InterMath | Workshop Support | Write Up Template

Title
Double the fun

Problem Statement
What happens to the surface area of a cube when all the dimensions are doubled? Tripled?

What happens to the volume of a cube when all the dimensions are doubled? Tripled?

If you want to increase the volume of a cube by 216 cubic units, how should you change the dimensions of the cube?

Repeat the investigation above using a rectangular prism and a cylinder.

Problem setup
What is the relationship between the dimensions of a cube and its surface area and volume? When the dimensions are increased how does the volume and surface area changed? When the dimensions are decreased how does the volume and surface area change? If you want to double the volume of the cube how will you change the dimensions? This problem is similar to ones I have done in this class in that we learn about the relationships between shapes and their properties through manipulations. It is also similar to the Biggie Size It problem where the dimensions of a triangle are manipulated.

Plans to Solve/Investigate the Problem
To solve this problem I will first try to visualize a cube getting bigger and smaller and think about what that really means. Next I will begin sketching cubes and assigning them dimensions. I will then calculate their volumes and surface areas and see if I can make any conjectures about the relationships between them and the dimensions. After working with the sketches I will then construct a cube and calculate its volume and surface area using Geometer’s Sketchpad. Once I have the cube constructed I will drag on its sides and verticies until the dimensions have been doubled and then tripled. I will pay close attention to their respective volumes and surface areas. Finally, I will make conjectures about the relationship between the dimensions of a shape and its surface area and volume.

Investigation/Exploration of the Problem
I will start by visualizing a cube and what would happen to its volume and surface area if the dimensions are doubled and then tippled. Next I will sketch construct three cubes on Sketchpad and calculate their surface area and volume. It does not matter what the dimensions of the first cube are as long as the second has twice the dsminsions and the third has three times the dimensions. The first cube has dimensions of 2.67cm, its surface area is and its volume is. The second cube had all their dimensions doubled to 5.34cm, its area is and its volume is. Finally, the third cube’s dimensions were tripled to 8.01cm, its area is and its volume is. From my representations I have concluded that when the dimensions are doubled the volume is magnified eight times. When dimensions are tripled the volume is magnified by twenty-seven.

Extensions of the Problem
What happens to the surface area and volume of a rectangular prism and cylinder when only one dimension is modified? How about if two of the dimensions are modified?

Author & Contact
Nathan Little

nlittle@aug.edu

Link(s) to resources, references, lesson plans, and/or other materials
My constructions
Intermath

U Important Note: You should compose your write-up targeting an audience in mind rather than just the instructor for the course. You are creating a page to publish it on the web.