LEARN NC

Problem centered math

Developed by LEARN NC with Grayson Wheatley

3.1 Math problems for grade 8 geometry

By Grayson Wheatley

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 objectives of the Spatial Sense, Measurement, and Geometry strand (Goal 2) of the North Carolina Standard Course of Study for Mathematics, Grade 8.

The problems are available in PDF format except where otherwise noted. (The free Adobe Acrobat Reader is required to view PDF files.) To view a problem, click on its title; to view the text of relevant curriculum objectives, click on the numbers of the objectives.

Relevant goals and objectives are listed in parentheses after the title of each problem set and refer to the 1999 version of the mathematics Standard Course of Study.

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 24cm × 27cm rectangle exactly with 2cm square tiles. In reality, you would need 168 tiles, not 162. You could change the problem to a 24cm × 30cm rectangle, in which case it would require 180 2cm square tiles or 80 3cm 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!

Fencing Problems

A set of 8 related problems involving building of fences around various sizes of garden plots. Solutions provided. By Randy Harter, Buncombe County Schools.

Gardening Problems

A set of 3 related problems involving manipulation of sizes of garden plots and construction of boxes from sheets of given sizes. Illustrations provided. By Pat Sickles, Durham Public Schools.

A Pile of Gold

Comparison of volumes of containers of various shapes. Illustrations provided. By Holley Merschat, North Buncombe Middle School, Buncombe County.

Circles and Regular Polygons

Using a spreadsheet to explore relationships between sizes of inscribed and circumscribed circles. By Randy Harter, Buncombe County Schools.

PE Storage Box

A set of 7 related problems involving the visualization and manipulation of boxes of different sizes and their contents. By Jeanne Joyner, North Carolina Department of Public Instruction.

Manipulating Pyramids

Finding surface area and volume of a pyramid from given dimensions; understanding the relationship between changes in various dimensions of the pyramid. Illustration provided. By Wayne Drummond, Owen Middle School, Buncombe County.

Pythagorean Problems

A set of 5 real-life problems requiring use of the Pythagorean Theorem. Solutions provided. By Brent Bustle, Troutman Middle School, Iredell-Statesville Schools.

Explaining the Area of a Triangle

From a triangle with given dimensions, explain three different ways of computing the area. Illustration provided. By Wayne Drummond, Owen Middle School, Buncombe County.

Overlapping Triangles

Comparison of areas of various triangles in a complex diagram. Illustration provided. By Wayne Drummond, Owen Middle School, Buncombe County.

What’s My Rule?

A game to help students understand the concept of adding squares and their square roots. This makes a good way to begin a lesson on the Pythagorean theorem. By Brent Bustle, Troutman Middle School, Iredell-Statesville Schools.

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.