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* *Reflection: High Quality Task
Circles Unit Assessment - Section 2: Circles Unit Assessment

This problem is rich with opportunities for students to do a lot of sense making. Students have to make sense of what it means to have an arc that is 1/9 of a circle, 12 meters long, and can fit eight desks. Students need to be able to interpret the question “how far are the desks from the center of the circle?” and to understand that the distance at hand is represented by the radius, not diameter, of the circle containing the arc. Students’ work on part (a) of this problem often reveals the strengths of their thinking and the possible breakdowns in their understanding.

What I love about part (b) is that most students can successfully work their way through the problem if they really make sense of what it means to have an arc with the same central angle but half the distance from the center. My favorite approach is not the one that requires students to calculate that each desk takes up 1.5 meters of an arc, but rather, the one that leads students to simply conclude that the arc length is directly proportional to the radius and will therefore hold half the number of desks.

*Proportional Reasoning and Sense Making*

*High Quality Task: Proportional Reasoning and Sense Making*

# Circles Unit Assessment

Lesson 8 of 8

## Objective: Students will be able to solve problems by identifying relationships among radii, chords, tangents, angles and arcs.

Before we start this activity, I tell students that they will be taking a critical look at student work, which means they most be especially respectful when voicing their critique. I pass out an 11"x17" paper to each group, which has samples of student work from the Arcs and Angles menu photocopied onto it. My expectations are that each student in the group will write in their own color, recording the elements of the justifications they thought were effective (or ineffective). I also remind groups that Recorder/Reporters should be prepared to share on the group's behalf.

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#### Circles Unit Assessment

*65 min*

This test features a variety of test questions: true/false, fill in the blank, and vocabulary matching, to assess students’ foundational knowledge of circles properties—these are the kinds of questions that requires students to attend to precision (**MP6**). Problems #4-7 give students the opportunity to apply their understanding of chord, tangent, and angles/arcs properties while solving problems.

I include Problem #8 because it requires students to use higher order thinking skills; students must integrate several different concepts to solve the problem (any parallelogram inscribed in a circle must be a rectangle, properties of rectangle diagonals and equilateral triangles, arc length) while looking for and making use of structure—this problem definitely asks students to make sense of the problem and persevere while solving it (**MP1**, **MP7**). The two proofs on the test require students to thoughtfully justify their reasoning while writing the proof; the second proof, in particular, encourages a variety of strategies to think through the proof.

#### Resources

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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: Foundational Circles Vocabulary and Chord Properties
- LESSON 2: Tangent Properties
- LESSON 3: Chord and Tangent Group Challenge
- LESSON 4: Arcs and Angles: Central and Inscribed Angles
- LESSON 5: Circumference-Diameter Ratio and Arc Length
- LESSON 6: Review: Arcs, Angles, Chords, Tangents, and Proof
- LESSON 7: Prove Circles Conjectures
- LESSON 8: Circles Unit Assessment