Lesson 6 of 6
Objective: SWBAT solve problems related to angle measures and angle sums in regular polygons.
To begin today's lesson, I hand out lined paper and ask my students explain, in the form of an essay, the different methods that can be used to prove that triangles are congruent. I ask them to also include a discussion of those methods that cannot be used. I mention that they may include diagrams in their essay.
I have included two samples of student work. When I asked my class to do this assignment this year, the students had not yet learned Hypotenuse-Leg, so that method will be missing from their essays.
Review of Polygons
This short problem set, Review of Polygons, serves as a warm-up for our work on regular polygons. It reviews what we have learned in our previous lesson about polygons. The activity was designed to help manage the transition from the essay to the following exploration.
Introducing Regular Polygons
Once everyone has finished the Essay and/or the Review of Polygons, I will hand out the next set of problems, Regular Polygons. This problem set begins with the definition of a regular polygon. I plan to work the first one or two problems out with the class. Then, I will ask the students to work cooperatively in their groups.
As students work, I circulate throughout the room, listening to conversations. Occasionally, I will stop the class for a group discussion. I hope to focus on an interesting approach that I might have noticed. But, I will also intervene if there is widespread difficulty. These problems require a good deal of sense making (MP1), reasoning quantitatively (MP2), and focusing on the structure of the figures (MP7). My students are generally good at some of these skills, but not necessarily all of them. Group work helps students to improve their skills in each area. I will also provide (or allow) calculators for these problems (MP5).
The last few problems hearken back to our unit on parallel lines. I like to include references to parallel lines throughout the year, because my students find this topic to be challenging.
When we have finished our discussion of the Regular Polygon problem set, I will hand out two proofs that involve regular polygons. Many students find these two proofs to be quite easy. In order to increase the level of challenge, I have added a twist. The proofs do not end with proving corresponding parts congruent. Both proofs extend one step further:
Proof 1: proving a segment is an angle bisector
Proof 2: proving that a triangle is isosceles
I expect many of my students will use "Corresponding parts of congruent triangles" incorrectly to explain these conclusions. After they have attempted both proofs, we will take the time to discuss this nuance.
Handing Out the Homework
When I hand out tonight's homework assignment, I take a little time to set up the last two proofs on the assignment. These proofs introduce have several interesting wrinkles:
- Overlapping Triangles
- Reflexive Angles
- Proving two sets of triangles congruent.
When I setup Proof #2, the one that involves overlapping triangles, I ask the students to look at the diagram and read the givens. I plan to ask a series of questions:
- What triangles are we going to work on in this proof? How do you know? At this point I have them outline the two triangles in colored pencil.
- What pairs of parts are we going to work on? I have found that the students quickly volunteer the given sides and the right angles, but need some time to recognize the need to work with the reflexive angle. Eventually a student will volunteer that angle A is the same angle in both triangles, and this allows me to introduce the notion of a shared, reflexive angle.
For Proof #3 we discuss the fact that there are multiple sets of triangles in the diagram. We verbally run through how we might approach this proof (e.g., starting with the largest triangles first, then working on a set of smaller ones). Again, colored pencils prove to be helpful for students to annotate the diagrams and prepare for the assignment.