geometry big ideas math 2017-2018 sy - philasd.org · geometry big ideas math 2017-2018 sy ......

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018 1 Geometry Big Ideas Math 2017-2018 SY Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4 Dates September 5 – October 31 BM Window Opens Nov 1 Term 1 ends Nov 13 November 1 – January 26 BM Window Opens Jan 29 Keystone Exam Window Opens Jan 8 Term 2 ends Jan 29 January 29 – April 6 BM Window Opens April 9 Term 3 ends Apr 9 April 9 – June 12 Keystone Exam Window Opens May 14 Cycle 4 ends Jun 12 Total School Days 39 Total Days Including 1 Half Day 51 Total Days Including 4 Half Days 46 Total Days Including 3 Half Days 46 Total Days Including 13 Half Day Chapters 1 – Basics of Geometry 2 – Reasoning and Proof 3 – Parallel and Perpendicular Lines 4-1 – 4-4 – Transformations 4-5 – 4-6 – Transformations 5 – Congruent Triangles 6 – Relationships within Triangles 7 – Quadrilaterals and Other Polygons 8 – Similarity 9 – Right Triangles and Trigonometry 10 – Circles 11-1 – 11-6 – Surface Area and Volume 11-7 – 11-8 – Surface Area and Volume 12 – Probability Notes A Benchmark Cycle is defined as the time allotted to teach the content on each benchmark, assuming the benchmark is taken on the first day of the window. This means that, even though you may give the benchmark later in the window, you should move into new content as of the first date in the BM Window or you will fall behind the suggested pacing.

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Page 1: Geometry Big Ideas Math 2017-2018 SY - philasd.org · Geometry Big Ideas Math 2017-2018 SY ... G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, ... cylinders, cones,

THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

1

Geometry Big Ideas Math 2017-2018 SY

Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4

Dat

es

September 5 – October 31 BM Window Opens Nov 1 Term 1 ends Nov 13

November 1 – January 26 BM Window Opens Jan 29 Keystone Exam Window

Opens Jan 8 Term 2 ends Jan 29

January 29 – April 6 BM Window Opens April 9 Term 3 ends Apr 9

April 9 – June 12 Keystone Exam Window

Opens May 14 Cycle 4 ends Jun 12

Tota

l Sc

ho

ol

Day

s

39 Total Days Including 1 Half Day

51 Total Days Including 4 Half Days

46 Total Days Including 3 Half Days

46 Total Days Including 13 Half Day

Ch

apte

rs

1 – Basics of Geometry 2 – Reasoning and Proof 3 – Parallel and Perpendicular

Lines 4-1 – 4-4 – Transformations

4-5 – 4-6 – Transformations 5 – Congruent Triangles 6 – Relationships within

Triangles 7 – Quadrilaterals and Other

Polygons 8 – Similarity

9 – Right Triangles and Trigonometry

10 – Circles 11-1 – 11-6 – Surface Area and

Volume

11-7 – 11-8 – Surface Area and Volume

12 – Probability

Notes

A Benchmark Cycle is defined as the time allotted to teach the content on each benchmark, assuming the benchmark is taken on the first day of the window. This means that, even though you may give the benchmark later in the window, you should move into new content as of the first date in the BM Window or you will fall behind the suggested pacing.

Page 2: Geometry Big Ideas Math 2017-2018 SY - philasd.org · Geometry Big Ideas Math 2017-2018 SY ... G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, ... cylinders, cones,

THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

2

Table of Contents

Block Schedule Options ...................................................................................................................................................................................................... 3

Benchmark Cycle 1 Standards ............................................................................................................................................................................................. 4

Benchmark Cycle 1 Scope and Sequence ........................................................................................................................................................................... 7

Benchmark Cycle 2 Standards ........................................................................................................................................................................................... 11

Benchmark Cycle 2 Scope and Sequence ......................................................................................................................................................................... 14

Benchmark Cycle 3 Standards ........................................................................................................................................................................................... 19

Benchmark Cycle 3 Scope and Sequence ......................................................................................................................................................................... 22

Cycle 4 Standards .............................................................................................................................................................................................................. 25

Cycle 4 Scope and Sequence ............................................................................................................................................................................................. 27

PA Core Standards by Cycle .............................................................................................................................................................................................. 30

Geometry Eligible Content Taught ................................................................................................................................................................................... 31

Document Information Page ............................................................................................................................................................................................ 34

Page 3: Geometry Big Ideas Math 2017-2018 SY - philasd.org · Geometry Big Ideas Math 2017-2018 SY ... G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, ... cylinders, cones,

THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

3

Block Schedule Options

Note: With block schedules, generally follow the suggested sequence of lessons. Asterisked lessons are approximately 90-minutes in length. Non-asterisked lessons (approximately 45-minutes in length) may be combined, but consider combining them meaningfully, instead of simply teaching two-lessons back-to-back. When single non-asterisked lessons precede or follow an asterisked one, consider using Concept Bytes, Mathematical Modeling, or Common Core Performance Tasks to start or complete the lesson (using 45 available minutes to construct a full 90-minute lesson).

Benchmark Cycle 1 Benchmark Cycle 2 Benchmark Cycle 3 Cycle 4

Suggested Pacing

A /

B

Blo

ck

Use the suggested pacing, but combine 45-minute (non-asterisked) lessons, in accordance with the general note above.

Full

Blo

ck

Fall

9/5 – 9/29 Ch. 1 to Ch. 4-4

10/2 – 10/31 Ch. 4-5 to Ch. 8

11/1 – 12/4 Ch. 9 to Ch. 11-6

12/5 – 1/29 Ch. 11-7 to Ch. 12 Remediation and Extension

Full

Blo

ck

Spri

ng

1/30 – 2/23 Ch. 1 to Ch. 4-4

2/26 – 3/28 Ch. 4-5 to Ch. 8

4/3 – 4/30 Ch. 9 to Ch. 11-6

5/1 – 6/12 Ch. 11-7 to Ch. 12 Remediation and Extension

Benchmark Schedule

A /

B

Blo

ck

Follow Benchmark schedule indicated in the Scope and Sequence document that follows this page.

Full

Blo

ck

Fall

Benchmarks 1 and 2 (No need to complete

in one sitting)

None (Benchmark 3 not taken)

Full

Blo

ck

Spri

ng

Benchmarks 1 and 2

(No need to complete in one sitting)

None (Benchmark 3 not taken)

Page 4: Geometry Big Ideas Math 2017-2018 SY - philasd.org · Geometry Big Ideas Math 2017-2018 SY ... G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, ... cylinders, cones,

THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

4

Benchmark Cycle 1 Standards

PA Core Standards PA Core Eligible Content

CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context.

A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.

A1.2.1.1.2 Determine whether a relation is a function, given a set of points or a graph.

A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).

G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular figure.

G.2.2.2.2 Find the measurement of a missing length, given the perimeter, circumference, or area.

G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon.

G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite figure.

G.2.2.2.5 Find the area of a sector of a circle.

CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools.

M08.C-G.1.1.1 Identify and apply properties of rotations, reflections, and translations. Example: Angle measures are preserved in rotations, reflections, and translations.

M08.C-G.1.1.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures, using coordinates.

G.1.2.1.1 Identify and/or use properties of triangles.

G.1.2.1.2 Identify and/or use properties of quadrilaterals.

G.1.2.1.3 Identify and/or use properties of isosceles and equilateral triangles.

G.1.2.1.4 Identify and/or use properties of regular polygons.

G.1.2.1.5 Identify and/or use properties of pyramids and prisms.

G.2.2.1.1 Use properties of angles formed by intersecting lines to find the measures of missing angles.

G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles.

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

5

PA Core Standards PA Core Eligible Content

CC.2.3.HS.A.1 Use geometric figures and their properties to represent transformations in the plane.

G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids.

G.1.3.1.2 Identify and/or use proportional relationships in similar figures.

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.3 Find the measurement of a missing length given the surface area or volume.

CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.

G.1.2.1.1 Identify and/or use properties of triangles.

G.1.2.1.2 Identify and/or use properties of quadrilaterals.

G.1.2.1.3 Identify and/or use properties of isosceles and equilateral triangles.

G.1.2.1.4 Identify and/or use properties of regular polygons.

G.1.2.1.5 Identify and/or use properties of pyramids and prisms.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

G.2.2.1.1 Use properties of angles formed by intersecting lines to find the measures of missing angles.

G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles.

G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular figure.

G.2.2.2.2 Find the measurement of a missing length, given the perimeter, circumference, or area.

G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon.

G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite figure.

G.2.2.2.5 Find the area of a sector of a circle.

Page 6: Geometry Big Ideas Math 2017-2018 SY - philasd.org · Geometry Big Ideas Math 2017-2018 SY ... G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, ... cylinders, cones,

THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

6

PA Core Standards PA Core Eligible Content

CC.2.3.HS.A.4 Apply the concept of congruence to create geometric constructions.

No Eligible Content Alignment

CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle.

G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures.

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.

G.2.1.2.1 Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane.

G.2.1.2.2 Relate slope to perpendicularity and/or parallelism (limit to linear algebraic equations).

G.2.1.2.3 Use slope, distance, and/or midpoint between two points on a coordinate plane to establish properties of a two‐dimensional shape.

CC.2.3.HS.A.13 Analyze relationships between two‐dimensional and three‐dimensional objects.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle. G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures. G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

7

Benchmark Cycle 1 Scope and Sequence

Chapter 1: Basics of Geometry

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Objective(s)(s) Eligible Content

9/5

-9/2

0: 1

Day

Per

Les

son

+ 6

Day

s

9/5

-9/8

1/3

0-2

/2

1-1 Points, Lines, and Planes Name points, lines and planes; name segments and rays; sketch intersections of line and planes; solve real-life problems involving lines and planes.

G.1.1.1.1, G.1.1.1.2, G.1.1.1.3, G.1.1.1.4, G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.2.1.2.1, G.2.1.2.2, G.2.1.2.3, G.2.2.1.1, G.2.2.1.2

1-2 Measuring and Constructing Segments

Use the Ruler-Postulate; copy segments and compare segments for congruence; use the Segment Addition Postulate.

1-3 Using Midpoint and Distance Formulas

Find segment lengths using midpoints and segment bisectors; use the Midpoint Formula; use the Distance Formula.

1-4 Perimeter and Area in the Coordinate Plane

Classify polygons; find perimeter and areas of polygons in the coordinate plane.

1-5* Measuring and Constructing Angles

Name angle; measure and classify angles; identify congruent angles; use the Angle Addition Postulate to find angle measures; bisect angles.

1-6* Describing Pairs of Angles Identify complementary and supplementary angles; identify linear pairs and vertical angles.

Note: Chapter 1 revisits material with which students may be familiar through previous experiences in geometry. Review this material, as necessary, while concentrating on the more formal notation and explanations expected in high school geometry. In this chapter, essential questions considered are: How can you represent a three-dimensional figure with a two-dimensional drawing? What are the building blocks of geometry? How can you describe the attributes of a segment or angle? Note, in addition, that the PA Core Standard CC.2.3.HS.A.4 involves building and studying geometric constructions, but these do not appear within the descriptions of the eligible content for assessment. Understanding geometric constructions, while not part of the eligible content, can help students use technology or develop a concrete understanding of abstract ideas.

* Lessons marked with an asterisk may require two 45-minute class periods, for which flex days may be used.

Consider beginning the year by laying foundations for problem-solving via: WODB.ca, Estimation180.com, or solveme.edc.org/Mobiles.html (which can help maintain students algebra skills). Note that, while it is not always explicitly shown under the Eligible Content alignment column, Geometry must be infused with opportunities to “write, analyze, complete, or identify formal proofs” (G.1.3.2.1) throughout the year. Review foundational knowledge and check for understanding, as needed, through “Do Nows” or “Bell Ringers,” homework, and teacher-created assessments.

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

8

Chapter 2: Reasoning and Proof

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Objective(s) Eligible Content

9/2

5-1

0/6

: 1 D

ay P

er L

esso

n +

4 D

ays

9/1

1-9

/15

2/5

-2/9

2-1 Conditional Statements

Write conditional statements; use definitions as conditional statements; make truth tables.

G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.1.3.2.1, G.2.2.1.1, G.2.2.1.2

2-2* Inductive and Deductive Reasoning

Use inductive and deductive reasoning.

2-3 Postulates and Diagrams Identify postulates using diagrams; sketch and interpret diagrams.

2-4* Algebraic Reasoning

Use Algebraic Properties of Equality to justify the steps in solving; use the Distributive Property to justify the steps in solving an equation; use properties of equality involving segment lengths and angle measures.

2-5 Proving Statements about Segments and Angles

Write two-column proofs; name and prove properties of congruence.

2-6* Proving Geometric Relationships Write flowchart proofs to prove geometric relationships; write paragraph proofs to prove geometric relationships.

Note: Chapter 2 constitutes a formal study of logic and proof. The essential question of this chapter is: How can you make a conjecture and prove that it is true? The material in this chapter not only supports work throughout Geometry, but it is essential for later coursework in Algebra II and beyond, as well. Be sure to give students opportunities to construct viable arguments and to critique the reasoning of others (SMP3) during discussions throughout this chapter.

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

9

Chapter 3: Parallel and Perpendicular Lines

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Objective(s) Eligible Content

10

/9-1

0/2

0: 1

Day

Pe

r Le

sso

n +

5 D

ays

9/1

8-9

/26

2/1

2-2

/16

3-1 Pairs of Lines and Angles

Identify lines and planes; identify parallel and perpendicular lines; identify pairs of angles formed by transversals.

G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.2.1.2.1, G.2.1.2.2, G.2.1.2.3, G.2.2.1.1, G.2.2.1.2

3-2* Parallel Lines and Transversals Use properties of parallel lines; prove theorems about parallel lines; solve real-life problems.

3-3* Proofs with Parallel Lines Use the Corresponding Angles Converse; construct parallel lines; prove theorems about parallel lines; use the Transitive Property of Parallel Lines.

3-4 Proofs with Perpendicular Lines Find the distance from a point to a line; construct perpendicular lines; prove theorems about perpendicular lines; solve real-life problems involving perpendicular lines.

3-5* Equations of Parallel and Perpendicular Lines

Use slope to partition directed line segments; identify parallel and perpendicular lines; write equations of parallel and perpendicular lines; use slope to find the distance from a point to a line.

Note: In this chapter, students concentrate on: How do you prove that two lines are parallel? What is the sum of the measures of the angles in a triangle? How do you write an equation of a line in the coordinate plane? Students should be familiar with angles and parallel lines from Grade 7. In this chapter, students develop arguments about familiar geometric objects, now using the tools of logic (Chapter 2). Higher rigor is supported when arrangements of lines are presented and students are asked, “What math questions can we generate?”

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

10

Note: Chapter 4 concentrates on a systematic study of transforming objects in the plane. Essential questions investigated are: How can you change a figure’s position without changing its size and shape? How can you represent a transformation in the coordinate plane? How do you recognize symmetry in a figure? Investigating symmetry and types of transformations has been part of students’ geometric study since elementary school. Here and in the future, they begin to see these relationships through the lens of functions.

Benchmark 1 Window: Nov 1 – Nov 17

Chapter 4: Transformations

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Objective(s) Eligible Content

10

/23

-10

/31

: 1 D

ay P

er L

esso

n +

2.5

Day

s

9/2

7-9

/29

2/2

0-2

/23

4-1* Translations

Perform translations; perform compositions; solve real-life problems involving compositions.

G.1.3.1.1, G.1.3.1.2

4-2 Reflections Perform reflections; perform glide reflections; identify lines of symmetry; solve real-life problems involving reflections.

4-3* Rotations Perform rotations; perform compositions with rotations; identify rotational symmetry.

4-4 Congruence and Transformations Identify congruent figures; describe congruence transformations; use theorems about congruence transformations.

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

11

Benchmark Cycle 2 Standards

PA Core Standards PA Core Eligible Content

CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context.

A1.2.1.1.1 Analyze a set of data for the existence of a pattern and represent the pattern algebraically and/or graphically.

A1.2.1.1.2 Determine whether a relation is a function, given a set of points or a graph.

A1.2.1.1.3 Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).

G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular figure.

G.2.2.2.2 Find the measurement of a missing length, given the perimeter, circumference, or area.

G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon.

G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite figure.

G.2.2.2.5 Find the area of a sector of a circle.

CC.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

G.2.1.1.1 Use the Pythagorean theorem to write and/or solve problems involving right triangles.

G.2.1.1.2 Use trigonometric ratios to write and/or solve problems involving right triangles.

CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools.

M08.C-G.1.1.1 Identify and apply properties of rotations, reflections, and translations. Example: Angle measures are preserved in rotations, reflections, and translations.

M08.C-G.1.1.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures, using coordinates.

G.1.2.1.1 Identify and/or use properties of triangles.

G.1.2.1.2 Identify and/or use properties of quadrilaterals.

G.1.2.1.3 Identify and/or use properties of isosceles and equilateral triangles.

G.1.2.1.4 Identify and/or use properties of regular polygons.

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

12

PA Core Standards PA Core Eligible Content

G.1.2.1.5 Identify and/or use properties of pyramids and prisms.

G.2.2.1.1 Use properties of angles formed by intersecting lines to find the measures of missing angles.

G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles.

CC.2.3.HS.A.1 Use geometric figures and their

properties to represent transformations in the plane. G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids.

CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence.

G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids.

CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.

G.1.2.1.1 Identify and/or use properties of triangles.

G.1.2.1.2 Identify and/or use properties of quadrilaterals.

G.1.2.1.3 Identify and/or use properties of isosceles and equilateral triangles.

G.1.2.1.4 Identify and/or use properties of regular polygons.

G.1.2.1.5 Identify and/or use properties of pyramids and prisms.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

G.2.2.1.1 Use properties of angles formed by intersecting lines to find the measures of missing angles.

G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles.

G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular figure.

G.2.2.2.2 Find the measurement of a missing length, given the perimeter, circumference, or area.

G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon.

G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite figure.

G.2.2.2.5 Find the area of a sector of a circle.

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THE SCHOOL DISTRICT OF PHILADELPHIA, OFFICE OF CURRICULUM, INSTRUCTION, AND ASSESSMENT, CHRISTOPHER SHAFFER, DEPUTY, 2017-2018

13

PA Core Standards PA Core Eligible Content

CC.2.3.HS.A.5 Create justifications based on transformations to establish similarity of plane figures.

G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids.

G.1.3.1.2 Identify and/or use proportional relationships in similar figures.

CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures.

G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids.

G.1.3.1.2 Identify and/or use proportional relationships in similar figures.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle.

G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures.

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.

G.2.1.2.1 Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane.

G.2.1.2.2 Relate slope to perpendicularity and/or parallelism (limit to linear algebraic equations).

G.2.1.2.3 Use slope, distance, and/or midpoint between two points on a coordinate plane to establish properties of a two‐dimensional shape.

CC.2.3.HS.A.14 Apply geometric concepts to model and solve real‐world problems.

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.3 Find the measurement of a missing length, given the surface area or volume.

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Benchmark Cycle 2 Scope and Sequence

Note: Chapter 4 concentrates on a systematic study of transforming objects in the plane. Essential questions investigated are: How can you change a figure’s position without changing its size and shape? How can you represent a transformation in the coordinate plane? How do you recognize symmetry in a figure? Investigating symmetry and types of transformations has been part of students’ geometric study since elementary school. Here and in the future, they begin to see these relationships through the lens of functions.

Chapter 4: Transformations (cont.)

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4-5* Dilations Identify and perform dilations; solve real-life problems involving scale factors and dilations.

G.1.3.1.1, G.1.3.1.2 4-6 Similarity and Transformations Perform similarity transformations; describe similarity

transformations; prove that figures are similar.

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Note: Chapter 5 begins a formal study of congruence. Essential questions of this chapter are: How do you identify corresponding parts of congruent triangles? How do you show that two triangles are congruent? How can you tell whether a triangle is isosceles or equilateral? Students learn that objects may not simply look congruent, but that they must be proven so by logic. Chapter 5-8 on proofs in the coordinate plane can help support connections between geometry and algebra and maintain algebra skills.

Chapter 5: Congruent Triangles

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5-1* Angles of Triangles Classify triangles by sides and angles; find interior and exterior angle measures of triangles.

G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.1.3.1.1, G.1.3.1.2, G.1.3.2.1, G.2.2.1.1, G.2.2.1.2, G.2.2.2.1, G.2.2.2.2, G.2.2.2.3, G.2.2.2.4, G.2.2.2.5, G.2.3.1.1, G.2.3.1.2, G.2.3.1.3

5-2 Congruent Polygons Identify and use corresponding parts; use the Third Angles Theorem.

5-3 Proving Triangle Congruence by SAS

Use the Side-Angle-Side (SAS) Congruence Theorem; solve real-life problems.

5-4* Equilateral and Isosceles Triangles

Use the Base Angles Theorem; use isosceles and equilateral triangles.

5-5* Proving Triangle Congruence by SSS

Use the Side-Side-Side (SSS) Congruence Theorem; use the Hypotenuse-Leg (HL) Congruence Theorem.

5-6* Proving Triangle Congruence by ASA and AAS

Use the SAS and the AAS Congruence Theorem.

5-7 Using Congruent Triangles Use congruent triangles; prove constructions.

5-8* Coordinate Proofs Place figures in a coordinate plane; write coordinate proofs.

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Chapter 6: Relationships within Triangles

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6-1* Perpendicular and Angle Use perpendicular bisectors to find measures; use angle bisectors to find measures and distance relationships; write equations for perpendicular bisectors.

G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.1.3.2.1, G.2.2.1.1, G.2.2.1.2, G.2.2.2.1, G.2.2.2.2, G.2.2.2.3, G.2.2.2.4, G.2.2.2.5

6-2* Bisectors of Triangles Use and find the circumcenter of a triangle; use and find the incenter of a triangle.

6-3* Medians and Altitudes of Triangles

Use medians and find the centroids of triangles; use altitudes and find the orthocenters of triangles.

6-4* The Triangle Midsegment Theorem

Use midsegments of triangles in the coordinate plane; use the Triangle Midsegment Theorem to find distance.

6-5* Indirect Proof and Inequalities in One Triangle

Write indirect proofs; list sides and angles of a triangle in order by size; use the Triangle Inequality Theorem to find possible side lengths of triangles.

6-6 Inequalities in Two Triangles Compare measures in triangles; solve real-life problems using the Hinge Theorem.

Note: In Chapter 6, students connect coordinate geometry with their study of triangles. These skills are essential for working with objects in Algebra II and beyond. They explore questions such as: How do you use coordinate geometry to find relationships within triangles? How do you write indirect proofs? How do you solve problems that involve measurements of triangles?

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Chapter 7: Quadrilaterals and Other Polygons

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7-1 Angles of Polygons

Use the interior angle measures of polygons; use the exterior angle measures of polygons.

G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.1.3.2.1, G.2.2.1.1, G.2.2.1.2, G.2.2.2.1, G.2.2.2.2, G.2.2.2.3, G.2.2.2.4, G.2.2.2.5

7-2* Properties of Parallelograms Use properties to find side lengths and angles of parallelograms; use parallelograms in the coordinate plane.

7-3* Proving That a Quadrilateral is a Parallelogram

Identify and verify parallelograms; show that a quadrilateral is a parallelogram in the coordinate plane.

7-4* Properties of Special Parallelograms

Use properties of special parallelograms; use properties of diagonals of special parallelograms; use coordinate geometry to identify special types of parallelograms.

7-5* Properties of Trapezoids and Kites

Use properties of trapezoids; use the Trapezoid Midsegment Theorem to find distance; use properties of kites; identify quadrilaterals.

Note: Chapter 7 addresses these focal questions: How can you classify quadrilaterals? How can you find the sum of the measures of polygon angles? How can you use coordinate geometry to prove general relationships?

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Chapter 8: Similarity

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8-1* Similar Polygons

Use similarity statements; find corresponding lengths in similar polygons; find perimeters and areas of similar polygons; describe whether polygons are similar.

G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.1.3.1.1, G.1.3.1.2, G.1.3.2.1, G.2.2.1.1, G.2.2.1.2, G.2.2.2.1, G.2.2.2.2, G.2.2.2.3, G.2.2.2.4, G.2.2.2.5

8-2* Proving Triangle Similarity by AA Use the Angle-Angle Similarity Theorem; solve real-life problems.

8-3* Proving Triangles Similarity by SSS and SAS

Use the Side-Side-Side Similarity Theorem; use the Side-Angle-Side Similarity Theorem; prove slope criteria using similar triangles.

8-4* Proportionality Theorems Use the Triangle Proportionality Theorem and its converse; use other proportionality theorems.

Note: Chapter 8 deepens students’ understanding of similarity by investigating key questions such as: How do you use proportions to find side lengths in similar polygons? How do you identify corresponding parts of similar triangles? How do you show two triangles are similar? Students connect their understanding of proportions with their understanding of similarity. Similarity is the foundational idea underlying trigonometry and other topics studied later in Geometry, in Algebra II, and beyond.

Benchmark 2 Window: Jan 29 – Feb 13

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Benchmark Cycle 3 Standards

PA Core Standards PA Core Eligible Content

CC.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

G.2.1.1.1 Use the Pythagorean theorem to write and/or solve problems involving right triangles.

G.2.1.1.2 Use trigonometric ratios to write and/or solve problems involving right triangles.

CC.2.3.8.A.1 Apply the concepts of volume of cylinders, cones, and spheres to solve real‐world and mathematical problems.

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.3 Find the measurement of a missing length, given the surface area or volume.

CC.2.3.8.A.3 Understand and apply the Pythagorean Theorem to solve problems.

G.2.1.2.1 Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane.

G.2.1.2.2 Relate slope to perpendicularity and/or parallelism (limit to linear algebraic equations).

G.2.1.2.3 Use slope, distance, and/or midpoint between two points on a coordinate plane to establish properties of a 2-dimensional shape.

CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.

G.1.2.1.1 Identify and/or use properties of triangles.

G.1.2.1.2 Identify and/or use properties of quadrilaterals.

G.1.2.1.3 Identify and/or use properties of isosceles and equilateral triangles.

G.1.2.1.4 Identify and/or use properties of regular polygons.

G.1.2.1.5 Identify and/or use properties of pyramids and prisms.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

G.2.2.1.1 Use properties of angles formed by intersecting lines to find the measures of missing angles.

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PA Core Standards PA Core Eligible Content

G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles.

G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular figure.

G.2.2.2.2 Find the measurement of a missing length, given the perimeter, circumference, or area.

G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon.

G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite figure.

G.2.2.2.5 Find the area of a sector of a circle.

CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.

G.2.1.1.1 Use the Pythagorean theorem to write and/or solve problems involving right triangles.

G.2.1.1.2 Use trigonometric ratios to write and/or solve problems involving right triangles.

CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle.

G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures.

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

G.2.2.3.1 Describe how a change in the linear dimension of a figure affects its perimeter, circumference, and area (e.g., How does changing the length of the radius of a circle affect the circumference of the circle?).

CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

G.2.2.3.1 Describe how a change in the linear dimension of a figure affects its perimeter, circumference, and area (e.g., How does changing the length of the radius of a circle affect the circumference of the circle?).

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PA Core Standards PA Core Eligible Content

CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.

G.2.1.2.1 Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane.

G.2.1.2.2 Relate slope to perpendicularity and/or parallelism (limit to linear algebraic equations).

G.2.1.2.3 Use slope, distance, and/or midpoint between two points on a coordinate plane to establish properties of a 2-dimensional shape.

CC.2.3.HS.A.12 Explain volume formulas and use them to solve problems.

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.3 Find the measurement of a missing length, given the surface area or volume.

CC.2.3.HS.A.13 Analyze relationships between two‐dimensional and three‐dimensional objects.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle.

G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures.

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

G.2.3.2.1 Describe how a change in the linear dimension of a figure affects its surface area or volume (e.g., How does changing the length of the edge of a cube affect the volume of the cube?).

CC.2.3.HS.A.14 Apply geometric concepts to model and solve real‐world problems.

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.3 Find the measurement of a missing length, given the surface area or volume.

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Benchmark Cycle 3 Scope and Sequence

Chapter 9: Right Triangles and Trigonometry

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9-1* The Pythagorean Theorem Use the Pythagorean Theorem; use the Converse of the Pythagorean Theorem; classify triangles.

G.1.3.2.1, G.2.1.1.1, G.2.1.1.2, G.2.1.2.1, G.2.1.2.2, G.2.1.2.3

9-2 Special Right Triangles Find side lengths in special right triangles; solve real-life problems involving special right triangles.

9-3* Similar Right Triangles Identify similar triangles; sole real-life problems involving similar triangles; use geometric means.

9-4* The Tangent Ratio Use the tangent ratio; solve real-life problems involving the tangent ratio.

9-5* The Sine and Cosine Ratios Use the sine and cosine ratios; find the sine and cosine of angle measures in special right triangles; solve real-life problems involving sine and cosine ratios.

9-6* Solving Right Triangles Use inverse trigonometric ratios; solve right triangles.

9-7* Law of Sines and Law of Cosines Find areas of triangles; use the Law of Sines to sole triangles; use the Law of Cosines to solve triangles.

Note: In Chapter 9, students address these questions: How do trigonometric ratios relate to similar right triangles? How do you find a side length or an angle measure in a right triangle? Drawing on their understanding of triangles, students investigate trigonometry. This may be an introduction to trigonometry for many students, which is an important foundation for studying new families of functions in Algebra II and beyond. Working with special right triangles is an example of making sense of and using structure (SMP7). Continue to offer students open-middle problems, like those in the Solve It! and ask students to explain their thinking by drawing pictures and offering verbal explanations.

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Chapter 10: Circles

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10-1*

Lines and Segments That Intersect Circles

Identify special segments and lines; draw and identify common tangents; use properties of tangents.

G.1.1.1.1, G.1.1.1.2, G.1.1.1.3, G.1.1.1.4, G.1.3.2.1

10-2* Finding Arc Measures Find arc measures; identify congruent arcs; prove circles are similar.

10-3* Using Chords Use chords of circles to find lengths and arc measures.

10-4* Inscribed Angles and Polygons Use inscribed angles; use inscribed polygons.

10-5* Angle Relationships in Circles Find angle and arc measures; use circumscribes angles.

10-6* Segment Relationships in Circles Use segments of chords, tangents, and secants.

10-7* Circles in the Coordinate Plane Write and graph equations of circles; write coordinate proofs involving circles; solve real-life problems using graphs of circles.

Note: In Chapter 10, students use what they have learned about triangles and other planar objects to investigate circles and their properties. Questions addressed include: How can you prove relationships between angles and arcs in a circle? When lines intersect a circle or within a circle, how do you find the measures of resulting angles, arcs, and segments? How do you find the equation of a circle in the coordinate plane? Students have studied circles since elementary school, but now they use their knowledge of triangles to attend to precision. Their understanding of circles is essential for studying new families of functions in trigonometry in Algebra II and beyond. Look for opportunities to make connections in this chapter to prior content. Ask students to explain the reasons for these connections and offer them opportunities to make conjectures with paper folding and geometry software.

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Chapter 11: Surface Area and Volume

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11-1* Circumference and Arc Length

Use the formula for circumference; use are lengths to find measures; solve real-life problems; measure angles in radians.

G.1.2.1.1, G.1.2.1.2, G.1.2.1.3, G.1.2.1.4, G.1.2.1.5, G.1.3.2.1, G.2.1.2.1, G.2.1.2.2, G.2.1.2.3, G.2.2.1.1, G.2.2.1.2, G.2.2.2.1, G.2.2.2.2, G.2.2.2.3, G.2.2.2.4, G.2.2.2.5, G.2.2.3.1

11-2* Areas of Circles and Sectors Use the formula for the area of a circle; use the formula for population density; find areas of sectors; use areas of sectors.

11-3* Areas of Polygons Find areas of rhombuses and kites; find angle measures in regular polygons; find areas of regular polygons.

11-4 Three-Dimensional Figures Classify solids; describe cross sections; sketch and describe solids of revolution.

11-5* Volumes of Prisms and Cylinders Find volumes of prisms and cylinders; use the formula for density; use volumes of prisms and cylinders.

11-6* Volumes of Pyramids Find volumes of pyramids; use volumes of pyramids.

Note: In Chapter 11, students investigate: How do you find the area of a polygon or find the circumference and area of a circle? How do perimeters and areas of similar figures compare? This chapter also involves studying three-dimensional geometry, including making connections between area and surface area and volume. Students have studied three-dimensional objects, especially their properties and names, since elementary school. In this chapter, they build upon this understanding. Students should have opportunities to identify and measure real-world three-dimensional objects, as well as construct three-dimensional objects using simple materials and make conjectures about them.

Benchmark 3 Window: Apr 9 – Apr 25

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Cycle 4 Standards

PA Core Standards PA Core Eligible Content

CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle.

G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures.

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles.

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/proofs by contradiction).

CC.2.3.HS.A.13 Analyze relationships between two‐dimensional and three‐dimensional objects.

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle.

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle.

G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures.

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder.

CC.2.3.HS.A.14 Apply geometric concepts to model and solve real‐world problems.

G.2.2.4.1 Use area models to find probabilities.

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.

G.2.3.1.3 Find the measurement of a missing length, given the surface area or volume.

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PA Core Standards PA Core Eligible Content

CC.2.4.HS.B.6 Use the concepts of independence and conditional probability to interpret data.

A2.2.3.2.1 Use combinations, permutations, and the fundamental counting principle to solve problems involving probability.

A2.2.3.2.2 Use odds to find probability and/or use probability to find odds.

A2.2.3.2.3 Use probability for independent, dependent, or compound events to predict outcomes.

CC.2.4.HS.B.7 Apply the rules of probability to compute probabilities of compound events in a uniform probability model.

A1.2.3.2.1 Estimate or calculate to make predictions based on a circle, line, bar graph, measure of central tendency, or other representation. A2.2.3.2.1 Use combinations, permutations, and the fundamental counting principle to solve problems involving probability.

A2.2.3.2.2 Use odds to find probability and/or use probability to find odds.

A2.2.3.2.3 Use probability for independent, dependent, or compound events to predict outcomes.

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Cycle 4 Scope and Sequence

Chapter 11: Surface Area and Volume

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Objective(s) Eligible Content

4/9

-4/1

2: 1

Day

Per

Les

son

+ 2

Day

s

12

/5-1

2/6

5/1

-5/2

11-7* Surface Areas and Volumes of

Cones Find surface areas of right cones; find volumes of cones; use volume of cones.

11-8* Surface Areas and Volumes of

Spheres Find surface areas of spheres; find volumes of spheres.

Note: In Chapter 11, students investigate: How do you find the area of a polygon or find the circumference and area of a circle? How do perimeters and areas of similar figures compare? This chapter also involves studying three-dimensional geometry, including making connections between area and surface area and volume. Students have studied three-dimensional objects, especially their properties and names, since elementary school. In this chapter, they build upon this understanding. Students should have opportunities to identify and measure real-world three-dimensional objects, as well as construct three-dimensional objects using simple materials and make conjectures about them.

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Chapter 12: Probability

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Objective(s) Eligible Content

4/1

3-4

/30

: 1 D

ay P

er L

esso

n +

6 D

ays

12

/7-1

2/1

4

5/3

-5/1

0

12-1 Sample Spaces and Probability Find sample spaces; find theoretical probabilities; find experimental probabilities.

A1.2.3.2.1, A2.2.3.2.1, A2.2.3.2.2, A2.2.3.2.3, G.2.2.4.1, G.2.3.1.1, G.2.3.1.2, G.2.3.1.3,

12-2* Independent and Dependent Events

Determine whether events are independent events; find probabilities of independent and dependent events; find conditional probabilities.

12-3* Two-Way Tables and Probability Make two-way tables; find relative and conditional relative frequencies; use conditional relative frequencies to find conditional probabilities.

12-4 Probability of Disjoint and Overlapping Events

Find probabilities of compound events; use more than one probability rule to solve real-life problems.

12-5* Permutations and Combinations Use the formula for the number of permutations; use the formula for the number of combinations.

12-6* Binomial Distributions Construct and interpret probability distribution; construct and interpret binomial distribution.

Note: In Chapter 12, students investigate these questions: What is the difference between experimental probability and theoretical probability? What does it mean for an event to be random? What is the difference between a frequency table and a contingency table? Note that several sections of this chapter go beyond the PA Core eligible content expected in Geometry. Some of this material may motivate students, though, and may help tie areas of mathematics together. Consider including this material, as time permits. Consider previewing this content by refreshing students’ memories to “investigate chance processes and develop, use, and evaluate probability models” (7.M07.D-S.3), as well as Algebra I probability content. Students should be familiar with many of the ideas underlying this chapter, because data analysis and probability are important threads throughout K-12 mathematics. Chapter 12 deepens students understanding and supports opportunities to model with mathematics. Students can ask and investigate more rich questions of probabilistic situations with these tools. Consider using simulations to model with mathematics and to see how predictions are often counterintuitive in probabilistic situations.

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Remediation and Extension Options

Option 1: Revisit Common

Core Performance

Tasks

Each chapter contains a Common Core Performance Task (CCPT), intended to deepen students’ understanding of the content, as they proceed through the lessons. The CCPT may be previewed before beginning a chapter, and students can share their early ideas on how to approach it. Later, as they develop and strengthen tools for solving the CCPT, they can return to it and approach it in a different light. CCPTs can be used as summative assessments for individual chapter or you can use them to summarize and wrap-up the school year.

Option 2: Reteach or Preteach

Consider teaching the Concept Bytes or Algebra Reviews that you may not have covered yet, which can be a springboard for work in subsequent years. You may also want to check with your colleagues about what content you could pre-teach or what content they would like you to cover or revisit. Some ideas might include solving linear equations, plotting points of quadratic functions and making conjectures about them, or characteristics of functions.

Option 3: Mathematical Modeling in Three Acts

Students may remember Three-Act mathematical modeling activities from Grades 6-8. During the remainder of the year, consider offering students opportunities to engage authentically in the Standards for Mathematical Practice through the Mathematical Modeling activities available on Pearson Realize. You can find additional three-act modeling activities at: https://whenmathhappens.com/3-act-math/, https://gfletchy.com/3-act-lessons/, or within this Google Sheet.

Option 4: Miscellaneous

(Financial Literacy, Quantitative Reasoning,

Statistics, etc.)

The Federal Reserve Bank of Philadelphia has posted free lesson plans for high school teachers on financial literacy at: https://philadelphiafed.org/education/teachers/lesson-plans?tabNum=3. In addition, Census.gov has a number of data analysis activities, using real U.S. Census data, at: https://census.gov/schools/activities/math.html.html. Finally, consider other cross-curricular quantitative reasoning activities, such as those available at: https://ww2.kqed.org/education/.

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PA Core Standards by Cycle

CYCLES

PA Core Standards

Eligible Content ↓ C

C.2

.2.H

S.C

.1

CC

.2.2

.HS.

C.9

CC

.2.3

.8.A

.1

CC

.2.3

.8.A

.2

CC

.2.3

.8.A

.3

CC

.2.3

.HS.

A.1

CC

.2.3

.HS.

A.2

CC

.2.3

.HS.

A.3

CC

.2.3

.HS.

A.5

CC

.2.3

.HS.

A.6

CC

.2.3

.HS.

A.7

CC

.2.3

.HS.

A.8

CC

.2.3

.HS.

A.9

CC

.2.3

.HS.

A.1

1

CC

.2.3

.HS.

A.1

2

CC

.2.3

.HS.

A.1

3

CC

.2.3

.HS.

A.1

4

1 2 3 4

MO

DU

LE 1

G.1.1.1.1 x x x x

G.1.1.1.2 x x x x G.1.1.1.3 x x x x G.1.1.1.4 x x x x G.1.2.1.1 x x x G.1.2.1.2 x x x G. 1.2.1.3 x x x G. 1.2.1.4 x x x G. 1.2.1.5 x x x G.1.3.1.1 x x G.1.3.1.2 x x G.1.3.2.1 x x x x

MO

DU

LE 2

G.2.1.1.1 x x

G.2.1.1.2 x x

G.2.1.2.1 x x x

G.2.1.2.2 x x x

G.2.1.2.3 x x x

G.2.2.1.1 x x x G.2.2.1.2 x x x G.2.2.2.1 x x x G.2.2.2.2 x x x G.2.2.2.3 x x x G.2.2.2.4 x x x G.2.2.2.5 x x x

G.2.2.3.1 x

G.2.2.4.1 x

G.2.3.1.1 x x x x G.2.3.1.2 x x x x G.2.3.1.3 x x x x

G.2.3.2.1 x

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Geometry Eligible Content Taught

PA Eligible Content Cycle

1 Cycle

2 Cycle

3 Cycle

4

G.1.1.1.1 Identify, determine, and/or use the radius, diameter, segment, and/or tangent of a circle. x x x x

G.1.1.1.2 Identify, determine, and/or use the arcs, semicircles, sectors, and/or angles of a circle. x x x x

G.1.1.1.3 Use chords, tangents, and secants to find missing arc measures or missing segment measures. x x x x

G.1.1.1.4 Identify and/or use the properties of a sphere or cylinder. x x x x

G.1.2.1.1 Identify and/or use properties of triangles. x x x

G.1.2.1.2 Identify and/or use properties of quadrilaterals. x x x

G.1.2.1.3 Identify and/or use properties of isosceles and equilateral triangles. x x x

G.1.2.1.4 Identify and/or use properties of regular polygons. x x x

G.1.2.1.5 Identify and/or use properties of pyramids and prisms. x x x

G.1.3.1.1 Identify and/or use properties of congruent and similar polygons or solids. x x

G.1.3.1.2 Identify and/or use proportional relationships in similar figures. x x

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PA Eligible Content Cycle

1 Cycle

2 Cycle

3 Cycle

4

G.1.3.2.1 Write, analyze, complete, or identify formal proofs (e.g., direct and/or indirect proofs/ proofs by contradiction). x x x x

G.2.1.1.1 Use the Pythagorean theorem to write and/or solve problems involving right triangles. x x

G.2.1.1.2 Use trigonometric ratios to write and/or solve problems involving right triangles. x x

G.2.1.2.1 Calculate the distance and/or midpoint between two points on a number line or on a coordinate plane. x x x

G.2.1.2.2 Relate slope to perpendicularity and/or parallelism (limit to linear algebraic equations). x x x

G.2.1.2.3 Use slope, distance, and/or midpoint between two points on a coordinate plane to establish properties of a two-dimensional shape. x x x

G.2.2.1.1 Use properties of angles formed by intersecting lines to find the measures of missing angles. x x x

G.2.2.1.2 Use properties of angles formed when two parallel lines are cut by a transversal to find the measures of missing angles. x x x

G.2.2.2.1 Estimate area, perimeter, or circumference of an irregular figure. x x x

G.2.2.2.2 Find the measurement of a missing length, given the perimeter, circumference, or area. x x x

G.2.2.2.3 Find the side lengths of a polygon with a given perimeter to maximize the area of the polygon. x x x

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PA Eligible Content Cycle

1 Cycle

2 Cycle

3 Cycle

4

G.2.2.2.4 Develop and/or use strategies to estimate the area of a compound/composite figure. x x x

G.2.2.2.5 Find the area of a sector of a circle. x x x

G.2.2.3.1 Describe how a change in the linear dimension of a figure affects its perimeter, circumference, and area (e.g., How does changing the length of the radius of a circle affect the circumference of the circle?). x

G.2.2.4.1 Use area models to find probabilities. x

G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet. x x x x

G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet. x x x x

G.2.3.1.3 Find the measurement of a missing length given the surface area or volume. x x x x

G.2.3.2.1 Describe how a change in the linear dimension of a figure affects its surface area or volume (e.g., How does changing the length of the edge of a cube affect the volume of the cube?). x

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Document Information Page Overview of Contents of Document

The Cover Page lays out the Topics taught within each Cycle as well as the corresponding dates.

The Benchmark Cycle X Standards pages that precede each cycle outline all of the standards that are taught within that cycle. These are, therefore, all of the standards that may be on that Cycle’s Benchmark.

The Benchmark Cycle X Scope and Sequence pages provide suggested pacing that allows for 1 day per lesson plus some flex days. We recommend proactively using the flex days for: reviewing pre-requisite content, splitting lessons over multiple days, assessing, reteaching, and doing projects. Of course, some of these will also be taken by field trips and other school activities.

The PA Core Standards and Eligible Content by Cycle page lists all of the standards and indicates in which cycle(s) they are taught.

What is a Cycle?

We want to offer clarity on what appears on each benchmark. Additionally, there should be sufficient time to teach that content before it is tested. Because each school administers the benchmark on a different day, not necessarily corresponding with the last day of the Term, we have created Cycles. Each Cycle contains the content that is to be taught and tested on a given benchmark. Please refer to the dates on the Cover Page to ensure you are aware of the beginning and ending dates for each Cycle.

What If I Fall Behind?

We trust you to make decisions about what is best for your students. This pacing will prepare you for the Benchmarks and the Keystone Exam, but it is a suggested, not mandated, pacing. You may also wish to move at a faster pace. Do not feel you should slow down to match this guide.

If you are concerned about content that you may not reach before the end of the year, consider implementing number talks and other short routines and games. For example, a lot of mathematics vocabulary and concepts could be taught through Which Once Doesn’t Belong. Rather than pushing to “cover” content, or using test prep resources, content can be infused through short but meaningful structures.

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The Standards for Mathematical Practice are:

1. Make sense of problems and persevere in solving them

2. Reason abstractly and quantitatively

3. Construct viable arguments and critique the reasoning of others

4. Model with mathematics

5. Use appropriate tools strategically

6. Attend to precision

7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

You should try to infuse these Standards into your lessons regularly. Rather than thinking of them as a box to check, think about how you are providing students with opportunities to develop as mathematical thinkers and doers.

The Common Core has identified the following as the critical areas for Geometry:

(1) Students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions, and they use their understanding of triangle congruence and constructing formal proofs about triangles, quadrilaterals, and other polygons;

(2) They identify criteria for similarity of triangles, use similarity to solve similarity and right triangle trigonometry problems; (3) Students extend their experiences with two- and three-dimensional objects, including considering the shapes of cross-sections and

rotations of objects about a line; (4) Students use a rectangular coordinate system to verify geometric relationships; (5) Students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems

about chords, secants, and tangents dealing with segment lengths and angle measures, among others; and (6) They use set theory to compute and interpret theoretical and experimental probabilities for compound events, mutually exclusive events,

independent events, and conditional probability; students should make use of geometric probability models wherever possible.

These do not necessarily reflect content in the PA Core Standards, but they are generally the essential content for preparing students for their studies of higher mathematics, including Algebra II and beyond.

The School District of Philadelphia’s Vision for Mathematics Teaching and Learning:

All students think mathematically, and they will be empowered to own, share, and do mathematics.

Our Guiding Principles:

Equitable Discourse Rich + Meaningful Tasks Purpose-Driven Work

Questioning and Curiosity Valuing Diverse Thinking

If you have any questions or would like more information, please contact us at [email protected].