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. Please add __your__ ideas on how to teach these concepts in the comments section.

**Keeping Cool: When Should You Buy Block Ice or Crushed Ice?**

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.

**How High? Geometry (Grades 6-8)**

Using an excellent online simulation, students pour 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 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.

**Popcorn: If You Like Popcorn, Which One Would You Buy?**

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.

**Surface Area and Volume**

With this applet students explore 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 or have students practice computing these measurements.

**Pyramid Applet**

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.

**Three Dimensional Box Applet: Working with Volume**

With this applet, students create boxes online; for each box, its dimensions, surface area, and volume are displayed onscreen. Since various sizes of boxes can be created, data can be quickly collected and the relationship between volume and surface area explored. A visual and “hands-on” experience!

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**Middle School Portal 2**publications. You can also email us at msp@msteacher.org. Post updated 11/10/2011.

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Excellent examples and links, thanks. Geometry is one of few areas of mathematics where visualization helps a great deal. I would like to suggest the site NeoK12.com for its online math lessons including few videos on geometry, in addition to other topics.

Muy bueno para aprender a visualizar y tener nocion espacial.

this is good

I’ve also found some of the blogs I read (or used to frequent)have a certain group of the “usual” commenters. After finding one very “agreeable” commenter on one blog, I then found the same commenter on another blog taking a polarized stance from the opinion they had agreed to on the other blog!