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 360º around a vertex. Regular hexagons and polygons with more than six sides all have 120º 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