MATH MODELING

A Fourth Year Course

North Salem Middle High School

Teaching and learning since 1985

You name it …. I probably taught it!

Been searching for ways to make mathematics meaningful, and to put the meaning into mathematics.

Ellen Falk

Problem Based Learning◦ Involvement that leads to questioning and

comprehending. ◦ Investigations and meaningful tasks◦ Construct Knowledge through meaningful tasks◦ Culminates and a real life task or problem to solve

5 E’s◦ Engage, explore, explain, elaborate, evaluate.

Hear, See, DoI forget, I

remember,I understand !

A person gathers , discovers or creates knowledge in the course of some purposeful activity set in a meaningful context.

Improve understanding.

Hear, See, Do

Provide meaning to mathematics through activities that have a real purpose-

Provide an answer to the question: When am I ever going to use this?

Solve problems in a STEM context.

Bring meaning through purposeful activities

Pose meaningful questions.

Provide the background and knowledge students will need to solve their problem.

Context

“They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has notserved its purpose.”

Rules of Engagement

FOCUS

MATH WITH MATH

M

ATH

I

N

CO

NTEX

T

LOSS of :Width, Motivation, Applications

Loss of: DepthEfficiencyElegance

Performance Tasks•Designed to reveal a learner's understanding of a problem/task and her/his mathematical approach to it.

•Can be a problem, project, or performance.

• Individual, group or class-wide exercise.

•A good performance task usually has eight characteristics (outlined by Steve Leinwand and Grant Wiggins and printed in the NCTM Mathematics Assessment book).

•Good tasks are: essential, authentic, rich, engaging, active, feasible, equitable and open.

Project Based Learning Investigations and meaningful tasks.

Construct knowledge through inquiry.

Culminates in a realistic hands –on project.

5 Es Instructional Model.

Problem: When will this particular species be delisted from endangered to threatened? Will it happen in your life time?

Exponential Functions. Model population decay and growth of the Kemp

Ridley Sea Turtle with technology. Data provided by a turtle demographer from Duke

University- Dr. Selina Heppell. Construct an internet scavenger hunt to find

details about the Kemp Ridley Sea turtle.

Kemp Ridley Sea Turtles

Using Technology

A Scatter Plot of the data Point of intersection represents the solution.

-Satellite tracking of Sea Turtles allowed students to follow the behavior of a particular turtle for as long as data was available.

-As the project evolved pieces like this were added to improve the overall experience.

-It made it real.

A Presentation from 1998

Students predicted that in the year 2013 the Kemp Ridley would be

delisted.

Starting PointsHow can I make this topic more

meaningful to students and relevant to other disciplines?

An Idea. Started with a question concerning the use of exponential functions

to study population of endangered animal species. Just thought that studying animals would be more fun than the growth of cell phones.

My Research. Extensive use of the internet led me to sea turtles and an obscure

posting on a website led me to Dr. Heppell. Great sources : www.signalsofspring.netwww.seaturtles.org

Some Issues. Students did not initially expect to be spending time in a math class

learning about a particular sea turtle as extensively as they did. And did not expect to be writing as much as they were expected to.

Problem: You and your partner are surveyors and are asked to provide an accurate survey of a plot of land of your choosing.

Geometry- Polygons, convex and concave, parallel lines, alternate interior angles.

Orienteering Using a compass to create the plot and test the region.

Trigonometry Pythagorean Theorem, Right Triangle Trig, Law of Sines and

Cosines, Area and Triangulation.

What’s Your Bearing?

Out in the Field

To test their orienteering skills, we go out into the wild!

Surveying their plot of land.

Students create their plot using a compass

and pacing off distance.

A detailed map is created with appropriate scale, surveyor bearings and area.

A great real-life application of trigonometry.

Demonstrating Understanding

Applying the trig. Reflecting on the results

So What’s the “math” Pythagorean Thm Alternate int. angles, corresponding

angles Triangle-Angle_Sum Thm Parallel lines Soh Cah Toa Law of Sines Law of Cosines Area of triangles Non right triangles-icky ones too! Measurement and measuring tools Dimensional analysis ?

Problem: Design and build a car so as to determine its acceleration using a variety of methods.

Functions Constant, Linear, Quadratic. Function notation as it

applies to physics. Technology

Authentic Data Collection, graphing calculators, motion detectors.

Physics 1-Dimensional Kinematics

SPEED RACER

Students often just

want to get to building without

thoughtful planning

…keep them on track.

Kelvin.com is a wonderful source for technology and finding cool things to build. You can get great ideas there too!

Building the Car

Variation in Design

It’s a team effort. After data is collected students decide through applying their new skills and knowledge if the data is “good” data.

The Set Up

Data Analysis

How do you know you have “good” data?

The following are from student reports.

Collecting & Analyzing DataAcceleration Graph Distance time graph Velocity time graph

Constant graph, as time increases, acceleration remained the same.

As time increases on a distance time graph, so does the distance, quadratically.

Linear graph, when time increases, velocity does also at a constant rate.

Distance Time Graph

D(T)= ½aT^2 + V0T + D0 a (lead coefficient) = acceleration V0 = initial

velocityT = time D0 = initial distance

My DataD(T)= (.31)T^2 + (-.51)T + .62

Acceleration = .62 m/s/s

Doubled lead coefficient to find this.

Velocity Time Graph

V(T) = aT + V0

a = acceleration V0 = initial velocity T = time

My Data V(T) = .63T + (-.534)Slope = .63 m/s/sAcceleration = change in

velocity/change in time

Average Acceleration

_X = ave acceleration

Constant function Average Acceleration = .62

m/s/s

Look at the next slide carefully…

What do you notice?

What do you think happened?

Unexpected Results ?

Distance(Time)D(T)= -.312T2+2.136T-.993Quadratic EquationAcceleration = a(2) = -.624 m/s

What math Do YOU see? ? ? ? ?

Identify the “math”

What is meant by mathematical modeling?

How do we construct meaningful tasks?

Real, relevant, reliable, reusable.

Where do I start?

How many do we need?

Who is your audience?

What are your topics?

Integrate STEM activities

Modify!

ELA

What Should Your Course look like?

Closing-

Mathematical Modeling can answer the age old question…

“When am I ever going to use this?”

Mathematical Modeling can generate new questions.

“Why didn’t this work?” or “ Why did this work?”

Videos Dan Meyer-math class needs a makeover. RSA Animate-Ken Robinson

Hans Rosling : Population Growth over 200 years.

David McCandless turns complex data sets (like worldwide military spending, media buzz, into beautiful, simple diagrams that tease out unseen patterns and connections.

Taylor Mali- just because

At the heart of it…