Geometry Hs Mathematics Unit 07 Lesson 01

[Mrx Wylie]

Thepurpose of this activity is for students to develop intuition that the set of points equidistant to two given points forms a perpendicular bisector by asking them to play the role of the points. Students will formalize this conjecture in the lesson synthesis and prove it in a subsequent lesson.

Locate an area in the classroom or nearby where several students can stand together and be seen by all students. Mark two points on the floor about 2 meters apart with masking tape and clear a space between and around the points. Label one point \(A\) and one point \(B\). Invite one student to stand at \(A\) and one student to stand at \(B\).

Your teacher will mark points \(A\) and \(B\) on the floor. Decide where to stand so you are the same distance from point \(A\) as you are from point \(B\). Think of another place you could stand in case someone has already taken that spot.

Invite students to look at their sketch of points whose distance from \(A\) is the same as their distance from \(B\). Ask them what they notice about the points. Ask them what they wonder about the points.

Tell students that in mathematics, things people wonder are often referred to as conjectures. A conjecture is a statement that we wonder whether it is true. Ask students to make a conjecture about the collection of all the points whose distance from \(A\) is the same as their distance from \(B\), and select 2 or 3 to share.

Define the perpendicular bisector as a line through the midpoint of a segment that is perpendicular to that segment. Informally, explain that bi means two and sect means cut, and so a perpendicular bisector is literally a line perpendicular to a segment that cuts it into two congruent pieces.

Students might point out that in one, the angles are not right angles; therefore, the lines are not perpendicular. In the other, students might state that the lines are perpendicular but segment \(FG\) is not bisected since \(H\) is not the midpoint of \(FG\).

Display two copies of segment \(PQ\) for all to see. Explain that the displayed segments cannot be folded. Invite a student who did a freehand drawing to demonstrate next. Follow with a student who used a compass and straightedge to make a construction.

If no student uses a compass to construct or check, encourage the class to consider how to use that tool. Display the image from the warm-up again and invite students to explain how to use that construction to find a perpendicular bisector.

Emphasize that both paper folding and construction with a compass and straightedge are valid, accurate methods, but freehanding only works for a sketch. Choosing which one to use will depend on the problem and tools available.

Remind students that a conjecture is a statement they think might be true. Display the conjecture that the perpendicular bisector of a segment is the set of points that are the same distance from the endpoints of that segment. Here are some questions for discussion:

A perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. Recall that a right angle is the angle made when we divide a straight angle into 2 congruent angles. Lines that intersect at right angles are called perpendicular.

A conjecture is a guess that hasn't been proven yet. We conjectured that the perpendicular bisector of segment \(AB\) is the set of all points that are the same distance from \(A\) as they are from \(B \). This turns out to be true. The perpendicular bisector of any segment can be constructed by finding points that are the same distance from the endpoints of the segment. Intersecting circles centered at each endpoint of the segment can be used to find points that are the same distance from each endpoint, because circles show all the points that are a given distance from their center point.

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The full experience and value of eMATHinstruction courses are achieved when units and lessons are followed in order. Students learn skills in earlier units that they will then build upon later in the course. Lessons can be used in isolation but are most effective when used in conjunction with the other lessons in this course. All Lesson/Homework files and videos are available for free. Other resources, such as answer keys and more, are accessible with a paid membership.

Each month August through May we release new resources for this course that are accessible with a membership. We release new resources in unit order throughout the school year. You can see a list of our new releases by visiting our blog and selecting the most recent newsletter.

The purpose of this Math Talk is to elicit strategies and understandings students have for rigid transformations. These understandings help students develop fluency and will be helpful later in this unit when students will need to be able to define transformations rigorously and use transformations in proofs. While participating in this activity, students need to be precise in their word choice and use of language (MP6). Students will continue developing transformation vocabulary throughout the unit, it is not necessary for students to use phrases such as directed line segment at this point. It is okay if there is not enough time to discuss all 4 problems.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

The goal of this discussion is to identify parallel lines. Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

The purpose of this activity is to extend what students know about constructing a perpendicular line through a point on the given line to a new situation in which the constructed perpendicular line goes through a point not on the given line.

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