# Maximum Volume - How is the Box Changing?

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Many students don't get an opportunity to see how cutting out different sized squares from a sheet of centimeter grid paper and then folding the sheet of paper to make an open box produces rectangular boxes with different volume. Some students think as along as the same sheet of paper is being used, the volume stays the same.

By engaging students in this activity of cutting out a 1 cm by 1 cm square from each corner of a sheet of centimeter grid paper, folding the remaining sheet into an open box, and calculating the volume students begin building a visual understanding for what is happening to the volume. Students continue cutting 2 cm by 2 cm squares, 3 cm by 3 cm square, ... until they have cut the largest square possible from each corner. Students calculate the volume of each open box, and notice how the volume of the boxes are changing. Students notice that the volume of the open boxes increases and then at a certain point the volume begins decreasing. After students have collected the data of the size of the removed square and the corresponding volume, students use their graphing calculator to create a scatter plot and write an equation to represent the volume of the box.

The activity includes a copy of the directions, a sheet of centimeter grid paper, a student activity sheet, and a class record sheet.

This is a great exercises to do in Algebra 2 or Pre-calculus before students again see the same problem in Calculus.

Answers are given within the directions.

Total Pages
6 pages
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Teaching Duration
55 minutes
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