Transcript Powerpoint - College of Education

Mathematical Modeling: Tools to
Link CCSSM Content & Practices
to Science Standards
SCCTM Annual Conference
Greenville, SC
October 26, 2013
Ed Dickey
College of Education
Plan
• What is mathematical modeling?
• How does apply to Common Core
Mathematics and Next Generation Science?
• Why do we care about modeling?
• What tools help us create mathematical
models?
• How might we teach mathematical modeling?
Common Core Standards
• Sponsored by the Council of Chief State School
Officers (CCSS) and the National Governors
Association (NGA)
• First significant attempt to systematically align
K-12 standards across the U.S.
• Building on NCTM’s standards documents from
1980, 1989, 2000, 2006, and 2009
• NCTM among groups providing feedback
Common Core Standards
• Different from most current state standards
• Based on most recent research regarding students’
learning trajectories related to mathematics content
• Includes detailed description of the way
mathematics is learned and used by students
(Mathematical Practice)
Common Core Standards
• Initially 48 states and three territories
signed on
• Final Standards released June 2, 2010, at
www.corestandards.org
• Adoption required for Race to the Top
funds
• As of October 2013, 45 states have
officially adopted (plus DC, 4 territories &
Dept of Defense Schools)
South Carolina
• State Board of Education adopted the Common Core for
SC on July 14, 2010
• In November 2010, Mick Zais was elected
Superintendent of Education and with Governor Haley
chose not to apply for Race to the Top funds
• In February 2012, the SC Board of Education approved
joining the Smarter Balanced Assessment Consortium
(www.smarterbalanced.org) of which SC is now a
governing state.
South Carolina
• SC Department of Education
ed.sc.gov/agency/pr/standards-andcurriculum/South_Carolina_Common_Core.cfm
• We are in a Bridge Year (2013-2014) in which CCSSM
is being used for instructional purposes.
• 2014-2015 will be the Full Implementation of CSSSM
and SBAC assessment
• BUT… in an August 6, 2013, letter to the EOC Chair,
Superintendent Zais wrote…
• Since Sept 9, 2013, the EOC has been considering ACT
as alternative to SBAC
CCSSM Mathematical Practices
• Common Core includes a set of Standards
for Mathematical Practice that all teachers
should develop in their students.
• Similar to NCTM’s Mathematical Processes
from the Principles and Standards for
School Mathematics.
• Mathematics Proficiencies from the National
Research Council report Adding It Up
• Practices MUST be assessed
Next Generation Science Standards:
Science and Engineering Practices
• “… behaviors that scientists engage in as they
investigate and build models and theories about the
natural world.
• “… to better explain and extend what is meant by
‘inquiry’ in science and the range of cognitive, social, and
physical practices that it requires.
• … behaviors that engineers engage in as they apply
science and mathematics to design solutions to problems.”
Importance of
Mathematical Practices
• https://www.youtube.com/watch?v=m1rxkW8ucAI
8 CCSSM Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
8 CCSSM Mathematical Practices
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning.
8 Science and Engineering Practices
1. Asking Questions (for science) and Defining
Problems (for engineering)
2. Developing and Using Models
3. Planning and Carrying Out Investigations
4. Analyzing and Interpreting Data
8 Science and Engineering Practices
5. Using Mathematical and Computational Thinking
6. Constructing Explanations (for science and
Designing Solutions (for engineering)
7. Engaging in Argument from Evidence
8. Obtaining, Evaluating, and Communicating
Information
Graphic Organizer
Reasoning and
Explaining
Overarching
habits of mind
of a productive
mathematical
thinker
Modeling and
using tools
Seeing structure
and generalizing
From Bill McCallum:
http://commoncoretools.files.wordpress.com/2011/03/practices.pdf
Graphic Organizer
Overarching
habits of mind
of a productive
scientist or
engineer
Mathematical Modeling
•
Dr. Christian Hirsch, Western Michigan Univ.
Science and Engineering Practices
FoxTrot
Mathematical Modeling
• is a description of a system using mathematical concepts
and language. The process of developing a
mathematical model is called mathematical modeling.
(from http://www.answers.com/topic/mathematicalmodel ).
• Natural sciences: physics (theories expressed using
mathematical models, Newton, Einstein)
• Business and engineering use models to examine output
based on input variables: decision variables, state
variables, exogenous variables, and random variables
Types of Models
• Theoretical vs Data
• Linear vs. nonlinear (algebra through precalculus)
• Deterministic vs probabilistic (stochastic, discrete math,
systems, and probability)
• Static vs dynamic: static doesn’t account for time
(difference equations and differential equations)
• Discrete vs. continuous
Tutorial
• http://www.causascientia.org/math_stat/Tutorial.pdf
• Kepler and the motion of the planets
• Predicting the Stock Market
• Data = Information + Error
Two Approaches to Construct a Model:
1. Stochastic: outcome based on random variable
2. Deterministic: outcome precisely determined through
relatationship among states or event.
Sample Models
• +plus Magazine.. Internet magazine from the UK
• http://plus.maths.org/content/os/issue44/package/index
• Over 60 articles with classroom ideas for modeling
Smarter Balanced
• From Item Preview: http://bit.ly/16aWVjI
PARCC
• From Item Preview:
http://ccsstoolbox.agilemind.com/parcc/highschool_3.html
School of Hard Sums
• British Game Show… pits brains vs brawn and students
of math
• http://www.youtube.com/watch?v=iNJoQwMnCMc
Romeo to Juliet over 2 Rivers
• Quickest way for Romeo to reach Juliet but bridges must
be perpendicular to rivers
Graphing Stories
• http://www.graphingstories.com/
Try one….
• Height of Waist Off Ground (by Adam Poetzel)
Three Acts for Modeling
• https://docs.google.com/spreadsheet/pub?key=0AjIqyK
M9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE&
output=html
Bolt Running
Bolt
Act One
1. Guess how many miles per hour he's running.
2. Write a guess that's too high. Too low.
Act Two
3. What information is important to know here? (The
distance of the race. Bolt's time.)
4. Find out kilometers per hour.
Act Three Sequel
5. If Bolt ran 1,000 meters, what would happen to his
speed?
NFL Bobby Gill
• 25 mph Treadmill
What to LOOK FOR
in Lessons on Modeling
• Determine equation or function that represents the
situation
• Illustrate mathematical relationships using diagrams,
tables, graphs, flowcharts, or formulas
• Apply assumptions to make a problem simpler
• Check to see whether an answer makes sense within the
context of a situation
• Change a model when necessary
Source: Mathematics Coaching by Bay-Williams, McGatha, Kobett & Wray
Dare Devil Problem
• A Dare Devil wants to jump 16 buses, find the best speed
and ramp angle
Dare Devil Problem
• Computer Algebra for Symbolic Model
• Assumptions:
– each bus 10 ft wide
– identical ramps to rise up on one side and
– land (safely) on other side.
• Variables:
– Velocity of motorcyle, v, in mph
– Angle of ramp, θ, in degrees
– Time, t, in seconds
– Position parameters, x and y, in feet
TI Nspire CAS
Dare Devil Problem
• Spreadsheet for finding solutions
– Does a spreadsheet evaluate trig functions using
degrees or radians?
– Should the problem use degrees or radian measure?
Over the Hill
• Students determine locations on a hillside for a cell phone
tower erected to provide a signal to people on the other
side of the hill.
• They identify necessary information, represent the
problem with a scale model, and answer questions in
context.
Over the Hill
• What information is needed?
• Think algebraically or geometrically
• How can you mathematize the problem?
Over the Hill
• Task Guide and Student Activity Sheet available
from NCTM
• http://www.nctm.org/uploadedFiles/Journals_and_Books
/Books/FHSM/RSM-Task/RSM_OverTheHill.pdf
Over the Hill Applet
• GeoGebra Applet also available at
• http://mathrsm.net/applets/hill/index.html
Computer Simulation as Modeling
•
•
•
•
•
•
Must ALL models be based on equations or functions?
Can a computer simulation serve as a model?
https://en.wikipedia.org/wiki/Computer_simulation
Bret Victor “Kill Math” series http://worrydream.com/
http://worrydream.com/#!/SimulationAsAPracticalTool
Can we create useful mathematical models without
algebra symbols, equations, and formulas?
Problem from a high school text
A skateboarder holds on to the merry-goround pictured to the right. The platform of
the merry-go-round has a 7-foot radius
and makes a complete turn every 6
seconds. The skateboarder lets go at the 2
o'clock position in the picture, at which time
she is 30 feet from the padded wall. How
long will it take the skateboarder to hit the
wall?
Traditional Solution
•
•
•
•
BC = 30/sqrt(3) = 10 sqrt(3) feet
AB = 20 sqrt(3) feet
Circum = 2π (7) = 43.98 ft.
So the skateboarder is traveling at
43.98 / 6 = 7.330 feet per second.
• So the time it takes the skateboarder to
reach the wall is
20 sqrt(3) / 7.330 = 4.726 seconds
Solution by Simulation
… with extensions…
2014 Institutes
PK-5, 6-8, 9-12, School Leaders
February 14-15, Orlando, FL
High School Mathematical Practices:
July xxx, ???
K-5 Number and Operations Institute
July xxx, ???
6-8 Algebra Readiness Institute:
July xxx, ???
SCCTM
Math Common Core Resources
• http://www.nctm.org/standards/mathcommoncore/
• Jennifer Bay-Williams, Maggie McGatha with Beth
Kobett and Jon Wray (2014). Mathematics Coaching:
Resources and Tools for Coaches and Leaders, K-12
Pearson Education
• http://www.pearsonhighered.com/educator/product/Ma
thematics-Coaching-Resources-and-Tools-for-Coachesand-Leaders-K12/9780133007008.page
Web Resources
• Common Core: http://www.corestandards.org
• Smarter Balanced AC: http://www.smarterbalanced.org/
– SBAC Modeling Item: http://bit.ly/16aWVjI
– PARCC Item:
http://ccsstoolbox.agilemind.com/parcc/highschool_3.html
• South Carolina Common Core: http://tinyurl.com/kg7og8t
• Superintendent Zais Letter: http://tinyurl.com/k4p2vy5
Web Resources
• Next Generation Science Standards:
http://www.nextgenscience.org/
• Mathematical Modeling:
http://www.answers.com/topic/mathematical-model
• Modeling Tutorial:
http://www.causascientia.org/math_stat/Tutorial.pdf
• +plus Magazine: http://plus.maths.org/content/os/issue44/package/index
• Graphing Stories: http://www.graphingstories.com/
• 3 Acts for Modeling: http://tinyurl.com/acdlafm
Web Resources
• Over the Hill:
http://www.nctm.org/uploadedFiles/Journals_and_Books
/Books/FHSM/RSM-Task/RSM_OverTheHill.pdf
• Over the Hill Applet:
http://mathrsm.net/applets/hill/index.html
• Bret Victor “Kill Math” Series: http://worrydream.com/
• Simulation as a Modeling Tool:
http://worrydream.com/#!/SimulationAsAPracticalTool
Video Resources
• Importance of Mathematical Practices:
https://www.youtube.com/watch?v=m1rxkW8ucAI
• School of Hard Sums:
http://www.youtube.com/watch?v=iNJoQwMnCMc
• Bolt 200m Video:
https://s3.amazonaws.com/threeacts/bolt.zip
Thank you…
6Q96
[email protected]
www.ite.sc.edu/dickey.html