Walking a Math Trail
By Michael Naylor

Math trails can be found everywhere – from how many squares can be found in your school's windows to how many steps it will take to the get to the top of the jungle gym.
Artwork: Laura Bethel-Sehn

Math is all around us, and taking your students on a Math Trail is a great way to get them to see the world through the eyes of mathematics. A Math Trail is an opportunity to take your students out of the classroom to solve a variety of math problems related to shapes, structures and numbers in their environment.

A Math Trail can take place on the playground, in the cafeteria or gym, in the school library or any other place where students can roam freely. When working in pairs or small groups, students are given a menu of 10 or 20 problems related to objects or features of the area. They select problems that interest and challenge them, work for a generous amount of time and then come back to the classroom to discuss methods and possible solutions.

Planning the route
It's easy to make your own Math Trail. First, choose the area you want your students to explore. Then walk around the area yourself, looking for appropriate mathematical questions you could ask your students. When you're writing your questions, it helps to think about the NCTM's five content strands: Number Sense, Geometry, Measurement, Algebraic Reasoning and Probability and Statistics. I've listed some sample activities at the end of this column to get you started.

Give your students copies of the problems and let them know that they should only choose problems that interest them; they do not need to complete all of the problems.

Talk about it
Be sure to schedule time to discuss the problems. Call on different groups to share methods and solutions. Your students will have many different answers to the same problems, and that's an opportunity to ask them to explain their reasoning and justify their answers. You'll generate some rich mathematical discussion. If you like, you can have groups write up their method and solution to their favorite problem.

A great assignment is to have students write their own Math Trail questions. You can add student-generated questions to your own question pool to use again later and with future students. Your trails will become better and better each time you do them!

Sample problems
Though many of these problems fit into more than one category, I've categorized them so you get a feel for the many different possibilities.

If you were to hang streamers between the lampposts on the playground, how many streamers do you think that would that be?

Number Sense and Estimation

• When in Rome. Where can you find Roman numerals? What numbers do they represent? Can you find any Hindu-Arabic numerals?
• Sixes. How many different sets of six things can you find?
• How many stairs? How many stairs are in the school?
• Recess Running. About how many steps do you take during recess?
• A Lot of Bricks! How many bricks are there in the school? What does the school weigh?

Geometry

• Window Squares. The windows are divided into a 3 x 5 array of squares. How many squares of all sizes can you find in this array?
• Circular Rockery. Into how many sections is the concrete perimeter of the fountain divided? If these sections could be removed and used to make a continuous path along a diameter of the fountain, how many sections would it take?
• Tilings. Find and sketch at least three different tessellating patterns that use either one shape or a combination of shapes to tile an area.
• Shapes Everywhere. What shapes do you see in and around the cafeteria? Are there some shapes that appear more often than others? Can you find a trapezoid, a hexagon, an irregular decagon and a parallelogram? Sketch and name some shapes you see and where they can be found.

Measurement

• Square or Not. Is the playground area a square? If so, how can you tell? If not, how close is it to being a square?
• Side by Side. How many people standing shoulder to shoulder could circle the entire school? How did you decide?
• Square Tables. The smaller tables in the cafeteria have four bench seats that are attached to them. Which area is greater: the table top, or the combined areas of the tops of the four seats?
• Tall Order. How could you closely estimate the height of the school's tower? What is your estimate?

Algebraic Reasoning

• Stepping Out. The ladder on the jungle gym has six stairs. If you take either one step or two steps a time, how many different combinations of steps will take you to the top? (for example, you could step 2, 2, 2; or 1, 1, 1, 2, 1; etc.)
• Ramps. Do the two ramps next to the gym have the same slope, or is one steeper than the other? How can you tell?
• Circle of Lamps. Surrounding the playground is a circle of lampposts. If you were to hang one streamer between each possible pair of lampposts in this circle, how many streamers would that be? Is it possible to create this same design using just one very long streamer that doesn't "backtrack" or "double-up" any segments of the design?

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.

November/December 2004, Vol.35, No.3