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* *Reflection: Unit Exams
Final Exam Part 1: Open Response - Section 1: Final Exam Part 1: Open Response

At this point in the year, I was happy to see students adding lines to make sense of the problem. There are certain problems on which students might opt out of showing work, not for lack of thinking about the problem, but because they do not always have the problem solving strategies to make sense of what is given to them. There were some students who did not ultimately solve the problem correctly, but even they were able to show an understanding of key geometric ideas. For example, some students drew the radii from points V, S, L, and N to the center of the circle as a way to look for congruent triangles and to establish some key relationships between corresponding sides. On this particular problem, the most successful students found similar triangles and used inscribed angle relationships to solve.

*Seeing and Making Use of Structure*

*Unit Exams: Seeing and Making Use of Structure*

# Final Exam Part 1: Open Response

Lesson 3 of 4

## Objective: Students will be able to demonstrate what they know and understand about foundational geometry ideas, constructions, and proof, on the open response portion of the final exam.

In the first part of the final exam, which is all open response, students work through several problems that target their understanding of foundational geometry topics in novel ways. Several of the problems, like problems 4, 5, and 6, require students to synthesize several tools and geometric concepts in order to solve. Some of the problems are my own, but some have been adapted from an NCTM calendar of problems.

- Problem 1: compare the areas of circles and annuluses
- Problem 2: find the perimeter of a heptagon made by right triangles
- Problem 3: determine the number of sides of a regular polygon using angle relationships
- Problem 4: use square properties, the Pythagorean Theorem, and trigonometry to find the area of a slanted rectangle
- Problem 5: apply triangle similarity, trigonometry, proportional reasoning to solve a problem
- Problem 6: construct a regular hexagon and solve using areas of sectors
- Problem 7: prove intersecting chords in a circle create similar triangles
- Problem 8: prove a regular hexagon can be divided into a rectangle and two congruent isosceles triangles such that the rectangle’s area is two-thirds the area of the regular hexagon

*expand content*

- 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