In Advanced Algebra, students are required to do one project each marking period. These projects are usually done alone or in pairs from projects listed in the textbook. I thought that doing more of a class project would be a neat thing to do for the first marking period. During the first marking period, students study concepts such as relations, functions, variation and graphs, and fitting a model to data. Students also use their graphing calculators throughout the course. I thought that a real-world investigation that would tie these concepts together and incorporate other areas such as science, social studies and language arts would be interesting, fun, and a great learning experience. Trees are very plentiful in our area so I thought they would be good subjects to investigate. The question was "which tree species should we study?" At first, the sugar maple came to mind, but I was inspired by a stand of virgin white cedars in "the valley of the giants" at the Sleeping Bear Dunes National Lakeshore on South Manitou Island in Lake Michigan. These were huge trees, one of them being about 5.4 meters in circumference. The cedars are believed to be over 500 years old (Jans, 1999). One of the largest trees was actually dead and only partially standing and I wondered, "How tall was this giant?" So I decided that we would study cedars, but any species of tree would do. Here is a summary of the project.

This project will give students the opportunity to be mathematical and scientific researchers, that is, they will be hypothesizing, observing, measuring, and collecting and analyzing data. They will use their prior knowledge and recent instruction to find a rule relating the two variables DBH and height. They will then use their rules to make predictions.

At the conclusion of this project students will be able to: 1) identify several types of trees common to the Upper Peninsula, 2) determine DBH and height of trees, 3) use their knowledge of patterns, relationships, functions, data analysis and technology to find a rule relating DBH and height and 4) make predictions based on their findings.

The projects will start with a pretest covering right triangle trigonometry, similar triangles, DBH, tree identification, and prediction of what a plot of DBH vs. height data would look like. In this project, students will be trying to find a relationship between DBH and height of white cedar trees. The goal is to be able to predict the heights of cedars by finding the DBH and plugging it into a formula. The surroundings of the trees will also be considered. Students will be responsible for collecting data in addition to analyzing it. Prior to data collection, a number of things must be done.

We can either walk through the forest or on the nearby Michigan Tech trails. I will point out examples of the trees and have students make sketches and take notes on the general shape, bark, foliage, etc. We will also identify several other types of trees in the area and learn to distinguish cedars from those trees. Cedars are pretty unique, so I do not think it will be a problem. In addition to being able to identify the cedar, students will need to be able to gather the data we want - DBH and height.

Depending on the availability of measuring devices, there are several ways DBH can be determined. Using a diameter tape is the quickest. Simply wrap the tape around the tree at breast height, which is about 1.4 meters from the ground, and read off the diameter. If a diameter tape isn't available, a flexible measuring tape or a piece of string can be used to measure the circumference (c) of the tree. Circumference is equal to pie (p) times diameter (d). Thus d = c/p. So divide the measured circumference by p (or approximately 3.14) and diameter is obtained. Next we will learn a couple of methods for estimating the height of a tree.

If you are measuring a short tree (3 meters or less) you can simply use a meterstick to measure from the ground to the top of the tree. However, the heights of taller trees must be estimated. There are two methods that I will teach the students. Method one is the more accurate of the two and involves the use of a clinometer. A clinometer can be used to measure the angle of elevation (a) from a persons eye level to the top of a tree. Refer to diagram 1 below for this method.

Once a is found, the distance (d) from the person to the base of the tree must be measured. Once it is measured, we can find x by using the tangent function. Tan (a) = x/d so x = d*tan (a). Then you must add the height (t) of the person's eyes (from the ground). So the total height (h) of the tree will be d*tan (a) + t. So, for method one, you need a clinometer (to measure a) and a measuring tape or meterstick (to measure d).

The second method I will teach the students is one I learned at an inservice on October 13, 2000, at the Lake Linden-Hubbell School Forest from Bill Jarvis, a science teacher in the district. It does not require the use of a clinometer and is not as accurate, but it is good enough for our purposes. The equipment you do need is a measuring tape and a straight stick that is a little longer than your arm. A meterstick is ideal. You also need to be standing so that you have good view of the tree, especially the top and bottom. Diagram 2 gives a basic picture of the situation.

You want to stand facing the tree with arm outstretched holding the sick straight up (parallel to the tree). The distance from the top of the stick to your fist should be the same as the distance from your eye to your fist. For safety, the distance from the notch on top of your shoulder to your fist is about the same (see Diagram 3).

Move forwards or backwards until the stick (from fist to top) exactly covers the tree from the base to the top. Make sure you keep your arm straight out at all times. Then measure the distance from where you are standing to the base of the tree. This is the approximate height of the tree. How does this work? Well, this method uses similar triangles (see Diagram 4). In

Diagram 4, triangle ABC is similar to triangle ADE.

The distance from eye to fist (AC) and the distance from the fist to the top of the stick (BC) are equal; therefore, the distance from the eye to the base of the tree (AE) equals the height of the tree (DE). We use the distance along the ground to the base of the tree instead of AE, because it is nearly the same and easier to measure.

Those are the methods we will use to approximate heights of trees. We will take a couple of class periods and practice these two methods outside, finding heights of nearby trees, buildings or flagpoles. After that, students will be ready to collect data on their own. They may do this anywhere they can find cedar trees and are expected to do most of the data collecting outside of class. I will suggest that they get together in pairs or small groups to gather the data. It will make measuring a lot easier, although it can be done alone. I will have several clinometers, tape measures, diameter tapes and metersticks available for check out. I also expect that some students will have metersticks and/or tape measures at home. Students can also build their own clinometers using a protractor, straw, string, coin or washer and some tape. In lieu of a measuring tape, students can measure distances along the ground using their pace or walking heel-to-toe and counting steps. I want to make sure to discuss these options so that a lack of materials isn't an excuse.

I expect to have about 20-25 students in my class. I would like each of them to collect data for 5 trees. These would make 100-125 pairs of data to analyze. I will also consider giving extra credit for additional data. I will give students 2-3 weeks to collect data. This time frame should allow each student to have the opportunity to check out measuring devices. Students will need to fill out information sheets for each tree including their name, location of tree, DBH, height and method used. They will also need to describe the surroundings of the tree. Is it in the middle of a wooded area, standing alone in a clearing, or on the border of woods and a clearing? I need to remember to tell them not to measure cedars that have been trimmed or shaped, such as cedar hedges. After all data is in, I will compile it and give it out to each student. They will then be assigned groups. Each group will be responsible for analyzing the data and coming up with a model for a relationship between DBH and height. Each group will make a poster, write a brief report and give a presentation on their findings. In their reports, they will be asked to predict the height of the giant virgin cedar tree that I talked about at the beginning of this paper. The reports will also include a section on why having models for finding heights of trees is important. A post-test, exactly the same as the pre-test, will be given at the conclusion of the project.

This project gives students a chance to experience real-world applications of math and science and addresses numerous standards and benchmarks from the Michigan Curriculum Frameworks. This project covers benchmarks in the first five Mathematics Content Strands and it also addresses benchmarks within the first three Science Content Strands (Michigan Department of Education, 1996).

We started in on the project a little later then I wanted to. I am teaching, for the first time, one of the four sections of Advanced Algebra and am expected to keep pace with the other classes. We cover a section per day with very few exceptions. I was told we have to keep up that pace in order to cover everything the students need to know before the MEAP and ACT tests are given. Therefore, we only actually went outside one day to identify cedar trees and practice measuring them, while I would have loved to spend several days doing it. We talked about the project many times and took the pre-test a couple of weeks before actually going outside. When we finally had a suitable day to go out, I showed them some cedar trees in the back of our school and had them practice measuring them. They had fun and thought the clinometers were neat.

Then I divided the class into 5 groups of 4 to 5 students and gave them their instructions. Each person needed to collect data on 5 trees and extra credit was given for each additional tree (up to 5). They then needed to analyze their group data and come up with the best equation they could find to express the relationship between DBH and height. They also had to analyze the data of the entire class and again find the best equation they could. We discussed how to use their graphing calculators to find various regression equations and the corresponding correlation coefficients. Then they had to experiment and find the best equation they could.

Some groups found that linear equations fit best for their group data and some found that other types of regression equations worked better. However, when all of the class data was analyzed, all groups found that the best equation was a power regression equation which is of the form y = a*xb where a and b are positive real numbers, x represents DBH (in cm) and y represents height (in meters). Here is a list of the actual equations they came up with: y = 2.99x.49, y = 3.1x.3, y = 2.64x.41, y = 2.95x.384, and y = 2.7x.4. The correlation coefficients ranged between .5 and .62. The ideal coefficient would be 1. There are several reasons for the differences in their results. One is that I told them that they could omit data that didn't seem reasonable. Some groups did and some didn't. Another reason for differences is that some students didn't turn in their data on time and it didn't get included in some of the analysis done by the groups while other groups did include it. The predictions for the height of the giant cedar tree from South Manitou Island ranged from 14 to 37 meters, while the majority of the groups predicted it to be around 21 meters. The actual Michigan Big White-cedar Tree on record has a circumference of 5.5 meters and a height of 34 meters and is located in Leelanau County, which is where South Manitou Island is (Barnes and Wagner, 100). While most of the group equations predict a height of 21 meters, they don't take into consideration data from that part of Michigan or the idea of Island Gigantism.

In talking to the students, they did enjoy doing the project and thought it was neat to relate the concepts they were studying in math to science and nature, but wished we could have taken a break from the textbook during that time and spent more class periods outside collecting data. I agree. Their scores drastically improved from a 32% average on the pre-test to a 94% on the post-test. I want to do this project again next year and will try to find a way to lessen their load during that time.