Session 10, Part C:
Problems That Illustrate Geometric Reasoning (55 minutes)

In This Part: Geometric Reasoning Problems, Part 1 | Geometric Reasoning Problems, Part 2

In this part, you'll look at several problems that are appropriate for students in grades 6-8. As you look at the problems, answer these questions:

 a. What is the geometry content in this problem? b. What skills do students need to work through this problem? What skills will this problem help them develop for later work? c. What level of geometric thinking is expected of students in the problem? Does it ask students to bridge levels? d. What other questions might extend students' thinking about the problem? e. Describe a lesson that you could develop based on the content of this problem.

Note 6

Problem C1

One way to test whether two figures are congruent is to try fitting one exactly on top of the other. Sometimes, though, it's not easy to cut out or trace figures, so it's helpful to have other tests for congruency.

Each problem below suggests a way to test for the congruence of two figures. Decide whether each test is good enough to be sure the figures are congruent. Assume you can make exact measurements. If a test isn't good enough, give a counterexample -- that is, an example for which the test wouldn't work.

 1 For two line segments, measure their lengths. If the lengths are equal, the line segments are congruent. 2 For two squares, measure the length of one side of each square. If the side lengths are equal, the squares are congruent. 3 For two angles, measure each angle with a protractor. If the angles have equal measures, they are congruent. 4 For two rectangles, find their areas. If the areas are equal, the rectangles are congruent.

Problem C2

Not all of the following statements are true. For the ones that you think are false, make up a counterexample. Then make up two statements of your own, one true and one false.

 • If something is a cube, then it is a prism. • If something is a prism, then it is a cube. • If something is a square, then it is a rectangle. • If something is a rhombus, then it is a square. • All parallelograms have congruent diagonals. • All quadrilaterals with congruent diagonals are parallelograms. • If two triangles have the same perimeter, then they are congruent. • If two rectangles have the same area, then they are congruent. • All prisms have a plane of symmetry.

 Problem C3 A rectangle has been divided into two congruent parts. What could the parts be?

 Problem C1 adapted from IMPACT Mathematics, Course 2, developed by Educational Development Center, Inc. p. 453. © 2000 Glencoe/McGraw-Hill. Used with permission. www.glencoe.com/sec/math Problems C2-C3 adapted from Van de Walle, John A. Geometric Thinking and Geometric Concepts. In Elementary and Middle School Mathematics: Teaching Developmentally, 4th ed. p. 343. Copyright © 2001 by Pearson Education. Used with permission from Allyn and Bacon. All rights reserved.
 Session 10, Grades 6-8: Index | Notes | Solutions | Video