Geometry Unit 1: Foundations of Geometry, Lines &
Angles (Gr. 9 - 11)
Content Area:
Course(s):
Time Period:
Length:
Status:
Mathematics
Geometry
1st Marking Period
7 Weeks
Published
Unit Overview As this unit begins, students will develop an understanding of basic geometric terminology, constructions, line and angle relationships. Following this, students will begin to discuss logical reasoning as they explore patterns, evaluate the validity of statements and provide counterexamples as appropriate. In addition, students will learn to recognize and write conditional statements, their converses, and biconditional statements. Building upon all of this, students will explore the properties of parallel and perpendicular lines and begin to explore deductive reasoning and write formal geometric proofs. Transfer Students will be able to independently use their learning to... •
Relate geometric terminology to real‐world settings. •
Create formal constructions of basic geometric figures. •
Use available information to reach logical conclusions and construct logical arguments. •
Critique the reasoning of others, providing counterexamples as appropriate. •
Understand and apply the attributes of, and relationships between, lines, transversals, and angles, both analytically and synthetically (in and outside of the coordinate plane). Meaning Understandings Students will understand that... •
The terms point, line, and plane are undefined terms upon which all other geometric concepts are based. •
The relationships that exist between specific angle pairs are used in solving for unknown measures. •
Mathematical statements are either true or false. •
Statements can be proven false with one counterexample. •
Logical mathematical arguments, known as proofs, are used to prove that certain statements are true, and to establish rules in geometry. •
The phrase "if and only if" implies that both a conditional statement and its converse are true. •
Specific angle relationships can be used to distinguish between parallel and non‐parallel lines. Conversely, when parallel lines are intersected by a transversal, unique relationships exist between specific angle pairs. •
When working in the coordinate plane, the slopes of two lines can be used to determine whether the lines are parallel, perpendicular, or neither. Essential Questions Students will keep considering... •
How can geometric figures and their properties be described through careful use of geometric language? •
How can the unique properties of geometric figures be used to determine new information? •
How can available information and logical reasoning be used to develop and prove conjectures? •
When can mathematical properties effectively be written as biconditional statements? •
How can formal geometric constructions be created using a variety of tools? Application of Knowledge and Skill Students will know... •
The meaning of basic geometric terminology and symbols. •
Postulates describing the relationships between points, lines, and planes. •
That bisectors divide angles or segments into two congruent parts. •
The relationships between linear pairs, vertical, supplementary, and complementary angles. •
The distance and midpoint formulas. •
What is meant by writing conjectures. •
That conditional statements and their converses contain both a hypothesis and a conclusion. •
That biconditional statements are used to combine conditional statements and their converses when both statements are true. •
That counterexamples can be used to disprove statements. •
That formal geometric proofs are logical arguments which contain reasons to support each statement that is made. •
That most rules in geometry come in the form of theorems, which are statements that have been proven. •
Which angle pairs are congruent and which are supplementary when parallel lines are intersected by a transversal. •
That specific angle relationships formed when two lines are interesected by a transversal can be used to show that two lines are parallel. •
The properties of perpendicular lines. •
The relationships between the slopes of parallel and perpendicular lines in the coordinate plane. Students will be skilled at... •
Using geometric symbols and terminology correctly. •
Using given information and diagrams to make accurate conclusions about points, lines, and planes. •
Using the segment and angle addition postulates to write accurate equations and solve for unknowns. •
Using the relationships between linear pairs, vertical. supplementary, complementary, and bisected angles to write accurate equations and solve for unknowns. •
Making formal geometric constructions to copy a segment, copy an angle, bisect a segment, bisecting an angle, construct perpendicular lines, including the perpendicular bisector of a line segment, and construct a line parallel to a given line through a point not on the line. •
Using the distance formula to calculate the length of a segment or measurement needed to find the area or perimeter of a polygon. •
Calculating the areas and perimeters of simple polygons. •
Using the midpoint formula to find the coordinates of an unknown midpoint or endpoint of a endpoint of a segment. •
Creating and using examples to write valid conjectures. •
Reading and interpreting conditional and biconditional statements. •
Judging the validity of statements and writing counterexamples to disprove them as appropriate. •
Writing simple geometric proofs involving basic angle relationships, parallel and perpendicular lines. •
Solving for unknowns when parallel lines and a transversal are given, writing and using equations as needed. •
Using known angle relationships to determine when two lines are parallel or perpendicular. •
Determing the slope of a line given a sketch of the line, the coordinates of two points on the line, or an equation of the line. •
Using their slopes to establish relationships between lines in the coordinate plane. •
Using point‐slope and slope‐intercept form to write the equation of a line. Academic Vocabulary •acute angle •adjacent angle •alternate exterior angles •alternate interior angles •angle •angle bisector •area •base •between •biconditional statement •bisect •collinear •complementary angles •conclusion •conditional statement •congruent •congruent angles •congruent segments •conjecture •construction •converse •coordinate •coordinate plane •coplanar •corresponding angles •counterexample •deductive reasoning •definition •degree •distance •distance from a point to a line •endpoint •exterior of an angle •height •hypotenuse •hypothesis •image •inductive reasoning •interior of an angle •leg •length •line •linear pair •measure •midpoint •obtuse angle •opposite rays •parallel lines •parallel planes •perimeter •perpendicular bisector •perpendicular lines •plane •point •point‐slope form •polygon •postulate •preimage .
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•proof •quadrilateral •ray •right angle •rise •run •same‐side exterior angles •same‐side interior angles •segment •segment bisector •skew lines •slope •slope‐intercept form •straight angle •supplementary angles •theorem •transversal •triangle •two‐column proof •undefined term •vertex •vertical angles .
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Learning Goal 1.1 Students will understand basic geometric terminology, create and use sketches and solve problems involving angle and segment measures, and will create formal geometric constructions to illustrate basic concepts. Objective 1.1.1 (Level of Difficulty: 1 - Recall) SWBAT identify, name, and sketch points, lines, segments, rays, and planes, as well as explain the basic relationships that exist among them. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐CO.A.1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 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. 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. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Objective 1.1.2 (Level of Difficulty: 2 - Skill) SWBAT calculate segment lengths and angle measures, as well as use the Segment and Angle Addition Postulates to solve for unknown measurements. MA.9‐12.CCSS.Math.Content.HSG‐CO.A.1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP7 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. Reason abstractly and quantitatively. Use appropriate tools strategically. Look for and make use of structure. Objective 1.1.3 (Level of Difficulty: 3 - Strategic Thinking) SWBAT make formal geometric constructions of: •
congruent segments •
segment bisectors •
congruent angles •
angle bisectors MA.9‐12.CCSS.Math.Content.HSG‐CO.A.1 MA.9‐12.CCSS.Math.Content.HSG‐CO.D.12 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 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. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Attend to precision. Objective 1.1.4 (Level of Difficulty: 3 - Strategic Thinking) SWBAT identify adjacent, vertical, complementary, and supplementary angles, and will be able to use given information to solve for unknowns in related problems. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 1.1.5 (Level of Difficulty: 3 - Strategic Thinking) SWBAT apply the distance and midpoint formulas to solve related problems. Given the coordinates of their vertices, students will solve for areas and perimeters of basic polygons, using the distance formula as needed. MA.9‐12.CCSS.Math.Content.HSG‐GPE.B.7 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Make sense of problems and persevere in solving them. Model with mathematics. Look for and make use of structure. Learning Goal 1.2 Students will be able to identify patterns, write and evaluate conjectures in the form of conditional and biconditional statements, and provide counterexamples as appropriate. Objective 1.2.1 (Level of Difficulty: 3 - Strategic Thinking) SWBAT develop examples to illustrate mathematical properties, identify patterns and use inductive reasoing to make conjectures. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 LA.9‐10.CCSS.ELA‐Literacy.WHST.9‐10.1e
MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 MA.K‐12.CCSS.Math.Practice.MP8 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. Provide a concluding statement or section that follows from or supports the argument presented. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Objective 1.2.2 (Level of Difficulty: 4 - Extended Thinking) SWBAT develop counterexamples to disprove given conjectures. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 1.2.3 (Level of Difficulty: 3 - Strategic Thinking) SWBAT identify, write, and analyze the truth values of conditional statements and their converses. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 LA.9‐10.CCSS.ELA‐Literacy.CCRA.R.4 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 1.2.4 (Level of Difficulty: 3 - Strategic Thinking) SWBAT write and analyze biconditional statements. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 LA.9‐10.CCSS.ELA‐Literacy.CCRA.R.4 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and analyze how specific word choices shape meaning or tone. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Learning Goal 1.3 Students will prove simple theorems involving lines and angles. Objective 1.3.1 (Level of Difficulty: 4 - Extended Thinking) SWBAT use properties of equality and congruence to justify mathematical statements and write simple algebraic proofs. MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 1.3.2 (Level of Difficulty: 4 - Extended Thinking) SWBAT use a diagram and given information to construct a simple two‐column deductive proof relating lines and angles. (Including proof that vertical angles are congruent.) MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Learning Goal 1.4 Students will be able to prove and apply theorems involving parallel lines, transversals, and perpendicular lines. Objective 1.4.1 (Level of Difficulty: 1 - Recall) SWBAT identify each of the following: •
parallel lines •
perpendicular lines •
skew lines •
transversals •
corresponding angles •
alternate interior angles •
altermate exterior angles •
same‐side interior angles •
same‐side exterior angles LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐CO.A.1 MA.K‐12.CCSS.Math.Practice.MP6 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. 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. Attend to precision. Objective 1.4.2 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove theorems about angles formed by parallel lines and a transversals, as well as use these theorems to solve for unknown measures. Theorems should include: •
When a transversal crosses parallel lines, corresponding angles are congruent. •
When a transversal crosses parallel lines, alternate interior angles are congruent. •
When a transversal crosses parallel lines, alternate exterior angles are congruent. •
When a transversal crosses parallel lines, same‐side interior angles are supplementary. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.9 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 Prove theorems about lines and angles. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Attend to precision. MA.K‐12.CCSS.Math.Practice.MP7 MA.K‐12.CCSS.Math.Practice.MP8 Look for and make use of structure. Look for and express regularity in repeated reasoning. Objective 1.4.3 (Level of Difficulty: 4 - Extended Thinking) SWBAT use angles formed when a transversal intersects coplanar lines to prove that two lines are parallel, as well as use these relationships to solve for unknown measures. Theorems should include: •
If two coplanar lines are cut by a transversal so that alternate interior angles are congruent, then the two lines are parallel. •
If two coplanar lines are cut by a transversal so that alternate exterior angles are congruent, then the two lines are parallel. •
If two coplanar lines are cut by a transversal so that same‐side interior angles are supplementary, then the two lines are parallel. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.9 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 MA.K‐12.CCSS.Math.Practice.MP8 Prove theorems about lines and angles. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Objective 1.4.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT formally construct a line parallel to a given line through a point not on the line. MA.9‐12.CCSS.Math.Content.HSG‐CO.D.12 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP5 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Objective 1.4.5 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove and apply theorems about perpendicular lines. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.9 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about lines and angles. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 1.4.6 (Level of Difficulty: 4 - Extended Thinking) SWBAT formally construct: •
perpendicular lines •
perpendicular bisectors of given line segments. MA.9‐12.CCSS.Math.Content.HSG‐CO.D.12 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. MA.K‐12.CCSS.Math.Practice.MP5 Use appropriate tools strategically. Learning Goal 1.5 Students will solve problems involving parallel and perpendicular lines in the coordinate plane. Objective 1.5.1 (Level of Difficulty: 2 - Skill) SWBAT use slopes of lines to identify parallel and perpendicular lines in the coordinate plane. MA.9‐12.CCSS.Math.Content.HSG‐GPE.B.5 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 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). Reason abstractly and quantitatively. Model with mathematics. Look for and make use of structure. Learning Goal 1.5.2 (Level of Difficulty: 2 - Skill) SWBAT write the equation of a line parallel or perpendicular to a given line that passes through a given point. MA.9‐12.CCSS.Math.Content.HSG‐GPE.B.5 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 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). Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Look for and make use of structure. 21st Century Life and Careers WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.A.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.F.2 WORK.9‐12.9.3.12.C.6 The ability to recognize a problem and apply critical thinking and problem‐solving skills to solve the problem is a lifelong skill that develops over time. Apply critical thinking and problem‐solving strategies during structured learning experiences. Critical thinking and problem solving in the 21st century are enhanced by the ability to work in cross‐cultural teams in face‐to‐face and virtual environments. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Leadership abilities develop over time through participation in groups and/or teams that are engaged in challenging or competitive activities. Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences. Develop job readiness skills by participating in structured learning experiences and employment seeking opportunities. Summative Assessment •Projects •Quizzes •Student Portfolios •Tests •Unit 1 Assessment (Common Assessment) .
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Formative Assessment and Performance Opportunities •"I have...Who has..." Review Activities •Academic Games •Carousel Activities •Class Discussions •Classwork •Closure Activities •Concept Sorting Activities •Do Nows •Exit Tickets •Four Corners Activities •Graphic Organizers •Homework •Placemat Activities .
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•Question‐All‐Writes •Quiz‐Quiz‐Trade Activities •Station Activities •Student Interviews •Student Response Systems •Student Self‐Ratings •Teacher Observation •Teacher Questioning •Think, Pair, Share Discussions •Thumbs Up/Down •Whip Around •Whiteboard Use .
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Differentiation/Enrichment •504 Accomodations •Challenge Problems •IEP Modifications •Learning Centers/Stations •Leveled Practice Opportunities •Scaffolding Questions •Small Group Instruction •Stundent Companion Website Resources •Technology •Use of Manipulatives (Paper Strips, Exploragons, etc.) .
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Unit Resources •
Textbook: Geometry, Common Core Ed. (Holt McDougal, 2012) •
Textbook Resource Kit & Companion Website: https://my.hrw.com/ •
Geometer's Sketchpad •
Kuta Software Additional Websites: •
Dan Meyer's 3‐Act Math Tasks: https://docs.google.com/spreadsheet/pub?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE
&output=htmlG •
Engage NY: Geometry Lesson Notes & Handouts: https://www.engageny.org/resource/high‐school‐geometry •
Geometry Teacher Mike Patterson's Common Core Teaching Notes: http://www.geometrycommoncore.com/ •
Khan Academy: https://www.khanacademy.org/ •
NCTM Illuminations Website: Resources for Teaching Math: http://illuminations.nctm.org/Default.aspx •
PARCC Educator Resources: http://www.parcconline.org/for‐educators •
The Geometer's Sketchpad Resource Center: http://www.dynamicgeometry.com/ Geometry Unit 2: Transformations, Triangles &
Congruence (Gr. 9 - 11)
Content Area:
Course(s):
Time Period:
Length:
Status:
Mathematics
Geometry
2nd Marking Period
8 Weeks
Published
Unit Overview Through this unit, students will investigate transformations, both in and outside of the coordinate plane. They will develop definitions of various types of transformations, sketch given transformations, and describe the transformations that can be used to map one figure onto another. Building upon this work, students will define and identify the properties of congruent figures. The study of congruent figures will focus on triangles, as students explore and justify different methods that can be used to prove triangles and their corresponding parts congruent. Students will continue their study of triangles as they investigate the properties of triangle inequalities and learn to make conclusions about triangles based on these. Finally, students will learn to identify perpendicular bisecors, angle bisectors, medians and midsegments of triangles, as well as use their properties to solve related problems. Transfer Students will be able to independently use their learning to... •
Transform figures in a plane. •
Determine when figures are congruent. •
Use the properties of congruence to reach accurate conclusions. •
Apply geometric properties in solving design problems. Meaning Understandings Students will understand that... •
Geometric relationships and definitions can be used to construct geometric figures and solve real‐world problems. •
Isometries are transformations that do not change the size or shape of a figure. •
An object in a plane can be oriented infinitely many different ways while maintaining its size, shape, and original properties. •
Specific instructions can be used to produce a desired image through the manipulation of a given image. •
Knowing the properties of triangles enables us to reach conclusions and solve for unknown measures related to triangles. •
The properties of geometric figures can be proven. •
The properties of special points, segments, and lines related to triangles. Essential Questions Students will keep considering... •
How can geometric figures and their properties be described by careful use of geometric language? •
How can a desired image be produced through through the manipulation of a given figure in a plane? •
What relationships exist between the sides and/or angles of a triangle? •
When is it possible to form a triangle with given constraints? •
How can the properties of geometric figures be determined using the coordinate plane? •
How are perpendicular bisectors, angle bisectors, and medians of triangles related? Application of Knowledge and Skill Students will know... •
That transformations are functions that take points in the plane as inputs and give other points as outputs. •
The definitions of reflections, rotations, translations, dilations, and symmetry. •
The properties of isometries. •
The classifications and properties of triangles. •
The five basic methods used to prove triangles congruent. •
That corresponding parts of congruent triangles are congruent. •
The properties of perpendicular and angle bisectors. •
The properties of perpendicular bisectors, angle bisectors, medians, and midsegments of triangles. •
The properties of triangle inequalities in one triangle. Students will be skilled at... •
Drawing specified transformations of figures. •
Specifying a sequence of transformations that will carry a given figure onto another. •
Identifying symmetry in figures. •
Classifying triangles according to their angle measures and side lengths. •
Applying the properties of triangles to solve for unknonws. •
Identifying congruent figures. •
Proving triangles congruent using SSS, SAS & ASA postulates, as well as the AAS & HL theorems. •
Proving that specified corresponding parts of congruent triangles are congruent. •
Applying the properties of perpendicular bisectors and angle bisectors in solving problems. •
Applying the properties of perpendicular bisectors, angle bisectors, medians and midsegments of triangles to solving problems. •
Determining whether or not it is possible to form a triangle with three given side lengths. •
Using properties of triangle inequality to make conclusions about the side lengths and angle measures of given triangles. Academic Vocabulary •acute triangle •altitude of a triangle •auxiliary line •base •base angle •center of dilation •centroid of a triangle •circumcenter of a triangle •circumscribed •compositions of transformations •concurrent •congruent polygons •corollary •corresponding angles •corresponding sides •CPCTC •dilation •enlargement •equiangular triangle •equidistant •equilateral triangle •exterior •exterior angle •glide reflection •glide reflection symmetry •incenter of a triangle .
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•included angle •included side •inscribed •interior •interior angle •isometry •isosceles triangle •legs of an isosceles triangle •line of symmetry •line symmetry •median of a triangle •midsegment of a triangle •obtuse triangle •orthocenter of a triangle •point of concurrency •reduction •reflection •remote interior angles •right triangle •rigid transformation •rotation •rotational symmetry •scalene triangle •symmetry •transformation •translation •translation symmetry •triangle rigidity •vertex angle .
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Learning Goal 2.1 Students will identify the properties of, sketch, and describe transformations. Objective 2.1.1 (Level of Difficulty: 1) SWBAT describe transformations as functions that take points in the plane as inputs and give other points as outputs. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐CO.A.2 MA.K‐12.CCSS.Math.Practice.MP7 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. 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). Look for and make use of structure. Objective 2.1.2 (Level of Difficulty: 2 - Skill) SWBAT define (in terms of angles, circles, perpendicular lines, parallel lines, and line segments), identify, draw, and describe the effect of each of the following: •
Reflections •
Translations •
Rotations MA.9‐12.CCSS.Math.Content.HSG‐CO.A.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). MA.9‐12.CCSS.Math.Content.HSG‐CO.A.4 MA.9‐12.CCSS.Math.Content.HSG‐CO.A.5 MA.9‐12.CCSS.Math.Content.HSG‐CO.B.6 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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. 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. Make sense of problems and persevere in solving them. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Objective 2.1.3 (Level of Difficulty: 2 - Skill) SWBAT identify and draw compositions of transformations. MA.9‐12.CCSS.Math.Content.HSG‐CO.A.2 MA.9‐12.CCSS.Math.Content.HSG‐CO.A.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 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). 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. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Objective 2.1.4 (Level of Difficulty: 3 - Strategic Thinking) SWBAT specify a sequence of transformations that will carry a given figure onto another. MA.9‐12.CCSS.Math.Content.HSG‐CO.A.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 2.1.5 (Level of Difficulty: 2 - Skill) SWBAT identify and describe symmetry in geometric figures. MA.9‐12.CCSS.Math.Content.HSG‐CO.A.3 MA.9‐12.CCSS.Math.Content.HSG‐CO.A.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP7 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. 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. Make sense of problems and persevere in solving them. Look for and make use of structure. Objective 2.1.6 (Level of Difficulty: 2 - Skill) SWBAT identify three‐dimensional objects generated by rotations of two‐dimensional objects. MA.9‐12.CCSS.Math.Content.HSG‐GMD.B.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.K‐12.CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. MA.K‐12.CCSS.Math.Practice.MP2 Reason abstractly and quantitatively. MA.K‐12.CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. MA.K‐12.CCSS.Math.Practice.MP7 Look for and make use of structure. Objective 2.1.7 (Level of Difficulty: 2 - Skill) SWBAT identify and draw dilations. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐CO.A.2 MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.1 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. 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). Verify experimentally the properties of dilations given by a center and a scale factor: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Objective 2.1.8 (Level of Difficulty: 3 - Strategic Thinking) SWBAT compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). MA.9‐12.CCSS.Math.Content.HSG‐CO.A.2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 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). Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Learning Goal 2 SWBAT solve problems involving triangles. Objective 2.2.1 (Level of Difficulty: 4 - Extended Thinking) SWBAT construct an equilateral triangle. MA.9‐12.CCSS.Math.Content.HSG‐CO.D.12 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Objective 2.2.2 (Level of Difficulty: 1 - Recall) SWBAT classify triangles by their angles measures and side lengths. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Attend to precision. Look for and make use of structure. Objective 2.2.3 (Level of Difficulty: 2 - Skill) SWBAT use properties of triangles to solve for unknown measures, including problems involving: •
Interior angles •
Exterior angles •
Isosceles triangles •
Equilateral triangles MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about triangles. Make sense of problems and persevere in solving them. Model with mathematics. Look for and make use of structure. Objective 2.2.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove The Triangle Sum Theorem. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about triangles. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 2.2.5 (Level of Difficulty: 3 - Strategic Thinking) Students will know and be able to apply triangle inequality theorems: •
If two sides of a triangle are not congruent, then the larger angle is opposite the longer side. •
If two angles of a triangle are not congruent, then the longer side is opposite the larger angle. •
The sum of two side lengths of a triangle must be greater than the third side length. •
The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about triangles. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Look for and make use of structure. Learning Goal 2.3 Students will be able to understand congruence in terms of rigid motions. Objective 2.3.1 (Level of Difficulty: 2 - Skill) SWBAT define congruence in terms of rigid motions and will use this definition to determine whether given figures are congruent. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐CO.B.6 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. 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 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 decide if they are congruent. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Objective 2.3.2 (Level of Difficulty: 2 - Skill) SWBAT show that two figures are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. MA.9‐12.CCSS.Math.Content.HSG‐CO.B.7 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Attend to precision. Look for and make use of structure. Objective 2.3.3 (Level of Difficulty: 4 - Extended Thinking) SWBAT justify and use each of the following methods to prove that triangles are congruent: •
SSS Postulate •
SAS Postulate •
ASA Postulate •
AAS Theorem •
HL Theorem MA.9‐12.CCSS.Math.Content.HSG‐CO.B.8 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 MA.K‐12.CCSS.Math.Practice.MP8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Objective 2.3.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT use CPCTC to prove relationships within congruent triangles. MA.9‐12.CCSS.Math.Content.HSG‐CO.B.8 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 2.3.5 (Level of Difficulty: 4) SWBAT prove the slope criteria for parallel and perpendicular lines. MA.9‐12.CCSS.Math.Content.HSG‐GPE.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 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). Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Objective 2.3.6 (Level of Difficulty: 4) SWBAT prove the Isosceles Triangle Theorem and its converse. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 MA.K‐12.CCSS.Math.Practice.MP8 Prove theorems about triangles. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Look for and express regularity in repeated reasoning. Learning Goal 2.4 Students will be able to state and apply properties of perpendicular bisectors, angle bisectors, medians, and midsegments of triangles. Objective 2.4.1 (Level of Difficulty: 4) SWBAT prove and apply theorems about perpendicular bisectors. (Including prove that points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.) MA.9‐12.CCSS.Math.Content.HSG‐CO.C.9 MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about lines and angles. Prove theorems about triangles. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Attend to precision. Look for and make use of structure. Objective 2.4.2 (Level of Difficulty: 4) SWBAT prove and apply theorems about angle bisectors. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.9 MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 Prove theorems about lines and angles. Prove theorems about triangles. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Attend to precision. Objective 2.4.3 (Level of Difficulty: 3 - Strategic Thinking) SWBAT identify and apply the properties of perpendicular and angle bisectors of a triangle, including be able to identify the incenter of a triangle and use this to inscribe a circle in the triangle. MA.9‐12.CCSS.Math.Content.HSG‐CO.D.12 MA.9‐12.CCSS.Math.Content.HSG‐C.A.3 MA.9‐12.CCSS.Math.Content.HSG‐MG.A.3 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Attend to precision. Objective 2.4.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT identify medians of a triangle, and to prove that the medians of a triangle are concurrent. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.9‐12.CCSS.Math.Content.HSG‐CO.D.12 MA.9‐12.CCSS.Math.Content.HSG‐MG.A.3 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 Prove theorems about triangles. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Use appropriate tools strategically. Objective 2.4.5 (Level of Difficulty: 3 - Strategic Thinking) SWBAT prove and apply theorems about the Triangle Midsegment Theorem. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.10 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about triangles. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Look for and make use of structure. 21st Century Life and Careers WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.A.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.F.2 WORK.9‐12.9.3.12.C.6 The ability to recognize a problem and apply critical thinking and problem‐solving skills to solve the problem is a lifelong skill that develops over time. Apply critical thinking and problem‐solving strategies during structured learning experiences. Critical thinking and problem solving in the 21st century are enhanced by the ability to work in cross‐cultural teams in face‐to‐face and virtual environments. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Leadership abilities develop over time through participation in groups and/or teams that are engaged in challenging or competitive activities. Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences. Develop job readiness skills by participating in structured learning experiences and employment seeking opportunities. Summative Assessment •Projects •Quizzes •Student Portfolios •Tests •Unit 1 Assessment (Common Assessment) .
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Formative Assessment and Performance Opportunities •"I have...Who has..." Review Activities •Academic Games •Carousel Activities •Class Discussions •Classwork •Closure Activities •Concept Sorting Activities •Do Nows •Exit Tickets •Four Corners Activities •Graphic Organizers •Homework •Placemat Activities •Question‐All‐Writes •Quiz‐Quiz‐Trade Activities •Station Activities •Student Interviews •Student Response Systems •Student Self‐Ratings •Teacher Observation .
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•Teacher Questioning •Think, Pair, Share Discussions •Thumbs Up/Down •Whip Around •Whiteboard Use .
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Differentiation/Enrichment •504 Accomodations •Challenge Problems •IEP Modifications •Learning Centers/Stations •Leveled Practice Opportunities •Scaffolding Questions •Small Group Instruction •Stundent Companion Website Resources •Technology •Use of Manipulatives (Paper Strips, Exploragons, etc.) .
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Unit Resources •
Textbook: Geometry, Common Core Ed. (Holt McDougal, 2012) •
Textbook Resource Kit & Companion Website: https://my.hrw.com/ •
Geometer's Sketchpad •
Kuta Software Additional Websites: •
Dan Meyer's 3‐Act Math Tasks: https://docs.google.com/spreadsheet/pub?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE
&output=htmlG •
Engage NY: Geometry Lesson Notes & Handouts: https://www.engageny.org/resource/high‐school‐geometry •
Geometry Teacher Mike Patterson's Common Core Teaching Notes: http://www.geometrycommoncore.com/ •
Khan Academy: https://www.khanacademy.org/ •
NCTM Illuminations Website: Resources for Teaching Math: http://illuminations.nctm.org/Default.aspx •
PARCC Educator Resources: http://www.parcconline.org/for‐educators •
The Geometer's Sketchpad Resource Center: http://www.dynamicgeometry.com/ Geometry Unit 3: Polygons, Quadrilaterals &
Similaritry (Gr. 9 - 11)
Content Area:
Course(s):
Time Period:
Length:
Status:
Mathematics
Geometry
8 weeks
8 Weeks
Published
Unit Overview This unit opens with a review of polygon vocabulary and the investigation of the interior and exterior angle measures of convex polygons and, more specifically, regular polygons. From there, students will move on to study quadrilaterals. They will investigate the properties of parallelograms, how these can be used to solve problems related to parallelograms, and the unique properties of special parallelograms, trapezoids and kites. Students will be asked to prove that specific quadrilaterals are parallelograms, rectangles, rhombi, or squares. From here, students will move on to investigate dilations and the properties and uses of similar polygons. Transfer Students will be able to independently use their learning to... •
Determine the properties of polygons and quadrilaterals that are pertinent to solving various problems. •
Identify dilations and similar polygons as they occur in nature, art, and other applications. •
Sketch dilations and similar polygons to meet given criterion. •
Use proportional reasoning to solve applied problems as they arise. Meaning Understandings Students will understand that... •
Geometric relationships and definitions can be used to construct geometric figures and solve real‐world problems. •
The more sides that a convex polygon has, the greater the sum of its interior angles. •
The sum of the exterior angles of a convex polygon is consistent, regardless of the number of sides that the polygon has. •
The unique properties of parallelograms and other special quadrilaterals allow us to solve for unknowns involving these quadrilaterals. •
All figures can be dilated to produce similar figures. •
When one figure is dilated to produce another, similar figures result. •
Similar polygons have congruent corresponding angles, and side lengths that are in proportion to one another. •
There are shortcuts that can be used to show that two triangles are similar. Essential Questions Students will keep considering... •
How can geometric figures and their properties be described by careful use of geometric language? •
Why is it useful to classify geometric figures? •
How can various geometric properties be verified by using the coordinate plane? •
How can a desired image be produced through through the manipulation of a given figure in a plane? •
How are transformations used in various careers and in the real world? Application of Knowledge and Skill Students will know... •
The names of polygons with ten or fewer sides. •
The formulas used to determine angle measures related to polygons. •
The definitions and unique characteristics of parallelograms, rectangles, rhombi, squares, trapezoids and kites. •
That transformations produce similar figures. •
The definition and properties of similar polygons. •
That the AA Postulate, SAS and SSS Similarity Theorems can be used to prove triangles similar. •
The Triangle Proportionality Theorems and the Triangle Angle Bisector Theorem. •
That ratios and proportions are useful in a variety of applications. •
That all circles are similar. Students will be skilled at... •
Calculating the interior and exterior angle sums of polygons, as well as individual angle measures in regular polygons. •
Classifying polygons based on given angle measurements. •
Identifying parallelograms, rectangles, rhombi, squares, trapezoids, kites, and similar figures. •
Using the properties of parallelograms and special parallelograms to solve for unknowns. •
Using given information to prove that certain quadrilaterals are parallelograms, rectangles, rhombi, or squares. •
Using the properties of trapezoids and kites to solve for unknowns. •
Identifying the scale factor for similar polygons. •
Verifying that given polygons are similar. •
Proving that triangles are similar. •
Solving for unknown measurements in similar polygons. •
Using proportional relationships to solve for unknowns. Academic Vocabulary •base angle of a trapezoid •base of a trapezoid •concave •convex •decagon •diagonal •dilation •directed line segment •heptagon •hexagon •indirect measurement •isosceles trapezoid •kite •leg of a trapezoid •midsegment of a trapezoid •n‐gon •nonagon •octagon •parallelogram •pentagon •proportion •quadrilateral •ratio •rectangle •regular polygon •rhombus •scale •scale drawing •scale factor •side of a polygon •similar •similar polygons •similarity ratio •similarity transformations •square .
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•trapezoid •vertex of a polygon .
Learning Goal 3.1 Students will prove and apply theorems about polygons and quadrilaterals. Objective 3.1.1 (Level of Difficulty: 1 - Recall) SWBAT classify polygons according to their sides and angles. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.K‐12.CCSS.Math.Practice.MP6 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. Attend to precision. Objective 3.1.2 (Level of Difficulty: 3 - Strategic Thinking) SWBAT develop and use formulas to determine the following angle measures in given polygons. Given these angle measures, SWBAT determine the associated polygon. •
The sum of the measures of the interior and exterior angles •
The measure of one interior or exterior angle of a regular polygon MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP8 Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and express regularity in repeated reasoning. Objective 3.1.3 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove properties of parallelograms, as well as apply these properties in solving for unknowns. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.11 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 Prove theorems about parallelograms. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Objective 3.1.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT use the properties of given quadrilaterals to determine whether they are parallelograms. When appropriate, SWBAT prove that given quadrilaterals are parallelograms. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.11 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about parallelograms. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Objective 3.1.5 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove properties of rectangles, rhombuses, and squares, as well as use these properties in solving for unknowns. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.11 MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about parallelograms. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Objective 3.1.6 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove that a given quadrilateral is a rectangle, rhombus, or square. MA.9‐12.CCSS.Math.Content.HSG‐CO.C.11 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 Prove theorems about parallelograms. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Objective 3.1.7 (Level of Difficulty: 2 - Skill) SWBAT identify and use the properties of trapezoids to solve problems. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 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. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Look for and make use of structure. Objective 3.1.8 (Level of Difficulty: 2 - Skill) SWBAT identify and use the properties of kites to solve problems. MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.5 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Look for and make use of structure. Objective 3.1.9 (Level of Difficulty: 2 - Skill) Given a rectangle, parallelogram, trapezoid, or regular polygon, SWBAT describe the rotations and reflections that carry it onto itself. MA.9‐12.CCSS.Math.Content.HSG‐CO.A.3 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP7 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Use appropriate tools strategically. Look for and make use of structure. Learning Goal 3.2 Students will understand similarity in terms of similarity transformations, and will solve problems involving similar polygons. Objective 3.2.1 (Level of Difficulty: 4 - Extended Thinking) SWBAT verify experimentally the properties of dilations given by a center and a scale factor: •
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. •
The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MA.K‐12.CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. MA.K‐12.CCSS.Math.Practice.MP2 Reason abstractly and quantitatively. MA.K‐12.CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. MA.K‐12.CCSS.Math.Practice.MP5 Use appropriate tools strategically. Objective 3.2.2 (Level of Difficulty: 2 - Skill) SWBAT define similarity in terms of similarity transformations. Based on this definition, students will determine when two polygons are similar. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.2 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. 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 sides. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 3.2.3 (Level of Difficulty: 2 - Skill) SWBAT identify properties of similar polygons, and apply these properties in solving for unknowns. MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.2 MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP7 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 sides. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Use appropriate tools strategically. Look for and make use of structure. Objective 3.2.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove that all circles are similar. MA.9‐12.CCSS.Math.Content.HSG‐C.A.1 MA.K‐12.CCSS.Math.Practice.MP3 Prove that all circles are similar. Construct viable arguments and critique the reasoning of others. Objective 3.2.5 (Level of Difficulty: 3 - Strategic Thinking) SWBAT use the properties of similarity transformations to establish the AA, SSS, and SAS criterion for two triangles to be similar. SWBAT prove that given triangles are similar. MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.2 MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.3 MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.5 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP6 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 sides. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Attend to precision. Objective 3.2.6 (Level of Difficulty: 3 - Strategic Thinking) SWBAT prove and apply the Triangle Proportionality Theorem, its converse, and the Triangle Angle Bisector Theorems in solving for unknown lengths. MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.4 MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about triangles. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Look for and make use of structure. Objective 3.2.7 (Level of Difficulty: 2 - Skill) SWBAT use ratios and scale drawings to make indirect measurements and solve problems. MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Look for and make use of structure. Objective 3.2.8 (Level of Difficulty: 2 - Skill) SWBAT apply properties of similarity to figures in the coordinate plane. MA.9‐12.CCSS.Math.Content.HSG‐CO.A.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). MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. MA.9‐12.CCSS.Math.Content.HSG‐SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor. MA.9‐12.CCSS.Math.Content.HSG‐MG.A.3 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). MA.K‐12.CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. MA.K‐12.CCSS.Math.Practice.MP2 Reason abstractly and quantitatively. MA.K‐12.CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. Objective 3.2.9 (Level of Difficulty: 4 - Extended Thinking) SWBAT use coordinate proof to prove that figures in the coordinate plane are similar. MA.9‐12.CCSS.Math.Content.HSG‐GPE.B.4 MA.K‐12.CCSS.Math.Practice.MP1 Use coordinates to prove simple geometric theorems algebraically. Make sense of problems and persevere in solving them. MA.K‐12.CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. Objective 3.2.10 (Level of Difficulty: 3 - Strategic Thinking) SWBAT find the point on a directed line segment between two given points that partitions the segment in a given ratio. MA.9‐12.CCSS.Math.Content.HSG‐GPE.B.6 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP7 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Make sense of problems and persevere in solving them. Look for and make use of structure. 21st Century Life and Careers WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.A.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.F.2 WORK.9‐12.9.3.12.C.6 The ability to recognize a problem and apply critical thinking and problem‐solving skills to solve the problem is a lifelong skill that develops over time. Apply critical thinking and problem‐solving strategies during structured learning experiences. Critical thinking and problem solving in the 21st century are enhanced by the ability to work in cross‐cultural teams in face‐to‐face and virtual environments. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Leadership abilities develop over time through participation in groups and/or teams that are engaged in challenging or competitive activities. Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences. Develop job readiness skills by participating in structured learning experiences and employment seeking opportunities. Summative Assessment •Projects •Quizzes •Student Portfolios •Tests •Unit 1 Assessment (Common Assessment) .
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Formative Assessment and Performance Opportunities •"I have...Who has..." Review Activities •Academic Games •Carousel Activities •Class Discussions •Classwork •Closure Activities •Concept Sorting Activities •Do Nows •Exit Tickets •Four Corners Activities •Graphic Organizers •Homework •Placemat Activities •Question‐All‐Writes •Quiz‐Quiz‐Trade Activities •Station Activities •Student Interviews •Student Response Systems •Student Self‐Ratings •Teacher Observation •Teacher Questioning •Think, Pair, Share Discussions •Thumbs Up/Down •Whip Around •Whiteboard Use .
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Differentiation/Enrichment •504 Accomodations •Challenge Problems •IEP Modifications •Learning Centers/Stations •Leveled Practice Opportunities •Scaffolding Questions •Small Group Instruction •Stundent Companion Website Resources •Technology .
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•Use of Manipulatives (Paper Strips, Exploragons, etc.) .
Unit Resources •
Textbook: Geometry, Common Core Ed. (Holt McDougal, 2012) •
Textbook Resource Kit & Companion Website: https://my.hrw.com/ •
Geometer's Sketchpad •
Kuta Software Additional Websites: •
Dan Meyer's 3‐Act Math Tasks: https://docs.google.com/spreadsheet/pub?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE
&output=htmlG •
Engage NY: Geometry Lesson Notes & Handouts: https://www.engageny.org/resource/high‐school‐geometry •
Geometry Teacher Mike Patterson's Common Core Teaching Notes: http://www.geometrycommoncore.com/ •
Khan Academy: https://www.khanacademy.org/ •
NCTM Illuminations Website: Resources for Teaching Math: http://illuminations.nctm.org/Default.aspx •
PARCC Educator Resources: http://www.parcconline.org/for‐educators •
The Geometer's Sketchpad Resource Center: http://www.dynamicgeometry.com/ Geometry Unit 4: Right Triangles, Area & Volume (Gr.
9 - 11)
Content Area:
Course(s):
Time Period:
Length:
Status:
Mathematics
Geometry
3rd Marking Period
7 Weeks
Published
Unit Overview This unit opens with a review of simplifying radical expressions. From there, students will review, prove, and solve problems involving the Pythagorean Theorem as well as its converses. They will move on to examine the relationships within special right triangles and use these to solve for unknowns. Students will next study trigonometric functions and their application in solving real‐world problems. During the second part of the unit, students will review known area, perimeter, and volume formulas, and build new ones. Emphasis will be placed on using these formulas in modeling situations. Transfer Students will be able to independently use their learning to... •
Solve for unknown measurements in right triangles. •
Solve applied problems involving right triangles, sketching diagrams as needed. •
Use perimeter, area, circumference, and volume formulas to sove for quantities of interest. Meaning Understandings Students will understand that... •
Values reported using radical notation are more accurate than those given in rounded decimal form. •
The Pythagorean Theorem is used to solve for unknown side lengths in right triangles when two side lengths are given. •
The patterns that exist in Special Right Triangle allow us to solve easily for unknown measuresments in these triangles. •
Right triangle trigonometry is useful in solving a variety of problems. •
Formulas can be applied to solve for any single unknown quantity that they contain. •
The areas and volumes of composite and irregular figures can be found by breaking them down into simpler figures and adding/subtracting their individual areas/volumes. •
Different cross sections of solids may yield different two‐dimensional shapes. •
Two‐ and three‐dimensional geometric figures can be used to model real‐world situations and figures. Essential Questions Students will keep considering... •
How can geometric concepts and figures be used to model real‐world phenomena? •
What relationships exist between the side lengths and angle measures of right triangles? •
How can right triangle relationships be used to calculate inaccessible measurements? •
How are the formulas for area, perimeter, circumference, and volume of geometric figures applied in solving problems? •
How can I find the area, perimeter, or volume of figures composed of a combination of various geometric figures? •
How can geometric figures be used in solving real‐world problems? Application of Knowledge and Skill Students will know... •
That when the altitude is drawn to the hypotenuse of a right triangle, the triangles formed are similar to each other, and to the original right triangle. •
The Pythagorean Theorem and its converses. •
The unique relationships that exist within 45‐45‐90 and 30‐60‐90 triangles. •
The definitions of the sine, cosine, and tangent ratios. •
That the sine and cosine of complementary angles are equal to one another. •
That right triangles are useful in solving a variety of real‐world problems. •
The formulas used to find areas, perimeters, circumference, and volume of geometric figures. •
That pi represents the ratio between the circumference and diameter of a circle. Students will be skilled at... •
Simplifying radical expressions. •
Using the Pythagorean Theorem to solve for unknown lengths in right triangles. •
Classifying triangles as acute, right, obtuse, or not possible when given their side lengths. •
Using Special Right Triangle relationships to solve for unknown measurements. •
Using trigonometric ratios to solve for unknown side lengths and angle measures. •
Choosing the most efficient method to solve various right triangles. •
Creating sketches appropriate to solving applied problems involving right triangles. •
Calculating the area, perimeter/circumference or volume of a given figure. •
Using given measurements and quantities to solve for unknown dimensions of geometric figures. •
Identifying various parts of solid figures. •
Applying geometric concepts in modeling situations. Academic Vocabulary •30‐60‐90 Triangle •45‐45‐90 Triangle •angle of depression •angle of elevation •apothem •center of a circle •center of a regular polygon •center of a sphere •central angle of a regular polygon •circle •composite figure •cone •cosine •cross section •cube •cylinder •edge •face •geometric mean •great circle •hemisphere •net •prism •pyramid •Pythagorean Theorem •radius of a sphere •sine •sphere .
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•tangent •trigonometric ratios •vertex of a solid •volume .
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Learning Goal 4.1 Students will use the Pythagorean Theorem and Special Right Triangle relationships to solve application problems involving right triangles. Objective 4.1.1 (Level of Difficulty: 2) SWBAT simplify, add, subtract, multiply, and divide radical expressions. MA.9‐12.CCSS.Math.Content.HSN‐RN.A.2 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Rewrite expressions involving radicals and rational exponents using the properties of exponents. Attend to precision. Look for and make use of structure. Objective 4.1.2 (Level of Difficulty: 2 - Skill) SWBAT use the geometric mean to find segment lengths in right triangles. MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.4 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Prove theorems about triangles. Make sense of problems and persevere in solving them. Attend to precision. Look for and make use of structure. Objective 4.1.3 (Level of Difficulty: 3 - Strategic Thinking) SWBAT apply the Pythagorean Theorem to solving for unknown segment lengths in right triangles and applied problems. MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.8 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Make sense of problems and persevere in solving them. Model with mathematics. Look for and make use of structure. Objective 4.1.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT prove the Pythagorean Theorem using triangle similarity. MA.9‐12.CCSS.Math.Content.HSG‐SRT.B.4 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 Prove theorems about triangles. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Objective 4.1.5 (Level of Difficulty: 2 - Skill) Given their side lengths, SWBAT use the Converses to The Pythagorean Theorem to classify triangles as acute, right, obtuse, or not possible. MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.8 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Look for and make use of structure. Objective 4.1.6 (Level of Difficulty: 3 - Strategic Thinking) SWBAT identify and determine unknown side lengths in 45‐45‐90 and 30‐60‐90 triangles. MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.6 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP8 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Make sense of problems and persevere in solving them. Model with mathematics. Attend to precision. Look for and express regularity in repeated reasoning. Learning Goal 4.2 Students will define and use trigonometric ratios to solve application problems involving right triangles. Objective 4.2.1 (Level of Difficulty: 1 - Recall) SWBAT recognize that by similarity, side ratios in right triangles are properties of the angles in the triangle, and define of trigonometric ratios for acute angles. LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.6 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 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. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Attend to precision. Look for and make use of structure. Objective 4.2.2 (Level of Difficulty: 2 - Skill) SWBAT use the sine, cosine, and tangent ratios and their inverses to solve for unknown side lengths and angle measures in right triangles. MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.8 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Make sense of problems and persevere in solving them. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Objective 4.2.3 (Level of Difficulty: 2 - Skill/Concept) SWBAT explain and use the relationship between the sine and cosine of complementary angles. MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.7 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP5 Explain and use the relationship between the sine and cosine of complementary angles. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Objective 4.2.4 (Level of Difficulty: 4 - Extended Thinking) SWBAT sketch right triangles and use trigonometric relationships to solve real‐world problems. MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.8 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Make sense of problems and persevere in solving them. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Objective 4.2.5 (Level of Difficulty: 3 - Strategic Thinking) SWBAT identify and use the most efficient method to solve right triangles in applied problems. MA.9‐12.CCSS.Math.Content.HSG‐SRT.C.8 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP7 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Look for and make use of structure. Learning Goal 4.3 Students will explain and apply formulas for finding the perimeter, circumference, and areas of figures. Objective 4.3.1 (Level of Difficulty: 2 - Skill) SWBAT calculate the area and perimeter of each of the figures listed below. Given the area and one or more measurements of each of the figures listed, SWBAT calculate unknown measurements. •
Triangles •
Parallelograms •
Trapezoids •
Rhombi •
Kites •
Circles •
Regular polygons MA.9‐12.CCSS.Math.Content.HSA‐SSE.A.1 MA.9‐12.CCSS.Math.Content.HSA‐CED.A.4 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP6 Interpret expressions that represent a quantity in terms of its context. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Model with mathematics. Attend to precision. Objective 4.3.2 (Level of Difficulty: 3 - Strategic Thinking) SWBAT describe the ratio represented by pi. Based upon this, SWBAT give informal arguments for the area and circumference of circles. MA.9‐12.CCSS.Math.Content.HSG‐GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. MA.K‐12.CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Attend to precision. Objective 4.3.3 (Level of Difficulty: 2 - Skill) SWBAT find the areas of composite figures, and will be able to use composite figures to estimate the areas of irregular shapes. MA.9‐12.CCSS.Math.Content.HSG‐MG.A.3 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Make sense of problems and persevere in solving them. Model with mathematics. Use appropriate tools strategically. Attend to precision. Objective 4.3.4 (Level of Difficulty: 3 - Strategic Thinking) SWBAT find the perimeters and areas of figures in the coordinate plane. MA.9‐12.CCSS.Math.Content.HSG‐GPE.B.7 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Attend to precision. Look for and make use of structure. Objective 4.3.5 (Level of Difficulty: 4 - Extended Thinking) SWBAT use geometric shapes, their measures, and their properties to describe two‐dimensional objects. MA.9‐12.CCSS.Math.Content.HSG‐MG.A.1 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Model with mathematics. Look for and make use of structure. Objective 4.3.6 (Level of Difficulty: 3 - Strategic Thinking) SWBAT apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). MA.9‐12.CCSS.Math.Content.HSG‐MG.A.2 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Model with mathematics. Look for and make use of structure. Learning Goal 4.4 Students will relate two and three‐dimensional figures, and will explain volume formulas and use them to solve problems. Objective 4.4.1 (Level of Difficulty: 3 - Strategic Thinking) SWBAT classify prisms, cylinders, cones, pyramids, and spheres, and will identify: •
the shapes of two‐dimensional cross‐sections of these figures •
the three‐dimensional figures that can be formed by given nets LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 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. MA.9‐12.CCSS.Math.Content.HSG‐GMD.B.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.K‐12.CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. MA.K‐12.CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. MA.K‐12.CCSS.Math.Practice.MP6 Attend to precision. MA.K‐12.CCSS.Math.Practice.MP7 Look for and make use of structure. Objective 4.4.2 (Level of Difficulty: 3 - Strategic Thinking) SWBAT explain and apply the formulas used to find the volumes of prisms, cylinders, cones, and pyramids. MA.9‐12.CCSS.Math.Content.HSG‐GMD.A.3 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP6 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Model with mathematics. Attend to precision. Objective 4.4.3 (Level of Difficulty: 3 - Strategic Thinking) SWBAT apply the formulas used to find the area and volume of a sphere. MA.9‐12.CCSS.Math.Content.HSG‐GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. MA.9‐12.CCSS.Math.Content.HSG‐GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. MA.9‐12.CCSS.Math.Content.HSG‐MG.A.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.9‐12.CCSS.Math.Content.HSG‐MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). MA.K‐12.CCSS.Math.Practice.MP1 Make sense of problems and persevere in solving them. MA.K‐12.CCSS.Math.Practice.MP3 Construct viable arguments and critique the reasoning of others. MA.K‐12.CCSS.Math.Practice.MP4 Model with mathematics. MA.K‐12.CCSS.Math.Practice.MP6 Attend to precision. Objective 4.4.4 (Level of Difficulty: 3 - Strategic Thinking) SWBAT 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.9‐12.CCSS.Math.Content.HSG‐MG.A.1 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. 21st Century Life and Careers WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.A.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.1 The ability to recognize a problem and apply critical thinking and problem‐solving skills to solve the problem is a lifelong skill that develops over time. Apply critical thinking and problem‐solving strategies during structured learning experiences. Critical thinking and problem solving in the 21st century are enhanced by the ability to work in cross‐cultural teams in face‐to‐face and virtual environments. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.F.2 WORK.9‐12.9.3.12.C.6 Leadership abilities develop over time through participation in groups and/or teams that are engaged in challenging or competitive activities. Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences. Develop job readiness skills by participating in structured learning experiences and employment seeking opportunities. Summative Assessment •Projects •Quizzes •Student Portfolios •Tests •Unit 1 Assessment (Common Assessment) .
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Formative Assessment and Performance Opportunities •"I have...Who has..." Review Activities •Academic Games •Carousel Activities •Class Discussions •Classwork •Closure Activities •Concept Sorting Activities •Do Nows •Exit Tickets •Four Corners Activities •Graphic Organizers •Homework •Placemat Activities •Question‐All‐Writes •Quiz‐Quiz‐Trade Activities •Station Activities •Student Interviews •Student Response Systems •Student Self‐Ratings •Teacher Observation •Teacher Questioning •Think, Pair, Share Discussions •Thumbs Up/Down •Whip Around •Whiteboard Use .
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Differentiation/Enrichment •504 Accomodations •Challenge Problems •IEP Modifications •Learning Centers/Stations •Leveled Practice Opportunities •Scaffolding Questions •Small Group Instruction •Stundent Companion Website Resources •Technology •Use of Manipulatives (Paper Strips, Exploragons, etc.) .
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Unit Resources •
Textbook: Geometry, Common Core Ed. (Holt McDougal, 2012) •
Textbook Resource Kit & Companion Website: https://my.hrw.com/ •
Geometer's Sketchpad •
Kuta Software Additional Websites: •
Dan Meyer's 3‐Act Math Tasks: https://docs.google.com/spreadsheet/pub?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE
&output=htmlG •
Engage NY: Geometry Lesson Notes & Handouts: https://www.engageny.org/resource/high‐school‐geometry •
Geometry Teacher Mike Patterson's Common Core Teaching Notes: http://www.geometrycommoncore.com/ •
Khan Academy: https://www.khanacademy.org/ •
NCTM Illuminations Website: Resources for Teaching Math: http://illuminations.nctm.org/Default.aspx •
PARCC Educator Resources: http://www.parcconline.org/for‐educators •
The Geometer's Sketchpad Resource Center: http://www.dynamicgeometry.com/ Geometry Unit 5: Circles (Gr. 9 - 11)
Content Area:
Course(s):
Time Period:
Length:
Status:
Mathematics
Geometry
4th Marking Period
4 Weeks
Published
Unit Overview This unit opens with a review of circle vocabulary that is already familiar to students and the introduction of new terminology, symbols, and properties related to circles. Students will learn about the properties of tangents of circles and the relationships between the angles, arcs, and lengths of segments formed by radii, chords, secants and tangents of circles. In addition to this, students will use proportional reasoning to find the arc lengths and areas of sectors of circles, leading to the definition of radian measure. Students will also learn to graph and write equations of circles in the coordinate plane. Transfer Students will be able to independently use their learning to... •
Use the relationships between the angles, arcs, and segments of circles to solve applied problems. •
Construct polygons inscribed in circles. •
Write and graph equations of circles. Meaning Understandings Students will understand that... •
Arcs lengths are portions of the circumference of a circle. •
Sector areas are portions of the area of a circle. •
Relationships exist between the angle and arc measures of circles. •
Relationships exist between the arc measures and lengths of segments associated with circles. •
Angles can be measured in degrees or radians. •
The equation of a circle is derived from the Pythagorean Theorem. Essential Questions Students will keep considering... •
How can geometric concepts and figures be described by careful use of geometric language? •
How can geometric concepts and figures be used to model real‐world phenomena? •
How are the relationships between the angles, arcs, and segments of circles applied in solving problems? •
How should I decide what method to use when solving problems involving circles? •
How can I inscribe regular polygons in circles? •
How can I write the equation of a circle in the coordinate plane? Application of Knowledge and Skill Students will know... •
The definitions of the terminology related to circles. •
That the relationships that exist between the angles, segments, and arcs associated with circles can be used to solve for unknown measures. •
That arcs and sectors of circles represent a portions of the circle. •
That equations can be written to describe circles in the coordinate plane. Students will be skilled at... •
Using the relationships between angles, arcs and segments of circles to solve for unknown measures. •
Calculating arc lengths and areas of sectors of circles. •
Writing and graphing equations of circles in the coordinate plane. Academic Vocabulary •adjacent arcs •arc •arc length •central angle •chord •common tangent •concentric circles •congruent arcs .
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•congruent circles •exterior of a circle •external secant segment •inscribed angle •intercepted arc •interior of a circle •major arc •minor arc •point of tangency •secant •secant segment •sector of a circle •sector of a circle •segment of a circle •semicircle •subtend •tangent circles •tangent of a circle •tangent segments .
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Learning Goal 5.1 Students will understand and apply theorems about circles. Objective 5.1.1 (Level of Difficulty: 1) SWBAT define and identify each of the following terms associated with circles, as well as their symbolic notations: •
tangents •
secants •
chords •
arcs •
minor arcs •
major arcs •
semicircles •
central angles •
sectors of circles •
inscribed angles LA.9‐10.CCSS.ELA‐Literacy.RST.9‐10.4 MA.9‐12.CCSS.Math.Content.HSG‐C.A.2 MA.K‐12.CCSS.Math.Practice.MP6 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. Identify and describe relationships among inscribed angles, radii, and chords. Attend to precision. Objective 5.1.2 (Level of Difficulty - 3: Strategic Thinking) SWBAT describe and use properties of tangents to verify relationships within circles and solve for unknowns. MA.9‐12.CCSS.Math.Content.HSG‐C.A.2 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP4 Identify and describe relationships among inscribed angles, radii, and chords. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Objective 5.1.3 (Level of Difficulty - 3: Strategic Thinking) SWBAT construct a tangent line from a point outside a given circle to the circle. MA.9‐12.CCSS.Math.Content.HSG‐CO.D.12 MA.9‐12.CCSS.Math.Content.HSG‐C.A.4 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Construct a tangent line from a point outside a given circle to the circle. Make sense of problems and persevere in solving them. Use appropriate tools strategically. Attend to precision. Objective 5.1.4 (Level of Difficulty: 2 - Skill) SWBAT describe and apply the relationships between arcs, central angles, chords, and radii in solving for unknowns. MA.9‐12.CCSS.Math.Content.HSG‐C.A.2 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 Identify and describe relationships among inscribed angles, radii, and chords. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Objective 5.1.5 (Level of Difficulty: 2 - Skill) SWBAT use the relationships between inscribed angles and their intercepted arcs to solve for unknowns. Theorems to include: •
The measure of an inscribed angle is half the measure of its intercepted arc. •
Inscribed angles on a diameter are right angles. •
Opposite angles of a quadrilateral inscribed in a circle are supplementary. (And proof of this) MA.9‐12.CCSS.Math.Content.HSG‐C.A.2 MA.9‐12.CCSS.Math.Content.HSG‐C.A.3 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 Identify and describe relationships among inscribed angles, radii, and chords. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Objective 5.1.6 (Level of Difficulty: 2 - Skill) SWBAT find the measures of angles and arcs defined by secants, chords and tangents of circles. MA.9‐12.CCSS.Math.Content.HSG‐C.A.2 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP7 Identify and describe relationships among inscribed angles, radii, and chords. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Look for and make use of structure. Objective 5.1.7 (Level of Difficulty: 3 - Strategic Thinking) SWBAT construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. (Note: See activity on pages 392 ‐ 393 of the textbook.) MA.9‐12.CCSS.Math.Content.HSG‐CO.D.13 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP5 MA.K‐12.CCSS.Math.Practice.MP6 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Use appropriate tools strategically. Attend to precision. Objective 5.1.8 (Level of Difficulty: 2 - Skill) SWBAT find the lengths of segments formed when tangents, secants, and chords intersect circles, and will use these relationships to solve problems. MA.9‐12.CCSS.Math.Content.HSG‐C.A.2 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Identify and describe relationships among inscribed angles, radii, and chords. Make sense of problems and persevere in solving them. Model with mathematics. Attend to precision. Look for and make use of structure. Learning Goal 5.2 SWBAT find arc lengths and areas of sectors of circles. Objective 5.2.1 (Level of Difficulty: 2 - Skill) SWBAT find arc lengths and areas of sectors of circles. http://www.youtube.com/watch?v=0JHEqBLG650 MA.9‐12.CCSS.Math.Content.HSG‐C.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP4 MA.K‐12.CCSS.Math.Practice.MP7 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. Make sense of problems and persevere in solving them. Model with mathematics. Look for and make use of structure. Objective 5.2.2 (Level of Difficulty: 2 - Skill) SWBAT define radian measure and convert between radian and degree measure. MA.9‐12.CCSS.Math.Content.HSG‐C.B.5 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Learning Goal 5.3 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. Make sense of problems and persevere in solving them. Attend to precision. Look for and make use of structure. Students will translate between the geometric description and the equation for a circle. Objective 5.3.1 (Level of Difficulty: 4 - Extended Thinking) SWBAT use the Pythagorean Theorem to derive the equation of a circle. Given their center and radius, SWBAT write equations of circles in the coordinate plane. MA.9‐12.CCSS.Math.Content.HSG‐GPE.A.1 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP3 MA.K‐12.CCSS.Math.Practice.MP7 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Look for and make use of structure. Objective 5.3.2 (Level of Difficulty: 2 - Skill) SWBAT complete the square to find the center and radius of a circle given by an equation. SWBAT graph circles in the coordinate plane. MA.9‐12.CCSS.Math.Content.HSG‐GPE.A.1 MA.K‐12.CCSS.Math.Practice.MP1 MA.K‐12.CCSS.Math.Practice.MP2 MA.K‐12.CCSS.Math.Practice.MP6 MA.K‐12.CCSS.Math.Practice.MP7 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Attend to precision. Look for and make use of structure. 21st Century Life and Careers WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.A.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.1 WORK.9‐12.9.1.12.2 WORK.9‐12.9.1.12.F.2 WORK.9‐12.9.3.12.C.6 The ability to recognize a problem and apply critical thinking and problem‐solving skills to solve the problem is a lifelong skill that develops over time. Apply critical thinking and problem‐solving strategies during structured learning experiences. Critical thinking and problem solving in the 21st century are enhanced by the ability to work in cross‐cultural teams in face‐to‐face and virtual environments. Collaboration and teamwork enable individuals or groups to achieve common goals with greater efficiency. Leadership abilities develop over time through participation in groups and/or teams that are engaged in challenging or competitive activities. Demonstrate a positive work ethic in various settings, including the classroom and during structured learning experiences. Develop job readiness skills by participating in structured learning experiences and employment seeking opportunities. Summative Assessment •Projects •Quizzes •Student Portfolios •Tests •Unit 1 Assessment (Common Assessment) .
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Formative Assessment and Performance Opportunities •"I have...Who has..." Review Activities •Academic Games •Carousel Activities •Class Discussions •Classwork •Closure Activities •Concept Sorting Activities •Do Nows •Exit Tickets •Four Corners Activities •Graphic Organizers •Homework .
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•Placemat Activities •Question‐All‐Writes •Quiz‐Quiz‐Trade Activities •Station Activities •Student Interviews •Student Response Systems •Student Self‐Ratings •Teacher Observation •Teacher Questioning •Think, Pair, Share Discussions •Thumbs Up/Down •Whip Around •Whiteboard Use .
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Differentiation/Enrichment •504 Accomodations •Challenge Problems •IEP Modifications •Learning Centers/Stations •Leveled Practice Opportunities •Scaffolding Questions •Small Group Instruction •Stundent Companion Website Resources •Technology •Use of Manipulatives (Paper Strips, Exploragons, etc.) .
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Unit Resources •
Textbook: Geometry, Common Core Ed. (Holt McDougal, 2012) •
Textbook Resource Kit & Companion Website: https://my.hrw.com/ •
Geometer's Sketchpad •
Kuta Software Additional Websites: •
Dan Meyer's 3‐Act Math Tasks: https://docs.google.com/spreadsheet/pub?key=0AjIqyKM9d7ZYdEhtR3BJMmdBWnM2YWxWYVM1UWowTEE
&output=htmlG •
Engage NY: Geometry Lesson Notes & Handouts: https://www.engageny.org/resource/high‐school‐geometry •
Geometry Teacher Mike Patterson's Common Core Teaching Notes: http://www.geometrycommoncore.com/ •
Khan Academy: https://www.khanacademy.org/ •
NCTM Illuminations Website: Resources for Teaching Math: http://illuminations.nctm.org/Default.aspx •
PARCC Educator Resources: http://www.parcconline.org/for‐educators •
The Geometer's Sketchpad Resource Center: http://www.dynamicgeometry.com/