Intent In this activity, students explore conjectures about the sum of the angles in triangles and quadrilaterals and gain further practice in the use of protractors.

Mathematics The central mathematical idea underlying the next three activities—Degree Discovery, Polygon Angles, and An Angular Summary—is that there is a functional relationship between the number of sides of a polygon and the sum of the degree measures of its interior angles: sum of angle measures is equal to 180 degrees multiplied by (number of sides – 2). Students get a good deal of practice with measuring angles in Degree Discovery. In fact, upon completing this activity, they should have measured at least 20 angles. The design of the activity provides students with feedback on correct protractor use—students tend to check their readings when patterns aren’t emerging or when one polygon stands out differently from the others.

Progression Students will draw several triangles, measure and sum the angles in them, and then do the same for quadrilaterals. Their observations are noted in class and initiate a sequence of activities in which students derive and prove the angle sum formula for polygons. Degree Discovery works particularly well as a homework assignment and sets up class work on Polygon Angles. An Angular Summary serves as a wrap-up assignment for this part of the course.

Approximate Time 20 minutes for activity (at home or in class) 15 minutes for discussion

Classroom Organization Individuals, then groups, followed by whole-class discussion

Materials Protractors

Doing the Activity To transition, mention that in Pattern Block Investigations, students found the measures of the interior angles of the special polygons represented by the six pattern blocks. Now they will be asked to make conjectures about the sum of the angle measures of any triangle and then of any quadrilateral. Point out that students are to draw a variety of triangles. Ask the class to come to an agreement on how many each person should draw (we suggest three at least).

Discussing and Debriefing the Activity Give students a short time to share results and ask questions within their groups. Then invite some students to share their observations about triangles. Since they are using approximate measurements, their angle sums may not be exactly 180°, but they should see that regardless of the shape of a triangle, the angles always seem to add up to about 180°. This might lead to the conjecture that the angle sum for any triangle is exactly 180°, perhaps using the analysis in Pattern Block Investigations for the triangle pattern block as support. But this is a strong statement, one that cannot be proved by measuring, as no measurement is ever exact. What if the real answer were 181° or 179.5°? And what if the result is different for some triangles? This activity is designed to raise, rather than settle, these questions. A thoughtful argument could arise from the green pattern block triangle. Students should have noted that six of these blocks seem to fit together around a single point, so the angles are apparently 60° each. However, this too is not a conclusive argument, as students have no way yet to be sure that the blocks fit together perfectly. Use the word conjecture to describe the hypothesis that the angle sum for every triangle is 180°. In the discussion of the next activity, Polygon Angles, students will be told that the angle sum for triangles is always 180° and that they will see a proof for this later in the year (in the unit Shadows). For now, leave the issue unresolved, so that students are not yet certain whether their conjecture is true. For Question 2, let students share their conclusions about angle sums for quadrilaterals. They will probably see that the sum always appears to be approximately 360°. Bring out that this observation, like the one for triangles, is only a conjecture (at least for now), since the measurements are only approximations. Have students discuss how their work on Pattern Block Investigations relates to this conjecture. Their results for the four quadrilaterals should confirm the conjecture. Some students might offer that if the angle sum for triangles is 180 degrees, then it makes sense for the sum for a quadrilateral to be 360 degrees, because a quadrilateral can be seen as two non-overlapping triangles. This connection will be explored more deeply in the next activity.

Key Questions What if your protractor measurements are not exact? Why might measurement results vary for some triangles?