Outbreak Architect: Modeling Exponential Growth to Flatten the Curve
Inquiry Framework
Question Framework
Driving Question
The overarching question that guides the entire project.How can we, as public health strategists, use mathematical modeling to predict the spread of a virus and design community interventions that successfully flatten the exponential curve to protect our healthcare capacity?Essential Questions
Supporting questions that break down major concepts.- How can mathematical models help us predict and control the spread of an infectious disease?
- What distinguishes exponential growth from linear growth, and why is this distinction critical during a public health crisis?
- How do variables like the 'initial number of cases' and the 'transmission rate' influence the shape and steepness of an exponential curve?
- In what ways can community interventions (like social distancing or vaccination) be represented as transformations or changes to the base of an exponential function?
- How do we determine when a mathematical model is 'flattening,' and what does that mean for a community's healthcare capacity?
- What are the limitations of using a simple exponential model to represent real-world human behavior and biological spread?
Standards & Learning Goals
Learning Goals
By the end of this project, students will be able to:- Construct exponential growth functions ($f(t) = a \cdot b^t$) to model the spread of a virus based on varying initial infection rates and transmission factors.
- Compare and contrast linear and exponential growth models to explain why exponential spread creates unique challenges for healthcare infrastructure.
- Analyze the impact of parameter changes (e.g., reducing the growth factor 'b') to simulate public health interventions like social distancing and vaccination.
- Interpret graphical data to identify the point at which healthcare capacity is exceeded and justify 'flattening the curve' strategies using mathematical evidence.
- Evaluate the limitations of simple exponential models by identifying real-world variables (e.g., immunity, population density) that require more complex modeling approaches.
Common Core State Standards for Mathematics (High School)
Common Core Mathematical Practices
Next Generation Science Standards (NGSS)
Entry Events
Events that will be used to introduce the project to studentsThe Viral Saturation Point
Students enter a classroom transformed into a high-stakes 'Social Command Center.' They are shown the real-time analytics of a local 'viral' video or meme and asked to use exponential modeling to predict the exact hour it will reach 'saturation' (every person in the city having seen it), before the teacher reveals that a biological pathogen follows the exact same math—only with deadly consequences.Portfolio Activities
Portfolio Activities
These activities progressively build towards your learning goals, with each submission contributing to the student's final portfolio.The Red Line: Finding the Breaking Point
Students graph their exponential model against a 'Healthcare Capacity Line.' They must determine the exact day the 'Red Line' (hospital capacity) is crossed. This activity introduces the concept of a 'Carrying Capacity' as a limit to the system, forcing students to see the human cost of the unchecked curve.Steps
Here is some basic scaffolding to help students complete the activity.Final Product
What students will submit as the final product of the activityA 'Crisis Threshold Graph' featuring the exponential curve, the horizontal capacity line, and a 'Crisis Point' annotation marking the day the community runs out of beds.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.PRACTICE.MP4 (Model with mathematics) and NGSS.HS-LS2-1 (Carrying capacity). This activity forces students to interpret the horizontal line as a resource constraint (hospital beds).Curve Crushers: Designing the Intervention
In this high-stakes simulation, students must 'flatten the curve.' They are given intervention cards (e.g., 'Mask Mandate' reduces $b$ by 15%, 'Strict Quarantine' reduces $b$ by 40%). Students will apply these changes to their original function to see if they can keep the curve below the 'Red Line' established in the previous activity.Steps
Here is some basic scaffolding to help students complete the activity.Final Product
What students will submit as the final product of the activityAn 'Intervention Impact Report' showing two overlaid graphs (Original vs. Mitigated) and a mathematical analysis of how the change in the 'b' value saved the community's healthcare system.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.HSF.BF.B.3: Identify the effect on the graph of replacing $f(x)$ by $f(x) + k$, $k f(x)$, $f(kx)$, and $f(x + k)$. It specifically focuses on how changing the base (b) acts as a transformation that stretches or compresses the growth over time.The Strategic Defense Portfolio: Briefing the Council
Students compile their findings into a final strategic portfolio. They must not only present their math but also critique it. They will identify why a simple $a \cdot b^t$ model eventually fails (e.g., people getting better/immune, finite population) and present their 'Flattened Curve' model to a mock 'City Council.'Steps
Here is some basic scaffolding to help students complete the activity.Final Product
What students will submit as the final product of the activityThe 'Outbreak Architect Portfolio,' containing the model, the intervention data, the capacity analysis, and a 'Limitations Brief' explaining the difference between theoretical exponential growth and real-world logistic growth.Alignment
How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.PRACTICE.MP4 and provides the final evaluation for the learning goal of identifying real-world limitations of simple models. It synthesizes all previous standards into a final communication piece.Rubric & Reflection
Portfolio Rubric
Grading criteria for assessing the overall project portfolioOutbreak Architect: Exponential Modeling & Public Health Rubric
Modeling Accuracy & Function Theory
Focuses on the core mathematical ability to build and identify exponential functions as outlined in CCSS.MATH.CONTENT.HSF.LE.A.1.C and HSF.LE.A.2.Mathematical Modeling & Construction
Ability to construct and distinguish exponential growth functions ($f(t) = a \cdot b^t$) from linear models based on provided pandemic data.
Exemplary
4 PointsConstructs a highly accurate exponential model with precise identification of initial value (a) and growth factor (b). Provides a sophisticated mathematical explanation of why exponential growth, rather than linear, is required for this context, citing specific rate-of-change characteristics.
Proficient
3 PointsConstructs an accurate exponential function that models the provided data. Correct identifies 'a' and 'b' and provides a clear explanation of the difference between linear and exponential growth in the context of viral spread.
Developing
2 PointsConstructs an exponential function with minor errors in the growth factor or initial value. Provides a basic explanation of exponential growth but may struggle to clearly distinguish it from linear growth mathematically.
Beginning
1 PointsAttempts to construct a function but uses an incorrect format (e.g., linear) or fails to identify 'a' and 'b' correctly. Demonstrates significant misconceptions about the nature of exponential growth.
Data Interpretation & Carrying Capacity
Evaluates the student's ability to model with mathematics (MP4) and understand carrying capacity (NGSS.HS-LS2-1).Constraint Analysis & Graphing
The ability to graph functions against a resource constraint (hospital capacity) and interpret the real-world implications of the intersection point.
Exemplary
4 PointsGraphs the 'Red Line' and exponential curve with professional precision. Identifies the 'Day of Saturation' accurately and provides a deep analysis of the human and systemic consequences of exceeding this carrying capacity.
Proficient
3 PointsSuccessfully graphs the function and the capacity line. Correct calculates the intersection point and explains what it represents for the community's healthcare system.
Developing
2 PointsGraphs both elements but with scaling or plotting errors. Identifies an intersection point but provides a limited explanation of its significance regarding hospital capacity.
Beginning
1 PointsFails to correctly graph the capacity line or the curve. Cannot identify or explain the significance of the intersection point where the model exceeds capacity.
Intervention Analysis & Curve Flattening
Aligned with CCSS.MATH.CONTENT.HSF.BF.B.3, focusing on how changes to the function parameters affect the output.Parameter Manipulation & Transformation
Ability to apply transformations to the exponential function by modifying the growth factor (b) and analyzing the resulting 'flattening' of the curve.
Exemplary
4 PointsCalculates complex mitigated growth factors with perfect accuracy. Demonstrates a sophisticated understanding of how changing the base (b) transforms the curve. Provides a detailed comparative analysis of 'days gained' and lives saved.
Proficient
3 PointsCorrectly calculates a new growth factor based on intervention cards. Accurately graphs the new 'flattened' curve and describes the transformation in terms of the function's steepness and growth rate.
Developing
2 PointsApplies an intervention to the growth factor but makes calculation errors. The resulting graph shows a flatter curve, but the student struggles to explain the mathematical relationship between the change in 'b' and the transformation.
Beginning
1 PointsStruggles to modify the growth factor or graph the new function. Shows little understanding of how interventions translate into mathematical transformations of the base function.
Strategic Synthesis & Model Critique
Assesses the student's ability to communicate complex mathematical ideas and evaluate the validity of their models in a real-world context.Communication & Critical Evaluation
Synthesis of findings into a professional briefing, including a critical evaluation of the model's real-world limitations (exponential vs. logistic growth).
Exemplary
4 PointsPresents a professional-grade portfolio with compelling mathematical justifications. Provides a profound reflection on logistic growth, herd immunity, and the ethical implications of modeling, showing mastery of model limitations.
Proficient
3 PointsCompiles a clear and organized briefing that uses mathematical evidence to justify interventions. Correct identifies at least two real-world factors (like immunity or population size) that limit simple exponential models.
Developing
2 PointsThe portfolio is organized but lacks strong mathematical justification for the interventions. Mentions limitations of the model but provides a shallow or purely descriptive explanation without scientific/mathematical depth.
Beginning
1 PointsThe final portfolio is incomplete or disorganized. Fails to address the limitations of the model or provides a justification for interventions that is not supported by the mathematical data.