Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes.

These activities involve middle school learners in scenarios that challenge them to apply their understanding of measurement of solids. They will also work on circumference and circle area problems, included in this area of emphasis.

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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. (From Village math)

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

Windshield wipers : it's raining! Who sees more? The driver of the car or the truck? http://www.figurethis.org/challenges/c32/challenge.htm

In this activity, students compare the areas cleaned by different wiper designs. An animation shows the movement of the two windshield wipers, each cleaning off a different geometric shape on the window. Students are encouraged to draw the shape cleaned by each wiper and find its area. MSP full record

Big tree : have you ever seen a tree big enough to drive a car through? http://www.figurethis.org/challenges/c15/challenge.htm

Students must consider the girth and height of ten National Champion trees and determine which, if any, of the trees is large enough to drive a car through. The solution relies on knowledge of the formula for finding the circumference of a circle. 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

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

Cylinders and Scale Activity http://nsdl.exploratorium.edu/nsdl/showRecord.do?id=10911

Using a film canister as a pattern, students create a paper cylinder. They measure its height, circumference, and surface area, then scale up by doubling and even tripling the linear dimensions. They can track the effect on these measurements, on the surface area, and finally on the amount of sand that fits into each module (volume). The lesson is carefully described and includes handouts. MSP full record

Measuring the Circumference of the Earth http://www.k12science.org/noonday/index.html

Here is a real-world project that will engage your class in measuring the circumference of the Earth! You will find all the information you need to enable students to re-create the measurement as done by the Greek librarian Eratosthenes over 2,000 years ago. The procedure is based on measurements of shadows taken at high noon local time on a designated day in March; results from several schools are posted online and used to calculate the circumference. Included are detailed explanations and illustrations of the mathematics involved. MSP full record

NCTM. (2006). Curriculum Focal Points for Kindergarten Through Grade 8 Mathematics: A Quest for Coherence. Reston, VA: Author

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Copyright May 2008 — 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.