This problem is practical, and therefore useful in the mathematics classroom. Students often struggle to understand what is being asked in a mathematics problem because of the lack of real-world implications. There are real-world occurrences when finding the dimensions that produce the largest area or volume prove to be beneficial. This problem can be used to show students how volume changes depending on the length, width, and height of the shape. In this example, you can show how the height of the box effects the length and width of the box, and you can show how variables can be used to represent unknown values.

This problem can be changed in numerous ways. You can change the dimensions, like in part two. Giving students different length and width values changes the amount of paper available, and therefore changes the number of possible solutions. From those solutions, students must find the one that represents the largest volume. You can also change what values are given. In this example, you are given the length and width values. However, the process changes completely when students are given the length and height instead. Having to find the width, students would represent it as a variable and consider the height, the square cut outs in each corner.

Continuing with the numerous adaptations, you can ask students to find the second largest or smallest volume. They would be using the same data, but they would analyze it differently. Students may also enjoy a more hands on task. For this, you can instruct students to cut out rectangles with various length, width, and height values. Then have them follow the instructions of cutting and folding. Once they have cut where needed and folded where needed, students can tape the corners of their boxes together and physically see the effects of the dimensions on the box’s volume.

Final thoughts. 

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