Students explore the sum of interior angles in a triangle in 3 ways: tearing corners off of paper triangles, the walking-and-turning exercise, and a proof using the parallel postulate. This activity (collection of activities) gives different arguments for the same geometry fact, and invites a discussion about the difference between examples and proofs, and of different levels of rigor for different grade levels.
First demonstrate by folding paper models, then with a proof using congruence of triangles or symmetry. As with previous example, gives multiple explanations for same fact and invites discussion of different levels of rigor. Similar activities for other properties of shapes (e.g. diagonals in quadrilaterals) could be developed.
Explore shapes with questions such as “why is this a …?” Encourages focus on definitions and derived properties, invites different levels of proof. Also encourages correct use of geometry terms.
Present a cube made of smaller cubes with k on each side. If the outside of the large cube is painted, how many square faces of the smaller cube will be painted? This question can be explored with different levels of rigor, starting with small values of k and counting faces, eventually giving a complete proof for general k. Students can give both geometric and algebraic justifications.
Count examples, formulate conjecture for general case, prove conjecture. Give proofs which are completely algebraic, or completely geometric, and discuss relationship.
Students explore and form conjecture about the four angles formed by two intersecting lines. Students then develop informal and formal arguments for their conjecture. This activity provides a good link between geometric and algebraic thinking. Can also give an argument using transformations (rotation).
Demonstrate/explain the proof in multiple ways: video or demonstration of water filling triangle/squares, geometric rearrangement, dissection argument, or other proofs. Giving some geometric and some algebraic proofs provides opportunity for a discussion about the connection between the two subjects.
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