You are the manager of a band that has been hired to perform at a party, and you want to create the largest dance floor possible for the attendees. If you only have a given

length of rope to delineate the space, how can you make the most of your space?

**Previous knowledge:** Students should be able to write the

equation of a quadratic

function in

vertex form.

**Objective:** At the completion of this lab, students will be able to write the

equation of a

parabola to model the

length (or width) versus

area of a specified perimeter.

**Materials:** graph paper and graphing calculator.

**Group Size:** 2

**Procedure:**- Each group will be given a specific length for their rope (perimeter). For the lab to work easily the teacher should assign each group a specific perimeter that is a multiple of four.

- Set up a table as follows, including all possible whole number lengths and widths (including 0) and find the corresponding area:
- On your graph paper, create a scatter plot using length as the independent variable and area as the dependent variable. Make sure you choose an appropriate interval and label for each axis.

- Using the vertex of your parabola, write the equation of the quadratic function that fits your scatter plot in vertex form.

- Rewrite the equation from #4 in standard form. Make sure you show all of your work.

- Enter the Lengths from your table into List 1 and their corresponding Areas into List 2. Give the window you set to see the entire scatter plot:
Xmin ______ Xmax ______ Xscl ______

Ymin ______ Ymax ______ Yscl ______

- Check that the equations you wrote are correct by putting the vertex form of the equation in Y1 and the standard form of the equation in Y2 (in the Y= screen) and graph them. Do you get the same parabola? Do they fit your data? If not, you need to go back and fix your work!

- Each group should display their data in a class chart in the front of the room.

- Which length gave you the maximum area? How does this length relate to your perimeter?

- On your TI-83, put the widths in L3. Make a new scatter plot using the widths as your independent variable and the corresponding areas as the dependent variable.

- How do the equations you wrote in #4 and #5 fit this data? Why?

- What width gave you the maximum area? How does this relate to the length that gave you the maximum area?

- A class discussion should take place of the length and width that yielded the maximum area for each group and where that data piece was located on the graph.

- Why did we say that the easiest perimeter to work with was one that was a multiple of four?

**Homework: Make sure that you show your work.**- Pick a perimeter that is not a multiple of 4.

- Make a table like you did in #2 using integral lengths for your length and width.

- On your graph paper, make a scatter plot with the length as the independent variable and the area as the dependent variable.

- Draw the parabola that fits the data on your graph.

- What is the vertex of this parabola? Is it one of your data points? Why or why not?

- Write the equation of the parabola in both vertex and standard form:

- Find the length and width that would give you the maximum area:

- What is the optimum shape that yields the largest area for any given perimeter?