Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Session 8:
Homework

Problem H1

The shapes below are pyramids. A pyramid is named for the shape of its base. The left shape is a triangular pyramid, the center shape is a square pyramid, and the right shape is a pentagonal pyramid. The sides of all pyramids are triangles.

 a. As the number of sides in the base of a pyramid increases, what happens to the shape of the pyramid? b. As the number of sides in the base of a pyramid increases, what happens to the volume of the pyramid?

 Problem H2 Spectacular Sports manufactures high-quality basketballs. The company packages its basketballs in 1 ft3 cardboard boxes. The basketballs fit nicely in the boxes, just touching the sides. To keep the ball from being damaged, Spectacular fills the empty space in the box with foam. How much foam goes into each basketball box?

Problem H3

Start with four identical sheets of paper with familiar dimensions (e.g., 8 1/2 by 11 in.). Use two of the sheets to make two different cylinders by taping either the long sides or the short sides of the paper together. Imagine that each cylinder has a top and a bottom. Take the other two sheets of paper and fold them to make two different rectangular prisms. Imagine that these rectangular prisms also have a top and a bottom. Note 7

 a. Which of the four containers has the greatest volume? Explain your reasoning. b. Which container has the greatest surface area? Explain your reasoning. c. Take a cylindrical and a rectangular container of the same height. Which one has a greater volume?

 Problem H4 When folded, what are the dimensions of each of the boxes below? What are the volumes?

Problem H5

Historically, units of measure were related to body measurements. Yet as we saw in Session 2, these measures were most often units of length, such as arm span, palm, and cubit. The cubit, used first by ancient Egyptians, is the distance from a person's elbow to the tip of the middle finger. The Egyptians standardized the cubit and called their standard measure the royal cubit. The measure of volume in ancient Egypt was a cubic cubit.

 a. Use your arms to estimate the size of a royal cubit. b. Estimate how many royal cubic cubits are in 1 m3. How many cubic centimeters are in 1 m3?

 Problem H6 The cubic fathom is a unit of measure that was used in the 1800s in Europe to measure volumes of firewood. A fathom is the distance of your two outstretched arms from fingertip to fingertip. Estimate the size of your cubic fathom. About how much firewood would fit into your cubic fathom? Explain your reasoning.

Zebrowski, Ernest (1999). A History of the Circle (pp. 77-81). Piscataway, N.J.: Rutgers University Press.
Reproduced with permission from the publisher. © 1999 by Rutgers University Press. All rights reserved.

 Download PDF File: A History of the Circle

To learn more about the research on the role of spatial structuring in the understanding of volume, read the following article:

Battista, Michael T. (March-April 1998). How Many Blocks? Mathematics Teaching in the Middle School, 3 (6), 404-11.
Reproduced with permission from Mathematics Teaching in the Middle School. © 1998 by the National Council of Teachers in Mathematics. All rights reserved.

 Download PDF File: How Many Blocks? Continued