**Olympic Track Times**

Christine Bolte Jehan Khan Susan Pavetto

**Class**: Algebra I

**Materials:** Tables for Olympic track times, graph paper (or other graphing devices).

**Goals**: To enhance the students' understanding of linear graphs. To estimate a line of best fit. To use linear graphs as a method of predicting future or past events. To decide if a linear model is the best/only model available.

**Time Required**: One class period (provided data is given).

**Background**: Plotting points on coordinate axes and finding a line of best fit This project could be utilized to review or introduce this technique.

**Setting**: The idea of projecting into the future based on historical data is an important one. Particularly in the field of economics is it important to be able to predict with some degree of success the liklihood of certain events happening (e.g., stock market trends, consumer buying habits). However, fascination with the future does not end with the business world. In this problem, the arena of Olympic track is used to motivate the desire to fit a function to a set of data to enable predictions to be made concerning past and future times in the men's 1500 meter run.

**Problem**: Data has been provided for the Olympic times for the men's 1500 meter run from 1896 to 1992. Notice there are no times for 1916, 1940 and 1944 due to the Wars.

1. Use the data to graph year versus time to the nearest hundredth of a minute on coordinate axes.

2. Draw a line of best fit.

3. Use the graph to predict the winning time for the 1916 men's 1500 meter run.

4. Predict the 1944 men's 1500 meter run.

5. Predict the time that will be recorded for the 2000 Olympics.

6. Justify why your model might be inappropriate for this data.

**Men's Olympic 1500 Meter Times**

(Times given in minutes and seconds)

__Date__ __Time__

1896 4m33.2s

1900 4m6.2s

1904 4m5.4s

1908 4m3.4s

1912 3m56.8s

1920 4m1.8s

1924 3m53.6s

1928 3m53.2s

1932 3m51.2s

1936 3m47.8s

1948 3m49.8s

1952 3m45.2s

1956 3m41.2s

1960 3m35.6s

1964 3m38.1s

1968 3m34.9s

1972 3m36.3s

1976 3m39.17s

1980 3m38.4s

1984 3m32.53s

1988 3m35.96s

1992 3m40.12s

**Evaluation**: Because this is such a short project, we recommend a small point value.

__Question __ __Point Value__

#1 4 points

#2 3 points

#3 2 points

#4 2 points

#5 2 points

#6 2 points

**Teacher notes:** Be sure the students convert the times to just minutes before graphing. (You could use seconds, but we prefer minutes.) We recommend two people in one group. Local papers can be a good source of data for local sporting events. Graphing calculators are a plus. Transparencies can be used to compare the times of men and women.

This project provides an excellent opportunity to discuss with the class the relative merits of the model just produced. Often models do a good job of predicting over the short run, but do not hold up over the long run. This can be seen by projecting the 1500 meter run to a time when it would be run in zero seconds! You may want to have students project to some year still in their potential life-time, say 2060, and see if they think the projected time is realistic.

**Extensions**: Any Olympic sporting event will work. You can also use the sporting events (specifically sectionals) at your high school to predict future values. Men's and women's times or distances could be compared to find out at what point their times may be the same. It is also interesting to investigate lines of best fit with positive slope, say for high jump heights. Then students' interpretations of positive and negative slope could be explored.

**Olympic Track Times**

**Sample Solution**

Using the computer software Derive, the data may be entered as a 22 x 2 matrix, and then plotted on a graph. A line of best fit may then be found using the FIT function and then graphed. Appropriate values representing the years 1916, 1944, and 1996 may be substituted into the linear equation y = -.03 x + 4.19 in order to predict the times for these years.

__Year__ __Predicted Time__

1916 4.02 min.

1944 3.83 min.

1996 3.46 min.

However, this model is inappropriate for the long run because it is linear. It would be impossible to run the 1500 meter race in zero seconds!

**Derive Program that Finds Line of Best Fit:**

1: DECLARE, MATRIX, ROWS = 22, COLUMNS = 2, and enter the 44 pieces of data.

2: AUTHOR and FIT([x, ax + b], F4 to enter matrix just entered)

3: APPROXIMATE to get 4.19233 - 0.0279975 x (see graph below of data points and line of best fit.

4: MANAGE, SUBSTITUTE, and 6 for x to get 4.19233 - 0.0279975 6, the expression for what would have been the sixth running (1916).

5: APPROXIMATE to get 4.02434

6: MANAGE, SUBSTITUTE, and 13 for x to get 4.19233 - 0.0279975 13, the expression for what would have been the 13th running (1944).

7: APPROXIAMTE to get 3.82836

8: MANAGE, SUBSTITUTE, and 26 to get 4.19233 - 0.0279975 26, the expression for the 26th data point (1996).

9: APPROXIMATE to get 3.46439

**Olympic Track Times**

**Student Lab Sheet**

Names_________________

_________________

Data has been provided for the Olympic times for the men's 1500 meter run from 1896 to 1992. Notice there are no times for 1916, 1940 and 1944 due to the wars. Answer the following using the data provided below.

1. Use the data to graph years versus time to the nearest hundredth of a minute on a coordinate axis.

2. Draw a line of best fit.

3. Use the graph to predict the winning time for the 1916 men's 1500 meter run.

4. Predict the 1944 men's 1500 meter run.

5. Predict the times that will be recorded for the last Olympics, and for the next two Olympics

6. What are the strengths and weaknesses of your model?

**Men's Olympic 1500 Meter Times**

(Times given in minutes and seconds)

__Date__ __Time__

1896 4m33.2s

1900 4m6.2s

1904 4m5.4s

1908 4m3.4s

1912 3m56.8s

1920 4m1.8s

1924 3m53.6s

1928 3m53.2s

1932 3m51.2s

1936 3m47.8s

1948 3m49.8s

1952 3m45.2s

1956 3m41.2s

1960 3m35.6s

1964 3m38.1s

1968 3m34.9s

1972 3m36.3s

1976 3m39.17s

1980 3m38.4s

1984 3m32.53s

1988 3m35.96s

1992 3m40.12s

1996

2000

2004

*Funded in part by the National Science Foundation and Indiana University 1995*