Population Boom: A Calculus-Based Simulator
Created byJames Dye
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Population Boom: A Calculus-Based Simulator

Grade 12Math3 days
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we use related rates in calculus to design a system for tracking and predicting population growth, considering various influencing factors, and accounting for the limitations of mathematical models in real-world scenarios?

Essential Questions

Supporting questions that break down major concepts.
  • How can calculus concepts, specifically related rates, be used to model population changes?
  • What factors influence population growth and decline, and how can these be represented mathematically?
  • How can we use mathematical models to predict future population trends and analyze potential outcomes?
  • What are the limitations of using calculus to model real-world populations, and how can these limitations be addressed?
  • How can technology be used to enhance the process of tracking and analyzing population dynamics?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Students will be able to apply related rates to model population growth in various scenarios.
  • Students will be able to analyze the impact of different factors on population dynamics using calculus.
  • Students will be able to evaluate the limitations of mathematical models in predicting real-world population trends.
  • Students will be able to use technology to simulate and visualize population changes over time.

AP Calculus AB

APCalcAB-RelRates1
Primary
Apply related rates to model real-world relationships.Reason: This standard directly aligns with the project's focus on using related rates to model population growth.

Common Core Math Standards

MathModeling1
Supporting
Use mathematical models to represent and analyze real-world phenomena.Reason: This standard supports the project's aim to model population dynamics.

Entry Events

Events that will be used to introduce the project to students

Island Biosphere

Students are presented with a fictional scenario: A rare species of bird has been discovered on a remote island. Scientists need to understand its population dynamics to ensure its survival. Initial data suggests the population is growing rapidly, but resources are limited. Students are tasked with developing a model to project the bird population over time, considering factors like birth rate, death rate, carrying capacity, and environmental changes.
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Portfolio Activities

Portfolio Activities

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

Population Dynamics Primer

Students will build a foundational understanding of population dynamics and related rates by exploring key concepts and terminology. They will define terms such as birth rate, death rate, carrying capacity, and exponential growth. They will work through basic related rates problems involving changing variables.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Define key terms related to population dynamics (birth rate, death rate, carrying capacity, exponential growth).
2. Work through practice problems involving related rates and changing variables.
3. Discuss how changing birth rates and death rates impact overall population growth.

Final Product

What students will submit as the final product of the activityA glossary of key terms related to population dynamics and a completed worksheet with practice related rates problems.

Alignment

How this activity aligns with the learning objectives & standardsAPCalcAB-RelRates1, MathModeling1
Activity 2

Building a Basic Population Model

Students will develop a simple mathematical model to represent the growth of the bird population on the island. They will use a basic exponential growth equation, incorporating birth and death rates. They will then use related rates to calculate how the population changes over time.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Develop a simplified exponential growth equation to represent the bird population.
2. Use related rates to calculate population change over time based on given birth and death rates.
3. Graph the projected population growth over a defined period.

Final Product

What students will submit as the final product of the activityA documented mathematical model (equation and explanation) and a graph illustrating projected population growth over a set period.

Alignment

How this activity aligns with the learning objectives & standardsAPCalcAB-RelRates1, MathModeling1
Activity 3

Considering Carrying Capacity

Students will refine their population model by incorporating the concept of carrying capacity. They will learn how to adjust their related rates calculations to account for the limiting factor of resources.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Research and define carrying capacity in the context of population dynamics.
2. Adjust the existing population model equation to include carrying capacity as a limiting factor.
3. Recalculate population projections using related rates, considering the carrying capacity constraint.
4. Compare and contrast the original population projection graph with the revised graph that incorporates carrying capacity, analyzing the differences.

Final Product

What students will submit as the final product of the activityA revised mathematical model incorporating carrying capacity and a revised graph showing adjusted population projections.

Alignment

How this activity aligns with the learning objectives & standardsAPCalcAB-RelRates1, MathModeling1
Activity 4

Environmental Impacts on Population Dynamics

Students will explore the impact of environmental changes on population dynamics. They will select an environmental factor (e.g., a new predator, a change in food supply) and modify their model to reflect its influence. They will recalculate and analyze the new projected population changes.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Select an environmental factor and research its potential impact on bird populations.
2. Modify the population model to incorporate the chosen environmental factor and its effects.
3. Use related rates to recalculate population projections, factoring in the environmental change.
4. Present findings, explaining the chosen factor, the modifications made to the model, and the impact on population predictions.

Final Product

What students will submit as the final product of the activityA modified mathematical model integrating an environmental factor and a presentation explaining the chosen factor and its impact on the population.

Alignment

How this activity aligns with the learning objectives & standardsAPCalcAB-RelRates1, MathModeling1
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Population Dynamics with Calculus Rubric

Category 1

Conceptual Understanding

Assesses the student's grasp of related rates and their application to population dynamics.
Criterion 1

Related Rates Comprehension

Measures how well students understand and apply related rates to model population dynamics.

Exemplary
4 Points

Demonstrates a sophisticated understanding of related rates and their application in modeling population dynamics, using precise and insightful explanations.

Proficient
3 Points

Shows a thorough understanding of related rates and can apply them appropriately to population dynamics scenarios.

Developing
2 Points

Shows an emerging understanding of related rates with occasional errors in application to population dynamics.

Beginning
1 Points

Shows initial understanding with frequent errors in applying related rates to population dynamics.

Criterion 2

Mathematical Modeling

Assesses proficiency in using calculus to construct accurate mathematical models for population growth.

Exemplary
4 Points

Creates sophisticated mathematical models that represent population growth accurately, demonstrating innovative and precise use of calculus.

Proficient
3 Points

Constructs accurate mathematical models for population growth using calculus, with most elements correctly applied.

Developing
2 Points

Develops mathematical models with basic accuracy, showing limited understanding of calculus applications.

Beginning
1 Points

Produces inaccurate or incomplete models, struggling to apply calculus to population growth.

Category 2

Analytical Thinking

Evaluates the ability to critically analyze and interpret mathematical results in the context of population dynamics.
Criterion 1

Factor Analysis

Measures students' ability to identify and integrate various factors affecting population dynamics in their models.

Exemplary
4 Points

Thoroughly analyzes and integrates multiple factors into population models, showing deep insight into their impacts.

Proficient
3 Points

Effectively identifies and integrates relevant factors, showing clear understanding of their impacts in population models.

Developing
2 Points

Identifies factors but integrates them inconsistently or superficially within population models.

Beginning
1 Points

Struggles to identify relevant factors or to integrate them effectively into population models.

Criterion 2

Graphical Interpretation

Assesses the ability to interpret and analyze graphs related to population projections and dynamics.

Exemplary
4 Points

Interprets graphs with sophistication, providing insightful analysis of population projections and dynamics.

Proficient
3 Points

Accurately interprets graphs and provides solid analysis of population projections and dynamics.

Developing
2 Points

Interprets graphs with some accuracy but demonstrates limited analytical depth.

Beginning
1 Points

Struggles to interpret graphs accurately, providing minimal analysis.

Category 3

Collaborative and Communication Skills

Assesses the ability to work collaboratively and communicate findings effectively.
Criterion 1

Presentation Skills

Evaluates clarity, coherence, and persuasiveness in presenting mathematical models and findings.

Exemplary
4 Points

Presents findings with exceptional clarity and coherence, persuasively communicating complex mathematical ideas.

Proficient
3 Points

Communicates findings clearly and coherently, effectively conveying mathematical ideas.

Developing
2 Points

Presents ideas with basic clarity, occasionally lacking coherence or depth.

Beginning
1 Points

Struggles to communicate ideas clearly, lacking coherence and depth.

Criterion 2

Collaboration

Measures the student's ability to collaborate effectively in group settings.

Exemplary
4 Points

Exhibits leadership and works collaboratively, greatly enhancing group dynamics.

Proficient
3 Points

Collaborates effectively with peers, contributing positively to group efforts.

Developing
2 Points

Collaborates with peers but with varying effectiveness, sometimes hindering group efforts.

Beginning
1 Points

Struggles to collaborate effectively, requiring support to engage with peers.

Reflection Prompts

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

Reflect on the process of building your population model. What were the most challenging aspects of applying related rates to this real-world scenario?

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Question 2

How did the inclusion of carrying capacity change your population predictions? Explain the significance of this concept in understanding population dynamics.

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Question 3

What are the limitations of using mathematical models to predict real-world population trends? Discuss the potential sources of error and uncertainty in your model.

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Question 4

How effective was the use of technology in helping you simulate and visualize population changes? What tools did you use, and how did they enhance your understanding of population dynamics?

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Question 5

On a scale of 1 to 5, how confident are you in your ability to apply related rates to model real-world phenomena after completing this project?

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