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Modeling Carrying Capacity with Probability, Graphing, and Ecology

In this interactive simulation, students explore the concept of carrying capacity by modeling changes in a deer population over 20 years. Using dice rolls and ecological scenarios, students apply probability, data collection, and graphing to analyze how biotic and abiotic factors affect population dynamics. The activity integrates biology and math by requiring students to interpret trends, calculate average population values, and visually represent ecosystem stability. Through guided questions, students also reflect on human impacts and wildlife management strategies.

Madison O’Neal

A High School STEM Lesson

The Arizona STEM Acceleration Project

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Notes & List of Materials

Student worksheet from the lab PDF�

6-sided dice (1 per group)�

Graph paper or graphing software (Desmos, Excel, Google Sheets)�

Colored pencils or markers�

Calculator

Narrative of what happens each year (slides)

Taught in-class as a more hands on follow up about population growth, logistic growth, and ecological environment management.

Relevant Materials:

Worksheet (Link)
Scenarios (Link)

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Standards

NGSS - Life Sciences:

HS-LS2-1: Use mathematical and/or computational representations to support explanations of factors that affect carrying capacity of ecosystems.�
HS-LS2-2: Use mathematical representations to support and revise explanations based on evidence about factors affecting biodiversity.�

AZ Standards:

Essential HS.L2U3.18: Obtain, evaluate, and communicate about the positive and negative ethical, social, economic and political implications of human activity on the biodiversity of an ecosystem.

Plus HS+B.L2U1.1: Develop a model showing the relationship between limiting factors and carrying capacity, and use the model to make predictions on how environmental changes impact biodiversity.

Plus HS+B.L2U1.2: Engage in argument from evidence that changes in environmental conditions or human interventions may change species diversity in an ecosystem.

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Objectives:

Students will simulate population changes in a deer ecosystem using probability-based events.

Students will graph population data over time and identify patterns that reflect carrying capacity.

Students will calculate estimated and calculated carrying capacity and compare short-term vs. long-term averages.

Students will analyze the effects of limiting factors (e.g., predators, disease, human impacts) on population stability.

Students will connect math and biology by applying probability, statistics, and graphing skills to ecological concepts.

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Agenda - 1 Class Period (50 minutes)

0–5 min | Engage / Driving Question

Post the driving question: “How can we use math to model and predict the carrying capacity of an ecosystem?”�
Quick class brainstorm: What factors might cause deer populations to grow or shrink?�

5–20 min | Simulation (Data Collection)

Students work in pairs or small groups.�
Roll dice, apply multipliers, and record deer population data for 20 years in the data table.�
Teacher circulates, checks math for first few rounds.�

20–35 min | Graphing & Carrying Capacity

Students transfer data into compact table.�
Create line graph (by hand or digitally).�
Draw and label Estimated Carrying Capacity (ECC) line.�
Calculate 10-year and 20-year averages (CCC) and record.

35–45 min | Analysis Questions (Group/Independent Work)

Students answer 2–3 of the guiding questions (limiting factors, why population fluctuates, human impact).�
Groups share quick insights.�

45–50 min | Exit Ticket & Wrap-Up

Exit Ticket: Write 1 biology concept + 1 math concept you used today and how they connected.

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Intro/Driving Question/Opening

How can wildlife managers use data and models to keep ecosystems in balance?

How can math help us predict the balance of life in an ecosystem?

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Hands-on Activity Instructions

Grouping: Students in pairs, or working individually
Instructions:

Guide students step-by-step through the narrative. For each slide, give the scenario, have students roll their die, and then give them time to do the math. Halfway through, students will check how they are doing by calculating the carrying capacity. Then, ask them what they need to focus on to successfully achieve the carrying capacity they predicted. Continue through the last 10 years. Complete the assignment graph & analysis questions independently.

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Assessment

Formative:

Group participation in the simulation
Accuracy in population calculations and graphing

Summative

Completion and correctness of analysis questions
Quality of discussion and reflection on population trends
Worksheet (Link)

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Differentiation

Remediation

Extension/Enrichment

For math learners: Use color-coded slope changes to help identify patterns.�

For ELL/struggling readers: Provide simplified definitions or sentence starters for analysis questions.�

For advanced learners: Challenge them to predict population trends under modified conditions using mathematical models or spreadsheet formulas.

Optional Extensions

Add a statistics mini-lesson on standard deviation or trendlines.�
Connect to real-world wildlife management case studies (e.g., wolves in Yellowstone).�
Tie into AP Bio content: logistic vs. exponential growth, r/K selection.