How Tall Is That Tree? Practical Math

I taught in a school district at the confluence of the Muskingum and Ohio Rivers. Remains of ancient Hopewell mounds remained in the town, though many of the structures there when the area was first settled as part of the Northwest Territory had been turned into bricks during the early 19th century. One elementary school I taught in was adjacent the public library which had been built on top of one mound. The tallest Hopewell burial mound was in the town cemetery.

Early excavations of burial mounds revealed evidence that the Hopewells traded with groups from the Atlantic Ocean to the Great Plains and from Lake Superior to Gulf of Mexico. (Excavations stopped when the mounds were recognized as burial site.) The Hopewells traveled by water, and dugouts were the mode of transportation. The East had been heavily wooded, a forest so dense it was said that a squirrel could travel from Long Island to the Mississippi River jumping tree to tree. Hopewell shelters were also built of wood and bark. By the middle of the 19th century, millions of acres of forest had been cleared. On a lecture tour to the region, Emerson recorded huge uprooted trunks were used as fencing.

One spring as part of an outdoor education program for sixth graders, I decided to teach a class on how to determine the height of a standing tree. It seemed a useful demonstration of applied math and a good way to gain insight into how Native Americans might have judged a tree adequate for their purpose. Felling a tree by hand, especially one big enough to make a dugout, is a major investment in labor. I’m sure the Hopewells practiced what many carpenters preach today: measure twice, cut once.

I chose to teach two quick and dirty methods, as demonstrated in How to Measure a Tree. The only tool required is a yardstick or tape measure. These methods used mathematical concepts but ,, except for linear measurement, in an unobtrusive way.

Two other methods provide more accurate results, though neither method would have been used by Native Americans. The first method uses the lengths of shadows and height of a known object and proportion to solve for the unknown in two similar triangles. How Big is Your Tree is a handy one page PDF that covers the same concepts. It also describes measuring tree circumference which would have been a consideration when felling a tree to make a dugout.

The second method requires students to use or to fashion a clinometer with a straw, protractor, and weighted cord. Some of my sixth graders, especially those who struggled in math, were put off by the word trigonometry, so I turned the problem on its side and talked about how surveyors used the ratio of sides of triangles to their angles.

Still another method, and one I didn’t know of at the time, uses a mirror and a tape measure to solve for the unknown side of two similar triangles.

The website Science and Plants for Schools has How To Find Out The Height Of A Tree which explains five ways to estimate and measure the height of a tree. A student worksheet asks students to determine which is the most accurate method and why.

The Tree Height Calculator tool allows a student to enter her measured height and her shadow length and shadow length of the tree to calculate tree height. This tool can be used to check accuracy of calculations but not of the measurements used.

Having students try all these methods will accomplish at least two things. First, they demonstrate that there is often more than one way to solve a problem. And second, they serve as examples of math being used in real life. We may not need to build a dugout, but if you’re like me who lives in a house surrounded by trees growing in thin soil on top of shale, you might like to know which trees are tall enough to fall on your house in a storm. All the methods can be used to calculate the height of other objects as well.

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