Zero-G Polynomials: Engineering Roller Coaster Tracks
Created byAlicia Corbley
0 views1 downloads

Zero-G Polynomials: Engineering Roller Coaster Tracks

Grade 8Math3 days
In this project, 8th-grade students step into the role of lead engineers to design a safe and thrilling roller coaster using the power of polynomial functions. Students analyze how algebraic components like coefficients and degrees translate into physical track features, performing operations to synthesize multiple "track modules" into a continuous ride. By applying factoring techniques to identify ground-level "zeros," students ensure the mathematical integrity and safety of their designs. The experience culminates in the creation of a "Master Zero-G Blueprint" and a technical pitch that demonstrates the real-world application of algebraic modeling.
PolynomialsFactoringEngineeringModelingRoller CoastersAlgebraic OperationsSafety Design
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, as lead engineers, use polynomial operations and factoring to design a safe and thrilling roller coaster that maps out precise ground-level transitions?

Essential Questions

Supporting questions that break down major concepts.
  • How do the components of a polynomial expression (coefficients, degrees, terms) translate into the physical features and "thrill factor" of a roller coaster?
  • How does the process of factoring help us predict and control where a coaster track meets the ground level?
  • In what ways do mathematical operations (addition, subtraction, and multiplication) allow us to combine different track sections into one continuous ride?
  • How can we use the zeros of a polynomial function to create an accurate and safe visual model of our roller coaster?
  • Why is the precision of our algebraic factoring critical to the safety of the riders and the integrity of the track design?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Perform addition, subtraction, and multiplication of polynomials to combine and modify different sections of a roller coaster track design.
  • Apply factoring techniques to quadratic and higher-order polynomial expressions to solve for 'zeros,' identifying the exact coordinates where the track meets the ground level.
  • Translate algebraic components (coefficients, degrees, and terms) into physical track features, such as height, direction, and complexity.
  • Construct an accurate visual and mathematical model of a roller coaster path using the identified zeros and function behavior.
  • Analyze the relationship between algebraic precision and engineering safety by verifying that track transitions are continuous and mathematically sound.

Common Core State Standards (Math)

CCSS.Math.Content.HSA-APR.A.1
Primary
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Reason: Students must use polynomial operations to synthesize various track segments into a singular, continuous function for their coaster design.
CCSS.Math.Content.HSA-APR.B.3
Primary
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Reason: This is the core of the project: using the zeros (roots) to determine where the coaster touches the ground and using those points to map the track path.
CCSS.Math.Content.HSA-SSE.A.1.a
Secondary
Interpret parts of an expression, such as terms, factors, and coefficients.Reason: Students need to interpret how changing a coefficient or term in their polynomial affects the 'thrill factor' or physical shape of the coaster.
CCSS.Math.Content.HSA-SSE.B.3.a
Primary
Factor a quadratic expression to reveal the zeros of the function it defines.Reason: Factoring is the primary method students will use to identify the safety ground-level points required by the project brief.

Common Core State Standards (Math Practices)

CCSS.Math.Practice.MP4
Supporting
Model with mathematics.Reason: The project requires students to take a real-world engineering challenge and apply algebraic functions to create a working model.

Entry Events

Events that will be used to introduce the project to students

The Secret Formula Leak

Students are introduced to a 'leaked' memo from a rival theme park claiming they have discovered a 'secret formula' for the smoothest ride transition using higher-degree polynomials. The challenge is to 'reverse-engineer' the rival's secret by factoring their equations and mapping out the 'Zero Points' to see if their claims of a perfect ride are mathematically sound.
📚

Portfolio Activities

Portfolio Activities

These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.
Activity 1

Decoding the Thrill: The Anatomy of an Expression

In this opening activity, students act as junior analysts decoding the 'leaked' memo from the rival theme park. They will break down a complex polynomial expression to understand how each part of the math translates to a physical feature of the roller coaster (e.g., the leading coefficient determining the steepness of the first drop).

Steps

Here is some basic scaffolding to help students complete the activity.
1. Examine the 'Secret Formula' polynomial provided in the entry event memo.
2. Identify and define the coefficients, constants, terms, and the degree of the polynomial.
3. Calculate the 'Base Height' of the coaster by evaluating the expression when the track begins (at x=0).
4. Use polynomial addition or subtraction to adjust the coaster's height based on a 'Safety Regulation' prompt (e.g., 'Lower the entire track by 10 units').

Final Product

What students will submit as the final product of the activityAn 'Anatomy of a Coaster' Infographic that labels a polynomial equation with its physical real-world counterparts (Height, Steepness, Number of Hills).

Alignment

How this activity aligns with the learning objectives & standardsThis activity directly addresses A.SSE.A.1A by requiring students to interpret terms, coefficients, and degrees as physical engineering constraints. It also touches on A.APR.A.1 by using addition and subtraction to calculate net height adjustments.
Activity 2

The Multi-Section Merge: Forging the Track

Engineers often design a coaster in sections. In this activity, students will take two 'Track Modules' (represented as binomials) and multiply them to create a continuous 'Thrill Curve.' This teaches them how polynomial multiplication allows for more complex, higher-degree paths that a simple linear track couldn't achieve.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Select two binomial 'Track Modules' from a provided list (e.g., (x - 4) and (x + 2)).
2. Multiply the modules using the distributive property or FOIL method to create a single polynomial expression.
3. Predict how many times this new track section will touch the ground based on the degree of the resulting polynomial.
4. Verify the expansion by checking work with a partner to ensure no 'structural' (mathematical) errors.

Final Product

What students will submit as the final product of the activityA 'Merged Track Report' showing the step-by-step multiplication of binomial segments into a single quadratic or cubic function.

Alignment

How this activity aligns with the learning objectives & standardsThis activity aligns with A.APR.A.1, focusing on the multiplication of polynomials to create more complex, continuous functions from simpler track segments.
Activity 3

Zero-G Detective: Factoring for Safety

Safety is paramount. Students must identify the 'Zero Points'—the exact coordinates where the coaster track meets the ground. Students will take their merged equations from the previous activity and use factoring techniques to work backward and find the roots of the function.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Take the merged polynomial functions created in Activity 2.
2. Apply factoring techniques (GCF, factoring trinomials) to return the polynomial to its factored form.
3. Solve for 'x' to find the 'Zero Points' where the track height is exactly zero.
4. Mark these points on a coordinate plane as the only safe places for the track to touch the ground.

Final Product

What students will submit as the final product of the activityA 'Ground-Level Safety Map' listing the factored forms of their equations and the resulting x-intercepts (Ground Points).

Alignment

How this activity aligns with the learning objectives & standardsThis activity targets A.SSE.B.3A by having students factor quadratic expressions to reveal the zeros. It also supports A.APR.B.3 by identifying the critical points needed for a graph.
Activity 4

The Final Descent: Mapping the Path

Using the data gathered in the previous three activities, students will now construct the final visual blueprint of their roller coaster. They will use their identified zeros, the y-intercept (starting height), and their knowledge of polynomial behavior to draw a safe and thrilling ride profile.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Plot the 'Zero Points' (x-intercepts) and the 'Starting Gate' (y-intercept) on a large coordinate grid.
2. Sketch the 'Thrill Path' between the zeros, ensuring the curve is smooth and continuous.
3. Label the 'Max Thrill' points (local maxima) and 'G-Force Dips' (local minima) on the graph.
4. Write a 'Safety Certification' paragraph explaining how factoring helped ensure the track doesn't crash into the ground unexpectedly.

Final Product

What students will submit as the final product of the activityThe 'Master Zero-G Blueprint'—a detailed graph of the roller coaster on poster paper, accompanied by a technical pitch explaining why the design is mathematically sound.

Alignment

How this activity aligns with the learning objectives & standardsThis final activity fulfills A.APR.B.3 by using zeros to construct a rough graph and MP.4 by modeling a real-world scenario with mathematics.
🏆

Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Zero-G Coaster: Polynomial Engineering Rubric

Category 1

Algebraic Anatomy & Interpretation

Focuses on the student's ability to decode algebraic expressions and understand the relationship between mathematical symbols and physical constraints.
Criterion 1

Structural Interpretation (A.SSE.A.1A)

Ability to identify and interpret polynomial parts (coefficients, degrees, terms, constants) and translate them into physical roller coaster features (height, steepness, number of hills).

Exemplary
4 Points

Accurately identifies all polynomial components and provides sophisticated, insightful explanations of how each part dictates specific physical coaster behaviors. Demonstrates a mastery of how the leading coefficient and degree impact the overall 'thrill' and shape.

Proficient
3 Points

Correctly identifies polynomial components and provides clear, logical connections to physical coaster features. Understanding of the relationship between math and engineering is thorough and consistent.

Developing
2 Points

Identifies most polynomial components correctly, but the translation to physical features is inconsistent or lacks detail. Some confusion may exist regarding how degrees or coefficients affect the coaster's shape.

Beginning
1 Points

Struggles to identify basic polynomial parts or provides incorrect physical interpretations. Connection between the algebraic expression and the coaster design is missing or flawed.

Category 2

Polynomial Operations & Synthesis

Evaluates the student's technical ability to manipulate polynomial expressions through arithmetic operations.
Criterion 1

Computational Fluency (A.APR.A.1)

Precision and accuracy in performing polynomial addition, subtraction, and multiplication to synthesize track segments into a continuous function.

Exemplary
4 Points

Calculations are flawless across all operations. Work is presented in a highly organized, step-by-step manner that clearly shows the synthesis of 'Track Modules' into a complex, higher-order function. Adjustments for 'Safety Regulations' are mathematically perfect.

Proficient
3 Points

Calculations are accurate with only minor, non-conceptual errors. Successfully multiplies binomials and adjusts functions for height correctly. Demonstrates a solid grasp of polynomial closure and operations.

Developing
2 Points

Shows basic understanding of operations but makes multiple calculation or distributive property errors. Has difficulty correctly synthesizing multiple binomials into a singular polynomial.

Beginning
1 Points

Significant errors in polynomial multiplication (e.g., failing to FOIL or distribute). Unable to combine track sections into a functional continuous expression.

Category 3

Factoring for Safety & Precision

Focuses on the critical skill of factoring to solve for roots and ensure the mathematical integrity of the track's interaction with the 'ground.'
Criterion 1

Factoring & Zero Identification (A.SSE.B.3A & A.APR.B.3)

Ability to factor quadratic expressions and solve for zeros to determine precise ground-level transition points for the coaster track.

Exemplary
4 Points

Expertly applies multiple factoring techniques (GCF, trinomials, etc.) to reveal zeros. Identifies all ground-level points with 100% accuracy and provides a sophisticated rationale for why these points are safety-critical.

Proficient
3 Points

Correctly factors polynomial expressions to find zeros. Ground-level points are accurately identified and mapped. Understands the relationship between the factored form and the x-intercepts.

Developing
2 Points

Attempts factoring but struggles with specific methods (e.g., sign errors in trinomials). Some zeros are identified correctly, while others may be inaccurate or missing.

Beginning
1 Points

Unable to factor expressions or identify zeros. Fails to find the 'Ground-Level Safety Map' points required for the design.

Category 4

Blueprint Design & Engineering Pitch

Assesses the student's ability to synthesize mathematical findings into a final real-world application and communicate their reasoning effectively.
Criterion 1

Modeling & Technical Communication (MP.4)

The creation of a visual blueprint and technical pitch that integrates mathematical data (zeros, intercepts, curves) into a coherent, engineered design.

Exemplary
4 Points

Produces an outstanding 'Master Blueprint' that is highly accurate, professional, and visually engaging. The technical pitch uses precise mathematical vocabulary to defend the safety and 'thrill' of the design, showing deep metacognition.

Proficient
3 Points

Constructs an accurate graph showing zeros, y-intercepts, and smooth curves. The safety certification paragraph clearly explains the mathematical reasoning behind the track path.

Developing
2 Points

The blueprint is partially complete; zeros are plotted, but the curve behavior (hills/valleys) may not align with the degree of the polynomial. The technical explanation is brief or lacks mathematical depth.

Beginning
1 Points

The blueprint is incomplete or contains significant inaccuracies that would result in an 'unsafe' coaster. Mathematical modeling is weak, and the technical pitch is missing or fails to use relevant concepts.

Reflection Prompts

End-of-project reflection questions to get students to think about their learning
Question 1

How confident do you feel in your ability to use factoring to identify 'Zero Points' (ground-level transitions) in a polynomial function?

Scale
Required
Question 2

In your Safety Certification paragraph, you explained why your design is mathematically sound. Beyond just getting the 'right answer,' why is the precision of factoring critical to the life-safety of people using your engineering designs?

Text
Required
Question 3

Which part of the roller coaster design process required the most mental effort to translate from a mathematical expression into a physical track feature?

Multiple choice
Required
Options
Translating 'degree' into the number of hills/turns
Multiplying 'Track Modules' (binomials) into a single function
Factoring trinomials to find the exact ground-level coordinates
Adjusting the 'Base Height' using polynomial addition/subtraction
Question 4

At the start of this project, you were asked to 'reverse-engineer' a rival's secret formula. How has your perspective on complex-looking polynomials changed now that you know they are simply made up of individual 'track modules' (factors)?

Text
Optional