Design a Roller Coaster with Systems of Equations

Created byChristine Danhoff

2 views0 downloads

Design a Roller Coaster with Systems of Equations

Grade 9Math1 days

In this project-based learning experience, 9th-grade students apply systems of equations to design a roller coaster that balances thrill, safety, and functionality. Through inquiry-based activities, they learn to represent roller coaster components with mathematical equations and model them for optimization and safety analysis. The project incorporates hands-on experiences, such as graphing the coaster path and analyzing safety constraints, fostering mathematical reasoning and critical thinking. As a culmination, students present their optimized designs, demonstrating their understanding of real-world mathematical modeling and problem-solving skills.

Systems of EquationsRoller Coaster DesignMathematical ModelingSafety ConstraintsOptimizationCritical Thinking

Want to create your own PBL Recipe?Use our AI-powered tools to design engaging project-based learning experiences for your students.

📝

Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we apply systems of equations to design a roller coaster that is safe, functional, and thrilling, while considering key components, safety limits, and real-world mathematical modeling?

Essential Questions

Supporting questions that break down major concepts.

How do systems of equations apply in real-world situations such as designing a roller coaster?
What are the key components of a roller coaster and how can mathematical equations represent these components?
How can we use mathematical modeling to ensure the safety and functionality of a roller coaster design?
What mathematical strategies can be used to optimize the design for speed and thrill while maintaining safety limits?
How can different solutions to a system of equations influence the design and structure of a roller coaster?

Standards & Learning Goals

Learning Goals

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

Understand and apply systems of equations in real-world situations, particularly in the context of roller coaster design.
Identify and represent key components of a roller coaster using mathematical equations.
Use mathematical modeling to evaluate the safety and functionality of a roller coaster design.
Analyze and optimize roller coaster designs for speed and thrill while maintaining safety limits through mathematical strategies.
Explore how different solutions to a system of equations can influence the design and structure of a roller coaster.
Develop problem-solving and critical-thinking skills through designing and evaluating mathematical models.

Common Core Standards

CCSS.MATH.CONTENT.HSA.REI.C.6

Primary

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Reason: The project's core involves solving systems of equations, a central standard covered by this part of the Common Core.

CCSS.MATH.CONTENT.HSA.CED.A.2

Primary

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Reason: Students will need to create and graph equations to represent roller coaster components, aligning with this standard.

CCSS.MATH.CONTENT.HSA.CED.A.3

Primary

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.Reason: The project requires understanding constraints such as safety limits, which ties to this standard.

Next Generation Science Standards

NGSS.HS.ETS1-2

Secondary

Design a solution to a complex real-world problem by breaking it down into smaller, manageable problems that can be solved through engineering.Reason: The project involves design and engineering, requiring students to apply their understanding to solve real-world problems, aligning with this NGSS standard.

Common Core Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1

Supporting

Make sense of problems and persevere in solving them.Reason: Students will engage in problem-solving throughout the project, demanding resilience and critical thinking.

CCSS.MATH.PRACTICE.MP4

Supporting

Model with mathematics.Reason: Students will be using mathematical modeling as a central part of their design process, aligning with this standard.

Entry Events

Events that will be used to introduce the project to students

Equation Escape Room

Students enter a digital escape room where each puzzle requires solving systems of equations to progress. The final challenge: design the ultimate roller coaster that is both exhilarating and safe using their newfound knowledge, blending gamification with real-world problem-solving.

📚

Portfolio Activities

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

Activity 1

Graphing the Coaster Path

This activity focuses on graphing the equations and inequalities developed previously to create an accurate representation of a roller coaster's path. Students will learn to use graphing tools to visualize their designs, ensuring that the ride meets both thrill and safety requirements.

Steps

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

1. Review the equations and inequalities created in the previous activity.

2. Introduce graphing tools and software that can be used to visualize these equations and inequalities.

3. Guide students to graph their roller coaster designs on a coordinate plane, ensuring accuracy in the representation.

4. Discuss the importance of scale, labels, and interpretation of intersecting lines as potential collision or problem points.

Final Product

What students will submit as the final product of the activityA graphically represented roller coaster path using equations and inequalities on a coordinate plane.

Alignment

How this activity aligns with the learning objectives & standardsSupports CCSS.MATH.CONTENT.HSA.REI.C.6 by solving systems of equations approximately using graphs.

Activity 2

Safety Constraints and Viability Analysis

Students will analyze their roller coaster designs based on safety constraints and determine the viability of their models. They will learn to apply system solutions meaningfully, evaluating if they meet both thrilling experience and safety standards.

Steps

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

1. Identify typical safety constraints for roller coasters, such as maximum drop height and speed limits.

2. Represent these constraints using systems of equations and inequalities.

3. Analyze solutions of these systems to interpret the viability of their models.

4. Revise the roller coaster design if needed to meet all safety constraints and optimize the thrill.

Final Product

What students will submit as the final product of the activityA viability report detailing which components of the roller coaster model meet safety standards and how others might be adjusted for improvement.

Alignment

How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.HSA.CED.A.3 by ensuring students represent constraints through equations/inequalities, interpreting solutions in context.

Activity 3

Design Optimization Challenge

With safety and functionality confirmed, students are now challenged to optimize their roller coaster designs for maximum thrill. They will use mathematical strategies and modeling to find the perfect balance between excitement and safety.

Steps

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

1. Review previous roller coaster models and identify areas for potential improvement.

2. Introduce optimization techniques, focusing on adjusting variables within systems of equations for maximum thrill.

3. Apply these techniques to tweak elements like slope, drop angles, and loop sizes, ensuring adherence to safety constraints.

4. Use mathematical modeling to test the impact of adjustments and finalize an optimal design.

Final Product

What students will submit as the final product of the activityAn optimized roller coaster design that balances thrill and safety, supported by mathematical strategies and models.

Alignment

How this activity aligns with the learning objectives & standardsSupports CCSS.MATH.PRACTICE.MP4 by engaging students in modeling with mathematics and optimization techniques.

Activity 4

Final Presentation and Reflection

Students will present their complete roller coaster designs, emphasizing the mathematical journey from conception to final optimization. They will reflect on how different solutions to systems of equations influenced their designs and the critical thinking skills developed throughout the project.

Steps

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

1. Prepare a presentation of the roller coaster design, focusing on the mathematical process and solutions utilized.

2. Include the graphs, models, optimization decisions, and safety analyses in the presentation.

3. Reflect on the learning experience, highlighting the impact of systems of equations and problem-solving skills developed.

4. Present the work in a class symposium, receiving feedback and discussing the outcomes.

Final Product

What students will submit as the final product of the activityA comprehensive presentation of roller coaster designs and a reflective piece on the mathematical and critical thinking skills applied.

Alignment

How this activity aligns with the learning objectives & standardsAligns with NGSS.HS.ETS1-2 by having students design solutions to complex problems and with CCSS.MATH.PRACTICE.MP1 by developing perseverance in problem-solving.

🏆

Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Roller Coaster Design & Analysis Rubric

Category 1

Mathematical Understanding

Assessment of students' comprehension and application of mathematical concepts through systems of equations in the design process.

Criterion 1

Conceptual Application

Evaluate how well students apply mathematical concepts such as systems of equations to model real-world scenarios.

Exemplary

4 Points

Demonstrates sophisticated application of systems of equations in a real-world context, with innovative solutions.

Proficient

3 Points

Demonstrates thorough application of systems of equations, providing appropriate solutions.

Developing

2 Points

Shows limited application of systems of equations, with some errors present.

Beginning

1 Points

Struggles to apply systems of equations in developing solutions.

Criterion 2

Mathematical Precision

The accuracy and detail of the mathematical work presented, such as calculations and graphing.

Exemplary

4 Points

Displays meticulous precision in mathematical reasoning and graphing, with no errors.

Proficient

3 Points

Shows accurate and precise mathematical reasoning with few errors.

Developing

2 Points

Exhibits some accuracy in mathematical reasoning, with several errors.

Beginning

1 Points

Displays minimal accuracy, with frequent errors in reasoning and graphing.

Category 2

Safety and Functionality Analysis

Evaluation of how well students identify and incorporate safety and functionality constraints in roller coaster design.

Criterion 1

Constraint Identification

Ability to identify and represent safety and functionality constraints using equations and inequalities.

Exemplary

4 Points

Expertly identifies and represents all relevant constraints, with innovative use of equations and inequalities.

Proficient

3 Points

Correctly identifies and represents key constraints with equations and inequalities.

Developing

2 Points

Identifies and represents some constraints, but incomplete or inaccurate at times.

Beginning

1 Points

Struggles to identify or represent constraints accurately.

Criterion 2

Solution Viability and Analysis

Analyze the viability of design solutions concerning identified constraints and propose improvements.

Exemplary

4 Points

Provides a comprehensive and innovative analysis of design solutions, exceeding safety standards.

Proficient

3 Points

Delivers effective analysis of design, meeting safety standards with logical improvements.

Developing

2 Points

Presents basic analysis of design solutions, with errors or omissions in meeting standards.

Beginning

1 Points

Minimal analysis with significant errors in understanding safety standards.

Category 3

Creative and Critical Thinking

Evaluation of the creativity and problem-solving skills demonstrated in designing a functional and thrilling roller coaster.

Criterion 1

Creativity and Innovation

The ability to innovate and create unique designs that represent systems of equations realistically and creatively.

Exemplary

4 Points

Designs an exceptionally creative roller coaster, showcasing advanced innovation in mathematical concepts.

Proficient

3 Points

Illustrates creativity and innovation in roller coaster design, using mathematical concepts well.

Developing

2 Points

Shows some creativity in design, with room for increased innovation and sophistication.

Beginning

1 Points

Minimal creativity and innovation in design, struggling to apply concepts creatively.

Criterion 2

Problem Solving and Adaptability

Assessment of how well students address challenges and adapt their designs in response to feedback.

Exemplary

4 Points

Demonstrates outstanding problem-solving skills, adapting designs creatively with thorough consideration of feedback.

Proficient

3 Points

Shows good problem-solving abilities, effectively adapting designs following feedback.

Developing

2 Points

Engages with problem-solving to a limited extent, with inconsistent adaptation strategies.

Beginning

1 Points

Struggles with problem-solving and adapting designs effectively.

Category 4

Presentation and Reflection

Assessment of the clarity and depth of students' presentations and reflections on their learning experience.

Criterion 1

Presentation Clarity and Organization

The clarity, organization, and professionalism of the presentation, including the communication of mathematical processes.

Exemplary

4 Points

Presents with exceptional clarity and organization, communicating mathematical processes effectively and professionally.

Proficient

3 Points

Provides a clear and organized presentation with effective communication of processes.

Developing

2 Points

Offers a presentation with some clarity issues, lacking in organization occasionally.

Beginning

1 Points

Presentation lacks clarity and organization, struggling to communicate ideas effectively.

Criterion 2

Reflective Insight

Depth of reflection on the learning process, including understanding of mathematical modeling and critical thinking development.

Exemplary

4 Points

Provides profound reflective insights, thoroughly analyzing the learning journey and growth in critical thinking.

Proficient

3 Points

Offers effective reflection, demonstrating awareness of learning growth and critical thinking development.

Developing

2 Points

Reflects on learning with limited insight or depth, touching on growth sporadically.

Beginning

1 Points

Provides superficial reflection with minimal engagement in analyzing learning experiences.

Reflection Prompts

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

Question 1

How did engaging with systems of equations help you understand the complexities of designing a roller coaster?

Text

Required

Question 2

On a scale from 1 to 5, how confident do you now feel in applying mathematical equations to real-world problems?

Scale

Required

Question 3

What was the most challenging part of using mathematical modeling in your roller coaster design, and how did you overcome it?

Text

Required

Question 4

Which mathematical strategy used in the project did you find the most effective, and why?

Text

Required

Question 5

Do you believe your final roller coaster design balances thrill and safety effectively? Why or why not?

Text

Required

Question 6

What feedback did you receive during the class symposium on your roller coaster design? How did it impact your perspective on your design work?

Text

Optional

Question 7

If you could start the project over, what would you do differently with what you know now?

Text

Optional