##
* *Reflection: High Quality Task
Tri-Mind: Perpendiculars and Squares - Section 5: Tri-Mind Differentiated Individual Product

I made a lot of changes to the Tri-Mind this year given the transformations thread I have been trying to weave in throughout the curriculum. The ultimate learning objectives of the Tri Mind are to copy segments angles and to construct perpendiculars, so I was happy with my decision to let students choose from a special quadrilaterals lens, transformations lens, or creative lens while working on the Tri-Mind. Students chose what interested them the most, and it showed in the work they submitted, which had required detailed explanations and illustrations.

The Tri-Mind was not without its challenges, however. Option 1 (special quadrilaterals) and Option 2 (transformations) posed non-routine constructions problems to students, especially because I used this task on only the third day of the unit.

For students who chose Option 1, I found that encouraging them to sketch each special quadrilateral beforehand helped them to visualize the constructions they would need to perform for each problem. For students who chose Option 2, I found that activating their prior knowledge from the transformations unit and using the same language from that unit really helped. For example, I reminded students of how they rotated figures in the past by connecting a vertex to the point of rotation and measuring the angle from that segment—this helped e perpendicular bisector, which t perpendicularly bsiected over the line, which must be along the perpendicular bisector, which them see exactly where they would need to copy the given angle to locate the vertex’s image for the rotation. Similarly, I helped students see how they could reflect a figure over a given line by having them imagine the reflection of one of the figure’s vertices over the line and asking students what they think is “special” about the segment connecting those points—this helped them see that the segment connecting corresponding points must be perpendicularly bisected by the line of reflection, which then helped them to figure out how to perform the construction.

*High Quality Task: Raising Expectations and Infusing Constructions with Transformations*

# Tri-Mind: Perpendiculars and Squares

Lesson 3 of 11

## Objective: Students will be able to construct perpendicular bisectors and squares.

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?"

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**How far away am I from the wall?**

**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?**

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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.

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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.

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review

- LESSON 1: Introducing Constructions: Copy a Segment and Angle
- LESSON 2: Bisect Segments and Construct Perpendiculars
- LESSON 3: Tri-Mind: Perpendiculars and Squares
- LESSON 4: Bisect Angles
- LESSON 5: Construct Parallel Lines Through a Point Not on the Line
- LESSON 6: Use Constructions to Show Slope Criteria for Parallel and Perpendicular Lines
- LESSON 7: Construct Points of Concurrency
- LESSON 8: Constructions Teaching Project: Day 1 of 3
- LESSON 9: Constructions Teaching Project: Day 2 of 3
- LESSON 10: Constructions Teaching Project: Day 3 of 3
- LESSON 11: Constructions Unit Assessment