geometry-2 - Math Leadership Corps

Geometry (CC) Grade 10|Mathematics|Common Core|2013-2014 Friday, May 16, 2014, 11:10AM Course Description: Introduction to the MAISA CCSS Mathematics Curriculum Materials The curriculum materials are designed to:    be professional learning tools to improve educators’ understanding of the CCSS; organize the CCSS into mathematically coherent and sequenced units of study that make visible connections among mathematical ideas. They are not designed to prescribe a single pathway through a particular unit; and provide a context for conversations among colleagues (e.g., Professional Learning Communities) within and across grades. Lesson and assessment topics within the units of study are selected to highlight content that might be new, different, or challenging for teachers and students. This highlighted content may be used to spark important planning and problem solving discussions related to the CCSS implementation. **Please read the following attachment addressing FAQs before using the MAISA CCSSI Mathematics Units ** Using the MAISA CCSS Units of Study for Mathematics - FAQ Overar ching Questi ons and Enduri ng Unders tanding s 1How Langu can age of coordi Geom nate etry system (Week s and geomet 1, 5 Week ric constr s) uctions help us explore and extend Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds In this CCSS: The first Mathema standar geometry tics, ds unit, CCSS: listed in students HS: this begin to Geometr section formalize y, have the Congrue been informal nce modifie language HSGd to be and CO.C. appropr geometri Prove iate for c geometri this relations unit. Te 1. H o w c a n c o o r d i n a Intellectual Processes Standards of angle relations 10_1 Mathematical hips Formativ Practice Students will inductiv e e Assessm have opportunities reasonin ent g Overvie to: conditio w nal  constr 10_1 stateme uct nts (if- Formativ viable e then argume stateme Assessm nts and ent nts, critique Student Less on Sequ ence Resources Unit Resources 10_1 Highl ight Less on Over view Overar ching Questi ons and Graphic Enduri Organizer ng Unders tanding s our unders tanding of line and angle relation ships? What are some potenti al ways to prove geomet Language of Geometry ric propert Highlight Lesson ies and proble ms? Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds hips they c xt in bring theorems gray with font is them to  H part of the S the CCS course G Swhich M stand will lead C ard but towards O does more not . formal C apply to proofs this . througho 9 unit. Te ut the xt in . geometry P bracket course*. s r For o denotes example, v a students e modific have ation t used h that has angle e been relations o made to hips to the r find e standar unknown m d. angles in s middle a CCSS: school b Mathem (includin o atics C g triangle u CSS: HS: sum and t Geomet angles li formed n ry Circl by e es parallel s t e s b e u s e d t o f i n d t h e s l o p e o f a s e g m e n t , t h inverse, Handout convers s e, contrapo sitive) deductiv e reasonin g countere xample hypothe sis/concl usion circles construc tions axiomati c system Euclidea n geometr y properti es of parallel lines transver sals midpoint perpendi cular bisector angle bisector Intellectual Processes  the reasoni ng of others j ustifyin g both their reasoni ng and critique s with ge ometric definitio ns; attend to precisi on usin g the languag e of geomet ry to express mathem atical ideas, develop and use clear definitio ns, and make and describ Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds lines cut by a transvers al) and reasoned about supplem entary and complem entary angles. In geometry , newly defined terms like "linear pair" allow students to provide precise justificati ons for problem solving. They may begin with less formal reasonin g like, a n d a n g l e s . T h e o r e m s i n c l u d e : v e rt i c a l a n HSGC.A. Underst and and apply theorem s about circles HSGC.A.3. Constru ct the inscribe d and circumsc ribed circles of a triangle, and prove propertie s of angles for a quadrilat eral inscribe d in a circle. CCSS: Mathem atics C CSS: HS: e m i d p o i n t o f a s e g m e n t , a n d l e n g t h o f a s e g m triangle centers (circumc enter, incenter, orthocen ter) distance formula perpendi cular and parallel line relations hips Pythago rean Theore m Intellectual Processes  e geomet ric constru ctions; and reason abstrac tly and quantit atively by conn ecting algebrai c and geomet ric represe ntations of objects. Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds "because they are linear pairs" and progress to using condition al statemen ts like, "if two angles form a linear pair, then they are supplem entary." Not only should students be able to justify their conjectur es and argumen ts by a variety of methods, but they should also be able to analyze g l e s a r e c o n g r u e n t; w h e n a tr a n s v e r s a l c r o s s Geomet ry Cong ruence HSGCO.A. Experim ent with transfor mations in the plane HSGCO.A.1. Know precise definitio ns of angle, circle, perpendi cular line, parallel line, and line segment , based on the undefine d notions of point, line, distance along a line, and e n t ? 2. H o w c a n s l o p e s b e u s e d t o d e t e r m i n e w h e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds and critique argumen ts to determin e their validity. In this unit, students will be extendin g their knowled ge of angle relations hips and making deductiv e argumen ts to informall y prove relations hips with a variety of angle pairs. These include vertical angles, linear pairs, e s p a r a ll e l li n e s , a lt e r n a t e i n t e ri o r a n g l e s a distance around a circular arc. t h e r t w o l i n e s a r e p a r a l l e l , p e r p e n d i c u l a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds supplem entary angles, complem entary angles, and angles formed by parallel lines intersect ed by a transvers al. These relations hips provide a context through which students can begin to build foundatio ns for reasonin g. The need to reason logically occurs daily. Principle r e c o n g r u e n t a n d c o r r e s p o n d i n g a n g l e s a r e r , o r n e i t h e r ? 3. W h a t m e t h o d ( s ) c a n b e u s e d t o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s of reasonin g are useful whether developi ng a strategy to win a game or the sequenc e of steps needed to repair a lamp. Inductive reasonin g relies on making observati ons to test conjectur es and arrive at conclusio ns. For example, students might draw intersecti ng lines c o n g r u e n t; p o i n t s o n a p e r p e n d i c u l a r b i s e c c o n s t r u c t t w o p a r a l l e l l i n e s c u t b y a t r a n s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds and observe that "if angles are vertical angles, then they are congruen t." How many observati ons suffice in "proving" the conjectur e will always be true? The limitation s of inductive reasonin g indicate the need for deductiv e reasonin g, a method that t o r o f a li n e s e g m e n t a r e e x a c tl y t h o s e e q u i d i v e r s a l ? H o w c a n t h e r e l a t i o n s h i p b e t w e e n t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds establish s es when t a a conjectur n e is t always fr true. This o unit also m attends t to the h structure e of if-then s statemen e ts and g how m language e structure n affects t’ reasonin s g. e Students n might d also p examine o the i validity of n related t condition s al . statemen ts. For HSGexample, CO.D. the Make inverse, geometri "if angles c are not h e a n g l e s f o r m e d b e j u s t i f i e d ? 4. W h a t a r e s o m Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds congruen construct t, then ions they are not  vertical H angles"; S the G converse , "if C angles O are . congruen D t, then . they are 1 vertical 2 angles", . and the M contrapo a sitive, "if k angles e are not f congruen o t, then r they are m not a vertical l angles". g Students e are likely o to use m countere e xamples tr to show i that not c all of c these o e w a y s t o i n f o r m a l l y p r o v e t h a t t w o l i n e s a r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds statemen ts are true. The relations hips between angles also provide a basis for construct ions of parallel and perpendi cular lines and a rationale for why the construct ions work. While students have informall y construct ed geometri c figures in earlier grades, n s tr u c ti o n s w it h a v a ri e t y o f t o o l s a n d m e t h o d s e p a r a l l e l ? 5. W h a t i s t h e d i f f e r e n c e b e t w e e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds in high school the construct ions become more precise and justifiable . They use a variety of tools (e.g., compass es, rulers, tracing paper and dynamic software) to make and utilize construct ions to explore and critique conjectur es. Establish ing methods ( c o m p a s s a n d s tr a i g h t e d g e , s tr i n g , r e fl e c ti v n i n d u c t i v e a n d d e d u c t i v e r e a s o n i n g ? W h a t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds for construct ions and the usefulne ss of a variety of tools in this first unit provides strategie s for students to employ in future explorati ons. Through out the study of geometry , the use of a computer software program or graphing calculato r with geometri c drawing capabiliti e d e v i c e s , p a p e r f o l d i n g , d y n a m i c g e o m e tr i a r e t h e v a l u e a n d l i m i t a t i o n s o f e a c h t y p e o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds es that allows students to draw and manipula te figures, make conjectur es, and reach conclusio ns should be encourag ed. Coordina te represen tations of geometri c figures provide a link between algebra and geometry and another method to make deductiv e c s o ft w a r e , e t c ). C o p y i n g a s e g m e n t; c o p y i n g a f r e a s o n i n g ? 6. W h a t t o o l s a r e h e l p f u l i n c o n s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds argumen ts about geometri c figures in the plane. Students will use also use coordinat es to calculate slopes, midpoint s and segment lengths. *In this unit, the foundatio ns for proof are built. The use of deductiv e reasonin g with informal proofs provides the backgrou nd for students to begin n a n g l e ; b i s e c ti n g a s e g m e n t; b i s e c ti n g a n a n g t r u c t i n g g e o m e t r i c o b j e c t s ? H o w a r e t h e s e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds developi ng more formal proof. Through out the course, students will move from explainin g their reasonin g informall y to more formal explanati ons. Using valid argumen ts and reasonin g patterns with correct assumpti ons and careful explanati on of the reasonin g is important l e ; c o n s tr u c ti n g p e r p e n d i c u l a r li n e s , i n c l u c o n s t r u c t i o n s u s e f u l ? Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . Students build flexibility with the forms used to prove theorems . Paragrap h proofs, flow chart proof, twocolumn proofs, transfor mational proofs, coordinat e proofs, argumen ts affirming the hypothes is, chains of if-then statemen ts, and use of countere xamples are some d i n g t h e p e r p e n d i c u l a r b i s e c t o r o f a li n e s e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds possible methods students could use to prove a conjectur e. Unit Overvie w (Word) Unit Overvie w (PDF) g m e n t; a n d c o n s tr u c ti n g a li n e p a r a ll e l t o a g i v e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n li n e t h r o u g h a p o i n t n o t o n t h e li n e .  H S G C O Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . D . 1 3 . C o n s tr u c t a n e q u il a t e r a l tr i a n g l e , a s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds q u a r e a n d a r e g u l a r h e x a g o n i n s c ri b e d i n a c ir Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds c l e . CCSS: Mathema tics, CCSS: HS: Geometr y, Expressi ng Geometri c Propertie s with Equation s HSGGPE.B. Use coordinat es to prove simple geometri c theorems algebraic ally  H S Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds G G P E . B . 5 . P r o v e t h e s l o p e c ri t e ri a f o r p a r a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ll e l a n d p e r p e n d i c u l a r li n e s a n d u s e t h e m t o s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o l v e g e o m e tr i c p r o b l e m s ( e . g ., fi n d t h e e q u a ti Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o n o f a li n e p a r a ll e l o r p e r p e n d i c u l a r t o a g i v Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e n li n e t h a t p a s s e s t h r o u g h a g i v e n p o i n t) .  H Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds S G G P E . B . 6 . F i n d t h e p o i n t o n a d ir e c t e d li n e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s e g m e n t b e t w e e n t w o g i v e n p o i n t s t h a t d i v i Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds d e t h e s e g m e n t i n a g i v e n r a ti o .  H S G G P E . B . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds 7 . U s e c o o r d i n a t e s t o c o m p u t e p e ri m e t e r s o f Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds p o l y g o n s a n d a r e a s f o r tr i a n g l e s a n d r e c t a n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds g l e s , e . g . u s i n g t h e d i s t a n c e f o r m u l a . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Graphic Enduri Organizer ng Unders tanding s 2What Trans impact forma does tional each Geom type of etry transfo (Week rmatio n 6, 7 Week (reflecti on, s) rotatio n, translat ion, and dilation ) have on the locatio Graphic n, size, Organizer and orienta tion of geomet ric objects ? Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds In CCSS: Not 1. Wh seventh Mathema applicabl at grade, tics, e. im students CCSS: pa began an HS: ct informal Geometr do study of y, es similarity Congrue ch as they nce an examine HSGgin d CO.A. g proportio Experim the nal ent with ce relations transfor nte hips and mations r of created in the dila and used plane tio scale n drawings  ha . They ve H also on S used a the G variety of loc methods ati C to on O informall of . y the A construct im . figures. ag 1 In eighth e? . grade 2. Wh K students ich n develope tra o d nsf w transfor or p mation ma r Intellectual Processes Standards of angle of Mathematical rotation center of Assessm Practice Students will dilation ent center of Overvie have opportunities rotation w to: composi tion/seq Student uence of Handout  look two or _Transfo for and more make rmation transfor Assessm use of mations ent structu congrue re nce when re Artifactdilation presenti Student ng Work geometri transfor Samples c constr mations : Using uctions algebrai Student image/pr cally Work to e-image and Reengag lines of geomet e dilation rically Students reflectio and exa in the n mining Assessm rotation how ent scale these factor or represe magnitu Artifactntations Student de of intercon dilation Work nect similarity Samples and similarity : Using build on transfor Student one mations Work to Less on Sequ ence Resources Instructional Resources Mathematics Assessment Project (MAP) Less Formative Assessment Lessons on Transformations: Representing and Over Combining Transformations: this lesson unit view is intended to help you assess how well students are able to recognize and visualize Prof transformations of 2D shapes as well as essi translate, reflect, and rotate shapes, and onal combine these transformations. It also aims Lerni to encourage discussion on some common ng misconceptions about transformations. Task http://map.mathshell.org/materials/lesso Stud ns.php ent Representing and Combining Work Transformations: This lesson unit is Sam intended to help you assess how well ple students are able to: Recognize and unit visualize transformations of 2D shapes; Translate, reflect, and rotate shapes, and 3 combine these transformations. It also aims Prof to encourage discussion on some common essi misconceptions about transformations. onal http://map.mathshell.org/materials/lesso Lear ns.php?taskid=223 ning NCTM Illuminations ( Task http://illuminations.nctm.org) – Teac  Symmetries I: Students will learn her about the mathematical properties Refle of rotation. ction http://illuminations.nctm.org/Les Vide sonDetail.aspx?ID=U138 o: Exa Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds based definition s of congruen ce and similarity. They transfor med figures into congruen t figures with rotations, reflection s, and translatio ns and verified experime ntally the related propertie s of these rigid transfor mations. They also created similar figures using dilations e c i s e d e fi n it i o n s o f a n g l e , c ir c l e , p e r p e n d i tio ns pre ser ve dist an ce ? sh ap e? ori ent ati on ? an gle ? 3. Ho w do es the val ue of the sca le fac tor in a dila Intellectual Processes symmetr Reengag e y (rotation Students  al and in the reflectio Assessm ent nal) transfor mation translati on triangle similarity  another ; model with mathe matics using tr ansform ations to model and interpre t physical , and mathem atical phenom ena; and use approp riate tools strategi cally to make geomet ric constru ctions and explore and deepen Less on Sequ ence Resources mini  ng the Laun ch of  the Less on  Symmetries II: Students investigate reflection. http://illuminations.nctm.org/Les sonDetail.aspx?ID=U139 Symmetries III: Students will learn how translations work and what happens when two or more translations are applied one after another http://illuminations.nctm.org/Les sonDetail.aspx?id=L474 Symmetries IV: Students will learn about glide reflection. http://illuminations.nctm.org/Les sonDetail.aspx?id=L475  Transformations and Frieze Patterns: http://illuminations.nctm.org/Les sonDetail.aspx?ID=U181 Texas Instruments ( http://education.ti.com)  Connecting Translations, Reflections, and Rotations(TI-84): In this activity, students will investigate the relationship among the three types of rigid transformations - translations, rotations, and reflections. http://education.ti.com/xchange/ Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds and used coordinat es to describe the effects of all transfor mations on twodimensio nal figures. When given similar or congruen t figures, students describe da sequenc e of transfor mations to exhibit the similarity and congruen ce. In high school students extend and c u l a r li n e , p a r a ll e l li n e , a n d li n e s e g m e n t, b a s tio n infl ue nc e the siz e of the im ag e? 4. Ho w ca n ge om etri c tra nsf or ma tio ns be rep res ent ed alg ebr aic Intellectual Processes underst anding of topics related to transf ormatio ns. Less on Sequ ence Resources  US/Math/Geometry/4046/CabriJr_ Interactive_Act%2009.pdf Scale Factor (TI-84): Students dilate polygons and find the perimeter and area of both the preimage and image. http://education.ti.com/calculator s/timath/US/Activities/Detail?sa= 5024&id=10233 Also for (TI-Nspire): In this activity, students will dilate polygons and find the perimeter and area of both the pre-image and image. Then they find the ratios of the perimeters and areas. When they change the scale factor, all of these values automatically update and students can see that the ratio of the perimeters equals the scale factor and the ratio of the areas equals the square of the scale factor. Students will see that this is true for both enlargements and reductions. http://education.ti.com/calculators/downl oads/US/Activities/Detail?ID=8299&MICR OSITE=ACTIVITYEXCHANGE Interactive Geometry Activities and Investigations: http://education.ti.com/educationportal/s ites/US/nonProductSingle/activitybook_c abrijr_interactive.html  Cabri™ Jr.: Interactive Geometry Activities and Investigations: Explore geometry through the 29 Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds formalize the study of transfor mations. Using precise definition s develope d in high school, students develop new definition s for rigid motions (i.e., translatio ns, reflection s, and rotations) . In addition, they verify experime ntally the propertie s of dilations subsequ ently comparin e d o n t h e u n d e fi n e d n o ti o n s o f p o i n t, li n e , d i s t ally ? 5. Ho w mi ght yo u inv esti gat e the co mp osit ion of two or mo re tra nsf or ma tio ns to det er mi ne whi ch co Intellectual Processes Less on Sequ ence Resources activities in this book. The focus here is on concepts rather than keystrokes and allows students to explore five areas of geometry in depth. http://education.ti.com/education portal/sites/US/nonProductSingle /activitybook_cabrijr_interactive. html NCTM Reasoning and Sense Making Task Bank: (www.nctm.org/hsfocus) Taking a Spin: Although students are often asked to find the angles of rotational symmetry for given regular polygons, in this task they are asked to find the regular polygons for a given angle of rotational symmetry, a reversal that yields some surprising results. This task would be most appropriate with students who have at least some experience in exploring rotational symmetry. http://www.nctm.org/uploadedFiles/Jour nals_and_Books/Books/FHSM/RSMTask/RSM_Task-Taking_a_Spin.pdf NCTM Navigations Series Day, R., Kelley, P., Krussel, L., Lott,J. W., & Hirstein, J. (2002). Navigating through Geometry in Grades 9–12 (with CD-ROM). Reston,VA: NCTM. Approaching geometry through a transformational lens, this book concentrates on topics such as the use of transformations, coordinates and matrices, and congruence and similarity. Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds g transfor mations that preserve angle and distance with those that do not. Students also use the transfor mations based definition s of similarity and congruen ce to decide if figures are similar or congruen t and then describe a sequenc e of transfor a n c e a l o n g a li n e , a n d d i s t a n c e a r o u n d a c ir c u mb ina tio ns are co m mu tati ve ? Intellectual Processes Less on Sequ ence Resources A Useful Activity Students are able to practice or demonstrate understanding of transformations by moving shapes on a coordinate grid to new locations following specific requirements. http://www.teachnetuk.org.uk/2006%20Projects/MathsTransformations/Tristan%20Jones/Home .htm Geometer’s Sketchpad  Investigate reflections, translations, dilations, glide reflections, rotations of 90º, 180º, 270º, rotations of any degree, and composition of reflections with this menu driven exploration. http://mathbits.com/MathBits/GS P/Transformations.htm  Investigate the result of two different geometric transformations when combined. Determine whether (or under what conditions) the order of transformations may be reversed, and which compositions may be written as a single transformation. http://www.uni.illinois.edu//~hcru ssel/CompositionOfTransformati ons.htm This unit is intended to guide students through the study of transformational geometry of isometries using Geometer's Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds mations to justify their thinking. Students have used patty  paper, tracing paper, or graph paper in seventh and eighth grade to draw preimages and images that result after a single transfor mation or sequenc e of transfor mations. In high school, drawing figures is Intellectual Processes Less on Sequ ence Resources SketchPad as the major tool for investigation and production. Students should work through each day's assignment, printing their results for teacher approval. After the investigations and productions have been performed, students will be asked to take a final assessment that will require them to use all that they have learned about transformations of isometries. Homework assignments each day will be answered in a journal that students must keep to write about the processes they have experienced and the discoveries l a r a r c . H S G C O . A . 2 . R e p r e s e n t tr a n s f o r m they have made. http://jwilson.coe.uga.edu/emt66 8/EMAT6680.Folders/Maddox/Tra nsformational.Geo.html Other Computer Applets  A series of 23 interactive online math lessons such as Basic Transformation Concepts, Reflections, Rotations, Translations, and Dilations http://enlvm.usu.edu/ma/nav/toc.j sp?sid=__shared&cid=emready @transformations&cf=activity  Use the virtual pantograph in this applet to create similar figures. Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds also extended with the use of formal geometri c construct ions with a variety of tools. Although students may have had experien ce with technolo gy tools, the use of a computer software program s or graphing calculato rs with geometri c drawing capabiliti es should be encourag a ti o n s i n t h e p l a n e u s i n g , e . g ., tr a n s p a r e n c i Intellectual Processes Less on Sequ ence Resources http://www.ies.co.jp/math/produc ts/geo1/applets/panta/panta.html    A free interactive math textbook on the web, defining and giving examples for transformational geometry http://www.mathopenref.com/tran slate.html A variety of interactive geometry activities, with some especially good activities on 3-D and 2-D views of objects http://www.internet4classrooms. com/eoc_geometry.htm CPMP-Tools   Course 1 – Tilings with Triangles or Quadrilaterals Tilings with Penrose Tiles Course 2 - Animate Shuttle Roll Over Geogebra  Pre-image and image points and coordinates are shown. Student works to determine what type of transformation was done. http://www.geogebra.org/en/uplo ad/files/knwilson/Transformation s.html Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ed througho ut the study of geometry . This allows students to draw and manipula te figures, make conjectur es, and reach conclusio ns. In particular , students not only represen t transfor mations in the plane, but also use coordinat es to develop function rules for e s a n d g e o m e tr y s o ft w a r e ; d e s c ri b e tr a n s f o r m a Intellectual Processes Less on Sequ ence Resources  The scale factor is dynamic and students are asked to observe how the scale factor effects the lengths of the pre-image and image. http://www.geogebra.org/en/wiki/i ndex.php/Dilations Manipulatives Mira Geoboard Linkage strips Patty paper Assessments This assessment is pre-built in Geometer’s Sketchpad. It has a number of figures and a design. The students must use transformations to move the figures onto the design with as few transformations as possible. http://jwilson.coe.uga.edu/emt668/EMAT 6680.Folders/Maddox/student%20directi ons/directions.html The quiz provided here has the students do a variety of activities on the computer. Some are manipulating figures to show transformations, others are answering questions regarding properties and results of doing transformations. http://enlvm.usu.edu/ma/nav/activity.jsp ?sid=__shared&cid=emready@transform ations&lid=1 Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds each transfor mation thus connecti ng the geometri c and algebraic represen tations. Later, students may also use matrices to transfor m figures in the coordinat e plane. Concept s and strategie s from this transfor mations geometry unit will be applied in later units. For example, ti o n s a s f u n c ti o n s t h a t t a k e p o i n t s i n t h e p l a Intellectual Processes Less on Sequ ence Resources This resource could serve as a formative assessment task on transformations and does not require the use of any technology. http://www.lewiscentral.org/pages/uploa ded_files/Geometry%201%20Practice%2 0Quiz%20%237.pdf Professional Resources NCTM (www.nctm.org) Focus in High School Mathematics: Reasoning and Sense Making: This publication elevates reasoning and sense making to a primary focus of secondary mathematics teaching. It shifts the teachers’ role from acting as the main source of information to fostering students’ reasoning to make sense of the mathematics. http://www.nctm.org/catalog/product.asp x?ID=13494 Focus in High School Mathematics: Reasoning and Sense Making In Geometry: Classically, geometry has been the subject in which students encounter mathematical proof based on formal deduction. Attention to proof in the geometry curriculum is strengthened by a focus on reasoning and sense making. The authors examine the four key elements (conjecturing about geometric objects, construction and evaluation of geometric arguments, multiple geometric approaches, and geometric connections and modeling) identified in Focus in High School Mathematics: Reasoning and Sense Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds students will use transfor mations to prove theorems about triangles, parallelo grams, lines, and angles. They will also use similarity and congruen ce to solve problems and develop definition s of trigonom etric ratios. In algebra, students transfor m function rules and observe correspo n e a s i n p u t s a n d g i v e o t h e r p o i n t s a s o u t p u t Intellectual Processes Less on Sequ ence Resources Making in more detail and elaborates on the associated reasoning habits. http://www.nctm.org/catalog/product.asp x?ID=13525 Articles from National Council of Teachers of Mathematics (www.nctm.org) Articles available as free downloads to NCTM members, or for a fee to nonmembers Mathematics Teacher, Focus Issue: Geometry, 105(4), November 2011 Bosse, M. and Adu-Gyamfi, K. (2011). Stop Teaching and Let Students Learn Geometry, Mathematics Teacher , 105(4), 293-7. Retrieved November 6, 2011 from http://www.nctm.org/publications/article. aspx?id=31287 Sheats Harkness, S. (2005). Geometry of Transformations: Teacher and Unit Under Construction. Mathematics Teacher, 99 (2), 88-92. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary. asp?URI=MT2005-09-88a&from=B Devaney, R. L. (2004). Fractal Patterns and Chaos Games. Mathematics Teacher, 98 (4), 228-233. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary. asp?URI=MT2004-11-228a&from=B Siegrist , R. (2009). Activities for Students: Inquiry into Fractals. Mathematics Teacher, 103 (3), 206-211. Retrieved December 6, 2010 from Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds nding changes in graphs of quadratic , exponent ial, and polynomi al functions . Unit Overvie w (Word) Unit Overvie w (PDF) s . C o m p a r e tr a n s f o r m a ti o n s t h a t p r e s e r v e d i Intellectual Processes Less on Sequ ence Resources www.nctm.org/eresources/article_summary. asp?URI=MT2009-10-206a&from=B Edwards, M.T. (2005). Activities for Students: Using Overhead Projectors to Explore Size Change Transformations. Mathematics Teacher, 98 (7), 498-505. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary. asp?URI=MT2005-03-498a&from=B Cooper, B. D. and Barger, R. (2009). Listening to Geometry. Mathematics Teacher, 103 (2), 108-115. Retrieved December 6, 2010 from www.nctm.org/eresources/article_summary. asp?URI=MT2009-09-108a&from=B Click here to link to instructional and professional resources. Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s t a n c e a n d a n g l e t o t h o s e t h a t d o n o t ( e . g ., tr Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a n s l a ti o n v e r s u s h o ri z o n t a l s tr e t c h ).  H S G C Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds O . A . 3 . G i v e n a r e c t a n g l e , p a r a ll e l o g r a m , tr Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a p e z o i d , o r r e g u l a r p o l y g o n , d e s c ri b e t h e r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o t a ti o n s a n d r e fl e c ti o n s t h a t c a r r y it o n t o it s e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds lf .  H S G C O . A . 4 . D e v e l o p d e fi n it i o n s o f r o t a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ti o n s , r e fl e c ti o n s , a n d tr a n s l a ti o n s i n t e r m s o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds f a n g l e s , c ir c l e s , p e r p e n d i c u l a r li n e s , p a r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a ll e l li n e s , a n d li n e s e g m e n t s .  H S G C O . A . 5 . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds G i v e n a g e o m e tr i c fi g u r e a n d a r o t a ti o n , r e fl e c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ti o n , o r tr a n s l a ti o n , d r a w t h e tr a n s f o r m e d fi g u Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r e u s i n g , e . g ., g r a p h p a p e r, tr a c i n g p a p e r, o r g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e o m e tr y s o ft w a r e . S p e c if y a s e q u e n c e o f tr a n s f Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o r m a ti o n s t h a t w il l c a r r y a g i v e n fi g u r e o n t o a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n o t h e r. HSGCO.B. Understa nd congruen ce in terms of rigid motions  H S G C O . B . 6 . U s e g e o m Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e tr i c d e s c ri p ti o n s o f ri g i d m o ti o n s t o tr a n s f o r m Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds fi g u r e s a n d t o p r e d i c t t h e e ff e c t o f a g i v e n ri g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i d m o ti o n o n a g i v e n fi g u r e ; g i v e n t w o fi g u r e s , Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds u s e t h e d e fi n it i o n o f c o n g r u e n c e i n t e r m s o f ri Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds g i d m o ti o n s t o d e c i d e if t h e y a r e c o n g r u e n t. Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds HSGCO.D. Make geometri c construct ions  H S G C O . D . 1 2 . M a k e f o r m a l g e o m e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds tr i c c o n s tr u c ti o n s w it h a v a ri e t y o f t o o l s a n d m e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t h o d s ( c o m p a s s a n d s tr a i g h t e d g e , s tr i n g , r e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds fl e c ti v e d e v i c e s , p a p e r f o l d i n g , d y n a m i c g e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o m e tr i c s o ft w a r e , e t c ). C o p y i n g a s e g m e n t; c o p Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds y i n g a n a n g l e ; b i s e c ti n g a s e g m e n t; b i s e c ti n g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a n a n g l e ; c o n s tr u c ti n g p e r p e n d i c u l a r li n e s , Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i n c l u d i n g t h e p e r p e n d i c u l a r b i s e c t o r o f a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds li n e s e g m e n t; a n d c o n s tr u c ti n g a li n e p a r a ll e l t o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a g i v e n li n e t h r o u g h a p o i n t n o t o n t h e li n e . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds CCSS: Mathema tics, CCSS: HS: Geometr y, Similarity , Right Triangles ,& Trigono metry HSGSRT.A. Understa nd similarity in terms of similarity transfor mations  H S G S R T . A . 1 Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . V e ri f y e x p e ri m e n t a ll y t h e p r o p e rt i e s o f d il a ti Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o n s :  H S G S R T . A . 1 a . A d il a ti o n t a k e s a li n e n o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t p a s s i n g t h r o u g h t h e c e n t e r o f t h e d il a ti o n t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o a p a r a ll e l li n e , a n d l e a v e s a li n e p a s s i n g t h r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o u g h t h e c e n t e r u n c h a n g e d .  H S G S R T . A . 1 b Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . T h e d il a ti o n o f a li n e s e g m e n t i s l o n g e r o r s h o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds rt e r i n t h e r a ti o g i v e n b y t h e s c a l e f a c t o r.  H S Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds G S R T . A . 2 . G i v e n t w o fi g u r e s , u s e t h e d e fi n it Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i o n o f s i m il a ri t y i n t e r m s o f s i m il a ri t y tr a n s f o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r m a ti o n s t o d e c i d e if t h e y a r e s i m il a r; e x p l a i n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds u s i n g s i m il a ri t y tr a n s f o r m a ti o n s t h e m e a n i n g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o f s i m il a ri t y f o r tr i a n g l e s a s t h e e q u a li t y o f a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ll p a ir s o f a n g l e s a n d t h e p r o p o rt i o n a li t y o f a ll Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Intellectual Processes Less on Sequ ence Resources p a ir s o f s i d e s . 3What Trian conditi gles ons (Week must 13, 6 be met Week to show s) similari ty or congru ence of triangle s? Wh at are the propert ies of Triangles median Graphic s, altitude s, angle The study of triangles includes proving many propertie s that students may already be familiar with. Th e angle sum of a triangle being 1800 and the relations hip CCSS: Mathema tics, CCSS: HS: Algebra, Arithmeti c with Polynomi als & Rational Function s HSAAPR.C. Use polynomi al identities to solve problems . 1. Wh at are the sim ilari ties an d diff ere nc es bet we en sim ilar an d co ngr Standards of altitudes Mathematical angle bisector Assessm Practice Students will s ent base template have opportunities angles for centroid triangle to: circumc types enter  constr correspo student uct nding workshe viable parts ets for argume hypoten triangles nts rela use types ted to incenter similar leg sample and medians solution congrue orthocen s for nt ter triangle triangle perpendi types s. cular Blank Unit Resources Artifact – Recording Sheet: Using Unit Highl Resources to Support Student ight Accessibility and Understanding Less on Highl ight Less on Stud ent Hand out Highl ight Less on Supp Overar ching Questi ons and Enduri ng Unders tanding s bisecto rs, and perpen dicular bisecto rs of the sides of a triangle ? Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds between an  exterior angle and the sum of the remote interior angles are familiar and connecte d ideas that can now be proved in a more formal manner using knowled ge from previous explorati ons. In seventh grade students explored proportio nal relations hips and learned H S A A P R . C . 4 . P r o v e p o l y n o m i a l i d e n ti ti e s ue nt tria ngl es ? 2. Us ea ge om etr y sof twa re pro gra m or co mp ass an d str aig hte dg e to co nst ruc t the me Intellectual Processes bisector Assessm  ent for s Pythago Possible teacher rean Theore use in Indepen m triangle dent congrue Practice ncy (SSS, Triangle types SAS, AAS, ASA, Assessm HL) ent triangle Template similarity -Triangle (SSS, congrue SAS, nce AA) Grid vertex paper for angles Assessm  ent coordina triangle te proofs congrue nce Professi onal Learning Tasks Student Work Samples -Triangle congrue Make sense of proble ms and persev ere in solving them: build new mathem atical knowle dge of triangle s through problem solving Model with mathe matics: use represe ntations of triangle s to model and interpre t physical , social, Less on Sequ ence ortin g Docu ment (Illu mina tions ) Artif act – Stud ent Work Sam ples: Usin g Stud ent Work Sam ples to Reen gage Stud ents in the Less on Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds about similar figures. I n eighth grade translatio ns, reflection s, rotations, and dilations were explored and the effect of each of these transfor mations on lines, angles, and twodimensio nal figures was studied. Students made connecti ons between using these transfor a n d u s e t h e m t o d e s c ri b e n u m e ri c a l r e l a ti o n s h dia ns, altit ud es, per pe ndi cul ar bis ect ors , an d an gle bis ect ors of a tria ngl e. Ma ke co nje ctu res ab out the ir nce Discours e Professi onal Learning Tasks Student Work Samples Triangle Congrue nce Misconc eption Intellectual Processes and mathem atical phenom ena Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds mations to demonstr ate congruen ce and similarity of triangles and other twodimensio nal figures. Discoveri ng and applying combinat ions of sides and angles that are sufficient condition s for similarity or congruen ce of two triangles (for similarity: AA, SSS, i p s . F o r e x a m p l e , t h e p o l y n o m i a l i d e n ti t y int ers ecti on poi nts an d an y ad diti on al ch ara cte risti cs. Incl ud e the eff ect of ch an gin g the typ e of tria ngl Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds SAS, and for congruen ce: SSS, SAS, ASA, AAS, HL) provides experien ce in making conjectur es. The results of these relations hips can be used to reason further about additiona l propertie s of triangles, isosceles triangles, and many quadrilat erals. St udents will apply their previous ( x ² + y ² ) ² = ( x ² – y ² ) ² + ( 2 x y ) ² c a n b e u s e d t o g e fro m ac ute to rig ht to obt us e. Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds understa e nding n about e transfor r mations a along t with their e study of P transfor y mations t in eighth h grade to a write g explanati o ons or r proofs e showing a how a n single tr transfor i mation or p sequenc l e of e transfor s mations . leads from one CCSS: figure to Mathema a second tics, congruen CCSS: t or HS: similar Geometr figure. y, Proofs Congrue related to nce triangles Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds can HSGagain CO.B. take Understa many nd different congruen forms ce in including terms of coordinat rigid e proofs. motions In  addition to congruen ce relations hips, similarity is an important area of study in triangles. In fact, it is reasonab le to begin with the propertie s of similarity and then move to congruen ce H S G C O . B . 6 . U s e g e o m e tr i c d e s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds propertie s as a special case of similarity. The propertie s of congruen ce and similarity should be used to solve problem situation s. Students should know the propertie s of medians, angle bisectors , perpendi cular bisectors , and altitudes of triangles and their concurre c ri p ti o n s o f ri g i d m o ti o n s t o tr a n s f o r m fi g u r e s a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds nt points. Methods of calculatin g area and perimete r of triangles should be recalled and new methods applied. Through out the study of geometry , the use of a computer software program or graphing calculato rs with geometri c drawing capabiliti es that allow n d t o p r e d i c t t h e e ff e c t o f a g i v e n ri g i d m o ti o n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds students to draw and manipula te figures to make conjectur es and reach conclusio ns should be encourag ed. Usin g coordinat e methods and proofs helps make connecti ons between the ideas of geometry and algebra. Unit template 8-2-12 (word) o n a g i v e n fi g u r e ; g i v e n t w o fi g u r e s , u s e t h e d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Unit template 8-2-12 (pdf) e fi n it i o n o f c o n g r u e n c e i n t e r m s o f ri g i d m o ti o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n s t o d e c i d e if t h e y a r e c o n g r u e n t.  H S G C O . B Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . 7 . U s e t h e d e fi n it i o n o f c o n g r u e n c e i n t e r m s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o f ri g i d m o ti o n s t o s h o w t h a t t w o tr i a n g l e s a r e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds c o n g r u e n t if a n d o n l y if c o r r e s p o n d i n g p a ir s o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds f s i d e s a n d c o r r e s p o n d i n g p a ir s o f a n g l e s a r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e c o n g r u e n t.  H S G C O . B . 8 . E x p l a i n h o w t h e c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ri t e ri a f o r tr i a n g l e c o n g r u e n c e ( A S A , S A S , a n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds d S S S ) f o ll o w fr o m t h e d e fi n it i o n o f c o n g r u e n c e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i n t e r m s o f ri g i d m o ti o n s . HSGCO.C. Prove geometri c theorems  H S G C O . C Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . 1 0 . P r o v e t h e o r e m s a b o u t tr i a n g l e s . T h e o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r e m s i n c l u d e : m e a s u r e s o f i n t e ri o r a n g l e s o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds f a tr i a n g l e s u m t o 1 8 0 ° ; b a s e a n g l e s o f i s o s c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e l e s tr i a n g l e s a r e c o n g r u e n t; t h e s e g m e n t j o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i n i n g m i d p o i n t s o f t w o s i d e s o f a tr i a n g l e i s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds p a r a ll e l t o t h e t h ir d s i d e a n d h a lf t h e l e n g t h ; Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t h e m e d i a n s o f a tr i a n g l e m e e t a t a p o i n t. CCSS: Mathema tics, Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds CCSS: HS: Geometr y, Similarity , Right Triangles ,& Trigono metry HSGSRT.A. Understa nd similarity in terms of similarity transfor mations  H S G S R T . A . 2 . G i Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds v e n t w o fi g u r e s , u s e t h e d e fi n it i o n o f s i m il a ri t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds y i n t e r m s o f s i m il a ri t y tr a n s f o r m a ti o n s t o d e c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i d e if t h e y a r e s i m il a r; e x p l a i n u s i n g s i m il a ri t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds y tr a n s f o r m a ti o n s t h e m e a n i n g o f s i m il a ri t y f o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r tr i a n g l e s a s t h e e q u a li t y o f a ll p a ir s o f a n g l e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s a n d t h e p r o p o rt i o n a li t y o f a ll p a ir s o f s i d e s . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds  H S G S R T . A . 3 . U s e t h e p r o p e rt i e s o f s i m il a ri Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t y tr a n s f o r m a ti o n s t o e s t a b li s h t h e A A c ri t e ri o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n f o r s i m il a ri t y o f tr i a n g l e s . HSGSRT.B. Prove theorems involving similarity  H S G S Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds R T . B . 4 . P r o v e t h e o r e m s a b o u t tr i a n g l e s u s i Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n g s i m il a ri t y tr a n s f o r m a ti o n s . T h e o r e m s i n c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds l u d e : a li n e p a r a ll e l t o o n e s i d e o f a tr i a n g l e d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i v i d e s t h e o t h e r t w o p r o p o rt i o n a ll y , a n d c o n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds v e r s e l y ; t h e P y t h a g o r e a n t h e o r e m p r o v e d u Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s i n g tr i a n g l e s i m il a ri t y .  H S G S R T . B . 5 . U s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e tr i a n g l e c o n g r u e n c e a n d s i m il a ri t y c ri t e ri a t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o s o l v e p r o b l e m s a n d t o p r o v e r e l a ti o n s h i p s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i n g e o m e tr i c fi g u r e s . CCSS: Mathema tics, CCSS: HS: Geometr y, Circles HSGC.A. Understa nd and apply theorems about circles  H Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds S G C . A . 3 . C o n s tr u c t t h e i n s c ri b e d a n d c ir c u m Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s c ri b e d c ir c l e s o f a tr i a n g l e , a n d p r o v e p r o p e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds rt i e s o f a n g l e s f o r a q u a d ri l a t e r a l i n s c ri b e d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i n a c ir c l e . CCSS: Mathema tics, CCSS: HS: Geometr y, Expressi ng Geometri c Propertie s with Equation s HSGGPE.B. Use coordinat es to prove simple geometri c theorems Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds algebraic ally  H S G G P E . B . 4 . U s e c o o r d i n a t e s t o p r o v e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s i m p l e g e o m e tr i c t h e o r e m s a l g e b r a i c a ll y . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds F o r e x a m p l e , p r o v e o r d i s p r o v e t h a t a fi g u r e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds d e fi n e d b y f o u r g i v e n p o i n t s i n t h e c o o r d i n a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t e p l a n e i s a r e c t a n g l e ; p r o v e o r d i s p r o v e t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds h a t t h e p o i n t ( 1 , √ 3 ) li e s o n t h e c ir c l e c e n t e r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e d a t t h e o ri g i n a n d c o n t a i n i n g t h e p o i n t ( 0 , Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds 2 ).  H S G G P E . B . 7 . U s e c o o r d i n a t e s t o c o m p Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds u t e p e ri m e t e r s o f p o l y g o n s a n d a r e a s f o r tr i a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n g l e s a n d r e c t a n g l e s , e . g . u s i n g t h e d i s t a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n c e f o r m u l a . CCSS: Mathema tics, CCSS: HS: Geometr y, Modeling with Geometr y HSGMG.A. Apply geometri c concepts in modeling situation s  H Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds S G M G . A . 3 . A p p l y g e o m e tr i c m e t h o d s t o s o l v Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e d e s i g n p r o b l e m s ( e . g ., d e s i g n i n g a n o b j e c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t o r s tr u c t u r e t o s a ti s f y c o n s tr a i n t s o r m i n i m Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i z e c o s t; w o r k i n g w it h t y p o g r a p h i c g ri d s y s t e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Intellectual Processes Less on Sequ ence Resources m s b a s e d o n r a ti o s ). 4How Quadr are ilatera various ls and quadril Other aterals Polyg related ons and (Week what 19, 5 are Week propert ies of s) their sides, angles, diagon als, Quadrilateral and areas? Graphic The CCSS: general Mathema classifica tics, tion of CCSS: quadrilat HS: erals can Geometr be y, divided Congrue into more nce specific HSGcategorie CO.B. s as Understa character nd istics congruen beyond ce in the terms of number rigid of sides motions are 1. Wh at i nfo rm ati on is ne ed ed to cal cul ate the are a of a Attend to areas of precision: use quadrilat erals Formativ the language of Highl kite reasoning, e ight parallelo Assessm logic, and proof less gram to express ent on properti mathematical es of Assessm ideas precisely Thin sides, ent Task k of Make sense of a angles, problems and Qua and Indepen persevere in diagonal drilat dent solving them: erals s of Practice various apply and adapt quadrilat a variety of Diag Artifact – appropriate erals onal Student quadrilat strategies to s Work eral solve problems and Sample: Unit Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds added. I ncluded  in these categorie s are parallelo grams, rectangle s, squares, trapezoid s, rhombus es, and kites. St udents should describe and justify the relations hips between these categorie s. In doing proofs with these figures, the idea of necessar y and H S G C O . B . 6 . U s e g e o m e tr i c d e s c ri p ti o n s o f reg ula r pol yg on ? 2. Us e an org ani zati on al tab le or Ve nn dia gra m to sh ow the rel ati on shi ps am on g the Intellectual Processes rectangl Using related to Student quadrilaterals e relations Work to Reengag Model with hips between e in the mathematics: quadrilat Assessm create and use ent erals representations rhombus to organize, square record, and sum of communicate exterior mathematical and ideas about interior quadrilaterals angles of a quadrilat eral trapezoi d coordina te proofs Less on Sequ ence Qua drilat erals activ ity shee t Diag onal s and Qua drilat erals OVH Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds sufficient condition s becomes important . Propert ies of quadrilat erals, including side lengths, angle measure s, diagonal lengths, perimete r, and area provide the basis for many problem scenario s and proofs. Proofs may take a variety of forms including twocolumn, paragrap h, a ri g i d m o ti o n s t o tr a n s f o r m fi g u r e s a n d t o p r e d i diff ere nt cla ssif icat ion s of qu adr ilat era ls. 3. Wh at is the im pa ct on the per im ete r an d are a of a reg ula r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds sequenc e of Ifthen statemen ts, using transfor mations, or using coordinat e methods. Beyond the study of triangles and quadrilat erals is the study of polygons in general. Connecti ons to propertie s of triangles and quadrilat erals assist in developi ng relations c t t h e e ff e c t o f a g i v e n ri g i d m o ti o n o n a g i v e n fi pol yg on wh os e siz e ha s be en ch an ge d by a fac tor of k? 4. Ex plai n an d just ify the su m of the int eri Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds hips between diagonal s, perimete r, and area in polygons . In particular , regular polygons are generally the focus of this study, and inscribin g and circumsc ribing polygons within or about a circle is included. Unit template 8-3-12 (word) Unit template g u r e ; g i v e n t w o fi g u r e s , u s e t h e d e fi n it i o n o f or an gle s of a qu adr ilat era l, an d the su m of the ext eri or an gle s of a qu adr ilat era l. 5. Us ea ge om etr Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds 8-3-12 (pdf) c o n g r u e n c e i n t e r m s o f ri g i d m o ti o n s t o d e c i d y dra win g pro gra m to inv esti gat e the thr eedi me nsi on al fig ure s tha t res ult fro m rot ati ng var iou s qu Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e if t h e y a r e c o n g r u e n t. HSGCO.C. Prove geometri c theorems  H S G C O . C . 1 adr ilat era ls ov er line s. Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds 1 . P r o v e t h e o r e m s a b o u t p a r a ll e l o g r a m s . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds T h e o r e m s i n c l u d e : o p p o s it e s i d e s a r e c o n g r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds u e n t, o p p o s it e a n g l e s a r e c o n g r u e n t, t h e d i a g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o n a l s o f a p a r a ll e l o g r a m b i s e c t e a c h o t h e r, a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n d c o n v e r s e l y , r e c t a n g l e s a r e p a r a ll e l o g r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a m s w it h c o n g r u e n t d i a g o n a l s . HSGCO.D. Make geometri c construct ions  H Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds S G C O . D . 1 2 . M a k e f o r m a l g e o m e tr i c c o n s tr u c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ti o n s w it h a v a ri e t y o f t o o l s a n d m e t h o d s ( c o m p Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a s s a n d s tr a i g h t e d g e , s tr i n g , r e fl e c ti v e d e v i Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds c e s , p a p e r f o l d i n g , d y n a m i c g e o m e tr i c s o ft w Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a r e , e t c ). C o p y i n g a s e g m e n t; c o p y i n g a n a n g l Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e ; b i s e c ti n g a s e g m e n t; b i s e c ti n g a n a n g l e ; c o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n s tr u c ti n g p e r p e n d i c u l a r li n e s , i n c l u d i n g t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds h e p e r p e n d i c u l a r b i s e c t o r o f a li n e s e g m e n t; Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a n d c o n s tr u c ti n g a li n e p a r a ll e l t o a g i v e n li n e t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds h r o u g h a p o i n t n o t o n t h e li n e . CCSS: Mathema tics, CCSS: HS: Geometr y, Expressi ng Geometri c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Propertie s with Equation s HSGGPE.B. Use coordinat es to prove simple geometri c theorems algebraic ally  H S G G P E . B . 4 . U s e c o o r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds d i n a t e s t o p r o v e s i m p l e g e o m e tr i c t h e o r e m s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a l g e b r a i c a ll y . F o r e x a m p l e , p r o v e o r d i s p Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r o v e t h a t a fi g u r e d e fi n e d b y f o u r g i v e n p o i n t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s i n t h e c o o r d i n a t e p l a n e i s a r e c t a n g l e ; p r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o v e o r d i s p r o v e t h a t t h e p o i n t ( 1 , √ 3 ) li e s o n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t h e c ir c l e c e n t e r e d a t t h e o ri g i n a n d c o n t a i n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i n g t h e p o i n t ( 0 , 2 ).  H S G G P E . B . 7 . U s e c o o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r d i n a t e s t o c o m p u t e p e ri m e t e r s o f p o l y g o n s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a n d a r e a s f o r tr i a n g l e s a n d r e c t a n g l e s , e . g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . u s i n g t h e d i s t a n c e f o r m u l a . CCSS: Mathema tics, CCSS: HS: Geometr y, Modeling with Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Geometr y HSGMG.A. Apply geometri c concepts in modeling situation s  H S G M G . A . 3 . A p p l y g e o m e tr Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i c m e t h o d s t o s o l v e d e s i g n p r o b l e m s ( e . g ., d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e s i g n i n g a n o b j e c t o r s tr u c t u r e t o s a ti s f y c o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n s tr a i n t s o r m i n i m i z e c o s t; w o r k i n g w it h t y p o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Intellectual Processes Less on Sequ ence Resources g r a p h i c g ri d s y s t e m s b a s e d o n r a ti o s ). 5Right Trian gle How can the unkno wn Trigono metry is a very useful CCSS: Mathema tics, CCSS: 1. Wh at is the Make sense of 30º- 60º90º Formativ problems and Highl triangles e ight Unit Resources Overar ching Questi ons and Graphic Enduri Organizer ng Unders tanding s Trigo measur nomet es of ry the (Week angles 24, 5 or Week sides of a s) triangle be found? How can trigono metry help Graphic find these Organizer missin g parts in many realworld situatio ns? Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds area of HS: mathema Function tics. Its s, methods Trigono are metric widely Function used in s navigatio HSFn, TF.A. surveyin Extend g, the building, domain engineeri of ng, trigonom computer etric graphics, functions machinin using the g and unit more. Ri circle. ght triangle  trigonom H etry uses S the F propertie s of T similarity F to . establish A trigonom . etric 3 ratios of . sine, ( cosine, + and ) tangent U rel ati on shi p bet we en the sid e len gth s of a rig ht tria ngl e an d the sin e, cos ine , an d tan ge nt rati Intellectual Processes Less on Sequ ence Less 45º- 45º- Assessm persevere in ent Right solving them: on 90º Tem triangles Triangle build new area of Trig mathematical plate a Right knowledge of triangle Triangle triangles New using through Highl Trig Retrigonom engagem problem solving ight etric Less ent formula Student Model with on cosine page.doc mathematics: Stud Law of x recognize and ent Cosines apply right work Right triangle Law of shee Triangle trigonometry in Sines t Trig Re- contexts outside sine tangent engagem of mathematics ent Key 1.jpg Construct Right viable Triangle arguments and Trig Re- critique the engagem reasoning of ent Key others: analyze evaluate the 2.jpg Right mathematical Triangle thinking and Trig Re- strategies engagem of others ent Key 3.jpg Right Triangle Trig Reengagem Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds for angles in right triangles. These ratios provide methods to solve problems involving unknown side lengths or angles of a right triangle. These methods also allow students to determin e otherwis e inaccessi ble distance s and angle measure ments. Because all s e s p e c i a l tr i a n g l e s t o d e t e r m i n e g e o m e tr i c os ? 2. Wh y do trig on om etri c rati os yiel d the sa me val ue s for diff ere nt siz es of rig ht tria ngl es ? 3. Wh at is ent Key 4.jpg Right Triangle Trig Reengagem ent Key 5.jpg Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds triangles are not right triangles, and regions that are not right triangles often model realworld situation s, the Law of Sines and Law of Cosines are develope d. Havin g students apply these two laws extends the ability to solve problems to triangles in a ll y t h e v a l u e s o f s i n e , c o s i n e , t a n g e n t f o r the rel ati on shi p bet we en the sid es in a 45º 45º 90º tria ngl e? In a 30º 60º 90º tria ngl e? 4. Wh en is it ap Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds general. A valuable tool for finding area comes from using the sine function to develop an area formula for triangles. This easily develops from drawing an altitude in a triangle and using sine ratios. Unit template 8-6-12 (word) Unit p / 3 , p / 4 a n d p / 6 , a n d u s e t h e u n it c ir c l e t o e x p pro pri ate to us e the La w of Sin es to sol ve a pro ble m? Th e La w of Co sin es ? Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds template 8-6-12 (pdf) r e s s t h e v a l u e s o f s i n e , c o s i n e s , a n d t a n g e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n t f o r x , p + x , a n d 2 p – x i n t e r m s o f t h e ir v a l u e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s f o r x , w h e r e x i s a n y r e a l n u m b e r. CCSS: Mathema tics, CCSS: HS: Geometr y, Similarity Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds , Right Triangles ,& Trigono metry HSGSRT.C. Define trigonom etric ratios and solve problems involving right triangles  H S G S R T . C . 6 . U n d e r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s t a n d t h a t b y s i m il a ri t y , s i d e r a ti o s i n ri g h t tr Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i a n g l e s a r e p r o p e rt i e s o f t h e a n g l e s i n t h e tr Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i a n g l e , l e a d i n g t o d e fi n it i o n s o f tr i g o n o m e tr Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i c r a ti o s f o r a c u t e a n g l e s .  H S G S R T . C . 7 . E Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds x p l a i n a n d u s e t h e r e l a ti o n s h i p b e t w e e n t h e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s i n e a n d c o s i n e o f c o m p l e m e n t a r y a n g l e s . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds  H S G S R T . C . 8 . U s e tr i g o n o m e tr i c r a ti o s a n d t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds h e P y t h a g o r e a n T h e o r e m t o s o l v e ri g h t tr i a n g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds l e s i n a p p li e d p r o b l e m s . HSGSRT.D. Apply trigonom etry to general triangles  H S G S R Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds T . D . 9 . ( + ) D e ri v e t h e f o r m u l a A = ½ a b s i n © f o r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t h e a r e a o f a tr i a n g l e b y d r a w i n g a n a u x il i a r y Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds li n e fr o m a v e rt e x p e r p e n d i c u l a r t o t h e o p p o s it Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e s i d e .  H S G S R T . D . 1 0 . ( + ) P r o v e t h e L a w s o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds f S i n e s a n d C o s i n e s a n d u s e t h e m t o s o l v e p r o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds b l e m s .  H S G S R T . D . 1 1 . ( + ) U n d e r s t a n d a n d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a p p l y t h e L a w o f S i n e s a n d t h e L a w o f C o s i n e s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t o fi n d u n k n o w n m e a s u r e m e n t s i n ri g h t a n d n o n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ri g h t tr i a n g l e s ( e . g ., s u r v e y i n g p r o b l e m s , Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Intellectual Processes Less on Sequ ence Resources r e s u lt a n t f o r c e s ). 6What Circle are the s propert (Week ies and 29, 5 represe Week ntation s of s) special lines, segme nts, and angles as they relate to a circle Graphic - Circles and how This unit CCSS: on Mathema circles tics, provides CCSS: the HS: opportuni Function ty for s, students Trigono to metric explore Function relations s hips, HSFmake TF.A. conjectur Extend es, and the reason domain deductiv of ely to trigonom verify etric 1. Ho w is Π( pi) rel ate d to the par ts of a circ le? Standards of radius Mathematical diameter center Formativ Practice Students will chord e central Assessm have opportunities angle ent inscribe Overvie to: d angle w circums  attend cribed to angle precisi tangent on care point of fully tangenc commu y nicating secant and equation precisel of a y using circle the Unit Resources Highl ight Less on Over view Overar ching Questi ons and Enduri ng Unders tanding s can these propert ies be used to solve proble ms? Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds these functions conjectur using the es. Revi unit siting the circle. familiar topics of  circumfer H ence and S area of F circles is the basis T for F exploring . arcs and A sectors . of circles 1 and their . correspo U nding n lengths d and e areas. P r roperties s of chords t of a a circle n and lines d tangent r to a a circle are d used to i solve a problems n . The m relations e hips a 2. Wh y is the me as ure of an ins cri be d an gle hal f of the me as ure of a ce ntr al an gle ins cri be d in the sa me arc arc length locus degree and radian measure s and connecti ons circumfe rence area sector area of a sector Intellectual Processes   languag e of circles; model with mathe matics using c ircles to organiz e, record, and commu nicate mathem atical ideas; and constr uct viable argume nts and critique the reasoni ng of others: Underst and and use stated assump tions, definitio Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds between central angles, inscribed angles, circumsc ribed angles, and triangles in a circle are explored and proved deductiv ely. The equation of a circle in the coordinat e plane links the geometri c represen tation of a circle to its algebraic represen tation and defines a circle as s u r e o f a n a n g l e a s t h e l e n g t h o f t h e a r c o n t h arc ? 3. Wh at is the rel ati on shi p of the are a of a sec tor of a circ le to the are a of tha t circ le? 4. Wh at is the Intellectual Processes ns, and previou sly establis hed results in constru cting argume nts Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds the locus of points a fixed distance from a fixed point. Unit Overvie w (Word) Unit Overvie w (PDF) e u n it c ir c l e s u b t e n d e d b y t h e a n g l e . CCSS: Mathema tics, CCSS: HS: Geometr rel ati on shi p bet we en the arc len gth on a circ le an d the circ um fer en ce of tha t circ le? 5. Giv e an ex am ple of Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds y, Congrue nce HSGCO.A. Experim ent with transfor mations in the plane  H S G C O . A . 1 . K n o w p r e c i s e d an eq uat ion of a circ le wh os e ce nte r is at the ori gin . H ow ca n yo u tra nsf or m this eq uat ion so the siz e is Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e fi n it i o n s o f a n g l e , c ir c l e , p e r p e n d i c u l a r li the sa me , but the ce nte r is not at the ori gin ? Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n e , p a r a ll e l li n e , a n d li n e s e g m e n t, b a s e d o n t h Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e u n d e fi n e d n o ti o n s o f p o i n t, li n e , d i s t a n c e a l Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o n g a li n e , a n d d i s t a n c e a r o u n d a c ir c u l a r a r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds c . CCSS: Mathema tics, CCSS: HS: Geometr y, Circles HSGC.A. Understa nd and apply theorems about circles  H S G C . A . 1 . P r o v e t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds h a t a ll c ir c l e s a r e s i m il a r.  H S G C . A . 2 . I d e n ti Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds f y a n d d e s c ri b e r e l a ti o n s h i p s a m o n g i n s c ri b e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds d a n g l e s , r a d ii , a n d c h o r d s . I n c l u d e t h e r e l Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a ti o n s h i p b e t w e e n c e n tr a l, i n s c ri b e d a n d c ir c u Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds m s c ri b e d a n g l e s ; i n s c ri b e d a n g l e s o n a d i a m e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t e r a r e ri g h t a n g l e s ; t h e r a d i u s o f a c ir c l e i s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds p e r p e n d i c u l a r t o t h e t a n g e n t w h e r e t h e r a d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i u s i n t e r s e c t s t h e c ir c l e .  H S G C . A . 3 . C o n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s tr u c t t h e i n s c ri b e d a n d c ir c u m s c ri b e d c ir c l e s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o f a tr i a n g l e , a n d p r o v e p r o p e rt i e s o f a n g l e s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds f o r a q u a d ri l a t e r a l i n s c ri b e d i n a c ir c l e .  H S Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds G C . A . 4 . ( + ) C o n s tr u c t a t a n g e n t li n e fr o m a p o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds i n t o u t s i d e a g i v e n c ir c l e t o t h e c ir c l e . HSGC.B. Find arc Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds lengths and areas of sectors of circles  H S G C . B . 5 . D e ri v e u s i n g s i m il a ri t y t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds h e f a c t t h a t t h e l e n g t h o f t h e a r c i n t e r c e p t Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e d b y a n a n g l e i s p r o p o rt i o n a l t o t h e r a d i u s , Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a n d d e fi n e t h e r a d i a n m e a s u r e o f t h e a n g l e a s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t h e c o n s t a n t o f p r o p o rt i o n a li t y ; d e ri v e t h e f Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o r m u l a f o r t h e a r e a o f a s e c t o r. CCSS: Mathema tics, CCSS: HS: Geometr y, Expressi ng Geometri Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds c Propertie s with Equation s HSGGPE.A. Translate between the geometri c descripti on and the equation for a conic section  H S G G P E . A . 1 . D e ri Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds v e t h e e q u a ti o n o f a c ir c l e o f g i v e n c e n t e r a n d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r a d i u s u s i n g t h e P y t h a g o r e a n T h e o r e m ; c o m Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds p l e t e t h e s q u a r e t o fi n d t h e c e n t e r a n d r a d i u Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s o f a c ir c l e g i v e n b y a n e q u a ti o n . CCSS: Mathema tics, CCSS: HS: Geometr y, Geometri c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Measure ment & Dimensio n HSGGMD.A. Explain volume formulas and use them to solve problems  H S G G M D . A . 1 . G i v e a n i n f Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o r m a l a r g u m e n t f o r t h e f o r m u l a s f o r t h e c ir c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds u m f e r e n c e o f a c ir c l e , a r e a o f a c ir c l e , v o l u m Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e o f a c y li n d e r, p y r a m i d , a n d c o n e . U s e d i s s e c Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds ti o n a r g u m e n t s , C a v a li e ri ’ s p ri n c i p l e , a n d i n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Intellectual Processes Less on Sequ ence Resources f o r m a l li m it a r g u m e n t s . 7What Model are the ing connec with tions 3betwee Dime n two nsion dimens al ional Figur and es three (Week dimens 34, 4 ional Week figures ? s) Spatial- CCSS: visualizat Mathema ion skills tics, are CCSS: important HS: in a wide Num/Qu variety of antity, occupati Quantitie ons, as s well as HSNmany Q.A. everyday Reason tasks. An quantitati important vely and 1. Ho w are pyr am ids like pri sm s? Ho w are the altitude appropri ate use of units in area and volume calculati ons cones cross section cylinder s Attend to precision: Formativ communicate mathematical e Assessm thinking coherently and ent Template clearly Student Model with Handout Mathematics: analyze 3-D Formativ geometric relationships e Assessm and draw Unit Resources Geo metr yUnit 8 Thre e Dime nsio nal Figur es Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer 3D Figures Unit Graphic Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds aspect of use units threeto solve dimensio problems nal . figures is the  ability to H visualize S the N shape and Q propertie . s from a A two . dimensio 1 nal . represen U tation. T s he e reverse, u making a n twoit dimensio s nal a represen s tation of a a threew dimensio a nal y figure, is t equally o important u . Using n physical d models, e y diff ere nt? Ho w are co ne s like cyli nd ers ? Ho w are the y diff ere nt? 2. Wh at ch ara cte risti cs of a thr ee di ent faces impact of scale Student factor on Handout the surface Indepen area dent and Practice volume of a figure isometry net orthogra phic planes of symmetr y polyhedr on prisms pyramid s relations hip between the volumes of prisms and pyramid s; cylinder Intellectual Processes Less on Sequ ence mathematical conclusions Highl ight Reason Less abstractly and on quantitatively: Hand apply and adapt outs a variety of appropriate Paint strategies to Can solve Teac problems about her threeNote dimensional s figures Exce l Docu ment for Paint Can Inve stiga tion Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds drawings , and software will help students improve visualizat ion and spatial reasonin g and assist them in finding lines and planes of symmetr y and cross sections of threedimensio nal figures. In addition, students can experien ce the effect of rotating a twodimensio nal figure r s t a n d p r o b l e m s a n d t o g u i d e t h e s o l u ti o n o me nsi on al obj ect are ne ces sar y to dra wa ske tch of the obj ect ?D ra w the obj ect . 3. Giv en at hre edi me nsi s and cones rotationa l symmetr y of a 3D figure slant height solids of revolutio n spheres surface area and volume of figures vertex Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds and conceptu alize, create, and make calculatio ns about the resulting threedimensio nal figure. Computi ng surface areas and volumes is another compone nt of the study of threedimensio nal figures. Previous studies have included the f m u lt is t e p p r o b l e m s ; c h o o s e a n d i n t e r p r e on al ge om etri c obj ect , ho w ca n yo u dra wa two di me nsi on al net tha t rep res ent this obj ect ? Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds calculatio n of surface area and volume of many threedimensio nal figures. I t is extended in this unit to include 3 -d figures w ith regular p olygons as faces. These skills can be applied to many realworld applicati ons such as packagin g and optimizin t u n it s c o n s i s t e n tl y i n f o r m u l a s ; c h o o s e a n 4. Wh at are the po ssi ble cro ss sec tio ns of var iou s pol yh edr a? 5. Giv en a two di me nsi on al fig ure , ske tch Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds g use of materials needed to make a particular shape, which in many cases will be more complex or a combinat ion of those previousl y studied. Calculati ons of appropri ate units and conversi ons between them should also be included. Student s will also be d i n t e r p r e t t h e s c a l e a n d t h e o ri g i n i n g r a p the res ult of rot ati ng the fig ure ab out a line . 6. Wh at is the rel ati on shi p bet we en the vol um e of a pri sm Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds able to h give s argumen a ts n supportin d g area d and a volume t formulas. a Student d s will i investiga s te the p impact of l changing a a scale y factor on s the . surface area and CCSS: volume Mathema and tics, explain CCSS: these HS: relations Geometr hips. y, Geometri c Unit Measure Templat ment & Dimensio e.doc (WORD) n HSGUnit GMD.A. Templat Explain an da pyr am id wit h co ngr ue nt ba ses an d hei ght s? Wh at is the rel ati on shi p bet we en the vol um e of Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e.pdf (PDF) volume formulas and use them to solve problems  H S G G M D . A . 1 . G i v e a n i n f o r m a l a a cyli nd er an da co ne wit h co ngr ue nt ba ses an d hei ght s? Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r g u m e n t f o r t h e f o r m u l a s f o r t h e c ir c u m f e r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e n c e o f a c ir c l e , a r e a o f a c ir c l e , v o l u m e o f a Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds c y li n d e r, p y r a m i d , a n d c o n e . U s e d i s s e c ti o n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a r g u m e n t s , C a v a li e ri ’ s p ri n c i p l e , a n d i n f o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r m a l li m it a r g u m e n t s .  H S G G M D . A . 2 . ( + ) G i Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds v e n a n i n f o r m a l a r g u m e n t u s i n g C a v a li e ri ’ s Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds p ri n c i p l e f o r t h e f o r m u l a s f o r t h e v o l u m e o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds f a s p h e r e a n d o t h e r s o li d fi g u r e s .  H S G G M D Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds . A . 3 . U s e v o l u m e f o r m u l a s f o r c y li n d e r s , p Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds y r a m i d s , c o n e s a n d s p h e r e s t o s o l v e p r o b l Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e m s . HSGGMD.B. Visualize the relation between twodimensio nal and threedimensio nal objects  H S G G M D . B . 4 . I d e Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds n ti f y c r o s s s e c ti o n a l s h a p e s o f s li c e s o f t h Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r e e d i m e n s i o n a l o b j e c t s , a n d i d e n ti f y t h Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds r e e d i m e n s i o n a l o b j e c t s g e n e r a t e d b y r o Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t a ti o n s o f t w o d i m e n s i o n a l o b j e c t s . CCSS: Mathema tics, Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds CCSS: HS: Geometr y, Modeling with Geometr y HSGMG.A. Apply geometri c concepts in modeling situation s  H S G M G . A . 1 . U s e g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds e o m e tr i c s h a p e s , t h e ir m e a s u r e s a n d t h e ir p r Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o p e rt i e s t o d e s c ri b e o b j e c t s ( e . g ., m o d e li n g Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds a tr e e tr u n k o r a h u m a n t o r s o a s a c y li n d e r ).  H S Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds G M G . A . 2 . A p p l y c o n c e p t s o f d e n s it y b a s e d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o n a r e a a n d v o l u m e i n m o d e li n g s it u a ti o n s ( e . Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds g ., p e r s o n s p e r s q u a r e m il e , B T U s p e r c u b i c f Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds o o t) .  H S G M G . A . 3 . A p p l y g e o m e tr i c m e t h o d Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds s t o s o l v e d e s i g n p r o b l e m s ( e . g ., d e s i g n i n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds g a n o b j e c t o r s tr u c t u r e t o s a ti s f y c o n s tr a i n Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds t s o r m i n i m i z e c o s t; w o r k i n g w it h t y p o g r a p h Intellectual Processes Less on Sequ ence Resources Overar ching Questi ons and Enduri ng Unders tanding s Graphic Organizer Content Unit Essential/ Key Assessm Unit Expectat Level Focus Concep ent Abstract ions/Sta Standar Questions ts Tasks ndards ds Intellectual Processes Less on Sequ ence Resources i c g ri d s y s t e m s b a s e d o n r a ti o s ). Atlas Version 8.0.3LP © Rubicon International 2014. All rights reserved

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