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Hint: You will need to make use of the center point of the circles, so mark them down on paper after you have drawn any circles. The circles can be bigger or smaller as you choose. The important thing is to determine the relationship between the radii of the circles.

3. Draw a line and then two points on the line.

Then draw a circle using one point as a center point and the other point as a marking the radius. In other words, use the distance between the two points as the radius. Now draw another circle using the second point as the center point and the first point as indicating the radius. Look at the picture.

a) Now draw a line through the points where the two circles intersect. What do you notice?

b) If you connect the two initial points to the point above them where the two circles intersect (cross), what kind of figure do you get?

Part 3: Interactive Game

Students will take part in this fun and involving game. We will have to move all the desks to the sides of the room, and the students will sit in a large circle in the middle of the floor. I will have a long piece of rope, and call on students to “Show me a diameter” or “Show me the circumference” using the piece of rope. Students will be able to ask others for help if they need it. I will also be explaining definitions of the parts of a circle as we go, and asking questions like, “So what is the area between the chord and the outside of the circle called? –Segment” Homework: Students will have the rest of the time of class to write in their math journals about what they learned or to start working on their homework (HW 1).

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Lesson II: Arcs, Chords, Tangents Objectives:

(1) Develop a further understanding of arcs, chords, tangents (2) Interact with chords through technology, using Geometer’s Sketchpad

Parts of Lesson: (1) Arcs, (2) Chords, (3) Group Activity, (4)Tangents Teaching Style: Lecture (Part 1, Part 2, Part 4), Discovery Learning (Part 3) Part 0: Warm Up: Write down all the parts of a circle (and draw diagrams) that you remember from yesterday’s lesson. What relationships between the parts can you think of?

As a class, we will go over this briefly, and I will explain that we are going to look more deeply at arcs, chords, and tangents in today’s lesson. This material will be useful for solving proofs in the future, and, although a bit more specific than the “general idea,” it is still very important and will be tested upon.

Part 1: Arcs: An arc is a part of the circumference of a circle and defined as: 'an arc of a circle is the part of the circle between two points on the circle'. The longer arc is called the major arc while the shorter one is called the minor arc. Arc is measured in degrees and length. If the measure of minor arc is i.e. the measure of the central angle intercepted by the minor arc, then the measure of major arc is (360 - ) i.e. the measure of the central angle intercepted by the minor arc, then the measure of major arc is (360 - ). E.g. if measure of a minor arc is 100 , then major arc is (360 - 100 ) = 260 .

Types of Arcs: major arc, minor arc, and semicircle • If m AOB < 180 , points A and B and the points of circle in the interior of AOB make up minor arc AB, written as • Points A and B and the points of the circle not in the interior of AOB make up major arc AB, written

as or • If m AOC = 180 , points A and C separate circle O into two equal parts, each of which is called a semicircle. In

(Fig: 1), and two different semicircles

Congruent Arcs: Central angle is an angle whose vertex is the center of a circle ‘O’. Any central angle intercepts the circle at two points, thus defines the arc (Fig: 2).

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Congruent arcs are arcs of the same circle or of congruent circles that are equal in measure. In (Fig: 3, above), if O O' and mCD = mC'D' = 60 , then Arc Addition Postulate If AB and BC are two arcs of the same circle having a common endpoint and no other points in common, then AB + BC = and mAB + mBC = m . Theorems

• In a circle or in congruent circles, if central angles are congruent, then their intercepted arcs are congruent (Figure below).

if O O', AOB COD, and AOB A'O'B'. then and

• In a circle or in congruent circles, central angles are congruent if their intercepted arcs are congruent. • In a circle or in congruent circles, central angles are congruent if and only if their intercepted arcs are

congruent. E.g. (In figure below) P, Q, S, and R are points on circle O, mPOQ = 100 ,m QOS = 110 , and mSOR = 35 . Find m ROP.

m ROP will be- [360 – (100 + 110 + 35 )] = 115 , as the final answer.

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Part 2: Chords: A chord of a circle is a line segment whose endpoints are points on the circle. E.g. in the Fig: 4, AB and AOC are chords of circle O. Thus, a diameter is a special chord of a circle that has the center of the circle as one of its points. In the Fig: 4, AOC is the diameter.

Important Properties of chords • every chord defines an arc whose endpoints are the same as those of the chord and AD = DB. E.g. diameter and semicircle are the chord and arc that share the same endpoints. • the central angle forms an isosceles triangle, with chord as one side and the other two sides are rays that make the central angle. • the only diameter perpendicular to the given chord, is the perpendicular bisector also of that chord.

Part 3: Group Activity Students will break into groups of two or three and work on the group activity entitled ‘Chords in a Circle’ for Geometer’s Sketchpad. The use of Geometer’s Sketchpad will allow students to discover relations and conclusions on their own, being able to interact with the concepts with a more hands-on approach.

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Part 3: Tangents: A tangent to a circle is a line in plane of the circle that intersects the circle in one and only one point (Fig: 18). A secant of a circle is a line that intersects the circle in two points.

Postulate At a given point on a given circle, one and only one line can be drawn that is tangent to the circle. Theorems: • if a line is perpendicular to a radius at a point on the circle, then the line is tangent to the circle. • if a line is tangent to a circle, then it is perpendicular to the radius at a point on the circle. Common Tangents A line that is tangent to each of two circles is called a common tangent. Homework: HW 2: Display your understanding of 5 of the following by the use of pictures, diagrams, and/or proofs: -Congruent Arcs, Arc Addition Postulate, The 3 Arc Theorems, Important Properties of Chords, Tangent Postulate, The 2 Tangent Theorems, Common Tangents

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Lesson III: Discovering Pi Objective: (1) To get the students more interested in the idea of pi and this lesson plan about circles. (2) To get the students to understand the concept of pi, how it is a ratio comparing diameter and circumference Overview: (1) “Discovering Pi”, (2) Some Problems, (3) Historical Information about Pi, (4) More Pi Fun Teaching Style: Scaffolding/Cooperative Learning (Part 1), Lecture -with class involvement (Parts 2, 3, 4) Part 1: Discovering Pi

AUTHOR: Jack Eckley, Sunset Elem., Cody, WY Date: 1994

Subject(s): Mathematics/Geometry

OVERVIEW: Many students tend to memorize, without understanding, formulas that we use in geometry or other mathematic areas. This particular activity allows students to discover why pi works in solving problems dealing with finding circumference.

OBJECTIVES: The students will:

1. Measure the circumference of an object to the nearest millimeter. 2. Measure the diameter of an object to the nearest millimeter. 3. Explain how the number 3.14 for pi was determined. 4. Demonstrate that by dividing the circumference of an object by its diameter you end up with pi. 5. Discover the formula for finding circumference using pi, and demonstrate it.

RESOURCES/MATERIALS:

• round objects such as jars, lids, etc. • measuring tapes, or string and rulers • paper • pencil • calculator

ACTIVITIES AND PROCEDURES:

1. Divide class into groups of three or four. 2. Give materials to student teams. 3. Have student teams make a table or chart that shows name or number of object, circumference, diameter,

and ?. 4. Have students measure and record each object's circumference and diameter, then divide the circumference

by the diameter and record result in the ? column. 5. Have students find the average for the ? column and compare to other groups in the class to determine a

pattern. Students can then find the average number for the class. 6. Explain to the students that they have just discovered pi, which is very important in finding the

circumference of an object. (You may wish to give some historical information about pi at this time or have students research the information.)

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7. Have students come up with a formula to find the circumference of an object knowing only the diameter of that object, and the number that represents pi. Students must prove their formula works by demonstration and measuring to check their results.

TYING IT ALL TOGETHER:

1. Have students write their conclusions for the activities they have just done. Students may also share what they have learned with other members of the class.

2. Give students three problems listing only the diameter of each object and have them find the circumference. 3. Encourage students to share learned knowledge with parents

Part 2: Some Problems

Say, “As we just learned, pi is a constant value that describes the ratio of a circle’s diameter to its circumference. We see that C/d = or C = d. Using this information solve for the circumferences of the circles with the given information: (solve both keeping in your answer, and with using =3.14”

1. D=5 cm 2. D=3 in 3. R=.5 cm (Have the students work on these problems by themselves, then check with the person sitting next to them. Then, we will go over them as a class.) Say, “ is used in several other formulas, such as area of a circle, area of a sector, volume of a sphere, surface area of a cylinder, etc. We will be learning about some of these other formulas (area of a circle, area of a sector, circumference, length of an arc) in the next few days.”

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Part 3: Historical Information about Pi

Just for fun, I will give the students the following information to get them more involved in the idea of pi.

Facts about Pi

, a fundamental constant of nature, is one of the most famous and most remarkable numbers you have ever met.

The Egyptians and the Babylonians are the first cultures that discovered about 4,000 years ago. Here is a small table that shows some of the very old discoveries of :

Culture/Person Approximate Time Value Used

Babylonians 2000 BC 3 + 1/8 = 3.125

Egyptians 2000 BC 3.16045

China 1200 BC 3

Bible mentions it 550 BC 3

Archimedes 250 BC 3.1418

Hon Han Shu 130 sqrt (10) = 3.1622

Ptolemy 150 3.14166

William Jones, a self-taught English mathematician born in Wales, is the one who selected the Greek letter for the ratio of a circle's circumference to its diameter in 1706.

is an irrational number. That means that it can not be written as the ratio of two integer numbers. For example, the ratio 22/7 is a popular one used for but it is only an approximation which equals about 3.142857143... Another more precise ratio is 355/113 which results in 3.14159292... This was given to me by a student. Another characteristic of as an irrational number is the fact that it takes an infinite number of digits to give its exact value, i.e. you can never get to the end of it.

Since 4,000 years ago and up until this very day, people have been trying to get more and more accurate values for pi. Presently supercomputers are used to find the value of with as many digits as possible. Pi has been calculated with a precision containing more than one billion digits, i.e., more that 1,000,000,000 digits!

Here are three different ways to approximate the value of :

1. / 2 ~= (2*2*4*4*6*6*8*8*...) / (1*3*3*5*5*7*7*9*...) 2. / 4 ~= 1 -1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ... 3. ~= 3 + 1/10 + 4/102 + 1/103 + 5/104 + 9/105 + ...

The symbol "~=" means approximate. They are not equalities but can be very close. Try them out in your calculator! It is fun!

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Part 4: More Pi Fun On an projection screen, I will pull up the following websites and go through them with the class. The first website is where you can find the digits of your birthday in pi (as well as its location). I will select several students in the class and put their birthdays into the website, and the class can watch. The second website shows clever T-shirts designed with images of pi. I think that both of these websites will engage the students further into the lesson and get them more interested in the idea of pi. At the end of the class, I will write the links on the board to make them available for the students to use at home. http://www.angio.net/pi/piquery is a website where you can find your birthday in pi.

http://www.zazzle.com/pi+tshirts is a website showing funny pi T-shirts. (“pumpkin pi”, “cow pi”, “cutie pi”)

Homework: HW 3: Continue investing pi online. Find one interesting pi fact, and we will share them as a class tomorrow. For example, you can read about the world record holder for the most digits of pi memorized. Write down your pi facts in your math journal. http://www.pi-world-ranking-list.com/lists/details/luchaointerview.html

 

Lesson IV

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You may recall from earlier that the ratio of the circumference of a circle to the length of its diameter (d), is . We see that the formula for circumference is C = d, or (since we know d = 2r) , C = 2 r. The formula for area of a circle is A = r^2 Basic Problems (to be done on board): Find Circumference:

1. D= 2 cm 2. R=4 cm Find Area

1. D=2 cm 2. R=4 cm Part 3: Arcs: Degree Measure of an Arc: An arc of a circle is the part of the circle between two points on the circle. An arc of a circle is called an intercepted arc, or an arc intercepted by an angle, if each end point of the arc is on a different ray of the angle and the other points of the arc are in the interior of the angle. The degree measure of an arc is equal to

the measure of the central angle that intercepts the arc. E.g. . There are 360 total in a circle, so the degree measure of an arc can be anywhere from 0 to 360 .

Length of an Arc:

We see that an arc is a fraction of the whole circumference of the circle. Thus, if is the measure of central angle, then the length of the arc intercepted by the angle is given by / 360 * 2 r. For example, if = 90 and r=6, arc length would be (1/4)*2* *6 = (1/4)(12) =3 . Example Problems: (students will have time to do these on their own, then we will discuss as a class) Find the arc lengths of circles with the following information.

1. =45, r = 10 cm 2. =60, r = 8 in. 3. =110, d= 7 m. Part 4: Sectors: A sector of a circle is a pie shaped portion of the circle area, and it is between two segments coming out of the center of the circle (Fig: 16). Another way to say, it is the region enclosed by the central angle of a circle and the circle itself. A segment of a circle is the region enclosed by a chord and the arc that the chord defines (Fig: 17). The figure given below will help you understand it easily: For example: Find the area of the segment of a circle in Fig: 17,

 

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Lesson V: Using a Compass Objectives: (1) Introduction to compasses and how to use them (2) Obtain a better understanding of compasses, using them to construct circles with 3 points and angle bisectors Parts of Lesson: (1) Introduction to compasses, (2) Group work: compass practice and activities Teaching Style: Lecture (Part 1), Cooperative Learning/Group work (Part 2) Part 0: Warm Up For the first part of class, I will give some time for students to just play around with the compasses and see if they can figure out how to use them on their own. They will be able to talk to the people sitting around them (about compasses) if they wish (this will be a discovery/cooperative learning style of teaching). Part 1: Introduction to compasses First, I will provide a lesson on what a compass is and how to properly use it to draw circles. A compass looks like this:

Explain these directions: We can use an instrument called a compass to draw circles. To use a compass, fasten a pencil in the pencil hold and adjust the hinge so that the distance between the compass needed and the pencil tip is the desired radius length. Then, put the compass needle on a piece of paper where you want the origin to be, bring the pencil tip so that it is touching the paper, and rotate the pencil around the origin until a complete circle is made. We see that all the points we have drawn are equidistant from the origin (so, by definition, we have a circle).

Tell the students to practice drawing circles of various radii or other qualifications.. one with radius 3 cm, one with radius 4cm, one with diameter 5 cm, one with circumference 10 cm.

Part 2: Group Work

The students will then break into groups and work on the following two assignments, discovering how to construct a circle through 3 points and how to find the center of a circle using a compass. In order to do these activities, the students will also have to learn how to construct a perpendicular bisector (information attached). The extra practice using a compass will be beneficial for students, and hopefully they will increase the students’ increase in the subject matter.

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(1) Constructing a Circle through 3 points

After doing this Your work should look like this

We start with three given points. We will construct a circle that passes through all three.

1. (Optional*) Draw straight lines to create the line segments AB and BC. Any two pairs of the points will work. * We draw the two lines to make it clear when we later draw their perpendicular bisectors, but it is not strictly necessary for them to actually be there to do this.

2. Find the perpendicular bisector of one of the lines. See “Constructing the Perpendicular Bisector of a Line Segment.”

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After doing this Your work should look like this

3. Repeat for the other line.

4. The point where these two perpendiculars intersect is the center of the circle we desire.

5. Place the compass point on the intersection of the perpendiculars and set the compass width to one of the points A,B or C. Draw a circle that will pass through all three.

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After doing this Your work should look like this

6. Done. The circle drawn is the only circle that will pass through all three points.

(2) Finding the center of a circle

After doing this Your work should look like this

We start with a given circle.

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After doing this Your work should look like this

1. Using a straightedge, draw any two chords of the circle. For greatest accuracy, avoid chords that are nearly parallel.

2. Construct the perpendicular bisector of one of the chords using the method described in “Constructing a perpendicular bisector of a line segment.”

3. Repeat for the other chord

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After doing this Your work should look like this

4. The point where the two lines intersect is the center C of the circle.

Constructing the perpendicular bisector of a line segment: This will be necessary to complete the other two activities.

After doing this Your work should look like this

Start with a line segment PQ.

1. Place the compass on one end of the line segment.

2. Set the compass width to a approximately two thirds the line length. The actual width does not matter.

3. Without changing the compass width, draw an arc on each side of the line.

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After doing this Your work should look like this

4. Again without changing the compass width, place the compass point on the the other end of the line. Draw an arc on each side of the line so that the arcs cross the first two.

5. Using a straightedge, draw a line between the points where the arcs intersect.

6. Done. This line is perpendicular to the first line and bisects it (cuts it at the exact midpoint of the line).

Homework: HW 5: Write about the process of how to use a compass in your math journal. What did you learn from today’s activities? How could you use a compass in a real-world application? What would be another tool that could be used to draw circles?

 

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2. 3. 100x2 + 100y2 – 100x + 240y – 56 = 0. 5. 4x2 + 4y2 – 4x + 8y = 11 3. x2 + y2 +6x – 4y = 12

Graphing Circles: Once we solve for the center and radius of our circle, we can easily plot them on a graph. Simply plot the center point of a circle, and then use your compass to draw a circle with the appropriate radius. Part 2: Group work problems: The students will split up and work on the following problems, developing their understanding of the subject matter and learning how to solve these types of problems. Example 1: Look at the figure below: The circle is located at (-1, -2) and the radius is 1.5 units. What is the equation of the circle in standard form?

Example 2: Now look at a different case where equation of the circle is not in standard form. Given: Find the center and radius of the circle, x2 + y2 + 6y + 8 = 0, and graph it. Hint: Write the equation of the circle in standard form (x - h)2 + (y - k)2 = r2 using the ‘completing the square’ method.

Example 3: Given- the circle equation is, (x + 3)2 + (y - 4)2 = 16. Find out, if the point A (4, 5) is inside, outside or on the circle.

After the students solve the problems, we will discuss them as a class. I, as the teacher, can work out problems that need special attention on the board for everyone to see. Example 3 will probably need attention, and I can explain the steps (find the radius of the circle, find the distance between the center of the circle and pt A and determine if this distance is less than, equal to, or greater than the radius) Homework: With any remaining time in class, students will be asked to write in their math journals what they have learned today, and then to begin working on their homework (HW 6).

26  

Lesson VII: Circumscribed Polygons and Inscribed Angles

Objectives: (1) Understand and be able to differentiate between inscribed and circumscribed polygons (2) With past knowledge, come up with ideas on how to find areas of such figures (3) Explore inscribed angles further with the online activity Parts of Lesson: (1) Inscribed and Circumscribed Polygons, (2) Group Work, (3) Online Group Activity Teaching Style: Lecture (Part 1), Cooperative Learning (Part 2, Part 3), Discovery (Part 3) Part 0: Go over yesterday’s homework, work out any specific problems that students had trouble with. Especially give attention to the completing the squares problems. Part 1: Inscribed and Circumscribed Polygons

Polygons Inscribed in a Circle If all of the vertices of a polygon are points of a circle, then the polygon is said to be inscribed in the circle. In other words, it can also be expressed that the circle is circumscribed about the polygon.

As you may recall from earlier geometry, a polygon is a closed plane figure bounded by straight line segments as sides. A regular polygon may be a polygon which is equiangular (all angles are equal in measure) or equilateral (all sides have the same length). We will see that every regular polygon has an inscribed circle. Examples of the simplest regular polygons are the equilateral triangle, the square, the regular pentagon etc. An inscribed regular polygon is a polygon placed inside a circle such that each vertex of the polygon touches the circle and each of its sides is a chord. A circumscribed regular polygon is a polygon whose segments are tangent to a circle. Memory tricks: An INscribed regular polygon is a polygon INside a circle. (Fig. 2) A circumscribed regular polygon goes AROUND a circle. (“circum” means “around”.. think of the word “circumference”) (Fig. 3)

Inscribed Angle: An inscribed angle of a circle is an angle whose vertex is on the circle and whose sides contain chords of the circle (Fig: 5).

 

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(This problem can be discussed in front of the class as well. I can discuss the following steps to go about this problem and the rationale to coming to the answer. Step 1: In case of polygon inscribed in a circle, each of its sides is a chord. Step 2: For SQP, SQR, SRP, QRP and Quadrilateral SPQR, each side is a chord of the circle. Step 3: For SRO, SPO, QPO, QRO, some of the sides are not the chords of the circle. Step 4: Therefore, SQP, SQR, SRP, QRP and Quadrilateral SRQP are the inscribed polygons. ) Inscribed Angle Theorem: The measure of an inscribed angle of a circle is equal to one-half the measure of its intercepted arc. Proof If one of the sides of the inscribed angle contains a diameter of the circle.

Consider first an inscribed angle (Fig: 6), ABC, with diameter of circle O. m OAB = m OBA = x => m AOC = x + x = 2x Also, m AOC = m = 2x => m ABC = x = ½ m

Example 4 : find the measure of arc in Fig: 7. By the theorem stated above, A and C are supplementary. Therefore, C equals 95 . (Solution: From the theorem, measure of an arc is double that of its inscribed angle. Therefore, arc 190 , as the final answer. )

29  

Part 3: Online Group Activity Group activity: This activity uses a java applet that allows students to play with parts of a circle (changing radius, points, etc) and further investigate the idea of inscribed angles. “Discovery” questions are the provided for the students to investigate (“What happens when this happens?”, etc) http://www.analyzemath.com/Geometry/CentralInscribedAngle/CentralInscribedAngle.html  Homework: With any extra time, students will be asked to record their feelings about today’s lesson and then start working on their homework (HW 7).