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
Lesson Questions 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&nbst;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 (x, y) 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 Lesson Questions overhead master.
