Augmentation

Augmentation

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Introduction
 Fathom allows users to begin modeling with students at the very low-level stages of SAMR. With a little creativity, several data sets can be used to create modeling activities. Augmentation is a difficult step to create modeling activities with, since we are just simply using the tools provided within the Fathom program to enhance our data construction and modeling experience. However, the tools provided by Fathom are so strong that, given an size set of quantitative data, we are able to generate modeling activities for our students. Under the "File" you will find a prepared document. Use the instructions below to see how Fathom can be used at the Augmentation step as well as how a modeling activity can be prepared at this step.

File
Download this file to complete the following activity:



Note: You'll need either the Fathom disc or the free trial of Fathom in order to complete this activity. Click [|here] to download the free trial.

Activity
This investigation presents students with available Olympic data and asks them to develop a statistical model that can help answer the question. The data were collected at each of the modern Olympics that included the 200-meter dash. Students think about how this observed data could analyzed. They are often amazed by the patterns that emerged and the implications.

The Summer Olympic games are generally held every four years. They provide an opportunity for athletes from all over the world to compete. As expected, this level of competition results in the best athletes demonstrating their abilities to run, jump, throw, wrestle, box and excel in many other individual and team sports. The track and field events are particularly noteworthy, as the Olympics provide a prime-time audience with access to competitions not normally showcased. The running evens are frequently won or lost by a fraction of a second. The gold medal times for the 200-meter dash are found in the Men and Women's collection from 1900 to 2004 for this race.

Questions to Think About (Modeling)
- Can you use these data to predict the future times of men and women in this event? - Do you think women will ever run faster than men, and if yes, when? - Construct an argument to support your answer by using this data.

Using Fathom to do This
<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">1. Open the Fathom document located under the "File" header, named "augmentation.ftm"

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">2. Highlight the collection named "Men's Olympic 200-Meter Dash Times" and drag down a table from the menu bar. This should display a table of the men's data in two columns "Time" and "Year".

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">3. Highlight the collection named "Women's Olympic 200-Meter Dash Times" and drag down a table from the menu bar. This should display a table of the women's data in two columns "Time" and "Year".

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">4. Situate these two tables next to one another. Have students explain what the two data sets are summarizing.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">5. Discuss with students to see if there are any interesting relationships that they see in the data sets. <span style="display: block; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%; text-align: center;">Click here to see a screenshot of this activity so far.

<span style="display: block; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%; text-align: left;">6. Open the inspector for Men's Olympic 200-Meter Dash Times. Drag down an empty graph. Put the attribute "Year" on the x-axis and the attribute "Time" on the y-axis.6.Open the inspector for Women's Olympic 200-Meter Dash Times. Drag down an empty graph. Put the attribute "Year" on the x-axis and the attribute "Time" on the y-axis.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;"> 7. Discuss with the students the distribution of data of these two graphs. Ask questions about the both scatter plot's distributions regarding whether the time is increasing, decreasing, and the void of women's data in the years between 1900-1948. Click here to see a screenshot of this activity so far.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">8. Have students generate algebraic expressions that they think will fit the data based on the observations of these scatter plots.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">9. Have students generate regression lines for each of the scatter plots. This is done by right-clicking on the scatter plot and scrolling down until you see the option "Least Squares Line".

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">10. Have students compare the two algebraic expressions at the bottom of each plot.* Do these equations make sense?

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">11. Have students reason whether or not the women's time will ever beat the men's times. What elements must they consider? <span style="display: block; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%; text-align: center;">Click here to see the added regression lines.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">12. Have students sketch what they think //would// be a better model than a Least Squares Line. Have students sketch where they think the body's limitations meet. Have students to do research on the computer to determine a maximum speed at which a human can run.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">13. Have students try and produce exponential models. and think about how this can be done in Fathom. Have them think about how the data needs to be different.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">14. Next, delete scatter plots or reopen the file under a new name.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">15. Open "Gap Times" and drag down a scatter plot.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">16. Put "Year" on the x-axis and "Difference" on the y-axis.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">17. Have students add a Least Squares Line. <span style="display: block; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%; text-align: center;">Click here to see Gap Times modeled.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">18. have them rationalize whether or not this is a good fit.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">19. have students predict some better models, or if there is a correlation.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">20. Have students find their own data sets for various Olympic events. Perhaps they could compare countries for a different event. Have them predict future outcomes, if their Least Squares lines seem reasonable.*

Credit
<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">This activity is adapted from //Focus in High School Mathematics: Reasoning and Sense Making// by J. Michael Shaughnessy, Beth Chance, and Henry Kranendonk.

<span style="font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;"><span style="color: #ff0000; font-family: 'Trebuchet MS',Helvetica,sans-serif; font-size: 120%;">Citation: Shaughnessy, J.M, Chance, B., and Kranendonk, H. (2009). //Focus in High School Mathematics: Reasoning and Sense Making.// National Council of Teachers of Mathematics. Reston, Vermont.