 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum           A B C Solutions for Session 10, Part B

See solutions for Problems: B1 | B2 | B3 | B4 | B5 | B6    Problem B1

Answers will vary. Some possible responses:

 • (Level 1 thinking) The students easily calculate the areas of the squares and triangles in different positions and recognize the triangles as congruent even though they are positioned differently. • (Level 2 thinking) By summing areas to find the total and equating the areas found in two different ways, students are showing logical thinking about geometric objects. • (Level 3 thinking) This is harder to see. The teacher is clearly trying to move them through a multi-step argument, but students may not all be aware that they are using previous results (area formulas, algebraic facts, solving equations) to prove something new.   Problem B2 There were many adaptations. Here are some: The proof was adapted to be one that was easier for students to follow. (It was based on the Garfield proof in this course, but was adapted to remove the need for computing with fractions.) Students worked through numerical and numeric/variable mixed problems before working with variables only. The teacher works with students as a whole class (on the proof itself) rather than asking them to work through individually or with pairs.   Problem B3 Answers will vary. The thinking often goes like this: "If it's going to make a solid shape, then 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. Putting all of this together to convince yourself that only five such solids are possible constitutes level 3 thinking.   Problem B4

 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 in Session 9, Problem A8. 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 to use those facts in making deductions.   Problem B5 Answers will vary. Students will probably gain understanding of three-dimensional figures and how they differ 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 B6 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.     