Englewood Public School District Second Marking Period

Unit 2: , , and

Overview: During this unit, students will learn how to prove two triangles are congruent, relationships between angle measures and side lengths within a , and about different types of quadrilaterals.

Time Frame: 35 to 45 days (One Marking Period)

Enduring Understandings: • Congruent triangles can be visualized by placing one on top of the other. • Corresponding sides and angles can be marked using tic marks and angle marks. • Theorems can be used to prove triangles congruent. • The definitions of isosceles and equilateral triangles can be used to classify a triangle. • The Midpoint Formula can be used to find the midsegment of a triangle. • The Distance Formula can be used to examine relationships in triangles. • Side lengths of triangles have a relationship. • The negation statement can be proved and used to show a counterexample. • The diagonals of a can be used to derive the formula for the angle measures of the polygon. • The properties of parallel and perpendicular lines can be used to classify quadrilaterals. • Coordinate geometry can be used to classify special . • Slope and the distance formula can be used to prove relationships in the coordinate plane.

Essential Questions: • How do you identify corresponding parts of congruent triangles? • How do you show that two triangles are congruent? • How can you tell if a triangle is isosceles or equilateral? • How do you use coordinate geometry to find relationships within triangles? • How do you solve problems that involve measurements of triangles? • How do you write indirect proofs? • How can you find the sum of the measures of the angles of a polygon? • How can you classify quadrilaterals? • How can you use coordinate geometry to prove general relationships?

Standards Topics and Objectives Activities Materials/Resources Assessments Topics NJSLA Geometry Lessons: Pearson Realize Chapters 4, Formative G-CO.C.9 Prove theorems • Determining 5, and 6 Assessments: about lines and angles. Congruence, Corresponding Congruence: The Case Textbook Pages 243, parts of congruent figures, Study of a Triangle NJSLA Lessons (Shared 277, 278, 316, 345, 346, G-CO.C.10 Prove theorems triangle congruence • Proving Folder) 398, 425, 426 about triangles. theorems, isosceles and Theorems equilateral triangles, • Proving Triangle Illustrative Mathematics Math journal G-CO.C.11 Prove theorems midsegments of triangles, Theorems https://www.illustrativemath (NJSLSA.W10, about parallelograms. altitudes and medians of ematics.org/ NJSLSA.L1, triangles, perpendicular and Is this a ? NJSLSA.L6) G-CO.D.12 Make formal angle bisectors, inequalities https://www.illustrativemat National Library of Virtual geometric constructions in triangles, quadrilaterals, hematics.org/content- Manipulatives Summative with a variety of tools and parallelograms, , standards/HSG/CO/B/tasks http://nlvm.usu.edu/ Assessments: methods rhombi, , , /1302 (MP4) Multiple choice / short kites, and quadrilaterals in Alabama Learning answer assessments G-CO.D.13 Construct an the coordinate plane Are the Triangles Exchange (CRP2, CRP4, CRP8) , a Congruent? http://alex.state.al.us/search. , and a regular Objectives https://www.illustrativemat php?fa_submit=ALLPLAN Chapter quizzes/tests inscribed in a hematics.org/content- S • Pearson Realize circle. Students will standards/HSG/CO/B/tasks • MathXL • prove theorems about /33 (MP4) NYC Department of G-SRT.B.5 Use congruence lines and angles Education Benchmark and similarity criteria for • prove vertical angles Properties of Congruent http://schools.nyc.gov/defau Assessment: triangles to solve problems are congruent Triangles lt.htm Midterm Assessment and to prove relationships • prove that when a https://www.illustrativemat in geometric figures. transversal crosses hematics.org/content- Mathematics Assessment Alternative parallel lines, standards/HSG/CO/B/7/tas Project Assessments: G-C.A.3 Construct the alternate interior ks/1637 (MP1) http://map.mathshell.org/ Learning centers: each inscribed and angles are congruent learning center focuses circumscribed circles of a • prove that when a SSS Congruence Criteria Texas Instruments on a different type of triangle, and prove transversal crosses https://www.illustrativemat https://education.ti.com/en/u problem properties of angles for a parallel lines, hematics.org/content- s/home inscribed in a corresponding angles standards/HSG/CO/B/8/tas Create posters circle. are congruent ks/1930 (MP4) Desmos illustrating the main • prove measures of the https://teacher.desmos.com/ objectives of the unit G-GPE.B.4 Use coordinates interior angles of a When does SSA work for (CRP6) to prove simple geometric triangle sum to 180 Triangle Congruence? Worksheets for every topic: theorems algebraically. degrees. https://www.illustrativemat http://kutasoftware.com/free • Prove theorems about hematics.org/content- .html G-GPE.B.7 Use coordinates triangles standards/HSG/CO/B/8/tas (CRP2, CRP4, CRP8) to compute of • prove theorems about ks/340 (MP3) polygons and areas of parallelograms Algebra assessments, triangles and rectangles, • prove opposite sides Why does ASA work? interactive, videos, games, e.g., using the distance in a parallelogram are https://www.illustrativemat lessons, homework: formula. congruent hematics.org/content- https://www.opened.com/se • prove opposite angles standards/HSG/CO/B/8/tas arch?area=mathematics&gr Mathematics Practices in a parallelogram are ks/339 (MP4) ade=9&offset=0&resource_ MP1 Make sense of problems congruent type=interactive-assessment and persevere in solving them. • use a variety of Why does SAS work? (CRP2, CRP4, CRP8, MP2 Reason abstractly and geometric tools and https://www.illustrativemat 8.1.12.A.3) quantitatively. methods to construct hematics.org/content- MP3 Construct arguments perpendicular lines, standards/HSG/CO/B/8/tas Geometry NJSLS and critique the reasoning of including ks/109 (MP7) worksheets: others. perpendicular https://www.ixl.com/math/g MP4 Model Mathematics. bisectors Why does SSS work? eometry MP6 Attend to precision. • use a variety of https://www.illustrativemat (CRP2, CRP4, CRP8) MP7 Look for and make use geometric tools and hematics.org/content- of structure. methods to construct standards/HSG/CO/B/8/tas Khan Academy – videos, MP8 Look for and express a line parallel to a ks/110 (MP4) lessons, assessments regularity in represented given line through a www.khanacademy.org reasoning. point not on the line Congruent Angles in (CRP2, CRP4, CRP8, • construct an Isosceles Triangles CRP11, 8.1.12.A.3) equilateral triangle https://www.illustrativemat inscribed in a circle hematics.org/content- Worksheets / assessment standards/HSG/CO/C/10/ta items for all topics based on • construct a regular sks/1921 (MP4) standards: hexagon inscribed in http://jmap.org/JMAP_RES a circle Midpoints of Triangle OURCES_BY_TOPIC.htm

• construct a square Sides #AI(CRP2, CRP4, CRP8) inscribed in a circle https://www.illustrativemat • use the definition of hematics.org/content- similarity in terms of standards/HSG/CO/C/10/ta similarity sks/1872 (MP2) transformations to decide if two given figures are similar Is this a Parallelogram? • explain, using https://www.illustrativemat similarity hematics.org/content- transformations, the standards/HSG/CO/C/11/ta meaning of similarity sks/1321 (MP1) for triangles as the equality of all Midpoints of the Sides of a corresponding pairs Parallelogram of angles and the https://www.illustrativemat proportionality of all hematics.org/content- corresponding pairs standards/HSG/CO/C/11/ta of sides sks/35 (MP4) • construct the inscribed and What do you know about circumscribed circles the Bermuda Triangle? of a triangle http://alex.state.al.us/lesson • prove properties of _view.php?id=33167 angles for a (MP1) quadrilateral inscribed in a circle Triangle Midsegments • prove simple https://education.ti.com/en/ geometric theorems us/activity/detail?id=BD84 algebraically using 0BAD7E754B7BB0B5A2 coordinates ECB79A2107&ref=/en/us/ • compute of activity/search/advanced polygons using (MP6) coordinates • compute areas of Exterior and Remote rectangles using Interior Angles coordinates https://education.ti.com/en/ • compute area of us/activity/detail?id=A0F8 triangles using CB43E61B430B8882100C coordinates 56D8AA09&ref=/en/us/act ivity/search/advanced (MP2)

Properties of Parallelograms https://education.ti.com/en/ us/activity/detail?id=5CDF 181BDFBE42BB8FC1070 63B5EBA08&ref=/en/us/a ctivity/search/advanced (MP2)

Triangle Sides and Angles https://education.ti.com/en/ us/activity/detail?id=64E71 D93C94F43C0968730A66 24B30BF&ref=/en/us/activ ity/search/advanced(MP7)

Properties of Special Quadrilaterals https://education.ti.com/en/ us/activity/detail?id=3EFE CCEBAFA04B6990D369 A41FA4F83F&ref=/en/us/ activity/search/advanced (MP2)

Creating a Parallelogram https://education.ti.com/en/ us/activity/detail?id=CA35 3F7DA7034A878EDA599 7128351B2&ref=/en/us/act ivity/search/advanced (MP6)

Properties of Trapezoids and Kites https://education.ti.com/en/ us/activity/detail?id=F33C 9C7C2E8842BEB300C582 7F081E93&ref=/en/us/acti vity/search/advanced (MP8)

Congruent or Not? https://education.ti.com/en/ us/activity/detail?id=E0879 6D0158C40B2A507501D0 B66E8A7&ref=/en/us/activ ity/search/advanced(MP6)

Rhombi, Kites, and Trapezoids https://education.ti.com/en/ us/activity/detail?id=8CC6 29DAF55548EF83831A82 42DB4AD5&ref=/en/us/ac tivity/search/advanced (MP3)

SSA Ambiguity https://education.ti.com/en/ us/activity/detail?id=6AB9 AE39801340BA9B6354D 91819176F&ref=/en/us/act ivity/search/advanced (MP1)

Everything you need to know about math journals: https://thecornerstonefortea chers.com/math-journals/ (NJSLSA.W10, NJSLSA.L1, NJSLSA.L6)

Discuss math skills that support various STEAM careers (9.2.12.C.3)

Additional texts: www.newsela.com www.readworks.org www.commonlit.org

Key Vocabulary: Base angle of an , base of an isosceles triangle, congruent polygons, corollary, hypotenuse, legs of an isosceles triangle, legs of a , vertex angle of an isosceles triangle, altitude of a triangle, centroid, circumcenter, equidistant, incenter, indirect proof, median, midsegment of a triangle, orthocenter, coordinate proof, , equilateral polygon, isosceles , , midsegment of a trapezoid, parallelogram, rectangle, , , trapezoid Accommodations and Modifications:

Students with special needs: Support staff will be available to aid students related to IEP specifications. 504 accommodations will also be attended to by all instructional leaders. Modifications, alternative assessments, and scaffolding strategies will be used to support this learning. The use of Universal Design for Learning (UDL) will be considered for all students as teaching strategies are considered. Additional staff should be included so all students can fully participate in the standards associated with this curriculum.

ELL/ESL students: Students will be supported according to the recommendations for “can do’s” as outlined by WIDA - https://wida.wisc.edu/teach/can-do/descriptors

Students at risk of school failure: Formative and summative data will be used to monitor student success at first signs of failure. Student work will be reviewed to determine support. This may include parent consultation, basic skills review and differentiation strategies. With considerations to UDL, time may be a factor in overcoming developmental considerations. More time will be made available with a certified instructor to aid students in reaching the standards.

Gifted and Talented Students: Students excelling in mastery of standards will be challenged with complex, high level challenges.

English Language Learners: Special Education: At-Risk: Gifted and Talented:

• Teaching modeling • Utilize modifications & • Use visual demonstrations, • Inquiry based instruction • Peer modeling accommodations delineated illustrations and models • Independent study • Word walls in the students’ IEP • Give directions / • Higher order thinking • Give directions in small steps • Work with paraprofessional instructions verbally and in skills and in as few words as • Work with a partner simple written format • Adjusting the pace of the possible • Shorten assignments to • Peer support lessons • Provide visual aids focus on mastery or key • Increased one – on – one • Real world scenarios • Group similar problems concepts time • Student driven instruction together • Maintain adequate space • Teachers may modify • Allow students to • Repeat directions when between desks instructions by modeling complete an independent necessary • Keep workspaces clear of what the student is expected project as an alternative • Provide a vocabulary list unrelated materials to do test with definitions • Provide fewer problems to • Instructions may be printed • Use of alge-tiles when attain passing grades out in large print and hung needed • Tape a number line to the up for the students to see • Use of number line when student’s desk during the time of the needed • Create a math journal that lesson they can use during class, • Review behavior on assignments and (if expectations and make teacher allows) on adjustments assessments • Create a math journal that • Provide extra time to they can use during class, complete a task when on assignments and (if needed teacher allows) on • Provide definitions of assessments different graphs / charts • Allow students to complete with illustrations an independent project as an • Allow tests to be taken in a alternative test separate room • Use of alge-tiles when • Allow students to use a needed calculator when appropriate • Use of number line when • Divide test into small needed sections of similar questions or problems • Use of alge-tiles when needed • Use of number line when needed

Interdisciplinary Connections: ELA NJSLSA.L1. Demonstrate command of the conventions of standard English grammar and usage when writing or speaking NJSLSA.L6: Acquire and use accurately a range of general academic and domain-specific words and phrases sufficient for reading, writing, speaking and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when encountering an unknown term important to comprehension or expression. NJSLSA.W10. Write routinely over extended time frames (time for research, reflection, and revision) and shorter time frames (a single sitting or a day or two) for a range of tasks, purposes, and audiences. 21st Century Standards 9.2.12.C.3 Identify transferable career skills and design alternate career plans. Career Ready Practices: CRP2: Apply appropriate academic and technical skills CRP4: Communicate clearly and effectively and with reason CRP6: Demonstrate creativity and innovation CRP8: Utilize critical thinking to make sense of problems and persevere in solving them CRP11: Use technology to enhance productivity Technology Standards: 8.1.12.A.3: Collaborate in online courses, learning communities, social networks or virtual worlds to discuss a resolution to a problem or issue.

Major Supporting Additional (Identified by NJSLA Model Content Frameworks)