7-26 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 non-linear examples
  • Involve students in active simulations (coin-flipping, dice rolling, etc.)
  • Compare plots of distance, speed, acceleration

Examples in a Geoscience context:

  1. 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.
  2. 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.
  3. Illustrate half-lives 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:

  1. Don't be abstract - Go outdoors and take measurements of distance between real objects and establish direction
  2. Use a magnetic compass (quality will determine accuracy and precision) and your pace or tape measure.
  3. Use protractor and scale to draw a representation of real situation. (Add points to map as a function of precision - users)
  4. 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:

  1. Decide
FINISH

Approximation

Kohn, Kroeger, Macdonald, Shell-Gellasch

Reasonableness

  • Order of magnitude
  • Realistic (range of possible values)
Purpose
  • Level of accuracy needed
  • Level of accuracy possible
Implementation
  • 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)
Evaluation: consider "outliers"

Data Analysis/ Intro. Statistics

Carolyn Dobler, Chris Gellash, Patty Crews, Marvin Bennett and Rick Ford

Critical steps/strategies in teaching this concept:

  1. What is the problem / research question?
  2. 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 m2
        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)

      Multiply by a "convenient one" to convert units
      • 70 miles/hour * 1 hour/3600 seconds * 1.6 km/1 mile * 1000m/1 km = 31m/s

      Using dimensional analysis to solve simple problems
      • 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

    Example in a geoscience context:

    1. (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
    2. 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 3-pt. Problem

    • Data: 5 points, (xi, yi, zi)

    • Model: axi + byi + gzi = di

      • Minimize: (from i = 1 to 5) ==> S [zi - ((di - axi - byi) / 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 - multi-stage 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:

    1. Decide upon what type of graphs you want students to work with:
      (bar and scatterplot vs. [absolute?] lines - drawn on poster
    2. 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 Energy consumption
    Atmospheric carbon dioxide concentrations
    Draw down in well
    Land subsidence
    Radiometric decay
    Linear:
    Travel-time of seismic waves (approximately)
    Topographic slope (gradient)
    Water-table slope
    Miscellaneous: Bowen's reaction series
    Sea-level curves
    Climatic data/variation
    Pressure-temperature curves

    Helping Students Visualize Better in 3-D

    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)

    1. Axis of Earth
    2. Meridian and coordinates on surface of Earth
    3. 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 (90-a)

      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 sea-level change

    Strategies: (use tide as example)

    • Display measured tidal curves (60-day record)
    • Question: What are the tidal ranges (maximums, minimums, "average")? What is the periodicity of the tides?

    • 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

      • Let's take a step further than just visual observation
      • Let's try to describe tides with mathematical functions
      Introducing math concepts and Exercises I

      • The math tool - simple: y = sin x
        • Includes graph of y = sin x
        • (we got the cycle!!)
        Let's be a bit more "mathematical"

        • F(T) = A sin ((t * T) + f)
        • A = tide magnitude; t = tide frequency; T = time - the variable; f = a phase angle
        Let's take a step back and compare the above sine curve with "real" tides

        • Apparently, nature is more complicated than just one sine curve
        • What do we do?
        • Add several sine curves together.

    Exercise II: Let's Predict Tides (with 6 components)

    • Given: T= (2*p / t) - period of the component
    • Create sine curves (in excel) with
      • F1(T) =A1 sin (t1 T + f1)
      • F2 (T) = A2 sin (t1 T + f1) -- Note: only A changed
      • F3 (T) = A3 sin (t2 T + f1) -- Note: both A and t changed
      • F4 (T) = A4 sin (t2 T + f1) -- Note: only A changed
      • F5 (T) = A5 sin (t1 T + f2) -- Note: A and changed
      • F6 (T) = A6 sin (t3 T + f3) -- 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)