Students will probably have heard about some ratios that are said to
exist in the body. For examples, it is often said that if you open your
arms wide and measure the length from the fingertips of one hand to the
fingertips of the other and compare this to your height, the two
measurements are equal. Another example is the claim that if you
measure your height and compare it to the distance from the floor to
your navel, then the resulting ratio is approximately 1.618:1, which is
also known as the golden ratio.
If students measure these ratios on their own bodies, they may
notice that the ratios are not exactly what the students predict, but
they are probably pretty close. One question that will naturally arise
from this is, Which body ratio will be closest to being the same for
every person? This activity allows students to find the answer. All of
the ratios they will explore compare body measurements to height. This
will allow students to discover which body measurement is the best
predictor of height.
Data Collection
First, you need to decide how many body measurements each student will record. The Class Reporting Activity Sheet identifies 10 body measurements, and allows your class
to choose up to 4 additional body measurements to work with. Decide as
a class what other measurements might be good predictors.
Class Reporting Activity Sheet
Students should work in pairs to take each other's measurements.
Before students begin working, assign each student a row number on the
activity sheet in which to record their personal data. If you have more
than 20 students, copy the activity sheet and adjust the numbering on
the second page.
Create an overhead using the Class Reporting Activity Sheet for students to report their individual data. Once the
overhead is filled in, each student or pair should copy all the data
onto their own activity sheet.
Analyzing the Data
Allow students to continue working in the same pairs. Assign each pair one body measurement to investigate. Distribute the How Tall? Activity Sheet to all students. Assign each pair a body measurement to
compare to height. If you have fewer body measurements than student
pairs, assign a particular measurement to two or more pairs, and have
those pairs compare their results before reporting to the class. If you
have more body measurements than students, assign more than one
measurement to each student pair.
How Tall? Activity Sheet
To calculate the correlation coefficient, r, and linear equation, y = mx + b, students can use the Line of Best Fit applet or a graphing calculator, depending on the technology available in your classroom.
Line of Best Fit Applet
As students begin calculating r and y = mx + b, ask them which measurement should be the independent (x) variable, and which should be the dependent (y)
variable. Biologically speaking, your height does not determine the
length of your foot, and the length of your foot does not determine
your height—they are both controlled by your genetic makeup. There may
be a correlation between these values, but one does not determine the
other. However, in this activity, students are predicting height. The
goal is to determine height when you already know another measurement,
making height dependent on this other measurement. The other
measurement (e.g., length of your foot) is therefore the independent (x) variable.
Discuss the meaning of r, the correlation coefficient, with students. It measures how close data are to forming a perfect line. The closer the value of r is to 1 or –1, the closer the data are to forming a line. If r
is 0, the data have no correlation, meaning that the variables are not
related to each other and there is no single line that can be drawn
through all the data points. The graph that produces the r-value closest to 1 represents the best predictor of height for your class.
Student groups should report the r-values and linear equations for their measurement(s) on the How Tall? Overhead. Student can then copy the data onto their How Tall? Activity Sheets. If you wish to collect these activity sheets later,
make sure students mark which measurement(s) they were responsible for.
How Tall? Overhead
Give students some time to discuss Question 1 on the How Tall? Activity Sheet, and then discuss the question as a class. The measurement that produced the r
value closest to 1 is the answer because that measurement and height
have the highest correlation. You may also ask these questions during
the discussion:
- What happens if there are 2 r-values that are similar?
[If 2 r-values are exactly equal, your class could
conclude that they are equally good at predicting height. An
interesting follow-up question in this situation is, What is the
correlation coefficient between these 2 body measurements?]
- If there are 2 r-values that are approximately equal,
how much of a difference would be needed in order to determine which of
the 2 measurements is actually a better predictor? A thousandth? A
hundredth? A tenth?
[This is a very difficult question to answer in an
absolute sense. You may wish to encourage a discussion about how
accurate students believe their measurements were. Could they have been
off by a centimeter? a millimeter? It would be reasonable for students
to believe that their measurements are accurate to the nearest
centimeter. Looking at their r-values, then, they could conclude that values similar to the nearest hundredth could be considered as being equal.]
To test your class's results, invite students (of similar age) in
from another class and have your students measure them on the specific
body measurement that was the best predictor of height. Have your
students then use the linear equation to predict the heights of the
visitors. Your students can then measure the heights of the visitors to
see how accurate their height predictor is. Have students record their
values and answer Questions 2 and 3 on the How Tall? Activity Sheet. If it is difficult to bring in other students, have
students complete this part of the assignment with a friend or family
member at home.
If the results of measuring another student are not what was
expected, take time to discuss why this might be. Some possibilities
are:
- Humans grow in a non-uniform way—different body parts grow at different rates. This is called allometric growth.
- The height predictor is making a prediction based on a
population of a classroom, not the characteristics of a single
individual.
- Gender differences were not taken into account in this
activity. There may be a different body measurement that is a better
predictor for one gender than the other.
Questions for Students
1. What is the meaning of r?
[The r-value is called the correlation coefficient. It measures how close data are to forming a perfect line.]
2. How do you know whether 2 measurements are related to each other?
[The closer r is to 1 or –1, the closer the data are to forming a line, and the more one variable is related to another. If r is 0, the data have no correlation, meaning that the values plotted on the graph are not related to each other.]
3. If you were to graph hair length vs. height and found that r = 0.3, what would this tell you?
[There is very little relationship between a person's hair length and their height.]
4. Do ratios exist in the body that can be used to accurately predict a body measurement?
[The answer is the relationship that gives the r-value closest to 1 for your group of students. If there were no relationships that gave an r-value
close to 1, then students could conclude that ratios in body
measurements cannot be used to predict specific body measurement
lengths.]
Teacher Reflection
- Were students engaged by the idea that ratios exist in the body?
- How did your lesson address auditory, tactile and visual learning styles?
- This lesson has many transitions embedded in its structure.
What worked with classroom behavior management? What didn't work? How
would you change what didn’t work?