Investigating Parabolas: Hanging Chains

From NCTM "Illuminations". Detailed lesson plan using a parabola formed by a small chain attached to a chalk/white board. Students choose points along the curve and use them to identify an equation. Repeating the process, students will discover how the equation changes when the chain is shifted. The resource includes pre- and post-questions for students, assessment suggestions, and possible problem extensions. Allows students to analyze the effect of changing various parameters on functions and their graphs.

Weblink address (URL):

Standards & Objectives

Learning objectives:

Students will:

Make real-world connections, recognizing that catenaries are naturally occurring shapes that can be approximated by parabolic functions.
Substitute points on a graph into a function form to find the equation of a graph.

Essential and guiding questions:

What is the difference between a quadratic function and a linear function? How can you detect the difference from their equations? …from their graphs?
In the equation of a parabola y = ax2 + bx + c, why are a, b, and c considered constants, but x and y are variables? Aren’t all five of them letters that represent numbers?
If a graph has an equation of the form y = mx3 + kx, how many points on the graph would you need to know in order to find the values for m and k? Explain how you know.

Lesson Variations

Blooms taxonomy level:

Understanding

Extension suggestions:

Using the same chain, vary the position to begin to predict the result on the quadratic coefficients a, b and c. To shift the parabola up or down, and to assure the shape does not change, move the x‑axis. Otherwise, measure the distance between the ends so that vertical moves keep the same shape. Such shifts should only affect c. In particular, see what happens to the coefficient a. To observe horizontal shifts, complete the square of each quadratic equation, so that the equation is in the following form, y= a(x-(b/2a))^2+ (c-(b^2/4a)) then shift horizontally. (This shift is probably easier done by moving the y‑axis rather than moving the chain.)
Have students search the Internet and report on catenaries and parabolas, and how they are similar and different. 3. Have students explore the effect of horizontally stretching or compressing the parabola. Leave one end taped in its position, and move the other end. What happens to coefficients, particularly a, when the two ends are twice as far apart, half as far apart, and so on.

Helpful Hints

Materials:

Small chain (a metal necklace, or thin chain from hardware store)
Pre-Activity and Summary Questions- Overhead Master
Graphing calculator
Masking tape

References

Contributors:

Jennifer Bruce

Originator:

Ayers Institute for Learning & Innovation