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ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary |
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ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary |
Please keep in mind that this is a research project, and there may sometimes be glitches with the interactive software. Please let us know of any problems you encounter, and include the computer operating system, the browser and version you're using, and what kind of connection you have (dial-up modem, T1, cable).
Marathon Graphing
Introduction
Here are record-breaking times for the men's marathon from all over the world. We've graphed the times and a line that models the data. You can use the line to see trends in the data.
Men's Times Year Athlete Time
1935 Kitei Son 2h 26m 42s 1947 Yun Bok Suh 2h 25m 49s 1952 Jim Peters 2h 20m 43s 1954 Jim Peters 2h 17m 40s 1963 Buddy Edelen 2h 14m 28s 1969 Derek Clayton 2h 8m 34s 1981 Robert de Castella 2h 8m 18s 1985 Carlos Lopes 2h 7m 12s 1999 Khalid Khannouchi 2h 5m 42s Use the applet to graph the women's times for the marathon so you can answer the questions below.
Women's Times Year Athlete Time (Approx.
Minutes)1964 Dale Grieg 3h 27m 45s 207 1970 Caroline Walker 3h 2m 53s 182 1971 Adrienne Beames 2h 46m 30s 166 1974 Jackie Hansen 2h 43m 55s 164 1975 Jackie Hansen 2h 38m 19s 158 1977 Christa Vahlensieck 2h 34m 47s 155 1978 Grete Waitz 2h 32m 30s 152 1980 Grete Waitz 2h 25m 42s 145 1983 Joan Benoit 2h 22m 43s 143 1985 Ingrid Kristiansen 2h 21m 6s 141 1998 Tegla Loroupe 2h 20m 47s 140 1999 Tegla Loroupe 2h 20m 43s 140
Click To Show Applet Window Questions:
Use the graph you made in the applet to answer the following questions:
- Based on the graph, what do you think the women's winning time should have been this year at the Sidney 2000 Olympic Games?
- Predict when women will run the marathon as fast as men.
- What do you think the women's world record marathon time was in 1926? Use this Web link from the Centre for Innovation in Mathematics Teaching (CIMT) to look up times from 1926 to see how well they fit your prediction.
- Using lines to model data can be a powerful tool, but we have to be careful to be aware of their limitations. What real-world limitations do you think a line has in modeling the data?
Teacher Support Page Because when this PoW first went live there were technical problems that prevented students from submitting their solutions, we ran it again at the end of December. Of course then it was holiday time for many people, so there were very few submissions.
There were only four submissions to this puzzle, and no one received credit for it. The first three questions asked for predictions that used the line to model the data. Since the data weren't really in a line -- making it a little tricky to find a best-fit line -- we were pretty relaxed about what we'd accept. Two of the four students got these numbers right. A third student got the first two right.
Question 4 asked about real-world limitations in using a line to model data. No one answered this question in a way we thought was reasonable. Here are the answers we thought we'd see:
- Based on the graph what do you think the women's winning time should have been this year at the Sidney 2000 Olympic Games?
MANY possible solutions...
Fitting a line, the answer should be somewhere in the range of 115 to 130 minutes.If the students use a linear regression, they will come up with y = -1.6x +3236.5, and get a time of about 37 minutes!
If students look at the graph and notice that the times have flattened out considerably in the past few races, they may answer somewhere slightly under 140 minutes.... It seems that we have hit a "wall" in terms of our ability to continue to get faster.
- Predict when women will run the marathon as fast as men.
Using the fitted line, this would happen somewhere between 2005 and 2020. If you look at the flattening-out theory, though, it seems as if it would be further out in the future. I'm not even sure we can predict that, because when I flatten out the women's line, the two lines look close to parallel.
- What do you think the women's world record marathon time was in 1926?
In 1926, the women's record time was 3 hours, 40 minutes, 22 seconds, which is 220.36 seconds. Using a fitted line, an estimate of 280 minutes is not unreasonable.
- Using lines to model data can be a powerful tool, but we have to be careful to be aware of their limitations. What real-world limitations do you think a line has in modeling the data?
There are many. The most glaring is that data can look linear in a small window. But when you look at the big picture, other considerations creep in. People have physical limitations, and at some point, they won't be able to run much faster than the previous records, and a line won't fit the data. ....