Park City Mathematics Institute Geometry Project Abstract
Modifying and Creating Rigid Motion Tasks
Authors: Ramona Fittipaldi, Abigail Kirchman, Vince Muccioli, Arundhati Velamur

We have revised or created six geometry tasks about rigid motions. With revised tasks, we include the original task so teachers can consider their own process of revisions. Our goal was to modify or create tasks to increase their cognitive demand and accessibility and to incorporate new ideas from Common Core standards regarding rigid motions. Each task additionally includes teacher notes (questions, hints, and extensions) to support implementation. Tasks include:
• Task 1: Provides students an opportunity to explore the ideas of rigid motion before they know the formal definitions.
• Task 2: Helps students develop an understanding of reflections with folding the figure.
• Task 3: Explores rotations and motivates students to understand the need for a center, angle measurement, and direction when describing a rotational symmetry.
• Task 4: Allows students to observe, describe, and apply translations in a wallpaper pattern.
• Task 5: Uses Geogebra in order to provide students an opportunity to compose reflections to see other reflections.
• Task 6: Uses the idea of "Which doesn't belong?" Students explore groups of pictures and identify distinguishing characteristics.

Proof in The Thinking Classrooms
Authors: Jace Arends, Carla Parker, Matthew Rosenberg

As a group, we examined the question of what proof should look like in high school geometry classrooms. We developed a series of lessons to introduce students to both the concepts of building a rigorous geometric argument and to the structure of the formal two column proof. In designing these lessons we wanted to incorporate the curricular shifts from the Common Core State Standards in geometry and incorporate the practices of a thinking classroom as discussed in our "Reflecting on Practice" sessions.

(Visuals)
This is a problem-based task meant to introduce students to geometric reasoning and justification. In this task students determine the best route between 3 blips on a radar using their noticings, wonderings. Student groups are tasked with describing and justifying their route to their superior.
Noticings and Wonderings This activity is designed to help build up a routine of noticing, wondering, and making conjectures. In groups, students will examine a series of geometric diagrams. For each diagram students will attempt
Proof Comparison Task Students examine a diagram of a triangle with 3 midsegments and make a list of noticings, wonderings, and conjectures. After building out these lists, students review, compare, and contrast the exemplar proofs of triangle congruence within the mid segment diagram.
Formal Systems This lesson introduces students to two-column proofs through a formal system of rules involving letter strings. Students build letter strings and justify each step with the rule used. This task has been adapted from a Justin Lanier's activity.

Hexagrams: Year-Long Wondering, Sensemaking, and Connections
Authors: Daniel Nissani, Will Stafford

A hexagram inscribed in a circle is a diagram that posses near endless possibilities for exploration within the geometry curriculum. A series of 11 tasks and 2 mini projects was developed for implementation throughout the school year, all around the same diagram and its different iterations. By revisiting and investigating the diagram throughout the year, students are continuously exposed to the idea of the richness of mathematics continues to get deeper and deeper the more questions we ask. Further, it provides an opportunity for students to make connections between different key concepts that they study throughout the year.

The tasks, along with associated teacher guides, included in this packet are

Task 1 - Points, Lines, Planes
Task 4 - Rigid Motions and Congruence

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