Lesson 6

Construction Techniques 4: Parallel and Perpendicular Lines

6.1: Math Talk: Transformations (10 minutes)

Warm-up

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.

Launch

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.

Student Facing

Each pair of shapes is congruent. Mentally identify a transformation or sequence of transformations that could take one shape to the other.

Two identical circles, one point in each circle, left is circle P and right is circle Q.
Two identical segments, parallel to each other, one is AB and other is CD.
Two identical triangles, DEF and JKL, JKL is rotated almost upside down.
Two identical segments not parallel to each other, one is ST and other is MN.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

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:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?“
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

If students do not mention parallel lines, ask, “Why don't we need to use a rotation for this pair?“ (The lines are parallel.) 

Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. 
Design Principle(s): Optimize output (for explanation)

6.2: Standing on the Shoulders of Giants (10 minutes)

Activity

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.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. Display the image for all to see:

Point B on line l.

Remind students that in the previous lesson, they used straightedge and compass moves to create a line perpendicular to \(\ell\) that goes through point \(B\), which was on line \(\ell\). Now display an image of the construction of a perpendicular line through \(B\) for all to see throughout the activity:

Construction of perpendicular bisector through B.

Display a list of constructions students already know. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like (only include the first 6 constructions now):

  • circles of a certain radius
  • lines and line segments through two points
  • regular hexagons
  • equilateral triangles
  • a perpendicular bisector of a given segment
  • a perpendicular line through a point on the given line
  • a perpendicular line through a point not on the given line (added in this lesson)
  • a parallel line through a point not on the given line (added in this lesson)

After quiet work time, ask students to compare their responses to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as dynamic geometry software. 
Supports accessibility for: Visual-spatial processing;Conceptual processing; Organization

Student Facing

Here is a line \(m\) and a point \(C\) not on the line. Use straightedge and compass tools to construct a line perpendicular to line \(m\) that goes through point \(C\). Be prepared to share your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. The line segment \(AB\) has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment \(AB\). How far is this new point from \(A\)?

  2. We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment \(AB\). How far is this new point from \(A\)?

  3. If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?

Horizontal line segment A B. Point A on the left, Point B on the right.
 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 2. Display the image for all to see:

Point B on line l.

Remind students that in the previous lesson, they used straightedge and compass moves to create a line perpendicular to \(\ell\) that goes through point \(B\), which was on line \(\ell\). Now display an image of the construction of a perpendicular line through \(B\) for all to see throughout the activity:

Construction of perpendicular bisector through B.

Display a list of constructions students already know. This display should be posted in the classroom for the remaining lessons within this unit. It should look something like (only include the first 6 constructions now):

  • circles of a certain radius
  • lines and line segments through two points
  • regular hexagons
  • equilateral triangles
  • a perpendicular bisector of a given segment
  • a perpendicular line through a point on the given line
  • a perpendicular line through a point not on the given line (added in this lesson)
  • a parallel line through a point not on the given line (added in this lesson)

After quiet work time, ask students to compare their responses to their partner’s and decide whether they are both correct, even if they are different. Follow with a whole-class discussion.

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as dynamic geometry software. 
Supports accessibility for: Visual-spatial processing;Conceptual processing; Organization

Student Facing

Here is a line \(m\) and a point \(C\) not on the line. Use straightedge and compass moves to construct a line perpendicular to line \(m\) that goes through point \(C\). Be prepared to share your reasoning.

Point C lies roughly above the middle on horizontal line m.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. The line segment \(AB\) has a length of 1 unit. Construct its perpendicular bisector and draw the point where this line intersects our original segment \(AB\). How far is this new point from \(A\)?

  2. We now have 3 points drawn. Use a pair of points to construct a new perpendicular bisector that has not been drawn yet and label its intersection with segment \(AB\). How far is this new point from \(A\)?

  3. If you repeat this process of drawing new perpendicular bisectors and considering how far your point is from A, what can you say about all the distances?

Horizontal line segment A B. Point A on the left, Point B on the right.
 

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may struggle more than is productive. Ask these students to draw a line segment and construct the perpendicular bisector of it. In that construction, the perpendicular bisector will go through an intersection point of two circles. Ask, “What happens if you create a circle centered at that intersection point that goes through an endpoint of the segment? Why does that happen? How can you use this idea in this new activity?”

Activity Synthesis

Focus on the process of using a previous construction to generate new constructions. Here are some questions for discussion:

  • “How was this construction different from perpendicular line constructions you have done before? How did thinking about the differences help you plan what to do?” (I didn't have a segment to bisect, because the point wasn't on the line. I realized I could still make a segment using the new point.)
  • “How does knowing some constructions help you do other, more complicated constructions?” (I can use the same moves, but just change them a little bit.)
Writing, Speaking, Conversing: MLR 1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine the reasoning used to create their construction. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help individuals strengthen their ideas and clarify their language. For example, “Can you explain how…?”, “Why did you choose to construct a circle there?”, “What do you know about radii that helps here?”, or “What do you mean by…?”. Students can borrow ideas and language from each partner to strengthen their final explanation. Design Principle(s): Optimize output (for justification); Support sense-making

6.3: Parallel Constructions Challenge (15 minutes)

Activity

The purpose of this activity is for students to think strategically about how to apply previous constructions to a new construction.

It is not expected that students will use any method other than two consecutive perpendicular lines, but if students come up with other methods, consider discussing those methods during the lesson synthesis.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Add an additional item to the inventory of different constructions students have learned.

  • a perpendicular line through a point on the given line

Remind students that they can use this inventory to think about how to use constructions they know to build something new. 

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students with organizational skills in problem solving. For example, first invite students to construct a line perpendicular to line \(m\) that goes through point \(C\). Then ask students to consider the relationship between the line perpendicular to \(m\) and the line parallel to line \(m\). Finally, ask students to construct a line parallel to line \(m\) that goes through point \(C\)
Supports accessibility for: Organization; Attention

Student Facing

Here is a line \(m\) and a point \(C\) not on the line. Use straightedge and compass moves to construct a line parallel to line \(m\) that goes through point \(C\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Add an additional item to the inventory of different constructions students have learned.

  • a perpendicular line through a point on the given line

Remind students that they can use this inventory to think about how to use constructions they know to build something new. 

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students with organizational skills in problem solving. For example, first invite students to construct a line perpendicular to line \(m\) that goes through point \(C\). Then ask students to consider the relationship between the line perpendicular to \(m\) and the line parallel to line \(m\). Finally, ask students to construct a line parallel to line \(m\) that goes through point \(C\)
Supports accessibility for: Organization; Attention

Student Facing

Here is a line \(m\) and a point \(C\) not on the line. Use straightedge and compass moves to construct a line parallel to line \(m\) that goes through point \(C\)

Point C lies roughly above the middle on horizontal line m.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may struggle more than is productive. Ask these students to consider what they just learned to construct starting from the point and line. (A perpendicular line.) Invite them to consider the relationship between the line they could construct and the parallel line they want to construct. (Those are also perpendicular.)

Activity Synthesis

The purpose of the discussion is to focus on the process of using previous constructions to generate new constructions. Ask students, “How does knowing some constructions help you do other, more complicated constructions?” (In this construction, I repeated a construction I already knew twice.)

Add an additional item to the display of constructions students already know:

  • A parallel line through a point not on the given line

Lesson Synthesis

Lesson Synthesis

Reference the display of figures students know how to construct. Invite students to discuss what other figures they could draw using this inventory. (Quadrilaterals with parallel sides or right angles including trapezoids, parallelograms, rectangles, and squares.)

6.4: Cool-down - Find the Missing Endpoint (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

When we write the instructions for a construction, we can use a previous construction as one of the steps. We now know 2 new constructions that are made up of a sequence of moves.

  • Perpendicular lines are lines that meet at a 90 degree angle.
  • Parallel lines are lines that don’t intersect. One way to make parallel lines is to draw 2 lines perpendicular to the same line.
Two figures of perpendicular lines.