Many math textbooks include applications of mathematics. It is important that students see that mathematics is a useful and practical subject, and worth their time and attention. Studies have shown that students have a greater interest in learning when they have clear and concrete examples of the applications of the mathematics that they are studying.

The concept of indirect measurement is one that is found in most geometry textbooks. In this project, students have real hands-on experience with the properties of right triangles. Creating a poster of their work gives them additional practice, a more concrete experience, and, perhaps most important, a sense of "ownership" of the material.

On reading or hearing a theorem, a student might remember it. If the student writes the theorem on paper, he or she is more likely to remember it. Applying the theorem further increases understanding as well as retention. Physically measuring and then creating a presentation of the theorem embeds the concept in memory, and furthers a true understanding of the concept.

The "Geometrees" Project is an application of similar triangles. Many textbooks use indirect measurement as examples of the applications of similar triangles (and later in many geometry courses, as applications of trigonometry). In the textbook examples, problem statements include data and a diagram, and the student is asked to permform the necessary calculations to determine the answer.

In this "hands-on" project, the students have the opportunity to go outside the classroom, select a tall tree, and gather their own data, which they will then use to calculate the height of a tree on campus. This project works best if the students work in groups of three or four. The students choose a tree, with a prominent shadow (away from other shadows or obstacles and on level ground). Then they measure the length of the shadow cast by the tree, using a tape measure, and record this information. Then a student holds a yardstick vertically near the tree, and another student measure the shadow cast by the yardstick. (If a yardstick is not available, any straight stick will do, but a tape measure will be needed to measure the stick.)

As the text is unreadable at this size, I have re-written a summary of their explanation below the picture.

Since the triangles are similar, we can now write a proportion: x/1360 = 63/140

In this equation, x is the height of the tree, 300 is the distance in inches from the base of the tree to the end of its shadow, 60 is the height of the observer, and 55 is the length of the observer's shadow.

There are two triangles formed. One is on the left, formed by the height of the tree and the length of the it's shadow. The other is on the right, formed by the height of the surfer and the length of his shadow. We set up a proportion based on these two similar triangles. (The triangles are similar by the Angle-Angle Similarity Theorem, using the right angles of height, and the congruent angles formed by the angle of the rays of the sun which cause the shadows.) So, if we call the height of the tree x, we can write the equation x/1360 = 63/140 and we found the height of the tree to be approximately 51 feet.

Then the students did a second project using a different method, though still based on similar triangles. This method was called "The Mirror Method".

The students should then make accurate drawings to record the data. If the students create a poster displaying their work on this project, they will always remember it, and really "own" their learning. The example below was created by a group of my geometry students, using the "Mirror Method". As before, the text is unreadable at this size so I have re-written a summary of their explanation below the picture.

"We used a mirror and a yardstick to measure a tree with similar triangles. The reason that the two triangles are similar is the "angle-angle similarity theorem: if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. One pair of angles is at the foot of the tree and at the foot of the person (although the tree is not straight, it's height is measured on a vertical straight line, as is the guy's height so these are both right angles). The other pair of angles are at the point in the diagram that is shared by the two triangles. Those two angles are congruent because "the angle of incidence is equal to the angle of reflection", a law of physics.

Since the triangles are similar, we can now write a proportion: x/600 = 60/55

In this equation, x is the height of the tree, 600 is the distance in inches from the base of the tree to the mirror, 60 is the height of the observer, and 55 is the distance from the observer to the mirror.

Using this method, we calculated the height of the tree to be approximately 54 feet. This is 3' more than our previous calculations, which shows you that it is a fairly accurate method, but not completely accurate."

In summary, the students found this to be an interesting and worthwile project, They enjoyed the opportunity to go outside the classroom, and to do something that had a real and concrete application in "real life". They also found that working together was very helpful, and asked if we could do this kind of thing again sometime soon. And, when questions about similar triangles showed up on tests, the students found they had no trouble remembering the concepts!