## Hanging Chains

• Lesson
9-12
1

Both ends of a small chain will be attached to a board with a grid on it to (roughly) form a parabola. Students will choose three 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.

To get students ready for the main activity of this lesson, project the Pre‑Activity Questions that appear on the first two pages of the following Overhead Master.

Conduct a brief class discussion using these questions. Spend as much or as little time on each question as necessary to ensure that students are ready for the main activity.

Project a set of coordinate axes onto a screen or whiteboard, or draw a set of axes on the chalkboard. (A whiteboard is preferable.) With masking tape, tape the ends of a chain so that it hangs over the axis system, as shown in the diagram below.

Have the entire class discuss what type of shape the chain appears to form—linear or quadratic. Once they settle on the most likely shape (quadratic) have them find or discuss the general equation, y = ax2 + bx + c. (Note that the actual shape of the curve formed by a hanging chain is a catenary, but it is not necessary to discuss this aspect with students. Such a discussion can be saved for later, unless a student who already knows about catenaries brings it up. For the purpose of this activity, it can be assumed that the shape formed by the necklace is a parabola, and its curve can be approximated by a quadratic function.)

Have the entire class select three points on the hanging chain. To simplify the next step, suggest that students select the y-intercept as one of the points; using the figure above, for instance, students should select (0, ‑4). In addition, it will help if students select the other two points so that they lie on a grid line, if possible; for the figure above, students might select (6, 0) as one of these other points. Write the selected points on the board.

Have students work in groups to substitute the x‑ and y‑values of each of the three points into the equation y = ax2 + bx + c to set up three equations with three unknowns. Students should then solve the resulting system of equations. (If the y‑intercept was one of the selected points, have students substitute the appropriate value for c to reduce the question to solving a system of two equations with two unknowns.) Allow students to compare answers with other groups to see if their work resulted in the same equation. If differences are slight, discuss why there might be differences due to rounding; if difference are significant, have students identify their errors.

Substitute the values that students found for a, b and c into y = ax2 + bx + c to write the equation of a parabola.

This would be a good time to pose the question, "In the equation of a parabola, y = ax2 + bx + c, why are a, b, and c considered constants but x and y variables? After all, a, b, c, x and y are all letters that represent numbers, right?" At this point, students should recognize that each group got the same values for a, b and c, but that (xy) vary depending on the particular point on the parabola. It would also be good to pose the question, "What could be done to change the values of a, b and c?" [Hang the chain differently on the coordinate axes.)

Enter the quadratic function using the students’ result into a graphing calculator. (the coordinate axes were projected using a whiteboard, the graph can be projected onto the same whiteboard. This will allow students to see how the graph compares to the hanging chain.) If the chain and graph do not have a very similar shape, the students should attempt to find errors in their calculations.

Change the shape of the chain and repeat the above steps, except this time, have each group of students select their own three points. Compare student results to other groups. Did they get approximately the same equation no matter which points they chose? This is another good time to discuss why a, b and c are considered constants and x and y are variables.

Conclude the lesson by displaying the summary questions that appear on the last page of the Overhead Master.

Assessment Options

1. Compare the work of student groups. Do all of the student groups get the same result when solving the same system of equations? If not, how do they resolve differences in results?
2. Listen to the student discussions. Do students correctly articulate how to solve a system of equations? Do students correctly articulate how to use the solution of the system of equations to write an equation of a parabola? Do they grasp the concepts of variable and constant? LI>Look at the student graphs. Do they approximately match the shape of the chain? Do students recognize whether their graph is approximately the shape of the chain?

Extensions

1. 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,

then shift horizontally. (This shift is probably easier done by moving the y‑axis rather than moving the chain.)

2. 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.

Questions for Students

1. What is the difference between a quadratic function and a linear function? How can you detect the difference from their equations? …from their graphs?

[The degree of a quadratic function is 2, and it has the general form y = ax2 + bx + c. The degree of a linear function is 1, and it has the general form y = ax + b. The graph of a quadratic function is a parabola, but the graph of a linear function is a straight line.]

2. 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?

[For a particular location of the chain—that is, for a specific parabola—the values of a, b, and c do not change, but the values of x and y change from point to point.]

3. 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.

[Two points on the graph would be needed, because there are two constants in the equation, m and k.]

Teacher Reflection

• Were students clear about what they were expected to do?
• In what way were students challenged?
• Were students able to generalize the process of substituting in coordinates of points to a different form of equation?
• Did students have adequate background information to do the work? Did they know how to solve a system of equations when the values were messy? Did they know how to correctly substitute for the appropriate variable? Did they know the syntax to correctly enter a function into a graphing calculator?
• How did students demonstrate their understanding of the material, especially of the difference between a constant and a variable?

### 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.

### NCTM Standards and Expectations

• Interpret representations of functions of two variables.
• Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency—mentally or with paper and pencil in simple cases and using technology in all cases.
• Identify essential quantitative relationships in a situation and determine the class or classes of functions that might model the relationships.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.