726 AM Posters
Rate of Change
Strategies that Work:
 Use familiar concepts (e.g. speed, human growth) to start
 Emphasize proper units/magnitude
 Move from verbal to data table to plot to equation
 Move from linear to nonlinear examples
 Involve students in active simulations (coinflipping, dice rolling, etc.)
 Compare plots of distance, speed, acceleration
Examples in a Geoscience context:
 Determine average plate motion by measuring distance between two Hawaiian islands of known age. Determine average for several pairs of islands and discuss whether plate motion is a constant rate.
 Determine the rate of retreat of St. Anthony Falls based on location in historical record and location today on topographic maps. Assuming retreat began at the confluence of the Minnesota and Mississippi Rivers and that the glacier was present at this location, and a constant rate of retreat, calculate when the glacier began to retreat from the Minneapolis area.</span>
 Illustrate halflives and exponential decay by giving each student a coin. All flip coins, if a student has "heads" they are not "decayed" and can flip again. Count students with "heads," flip again, and repeat. Data can be plotted, etc.
Remote Sensing Examples
Matrices and Eigen vectors:
 Question: Why? Is it worth the math?
 Strategy that works: Principle components (also teaches covariance)
 See diagram on poster
 Challenge: Getting math poor/phobic/atrophied students to appreciate it
Systems of equations:
 Strategy that works: Endmember analysis/classification
 (Spectral unmixing)
 See diagram/example on poster
 Challenges: undersolving and oversolving
 Simultaneous equations
Vectors
Jordan, Hutchings, Hutchings
Focus: Using vectors for navigation and mapping
Strategies that work:
 Don't be abstract  Go outdoors and take measurements of distance between real objects and establish direction
 Use a magnetic compass (quality will determine accuracy and precision) and your pace or tape measure.
 Use protractor and scale to draw a representation of real situation. (Add points to map as a function of precision  users)
 Triangulate, triangulate, triangulate OR trilaterate, trilaterate, trilaterate volcano, your house (or any other spatially distributed feature)
Graphical Literacy
Moe M., Carol D., Janet A., Cathy S.
Strategies that work:
 Decide
Approximation
Kohn, Kroeger, Macdonald, ShellGellasch
Reasonableness
 Order of magnitude
 Realistic (range of possible values)
 Level of accuracy needed
 Level of accuracy possible
 Conditions
 Audience
 Methods
Evaluation and refinement
Example: Approximate rainfall within a drainage basin.
 Purpose: determining water budget for a given area
 Given: rain gauge data
 Implementation
 Contour maps
 Statistical averaging
 Weighted average
 Histogram (1 and 2 dimensional)
Data Analysis/ Intro. Statistics
Carolyn Dobler, Chris Gellash, Patty Crews, Marvin Bennett and Rick Ford
Critical steps/strategies in teaching this concept:
 What is the problem / research question?
 Which data will be useful in answering the question?
 Issues related to data collection:
 Sampling strategies / constraints
 Sample size
 Representative samples
 Samples over time (time series)
 Data quality  accuracy and precision;
 Quality assurance and control
 Nature of the phenomena being studied (geoscience context)
 Data description
 Averages, mean, mode, median, variance, etc.
 Weighted mean  important in many geoscience contexts
 Correlations / regression
 Graphs
 Drawing conclusions from our data (statistical inference)
 Estimation, hypothesis testing, prediction
Units and Significant Digits
Check errors in work using significant digits
Motivate significant digits from error analysis
Convert units
Dimensional analysis
Strategies that work
 Order of magnitude reasoning to check work
 Rivers on earth do not flow at 500mph
 Hailstones do not grow to the size of basketballs
 Example showing how errors propagate (significant digits)

Q=Av
A= 2.3 +/ 10^{2} m^{2}
v =1.5 +/ 10^{2} m/s
(10^{2} represents uncertainty)
Q= (2.3 +/ 10^{2})(1.5 +/ 10^{2}) m3 / s = 3.45 +/ .015 +/ .023 +/ 10^{4} m3 / s
=3.4 (must throw out the 5)  70 miles/hour * 1 hour/3600 seconds * 1.6 km/1 mile * 1000m/1 km = 31m/s
 See diagram on chart
Identify important parameters (g, t, z)
Identify units ( L / T2, T, L)
Laws of the universe don't care about units:
Construct nondimensional group: z / (g * t2) = constant
z=cgt2
Multiply by a "convenient one" to convert units
Using dimensional analysis to solve simple problems
Example in a geoscience context:

(a) Use dimensional analysis to obtain the drag law for a spherical particle
dropped through a viscous fluid (water or magma)
(b) Conduct physical experiments to determine the viscosity coefficient  Repeat the hydraulic example using order of magnitude reasoning:
Q= Av: A is order 1, v is order 1, so Q must be order 1
i.e. Q cannot be greater than 10 or less than 1
Optimization in Geology
Co, Bruce, Will
Math
 Definition:
 Min/max functions, typically misfits between data and a model
 Issues:
 Constraints
Global vs. local minima  Difficulties:
 "Word Problem"
No such thing as perfect data
Geology
 Definition:
 Finding the best earth model to explain a given set of data
 Issues:
 Realism of results
Sufficiency of model  Difficulties:
 Lack of perfect answer
Equivalence of solutions
Strategies that work:
Concrete examples
Diagrams, photos, road cuts, graphic solutions
Guidance and comfort
Example: Overdetermined 3pt. Problem
 Data: 5 points, (x_{i}, y_{i}, z_{i})
 Model: ax_{i} + by_{i} + gz_{i} = d_{i}

Minimize: (from i = 1 to 5) ==> S [z_{i}  ((d_{i}  ax_{i}  by_{i}) / g)]z
NB: system is unsolvable (singular matriatic) if five points are Colinear
NB2: Measurements errors add uncertainty

Examples [to what?]
Problem: Are containments leaking from an underground storage tank?
 Data:
 Nature of phenomenon  soil and water
Variables to measure: hydrocarbon, lead  Data collection issues:
 Sampling strategy:
 Sample size  number and placement of wells
 Sample design  multistage to determine ground water flow direction
 Place additional wells in down gradient direction
 Background sample
 Data quality:
 Data collection  field collection protocol
 Chain of custody
 Certified lab quality
 Data description:
 Plot data on map
Inspect for variability  Compare concentrations with background value to define presence of a plume
 Draw contours
 Data Conclusions:
 If values are unusal, are they significant?
If values are not unusual, how confident are you that the site is clean?
Graphical Literacy
Strategies that work:
 Decide upon what type of graphs you want students to work with:
(bar and scatterplot vs. [absolute?] lines  drawn on poster  Be clear about how students should work with graphs
 simply read information
 find slopes and calculate percent change
 fit a line to a "bunch of points" (i.e. data)
 interpret and predict (requires use of symbols)
 differentiate between different types of graphs and define behavior of system (is it linear or exponential? Power function made linear on log graph)
 understand scatter and error
Examples in Geoscience:
 Exponential:
 Number of aftershocks / time
Population Growth</br> Energy consumption
Atmospheric carbon dioxide concentrations
Draw down in well
Land subsidence
Radiometric decay  Linear:
 Traveltime of seismic waves (approximately)
Topographic slope (gradient)
Watertable slope
Miscellaneous: Bowen's reaction series
Sealevel curves
Climatic data/variation
Pressuretemperature curves
Helping Students Visualize Better in 3D
Vince Schielack, Jim Sochacki, Denyse Lemaire
Our Goals:
 Increase awareness of angle measurement;
 Help students understand how arcs of longitude decrease in length as one moves toward the poles;
 Have students ocmputer the chagnes in arcs of latitude as one nears the pole;
 Have students use Geometer's Sketch Pad
Problem:
 Students have a hard time visualizing volumes
 When studying geographic coordinates, students often get confused about angles
 Rotating meridians adds insult to injury
See diagram depicting earth with tropics, arctic circles, equator marked
3D Visualization of Earth
(Coordinates of Rotation)
Strategies: (Diagrams included for each step on original)
 Axis of Earth
 Meridian and coordinates on surface of Earth
 If a point moves in miles along the equator, how far would another point at a degrees of latitude move at the same time?

x / r = sin (90a)
x = r cos (a)
Degrees Latitude: x/r Proportion => 90:0, 80:0.1736, 70:0.3420, 60:0.5, 50:0.6428, 40:0.7660, 30:0.8660, 20:0.9397, 10:0.9848, 0:1.0
Cycles of changes
Specific topics: e.g. tides, cycles of sealevel change
Strategies: (use tide as example)
 Display measured tidal curves (60day record)
 Question: What are the tidal ranges (maximums, minimums, "average")? What is the periodicity of the tides?
 Let's take a step further than just visual observation
 Let's try to describe tides with mathematical functions

The math tool  simple: y = sin x</span>
Includes graph of y = sin x
(we got the cycle!!)
F(T) = A sin ((t * T) + f)
A = tide magnitude; t = tide frequency; T = time  the variable; f = a phase angle
Apparently, nature is more complicated than just one sine curve
What do we do?
Add several sine curves together.
Lead to: Tidal ranges vary with time. Tides vary with more than one period. Therefore, tidal cycles are quite complicated.
We need to be more quantitative
Introducing math concepts and Exercises I
Let's be a bit more "mathematical"
Let's take a step back and compare the above sine curve with "real" tides
Exercise II: Let's Predict Tides (with 6 components)
Given: T= (2*p / t)  period of the component
Create sine curves (in excel) with
F_{1}(T) =A_{1} sin (t_{1} T + f_{1})
F_{2} (T) = A_{2} sin (t_{1} T + f_{1}) – Note: only A changed
F_{3} (T) = A_{3} sin (t_{2} T + f_{1}) – Note: both A and t changed
F_{4} (T) = A_{4} sin (t_{2} T + f_{1}) – Note: only A changed
F_{5} (T) = A_{5} sin (t_{1} T + f_{2}) – Note: A and changed
F_{6} (T) = A_{6} sin (t_{3} T + f_{3}) – Note: A, t, and f all changed
Hint: same Lt
How do all your curves look?
Question: Similarities? Differences?
Now, the big moment: Add all the F(t)'s together.
Hint: make sure they are scaled the same!
You just developed a predictive tool for Tampa Bay tides!
Examples in a geoscience context: Oceanography (intro)
Sedimentary Geology (sedimentary processes)