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Circles

Unit Plan

Circles

Objectives

In this unit, students investigate characteristics of circles. Students will:

  • find the circumference of circles in a variety of settings.
  • find arc length.
  • identify and measure central angles, major arcs, minor arcs, semicircles, inscribed angles, intersecting segment lengths, and angles formed by intersecting segments using a variety of formulas.
  • recognize and use relationships between arcs and chords.
  • apply properties of tangents.
  • apply all objectives in a variety of settings, both textbook and real-world.

Essential Questions

  • What are the different characteristics of circles and how can they be used to solve problems?

Related Unit and Lesson Plans

Related Materials & Resources

The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan.

 

  • Worksheet for practice on finding the circumference of a circle.

http://www.helpingwithmath.com/printables/worksheets/geo0701circle01.htm

  • This Web site on Arcs and Chords gives explanations and examples on the relationships between arcs and chords, as shown on slides 17–26 in the Lesson 1 PowerPoint.

http://www.cliffsnotes.com/study_guide/Arcs-and-Chords.topicArticleId-18851,articleId-18827.html

  • Circle Worksheets and Activities from this Web site contain extra examples, practice, and explanations of information and concepts from Lesson 2; for example, the link “Tangent, secant, arcs, and angles of a circle.” There are options you can use for other components of this unit as well.

http://www.mathwarehouse.com/classroom/worksheets/circles/circles-worksheets-drills-and-activities.php

  • The Side Length of Tangent and Secant of a Circle section on this Web site provides examples and guided practice problems about the segment product theorems in Lesson 3, which could be used for extra practice.

http://www.mathwarehouse.com/geometry/circle/tangent-secant-side-length.php

  • A Secant and Secant Segments Lab offers a PDF file that investigates the secant segment theorem provided in Lesson 3. This partner activity allows students to discover the relationship within the formula.

http://www.google.com/url?sa=t&source=web&cd=17&ved=0CDUQFjAGOAo&url=http%3A%2F%2Fwww.math.uakron.edu%2Famc%2FGeometry%2FHSGeometryLessons%2FSecantandSecantSegments.pdf&ei=iZppTIu5Msndnge_q5TBBQ&usg=AFQjCNFiP_hyEkbZKkKus_qMPmZYNH0uJg&sig2=MKSjTfrsxCRjj8GUBENScg

  • The Tangent of a Circle site offers extra practice and examples of the tangent theorem, discussed as theorem 1 in Lesson 3.

http://www.mathwarehouse.com/geometry/circle/tangent-to-circle.php

Formative Assessment

  • View

    Multiple-Choice Items:

    1. Find the measure of arc XY if  is the diameter of circle P.

     

    A

    113°

    B

    67°

    C

    74°

    D

    37°

     

     

     

    2. If circle Y is congruent to circle Z, find the length of .

     

    A

    114

    B

    3

    C

    18

    D

    11

     

     

    3. Based on the diagram, find the measure of arc WZ.

     

    A

    5°

    B

    92°

    C

    23°

    D

    46°

     

    4. Find the measure of angle x.

     

    A

    23°

    B

    31°

    C

    64°

    D

    128°

     

    5. Find the measure of angle XWY.

     

    A

    19°

    B

    330°

    C

    38°

    D

    74°

     

    6. Find the measure of angle X.

     

    A

    85°

    B

    40°

    C

    23°

    D

    46°

     

     

    7. Find x.

     

    A

    7.6

    B

    4.8

    C

    6.4

    D

    5.7

     

     

    8. Circle A is circumscribed by a square. The radius of the circle is 6 and the length of  is 4. Find the perimeter of the square.

     

    A

    8

    B

    93.3

    C

    32

    D

    64

     

     

    9. Solve for x.

     

    A

    30

    B

    16

    C

    18

    D

    18.2

    Multiple-Choice Answer Key:

    1. B

    2. D

    3. B

    4. C

    5. A

    6. C

    7. D

    8. D

    9. B

     

    Short-Answer Items:

    10. The South Rose Window of Notre Dame Cathedral in Paris, France (shown below) has a diameter of approximately 42.3 feet. Calculate the approximate circumference around the window to the nearest tenth of a foot.

     

     

     

    11. The following circles share a common center. Find the measure of arc x.

     

     

    12. When you look through a microscope, the viewing area is circular. If a scientist is viewing a bacterium in a microscope in a viewing area with the diameter of 2.5 mm, find the length of the bacterium if the bacterium is located 0.2 mm from the bottom of the viewing area and stretches across from the diameter to the edge of the viewing area.

     

     

     


    Short-Answer Key and Scoring Rubrics:

    10. The South Rose Window of Notre Dame Cathedral in Paris, France has a diameter of approximately 42.3 feet. Calculate the approximate circumference around the window to the nearest tenth of a foot.

    Solution: 132.9 feet

    Points

    Description

    2
    • The student correctly calculates the circumference.
    • The student shows all work.

    1
    • The student’s work supports understanding of the formula and number placement, but the student makes a computational error.

    OR

    • The student uses the correct formula, but places 42.3 as the radius or 84.6 as the diameter.

    0
    • The student demonstrates no understanding of proper procedures or concepts.

     

     

    11. The following circles share a common center. Find the measure of arc x.

     

    Solution: 30°

    Points

    Description

    2
    • The student’s work demonstrates understanding of which formulas/concepts to apply to solve the problem.
    • The student supplies the correct answer.

    1
    • The student has some gaps in the work, but demonstrates an ability to supply the correct answer.

    OR

    • The student’s work supports understanding of concepts, but a computational error yields an incorrect answer.

    0
    • The student demonstrates no understanding of proper procedures or concepts.

     

     

     

    12. When you look through a microscope, the viewing area is circular. If a scientist is viewing a bacterium in a microscope in a viewing area with the diameter of 2.5 mm, find the length of the bacterium if the bacterium is located 0.2 mm from the bottom of the viewing area and stretches across from the diameter to the edge of the viewing area.

    Solution: Approximately 0.68 mm × 2 = 1.36 mm

     

     

    Points

    Description

    2
    • The student shows all work and uses correct formulas and procedures.
    • The student calculates the correct answer.
    • The student draws a diagram to represent the problem.

    1
    • The student has gaps in the work, such as no picture, but calculates the correct answer.

    OR

    • The student’s picture and work support knowledge of the correct formula/concept, but the student performs a computational or setup error.

    OR

    • The student shows gaps in understanding and partial understanding of concept(s), and answers are not completely correct.

    0
    • The student demonstrates no understanding of proper procedures or concepts.


    Performance Assessment:

     

    The diagram shows the paths of four soccer players during a game. If Bill travels the path marked in red, Sally in pink, John in blue, and Nathan in yellow, answer the following questions. Assume the diagram is based on professional field dimensions, where the center circle has a diameter of 10 yards.

    1. Name each player’s path as a chord, secant, or tangent.
    2. Find the circumference of the center circle.
    3. If Bill’s path is perpendicular to the half-line, what conclusion can you draw about his path through the circle?
    4. What is the distance Bill travels across the circle if his path is 3 yards from the bottom of the circle when he crosses the center line?
    5. If John makes a run from the edge of the circle to point A, 4 yards away from the center circle, how far did John run?
    6. Given that Nathan ran across the circle, passing the center line 3 yards from the side of the circle, and the distance he traveled from the center line to the circle was 3.75 yards, how far did Nathan run?
    7. If Nathan and Bill both begin their runs at point C and run away from one another at a 43 degree angle, determine the arc measure between the two as they pass across the edge of the circle.
    8. Based on the information in question G, what is the distance on the circle that separates Nathan and Bill’s paths?
    9. If Bill and Sally run toward the ball located at point F at a 20 degree angle, and the measure of the arc created from where Sally reaches the circle and Bill’s starting point is 85 degrees, what is the measure of the arc created between Bill and Sally’s paths?

    Performance Assessment Answer Key:

    1. Bill, secant; Sally, tangent; John, tangent; Nathan, chord
    2. 10π yards or 31.4 yards
    3. Bill’s path is bisected by the diameter of the center circle.
    4. 4.58 yards
    5. 7.48 yards
    6. 9.35 yards
    7. 86º
    8. 7.5 yards
    9. 45º

     

     

    Performance Assessment Scoring Rubric:

    Points

    Description

    4
    • All answers are accurate.
    • All questions are answered in a well thought-out format.
    • The student shows all work.
    • The student demonstrates a full understanding of all concepts.

    3
    • Answers contain only a few computational errors.
    • The student shows only a few gaps in his/her work.
    • Some answers are a little brief.
    • The student demonstrates an overall understanding of concepts with only minor errors.

    2
    • A few of the answers contain inaccurate processes.
    • The student’s work has computational and/or algebraic errors.
    • The student does not show enough work consistently.
    • The student demonstrates an understanding of some components, but not all concepts.

    1
    • Answers contain many algebraic/computational errors.
    • The student skips many pieces of the assessment.
    • The student’s work has many gaps and/or is missing.
    • The student demonstrates very little understanding of concepts.

    0
    • All components of the problems are incomplete.

    OR

    • No components of the problems are completed accurately.
    • The student demonstrates little or no understanding of the concepts assessed.
    • The student has not shown any work.

     

DRAFT 10/13/2011
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