Two 50-minute class periods
- Some familiarity with right triangle trigonometry is helpful
- Some facility with graphing calculators and/or Geometer's Sketchpad®is helpful
Ohio Standards Alignment
- 8.5" x 11" paper
- centimeter rulers
- Graphing calculators and/or Geometer's Sketchpad® if available
Functions, right triangles, trigonometry, maximum value
The "Folded Triangle" problem is rich by virtue of its connections to so many different levels of secondary school mathematics. Students in a first-year algebra or geometry course can investigate the problem just as meaningfully as students enrolled in a senior-level calculus course. Moreover, the problem can be modeled inexpensively and conveniently with readily available hands-on manipulatives, namely, a standard sheet of 8.5 "x 11" copier paper. The problem has a delightful solution that is completely obvious in hindsight, yet deliciously counter-intutive at first glance.
Start with an 8.5 "x 11" rectangular piece of paper in front of you on the desk. Pick up the upper right-hand corner of the paper and, without otherwise moving the paper, place the corner somewhere along the bottom edge of the paper. Holding the corner on the bottom edge, crease the looped paper so that the fold keeps the corner in place. You have now created a right triangle in the lower right corner of the paper.
What is the maximum area of the triangle formed?
Distribute several pieces of 8.5" x 11" paper (copier paper works nicely) to individual students. In a "theater in the round" fashion (with desks arranged in a semi-circle), model the problem with a piece of paper, paying careful attention to the names of various corners of the paper (as they are labeled in Figure 1). Ask students to make conjectures about the type of triangle that will maximize area. Typically, students will make guesses such as the following:
Equilateral Triangle (clearly not possible since EFC is a right triangle, but a good guess; when a triangle is constrained by a fixed perimeter, an equillateral triangle will maximize area)
Golden Triangle (one in which the ratio of the long leg to the short leg is equal to phi, the Golden Ratio)
While each of these conjectures has its merits, none is correct. Record the conjectures of your students in front of the room. If Geometer's Sketchpad® is available, use the dynamic model of the problem to quickly illustrate that these initial conjectures are incorrect by sliding the location of point F along the base BC. (This will cause many students to become more curious about the correct answer).
If Sketchpad is not available, students can explore their initial conjectures by collecting data "by hand" with centimeter rulers and protractors (see steps further described below).
Once students have had an opportunity to make conjectures about the problem solution, have them fold their own paper to create their own "folded" triangle (depicted as triangle EFC in Figure 1). Distribute centimeter rulers and protractors and have students measure the height, width, and angle EFC of their own "folded" triangles. Encourage several students to fold their paper "thoughtfully" in order to generate triangles identified by students as possible solutions to the problem.
Record the data in front of the classroom, optionally typing the data into lists in a graphing calculator for more detailed analysis. You may wish to have students write their data on the board on their own as soon as they have successfully collected their data. Once all students have had an opportunity to create and record data for their own "folded" triangle, calculate the area for each case (this is nicely done using lists on a graphing calculator, with list L1 storing Length Data, list L2 storing Height Data, and L3 defined as (1/2)*L1*L2. Areas are calculated almost instantaneously). Verify that isoceles and golden triangles do not maximize area.
Next, provide time for students to analyze properties of the right triangle EFC that maximizes area.
Students in a first-year algebra course (or prealgebra course) may wish to perform various regressions on the collected data in an effort to build a "best fit" function that fits the data as "good" as possible. From such a model, a maximum area can be determined (along with corresponding dimensions of the triangle that generates the maximum area).
You may wish to encourage students, particularly those in an algebra course, to label segment FC as x and segment EC as y and brainstorm ways to maximize the value of (1/2)•x•y. From there, a variety of algebraic approaches may be used, including trigonometry-based and function-building methods. Students enrolled in an introductory geometry course may wish to build a dynamic sketch of the problem situation using Geometer's Sketchpad® to determine the properties of the maximal "folded" triangle.
Peruse each of the sample strategies provided in the Solutions for insight into approaches you may wish to encourage with your students.
We recommend that two 50-minute class periods be devoted to the "Folded Triangle" problem. The first day is typically spent introducing the problem, collecting data, discussing approaches, building models of the problem in Geometer's Sketchpad, and so forth. During the second day, students share solutions in a collaborative fashion with an emphasis on proving the 30º-60º-90º conjecture using a variety of approaches.
For a sketch of the "Folded Triangle" problem, click here.
A PDF file of folded-triangle materials (including classroom-ready handouts) can be downloaded here.
Adapted by Michael Todd Edwards from Edwards, M. T. , & Reeder, J. (2004, Spring). Uncovering unexpected mathematical connections with the "folded" triangles problem. Ohio Journal of School Mathematics, 49, 3-16.