the cycloid: from geometry to calculus · 2015-04-28 · the cycloid: from geometry to calculus...
TRANSCRIPT
![Page 1: The Cycloid: From Geometry to Calculus · 2015-04-28 · The Cycloid: From Geometry to Calculus NCTM National Conference, April 2015 Maria Hernandez, NC School of Science and Math](https://reader030.vdocument.in/reader030/viewer/2022040415/5f2d7802e4def4399112937b/html5/thumbnails/1.jpg)
The Cycloid: From
Geometry to Calculus
NCTM National Conference, April 2015
Maria Hernandez, NC School of Science and Math
Taylor Gibson, NC School of Science and Math
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● Introduce the project and parametric equations
● Demo data collection via LoggerPro
● Create models for the collected data
● Evaluate models
● Explore the geometric representations
● Share resources to extend the project
Goals for Our Session
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The Cycloid
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Intro to Parametric Models
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Creating Models From Data
Logger Pro is a tool to capture data from a video.
Our data will be shared in other forms for your use.
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Data Collected from Video
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Below we see the data (time, vertical position of dot)
Vertical Model
2( ) 5cos 5
1.21v t t
What is the contextual meaning of the constants?
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Below we see the data (time, horizontal position of dot)
Horizontal Model
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𝑥 𝑡 = 25.5𝑡
Linear Model and Residuals for Linear Model
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Residuals for Linear Model and Model for Residuals
2( ) 4sin 4
1.21R t t
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What are some possible choices?
Putting the Model Together
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What do the constants mean in context?
Summing the Models
2( ) 25.5 4sin
1.21h t t t
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Final Model for Position of Dot
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• What are the strengths and weaknesses
of our models?
• How can evaluating our models help
students make connections between the
mathematics and the physical
phenomenon?
Evaluating the Models
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Modeling With Geometry
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Modeling with Geometry
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Modeling with Geometry
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Help me determine 𝑠, the distance rolled
Geometric Model
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Modeling with Geometry
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𝑥 𝛼 = 𝑟𝛼 − 𝑟 sin 𝛼𝑦 𝛼 = 𝑟 − 𝑟 cos 𝛼
Final Geometric Model
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"One of my favorite aspects of this project was that it was (with your input
and guidance when we were stuck) a self-discovery." - Thomas
"I think the coolest part of the cycloid project was that it made you think
differently than normal; My first thought would have never been a
combination of a cosine and linear function. I didn't even know that was a
thing until this project." - Vishwa
"Hopefully, in the future, we'll be able to do a few more independent
modeling investigations." -Jules
"I felt like I really wanted to learn this answer in this lab because it was
something new that I got to discover by myself. Overall, the cycloid lab was
very interesting and a very fun and inventive way to introduce the class into
combining equations." - Matthew
Student Reflections
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• Modeling Cycle - How does the project
fit into a focus on math modeling?
• What’s powerful about it for students?
• What’s powerful about it for teachers?
Questions to Consider
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• Length of path for an arch o 4∙diameter
• Speed of the dot (vectors)
• Area under an archo 3 ∙ (Area of circle)
• Tangent line through doto Passes through top of circle
• Epicycloids/Hypocycloids
Calculus and Geometry Extensions
By Sam Derbyshire (Original file got from WP:EN) [GFDL (http://www.gnu.org/copyleft/fdl.html) or
CC-BY-SA-3.0 (http://creativecommons.org/licenses/by-sa/3.0/)], via Wikimedia Commons
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• NCSSM Swing Lab Materialshttp://www.dlt.ncssm.edu/stem/content/swing-lab-documents
• Cycloid Materials from TCM Conferencehttp://www.ncssm.edu/courses/math/tcm/TCM2014/talks/
• Inspirations Video by Cristobal Vila https://www.youtube.com/watch?v=oVthC6neqVc
References and Resources
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1. Bridges Conference: Mathematics, Music, Art,
Architecture and CultureBaltimore, MD July 29 - Aug 1, 2015
2. Anya Greer Math, Science and Technology ConferencePhillips Exeter Academy, June 21 - 26, 2015
3. MAA/PREP Teaching Mathematical Modeling as
Creating Mathematical DiscoveryLincoln, NB July 20 - 25
Conferences
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The NC School of Science and MathematicsDurham, NC, January 29 – 30, 2016
http://www.ncssm.edu/courses/math/tcm/TCM2015/
NCSSM Teaching Contemporary
Mathematics Conference
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Questions?
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http://www.oshr.nc.gov/jobs/• Search Job Positions
• Departments: NCSSM
Open Position @ NCSSM