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
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.
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 andy 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.]
- 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