Learning Math: Measurement
Session 8, Homework
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.
- As the number of sides in the base of a pyramid increases, what happens to the shape of the pyramid?
- As the number of sides in the base of a pyramid increases, what happens to the volume of the pyramid?
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.
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.
- Use your arms to estimate the size of a royal cubit.
- Estimate how many royal cubic cubits are in 1 m3. How many cubic centimeters are in 1 m3?
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.
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.
- As long as the base is a regular polygon, its shape begins to approximate a circle. The triangular sides become increasingly small. In time, the pyramid becomes indistinguishable from a cone.
- The volume formula is still V = (1/3)Bh, but since the base becomes a circle, the volume becomes V = (1/3)πr2h.
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.
- The shorter, wider cylinder has the greatest volume. In the formula for the volume of a cylinder, V = πr2h, notice that the radius is squared and the height is not. Using the larger dimension of the paper as a circumference of the base produces a larger radius, and in turn, an exponentially larger volume, and vice versa: Using a smaller dimension of the paper as a circumference of the base produces a smaller radius, and in turn, an exponentially smaller volume.
- Again, the shorter, wider cylinder has the greatest surface area. All will have equal lateral surface area (not including the top and bottom), since the same paper is being used. For total surface area, add the area of the bases — so the problem boils down to which has the largest base area. As described in the solution to part (a), the shorter, wider cylinder has the largest base area.
- The cylindrical container has the greater volume, as long as they both have the same lateral surface. If only the heights are known, then there is no comparison to make — one could have a much larger base area than the other.
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.
- Answers will vary. Your answers should reveal a royal cubit to be about 52.4 cm or 20.62 in.
- Since a cubit is just over 0.5 m, we would expect there to be just under 2 cubits in each dimension in 1 m3. This means the volume is just under 8 cubic cubits. Based on the value of 52.4 cm in 1 cubit, there are 6.95 cubic cubits in a cubic meter. There are 1,000,000 cm3 in a cubic meter.
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.
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.