N YS CO MMO N COR E MAT

N YS CO MMO N COR E MAT

N YS CO MMO N COR E MAT H E MAT I C S C U RR I C U LU M A Story of Functions A Story of Functions Grade 10 Geometry Module 5 Circles With and Without Coordinates 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Participant Poll

Classroom teacher Math trainer or coach Principal or school leader District representative / leader Other 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Session Objectives Participants will understand the development of mathematical concepts surrounding circles and line segments that intersect circles. Participants will enrich their knowledge and experience in order to implement Module 5 with confidence and success. 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Agenda Session 1 Module Overview and Topic A Session 2 Complete Topic A and begin Topic B Session 3 Complete Topic B and the Mid-Module-Assessment Session 4 Topics C and D

Session 5 Complete Topics D, E, and the End-of-Module Assessment 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Module 5: Circles With and Without Coordinates This module is allotted 25 instructional days. Topic A: Central and Inscribed Angles (G-C.A.2, G-C.A.3) Topic B: Arcs and Sectors (G-C.A.1, G-C.A.2, G-C.B.5) Topic C: Secants and Tangents (G-C.A.2, G-C.A.3) Topic D: Equations for Circles and Their Tangents (G-GPE.A.1, G-GPE.B.4)

Topic E: Cyclic Quadrilaterals and Ptolemys Theorem (G-C.A.3) 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Module 5: Circles With and Without Coordinates Topic A: Central and Inscribed Angles (6 lessons) Thales theorem, general inscribed-central angle theorem Students solve unknown angle problems. Students inscribe triangles and rectangles in circles and study their properties. Topic B: Arcs and Sectors (4 lessons)

Relationships between chords, diameters, and angles are established. Students develop a formula for arc length and area of a sector. Students practice their skills by solving unknown area problems. Mid-Module Assessment (2 days) 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Module 5: Circles With and Without Coordinates Topic C: Secants and Tangents (6 lessons) Secant angle theorems, tangent-secant angle theorems. Students explore relationships in diagrams of secant and tangent lines. Students use similar triangles to discover length relationships in diagrams. Topic D: Equations for Circles and Their Tangents (3 lessons) Students develop the equation for a circle using the Pythagorean theorem and translation. Topic E: Cyclic Quadrilaterals and Ptolemys Theorem (2 lessons)

Focus on properties of quadrilaterals inscribed in circles. Bring together congruence and similarity, the Pythagorean theorem, facts about triangles, and trigonometry. End-of-Module Assessment (2 days) 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Topic A: Central and Inscribed Angles Standards G-C.A.2, G-C.A.3: Identify and describe relationships among inscribed angles, radii, and chords, construct inscribed and circumscribed circles of a triangle, and prove properties of angles for inscribed quadrilaterals. Key Concepts 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.

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Topic A: Central and Inscribed Angles Lesson 1: Lesson 2: Lesson 3: Lesson 4: Lesson 5: Lesson 6: 2015 Great Minds. All rights reserved. greatminds.net

Thales Theorem Circles, Chords, Diameter, and Their Relationships Rectangles Inscribed in Circles Experiments with Inscribed Angles Inscribed Angle Theorem and Its Applications Unknown Angle Problems with Inscribed Angles in Circles N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 1: Thales Theorem Using observations from a pushing puzzle, students explore the converse of Thales theorem: If triangle ABC is a right

triangle, then A, B, and C are three distinct points on a circle with a diameter . Students prove the statement of Thales theorem: If A, B, and C are three different points on a circle with a diameter then angle ABC is a right angle. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Opening Exercise Paper push a right angle between two fixed points to find points on a circle.

What curve is traced? Where is the center of the semicircle? What is the radius? Can we prove that all of the points (C, D, E) lie on the circle? Lesson 1

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenge Lesson 1 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 1: Proving Thales Theorem

Draw circle P with distinct points A, B, and C on the circle and diameter . Prove that is a right angle. Lesson 1 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exercise 2 Lesson 1 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Lesson 2: Circles, Chords, Diameters and Their Relationships Students identify the relationships between the diameters of a circle and other chords of the circle. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Opening Exercise and Discussion Given , construct its perpendicular bisector. Draw a different line bisecting segment . What are some similarities and differences of the bisectors? DrawEQUIDISTANT a circle of :any radiusisand Draw a chord thedifferent circle and label its and . A point saidcenter to be .equidistant fromontwo

points andendpoints if . Points and can be replaced in the definition above with other figures (lines, etc.) as long as the distance to the figures is Construct the perpendicular bisector of chord . What do you observe? given meaning first. In this lesson, we will define the distance from the center of a circle to a chord. This definition will allow us to talk about the center of a circle as being equidistant from two chords. Draw chord on the circle and construct the perpendicular bisector of chord . What do you observe? Lesson 2 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Students prove in groups 1. 2. 3. If a diameter of a circle bisects a chord, then it must be perpendicular to the chord. If a diameter of a circle is perpendicular to a chord, then it bisects the chord. In a circle, if two chords are congruent, then the center of the circle is equidistant from the two chords. 4. In a circle, if two chords are equidistant from the center, then the two chords are congruent.

5. In a circle, congruent chords define central angles that are equal in measure. 6. In a circle, if two chords define central angles that are equal in measure, then they are congruent. Lesson 2 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Lesson 3: Rectangles Inscribed in Circles Students inscribe a rectangle in a circle. Students understand the symmetries of inscribed rectangles across a diameter. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Opening Exercise Using only a compass and straight edge, follow the steps below to find the center of a given circle. Lesson 3

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenge Construct a rectangle that is inscribed in a given circle. How can we use the construction in the Opening Exercise to accomplish this goal? What do we know about the properties of rectangles? Lesson 3

2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenge Construct a rectangle that is inscribed in a given circle. Strategy #1: We know that the diagonals of a rectangle are equal in length and bisect each other.

The diameters/hypotenuses satisfy this criteria so their endpoints are the vertices of an inscribed rectangle. Lesson 3 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenge Construct a rectangle that is inscribed in a given circle. Strategy #2:

A rectangle is composed of two congruent right triangles. A congruence consisting of a 180 rotation about the midpoint of the hypotenuse maps one triangle onto the other. Rotate one of your triangles about the center of the circle 180. Students may find alternative strategies. Lesson 3 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exercises 1 5 Take a moment to review exercises 1 5. It is in these exercises that students consider the symmetries on inscribed rectangles across a diameter. Note that Exercise 5 is a challenge and can either be assigned to advanced learners or covered as a teacher-led example. Lesson 3 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 4: Experiments with Inscribed Angles Students explore the relationship between inscribed angles and central angles and their intercepted arcs. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenges 1-4 1.

2. 3. 4. Paper pushing exercise. Paper pushing exercise. Comparing inscribed angles. Comparing inscribed angles to a central angle. Lesson 4 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Lesson 5: Inscribed Angle Theorem and Its Applications Students prove the inscribed angle theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Students recognize and use different cases of the inscribed angle theorem embedded in diagrams. This includes recognizing and using the result that inscribed angles that intersect the same arc are equal in measure. 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. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Opening Exercises Lesson 5 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Examples 1 and 2 Based on our observation from the Opening Exercise, the following theorem can be developed: Theorem: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Case 1: The vertex of the central angle lies on a side of the inscribed angle. This was proven in the Opening Exercise. Lesson 5 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Examples 1 and 2 Based on our observation from the Opening Exercise, the following theorem can be developed: THEOREM: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle. Case 2: The vertex of the central angle lies in the interior of the inscribed angle. Case 3: The vertex of the central angle lies in the exterior of the inscribed angle. Lesson 5 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Lesson 6: Unknown Angle Problems with Inscribed Angles in Circles Students use the inscribed angle theorem to find the measures of unknown angles. Students prove relationships between inscribed angles and central angles. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Opening Exercise In a circle, a chord and a diameter are extended outside of the circle to meet at a point . If , and , find Lesson 6 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge THEOREM: If , and are four points with and on the same side of line , and and are congruent, then , and all lie on the same circle.

Lesson 6 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge THEOREM: If , and are four points with and on the same side of line , and and are congruent, then , and all lie on the same circle. First lets consider the case where is outside the circle. Label the intersection of and the circle point . Angles and are inscribed angles that intercept the same arc, so both angles have the same measure, .

Lesson 6 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge THEOREM: If , and are four points with and on the same side of line , and and are congruent, then , and all lie on the same circle. Angle in triangle has degree measure . By the exterior angle theorem, . Using substitution, . However since , this is a contradiction, so cannot lie outside the circle. Lesson 6

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge THEOREM: If , and are four points with and on the same side of line , and and are congruent, then , and all lie on the same circle. Next lets consider the case where lies in the interior of the circle. Extend and marks its intersection with the circle . Angles and are both inscribed angles that intercept the same arc, so their measures are equal, . Lesson 6 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge THEOREM: If , and are four points with and on the same side of line , and and are congruent, then , and all lie on the same circle. In triangle , angle has degree measure . Using the exterior angle theorem, . Using substitution, . However since , this is a contradiction, so cannot lie in the circle. Since cannot lie outside the circle, nor can it lie inside the circle, must lie on the circle, and thus, the theorem is proved.

Lesson 6 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Topic B: Arcs and Sectors Standards G-C.A.1, G-C.A.2, G-C.B.5: Prove that all circles are similar, identify and describe relationships among inscribed angles, radii, and chords. Use similarity to define arc length, define radian measure, derive area formula for sectors.

Key Concepts Properties of congruence, similarity, and isosceles triangles are applied to proofs; congruent chords lie in congruent arcs, all circles are similar. Students extend the protractor axiom (angles add) to arcs. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Topic B: Arcs and Sectors Lesson 7: The Angle Measure of an Arc Lesson 8: Arcs and Chords Lesson 9: Arc Length and Area of Sectors

Lesson 10: Unknown Length and Area Problems 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 7: The Angle Measure of an Arc Define the angle measure of arcs, and understand that arcs of equal angle measure are similar. Restate and understand the inscribed angle theorem in terms of arcs: The measure of an inscribed angle is half the angle measure of its intercepted arc.

Only in this lesson is the theorem stated in full: The measure of an inscribed angle is half the angle measure of its intercepted arc. Explain and understand that all circles are similar. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Discussion and Example 1: Jigsaw Think about how you would deliver the content in the discussion/example in your classroom. Be prepared to share key points and instructional tips with the whole group (2 3 min per part).

Group 1: Group 2: Group 3: Group 4: Part 1 of Discussion Part 2 of Discussion Part 3 of Discussion Example 1 Lesson 7 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Lesson 8: Arcs and Chords Students know that congruent chords have congruent arcs, and that the converse is also true. Students know that arcs between parallel chords are congruent. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenge

Prove that congruent chords have congruent arcs. Lesson 8 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 1 THEOREM: In a circle, arcs between parallel chords are congruent. Prove this theorem in two ways. Lesson 8 2015 Great Minds. All rights reserved. greatminds.net

A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 9: Arc Length and Area of Sectors When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector. ARC: An arc is any of the following three figuresa minor arc, a major arc, or a semicircle. LENGTH OF AN ARC: The length of an arc is the circular distance around the arc. MINOR AND MAJOR ARC: In a circle with center O, let A and B be different points that lie on the circle but are not the endpoints of a diameter. The minor arc between A and B is the set containing A, B, and all points of

the circle that are in the interior of AOBAOB. The major arc is the set containing A, B, and all points of the circle that lie in the exterior of AOBAOB. RADIAN: A radian is the measure of the central angle of a sector of a circle with arc length of one radius length. SECTOR: Let AB be an arc of a circle with center O and radius r. The union of the segments OP, where P is any point on the AB, is called a sector. AB is called the arc of the sector, and r is called its radius. SEMICIRCLE: In a circle, let A and B be the endpoints of a diameter. A semicircle is the set containing A, B, and all points of the circle that lie in a given half-plane of the line determined by the diameter. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 1 Lesson 9

2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 1: Discussion Lesson 9 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 1: Discussion

Lesson 9 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 2 In this example, the formula to find the area of a sector is developed. Lesson 9 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 10: Unknown Length and Area Problems Students apply their understanding of arc length and area of sectors to solve problems of unknown length and area. 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

Opening Exercise In the following figure, a cylinder is carved out from within another cylinder of the same height; the bases of both cylinders share the same center. a. Sketch a cross section of the figure parallel to the base. This figure is referred to as an annulus. Lesson 10 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Opening Exercise

In the following figure, a cylinder is carved out from within another cylinder of the same height; the bases of both cylinders share the same center. b. Mark and label the shorter of the two radii as and the longer . Show how to calculate the area of the shaded region, and explain the parts of the expression. Lesson 10 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exercises 1-13

Take a few minutes to review and complete some of the exercises in this lesson. The problems vary in level of difficulty and can be used to differentiate in your classroom. Lesson 10 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Mid-Module Assessment

Take a few moments to review the problems provided on the Mid-Module Assessment Task. Choose one problem to complete in its entirety. Compare your response to the provided sample work. Compare your response to the Mastery Rubric. What student response(s) would demonstrate STEP 1 in mastery? STEP 2? STEP 3? STEP 4? Where do you feel your response lies?

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Topic C: Secants and Tangents Standards G-C.A.2, G-C.A.3: Identify and describe relationships among inscribed angles, radii, and chords. Construct the inscribed and circumscribed circles of a triangle, prove properties of angles for an inscribed quadrilateral. Key Concepts

Students 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. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Topic C: Secants and Tangents Lesson 11: Lesson 12:

Lesson 13: Lesson 14: Lesson 15: Lesson 16: 2015 Great Minds. All rights reserved. greatminds.net Properties of Tangents Tangent Segments The Inscribed Angle Alternate A Tangent Angle Secant Lines; Secant Lines That Meet Inside a Circle Secant Angle Theorem, Exterior Case Similar Triangles in Circle-Secant (or Circle-SecantTangent) Diagrams. N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Lesson 11: Properties of Tangents Students discover that a line is tangent to a circle at a given point if it is perpendicular to the radius drawn to that point. Students construct tangents to a circle through a given point. Students prove that tangent segments from the same point are equal in length.

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Opening Sets the stage for Topic C and presents the vocabulary secant line, tangent line and tangent segment. Students draw a circle and line, teacher groups drawings and discusses.

Lesson 11 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge Students draw a circle and a tangent line and note the point of tangency as P. Draw a circle and a radius intersecting the circle at a point P. Construct a line through P, perpendicular to the radius. Students measure the angle formed by the radius and tangent and compare findings with neighbors.

A tangent line to a circle is perpendicular to the radius of the circle drawn to the point of tangency. If a line through a point on a circle is perpendicular to the radius drawn to that point, the line is tangent to the circle. Lesson 11 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Example 1 Students use constructions to show that tangent segments from the same point are equal in length. Draw a point R, exterior to the circle. Construct a line through point R tangent to the circle. Draw AR and find its midpoint, M.

Draw an arc of radius MA. Label points of intersection with circle as B and C. Verify perpendicularity. What is true about MB, MA, MR and MC? Lesson 11 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

Exercise 1 Lesson 11 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 12: Tangent Segments Students use tangent segments and radii of circles to conjecture and prove geometric statements, especially those that rely on the congruency of

tangent segments to a circle from a given point. Students recognize and use the fact if a circle is tangent to both rays of an angle, then its center lies on the angle bisector. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 1 In each diagram, try to draw a circle with center that is tangent to both rays of .

Why did this make What is special about a(b)difference? that was not true for (a) and (c)? D was in the of theof angle

in (b). then the center of the circle lies on the angle bisector. Conjecture: IfPoint a circle is more tangent tomiddle both rays an angle, Lesson 12 2015 Great Minds. All rights reserved. greatminds.net Point D is on the angle bisector. N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

Exercise 1 Given a circle with center and tangent to and ; ; ; show . Lesson 12 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exercise 2 An angle is show below. a. Draw at least three different circles that are tangent to both rays of the

given angle. b. How does the distance between the centers of the circles and the rays of the angle compare to the radius of the circle? Lesson 12 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exercise 3 Construct as many circles as you can that are tangent to both the given angles at the same time. You can extend the rays as needed. These two angles share a side.

Lesson 12 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exercise 4 In a triangle, let be the location where two angle bisectors meet. Must be on the third angle bisector as well? Explain your reasoning. Lesson 12 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Lesson 13: The Inscribed Angle Alternate A Tangent Angle Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is tangent). Students solve a variety of missing angle problems using the inscribed angle theorem. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Exploratory Challenge Connects to Lesson 12 Problem Set about the relationship between the measure of an arc and an angle. Goal of challenge is to establish and prove the following conjecture: An inscribed angle formed by a secant and tangent line is half of the angle measure of the arc it intercepts. Lesson 13 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example

Now we develop another theorem in the family of inscribed angle theorems: the angle formed by the intersection of the tangent line and a chord of the circle on the circle, and the inscribed angle of the same arc, are congruent. The exercises that follow this example provide time for students to practice applying the inscribed angle theorem to solve missing angle problems. Lesson 13 2015 Great Minds. All rights reserved. greatminds.net

Click re N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 14: Secant Lines; Secant Lines That Meet Inside a Circle Students understand that an angle whose vertex lies in the interior of a circle intersects the circle in two points and that the edges of the angles are contained within two secant lines of the circle.

Students discover that the measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Discussion Draw a circle on a plain sheet of paper and draw two intersecting lines on a clean sheet of patty paper. Lesson 14

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exercise 1 In circle , is a radius and . Find and explain how you know. Lesson 14 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exercise 2 In the circle shown, . Find and . Explain your answer.

Lesson 14 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 1 SECANT ANGLE THEOREM INTERIOR CASE: The measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle. Lesson 14

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 15: Secant Angle Theorem, Exterior Case Students find the measures of angles, arcs, and chords in figures that include two secant lines meeting outside a circle, where the measures must be inferred from other data. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

Opening Exercise 2 Lesson 15 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Opening Exercise 2 Lesson 15 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenge Lesson 15 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example Lesson 15 2015 Great Minds. All rights reserved. greatminds.net

A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Lesson 15 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Closing Lesson 15 2015 Great Minds. All rights reserved. greatminds.net

A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 16: Similar Triangles in Circle-Secant (or Circle-Secant-Tangent) Diagrams Students find missing lengths in circle-secant or circle-secant-tangent diagrams. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Exploratory Challenge 1 Measure the lengths of the chords in each circle (in centimeters) and record the values in the table. Lesson 16 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge 2 Measure the lengths of the chords in centimeters and record them in the

table. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge 3 Prove mathematically why the relationships that we found in Exploratory Challenges 1 and 2 are valid. Lesson 16 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Exploratory Challenge 3 Prove mathematically why the relationships that we found in Exploratory Challenges 1 and 2 are valid. Lesson 16 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exploratory Challenge 3 Lets extend this reasoning to a tangent line. Lesson 16

2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Closing Lesson 16 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions

Topic D: Equations for Circles and Their Tangents Standards G-GPE.A.1, G-GPE.A.4: Derive the equation of a circle of given center and radius using the Pythagorean theorem, complete the square to find the center and radius of a circle given by an equation. Use coordinates to prove simple geometric theorems algebraically. Key Concepts Students 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. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Topic D: Equations for Circles and Their Tangents Lesson 17: Writing the Equation for a Circle Lesson 18: Recognizing Equations of Circles Lesson 19: Equations for Tangent Lines to Circles 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 17: Writing the Equation for a Circle Students write the equation for a circle in center-radius form, using the Pythagorean theorem or the distance formula. Students write the equation of a circle given the center and radius. Students identify the center and radius of a circle given the equation. 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 1 If we graph all of the points whose distance from the origin is equal to 5 units, what shape is formed? Find the coordinates of 8 points that lie on the circle. How can we verify that the points that appear to lie on the circle actually do?

We can conclude that a point (x, y) lies on the circle if the point satisfies the equation: Lesson 17 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 2 Lesson 17 2015 Great Minds. All rights reserved. greatminds.net

A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 2 Lesson 17 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Lesson 17 2015 Great Minds. All rights reserved. greatminds.net

A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 18: Recognizing Equations of Circles Students complete the square in order to write the equation of a circle in center-radius form. Students recognize when a quadratic in x and y is the equation for a circle.

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Examples 1 and 2 1. The following is the equation of a circle with a radius of 5 and center (1,2). Explain why. 2. What is the center and radius of the following circle? Lesson 18 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Example 3 Could an equation of the form represent a circle? To answer this question, consider the center-radius form of a circle . Lesson 18 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 3

How do we determine if an equation of the form represents a circle? Lesson 18 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 3 Does the equation represent a circle? If so, identify the center and radius. If not, explain why. Lesson 18 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 19: Equations for Tangent Lines to Circles Given a circle, students find the equations of two lines tangent to the circle with specified slopes. Given a circle and a point outside the circle, students find the equation of the line tangent to the circle from that point. 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 1 Lesson 19 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 1 Lesson 19 2015 Great Minds. All rights reserved. greatminds.net

A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Example 2 Lesson 19 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Topic D Now that youve seen the content of Topic D,

consider the following classroom implications: Which algebra skills are required in this topic? How can you prepare students for the symbolic manipulation in this topic? What scaffolds do you feel would be appropriate to aide students understanding? How much work will you require of your students this year? Next year? Subsequent years? 2015 Great Minds. All rights reserved. greatminds.net

A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Topic E: Cyclic Quadrilaterals and Ptolemys Theorem Standards G-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Key Concepts

Students 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., Ptolemys theorem. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Topic E: Cyclic Quadrilaterals and Ptolemys Theorem Lesson 20: Cyclic Quadrilaterals

Lesson 21: Ptolemys Theorem 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Lesson 20: Cyclic Quadrilaterals Students show that a quadrilateral is cyclic if and only if its opposite angles are supplementary.

Students derive and apply the area of cyclic quadrilateral ABCD as (AC) (BD)(sin(w)), where w is the measure of the acute angle formed by diagonals AC and BD. Students first saw this area formula in Module 2 when trigonometric functions were introduced. 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Opening Exercise Given cyclic quadrilateral in the image below, prove that . What does the term cyclic quadrilateral mean? Lesson 20

2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 1 Prove the converse: Given a quadrilateral with one pair of opposite angles being supplementary, prove that the quadrilateral is cyclic. Case 1 Lesson 20

2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exercise 1 Prove the converse: Given a quadrilateral with one pair of opposite angles being supplementary, prove that the quadrilateral is cyclic. Case 2 Lesson 20 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Example 1 and Exercise 1 Prove the converse: Given a quadrilateral with one pair of opposite angles being supplementary, prove that the quadrilateral is cyclic. Case 1 Case 2 We can conclude that must be on the circle. Lesson 20 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Exercise 3 A cyclic quadrilateral has perpendicular diagonals. What is the area of the quadrilateral in terms of and . Lesson 20 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M Exercises 4-5 4. Show that the triangle in the diagram has area . 5. Show that the triangle with obtuse angle has area .

Lesson 20 2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exercise 6 6. Show that the area of the cyclic quadrilateral shown in the diagram is . Lesson 20 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M

A Story of Functions Lesson 21: Ptolemys Theorem Students determine the area of a cyclic quadrilateral as a function of its side lengths and the acute angle formed by its diagonals. Students prove Ptolemys theorem, which states that for a cyclic quadrilateral ABCD, (AC)(BD) = (AB)(CD) + (BC)(AD). Students explore applications of the result. 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Exploratory Challenge A Greek mathematician Claudius Ptolemy discovered a relationship between the side lengths and the diagonals of any quadrilateral inscribed in a circle. Ptolemys theorem says that for a cyclic quadrilateral ABCD, (AC)(BD) = (AB)(CD) + (BC)(AD)

Lesson 21 2015 Great Minds. All rights reserved. greatminds.net N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M End-of-Module Assessment Do the problems in the End-of-Module Assessment As you work, think about the following: Which lesson(s) does this assessment item tie to? Is there vocabulary that students may struggle with? Can this item be used as part of a quiz for Topic C or Topic D?

2015 Great Minds. All rights reserved. greatminds.net A Story of Functions N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Key Themes of Module 5: Circles With and Without Coordinates

Many skills and concepts learned in previous modules have been revisited. Circles bring about a wealth of new and interesting geometric relationships among figures such as angles, segments, and inscribed polygons. These relationships are rooted in two intersecting lines and the different cases in which those lines intersect a circle. Thales theorem provides a necessary entry point to unveil these vast relationships. Once established on the plane, the circle itself and its relationships can be examined algebraically under coordinate systems. These relationships, together with other previously studied skills, can help us to recognize and prove even greater relationships such as Ptolemys theorem. 2015 Great Minds. All rights reserved. greatminds.net

N YS C O M M O N C O R E M AT H E M AT I C S C U R R I C U L U M A Story of Functions Biggest Takeaway What is your biggest takeaway with respect to Module 5? How can you support successful implementation at your school/s given your role? 2015 Great Minds. All rights reserved. greatminds.net

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