Mathematical Modeling: Tools

to Link CCSSM Content &

Practices to Science

Standards

Ed DickeyCollege of Education

SCCTM Annual Conference

Greenville, SCOctober 26, 2013

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-and-curriculum/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

Overarching habits of mind of a productive mathematical thinker

Reasoning and Explaining

Seeing structure and generalizing

Modeling and using tools

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/mathematical-model ).

• 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 + ErrorTwo 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=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE&output=html

Bolt Running

Bolt

Act One1.Guess how many miles per hour he's running.2.Write a guess that's too high. Too low.Act Two3. What information is important to know here? (The distance of the race. Bolt's time.)4. Find out kilometers per hour.Act Three Sequel5. 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_simulatio

n

• 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-go-round 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…

SCCTM

2014 Institutes

PK-5, 6-8, 9-12, School LeadersFebruary 14-15, Orlando, FL

High School Mathematical Practices: July xxx, ???

K-5 Number and Operations InstituteJuly xxx, ???

6-8 Algebra Readiness Institute:July xxx, ???

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/Mathematics-Coaching-Resources-and-Tools-for-Coaches-and-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

ed.dickey@sc.edu

www.ite.sc.edu/dickey.html