Overarching Objectives of this March 2015 Network Team Institute
 Participants will be able to identify, practice, and use best instructional moves and scaffolds for chosen common core standards.
High-Level Purpose of this Session
 Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for teaching these modules.
 Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within this module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.
 Standards alignment the major work of the grade in order to fully implement the curriculum.
 Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.
Related Learning Experiences
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This session is part of a sequence of Module Focus sessions examining the Geometry modules within the curriculum, A Story of Functions.
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Thales’ theorem, inscribed angles, an inscribed right angle subtends the arc of a semicircle, the measure of an inscribed angle is half the measure of the central angle that subtends the same arc, inscribed angles that subtend the same arc have equal measures.
Properties of congruence, similarity, and isosceles triangles are applied to proofs; congruent chords lie in congruent arcs, all circles are similar.
Students will extend the protractor axiom (angles add) to arcs.
Students will continue to use properties of congruence and triangles in general to describe relationships formed by tangent and secant lines while incorporating recent knowledge related to inscribed angles and intercepted arcs.
Students will derive the equation for a circle centered at the origin by analyzing how to find the coordinates of points that lie on a circle. Students extend their understanding of equations of circles whose center is not at the origin using the rigid motion translation. Using knowledge of slope, students write equations for lines that are tangent to a circle.
Students will prove that a quadrilateral is cyclic if the opposite angles of the quadrilateral are supplementary. Students learn that the area of a cyclic quadrilateral is a function of its side lengths and an acute angle formed by its diagonals, i.e., Ptolemy’s theorem.
What do we want participants to be able to do as a result of this
session?
How will we know that they are able to do this?

 Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within this module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.
 Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade in order to fully implement the curriculum.
 Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their students while maintaining the balance of rigor that is built into the curriculum.
Participants will be able to articulate the key points listed above.
Section
Introduction
Topic A: Central and
Inscribed Angles
Topic B: Arcs and
Sectors
Mid-Module
Assessment
Topic C: Secants and
Tangents
Time Overview
117 min Explores central and inscribed angles.
Prepared Resources
16 min Introduces Geometry Module 5.  Geometry Module 5 PPT
 Geometry Module 5
Facilitator Guide


Geometry Module 5 PPT
Geometry Module 5
Facilitator Guide
79 min Explores how properties of congruence, similarity, and isosceles triangles are applied to proofs and extend the protractor axiom arcs.


Geometry Module 5 PPT
Geometry Module 5
Facilitator Guide
16 min Allows participants to complete a
Mid-Module Assessment and engage in a follow-up discussion.
122 min Explores how to use properties of congruence and triangles in general to describe relationships formed by tangent and secant
 Geometry Module 5 PPT
 Geometry Module 5
Facilitator Guide
 Geometry Module 5 PPT
 Geometry Module 5
Facilitator Guide
Facilitator Preparation
Review Geometry Module 5
Review Topic A
Review Topic B
Review End of Module
Assessment
Review Topic C

Topic D: Equations for Circles and Their
Tangents
Topic E: Cyclic
Quadrilaterals and
Ptolemy’s Theorem lines while incorporating recent knowledge related to inscribed angles and intercepted arcs.
79 min Explores deriving the equation for a circle at the origin by analyzing how to find the coordinates of points that lie on a circle.
 Geometry Module 5 PPT
 Geometry Module 5
Facilitator Guide
66 min Explores proving that a quadrilateral is cyclic if the opposite angles of the quadrilateral are supplementary.
 Geometry Module 5 PPT
 Geometry Module 5
Facilitator Guide
End of Module
Assessment and
Conclusion
35 min Allows participants to complete an
End of Assessment and engage in a follow-up discussion.
 Geometry Module 5 PPT
 Geometry Module 5
Facilitator Guide
Review Topic D
Review Topic E
Review End of Module
Assessment
In this section, you will be introduced to Geometry Module 5.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions GROUP

3 min 1.
3 min 2.
“Welcome to this Grade 10 segment of the NTI. Today we will take a look at
Module 5: Circles With and Without Coordinates.”
[Presenter introductions and bios.]
“You will find a variety of tools and materials where you are seated. First, you have a copy of the presentation to keep any important notes from the session. You will also find [in your binder] a Participant Handout that contains all of the problems and diagrams that we will work on throughout the session. Among other things you will find copies of the Mid-Module and End-of-Module Assessment bundles.”
[Ensure that participants have the tools and materials that they need; distribute items as needed.]
[Optional]
“I/We have provided a parking lot poster [make note of its location] for any comments or questions that you do not want to ask aloud. Please use a post-it note to leave your comment/question for sorting purposes.”
“In order for us to better address your individual needs, it is helpful to know a little about you collectively.
Pick one of these categories that you most identify with. As we go through these, feel free to look around the room and identify other participants in the same role that you may want to exchange ideas with over lunch or on breaks.”
“By a show of hands, who in the room is a classroom teacher?”
Math trainer or math coach?
Principal or school-level leader?
District-level leader?
“And who among you feel like none of these categories really fit for you?”
(Perhaps ask a few of these participants what their role is).
“Regardless of your role, what you all have in common is the need to understand the curriculum well enough to make informed decisions about

2 min 3.
1 min 4.
implementing it. A significant piece of that will happen through experiencing the major work of this curriculum and then participating in the commentary and discussion that comes from the classroom teachers and others in the group.”
“Our objectives for this session are to:
• Examine the development of mathematical understanding across the module using a focus on Conceptual Development.
• Experience the mathematical content that we are going to be teaching in order to implement that material confidently and with meaning so that we can genuinely engage our students in that content in the classroom and beyond.”
“The overarching goal of this session is to take a look at the content in the lessons of Module 5 and see how the concepts build upon each other and upon previous modules. My/Our hope is that the themes of the module are clear and that what has changed under the Common Core State Standards is very apparent.”
“Here is our agenda for today’s and tomorrow’s individual sessions.”
(Review the agenda on the slide)
_______________________________________________________________________________________
[OPTIONAL: Use the notes below if the EngageNY / Eureka curriculum is being introduced to participants for the first time.]
“If needed, we will start today by orienting ourselves to what the module materials consist of, however, I/we would like to spend the majority of the session discussing the content and its overarching development throughout Module 5. The hope is that you leave with an understanding of where the major shifts in Geometry are, and use examples to make sense of those changes.”
[Take a fist-to-five poll of the participants as to the level of knowledge of the module materials. If necessary, take the time to cover the Module
Overview, Topic Openers, Lesson Types, Lesson structures, and assessment materials.]

1 min 5.
2 min 6.
2 min 7.
Let’s begin with an orientation to the materials for those that are new to the materials (Skip if participants are already familiar with the materials).
“Let’s begin with a quick overview of the Module.” [read title]
“Module 5 is a 25-day module consisting of 21 lessons. The module has 5 topics.” (click to advance and read the topic titles and associated focus standards).
“Next, we’ll take a closer look at what the topics entail.”
[Click to advance and read the key concepts for the topics listed on the slide].
[Click to advance and read the key concepts for the topics listed on the slide].

Time Slide
#
Slide #/ Pic of Slide
2 min 8.
In this section, you will explore central and inscribed angles.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide
 Participant Handout
 Plain white paper
 3 x 5 Index Cards
 Compass
 Straight edge
 Trapezoidal cardstock cutouts
 Diagram completed in Exploratory Challenge 1
 Circle with inscribed angle template for M5 Lesson 4 - 2 per participant
 Scissors
Script/ Activity directions
Discuss the focus standards for Topic A and the key concepts that participants can expect to see as they progress through the topic.
GROUP

1 min 9.
1 min 10.
5 min 11.
“Topic A is comprised of six lessons, shown here.”
[Click to advance and read each lesson title one at a time].
[Read the title and the objectives of the lesson.]
Required tools and materials:
• Participant Handout
• Plain white paper (or use the space provided in the Participant
Handout)
• 3”x 5” or larger index card (a right angle is required with each side a minimum 3” in length)
•
“The purpose of this Opening Exercise is to explore the converse of
Thales’ theorem (named later in the lesson).”
[Instructions are also provided in the Participant Handout] Ask participants to, “Fix two points, less than 3 inches apart, named A and
B. Then take a notecard or other setsquare and “push” it through the space between A and B until the sides of the notecard or setsquare touch both points. When the notecard touches both points A and B, mark a point on the paper at the location of the vertex of the right angle

12.
(the corner of the card). Repeat this process several times noting each new point as C, D, E, etc.”
[After participants have drawn 5 or more points, click to reveal the questions on the slide.]
• What curve is traced?
A semicircle.
• Where is the center of the semicircle?
The center of the semicircle will be at the midpoint of segment AB.
• What is the radius?
The radius will be half the length of the segment AB.
• Can we prove that all of the points lie on the circle?
By definition, a circle is the set of all points that are a fixed distance r>0 from a given point. We must show that all segments formed by the center of the circle and the points C, D, E, etc. are equal in length to prove the
points lie on the circle.

6 min 13.
5 min 14.
Required tools and materials:
• Document camera
“The Exploratory Challenge in this lesson proves the theorem shown at the top of the slide.”
[Read the theorem]
[Switch to the document camera to lead participants through the
Exploratory Challenge.]
[Switch back to the presentation and click to advance to further reasoning.]
“Based on the theorem we just proved, we can state the following relationship.”
[Read the theorem and then advance the slide for the proof.]
Required tools and materials:
• Participant Handout
• Compass
• Straight edge
• Document camera
“Use the steps given in the Participant Handout to prove Thales’ theorem. Be prepared to share your work.”
[Provide participants time to work. While circulating, find a volunteer to share their work with the whole group.]

2 min 15.
1 min 16.
4 min 17.
Required tools and materials:
• Participant Handout
“Because of the experiences students had in the lesson, this exercise should be easy! How can we determine the measure of the angle in part a? Part b? Part c?”
[Ask participants for responses.]
Read the title and student outcome aloud.
Required tools and materials:
• Participant Handout
• Plain white paper
• Compass
• Straight edge
“Given segment 𝐴𝐵 , construct its perpendicular bisector. Then draw a different line bisecting segment 𝐴𝐵 .”
[Give participants 1 minute for the construction]
“Name one similarity and one difference of the two constructed lines.”
Sample responses: They both pass through the midpoint of 𝐴𝐵 , the points in the perpendicular bisector are the same distance from each endpoint A and B, however the points that make up the second line are not, only the

midpoint; the perpendicular bisector meets the segment at congruent
angles and the other bisector does not.
[Click to advance and read the definition of equidistant and the notes that follow.]
“The perpendicular bisector is the set of all points that are equidistant from points A and B.”
[Click to advance] “Next, on a sheet of white paper, draw a circle of any radius and center P. Draw a chord on the circle and label its endpoints
A and B.”
[Click to advance] “Construct the perpendicular bisector of chord 𝐴𝐵 .
What do you observe?”
The perpendicular bisector appears to pass through center P.
[Click to advance] “Draw chord 𝐶𝐷 on the circle and construct its perpendicular bisector. What do you observe?”
The perpendicular bisector appears to also pass through the center P.
“What can we say about any two points on a circle in relation to the center of the circle?”
The center of the circle is equidistant from any two points on the circle.
“Look at the diagrams created by your neighbors. They are obviously different circles and different chords, but what do they all have in common?”
The perpendicular bisectors pass through the center of the circle in all
cases.
“The perpendicular bisector of any chord in a circle must contain the center of the circle. The center of the circle is equidistant to any two points on the circle that are endpoints of a chord.”

17 min
18.
1 min
6 min
19.
20.
Required tools and materials:
• Compass
• Straight edge
• White paper
• Document camera
[Divide Exercises 1-6 among 6 groups of participants to prove. Provide
4 minutes to work. Then ask groups to present their proof to the group including any given assumptions, the goal, and the steps used in reaching the goal. Allow each group a maximum of 2 minutes to present their proof to the group. ]
“In the Closing, students begin completing a graphic organizer in which they describe a given diagram and write the theorem or relationship that corresponds with it. You will find the graphic organizer on yellow paper where you are seated.”
Read the title and the objectives of the lesson.
Required tools and materials:
• White paper
• Compass
• Straight edge
[Read the direction on the slide aloud, then provide participants time to do the construction (or not if time is an issue). Click to show the sample construction.]

2 min 21.
2 min 22.
2 min 23.
“Think about how this construction may lead to achieving our goal of inscribing a rectangle in a circle.” [Click to advance slide.]
[Read the text on the slide. Ask participants for responses.]
Sample Response: Knowing the properties of rectangles, namely that the
diagonals are equal in length and bisect each other.”
[Click to advance slide.]
[Click to show the strategy bullet points.]
“The first strategy shown here uses the properties of rectangles to construct the rectangle. Notice that the hypotenuses of the right triangles we constructed in the opening exercise are the diagonals of the rectangle.”
[Click to animate the rectangle.]
“Another strategy, shown here, is to simply rotate the first triangle constructed by 180˚ around the center of the circle.”
[Click to animate the rectangle, then click to bring in the last line.]
“Of course students may find alternative strategies. For example, students may construct a chord, then construct perpendicular segments through the endpoints to achieve the construction of the rectangle.”

2 min 24.
1 min 25.
8 min 26.
Required tools and materials:
• Participant Handout
[Allow time for participants to take a quick look at the exercises. If you can allot more time, ask teachers to discuss the exercises with their table partners.]
[Read the title and student outcome aloud.]
“Note that the Inscribed angle theorem will take several lessons to define in its entirety. This lesson focuses on the relationship between an inscribed angle and a central angle that intercept the same arc. At this time, students have not studied arc measure which is required for the full understanding of the theorem.“
Required tools and materials:
• Participant Handout
• White paper (or use the extra space provided in the Participant
Handout)
• Trapezoidal cardstock cutouts (both bases must be at least 3 inches in length)
“Please take one of the cardstock trapezoids from the center of your table and label one of its acute angles with the variable a. Tell me something about the angle relationships in the trapezoid.”
Consecutive angles along the parallel sides of the trapezoid are
supplementary.
[Presenter please note: The prompt in the teacher’s materials seems to assume an isosceles trapezoid; this is being addressed.]

5 min 27.
“Label the supplement to the acute angle you chose as 180-a.”
[Direct participants to complete Exploratory Challenge 1 using the acute angle that they labeled (3 minutes), then ask the following questions:]
“What shape do the plotted points form?”
They seem to form the major arc of a circle.
“Can you continue and plot the points of the minor arc of the circle?
How?”
Use the supplement to the acute angle to find the points in the minor arc.
[Direct participants to plot several points along the remaining minor arc of the circle.] (2 minutes)
“How does this relate to our work with Thales’ theorem in Lesson 1?”
In Lesson 1, we were able to create a semi-circle with the right angle corner of a sheet of paper which follows Thales’ theorem that a triangle created by connecting the endpoints of a diameter of a circle is a right
triangle. Here we’ve changed to an acute angle and its supplement.
Required tools and materials:
• Participant Handout
• Diagram completed in Exploratory Challenge 1
• Straight Edge
[Direct participants to complete parts (a) and (b) from Exploratory
Challenge 2.] (3 minutes)
[Ask participants to share their observations in parts (a) and (b) with the group.] (2 minutes)

5 min 28.
3 min 29.
Required tools and materials:
• Participant Handout
• Circle with inscribed angle template for M5 Lesson 4 - 2 per participant
• Straight edge
• Scissors
[Direct participants to complete Exploratory Challenge 3 using the instructions and the circle templates provided.]
[Note to presenter: Direct groups to plot point D on the major arc of the circle. An optional extension to this challenge would be to have one group plot point D on the minor arc, then compare the angle measures between groups at the end to show that the angles formed are supplementary.]
[Ask participants to share aloud their observations.]
The inscribed angles, even though their vertices lie on different parts of
the circle, all seem to have the same measure.
Required tools and materials:
• Participant Handout
• Straight edge
[Direct participants to complete parts (a) through (f) of Exploratory
Challenge 4.] (2 minutes)
[Ask participants to share their answers and prediction in part (f). Ask if they can justify their prediction.]
Sample response: One option is to use two inscribed angle cutouts from the previous Exploratory Challenge to compose the central angle in the
smaller circle. This could also be done with patty paper or transparency.

1 min 30.
5 min 31.
“The observation made regarding the relationship of a central angle and an inscribed angle that intercept the same arc is left for students to prove in Problem set #6.”
[Read the title and the objectives of the lesson.]
“This lesson covers only part of the inscribed angle theorem – the central angle version. The case for a central angle whose measure is greater than 180 degrees is left for lesson 7.”
Required tools and materials:
• Participant Handout
• Document camera
“Let’s take a look at the opening exercises.”
[Switch to the document camera and work through both exercises.]
[Come back to the slide and advance the animation.]
“Notice that in the solution shown in the current edition, it says that we are given the measure of angle x. We saw that was not true in our work on the document camera. We were actually given the measure of ACD and because base angles of isosceles triangles are congruent, we can conclude that angle D also has a measure of x.”

2 min 32.
10 min
33.
1 min 34.
[Read the text on the slide, then click to show Case 1.]
“Does this case and diagram look familiar?”
[Click to advance]
“It should because we just proved this in the opening exercise!”
“There are two other cases we need to consider.”
[Click to advance and read case 2, then click to advance again and read case 3).
“We will work together on these two cases using the document camera.”
[Switch to the document camera and complete examples 1 and 2 with the group.]
[Switch back to the presentation when completed.]
[Read the title and student outcomes aloud.]

4 min 35.
8 min 36.
Required tools and materials:
• Participant Handout
[Direct participants to complete the Opening Exercise on the slide independently, then share their strategy with a shoulder partner.] (3 minutes)
“Why do you think this is done as an Opening Exercise in this lesson?”
This serves as an informal assessment of student knowledge/skills of
Topic A and can help the teacher navigate students away from
misconceptions.
Required tools and materials:
• Sheet of white paper
• Compass
• Straight edge
[Read the theorem aloud. Direct participants to plot three noncollinear points A, B, and C on a sheet of white paper, then draw the circle on the three points. Ask one participant to share how they constructed their circle.]
Construct the perpendicular bisector of two segments that join points A,
B, and C. The intersection of the perpendicular bisectors is a point that is equidistant so points A, B, and C, and is, therefore, the center of the circle.
The distance from the center to any of the three points is the radius of the
circle.
“What is the rest of our given information?”
There is a point B’ on the same side of line AC as B and angles AB’C and
ABC have equal measure.
“What do we need to show in this theorem?”
Show that B’ lies on the same circle with A, B, and C.
“Where could B’ lie?”
It could lie outside the circle, inside the circle, or on the circle.

2 min 37.
“To show that B’ must be on the circle, we will show that the other two cases are impossible.”
[Divide table groups so that half show the contradiction when B’ lies outside the circle, and the other half shows the contradiction when B’ lies inside the circle.] (5 min)
[Ask two volunteers to share their work with the group using the document camera. The proofs are provided on the next four slides for each case.]
Required tools and materials:
• Sheet of white paper
• Compass
• Straight edge
“This slide contains the contradiction that exists where B’ is assumed to lie outside the circle.”
[Click to advance through the reasoning and continue through the next slide.]
“Angle PCB’ > 0 since B’ and P are distinct points. The inscribed angle theorem and the exterior angle theorem show that angle PCB’ = 0, and this contradicts our stated assumption, so B’ cannot lie outside the circle.”

1 min 38.
Required tools and materials:
• Sheet of white paper
• Compass
• Straight edge
[Click to advance through and follow the reasoning.]
1 min 39.
Required tools and materials:
• Sheet of white paper
• Compass
• Straight edge
“This slide contains the contradiction that exists when B’ is assumed to be inside the circle.”
[Click to advance through this slide and the next.]
1 min 40.
Required tools and materials:
• Sheet of white paper
• Compass
• Straight edge
[Click to advance through and follow the reasoning.]
“Since both cases, where B’ is outside the circle and where B’ is inside the circle, are shown to be impossible by contradiction, B’ must lie on the circle with points A, B, and C.”
In this section, you will explore how properties of congruence, similarity, and isosceles triangles are applied to proofs and extend the protractor axiom arcs.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide

Time Slide
#
Slide #/ Pic of Slide
2 min 41.
 Participant Handout
 Chart Paper (optional)
 Plain white paper
 Compass
 Straight edge
 Document camera
Script/ Activity directions
[Discuss the focus standards for Topic B and the key concepts that participants can expect to see as they progress through Topic B.]
GROUP
1 min 42.
“Topic B is comprised of four lessons, shown here.”
[Click and read one at a time]

2 min 43.
20 min
44.
45.
[Read the title and the objectives of the lesson.]
“We will not be reviewing the proof that all circles are similar in this lesson.
Essentially we have students reason that the size of the circle is a function of the radius. Therefore a dilation will map one circle onto another, no matter the size, and we can conclude that all circles are similar.”
Required tools and materials:
• Participant Handout
• Chart Paper (optional)
[The handout portion for this lesson divides the content into parts for the groups. Assign two tables that are close in proximity to review the content for each part of the discussion/example. Instruct tables that are assigned the same part to share ideas with one another prior to sharing out whole group. Provide chart paper as needed.]
Hidden Slide
[Use this slide if needed to highlight key points from the discussion.]

46.
47.
48.
Hidden Slide
[Use this slide if needed to highlight key points from the discussion.]
Hidden Slide
[Use this slide if needed to highlight key points from the discussion.]
Hidden Slide
[Use this slide if needed to highlight key points from the discussion.]

1 min 49.
2 min 50.
[Read the title and student outcomes aloud.]
“Students begin this challenge with a discussion about chords, arc, and arcs subtended by chords.”
[Advance to show the congruent arc/congruent chord CD]
“The task of proving that congruent chords have congruent arcs, and the converse, that congruent arcs have congruent chords, is left to students.”
“How will students prove these two relationships?”
If the radii are draw to the endpoints of the congruent chords, then the triangles formed can be shown congruent by SSS. Conversely, if the arcs are congruent, then the measures of the central angles are congruent, and the triangles can be proven congruent by SAS. Either way,
corresponding parts of the triangles are congruent.
[Advance to show the radii on the diagram on appropriate participant cue.]

7 min 51.
2 min 52.
Required tools and materials:
• Plain white paper
• Compass
• Straight edge
[Read the theorem aloud and direct participants to prove the theorem in two different ways. They should draw the diagram on white paper and show their proof.]
[As an alternative, give participants just a few moments to prove the theorem in one way, then look for how many variations there are and present those ideas using the document camera.]
Possible strategies: Construct a perpendicular bisector of DE or BC followed by a reflection over the perpendicular bisector. A circle reflected over a diameter maps points on the circle to points on the circle, and endpoints of the chords are, therefore, reflected onto each other. Second possible strategy: Draw diagonal DC (or BE) and use reasoning about
angles formed by parallel lines, transversal, and inscribed angle theorem.
[Read the title and the objective of the lesson. Note that there is vocabulary that is reviewed (arc, minor and major arc, semicircle), but that new terms are being introduced (arc length, radian, and sector).]

8 min 53.
54.
Required tools and materials:
• Participant Material packet
• Document camera
[Read the text on the slide]
[Switch to the document camera to work through example 1 with the group.
Before beginning, ask the following questions]
“Do you believe arcs can have a length? Explain your reasoning.”
Yes, arcs can have length. Since we know adjacent arcs can be added together, if we consider a major and minor arc together, the total length would be 2(pi)r. Then arc lengths can be 0 < arc length < 2(pi)r. Since arc lengths are portions of circles, an arc length cannot be equal to
2(pi)r.
“Let’s say we have an arc that measures 60˚ in a circle. What fraction of the circles circumference is taken up by this arc? Explain.”
The arc takes up 60/360 of the total circumference because the entire circle is 360˚. The arc of measure 60˚ can then be thought of as 60 out of
360 or 1/6.

3 min 55.
56.
“Following the work of the example, we take a closer look at parts b and e.
Let’s start with part b.
[Click to advance slide]. “Take a look at the work we did to determine arc lengths. What variable is determining the arc length as the central angle remains constant?
The radius determines the length of the arc, because all circles are similar
(lesson 7). That means there exists a constant of proportionality.
[Click and state the constant of proportionality.]
“This demonstrates that arcs of the same degree measure are similar.
Now if we are looking at a central angle of 1˚ for a unit circle
[Click to advance], then we can say the arc length is (pi)/180. In general though, the arc length for any circle with a central angle of 1˚ will be (pi)/180 times the radius.”

3 min 57.
3 min 58.
1 min 59.
[Click and read through the text on the slide. When you get to sector say:]
“Following Example 1 there is a quick discussion about the definition of a sector. Once defined, ask students to think about how they may determine the area of the sector.”
“The development of the formula for the area of a sector mimics what was done to determine arc length. Students complete some concrete problems first. The goal is for them to see that now that we are talking about the area of a sector, we take the fraction of the circle taken up for the arc measure and multiply it by the area formula for circles.
Essentially finding the fractional portion of the circles area based on the arc measure.”
[Read the title and student outcome aloud.]

2 min 60.
2 min 61.
20 min
62.
[Read the problem aloud and part (a).]
“What will the cross section look like?” (Pause for participant response)
[Click to reveal the cross section]
“The cross section is two concentric circles, but it also includes the area bounded by those concentric circles. This figure is referred to as an annulus.”
[Click to advance]
[Read part (b) aloud.]
[Click to advance and reveal radii of concentric circles, then pause for participant responses to the question.]
Participants will likely provide the difference of the area of the circles.
[The formula can be simplified using the distributive property. Click to reveal area of the annulus]
Required tools and materials:
• Participant Handout (Lesson 10 Exercises 1-13)
[Direct participants to review and complete some of the exercises provided in Lesson 10. Explain that the problems vary in difficulty and include a variety skills learned throughout this module and beyond.]
(provide 12-15 minutes to work)
[Seek volunteers to share their work using the document camera.] (5-8 minutes)

In this section, you will complete a Mid-Module Assessment and follow up discussion.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide
 Mid-Module Assessment Handout
Time Slide # Slide #/ Pic of Slide
16 min 63.
Script/ Activity directions
Required tools and materials:
• Mid-Module Assessment Handout
[Advance through the bulleted directions on the slide to guide participants in evaluating the Mid-Module Assessment. Provide 3 minutes to review all problems, 5 minutes to complete a problem of their choice, 5 minutes to review the sample work and rubric, and 3 minutes to discuss STEP 1, 2, 3, and 4 responses.]
In this section, you will explore how to use properties of congruence and triangles in general to describe relationships formed by tangent and secant lines while incorporating recent knowledge related to inscribed angles and intercepted arcs.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide
 Participant Handout
 Compass
 Straight edge
 Plain white paper
 Patty paper
 Document camera
 Ruler (with metric scale)
Time Slide
#
Slide #/ Pic of Slide Script/ Activity directions
GROUP
GROUP

2 min 64.
1 min 65.
1 min 66.
[Discuss the focus standards for Topic C and the key concepts that participants can expect to see as they progress through Topic C.]
“Topic C is comprised of six lessons, shown here.”
[Click and read one at a time.]
[Read the title and the objectives of the lesson.]

2 min 67.
4 min 68.
5 min 69.
“Lesson 11 begins with an opening exercise that sets the stage for Topic
C. Students are asked to draw a circle and a line, I would do this on a post-it note. Students bring their drawings to the board. The drawings are then grouped by the number of times the line intersects the circle.
This allows the teacher to review the vocabulary terms secant line,
tangent line and tangent segment.”
“The opening exercise leads to the exploratory challenge where the term Point of Tangency is introduced.”
[Read the first bullet aloud then click to advance.]
“Students then use protractors to measure the angle formed. Once measured they compare the result with that of their neighbors. Of course they will see that no matter where P was on the circle, the angle formed by the radius and the tangent line is equal to 90˚.
[Click to advance]
“We then make clear the relationship of the tangent line to the radius.”
[Read last two bullets aloud.]
“In Example 1, students use constructions to draw some conclusions about the lengths of tangent segments. Specifically, this Example sets students up to understand that tangent segments from the same point are equal in length.
[Click to advance] “Students are asked to draw a circle and a radius to some point P.
[Click to advance] “Next, students draw a tangent line through point P using construction skills learned in Module 1.
[Click to advance] “Next, students are asked to draw a point R, exterior to the circle and construct a line through point R tangent to the circle.

5 min 70.
1 min 71.
This is no easy task! How would you do it?
[Pause for responses]
[Click to advance] “If necessary, the following steps are provided to help students with the construction. You can challenge students by asking them why these steps work. Think back to Lesson 1 and Thales’ theorem and you’ll have your answer. That said, we can conclude that
MB, MA MR and MC are equal lengths because they are radii of the same circle. Keep this in mind for what we do next.”
Required tools and materials:
• Participant Handout
“Take a minute to complete exercise 1. This is where students see that tangent segments that meet at the same point exterior to the circle will be equal in length.”
[Provide time for participants to work. If time, ask for a volunteer to share their solution.]
[Read the title and student outcome aloud.]

3 min 72.
10 min
73.
3 min 74.
Required tools and materials
• Participant Handout
[Read the problem aloud. Give participants 1 minute to complete the given task and respond their observations.]
[Click to advance sample circles]
[Follow up with the questions on the slide.]
“Does this suggest any conjectures?”
If a circle is tangent to both rays of an angle, then the center of the circle
lies on the angle’s bisector.
Required tools and materials:
• Participant Handout
[Read the problem aloud then provide participants 5 minutes to complete the proof with a partner. Ask for a volunteer to explain what the theorem states in words, then share their proof using the document camera and explain their strategy. Look for any other creative variations of the proof and ask those participants to share those as well.]
Required tools and materials:
• Participant Handout
• Compass
• Straight edge
[Direct participants to draw at least three different circles that are tangent to both rays of the given angle.]
“How many circles are possible?”
Infinite
“How did you locate the center(s) and radius (radii) of your circle(s)?”

3 min 75.
Select any point along the angle bisector, construct a perpendicular to
one side of the angle in order to obtain the radius of the circle.
“Why do we know that the radius is the distance from the center to the tangent line?”
The radii are perpendicular to the rays.
“What do you think is the main idea of this exercise thus far?”
To know that the center of any circle that is tangent to both rays of an
angle must lie on the angle’s bisector.
Required tools and materials:
• Participant Handout
• Compass
• Straight edge
[Provide 2 minutes for participants to complete the task on the slide then ask the following:]
“Explain how many circles can be drawn to meet the given conditions.”
There is only one circle. The center of the circle has to be on the angle bisector of both angles; the angle bisectors only intersect at one point.
There is only one circle with that point as a center that is tangent to the rays. It is the circle with the radius obtained by drawing the
perpendicular from the point to one side of angle A (or C).
“What does this Exercise seem to be leading up to?”
The inscribed circle in a triangle whose center is the centroid, or the point
of concurrency of the three angle bisectors in a triangle.

4 min 76.
2 min 77.
Required tools and materials:
• Plain white paper
• Straight edge
• Compass
[Read the Exercise on the slide aloud. Participants will likely know the answer to this question, however provide the option to draw triangles of different classifications and test their ideas, as students would, by bisecting the angles in their triangles. Then ask for a volunteer to explain and show why 𝑃 must lie on the third angle bisector.]
The angle bisectors of any two consecutive angles are the same distance from both rays of both angles. Let the vertices of the triangle be A, B, and
C. If P is on the angle bisector of angles A and B, then P is the same distance from AB and AC, and is also the same distance from BA and BC, and so by transitivity is the same distance from CA and CB. This being the
case, P is then on the angle bisector of angle C.
“This exercise extends the reasoning from Exercise 3 to conclude that all three angle bisectors of a triangle meet at a single point. This point is the center of the inscribed circle, or the centroid.”
“A fifth exercise further applies what was learned in this exercise.”
[Read the title and the objectives of the lesson.]

7 min 78.
4 min
1 min
79.
80.
“In the last problem of the problem set of the previous lesson students explore the relationship between the measure of an arc and an angle.
In the exploration part of this challenge students use protractors and rulers to develop a conjecture about the relationship between the arc with degree measure b and the angle measure a, as shown in the diagram.”
[Click to advance] “We recognize the relationship as the tangent-secant theorem. How might we be able to prove this relationship a = 1/2 b?”
[Pause for responses, click and read proof.]
“The next question in mind should be ‘did we get lucky or does this relationship exist no matter where point c is?’”
[Click to advance to the next slide.]
Required tools and materials:
• The hyperlinked image on the slide must be correctly hyperlinked to the interactive geometry software file of your choosing.
[Read the bullet point and theorem. Click the hyperlink to model the theorem using dynamic geometry software. Move point E to different locations to show that the angles remain equal. Come back to the slide and click to reveal the last discussion point.]
[Read the title and student outcome aloud.]

5 min 81.
Required tools and materials.
• Plain white paper
• Compass
• Straight edge
• Patty paper
• Document camera
[Direct participants to draw a circle on a plain sheet of white paper, and then draw a pair of intersecting lines (or segments rather) on a sheet of patty paper.]
[Switch to the document camera to model the patty paper activity]
“We can use the patty paper to visually gauge how the size of the angle made by the two lines relates to the arc measures that the secants cut on the circle. With the intersection of the lines on the circle, the angle formed by the secants is an inscribed angle, and is therefore equal to half the measure of the intercepted arc. If we move the intersection point to the inside of the circle, we can see that the measure of the previously considered arc decreases, however the measure of a new arc intercepted by the vertical angle is increasing in measure. How do you think the measure of the secant angle relates to the intercepted arcs?”
Students may or may not conjecture that the secant angle’s measure is
the average of the arc measures.
“Next, let’s translate the intersection point to the exterior of the circle.
Notice that the original arc that we considered is increasing in measure, and the arc in the vertical angle is decreasing until the point reaches the exterior of the circle, at which time the arc measure begins increasing again. How do you think the measure of the secant angle relates to the intercepted arcs now?”
Student may or may not conjecture that the secant angle’s measure is
half the difference of the intercepted arc measures.

3 min 82.
3 min 83.
8 min 84.
Required tools and materials:
• Participant Handout
[Read the Exercise on the slide aloud and ask participants to devise a strategy for finding the measure of the unknown angle. Ask for a volunteer to provide their strategy and answer to the problem.]
[Note to the presenter: There is a typo in the Teacher’s Materials for this problem. The sample solution shows m∠MOP = 1° , which is of course incorrect. The correct measure of the angle is 19° .]
Required tools and materials:
• Participant Handout
[Read the Exercise on the slide aloud and ask participants to devise a strategy for finding the measures of the unknown angle and arc. Ask for a volunteer to provide their strategy and answer to the problem.]
[Note to the presenter: There is a typo in the Teacher’s Materials also for this Exercise. The Exercise gives mCE
, however the arc’s measure should be 55° . Also, the sample response shows the measure of angle DEF to be 27° , however it should be 27.5° .]
Required tools and materials:
• Participant Handout
[Read the Secant angle theorem on the slide, then provide 5 minutes for participants to develop a proof to the theorem to share on the document camera.]

1 min 85.
2 min 86.
87.
[Read the title and the objective of the lesson.]
“You saw in the previous lesson that this theorem for the interior case was proven. The opening exercise provides time for the students to practice writing their own proof. Scaffolded questions are used to guide students’ thinking.”

2 min 88.
89.
5 min 90.
“Here students explore the relationship of the larger and shorter arcs to the angle formed by the intersection of the secant lines. Again students are using a protractor and ruler to try to develop the relationship. It is likely that you will need to discuss “human error” with students and present the theorem so that they are clear about the relationship before moving on to the next example.”
“Our goal in this example is to extend the relationship we observed in the exploratory challenge to include tangent lines.”
[Click to advance] “From the exploratory challenge, we have the relationship shown here about secant lines.”
[Click to advance] “Now if we rotate one of the secant lines so that it becomes tangent to the circle, does the relationship still hold?
Yes, and we can write the formula to determine the measure of angle C.
[Click to advance] “The last point for this part of the exercise is to make clear that they symbols in the formula change, but the relationship remains constant.”

3 min 91.
3 min 92.
2 min 93.
“We continue the work with the example by now rotating the other secant line so that it becomes a tangent. This issue here is that shorter arc and the longer arc have the same name. We need to differentiate them somehow. We do so by noting a point on the circle and renaming the longer arc with 3 letters.”
“We continue the work with the example by now rotating the other secant line so that it becomes a tangent. This issue here is that shorter arc and the longer arc have the same name. We need to differentiate them somehow. We do so by noting a point on the circle and renaming the longer arc with 3 letters.”
[Click to advance and follow discussion on the slide]
“Here the closing is slightly different than the typical. Students have 5 diagrams in a graphic organizer. For each diagram they have to state how the two shapes overlap as well as the relationship between the arcs and angles formed.”

1 min 94.
7 min 95.
“In Lesson 16 we take a shift from the angle relationships that exist where tangents and secants of a circle are concerned and look now at the relationship of the lengths of segments.”
[Read the title and student outcome aloud.]
Required tools and materials:
• Participant Handout
• Ruler (with metric scale)
[Direct participants to measure and complete columns for a, b, c, and d of the table in Exploratory Challenge 1.]
“There is a great margin for error in measurement and the relationship between the segment lengths, therefore, will not be as obvious as one might think. The products of the segment lengths are relatively close, however they will likely not be ridiculously close.”
“What are some considerations that we can make with this Exploratory
Challenge to help students recognize the relationship that exists?”
Consider having students work in groups and take an average of their
measurements before trying to develop a relationship.
Have students round their measurements to the nearest tenth or nearest
mm.
Provide the relationship and ask students to determine whether or not the relationship seems valid and why, noting human error, size of points
in the diagram, measuring tool, rounding, etc.

4 min 96.
6 min 97.
Required tools and materials:
• Participant Handout
• Ruler (with metric scale)
[Direct participants to measure and complete columns for a, b, c, and d of the table in Exploratory Challenge 2.]
“Will our considerations from Exploratory Challenge 1 work in this challenge as well?”
“The goal of these two Exploratory Challenges is to get students to recognize that there is in fact a relationship between the lengths of segments composed in secants and tangents on a circle. The goal of the next two Exploratory Challenges is to reveal why these relationships exist and prove that they exist in all cases.”
Required tools and materials:
• Participant Handout
• Compass
• Straight edge
[Direct participants to discuss the diagram shown on the slide with a partner and prove why the relationship found in Exploratory Challenge
1 is valid. Seek volunteers to share their reasoning using the document camera.]
Draw chords BD and EC. Triangles BFD and EFC are similar by AA
(vertical angles and inscribed angles of the same arc). Similar triangles have proportional corresponding side lengths which provides the proportion 𝐵𝐹
=
𝐹𝐸
𝐹𝐶 = 𝐷𝐹 ⋅ 𝐹𝐸 .
𝐷𝐹
𝐹𝐶
. The cross product of this proportion provides 𝐵𝐹 ⋅

4 min 98.
3 min 99.
Required tools and materials:
• Participant Handout
• Compass
• Straight edge
[Direct participants to discuss the diagram shown on the slide with a partner and prove why the relationship found in Exploratory Challenge
2 is valid. Seek volunteers to share their reasoning using the document camera.]
Draw chords DE and BF. Triangles BFC and DEC are similar by AA
(shared angle C and inscribed angles of the same arc). Similar triangles have proportional corresponding side lengths which provides the proportion 𝐶𝐵
=
𝐶𝐷
𝐶𝐵 = 𝐶𝐹 ⋅ 𝐶𝐷 .
𝐶𝐹
𝐶𝐸
. The cross product of this proportion provides 𝐶𝐸 ⋅
Required tools and materials:
• Patty paper (previously used on the Opening Discussion of
Lesson 14)
• Plain white paper
• Compass
[Switch to document camera]
“Draw a circle on a plain white sheet of paper. Use the patty paper with the intersecting lines and slide the paper over the circle so that the lines intersect outside the circle, and both lines are secants to the circle.
Slowly slide the patty paper so that one of the secants approaches the circle until it becomes a tangent to the circle. What relationship exists among the lengths a, b, and c? Explain.
As the secant approaches a position of turning into a tangent, the length of the secant segment in the circle’s interior gets shorter and shorter and approaches a length of zero. At the instant that the secant changes to a tangent, the secant segment inside the circle no longer exists and so has a

length of zero. This then gives the relationship 𝑎(𝑎 + 0) = 𝑏(𝑏 + 𝑐) ,
leading to: 𝑎 2 = 𝑏(𝑏 + 𝑐) or 𝑎 = √𝑏(𝑏 + 𝑐)
2 min 100.
“At the end of Lesson 16, students complete the sections in their graphic organizer that pertain to the relationships of segment lengths of secants intersecting a circle.”
In review: [Click to reveal answers]
• When two secants intersect in a circles interior such that one segment is cut into length a and b, and the other segment cut into lengths c and d by their intersection, then the relationship of a, b, c, and d is 𝑎𝑏 = 𝑐𝑑 . [click to advance]
• When two secants intersect on a circle’s exterior such that one segment is cut into lengths a and b by the intersection with the circle, and the other secant is cut into segments with lengths c and d by the intersection with the circle, then the relationship of
a, b, c, and d is 𝑎(𝑎 + 𝑏) = 𝑏(𝑏 + 𝑐) . [click to advance]
In this section, you will explore deriving the equation for a circle at the origin by analyzing how to find the coordinates of points that lie on a circle.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide
 Participant Handout
 Document Camera
Time Slide
#
Slide #/ Pic of Slide Script/ Activity directions GROUP

2 min 101.
1 min 102.
2 min 103.
[Discuss the focus standards for this Topic and the key concepts that participants can expect to see throughout Topic D.]
“Topic D is comprised of three lessons, shown here”
[Click and read one at a time.]
[Read the title and the objectives of the lesson.]

5 min 104.
0 min 105.
“In this example, we begin with a question that gets students thinking about the definition of a circle.”
[Read the question on the slide and then click to advance.]
“Next we task the students with identifying 8 points that lie on the circle. Four of them are easy to locate, go 5 units up and down the yaxis, and 5 units to the left and right on the x-axis, from the origin. Of course students will identify the other points that appear to lie on the circle. For example, (3,4), (-4, 3), or (-3, -4) and there are others.”
[Click to advance] “We want students to see the relationship between the equation for a circle and the Pythagorean theorem so we ask students to verify that the points that appear on the circle are in fact on the circle. In other words, how can we determine if the length of the radius from the center to a point we named is in fact equal to 5?”
[Click to advance] “Of course! We use the Pythagorean theorem. After verifying a the points lie on the circle we can conclude that a point lies on the circle if it satisfies the equation x^2+y^2=5^2.”
Hidden Slide

5 min 106.
107.
“Do all circles have centers at the origin? No, so we now consider the equation of whose center lies off the origin, in this case at (2,3). This circle has a radius of 5. How does this circle compare to the circle we looked at in Example 1? [Pause for responses]
[Click to advance and reiterate what the participants said, then click to advance again.]
“So are the circles congruent?”
Yes
“That means there exists a congruence that maps one circle onto the other. How can we map this circle onto the circle whose center was at the origin?”
We can translate the plane that contains this circle 2 units right and 3
units down to map it onto the circle from Example 1.

3 min 108.
109.
4 min 110.
“When we translate we are decreasing the x value by 2 units and decreasing the y value by 3 units. We know from Module 1 that all points in the plane are moved in the same way, not just the center.”
[Click to advance] “Therefore if (x,y) is a point on the circle whose center is at (2,3) with radius 5, then the coordinates of all of those points after the translation are located at (x-2, y-3).”
[Click to advance] “Combining this understanding with that from the first example, we write the equation of the circle as (x-2)^2 +(y-3)^2 =
5^2.”
“Still with Example 2, we want to make sure students are clear as to why this form of the equation for a circle is called “center-radius form”.
For that reason we bring students attention to the numbers 2, 3, and 5 in the equation and in the context of the problem.”
[Click to advance] “Now we begin generalizing. What if the center was at (a, b)? What would change in the equation? The 2 and 3 would be replaced by the a and the b.
[Click to advance] “Further generalizing, what if the radius were r? All we need to do is replace the 5 with r.”

1 min 111.
8 min 112.
[Click to advance] “Now we finally have the center-radius form for the equation for an arbitrary circle in the plane with center (a, b) and radius r.”
[Read the title and student outcomes aloud.]
“As with any algebraic relationship, the equation of a circle can take on a variety of forms. In this lesson we focus on how to recognize if a given equation is the equation of a circle or not.”
Required tools and materials:
• Participant Handout
[Direct participants to complete Example 1 shown on the screen. Ask for a participant to share and defend their response.]
By grouping the polynomials and factoring, the quadratic trinomials, you can write the equation in the form (𝑥 − 1) 2 + (𝑦 − 2) 2 = 5 2 , which
reveals a center of (1,2) and a radius of 5.
[Click to reveal Example 2. Direct participants to complete Example 2 shown on the screen. Ask for a participant to share and defend their response.]
By completing the square twice, the equation can be rewritten as
(𝑥 + 2) 2 + (𝑦 − 6) radius of 9.
2 = 81
. This reveals a center point of (-2,6) and a
“The point of these examples is to show students that with some manipulation, the equations can be put in center-radius form and then become easily recognizable as equations of circles.”

5 min 113.
5 min 114.
Required tools and materials:
• Participant Handout
“The ultimate goal is for students to be able to recognize when an equation is, in fact, the equation of a circle.”
“Can the equation be rewritten in center-radius form? How?” –
[Provide 1-2 min for participants to explore this question, then seek volunteers to respond.]
By expanding the left side of the center-radius form, we get 𝑥 2 − 2𝑎𝑥 + 𝑎 2 + 𝑦 2 − 2𝑏𝑦 + 𝑏 2 = 𝑟 2 . In considering that 𝑎, 𝑏, and 𝑟 are constants,
𝐴 = 2𝑎 , 𝐵 = 2𝑏 , and 𝐶 = 𝑎 2 + 𝑏 2 − 𝑟 2 , then the given form of the
equation is in fact an equivalent form for a circle.
“Does the equation necessarily always represent a circle? Why or why not?” [Provide 1-2 min for discussion, then seek participant response]
The sum of two squared numbers cannot be negative, so the value 𝑟 2 ≥ 0 .
If the value of 𝑟 2 = 0 , then the radius of the “circle” is zero, which means the circle is not really a circle, but rather just a single point, (−𝑎, −𝑏) .
Only if the value of 𝑟 2 > 0 does the equation represent a circle.
Required tools and materials:
• Participant Handout
• Document camera
[Switch to the document camera and model the manipulation of the given equation into center-radius form]
“Is there a way to determine if the equation given in this form will or will not represent a circle? Let’s work the opposite way and rewrite this equation in center-radius form.”

6 min 115.
“Complete the square to get (x + center of the circle is (−
A
2
, −
A
)
2
2
+ (y +
B
)
2
2
=
A
2
+B
2
−4c . So the
4
B
) , and the radius of the circle is
2
√
A 2 +B 2 −4C .”
4
“The fractional value on the right side of the equation can be used like the discriminant in that it tells us whether the equation represents an empty set, a single point, or a set of points. If the fractional value is negative, then there are no points (𝑥, 𝑦) that satisfy the equation. If the value of the fraction is zero, then there is only one point that satisfies the equation, (−
𝐴
2
, −
𝐵
) . Finally, if the fraction is a positive value, then
2 the circle exists with center (−
𝐴
2
, −
𝐵
) and radius of √
2
𝐴 2 +𝐵 2 −4𝐶
4
.”
FYI: There’s a lot of work related to completing the square that isn’t shown in the teacher’s materials. It is up to teachers as to how much of the symbolic manipulation students should do.
Required tools and materials:
• Participant Handout
[Read the problem on the slide aloud, then direct participants to complete the problem independently using what was discussed on the last slide. Provide 3 minutes to work, then 2 minutes to discuss, then ask for a volunteer to share their answer]
Sample response: The equation takes the form (𝑥 + 1) 2 + (𝑦 + 2) 2 = 0 , which means that the equation has only one solution,
(−1, −2)
.
Using the fraction from the last slide, its value is
2
2
+4
2
−4(5)
, which when
4 evaluated is 0. So the equation represents a single point,
(−
2
2
, −
4
2
) =
(−1, −2) .

2 min 116.
117.
6 min 118.
[Read the title and the objectives of the lesson.]
“Though students have had some experience before, they may need a moment to consider the validity of the following statement.”
[Read the text on the slide]
“That is, students may need to visualize that there can be two tangent lines to a circle where each has the same slope. They have seen tangent lines that intersect quite a bit, so this is slightly different thinking. Once certain that two tangent lines in fact exist, we have to consider the slope of the radii that are tangent to these lines.”
[Click to advance] “We know from our work in Module 4 that the slope of the radii must be 2.
[Click to advance] “Then using an arbitrary point A on the circle, together with the center, we can write a slope equation equal to 2.

119.
4 min 120.
[Click to advance] “Next we rewrite the slope equation. Notice that we now know what y-5 is equal to. If we square both sides of the equation we get something useful. We can take the value of (y-5)^2 and substitute it into the original equation of the circle. Now we have an equation in one variable that we can solve.”
[Click to advance]
“So when we solved for x we got two answers, which makes sense because we are looking for two points of tangency: one for each line.”
[Click to advance] “We can determine the y-values that go with our solutions for x by substituting, one at a time, into the slope equation we wrote before. This will give us the precise points we are looking for. If we substitute into the equation of the circle we will get multiple solutions which we will then have to tease out to see which ones we need.”
[Click to advance] “We want students to verify that the points we found lie on the circle using its equation.”
[Click to advance] “Finally, with a point and slope known, we can write the equation for each of the tangent lines.”

10 min
121.
Required tools and materials:
• Document camera
• Participant Handout
[Switch to the document camera and go through the example with the participants.]
10 min
122.
“Since there is so much algebraic manipulation in this topic, it seemed appropriate to stop and reflect on how much of the symbolic notation we want to require of our students. Take a moment to review the content of the last three lessons and discuss the following classroom implications at your table”
[Read the bulleted prompts. After allowing time for small group discussion, ask groups to share the comment they felt was most important.]
In this section, you will explore proving that a quadrilateral is cyclic if the opposite angles of the quadrilateral are supplementary.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide
 Participant Handout
 Document camera
Time Slide
#
Slide #/ Pic of Slide Script/ Activity directions GROUP

2 min 123.
1 min 124.
1 min 125.
[Discuss the focus standard for Topic E and the key concepts that participants can expect to see through the remaining lessons in this module.]
“Topic E is comprised of the final two lessons of the module and
Geometry curriculum, shown here.”
[Click to advance and read one at a time].
[Read the title and student outcomes aloud.]

6 min 126.
10 min
127.
Required tools and materials:
• Participant Handout
[Read the Opening Exercise aloud, then point out that we need to develop an understanding of the term cyclic quadrilateral first.]
“What does the term cyclic quadrilateral mean?”
A quadrilateral whose vertices all lie on the same circle.
“What will we need to show that 𝑥 + 𝑦 = 180° per the Opening
Exercise?”
[Provide 3 minutes to develop a proof, then ask for volunteer to share using the document camera]
Sample response:
Inscribed angle theorem. 𝐵𝐴𝐷 and 𝐵𝐶𝐷 are non-overlapping arcs that form a complete circle and so their sum is 360° . Therefore, 𝑚𝐵𝐴𝐷 ̂ 𝑦 =
1 𝑚𝐵𝐶𝐷 by the inscribed angle theorem 𝑦 =
2
1
(360 − 𝑚𝐵𝐴𝐷 by substitution
2 𝑦 = 180 −
1 𝑚𝐵𝐴𝐷 by the distributive property
2 𝑦 = 180 − 𝑥 by substitution 𝑥 + 𝑦 = 180 addition property of equality
[Note that there is NOT enough room in the student materials to record this proof]
“Being direct, what exactly can be assumed here and what do we have to show?” [Ask for volunteer response]
Three of the four vertices of a quadrilateral define a circle. If the quadrilateral is given such that a pair of opposite angles are
supplementary, then the fourth vertex must lie on the same circle.
Proof strategy:

7 min 128.
1 min 129.
Three non-collinear points determine a circle since any triangle can be
inscribed in a circle.
There are three cases for which the fourth vertex of the quadrilateral
might take on.
i. The vertex lies outside the circle
ii. The vertex lies inside the circle
iii. The vertex lies on the circle
We must show that the fourth vertex does not lie outside the circle, nor
inside, to prove that it is a cyclic quadrilateral.
“Start by proving that the first case is impossible.” [Provide 5 minutes for participants to develop a proof that the first case is impossible, then ask for volunteers to share their proof]
[Note that there is NOT enough room in the student materials to record this proof]
“Next, let’s prove that the second case is also impossible.”
[Provide 5 minutes for participants to develop a proof that the second case is impossible, then ask for volunteers to share their proof]
“To complete the reasoning for the proof of the converse, since we’ve shown that case 1 and case 2 are impossible, then it must be the case that 𝐷 lies on the same circle with 𝐴, 𝐵, and 𝐶 .”
“Proving the original theorem and its converse leads us to confirmation of the following: A quadrilateral is cyclic if and only if its opposite angles are supplementary.”

4 min 130.
2 min 131.
10 min
132.
Required tools and materials:
• Participant Handout
• Document camera
[Read the problem on the slide then direct participants to complete the task. Ask for a volunteer to share their work on the document camera]
“In Exercises 4 and 5, students revisit the area formula for an acute triangle that they experienced with trigonometry at the end of Module
2. In Exercise 5, however, they extend their understanding of the area formula as it is applied to an obtuse triangle. This setup will be necessary for Exercise 6 and Lesson 21 on Ptolemy’s Theorem.”
Required tools and materials:
• Document camera
• Participant Handout
“Using their discovery from Exercises 4-5, students can now develop an area formula for a cyclic quadrilateral in terms of its diagonals. Use skills that have been studied recently and throughout the year to show the relationship shown on the screen.”
[Provide 6-8 min for participants to work and discuss the problem, then ask for a volunteer to share their reasoning using the document camera]
Sample response:
Decompose the cyclic quadrilateral along its diagonal into 4 triangles.
Two of these triangles have acute angle 𝑤 and the other two triangles are obtuse with the supplement of angle 𝑤 , which is (180 − 𝑤)° . The areas

2 min 133.
of the individual triangles are then
1
2 and 1
2 𝑏𝑑 sin(𝑤) . 𝑎𝑑 sin(𝑤) ,
1
2 𝑎𝑐 sin(𝑤) ,
The sum of the areas is the area of the quadrilateral, so
1
𝐴𝑟𝑒𝑎 𝑡𝑜𝑡𝑎𝑙
=
2 𝑎𝑑 sin(𝑤) +
𝐴𝑟𝑒𝑎 𝑡𝑜𝑡𝑎𝑙
=
1
1
2 2 sin 𝑤 (𝑎𝑑 + 𝑎𝑐 + 𝑏𝑐 + 𝑏𝑑)
2
1 𝑎𝑐 sin(𝑤) +
1 𝑏𝑐 sin(𝑤) +
𝐴𝑟𝑒𝑎 𝑡𝑜𝑡𝑎𝑙
=
2 sin(𝑤) (𝑎(𝑑 + 𝑐) + 𝑏(𝑐 + 𝑑))
1
2
1
𝐴𝑟𝑒𝑎 𝑡𝑜𝑡𝑎𝑙
=
2 sin(𝑤) ((𝑎 + 𝑏)(𝑐 + 𝑑))
1
2 𝑏𝑐 sin(𝑤) 𝑏𝑑 sin(𝑤)
,
[Read the title and the objectives of the lesson.]
20 min
134.
Required tools and materials:
• Document camera
• Participant Handout
[Provide the description of Ptolemy’s theorem with the information on the slide.]
[Switch to the document camera and go through the exploratory challenge with the participants.]

In this section, you will complete an End of Module Assessment and participate in a follow-up discussion.
Materials used include:
 Geometry Module 5 PPT
 Geometry Module 5 Facilitator’s Guide
 End-of-Module Assessment Handout
 End-of-Module Assessment Sample Work
 Progressions Toward Mastery Rubric
Time Slide
#
Slide #/ Pic of Slide Script/ Activity directions GROUP
30 min
135.
3 min 136.
Required tools and materials:
• End-of-Module Assessment
• Progressions Toward Mastery Rubric
• End-of-Module Assessment Sample Work
[Provide 25-30 minutes for participants to complete the End-of-Module
Assessment. Then conduct a 5-10 minute discussion about the lessons from which the assessment items stem from, the provided sample work, scaffolding issues, use of items as Topic C,D quiz questions, etc.]
[Advance through the bulleted points to summarize the key points and themes of Module 5.]

2 min 137.
“Take a few minutes to reflect on this session. You can jot your thoughts on your copy of the PowerPoint. What are your biggest takeaways?”
[Click to advance] “Now, consider specifically how you can support successful implementation of these materials at your schools given your role as a teacher, school leader, administrator or other representative.
Use the following icons in the script to indicate different learning modes.
Video Reflect on a prompt Active learning Turn and talk
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Geometry Module 5 PPT
Geometry Module 5 Facilitator’s Guide
Participant Handout
Mid-Module Assessment Handout
End-of-Module Assessment Handout
End-of-Module Assessment Sample Work
Progressions Toward Mastery Rubric
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How to Implement A Story of Functions
A Story of Functions Year Long Curriculum Overview