2013-2014 COURSE NAME: Geometry

2013-2014 COURSE NAME: Geometry `

1

COURSE CODE: 1206310 COURSE NAME: Geometry

UNIT TITLE: GEOMETRY TOOLS

UNIT ESSENTIAL QUESTION:

SEMESTER: 1st

Grading Period: 1st

CONCEPT CONCEPT CONCEPT

Reason quantitatively and use units to solve problems.

1-1 Points, Lines and Planes

1-2 Linear Measure

STANDARD(S) STANDARD(S) STANDARD(S)

MACC.912.N-Q.1.1 Reason quantitatively and use units to solve problems. MACC.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling. MACC.912.N-Q.1.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

LACC.910.RST.2.4 - Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics. (1) MACC.912.G-CO.1.1- Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (2)

MACC.912.G-CO.1.1- Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (1) MACC.912.G-GPE.2.6 : Find the point on a directed line segment between two given points that partitions the segment in a given ratio (2) MACC.912.G-CO.4.12 : Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (3)

LESSON ESSENTIAL QUESTION LESSON ESSENTIAL QUESTION LESSON ESSENTIAL QUESTION

Describe a real-world situation in which you would need to find a perimeter. Then describe a situation in which you would need to find an area.

How can you represent a 3-dimensional figure with a 2-dimensional drawing?

How can you use algebraic equations to determine the measure of a line segment?

How do you use one- and two-dimensional coordinate systems to represent points, lines, rays, line segments, and figures.

http://www.cpalms.org/Standards/PublicPreviewBenchmark5598.aspx











2

VOCABULARY VOCABULARY VOCABULARY

Perimeter • Area • Problem-Solving Undefined term • Point • Line • Plane • Collinear • Coplanar • Intersection • Definition • Defined term • space

• Line segment • Betweenness of points • Between • Congruent segments • Construction

RESOURCES Textbook: Glencoe McGraw-Hill Geometry Problem-Solving Task (illuminations.nctm.org) www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching. Glencoe Lesson 1-6

Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners

glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention Workbook and Word Problem Practice http://tylertarver.com (video) Definition Song (video)

Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention Workbook and Word Problem Practice http://tylertarver.com (video) teachertubemath.com (betweenness of points - video)

Additional Information

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (1) Cognitive Complexity: Level 1: Recall (2) NGSSS Benchmarks: LA.1112.1.6.1 – The student will use new vocabulary that is introduce and taught directly. MA.912.G.8.1 – Analyze the structure of Euclidean geometry as an axiomatic system. Distinguish between undefined terms, definitions, postulates and theorems.

Cognitive Complexity: Level 1: Recall (1) Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2) Cognitive Complexity: Level 1: Recall (3) NGSSS Benchmarks: MA.912.G.1.2 – Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass or a drawing program, explaining and justifying the process used. MA.912.G.8.6 - Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures.

http://illuminations.nctm.org/Lessons/Architect/Architect-AS-ProbSolvTasks.pdf

http://tylertarver.com/

http://www.youtube.com/watch?v=82HDnvFTZwI

http://tylertarver.com/

http://www.teachertubemath.com/


3


1-3 Distance and Midpoint

1-4 Angle Measure 1-5 Angle Relationship Virtually Possible


MACC.912.G-CO.1.1- Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. (1) MACC G.GPE.2.4 Use coordinates to prove simple geometric theorems algebraically. MACC G.GPE.2.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (2) MACC G.GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. (3) MACC.912.G-CO.4.12 : Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (4)

MACC.912.G-CO.1.1- Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

MACC.912.G-CO.1.1- Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.(1) MACC.912.G-CO.4.12 : Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (2)


How do you find the distance between two points? • How do you find the midpoint of a segment?

How do you measure and classify angles? • How do you identify and use congruent angles and the bisector of an angle?

Why is it important to be able to identify and use special pairs of angles?

http://www.cpalms.org/Resources/PublicPreviewResource28789.aspx

















4


• Distance • Midpoint • Segment bisector

Ray • Opposite rays • Angle • Side • Vertex • Interior • Exterior • Degree • Right angle • Acute angle • Obtuse angle • Angle bisector

Adjacent angles • Linear pair • Vertical angles • Complementary angles • Supplementary angles • Perpendicular

RESOURCES

Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice Distance and Midpoint (Notes) www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.

Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.



Cognitive Complexity: Level 1: Recall (1),(3) Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2),(4) NGSSS Benchmarks: MA.912.G.1.1 – Find the lengths and midpoints of line segments in two-dimensional coordinate systems. MA.912.G.1.2 - Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straightedge and compass or a drawing program, explaining and justifying the process used.

Cognitive Complexity: Level 1: Recall NGSSS Benchmarks: MA.912.G.1.2 - Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straightedge and compass or a drawing program, explaining and justifying the process used. MA.912.G.8.4 - Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture.

Cognitive Complexity: Level 1: Recall (1) Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2) NGSSS Benchmarks: MA.912.G.1.2 - Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straightedge and compass or a drawing program, explaining and justifying the process used. MA.912.G.8.2 - Use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards.

http://www.mathplanet.com/education/geometry/points,-lines,-planes-and-angles/finding-distances-and-midpoints


5


1-6 Two-Dimensional Figures 2-D Figures

1-7 Three-Dimensional Figures 3-D Figures


MACC.912.G-CO.1.1- Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.(1) MACC.912.G-CO.4.12 : Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (2) MACC N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.(3)

MACC.912.G-CO.1.1- Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.(1) MACC.912.G-CO.4.12 : Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. (2) MACC N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.(3)


How do you find the perimeter, circumference, and area of two- dimensional figures?

How do you find the surface area and volume of three-dimensional figures?


Polygon • Vertex of a polygon • Concave • Convex • N-gon • Equilateral polygon • Equiangular polygon • Regular polygon • Perimeter • Circumference • area

Polyhedron • Face • Edge • Vertex • Prism • Base • Pyramid • Cylinder • Cone • Sphere • Regular polyhedron • Platonic solid • Surface area • volume

http://www.coolmath.com/reference/geometry-trigonometry-reference.html

http://www.coolmath.com/reference/geometry-trigonometry-reference.html












6

RESOURCES




Cognitive Complexity: Level 1: Recall (1),(3) Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2) NGSSS Benchmarks: MA.912.G.2.5 - Explain the derivation and apply formulas for perimeter and area of polygons (triangles, quadrilaterals, pentagons, etc.). MA.912.G.2.6 – Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.

Cognitive Complexity: Level 1: Recall (1),(3) Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2) NGSSS Benchmarks: MA.912.G.7.1 - Describe and make regular, non-regular, and oblique polyhedra, and sketch the net for a given polyhedron and vice versa. MA.912.G.7.2 - Describe the relationships between the faces, edges, and vertices of polyhedra.


7

English Language Arts Standards (Supported by the Standards for Mathematical Practices) LACC.910.SL.1.1 Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9–10 topics, texts, and issues, building on others’ ideas and expressing their own clearly and persuasively. a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision-making (e.g., informal consensus, taking votes on key issues, presentation of alternate views), clear goals and deadlines, and individual roles as needed. c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. LACC.910.SL.1.2 Integrate multiple sources of information presented in diverse media or formats (e.g., visually, quantitatively, orally) evaluating the credibility and accuracy of each source. LACC.910.SL.1.3 Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. LACC.910.SL.2.4 Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task. LACC.910.RST.1.3 Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. LACC.910.RST.2.4 Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics. LACC.910.RST.3.7 Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.


8

LACC.910.WHST.1.1 Write arguments focused on discipline-specific content. a. Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims, reasons, and evidence. b. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates the audience’s knowledge level and concerns. c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. d. Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing. e. Provide a concluding statement or section that follows from or supports the argument presented. LACC.910.WHST.2.4 Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.

Standards for Mathematical Practices (SMPs) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitively. 3. Construct a viable argument 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.


9


UNIT TITLE: Reasoning and Proof


SEMESTER: 1st

Grading Period: 1


2-1 Inductive Reasoning and Conjecture

2-3 Conditional Statements 2-4 Deductive Reasoning


MACC.912.G-CO.3.9 - Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. MACC.912.G-CO.3.10 - Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MACC.912.G-CO.3.11- Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.



How do you use logical reasoning to prove statements are true and find counterexamples to disprove statements that are false?


10


How is conjecture that is based on inductive reasoning used to solve real-world problems?

How do you write the converse, inverse, and contrapositive of if-then statements?

How can you use deductive reasoning to solve real-world problems?


Inductive reasoning • Conjecture • Counterexample

Conditional statement • If-then statement • Hypothesis • Conclusion • Related conditionals • Converse • Inverse • Contrapositive • Logically equivalent

Deductive reasoning • Valid • Law of Detachment • Law of Syllogism

RESOURCES

Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching. http://www.cvsd.org/centralvalley/classes/math/tirby/cmic_1b.asp (Mrs. Tiera Irby Class @ Central Valley HS)



http://www.cvsd.org/centralvalley/classes/math/tirby/cmic_1b.asp







11


Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmarks LA.910.1.6.5 – The students will relate new vocabulary to familiar words. MA.912.G.8.3 - Determine whether a solution is reasonable in the context of the original situation.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmark MA.912.D.6.1 – Use truth tables to determine truth values of propositional statements.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmark MA.912.D.6.4 - Use methods of direct and indirect proof and determine whether a short

proof is logically valid.


12


2-6 Algebraic Proof http://www.youtube.com/watch?v=U1STnlVGls8

2-7 Proving Segment Relationships Joking with Proofs http://illuminations.nctm.org

2-8 Proving Angle Relationships




MACC.912.G-CO.3.9 - Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.


How do you use properties of equality to write geometric proofs?

What are the basic postulates about points, lines, and planes needed to write paragraph proofs?

How do you write proofs involving supplementary, complementary, congruent, and right angles?

http://www.youtube.com/watch?v=U1STnlVGls8

http://illuminations.nctm.org/LessonDetail.aspx?id=L823


13


Algebraic proof • Two-column proof • Formal proof

Ruler postulate • Segment addition postulate

Supplementary angles • Complementary angle • Protractor Postulate • Angle Addition Postulate • Properties of angle congruence • Symmetric property of congruence

RESOURCES







14


Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmarks MA.912.G.8.4 Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. MA.912.G.8.5 Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two-column, and indirect proofs.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmarks MA.912.G.8.5 Write geometric proofs, including proofs by contradiction and proofs

involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two column, and indirect proofs.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmarks MA.912.G.8.5 Write geometric proofs, including proofs by contradiction and proofs

involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two column, and indirect proofs.


15



16




17


UNIT TITLE: Parallel and Perpendicular Lines


SEMESTER:

Grading Period:

CONCEPT CONCEPT CONCEPT 3-1 Parallel Lines and Transversals 3-2 Angles and Parallel Lines

3-5 Proving Lines Parallel






What information can be used to identify the relationships between two lines or two planes?

What theorems can used to determine the relationship between specific pairs of angles?

How do you prove that two lines are parallel using angle relationships?








18


Parallel lines • Skew lines • Parallel planes • Transversal • Interior angle • Exterior angles • Consecutive interior angles • Alternative interior angles • Alternative exterior angles • Corresponding angles

Corresponding angles postulate • Parallel lines and angle pairs • Alternate interior angles theorem

Converse of corresponding angles postulate • Parallel postulate

RESOURCES

http://www.cvsd.org/centralvalley/classes/math/tirby/cmic_1b.asp (Mrs. Tiera Irby Class @ Central Valley HS) Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.


http://www.cvsd.org/centralvalley/classes/math/tirby/cmic_1b.asp (Mrs. Tiera Irby Class @ Central Valley HS)

Textbook: Glencoe McGraw-Hill Geometry Additional Concepts: 3-3 Slopes of Lines & 3-4 Equations of Lines http://www.mathopenref.com/coordintro.html

Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.





http://www.mathopenref.com/coordintro.html


19


Cognitive Complexity: Level 1: Recall (1) Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2) NGSSS Benchmark

LA.1112.1.6.1 – The student will use new vocabulary that is introduced and taught directly. MA.912.G.1.3 - Identify and use the relationships between special pairs of angles formed by parallel lines and transversals.

Standard(s): MA.912.G.1.3 - Identify and use the relationships between special pairs of angles formed by parallel lines and transversals.

Standard(s): MA.912.G.8.3 Determine whether a solution is reasonable in the context of the original situation.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmark MA.912.G.1.3 - Identify and use the relationships between special pairs of angles formed by parallel lines and transversals.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmark MA.912.G.1.2 - Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straightedge and compass or a drawing program, explaining and

justifying the process used.


20


3-6 Perpendicular and Distance




How do you find the distance between a point and a line?

VOCABULARY VOCABULARY VOCABULARY Equidistant • Perpendicular postulate







21

RESOURCES



Cognitive Complexity: Level 1: Recall (1) Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2) NGSSS Benchmark MA.912.G.1.2 - Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straightedge and compass or a drawing program, explaining and

justifying the process used.




22



23




24


UNIT TITLE: Congruent Triangles


SEMESTER: 1st

Grading Period: 4th


4-1 Classifying Triangles 4-2 Angles of Triangles 4-3 Congruent Triangles


MACC.912.G-MG.1.1 : Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).


MA CC G.SRT2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


What are the characteristics of the different classifications of triangles?

How does the Triangle Angle-Sum Theorem show the relationship among the interior angle measures of any triangle?

How do you prove triangles congruent using the definition of congruence?

VOCABULARY VOCABULARY VOCABULARY Equiangular triangle • Obtuse triangle • Right triangle • Equilateral triangle • Isosceles triangle • Scalene triangle

Auxiliary line • Exterior angle • Remote interior angles • Flow proof

Congruent • Congruent polygons • Corresponding parts











25

RESOURCES





Cognitive Complexity: Level 1: Recall (1) NGSSS Benchmarks MA.912.G.4.1 - Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. MA.912.G.8.6 - Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures.

Cognitive Complexity: Level 1: Recall (1) NGSSS Benchmarks MA.912.G.2.2 - Determine the measures of interior and exterior angles of polygons, justifying the method used. MA.912.G.8.5 - Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two-column, and indirect proofs.

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts (2) NGSSS Benchmarks MA.912.G.4.4 - Use properties of congruent and similar triangles to solve problems involving lengths and areas. MA.912.G.4.6 - Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.




26


4-4 Proving Triangles Congruent-SSS, SAS 4-5 Proving Triangles Congruent-ASA, AAS 4-6 Isosceles and Equilateral Triangles




MACC G.CO.3.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MACC G.CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. MA CC G.SRT2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


How do you use the SSS and SAS postulate to test for triangle congruence?

How do you use the ASA and AAS postulate to test for triangle congruence?

What are the characteristics of an isosceles triangle?


Included angle • Side-side-side (SSS) congruence • Side-angle-side (SAS) congruence

Included side • Angle-side-angle (ASA) congruence • Angle-angle-side (AAS) congruence • Angle-angle-side theorem

Legs of an isosceles triangle • Vertex angle • Base angle


27

RESOURCES





Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmarks MA.912.G.4.6 - Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. MA.912.G.4.8 – Use coordinate geometry to prove properties of congruent, regular, and similar triangles.

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmarks MA.912.G.4.6 - Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. MA.912.G.4.8 – Use coordinate geometry to prove properties of congruent, regular, and similar triangles.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmarks LA.910.1.6.5 – The student will relate new vocabulary to familiar words. MA.912.G.4.1 - Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular.








28


4-7 Congruence Transformations

4-8 Triangles and Coordinate Proof


MACC G.CO.3.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MA CC G.SRT2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

MACC G.CO.3.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. MA CC G.SRT2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


How do you verify congruence after a congruence transformation?

How are coordinate proofs used to prove geometric concepts.


Transformation • Congruence transformation • Preimage • Isometry • Reflection • Translation • rotation

coordinate proof


29

RESOURCES




Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmarks MA.912.G.2.4 - Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. MA.912.G.2.6 – Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane. MA.912.G.4.3 - Construct triangles congruent to given triangles.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmarks MA.912.D.6.4 - Use methods of direct and indirect proof and determine whether a short proof is logically valid. MA.912.G.4.8 – Use coordinate geometry to prove properties of congruent, regular, and similar triangles.


30


UNIT TITLE: Relationships in Triangles


SEMESTER: 1

Grading Period: 2nd


5-1 Bisectors of Triangles 5-2 Median and Altitude of Triangles 5-3 Inequalities in one Triangle


MACC G.C.1.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

MACC G.CO.3.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point

MACC G.CO.3.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point


How do you identify and use perpendicular bisectors in triangles?

How do you identify and use medians and altitudes in triangles?

How do you recognize and apply properties of inequalities to the measures of the angles of a triangle?


Perpendicular bisector • Concurrent lines • Point of concurrency • Circumcenter • incenter

median • centroid • altitude • orthocenter

exterior angle inequality • angle-side relationships in triangles

How can the historical development of geometric systems show how mathematics was developed for a variety of purposes?


31

RESOURCES





Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning NGSSS Benchmarks MA.912.G.4.1 - Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. MA.912.G.4.2 - Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter.

Cognitive Complexity: Level 3: Strategic Thinking & Complex Reasoning Cognitive Complexity: Level 2: Basic Application of Skills & Concepts Cognitive Complexity: Level 1: Recall NGSSS Benchmarks LA.1112.6.1 – The student will use new vocabulary that is introduced and taught directly. MA.912.G.4.2 - Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter.

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmarks LA.910.1.6.5 – The student will relate new vocabulary to familiar words. MA.912.G.4.7 - Apply the inequality theorems: triangle inequality, inequality in one triangle, and the Hinge Theorem.


32


5-5 The Triangle Inequality

5-6 Inequalities in Two Triangles


MACC G.CO.3.10 Prove theorems about

triangles. Theorems include: measures of

interior angles of a triangle sum to 180

degrees; base angles of isosceles triangles

are congruent; the segment joining midpoints

of two sides of a triangle is parallel to the

third side and half the length; the medians

of a triangle meet at a point

MACC G.CO.3.10 Prove theorems about

triangles. Theorems include: measures of

interior angles of a triangle sum to 180

degrees; base angles of isosceles triangles

are congruent; the segment joining midpoints

of two sides of a triangle is parallel to the

third side and half the length; the medians

of a triangle meet at a point


How do you prove triangle relationships using the Triangle Inequality Theorem?

How do you apply the Hinge Theorem or its converse to make comparisons in two triangles?


The Triangle Inequality • The Hinge Theorem

The Hinge Theorem

RESOURCES




33


Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmarks LA.1112.1.6.2 – The student will listen to, read, and discuss familiar and conceptually challenging text. MA.912.G.4.7 - Apply the inequality theorems:

triangle inequality, inequality in one triangle, and the Hinge Theorem.

Cognitive Complexity: Level 2: Basic Application of Skills & Concepts NGSSS Benchmark MA.912.G.4.7 Apply the inequality theorems:

triangle inequality, inequality in one triangle, and the Hinge Theorem.


34

COURSE NAME: Geometry

UNIT TITLE: Quadrilaterals


SEMESTER: 1st

Grading Period: 2nd

CONCEPT CONCEPT CONCEPT 6-1 Angles of Polygons 6-2 Parallelograms

6-3 Tests for Parallelogram



MACC.912.G-MG.1.1 : Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). MACC G.SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures

MACC G.CO.3.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other and its converse, rectangles are parallelograms with congruent diagonals. MACC G.SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


How do you find and use the sum of the measures of the interior and exterior of a polygon?

How do you recognize and apply the properties of the sides, angles, and diagonals of parallelograms?

What are the conditions that ensure a quadrilateral is a parallelogram?


Diagonal • Polygon Interior Angles Sum

Parallelogram • Properties of Parallelograms • Diagonals of Parallelograms

Conditions for Parallelograms

How do you use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons?










35

RESOURCES

FLDOE PearsonAccess ePat http://jcs.k12.oh.us/teachers/kstepp/Geometry/ Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.




NGSSS Benchmarks MA.912.G.2.2 - Determine the measures of interior and exterior angles of polygons, justifying the method used. MA.912.G.3.4 - Prove theorems involving

quadrilaterals.

NGSSS Benchmarks MA.912.G.3.1 - Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite. MA.912.G.3.4 - Prove theorems involving

quadrilaterals.

NGSSS Benchmarks MA.912.G.3.1 - Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite.

MA.912.G.3.3 - Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals.


36


6-4 Rectangles 6-5 Rhombi and Squares 6-6 Trapezoids and Kites






How do you determine whether parallelograms are rectangles?

How do you determine whether quadrilaterals are rectangles, rhombi, or squares?

What properties can be used to find missing measures in trapezoids and kites?


rectangle

Rhombus • Square

Trapezoid • Bases • Legs of a trapezoid • Base angles • Isosceles trapezoid • Midsegment of a trapezoid • kite


37

RESOURCES

Mr. Techner’s Math http://nrsteachers.org/teschner/ Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.




NGSSS Benchmarks MA.912.G.3.1 - Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite. MA.912.G.3.3 - Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals.




38


UNIT TITLE: Proportions and Similarity


SEMESTER: 2nd

Grading Period: 3rd


7-1 Ratios and Proportions 7-2 Similar Polygons

7-3 Similar Triangles


MACC G.SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


MACC G.GPE.2.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). MACC G.SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.


How do you use a ratio to compare two quantities?

How do you use proportions to identify similar polygons?

How do you use the AA Similarity Postulate and the SSS and SAS Similarity Theorem to identify similar triangles?


Ratio • Extended ratios • Proportion extremes • Means • Cross products • Fibonacci sequence • golden ratio

Similar polygons • Scale factor

Angle-Angle (AA) Similarity • Properties of Similarity

How do you formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models?


39

RESOURCES

Teschner's Math http://nrsteachers.org/teschner/GeoCH8.htm Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.




NGSSS Benchmarks MA.912.D.11.5 – Explore and use other sequences found in nature such as the Fibonacci sequence and the golden ratio.

NGSSS Benchmarks MA.912.G.2.3 - Use properties of congruent and similar polygons to solve mathematical or real-world problems. MA.912.G.3.3 - Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals.

NGSSS Benchmarks MA.912.G.4.6 - Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. MA.912.G.4.8 – Use coordinate geometry to prove properties of congruent, regular, and similar triangles.


40


7-4 Parallel Lines and Proportional Parts

7-5 Parts of Similar Triangles

7-6 Similarity Transformation


MACC G.SRT.2.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally and its converse; the Pythagorean Theorem proved using triangle similarity.


MACC G.SRT.2.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. MACC G.CO.1.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).


How can you determine if two triangles are proportional by using the Angle-Angle Similarity Postulate?

How do you use properties of congruent and similar polygons to solve mathematical or real-world problems?

How do you verify similarity after a similarity transformation?

VOCABULARY VOCABULARY VOCABULARY Midsegment of a triangle

Special Segments of Similar Triangles • Triangle Angle Bisector Theorem

Dilation • Similarity transformation • Center of dilation • Factor of a dilation • Enlargement • Reduction


41

RESOURCES





NGSSS Benchmarks MA.912.G.4.5 - Apply theorems involving segments divided proportionally. MA.912.G.4.6 - Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.

NGSSS Benchmarks MA.912.G.2.3 - Use properties of congruent and similar polygons to solve mathematical or real-world problems. MA.912.G.4.6 - Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.

NGSSS Benchmarks MA.912.G.2.4 - Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. MA.912.G.2.6 – Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.


42


7-7 Scale Drawings and Models




How do you use a scale factor to solve a real-world problem?

VOCABULARY VOCABULARY VOCABULARY Scale model • Scale drawing • Scale • Scale factor

RESOURCES



NGSSS Benchmark MA.912.G.8.4 - Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture.


43


UNIT TITLE: Right Triangles and Trigonometry


SEMESTER: 2nd

Grading Period: 3rd

CONCEPT CONCEPT CONCEPT 8-1 Geometric Mean

8-2 The Pythagorean Theorem and its Converse

8-3 Special Right Triangles



MACC G.SRT.2.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally and its converse; the Pythagorean Theorem proved using triangle similarity. MACC G.SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

MACC G.SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.


How do you solve problems involving relationships between parts of a right triangle and the altitude to its hypotenuse?

How do you use the Pythagorean Theorem to solve real-world problems?

How can you use the properties of special right triangles to solve real-world problems?


Geometric mean

Pythagorean triple

45-45-90 Triangle Theorem • 30-60-90 Triangle Theorem

How do you use properties of right triangle and trigonometric ratios to solve meaningful problems?


44

RESOURCES

http://teachers.henrico.k12.va.us/math/igo/07RightTriangles/7_1.html (Lecture Notes, Classwork, and Practice) Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.




NGSSS Benchmarks MA.912.G.4.5 - Apply theorems involving segments divided proportionally. MA.912.G.5.2 - State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.

NGSSS Benchmarks MA.912.G.5.1 - Prove and apply the Pythagorean Theorem and its converse. MA.912.G.5.4 - Solve real-world problems involving right triangles.

NGSSS Benchmarks MA.912.G.5.3: Use special right triangles (30° - 60° - 90° and 45° - 45° - 90°) to solve problems. MA.912.G.5.4: Solve real-world problems involving right triangles.

http://teachers.henrico.k12.va.us/math/igo/07RightTriangles/7_1.html

http://teachers.henrico.k12.va.us/math/igo/07RightTriangles/7_1.html


45


8-4 Trigonometry

8-5 Angles and Elevation and Depression

8-7 Vectors


MACC G.MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). MACC G.SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. MACC G.SRT.3.7 Explain and use the relationship between the sine and cosine of complementary angles.

MACC G.SRT.3.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

NGSSS Benchmark MA.912.D.9.3 – Use vectors to model and solve application problems.


How do you use trigonometric ratios to find angle measures in right triangles?

How do you use angles of elevation and depression to find the distance between two objects?

How do you find magnitudes and direction of vectors?


Trigonometry • Trigonometric ratio • Sine • Cosine • Tangent • Inverse sine • Inverse cosine • Inverse tangent • Law of Sines • Law of Cosines

Angle of elevation • Angle of depression

Vector • Standard position • Component form • Magnitude • Direction • Resultant • Parallelogram method • Triangle method


46

RESOURCES

8-6 The Law of Sines and Law of Cosines Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.




NGSSS Benchmarks MA.912.T.2.1 - Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles MA.912.G.5.4 - Solve real-world problems involving right triangles.

NGSSS Benchmarks LA.1112.1.6.2 – The student will listen to, read, and discuss familiar and conceptually challenging text. MA.912.G.5.4 Solve real-world problems involving right triangles

NGSSS Benchmarks LA.1112.1.6.1 – The student will use new vocabulary that is introduced and directly taught. MA.912.D.9.3 – Use vectors to model and solve application problems.


47


UNIT TITLE: Transformations and Symmetry


SEMESTER: 2nd

Grading Period: 4th

CONCEPT CONCEPT CONCEPT 9-1 Reflections

9-2 Translations

9-3 Rotations


MACC G.CO.2.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. MACC G.CO.1.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. MACC G.CO.1.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). MACC G.CO.1.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.




How do you use congruence transformation to make conjectures and justify properties of geometric figures?


48

How do you use the line of reflection to draw reflections in the coordinate plane?

How do you draw translations in the coordinate plane?

How do you draw rotations in the coordinate plane?


Line of reflection

Reflection, line of reflection

Center of rotation • Angle of rotation

RESOURCES





NGSSS Benchmarks MA.912.G.2.4 - Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons.

MA.912.G.2.6 – Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.



NGSSS Benchmarks MA.912.G.2.4 - Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. MA.912.G.2.6 – Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.


49


9-4 Composition of Transformations

9-5 Symmetry

9-6 Dilations


MACC G.CO.2.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. MACC G.CO.2.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MACC G.CO.2.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

MACC G.SRT.1.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of side MACC G.SRT.1.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

MACC G.SRT.1.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of side


How do you draw glide reflections and other compositions of isometries in the coordinate plane?

How do you identify line and rotational symmetries in two-dimensional and three-dimensional figures?

How do you draw dilations in the coordinate plane?

VOCABULARY VOCABULARY VOCABULARY Composition of transformation • Glide reflection

Symmetry • Line symmetry • Line of symmetry • Rotational symmetry • Center of symmetry • Order of symmetry • Magnitude of symmetry

Dilations


50

RESOURCES







NGSSS Benchmarks LA.910.1.6.5 – The student will relate new vocabulary to familiar words.

MA.912.G.2.4 - Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons.




51


UNIT TITLE: Circles


SEMESTER: 2

Grading Period: 4th


10-1 Circles and Circumference

10-2 Measuring Angles and Arcs

10-4 Inscribed Angles


MACC G.CO.1.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MACC G.C.2.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. MACC G.C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.



How do you use proportional reasoning to find areas of sectors and arc lengths of circles?


52


How do you solve problems involving the circumference of a circle?

What are the characteristics of central angles, major arcs, minor arcs, and semicircles?

How do you find measures of angles of inscribed polygons?


circle • Center • Radius • Cord • Diameter • Congruent circles • Concentric circles • Circumference • Pi • Inscribed • circumscribed

• central angle • arc • minor arc • major arc • semicircle • congruent arcs • adjacent arcs

Inscribed angle • Intercepted arc

RESOURCES


10-3 Arcs and Chords (Glencoe textbook) Mr. Techner’s Math http://nrsteachers.org/teschner/ Textbook: Glencoe McGraw-Hill Geometry Differentiated Instruction – Approaching –Level, On-Level, Beyond-Level, English Learners glencoe.com (personal tutor) Textbook: Scaffolding Questions Glencoe - Student Guide and Intervention workbook and Word Problem Practice www.teachingtoday.glencoe.com – Gives secondary teachers practical strategies and material that inspire excellence and innovation in teaching.



NGSS Benchmarks MA.912.G.6.1 - Determine the center of a given circle. Given three points not on a line, construct the circle that passes through them. Construct tangents to circles. Circumscribe and inscribe circles about and within triangles and regular polygons. MA.912.G.6.2 - Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles.

NGSS Benchmarks MA.912.G.6.2 - Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles. MA.912.G.6.4 - Determine and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).

NGSS Benchmarks MA.912.G.6.1 - Determine the center of a given circle. Given three points not on a line, construct the circle that passes through them. Construct tangents to circles. Circumscribe and inscribe circles about and within triangles and regular polygons. MA.912.G.6.4 - Determine and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).


53


10-5 Tangents

10-7 Special Segments in a Circle

10-8 Equations of Circles




MACC G.CO.1.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MACC G.C.2.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.


How do you solve problems involving circumscribed polygons?

How do you find measures of segments that intersect in the interior or exterior of a circle?

How do you write and graph the equation of a circle?

VOCABULARY VOCABULARY VOCABULARY Tangent • Point of tangency • Common tangent • secant

Chord segment • Secant segment • External secant segment • Tangent segment

Compound locus


54

RESOURCES

10-6 Secants, Tangents, and Angle (Glencoe textbook)





NGSS Benchmarks MA.912.G.6.1 - Determine the center of a given circle. Given three points not on a line, construct the circle that passes through them. Construct tangents to circles. Circumscribe and inscribe circles about and within triangles and regular polygons. MA.912.G.6.2 - Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles.

MA.912.G.6.4 - Determine and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).

NGSS Benchmarks MA.912.G.6.3 – Prove theorems related to circles, including related angles, chords, tangents, and secants.

MA.912.G.6.4 - Determine and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).

NGSS Benchmarks MA.912.G.6.6 - Given the center and the radius, find the equation of a circle in the coordinate plane or given the equation of a circle in center-radius form, state the center and the radius of the circle. MA.912.G.6.7 - Given the equation of a circle in center-radius form or given the center and the

radius of a circle, sketch the graph of the circle.


55


UNIT TITLE:

Areas of Polygons and Circles


SEMESTER: 2nd

Grading Period: 4th

CONCEPT CONCEPT CONCEPT 11-1 Areas of Parallelograms and Triangles 11-2 Areas of Trapezoids, Rhombi, and Kites

11-3 Areas of Circles and Sectors


MACC G.MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). MACC G.GPE.2.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

MACC G.MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

MACC G.CO.1.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. MACC G.C.2.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. MACC G.C.1.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. MACC G.C.1.1 Prove that all circles are similar.

How do you find the area of regular polygons, circles, and composite figures using proportional reasoning?


56


How do you find perimeters and areas of parallelograms and triangles?

How do you find areas of trapezoids, rhombi and kites?

How do you use the formula for the circumference of a circle to develop the formula for the area of a circle?


Base of a parallelogram • Height of a parallelogram • Base of a triangle • Height of a triangle

Height of a trapezoid

Sector of a circle • Segment of a circle

RESOURCES





NGSSS Benchmarks MA.912.G.2.5 - Explain the derivation and apply formulas for perimeter and area of polygons (triangles, quadrilaterals, pentagons, etc.).

MA.912.G.2.6 - Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.



NGSSS Benchmarks MA.912.G.2.6 - Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.

MA.912.G.6.5 - Solve real-world problems using measures of circumference, arc length, and areas of circles and sectors.


57


11-4 Areas of Regular Polygons and Composite Figures

11-5 Areas of Similar Figures


MACC G.CO.4.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. MACC G.MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

MA CC G.GMD.1.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.


How do you find areas of regular polygons and composite figures?

How do you use scale factors to find areas of similar figures?

VOCABULARY VOCABULARY VOCABULARY Center of a regular polygon • Radius of a regular polygon • Apothem • Central angle of a regular polygon • Composite figure

Scale factor


58

RESOURCES






NGSSS Benchmarks MA.912.G.2.6 - Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane.

MA.912.G.2.7 – Determine how changes in dimensions affect the perimeter and area of common geometric figures.


59


UNIT TITLE: Extending Surface Area and Volume


SEMESTER: 2

Grading Period: 4th

CONCEPT CONCEPT CONCEPT 12-1 Representations of Three-Dimensional Figures http://www.youtube.com/watch?v=rtgJRrQ1nRQ

12-2 Surface Areas of Prisms and Cylinders

12-3 Surface Areas of Pyramids and Cones


MACC G.MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). MACC G.GMD2.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

MA CC G.GMD.1.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. MACC G.MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).



How do use isometric dot paper to draw isometric views, or corner views, or three-dimensional geometric solids on two-dimensional paper?

How do you find areas and surface areas of pyramids and cones?

How do you find areas and surface areas of pyramids and cones?

VOCABULARY VOCABULARY VOCABULARY Isometric view • Cross section

Lateral face • Lateral edge • Base edge • Altitude • Height • Lateral area • axis • composite solid

regular pyramids • slant height • right cone • oblique cone

What is the effect on area and volume when one or more dimensions of a figure are changed?


60

RESOURCES





NGSSS Benchmark MA.912.G.7.3 – Identify, sketch, and find areas and/or perimeters of cross sections of solid objects.

NGSSS Benchmarks MA.912.G.7.5 - Explain and use formulas for lateral area, surface area, and volume of solids. MA.912.G.7.7 - Determine how changes in dimensions affect the surface area and volume of common geometric solids.



61


12-4 Volumes of Prisms and Cylinders

12-5 Volumes of Pyramids and Cones

12-6 Surface Areas and Volumes of Spheres






How do you find volumes of prisms and cylinders?

How can knowing the volume of pyramids and cones be helpful in solving real-world problems?

How do you find surface areas and volumes of spheres?

VOCABULARY VOCABULARY VOCABULARY Volume of a Prism • Volume of a Cylinder

Volume of a pyramid • Volume of a cone Great circle • Pole • hemisphere

RESOURCES





62






63


12-7 Spherical Geometry

12-8 Congruent and Similar Solids


MACC G.MG.1.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). MACC G.MG.1.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).


What are the differences between Euclidean geometry and Spherical geometry?

How do you identify congruent or similar solids?

VOCABULARY VOCABULARY VOCABULARY Euclidean geometry • Spherical geometry • Non-Euclidean geometry

Similar solids • Congruent solids

RESOURCES




64


NGSSS Benchmarks LA.1112.1.6.2 – The student will listen to, read, and discuss familiar and conceptually challenging text. MA.912.G.7.4 - Identify chords, tangents, radii, and great circles of spheres.

NGSSS Benchmark MA.912.G.7.6 - Identify and use properties of congruent and similar solids.

Documents

2013-2014 COURSE NAME: Geometry