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Segments in Circles

Lesson Plan

Segments in Circles

Objectives

Students will solve problems involving segment lengths within circles. Students will:

  • discover the segment product properties of circles through the use of technology.
  • apply the perpendicular property of tangents and the Pythagorean theorem to find missing segment lengths.
  • apply theorems and formulas involving secant, tangent, and chord segments to solve problems involving algebra.

Essential Questions

  • What are the different characteristics of circles and how can they be used to solve problems? [IS.4 - All Students]

Vocabulary

[IS.1 - All Students] [IS.2 - All Students] 

  • Chord: A line segment whose endpoints are on a circle. [IS.3 - All Students]
  • Circumscribed Polygon: A polygon such that every side of the polygon is tangent to the curve and that the curve is contained in the polygon.
  • Converse of the Pythagorean Theorem: If in a triangle, a2 + b2 = c2 and a, b, and c are the sides of the triangle, then the triangle is a right triangle; if c2 > a2 + b2, then the triangle is an obtuse triangle; if c2 < a2 + b2, then the triangle is an acute triangle.
  • Inscribed Polygon: A polygon such that every vertex of the polygon is incident upon the curve and the polygon is contained inside the curve.
  • Intersecting Chord Theorem: If two chords intersect in a circle, then the products of the lengths of the chords segments are equal.
  • Perpendicular Lines: Two lines, segments, or rays that intersect to form right angles.
  • Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse; in any right triangle where the length of one leg is a, the length of the second leg is b, and the length of the hypotenuse is c, as in: c2 = a2 + b2.
  • Secant (of a circle): A line that intersects a circle in exactly two points.
  • Secant Segment Theorem: If two secants intersect in the exterior of a circle, then the product of the measure of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.
  • Segment: A part of a line with two endpoints.
  • Tangent (of a circle): A line that touches a circle in exactly one point. Tangent theorem 1 states that a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency. Tangent theorem 2 states that if two line segments from the same exterior point are tangent to the same circle, then they are congruent.
  • Tangent Secant Segment Theorem: If a tangent and a secant intersect in the exterior of a circle, then the square of the measure of the tangent is equal to the product of the measures of the secant segment and its external secant segment.

Duration

8 hours–8 periods

Prerequisite Skills

Prerequisite Skills haven't been entered into the lesson plan.

Materials

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.

  • This site provides more examples and guided-practice problems covering the segment product theorems from Lesson 3, and could be used for extra practice.

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

  • This site has links to a PDF file containing an investigation lab for the secant segment theorem provided in Lesson 3. This partner activity allows students to discover a relationship, rather than just be told a 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

  • Extra practice and examples to cover the tangent theorem, discussed as theorem 1 in Lesson 3.

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

  • Hands-on activity for discovering that two tangents drawn to the same external point from a circle are equal.

http://www.docstoc.com/docs/14217138/Lengths-of-tangents-drawn-from-an-external-point-to-a-circle-are-equal

  • Calculator files for the Technology Explorations
    • TI-84 calculators:

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=12512

  • TI-Nspire™ calculators:

http://education.ti.com/educationportal/activityexchange/Activity.do?cid=US&aId=12513

Formative Assessment

  • View
    • Evaluate student work in guided-practice examples by checking correspondence between circle components and measurements, appropriate operations, and accuracy of computation.

Suggested Instructional Supports

  • View
    Scaffolding, Active Engagement, Modeling, Explicit Instruction
    W: As students progress through this unit, they encounter many theorems and concepts that involve angle and arc relationships. At this point, students may begin to wonder about segment lengths as well. This lesson investigates the secant, chord, and tangent segment relationships found in circles. Students may notice that some diagrams look similar to problems encountered in Lessons 1 and 2, but the focus is on segments that are created rather than on angles. This is a logical next step in the geometry learning process.  
    H: Lesson 3 provides student engagement through the use of technology activities. Introducing the lesson through a Technology Exploration will help to hook and hold students’ interest in the lesson topics. It also provides students an opportunity to take responsibility for learning into their own hands. 
    E: The structure of this lesson is designed to equip students with the skills needed to complete similar concepts independently and apply learned knowledge in new situations. Using the PowerPoint presentation and handouts will help students keep organized and on track during the lesson. The use of extensions and critical-thinking activities allows students to make meaning of their learning so they may apply it in future learning situations.  
    R: Students revise their thoughts based on teacher feedback during instruction and technology activities. Feedback is critical to assist students in finding where they need to revise their thought processes.  
    E: The Technology Exploration activities provide students a great way to express their understanding as well as tools for self-evaluation. As students progress through the activities, they are able to understand why things are not working or make decisions based on their own progress. You should guide them to the correct path in their learning process. Students will express their understanding through the Extension activities near the end of the lesson. 
    T: This lesson provides two approaches to teaching the concepts. Use the approach that best fits the needs of your classroom. Extension activities are provided to address the needs of a variety of classroom settings. The PowerPoint presentation and the Concept Builder worksheet help students gain experience in a variety of learning styles and can be helpful when working with other teachers/aides in the classroom.  
    O: Using the Technology Explorations option, students can move from an investigative approach to learning to an independent level with minimal direct instruction. Be sure to provide assistance to any students who miss pieces of the learning when using this method. If proper checking for understanding does not take place, students may have gaps in their understanding that will show up quickly when they attempt to do independent activities. Option 2 of the lesson takes a direct-instruction approach to teaching, but allows students the opportunity to practice and apply their learning through the same critical-thinking and extension activities.  

     

    IS.1 - All Students
    Consider using graphic organizers (e.g., Frayer Model, Verbal Visual Word Association, Concept Circles) to review key vocabulary prior to the lesson. Post terms on a word wall in classroom  
    IS.2 - All Students
    Constantly reinforce and show models of the big ideas of the content.  Show connections between other subject and topic areas.  
    IS.3 - All Students
    Model the use of vocabulary words throughout the lessons.  Incorporate strategies that promote the use of the vocabulary words as well as models and pictures.  
    IS.4 - All Students
    Provide samples and graphic organizers to assist with instruction.  Show real life models.  
    IS.5 - All Students
    Consider pre-teaching the concepts critical to this lesson, including the use of hands-on materials. Throughout the lesson (based on the results of formative assessment), consider the pacing to be flexible to the needs of the students. Also consider the need for re-teaching and/or review both during and after the lesson as necessary.  
    IS.6 - All Students
    Ensure that students have had sufficient instruction on working with scientific calculators.  
    IS.7 - All Students
    Re-teaching and reinforcement of the concepts may need to happen.  

Instructional Procedures

  • View

    Option 1: Technology Approach

    The use of technology by students is an option that involves calculator explorations to teach the important theorems of this lesson. The Circle Product Theorems activities can be done in groups, pairs, individually, or as a class. Choose whichever method best meets the needs of your students. The activities are designed to be done on either TI-84 or
    TI-Nspire™ calculators. Use the activity that corresponds to the platform in your classroom. Use the links below to download all necessary documents associated with the activity. When students are doing the activity in class, make sure you answer questions and assist students as needed.

    After the activity, students should understand the theorems associated with segments created by secants, [IS.7 - All Students] chords, and tangents within a circle and how they can be used to solve problems. Use the PowerPoint presentation (M-G-6-3_Lesson 3 PowerPoint.pptx) and example problems in Lesson 3 to fill in gaps in understanding and to check for students’ grasp of the concepts. Ensure that students have the appropriate theorems and definitions in their Concept Builder worksheets (M-G-6-1_Concept Builder.xls) as well as the skills necessary to use the theorems for solving problems.

    Option 2: Direct Instruction

    Follow the Lesson 3 PowerPoint presentation to provide students the information and examples necessary to perform the concepts independently (M-G-6-3_Lesson 3 PowerPoint.pptx).

    Lesson 3 begins with a discussion of tangent segments. Present students with slides 2–5. Slide 2 explains the theorem relating to tangents. Be certain to give students time to record this information in their Concept Builder worksheets (M-G-6-1_Concept Builder.xls). Prior to modeling the examples on slides 3–5, hand a printout of these slides to students. Printouts allow students to follow along with the modeling.

    Example 1: Slide 3:

     

    Note: This example requires students to remember the Pythagorean theorem. Assist students in recalling this formula and explain that since the tangent creates a perpendicular with the radius, a right triangle can be formed.

     

    Example 2: Slide 4: x =3

     


    Example 3: Slide 5:

     

    Slides 6–8 continue the discussion on tangents. Display the theorem on slide 6, giving students time to record and process the information presented. Provide students with the printouts of slides 7 and 8, and then model these examples for students while they follow along on their worksheets.

    Example 4: Slide 7: x = 4

     

     

    Example 5: Slide 8: 40

     

    Slides 9–11 cover the intersecting chord theorem. Display slide 9 for students to record and process, using the Concept Builder worksheet. Provide students with printouts of slides 10 and 11, and then model these examples for students while they follow along on their worksheets.

    Example 6: Slide 10:

     

     

     

    Example 7: Slide 11: 

     

     

    Slides 12 and 13 deal with segments created by two secants intersecting a circle and intersecting each other outside of a circle. Display for students the theorem on slide 12 to be copied into their Concept Builder worksheets. Provide students with a printout of slide 13 to use while following along with the modeled example.

    Example 8: Slide 13: x = 3

     

    In Example 8, students must solve a quadratic equation by factoring to get the solution to the problem. Solving the quadratic equation indicates two solutions are possible. Discuss with students which of the two solutions is the correct answer for the problem presented: (You cannot have negative lengths in geometry.)


     

    Slides 14 and 15 deal with segments created by a secant and a tangent intersecting a circle and intersecting each other outside of a circle. Display the theorem on slide 14 to be copied into students’ Concept Builder worksheets. Provide students with a printout of slide 15 to use while following along with the modeled example.

     

    Example 9: Slide 15: x = 5.2

     

    Note: In Example 9, students must solve a quadratic equation using the quadratic formula to get to the solution. Solving the quadratic equation indicates two solutions are possible. Discuss with students which of the two solutions is the correct answer for the problem. (You cannot have negative lengths in geometry.)


     

    Extension:

    The picture below shows an architectural structure that involves many circles and lines. From this perspective, there is a center circle intersected by two secant structures. These have been traced in yellow and red. The chords within the circle measure 8 feet and 6 feet. If the longer pole is 35 feet long, find the length of the other pole. (Solution: 33.9 feet)

    Note: The quadratic formula is necessary to solve this problem; one answer is negative and thus cannot be the solution to this problem.

     

Related Instructional Videos

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DRAFT 10/13/2011
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