From Perpendiculars to Parallels

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Objective

SWBAT construct the perpendicular or parallel to a line through a given point. Students will understand how angle bisectors, segment bisectors, and perpendiculars are all related.

Big Idea

Students consolidate and extend their knowledge of constructions related to bisectors: "A perpendicular to a perpendicular is a parallel."

Lesson Open

9 minutes

I begin the lesson by displaying the warm-up prompt using the slide show for the lesson.

The prompt asks students to construct a line AB and a point C on the line between A and B.  Next, they are asked to construct the bisector of angle <ACB.  The purpose of the warm-up is to introduce the idea (to those who have not realized it) that the sides of an angle can form a straight line, that such a straight angle can be bisected, and that a perpendicular is precisely such a bisector: the bisector of a straight angle.

Students may balk at first.  How can they construct the angle bisector of a line?  I ask students to recall the definition of a line: a pair of rays that share a common initial point.  Doesn't angle <ACB meet the definition (MP7)?

The construction of the angle bisector of a straight angle is no different than the construction of the bisector of any other angle, but it may look different.  Structurally, however, there is no difference (MP7).

Since students cannot perform the construction on the front board, this warm-up will be completed individually, with students making their constructions right in their learning journals.  I circulate, offering help where needed.  As students finish the construction, I ask them to get out their notes on perpendiculars from the previous lesson.

I ask: How is the construction of the bisector of a straight angle similar to the construction of the perpendicular to a line through a given point?  In fact, the two constructions are no different, except that in the notes the given point is not located on the line to be bisected.  I ask students to imagine how the construction would look if the given point were located on the line, instead.  In fact, the construction of the perpendicular to a line through a given point is structurally identical to the construction of the bisector of a straight angle (MP7), regardless of whether the point is located on the line to be bisected or not.

I display the Agenda and Learning Targets for the lesson.  I tell the that today we will practice constructing perpendiculars, and we will learn more about their properties.

Constructing Perpendiculars

15 minutes

I distribute the handout for the activity and display the instructions.  During this part of the lesson, students practice the construction of the perpendicular to a line through a given point and investigate its properties.

The exercises ask students to verify the properties of a perpendicular in terms of congruent segments or angles.  Constructed properly, all four angles formed by the perpendicular and the original line are congruent.   I encourage students to verify congruence with construction techniques, using tracing paper, or using paper folding strategies (MP5).

The activity uses the Rally Coach format, so that students may provide each other support and check each other's work.

The exercises also provide opportunities to see whether students are performing the constructions accurately and labeling the figures correctly (MP6).

If students finish early, I may give them one of the problems from the Bisector Challenge activity

Summarizing Properties of Perpendiculars

15 minutes

I use the guided notes to summarize the properties of perpendiculars, which we define as bisectors of a straight angle.  Right angles, then, are the congruent halves of a bisected straight angle.

I then ask: Suppose you constructed the perpendicular to a line, then constructed a perpendicular to the perpendicular.  How would the perpendicular to the perpendicular be related to the first line?

I am prepared to start drawing a figure on the white board to help students imagine the lines in question.  I expect at least a few students to guess that the perpendicular to the perpendicular is parallel to the first line, either because they have heard something like this before, or because it looks like that in my drawing.

I tell students that we can construct a parallel to a line by constructing a perpendicular to a perpendicular.  In the next unit, we will learn just why this construction works.  For now, however, we will master the construction.  We will need it in a few lessons to investigate the properties of a rigid motion.

I demonstrate the construction of a parallel with the help of the notes.  My students follow along,  illustrating the steps of the construction in their guided notes.

Constructing Parallels

15 minutes

I distribute the handout for the activity and display the instructions.  During this part of the lesson, students practice the construction of the parallel to a line through a given point.

The activity uses the Rally Coach format, so that students may provide each other support and check each other's work.

Lesson Close and Homework

5 minutes

Individual Size-Up

The lesson close follows our Individual Size-Up Routine routine.  The prompt asks students to write in their learning journals: "The perpendicular to a perpendicular is a _______________" and complete the sentence.

Homework

For homework, I assign problems #21-22 of Homework Set 1 for this unit.  Problem #21 provides practice in the constructions that students learned in this lesson.  Problem #22 asks students to describe the symmetry of a regular polygon.  At this point, students can simply draw and label the lines of reflection and center of rotation right on the figure and refer to them in their descriptions of the rigid motions that carry the polygon onto itself.  In the next half of the unit, students will learn to reason precisely what those lines and points must be, based on the properties of rigid motions.

In addition to the two problems from the homework set, students should complete Portfolio Problem CR-1, Tricky Tiling.  By this point in the unit, they have learned what they need to answer that problem completely.

I also remind students to put together their unit learning portfolio, which they will turn in for a progress check.