A B C

Solutions for Session 10, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5

Problem B1

Answers will vary. Some possible responses:

 • (Level 1 thinking) In placing shapes into the circles, Stephanie and Cara don't worry about the orientation. They think of the particular shapes as representing all of the same type of figure. Also, they see the right angle at the vertex of the kite, even though it is oriented in an unfamiliar way, and appearing in an unfamiliar figure. • (Level 2 thinking) Students in the class make guesses about the labels and refine their guesses based on additional information. If the pentagon goes in the right circle, the label can't be quadrilaterals.

 Problem B2 Answers will vary. The thinking often goes like this: "If it's going to make a solid shape, there must be at least three polygons meeting at a vertex. If it's going to make a solid shape, then there must be less than a total of 360° around a vertex. Regular hexagons and polygons with more than six sides all have 120° or more at each vertex, so these shapes cannot be used." And so on.

Problem B3

Answers will vary. Some possible answers:

 a. Key pieces of geometry are definitions and properties of regular two- and three-dimensional figures, building polyhedra, angle relationships, and so on. We also explored Euler's formula. b. "If-then" thinking, reasoning through every possible case, and generalizing were all important parts of the activity. It was important to both know the geometry (what are the angle measures for polygons with different numbers of sides?) and use those facts in making deductions.

 Problem B4 Answers will vary. Students will probably gain understanding of three-dimensional figures and how they're different from polygons. They will likely gain valuable understanding and visualization skills from building and manipulating the solids, and from attempting to count faces, edges, and vertices. They may not have the prerequisite knowledge of angle measures in polygons as a solid foundation. Some students may also struggle with the generalizations. If six triangles don't work, how do we know seven triangles won't work? Why can we eliminate polygons with seven, eight, and more sides without even trying to build them?

 Problem B5 Answers will vary. Some ideas: Lots of experience with building generalizations in cases that are easier to check, and lots of experience with polygons will help. For example, students may experience some difficulty in thinking about the role of angles and their connection to building Platonic solids. To prepare them beforehand, you may want to work on the sum of the angles in a triangle and extend it onto making generalizations about other polygons through dividing them into triangles as you've seen in Session 3 of this course.

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