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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.
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?
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
When folded, what are the dimensions of each of the boxes below? What are the volumes?
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.
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.
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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.
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How Many Blocks?
Continued
Note 7
If you are working in a group, once you have determined which solid has the greatest volume and which solid has the greatest surface area, discuss why these configurations of the original sheet of paper produce these results.
Problem H1
Problem H2
If the diameter of the ball is 1 ft., then the radius is 0.5 ft. Also, the height of the box is 1 ft. So the volume of the sphere is (4/3) π (0.5)3 = 0.52 ft3 (rounded to hundredths using the π key on your calculator). The volume of the box is 1 ft3. To obtain the volume of box that is foam, subtract the volume of the sphere from the volume of the box (1 – 0.52 = 0.48 ft3). So 0.48 ft3 is filled with foam. You could convert everything to inches to solve this as well.
Problem H3
Problem H4
Box A is 1 by 1 by 6. Its volume is 6 cubic units.
Box B is 1 by 3 by 3. Its volume is 9 cubic units.
Box C is 2 by 2 by 4. Its volume is 16 cubic units.
Problem H5
Problem H6
Answers will vary. A fathom is roughly 2 m, so a cubic fathom is about 8 m3. If one piece of firewood measures 30 cm by 10 cm by 10 cm, one piece of firewood is 3,000 cm3. Since there are 1,000,000 cm3 in a cubic meter, then about 333 pieces of firewood fit in 1 m3, and about eight times that (roughly 2,600 pieces) fit in a cubic fathom.