## 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 .
- 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:
- 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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