A Math Trail to Every Classroom By Jan Cohen, Founder, UrbanMathTrails

A math trail through nature is a simple concept: use math to enhance student exploration, communication, understanding and appreciation of the spatial forms, patterns and quantitative underpinnings of the natural world. When students discover the connections between math and nature through observational learning, discovery and application, they embrace concepts and principles and retain knowledge and to a degree not achievable through traditional textbook-classroom-centered education.

No tools are needed, other than perhaps a tape measure, so students may roam freely, observe, consider, reason, and internalize a new appreciation of the math embedded in their natural environment. A myriad of topics may be explored: measurement, sorting and classifying, scale, symmetry, geometry, data and statistics, probability, algebra, etc., depending upon age, interest, season or geography.  Any natural habitat is suitable.

Find a location of natural interest and variety.  Walk, observe and consider the environment:

·         compare shapes, patterns and sizes;

·         make estimates and measure degrees, heights, circumferences, perimeters, areas, volumes, slopes and distances using standard and non-standard measurements; compare estimates to actual measurements;

·         evaluate ratios and proportion;

·         collect data, analyze statistically and assess probabilities; and,

·         devise scientific methods, make conjectures, test hypotheses.

Nature obeys rules, which students explore and express on a math trail, and nature brims with opportunities for fascinating mathematical investigations. The following are just a few examples to stimulate your thinking and, hopefully, inspire your planning of a math trail along the AT.

One important step when learning how to identify trees and plants is understanding phyllotaxy, or the arrangement of leaves around the stem. There are three basic types of leaf arrangements: alternate, opposite, and whorled. As you walk along the trail, notice the leaf arrangements on plants and keep a tally. What is the most common arrangement?

 Streams and ravines are part of the diversity and beauty of nature. This rustic foot bridge includes many geometric details. Find several sets of parallel lines. Estimate the angles of the posts. Estimate the length, width and height of the bridge. How many seasons do you think were observed to determine the height and span requirements of this stream crossing?

This tree fell down after a recent storm. Did you know you can estimate a tree’s age by measuring its trunk? Find a felled tree and measure its circumference. Given the circumference of trees grow at about ¹⁄₂ to ³⁄₄/ inch per year, how old was this tree when it fell?

Foresters use diameter at breast height, or DBH, as the standard for measuring trees. Measure the circumference of a tree at breast height, 4.5 feet above the ground. Find the diameter (divide circumference by pi) and multiply it by the tree’s growth factor to determine its approximate age.  Growth factor varies by species from about 2 to 7, but we can use 4 as an average of many species.  How old is the tree?  Keep a tally today.

A single frond of a fern resembles the whole fern, in miniature. This is called self-similarity. How many different types of ferns can you find? What other examples of self-similar structures in nature do you see along the trail?

A fractal is a term to describe self-similarity. It is a geometric pattern that repeats itself at different scales in a specimen.  What other examples of fractal geometry do see along the trail? Look at some trees and bushes and explain their fractal patterns.

Patterns prevail throughout the plant kingdom. Flowering plants exhibit numerical patterns. Count the petals on several flowers. Do the numbers adhere to the Fibonacci Sequence, 0, 1, 1, 2, 3, 5, 8…? What are the next two numbers in this sequence? As you walk along the trail, how many flowers can you find that conform to this sequence? Look for pinecones. They are a striking example of the Fibonacci sequence. Their scales are arranged in two intertwined spirals. Count the numbers in each direction. Does the pattern conform to the sequence?

Most plants exhibit some type of symmetry. Examine the symmetrical patterns of plant leaves along the trail. What type of symmetry do you observe in each leaf? Why do you think symmetry is so prominent in the natural world? What other examples of symmetry can you identify?

Daisies have rotational symmetry. As you rotate them in a circle, they always look the same. Since the typical number of petals on a daisy is 42, it has what’s called 42-fold rotational symmetry. Find daisies, dahlias or sunflowers and count the petals. This will tell you the number of distinct orientations in which they look the same. What other flowers do you see that have rotational symmetry?

Orchids are glorious examples of flowers that nearly all have bilateral symmetry. What is bilateral symmetry? Why is it also called mirror symmetry? What other examples of bilateral symmetry can you find along the trail?

See any spider webs? They create near-perfect circular webs that have near-equal-distanced radial supports coming out of the middle. How many radial supports do you count in the spider web? Estimate the interior angle formed between the radials?

Is it a sunny day? If so, you can estimate the height of a tree using its shadow. First, measure your shadow then measure your height. Measure the tree's shadow. Set up a proportion:

See any dried mud patches? Describe the crack pattern. Do you see any polygons? Is the cracking pattern uniform? Do you see any right angles? Straight angles? Acute angles? Any concave shapes? What other geometric observations can you make?

Math trails are a natural fit and essential element of outdoor education, providing opportunities to students to discover explanations of nature’s patterns and underlying logic, shape, quantity and arrangement. Regardless of age or skill level, math trails get students out of the classroom and into the great outdoors to help them develop mathematical and environmental literacy while having fun. They enable children to engage with mathematical experiences in the real world, gaining first-hand knowledge of how math can be used to interpret the world in which we live.

Jan Cohen is the founder of UrbanMathTrails, an education consulting firm that serves various institutions in the application of math in new contexts. Drawing upon an extensive background in math education, finance and architecture, and a passion for the outdoors and the arts, UrbanMathTrail programs inspire children to discover math in the environment around them, exciting their mathematical imagination and electrify their mathematical senses. See

https://www.urbanmathtrails.com for additional ideas and information.

National Trails Teaching Guide!

In honor of the 50th Anniversary of the Trail System Act, the Bureau of Land Management has created a teaching guide about National Scenic and Historic Trails.

Although these three lessons were designed with middle school in mind, they can also be used with upper elementary and high school students. Common core curriculum connections include english/language arts and social studies.

Check out the teaching guide, and be sure to let us know if you put it to use!

https://www.blm.gov/sites/blm.gov/files/documents/files/CI_trails_0.pdf

Student Mural

By Perry County Times

By Jim T. Ryan
Staff Writer

Driving Route 850 from Marysville to Shermans Dale will take you past a crossing for the Appalachian Trail (AT) on the footpath's meandering course through Perry County.

There, the white blazes mark the AT's path through fields and patches of trees, and a small parking area for hikers to step off from in Rye Twp.

A group of Susquenita Middle School students have made that parking area a little brighter with a bottle cap mosaic depicting a rising sun, mountains, trees and the AT symbol. 

"I've wanted to do a bottle cap mural for a long time and this seemed perfect," said Abby Fisher, a Susquenita art teacher who designed the mosaic and helped students put it together.

Six students in the life skills class, together with about 10 other middle school students and their teachers spent much of last school year collecting, sorting and arranging the various colored caps and lids from sodas, sports drinks, and coffee cans.

There were upwards of 30 students working on the project at various times

The project was more than just taking trash and turning it into a picture, the teachers said. It helped all the students in a wide variety of academic and social skills.

"The art was secondary to the social aspect of this for the students," Fisher said.

Jeff Henry, the life skills teacher, said it really helped his students work on cooperation and communication skills. 

Paul Marth, a Susquenita social studies teacher involved with the project, said working on an art project that would be featured at an AT trailhead fit in with his curriculum too. The trail runs close to the Susquenita campus in Duncannon and is an integral part of local and U.S. history.

"Our kids drive by it every day," he said.

Marth also is an avid hiker and was in contact with the Appalachian Trail Conservancy and Mountain Club of Maryland, a volunteer group that helps maintain part of the trail in Perry County. They wanted to do something creative with the back side of the message board, Marth said.

So he said, yes, the middle school could help.

The teachers paid for the hardware materials and they asked for donated bottle caps. The students sorted them, painted the board and used screws to fasten the approximately 1,000 caps to the board.

"They don't get to use power tools often, so that was really cool for them," Henry said.

The students took a class trip with their teachers on Sept. 29 to visit their art, now installed at the parking area along Route 850. And they confirmed their teacher's assessment: power tools -- awesome.

The best part was "sorting out the colors and using the drills," said student Austin Ciccocioppo.

"My favorite was the colors," student Delorean Michael said.

"Using the drills," student Codie Nelson said, laughing, when asked about his favorite part of the project.

The project is precisely what trail groups were looking for to spruce up the message kiosk at the parking area.

"It's sweet to look over my shoulder and see all that color while I'm mowing the parking area," said Christy Hoover of Carlisle.

Hoover is a volunteer with the Mountain Club of Maryland who has been helping to maintain the trail and parking area for the past 10 years.

"One of our goals is to get kids engaged with the trail and this is a great way to do that," said Ryan Seltzer, the corridor stewardship program manager with the Appalachian Trail Conservancy in Boiling Springs.

The high visibility of the mural will be a welcome sight to hikers coming off the AT, he said.

A plaque is being made that will be attached to the kiosk and identify the mural as a project of the Susquenita Middle School students.

Everyone -- most importantly, the students -- will take something away from the project, teachers said.

"The kids are going to know it'll be there for a while," Fisher said, "and they can take pride that they were involved."

Jim T. Ryan can be reached via e-mail at jtryan@perrycountytimes.com

Article link: https://www.pennlive.com/perry-county-times/index.ssf/2017/10/students_decorate_appalachian.html

Great resources from Nancy Reeder!

Nancy Reeder of Franklin, NC has developed a fantastic set of curriculum-based activities with an Appalachian Trail theme! Check them out here:

https://drive.google.com/drive/folders/1OBgARN8FdzmG_zPJftmN2cHsqDJpX0cA?usp=sharing

You can find this link anytime, and lots of other good stuff, under "Resource Links" on the right side of this blog.

From Nancy's Introduction to her project:

These activities with answer keys have been prepared so they are ready for implementation in your classroom at any time. They are divided into five subjects: Mathematics, History, Geography, Science, and Language.

The section from Springer Mountain, GA to Damascus, VA is emphasized, although some activities involve other parts of the trail. Class as well as research type lesson ideas are included to complement your general outdoor curriculum. They are intended to broaden your students’ knowledge of the Appalachian Trail.

Everything you need to complete the activity is included. Some of them are intended to be copied for each student, and some are intended to be put on your smart board, and discussed with your entire class, or be given to individuals to read and research.

There are various levels of activities geared for intermediate grade students as well as middle school students.

You can find the link to these resources anytime, and lots of other good stuff, under "Resource Links" on the right side of this blog.

FFEC Alumni Brings Students Back to Trail!

https://www.facebook.com/childrensforestnetwork/posts/598264507188521

Laurel Run Hiking Trip: Field Trip Day

Stephanie Tuttle
Fairfield Elementary School

Rockbridge County, VA

The day of the field trip came and all of my planning and preparation was about to be tested. The students were excited for the trip and we were able to load the busses and be on our way.

Boxerwood, The Department of Wildlife and Inland Fisheries, the STEM teacher, and parents all meet us at Laurel Run. The site was a great place for 120 students and their parents to explore and learn. Laurel Run had a huge pavilion with tables and picnic tables were scattered around the site. The students got off the buses and I divided the students into 4 groups. We then started the activities. The 5th grade teacher and I planned a variety of activities which cover all content areas. The students participated in a hike, water testing, nature journaling, an antonym search activity, and a STEM survival activity.

A member of the Boxerwood staff and myself led the hiking activity. The students came to us and we divided them into two groups, so the students could better experience nature. I took my groups first and the Boxerwood staff member started her hike 5 minutes after my group. We started the hike. The students were give cameras and directed to take a “selfie” of themselves and 3 “WOW” moment pictures. A “WOW” moment is a moment during the hike the students were amazed by what they were experiencing. As we hiked the students took pictures of the waterfall, rock formations, leaves, mushrooms and anything else amazing. Part way through the hike the students were given crayons and ask to match the crayon color to something they saw in nature. Overall the students enjoyed the hike expect for the gnats.

Another activity the students participated in was water testing. Boxerwood lead this activity and brought all of the materials needed. The students worked in small groups to test the healthiness of the water. They tested the pH, oxygenation, and the amount of pollution in the water. The students enjoyed this activity because it allowed them to compare the cleanliness of this water to other location they have tested with Boxerwood.

Students participated in a journaling and an antonyms search activity. The students were asked to journal about their experience throughout the day, the weather, and wrote a haiku about what they heard in nature. The students then worked with a partner to find objects in nature they described the antonyms they were given. The students were given egg cartons with two antonyms written on them like smooth and rough. They then had to find objects in nature to match their words. These were great language arts activities because the students were able to reflect on their experience and explore nature up close.
The final activities the students participated in was a STEM survival activity. The students had to figure out how long their stride was and do the math to determine the distance they could walk in a given amount of time as well as how many steps they take during each mile they walk. The students were then given a survival situation and asked to figure out how they would survive. The students came up with a variety of creative ways to survive.

This field trip was a success. The students experienced nature in their community. Overall, the students enjoyed the variety of activities. The 4th and 5th grade students enjoyed going on the trip together and getting to learn from each other. This trip was a success and I have many ideas on how to improve and make this trip better for next year.

Laurel Run Hiking Trip: Planning and Preparation

Stephanie Tuttle
Fairfield Elementary School

Rockbridge County, VA

This field trip started with the idea to take 4th and 5th grade students on a hike in their community. I wanted the students to better understand the beauty of the nature that surrounds them. To many students going to Walmart is the highlight of their week and I wanted them to see that their community offers so much more.

This trip would not have been possible without the help from Boxerwood. Boxerwood is an organization in Rockbridge County whose goal is to teach students how to care for the Chesapeake Bay Watershed. Boxerwood offered up a variety of sites to choose from located across the county. I choose the Laurel Run site at Goshen pass because most of the students live close to this area, but have never stopped and explored.

At first this location seemed to be the best and easiest choice. As the field trip approached I learned we needed an access pass from the Department of Game and Inland Fisheries. Each adult who went on the trip would have to buy a pass to come. I was worried our trip would not run smoothly if parents had to buy passes to come. Luckily one of the students’ parents is a game warden and was able to waive the fee for the pass. He also came out and went on the trip with us, a pleasant addition I did not originally plan for.

I wanted the activities provided to cover all content areas. Selecting the activities was the easy part. The activities the students would do were Nature Journaling, Egg Carton Antonyms, Water Testing, Survival Scenarios, Hiking Math, and Crayon Color matching. The 5th grade teacher and I had to design and created all of the activities the students participated in. This took more time than I originally expected. Planning for a field trip where you have to pack everything but the bathroom took organization and time. Working with the 5th grade science teacher was helpful because I was able to use her experience and knowledge of creating field trips to help plan our trip.

Overall planning the Laurel Run hiking trip would not have been possible without knowledgeable and experienced teachers to help guide me through the process.