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About Circles

I. Section Objectives

  • Distinguish between radius, diameter, chord, tangent, and secant of a circle.
  • Find relationships between congruent and similar circles.
  • Examine inscribed and circumscribed polygons.
  • Write the equation of a circle.

II. Cross- curricular-Nature

  • Look at the following examples of circles in nature.
  • These images are Figure 09.01.01
  • www.naturesmightypictures.blogspot.com/2006/06/circles-in-nature.html
  • While these don’t specifically name all of the parts of a circle, use these images to discover the different parts of a circle.
  • Where is the radius or the diameter?
  • Is there a polygon inscribed in any of the circles?
  • For example, look at the sunflower or the rose.
  • Are any of the circles similar?
  • For example, look at the patterns in the different images. Do you see any similar circles?
  • Have a discussion with the students that broadens their thinking about circles and the parts of a circle.
  • Then ask the students to find an example of circles in nature.
  • Bring it into class the next day.

III. Technology Integration

  • Students are going to be working to make connections between circles and real world activities.
  • How are circles used in different careers?
  • This first example is a designer who makes wheels.
  • This designer makes wheels that are used in performance racing.
  • As students watch this video, have them make notes on the different geometric elements that are mentioned in the video.
  • Then following the video, conduct a discussion on how geometry and wheel design are related.
  • www.thefutureschannel.com/hands-on_math/spoke_math.php

IV. Notes on Assessment

  • Assessment is completed through class discussion.
  • Work to have all students participate in the discussion.
  • Ask questions of the students and provide feedback as needed.

Tangent Lines

I. Section Objectives

  • Find the relationship between a radius and a tangent to a circle.
  • Find the relationship between two tangents draw from the same point.
  • Circumscribe a circle.
  • Find equations of concentric circles.

II. Cross- curricular-Design

  • Have students look at the following image from Wikipedia.
  • This is Figure 09.02.01
  • www.en.wikipedia.org/wiki/File:Concentric_(PSF).png
  • This is a picture of concentric circles.
  • Have students discuss the characteristics of concentric circles.
  • Are they similar?
  • How can we design a concentric circle?
  • Ask the students to create a black and white art design using concentric circles.
  • Students will need white paper, black pencil or marker, a compass.
  • Have the students work to create their own design.
  • Also have them insert one tangent line somewhere in the design.
  • Allow time for students to share their work when finished.

III. Technology Integration

  • This is a very fascinating website for students to explore.
  • www.cut-the-knot.org/Curriculum/Geometry/TangentTwoCirclesI.shtml
  • In working with this website, students will be manipulating the center of one of the circles.
  • They can click on the center and drag the center anywhere that they wish to.
  • When they do this, they will alter the diagram of the two circles and their tangents.
  • It is a great visual and very interactive.

IV. Notes on Assessment

  • Assess student work with the art design.
  • Is the design of the concentric circles accurate?
  • Are the circles organized around a common center?
  • Is there a tangent in the design?
  • Does the student understand what a tangent is based on what he/she has drawn?
  • Provide students with feedback as needed.

Common Tangents and Tangent Circles

I. Section Objectives

  • Solve problems involving common internal tangents of circles.
  • Solve problems involving common external tangents of circles.
  • Solve problems involving externally tangent circles.
  • Solve problems involving internally tangent circles.
  • Common tangent

II. Cross- curricular-Mad Tea Party

  • Use the following images from Wikipedia on the Mad Tea Party.
  • This is Figure 09.03.01
  • www.en.wikipedia.org/wiki/Mad_Tea_Party
  • Students can even complete the technology integration first to see some real pictures of the tea cups in action.
  • Tell the students that their task is to draw the design of the Mad Tea Party using circles that are connected.
  • The design of the Mad Tea Party consists of three small turntables, which rotate counter clock-wise, each holding six teacups, within one large turntable, rotating clockwise.
  • Students are to draw this design and how they hypothesize that the circles are or are not connected.
  • Do the students think that tangents play a role in this?
  • Why or why not?
  • Ask the students to write a short paragraph explaining their thinking about the ride.

III. Technology Integration

  • Have students complete a websearch for the Mad Tea Party at Disney World.
  • Students will see images and can even see a film clip of the ride on youtube.
  • Students can use this information to assist them in drawing the design of the ride.

IV. Notes on Assessment

  • Assess student work.
  • How did the students draw the design of the ride?
  • What was the student’s hypothesis about tangents?
  • Does the reasoning make sense?
  • Provide students with comments on their work.

Arc Measures

I. Section Objectives

  • Measure central angles and arcs of circles.
  • Find relationships between adjacent arcs.
  • Find relationships between arcs and chords.

II. Cross-curricular-Plate Design

  • Use the image of the dinner plate with the stripes by Cynthia Rowley.
  • This is Figure 09.04.01
  • www.prontohome.com/product/whim-by-cynthia-rowley-melamine-p_1213285046
  • Use this to show the students where there are arcs and chords.
  • Then show them major and minor arcs as well.
  • The assignment is for the students to design their own plate design using lines, chords and circles.
  • Tell the students that they are free to design the plate however they would like as long as they label the major arcs and the minor arcs.
  • They also need to figure out the measure of one of the arcs and explain how they completed this task.
  • Allow time for the students to share their work when finished.

III. Technology Integration

  • Have students go to www.brittanica.com the encyclopedia Brittanica’s website and search for Eratosthenes of Cyrene.
  • Have them research how he used arcs to figure out the circumference of the earth.
  • This may be challenging for the students to understand, so you may want to either allow them to work in small groups or to discuss this as a whole class.
  • Begin by having them take notes on their own, then begin the discussion.

IV. Notes on Assessment

  • Assess each plate design.
  • Is it creative?
  • Does it use the concepts of chords and arcs?
  • Are the major and minor arcs labeled?
  • Did the students figure out the measure of one of the arcs?
  • Is the work written out and explained?
  • Provide students with comments/feedback on their work.

Chords

I. Section Objectives

  • Find the lengths of chords in a circle.
  • Find the measure of arcs in a circle.

II. Cross- curricular-Archimedes

  • Use the image and information at the following website.
  • Display this image for the students to see.
  • This is Figure 09.05.01
  • Now have the students copy this image on a piece of paper.
  • In small groups, the students need to use this image to prove that the sum of the intercepted opposite arcs is equal.
  • Students can use the text to refer back to the information that they have learned.
  • They need to write five statements that demonstrate that this is a true statement.
  • Students should be prepared to present their findings.

III. Technology Integration

  • Begin with this statement, “It could be said that a spoke is the chord of a wheel.”
  • Use different wheel designs to demonstrate how this is true or untrue.
  • You may use a drawing program to draw and design support for your answer.
  • You may also use a collection of images to support your answer.
  • Be prepared to share your work when finished.

IV. Notes on Assessment

  • When the students present their findings, listen to their reasoning.
  • Challenge the others in the class to do the same thing.
  • How does it support what we know about perpendicular lines and angles?
  • How does it support what we have learned about arcs?
  • Does the reasoning of the group make sense or is something missing?
  • Is there a diagram to support their thinking?
  • Did the students complete any measurements?
  • Provide students with feedback on their work.

Inscribed Angles

I. Section Objectives

  • Find the measure of inscribed angles and the arcs they intercept.

II. Cross- curricular-Theater

  • This is a problem that needs to be solved. It will require the students to use angle measures.
  • This is picture of a seating chart for the Fichander Theater.
  • This is Figure 09.05.01
  • www.gotickets.com/venues/dc/fichandler_theatre.php
  • Be sure that each student has a copy of the image.
  • Show students how this is a theater in the round.
  • The seating is arranged in a circle.
  • The students need to use what they have learned about angles and arcs to determine which seats have the best angle to see the stage.
  • Note: Students may determine right away that all of the seats are equal due to their angles. Why is this? Have them prove their thinking.

III. Technology Integration

  • Students can go to the following website for a worksheet where they can practice finding the measure of inscribed angles.
  • This is a great site for simple practice and drill of skills already learned.
  • www.regentsprep.org/Regents/math/geometry/GP15/PcirclesN.htm
  • Students can also go to any of several websites to find further explanation of inscribed angles and of the measure of those angles.
  • Any of these sites will support students in expanding their understanding.

IV. Notes on Assessment

  • Walk around and observe students as they work.
  • Then have the students share their thinking about the theater problem.
  • Be sure students are able to articulate their reasoning by using content from geometry.
  • Diagrams are an excellent way for students to share their thinking.

Angles of Chords, Secants and Tangents

I. Section Objectives

  • Find the measures of angles formed by chords, secants and tangents.

II. Cross- curricular-Poetry

  • Students are assigned the task of writing a poem or rap about the theorems in the text.
  • Students can choose to write their poem about one of the theorems or all of the theorems.
  • Students could also write a poem that defines and explains the relationship between chords, secants and tangents.
  • It isn’t necessary to give too many directions for this assignment.
  • Let the students work in small groups, and they will illustrate their level of understanding of the material through the poem.
  • When finished, allow students time to present their work.

III. Technology Integration

  • Have students complete this chapter by completing a websearch on circles in architecture.
  • They can google this topic.
  • Have the students keep track of the sites that they visit.
  • They need to select three different images that best illustrates the content of the chapter.
  • The students need to write a paragraph explaining how each one illustrates the concepts of the chapter, and which concepts it illustrates.
  • Have students share their work when finished.

IV. Notes on Assessment

  • Assess student work through their presentations.
  • How well does the poem explain the theorem or theorems?
  • How well does the poem explain the definitions from the text?
  • Are the images that the student selected in line with the content from the chapter?
  • Did the student explain which concepts are illustrated in the image?
  • Is the information accurate?
  • Provide students with feedback on their work.

Segments of Chords, Secants and Tangents

I. Section Objectives

  • Find the lengths of segments associated with circles.

II. Cross- curricular- Circus math

  • This is a problem having to do with the circus.
  • Here is the problem.
  • A circus ring has a diameter of 42\;\mathrm{feet}.
  • A high wire is stretched across the diameter of the circle
  • A second wire is stretched across the diameter of the circle.
  • The two wires intersect at one point.
  • On the first wire, the lengths of the wire are ten feet and eight feet.
  • On the second wire, only one section of the wire is known and that is five feet.
  • What is the length of the second section of the wire?
  • Have students work in small groups on this problem.
  • It is a great idea to have students draw a diagram of the solution of the problem.
  • Solution:
  • 10 \times 8 = 5x
  • 80 = 5x
  • x = 16\;\mathrm{feet}
  • The diameter of the circle has no impact on the answer of this problem.

III. Technology Integration

  • Have students go to the following website to do some research about high wire acts in the circus.
  • www.reachoutmichigan.org/funexperiments/agesubject/lessons/newton/hwire.html
  • What kind of math is involved in this art?
  • Does the diameter of the wire impact the act?
  • Have students write a short report on what they have learned about math and the high wire.
  • Students can even complete the activity at the end of the web page and experience walking a “high wire” of sorts themselves.

IV. Notes on Assessment

  • Check the solution to the problem.
  • Did the students use a diagram?
  • Is the diagram accurate?
  • Were they able to solve the problem accurately?
  • Provide students with feedback and comments.

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