Created byJulie Padgett

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Logarithm Cityscape Design

Grade 12Math2 days

The Logarithm Cityscape Design project is a mathematics-based learning experience aimed at high school seniors, focusing on the application of logarithmic scales to model city skylines. Students explore the relationship between logarithms and exponential functions by designing and visually representing cityscapes, using diverse logarithmic bases to express real-world phenomena. Through a series of activities and projects, students enhance their understanding of mathematical modeling while developing skills in decoding and representing complex structures, culminating in a presentation of their unique cityscape designs.

LogarithmsExponential FunctionsCity SkylinesMathematical ModelingReal-World DataVisual RepresentationLogarithmic Scales

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Inquiry Framework

Question Framework

Driving Question

The overarching question that guides the entire project.How can we design a city skyline model that demonstrates the application of logarithmic scales in understanding and interpreting real-world structures, and what does this reveal about the relationship between logarithms and exponential functions?

Essential Questions

Supporting questions that break down major concepts.

What are logarithms and how do they relate to exponential functions?
How can logarithmic scales be used to model real-world phenomena, such as city skylines?
In what ways do logarithms help us understand and interpret data differently compared to linear scales?
How can the properties of logarithms be applied to create a visual representation of a cityscape?
What is the significance of choosing appropriate logarithmic bases when modeling real-world data?

Standards & Learning Goals

Learning Goals

By the end of this project, students will be able to:

Students will understand the relationship between logarithms and exponential functions, including how to express models with logarithms.
Students will be able to apply logarithmic scales to model real-world phenomena, such as city skylines.
Students will visually represent cityscapes using the properties of logarithms and understand the choice of logarithmic base.
Students will develop skills in mathematical modeling, applying math to solve real-life problems such as designing a city skyline based on specified scales.

Common Core Standards

F-LE.4

Primary

For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.Reason: This standard aligns as students will use logarithmic expressions to model city skylines as exponential models, exploring heights and growth patterns.

F-IF.8b

Primary

Use the properties of exponents to interpret expressions for exponential functions.Reason: Understanding the relationship between exponents and logarithms is key to the project, aligning directly with interpreting exponential functions.

N-RN.1

Supporting

Explain how expressions with rational exponents represent the power of a number. Use this knowledge when applying logarithmic scales.Reason: Students will need to understand rational exponents which relate closely to understanding logarithms.

Common Core Standards for Mathematical Practice

MP.4

Secondary

Model with mathematics: Apply mathematics to solve problems arising in everyday life, society, and the workplace.Reason: Students will model a city using mathematical concepts like logarithms, demonstrating practical applications of abstract math.

Entry Events

Events that will be used to introduce the project to students

Unexpected Soundscapes

Students listen to a series of city soundscapes that vary based on building heights. They explore how logarithms can measure and describe the changes in perceived sound levels, linking mathematical concepts to sensory experiences.

Mysterious City Maps

Students receive a set of city maps with skyline heights marked using strange symbols. Their task is to decode these symbols using logarithms, engaging their curiosity about how these mathematical concepts can represent real-world data.

Skyline Drone Footage

Aerial drone footage of famous city skylines is shown, with an overlay challenge: "How would you mathematically model these heights?" This invites students to explore and apply logarithmic functions to understand and reproduce these towering structures.

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Portfolio Activities

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

Activity 1

Logarithm Detective: Cracking the City Map Code

Students take on the role of detectives to decipher city map symbols using their understanding of logarithms. This activity introduces students to logarithmic concepts in a real-world context, setting the foundation for scaling city skyline heights using these principles.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Receive a set of fictional city maps with marked skyline heights represented by strange symbols.

2. Research and review logarithmic basics, focusing on the significance of base 2, 10, and e, and how these relate to exponential growth.

3. Decode the symbols using logarithmic expressions, matching the decoded heights to actual city structures. Document your process for matching symbols to heights, explaining your reasoning.

Final Product

What students will submit as the final product of the activityA decoded city map with documented symbols, their meaning, and a reflection on the application of logarithms in deciphering real-world data.

Alignment

How this activity aligns with the learning objectives & standardsAligns with F-LE.4 as students express models using logarithms; F-IF.8b as students explore the relationship between exponents and logarithms.

Activity 2

Skyline Architect: Constructing Exponential Growth

Students become architects to construct a visual model of a city skyline using exponential and logarithmic functions. This ensures comprehension of the relationship between these functions and prepares students for their final cityscape creation.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Utilize graphing technology to explore exponential models and express these as logarithms, reinforcing understanding.

2. Choose a city skyline from a given set and calculate each building's height using logarithmic functions, considering different bases.

3. Translate these logarithmic calculations into a scaled cityscape drawing, using graph paper or digital design tools.

Final Product

What students will submit as the final product of the activityA city skyline drawing that represents an accurate mathematical model of exponential heights, with annotations explaining the choice of bases and logarithmic applications.

Alignment

How this activity aligns with the learning objectives & standardsAligns with F-LE.4 in using technology to evaluate logarithms; MP.4 as a real-world model is constructed using mathematical concepts.

Activity 3

Cityscape Visionary: Designing Your Logarithmic City

Students act as city designers to create their unique cityscapes using logarithmic scales. This activity synthesizes all previously learned concepts, culminating in a comprehensive project that demonstrates their understanding of logarithmic applications.

Steps

Here is some basic scaffolding to help students complete the activity.

1. Design an original city skyline using logarithmic scales, determining the base that best fits your vision.

2. Ensure the cityscape reflects various scales and heights, accurately modeled with logarithmic functions.

3. Present your cityscape, discussing your choice of logarithmic bases and how they visually affected your model.

Final Product

What students will submit as the final product of the activityA designed cityscape model presented as a physical model or digital representation, with an accompanying presentation on logarithmic application and base choices.

Alignment

How this activity aligns with the learning objectives & standardsCombines F-LE.4, F-IF.8b, and MP.4 standards as students apply logarithmic scales in design, interpret exponential functions, and model in a real-world context.

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Rubric & Reflection

Portfolio Rubric

Grading criteria for assessing the overall project portfolio

Logarithmic Cityscape Rubric

Category 1

Mathematical Understanding

Evaluates the student's grasp of logarithmic concepts and their relationship with exponential functions, including identifying and applying correct formulas and bases.

Criterion 1

Application of Logarithms

Assesses how well the student applies logarithmic principles to decode and model city structures.

Exemplary

4 Points

Demonstrates a sophisticated understanding by accurately and innovatively applying logarithmic principles to complex city models.

Proficient

3 Points

Shows thorough understanding and appropriate application of logarithmic principles to city models with minor inaccuracies.

Developing

2 Points

Displays emerging understanding with inconsistent or partial application of logarithmic principles to city models.

Beginning

1 Points

Displays minimal understanding, struggling with correct application of logarithmic principles to city models.

Criterion 2

Mathematical Representation

Assesses the quality of mathematical representations, calculations, and scaling used to depict city skylines.

Exemplary

4 Points

Produces outstanding quality models with accurate calculations and scaling that clearly illustrate mathematical concepts.

Proficient

3 Points

Produces quality models with generally accurate calculations and scaling, illustrating mathematical concepts effectively.

Developing

2 Points

Produces models with varying degrees of accuracy in calculations and scaling, illustrating some mathematical concepts.

Beginning

1 Points

Produces incomplete models with inaccurate calculations and scaling, struggling to illustrate mathematical concepts.

Category 2

Creative Design and Innovation

Evaluates creativity and innovation in the design of the city skyline, and how well the mathematical concepts are integrated into the visual model.

Criterion 1

Originality in Design

Assesses the originality and creativity of the cityscape design, including the use of logarithmic principles to enhance the design.

Exemplary

4 Points

Shows exceptional creativity and originality, designing a cityscape that innovatively incorporates logarithmic principles.

Proficient

3 Points

Demonstrates creativity and originality, designing a visually appealing cityscape that incorporates logarithmic principles proficiently.

Developing

2 Points

Shows some creativity and originality, but designs a cityscape with limited integration of logarithmic principles.

Beginning

1 Points

Shows minimal creativity and originality, struggling to integrate logarithmic principles in the cityscape design.

Criterion 2

Integration and Complexity

Assesses integration and complexity of design elements using logarithmic scales and mathematical modeling.

Exemplary

4 Points

Skillfully integrates complex design elements using logarithmic scales, creating a cohesive and sophisticated skyline.

Proficient

3 Points

Successfully integrates design elements using logarithmic scales, creating a cohesive and well-organized skyline.

Developing

2 Points

Inconsistently integrates design elements using logarithmic scales, resulting in a less cohesive skyline.

Beginning

1 Points

Struggles to integrate design elements using logarithmic scales, resulting in an incoherent skyline.

Category 3

Communication and Presentation

Evaluates the effectiveness of the student's presentation and communication of mathematical ideas and design elements.

Criterion 1

Presentation Clarity

Assesses the clarity and coherence of the student's presentation, including the explanation of mathematical concepts and design choices.

Exemplary

4 Points

Presents information with exceptional clarity and coherence, expertly explaining mathematical concepts and design choices.

Proficient

3 Points

Presents information clearly and coherently, effectively explaining mathematical concepts and design choices.

Developing

2 Points

Presents information with some clarity, but struggles to effectively explain all mathematical concepts and design choices.

Beginning

1 Points

Presents information with minimal clarity, struggling to explain mathematical concepts and design choices.

Criterion 2

Audience Engagement

Evaluates the ability to engage the audience and effectively communicate complex ideas, enhancing understanding and interest.

Exemplary

4 Points

Engages the audience exceptionally, effectively communicating complex ideas and enhancing understanding and interest.

Proficient

3 Points

Engages the audience well, effectively communicating ideas and enhancing understanding and interest.

Developing

2 Points

Engages the audience at times, but struggles to communicate complex ideas effectively, limiting understanding.

Beginning

1 Points

Struggles to engage the audience and communicate complex ideas, hindering understanding.

Reflection Prompts

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

Question 1

Reflect on how understanding logarithms and exponential functions has influenced your ability to design and model real-world structures. What insights have you gained?

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Required

Question 2

On a scale from 1 to 5, how would you rate your confidence in applying logarithmic scales to model real-world phenomena after completing this project?

Scale

Required

Question 3

Which activity did you find most beneficial in helping you understand the application of logarithms to real-world problems, and why?

Multiple choice

Required

Options

Logarithm Detective: Cracking the City Map Code

Skyline Architect: Constructing Exponential Growth

Cityscape Visionary: Designing Your Logarithmic City

Question 4

What challenges did you face when designing your cityscape using logarithmic scales, and how did you overcome them?

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Optional

Question 5

In what ways did this project alter your perception of mathematics and its role in interpreting real-world data?

Text

Required