating zipline course using trigonometric ratios to determine cable lengths, angles, and appropriate materials while considering potential risks and construction challenges?

Essential Questions

Supporting questions that break down major concepts.

How can we use trigonometric ratios to calculate distances and angles in real-world scenarios?
What factors influence the design and safety of a zipline course?
How can we model and represent a zipline course using geometric principles?
How do we select appropriate materials and construction techniques for a zipline based on our calculations?
What are the potential risks and challenges involved in building a zipline, and how can we mitigate them?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:

Students will be able to apply trigonometric ratios (sine, cosine, tangent) to calculate unknown distances and angles in the context of designing a zipline course.
Students will be able to model and represent a zipline course using geometric principles, including angle relationships, triangles, and coordinate systems.
Students will be able to analyze and interpret the results of their calculations to make informed design decisions, considering factors like safety, material selection, and construction feasibility.
Students will be able to evaluate potential risks and challenges associated with zipline construction and propose mitigation strategies.

Common Core State Standards (CCSS)

CCSS.Math.Content.HSG.SRT.C.8

Primary

CCSS.Math.Content.HSG.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Reason: This standard directly addresses the application of trigonometric ratios to solve real-world problems, which is central to the zipline design project.

CCSS.Math.Content.HSG.MG.A.1

Supporting

CCSS.Math.Content.HSG.MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).Reason: This standard supports the geometric modeling aspect of the project, where students represent the zipline course using geometric shapes and principles.

CCSS.Math.Content.HSG.MG.A.3

Supporting

CCSS.Math.Content.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).Reason: This standard aligns with the design aspect of the project, where students use geometric methods to solve design problems related to the zipline course.

Next Generation Science Standards (NGSS)

NGSS.HS-ETS1-2

Supporting

NGSS.HS-ETS1-2. Design a solution to a complex real-world problem by breaking it down into smaller, more manageable problems that can be solved through engineering.Reason: The zipline design project involves a complex, real-world problem requiring students to break it into smaller problems involving trigonometry, material selection, and risk assessment.

Entry Events

Events that will be used to introduce the project to students

Adventure Park Challenge

A local adventure park challenges students to design a new zipline course using their knowledge of trigonometry. The park provides topographical maps, site surveys, and budget constraints, pushing students to develop practical, cost-effective, and thrilling zipline routes.

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Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.

Activity 1

Angle Explorers

Students will identify and measure angles of elevation and depression within the adventure park setting, using protractors and clinometers.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Visit the adventure park or use provided images.

2. Identify at least three locations for potential zipline start and end points.

3. Use a clinometer or protractor to measure the angles of elevation and depression between the chosen points.

4. Annotate photos or create sketches of the locations, clearly labeling the measured angles.

Final Product

What students will submit as the final product of the activityA series of annotated photographs or sketches of the park, with labeled angles of elevation and depression, along with their corresponding measurements.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSG.SRT.C.8, CCSS.Math.Content.HSG.MG.A.1

Activity 2

Zipline Mathematicians

Students calculate the lengths of zipline cables using trigonometric ratios based on measured angles and estimated heights/distances.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Choose one of the identified zipline routes.

2. Estimate the vertical or horizontal distance between the start and end points.

3. Use the measured angle and the estimated distance to set up a trigonometric equation (sine, cosine, or tangent) to solve for the cable length.

4. Solve the trigonometric equation to calculate the cable length.

5. Repeat steps 1-4 for at least two additional potential routes.

Final Product

What students will submit as the final product of the activityA table summarizing the calculations for each potential zipline route, including the angle of elevation/depression, estimated height/distance, and calculated cable length.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSG.SRT.C.8

Activity 3

Miniature Zipline Engineers

Students design a scaled model of their zipline course.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Choose a suitable scale for the model (e.g., 1:100).

2. Calculate the scaled lengths of the zipline cables and supporting structures.

3. Use materials like cardboard, string, and dowels to construct the model, ensuring the angles and lengths match the scaled calculations.

4. Label key components of the model, including cable lengths, angles, and support structures.

Final Product

What students will submit as the final product of the activityA physical scale model of the zipline course using materials like cardboard, string, and dowels, representing the calculated cable lengths, angles, and supporting structures.

Alignment

How this activity aligns with the learning objectives & standardsCCSS.Math.Content.HSG.MG.A.1, CCSS.Math.Content.HSG.MG.A.3, NGSS.HS-ETS1-2

Activity 4

Zipline Design Professionals

Students create a comprehensive design proposal for their zipline course.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Compile the calculations, diagrams, and model from the previous activities.

2. Research and select appropriate materials for the zipline cables, platforms, and supporting structures.

3. Estimate the cost of materials and construction.

4. Identify potential risks and challenges associated with zipline construction and operation.

5. Propose mitigation strategies to address the identified risks.

6. Compile all information into a formal design proposal document.

Final Product

What students will submit as the final product of the activityA formal design proposal document that includes calculations, diagrams, material specifications, cost estimations, risk assessments, and mitigation strategies.

Alignment

How this activity aligns with the learning objectives & standardsNGSS.HS-ETS1-2

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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Zipline Geometrics Project Rubric

Category 1

Trigonometric Application

Assessment of student ability to apply trigonometric ratios and the Pythagorean theorem to solve real-world problems in the design of a zipline course.

Criterion 1

Accuracy of Trigonometric Calculations

Measures the correctness of trigonometric calculations for angles and cable lengths.

Exemplary

4 Points

All calculations are accurate, demonstrating a sophisticated understanding of trigonometric concepts and their application.

Proficient

3 Points

Calculations are mostly accurate with minor errors, showing a thorough understanding of trigonometric concepts.

Developing

2 Points

Calculations contain frequent errors, indicating emerging understanding of trigonometric concepts.

Beginning

1 Points

Calculations are mostly incorrect, showing initial understanding of trigonometric concepts.

Criterion 2

Calculation Methodology

Evaluation of methodology used to solve and present trigonometric equations and problems.

Exemplary

4 Points

Uses a clear, methodical approach consistently to solve problems, demonstrating exceptional critical thinking.

Proficient

3 Points

Uses a mostly clear and methodical approach with some minor lapses or errors in problem-solving.

Developing

2 Points

Methodology is inconsistent, with several errors in the approach to problem-solving.

Beginning

1 Points

Approach is disorganized or incorrect, hindering accurate problem-solving.

Category 2

Geometric Modeling and Representation

Evaluation of the ability to model the zipline course using geometric principles and accurate scaling techniques.

Criterion 1

Geometric Accuracy and Detail

Assesses the precision of geometric modeling, including accurate scaling and detailed representation of angles and distances.

Exemplary

4 Points

The model is highly accurate and detailed, showcasing exceptional precision and creativity in geometric representation.

Proficient

3 Points

The model is accurate with good detail, reflecting a strong understanding of geometric principles.

Developing

2 Points

The model has some inaccuracies or lacks detail, indicating a developing understanding of geometric principles.

Beginning

1 Points

The model is inaccurate, with significant errors in geometric representation.

Criterion 2

Quality of Physical Model

Evaluates the construction and precision of the physical model, including scaled accuracy and structural integrity.

Exemplary

4 Points

The physical model is exceptionally well-crafted with precise scale and sturdy construction.

Proficient

3 Points

The physical model is well-constructed, mostly accurate to scale, and structurally sound.

Developing

2 Points

The physical model has structural weaknesses or inaccuracies, reflecting a developing skill in model construction.

Beginning

1 Points

The physical model is poorly constructed with significant inaccuracies and structural flaws.

Category 3

Design Proposal and Documentation

Assessment of the comprehensiveness and clarity of the final design proposal, including calculations, design rationale, and risk mitigation strategies.

Criterion 1

Proposal Clarity and Organization

Evaluates the coherence, clarity, and organization of the design proposal document.

Exemplary

4 Points

The proposal is exceptionally clear, well-organized, and comprehensive, reflecting sophisticated understanding and communication.

Proficient

3 Points

The proposal is clear, well-organized, and comprehensive, indicating thorough understanding and effective communication.

Developing

2 Points

The proposal is unclear or disorganized in several areas, showing developing communication skills.

Beginning

1 Points

The proposal is poorly organized or unclear, hindering understanding and demonstrating emerging communication skills.

Criterion 2

Risk Assessment and Mitigation

Evaluates the identification of potential risks and formulation of practical mitigation strategies.

Exemplary

4 Points

Thoroughly identifies potential risks with comprehensive, innovative mitigation strategies.

Proficient

3 Points

Effectively identifies potential risks with practical and mostly comprehensive mitigation strategies.

Developing

2 Points

Identifies some potential risks with limited or incomplete mitigation strategies.

Beginning

1 Points

Rarely identifies potential risks or provides ineffective mitigation strategies.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning

Question 1

Reflect on the entire zipline design process. What were your biggest challenges, and how did you overcome them?

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Question 2

How effectively did you apply trigonometric ratios to solve real-world problems in this project?

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Question 3

What were the most important factors you considered when selecting materials for your zipline design, and why?

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Question 4

How did the process of building a scale model enhance your understanding of the design process?

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Question 5

If you were to redesign your zipline course, what changes would you make, and why?

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Question 6

How did your understanding of trigonometric ratios and their applications evolve throughout this project?

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Question 7

What are the key takeaways from this project regarding the design, construction, and safety considerations of ziplines?

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Question 8

How did this project connect to your previous learning experiences in math and science?

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Optional

Question 9

How can the skills and knowledge gained in this project be applied to other real-world situations?

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Optional