SFUSD Mathematics Core Curriculum Development · PDF fileSFUSD Mathematics Core Curriculum Development Project ... Unit Objectives ... and area of circles to arcs and sectors using

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SFUSD Mathematics Core Curriculum, Grade 10, Unit G.6: Circles and Conics, 2014–2015

SFUSD Mathematics Core Curriculum Development Project

2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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Geometry

G.6 Circles and Conics

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Similar Circles (2 pages) 1 per pair Graph paper 4 Lesson Series 1 DG Lesson 3.7 (4 pages)

DG Lesson 3.7 Homework (2 pages) DG Lesson 6.1 Tangent Properties (4 pages) DG Lesson 6.1 Homework (2 pages) DG Lesson 6.2 Chord Properties (4 pages) DG Lesson 6.2 Homework (2 pages) DG Lesson 6.3 Arcs and Angles (2 pages) DG Lesson 6.3 Homework (2 pages)

1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per student

Compasses and straightedges Protractors

2 Apprentice Task Inscribing and Circumscribing a Right Triangle Circle Theorems Sample Responses to Discuss How Did You Work?

1 per student 1 per pair 1 per pair 1 per student

Compasses Rulers Protractors

3 Lesson Series 2 CPM CCG Lesson 10.1.2 (4 pages) DG Lesson 10.1.2 Homework (2 pages) DG Lessons 6.7 and 8.6 Finding the Arcs (2 pages) DG Lessons 6.7 and 8.6 Homework (2 pages) What Is a Radian? (2 pages)

1 per pair 1 per student 1 per pair 1 per student 1 per pair

Patty paper Compasses and straightedges String (cut into various lengths)

2 Expert Task Sectors of Circles (2 pages) Circles Sectors of Circles (revisited) (2 pages)

1 per student 1 per student 1 per student

Card Sets (1 per group) Glue sticks

4 Lesson Series 3 CPM CCG Lesson 12.1.1 (2 pages) CPM CCG Lesson 12.1.1 Homework (2 pages) CPM CCG Lesson 12.1.2 (4 pages) CPM CCG Lesson 12.1.2 Homework (2 pages) CPM CCG Lesson 12.1.3 (2 pages) Lesson 12.1.3 Resource Page CPM CCG Lesson 12.1.3 Homework CPM CCG Lesson 12.1.4 (3 pages) Lesson 12.1.4 Resource Page (2 pages) CPM CCG Lesson 12.1.4 Homework (2 pages)

1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per pair 1 per student 1 per pair 1 per student 1 per student

Compasses Graph paper Patty paper Scissors and tape or glue Dynamic Tool: Parabola Computer and projector

1 Milestone Task Circles and Conics Assessment (2 pages) 1 per student

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Unit Overview

Big Idea

Theorems about circles can be understood and applied through constructions and on the coordinate plane.

Unit Objectives

• Students explore, understand, and prove basic theorems about circles. • Students extend previous understanding of similar polygons to circles using translations and dilations. • Students use coordinate geometry to derive the equations of a circle and parabola.

Unit Description

This unit looks at circles from the perspectives of both Euclidean and coordinate geometry. Students explore and prove Euclidian theorems about circles, extend their understanding of circumference and area of circles to arcs and sectors using central angles measured in both degrees and radians, and use the Pythagorean Theorem to derive the equation of a circle as the locus of points in the xy-plane that are the same distance from a fixed point. Finally, students will extend their previous understandings of quadratic functions from Algebra I as well as the new idea of a locus to develop the formula of a parabola as the locus of points in the xy-plane that are the same distance from a fixed point and line.

CCSS-M Content Standards

Circles G-C Understand and apply theorems about circles

G.C.1 Prove that all circles are similar.

G.C.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Find arc lengths and areas of sectors of circles

G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angles as the constant of proportionality; derive the formula for the area of a sector. Expressing Geometric Properties with Equations G-GPE

G.GPE.1 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.

G.GPE.2 Derive the equation of a parabola given a focus and directrix.

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Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

The unit assumes knowledge of the Pythagorean Theorem and distance formula, similarity, writing quadratic functions in vertex form, circumference and area of circles. Completing the square should also have been addressed in Algebra I, but for many students, this unit may be their first exposure.

Theorems about circles can be understood and applied through constructions and on the coordinate plane. Analytic Geometry connects algebra and geometry, resulting in powerful methods of analysis and problem solving. Geometric shapes can be described by equations, making algebraic manipulation into a tool for geometric understanding, modeling, and proof. Geometric transformations of the graphs of equations correspond to algebraic changes in their equations.

High School Geometry In order for students to model situations in geometry involving conics, they must first have a deep understanding of conics and their properties. Soon, students will use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G-MG 1. Pre-Calculus/Calculus/Physics Eventually, students will derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of the distances from the foci is constant. G-GPE 3. Students will also construct a tangent line from a point outside a given circle to the circle in order to more fully understand the concepts of velocity and motion in physics.

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Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

1 day 4 days 2 days 3 days 2 days 4 days 1 day

Total: 17 days

Lesson Series 1

Lesson Series 2

Lesson Series 3

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Entry Task Similar Circles

Apprentice Task Inscribing and Circumscribing

Right Triangles

Expert Task Sectors of Circles

Milestone Task Circles and Conics Unit Test

CCSS-M Standards

G.C.1 G.C.2, G.C.3 G.C.5 G.C.1, G.C.2, G.C.3, G.C.5 G.GPE.1, G.GPE.2

Brief Description of Task

Given two ordered pairs for the centers of two distinct circles as well as another ordered pair for each representing a point on the circle, students will perform translations and a dilation to show that the circles are similar.

Students will use circle theorems to describe relationships between right triangles and the radii of their inscribed and circumscribed circles.

Students will match sectors of different circles based on a shared measurement: arc length, area, and/or radian measure.

Performance assessment.

Source CPM Supplement “Similar Circles” Shell Centre FAL #4 Shell Centre FAL #23 SFUSD teacher-created

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

G.C.1, G.C.2, G.C.3 G.C.5 G.CPE.1, G.CPE.2

Brief Description of Lessons

Students will use constructions to explore circle theorems for inscribed angles, chords, and tangents. They will then prove these theorems.

Students will explore Eratosthenes’ technique for finding the circumference and radius of the Earth. Following a PowerPoint presentation that introduces/recaps circle vocabulary and builds an intuitive understanding of arc length and sector area, students will develop arc length and area of sector formulas using degrees. Finally, students will explore radian measure and apply it to arc length and area problems.

Students will use the distance formula to derive the equation of a circle. Students will then use patty paper to explore the set of points equidistant from a point and a line, and then derive the equation of a parabola using the focus and directrix.

Sources Discovering Geometry, 4th Edition, Lessons 6.1, 6.2, 6.3 and 6.4

YouTube video of Carl Sagan discussing Eratosthenes Circle Arcs and Sectors PowerPoint CPM Core Connections Geometry, Lesson 10.1.2 Discovering Geometry, 4th Edition, Lessons 6.7 and 8.6 What is a Radian? worksheet modified from Project Maths (Ireland) http://www.projectmaths.ie/documents/

CPM Core Connections Geometry, Lessons 12.1.1, 12.1.2, 12.1.3 and 12.1.4

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Entry Task

Similar Circles

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will extend their understanding of similarity to circles.

CCSS-M Standards Addressed: G.C.1 Potential Misconceptions:

Launch: Give a warm-up asking students to show that two polygons in the plane are similar. Tell the class that they will be using the same technique on circles. During: Make sure that groups are justifying the similarity of two circles using complete sentences. Not all problems must be completed if students are showing understanding. Encourage generalization. Closure/Extension: Formalize the class’s findings: We know that all circles are similar because________________________. Ask fast groups to describe the translation and dilation for circle A with center (h, k) and radius r1 mapped to circle B with center (p, q) and radius r2.

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Similar Circles

How will students do this?

Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: translation, dilation

Sentence frames: Circles C and D are similar because circle ____ can be obtained by ____ and ____ circle ____.

Differentiation Strategies: Participation Structures (group, partners, individual, other): Groups of 3 to 4 students

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Lesson Series #1 Lesson Series Overview: Students will use constructions to explore circle theorems for inscribed angles, chords, and tangents. They will then prove these theorems. CCSS-M Standards Addressed: G.C.1, G.C.2, G.C.3 Time: 4 days

Lesson Overview - Day 1 Resources

Description of Lesson: Students will use basic constructions (angle bisector and perpendicular bisector) to explore inscribed and circumscribed circles. Notes: Be sure to have students discuss why the constructions work, either in teams or in a whole class discussion.

Discovering Geometry, 4th Edition Lesson 3.7, Investigations 2 and 3, pp. 179–181

Suggested homework: Lesson 3.7, pp. 182–183, Exercises 1-9, 14


Description of Lesson: Students will investigate properties of tangents to a circle and determine that tangents are perpendicular to the radius drawn to the point of tangency, and that tangent segments to a circle from a point outside the circle are congruent. Notes: Introduce vocabulary: tangent.

Discovering Geometry, 4th Edition Lesson 6.1, Investigations 1 and 2 Suggested homework: Lesson 6.1, pp. 314–315, Exercises 1–7, 9, 16


Description of Lesson: Students will investigate and discover properties relating central angles, chords, and arcs of circles Notes: Introduce vocabulary: central angle, arc measure, chord.

Discovering Geometry, 4th Edition Lesson 6.2, Investigations 2, 3, 4 (pp. 318-320)

Suggested homework: Lesson 6.2, selected problems from Exercises 1–12, 16–19

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Description of Lesson: Students will discover and prove the inscribed angle theorem and prove that the opposite angles of a cyclic quadrilateral are supplementary Notes: Introduce vocabulary: inscribed angle, cyclic quadrilateral.

Discovering Geometry, 4th Edition Lesson 6.3, Investigation 1 Lesson 6.4, Exercises 1–3

Suggested homework: Lesson 6.3, selected problems from Exercises 1–17 Optional pre-assessment from FAL on Inscribing and Circumscribing Right Triangles (the Apprentice Task for this unit)

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Apprentice Task Inscribing and Circumscribing Right Triangles



Math Objectives: ● Students will develop strategies for describing relationships between

right triangles and the radii of their inscribed and circumscribed circles.

● Students will apply circle theorems. CCSS-M Standards Addressed: G.C.2, G.C.3 Potential Misconceptions:

● Students may see relationships in the diagram that aren’t supported by circle theorems, i.e., thinking that if a side or angle looks bisected, then it can be generalized.

Launch: If FAL was assigned for homework, debrief as a class: what did students find challenging or unclear? If not assigned, put task on overhead and ask students to generate a list of relationships they would need to know in order to solve the problem. Let class know that the purpose of the following task is to address these questions and help them deepen their understanding of This lesson is designed to enable students to develop strategies for describing relationships between right triangles and the radii of their inscribed and circumscribed circles. During: Closure/Extension:

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Inscribing and Circumscribing Right Triangles


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.


Vocabulary: inscribed circle, circumscribed circle

Sentence frames: In a right triangle with sides a, b, and c, the radius of the circumscribed circle is ____. In a right triangle with sides a, b, and c, the radius of the inscribed circle is ____.

Differentiation Strategies: Participation Structures (group, partners, individual, other): Groups of 2 to 4 students.

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Lesson Series #2 Lesson Series Overview: Students will use their previous knowledge of circumference and area of circles to modify those formulas for arc length and area of sectors, using central angles first in degrees, then in radians once the meaning of a radian is explored. CCSS-M Standards Addressed: G.C.5 Time: 3 days


Description of Lesson: Students will view an excerpt from the PBS series Cosmos that introduces Eratosthenes’ measurement of the circumference of the Earth. They will then find the circumference themselves using the distance (arc length) between the ancient cities of Alexandria and Syene. Notes: Only problems 13 and 14 from CPM Core Connections Geometry Lesson 10.1.2 are essential.

YouTube clip from Carl Sagan’s Cosmos http://www.youtube.com/watch?v=G8cbIWMv0rI&safe=active CPM Core Connections Geometry, Lesson 10.1.2 (If you have CPM Geometry Connections textbooks, Lesson 10.1.2 is a similar lesson.) Suggested homework: Assorted problems from Discovering Geometry, 4th Edition, Lesson 6.5 (review of circumference)


Description of Lesson: Students will use the formulas for circumference and area of circles to derive related formulas for arc length and area of sectors. Notes: This lesson assumes that students are familiar with formulas for circumference and area of circles, including finding the radius if given the circumference or area.

PowerPoint - Circle Arcs and Sectors Discovering Geometry, 4th Edition, Lesson 6.7 Investigation (p. 350) and Lesson 8.6, Exercises 1–3 (p. 455)

Suggested homework: Discovering Geometry, Lesson 6.7 Exercises 1–7 odd, and Discovering Geometry Lesson 8.6, Exercises 7–11. (May be adjusted to suit needs of students.)

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Description of Lesson: Students will first explore the idea of radian measure informally, and then connect radian measure to the circumference formula of a circle. Finally, they will use this new angle measure to rewrite the formulas for arc length and area of a circle. Notes: This activity was modified from Project Maths from Ireland.

What is a R Activity (provided in unit folder) Materials Required: pre-cut lengths of string, each representing a circle’s radius length, for each collaborative group in the class Suggested homework: pre-assessment for FAL #23 (Sectors of Circles)

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Expert Task Sectors of Circles



Math Objectives: ● Students will compute perimeters, areas, and arc lengths of sectors. ● Students will find the relationships between arc lengths and areas of

sectors after scaling. CCSS-M Standards Addressed: G.C.5 Potential Misconceptions:

● Students may have difficulty using radian measure confidently since it was just introduced during the previous lesson.

Launch: Day 1: Begin by discussing homework (pre-assessment for FAL #23). Review the concepts from the previous lesson as needed. Day 2: Begin by reminding students where the class left off the previous day, and set pairs back to work. During: Circulate during matching activities and periodically ask partners to justify their choices. Closure/Extension: Assign “Sectors of Circles Revisited” as a post-assessment of task. May be competed in class or at home (at your discretion).

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Sectors of Circles


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 7. Look for and make use of structure.


Vocabulary: radian vs. radius Sentence frames: I know that Sector _____ has the same (area or arc length) as Sector _____ because _______________________.

Differentiation Strategies: Participation Structures (group, partners, individual, other): Whole-class discussion for introduction to task. Partners for domino activity.

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Lesson Series #3 Lesson Series Overview: Students will be able to derive the equation of a circle if given center and radius using the Pythagorean Theorem, complete the square to find the center and radius of a circle given by an equation, and derive the equation of a parabola given a focus and directrix. CCSS-M Standards Addressed: G.GPE.1, G. GPE.2 Time: 4 days


Description of Lesson: Given the center of a circle and the radius, students use the Pythagorean Theorem to derive the equation of the circle. Notes: Students may need extra practice regarding the Pythagorean Theorem; we suggest addressing this in the Warm-Up to activate prior learning. Be sure to introduce the idea of a locus so that students will more easily make the transition on Day 3 to parabolas.

CPM Core Connections Geometry, Lesson 12.1.1, Exercises 12-1, 12-2, 12-9, 12-10

Suggested homework: CPM Core Connections Geometry, Lesson 12.1.1 Exercises 12-6, 12-7, 12-11


Description of Lesson: Students use the standard form and graphing form of a quadratic function to find the center and radius of a circle. They complete the square to rewrite a function from standard form to graphing form. Notes: Students may need extra practice regarding completing the square; we suggest addressing this in the Warm-Up to activate prior learning.

CPM Core Connections Geometry, Lesson 12.1.2 Exercises 12-13 through 12-19 Suggested homework: CPM Core Connections Geometry, Lesson 12.1.2 Exercises12-24, 12-27, 12-31


Description of Lesson: Students find the focus and directrix of a parabola while investigating conic sections using tracing paper. They determine the relationship between the points on the parabola and its focus and directrix. Students will need 12.1.3 Resource Page and tracing paper.

CPM Core Connections Geometry, Lesson 12.1.3 Exercises 12-35 through 12-38

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Notes: Provide a formal definition of focus and directrix of a parabola. Stress the idea of a locus to help facilitate connections between the circle and parabola as well as possible further study of conic sections.

Suggested homework: CPM Core Connections Geometry, Lesson 12.1.3 Exercises 12-43, 12-45


Description of Lesson: Students use 12.1.4 Resource Page (focus-directrix graph paper) to graph parabolas. They derive the equation of a parabola using the focus and directrix. Notes: The Warm-Up should include the focus and directrix of a parabola.

CPM Core Connections Geometry, Lesson 12.1.4 Exercises 12-47, 12-48

Suggested homework: CPM Core Connections Geometry, Lesson 12.1.4 Exercises 12-51, 12-66

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Milestone Task Individual Unit Assessment



Math Objectives: ● Students will show understanding of the main ideas from the Circles

and Conics unit. ● Students will demonstrate progress on Standards for Mathematical

Practices. CCSS-M Standards Addressed: G.C.1, G.C.2, G.C.3, G.C.5, G.GPE.1, G.GPE.2 Potential Misconceptions

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Launch: Remind students that the purpose of this assessment is to allow them to demonstrate what they have learned in the unit, and that they need to explain their thinking for questions 7, 8, and 9. During: Students complete the Milestone Task individually. These questions mostly come from the test bank of Discovering Geometry. Closure/Extension: If students finish early, ask them to draw two arbitrary triangles. In one triangle, they should construct an inscribed circle. With the other triangle, they should construct a circle that circumscribes the triangle.

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Individual Unit Assessment


Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure.


Vocabulary: Make sure English Learners understand the term “equidistant.”

Sentence frames:

Differentiation Strategies: Participation Structures (group, partners, individual, other): Individual

Documents

SFUSD Mathematics Core Curriculum Development · PDF fileSFUSD Mathematics Core Curriculum Development Project ... Unit Objectives ... and area of circles to arcs and sectors using