A net is the two-dimensional representation of a three-dimensional object. For example, you can cut the net of a cube out of paper and then fold it into a cube.
Here are the nets of some "open" boxes -- boxes without lids.
If you were to cut out each net, fold it into a box, and fill the box with cubes, how many cubes would it take to fill the box? Make a quick prediction and then use two different approaches to find the number of cubes. You may want to cut out the actual nets (PDF file), fold them up, and tape them into boxes to help with your predictions.
b.
What strategies did you use to determine the number of cubes that filled each box?
Problem A2
Given a net, generalize an approach for finding the number of cubes that will fill the box created by the net. How is your generalization related to the volume formula for a rectangular prism (length • width • height)?
Problem A3
a.
Imagine another box that holds twice as many cubes as Box A. What are the possible dimensions of this new box with the doubled volume?
b.
What if the box held four times as many cubes as Box A? What are the possible dimensions of this new box with quadrupled volume?
c.
What if the box held eight times as many cubes as Box A? What are the possible dimensions of the new box with an eightfold increase in volume?
You may want to start by constructing a solid Box A (2 by 2 by 4) from cubes that can be connected together. Next, double one dimension of the solid and build a new solid. What is its volume? What happens to the volume of the original solid (Box A) if you double two of the dimensions? If you double all three of its dimensions? Try it. Close Tip
Problem A4
If you took Box B and tripled each of the dimensions, how many times greater would the volume of the larger box be than the original box? Explain why.
Divide the new volume by the original volume to see how many times greater it is. Can you figure out why? Close Tip
Problem A5
What is the ratio of the volume of a new box to the volume of the original box when all three dimensions of the original box are multiplied by k? Give an example.
