MTE 494 / 598 – Fall 2008

Technology and Mathematical Visualization

Homework

Date Due Assignment
Thursday,
August 28
 Email Introduction: Send me an email introducing yourself. Please give me some background on yourself, your academic experiences, why you have decided to become a teacher, and your expectations for this course. I would like to get to know each of you and to help shape your experiences in this class to maximize your goals as future teachers.
Thursday,
August 28		 Syllabus: Carefully read the course syllabus.
Tuesday,
September 2		 Mathematics Typesetting: Write up responses to the following four items using Equation Editor in Word or TeX. Explain in full sentences what you are doing at each step (avoid linked equations). Make sure your name is at the top.
  1. Derive the quadratic formula.
  2. Prove that √2 is irrational.
  3. Explain in detail how to use calculus to determine the volume of water in a unit sphere filled to ¼ its height.
  4. Explain in detail how to use calculus to draw an accurate graph of  by hand.
Thursday,
September 4
 Article: Read Does it compute? The relationship between educational technology and student achievement in mathematics, a 1998 report by H. Wenglinsky at the Educational Testing Service.
Thursday,
September 11
 Article: Read Pea, R. (1985). Beyond Amplification: Using the Computer to Reorganize Mental Functioning. Educational Psychologist, 20, 167-182. Write up responses to the following questions.
  1. What is the difference between a computer "as amplifier of cognition" and "as reorganizer of mental functions?" Discuss how this distinction applies to forms of technology other than computers, such as a calculator, a pen or pencil, a book, etc.?  How about to more abstract "cognitive technologies" like, written language, mathematical constructs, or even cultural practices?
  2. What are some implicit assumptions typically made about how computers are used in classrooms (e.g., why are they almost always around the outside of a room facing away from one another? From how many students do software programs typically assume, or accept, input? etc.) What are the (possibly unintended) effects of these assumptions?
  3. Pea refers to functional reorganization. To what does the word "functional" refer? Give an example of a physical tool which organizes manual labor in two different ways depending on how it is used. Second give an example of a cognitive tool which organizes mental inquiry in two different ways depending on how it is used. Finally give an example of an instructional technology which organizes classroom activity in two different ways depending on how it is used.
Thursday,
September 18		 Geometer's Sketchpad: Consider the class activities about visualizing optimization problems and tangent lines to the graph of a function. Describe the elements of covariational reasoning that you think are most relevant for students developing robust understandings of these concepts and what implications their consideration has for designing instruction. Re-design the simple sketchpad files that we created in class to incorporate these elements of covariational reasoning and describe in detail how you would engage students in a classroom activity using your new sketches.
Thursday,
September 25
 Geometer's Sketchpad:
  1. Consider the following problem:
    A square piece of paper ABCD is blue on the front side and yellow on the backside and has an area of 3 in2. Corner A is folded over to point A' which lies on the diagonal AC such that the total visible area is ½ blue and ½ yellow. How far is A' from the fold line?
    1. Construct an interactive sketchpad document to illustrate the problem allowing the user to place A' at any location on the diagonal.
    2. Generalize your sketch from Part a so that corner C can be moved to adjust the paper to be any rectangle. Make sure that the foldings appear properly for any size or shape of rectangle chosen.
    3. Solve the problem from Part a purely algebraically. Then solve it without any algebra, but using only measurements provided by your sketch.

  2. Construct a sketchpad document with an arbitrary quadrilateral PQRS (whose vertices can be moved). Construct quadrilateral LMNO whose vertices are the midpoints of the edges of PQRS. Generate, test, and prove or disprove at least two interesting properties about quadrilateral LMNO.

  3. Download the file SAS.gsp. Suppose you know that AB≅A'B', AC≅A'C' and ∠CAB≅∠C'A'B'. Using only geometric constructions in sketchpad construct a sequence of symmetries whose final result maps ΔABC onto ΔA'B'C' (i.e., you can select objects and perform any operation under the Transform and Construct menus). Explain how you know this would always work given the information provided.
Thursday, October 2
 Fathom: This How-To document has some helpful hints on function modeling in Fathom.
  1. Consider the following problem:
    Professor Thistlebush conducted an investigation to determine the number of frogs that live in a pond near the field station.  Because he could not catch all the frogs, he caught as many as he could, put a band around their left hind legs, and then put them back in the pond.  A week later he returned to the pond and again caught as many frogs as he could.  On his first trip to the pond, he caught and banded 55 frogs. On his second trip, he caught 72 frogs and 12 had bands around their leg. What is the best estimate of the frog population size?
    1. Work the problem algebraically. Describe what aspects of proportional reasoning are necessary to solve this problem.
    2. Create a Fathom simulation that has the number of frogs (cases) you found in Part a and an attribute that labels the frogs as banded or unbanded in the exact proportion implied in the problem. Have Fathom collect a sample of 72 frogs from your virtual pond and generate a summary table and histogram of the two types of frogs in this sample. Have Fathom repeat this sampling a large number of times and create a histogram of the proportion of banded frogs in the samples. Use this data to estimate a 95% confidence interval for the sample proportion.
    3. Look up the standard deviation for a sampling distribution of a proportion. Use this to compute a 95% confidence interval for the sample proportion. How close does this match the result from your simulation?
    4. Explain how you could use the results of this simulation to test whether a given data collection supports an assumption about the value of a proportion.

  2. Import the data from Yeast-1, Yeast-2, and Yeast-3 into separate Fathom documents. These files contain data collected in a Biology lab for an experiment involving a biomass (in grams) of yeast cells over 3, 6, and 9 hours of observation, respectively.
    1. Plot the data from Yeast-1. Write an equation which gives the mass of the yeast cells as a function of the number of hours passed. Your model will have some numbers in it as parameters. What do these numbers mean in the context of the yeast cells? What are their units? Superimpose a graph of your function on the graph in Fathom using sliders for the parameters and adjust the sliders until you think the model is most accurate.
    2. Plot the data from Yeast-2. What does this suggest about the way the yeast cells are growing? Write a new function that better models the data. Explain the meaning of the parameters in your new model. Superimpose a graph of your function on the data plot in Fathom using sliders for the parameters and adjust the sliders until you think the model is most accurate.
    3. Plot a logarithm of the mass vs. time and have Fathom create a least-squares line. Using the parameters from this line, what are the values of the corresponding parameters for the exponential model. Enter those values into your sliders for your exponential model and see if your can reduce the sum of the squares of the errors. What does this imply about exponential regression?
    4. Plot the data from Yeast-3. For an exponential growth model, the rate at which the mass increases is proportional to the existing mass. That is, if dm/dt is the rate of growth, then dm/dt=kt where k is a constant. Write an appropriate equation for the rate of growth that captures the data in the new data table. Find the general form of the solution to this differential equation. Explain the meaning of the parameters in your new model. Superimpose a graph of your function on the graph in Fathom using sliders for the parameters and adjust the sliders until you think the model is most accurate.
Thursday, October 9
 Fathom: This How-To document has some helpful hints on function modeling in Fathom.

Background: The data for this assignment was collected in a medical research lab on a sample of Iodine-123, a radioactive isotope used in thyroid ablation, using a mass spectrometer to sample the proportion of unstable I-123 which was then used to compute a mass. The data points give the amount of I-123 in micrograms at each hour over the course of six days. Note that I-123 decays when one of its protons “captures” an electron from an inner shell, forming a neutron and a neutrino. An electron from an outer shell then falls to the inner shell, releasing energy as x-rays. The loss of a proton-electron pair and the addition of a Neutron forms an isotope of Tellurium (Te-123).
  1. Import the data Iodine-123 into a new Fathom document. Use a table of values to get two different estimates for the half life of I-123, then use a graph to get two more estimates. Explain your method and why it works for both representations. Move to various other places of the same duration to check your results and report on what you find. Explain how the concept of a half life is related to exponential functions.
  2. After half the duration of a half-life, what proportion of the I-123 will have decayed? After the duration of two half-lives, what proportion of the I-123 will have decayed? What misconceptions do you think students are likely to have related to these questions?
  3. Determine a reasonable value for the initial amount of iodine. Express the amount of I-123 as a function of the number of half-lives that have passed since the beginning of the data collection. Express the number of half-lives as a function of time. Use composition of your functions from the previous questions to express the amount of I-123 as a function of time. Write a brief paragraph describing how the progression of these steps may help students understand the final formula you obtain. Include any thoughts on improvements to help students understand this.
  4. Add a graph of a function to your data plot using parameters (sliders) for half-life and initial amount and adjust them to get a good fit. Add appropriate units to your sliders. Use the graph of the function (not the data) to determine the constant of proportionality, k, between the rate of change and the amount of iodine. Use more than one location on the graph to verify your value. Explain your method. Using the units for rate and amount, determine the units for k.
  5. Using your value for k, add another function to the graph with the formula I0Bkt. (Note that you may have used a different name for your initial value than I0.) Explain how this formula represents repeated division by B over some duration of time. Determine that duration, explaining your method. Add a slider for B, and adjust it to get a good fit. Write a short paragraph explaining the importance of this value as the base of the exponential function.
  6. In Fathom, add a new data column to the table called “decays” that computes the mass of radioactive iodine that decays to Tellurium in each hour. Using the “Plot Value” command on your graph, enter the formula “sum(decays*time)/sum(decays)” and click “ok.” Carefully explain why this formula gives you the average lifespan of a microgram of radioactive I-123. How is this value related to the value of k? Write a third formula for the amount of iodine as a function of time that uses only the parameters of initial amount and average lifespan of a microgram.
Thursday, October 16
 Fathom: This How-To document has some helpful hints on function modeling in Fathom.
  1. Import the data Cart-Ramp into a new Fathom document. There are three data sets that we collected in class. Recall that the first ramp had height=2cm and length=94.5cm. The second ramp had height=5.5cm and length=103.8cm. The third ramp had height=11.6cm and length=100.2cm. Choose one data set and display the data in a table and on a graph. Look at the data plot and decide what time the motion started and create a new time variable whose value is zero at the start of motion. Superimpose a quadratic function on you graph with sliders for the parameters. Adjust the parameters until you get a good fit. Describe what each slider does to the graph, using the connection between the symbolic expression p(t) = at2 + bt + c and the graph.
  2. Determine the units for the three parameter based on the algebraic expression. Add units to your sliders. Given these units, hypothesize the physical meaning of these parameters. Explain. Are these meanings consistent with the effects of the parameters on the graph interpreted in terms of the physical context? Are they consistent with the numerical data in table? Explain.
  3. Have Fathom compute average velocities over each time interval. Explain whether the data in this column is consistent with your hypothesized interpretation of the parameters.
  4. Create a new plot of (average) velocity vs. time (keeping the position vs. time). Superimpose an appropriate function on this graph with sliders for the parameters and adjust them to obtain a good fit. How does changing these parameters affect the velocity graph? Determine the units for the parameters of the velocity model algebraically and add them to the sliders. How do these parameters relate to the ones generated from your position vs. time graph? Explain using multiple representations. Based on this, consolidate as many parameters as possible.
  5. Add a new variable and have Fathom compute the average acceleration over each time interval and plot acceleration vs. time. Superimpose a function on the graph with sliders for the parameters and adjust them to obtain a good fit. Do these representations of acceleration match your hypothesized interpretations of the original parameters? Again, consolidate as many parameters as possible.
  6. After consolidating parameters, you should only have three sliders, each with a particular meaning which you have explored numerically, graphically, algebraically, and contextually. If you move any of the sliders, all three graphs should change simultaneously. You should be able to adjust the sliders to get a pretty accurate model for the motion of the cart that you observed. Describe how each parameter affects each of the three graphs. Explain why changes in each graph correspond to the changes in the other graphs.
  7. On paper (not in Fathom), draw a general graph for velocity vs. time for an arbitrary constant acceleration a starting at rest at time 0. For two times, 0 and t, label v(0) and v(t). What are graphical and algebraic representations of the final velocity v(t) and the acceleration a? What is the average velocity from time 0 to time t? Base your response on the interpretation of average velocity as the constant velocity at which you would have to travel in order to realize the same change in position over the same change in time. Explain how this relates to your investigation of velocity in Fathom.
  8. In SI units, acceleration due to gravity at sea level is 9.8 m/s2 and in English units, the acceleration due to gravity is 32 ft/s2. What fraction of free fall acceleration did you observe as your cart rolled down the ramp? How does this your answer to part a compare with the ratio of the height of the ramp to the length of the ramp? Explain why this happens using only similar triangles (no trigonometry).
Thursday, October 23
 Conceptual Analysis:
  1. Re-read the Elements of Covariational Reasoning.
  2. For each of the eight questions in the previous homework assignment, discuss what aspects of covariational reasoning are provoked. Be specific and characterize the reasoning involved using all three dimensions described in the document (Mathematical Content, Representation, and Quantification).
  3. Discuss what aspects of covariational reasoning were not involved in responding to these questions but that could have been productively explored using the context of position-time data of the cart on a ramp and the technology provided.
Thursday, November 6
 Project 1 Due.
Tuesday, November 18
 Graphing Cala
culator.
  1. Download the file Vector Fields.pdf.
    1. Explain graphically whether it is possible for each of the three vector fields to represent linear maps v:R2→R2.
    2. Find formulas that could plausibly generate these vector fields.
    3. Show algebraically that your formula for the first vector field satisfies both linearity conditions.
    4. Show algebraically that your formula for the third vector field satisfies the condition v(cw)=cv(w) for all w in R2 and c in R.
    5. Demonstrate algebraically an example of vectors w1 and w2 such that v(w1+w2)≠v(w1)+v(w2).
    6. Explain the meaning of "constant rate of change"  for a linear function v:R2→R2.
  2. Generate a vector field v:R2→R2 with a zero at the origin that has +2 index. Draw a sketch of a continuous vector field on a sphere that has only one zero. (You may do this on paper or using Graphing Calculator.)
  3. Create a Graphing Calculator document starting with a function definition "f (x) =..." (so that any function f :R→R can be entered) that illustrates the meanings of amounts of change on both the function and on the tangent line at any given point leading to ideas of average and instantaneous rates of change. Create the same illustration in Geometer's Sketchpad. Write a discussion of the differences in what the two programs afford in creating this visualization.
Tuesday, December 2
 Project 2 Due. Choose a topic from secondary mathematics. Using technology, explore aspects of that topic that you have not considered previously. This may include generalization to higher dimensions, development in other number systems or algebras, exploration using multiple representations, simulations, etc. Write up an analysis of what you learned and how the technology aided your learning. Some examples of ideas to explore
  • Linearity for functions v:R2→R2 (we already did this pretty extensively in class, but there are additional things to explore such as the role of eigenvectors and eigenvalues, nullspace, )
  • The meaning of rate of change by exploring directional derivatives (again, we already explored some of this, but there is much more that could be done)
  • Determinants of matrices (change of coordinates, meaning in the Jacobian, meaning in Cramer's rule, the relationship of these things to the 1-dimensional case, etc.)
  • Optimization (multiple dimensions, linear programming, visualization and techniques for higher dimensions R2→R, R3→R, or even R4→R, relation to gradient vector fields, etc.)
  • Congruence theorems for polyhedra analogous to those for triangles and polygons
  • The chain rule (explanation and meaning in higher dimensions, conflict with the view of ratios of differentials in Leibniz notation, etc.)
  • A "multiplicative derivative" defined replacing addition (subtraction) with multiplication (division), and multiplication (division) with exponentiation (exponent to a fractional power)
  • Others: be creative and have fun!
Tuesday, December 9
 Project 3 Due.
Thursday, December  4 & Tuesday, December 9 Project 4 Presentations.
Thursday, December 11
 Final Exam. Answer any two of the following six questions. Email your responses and any auxilliary files to me by 2:00 pm on Thursday, December 11.
  1. Art and Val ran in a marathon, which is 26 miles and 385 yards. Art ran at a perfectly steady pace of eight minutes-per-mile the entire time. Val took exactly eight minutes and one second to complete every one-mile stretch (including, for example, the interval from 5.63 miles to 6.63 miles). Nevertheless, Val won the race by over one minute! Explain how this could happen. Using your choice of technology, construct a visualization that clearly illustrates that i) Art ran at a constant speed, ii) Val took exactly eight minutes and one second to complete every one-mile stretch, and iii) Val finished before Art.
  2. From the Weather webpage http://www.weather.com/weather/climatology/daily/USAK0012?climoMonth=1, download the sunrise and sunset times for each day of the year (you can only access one month at a time). Using your choice of technology, compute the number of minutes of daylight each day to the nearest minute. Develop a function model that matches this data. Explain the meaning of each variable and parameter in your model. Find values for the parameters in your model that minimize the sum of the squares of the errors. Create a residual plot and use the pattern you see here to add another component to your original model that improves its accuracy. Explain what is happening.
  3. Using the technology of your choice, create a simulation of Buffon's needle experiment in which you can control the number of needles, the length of the needles, and the separation between the lines. Use your simulation to approximate π. How few needles can you use to reliably approximate π to two decimal places?
  4. Consider the well-known problem of maximizing the volume of a box constructed from a rectangular sheet of paper with square pieces cut from the corners. Using your choice of technology, construct a visualization that clearly illustrates why the maximum occurs when the area of the bottom of the box is equal to the total area of the four sides.
  5. A straight highway passes nearby two cities, City A and City B. Use Sketchpad or Geogebra to create a document which generates a graph of the distance to City B vs. the distance to City A for points along this road. Describe all graphs that can possibly be generated for various configurations of such a road and two cities. In Graphing Calculator, create a parameterized curve in 3-dimensions for the distances to three separate Cities A, B, and C from points along a straight road. Describe all curves that can be generated for such a road and three cities.
  6. Write a program in VPython that sets as constants the masses of a star and a planet, an initial separation, and an initial velocity of the planet (with respect to the star which you may assume is stationary). Use Newton's law of gravitation to simulate the orbit of the planet about the star. Look up parameters for the Earth and Sun and get your simulation to work accurately for this system. Have VPython report the period and eccentricity of the orbit as a check.

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