**Exploring Proportionality through Measurement and
Geometry**

** **

Bruce Olberding (NMSU Math) and Maribeth Olberding (NMSU Math-Ed)

with the assistance of

Sharon Brooks (Math Teacher, West Monroe High) and Stephanie Welch (ULM Math-Ed undergraduate)

** **

January 22, 2004

** **

** **

**Introduction**

** **

One of the main goals of our project was to design an
application of readily accessible mathematics to a physical problem in a
non-trivial way. Often the first applications of mathematics to physical
situations that students (both secondary and undergraduate) see are not
particularly powerful examples. We attempted to design a project that would
provide a convincing and historically important application of elementary
mathematics by basing our project on a 3^{rd} century C.E. Chinese
surveying text, *The Sea Island Mathematics Manual*.[1]
In this project students are required to perform measurements of inaccessible
objects (for example, large buildings) by the use of sighting poles and the
Theorem of Similar Triangles. While it is not unusual for high school students
to see applications of this theorem to measurement, the surveying methods of Sea
Island require the use of two pairs of similar triangles, solving a system of
equations and some geometrical insight. By a more intensive use of the Theorem
of Similar Triangles, the students are able to make measurements of distant
objects without approaching them. Class discussions emphasize the historical
importance of this skill to help the students further their understanding of
the mathematical concepts.

** **

**I.** **Mathematical**
**Discussion**

**A. Applicability of mathematics to physical problems**

It is not unusual for high school students to have seen the theorem of similar triangles used to measure the height of, say, a telephone pole or tree.[2] This is an interesting example and it is well-suited to the secondary level, but it involves what are often unrealistic assumptions—in particular, one typically measures the distance from the telephone pole (or tree) to the observer or sighting pole, and this distance can be quite large. This works sufficiently well with some objects, like a telephone pole, but it requires that the object be quite narrow. By contrast, this method poses difficulties in the measurement of, for example, one of the Great Pyramids. (The base of the pyramid must be accounted for in the measurement.) Also, there are numerous applications where one can only perform “local” measurements in the near vicinity of the observer; in this case, one may not be able realistically to measure the distance between observer and the object of measurement.

Thus this typical application of similar triangles
has a number of shortcomings as an actual technique of measurement. From a
pedagogical point view, it also does not make a very deep use of the theorem of
similar triangles. In order to intensify the mathematics in the use of similar
triangles for measurement, and to help demonstrate its powerful applicability,
we have centered our project on a 3^{rd} century C.E. Chinese surveying
text: *The Sea Island Mathematics Manual*. This particular text outlines
methods for measuring a number of inaccessible objects (sea bluffs, deep
ravines, lengths of fortress walls, etc.). The main method requires the use of
*two* sighting poles, rather than the one that is often seen in textbook
examples of the use of similar triangles. The use of two sighting poles has
the advantage of allowing the observer to perform only local measurements. The
trade-off for this added power is that now the observer must deal with two sets
of pairs of similar triangles. Ultimately this produces a system of two linear
equations in two unknowns that must be solved. Hence the algebraic
manipulation required to solve the problem is more substantial than the
traditional textbook version of the measurement problem, which requires the
solution of a single linear equation in one unknown.

**B. Proportionality and similar triangles**

The basic tool in our use of measurement is the theorem of similar triangles. Students gain familiarity with this concept by employing it to solve problems involving measurement; then later in the activity, they explore how it can be used to provide a foundation for the notion of the tangent function. Also, on the last day of the activity, they explore an interesting proof of this theorem using the “out-in complementary principle” of Liu Hui’s surveying text.

**C. Systems of linear equations**

The Sea Island method of measuring inaccessible distances involves the analysis of two pairs of similar right triangles. This yields a system of two linear equations in two unknowns which must be solved in order to obtain the desired measurements. Thus the activity reinforces the importance of systems of linear equations in applications.

**D. Modeling **

After attempting some of the difficult measurements (e.g. measuring the height of a distant tower), the students better appreciates some of the obstacles to modeling a physical situation using mathematics. We designed our project so that the students would first attempt the measurement of a campus tower without knowledge of the Sea Island method. This poses a number of obstacles to modeling, and the first stage of the activity is meant to encourage students to document these difficulties. Once they see the obstacles involved, they better appreciate how mathematics can be used to help surmount some of the difficulties.

**E. Geometrical origin of the tangent function and an
example of an invariant**

On the last day of the activity the students explore
how, corresponding to a fixed angle θ, one may associate a ratio (the
tangent) that depends only on the given angle θ. That this ratio depends
only on the angle θ is justified using similar triangles. It is also
illustrates that the ratio is an *invariant* of an equivalence class
(under similarity) of a given right triangle.

**II. Pedagogical
Discussion**

**A. Applicability of mathematics**

It is extremely important for future teachers to be convinced of—and to have experience with—the applicability of mathematics. A familiarity with and appreciation for this applicability can be communicated to secondary students, who too often encounter the problem of wanting interesting applications of mathematics yet not having the mathematical or scientific sophistication necessary to understand such applications. With this in mind we have tried to design our activity so that most if not all of it can be transferred to a high school Geometry or Algebra II class.

At the same time, we have sought to make the project
interesting to mathematically sophisticated undergraduates. While the activity
really requires no mathematics beyond the high school level, it does demand a
mature approach to problem-solving. It also synthesizes several aspects of
geometry (similar triangles), linear algebra (systems of linear equations) and
trigonometry (the tangent function *as a function*) in a way that is not
typically seen in a single course in the undergraduate curriculum. We believe
that this synthesis of different subject areas in mathematics is very important
in the preparation of future secondary teachers.

We have also purposely designed a major part of the activity to take place “off the paper.” By having the students attempt difficult measurements outdoors, the activity helps to give future teachers a robust sense of what mathematics entails. Throughout the activity, the students appeared more comfortable with the abstraction of mathematics rather than its application. This is to be expected since most of one’s mathematical training involves such abstraction, but it is useful to keep in mind how much of mathematics was developed to model physical situations.

**B. Historical motivation**

In order to emphasize the historical importance of
indirect measurement, our team chose to make the centerpiece of the activity
the 3^{rd} century Chinese surveying text, Liu Hui’s *The Sea Island
Mathematics Manual*. By discussing this text and its cultural importance
with the students, the mathematics behind it is humanized to an extent that is
sometimes unusual in an undergraduate class. One of the main goals of the
activity is to provide a compelling application of mathematics, and we feel
that discussing this application in a historical context helps make this a
particularly nice application of relatively simple mathematics.

**C. ****NCTM
Standards **

** **

The publication of the
National Council of Teachers of Mathematics (NCTM) *Standards *documents
(1989, 1991, 1995, 1998, 2000), focused significant attention on curricula,
professional development, and assessment issues as they relate to the teaching
and learning of mathematics. The message contained in these documents had a
significant impact on the development and implementation of the plan for this
proposed project. As is noted in the *Principles and Standards* (2000),
the world today is dramatically different from what it was in 1989 when the
first *Standards* document was published. Student access to computers and
the World Wide Web is now common, whereas in 1989 the availability of handheld
scientific and graphing calculators was uncommon. Anything that has been algorithmized
can be done by a computer. This means that actually performing a calculation
is no longer a problem about which most people have to worry. Rather, what
requires attention is determining what calculation to do. In short, people now
need more understanding and less procedural skill. The practice of collecting
and representing data from real world settings and then applying a mathematical
concept to that skill is another leap in the direction that modern mathematics
has taken us. Our project focuses on the application of these skills by
learning from historical sources how to apply the knowledge of their modern day
world to that used in the 3rd century from *Sea Island Mathematics Manual*.

**III. Lesson
reports and student work**

This activity was conducted by Maribeth Olberding (team educator) in her junior-level Modern Geometry course in Spring 2002. The class consisted of 10 students, 5 of whom were mathematics education majors. Students were divided into three groups. They maintained these same groups throughout the activity. A MWF class, the activity took place over three class meetings during the span of one week. Prior to this activity, the students spent the first week of class by studying Oliver Byrne’s colorful version of Euclid’s Elements[3] as a quick (and rather novel) review of the nature of geometrical arguments.

*Day 1.* The activity began with a warm-up
problem meant to reinforce the notion that physical situations often require *indirect*
measurement. This warm-up problem took approximately 35 minutes. Despite the
amount of time devoted to this rather simple exercise, we believe it worthwhile
to get the students thinking about the difficulties sometimes involved in
indirect measurement.

*Warm-up activity:** Using a yardstick, find
the perimeter of the following objects: Group A, classroom door; Group B, one
panel of the classroom blackboard; Group C, classroom window. *

The students, as members of a junior level mathematics course, were suspicious of this exercise. Although they approached it with good humor, they appeared to be surprised by having to accept the imprecision in the use of yardsticks. One surprising outcome of this experiment is it revealed how, despite the training and sophistication of the class members, a very simple application of mathematical ideas to physical problems could raise naïve questions. It provided an interesting contrast to the confidence with which the students had approached the abstract setting of Euclid’s Elements the previous week.

Group A began the exercise with a long discussion of how the problem was “too easy.” Group C began with a curious discussion of whether it mattered if the measurement was made from the top to the bottom of the window, or vice versa. All groups, despite their initial discussions, quickly completed their assignments.

*In-class problem:** Find the length of the
diagonal of the door (blackboard, window). *

A student in Group A began this exercise by attempting to measure the diagonal using the yardstick. She was stopped by the members of her group, who asked the instructor, “Do we do this by hand?” The instructor simply shrugged and left the students to decide for themselves whether to use a yardstick or not. Group C, wanting to use trigonometry, debated whether or not this would be appropriate if the diagonal was at 45 degrees. (They apparently wanted to be certain that the angles involved corresponded to memorized values for the trigonometric functions!) Finally all groups settled on the Pythagorean theorem as not only easier, but more accurate, since the diagonal was hard to follow with a yardstick.

*Follow-up Problem (all groups):**Measure the
length of the diagonal of the classroom. *

In this problem, there was no debate. The students,
taking into account the size of the room and the clusters of desks blocking
potential measurements, immediately agreed that it was easiest *and more
accurate* to measure the diagonal via the Pythagorean theorem. The tile
floor of the classroom allowed them a further simplification: they measured one
tile, then counted tiles to arrive at a quick estimate of the measure of the
diagonal.

** Discussion (approximately 10 minutes):** The
students were asked to return to their seats, and the instructor related the
traditional legend of Thales measurement of the pyramids by shadow reckoning.
She drew a pyramid on the chalkboard with Thales a good distance away. She
asked the students to indicate possible problems for Thales in seeking to
measure the height of the pyramid. The students quickly pinpointed a number of
problems such as the awkward shape of the pyramid (how does one measure it’s
altitude?), and the difficulty of devising a long enough yardstick or tape
measure. The instructor then asked for suggestions on how Thales could measure
the pyramid. One participant in the class, a teacher seeking mathematics
certification, suggested similar triangles. The students reminded each other
of the theorem of similar triangles that, in their words, “corresponding sides
are proportional.” This also led to a discussion among the students of the
difference between similar and congruent triangles.

The instructor related a version of the legend of how Thales used the sun and shadows to measure the height of the pyramid. Thales paces a straight line away from the pyramid until he can sight the sun just over the tip of the pyramid. Having measured the distance from the pyramid, he measures now the length of his shadow. He knows also his height, so using a pair of similar triangles, he calculates the height of the pyramid.

To close the class, the instructor assigned the following group project to be completed by the next class period. The students maintained their same in-class groups for this project.

*Group Project 1 (Due next class period):**
Measure the height of the tower of the **Administration** **Building**
on the **ULM** campus. Discuss the reliability of your measurements
and any obstacles you encounter in making your measurement. *

The students asked questions about how they were supposed to accomplish this. The instructor deliberately gave no clues.

*Day 2. *The students began the next class period by
discussing the successes and failures of their attempt to measure the Administration
Building tower. The instructor asked whether the groups had been successful
in their measurements. The students, despite a diversity of methods (discussed
below) used to solve the problem, expressed confidence in their measurements.
They were reassured by the fact that all 3 groups found the height to be with
the range of 95 to 115 feet. When pushed further on this point, however, they
agreed that this was probably too significant of an error to make their
measurements useful.

*Group B*. This group tried three different methods to
measure the height of the tower.

(1) We just looked at it and estimated the height by our personal imagination. We figured it to be about 120 feet. This is obviously not very reliable or accurate, but it gets you started. To estimate it just by looking at it would be more accurate if you knew the height of a building close to it and could compare them.

(2) Next we counted the windows on the tower and multiplied that (18) by the length of one window (68 inches) which is about 102 feet. This is much more accurate, except the top window is perhaps longer or shorter than the others, but we are probably within 2 feet.

(3) Next we used the legend of Thales to get the height of a light pole in front of the tower. We lined up the end of one of our shadows with the end of the pole’s shadow. We then measured the length of our shadow as well the shadow of the pole. After we knew the height of the pole (19.8 feet), we measured the distance from the pole to the tower (84 feet). We then put our face on the ground in a straight line with the pole and edge of the tower. We moved back until our head lined the top of the pole with the top of the tower. [This discussion was accompanied with a diagram and a calculation that finds the height of the tower to be 106.2 feet.] This is a fairly accurate approach, except since shadows are not definite lines, you are dealing with estimates. You must wait until the shadows are longer to reduce the roundings [?] and make for more exact figures.

Method (3) is interesting in that the students sought to use shadow-reckoning but found that, while simple to understand, it was difficult to achieve in practice. Note that the students did not actually use their shadow measurements; evidently they felt compelled to carry them out in order to mimic what Thales did, even though the information proved extraneous for them.

*Group A*. This group tried 3 different ways. They
stated, “We did not use the legend of Thales or trigonometry because the shadow
was going across the top of [another] building and could not be measured.”
Note that this group did not recognize that Thales method could also be applied
*without* shadows.

(1) The students first tried the same method as (2) of Group B. Measuring a window and multiplying by the number of windows, they arrived at a height of 102 feet and 2 inches. They recognized that “The error in this method is having to estimate the height of the stucco panel at the top….We are also assuming that all the panels are exactly the same. If this is not true then our error will be greater.”

(2) For their second method, the students turned to the Internet. There they discovered (on a website entitled “Bizarre Stuff”!) how to use a device called a “hypsometer” to measure the height of a tree. The website discussed how to use a yardstick as a hypsometer. “Stand exactly 25 feet from the tree being measured. Hold the yardstick, with the zero end downward, 25 inches from your feet. Line up the bottom of the yardstick with the base of the tree. Without moving your head, look to the top of the tree. Where it crosses the yardstick, read of the measurement in inches. Each inch will equal one foot in the tree’s height.”[4] The students used the hypsometer to measure the tower as 114 feet high. They stated that “the error in this method is the accuracy of our measurement, both the 25 feet segment and the 25 inches from our eye to the yardstick. Looking up without moving your head also leaves room for error.”

(3) The third approach of Group A was particularly novel. “We attached ribbon to a bunch of helium balloons. [See picture.] The balloons were then raised to the top of the tower, and we measured the ribbon. This was the least accurate of our methods because of the wind. We did this twice and the wind was blowing in the opposite direction, so we decided to average our 2 tries.” They computed the height with this method to be 106 feet. “The more times you do this, the closer your answer will be.”

*Group C*. This group tried two methods. In the first
method, which was also but the other two groups, they measured the height of a
window on the tower and multiplying according to the number of windows. They
arrived at a height of 103.92 feet. In the second, they used similar triangles
in a fashion similar to that of Group B, method (3). They calculated the
height of the tower to be 114.9 feet.

** Conclusion:** We view this group project as a
good way to emphasize the difficulty of putting mathematical ideas into
practice. In fact, the students appeared the most confident in their estimates
that used a minimum of mathematical reasoning, namely measuring the height of a
window and multiplying by the number of windows. Despite the juxtaposition of
the story of Thales with the assignment of the group problem, the students
appeared distrustful of their estimates achieved using this method.

We were pleased with the novelty of some of the students’ solutions, such as the use of the hypsometer or the balloons. Also, we had not anticipated that the students would use the windows as a means to measure the tower. We were pleased in the students desire to try several methods in order to confirm their estimates. As mathematics educators, we would have liked to have seen the students introduce novel mathematically based approaches to this project. Yet viewed against the background of history, the introduction of new mathematically based approaches to measurement is a rather exceptional occurrence.[5]

** Sea-Island Discussion:** After discussing the
successes and failures of the measurement of the height of the tower, the
students were presented with a new twist on this same problem:

At this point the instructor introduced *The Sea
Island Mathematics Manual,* and pointed out that as early as 263 C.E. the
Chinese were employing sophisticated arguments involving sets of similar
triangles to measure inaccessible distances.[6]
The motivating example is that of Sea Island, a distant bluff separated from
the observer by a small sea. Liu Hui, the author of the text, outlines a
solution for this problem that involves only “local” measurements, that is,
measurements that can be accomplished in the near vicinity of the observer.
The basic idea is to plant two same height sighting poles, or “gnomons,” a
fixed and measurable distance apart (see figure below.)

The poles lie on a line emanating from Sea Island. Lying on the ground, the observer then sights the top of Sea Island over the top of each pole; the distances A and B from each sighting position to the sighting pole are then measured. The following five measurements are known: the height of the poles, the distance between the poles, and the two distances A and B. Note that all five of the distances are easily obtained in the immediate vicinity of the observer. Using the theorem of similar triangles, it is now possible to compute the height h of Sea Island; also, this same computation produces (rather than requires!) the distance d from the first sighting pole to Sea Island.

To reinforce the Sea Island method, the students were given the following in-class activity to simulate the use of sighting poles.

*In-Class Group Project (approximately 30 minutes):**
Using yardsticks as sighting poles, measure the following distances using the **Sea**
**Island** method: Group A: floor to the top of the chalkboard;
Group B: height of classroom door; Group C: floor to the top of window. *

Since these measurements had to be accomplished in a very small space, the students were given fishing line to simulate the line of sight. This also allowed the students to visualize the formation of the sets of similar triangles to be used in the computation of the heights.[7]

In completing the in-class assignment, all three groups encountered difficulties in obtaining their measurements. For example, one group pointed out that the measurements would be incorrect if the fishing line was not taut. Another group discussed whether or not the problem could be solved if the “shadows” of the sighting poles (i.e. the base of the small right triangle formed by the sighting pole and the fishing line) overlapped. Reaching no conclusion, they simply moved their sighting poles farther apart until the “shadows” no longer overlapped.

To end this class period the three groups were given a new project:

*Group Project 2**: (Due next class period)
(a) Use the Sea Island method to measure the height of the tower on the
administration building. (This is the tower that they were asked to measure
last time.) (b) Measure the height of the campus bridge over Bayou Desiard.
(The students were not told to use the **Sea** **Island**
method for this measurement. We wanted to see if they would adapt the method
to this rather different setting.)*

The students were also given a bonus challenge problem:

*Challenge Problem**: (Due next class
period) Measure the height of the administration building from the seventh
floor of the university library. All measurements must be obtained while
positioned on the seventh floor of the library. *

*Day 3.* The three groups returned the next class
period with completed projects. Groups A and B found the height of the tower
to be approximately 100 and 102 feet, respectively. Group C found the height
to be about 123 feet. All three groups used the Sea Island method correctly.
Group B commented that the “obstacles are keeping meter sticks perpendicular
and the grass isn’t completely flat.” In class the students also commented
that it also could be difficult sighting the top of the building while lying on
the ground.

All 3 groups were able to successfully adapt the Sea Island method to the measurement of the height of the bridge above Bayou Desiard. Group A found the height to be 15 feet 3 inches. Group B measured the height at 186.35 inches. Group C measured the height at 13.46 feet. Group A confirmed their measurement by lowering a stick on a string from the top of the bridge till it touched the water. They then measured the length of the string at 15 feet 1 inch. They stated, “We did this method because we wanted to be more sure of our measurement from Liu Hui’s [Sea Island] method.”

Only one group (Group C) attempted the challenge
problem. The challenge problem can be solved by a modification of the Sea Island
method. However, the yardsticks have to be used in a different way. Liu Hui
solves this problem in *The Sea Island Mathematics Manual* by the use of
carpenter squares, which can be formed by joining the ends of the two
yardsticks to form a right angle. Ultimately the problem reduces to two sets
of similar triangles (cf. page 49 of *Sea** **Island*).

Group C was not able to adapt the Sea Island method
to this measurement of the height of the administration tower from the 7^{th}
floor balcony of the library. However, they did use an imaginative method to
obtain the measurement.

We
measured the distance from the bottom of the 6^{th} floor to the bottom
of the 7^{th} floor (= 191.5 inches). This allowed us to determine the
height from the ground to the top of the 6^{th} floor (= 1149 inches).[1]
While standing on the 7^{th} floor balcony, we realized that we were
not quite as high as the tower. This presented us with a problem. How do we
estimate the height remaining [i.e. the difference in height between the
library’s 7^{th} floor balcony and the administration tower]? The
solution was not in front of us, but behind us. We found the reflective
windows of the library [which were behind them] and were able to use the image
before us as a gauge. Using the treeline as a reference point, we took notice
of where our eyes were lining up with the tower. We used Sam’s eyes because
his fell right at the top of the highest pane of stained glass [on the
administration tower]. His eyes are 65.25 inches off the ground. With only
the height of the concrete section of the very top of the tower unknown, we estimated
it to be half of a window pane [on the tower] by observation from the balcony.
We measured the height of a window pane and then put the numbers together (=
103.95 feet to the top of the administration tower).

The students located three sources of possible error: “(1) The height of each floor of the library should be consistent, however it may be a little off; (2) Even though the reflection in the mirror [reflective glass] appeared to be a true representation, it could have shown a slight distortion; (3) Estimating the top concrete section to be half of [administration tower’s] stained glass window pane.”

In the class discussion that followed, the students expressed confidence in their measurements of the Administration tower. However, they all recognized that small errors in their measurements using the sighting poles could translate into large errors in the computation of the height of the tower.

At the close of the discussion, the students
inevitably asked for the “right” answer, namely, what *is* the height of
the administration tower? They were unhappy to learn that they were the
experts on its height. While of course it would be possible to learn the
height of the tower from data the university keeps on its buildings, we felt it
better to leave this a mystery, since the point of the Sea Island Method is to
measure inaccessible distances and to gain confidence in the mathematics that
replaces direct measurement.

** Follow-up assignment: **In addition to the Sea Island
problem, there are 11 other surveying problems in the Sea Island Mathematics
Manual, some of which are quite challenging. The instructor assigned two of
these problems for homework:

*Individual Homework Assignments:** (a)
Design a method using two sighting poles to measure the height of a pine tree
on a hill; (b) Design a method using two sighting poles to measure the length
of a fortress wall. [8]
*

This discussion and the assignment of the individual
homework problems took approximately 10 minutes of the class period. The rest
of the class period was devoted to further exploration of similar triangles.
In particular, the instructor led the class through Liu Hui’s proof of the
theorem of similar triangles using his “out-in complementary principle.” (See
pp. 37-39 of *Sea Island* for a discussion of this proof.) This took
approximately five minutes.

The class then explored an invariant for similar triangles, namely the ratio of the lengths of two sides. Of course, this ratio coincides with the tangent of an angle of the triangle, but the instructor did not mention this connection, preferring instead for the students to discover this on their own. This was accomplished through the following series of in-class group activities.

*In-class activity:** On a blank piece of
paper positioned on its side, draw a horizontal line near the bottom of the
page. (Inch grid paper also works very well for this exercise; in what
follows, substitute “inch” for “centimeter.”) Draw a slanted line that meets
the left endpoint of the horizontal line at an acute angle θ. Using a
ruler, number marks every centimeter along the horizontal line. (See figure.)
Draw a vertical line from each of the centimeter marks to form a series of
trapezoids along the horizontal line. *

* *

*View the diagram as compromised of a series of similar
right triangles. Note that each triangle shares the angle θ (angle BAC in
the diagram). Construct a table of measurements A and B, where A represents
the length of the base of a triangle, and B represents the length of the side
opposite the angle θ. Finally, in a third column, compute the ratio B/A.
What do you notice? Can you explain what you observe? *

After the students completed this activity, the instructor gathered the data from the students and asked them to explain how each group arrived at a different constant ratio. The students recognized immediately that the ratio should depend on the angle θ in the diagram. As each group had begun with a different diagram, the constant ratios differed also. The instructor summarized the conclusion: To each angle θ, one may associate a unique number. It can be computed as the ratio of the lengths of two particular sides of a right triangle with angle θ. In this way, one can define a function f(θ) = B/A, where B/A is computed as above (and of course depends on the angle θ ). At this point the instructor posed the question of whether this defines a new function? After some discussion, the students concluded that this is the familiar tangent function.[9] The instructor summarized this by emphasizing that our familiar tangent function can be shown to be well-defined because of the theorem of similar triangles.

**III. Conclusion**

By having the students discover for themselves many of the difficulties involved in collecting data and modeling a physical situation, we believe they achieved a greater appreciation of the powerful tool that mathematics can be in measuring inaccessible distances. By exploring the theme of similar triangles in a number of ways, we hope the future teachers in the class will think in a richer way about a fundamental theorem of geometry. As the activity progressed, the students traded complicated measurements and simple use of the Theorem of Similar Triangles for less complicated measurements and deeper use of this theorem. At the same time, they saw that mathematical insight (in particular the use of two sighting poles, rather than one) led to a more powerful means in which to obtain measurements of distant objects. This gave students a sense that the measurements get easier and more accurate as the use of mathematics involved becomes more sophisticated. As instructors we were pleased with the enthusiasm which the students undertook their projects and with the interesting and inventive approaches they discovered to some challenging problems.

[1]
Frank Swetz, *The **Sea** **Island** Mathematical Manual:
Surveying and Mathematics in Ancient **China*, Penn State Press, 1992.

[2]
Or perhaps it is unusual in the sense that many students probably do not
actually perform this measurement for themselves. Instead, they may work a
problem from a text in which the data is already given. The mathematician
David Mumford writes that, “I tried (unsuccessfully) to get each high school in
which my children were enrolled to go outside during geometry and find out how
tall the oak in the yard really is.” (This quote is taken from the preface to
the pamphlet *Drawbridge Up*, by Hans Magnus Enzensberger, A. K. Peters,
1999.)

[3] A digital reproduction of Oliver Byrne’s 1847 text can be found online at http://www.sunsite.ubc.ca/DigitalMathArchive/Euclid/byrne.html. Byrne replaces mathematical notation with a system of color-coded references to angles, segments, triangles, etc.

[4] This device and its use were unknown to the members of our team. We think however that an excellent group project in the context of this lesson would be to prove that it works. Ultimately, it is unsatisfactory for the more difficult measurements we have the students perform next. In particular, the need to pace off the 25 feet from the tree (what if the tree is a distant bluff separated from us by a small sea?) is a deficiency in the use of the hypsometer.

[5]
The students, in an in-class discussion after completing the project, made some
interesting “sociological” observations. They remarked on the comical effect
of *a mathematician* lying on the ground in order to accomplish a
measurement. This is perhaps an example of the cultural perception of
mathematics (even among mathematics majors!) as an abstract, austere activity.

[6]
It is evidently unclear whether the Chinese used ratios in their computations
with similar triangles. They appear to have used, instead, calculations
involving areas of rectangles, where the rectangles inscribe the relevant right
triangles. While we did not have time to incorporate the Chinese method into
the activity, we believe it would make an excellent topic of exploration in
either a high school or college class. Interestingly, the Chinese seem to have
performed their calculations by cutting rectangles from diagrams and
manipulating these rectangles. The mathematics involved is clever but
elementary. Again, this is a nice way to emphasize a more robust conception of
mathematics than is often seen by undergraduate mathematics majors, who are
accustomed to manipulating formulas rather than strips of paper. For a
discussion of this technique (“the out-in complementary principle”), see pp.
39-41 of *Sea** **Island*.

[7] Instead of these contrived classroom measurements, the students could have been taken outside as a group and given a more realistic outdoor group project to reinforce the Sea Island method. However we preferred to have the students discover the experience (and difficulties) of performing the actual measurements outdoors on their own.

[1] Note that the students were
attempting here and throughout their solution to follow the instructions of the
Challenge Problem to make all measurements while on the 7^{th} floor.

[8]
The solution to these problems can be found in *Sea** **Island*,
pp. 43-46.

[9] It is interesting that the students did not anticipate the connection with the tangent function until the final discussion.