Tri-Mind: Perpendiculars and Squares

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Students will be able to construct perpendicular bisectors and squares.

Big Idea

In the Tri-Mind, a differentiated task, students will choose the format through which they will explain how to construct perpendicular bisectors and squares.

Review Features of High-Quality Constructions Explanations

5 minutes

As we begin to move on to more complicated constructions, I want to reiterate the features of high quality explanations with my students. 

High Quality Explanations...

  • Provide clear explanations on what to do
  • Provide directions that answer the question, “How do I perform this construction?”
  • Provide a clear explanation on why a “step” must be completed.
  • Enable the user to answer the question “Why did I have to perform these steps?"

Distance Discussion: Construct Perpendiculars

20 minutes
Next, I ask my students to imagine that the front wall of the classroom represents a line and that I represent a point that is not on the line.  I ask the students:
How far away am I from the wall?

I am, first making sure to ask them questions like, "would I measure the distance from me to the corner of the room?" or "would I measure the distance from me to the door to the classroom, which is in the opposite corner?"  Students resoundingly agree that when considering distance, we should measure the distance along the perpendicular, which is the shortest distance from the point to the line.

This motivates our discussion of how to construct a model of the distance from a point to a line. I want each construction our prior work during this unit.  Yet, I want my students to understand that we cannot construct a perpendicular bisector to a line. Nonethless, I want them to extend the work we did when constructing perpendicular bisectors to this construction.  

Again, we will start with tracing paper. I pass out tracing paper to groups and ask them to place a point not on a line and then to construct the perpendicular to the line through that point. Students will physically fold the tracing paper so that the line overlaps itself with the point on the fold.  I ask them to pause and think about how this relates to the construction for perpendicular bisectors (tracing paper and compass and straightedge construction).  I give students time to tinker around, asking them:

How would you construct a perpendicular to a line through a point using a compass and straightedge?  How can you connect this question to our construction in the last lesson, which was to construct perpendicular bisectors?

I circulate the room; if I see that students have discovered a way to perform the construction, I ask them to share out with the document camera, then I call on student volunteers to summarize or re-phrase what the student did and why he/she did it.  If, when circulating the room, students seem stumped, I will show the first step of the construction, give think time, and then ask for a volunteer to explain why we need to mark off two points equidistant from the point not on the line.  I will then ask them what they should do next, then debrief the construction. 

Check for Understanding (Pairs) and Debrief

10 minutes

For this Pair Check for Understanding, I use the document camera to show my work for constructing a perpendicular through a point given on a line, with each "step" numbered.  I ask students to write step-by-step directions for how to perform the construction and require them to use the words "equidistant" and "endpoints" in their explanations.

I remind students of the features of high-quality explanations we discussed at the beginning of the lesson, telling them that they need to make sure their explanations answer the questions, “How do I perform this construction?" and “Why did I have to perform these steps?"

I ask for at least two different pairs to read out their explanations so that everyone can hear them and check their explanations.  

Tri-Mind Differentiated Individual Product

45 minutes

I want to assess individual student's understanding, particularly at this point in the unit when we have performed several constructions.  I introduce the Constructions Tri-Mind to students, explaining to students that they can choose one of three options for demonstrating their understanding of the constructions we have done so far--the Tri-Mind gives me a way to differentiate the product students create to show the level at which they can meet this lesson's learning objectives.    

Before having students work, I make my expectations explicit, telling students they need to incorporate clear constructions markings (arcs, rays, lines), precise geometry vocabulary (point, ray, endpoint, adjacent, bisect, perpendicular, etc.), and step-by-step explanations of all constructions, even the most basic ones.  I tell students they should check their explanations by asking if anyone can follow their directions and perform the construction successfully.