Leonardo 

da Vinci 

Are His Body Proportion Theories Still Valid Today?

A WebQuest for 6th through 8th
Grade Math Students

Designed by Jim Reynolds
Santa Barbara Junior High School
Santa Barbara, CA 93103

jimrmath@hotmail.com

Introduction

Another theory proposed that the hips are halfway between your head and your toes.

Also, that the arms               
and hands at the               
sides fall halfway               
between the hips               
and knees                           

              Many artists study the proportion
              of the human body to make
              figures look real. Artists often                           make quick sketches or "studies"                       of models to learn about anatomy.

Learners

This lesson is anchored in seventh grade math and involves research with language arts to a lesser extent. This lesson can easily be extended to sixth and eighth grades and also in subjects such as art and science.

First, as an introduction to scatter plots, students will record a number of human examples and will study what these examples have in common. Second, with that new understanding, students will demonstrate their understanding by creating a scatter plot and looking for what kind of correlation there might be (positive, negative, or none).

In this assignment, students will show whether da Vinci's ratios are still valid today. Initially, we will compare the height of a person and their wingspan. Leonardo's theory was that they were very nearly the same in length. Then we will explore some of his other proportions theories (hips in the middle of the body, hands between hip and knee, foot length  tequalinghe distance between the waist and elbow).

Curriculum Standards

What the students learn as a result of this lesson. For example:

6th Grade
Math Standards Addressed

• 1.3   Understand how the inclusion or exclusion of outliers affects measures of central tendency. 
• 2.0   Students use data samples of a population and describe the characteristics and limitations of the samples.
• 3.1   Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
  

• 3.3   Develop generalizations of the results obtained and the strategies used and apply them in new problem situations.
  

Science Standards Addressed
   Investigation and Experimentation

• 7b   Select and use appropriate tools and technology (including calculators, computers, balances, spring scales, microscopes, and binoculars) to perform tests, collect data, and display data. 
• 7c   Construct appropriate graphs from data and develop qualitative statements about the relationships between variables.

English Language-Arts Addressed

• 1.6   Support opinions with detailed evidence and with visual or media displays that use appropriate technology.
• 1.7   Use effective rate, volume, pitch, and tone and align nonverbal elements to sustain audience interest and attention.
• 2.2b   Deliver informative presentations: Develop the topic with facts, details, examples, and explanations from multiple authoritative sources (e.g., speakers, periodicals, online information). 
• 2.5a   Deliver presentations on problems and solutions: Theorize on the causes and effects of each problem and establish connections between the defined problem and at least one solution.
•  2.5b   Deliver presentations on problems and solutions: Offer persuasive evidence to validate the definition of the problem and the .proposed solutions
  

7th Grade
Math Standards Addressed

• 1.5   Represent quantitative relationships graphically and interpret the meaning of a specific part of a graph in the situation represented by the graph.
  

   Mathematical Reasoning

• 1.1   Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns.
  

• 2.5   Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.
  

   Statistics, Data Analysis & Probability

• 1.2   Represent two numerical variables on a scatter plot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables (e.g., between time spent on homework and grade level).
  

Science Standards Addressed
   Investigation and Experimentation

• 7a   Select and use appropriate tools and technology (including calculators, computers, balances, spring scales, microscopes, and binoculars) to perform tests, collect data, and display data.
  
• 7e   Communicate the steps and results from an investigation in written reports and oral presentations. 

English Language-Arts

• 1.5   Arrange supporting details, reasons, descriptions, and examples effectively and persuasively in relation to the audience. 
•  1.6   Use speaking techniques, including voice modulation, inflection, tempo, enunciation, and eye contact, for effective presentations. 
• 2.3a   Deliver research presentations: Pose relevant and concise questions about the topic. 
• 2.4a   Deliver persuasive presentations: State a clear position or perspective in support of an argument or proposal.
   
  

8th Grade
Math Standards Addressed
Algebra

• 17.0   Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
  

Science Standards Addressed

• 9a   Plan and conduct a scientific investigation to test a hypothesis.
• 9b   Evaluate the accuracy and reproducibility of data.
• 9e   Construct appropriate graphs from data and develop quantitative statements about the relationships between variables.

English Language-Arts

• 1.6   Use appropriate grammar, word choice, enunciation, and pace during formal presentations. 
• 1.9   Interpret and evaluate the various ways in which visual image makers (e.g., graphic artists, illustrators, news photographers) communicate information and affect impressions and opinions.
• 2.3a   Deliver research presentations: Define a thesis. 
• 2.3d   Deliver research presentations: Organize and record information on charts, maps, and graphs. 
• 2.4a   Deliver persuasive presentations: Include a well defined thesis (i.e., one that makes a clear and knowledgeable judgment).
  

Many kinds of thinking and communications skills were encouraged by this lesson. Critical thinking, Creative production, Creative problem solving, Observation and categorization, Comparison, and Teamwork.

Process

This lesson will involve more than one class (for comparison). Is it all taught in one period per day. It should take one week or less to complete. It can be a single disciplinary (math), or a multidisciplinary (math, art, science, language arts).

When students are divided into groups, the guidelines on how you might do that are initially one learner to measure, one to record, and one to be measured. Later on, the data can be gathered from the whole class either on a computer or a plot on the wall.

Possible misconceptions or stumbling blocks that I anticipate would be that the learners' knowledge of scatter plots and how to read them is somewhat insufficient. I suggest a way to get around those by going to the fourth section of the student process listed below.

Skills a teacher needs in order to pull this lesson off are as follows. Knowledge of how to use measuring devices (either the metric or standard), gather data, plot data on the first quadrant (Cartesian coordinates), and finally, how to read the scatter plot results.

Variations

There are a few other possible scatter plot variations based on more of da Vinci's theories on the human bodily proportions given in the introduction and in the conclusion. Here they are briefly for you.

1. Hips are halfway between your head and your toes.
2. Arms and hands at the sides fall halfway between the hips and knees.
   
3.  Standard Visual Human Body Proportions
4.  Head proportions
5.  Fibonacci and da Vinci
6.  da Vinci Facial Proportions
7.  Other Proportions
8.  Backwards writing

These are the instructions in the task section of the student page.

1. Working in groups of three, students will measure each other's height and wingspan from fingertip to fingertip. Use this form .
2. These measurements will be posted on the board and into the computer designated for this task, by student ID number and period.
3. Students will use these measurements on a scatter plot. Use this form .
4. Each student group will create a product (Power Point, chart, plot) that  demonstrates their understanding by creating a scatter plot and looking for what kind of correlation there might be (positive, negative, or none). If needed, information on how to make scatter plots are on this site:  Scatter Plot
5. For a look at da Vinci's Vitruvian Man and the proportions of the human body, try this site: Vitruvian Man
6. Here is a site to see human body proportions: Standard Visual Human Body Proportions
   
7. For a look at da Vinci's head proportions, check out this site: Head proportions
8. Look at these sites for an idea of what can happen with proportions that Fibonacci did with two of da Vinci's drawings. Fibonacci and da Vinci
   
9. A class discussion debriefing the analysis and consolidating any discrepancies in analyses from possible outliers, and the correlations of the scatter plots.
10. Student groups will try to find other proportions on the body that have positive correlations on a minimum of three other students outside their own group.
11. Finally, the students will come up with a business that sells shirts and shorts. The use of the scatter plot information on all the above will dictate why they are selling which sizes of both.

Resources Needed

• Class sets of math, art, science books with information on Leonardo da Vinci)
• Specific software (Apple Works Spreadsheet, Excel, or some type of spreadsheet)
• Specific hardware (Computers)
• Specific reference material in the classroom or school library (da Vinci background)
• Video or audio materials (plug-in for videos at given selected sites)

One  teacher is needed to implement this lesson. It can be an interdisciplinary unit with English (how to make an oral presentation), art (how da Vinci drew human figures with the use of ovals and the proportions stated in this lesson), or science (a look at some or many of the da Vinci's machines and inventions). Of course, parents can get involved with the learners by helping investigate with the computer at home or at a library. Aides in the classroom can help by roaming the room giving help where needed.

    Evaluation of Presentation

The collaborative group work and the presentation to the class will be assessed according to the rubric. See the chart below to guide your performance and reach your goal.

Conclusion

Students should have found positive correlations between the heights and wingspans of themselves and their peers. 

 Other Proportions

Credits & Reference

I wish to thank Judy and Jerry Headley, Kay Butler-Nalin, Kris White, Mike Williams, and Jill Proudfoot for all their patience, support, and assistance in the development and maintenance of this WebQuest. Without their encouragement and expertise this educational opportunity for students would not be possible.

I also wish to thank the Santa Barbara County Office of Education for the use of their facilities during the development of this project.  

In creating this lesson I consulted the Mathematics Framework for California Public Schools.

"We all benefit by being generous with our work. Permission is hereby granted for other educators to copy this WebQuest, update or otherwise modify it, and post it elsewhere provided that the original author's name is retained along with a link back to the original URL of this WebQuest. On the line after the original author's name, you may add Modified by (your name) on (date) . If you do modify it, please let me know and provide the new URL."

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Last updated on June 27, 2002. Based on a template from The WebQuest Page