Time Allotment: Three 50-minute class periods.

Anytime we watch a great athletic event such as the Olympics, we are often told that with hard work and determination we too could be a great athlete, but, can we? When we see an outstanding young athlete, we often make predictions about his/her future, but on what basis are we making those predictions. For example, is it reasonable to expect that a 5' 3" young man with a size 7 shoe could possibly be the next Michael Jordan? On the other hand, could we really expect a 6' 1" young lady to be another Mary Lou Retton? Certainly, a person's natural body structure will be a determining factor. Are there other factors to consider as well?

Leonardo da Vinci viewed the human body as a perfect example of nature arranging the different body parts into relationships. These relationships may determine a child's potential as an athlete, they may be of practical interest to the producers of clothing or they may simply decide our choices and preferences for beauty in a human being.

Through the activities presented in this lesson, students will be able to test Leonardo's theory through the exploration of statistical concepts. The activities will involve the actual measurement and collection of data, fitting simple linear models to the data, and then analyzing and interpreting those models.

Students will be able to measure the various body parts using a tape measure; to create a data table; to create a scatter plot using appropriate scale; to develop the linear equation that describes the line of best fit (linear regression line); to analyze and interpret the results of their performance using a graphing calculator; and to compare the results of their observations for people of different age groups, gender groups, athletic interests, etc.

I. Number Sense and Operations:

1. Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

2. Understand meanings of operations and how they relate to one another.

3. Compute fluently and make reasonable estimates.

II. Patterns, Relations, and Algebra

1. Understand patterns, relations, and functions.

2. Represent and analyze mathematical situations and

structures using algebraic symbols.

3. Use mathematical models to represent and understand quantitative relationships.

4. Analyze change in various contexts.

III. Measurement

1. Understand measurable attributes of objects and the units,

systems, and processes of measurement.

2. Apply appropriate techniques, tools, and formulas to determine measurements.

IV. Data Analysis, Statistics, and Probability

1. Formulate questions that can be addressed with data and

collect, organize, and display relevant data to answer them.

2. Select and use appropriate statistical methods to analyze

data.

State Standards:

From the Massachusetts Curriculum Framework

10.N.1, 2, 4

10.P.1, 2, 4, 7

10.M.3

10.D.1, 2, 3

12.P.1, 6, 11, 12

12. M.2 and 12.D, 1, 2, 3

Video: Against All Odds: Statistics: Models for Growth (episode 7) (The Annenberg/CPB Collection)

Science of Talent Search and Long Term Training System

Written by Ladislav Pataki, Ph.D., CsC., a former Czechoslovakian Olympian who defected to the U.S., discusses the theory and methodology of sports training, including the need to identify mental and physical characteristics appropriate for a given sport.

Winning Secrets

Written by Pataki, as well, this article focuses on how to select the appropriate sport for a child with potential athletic ability by considering both genetic background and environment.

Leonardo da Vinci - Human Study

This site provides a visual tour of many of Leonardo da Vinci's drawings illustrating the theory of proportions of the human figure.

The Drawings of Leonardo da Vinci

This site provides a visual tour of many of Leonardo da Vinci's drawings, including Vitruvian Man.

Leonardo da Vinci

This site is a resource for students and teachers and is sponsored by the Museum of Science, Boston. It includes information about da Vinci, the scientist, inventor and artist as well as a virtual tour of the da Vinci exhibit that was held at the museum in 1997.

Leonardo da Vinci, The Anatomist

This biographical sketch of Leonardo da Vinci focuses on his work as an anatomist and the development of his anatomical drawings.

For each student:

For each pair of students:

Prepare a summary of the article Leonardo's View of the Human Body that can be distributed to the students.

Cue the video tape to the appropriate starting point.

Prior to the instruction of this material, bookmark all of the Web sites used in the lesson. Load in any plug-ins necessary to run the Web sites.

Set up the overhead projector attachment for the graphing calculator.

If necessary, arrange to have your students visit another classroom(s) so that at least 15 sets of data can be collected for the first data collection activity.

Vocabulary (The following vocabulary will be used during this lesson.)

independent variable -- the variable that attempts to explain the observed outcomes; the explanatory variable; the x-variable

dependent variable -- the variable that measures an outcome of a study; the response variable; the y-variable

scatter plot -- a plot of observations xI and yI as points on a coordinate axis system, where xI and yI are the values of the numerical variables x and y measured for the same individual or object.

positive association -- two variables are positively associated when above-average values of one tend to accompany above-average values of the other and below-average values of one tend similarly to accompany below average values of the other

negative association --two variables are negatively associated when above-average values of one accompany below-average values of the other, and vice-versa

slope -- the change in the y-variable with respect to the change in the x-variable

regression line -- a straight line that describes the dependence of one variable on another

least squares linear regression line -- the line that makes the sum of the squares of the deviations of the data points from the line in the vertical direction as small as possible

residual -- the difference between an observed value of the y-variable and the value predicted by the linear regression equation

The following steps will prepare your students for a lesson in gathering data and analyzing that data using a linear regression model.

Step 1: Ask the students, "When you buy a new shirt, do you expect both the neck and sleeve length to fit? Is there a relationship between the neck and sleeve length that helps the manufacturer determine which combinations to produce? How do you think these combinations are determined - perhaps by surveys? If so, which people should be measured and do you think that there are anatomical relationships that exist for all people?"

Focus for Media Interaction: To provide students with a specific task to complete while viewing, ask them to make a list of all indications on the drawing that Leonardo used to suggest proportional relationships.

*Explain that Leonardo produced many other diagrams illustrating his theory of proportions of the human figure.

Focus for Media Interaction: Ask students to make a list of all indications on the drawing that Leonardo used to suggest proportional relationships. Have students make note of which proportions are suggested by the drawings.

Step 2: Hand out the informational sheets on Leonardo's theory on the relationship of human body parts. Have students acquaint themselves with the proportions set forth in Leonardo's theories and compare them to those suggested in his drawings. Ask the students, "If we want to determine relationships that would help determine the size for your new shirt, which would be most helpful? If you wanted to know whether you have the necessary physical build to become an Olympic swimmer, what would you measure?" Step 3: Explain to the class that they will now design an activity to investigate Leonardo's theory of human body parts and their relationship to each other. Have the students discuss the importance of collecting information that is necessary to test the model suggested by Leonardo. For instance, is the measurement of the wrist to knuckles a good association with the measurement of the foot? Or, would the theory be better illustrated by the measurement of the neck versus overall height? Have the class choose at least two relationships for investigation - ex. height and arm span or arm length and hand length.

Step 4: Have the students list all the variables necessary to investigate the relationships and then use these variables to create a table to record their data. Have the students determine a set of guidelines for measuring these parts on any one person so that all measurements are taken consistently. Be sure to determine the units that will be used - inches or centimeters. This would be an excellent time to discuss the fact that NASA's latest mission to Mars failed because two different teams of scientists used two different measuring units and did not communicate that fact to each other.

Step 5. Divide the class into groups of two or three in order to collect the data. The students may measure each other and perhaps members of other classes or family members so that at least 15 sets of measurements are taken collectively (more is better).

Step 1: Explain to the students that they will now construct scatter plots to compare the variables in all of the relationships that they have chosen to study (one scatter plot will be used to investigate each relationship). Explain the mathematical process of arranging this information into a scatter plot and the mathematical means available to test the usefulness of the information, the line of regression, and the concept of residuals. Be sure to help the students determine which variable will be the independent and which will be the dependent variable in each scatter plot so that results within the class will be consistent.

Step 2: Each group of students should construct a scatter plot from their data on graph paper. A discussion of the number of points should follow, and the students should be asked if their points are enough to make proper conclusions. Then, data points of everyone in the class should be accumulated on a transparency for the overhead projector. Students should copy these new points onto their scatter plots. Have the students describe any patterns that they see.

Step 3: The students should be provided with pieces of spaghetti and asked them to place the spaghetti in a place of best fit on their graphs. It should be emphasized that the spaghetti represents a linear estimate of the information. The students should be able to estimate the equation of the line defined by the piece of spaghetti by estimating two points on the line and using the point-slope form for a linear equation. Have the students write their final equation in slope-intercept form, y = mx + b. Have the students interpret the slope and y-intercept within the context of the problem.

Step 4: Explain to the students that the line they have just constructed is called a linear regression line. Continue to explain that they will now see how the slope of such a linear regression line can be used to help diagnose and treat children with growth deficiencies.

For another Focus for Media Interaction provide them with a copy of the "Focus for Viewing Worksheet", Worksheet # 2. Tell the students that they are responsible for completing the worksheet while watching the video. Read through the questions with the students before the video begins. (Note: Students may simply be asked to jot down notes regarding the answers to the questions and then use these for class discussion afterwards.)

Insert video Against All Odds: Statistics: Models for Growth (episode 7) into your VCR. Cue the tape to after the animated title Against All Odds has become still and the screen begins to show the opening of the garden flowers. Your auditory prompt will be "...growth is the driving force of nature..." Pause when the screen shows the frame "3. Exponential Growth: Beats Linear Growth."

Ask students to recap the difference between linear and exponential growth and note their answers on the "Focus for Viewing" Worksheet. Resume the video. Pause when the screen shows the graph of growth patterns and the auditory prompt says "... and the 95th percentile the upper line." Ask students for what ages the growth chart on the video is appropriate and between what percentile readings is a growth rate considered normal. Have the students note their responses on the "Focus for Viewing" Worksheet. Resume the video. Pause when the speaker says "...Sarah was falling further and further away from the normal range." Ask students to identify the mathematical indications that Sarah's growth pattern was falling below the normal range. Have them note their answers on the "Focus for Viewing" Worksheet.

Resume the video. Pause when the screen shows the graph of Sarah's growth pattern after growth hormone therapy and the speaker says "... Sarah is growing faster than her peers." Ask the students to identify the mathematical indications that Sarah's growth pattern is returning to normal. and note these on the "Focus for Viewing" Worksheet.

Resume the video. Pause when the screen shows a graph of the growth rate for Jason and the auditory prompt says "...so his rate of growth is normal." Pause and discuss with the class what mathematical indications are shown on the graph that indicate Jason should not receive hormone therapy. Discuss how slope was the key to determining whether or not each child received hormone therapy. Have the students note their conclusions on the "Focus for Viewing" Worksheet. Resume the video. Stop when the screen shows Jason in a sailboat and the speaker says "...altering our natural height remains a fantasy."

Post-Viewing Activity: After viewing the video, review the questions on the "Focus for Viewing" Worksheet and the students' responses. Remind the students once more of the important role that the slope of a regression line plays in the interpretation of a linear regression line.

Step 4: Ask students how slope will help them to compare their results to Leonardo's suggested proportions. Have the students do the comparison and discuss their results. A discussion of any differences they observe should follow.

Step 1: Provide each student with a graphing calculator and ask each to enter the data collected by the class into the 'list' function.

Step 2: Using an overhead attachment help students determine the equation for the "best fit least squares line" using the graphing calculator. Paste the linear regression equation into the "y=" menu and graph the equation. They should compare their hand done results to the calculator results and discuss any discrepancies.

Step 3: Explain the concept of residuals and demonstrate how they are calculated. The students should be able to determine several residuals on their graphs and compare these to the calculator results.

Step 4: Discuss and explore how this experiment might be improved or changed to better justify Leonardo's theory. Also discuss the scientific and mathematical reasons that such exploration of the measurements might benefit mankind. This discussion may lead to other academic disciplines such as the health sciences, history, art, economics, etc.

Step 5: Have the students prepare a written summary of their results.

Mathematics: Have the students use a graphing calculator to determine the correlation coefficient, r, and the value of r2 for each of their linear regression lines. What information does each of these values provide within the context of the problem? How is the value of r related to the slope of each linear regression equation?

Mathematics/Science: Have students log on to Leonardo da Vinci, The Anatomist, www.geocities.com/CollegePark/1070/leonardo.html) to read about the methods Leonardo used to investigate his theory of proportions about the human body. What scientific methods did Leonardo use to investigate his theories on anatomy. Does any of this information help explain why their results differ from his?

Mathematics/Health Sciences: Have groups of students investigate the relationship of specific body parts for various groups such as swimmers, basketball players, wrestlers, dancers, the elderly, preschool children, etc. Have them compare their results with those collected for the general population in the earlier steps of this lesson. Students can access the following Web sites for additional information relating to the physical attributes of athletes.

Winning Secrets Web site

Science of Talent Search and Long Term Training System

Mathematics/Economics- Have students investigate the cost of manufacturing various goods: clothing, shoes, automobiles, athletic equipment, home furnishings, etc in the various sizes and shapes as demanded by the various sizes and shapes that characterize consumers.

1. Arrange for students to visit an art museum so that they might investigate several interpretations of the human form as displayed in paintings and sculptures.

2. Invite a physical therapist to speak to the class about the applications of the theory of human proportions in treating physical disabilities, training of athletes, etc.

3. Invite a clothing manufacturer or tailor to speak to the students about the way measurements are determined for clothing sizes.

1. "Data Collection" Worksheet #1.

2. Either a copy of the article on "Leonardo's View of the Human Body" or a summary of the article.

3. "Focus of Viewing Worksheet" Worksheet #2.