Exponential Growth and Logarithms: Populations and Earthquakes
Created byGeorge Mauser
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Exponential Growth and Logarithms: Populations and Earthquakes

Grade 11Math2 days
In this project, eleventh-grade students explore the application of exponential and logarithmic functions to understand real-world phenomena such as population growth and earthquake magnitudes. Through activities like creating an escape room scenario and utilizing earthquake simulators, students model and analyze these phenomena using mathematical concepts. The project emphasizes understanding and applying mathematical properties, graphing functions, and evaluating models' limitations, allowing students to gain a practical and theoretical grasp of these functions in real-world contexts.
Exponential FunctionsLogarithmsPopulation GrowthEarthquake MagnitudeMathematical ModelingReal-World ApplicationsGraphing Functions
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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we effectively use exponential and logarithmic functions to model and understand real-world phenomena such as population growth and earthquake magnitude?

Essential Questions

Supporting questions that break down major concepts.
  • What are exponential and logarithmic functions, and how do they relate to real-world phenomena like population growth and earthquakes?
  • How can we model population growth using exponential functions?
  • In what ways do logarithmic scales help us understand the magnitude of earthquakes?
  • What are the mathematical properties of exponential and logarithmic functions that make them suitable for modeling real-world situations?
  • How do changes in parameters of exponential functions affect the growth rate in a population study?
  • How can we use data to fit an exponential or logarithmic model in the context of real-world scenarios?
  • What are the limitations of using exponential and logarithmic models in predicting future trends in population growth or earthquake strengths?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:
  • Understand and apply exponential and logarithmic functions to model real-world phenomena such as population growth and earthquake magnitudes.
  • Analyze the impact of changing parameters within exponential functions on growth rates in population models.
  • Use logarithmic scales to interpret earthquake magnitude and comprehend their impact.
  • Graph and interpret exponential and logarithmic functions to explore their properties and implications in real-world contexts.
  • Evaluate the accuracy and limitations of using exponential and logarithmic models to predict future trends.

Common Core Standards

CCSS.MATH.CONTENT.HSF.IF.C.8.B
Primary
Use the properties of exponents to interpret expressions for exponential functions in terms of a context.Reason: This standard aligns with the project as students need to understand and use properties of exponential functions to model real-world phenomena such as population growth.
CCSS.MATH.CONTENT.HSF.LE.A.2
Primary
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs.Reason: Students will construct exponential functions to model population growth scenarios, which fits directly with this standard.
CCSS.MATH.CONTENT.HSF.LE.A.4
Primary
For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.Reason: As the project involves using logarithmic functions to understand earthquake magnitudes, this standard is highly relevant.
CCSS.MATH.CONTENT.HSF.IF.C.7.E
Secondary
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.Reason: Graphing is a key skill in visualizing and understanding the models students create during the project.
CCSS.MATH.CONTENT.HSS.ID.C.7
Supporting
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.Reason: Although focused more on linear models, understanding slope and intercept is fundamental to making comparisons and interpretations in exponential contexts.

Entry Events

Events that will be used to introduce the project to students

Escape Room: The Population Crisis

Create an escape room scenario where each puzzle solved with exponential and logarithmic equations brings students closer to solving a global population crisis. This hands-on and collaborative challenge encourages critical thinking and application of mathematical knowledge within engaging, real-world contexts.

Earthquake Simulator Challenge

Set up an earthquake simulation in the classroom, where students can use mathematical models to predict and analyze the impact of different magnitudes on a mock city. Ask them to use exponential and logarithmic functions to calculate the forces involved and explore how these natural phenomena relate to real-world data and safety engineering challenges.
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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

Logarithmic Earthquake Analysis

Students delve into logarithmic functions to understand and interpret earthquake magnitudes using real earthquake data.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Discuss the Richter scale and how logarithmic relationships convey magnitude differences between small and large earthquakes.
2. Guide students through expressing numbers as logarithms using base 10 to explore earthquake energy.
3. Provide datasets of earthquake magnitudes and have students interpret these using their understanding of logarithmic functions.

Final Product

What students will submit as the final product of the activityA detailed report interpreting the magnitude and energy differences of historical earthquakes using logarithmic functions.

Alignment

How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.HSF.LE.A.4 by using logarithms to understand exponential models, such as the Richter scale for earthquakes.
Activity 2

Graph Mastery Quest

Students will graph exponential and logarithmic functions to capture their behavior, focusing on intercepts and slopes to convey patterns.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Introduce students to graphing software (such as Desmos) to plot exponential and logarithmic functions.
2. Guide students in identifying key graph features, such as intercepts and asymptotes, using software tools.
3. Challenge students to graph their own population growth or earthquake magnitude models based on prior activities.

Final Product

What students will submit as the final product of the activityGraph plots that illustrate exponential growth and logarithmic decay, highlighting key features like intercepts.

Alignment

How this activity aligns with the learning objectives & standardsCovers CCSS.MATH.CONTENT.HSF.IF.C.7.E as students graph and analyze exponential and logarithmic functions to understand end behavior.
Activity 3

Exploring Exponential Relationships

Students explore how exponential functions can be used to model population growth. They will create foundational knowledge of exponents through interactive activities.

Steps

Here is some basic scaffolding to help students complete the activity.
1. Introduce the concept of exponential growth with real-world examples like bacteria growth or financial interest.
2. Engage students in calculating population growth using a simple exponential formula: P = P_0 * (1+r)^t.
3. Lead students to analyze how changes in the growth rate 'r' affect the overall population growth over time.

Final Product

What students will submit as the final product of the activityA worksheet with calculations and explanations of exponential population growth models.

Alignment

How this activity aligns with the learning objectives & standardsAligns with CCSS.MATH.CONTENT.HSF.LE.A.2 as students construct exponential functions to model population growth scenarios.
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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Exponential and Logarithmic Modeling Rubric

Category 1

Mathematical Understanding

Evaluates students' comprehension and application of exponential and logarithmic functions in modeling real-world scenarios.
Criterion 1

Application of Exponential Functions

Assesses how well students use exponential functions to model and explain population growth scenarios.

Exemplary
4 Points

Demonstrates advanced application, accurately using exponential functions to model complex population scenarios, and provides insightful analysis of parameter changes.

Proficient
3 Points

Accurately applies exponential functions to model population growth with clear explanation and consistent calculations.

Developing
2 Points

Uses exponential functions with partial accuracy, showing understanding of basic principles but lacks depth in explanation.

Beginning
1 Points

Struggles to apply exponential functions correctly, with incomplete or inaccurate calculations and explanations.

Criterion 2

Interpretation of Logarithmic Functions

Measures students' ability to use logarithmic functions to analyze and interpret earthquake magnitudes.

Exemplary
4 Points

Provides detailed and accurate interpretation of earthquake data using logarithmic functions, demonstrating sophisticated understanding of scale.

Proficient
3 Points

Correctly analyzes earthquake magnitudes using logarithmic functions, with clear and consistent interpretations.

Developing
2 Points

Shows basic interpretation skills, using logarithmic functions with occasional errors or incomplete analysis.

Beginning
1 Points

Struggles with interpreting earthquake magnitudes, showing limited understanding of logarithmic functions.

Category 2

Graphical Representation

Focuses on graphing skills and the interpretation of exponential and logarithmic functions.
Criterion 1

Graphing Accuracy and Clarity

Evaluates the precision and clarity of students' graph-based representations of mathematical models.

Exemplary
4 Points

Graphs are precisely plotted with clear labeling, accurately reflecting functions' behavior through intercepts and asymptotes.

Proficient
3 Points

Generates clear and accurate graphs, correctly interpreting features like intercepts and asymptotes.

Developing
2 Points

Produces graphs with some inaccuracies or unclear elements, missing some critical features.

Beginning
1 Points

Graphs are incomplete or inaccurate, lacking necessary elements to represent functions correctly.

Category 3

Model Evaluation and Limitations

Assesses students' ability to evaluate the effectiveness and limitations of exponential and logarithmic models in predicting real-world phenomena.
Criterion 1

Critical Evaluation of Models

Judges students' ability to assess model accuracy and limitations effectively.

Exemplary
4 Points

Provides in-depth evaluation of model effectiveness, critically analyzing limitations with comprehensive reasoning.

Proficient
3 Points

Effectively evaluates model limitations and potential, with clear reasoning supported by evidence.

Developing
2 Points

Recognizes some model limitations, but analysis lacks depth or is occasionally unsupported.

Beginning
1 Points

Limited evaluation of models, with minimal recognition of limitations or inaccuracies.

Reflection Prompts

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

How has your understanding of exponential and logarithmic functions evolved through this project, especially in relation to real-world phenomena like population growth and earthquakes?

Text
Required
Question 2

On a scale of 1 to 5, how confident do you feel about using exponential functions to model population growth now compared to before the project?

Scale
Required
Question 3

What were the key challenges you faced in using logarithmic functions to interpret earthquake magnitudes, and how did you overcome them?

Text
Optional
Question 4

Of the following activities, which one did you find most beneficial in understanding exponential and logarithmic functions: Escape Room, Earthquake Simulator, Graph Mastery Quest, or Exploring Exponential Relationships?

Multiple choice
Required
Options
Escape Room: The Population Crisis
Earthquake Simulator Challenge
Graph Mastery Quest
Exploring Exponential Relationships