Geometry : Unit #5 - Constructions : Lesson #4

Bisect Angles

Objective: Students will be able to construct angle bisectors using geometric concepts and tools.
Standards: HSG-CO.D.12 MP5 MP8
Subject(s): Math
60 minutes
1 Warm-Up: Proof Practice 2 - 15 minutes

I use the Proof Practice 2 Warm-Up to have my students focus on the given information from which they can draw logical conclusions. In Problem #2, I expect my students may struggle with using words to describe what they know to be true: complements of congruent angles are congruent.

I make sure to have at least one student present their ideas for each proof.  I project the diagram on the whiteboard so that students can write, use color to clarify their ideas, and invite other students in the audience to comment on their work.  Putting the proof into words is challenging for my students at this point in the year, which makes it particularly important for students to see multiple ways to logically show the angles are congruent (by using the idea that complements of congruent angles are congruent, or perhaps by making a strong algebraic argument). 


2 Notes: Medians, Altitudes, Midsegments - 15 minutes

I record the key ideas  for today's lesson in the Constructions Notes because I need students to have a formal, organized way of dealing with the many necessary concepts.  The collection of terms (e.g., Medians, Altitudes, Midsegments, and Perpendicular Bisectors) can often be very confusing for students, which is why recording meaningful notes about definitions and comparisons is a priority objective of this lesson.

3 Mixed Constructions Practice - 30 minutes

I use this Mixed Constructions Practice to give students a chance to review and refine their knowledge of the constructions they have learned in the unit thus far.  As they work, I encourage students to continue to ask questions and to explain constructions to each other in their groups. This practice is especially important because they will work with each other on their Group Constructions Quiz in the next lesson.  

To make sure that all students are accountable for the work, I ask each group to make sure that construction methods make sense to everyone in the group.  I say, "When your group completes a task and calls me over to check in, I want to be able to quiz anyone in the group to explain the method that I am checking."

4 Construction: Bisect Angles - 15 minutes

I pass out tracing paper to all of my students and ask them to construct an angle.  I then ask them how they would bisect their angle by folding, asking questions like, “How would you bisect an angle?" and "What would you need to make sure of?”  I call on a student volunteer to share out that the bisector would have to be the same distance away from each of the sides of the angle.  I ask students to consider this idea; I give students silent think time, maybe 30 seconds-one minute, to consider how they might construct an angle bisector using a compass and straightedge.

After trying out a few experiments, I ask a volunteer to share their construction ideas with the class by using the document camera.  Inevitably, I will have at least one or two other students who want to check their methods to see if they will work as well--I have found that it is really important to honor these requests as these related methods often reinforce the sense making aspect to performing the constructions.

Surfacing Misconceptions
Continuous Assessment

As I circulated the classroom, I listened for students’ reasoning as they tried to come up with a method for bisecting angles.  An idea came up to connect the two points on each side of the angle with a segment and to perpendicularly bisect that segment.  I decided to write up an Angle Bisector "Math Hospital" to provoke students’ talk around why making the initial arc on the angle is important (to locate two points equidistant from the vertex) and how this construction is essentially the same as the construction for a perpendicular bisector.


5 Homework: True or False? Show How You Know! - 5 minutes

For the homework assignment, I assign five True or False statements.  In this assignment, I want students to focus on the condition given--to sketch and/or construct several cases that fall within the condition--and then to come to a conclusion about whether the statement is true or false.