Math problems for grade 8 geometry - Problems and lessons

These problems were created or compiled by participants at LEARN NC’s May 2001 Problem-Based Mathematics workshop at the North Carolina Center for the Advancement of Teaching in Cullowhee, N.C. They address various aspects of Spatial Sense, Measurement, and Geometry.

Area, Surface Area, and Volume: Selected Problems

A set of 6 problems involving manipulation of sides of squares, rectangles, and rectangular solids and explaining resulting changes in area, surface area, and volume. Solutions provided. By Retella Jones and Grayling Williams, Durham County Schools. Based on problems from http://www.mathcounts.org and from the collection of Grayson Wheatley, Professor Emeritus, Florida State University.

Errata: The answer to problem #6 is technically correct but physically impossible — it isn’t possible to tile a 24-cm × 27-cm rectangle exactly with 2-cm square tiles. In reality, you would need 168 tiles, not 162. You could change the problem to a 24 cm × 30 cm rectangle, in which case it would require 180 2-cm square tiles or 80 3-cm square tiles. For mixed tiles, various answers are possible, but note that obtaining a physically possible answer will require more reasoning than just dividing area by area!

Visualizing Halves (PDF file|Web page)

Students examine diagrams to determine whether exactly half is shaded. Could be used as a warmup exercise. By Grayson Wheatley, Professor Emeritus, Florida State University.

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Problems and lessons

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1 The clinical interview
2 The problem-centered classroom
3 Problems and lessons
1. 3.1Math problems for grade 8 geometry
2. 3.2Lesson plans for problem centered math

What makes a good problem?

A problem should be non-routine — that is, a student can’t simply isolate the numbers and calculate but must give the numbers some meaning. - A problem should help students foster their mathematical reasoning.
The problem should include follow-ups that elaborate from a special case to a generality — for example, from finding the perimeter of a string of 2, 3, 4 adjacent hexagons to a string of n hexagons.
John A. Van de Walle, in "Reform vs. the Basics: Understanding the Conflict and Dealing with It," defines a good problem as “any task or exploration for which the solution has not been explained, that begins where kids are (with their ideas), that is challenging mathematically, and for which justification and explanations for answers, methods, and results are understood to be the responsibility of the students.”