Measuring a Solid

Teaching measurement is always a challenge; in three dimensions, the challenge increases. Many students never really understand volume or surface area, although they can memorize the formulas and even apply them on tests. These resources have been selected with an eye to helping students enter into the concepts of volume and surface area through practical problems, hands-on experiences, and applets they can manipulate to actually see how these measurements are affected by change in a figure’s dimensions.

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Keeping Cool: When Should You Buy Block Ice or Crushed Ice? http://www.figurethis.org/challenges/c62/challenge.htm

Which would melt faster: a large block of ice or the same block cut into three cubes? The prime consideration is surface area. A complete solution demonstrates how to calculate the surface area of the cubes as well as the large block of ice. Related problems involve finding surface area and volume for irregular shapes and examining the relationship between surface area and volume in various situations. MSP full record

Cordwood http://www.ankn.uaf.edu/publications/VillageMath/cordwood.html

This problem scenario, set in Alaska, asks students to find the volume of a shed, applying the standard formula, and then to determine the number of cords of wood needed to fill it. Finally, they must calculate the cost of the wood. The Village Math site features more than 25 math lessons that involve application-oriented problems relevant to life in Alaska today. MSP full record

Fill the Box http://www.centexeweek.org/html/fill_the_box.html

This activity has students explore dimensions and volume through practical problem solving. They must figure out a way to put the most blocks in a box, a real-life application in situations where the number and size of objects packed in a confined space can be critical. Like the Cordwood problem above, this activity demonstrates the very definition of volume. MSP full record

How High? Geometry (Grades 6-8) http://nlvm.usu.edu/en/nav/frames_asid_275_g_3_t_3.html

Using an online simulation, students investigate conservation of volume by pouring a liquid from one container to a container of the same shape, but of a larger size. Students choose from four shapes: rectangular prism, cylinder, cone, and pyramid. The smaller version of the selected shape is shown partially filled with liquid; the base dimensions of both containers are also given. Using this information, students use a slider to predict how high the liquid will rise when poured into the larger container. On "pouring" the liquid, students can compare their prediction with the results. Multiple problems are available for each of the shapes. MSP full record

Popcorn: If You Like Popcorn, Which One Would You Buy? http://www.figurethis.org/challenges/c03/challenge.htm

Students are directed to use popcorn to compare the volumes of tall and short cylinders formed with 8-by-11-inch sheets of paper. A simple but visual and motivating way of comparing volume to height in cylinders! The solution offered explains clearly all the math underlying the problem. MSP full record

Surface Area and Volume http://www.shodor.org/interactivate/activities/sa_volume/index.html

This applet enables students to form and rotate both rectangular and triangular prisms. They can set the dimensions (width, depth, and height), observing how each change in dimension affects the shape of the prism as well as its volume and surface area. This is a quick way to collect data for a discussion of the relationship between surface area and volume. Users can rotate the figure and call for its frontal, side, or back view—very interesting with a triangular prism! MSP full record

Pyramid Applet http://www.mste.uiuc.edu/users/pavel/java/pyramid/

This applet allows students to set the width, height and length of a pyramid. They then see the initial cutout (the net) and watch it fold into the pyramid specified. For better viewing, the pyramid can be rotated. At this point, the surface area and the volume are shown. No activities accompany the applet, except for the challenge to try to minimize the surface area while maximizing the volume. From the Office for Mathematics, Science and Technology Education (MSTE), a division of the College of Education at the University of Illinois at Urbana-Champaign. MSP full record

Three Dimensional Box Applet: Working with Volume http://www.mste.uiuc.edu/users/carvell/3dbox/default.html

With this applet, students create boxes online in order to explore the relationship between volume and surface area. The screen first shows a rectangular piece of graph paper. Students “cut” four squares of a size determined by the student from the corners of the rectangle. The cut surface then folds to form a box whose dimensions, surface area, and volume are displayed onscreen. Since various sizes of graph paper can be selected, data can quickly be collected and the relationship between volume and surface area explored. MSP full record

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Copyright September 2006 — The Ohio State University. This material is based upon work supported by the National Science Foundation under Grant No. 0424671. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

This work is licensed under a Creative Commons License.