Bisect Segments and Construct Perpendiculars
Lesson 2 of 11
Objective: Students will be able to bisect segments and construct perpendicular bisectors.
Warm-Up: More Proof Practice
I use the More Proof Practice Warm-Up to have my students focus on the given information from which they can draw logical conclusions. In problem #1b, students often want to conclude that angles are congruent, triangles are congruent, or that one half of the pentagon is congruent to the other half--all of these misconceptions are good for developing a rich discussion around deductive reasoning.
Before completing our construction for today, I plan to give notes on conditionals, converses, and bi-conditional statements. While the notion of "if...then..." statements is not entirely new to students, I have found that it helps students to formalize these ideas by taking notes.
After students evaluate whether conditional statements and their converses are true, they work on some practice, checking with their group mates before we go through each statement as a whole class.
Like the last lesson, I ask students to list the factors needed to really show they have performed a construction. For example, when constructing perpendicular bisectors, I ask students questions like, "How would you find a perpendicular bisector to a line segment?" and "What would you need to make sure of?” which will prompt students to state that the bisector must pass through the midpoint of the segment at a right angle.
I pass out tracing paper to students, asking the students how they would find the midpoint of the segment--a student will state that we can fold the segment so that it perfectly overlaps itself, which will give us not only the midpoint, but also the perpendicular bisector.
I ask students how they can construct the perpendicular bisector without tracing paper. Essentially, I want to know how students can transfer their understanding from the tracing paper construction to using the compass and straightedge to ensure the same results. My goal in this part of the discussion is for students to see that every point on the perpendicular bisector is equidistant from the segment's endpoints.
I call on a student to volunteer to come up to the document camera to explain how he/she performs the construction, explaining the why behind everything he/she does and how these steps guarantee that he/she has constructed a perpendicular bisector.
I debrief the main ideas of the construction by having the class take notes.
To end today' lesson I will ask students to complete the following as an Exit Ticket:
Construct a perpendicular bisector and choose from the words arc, intersect, length, segment, vertex, distance, endpoint, ray to explain why you need two pairs of arcs—assume each pair of arcs is made with the same compass setting—to construct a perpendicular bisector.