Off on a Tangent

Lesson Content

• Lesson Plan Template:

  General Lesson Plan

• Learning Objectives: What should students know and be able to do as a result of this lesson?

  
  

  Students will be able to:

  1. Understand and apply vocabulary, visual representations, descriptions, notation, definitions, concepts, and relationships associated with circles and their tangent lines.
  2. Use traditional geometric construction tools of compass and straightedge to make formal constructions, especially line segments, circles, and perpendicular bisectors of line segments.
  3. Use traditional geometric construction tools of compass and straightedge to construct a tangent line from a point outside a given circle to the circle.
  4. Follow a complex multi-step procedure to solve a real-world problem using content knowledge and skills, and a series of logical steps.
  5. Translate descriptive, quantitative, or technical information expressed in words in a text into visual form.

• Prior Knowledge: What prior knowledge should students have for this lesson?

  
  

  Students should know how to:

  • Recognize, draw, label, reference, and describe parts of and things associated with a circle.
  • Understand the radius of a circle is perpendicular to the tangent at the point of tangency.
  • Make formal geometric constructions with a variety of tools and methods, especially when constructing perpendicular bisectors of a line segment.
  • Determine cardinal directions of north, east, south, and west on a map by use of a north arrow or compass rose.

• Guiding Questions: What are the guiding questions for this lesson?

  
  
  1. What is the definition of a circle? (A circle is the set of all points in a plane equidistant from a given point.)
  2. What is the definition of a perpendicular bisector? (A perpendicular bisector is a line or ray that divides another line segment into two equal parts and forms a 90-degree angle with the line segment at the point of intersection.)
  3. What is the definition of a tangent line? (A tangent line is a line that intersects a circle or curve exactly once.)
  4. What is the definition of a secant line? (A secant line is a line that intersects a circle or curve twice.)
  5. What are the similarities and differences between a tangent line and a secant line? (They are both lines that intersect a circle or curve, but a tangent line shares only one point with the circle or curve, while a secant line shares two intersection points.)
  6. How do you recognize a tangent line? (It touches the circle or curve at only one point.)
  7. How do you recognize a secant line? (It touches the circle or curve at two points.)
  8. How many tangent lines does a circle have? (It has infinitely many.)
  9. How many tangent lines are there to a circle from a given point not on the circle? (two)
  10. What does the figure of speech "off on a tangent" mean? Give an example of its use. (To go off on a tangent means to veer away from what you were originally focused on, discussing, or doing. For example: "During our conversation about budgeting, she went off on a tangent about how much money she spent on her new computer and how fast it works.")
  11. What are some non-mathematical words or phrases to describe the relationship between a tangent line and a circle? (A tangent line brushes up against the circle, a tangent line skims the circle, a tangent line just barely touches the circle, a tangent line lightly "kisses" the circle, etc.)
  12. How do you know where north is on a map? (You look for a north arrow or N on the compass rose.)
  13. What do you know about the Mason-Dixon Line? (Answers will vary.)
  14. What is a peninsula? (A peninsula is piece of land that is almost entirely surrounded by water and is attached to a larger land area.)
  15. What does the word transpeninsular mean? (It means across the peninsula; the distance of the land from side to side of the peninsula.)
  16. What does the 39-degree parallel mean? (It is a circle (or line, depending upon your perspective) of latitude that is 39 degrees from, and parallel to, the Earth's equator.)

• Teaching Phase: How will the teacher present the concept or skill to students?

  
  

  The teacher will distribute and direct students to complete Off on a Tangent - Activity 1, which is related to vocabulary, visual representations, and descriptions associated with parts of a circle. This is meant to activate prior knowledge. Students are directed to complete it either individually, with a partner, or in small groups of the teacher's choosing. The teacher will circulate around the room and provide clarification, assistance, and praise. Discussion, debriefing, and consensus should occur. Misconceptions should be clarified.

  The teacher will distribute and direct students to complete Off on a Tangent - Activity 2, which is related to notation associated with parts of a circle. This is meant to activate prior knowledge. Students are directed to complete it either individually, with a partner, or in small groups of the teacher's choosing. The teacher will circulate the room and provide clarification, assistance, and praise. Discussion, debriefing, and consensus should occur. Misconceptions should be clarified.

• Guided Practice: What activities or exercises will the students complete with teacher guidance?

  
  

  The teacher will provide or direct students to use their own traditional geometric construction tools (compass and straightedge) to learn/review/practice construction of a circle, a perpendicular bisector of a line segment, and a tangent line to a circle from a point outside the circle. The teacher will lead the class through the processes and perhaps demonstrate with dynamic geometry software (Geometer's Sketchpad, GeoGebra, drawing tools in Microsoft Word, etc.), "chalkboard" sized tools, or one-on-one as needed. Verbal feedback and praise should be given as students correctly manipulate construction tools.

  The teacher will distribute Off on a Tangent - Activity 3, which provides step-by-step directions for construction of the perpendicular bisector of a line segment. Students should practice independently or in pairs with teacher guidance until mastery is attained. Verbal praise should be given for correct results.

  The teacher will distribute Off on a Tangent - Activity 4, which provides step by step directions for construction of a tangent line from a point outside a given circle to the circle. Students should practice independently or in pairs with teacher guidance until mastery is attained. Verbal praise should be given for correct results.

  This component is optional; use and method of implementation are at the teacher's discretion:

  • The teacher will advise students the geometric construction techniques just practiced and mastered will be applied to a real-world scenario and may wish to provide background information on the Mason-Dixon Line by use of supporting material.
  • The teacher will have students read an article by "," Ken Jennings dated July 9, 2012 and available from Conde' Nast Traveler. Students may read the article online. Students will be directed to either read silently or use "jump-in" reading, where different students read one paragraph each aloud to the class. The teacher will ask students to identify the main idea, the author's point of view, and supporting details.
  • Nations Online (nationsonline.org) has an excellent reference map of Delaware.
  • Students could also review other books or websites with information, images, and maps of the Mason-Dixon Line.

  The teacher will distribute Off on a Tangent - Activity 5 that tasks students to use a map of the Delmarva Peninsula, a written description, and geometric construction tools to draw the part of the Mason-Dixon Line that defines Delaware's boundary. Teacher will guide students through this activity and may use questions listed in the Guiding Questions section of this lesson to facilitate the process.

• Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the lesson?

  
  

  The teacher will advise students they will use geometric construction tools and techniques to independently complete Off on a Tangent - Activity 6, which asks students to divide Florida into two states. Students' interest may be piqued and background information checked by posing the question, "If you had to divide Florida into two states, where would you draw the boundary line and why?" The teacher may wish to provide or display a for geographical context.

  Teacher will distribute Off on a Tangent - Activity 6, which tasks students to use a map of the Florida Peninsula, a written description of the theoretical North-South Florida Line, and geometric construction tools and techniques to map the boundary between North and South Florida. Students will do this learning activity individually and demonstrate mastery of:

  1. Vocabulary, descriptions, notation, concepts, relationships and geometric constructions associated with circles, their parts, and tangent lines.
  2. The ability to follow a multi-step procedure to solve a real-world problem using a series of logical steps.
  3. Translating descriptive information into visual form.

• Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?

  
  

  The teacher will direct students to collect and numerically organize all six Off on a Tangent activities, and have them at hand, available for reference. Teacher will randomly state a term from Off on a Tangent - Activity 1 and randomly choose a student to state what he/she knows about the term/concept, and if and how the term/concept was used throughout the lesson. There are nine terms so at the teacher's discretion this may continue until all terms/concepts are discussed or be repeated until all students have contributed.

• Summative Assessment

  
  

  Off on a Tangent - Summative Assessment has two pages. The teacher may choose to use some or all of this assessment instrument. The first page contains 21 statements related to vocabulary, definitions, concepts, descriptions, and relationships associated with parts of circles, and their tangent and secant lines. Students are directed to determine the truth value of each statement and provide a written response. The second page has fill-in, sorting, and free response questions that require a higher level of cognitive process. Again, students are directed to complete these individually and provide written answers.

  The teacher may choose to photocopy only page one or do two-sided copies of both pages. Alternately, since questions on page two are listed in order of increasing difficulty, the teacher may choose to omit questions toward the bottom of the page by covering or cutting them off prior to photocopying.

• Formative Assessment

  
  

  The teacher will distribute Off on a Tangent - Activity 1, which is related to vocabulary, visual representations, and descriptions associated with parts of a circle. This is meant to activate prior knowledge. Students are directed to complete it either individually, with a partner, or in small groups to discuss, collaborate, compare, check, dispute, and verify answers. The teacher should circulate around the room and provide clarification, assistance, and praise where appropriate. Follow-up with whole class discussion, debriefing, and consensus. The teacher will take note of skill gaps, misconceptions, and specific difficulties so they may be addressed and remedied throughout the lesson.

  The teacher will distribute , which is related to notation associated with parts of a circle. This is meant to activate prior knowledge. Students are directed to complete it either individually, with a partner, or in small groups to discuss, collaborate, compare, check, dispute, and verify answers. The teacher should circulate around the room and provide clarification and assistance where needed. Follow-up with whole class discussion, debriefing, and consensus. The teacher will take note of skill gaps, misconceptions, and specific difficulties so they may be addressed and remedied throughout the lesson.

• Feedback to Students

  
  

  Students will receive feedback via internal reflection, oral discussion, written responses, and verbal praise throughout the lesson from themselves, the teacher, and classmates as follows:

  • Off on a Tangent - Activities 1, 2, 3, 4, 5, and 6: self (internal) throughout, classmates (oral) during, and teacher (oral and written) during and at completion of activity
  • Off on a Tangent - Summative Assessment: teacher (written) after submission

• Florida’s B.E.S.T. Benchmark Alignment Notes

  
  

  This lesson applies to a geometry class to practice and assess the vocabulary and relationships between circles, chords, secants, and tangents. Real-life applications involving geometric constructions are included.