Students take a numerical and tabular look at finding the maximum value of an open box constructed by folding a rectangular sheet of material with cutout square corners. They also understand the concepts of independent and dependent variables.

Before the Activity

See the attached PDF file for detailed instructions for this activity

Print pages 87 - 95 from the attached PDF file for your class

During the Activity

Distribute the pages to the class.

Follow the Activity procedures:

Use rectangles of grid paper, mark off equal-sized squares at each of the corners

Cut out the squares and fold to form open boxes

Notice that the size of the cut-out square determines the height of the box

Examine the relationship between the dimensions of the box

Calculate the length, width, and height of the box, and compute the volume

Repeat the steps for other boxes

For different sizes of cut-out squares, generate a table of volumes

Use the calculator, represent the height by a variable x, write an expression using x, specify its start and incremental values, and find the volume

Analyze the data to determine the relationship between the cut corners and the height of the box

Observe that the volume increases as the size of the cut square increases and then starts to decrease for further increase in the size of the square

Find the length of the square which forms a box with the largest volume

Note that the relationship between height and volume is not linear

After the Activity

Students complete the Questions sheet.

Review student results:

As a class, discuss questions that appeared to be more challenging