APPENDIX C: LESSON PLAN 4
Charting Data: An Activity From Leonardo da Vinci
Susan Pittman, President, E-Learning Connections, Inc.
Bonnie Vondracek, President, Vondracek Enterpises, Inc.
This lesson plan is used often in mathematics classrooms. Building on the problem-solving tips, skills, and strategies
presented during this Institute, redesign this lesson to address the gaps identified by the GED Testing Service.
Identified Skill Gaps of GED Candidates
GED candidates must be able to perform arithmetic computations quickly and accurately in each of the content areas
of the GED Mathematics Test. GED candidates should be able to use computational operations, such as finding ratios, to more accurately compare and contrast figures, as well as to translate such information into graphical formats.
The GED Mathematics Test also requires that students distinguish pertinent data from extraneous information presented.
Content Area/Theme Identified by GED Testing Service Data Analysis
Graphic Literacy and Calculation
NCTM Standards and Expectations
PROCESS STANDARDS

Build new mathematical knowledge through problem solving.

Make and investigate mathematical conjectures.

Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

Create and use presentations to organize, record, and communicate mathematical ideas.
NUMBER AND OPERATIONS

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
MEASUREMENT

Solve problems involving scale factors, using ratio and proportion.
DATA ANALYSIS AND PROBABILITY

Use observations about differences between two or more samples to make conjectures about the populations
from which the samples were taken.

Represent data using tables and graphs such as line plots, bar graphs, and line graphs.

Select, create, and use appropriate graphical representation of data, including histograms, box plots, and
scatterplots.

Predict the probability of outcomes of simple experiments and test the predictions.
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ALGEBRA

Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible,
symbolic rule.
Time Required for Activity—1–2 class periods
Objectives/Learning Goals
Students will be able to:

Explore and identify relationships based on data collected from multiple samples.

Identify patterns from data collected from multiple samples.

Collect and appropriately display data graphically using a table and graph.

Apply the correct mathematical operation for calculating ratios.

Use the appropriate type of graph to display data.

Analyze data and draw conclusions based on those data.

Communicate their mathematical thinking and conclusions to the teacher and fellow team members.
Prerequisite Knowledge
Students should be able to:

Conduct basic linear measurements.

Collaborate with team members to solve specific problems.

Complete a basic table or chart.

Construct a variety of graphs (line, chart, pie).

Plot information on graph paper.
Content/Cognitive Skills
This lesson develops students’ problem-solving skills by posing specific questions that are answered by obtaining
data, calculating data, and then documenting the data on a graph. Students are then asked to analyze and to draw
conclusions about the data collected.
Materials/Resources/Internet Sites/Handouts

Two measuring tapes (if not available, provide yardsticks and/or rulers for each team of students)

String

Scissors

At least 2 different color pens per group

Data Recording Form

Graph paper
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Activity Procedure
INTRODUCTION TO LESSON/ACTIVITY STARTER
Begin the lesson by asking students how they know what size clothing to purchase. Most people know what size they
wear. Ask students if they have ever gone to a store and tried on the “right size” only to find that it didn’t fit. Ask students whether or not they think that all clothing of the same size fits the same way. Ask students why they think different manufacturers make clothing of the same size with different measurements.
Explain that students will be working with different measurements and will then use their data to construct graphs and
draw conclusions about the data.
Divide the students into teams of four to five and provide each team with:

Two measuring tapes and/or yardsticks and/or rulers for each team of students

String (use yarn if string is not readily available, but remember that yarn may stretch and skew the measurements)

Data Recording Form

Graph paper
If possible, have both males and females in each group.
INSTRUCTIONAL OUTLINE
Instruct the students to measure the height and arm span of each person in their group. Share with students that a
person’s height should be measured without shoes for better accuracy and that arm span should be measured from
the tip of the longest finger on one hand to the tip of the longest finger on the other hand with the arms outstretched
horizontally. As they measure, they should cut a piece of string equal to the two measurements. This can be accomplished by measuring first and then measuring out the string OR by using the string as the measuring device, cutting
it, and then finding the length of the string. Either method should result in two pieces of string per person and two
measurements per person.
Have the students compare the difference between the lengths of string as they note the exact linear measurements
they find. Have them chart their findings on the Data Recording Form.
Monitor the students as they work. Check for accuracy in their measurements.
As students work through their measurements ask:

What do you notice about the relationship between your height and your arm span?
Students should respond that they are very similar or almost the same or equal. If students have drastic differences
between the two measurements, have them re-measure and check for accuracy in their measurements.

Did you find the same type of relationship between height and arm span among all the team members?
Students should respond that there were a few differences, but nothing significant. Again, if there are significant differences check for accuracy in measuring or in recording of the measurements.

Did you find any differences between the males and females on the team?
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Students should find little or no difference between the data on males and females.
Once students have finished this section of the chart, have them develop a graph to display the data. Students should
explore different ways that the data can be graphed. Most students should discover that a line graph is the best display to summarize how the two pieces of information are related and how they vary depending on one another (dependent and independent variables). Students will need to develop a legend that indicates height versus arm span.
They should use two colors when creating the graph: one for height and another color for arm span.

What conclusions can you draw based on the data you have collected?
Sample answers should include that measurements of arm span and height are similar. Have students re-measure
those individuals where there is a large discrepancy in the measurements.
Say: “Now that you have your data recorded, let’s find out what the ratio of height to arm span really is. Can anyone
tell me what a ratio is?”
Students may respond that a ratio is when you compare two things or that it shows how much bigger one thing is
than another.
Have students give you some examples of ratios that they see in real life.
Students may respond with things such as gas usage (miles/per gallon) or driving speed (miles/per hour).
Ask: “There are three ways that we can write ratios. What are they?”
Students may respond that you can write them like a fraction (3/4), or with words (3 to 4), or with standard notation
(3:4). If students have difficulty remembering one or more of the ways to write ratios, then provide them with the additional formats.
Ask: “Is it important which number goes first?”
Students should respond that it is important, because you have to remember what you are comparing to something
else. For example, if you are setting up a ratio to show the number of miles you can drive in one hour, you would
need to put the miles first and then the hour.
Now, have students calculate the ratio of their height to their arm span using the correct mathematical operation. Notice how students express the ratio. They may write a ratio as 1:2 or 1/2 or 1 to 2. Any of these notations is correct.
Ensure that students write the ratio in the correct order indicated on the chart. The order is very important to ensure
that the ratio is correct.
As students are completing their calculations, ask each group the following question.

Based on the data you have gathered, what is the ratio of your height to arm span?
Students should respond that the ratio is approximately 1:1. Discuss as a group the fact that most measurement tools
that we use do not always produce an exact measurement.
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If the group has discrepancies in their calculations of ratio, pull the group together and discuss the ratios that have
been calculated. Example: Group 1 has a ratio of 27/28 versus Group 2 who shows a ratio of 28/29. Both groups
have an approximate ratio of 1/1 or, in other words, they are almost equal.
Additional Practice
Have students return to their groups and complete the additional practice examples at the bottom of the Data Recording Form handout. Monitor the students as they make their measurements.
Ask students what relationships they found between the:

Height and foot measurements.

Height and shoulder measurements.
Students should respond that their height is approximately 6 times greater than the length of their foot and that their
height is approximately 4 times greater than the width of their shoulders.
Ask: “If you write a ratio showing the relationship between your height and foot measurement, which number will need
to go first in the ratio?”
Students should respond that their height will go first because they are determining the relationship between their
height and their foot measurements. The same will also apply in height to shoulder measurement ratios.
After students have completed their chart and graphs, have them explore other possible relationships. As students
work through exploring different relationships, share with them that early mathematicians also believed in relationships. One such individual was Vitruvius, who in the 1st century B.C. developed a set of proportional relationships for
different parts of the body.
Have students share their results with the class. Have them use their data to support their answers to the following
questions:

What was the conclusion your group was able to draw about the relationship between arm span to height?
Students should draw the conclusion that one’s arm span is equal to one’s height.

The great artist da Vinci used the proportions of the Vitruvian Man, the proportional relationships first developed
by Vitruvius in the first century. From the data you collected, what types of relationships did you discover about
different parts of the body?
Students should share the different relationships that they discovered.

What did you discover regarding the differences in the data for the proportions of males and females?
Students should find few, if any, differences in the data. They should have seen a pattern in which the comparison
between the measurements was very similar. However, they may have had one in which measurements were a little
different from all the rest.

What did you discover about differences in the data for the proportions of taller versus shorter people?
Students should find few, if any, differences in the data.
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Evaluation
Check students’ work as they are completing their measurements and calculations. Compare and contrast the different graphs to view the appropriateness of the graphics used as well as their accuracy. Have students document what
they learned from their investigation in their mathematical journal and then share their comments with the class.
Mathematical journals are an excellent method to assess students’ skills at communicating their problem-solving
strategies. For this activity, students have focused on finding out what the question is asking. Have students write a
short paragraph describing what the problem was and how they went about solving it.
Sample Line Graph
This lesson provides students with practice in developing line graphs, an essential skill required for the GED tests.
The following is an example of a line graph similar to one on the GED Mathematics Test. The graph requires students
to be able to read the information on each axis of the graph, interpret the data provided, and draw conclusions to answer the question. Students who can construct line graphs are much more likely to be able to apply the information
they know about graphs to new situations.
GED-Type Question
The profit, in thousands of dollars, that a company expects to make from the sale of a new video game is shown in
the graph.
What is the expected profit before any video games are sold?
(1) $0—(2) -$150— (3) -$250— (4) -$150,000— (5) -$250,000
$400
Profit in
Thousands
of Dollars
$200
$0
-$200
0
4,000
8,000
12,000
Video Games Sold
Question provided by GEDTS (2005)
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Extension Activity (optional)
Collect all of the data from the entire class. Have students create two sets of data, one for males and one for females.
Have students create a multi-line graph that depicts all data and is color-coded to show males versus females. Students should have four lines on the graph.
You may wish to have students complete another type of graph where more than one item is compared. Have students develop ratios for comparing certain body measurements, such as 6 to 1 for foot to height.
Da Vinci agreed with other proportional values that Vitruvius put forth in his treatise. Share with students some of the
following proportions and have the students provide mathematical evidence for why they agree or disagree with each
statement based on their investigation.

A palm is the width of four fingers.

A foot is the width of four palms.

A cubit is the width of six palms.

A man’s height is four cubits (twenty-four palms).

A pace is four cubits.

The length of a man’s outspread arms is equal to his height.

The distance from the top of the head to the bottom of the chin is one-eighth of a man’s height.

The maximum width of the shoulders is a quarter of a man’s height.

The distance from the elbow to the tip of the hand is one-fifth of a man’s height.

The distance from the elbow to the armpit is one-eighth of a man’s height.

The length of the hand is one-tenth of a man’s height.

The length of the ear is one-third of the length of the face.

The length of the foot is one-sixth of a man’s height.
Have students brainstorm where else they may need ratios. Have them make a list of ratios that they have come
across in daily life. Students may include items such as batting averages in baseball, paint formulas at home renovation centers, recipes, etc.
To the Trainer
You may wish to share with teachers the following background information, which will assist them in guiding students
through the discovery of proportional relationships.
Leonardo da Vinci was not only interested in art, but he was also very interested in the mathematical concept of proportions. Da Vinci studied the proportions of the human body based on an early work of Vitruvius, a Roman engineer
of the first century B.C. Vitruvius influenced Leonardo da Vinci’s work both in architecture and in his drawings of the
human figure.
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One of Leonardo’s drawings is called the Vitruvian Man. It is based on a
model of ideal proportions that Vitruvius established. The drawing shows a
square inscribed inside a circle. Inside is a man with outstretched arms and
legs. In fact, two pairs of each, which touch both the circumference of the
circle and the vertices of the square. By viewing the drawing, the conclusion can be reached that the length of a man’s arm span is equal to his
height. In other words, the ratio of the Vitruvian Man’s arm span to his
height equals 1. You may wish to find the drawing of the Vitruvian Man by
downloading the picture from Wilkepedia at:
http://en.wikipedia.org/wiki/Vitruvian_Man.
It is important that students have the opportunity to obtain data that they
can then graphically depict. Share with teachers the importance of having
students develop their own graphs and interpret graphics developed by
others. Ensure that appropriate legends and labels are used to identify
important information on all charts and graphs.
Vitruvian Man by Leonardo da Vinci.
Retrieved on May 6, 2006 from
http://en.wikipedia.org/wiki/Vitruvian_Man.
This lesson was developed using information and ideas from the following:

Alejandre, S. Leonardo da Vinci Activity. The Math Forum @ Drexel University. Retrieved May 12, 2006, from
http://mathforum.org/alejandre/frisbie/math/leonardo.html.

Constant Dimensions. Illuminations. National Council of Teachers of Mathematics. Retrieved May 12, 2006, from
http://illuminations.nctm.org/LessonDetail.aspx?id=L572.

Leonardo da Vinci Activity: Art, Nature, Ratios, and Graphing. MSTE. University of Illinois at Urbana-Champaign.
Retrieved May 12, 2006, from
http://www.mste.uiuc.edu/courses/mat764fa03/folders/jleel/Standards/da_Vinci_Teacher.html.

Leonardo da Vinci: Creative Genius. EDSITEment. Retrieved May 2, 2006, from
http://edsitement.neh.gov/view_lesson_plan.asp?id=624.

Humans Are Square. NASAexplores. Retrieved May 2, 2006, from
http://media.nasaexplores.com/lessons/04-065/9-12_2.pdf.

Vitruvian Man Ratios. NASAexplores. Retrieved May 2, 2006, from
http://media.nasaexplores.com/lessons/02-032/5-8_1.pdf.

Vitruvian Man. Wikipedia. Retrieved May 6, 2006, from http://en.wikipedia.org/wiki/Vitruvian_Man.

Welcome to Leonardo. Boston Museum of Science. Retrieved May 6, 2006, from
http://www.mos.org/sln/Leonardo/.
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LESSON PLAN 4: HANDOUT
Data Recording Form
Identify each person in the group and place their names in the first column. List the height and arm span in the appropriate columns. For the following measurements, measure each person’s height without shoes. Next, have each
person hold both arms out straight, horizontal to the floor. Measure the arm span from the tip of the longest finger on
one side to that of the same finger on the other side. Record all measurements to the nearest centimeter. Once the
data have been entered into columns one, two, and three, calculate the ratio of height to arm span and enter the ratio
into the final column.
NAME
HEIGHT
ARM SPAN
HEIGHT/ARM SPAN
Do you see any patterns in the data you have recorded? Write a brief description of the pattern that you see.
What do you think would happen to the pattern if you had more measurements?
Additional Practice
Is there a relationship between the length of your foot and your height? How about the width of your shoulders compared to your height? What is the relationship?
Do you think other relationships exist among other body measurements?
Extension Activity
Can you find other pairs of body measurements that show relationships? What are they?
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LESSON PLAN 4: HANDOUT
Graph Paper
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