Workshop Outline
for Thursday Sept 25, 97

Approximate times: Spent on: C o m m e n t s :
0:00-0:15 Administrativa Opening remarks, phone and email lists.
A "name game" or other short introductory activity of your choosing, getting the students to talk/participate.

Assign homework, with introductory comments on standards/expectations.

Encourage students to work together on HW and use the phone list.

Handouts: Suggestions on working in groups
Course Survey ; Course Summary Sheet;
Course Syllabus.

Make Additional Announcements (see attached). Explain: Exams are 5-7pm, students choose 5-6,5:30-6:30,or 6-7.

0:15-1:00 Lecture Cover: Sections 1.1 and 1.2.

Key points to stress:

  • Multiple interpretation of (or viewpoints about) functions
  • Being able to interpret, compare, analyze, and create graphical data -- e.g. scatter plots, tables, and especially graphs of functions
  • Student mathematical communication skills, e.g. translating informal descriptions into formalized ones, and vice versa.

  • Also mention increasing and decreasing functions, and symmetry.
    • 1:00-1:10 Break --
    1:10-1:55 Activity See comments below about Estimating the Mile Record activity

    Instructor Guidelines for the Activity
    "Estimating the Mile Record":

  • Setup: Have the students break up into groups of 3-4. Explain that this activity is an example of the usefulness of abstract mathematical modeling and analysis to real-world cases of information one wishes to better understand, and how to best make forecasts based on existing data.
  • Comments: The step-by-step instructions are fairly self-explanatory. The "Second Analysis" is open-ended, and the students can combine two, three, or more known records to get upper and/or lower bounds for the mile record. Do 4th Analysis only if there is plenty of time. For most sections, there won't be, in which case this last section of the activity will be done on Tuesday Sept 30th.
  • Part IV of the Activity "Estimating the Mile records" mentions concavity, although the concept will not have been formally covered. Many (most?) students will have an intuitive understanding of "concave up". If some are unsure, a simple picture (something vaguely resembling the graph of x2 for "concave up" and of -x2 for "concave down") should suffice.
  • Wrap-up: Issues to discuss include, What is Concavity, and what does it capture? Why is it so powerful? What is the real-life interpretation of the (reciprocal of) the slope of the line from the origin to a point on the graph?

    Answers
  • First Analysis: Lower bound of 212.2 s and upper bound of 291.4 s. The uncertainty gap is about 80 seconds.

  • Second Analysis: Here the students can make various combinations, such as adding the times for the 1,500 and 100 meter records to get a lower bound for the mile record (since a mile is longer than 1,500+100 = 1,600 meters), or they could add one half the 200 meter record instead of adding the 100 meter record for the lower bound, etc.

  • Third Analysis: 1,500 m record average speed: 1,500/212.2 = 7.0688 m/s. If the mile is run at this speed, we get a lower bound of 1,609.344/7.0688 = 227.67 s (approximately).
    For the 2,000 m record, the average speed is 2,000/291.4 = 6.8634 m/s. If the mile is run at this speed, we get an upper bound of 1,609.344/6.8634 = 234.48 s (approximately).
    The uncertainty gap is now about 7 seconds.

  • Fourth Analysis: The slope of the first secant is (291.4-212.2)/500 so it has the equation
    T-212.2 = [(291.4-212.2)/500] (x-1500)
    Plugging in x=1609.344, we get T=229.52 as the new upper bound.

    The second secant's slope is (212.2-133.9)/500 so its equation is

    T-133.9 = [(212.2-133.9)/500] (x-1500)
    Plugging in x=1609.344, we get T=229.32 as the new lower bound.

    The new uncertainty gap is just 0.2 or one-fifth of a second. Did the actual record lie between 229.32 and 229.52 seconds? It did! Walker's record which is mentioned, translates into 229.4 seconds.