Park City Mathematics Institute Investigating Geometry Project Abstract Drafts of Project Files (password required)
HANDS-ON LESSONS
Sassy Similarity
Rebecca Alleva, Evelyn Baracaldo, Anna Prusch, Joyce Rhee
This hands-on classroom activity allows students to explore the concept of similarity through the use of origami paper. Students will identify corresponding parts of similar figures and calculate ratios of corresponding sides to prove similarity. Prompts for class discourse are included as well as supplementary problems to check for understanding.

Pythagoras in the Folds
Anna Fox, Zack Bassman
Haga's theorem states that one carefully chosen fold creates three similar 3-4-5 right triangles on a piece of origami paper. Students use angle relationships, similarity, proportions, and the Pythagorean theorem to derive the lengths of the sides of these right triangles to verify Haga's theorem. Hands on activities provide a powerful tool for students to explore geometric concepts in a tactile, concrete fashion. This activity gives students the opportunity to identify, describe, and verify geometric relationships in the folded paper and generates collaborative learning and discussion. By physically creating a geometric object, students become personally invested in the outcome of the activity. At the end of the lesson, the students will have something that they can literally take home with them, and refer back to for further review. Folding instructions with teacher notes, and a student worksheet are included.

Bisecting Paralellograms
Assegid Derseh, Eric Flam
This hands-on classroom activity allows students to explore the concept of congruence through the use of folding paper in the shapes of squares, rectangles, rhombi, and parallelogram. Students will identify corresponding parts of congruent triangles and use them show that any segment, through the point of intersection of the diagonals of a parallelogram, bisects it. Prompts for class discourse are included as well as supplementary problems to check for understanding.

DYNAMIC SOFTWARE
Introduction to the Geometer's Sketchpad®: A Series of Challenges for Students
Seth Leavitt, Kaitlyn Peterson Spong, Felipe Rico, Todd Smallcanyon
Through a series of challenges, students learn to use Geometer's Sketchpad as a vehicle to explore in geometry. In each challenge, students are given a specific task and a restriction on the tools they are able to use. By engaging in the challenge, students will learn the function of the given tools and develop a need to learn and use more of the features of Sketchpad. The challenges are meant for students with no or limited prior exposure to the program.
CONNECTING ALGEBRA & GEOMETRY
Problem Solving within the Context of High School Geometry, Algebra I, and Algebra II
Sara Rezvi, Jessica Barker, Cameron Cassidy, Mary Gruber
Problem solving and communication are central elements of the NCTM Principles and Standards of School Mathematics. This project presents three different rich problems with multiple entry points and extensions that cross the typical Algebra 1, Geometry and Algebra 2 curricula and promote connections between the courses. The problems include the stair-step problem, clock problems and investigating r-gonal numbers. Students at all levels will be expected to look for patterns, problem solve, justify their solutions and clearly communicate their thinking to others. The problems are designed to give students multiple entry points, so that students of all levels will be engaged in mathematical thinking. Classroom discourse is emphasized and each problem includes questions to push student thinking and promote high level discourse among students.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.