Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 05/15/ A Study of Olympic Winning Times Objective: Students will graph data, determine a line that models the data, and solve a system of equations by graphing. Connections to Previous Learning: Students should be able to graph on a coordinate plane and write the equation of a line. Connections to AP: AP Calculus Topic: Analysis of Functions AP Statistics Topics: Graphical Displays; Bivariate Data Materials: Student Activity pages, graphing calculators, straight edge, graph paper, access to a data source or the data provided with this lesson Teacher Notes: This lesson is designed to be done after students have mastered writing the equation of a line, but before a formal study of systems of equations. Students can collect their own data from a sports event of their choice or use the data from the 0 meter dash which is given on a separate page. If students collect their own data, it must be an event that is performed competitively by both men and women, and one that will yield data from to the present. They can gather data from a source such as The World Almanac or an internet site such as http://www.infoplease.com (navigate to Sports > Olympics). Students are usually curious about the gap in the table no Olympics were held during World War II. In this lesson students are asked to predict beyond the range of the data so that they can gain an understanding of why extrapolation is not always reasonable. This would be a good time to revisit, or visit, piecewise functions as this data often breaks down into distinct sections. If graphing calculators are not available, the lesson can be modified to allow the students to manually fit lines to the scatterplot. Manually fitted lines may not model the data as well as the regression lines calculated on the calculator. *Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product. Copyright 200 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 1
A Study of Olympic Winning Times 1. Graph each data set on a coordinate graph below and title each graph. Use the horizontal axis for the year and the vertical axis for winning times. Label each axis appropriately. Let the year 100 be 0 on your horizontal axis..5.5.5.5.5.5.5 0 1 32 4 4 0.5.5.5.5.5.5.5 2 0 1 32 4 4 0 2 2. Explain why the year was placed on the horizontal axis and the winning times on the vertical axis. Be sure to include appropriate mathematical terminology in the explanation. 3. What number would be needed on the horizontal axis to correspond to the year 150? 4. Write functions to model the sets of data. Rewrite the functions using words in place of the variables. Copyright 200 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 2
5. What is the slope of your line for the men? For the women? Interpret the slopes in the context of the problem.. What is the y- intercept of your line for the men? For the women? Interpret the y-intercepts in the context of the problem.. Use your lines to predict the winning time in 14. How do the winning times compare to the data in the table?. Use your equation for the men to predict the winning times in 150. Is it a reasonable projection? Explain. What about for the year 100? The year 0?. Use your function for the women to predict the winning times in 150. Is it a reasonable projection? Explain. What about for the year 100? The year 0? Copyright 200 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 3
. Is the function a good representation of the possible data for the years before 100? Explain.. What winning time does your line predict for men for the year 201? Is it a reasonable prediction? Explain. What about for the year 2050? The year 3000?. What winning time does your line predict for women for the year 201? Is it a reasonable prediction? Explain. What about for the year 2050? The year 3000? 13. Using your functions, predict the year in which the men and women will run the 0 m dash in the same amount of time. Solve the system of equations using a method that you consider best, and justify your solution. What is the predicted winning time in that year? 14. What is a reasonable domain for these functions? What type of function might better represent how the winning times are changing over time? Explain. Copyright 200 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 4
A Study of Olympic Winning Times Data Page The following table gives the men and women s winning times for the 0m dash in the Olympics from -2004. Year Men s 0m Dash in seconds Women s 0m Dash in seconds..2 132.3. 13.3.5 14.3. 152.4.5 15.5.5 10.2.0 14.0.4 1.5.00.14.0 1.0.0 10.25.0 14.. 1.2.54..2 1.4.4 2000..* 2004.5.3 *In October 200, Marion Jones returned her 0 meter dash gold metal from the 2000 Olympics. Her winning time of.5 seconds was replaced by Ekaterini Thanon with a time of. seconds. (At the time this activity was printed, the decision to award the gold medal to Ekaterini Thanon was not finalized.) Copyright 200 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org 5
Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Answers: A Study of Olympic Winning Times 1. The students were not asked to graph the lines; however, they are included here to show the linear regression line for the data. Men s 0 meter Dash Women s 0 meter Dash Time in seconds y t 0 1 32 4 4 0 2 Years after 100 Time in seconds y t 0 1 32 4 4 0 2 Years after 100 2. The year is the variable that we are using to explain the winning times. We will consider this the explanatory variable. Scatterplots are graphed with the explanatory variable on the horizontal axis and the response variable on the vertical axis. 3. The year 150 is 50 years before zero since 100 is coded as zero. 150 is represented by 50 years. 4. Line of best fit for men: f( t) = 0.00t+.4 or the Olympic time for the men is 0.00 seconds per year (years after 100) +.4 seconds Line of best fit for women: f( t) = 0.01t+.31 or the Olympic time for the women is 0.01 seconds per year (years after 100) +.31 seconds. Manual fit lines should be close to these lines. 5. Slope for men: approximately 0.00 seconds per year; Slope for women: approximately 0.01 seconds per year. For every increase of one year, there is a decrease of approximately 0.00 seconds for the men and a decrease of approximately 0.015 seconds for the women.. y-intercept for men: approximately.4 seconds; y-intercept for women: approximately.31 seconds. The line would predict that in 100 the men would have won the race in approximately.4 seconds and the women in approximately.31 seconds. Copyright 200 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org
Answers. Men: In 14, the winning time is predicted to be approximately.1 seconds. Women: In 14, the winning time is predicted to be approximately.31 seconds. The men s time is higher by approximately 0.1 seconds; the women s time is lower by approximately 0.01 seconds.. Men: In 150, the time is predicted to be approximately.24 seconds. In 100, the predicted time is approximately 13.51 seconds. In the year 0, the predicted time would be approximately 2.434 seconds. Accept any well thought out, well explained answer for reasonable projection.. Women: In 150, the time is predicted to be approximately 13.05 seconds; in the year 100, it would be approximately 1. seconds, and in the year 0, it would be approximately 41. seconds. Accept any well thought out, well explained answer for reasonable projection.. A single line is not a good predictor. Extrapolation beyond a reasonable domain is not advisable. The data is probably not linear over a large domain.. Men: In the year 201, the predicted time would be.0 seconds; in 2050, it would be.31 seconds, and in 3000, it would be 0.5 seconds. Accept any well thought out, well explained answer for reasonable projection.. Women: In the year 201, the time is predicted to be.50 seconds; in 2050, the predicted time would be. seconds; in 3000, it would be 4.1 seconds. Accept any well thought out, well explained answer for reasonable projection. One reasonable explanation for 3000 is that it is not possible to run a race in a negative amount of time. 13. Using the best fit lines, the answer is (243.1,.521). This point means that in the year 2143 (or the Olympics of 2144) the men and women will both run the race in about.521 seconds. 14. Accept any well thought out, well explained answer for reasonable domain. Students might consider a piecewise function as a possibility or a function that approaches the x-axis as the years increase. Limits of the domains and range need to be discussed. Copyright 200 Laying the Foundation, Inc. Dallas, TX. All rights reserved. Visit: www.layingthefoundation.org