geometry pearson high school 2017-2018 sy · ch. 11 to ch. 13 ch. 1 to ch. 3 g 1/30 ......

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

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Geometry Pearson High School 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

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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 – Tools of Geometry 2 – Reasoning and Proof 3-1 – 3-6 – Parallel &

Perpendicular Lines

3-7 – 3-8 – Parallel & Perpendicular Lines (cont.)

4 – Congruent Triangles 5 – Relationships within

Triangles 6 – Polygons and Quadrilaterals

7 – Similarity 8 – Triangles and Trigonometry 9 – Transformations 10-1 – 10-5 – Area

10-6 – 10-8 – Area (cont.) 11 – Surface Area and Volume 12 – Circles 13 - 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.

Note that most chapters contain a mix of Module 1 and Module 2 Eligible Content (G.1 aligned items identify Module 1 and G.2 identify Module 2). In addition, there are some lessons that go beyond the PA Core Eligible Content but that are part of the PA Core Standards. Unaligned material, for possible extension, is marked with “Beyond Eligible Content.”


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Table of Contents

Block Schedule Options ....................................................................................................................................................... Error! Bookmark not defined.

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

Benchmark Cycle 1 Scope and Sequence ........................................................................................................................................................................... 6

Benchmark Cycle 2 Standards ........................................................................................................................................................................................... 10

Benchmark Cycle 2 Scope and Sequence ......................................................................................................................................................................... 12

Benchmark Cycle 3 Standards ........................................................................................................................................................................................... 17

Benchmark Cycle 3 Scope and Sequence ......................................................................................................................................................................... 18

Cycle 4 Standards .............................................................................................................................................................................................................. 22

Cycle 4 Scope and Sequence ............................................................................................................................................................................................. 25

PA Core Standards by Cycle .............................................................................................................................................................................................. 31

Geometry Eligible Content Taught ................................................................................................................................................................................... 32

Document Information Page ............................................................................................................................................................................................ 35


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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

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

Full

Blo

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Fall

9/5– 10/4 Ch. 1 to Ch. 3

10/5 – 10/31 Ch. 4 to Ch. 6

11/1 – 12/8 Ch. 7 to Ch. 10

12/11 – 1/29 Ch. 11 to Ch. 13

Full

Blo

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Spri

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1/30 – 2/28 Ch. 1 to Ch. 3

3/1 – 3/28 Ch. 4 to Ch. 6

4/3 – 5/2 Ch. 7 to Ch. 10

5/3 – 6/12 Ch. 11 to Ch. 13

Benchmark Schedule

A /

B

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

Full

Blo

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Fall

Benchmarks 1 and 2 (No need to complete

in one sitting)

None (Benchmark 3 not taken)

Full

Blo

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Spri

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Benchmarks 1 and 2

(No need to complete in one sitting)

None (Benchmark 3 not taken)


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Benchmark Cycle 1 Standards

PA Eligible Content PA Core Standards

M04.D-M.3.1.1 Measure angles in whole-number degrees using a protractor. With the aid of a protractor, sketch angles of specified measure.

(No aligned PA Core Standards for HS Geometry. Prerequisites.)

M04.C-G.1.1.1 Draw points, lines, line segments, rays, angles (right, acute, and obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

M04.D-M.3.1.2 Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems.

M06.C-G.1.1.5 Represent three-dimensional figures using nets made of rectangles and triangles.

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

CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles. CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles. CC.2.3.HS.A.13 Analyze relationships between two‐dimensional and three‐dimensional objects.

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

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

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

CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.13 Analyze relationships between two‐dimensional and three‐dimensional objects.

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.


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

CC.2.3.HS.A.1 Use geometric figures and their properties to represent transformations in the plane. CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence. CC.2.3.HS.A.5 Create justifications based on transformations to establish similarity of plane figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures.

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.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.

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

CC.2.3.8.A.3 Understand and apply the Pythagorean theorem to solve problems. CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.

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.

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

CC.2.3.8.A.2 Understand and apply congruence, similarity, and geometric transformations using various tools. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures.

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.

CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles.

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

(No aligned EC item) CC.2.3.HS.A.4 Apply the concept of congruence to create geometric constructions.

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


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Benchmark Cycle 1 Scope and Sequence

Chapter 1: Tools of Geometry

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic(s) Eligible Content

9/5

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1-1 Nets and Drawings for Visualizing Geometrya

Make nets and drawings of three-dimensional figures.

M06.C-G.1.1.5, 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

1-2* Points, Lines, and Planes Understand basic terms and postulates of geometry. M04.C-G.1.1.1

G.1.3.1.1

1-3 Measuring Segments Find and compare lengths of segments. G.1.3.1.1

1-4* Measuring Angles Find and compare the measures of angles.

M04.D-M.3.1.1, M04.D-M.3.1.2,

G.1.1.1.2, G.2.2.1.1

1-5* Exploring Angle Pairs Identify special angle pairs and use their relationships to find angle measures.

G.2.2.1.1

Concept Byteb

Compass Designs Find complements and supplements of angles that complete straight and right angles.

CC.2.3.HS.A.4 1-6* Basic Constructions Make basic constructions using a straight edge and a compass.

Concept Byte

Exploring Constructions Use geometry software to make points, lines, and planes and complete constructions.

a A Solve It! begins every lesson, providing students opportunities to strengthen their problem-solving skills. Students draw on prior knowledge and connect with key concepts from the lesson before refining solution methods. The Solve It! is an essential component of the program.

b Concept Bytes are to be used either before or after lessons (as indicated) to build conceptual understanding, support development of problem-solving skills, to engage with the PA Core Standards for Mathematical Practice, to make conjectures from paper constructions, and to integrate learning with technology (consider Google Sheets, TI calculators, Desmos, or Geogebra for these).

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


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Chapter 1: Tools of Geometry (cont.)

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic(s) Eligible Content (s

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1-7* Midpoint and Distance in the Coordinate Plane

Find the midpoint of a segment, and find the distance between two points in the coordinate plane.

G.2.1.2.1

Concept Byte

Partitioning a Line Segment Apply proportional reasoning to partition a directed line segment in the coordinate plane.

G.1.3.1.2, G.2.1.2.3

1-8* Perimeter, Circumference, and Area

Find the perimeter or circumference of basic shapes, and find the area of basic shapes.

G.1.1.1.1, G.1.2.1.1, G.1.2.1.2, G.2.1.2.1, G.2.2.2.1, G.2.2.2.4

Concept Byte

Comparing Perimeters and Areas Use a graphing calculator or spreadsheet software to find maximum values for the area and perimeter of a rectangular figure.

Focus: 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.

Coherence: As noted above, students should be familiar with many of the concepts and definitions in this chapter from elementary and middle grades mathematics, but may not recall the formal geometric notation.

Rigor: Studying geometry presents regular opportunities in thinking mathematically, or drawing on the Standards of Mathematical Practice (SMPs) of the PA Core Standards. In particular, students should have opportunities to ask questions, make sense of problems, and use reasoning.

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.

Also consider using flex days, on occasion, to complete or present the Common Core Performance Tasks or Mathematical Modeling in Three Acts activities (found on Pearson Realize), which support increased rigor. Review foundational knowledge and check for understanding through “Do Nows” or “Bell Ringers,” homework, and teacher-created assessments.


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Chapter 2: Reasoning and Proof

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic Eligible Content

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2-1* Patterns and Inductive Reasoning Use inductive reasoning to make conjectures.

G.1.3.2.1

2-2 Conditional Statements Reorganize conditional statements and their parts, and write converse, inverse, and contrapositives of conditional statements.

Concept Byte

Logic and Truth Tables Use logic and truth tables to determine truth values of compound statements.

2-3 Biconditionals and Definitions Write biconditionals and recognize good definitions.

2-4 Deductive Reasoning Use the Law of Detachment and the Law of Syllogisms.

2-5 Reasoning in Algebra and Geometry

Connect reasoning in algebra and geometry.

2-6* Proving Angles Congruent Prove and apply theorems about angles.

Focus: 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?

Coherence: 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.

Rigor: 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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Chapter 3: Parallel and Perpendicular Lines

Suggested Dates

Chapter- Lesson


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3-1 Lines and Angles

Identify relationships between figures in space, and identify angles formed by two lines and a transversal.

G.1.3.1.1, G.2.2.1.2

Concept Byte

Parallel Lines and Related Angles

Use geometry software to create, manipulate, and make conjectures about angles formed by two parallel lines and a transversal.

CC.2.3.HS.A.4, G.1.3.1.1, G.2.2.1.2

3-2* Properties of Parallel Lines Prove theorems about and use properties of parallel lines, including finding angle measures.

G.2.2.1.1, G.2.2.1.2

3-3* Proving Lines Parallel Determine whether two lines are parallel.

3-4 Parallel and Perpendicular Lines

Relate parallel and perpendicular lines.

Concept Byte

Perpendicular Lines and Planes

Make discoveries about the relationships of perpendicular planes and lines contained in those planes.

CC.2.3.HS.A.13

3-5* Parallel Lines and Triangles Use parallel lines to prove a theorem about triangles, find measures of angles of triangles.

G.1.2.1.1, G.2.2.1.1, G.2.2.1.2

Concept Byte

Exploring Spherical Geometry Explore spherical geometry. G.1.1.1.4

CC.2.3.HS.A.13, CC.2.3.HS.A.14

3-6* Construct Parallel and Perpendicular Lines

Construct parallel and perpendicular lines. CC.2.3.HS.A.4

Focus: 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. If time is needed, consider omitting Concept Bytes.

Coherence: In this chapter, students develop arguments about familiar geometric objects, now using the tools of logic (Chapter 2).

Rigor: Higher rigor is supported when arrangements of lines are presented and students are asked, “What math questions can we generate?”

Benchmark 1 Window: Nov 1 – Nov 17


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A1.1.2.2.1 Write and/or solve a system of linear equations (including problem situations) using graphing, substitution, and/or elimination. Note: limit systems to two linear equations.

CC.2.1.HS.F.5 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. CC.2.2.8.B.3 Analyze and solve linear equations and pairs of simultaneous linear equations. CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.

A1.1.2.2.2 Interpret solutions to problems in the context of the problem situation. Note: Limit systems to two linear equations.

A1.1.3.1.1 Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities).

CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method.

A1.1.3.1.2 Identify or graph the solution set to a linear inequality on a number line.

CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships.

A1.1.3.1.3 Interpret solutions to problems in the context of the problem situation. Note: linear inequalities only.

CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multi-step problems.



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

CC.2.3.HS.A.1 Use geometric figures and their properties to represent transformations in the plane. CC.2.3.HS.A.2 Apply rigid transformations to determine and explain congruence. CC.2.3.HS.A.5 Create justifications based on transformations to establish similarity of plane figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures.

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


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

CC.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate to geometric figures. CC.2.3.HS.A.6 Verify and apply theorems involving similarity as they relate to plane figures. CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles.

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

CC.2.3.8.A.3 Understand and apply the Pythagorean theorem to solve problems. CC.2.3.HS.A.11 Apply coordinate geometry to prove simple geometric theorems algebraically.

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.

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




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Chapter 3: Parallel and Perpendicular Lines (cont.)

Suggested Dates

Chapter- Lesson


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3-7* Equations of Lines in the Coordinate Plane

Graph and write linear equations.

G.2.1.2.2, G.2.1.2.3

3-8* Slopes of Parallel and Perpendicular Lines

Relate slope to parallel and perpendicular lines.

Focus: 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. If time is needed, consider omitting Concept Bytes.

Coherence: In this chapter, students develop arguments about familiar geometric objects, now using the tools of logic (Chapter 2).

Rigor: Higher rigor is supported when arrangements of lines are presented and students are asked, “What math questions can we generate?”


13

Focus: Chapter 4 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? If time is needed in your schedule, consider omitting the Concept Bytes after Chapter 4-1 and 4-3.

Coherence: Note that the Algebra Review can help support connections between geometry and algebra and maintain algebra skills. Students learn that objects may not simply look congruent, but that they must be proven so by logic.

Rigor: Make connections to real-world applications, to the extent possible, in order to help students model with mathematics.

Chapter 4: Congruent Triangles

Suggested Dates

Chapter- Lesson


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4-1* Congruent Figures Recognize congruent figures and their corresponding parts.

G.1.2.1.1, G.1.3.1.1

Concept Byte

Building Congruent Triangles Build and compare triangles in an effort to find shortcuts in identifying congruent triangles.

4-2* Triangle Congruence by SSS and SAS

Prove triangles congruent using the SSS and SAS Postulates.

4-3* Triangle Congruence by ASA and AAS

Prove two triangles congruent using the ASA Postulate and the AAS Theorem.

Concept Byte

Exploring AAA and SSA Use geometry software to explore whether triangles can be proved congruent using AAA or SSA.

4-4 Using Corresponding Parts of Congruent Triangles

Use triangle congruence and corresponding parts of congruent triangles to prove that parts of two triangles are congruent.

Concept Byte

Paper-Folding Conjectures Use paper-folding activities to reveal important properties of isosceles triangles.

G.1.2.1.1, G.1.2.1.3, G.1.3.1.1 4-5*

Isosceles and Equilateral Triangles

Use and apply properties of isosceles and equilateral triangles.

Algebra Review

Systems of linear equations Review how to solve a system of linear equations in two variables using the substitution method.

A1.1.2.2.1, A1.1.2.2.2

4-6* Congruence in Right Triangles Prove right triangles congruent using the Hypotenuse-Leg Theorem.

G.1.2.1.1, G.1.2.1.3, G.1.3.1.1 4-7*

Congruence in Overlapping Triangles

Identify congruent overlapping triangles, and prove these triangles congruent using other triangles.


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Focus: In Chapter 5, students connect coordinate geometry with their study of triangles. 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?

Coherence: In this chapter, students connect their understanding of Algebra I with their understanding of proof, to make arguments about shapes in the plane. These skills are essential for working with objects in Algebra II and beyond.

Rigor: Paper folding and geometry software (like Desmos or Geogebra) can help students make conjectures and gain a practical sense of geometric properties before using formal logical arguments.

Chapter 5: Relationships Within Triangles

Suggested Dates

Chapter- Lesson


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Concept

Byte Investigating Midsegments

Use geometry software to investigate properties of triangle midsegments.

G.1.2.1.1, G.1.3.1.2, G.2.1.2.3 5-1 Midsegments of Triangles Use properties of midsegments to solve problems.

5-2 Perpendicular and Angle Bisectors

Use properties of perpendicular bisectors and angle bisectors.

G.1.2.1.1, G.1.3.1.1, G.2.1.2.3

Concept Byte

Paper Folding Bisectors Use paper-folding activities to investigate angle bisectors of a triangle.

5-3 Bisectors in Triangles Identify properties of perpendicular bisectors and angle bisectors.

Concept Byte

Special Segments in Triangles Use geometry software to explore special segments in triangles.

G.1.2.1.1, G.1.3.1.1, G.1.3.1.2, G.2.1.2.2, G.2.1.2.3

5-4 Medians and Altitudes Identify properties of medians and altitudes of a triangle.

5-5* Indirect Proof Indirect reasoning to write proofs. G.1.3.2.1

Algebra Review

Solving Inequalities Review what the solution to an inequality represents, and review the steps to solving inequalities.

A1.1.3.1.2, A1.1.3.1.3

5-6* Inequalities in One Triangle Use inequalities involving angles and sides of triangles. G.1.2.1.1

5-7* Inequalities in Two Triangles Apply inequalities of two triangles.


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Chapter 6: Polygons and Quadrilaterals

Suggested Dates

Chapter- Lesson


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6-1*

The Polygon-Angle Sum Theorems

Find the sum of measures of the interior angles of a polygon, and find the sum of the measures of the exterior angles of a polygon.

G.1.2.1.1

6-2* Properties of Parallelograms Use relationships among sides, angles, and diagonals of parallelograms.

G.1.2.1.2 6-3 Proving that a Quadrilateral is a Parallelogram

Determine whether a quadrilateral is a parallelogram.

6-4 Properties of Rhombuses, Rectangles, and Squares

Define and classify special types of parallelograms, and use properties of diagonals of rhombuses and rectangles.

6-5* Conditions for Rhombuses, Rectangles, and Squares

Determine whether a parallelogram is a rhombus or a rectangle.

G.1.2.1.2, G.1.2.1.4

6-6* Trapezoids and Kites Verify and use properties of trapezoids and kites. G.1.2.1.2


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Chapter 6: Polygons and Quadrilaterals (cont.)

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic Eligible Content (s

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Algebra Review

Simplifying Radicals Recognize when a radical expression is in simplest form, and simplify radical expressions.

A1.1.1.3.1

6-7 Polygons in the Coordinate Plane

Classify polygons in the coordinate plane. G.2.1.2.1, G.2.1.2.2, G.2.1.2.3 6-8*

Applying Coordinate Geometry

Name coordinates of special figures by using their properties.

Concept Byte

Quadrilaterals in Quadrilaterals

Use geometry software to explore quadrilaterals formed when connecting the midpoints of a quadrilateral are joined.

CC.2.3.HS.A.4, G.2.1.2.1, G.2.1.2.2, G.2.1.2.3

6-9* Proofs Using Coordinate Geometry

Prove theorems using figures in the coordinate plane. G.2.1.2.1, G.2.1.2.2, G.2.1.2.3

Focus: Chapter 6 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?

Coherence: Students draw on the classification of geometric objects that they began in early elementary school, but now they formalize these ideas in high school. The Algebra Review will help with studying the Pythagorean Theorem in Geometry and quadratics in Algebra II.

Rigor: Arguments about quadrilaterals, using formal notation, help students with the precise language of geometry.

Benchmark 2 Window: Jan 29 – Feb 13


17



A1.1.3.1.1 Write or solve compound inequalities and/or graph their solution sets on a number line (may include absolute value inequalities).

CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method.

A1.1.3.1.2 Identify or graph the solution set to a linear inequality on a number line.

CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships.



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

CC.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.

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

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



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.8 Apply geometric theorems to verify properties of circles. CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles.



18


Chapter 7: Similarity

Suggested Dates

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7-1* Ratios and Proportions Write ratios and solve proportions. G.1.3.1.2

7-2* Similar Polygons Identify and apply similar polygons. G.1.3.1.1, G.1.3.1.2 Concept

Byte Fractals Introduce fractals and their properties.

7-3* Proving Triangles Similar Use the AA Postulate and the SAS and SSS Theorems, and use similarity to find indirect measurements. G.1.2.1.1,

G.1.3.1.2 7-4 Similarity in Right Triangles Find and use relationships in similar right triangles.

Concept Byte

The Golden Ratio Explore the golden ratio and how it relates to the Fibonacci sequence.

G.1.3.1.1, G.1.3.1.2

Concept Byte

Exploring Proportions in Triangles

Use geometry software to investigate proportions in triangles. G.1.2.1.1,

G.1.3.1.2 7-5 Proportions in Triangles

Use the Side-Splitter Theorem and the Triangle-Angle Bisector Theorem.

Focus: Chapter 7 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? If time is a concern, omit the Concept Bytes on fractals and the golden ratio. Note, also, that some examples in 7-4 require the quadratic formula. Therefore, only solve quadratic equations in this chapter that do not require factoring or the quadratic formula, since these are topics in Algebra II. If you choose to offer the Algebra Review on solving quadratics, be aware that many students may not have seen this material before now.

Coherence: Students connect their understanding of proportions with their understanding of similarity and self-similarity. Similarity is the foundational idea underlying trigonometry and other topics studied later in Geometry, in Algebra II, and beyond.

Rigor: As with previous units, encouraging students to formulate and tackle their own questions (such as in the Concept Byte on similar triangles), or asking students to solve proportions problems in various ways and explain their own reasoning (formal or algebraic, as well as informal or numerical), can support higher levels of rigor.


19

Chapter 8: Triangles and Trigonometry

Suggested Dates

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Concept

Byte The Pythagorean Theorem

Help students understand why the Pythagorean Theorem works.

G.2.1.1.1

8-1* The Pythagorean Theorem and Its Converse

Use the Pythagorean Theorem and Its Converse

8-2* Special Right Triangles Use the properties of 45-45-90 and 30-60-90 triangles.

G.2.1.1.2

Concept Byte

Exploring Trigonometric Ratios Use geometry software to explore the trigonometric ratios of sine, cosine, and tangent.

8-3* Trigonometry Use the sine, cosine, and tangent ratios to determine sine lengths and angle measures in right triangles.

Concept Byte

Complementary Angles and Trigonometric Ratios

Explain and use the relationship between the sine and cosine of complementary angles.

8-4 Angles of Elevation and Depression

Use angles of elevation and depression to solve problems.

8-5 Law of Sines Apply the Law of Sines. Beyond Eligible

Content 8-6 Law of Cosines Apple the Law of Cosines.

Focus: In Chapter 8, 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?

Coherence: 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.

Rigor: 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.


20

Chapter 9: Transformations

Suggested Dates

Chapter- Lesson


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Concept

Byte Tracing Paper Transformations

Perform translations, reflections, and rotations of geometric figures on coordinate planes by using tracings. G.1.3.1.1,

G.1.3.1.2 9-1* Translations Identify rigid motions, find translation images of figures.

Concept Byte

Paper Folding and Reflections Use paper-folding to see how a figure and its reflection image are related, and construct a reflection image using a compass and straightedge.

CC.2.3.HS.A.4, G.1.3.1.1, G.1.3.1.2

9-2 Reflections Find reflection images of figures.

G.1.3.1.1, G.1.3.1.2

9-3* Rotations Draw and identify rotations of figures.

Concept Byte

Symmetry Identify reflectional (line), rotational, and point symmetry using reflections and rotations.

9-4 Compositions and Isometries Compositions of isometries, including glide reflections, and classify isometries.

9-5* Triangle Congruence Identify congruence transformations, and prove triangle congruence using isometries.

Concept Byte

Exploring Dilations Dilate lines and geometric figures and make conjectures about properties of dilations.

9-6 Dilations Understand dilation images of figures.

9-7 Similarity Transformations Identify similarity transformations and verify properties of similarity.

Focus: Chapter 9 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?

Coherence: 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.

Rigor: Consider introducing the Common Core Performance Task before starting this chapter and revisiting it, throughout, so that students can try new strategies to make sense of the problem and to make an argument on how to solve it.


21

Chapter 10: Area

Suggested Dates

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Concept Byte

Transforming to Find Area Derive formulas for the areas of polygons. G.2.2.2.4

10-1* Areas of Parallelogram and Triangles

Find the area of parallelograms and triangles.

G.2.2.2.2 10-2* Areas of Trapezoids, Rhombuses, and Kites

Find the area of a trapezoid, rhombus, or kite.

10-3 Areas of Regular Polygons Find the area of a regular polygon.

10-4* Perimeters and Areas of Similar Figures

Find the perimeters and areas of similar polygons. G.2.2.3.1

10-5 Trigonometry and Area Find the areas of regular polygons and triangles using trigonometry.

G.2.2.2.4

Focus: In Chapter 10, 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?

Coherence: Students have studied area since elementary school. They formalize and extend their understanding here.

Rigor: Here, they attend to precision and make use of structure by finding areas of more complex shapes.

Benchmark 3 Window: Apr 9 – Apr 25


22

Cycle 4 Standards


A1.1.2.1.1 Write, solve, and/or apply a linear equation (including problem situations).

CC.2.2.HS.D.7 Create and graph equations or inequalities to describe numbers or relationships. CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.

A1.1.2.1.2 Use and/or identify an algebraic property to justify any step in an equation-solving process. Note: Linear equations only

CC.2.2.HS.D.8 Apply inverse operations to solve equations or formulas for a given variable. CC.2.2.HS.D.9 Use reasoning to solve equations and justify the solution method. CC.2.2.HS.D.10 Represent, solve, and interpret equations/inequalities and systems of equations/inequalities algebraically and graphically.

A1.1.2.1.3 Interpret solutions to problems in the context of the problem situation. Note: Linear equations only.

CC.2.1.HS.F.3 Apply quantitative reasoning to choose and interpret units and scales in formulas, graphs, and data displays. CC.2.1.HS.F.4 Use units as a way to understand problems and to guide the solution of multi-step problems.

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.

CC.2.4.HS.B.1 Summarize, represent, and interpret data on a single count or measurement variable. CC.2.4.HS.B.3 Analyze linear models to make interpretations based on the data. CC.2.4.HS.B.5 Make inferences and justify conclusions based on sample surveys, experiments, and observational studies.

A1.2.3.3.1 Find probabilities for compound events (e.g., find probability of red and blue, find probability of red or blue) and represent as a fraction, decimal, or percent.

CC.2.4.7.B.3 Investigate chance processes and develop, use, and evaluate probability models. CC.2.4.HS.B.4 Recognize and evaluate random processes underlying statistical experiments. CC.2.4.HS.B.7 Apply the rules of probability to compute probabilities of compound events in a uniform probability model.

A2.2.1.1.4 Identify and/or determine the characteristics of an exponential, quadratic, or

CC.2.1.HS.F.7 Apply concepts of complex numbers in polynomial identities and quadratic equations to solve problems.


23


polynomial function (e.g., intervals of increase/decrease, intercepts, zeros, and asymptotes).

CC.2.2.HS.C.1 Use the concept and notation of functions to interpret and apply them in terms of their context. CC.2.2.HS.C.2 Graph and analyze functions, and use their properties to make connections between the different representations. CC.2.2.HS.C.3 Write functions or sequences that model relationships between two quantities. CC.2.2.HS.C.5 Construct and compare linear, quadratic, and exponential models to solve problems. CC.2.2.HS.C.6 Interpret functions in terms of the situations they model. CC.2.3.HS.A.10 Translate between the geometric description and the equation for a conic section. CC.2.4.HS.B.2 Summarize, represent, and interpret data on two categorical and quantitative variables.

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

CC.2.4.HS.B.4 Recognize and evaluate random processes underlying statistical experiments. CC.2.4.HS.B.5 Make inferences and justify conclusions based on sample surveys, experiments, and observational studies. CC.2.4.HS.B.6 Use the concepts of independence and conditional probability to interpret data. CC.2.4.HS.B.7 Apply the rules of probability to compute probabilities of compound events in a uniform probability model.

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.

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

CC.2.3.HS.A.8 Apply geometric theorems to verify properties of circles. CC.2.3.HS.A.9 Extend the concept of similarity to determine arc lengths and areas of sectors of circles.


24


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

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

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

CC.2.2.HS.C.9 Prove the Pythagorean identity and use it to calculate trigonometric ratios. CC.2.3.HS.A.7 Apply trigonometric ratios to solve problems involving right triangles.

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



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

G.2.2.4.1 Use area models to find probabilities. 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.

CC.2.3.8.A.1 Apply the concepts of volume of cylinders, cones, and spheres to solve real‐world and mathematical problems. CC.2.3.HS.A.12 Explain volume formulas and use them to solve problems. CC.2.3.HS.A.14 Apply geometric concepts to model and solve real‐ world problems. 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.

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.13 Analyze relationships between two‐dimensional and three‐dimensional objects.


25

Cycle 4 Scope and Sequence

Chapter 10: Area (cont.)

Suggested Dates

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10-6 Circles and Arcs Find the measures of central angles and arcs, and find the circumference of arc length.

G.1.1.1.1, G.1.1.1.2, G.2.2.2.1

Concept Byte

Radian Measure Explore radian measure and convert between radians and degrees.

G.2.1.1.2

10-7 Areas of Circles and Sectors Find the areas of circles, sectors, and segments of circles. G.2.2.2.1, G.2.2.2.4, G.2.2.2.5

Concept Byte

Inscribed and Circumscribed Figures

Calculate, explore, and make conjectures about the circumference and area of a circle and the perimeter and area of regular polygons that are inscribed in and circumscribed about the circle.

G.2.2.2.1

10-8* Geometric Probability Use segment and area models to find probabilities of events. G.2.2.4.1

Focus: In Chapter 10, 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? If time is a concern, you could omit the Concept Byte after Chapter 10-7.

Coherence: Students have studied area since elementary school. They formalize and extend their understanding here.

Rigor: Here, they attend to precision and make use of structure by finding areas of more complex shapes.


26

Chapter 11: Surface Area and Volume

Suggested Dates

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11-1* Space Figures and Cross Sections Recognize polyhedrons and their parts, and visualize cross sections of space figures. G.2.3.1.1,

G.2.3.1.2, G.2.3.1.3 Concept

Byte Perspective Drawing Introduce perspective drawing.

Algebra Review

Literal Equations Review how to solve an equation for one variable in terms of another.

A1.1.2.1.1, A1.1.2.1.2, A1.1.2.1.3

11-2* Surface Areas of Prisms and Cylinders

Find the surface area of a prism and a cylinder. G.2.3.1.1, G.2.3.1.2, G.2.3.1.3

11-3* Surface Areas of Pyramids and Cones

Find the surface area of a pyramid and a cone.

11-4* Volumes of Prisms and Cylinders Find the volume of a prism and the volume of cylinder.

G.1.1.1.4, G.2.3.1.1, G.2.3.1.2, G.2.3.1.3


27

Chapter 11: Surface Area and Volume (cont.)

Suggested Dates

Chapter- Lesson

Lesson Title Lesson Topic Eligible Content (s

ee p

revi

ou

s)

Concept Byte

Finding Volume Derive formulas for finding the volumes of pyramids and cones.

G.2.3.1.1, G.2.3.1.2, G.2.3.1.3 11-5* Volumes of Pyramids and Cones Find the volume of a pyramid and a cone.

11-6* Surface Areas and Volumes of Spheres

Find the surface area and volume of a sphere.

G.1.1.1.4, G.2.3.1.1, G.2.3.1.2, G.2.3.1.3

Concept Byte

Exploring Similar Solids Use a spreadsheet to explore similar solids and how ratios of surface areas and ratios of volumes are related to scale factor.

G.2.3.2.1 11-7*

Areas and Volumes of Similar Solids

Compare and find the areas and volumes of similar solids.

Focus: Chapter 11 involves three-dimensional geometry. The following essential questions are addressed: How can you determine the intersection of a solid and a plane? How can you find the surface area and volume of a solid? How do the surface areas and volumes of similar solids compare?

Coherence: Students have studied three-dimensional objects, especially their properties and names, since elementary school. In this chapter, they build upon this understanding by making connections to area (Chapter 10).

Rigor: 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.


28

Chapter 12: Circles

Suggested Dates

Chapter- Lesson


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12-1* Tangent Lines Use properties of a tangent to a circle. G.1.1.1.1

Concept Byte

Paper Folding with Circles Use paper-folding activities to explore properties of chords.

G.1.1.1.2, G.1.1.1.3

12-2* Chords and Arcs Use congruent chords, arcs, and central angles, and use perpendicular bisectors of chords.

12-3 Inscribed Angles Find the measure of an inscribed angle, and find the measure of an angle formed by a tangent and a chord.

Concept Byte

Exploring Chords and Secants Explore various properties of chords and secants.

12-4 Angle Measures and Segment Lengths

Find measures of angles formed by chords, secants, and tangents, and find the length of segments associated with circles.

G.1.1.1.1

12-5 Circles in the Coordinate Plane Write the equation of a circle, and find the center and radius of a circle.

A2.2.1.1.4 Concept

Byte Equation of a Parabola Write the equation of a parabola using the focus and directrix.

12-6 Locus: A Set of Points Draw and describe a locus.

Focus: In Chapter 12, 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?

Coherence: 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.

Rigor: 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.


29

Chapter 13: Probability

Suggested Dates

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13-1

Experimental and Theoretical Probability

Calculate experimental and probability. A2.2.3.2.1,

13-2 Probability Distributions and Frequency Tables

Make and use frequency tables and probability distributions. A1.2.3.2.1

13-3* Permutations and Combinations Use permutations and combinations to solve problems. A1.2.3.3.1, A2.2.3.2.1

13-4 Compound Probability and Probability of Multiple Events

Identify independent and dependent events, and find compound probabilities.

A1.2.3.3.1, G.2.2.4.1,

A2.2.3.2.1, A2.2.3.2.2, A2.2.3.2.3

13-5 Probability Models Construct and use probability models. A2.2.3.2.1, A2.2.3.2.3

13-6 Conditional Probability Formulas Understand and calculate conditional probabilities.

A2.2.3.2.1, A2.2.3.2.2, A2.2.3.2.3

13-7* Modeling Randomness Understand random numbers, and use probabilities in decision making.

Concept Byte

Probability and Decision Making Use probability concepts in decision making.

Focus: In Chapter 13, 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.

Coherence: 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 13 deepens students understanding and supports opportunities to model with mathematics.

Rigor: 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.


30

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/.

https://whenmathhappens.com/3-act-math/

https://gfletchy.com/3-act-lessons/

https://docs.google.com/spreadsheets/d/1jXSt_CoDzyDFeJimZxnhgwOVsWkTQEsfqouLWNNC6Z4/edit#gid=0

https://philadelphiafed.org/education/teachers/lesson-plans?tabNum=3

https://census.gov/schools/activities/math.html.html

https://ww2.kqed.org/education/


31

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

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

MO

DU

LE 2

G.2.1.1.1 x

G.2.1.1.2 x x

G.2.1.2.1 x x

G.2.1.2.2 x

G.2.1.2.3 x x

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

G.2.2.3.1 x

G.2.2.4.1 x

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

G.2.3.2.1 x 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

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

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

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

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

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

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

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

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

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

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

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


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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

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

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

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


36

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].

geometry pearson high school 2017-2018 sy · ch. 11 to ch. 13 ch. 1 to ch. 3 g 1/30 ......

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