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Mathematical Curve Conjectures

This page and activity authored by James Rutledge, St. Petersburg College.
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This material was originally developed through Merlot
as part of its collaboration with the SERC Pedagogic Service.

Summary

In this activity, a six-foot length of nylon rope is suspended at both ends to model a mathematical curve known as the hyperbolic cosine. In a write-pair-share activity, students are asked to make a conjecture concerning the nature of the curve and then embark on a guided discovery in which they attempt to determine a precise mathematical description of the curve using function notation.

Learning Goals

To enable students to:

Context for Use

This activity can be carried out in either a small class or a large lecture setting. It is best presented as an introduction to the hyperbolic functions normally studied near the end of Calculus I or near the beginning of Calculus II. This project requires students to apply what they have learned about functions to a real-world scenario, viz., a suspended rope, in an attempt to construct an accurate mathematical model.

The activity takes approximately 30 minutes although less time may be taken if desired.

Teaching Materials

Materials
Activity description

Teaching Notes and Tips

Students may need to be reminded of the standard form of a parabola and may need a brief review of the parameters involved (the values of a, h and k).

I generally suspend the rope in front of a rectangular grid board so that students will be led to make use of it in their investigation; however, I purposefully do not position the lowest point on the rope on a lattice point on the grid. This allows students to come to the realization that positioning the lowest point of the rope on a lattice point and designating this lattice point as the origin of the system, i.e., the ordered pair (0,0), will be most helpful in determining a precise function description.

Once the rope has been positioned with the "vertex" at a designated origin, and once students have identified a second point on the curve, they can construct a mathematical description of their conjectured curve fairly easily; however, knowing how to verify that this is accurate is not so evident and they often need instructor guidance with this.

As a lead-in to the nature of the hyperbolic cosine function and its connection with f(x)=e^x, the instructor may want to have students make a conjecture concerning the nature of the curve represented by only one-half of the suspended rope (the right half is more familiar in appearance). This should be reminiscent of f(x)=e^x from Calculus I.

Assessment

I generally give the students a grade for participation in this activity and sometimes grade their activity sheets.

References and Resources


Subject

Mathematics

Resource Type

Activities:Classroom Activity:Short Activity:Think-Pair-Share, Activities

Grade Level

College Lower (13-14):Introductory Level, College Lower (13-14)

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