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Grades 9–11 Language Development for Success BOOK 1 Basic Terms, Angles, Proofs, Lines and Transversals, Triangles Sealaska Heritage Institute The contents of this program were developed by Sealaska Heritage Institute through the support of a Special Projects 'HPRQVWUDWLRQ*UDQWIURPWKH86'HSDUWPHQWRI(GXFDWLRQ2I¿FHRI,QGLDQ(GXFDWLRQ&)'$$+RZHYHUWKHFRQWHQWV do not necessarily represent the policy of the Department of Education and you should not assume endorsement. 2009 Integrating Culturally Responsive, Place-Based Content with Language Skills Development for Curriculum Enrichment DEVELOPED BY Stephanie Hoage Steve Morley Jim MacDiarmid Unit Assessments Bev Williams LAYOUT & FORMATTING Matt Knutson PRINTERS Capital Copy, Juneau, Alaska PROJECT ASSISTANT Tiffany LaRue 2009 Table of Contents BOOK 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Developmental Language Process . . . . . . . . . . . . . 7 Math and the Developmental Language Process . . . . . 9 Unit 1—Basic Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 11 8QLW²$QJOHV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8QLW²3URRIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8QLW²/LQHV7UDQVYHUVDOV . . . . . . . . . . . . . . . . . . . 8QLW²7ULDQJOHV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 INTRODUCTION 2YHUWKH\HDUVPXFKKDVEHHQZULWWHQDERXWWKHVXFFHVVHVDQGIDLOXUHVRIVWXGHQWVLQ VFKRROV7KHUHLVQRHQGWRWKHVROXWLRQVRIIHUHGSDUWLFXODUO\IRUWKRVHVWXGHQWVZKRDUH struggling with academics. There have been efforts to bring local cultures into the classURRPWKXVSURYLGLQJWKHVWXGHQWVZLWKIDPLOLDUSRLQWVRIGHSDUWXUHIRUOHDUQLQJ+RZHYHU most often such instruction has been limited to segregated activities such as arts and crafts or Native dancing rather than integrating Native culture into the overall learning SURFHVV 7ZR FRUH FXOWXUDO YDOXHV Haa Aaní, WKH UHIHUHQFH IRU DQG XVDJH RI WKH ODQG and Haa Shagóon, WKHW\LQJRIWKHSUHVHQWZLWKWKHSDVWDQGIXWXUH are known by both VWXGHQWVDQGSDUHQWVDQGFDQEHLQFOXGHGLQDFXUULFXOXPWKDWVLPXOWDQHRXVO\SURYLGHV DEDVLVIRUVHOILGHQWLW\DQGFXOWXUDOSULGHZLWKLQWKHHGXFDWLRQDOVHWWLQJ7KLVZLOOSURYLGH a valuable foundation for improved academic achievement. :KLOHWKHLQFOXVLRQRI1DWLYHFRQFHSWVYDOXHVDQGWUDGLWLRQVLQWRDFXUULFXOXPSURYLGHVD YDOXDEOHIRXQGDWLRQIRUVHOILGHQWLW\DQGFXOWXUDOSULGHLWPD\QRWRQLWVRZQIXOO\DGGUHVV improved academic achievement. 7KLVSURJUDPLVGHVLJQHGWRPHHWWKHDFDGHPLFUHDOLWLHVIDFHGE\KLJKVFKRROVWXGHQWV HYHU\GD\XVLQJDGHYHORSPHQWDOSURFHVVWKDWLQWHJUDWHVculture with skills development. The values of Haa Aaní and Haa Shagóon are reinforced through the various activities in the program. 'XULQJ PDWK OHVVRQV VWXGHQWV DUH H[SRVHG WR PDWKLQIRUPDWLRQ DQG WRNH\ YRFDEXODU\ WKDWUHSUHVHQWWKDWLQIRUPDWLRQ:KLOHWKHVWXGHQWVPD\DFTXLUHWKURXJKYDULRXVSURFHVVHVWKHPDWKHPDWLFDOLQIRUPDWLRQWKHYRFDEXODU\LVRIWHQOHIWDWDQH[SRVXUHOHYHODQGLV QRWLQWHUQDOL]HGE\WKHP2YHUWLPHWKLVOHDGVWRlanguage-delay that impacts negatively on a student’s on-going academic achievement. Due to language delayPDQ\1DWLYH$ODVNDQKLJKVFKRROVWXGHQWVVWUXJJOHZLWKWH[WVWKDW are beyond their comprehension levels and writing assignments that call for language WKH\GRQRWKDYH7RWKLVHQGLQWKLVUHVRXUFHSURJUDPHDFKNH\YRFDEXODU\ZRUGLQPDWK is viewed as a concept7KHZRUGVDUHLQWURGXFHGFRQFUHWHO\XVLQJSODFHEDVHGLQIRUPDWLRQDQGFRQWH[WV8VLQJWKLVDSSURDFKWKHVWXGHQWVKDYHWKHRSSRUWXQLW\WRDFTXLUHQHZ information in manageable chunks; the sum total of which represent the body of information to be learned in the math program. In many high school math classes it is assumed WKDWWKHDFDGHPLFYRFDEXODU\LVEHLQJLQWHUQDOL]HGGXULQJWKHOHDUQLQJSURFHVVZKLFKLV most often an erroneous assumption. — 5 — :KHQ WKH NH\ YRFDEXODU\FRQFHSWV KDYH EHHQ LQWURGXFHG WKH VWXGHQWV DUH WKHQ WDNHQ WKURXJK D VHTXHQFH RI OLVWHQLQJ VSHDNLQJ UHDGLQJ DQG ZULWLQJ DFWLYLWLHV GHVLJQHG WR instill the vocabulary into their long term memories - see the Developmental Language ProcessZKLFKIROORZV It should be understood that these materials are not a curriculum UDWKHU WKH\ DUH resource materials designed to encourage academic achievement through intensive language development in the content areas.These resource materials are culturally responsive in that they utilize teaching and learning styles effective with Native students. $VWKHVWXGHQWVSURJUHVVWKURXJKWKHVWHSVRIWKH3URFHVVWKH\PRYHIURPDFRQFUHWH LQWURGXFWLRQ RIWKH NH\ YRFDEXODU\ WRD V\PEROLF UHSUHVHQWDWLRQ RIWKH YRFDEXODU\ DQG ¿QDOO\WRWKHLUDEVWUDFWIRUPVUHDGLQJDQGZULWLQJ7KLVSURYLGHVDIRUPDWIRUWKHVWXGHQWV to develop language and skills that ultimately lead to improved academic performance. The Integration of Place-Based, Culturally Responsive Math Content and Language Development Introduction of Key Math Vocabulary Math, Vocabulary Development Listening, speaking, reading & writing Math Application Reinforcement Activities — 6 — The Developmental Language Process 7KH'HYHORSPHQWDO/DQJXDJH3URFHVV'/3LVGHVLJQHGWRLQVWLOOODQJXDJHLQWRORQJ term memory. The origin of the Process is rooted in the struggles faced by languageGHOD\HGVWXGHQWVSDUWLFXODUO\ZKHQWKH\¿UVWHQWHUVFKRRO 7KH3URFHVVWDNHVWKHVWXGHQWVFKLOGUHQWKURXJKGHYHORSPHQWDOVWHSVWKDWUHÀHFWWKH QDWXUDODFTXLVLWLRQRIODQJXDJHLQWKHKRPHDQGFRPPXQLW\,QLWLDOO\RQFHNH\ODQJXDJH LWHPVKDYHEHHQLQWURGXFHGFRQFUHWHO\WRWKHVWXGHQWVWKHYRFDEXODU\DUHXVHGLQWKH ¿UVWRIWKHODQJXDJHVNLOOV%DVLF/LVWHQLQJ7KLVVWDJHLQWKHSURFHVVUHSUHVHQWVinput DQGLVDFULWLFDOYHQXHIRUODQJXDJHDFTXLVLWLRQDQGUHWHQWLRQ$EDE\KHDUVPDQ\GLIIHUHQWWKLQJVLQWKHKRPHJUDGXDOO\WKHEDE\EHJLQVWRlistenWRZKDWKHVKHKHDUV$V a result of the inputSURYLGHGWKURXJK%DVLF/LVWHQLQJWKHEDE\WULHVWRUHSHDWVRPHRI WKHODQJXDJHKHDUG±WKLVLVUHSUHVHQWHGE\WKHVHFRQGSKDVHRIWKH3URFHVV%DVLF Speaking - the oral output stage of language acquisition. $VPRUHODQJXDJHJRHVLQWRDFKLOG¶VORQJWHUPPHPRU\KHVKHEHJLQVWRXQGHUVWDQG simple commands and phrases. This is a higher level of listening represented by the VWDJH/LVWHQLQJ&RPSUHKHQVLRQ:LWKWKHLQFUHDVHLQYRFDEXODU\DQGVHQWHQFHGHYHORSPHQWWKHFKLOGEHJLQVWRH[SORUHWKHXVHRIODQJXDJHWKURXJKWKHQH[WVWDJHLQ WKH3URFHVV&UHDWLYH6SHDNLQJ$OORIWKHVHVWHSVLQWKH3URFHVVUHÀHFWWKHQDWXUDO sequence of language development. The listening and speaking skill areas represent trueODQJXDJHVNLOOVPRVWFXOWXUHV LQFOXGLQJ$ODVND1DWLYHFXOWXUHVQHYHUZHQWEH\RQGWKHPWRGHYHORSZULWWHQIRUPV2UDO traditions are inherent in the listening and speaking skills. +RZHYHU(QJOLVKGRHVKDYHDEVWUDFWIRUPVRIODQJXDJHLQUHDGLQJDQGZULWLQJ0DQ\ Native children entering kindergarten come from homes where language is used differently than in classic Western homes. This is not a value judgment of child rearing pracWLFHVEXWDGH¿QLWHFURVVFXOWXUDOUHDOLW\7KHUHIRUHLWLVFULWLFDOWKDWWKH1DWLYHFKLOGEH introduced to the concepts of reading and writing before ever dealing with them as skills areas. It is vital for the children to understand that reading and writing are talk in print. The Developmental Language Process integrates the real language skills of listening DQGVSHDNLQJZLWKWKHUHODWHGVNLOOVRIUHDGLQJDQGZULWLQJ$WWKLVVWDJHLQWKH3URFHVV WKHVWXGHQWVDUHLQWURGXFHGWRWKHSULQWHGZRUGVIRUWKH¿UVWWLPH7KHVHDEVWUDFWUHSUHVHQWDWLRQVDUHQRZIDPLOLDUWKURXJKWKHOLVWHQLQJDQGVSHDNLQJDFWLYLWLHVDQGWKHUHODWLRQVKLSLVIRUPHGEHWZHHQWKHZRUGVDQGODQJXDJHEHJLQQLQJZLWK%DVLF5HDGLQJ $VPRUHODQJXDJHJRHVLQWRWKHFKLOGUHQ¶VORQJWHUPPHPRULHVWKH\EHJLQWRFRPSUHKHQGPRUHRIZKDWWKH\UHDGLQ5HDGLQJ&RPSUHKHQVLRQ — 7 — 0DQ\$ODVNDQVFKRRODWWLFVDUH¿OOHGZLWKUHDGLQJSURJUDPVWKDWGLGQ¶WZRUN±LQUHDOLW\ any of the programs would have worked had they been implemented through a lanJXDJHGHYHORSPHQWSURFHVV)RUPDQ\1DWLYHFKLOGUHQWKHSULQWHGZRUGFUHDWHVDQJVW SDUWLFXODUO\LIWKH\DUHVWUXJJOLQJZLWKWKHUHDGLQJSURFHVV2IWHQFKLOGUHQDUHDVNHGWR read language they have never heard. 1H[WLQWKH3URFHVVLV%DVLF:ULWLQJZKHUHWKHVWXGHQWVDUHDVNHGWRZULWHWKHNH\ ZRUGV)LQDOO\WKHPRVWGLI¿FXOWRIDOOWKHODQJXDJHVNLOOV&UHDWLYH:ULWLQJKDVWKHVWXGHQWVZULWLQJVHQWHQFHVRIWKHLURZQXVLQJWKHNH\ZRUGVDQGODQJXDJHIURPWKHLUORQJ term memories. This high level skill area calls upon the students to not only retrieve ODQJXDJHEXWWRSXWWKHZRUGVLQWKHLUFRUUHFWRUGHUZLWKLQWKHVHQWHQFHVWRVSHOOWKH words correctly and to sequence their thoughts in the narrative. The Developmental Language Process is represented in this chart: $WWKHHQGRIWKH3URFHVVWKHVWXGHQWVSDUWLFLSDWHLQHQULFKPHQWDFWLYLWLHVEDVHGRQ recognized and reasearch-based best practices. By this time the information and vocabXODU\ZLOOEHIDPLOLDUDGGLQJWRWKHVWXGHQWV¶IHHOLQJVRIFRQ¿GHQFHDQGVXFFHVV 7KH8QLW¶V$VVHVVPHQWLVDOVRDGPLQLVWHUHGGXULQJWKH([WHQVLRQ$FWLYLWLHVVHFWLRQRI the Process. This test provides the teacher with a clear indication of the students’ progUHVVEDVHGRQWKHREMHFWLYHVIRUEDVLFOLVWHQLQJEDVLFUHDGLQJUHDGLQJFRPSUHKHQVLRQ basic writing and creative writing. Since the DLP is a processDQGQRWDSURJUDPLWFDQEHLPSOHPHQWHGZLWKDQ\PDWHULDOVDQGDWDQ\JUDGHRUUHDGLQHVVOHYHO$VWXGHQW¶VDELOLW\WRFRPSUHKHQGZHOOLQlistening and readingDQGWREHFUHDWLYHO\H[SUHVVLYHLQspeaking and writingLVGHSHQGHQW upon how much language he/she has in long-term memory. — 8 — Math & The Developmental Language Process The Developmental Language Process can be applied effectively in the development of math concepts and their vocabulary. Not all math vocabulary lend themselves well to listening comSUHKHQVLRQFUHDWLYHVSHDNLQJDQGFUHDWLYHZULWLQJDFWLYLWLHVDQGWKHUHIRUHWKH3URFHVVFDQEH adapted to create a fast track in math. This schema represents the use of the Process in math: $FWLYLWLHVIRUOLVWHQLQJFRPSUHKHQVLRQFUHDWLYHVSHDNLQJDQGFUHDWLYHZULWLQJFDQEHXVHG depending upon the vocabulary being developed. 7KLVUHVRXUFHERRNLVGHVLJQHGWREHXVHGDSSUR[LPDWHO\RQFHSHUPRQWKIRUDVL[W\WRQLQHW\ PLQXWHOHVVRQ'XULQJWKLVWLPHWKHGHYHORSPHQWRImath vocabulary is the principle endeavRUQRWWKHWHDFKLQJRIWKHPDWKFRQFHSWV+RZHYHUWKHPDWKFRQFHSWVIRUPWKHEDVHVIRUODQguage development. Increased vocabulary development in math will ultimately lead to improved academic achievePHQWLQFUHDVHGVHOIHVWHHPDQGWRDKLJKHUVXFFHVVUDWHRQDFDGHPLFDVVHVVPHQWV — 9 — — 10 — UNIT 1 Basic Terms Sealaska Heritage Institute Grade Level Expectations for Unit 1 Unit 1 Basic Terms Alaska State Mathematics Standard C $VWXGHQWVKRXOGXQGHUVWDQGDQGEHDEOHWRIRUPDQGXVHDSSURSULDWHPHWKRGVWR GH¿QHDQGH[SODLQPDWKHPDWLFDOUHODWLRQVKLSV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG &H[SUHVVDQGUHSUHVHQWPDWKHPDWLFDOLGHDVXVLQJRUDODQGZULWWHQ SUHVHQWDWLRQVSK\VLFDOPDWHULDOVSLFWXUHVJUDSKVFKDUWVDQGDOJHEUDLF H[SUHVVLRQV &UHODWHPDWKHPDWLFDOWHUPVWRHYHU\GD\ODQJXDJH GLEs The student communicates his or her mathematical thinking by >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQG RUV\PEROLFDOO\WUDQVODWLQJDPRQJWKHVHDOWHUQDWLYHUHSUHVHQWDWLRQVRUXVLQJ DSSURSULDWHYRFDEXODU\V\PEROVRUWHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQG strategies and solutions >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQG RUV\PEROLFDOO\FRPPXQLFDWLQJPDWKLGHDVLQZULWLQJRUXVLQJDSSURSULDWH YRFDEXODU\V\PEROVRUWHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQG solutions Vocabulary & Definitions Introduction of Math Vocabulary Infinite ,Q¿QLWHPHDQVXQOLPLWHGRUXQHQGLQJ:KHQVRPHWKLQJLV LQ¿QLWHLWFDQQRWEHPHDVXUHGRUFRXQWHG,WJRHVRQIRUHYHU6SDFHDVVKRZQLQWKHSKRWRLVLQ¿QLWHLWKDVQR ERXQGDULHV7LPHVDQGQXPEHUVDUHDOVRLQ¿QLWH Dimension 'LPHQVLRQVDUHGLUHFWLRQVRIH[WHQVLRQ/HQJWKZLGWKDQGKHLJKWDUHGLPHQVLRQV$VWUHWFKHGVWULQJPRGHOVDOLQHZKLFKKDVRQH GLPHQVLRQOHQJWK$ZDOORUÀRRUPRGHOVDSODQHZKLFK KDVWZRGLPHQVLRQVOHQJWKDQGZLGWK6SDFHDER[RUD KRXVHKDVWKUHHGLPHQVLRQVOHQJWKZLGWKDQGKHLJKW,Q WKLVSKRWRRIDORQJKDOOZD\\RXFDQVHHRQHGLPHQVLRQDO WZRGLPHQVLRQDODQGWKUHHGLPHQVLRQDOREMHFWVUHSUHVHQWHG/LQHVDQGSODQHVKDYHQRWKLFNQHVVEXWLQWKH UHDOZRUOGDQ\PRGHORIDOLQHRUDSODQHKDVVRPHWKLFNness. We say that telephone wires or lines on the pavement model or representRQHGLPHQVLRQDOREMHFWVDQGWKDWWKHLFHRQDSRQGmodels a two-dimensional object. Space 6SDFHLVWKHVHWRIDOOSRLQWV,WLVWKHGLPHQVLRQDOSODFH LQZKLFKDQREMHFWFDQH[LVWRUHYHQWVFDQWDNHSODFH 6SDFHLVLQ¿QLWH:HFDQ¶WPHDVXUHDOORIVSDFHDQGZH FDQ¶WPHDVXUHWKHLQ¿QLWHQXPEHURISRLQWVLQDQ\JLYHQ VSDFH:HXVHFXELFFHQWLPHWHUVFXELFIHHWFXELFPHWHUV or other measures for the amounts of space contained in LQVLGHRIMDUVURRPVRURWKHUERXQGDULHV — 17 — Introduction of Math Vocabulary Point $SRLQWLVDSRVLWLRQLQVSDFH,WKDVQRVL]HDQGQRGLPHQVLRQVDQGLVLQ¿QLWHO\ VPDOO$GRWFDQEHXVHGWRUHSUHVHQWDSRLQWEXWDSRLQW is smaller than the smallest dot you can make. The picture shows the symbol that is used by Google Maps to indicate a point. Some other models of points might be the very tip of a SHQFLORUWKHFRUQHURIDER[ Line $OLQHLVWKHVWUDLJKWSDWKWKDWFRQQHFWVWZRSRLQWVDQGJRHVRQIRUHYHUEH\RQG WKHSRLQWVLQERWKGLUHFWLRQV,WLVPDGHXSRILQ¿QLWHO\PDQ\SRLQWVLQDVWUDLJKW DUUDQJHPHQW$OLQHKDVQRWKLFNQHVVLWKDVRQHGLPHQVLRQ 7KH\HOORZOLQHVLQWKHSKRWRPLJKWKHOSWRJLYHXVDPHQWDOSLFWXUHRIDOLQH even though in reality they do have thickness and they don’t really go on forever! The symbol for line has an arrow on each end. Segment $VHJPHQWLVDSLHFHRUVHFWLRQRIDOLQH,WKDVWZRHQGSRLQWV and contains all the points in between. It can be measured. This is a picture of a line segment with HQGSRLQWV$DQG%,WLVFDOOHGVHJPHQW$%RUVHJPHQW%$ 7KHHGJHVRIWKHER[LQWKHSKRWRDUHOLQHVHJPHQWV2WKHU WKLQJVLQRXUHQYLURQPHQWWKDWUHSUHVHQWOLQHVHJPHQWVDUHSROHV ZLUHVDQGHGJHVRIEXLOGLQJV — 18 — Introduction of Math Vocabulary Midpoint $PLGSRLQWLVWKHSRLQWLQWKHPLGGOHRIDOLQHVHJPHQW,W divides a line segment in half. ,QWKHSKRWR\RXFDQVHHDVHJPHQWUHSUHVHQWHGE\WKH KRUL]RQWDOSROHRQWKH¿VKUDFN7KHPLGSRLQWLVORFDWHG where the center support comes in contact with that pole. Collinear &ROOLQHDUPHDQV³RQWKHVDPHOLQH´&ROOLQHDUSRLQWVDUHWKUHHRUPRUHSRLQWV WKDWOLHRQWKHVDPHOLQH,I\RXRQO\KDYHWZRSRLQWVWKHUHLVDOZD\VRQHOLQH WKDWFDQFRQWDLQWKHPERWK 3RLQWV$%DQG&DUHFROOLQHDU 3RLQWV$%DQG&DUHQRWFROOLQHDU ,QWKHSKRWRWKHWZRERDWVDQGWKHEXR\DUHDOOLQDVWUDLJKW OLQHVRWKH\DUHFROOLQHDU — 19 — Introduction of Math Vocabulary Intersect /LQHVOLQHVHJPHQWVUD\VDQGRWKHU¿JXUHVDUHVDLGWR LQWHUVHFWLIWKH\PHHWRULQRWKHUZRUGVLIWKH\VKDUHD common point. That point is called the point of intersection. ,QWKHSKRWRWKHFURVVLQJRIWKHWZRURSHVRUOLQHVUHSUHVHQWVDQLQWHUVHFWLRQ <RXFDQ¿QGLQWHUVHFWLRQVUHSUHVHQWHGDWURDGFURVVLQJVRQ¿VKQHWVRQIHQFHV and many other locations. ,QWKHSLFWXUHEHORZ6HJPHQW$%LQWHUVHFWVVHJPHQW23DWSRLQW& Ray $UD\LVKDOIDOLQHRUSDUWRIDOLQHWKDWKDVDVWDUWLQJSRLQW DQGH[WHQGVLQ¿QLWHO\LQRQHGLUHFWLRQ7KLQNRIWKHVXQ¶VUD\V – they start at the sun but do not end. $UD\LVUHSUHVHQWHGE\DQDUURZWKDWJRHVLQRQHGLUHFWLRQ 7KLVLVDSLFWXUHRIUD\$% ,QWKHSLFWXUH\RXFDQVHHWKHVXQ¶VUD\V )ODVKOLJKWEHDPVDQGVHDUFKOLJKWVDOVRUHSUHVHQW rays. — 20 — Introduction of Math Vocabulary Angle $QDQJOHLVPDGHXSRIWZRUD\VZLWKDFRPPRQVWDUWLQJSRLQW:HDOVRWKLQNRI DQDQJOHDVWKHDPRXQWRI³WXUQLQJ´RUVSDFHEHWZHHQWKHWZR UD\V$QJOHVDUHPHDVXUHGLQGHJUHHVRUUDGLDQV7KLVLVWKH symbol for angle. The picture shows a representation of an angle drawn on WKHSDYHPHQW7KHKDQGVRIDFORFNDOVRIRUPDQJOHV and angles can be seen commonly on buildings and other structures. Congruent &RQJUXHQWPHDQVH[DFWO\HTXDOLQVL]HDQGVKDSH/LQHVHJPHQWV FDQEHFRQJUXHQWDQGVRFDQWZRGLPHQVLRQDODQGWKUHHGLPHQVLRQDO¿JXUHVDQGREMHFWV This is the symbol for congruent: The chairs in the picture are congruent since they have the same size and shape. Other congruent objects around you PLJKWEHVKHHWVRIQRWHERRNSDSHURUWH[WERRNV — 21 — Introduction of Math Vocabulary Plane $SODQHLVDÀDWVXUIDFHH[WHQGLQJLQ¿QLWHO\LQDOOGLUHFWLRQV,WLVWZRGLPHQVLRQDOLWKDVOHQJWKDQGZLGWKEXWQRWKLFNQHVV5RRIVZDOOVFHLOLQJVDQGIUR]HQ SRQGVDUHDOOPRGHOVRISODQHVEXWDSODQHLVWKLQQHUWKDQDQ\WKLQJ\RXFRXOG WRXFKRUVHHDQGLWJRHVRQIRUHYHU 7KHÀRRUVKRZQLQWKHSLFWXUHUHSUHVHQWVDSODQHVXUIDFH Coplanar &RSODQDUPHDQV³RQWKHVDPHSODQH´:KHQSRLQWVOLQHVRU¿JXUHVDUHFRSODQDULWPHDQVWKH\DUHRQWKHVDPHSODQH &RSODQDU3RLQWV 1RW&RSODQDU3RLQWV $%&DQG'DUHFRSODQDU $%DQG&DUHQRWFRSODQDU — 22 — Introduction of Math Vocabulary 7KHWLOHVRQWKHÀRRULQWKHSKRWRDUHFRSODQDUDQGDOORIWKHOLQHVDQGSRLQWV represented by their edges and intersections are coplanar. 2WKHUH[DPSOHVWKDWUHSUHVHQWFRSODQDUREMHFWVPLJKWEHSLFWXUHVRQDZDOORU windows on the side of a building. Parallel lines 3DUDOOHOOLQHVDUHOLQHVWKDWDUHRQWKHVDPHSODQHFRSODQDUEXWWKDWGRQRW intersect. You can think of them as going in the same direction and staying the same distance apart. The wires in the photo represent parallel lines. Other common representations PLJKWEHWKHHGJHVRIWKHERDUGVDORQJDERDUGZDONRUWKHRSSRVLWHHGJHVRID sheet of paper. Parallel Lines: Not Parallel Lines: — 23 — — 24 — Language and Skills Development Using the Math Vocabulary Terms Language & Skills Development LISTENING Use the activity pages from the Student Support Materials. SPEAKING READING Use the activity pages from the Student Support Materials. WRITING Use the activity pages from the Student Support Materials. Illustration Bingo Provide each student with a copy of the mini-illustration activity page from the student support materials. The students should cut out the illustrations. Each VWXGHQWVKRXOGWXUQKLVKHULOOXVWUDWLRQVIDFHGRZQRQWKHGHVN7KHQHDFKVWX GHQWVKRXOGWXUQ21(LOOXVWUDWLRQIDFHXS6D\DYRFDEXODU\ZRUG$Q\VWXGHQW or students who have the illustration for the vocabulary word you said face up RQWKHLUGHVNVVKRXOGVKRZWKHLULOOXVWUDWLRQV7KRVHLOOXVWUDWLRQVVKRXOGWKHQ be put to the side and the students should turn over another illustration. The ¿UVWVWXGHQWRUVWXGHQWVWRKDYHQRLOOXVWUDWLRQVOHIWRQWKHLUGHVNVZLQWKH URXQG7KHLOOXVWUDWLRQVPD\EHFROOHFWHGPL[HGDQGUHGLVWULEXWHGWRWKHVWXdents for the different rounds of the activity. One to Six 3URYLGHHDFKVWXGHQWZLWKWZREODQNÀDVKFDUGV(DFKVWXGHQWVKRXOGWKHQZULWH DQXPEHURQHDFKRIKLVÀDVKFDUGVEHWZHHQRQHDQGVL[RQHQXPEHUSHUFDUG :KHQWKHVWXGHQWV¶QXPEHUFDUGVDUHUHDG\WRVVWZRGLFHDQGFDOOWKHQXPEHUV VKRZLQJ$Q\VWXGHQWRUVWXGHQWVZKRKDYHWKRVHWZRQXPEHUVPXVWWKHQLGHQ WLI\ D YRFDEXODU\ LOOXVWUDWLRQ \RX VKRZ 7KH VWXGHQWV PD\ H[FKDQJH QXPEHU cards periodically during this activity. Right or Wrong Mount the sight words on the chalkboard. Point to one of the sight words and QDPH LW 7KH VWXGHQWV VKRXOG UHSHDW WKH VLJKW ZRUG +RZHYHU ZKHQ \RX SRLQW WRDVLJKWZRUGDQGVD\WKHZURQJZRUGIRULWWKHVWXGHQWVVKRXOGUHPDLQVLOHQW 5HSHDW WKLV SURFHVV XQWLO WKH VWXGHQWV KDYH UHVSRQGHG DFFXUDWHO\ WR DOO RI WKH sight words a number of times. Mirror Writing *URXSWKHVWXGHQWVLQWRWZRWHDPV+DYHWKH¿UVWSOD\HUIURPHDFKWHDPVWDQG LQ IURQW RI WKH FKDONERDUG *LYH HDFK RI WKH WZR SOD\HUV D VPDOO XQEUHDNDEOH mirror. Stand some distance behind the two players with illustrations for the VLJKW ZRUGV +ROG XS RQH RI WKH LOOXVWUDWLRQV :KHQ \RX VD\ ³*R´ WKH SOD\HUV with the mirrors must look over their shoulders to see the illustration you are KROGLQJ:KHQDSOD\HUVHHVWKHLOOXVWUDWLRQKHVKHPXVWZULWHWKHVLJKWZRUGIRU WKDWLOOXVWUDWLRQRQWKHFKDONERDUG7KH¿UVWSOD\HUWRGRWKLVFRUUHFWO\ZLQVWKH URXQG5HSHDWWKLVSURFHVVXQWLODOOSOD\HUVLQHDFKWHDPKDYHKDDQRSSRUWXQLW\ to respond. — 27 — Student Support Materials — 31 — — 33 — — 35 — — 37 — — 39 — — 41 — — 43 — — 45 — — 47 — — 49 — — 51 — — 53 — — 55 — — 57 — — 59 — — 60 — True-False Sentences (Listening and/or Reading Comprehension) 1. $QLQ¿QLWH number cannot be measured. 2. 7KHÀRRURIDKRXVHKDVWZRdimensions. The spaceLQDER[FDQQRWEHPHDVXUHG $point has no dimensions. $line can be curved. $OLQHsegment has two dimensions. 7. $VHJPHQWLVGLYLGHGLQKDOIE\LWVmidpoint. )RXUSRLQWVWKDWDUHRQWKHVDPHOLQHDUHcollinear. 9. Two lines can intersect in only one point. 10. $ray can be curved. 11. $Qangle is made up of two rays. 12. Two objects that are the same shape are congruent. $plane LVLQ¿QLWHLQRQO\WZRGLUHFWLRQV Two lines are always coplanar. Lines that meet in a common point are parallel. $QVZHUV77)7))777)7)))) 1. There are an LQ¿QLWHQXPEHURIWUHHVLQWKH7RQJDVV1DWLRQDO)RUHVW 2. $ER[KDVRQHdimension. $KRFNH\SXFNH[LVWVLQspace. $point is always round. $line goes on forever in two directions. $OLQHsegment has a measurable length. 7. $midpoint is at the end of a line segment. Two points that are on the same line are called collinear. 9. Parallel lines can intersect each other. 10. $ray has a starting point. 11. $Qangle cannot be measured. 12. $VRGDFDQLVcongruent to another identical soda can. $plane LVDOZD\VÀDW Lines on the same plane are coplanar. 15. Parallel lines are on the same plane. $QVZHUV))7)77)))7)7777 — 61 — Match the Halves 1. Parallel lines are on the same plane $KDVQROHQJWKZLGWKRUGHSWK 3RLQWVOLQHVDQG¿JXUHV B. and it can be measured. $SODQHLVDÀDWVXUIDFHWKDW &FRQQHFWVWZRSRLQWV &RQJUXHQWREMHFWVDUHH[DFWO\HTXDO D. cannot be counted and has no limit. $QDQJOHLVPDGHXSRI E. divides it in half. $UD\LVSDUWRIDOLQHWKDW )H[WHQGVLQ¿QLWHO\LQRQHGLUHFWLRQ ,QWHUVHFWLQJ¿JXUHV G. includes all points in threedimensions. H. are dimensions. 7KUHHRUPRUHSRLQWV 9. The midpoint of a line segment $OLQHVHJPHQWKDVWZRHQGSRLQWV $OLQHLVDVWUDLJKWSDWKWKDW I. two rays with a common starting point. J. are coplanar if they are on the same plane. .LVWZRGLPHQVLRQDODQGH[WHQGV LQ¿QLWHO\LQDOOGLUHFWLRQV $SRLQWLVDSRVLWLRQLQVSDFHWKDW L. and they do not intersect. 6SDFHLVLQ¿QLWHDQG M. share at least one common point. /HQJWKZLGWKDQGKHLJKW N. on the same line are collinear. 6RPHWKLQJWKDWLVLQ¿QLWH O. in both size and shape. $QVZHUV/-.2,)01(%&$*+' — 62 — 'H¿QLWLRQV ,Q¿QLWH. Unlimited or unending. Dimension'LUHFWLRQVRIH[WHQVLRQOLNHOHQJWKZLGWKDQGKHLJKW Space. The set of all points. Point$SRVLWLRQLQVSDFH Line$VWUDLJKWSDWKWKDWFRQQHFWVWZRSRLQWV Segment$SLHFHRUVHFWLRQRIDOLQHZLWKWZRHQGSRLQWV Midpoint$SRLQWWKDWGLYLGHVDOLQHVHJPHQWLQKDOI Collinear Points. Three or more points that lie on the same line. Intersect7RPHHWRUVKDUHDFRPPRQSRLQW Ray3DUWRIDOLQHWKDWH[WHQGVLQ¿QLWHO\LQRQHGLUHFWLRQIURPDVWDUWLQJSRLQW Angle. Two rays with a common starting point. Congruent([DFWO\HTXDOLQVL]HDQGVKDSH Plane$ÀDWWZRGLPHQVLRQDOVXUIDFHH[WHQGLQJLQ¿QLWHO\LQDOOGLUHFWLRQV Coplanar. On the same plane. Parallel lines&RSODQDUOLQHVWKDWGRQRWLQWHUVHFW — 63 — Which Belongs 1. $SRLQWOLQHVHJPHQWDQJOHKDVQROHQJWKZLGWKRUGHSWK 2. 7KHXQLYHUVHLVLQ¿QLWHGLPHQVLRQDOFRQJUXHQWEHFDXVHLWLVXQOLPLWHG and cannot be measured. The vertical corner-posts on the front of a building represent lines that are FROOLQHDUSDUDOOHOLQ¿QLWH 7ZRWULDQJOHVKDYHWKHVDPHVL]HDQGVKDSHVRWKH\DUHSDUDOOHO FRQJUXHQWFRSODQDU $PLGSRLQWHQGSRLQWVWDUWLQJSRLQWLVDSRLQWWKDWGLYLGHVDVHJPHQWLQ half. 7KHVFKRROEXLOGLQJKDVWKUHHGLPHQVLRQVDQJOHVPLGSRLQWVEHFDXVHLW can be measured in more than two directions. 7. $SDUWRIDOLQHWKDWVWDUWVEXWGRHVQRWHQGLVDVHJPHQWUD\DQJOH $GLPHQVLRQDOSODFHZKHUHDOLQHPLJKWH[LVWLVVSDFHDSODQHDSRLQW 9. 7KHJ\PÀRRUUHSUHVHQWVSDUWRIDSODQHOLQHGLPHQVLRQ 10. :KHQDOLQHDQGDOLQHVHJPHQWVKDUHDFRPPRQSRLQWWKH\KDYHD PLGSRLQWDQLQWHUVHFWLRQDGLPHQVLRQ 11. 7KHZLQGRZVRQDÀDWZDOODUHSDUDOOHOFRSODQDUGLPHQVLRQDO 12. $OLQHUD\OLQHVHJPHQWLVVWUDLJKWFRQQHFWVWZRSRLQWVDQGLVLQ¿QLWH 7KUHHSRLQWVRQWKHVDPHUD\DUHFROOLQHDUFRSODQDUFRQJUXHQW $QDQJOHDOLQHVHJPHQWDSODQHFDQEHPHDVXUHGLQGHJUHHV $UD\VHJPHQWSRLQWLVDSLHFHRIDOLQHWKDWFDQEHPHDVXUHG AnswersSRLQWLQ¿QLWHSDUDOOHOFRQJUXHQWPLGSRLQW GLPHQVLRQVUD\VSDFHSODQHDQLQWHUVHFWLRQ FRSODQDUOLQHFROOLQHDUDQDQJOHVHJPHQW — 64 — Multiple Choice :KDWGRHVLWPHDQLIVRPHWKLQJLVLQ¿QLWH" $,WFDQRQO\EHPHDVXUHGLQRQHGLUHFWLRQ B. It is unlimited and unending. &,WWDNHVDYHU\ORQJWLPHWRFRXQWLW +RZPDQ\GLPHQVLRQVDUHUHSUHVHQWHGE\WKHVXUIDFHRIDSLHFHRISDSHU" $GLPHQVLRQ B. 2 dimensions &GLPHQVLRQV :KLFKRIWKHVHVWDWHPHQWVDERXW³VSDFH´LVWUXH" $,WKDVGLPHQVLRQV %,WLVLQ¿QLWH &%RWK$DQG%DUHWUXH :K\FDQ¶W\RXPHDVXUHDSRLQW" $,WLVWRRVPDOO %,WLVLQ¿QLWH &,WKDVQRGLPHQVLRQV :KDWLVDQLPSRUWDQWFKDUDFWHULVWLFRIDOLQH" $,WVOHQJWKFDQEHPHDVXUHG B. It has one dimension &,WLVVWUDLJKW 7KHHGJHRIDWDEOHLVVWUDLJKWDQGKDVDOHQJWKRIIHHW:KLFKRIWKHIROORZLQJGRHVWKDWHGJHUHSUHVHQW" $$OLQHVHJPHQW %$SODQH &$'LPHQVLRQ :K\FDQ¶WDOLQHKDYHDPLGSRLQW" $,WGRHVQ¶WKDYHHQRXJKGLPHQVLRQV B. Its endpoints are too far apart. &,WLVLQ¿QLWHDQGFDQQRWEHPHDVXUHG :K\GRHV³FROOLQHDU´UHIHUWRWKUHHRUPRUHSRLQWV" $$Q\WLPH\RXKDYHMXVWWZRSRLQWV\RXFDQGUDZDOLQHWKURXJKWKHP %&ROOLQHDUREMHFWVDUHWKUHHGLPHQVLRQDO &6RWKDWLWGRHVQRWJHWFRQIXVHGZLWK³FRSODQDU´ — 65 — :KHQWZRURDGVLQWHUVHFWZKLFKFDQQRWEHWUXH" $7KH\PLJKWEHFXUYHGURDGV B. They are parallel roads &7KH\PHHWHDFKRWKHU :K\LVDVXQUD\OLNHDUD\LQJHRPHWU\" $,WVWDUWVVRPHZKHUHDQGWUDYHOVLQDVWUDLJKWGLUHFWLRQ B. Both are hot &,WKDVWZRGLPHQVLRQV :KLFKRIWKHIROORZLQJGRHVQRWUHSUHVHQWDQDQJOH" $7KHKDQGVRIDFORFN B. The corner of a piece of paper &$SLQSRLQW :KLFKRIWKHVHPLJKWEHFRQJUXHQW" $7ZRUD\V B. Two line segments &7ZRSODQHV :K\GRHVQ¶WWKHRFHDQVXUIDFHUHSUHVHQWDSODQH" $,WLVQRWLQ¿QLWH %,WLVQRWÀDW &%RWK$DQG%DUHWUXH :KLFKRIWKHVHKDVWREHFRSODQDU" $7ZRLQWHUVHFWLQJOLQHV %)RXUSRLQWV &7KUHHOLQHVHJPHQWV 7KHWZRHGJHVRIDSHUIHFWO\VWUDLJKWÀDWURDGZRXOGEH $&ROOLQHDUDQGFRQJUXHQW B. Parallel and coplanar &,Q¿QLWHDQGWZ $QVZHUV 1. B 2. B & & & $ 7. & $ 9. B 10. $ 11. & 12. B & $ B — 66 — Complete the Sentence 1. ________________ has three dimensions and is boundless and unlimited. 2. Three bowling pins arranged in a straight row might be called ______________ . The line of the roof formed an ______________with the line of the wall. $BBBBBBBBBBBBBBPLJKWEHUHSUHVHQWHGE\DEHDPRIOLJKW 7KHGLVWDQFHWKDWDSODQHH[WHQGVLQVSDFHLVBBBBBBBBBBBBBBBBBB 7KHÀDWVXUIDFHRIWKHLFHUHPLQGHGKLPRIDBBBBBBBBBBBBB 7. $VWUDLJKWOLQHKDVRQHBBBBBBBBBBBBBBBBBBBBDQGDWULDQJOHKDVWZR _____________. $SRVLWLRQRUORFDWLRQLQVSDFHLVFDOOHGDBBBBBBBBBBBBBBDQGLWKDVQR OHQJWKZLGWKRUGHSWK 9. $VHFWLRQRIDOLQHFDOOHGDBBBBBBBBBBBBBBBBBBBFDQEHPHDVXUHG 10. $VHJPHQWLVGLYLGHGLQKDOIE\LWVBBBBBBBBBBBBBBBBBBBBBB 11. $OORIWKHOLQHVRQWKHEDVNHWEDOOFRXUWÀRRUDUHBBBBBBBBBBBBBBBB 12. The two roads ________________________ near the upper part of the map. The two squares on the paper are ________________ because they are the same size. The straight railings on the bridge were ________________to each other. 7KHBBBBBBBBBBBBBBBEHWZHHQWKHWZRSRLQWVH[WHQGHGIRUHYHULQERWK directions. $QVZHUV 1. Space 2. &ROOLQHDU $QJOH 5D\ ,Q¿QLWH Plane 7. Dimension Point 9. Segment 10. Midpoint 11. &RSODQDU 12. Intersect &RQJUXHQW Parallel Line — 67 — Creative Writing Have the students write sentences of their own, based on the picture below. When finished, have each student read his/her sentences to the others. — 68 — Place-Based Practice Activity +DYHDVFDYHQJHUKXQWLQWKHFODVVURRPDWKRPHDQGRURXWGRRUVWR¿QGUHSUHVHQWDWLRQVRISRLQWVOLQHVSODQHVVHJPHQWVDQJOHVLQWHUVHFWLRQVDQGRWKHU terms from this unit. 'LYLGHWKHFODVVLQWRWHDPVDQGDOORZFODVVWLPHIRUWHDPVWR¿QGDQGOLVWDV PDQ\H[DPSOHVRIHDFKDVWKH\FDQ¿QG*LYHDWLPHOLPLW 25 $VVLJQWKHVFDYHQJHUKXQWDVKRPHZRUNWKHQJLYHHDFKWHDPWLPHWRFRPSLOH WKHLU³PDVWHUOLVW´DWWKHEHJLQQLQJRIFODVV +DYHWKHWHDPVH[FKDQJHOLVWVDQGFULWLTXH'LVFXVVDQ\H[DPSOHVWKDWUDLVH TXHVWLRQVDVDFODVV 2IIHUH[WUDFUHGLWRUDSUL]HIRUWKHWHDPZLWKWKHPRVWH[DPSOHVRUJLYHFUHGLW WRDQ\WHDPWKDWH[FHHGVDJLYHQQXPEHURIFRUUHFWH[DPSOHV — 69 — — 70 — Unit Assessment — 71 — — 72 — Unit 1 Geometry An Introduction to Math Vocabulary Name: ___________________ Date: ___________________ Match the definition in the column on the right with the key vocabulary in the column on the left. Place the letter of the definition in front of the word it matches. a. an infinitely small position in space with no size or dimension. 1) _____ space 2) _____ dimension b. the straight path that connects two points and goes on forever in both directions. 3) _____ point 4) _____ line c. the 3-dimensional place in which an object can exist or events can take place. 5) _____ ray 6) _____ angel d. directions of extension- length, width, and height e. half a line, or part of a line that has a starting point and extends infinitely in one direction. f. made up of two rays with a common starting point Multiple Choice: Read each statement below and choose the best answer from the choices provided. Circle the best choice. 7) The point in the middle of a line segment that divides a line segment in half is known as ___________. a) the halfway point b) the midpoint c) the intersect 8) A piece or section of a line with two endpoints, that contains all the points in between and can be measured is a __________________. a) midpoint b) collinear point c) segment — 73 — 9) When something is unlimited or unending, goes on forever and cannot be measured or counted it is ____________________. a) a segment b) infinite c) a ray 10) Telephone wires or lines on the pavement model or represent ________________ objects. a) rays b) intersects c) two dimensional 11) Collinear points is/are __________ or more points that lie on the same line, a) one b) two c) three Fill in the Blank: Complete each of the statements below with the word that fits best. Choose words from the Word Bank below. Word Bank Congruent infinite infinite intersect line plane plane point rays 12) A dot can be used to represent a 13) Space is on a map. . We can2019t measure all of space, and we can2019t measure the number of points in any given space. 14) A is made up of infinitely many points in a straight arrangement. It has no thickness; it has one dimension. 15) Lines, line segments, rays, and other figures are said to common point — 74 — if they meet, or share a 16) The sun2019s 2013 they start at the sun but do not end and are part of a line that has a starting point and extends infinitely in one direction. 17) When points, lines, or figures are coplanar, it means they are on the same 18) . means exactly equal in size and shape. 19) A is a flat surface that extends infinitely in all directions. It is two- dimensional; it has length and width but no thickness. Illustrations for Key Vocabulary: Choose the correct illustration to match the vocabulary word. 20) Look at the illustration below and write in the space provided, the key vocabulary word that it represents. ILLUSTRATION OF COPLANAR________________. 21) Look at the two illustrations below. Circle the illustrations with parallel lines. ILLUSTRATIONS ADDED for parallel and non parallel lines. — 75 — — 76 — bev williams 11.9999 Unit 1 Geometry An Introduction to Math Vocabulary Name: ___________________ Date: ___________________ Match the definition in the column on the right with the key vocabulary in the column on the left. Place the letter of the definition in front of the word it matches. 1) c space 2) d dimension 3) a point 4) b line 5) e ray 6) f angel a. an infinitely small position in space with no size or dimension. b. the straight path that connects two points and goes on forever in both directions. c. the 3-dimensional place in which an object can exist or events can take place. d. directions of extension- length, width, and height e. half a line, or part of a line that has a starting point and extends infinitely in one direction. f. made up of two rays with a common starting point Multiple Choice: Read each statement below and choose the best answer from the choices provided. Circle the best choice. 7) The point in the middle of a line segment that divides a line segment in half is known as ___________. a) the halfway point b) the midpoint c) the intersect 8) A piece or section of a line with two endpoints, that contains all the points in between and can be measured is a __________________. a) midpoint b) collinear point c) segment — 77 — 9) When something is unlimited or unending, goes on forever and cannot be measured or counted it is ____________________. a) a segment b) infinite c) a ray 10) Telephone wires or lines on the pavement model or represent ________________ objects. a) rays b) intersects c) two dimensional 11) Collinear points is/are __________ or more points that lie on the same line, a) one b) two c) three Fill in the Blank: Complete each of the statements below with the word that fits best. Choose words from the Word Bank below. Word Bank Congruent infinite infinite intersect line plane plane point rays 12) A dot can be used to represent a point on a map. 13) Space is infinite . We can2019t measure all of space, and we can2019t measure the infinite number of points in any given space. 14) A line is made up of infinitely many points in a straight arrangement. It has no thickness; it has one dimension. 15) Lines, line segments, rays, and other figures are said to intersect if they meet, or share a common point — 78 — 16) The sun2019s rays 2013 they start at the sun but do not end and are part of a line that has a starting point and extends infinitely in one direction. 17) When points, lines, or figures are coplanar, it means they are on the same plane . 18) Congruent means exactly equal in size and shape. 19) A plane is a flat surface that extends infinitely in all directions. It is two- dimensional; it has length and width but no thickness. Illustrations for Key Vocabulary: Choose the correct illustration to match the vocabulary word. 20) Look at the illustration below and write in the space provided, the key vocabulary word that it represents. ILLUSTRATION OF COPLANAR________________. Coplanar 21) Look at the two illustrations below. Circle the illustrations with parallel lines. ILLUSTRATIONS ADDED for parallel and non parallel lines. Correct answer not entered. — 79 — — 80 — UNIT 2 Angles Sealaska Heritage Institute Grade Level Expectations for Unit 2 Unit 2—Angles Alaska State Mathematics Standard A $VWXGHQWVKRXOGXQGHUVWDQGPDWKHPDWLFDOIDFWVFRQFHSWVSULQFLSOHVDQGWKHRULHV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG $FRQVWUXFWGUDZPHDVXUHWUDQVIRUPFRPSDUHYLVXDOL]HFODVVLI\DQGDQDO\]HWKH UHODWLRQVKLSVDPRQJJHRPHWULF¿JXUHV Alaska State Mathematics Standard C $VWXGHQWVKRXOGXQGHUVWDQGDQGEHDEOHWRIRUPDQGXVHDSSURSULDWHPHWKRGVWRGH¿QH DQGH[SODLQPDWKHPDWLFDOUHODWLRQVKLSV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG &H[SUHVVDQGUHSUHVHQWPDWKHPDWLFDOLGHDVXVLQJRUDODQGZULWWHQSUHVHQWDWLRQV SK\VLFDOPDWHULDOVSLFWXUHVJUDSKVFKDUWVDQGDOJHEUDLFH[SUHVVLRQV &UHODWHPDWKHPDWLFDOWHUPVWRHYHU\GD\ODQJXDJH GLEs The student demonstrates an understanding of geometric relationships by >@*LGHQWLI\LQJDQDO\]LQJFRPSDULQJRUXVLQJSURSHUWLHVRIDQJOHVLQFOXGLQJVXSSOHPHQWDU\RUFRPSOHPHQWDU\ The student demonstrates an understanding of geometric relationships by >@*LGHQWLI\LQJDQDO\]LQJFRPSDULQJRUXVLQJSURSHUWLHVRISODQH¿JXUHV VXSSOHPHQWDU\FRPSOHPHQWDU\RUYHUWLFDODQJOHV The student communicates his or her mathematical thinking by >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\WUDQVODWLQJDPRQJWKHVHDOWHUQDWLYHUHSUHVHQWDWLRQVRUXVLQJDSSURSULDWHYRFDEXODU\ V\PEROVRUWHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQGVROXWLRQV >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\FRPPXQLFDWLQJPDWKLGHDVLQZULWLQJRUXVLQJDSSURSULDWHYRFDEXODU\V\PEROVRU WHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQGVROXWLRQV Vocabulary & Definitions Introduction of Math Vocabulary Angle $QDQJOHLVPDGHXSRIWZRUD\VZLWKDFRPPRQVWDUWLQJ point. The symbol for angle is $QJOHVFDQEHVHHQDOODURXQG\RXRQEXLOGLQJVFORFNV URDGLQWHUVHFWLRQVDQGLQWKHZD\\RXEHQG\RXUDUPV DQGOHJV7KRVHDUHMXVWDIHZH[DPSOHV Angle measure 7KHVL]HRIDQDQJOHRUDPRXQWRIWXUQLQJRUVSDFH EHWZHHQWKHWZRUD\VRIDQDQJOHPHDVXUHGLQGHJUHHV or radians. Protractors and other devices are used to measure angles. 7RDFKLHYHJRRGIRUPDQGVSHHGWKLVVNLHUPLJKWZDQW WRNQRZWKHDQJOHPHDVXUHEHWZHHQKLVVNLVDQGWKH angle measure of his spine in relation to the ground. $VNVWXGHQWVLIWKH\FDQWKLQNRIRWKHUUHDVRQVWRPHDVXUHDQJOHVLHEXLOGHUV PHDVXUHDQJOHVIRUURRIV — 87 — Introduction of Math Vocabulary Degree $XQLWRIDQJOHPHDVXUH7KHUHDUHGHJUHHVLQDFLUFOHRU a complete revolution. The symbol for degree is o. $FRPSDVVPHDVXUHVGLUHFWLRQVXVLQJGHJUHHV Radian $QRWKHUXQLWIRUPHDVXULQJDQJOHV7KHDUH radians LQDFLUFOHRUDFRPSOHWHUHYROXWLRQo = radians. 5DGLDQVDUHPRVWRIWHQXVHGWRPHDVXUHWKHVSHHGDW which wheels and machine parts revolve. Vertex (of an angle) 7KHSRLQWZKHUHWZRUD\VPDNLQJXSDQDQJOHPHHWRUWKHFRUQHU point. 7KHWLSRIWKHVSLUHVKRZQLQWKHSKRWRUHSUHVHQWVDYHUWH[ $QRWKHUH[DPSOHPLJKWEHWKHFRUQHURIDSLHFHRISDSHUZKHUHWKHWZRHGJHV meet. — 88 — Introduction of Math Vocabulary Side (of an angle) Either of two rays that make up an angle. ,QWKHSKRWRWKHKDQGVRIDFORFNUHSUHVHQWWKH of an angle. Right angle $ULJKWDQJOHLVDoDQJOH:KHQOLQHVDUHSHUSHQGLFXODU WKH\IRUPULJKWDQJOHV5LJKWDQJOHVDUHHDV\WR¿QGWKH\ are very common in buildings. Usually the angle between two ZDOOVLVDULJKWDQJOHDQGWKHFRUQHUVRIZLQGRZVDQGGRRUV are right angles. The mast of this boat forms a right angle with the horizon. — 89 — sides Introduction of Math Vocabulary Perpendicular $WDULJKWoDQJOH ,QWKLVSLFWXUHWKHOLQHVRIJURXWEHWZHHQWKHWLOHVDUH SHUSHQGLFXODU2WKHUH[DPSOHVRISHUSHQGLFXODUOLQHVDUH XVXDOO\WKHDGMRLQLQJVLGHVRIGRRUVDQGZLQGRZV Acute angle $QDQJOHWKDWKDVDPHDVXUHOHVVWKDQo. $WWKHKDQGVRIDFORFNIRUPDQDFXWHDQJOH$FXWH DQJOHVFDQEHIRXQGRQURRIVRQVLJQVDQGLQWULDQJOHV,QWKLV SKRWRWKHODSWRSLVRSHQWRDQDFXWHDQJOH — 90 — Introduction of Math Vocabulary Obtuse angle $QDQJOHWKDWKDVDPHDVXUHPRUHWKDQDQGOHVVWKDQ $VWRSVLJQKDVREWXVHDQJOHV,QWKHSKRWRWKHVFLVVRUV are open to an obtuse angle. Adjacent angles $QJOHVWKDWDUHLPPHGLDWHO\QH[WWRHDFKRWKHU7KH\DUHLQWKHVDPHSODQH VKDUHDFRPPRQYHUWH[DQGDFRPPRQVLGHDQGGRQRWRYHUODS 2QWKHELF\FOHLQWKHSLFWXUHWKHUHDUHDGMDFHQWDQJOHVRIWKHIUDPHZLWKYHUWLFHV DWWKHFUDQN$VNVWXGHQWVLIWKH\FDQ¿QGRWKHUH[DPSOHVRIDGMDFHQWDQJOHVRQ the bicycle. — 91 — Introduction of Math Vocabulary Linear pair $SDLURIDGMDFHQWDQJOHVIRUPHGE\LQWHUVHFWLQJOLQHV 2QWKHSLFWXUHRILQWHUVHFWLQJMHWWUDLOVWKHUHDUHIRXUOLQHDU pairs of angles. Complementary angles Two acute angles that add up to 90o)RUH[DPSOHDo angle DQGDo angle are complementary. ,QWKHSKRWRRIDJDWHWKHGLDJRQDOVGLYLGHWKHFRUQHUVLQWRFRPplementary angles. — 92 — Introduction of Math Vocabulary Supplementary angles 7ZRDQJOHVZKRVHPHDVXUHVDGGXSWRo)RUH[DPSOHDo angle and a 70o angle are supplementary. $VNVWXGHQWVLIWKH\FDQ¿QGWKHVXSSOHPHQWDU\DQJOHV LQWKLVSKRWRRIDVNLQERDWIUDPHZKHUHWKHVXSSRUWV intersect. Vertical angles $SDLURIDQJOHVWKDWDUHRSSRVLWHHDFKRWKHU formed by two lines that intersect. $QJOHVDQGDUHYHUWLFDODQJOHVDQGDQJOHV DQGDUHYHUWLFDODQJOHV Many different vertical angles can be found on the photo of the quilt. Vertical angles can also EHIRXQGDWURDGLQWHUVHFWLRQVRQEXLOGLQJVDQGRWKHU objects. — 93 — — 94 — Language and Skills Development Using the Math Vocabulary Terms Language & Skills Development LISTENING Use the activity pages from the Student Support Materials. Illustration Bingo Provide each student with a copy of the mini-illustration activity page from the student support materials. The students should cut out the illustrations. Each VWXGHQWVKRXOGWXUQKLVKHULOOXVWUDWLRQVIDFHGRZQRQWKHGHVN7KHQHDFKVWX GHQWVKRXOGWXUQ21(LOOXVWUDWLRQIDFHXS6D\DYRFDEXODU\ZRUG$Q\VWXGHQW or students who have the illustration for the vocabulary word you said face up RQWKHLUGHVNVVKRXOGVKRZWKHLULOOXVWUDWLRQV7KRVHLOOXVWUDWLRQVVKRXOGWKHQEH SXWWRWKHVLGHDQGWKHVWXGHQWVVKRXOGWXUQRYHUDQRWKHULOOXVWUDWLRQ7KH¿UVW VWXGHQWRUVWXGHQWVWRKDYHQRLOOXVWUDWLRQVOHIWRQWKHLUGHVNVZLQWKHURXQG7KH LOOXVWUDWLRQVPD\EHFROOHFWHGPL[HGDQGUHGLVWULEXWHGWRWKHVWXGHQWVIRUWKH different rounds of the activity. SPEAKING Role’m Again! 0RXQWWKHYRFDEXODU\LOOXVWUDWLRQVRQWHKFKDONERDUG1XPEHUHDFKLOOXVWHUDWLRQ XVLQJWKHQXPEHUVWR*LYHGLFHWRWKHVWXGHQWV+DYHHDFKVWXGHQWUROOKLV her dice. The student should identify a picture with the same number showing on his/her die and then use that word in a sentence. READING Use the activity pages from the Student Support Materials. Sight Word Bingo Provide each student with a set of sight words from this unit. Each student should SODFHRQHZRUGRQKLVKHUGHVNKROGLQJWKHRWKHUVVHSDUDWHO\6KRZDYRFDEXODU\ SLFWXUH,IDVWXGHQWKDVWKHZRUGIRUWKDWSLFWXUHRQKLVKHUGHVNKHVKHVKRXOG show it to you; the student should then put that word to the side and place another RQHRQWKHGHVN&RQWLQXHLQWKLVZD\XQWLODVWXGHQWKDVQRZRUGVOHIW WRITING What’s Your Letter? Show one of the vocabulary pictures to the students. Each student should then ZULWH21(OHWWHUWKDWLVLQWKHZRUGIRUWKDWSLFWXUH7KHQFKHFNDOOWKHVWXGHQWV¶ letters to determine if all of the letters of the word were written. Have the stuGHQWVLGHQWLI\WKH³PLVVLQJOHWWHUV´LIWKHUHDUHDQ\ Use the activity pages from the Student Support Materials. — 97 — Student Support Materials — 101 — — 103 — — 105 — — 107 — — 109 — — 111 — — 113 — — 115 — — 117 — — 119 — — 121 — — 123 — — 125 — — 127 — — 129 — — 131 — — 132 — True-False Sentences (Listening and/or Reading Comprehension) 1. The two rays that make up an angle have a common starting point 2. $UXOHUFDQEHXVHGWRJHWDQangle measure. $degree LVRQHRIGLYLVLRQVRIDFLUFOH $radian is the unit of measure used on a compass Not all angles have a vertex. Every angle has three sides. 7. The adjacent sides of a square form right angles. 8. Perpendicular lines never intersect. 9. $VPDOOVKDUSDQJOHZRXOGEHDQacute angle. 10. $Qobtuse angle is larger than a right angle. 11. Adjacent angles could add up to 100 degrees. 12. $QJOHVLQDlinear pair are always acute angles. $QREWXVHDQJOHFDQEHcomplementary to another angle. 14. Supplementary angles must be acute angles 15. Vertical angles are formed by two intersecting lines $QVZHUV7)7)))7)777)))7 1. Angles can be measured in inches. 2. Angle measuresUHSUHVHQWWKHDPRXQWRI³WXUQLQJ´EHWZHHQWZRUD\V Line segments are measured in degrees. $FLUFOHLVPDGHXSRIRUDERXWradians. The common starting point for the two rays in an angle is called the vertex. The side of an angle is the same as one of the rays of the angle. 7. Right angles have measures of 100 degrees. :KHQWZROLQHVLQWHUVHFWWRIRUPULJKWDQJOHVWKH\DUHperpendicular. 9. $WWKHKDQGVRIDFORFNIRUPDQacute angle. 10. $Qobtuse anglePLJKWKDYHDPHDVXUHRIGHJUHHV 11. Two angles that are across from each other are adjacent angles. 12. In a linear pairWKHDQJOHPHDVXUHVDGGXSWRGHJUHHV $GHJUHHDQJOHDQGDGHJUHHDQJOHDUHcomplementary angles. $QJOHVLQDOLQHDUSDLUDUHsupplementary. 15. Vertical anglesDUHQH[WWRHDFKRWKHU $QVZHUV)7)777)7)))777) — 133 — Match the Halves 1. Vertical angles $LVJLYHQLQGHJUHHVRUUDGLDQV 7ZRUD\VZLWKDFRPPRQYHUWH[ B. is one of its rays. 7ZRDQJOHVZKRVHPHDVXUHVDGGXS WRo &DUHSHUSHQGLFXODU $QDQJOHPHDVXUH D. acute angles. ,QDQDQJOHWKHYHUWH[LV E. share a common ray. 7RPDNHXSDOLQHDUSDLUDQJOHV )PDNHXSHDFKDQGHYHU\DQJOH $QJOHVZLWKPHDVXUHVRIo are G. are measured in radians. 2QHRIGLYLVLRQVRIDFLUFOH H. the point shared by two rays. &RPSOHPHQWDU\DQJOHVDUHDOZD\V I. measure less than 90 $QJOHVUHODWHGWRWKHVSHHGRI revolution J. are opposite each other. 11. The side of an angle K. must be adjacent. $GMDFHQWDQJOHV L. is known as a degree. $QJOHVEHWZHHQoDQGo are M. right angles. /LQHVWKDWLQWHUVHFWWRIRUPULJKW angles N. obtuse angles. $FXWHDQJOHV O. are supplementary. o $QVZHUV 1 J 2F 3O 4A 5H 6K 7M 8L 9D 10G 11B 12E 13N 14C 15I — 134 — 'H¿QLWLRQV Angle: Two rays with a common starting point. Angle measure:7KHVL]HRIDQDQJOHRUDPRXQWRIWXUQLQJRUVSDFHEHWZHHQ WKHWZRUD\VRIDQDQJOHPHDVXUHGLQGHJUHHVRUUDGLDQV Degreeth of a circle; a unit of angle measure. Radian: 1/2; a unit of angle measure often used to measure the amount of revolution. VertexRIDQDQJOH7KHSRLQWZKHUHWKHWZRUD\VRIDQDQJOHPHHW SideRIDQDQJOH(LWKHURIWKHWZRUD\VWKDWPDNHXSDQDQJOH Right angle:$o angle. Perpendicular$WDo angle. Acute angle:$QDQJOHWKDWKDVDPHDVXUHOHVVWKDQo. Obtuse angle:$QDQJOHWKDWKDVDPHDVXUHPRUHWKDQDQGOHVVWKDQ Adjacent angles:$QJOHVWKDWDUHLPPHGLDWHO\QH[WWRHDFKRWKHU Linear pair$SDLURIDGMDFHQWDQJOHVIRUPHGE\LQWHUVHFWLQJOLQHV Complementary angles: Two acute angles that add up to 90o . Supplementary angles:7ZRDQJOHVZKRVHPHDVXUHVDGGXSWRo. Vertical angles$SDLURIDQJOHVWKDWDUHRSSRVLWHHDFKRWKHUIRUPHGE\WZR lines that intersect. — 135 — Which Belongs 1. 7KHVL]HRIDQDQJOHLVNQRZQDVDUDGLDQDQDQJOHPHDVXUHDGHJUHH 2. :KHQWZROLQHVLQWHUVHFWWKH\DOZD\VIRUPYHUWLFDODFXWHULJKWDQJOHV $XQLWRIDQJOHPHDVXUHWKDWFRPPRQO\LVDVVRFLDWHGZLWKWXUQLQJRUUHYROXWLRQVLVWKHGHJUHHUDGLDQGLDPHWHU ,IWZRDGMDFHQWDQJOHVDUHIRUPHGE\LQWHUVHFWLQJOLQHVWKH\DUHYHUWLFDO DQJOHVDOLQHDUSDLUFRPSOHPHQWDU\DQJOHV 7KHVLGHPHDVXUHFRPSOHPHQWRIDQDQJOHLVHLWKHURILWVWZRUD\V &RPSOHPHQWDU\6XSSOHPHQWDU\9HUWLFDODQJOHVKDYHPHDVXUHVWKDW add up to 90o. 7. :KHQWZRDGMRLQLQJUD\VDUHSHUSHQGLFXODUWKH\IRUPDQREWXVHDYHUWLFDODULJKWDQJOH $QDFXWHDULJKWDQREWXVHDQJOHKDVDPHDVXUHODUJHUWKDQo. 9. /LQHVUD\VRUVHJPHQWVWKDWDUHDWDoDQJOHVWRHDFKRWKHUDUHYHUWLFDOSHUSHQGLFXODUDGMDFHQW 10. 7ZRUD\VRIDQDQJOHPHHWDWWKHPLGSRLQWYHUWH[VLGH 11. $W30WKHKDQGVRIDFORFNUHSUHVHQWDQDFXWHDQREWXVHDULJKW angle. 12. 7KHUHDUHGHJUHHVUDGLDQVVLGHVLQDFLUFOH $QJOHVWKDWVKDUHDFRPPRQUD\DUHYHUWLFDOOLQHDUDGMDFHQWDQJOHV $OLQHDUSDLUDYHUWH[DQDQJOHKDVWZRUD\VWKDWPHHWDWDFRPPRQ point. ,IWZRDQJOHPHDVXUHVDGGXSWRoWKHDQJOHVDUHVXSSOHPHQWDU\ FRPSOHPHQWDU\DGMDFHQW 1. $QDQJOHPHDVXUH 2. Vertical 5DGLDQ $OLQHDUSDLU Side &RPSOHPHQWDU\ 7. $ULJKW $QREWXVH 9. Perpendicular 10. 9HUWH[ 11. $QDFXWH 12. Degrees $GMDFHQW $QDQJOH Supplementary — 136 — Multiple Choice 1. Which one of the following would not contain any representations of DQJOHV" DDQH[HUFLVHEDOO EDSDLURIGLFH FWKHURRIRIDKRXVH :KLFKRIWKHIROORZLQJGHVFULEHVDQDQJOHPHDVXUH" DWKHDPRXQWWKDWWKHVLGHVRIDQDQJOHDUHVSUHDGDSDUWLQGHJUHHV EWKHVL]HRIDQDQJOHLQFHQWLPHWHUV FWKHQXPEHURIPHWULFXQLWVLQDQDQJOH $Q\FLUFOHFRQWDLQV D degrees E radians FERWKRIWKHDERYH 7KHDQJOHRIDURRÀLQHPLJKWEHPHDVXUHGLQ DHUJVRUGHJUHHV EGHJUHHVRUUDGLDQV FIHHWRUUDGLDQV :KHQWZRVWUHHWVRQDPDSLQWHUVHFWDWDQDQJOHWKHSRLQWZKHUHWKH\PHHW would be DWKHVLGHRIWKHDQJOH EWKHFHQWHURIWKHDQJOH FWKHYHUWH[RIWKHDQJOH 7KHVLGHRIDQDQJOHPLJKWEHUHSUHVHQWHGE\ DDEODGHRQDQRSHQSDLURIVFLVVRUV EWKHFRUQHURIDQRWHERRN FWKHHGJHRIDSODWH ,QZKLFKRIWKHIROORZLQJSODFHVZRXOG\RX¿QGWKHPRVWULJKWDQJOHV" DWKHGRPHRIDFDWKHGUDO EDVFKRROEXLOGLQJ FWKHLQWHULRURIDFDU $WUHHLVSHUSHQGLFXODUWRWKHJURXQGLI DLWJURZVVWUDLJKWXS ELWLVJURZLQJLQDSODFHWKDWLVÀDW FERWKRIWKHDERYHDUHWUXH — 137 — ,I\RXPDNHDYHU\VKDUSULJKWWXUQDWDURDGLQWHUVHFWLRQ\RXZRXOGEH turning at DDQDFXWHDQJOH EDVXSSOHPHQWDU\DQJOH FDSHUSHQGLFXODUDQJOH $QREWXVHDQJOH DPXVWPHDVXUHOHVVWKDQo EFDQQRWKDYHDPHDVXUHRIPRUHWKDQo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o 1. a 2. a c b c a 7. b c 9. a 10. b 11. c 12. a b c c — 138 — Complete the Sentence 1. The two rays of an angle are also called the two _______________. 2. &RPSOHPHQWDU\DQJOHVDUHDOZD\VBBBBBBBBBBBBBBBBBBBBDQJOHV $QBBBBBBBBBBBBBBBBBBBBBBBBBLVWKHDPRXQWRIWXUQLQJRUVSDFH between two rays that form an angle. $QJOHVWKDWDUHIRUPHGE\LQWHUVHFWLQJOLQHVDUHBBBBBBBBBBBBBBBBBBBBLI they are not adjacent. :KHQDUD\PRYHVLQDFRPSOHWHFLUFOHLWFRYHUVDGLVWDQFHRI ___________. :KHQWZRUD\VEHJLQDWDFRPPRQSRLQWWKH\IRUPDQ ___________________. 7. The corners of a rectangle are _______________angles. ________________angles have measures that add up to equal a right angle’s measure. 9. ,QPRVWEXLOGLQJVWKHZDOOVDUHBBBBBBBBBBBBBBBBBBBWRWKHÀRRU 10. ,QDQDQJOHWKHBBBBBBBBBBBBBBBBBBBBBBBBBLVWKHSRLQWZKHUHWZR rays meet. 11. $QJOHVDUHBBBBBBBBBBBBBBBBBBBBLIWKH\DUHZLGHUWKDQULJKWDQJOHV DQGPHDVXUHOHVVWKDQo. 12. 2QDFRPSDVVDQJOHVDUHPHDVXUHGLQBBBBBBBBBBBBBBBBBBB :KHQVXSSOHPHQWDU\DQJOHVDUHQRWDGMDFHQWWKH\GRQ¶WIRUPD ____________. Two right angles would also be ___________________angles. BBBBBBBBBBBBBBBBBBBBBBDQJOHVDUHQH[WWRHDFKRWKHUDQGVKDUHD common ray. $QVZHUV 1. sides 2. acute angle measure vertical radians angle 7. right complementary 9. perpendicular 10. YHUWH[ 11. obtuse 12. degrees linear pair supplementary DGMDFHQW — 139 — Creative Writing Have the students write sentences of their own, based on the picture below. When finished, have each student read his/her sentences to the others. — 140 — Place-Based Practice Activity $VNVWXGHQWVWR¿QGDQGSKRWRJUDSKUHSUHVHQWDWLRQVRIGLIIHUHQWW\SHVRIDQJOHV LQWKHLURZQHQYLURQPHQW6WXGHQWVPD\ZRUNLQSDLUVRUVPDOOJURXSVVKDULQJ DFDPHUDRUWKH\PD\VNHWFKWKLQJVWKH\¿QGLIFDPHUDVDUHQRWDYDLODEOH The following types of angles should be included: 5LJKWDQJOHV $FXWHDQJOHV Obtuse angles 9HUWLFDO$QJOHV Linear pairs of angles $GMDFHQWDQJOHV Supplementary angles &RPSOHPHQWDU\DQJOHV Vertical angles 'HSHQGLQJRQWLPHDYDLODEOHKDYHHDFKVWXGHQWRUJURXSWU\WR¿QGDWOHDVWRQH RIHDFKDQJOHW\SHRUDVVLJQWZRRUWKUHHDQJOHW\SHVWRHDFKVWXGHQWJURXS $OORZFODVVDQGRUKRPHZRUNWLPHIRUVWXGHQWVWRWDNHWKHSKRWRVWKHQVKDUH and compile them in a display or a slide show. — 141 — — 142 — Unit Assessment — 143 3 — — 144 — Geometry: Unit 2-Angles Name: ___________________ Date: ___________________ Match the symbol on the right with the word it represent on the left. Place the letter of the correct symbol in front of the word it matches. a. illustration of complementary angle 1) _____ angles 2) _____ degree 3) _____ acute angle 4) _____ obtuse angle b. illustration of obtuse angle 5) _____ perpendicular 6) _____ linear pair c. illustration of adjacent angle 7) _____ adjacent angle 8) _____ complementary angle d. illustration of linear pair e. illustration of perpendicular lines f. illustration of acute angle g. Illustration of angle h. illustration of degree — 145 — Word Bank acute angle complementary degree obtuse perpendicular right vertex vertical 9) A stop sign has angles. 10) The lines of grout between tiles are . triangle. 11) On a clock with sweep hands, the time 12:05 creates a 12) The corner of a window or of a door creates a angle. 13) When you use a protractor to find out the size of an angle, or amount of turning or space between the two rays of an angle, measured in degrees or radians you calculating an measure. 14) A pair of angles that are opposite each other, formed by two lines that intersect are angles. 15) The corner of a piece of paper where the edges meet is the of an angle. Multiple Choice: Read each statement below and select the answer that fits best. Circle letter in front of you best choice. 16) Either of the two rays that make up an angle are the ____________ of an angle. a) vertex b) degree c) side 17) Another unit for measuring angles, often used to measure the speed at which wheels and machine parts revolve is the ____________. a) radian b) vertex c) degree — 146 — o 18) Two angles whose measures add up to 180o are called _______________. a) obtuse angles b) complementary angles c) adjacent angles 19) ___________ angles are pair of angles that are opposite each other, formed by two lines that intersect. a) vertical b) complementary c) supplementary 20) A unit of angle measure is known as a . Illustrations of key vocabulary: 21) Draw the triangles below each of the key vocabulary words they represent. acute angle obtuse angle vertical angle adjacent angle — 147 — — 148 — Geometry: Unit 2-Angles Name: ___________________ Date: ___________________ Match the symbol on the right with the word it represent on the left. Place the letter of the correct symbol in front of the word it matches. a. illustration of complementary angle 1) g angles 2) h degree 3) f acute angle 4) b obtuse angle e. illustration of perpendicular lines 5) e perpendicular f. illustration of acute angle 6) d linear pair g. Illustration of angle 7) c adjacent angle 8) a complementary angle b. illustration of obtuse angle c. illustration of adjacent angle d. illustration of linear pair h. illustration of degree Word Bank acute angle complementary degree obtuse perpendicular right vertex vertical 9) A stop sign has obtuse angles. 10) The lines of grout between tiles are perpendicular . 11) On a clock with sweep hands, the time 12:05 creates a acute triangle. 12) The corner of a window or of a door creates a right angle. — 149 — 13) When you use a protractor to find out the size of an angle, or amount of turning or space between the two rays of an angle, measured in degrees or radians you calculating an angle measure. 14) A pair of angles that are opposite each other, formed by two lines that intersect are vertical angles. 15) The corner of a piece of paper where the edges meet is the vertex of an angle. Multiple Choice: Read each statement below and select the answer that fits best. Circle letter in front of you best choice. 16) Either of the two rays that make up an angle are the ____________ of an angle. a) vertex b) degree c) side 17) Another unit for measuring angles, often used to measure the speed at which wheels and machine parts revolve is the ____________. a) radian b) vertex c) degree 18) Two angles whose measures add up to 180o are called _______________. a) obtuse angles b) complementary angles c) adjacent angles 19) ___________ angles are pair of angles that are opposite each other, formed by two lines that intersect. a) vertical b) complementary c) supplementary — 150 — 20) A unit of angle measure is known as a degree . Illustrations of key vocabulary: 21) Draw the triangles below each of the key vocabulary words they represent. acute angle obtuse angle vertical angle adjacent angle Illustrations of acute angle obtuse angle vertical angle adjacent angle — 151 — — 152 — UNIT 3 Proofs Sealaska Heritage Institute Grade Level Expectations for Unit 3 Unit 3—Proofs Alaska State Mathematics Standard C $VWXGHQWVKRXOGXQGHUVWDQGDQGEHDEOHWRIRUPDQGXVHDSSURSULDWHPHWKRGVWRGH¿QH DQGH[SODLQPDWKHPDWLFDOUHODWLRQVKLSV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG &H[SUHVVDQGUHSUHVHQWPDWKHPDWLFDOLGHDVXVLQJRUDODQGZULWWHQSUHVHQWDWLRQV SK\VLFDOPDWHULDOVSLFWXUHVJUDSKVFKDUWVDQGDOJHEUDLFH[SUHVVLRQV &UHODWHPDWKHPDWLFDOWHUPVWRHYHU\GD\ODQJXDJH GLEs The student communicates his or her mathematical thinking by >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\WUDQVODWLQJDPRQJWKHVHDOWHUQDWLYHUHSUHVHQWDWLRQVRUXVLQJDSSURSULDWHYRFDEXODU\ V\PEROVRUWHFKQRORJ\WRH[SODLQMXVWLI\ and defend strategies and solutions >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\FRPPXQLFDWLQJPDWKLGHDVLQZULWLQJRUXVLQJDSSURSULDWHYRFDEXODU\V\PEROVRU WHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQGVROXWLRQV The student demonstrates an ability to use logic and reason by >@36IROORZLQJDQGHYDOXDWLQJDQDUJXPHQWMXGJLQJLWVYDOLGLW\XVLQJLQGXFWLYHRUGHductive reasoning and logic; or making and testing conjectures >@36XVLQJPHWKRGVRISURRILQFOXGLQJGLUHFWLQGLUHFWDQGFRXQWHUH[DPSOHVWRYDOLdate conjectures Vocabulary & Definitions Introduction of Math Vocabulary conditional statement $FRQGLWLRQDOVWDWHPHQWLVWKHFRPELQDWLRQRIWZRVWDWHPHQWVLQDQ ³LIWKHQ´VWUXFWXUH)RUH[DPSOH³,ILWLVUDLQLQJWKHQWKH¿HOGLVZHW´ $VNVWXGHQWVWRKHOS\RXWKLQNRIPRUHH[DPSOHVRI³LIWKHQ´VWDWHments. hypothesis 7KHK\SRWKHVLVLVWKH¿UVWSDUWRUWKH³LI´SDUWRIDQ ³LIWKHQ´VWDWHPHQW8VHWKHSLFWXUHWRKHOSVWXGHQWV ¿QGWKHK\SRWKHVLVRIWKHSUHYLRXVVWDWHPHQW³,ILWLV UDLQLQJ«´LVWKHK\SRWKHVLVLQWKHFRQGLWLRQDOVWDWHPHQW³,ILWLVUDLQLQJWKHQWKH¿HOGLVZHW´ conclusion &RQFOXVLRQPHDQVWKHODVWSDUWRIDQ\WKLQJWKHWHUPLQDWLRQRUWKH end. It also means a reasoned deduction or inference. In geometry SURRIVDFRQFOXVLRQLVWKHSDUWRIDQ³LIWKHQ´FRQGLWLRQDOVWDWHPHQWWKDWFRPHVDIWHU³WKHQ´$VNVWXGHQWVLIWKH\NQRZWKHFRQFOXVLRQIRUWKHFRQGLWLRQDOVWDWHPHQW³,ILWLVUDLQLQJWKHQWKH¿HOGLV ZHW´DQGVKRZWKHSLFWXUH converse 7KHFRQYHUVHRIDQ³LI±WKHQ´FRQGLWLRQDOVWDWHPHQWLV WKHVWDWHPHQWWKDWUHVXOWVZKHQWKHWZRSDUWVK\SRWKHVLVDQGFRQFOXVLRQDUHVZLWFKHG7KHFRQYHUVHRI ³,ILWLVUDLQLQJWKHQWKH¿HOGLVZHW´LV³,IWKH¿HOGLV ZHWWKHQLWLVUDLQLQJ´ Show students the picture of the switch to help them remember that in the conYHUVHRIDVWDWHPHQWWKHSDUWVDUHVZLWFKHG — 159 — Introduction of Math Vocabulary negation To negate something means to rule it out or deny it. In JHRPHWU\SURRIVDQHJDWLRQLVDGHQLDORUDQRSSRVLWHVWDWHPHQWRIWHQFUHDWHGE\LQVHUWLQJWKHZRUG³QRW´)RUH[DPSOH ³,WLVQRWUDLQLQJ´LVWKHQHJDWLRQRI³,WLVUDLQLQJ´ inverse ,QYHUVHPHDQVUHYHUVHGLQSRVLWLRQRUGHURUGLUHFWLRQ,QJHRPHWU\ SURRIVWKHLQYHUVHRIDFRQGLWLRQDO³LIWKHQ´VWDWHPHQWLVWKHVWDWHPHQWWKDWUHVXOWVZKHQWKH³LI´SDUWK\SRWKHVLVDQGWKH³WKHQ´SDUWDUH ERWKQHJDWHG)RUH[DPSOHWKHLQYHUVHRI³,ILWLVUDLQLQJWKHQWKH¿HOG LVZHW´ZRXOGEH³,ILWLVQRWUDLQLQJWKHQWKH¿HOGLVQRWZHW´ ,QWKHSLFWXUHWKHVQRZERDUGLVWXUQHGWRWKHRSSRVLWHVLGHWRUHSUHsent the idea that you would write opposite statements when you negate both the hypothesis and the conclusion. contrapositive $FRQWUDSRVLWLYHLVDVWDWHPHQWWKDWUHVXOWVZKHQWKHWZR SDUWVK\SRWKHVLVDQGFRQFOXVLRQRIDQ³LIWKHQ´FRQGLWLRQDO VWDWHPHQWDUHVZLWFKHGDQGERWKDUHQHJDWHG)RUH[DPSOH LIDFRQGLWLRQDOVWDWHPHQWVD\V³,ILWLVUDLQLQJWKHQWKH¿HOGLV ZHW´LWVFRQWUDSRVLWLYHZRXOGEH³,IWKH¿HOGLVQRWZHWWKHQLW LVQRWUDLQLQJ´ 7KHSLFWXUHVKRZVVRPHWKLQJWKDWLVERWKXSVLGHGRZQDQGEDFNZDUGVWRUHSresent a contrapositive; because in a contrapositive you both switch the order of statements and create their opposites. — 160 — Introduction of Math Vocabulary conjecture $FRQMHFWXUHLVDVWDWHPHQWRURSLQLRQWKDWLVQRWEDVHGRQHYLGHQFH,WLVDQHGXFDWHGJXHVV6KDUHVRPHH[DPSOHVRIFRQjectures: ³,WZLOOUDLQQH[WZHHN´ ³+DOLEXWWDVWHVEHWWHUWKDQVDOPRQWRPRVWSHRSOH´ ³7RXULVWVDUHJRRGIRURXUWRZQ´ counterexample $FRXQWHUH[DPSOHLVDQH[DPSOHWKDWSURYHVDVWDWHPHQWWREH IDOVH)RULQVWDQFHWKHH[DPSOH³7KHUHLVDFHGDUWUHHLQP\ \DUG´SURYHVWKDWWKHVWDWHPHQW³$OOWUHHVLQ6RXWKHDVW$ODVNDDUH KHPORFNV´FDQQRWEHWUXH,WLVDFRXQWHUH[DPSOH$VVXPLQJRI FRXUVHWKDWWKH\DUGLVLQ6RXWKHDVW$ODVND 7KHSLFWXUHVKRZVDFRXQWHUH[DPSOHRIWKHVWDWHPHQW³$OOVZDQVDUHZKLWH´ 2UPD\EH³$OOVZDQVDUHEODFN´ — 161 — Introduction of Math Vocabulary deductive reasoning Deductive reasoning is a form of thinking that moves from the general to the SDUWLFXODU,QGHGXFWLYHDUJXPHQWVLIWKH¿UVWVWDWHPHQWWKHSUHPLVHRUK\SRWKHVLVLVWUXHWKHFRQFOXVLRQ0867EHWUXH$UJXPHQWVEDVHGRQODZVUXOHVRU RWKHUZLGHO\DFFHSWHGSULQFLSOHVDUHEHVWH[SUHVVHGGHGXFWLYHO\+HUHDUHDIHZ H[DPSOHVRIGHGXFWLYHUHDVRQLQJ 1HZWRQ¶V/DZVD\VWKDWHYHU\WKLQJWKDWJRHVXSPXVWFRPHGRZQDJHQHUDOVWDWHPHQWVRLI,WKURZWKLVEDOOLQWRWKHDLULWZLOOFRPHGRZQDSDUWLFXODUVLWXDWLRQ $ODVNDQUHVLGHQWVDUHUHVLGHQWVRIWKH8QLWHG6WDWHVDJHQHUDOUXOHRU GH¿QLWLRQVRP\PRPLVDUHVLGHQWRIWKH8QLWHG6WDWHVDSDUWLFXODU H[DPSOH $OOEHDUVDUHPDPPDOVDUXOH$OOPDPPDOVKDYHOXQJVDUXOH6R EHDUVPXVWKDYHOXQJVDVSHFL¿FH[DPSOH The picture of the gavel represents the idea that deductive DUJXPHQWVVWDUWZLWKODZVUXOHVDQGDFFHSWHGSULQFLSOHV inductive reasoning Inductive reasoning is the opposite of deductive reasoning. It is a form of thinkLQJWKDWPRYHVIURPWKHVSHFL¿FWRWKHJHQHUDO*HQHUDOL]DWLRQVDUHPDGH EDVHGRQLQGLYLGXDOLQVWDQFHV,QLQGXFWLYHUHDVRQLQJVWDWHPHQWVDUHPDGH that support a conclusion but that do not automatically insure that it will be true. ,IWKHVWDWHPHQWVDUHWUXHWKHQWKHFRQFOXVLRQLV3266,%/<WUXH$UJXPHQWV EDVHGRQH[SHULHQFHRUREVHUYDWLRQVDUHEHVWH[SUHVVHGLQGXFWLYHO\+HUHDUHD IHZH[DPSOHVRILQGXFWLYHUHDVRQLQJ 7KHODVWWKUHHWLPHVWKDW,WKUHZDEDOOLQWRWKHDLULWFDPHGRZQDJDLQ 6RLI,WKURZDEDOOLQWKHDLUQRZLWZLOOFRPHGRZQDJDLQ — 162 — Introduction of Math Vocabulary 0\PRPLVDUHVLGHQWRIWKH8QLWHG6WDWHVDQGVRDP,VRDOO$ODVNDQ residents are residents of the United States. (YHU\EHDUZHKDYHVHHQLVEODFNVRWKHQH[WEHDUZHVHHZLOOEHEODFN 7KLVSLFWXUHRIDSHUVRQREVHUYLQJEHDUVH[SUHVVHVWKHLGHDWKDW LQGXFWLYHUHDVRQLQJEHJLQVZLWKREVHUYDWLRQVDQGH[SHULHQFHV proof $SURRILQJHRPHWU\LVDGHPRQVWUDWLRQRIWKHWUXWKRID mathematical or logical statement based on postulates DQGWKHRUHPV,QDSURRIDVHTXHQFHRIVWHSVVWDWHPHQWVRUGHPRQVWUDWLRQVWKDWOHDGVWRDYDOLGFRQFOXsion. postulate $SRVWXODWHLVDVWDWHPHQWDFFHSWHGDVWUXHZLWKRXWSURRI $SRVWXODWHVKRXOGEHVRVLPSOHDQGGLUHFWWKDWLWVHHPVWR EHXQTXHVWLRQDEO\WUXH([DPSOHVRISRVWXODWHVXVHGLQ geometry are: ³7KURXJKDQ\WZRSRLQWVWKHUHLVH[DFWO\RQHOLQH´2U³$ FLUFOHFDQEHGUDZQZLWKDFHQWHUDQGDQ\UDGLXV´ 6KRZWKHSLFWXUHDQGWHOOVWXGHQWV³<RXPLJKWQRWUHDOO\TXHVWLRQWKLVPDQKHLV H[SUHVVLQJKLPVHOILQDYHU\VLPSOHDQGGLUHFWZD\´ — 163 — Introduction of Math Vocabulary theorem Theorem is a mathematical statement that can be proved true XVLQJWKHUXOHVRIORJLF$WKHRUHPLVSURYHQIURPSURSHUWLHV SRVWXODWHVRURWKHUWKHRUHPVDOUHDG\NQRZQWREHWUXH 7KLVSLFWXUHRIDVLJQUHSUHVHQWV³WKHRUHP´EHFDXVHWKHRUHPV are statements that have been proven according to rules. corollary $FRUROODU\LVDQDWXUDOFRQVHTXHQFHRUHDVLO\GUDZQFRQFOXVLRQ ,QJHRPHWU\LWLVDWKHRUHPRUSURSRVLWLRQWKDWIROORZVZLWKOLWWOH RUQRSURRIUHTXLUHGIURPRQHDOUHDG\SURYHQ$VSHFLDOFDVHRID more general theorem which is worth noting separately. The picture represents a corollary: If a person eats a lot of fat-ladHQIRRGDQDWXUDOFRQVHTXHQFHZRXOGEHWKDWWKH\JDLQZHLJKW indirect proof $QLQGLUHFWSURRILVSURRIE\FRQWUDGLFWLRQ$VWDWHPHQWFRQMHFWXUHLVSURYHGE\ assuming that the conjecture is false. If this assumption leads to a FRQWUDGLFWLRQVRPHWKLQJDEVXUGLWVKRZVWKDWWKHDVVXPHGIDOVH conjecture is impossible. Therefore the original statement must have EHHQWUXH,QDQLQGLUHFWSURRIDOORIWKHDOWHUQDWLYHVDUHUXOHGRXW )RUH[DPSOHLPDJLQHWKDW-RKQ-RH6XHDQG7HUU\DUHWKHRQO\ people on an isolated island and Joe has been shot with an arrow. 6XHPDNHVDFRQMHFWXUHWKDW7HUU\ZDVWKHVKRRWHU7RSURYHLW¿UVWVKH assumes that the conjecture is false: Terry was not the shooter. Joe could not KDYHVKRWKLPVHOIDQG6XHNQHZWKDWVKHGLGQRWVKRRWKLP6RLWKDGWREH John or Terry. John has a broken arm and can’t shoot a bow. That contradicts the idea that he was the shooter. The assumed false conjecture that Terry was QRWWKHVKRRWHULVQRZLPSRVVLEOH6R7HUU\PXVWKDYHEHHQWKHVKRRWHU 7KHSLFWXUHVKRZVD\RXQJPDQZKRLVWXUQHGDURXQGDQGEDFNZDUGVVRLW might represent an indirect method of proof; a method that starts with an opposite conjecture. — 164 — Introduction of Math Vocabulary verify 7RYHULI\VRPHWKLQJPHDQVWRWHVWDQGFRQ¿UPLWVWUXWK,QPDWKLWLVWKHSURFHVVRIFRQ¿UPLQJWKDWDVROXWLRQLVFRUUHFWE\PDNLQJVXUHLWVDWLV¿HVDQ\DQG DOORIWKHFRQGLWLRQVHTXDWLRQVDQGRULQHTXDOLWLHVLQDSUREOHP summarize 7RJLYHDFRQFLVHFRPSUHKHQVLYHVWDWHPHQWRUDQ DEULGJHGDFFRXQWRIVRPHWKLQJ$VXPPDU\FRQtains the important facts and main points without all of the details. $VXPPDU\LVVKRUWHUDQG³VPDOOHU´OLNHWKHOLWWOH dog. — 165 — — 166 — Language and Skills Development Using the Math Vocabulary Terms Language & Skills Development LISTENING Use the activity pages from the Student Support Materials. SPEAKING READING Use the activity pages from the Student Support Materials. WRITING Use the activity pages from the Student Support Materials. The Hidden Words Say a vocabulary word for the students. Tell the students to listen for that vocabulary word as you say a running story. Provide each student with writing paper and a pen. When the students hear the vocabulary word in the runQLQJVWRU\WKH\PXVWPDNHDFKHFNPDUNRQWKHLUSDSHUVHDFKWLPHWKHZRUG RFFXUV'HSHQGLQJXSRQWKHUHDGLQHVVRI\RXUVWXGHQWV\RXPD\ZLVKWRKDYH WKHPOLVWHQIRUWZRRUWKUHHZRUGV,QWKLVFDVHKDYHWKHVWXGHQWVPDNHD FKHFNPDUNIRURQHZRUGDQGD³;´DQGDQ³2´IRUWKHRWKHUZRUGV What’s the Date? %HIRUHWKHDFWLYLW\EHJLQVFROOHFWDQROGFDOHQGDURUFDOHQGDUVRIGLIIHUHQW\HDUV Say the name of a month to a student. The student should then say a date within that month. Look on the calendar to see which day the date represents. If the GDWHUHSUHVHQWVDGD\EHWZHHQ0RQGD\DQG)ULGD\WKHVWXGHQWVVKRXOGLGHQtify a vocabulary illustration you show or he/she should repeat a sentence you VDLGDWWKHEHJLQQLQJRIWKHURXQG+RZHYHULIWKHGDWHQDPHGE\WKHVWXGHQW LVD6DWXUGD\RU6XQGD\WKHVWXGHQWPD\³SDVV´WRDQRWKHUSOD\HU5HSHDWXQWLO many students have responded. The Lost Syllable 6D\DV\OODEOHIURPRQHRIWKHVLJKWZRUGV&DOOXSRQWKHVWXGHQWVWRLGHQWLI\WKH VLJKWZRUGRUZRUGVWKDWFRQWDLQWKDWV\OODEOH'HSHQGLQJXSRQWKHV\OODEOH\RX VD\PRUHWKDQRQHVLJKWZRUGPD\EHWKHFRUUHFWDQVZHU7KLVDFWLYLW\PD\DOVR EHGRQHLQWHDPIRUP,QWKLVFDVHOD\WKHVLJKWZRUGFDUGVRQWKHÀRRU*URXS the students into two teams. Say a syllable from one of the sight words. When \RXVD\³*R´WKH¿UVWSOD\HULQHDFKWHDPPXVWUXVKWRWKHVLJKWZRUGFDUGVDQG ¿QGWKHVLJKWZRUGWKDWFRQWDLQVWKHV\OODEOH\RXVDLG The Other Half &XWHDFKRIWKHVLJKWZRUGVLQKDOI*LYHHDFKVWXGHQWDVKHHWRIZULWLQJSDSHUD pen and one of the word-halves. Each student should glue the word-half on his/ her writing paper and then complete the spelling of the word. You may wish to have enough word-halves prepared so that each student completes more than RQHZRUG$IWHUZDUGVUHYLHZWKHVWXGHQWV¶UHVSRQVHV — 169 — Student Support Materials — 173 — — 175 — — 177 — — 179 — — 181 — — 183 — — 185 — — 187 — — 189 — — 191 — — 193 — — 195 — — 197 — — 199 — — 201 — — 203 — — 205 — — 207 — — 208 — True-False Sentences (Listening and/or Reading Comprehension) 1. ³,IWRPRUURZLV0RQGD\´FRXOGEHDQH[DPSOHRIDconclusion. 2. $conditional statementDOZD\VVWDUWVZLWK³LI´ ³,WZLOOWDNHPLQXWHVWRZDONWRVFKRRO´LVDQH[DPSOHRIDconjecture. $contrapositiveLVFUHDWHGE\QHJDWLQJWKH³LI´DQG³WKHQ´SDUWVRIDVWDWHPHQWDQGVZLWFKLQJ their order. To create the converseRIDVWDWHPHQW\RXQHJDWHERWKSDUWV $ÀRDWLQJFDQRHLVDcounterexampleRIWKHVWDWHPHQW³DOOERDWVÀRDW´ 7. $QH[DPSOHRIdeductive reasoningPLJKWEH³,WUDLQHGIRUWKHODVWGD\VVRLWZLOOUDLQ WRPRUURZ´ 8. Inductive reasoning LQYROYHVGUDZLQJFRQFOXVLRQVIURPREVHUYDWLRQVDQGH[SHULHQFHV 9. To write the inverseRI³,ILWVQRZVLWLVFROG´\RXZRXOGVD\³,ILWVQRZVLWLVQRWFROG´ 10. To negateVRPHWKLQJ\RXKDYHWRSURYHWKDWLWLVIDOVH 11. $proof is a demonstration of the truth of a statement. 12. $VWDWHPHQWWKDWLVDFFHSWHGDVWUXHZLWKRXWSURRILVDpostulate. $theorem is a statement that has been proven according to rules of logic. $corollary to a theorem is a special case of that theorem. No assumptions are made as part of an indirect proof. It is possible to verify every postulate in geometry. 17. When something is summarizedLWLVGHVFULEHGLQDFRQFLVHZD\ ,QVFLHQFHWKHZRUGhypothesisKDVH[DFWO\WKHVDPHPHDQLQJWKDWLWGRHVLQJHRPHWU\ $QVZHUV)777)777))7777))7) 1. 2. $conclusion always comes at the end of a conditional statement. $conditional statement is never true. $conjecture is usually incorrect. The contrapositiveRI³,I,JRWKHQ,ZLOOVWD\´ZRXOGEH³,I,VWD\WKHQ,GRQRWJR´ ³,ILWLVEOXHWKHQLWLVQRWUHG´LVWKHconverseRI³,ILWLVUHGWKHQLWLVQRWEOXH´ $counterexampleVKRZVWKDWVRPHWKLQJFDQQRWEHWUXHE\JLYLQJDQH[DPSOHRIVRPHWKLQJ WKDWGRHVQ¶W¿WWKHVWDWHPHQW 7. In deductive reasoning, DQDUJXPHQWVWDUWVZLWKDJHQHUDOUXOHRUODZDQGDSSOLHVWKDWWRD VSHFL¿FVLWXDWLRQ ,I\RXVDLG³7KHOHJDOVSHHGOLPLWLVPSKVRGULYLQJDWPSKLVOHJDO´\RXZRXOGEHXVLQJ inductive reasoning. 9. The inverse of a conditional statement is formed by negating both parts of the statement. 10. ³,DPQRWKDSS\´LVWKHnegationRI³,DPKDSS\´ 11. In a proof\RXFDQ¶WXVHWKHRUHPVWKDWKDYHDOUHDG\EHHQSURYHQ 12. $QH[DPSOHRIDpostulatePLJKWEH³$OOJHRPHWU\VWXGHQWVDUHVPDUWDQGJRRGORRNLQJ´ Every theorem automatically has a corollary. $FRQMHFWXUHDQGDtheorem are the same thing. In an indirect proofVWDWHPHQWVDUHSURYHGE\VKRZLQJWKDWWKHRSSRVLWHRIWKHVWDWHPHQW creates a contradiction. When you verifyVRPHWKLQJ\RXWHVWDQGFKHFNLWWRFRQ¿UPWKDWLWLVDFFXUDWH 17. 7RVXPPDUL]HVRPHWKLQJ\RXQHHGWRDGGH[WUDLQIRUPDWLRQ $hypothesisLVWKH¿UVWSDUWRIDQ³LIWKHQ´VWDWHPHQW $QVZHUV7)))777)77))))77 — 209 — Match the Halves $FRQFOXVLRQLV $:ULWLQJDFRQFLVHDEULGJHGDFFRXQW $FRQGLWLRQDOVWDWHPHQW $QHGXFDWHGJXHVVRUDQRSLQLRQPLJKW be called %$WKHRUHP &6KRZLQJWKDWWKHUHLVDFRQWUDGLFWLRQZKHQ you assume that it is false. $K\SRWKHVLVLV ',QVHUWLQJWKHZRUG³QRW´ +HVXPPDUL]HGWKHSURFHVVE\ E. The last part of a conditional statement :KHQ&DOYLQFKHFNHGDQGFRQ¿UPHGWKDW his proof was correct he was )$SRVWXODWH 7. Mildred demonstrated the truth of the statement by G. The inverse of the statement is created. $VLPSOHGLUHFWVWDWHPHQWDVVXPHGWREH true is H. Writing a proof $VWDWHPHQWWKDWKDVEHHQSURYHQXVLQJ VSHFL¿FUXOHVLV I. The contrapositive of the statement is created. $QLQGLUHFWSURRIVKRZVWKDWVRPHWKLQJLV -,VDVWDWHPHQWWKDWIROORZVDQ³LIWKHQ´ true by format. $QDWXUDOFRQVHTXHQFHRUVSHFLDOFDVHLV .,VDQH[DPSOHRIGHGXFWLYHUHDVRQLQJ $VWDWHPHQWPLJKWEHQHJDWHGE\ L. Verifying his work :KHQWKHSDUWVRIDFRQGLWLRQDOVWDWHPHQW M. Shows that a statement is false. are switched :KHQWKHSDUWVRIDFRQGLWLRQDOVWDWHPHQW 1,VDQH[DPSOHRILQGXFWLYHUHDVRQLQJ are both negated :KHQWKHSDUWVRIDFRQGLWLRQDOVWDWHPHQW 2$FRQMHFWXUH are switched and both are negated $QDUJXPHQWWKDWVWDUWVZLWKUXOHVRUODZV 3$FRUROODU\ DQGDSSOLHVWKHPWRDVSHFL¿FVLWXDWLRQ $QDUJXPHQWWKDWJHQHUDOL]HVIURP REVHUYDWLRQVDQGH[SHULHQFHV Q. The converse of the statement is created. $FRXQWHUH[DPSOH 57KH¿UVWSDUWRIDFRQGLWLRQDOVWDWHPHQW $QVZHUV(-25$/+)%&3'4*,.10 — 210 — 'H¿QLWLRQV conditional statement: WKHFRPELQDWLRQRIWZRVWDWHPHQWVLQDQ³LIWKHQ´VWUXFWXUH hypothesis:WKH¿UVWSDUWRUWKH³LI´SDUWRIDQ³LIWKHQ´VWDWHPHQW conclusion:WKHSDUWRIDQ³LIWKHQ´FRQGLWLRQDOVWDWHPHQWWKDWFRPHVDIWHU³WKHQ´ converse:WKHVWDWHPHQWWKDWUHVXOWVZKHQWKHWZRSDUWVK\SRWKHVLVDQGFRQFOXVLRQRI DQ³LIWKHQ´VWDWHPHQWDUHVZLWFKHG negation:DGHQLDORIWHQFUHDWHGE\LQVHUWLQJRUUHPRYLQJWKHZRUG³QRW´ inverse:WKHVWDWHPHQWWKDWUHVXOWVZKHQWKH³LI´SDUWK\SRWKHVLVDQGWKH³WKHQ´SDUW of a conditional statement are both negated. contrapositive:DVWDWHPHQWWKDWUHVXOWVZKHQWKHWZRSDUWVK\SRWKHVLVDQGFRQFOXVLRQRIDQ³LIWKHQ´FRQGLWLRQDOVWDWHPHQWDUHVZLWFKHGDQGERWKDUHQHJDWHG conjecture: a statement or opinion that is not based on evidence; an educated guess. counterexample:DQH[DPSOHWKDWSURYHVDVWDWHPHQWWREHIDOVH deductive reasoning: a form of thinking that starts with general rules and applies WKHPWRSDUWLFXODURUVSHFL¿FVLWXDWLRQV inductive reasoning:DIRUPRIWKLQNLQJWKDWXVHVVSHFL¿FREVHUYDWLRQVDQGH[SHULences to reach general conclusions. proof: a demonstration of the truth of a mathematical or logical statement based on SRVWXODWHVDQGWKHRUHPV,QDSURRIDVHTXHQFHRIVWHSVVWDWHPHQWVRUGHPRQVWUDtions that leads to a valid conclusion. postulate: a statement accepted as true without proof. theorem: DPDWKHPDWLFDOVWDWHPHQWWKDWFDQEHSURYHGWUXHXVLQJWKHUXOHVRIORJLF$ WKHRUHPLVSURYHQIURPSURSHUWLHVSRVWXODWHVDQGRURWKHUWKHRUHPVDOUHDG\NQRZQWR be true. corollary: DQDWXUDOFRQVHTXHQFHRUHDVLO\GUDZQFRQFOXVLRQ,QJHRPHWU\LWLVDWKHRrem or proposition that follows with little or no proof required from one already proven. $VSHFLDOFDVHRIDPRUHJHQHUDOWKHRUHPZKLFKLVZRUWKQRWLQJVHSDUDWHO\ indirect proof: SURRIE\FRQWUDGLFWLRQLQZKLFKDVWDWHPHQWFRQMHFWXUHLVSURYHGE\ ¿UVWDVVXPLQJWKDWWKHFRQMHFWXUHLVIDOVH verify: WRWHVWDQGFRQ¿UPWKHWUXWKRIVRPHWKLQJ summarize:WRJLYHDFRQFLVHFRPSUHKHQVLYHVWDWHPHQWRUDQDEULGJHGDFFRXQWRI VRPHWKLQJ$VXPPDU\FRQWDLQVWKHLPSRUWDQWIDFWVDQGPDLQSRLQWVZLWKRXWDOORIWKH details. — 211 — Which Belongs 1. ³,I,GRQ¶WKDYHP\VRFNVRQWKHQP\IHHWDUHQRWZDUP´ZRXOGEHWKHFRQWUDSRVLWLYHFRQYHUVHLQYHUVH RI³,I,KDYHP\VRFNVRQWKHQP\IHHWDUHZDUP´ 2. 7KHK\SRWKHVLVFRQFOXVLRQFRUROODU\LVWKHODVWSDUWRIDFRQGLWLRQDOVWDWHPHQW 5DFKHOSURYHGKHUVWDWHPHQWDERXWDQJOHVE\XVLQJSRVWXODWHVDQGRWKHUSURYHQVWDWHPHQWVVRVKH FRXOGVD\WKDWKHUVWDWHPHQWZDVDFRQMHFWXUHWKHRUHPFRQGLWLRQDOVWDWHPHQW 7KHVWDWHPHQW³7KHGDQFHZLOOEHFURZGHG´FDQQRWEHDFRQMHFWXUHWUXHVWDWHPHQWFRQGLWLRQDOVWDWHPHQW ³,ILWLVQRWKRWWKHQLWLVQRWFRRNHG´LVWKHLQYHUVHFRQYHUVHFRQWUDSRVLWLYHRI³,ILWLVFRRNHGWKHQLWLV KRW´ $³EOXHEHDU´LVDQHJDWLRQFRXQWHUH[DPSOHSURRIRIWKHVWDWHPHQW³$OOEODFNEHDUVDUHEODFNLQFRORU´ 7. 7RVXPPDUL]HYHULI\QHJDWHWKHVWHSVWKDWVKHXVHG%HWKZURWHWKHPLQDVKRUWHUPRUHFRQFLVH manner. ³$OLQHKDVDQLQ¿QLWHQXPEHURISRLQWV´ZRXOGEHDQH[DPSOHRIDFRQMHFWXUHSRVWXODWHWKHRUHP 9. 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B $ Complete the Sentence 1. ,QDQBBBBBBBBBBBBBBBBBBBBBBBBBDVWDWHPHQWRUFRQMHFWXUHLVSURYHQE\FRQWUDGLFWLRQ 2. ³,I,JHWDQHZSDLURIVKRHVWKHQ,ZLOOUXQIDVWHU´LVDQH[DPSOHRIDBBBBBBBBBBBBBBBBBB ³,I,DPZRUNLQJWKHQ,DPTXLHW´LVWKHBBBBBBBBBBBBBBBBBBBRIWKHVWDWHPHQW³,I,DPTXLHWWKHQ,DP ZRUNLQJ´ :KHQDVWDWHPHQWEHJLQVZLWKDQBBBBBBBBBBBBBBBBIROORZHGE\³WKHQ´LWLVQRWDOZD\VWUXH BBBBBBBBBBBBBBBBBBBBBBBBBLVXVHGZKHQ\RXNQRZRIVRPHJHQHUDOUXOHVRUODZVWKDWDSSO\WRDVSHFL¿F situation. 7KHBBBBBBBBBBBBBBBBBBBBLVWKH¿QDOSDUWRIDFRQGLWLRQDOVWDWHPHQW 7. ,I\RXPDNHDJHQHUDOL]DWLRQEDVHGRQ\RXUREVHUYDWLRQVDQGH[SHULHQFHV\RXDUHXVLQJ ____________________________. $VSHFLDOFDVHRUQDWXUDOFRQVHTXHQFHRIVRPHWKLQJLVLWVBBBBBBBBBBBBBBB 9. :KHQDVWDWHPHQWVD\V³,ILWLVGDUNWKHQLWLVQLJKW´LWVBBBBBBBBBBBBBBBBZRXOGVD\³,ILWLVQRWGDUNWKHQ LWLVQRWQLJKW´ 10. 7RZULWHWKHBBBBBBBBBBBBBBBBBBBBBBBRIDVWDWHPHQW\RXGHQ\LWRIWHQE\DGGLQJWKHZRUG³QRW´ 11. :KHQWKHRUGHURIDFRQGLWLRQDOVWDWHPHQWLVVZLWFKHGDQGERWKSDUWVDUHQHJDWHG\RXKDYHD ___________________________. 12. $GHPRQVWUDWLRQWKDWVRPHWKLQJLVWUXHLVDBBBBBBBBBBBBBBBBBB :KHQ\RX¿QGDVLWXDWLRQRUDWKLQJWKDWGLVSURYHVVRPHWKLQJ\RXKDYHIRXQGDQBBBBBBBBBBBBBBBBBBBB __________. $QBBBBBBBBBBBBBBBBBBBBBBLVDFFHSWHGWREHWUXHHYHQWKRXJKLWFDQQRWEHSURYHQ Suzanne _________________ her report because she knew that people would not have time to read the whole thing. $VWDWHPHQWWKDWKDVEHHQSURYHQWKURXJKDVHTXHQFHRIVWHSVXVLQJRWKHUWUXHVWDWHPHQWVLVD ___________________. 17. -DYLHUZDQWHGWRGRZHOORQWKHWHVWVRKHWRRNWKHWLPHWRBBBBBBBBBBBBBBBDOORIKLVDQVZHUV )UDQNPDGHDQBBBBBBBBBBBBBBBZKHQKHVDLGWKDWWKHVWRUHZRXOGEHRSHQ $QVZHUV 1. Indirect Proof 2. &RQGLWLRQDO6WDWHPHQW &RQYHUVH Hypothesis Deductive &RQFOXVLRQ 7. Inductive &RUROODU\ 9. Inverse 10. Negation 11. &RQWUDSRVLWLYH 12. Proof &RXQWHUH[DPSOH Postulate Summarized Theorem 17. Verify — 215 — Creative Writing Write about this picture of a moose stuck in telephone wires using the words from this unit. You might try VRPHFRQMHFWXUHVFRQGLWLRQDOVWDWHPHQWVDQGH[DPSOHVRIGHGXFWLYHDQGLQGXFWLYHUHDVRQLQJIRUVWDUWHUV — 216 — Place-Based Practice Activity $VNVWXGHQWVWRZULWHFRQGLWLRQDOVWDWHPHQWVWKDWUHODWHWRWKHLUGDLO\OLYHV DQGWKHHQYLURQPHQWDURXQGWKHP,QVPDOOJURXSVKDYHWKHVWXGHQWVH[FKDQJH VWDWHPHQWVDQGSUDFWLFHFRPLQJXSZLWKWKHFRQYHUVHLQYHUVHDQGFRQWUDSRVLtive of each statement. 'LVFXVVWKHZD\VWKDWFRQMHFWXUHVLQGXFWLYHUHDVRQLQJGHGXFWLYHUHDVRQLQJ and indirect proof might apply in students’ daily lives. Have students keep a log RUUHFRUGRIH[DPSOHVWKDWFRPHXSGXULQJWKHQH[WZHHN7KH\PD\EHDEOHWR ¿QGH[DPSOHVLQFRQYHUVDWLRQZLWKIULHQGVRUIDPLO\PHPEHUVLQERRNVLQWHOHYLVLRQVKRZVRUFRPPHUFLDOVRULQFODVVHV2IIHULQFHQWLYHVIRU¿QGLQJH[DPSOHV DQGSURYLGHFODVVWLPHIRUVWXGHQWVWRVKDUHWKHH[DPSOHVWKDWWKH\IRXQG 5HVHDUFKWKHZD\VWKDW1DWLYHSHRSOHLQ6RXWKHDVW$ODVNDGHYHORSHG knowledge about the world around them by using inductive reasoning. — 217 — — 218 — Unit Assessment — 219 9 — — 220 — Geometry: Unit 3 Proofs Name: ___________________ Date: ___________________ 1) "If it is raining, then the field is wet" is a/an _______________. a) hypothesis b) conditional statement c) conclusion 2) The first part statement, "If it is raining..." is the ________________ . a) hypothesis b) conditional statement c) conclusion 3) In the same statement, the part of the sentence that states "...the field is wet" is the _____________. a) hypothesis b) conditional statement c) conclusion Multiple Choice: Read the statement carefully and select the best choice for your answer. Circle your answer. The statements below are using the sentence, "If it is raining, the field is wet." As the sentence changes, choose the type of statement it would be in a geometry proof. 4) When the two parts of an IF-THEN statement are switched it is a/an ___________ statement. "If the field is wet then it is raining." a) converse b) inverse c) negation — 221 — 5) The ____________ of a IF-THEN statement is the statement that results when the IF part and the THEN part are both negated. If it is not raining, then the field is not wet. a) converse b) inverse c) negation 6) "It is not raining" is an example of a _________________. a) converse b) inverse c) negation Short Answer: For each item, fill in the blank with the word that fits best. Choose your words from the Word Bank. Word Bank conjecture contrapositive corollary counterexample hypothesis indirect summary theorem verify 7) A is a statement or opinion that is not based on evidence. It is an educated guess. "Halibut taste better than salmon to most people." 8) In a statement when two parts of the "If then" part of the sentence are switched and are BOTH negative, the statement is . 201CIf the field is not wet, then it is not raining.201D 9) An proof is proof by contradiction. This kind of proof is a statement that is proved by assuming that the conjecture, or guess is false. 10) When someone gives a abridged explanation. of something it is a concise, comprehensive statement or an 11) An easily drawn conclusion or natural consequence is a , and requires little or no proof from one already proved. For example, if person east a lot of fatty food, then a natural consequence would be that they gain weight. 12) To something means to test and confirm its truth. — 222 — 13) When a mathematical statement can be proved using the rules of logic, and has already been shown to be true, it is a . is one that proves a statement to be false. For instance, the statement about a yard 14) A in SE Alaska, 201CThere is a cedar tree in my yard201D proves that the following statement cannot be true---201CAll trees in Southeast Alaska are hemlocks.201D True or False: Read each statement below and decide if it is a true or a false statement. Circle the answer you think is the correct one. 15) The statement, All bears are mammals. All mammals have lungs. So bears must have lungs is an example of deductive reasoning. a) True b) False 16) A form of reasoning that moves from general to the particular is deductive reasoning. a) True b) False 17) Another example of deductive reasoning would be, Every bear we have seen is black, so the next bear we see will be black. a) True b) False 18) Inductive reasoning moves from the specific to the general. a) True b) False 19) An example of inductive reasoning is the following statement. My mom is a resident of the United States and so am I, so all Alaskan residents are residents of the United States. a) True b) False 20) With inductive reasoning, statements are made that support a conclusion but that does not automatically mean that the statements will be true. a) True b) False — 223 — — 224 — Geometry: Unit 3 Proofs Name: ___________________ Date: ___________________ 1) "If it is raining, then the field is wet" is a/an _______________. a) hypothesis b) conditional statement c) conclusion 2) The first part statement, "If it is raining..." is the ________________ . a) hypothesis b) conditional statement c) conclusion 3) In the same statement, the part of the sentence that states "...the field is wet" is the _____________. a) hypothesis b) conditional statement c) conclusion — 225 — Multiple Choice: Read the statement carefully and select the best choice for your answer. Circle your answer. The statements below are using the sentence, "If it is raining, the field is wet." As the sentence changes, choose the type of statement it would be in a geometry proof. 4) When the two parts of an IF-THEN statement are switched it is a/an ___________ statement. "If the field is wet then it is raining." a) converse b) inverse c) negation 5) The ____________ of a IF-THEN statement is the statement that results when the IF part and the THEN part are both negated. If it is not raining, then the field is not wet. a) converse b) inverse c) negation 6) "It is not raining" is an example of a _________________. a) converse b) inverse c) negation — 226 — Short Answer: For each item, fill in the blank with the word that fits best. Choose your words from the Word Bank. Word Bank conjecture contrapositive corollary counterexample hypothesis indirect summary theorem verify 7) A conjecture is a statement or opinion that is not based on evidence. It is an educated guess. "Halibut taste better than salmon to most people." 8) In a statement when two parts of the "If then" part of the sentence are switched and are BOTH negative, the statement is contrapositive . 201CIf the field is not wet, then it is not raining.201D 9) An indirect proof is proof by contradiction. This kind of proof is a statement that is proved by assuming that the conjecture, or guess is false. 10) When someone gives a summary of something it is a concise, comprehensive statement or an abridged explanation. 11) An easily drawn conclusion or natural consequence is a corollary , and requires little or no proof from one already proved. For example, if person east a lot of fatty food, then a natural consequence would be that they gain weight. 12) To verify something means to test and confirm its truth. 13) When a mathematical statement can be proved using the rules of logic, and has already been shown to be true, it is a theorem . — 227 — 14) A counterexample is one that proves a statement to be false. For instance, the statement about a yard in SE Alaska, 201CThere is a cedar tree in my yard201D proves that the following statement cannot be true---201CAll trees in Southeast Alaska are hemlocks.201D True or False: Read each statement below and decide if it is a true or a false statement. Circle the answer you think is the correct one. 15) The statement, All bears are mammals. All mammals have lungs. So bears must have lungs is an example of deductive reasoning. a) True b) False 16) A form of reasoning that moves from general to the particular is deductive reasoning. a) True b) False 17) Another example of deductive reasoning would be, Every bear we have seen is black, so the next bear we see will be black. a) True b) False 18) Inductive reasoning moves from the specific to the general. a) True b) False — 228 — 19) An example of inductive reasoning is the following statement. My mom is a resident of the United States and so am I, so all Alaskan residents are residents of the United States. a) True b) False 20) With inductive reasoning, statements are made that support a conclusion but that does not automatically mean that the statements will be true. a) True b) False — 229 — — 230 — UNIT 4 Lines & Transversals Sealaska Heritage Institute Grade Level Expectations for Unit 4 Unit 4—Lines and Transversals Alaska State Mathematics Standard A $VWXGHQWVKRXOGXQGHUVWDQGPDWKHPDWLFDOIDFWVFRQFHSWVSULQFLSOHVDQGWKHRULHV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG $FRQVWUXFWGUDZPHDVXUHWUDQVIRUPFRPSDUHYLVXDOL]HFODVVLI\DQGDQDO\]HWKH UHODWLRQVKLSVDPRQJJHRPHWULF¿JXUHVDQG Alaska State Mathematics Standard C $VWXGHQWVKRXOGXQGHUVWDQGDQGEHDEOHWRIRUPDQGXVHDSSURSULDWHPHWKRGVWRGH¿QH DQGH[SODLQPDWKHPDWLFDOUHODWLRQVKLSV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG &H[SUHVVDQGUHSUHVHQWPDWKHPDWLFDOLGHDVXVLQJRUDODQGZULWWHQSUHVHQWDWLRQV SK\VLFDOPDWHULDOVSLFWXUHVJUDSKVFKDUWVDQGDOJHEUDLFH[SUHVVLRQV &UHODWHPDWKHPDWLFDOWHUPVWRHYHU\GD\ODQJXDJH GLEs The student demonstrates an understanding of geometric relationships by >@*LGHQWLI\LQJDQDO\]LQJFRPSDULQJRUXVLQJSURSHUWLHVRIDQJOHVLQFOXGLQJVXSSOHPHQWDU\RUFRPSOHPHQWDU\ The student demonstrates an understanding of geometric relationships by >@*LGHQWLI\LQJDQDO\]LQJFRPSDULQJRUXVLQJSURSHUWLHVRISODQH¿JXUHV VXSSOHPHQWDU\FRPSOHPHQWDU\RUYHUWLFDODQJOHV DQJOHVFUHDWHGE\SDUDOOHOOLQHVZLWKDWUDQVYHUVDO The student communicates his or her mathematical thinking by >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\WUDQVODWLQJDPRQJWKHVHDOWHUQDWLYHUHSUHVHQWDWLRQVRUXVLQJDSSURSULDWHYRFDEXODU\ V\PEROVRUWHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQGVROXWLRQV >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\FRPPXQLFDWLQJPDWKLGHDVLQZULWLQJRUXVLQJDSSURSULDWHYRFDEXODU\V\PEROVRU WHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQGVROXWLRQV Vocabulary & Definitions Introduction of Math Vocabulary Equidistant (TXLGLVWDQWPHDQVHTXDOO\GLVWDQWRUDWWKHVDPHGLVWDQFH)RUH[DPSOHWKH two endpoints of a segment are equidistant from its midpoint. 2QDVQRZÀDNHHDFKRIWKHWLSVLVHTXLGLVWDQWIURPWKH center. Parallel lines 3DUDOOHOOLQHVDUHOLQHVWKDWDUHRQWKHVDPHSODQHFRSODQDUEXWWKDWGRQRW intersect. You can think of them as going in the same direction and staying the same distance apart. %RDUGVRQDGRFNUHSUHVHQWH[DPSOHVRISDUDOOHOOLQHV:KHQ \RXDUHOHDUQLQJWRVNL\RXPLJKWWU\WRNHHS\RXUVNLVSDUDOOHODVZHOO — 237 — Introduction of Math Vocabulary Skew Lines Skew lines are lines that do not intersect and that are not parallel. They cannot be coplanar. You can see skew lines represented in this picture that was WDNHQLQ$QFKRUDJHDIWHUWKHELJHDUWKTXDNHLQ 3LFWXUHLVIURP9,/'$8$) Transversals $WUDQVYHUVDOLVDOLQHWKDWFXWVDFURVVDQRWKHUVHWRI lines. ,QWKHSLFWXUHLIWKHWUHHVUHSUHVHQWOLQHVWKHIDOOHQWUHHWUXQN is the transversal of the two standing trees. — 238 — Introduction of Math Vocabulary Perpendicular lines Perpendicular lines are lines that intersect at a right oDQJOH ,QWKLVSLFWXUHWKHEDUVRQ the window are perpenGLFXODU2WKHUH[DPSOHV of perpendicular lines can EHIRXQGRQVLGHZDONV VWUHHWVODPSSRVWVIHQFHVDQGEXLOGLQJV Perpendicular bisector $SHUSHQGLFXODUELVHFWRULVDOLQHWKDWLVSHUSHQGLFXODUWR DVHJPHQWDQGWKDWSDVVHVWKURXJKLWVPLGSRLQW $VNVWXGHQWVLIWKH\FDQ¿QGSHUSHQdicular bisectors on this brick wall. Geometric figure $JHRPHWULF¿JXUHLVDQ\VHWRISRLQWVRQDSODQHRULQVSDFH ([DPSOHVRIJHRPHWULF¿JXUHVDUHSRLQWVVHJPHQWVUD\V FXUYHVSHQWDJRQVDQGFXEHV 3RLQWRXWH[DPSOHVRIJHRPHWULF¿Jures on this basket from Hoonah. — 239 — Introduction of Math Vocabulary Interior Angles $QLQWHULRUDQJOHLVDQDQJOHLQWKHLQWHULRU LQVLGHRIDJHRPHWULF¿JXUH $¿JXUHWKDWVKRZVWZROLQHVFXWE\DWUDQVYHUVDOLVDW\SHRIJHRPHWULF¿JXUHDQGWKHLQWHULRU angles are those that are between the two lines that are being cut. ,QWKLVGUDZLQJDQJOHV$%&DQG'DUHLQWHULRU angles. Use this picture of electrical wires to identify interior angles. Exterior Angles ([WHULRUDQJOHVDUHDQ\RIWKHIRXUDQJOHVWKDWGRQRW include a region of the space between two lines intersectHGE\DWUDQVYHUVDO,QWKLVGLDJUDPDQJOHVDQG DUHH[WHULRUDQJOHV )LQGSDUDOOHOOLQHVFXWE\DWUDQVYHUVDORQWKLVFURVVZRUGSX]]OHDQGLGHQWLI\WKHH[WHULRU angles. — 240 — Introduction of Math Vocabulary Alternate Exterior Angles $OWHUQDWHH[WHULRUDQJOHVDUHWZRH[WHULRUDQJOHV on opposite sides of a transversal that lie on differHQWSDUDOOHOOLQHV,QWKHGLDJUDPDOWHUQDWHH[WHULRU DQJOHVZRXOGEHµD¶DQGµK¶RUµE¶DQGµJ¶ This picture of a guitar shows VHYHUDOVHWVRIDOWHUQDWHH[WHULRUDQJOHV Alternate Interior Angles $OWHUQDWHLQWHULRUDQJOHVDUHWZRLQWHULRUDQJOHVZKLFKOLHRQ different parallel lines and on opposite sides of a transversal. 2QWKHGLDJUDPDQJOHVDQGDUHDOWHUQDWHLQWHULRUDQJOHV 6RDUHDQJOHVDQG Pick out alternate interior angles on WKLVSLFWXUHRI¿UHHVFDSHV — 241 — Introduction of Math Vocabulary Consecutive Interior Angles &RQVHFXWLYHPHDQV³IROORZLQJRQHDIWHUWKHRWKHULQRUGHU´ &RQVHFXWLYHLQWHULRUDQJOHVDUHWZRLQWHULRUDQJOHVO\LQJRQWKH same side of the transversal cutting across two lines. In the GLDJUDPDQJOHVGDQGIDUHFRQVHFXWLYHLQWHULRUDQJOHVDQG angles c and e are consecutive interior angles. )LQGFRQVHFXWLYHLQWHULRUDQJOHVRQWKLVIHQFH Corresponding Angles :KHQWZROLQHVDUHFURVVHGE\DQRWKHUOLQHZKLFKLVFDOOHG WKH7UDQVYHUVDOWKHDQJOHVLQPDWFKLQJFRUQHUVRUSRVLWLRQV DUHFDOOHGFRUUHVSRQGLQJDQJOHV,QWKLV¿JXUHWKHSDLUVRI FRUUHVSRQGLQJDQJOHVDUHDQGDQGDQGDQG DQG Many pairs of corresponding angles can be demonstrated on this picture of a bridge railing. — 242 — Introduction of Math Vocabulary Congruent angles &RQJUXHQWDQJOHVDUHDQJOHVWKDWKDYHWKHVDPHDQJOH measure. They are equal in size. +HUHLVDSLFWXUHRIDQROGFKDSHOEXLOGLQJLQ<DNXWDW0DQ\H[DPSOHVRIFRQgruent angles can be found on this building. — 243 — — 244 — Language and Skills Development Using the Math Vocabulary Terms Language & Skills Development LISTENING Use the activity pages from the Student Support Materials. SPEAKING READING Use the activity pages from the Student Support Materials. WRITING Use the activity pages from the Student Support Materials. Let’s Move Identify an appropriate body movement for each vocabulary word. This may LQYROYHPRYHPHQWVRIKDQGVDUPVOHJVHWF3UDFWLFHWKHERG\PRYHPHQWVZLWK WKHVWXGHQWV:KHQWKHVWXGHQWVDUHDEOHWRSHUIRUPWKHERG\PRYHPHQWVZHOO say a vocabulary word. The students should respond with the appropriate body movement. You may wish to say the vocabulary words in a running story. When DYRFDEXODU\ZRUGLVKHDUGWKHVWXGHQWVVKRXOGSHUIRUPWKHDSSURSULDWHERG\ PRYHPHQW5DWKHUWKDQXVLQJERG\PRYHPHQWVRULQDGGLWLRQWRWKHERG\PRYHPHQWV \RX PD\ ZLVK WR XVH ³VRXQG HIIHFWV´ IRU LGHQWLI\LQJ YRFDEXODU\ ZRUGV The students should perform the appropriate body movements/sound effects for the words you say. Right or Wrong? Mount the vocabulary illustrations on the chalkboard. Point to one of the illustrations and say its vocabulary word. The students should repeat the vocabulary ZRUGIRUWKDWLOOXVWUDWLRQ+RZHYHUZKHQ\RXSRLQWWRDQLOOXVWUDWLRQDQGVD\DQ LQFRUUHFWYRFDEXODU\ZRUGIRULWWKHVWXGHQWVVKRXOGUHPDLQVLOHQW5HSHDWWKLV process until the students have responded a number of times to the different vocabulary illustrations. Half Time %HIRUHWKHDFWLYLW\EHJLQVFXWHDFKRIWKHVLJKWZRUGVLQKDOI.HHSRQHKDOIRI each sight word and give the remaining halves to the students. Hold up one of your halves and the student who has the other half of that word must show his half DQVD\WKHVLJKWZRUG5HSHDWLQWKLVZD\XQWLODOOVWXGHQWVKDYHUHVSRQGHG$Q alternative to this approach is to give all of the word halves to the students. Say one of the sight words and the two students who have the halves that make up the sight word must show their halves. Depending upon the number of students in \RXUFODVV\RXPD\ZLVKWRSUHSDUHH[WUDVLJKWZRUGFDUGVIRUWKLVDFWLYLW\ Watch Your Half 3UHSDUHDSKRWRFRS\RIHDFKRIWKHYRFDEXODU\LOOXVWUDWLRQV&XWWKHSKRWRFRSLHG illustrations in half. Keep the illustration halves in separate piles. Group the students into two teams. Give all of the illustration halves from one pile to the players in Team One. Give the illustration halves from the other pile to the playHUVLQ7HDP7ZR6D\DYRFDEXODU\ZRUG:KHQ\RXVD\³*R´WKHVWXGHQWIURP HDFKWHDPZKRKDVWKHLOOXVWUDWLRQKDOIIRUWKHYRFDEXODU\ZRUG\RXVDLGVKRXOG UXVKWRWKHFKDONERDUGDQGZULWHWKHZRUGRQWKHERDUG7KH¿UVWSOD\HUWRGR WKLV FRUUHFWO\ ZLQV WKH URXQG 5HSHDW XQWLO DOO SOD\HUV KDYH SDUWLFLSDWHG 7KLV DFWLYLW\PD\EHSOD\HGPRUHWKDQRQFHE\FROOHFWLQJPL[LQJDQGUHGLVWULEXWLQJ the illustration halves to the two teams. — 247 — Student Support Materials — 251 — — 253 — — 255 — — 257 — — 259 — — 261 — — 263 — — 265 — — 267 — — 269 — — 271 — — 273 — — 275 — — 277 — True-False Sentences (Listening and/or Reading Comprehension) 1. 2QDUXOHUWKH´PDUNDQGWKH´PDUNDUHequidistantIURPWKH´PDUN 2. Parallel lines will always eventually intersect. 3. Skew lines are not necessarily straight lines. When a transversalFXWVDFURVVWZROLQHVWKHUHDUHWZRSRLQWVRILQWHUVHFWLRQ 5. Perpendicular lines make up two sides of every triangle. 6. Perpendicular bisectors divide line segments in half. 7. $OOJHRPHWULF¿JXUHVKDYHDQLQWHULRUDQGDQH[WHULRU $QREWXVHDQJOHFDQQRWEHDQinterior angle. 9. ,IDQJOHVDUHDGMDFHQWWRHDFKRWKHUWKH\FDQQRWEHexterior angles. 10. Vertical angles are a type of alternate exterior angles. 11. Alternate interior angles can form a linear pair. 12. Consecutive interior angles would never be adjacent to each other. :KHQWZROLQHVDUHFXWE\DWUDQVYHUVDOWKHUHDUHDOZD\VIRXUSDLUVRIcorresponding angles. If two angles each measure 90oWKH\DUHcongruent angles. $QVZHUV7))7)7)))))777 1. Sitka and Ketchikan are equidistant from Juneau. 2. If lines are parallelWKH\DUHDOVRFRSODQDU ,IOLQHVDUHLQWKHVDPHSODQHWKH\FDQQRWEHskew lines. The lines cut by a transversal have to be parallel lines. It is important for a house builder to use perpendicular lines. Lines and curves can both have perpendicular bisectors. 7. *HRPHWULF¿JXUHVFDQEHRQHWZRRUWKUHHGLPHQVLRQDO 8. Interior angles are in between two lines. 9. Exterior angles might be right angles. 10. Alternate exterior angles are always on opposite sides of a transversal. 11. Each angle in a pair of alternate interior angles has the transversal as one of its sides. 12. ,QDSDLURIFRQVHFXWLYHLQWHULRUDQJOHVWKHDQJOHPHDVXUHVDUHDOZD\VWKHVDPH 13. Corresponding anglesVKDUHWKHVDPHYHUWH[ $WULDQJOHDOZD\VKDVDWOHDVWWZRcongruent angles. $QVZHUV)77)7)77777))) — 279 — Match the Halves 1. The endpoints of a segment $DOOWKUHHOLQHVDUHLQWKHVDPHSODQH 2. The sides of a straight road B. are interior angles. /LQHVWKDWQHYHULQWHUVHFWDUHVNHZOLQHVLI they &GLYLGHVDVHJPHQWLQWRWZRHTXDO sections. ,IDWUDQVYHUVDOLQWHUVHFWVWZRSDUDOOHOOLQHV D. on opposite sides of a transversal. 7ZRDGMRLQLQJVLGHVRIDVTXDUH E. are alternate interior angles. $SHUSHQGLFXODUELVHFWRU )DUHHTXLGLVWDQWIURPWKHPLGSRLQW 6TXDUHVFLUFOHVDQGS\UDPLGV G. are consecutive interior angles. $Q\DQJOHVWKDWDUHLQVLGHJHRPHWULF¿JXUHV H. are on the same side of a transversal. ([WHULRUDQJOHVDUHQRWEHWZHHQ I. are not parallel lines. $OWHUQDWHH[WHULRUDQJOHVDUHH[WHULRUDQJOHV -DUHH[DPSOHVRIJHRPHWULF¿JXUHV 11. Interior angles that are not consecutive or adjacent K. the lines cut by a transversal. 12. Interior angles on the same side of a transversal L. are perpendicular lines. &RUUHVSRQGLQJDQJOHV M. are congruent angles. ,IDQJOHVKDYHWKHVDPHPHDVXUHWKH\ N. might represent parallel lines. $QVZHUV)1,$/&-%.'(*+0 — 280 — 'H¿QLWLRQV Equidistant: HTXDOO\GLVWDQWRUDWWKHVDPHGLVWDQFH Parallel linesOLQHVWKDWDUHRQWKHVDPHSODQHFRSODQDUEXWWKDWGRQRWLQWHUVHFW Skew Lines: lines that do not intersect and that are not parallel. Transversal: a line that cuts across another set of lines. Perpendicular lines: OLQHVWKDWLQWHUVHFWDWDULJKWoDQJOH Perpendicular bisector: a line that is perpendicular to a segment and that passes through its midpoint. *HRPHWULF¿JXUHany set of points on a plane or in space. Interior Angle: DQDQJOHLQWKHLQWHULRULQVLGHRIDJHRPHWULF¿JXUH Exterior Angles: any of the four angles that do not include a region of the space between two lines intersected by a transversal. Alternate Exterior Angles: WZRH[WHULRUDQJOHVRQRSSRVLWHVLGHVRIDWUDQVYHUVDOWKDW lie on different parallel lines. Alternate Interior Angles: two interior angles which lie on different parallel lines and on opposite sides of a transversal. Consecutive Interior Angles: two interior angles lying on the same side of the transversal cutting across two lines. Corresponding Angles: the angles in matching corners or positions, when two lines are crossed by a transversal. Congruent angles: angles that have the same angle measure. — 281 — Which Belongs 1. 6NHZSHUSHQGLFXODUSDUDOOHOOLQHVDUHQRWFRSODQDU 2. ,IDQJOHVDUHDGMDFHQWWKH\PLJKWEHLQWHULRUDQJOHVFRUUHVSRQGLQJDQJOHVFRQVHFXWLYHDQJOHV 3HUSHQGLFXODUVNHZSDUDOOHOOLQHVDOZD\VLQWHUVHFWHDFKRWKHU ,IDQJOHVDUHWKHVDPHVL]HWKH\PXVWEHFRUUHVSRQGLQJFRQJUXHQWYHUWLFDODQJOHV ([WHULRUFRQJUXHQWFRUUHVSRQGLQJDQJOHVDUHRQWKHRXWVLGHRIOLQHVFXWE\DWUDQVversal. /LQHDUSDLUVDUHH[DPSOHVRIFRUUHVSRQGLQJDQJOHVJHRPHWULF¿JXUHVDOWHUQDWHH[WHULRUDQJOHV 7. $OLQHDORQJDZDOOLQWHUVHFWVDOLQHDORQJDURRIVRWKHOLQHVFDQQRWEHSHUSHQGLFXODU SDUDOOHOLQWHULRU Two angles that are between lines cut by a transversal and on the same side of the WUDQVYHUVDODUHFRQVHFXWLYHLQWHULRUFRQJUXHQWFRUUHVSRQGLQJDQJOHV 9. 7KHPLGSRLQWRIRQHVLGHRIDUHFWDQJOHLVSDUDOOHOWRHTXLGLVWDQWIURPSHUSHQGLFXODU WRWKHFRUQHUVRQWKDWVLGH 10. /LQHDUSDLUVVXSSOHPHQWDU\DQJOHVDOWHUQDWHH[WHULRUDQJOHVPLJKWKDYHPHDVXUHV that add up to 90o. 11. &RUUHVSRQGLQJLQWHULRUH[WHULRUDQJOHVKDYHPDWFKLQJSRVLWLRQVLQDV\VWHPRI lines cut by a transversal 12. 7ZROLQHVFXWE\DWUDQVYHUVDOFDQQRWKDYHDQSHUSHQGLFXODUELVHFWRULQWHULRUDQJOHV H[WHULRUDQJOH ,IDQJOHVDUHRQRSSRVLWHVLGHVRIDWUDQVYHUVDOWKH\PLJKWEHDOWHUQDWHLQWHULRUFRQVHFXWLYHLQWHULRUFRUUHVSRQGLQJDQJOHV /LQHVFURVVHGE\DELVHFWRUSHUSHQGLFXODUWUDQVYHUVDOGRQRWKDYHWREHSDUDOOHO 1. Skew 2. Interior angles Perpendicular &RQJUXHQW ([WHULRU *HRPHWULF¿JXUHV 7. Parallel &RQVHFXWLYHLQWHULRU 9. Equidistant from 10. $OWHUQDWHH[WHULRU 11. &RUUHVSRQGLQJ 12. Perpendicular bisector $OWHUQDWHLQWHULRU Transversal — 282 — Multiple Choice :KLFKLV127DQH[DPSOHRIDSRLQWWKDWPXVWEHequidistant from the endpoints of a segment: DWKHPLGSRLQWRIWKHVHJPHQW EWKHLQWHUVHFWLRQRIWKHVHJPHQWZLWKDOLQHWKDWFURVVHVLW FWKHLQWHUVHFWLRQRIWKHVHJPHQWDQGLWVSHUSHQGLFXODUELVHFWRU 2. When two lines are parallelWKH\DUHDOVR DLQWKHVDPHSODQH ELQWHUVHFWLQJDWRQO\RQHSRLQW FH[WHQGLQJLQRQHGLUHFWLRQ Skew lines might be represented by DWKHVLGHVRIDSLFWXUHIUDPH EWZRWUHHVWKDWDUHWLOWHGDWGLIIHUHQWDQJOHV FWKHWZRUDLOVRIDVWUDLJKWÀDWUDLOURDGWUDFN Transversals are lines that always DFXWDFURVVDWOHDVWWZRRWKHUOLQHV EFXWDFURVVSDUDOOHOOLQHV FDUHSHUSHQGLFXODUWRWKHOLQHVWKH\FURVV :KHUHZRXOG\RXEHOLNHO\WR¿QGperpendicular lines represented; DWZRHGJHVRIDSLHFHRISDSHU EDUDLOLQJRQDGHFN FERWKRIWKHDERYH $perpendicular bisector might be described as follows: DDOLQHWKDWLVSHUSHQGLFXODUWRDVHJPHQWDQGLQWHUVHFWVLWVPLGSRLQW EDOLQHWKDWLVDWo to a segment and cuts it in half FDQ\VHJPHQWWKDWLVDWDULJKWDQJOHWRDQRWKHUVHJPHQW $Q\VHWRISRLQWVLQVSDFHLVDJHRPHWULF¿JXUHif DWKHSRLQWVDUHRQWKHVDPHSODQH EWKHSRLQWVGRQRWIRUPFXUYHV FWKHUHLVDWOHDVWRQHSRLQWLQWKHVHW $SDLURIinterior angles can never also be DFRUUHVSRQGLQJDQJOHV EDGMDFHQWDQJOHV FRQRSSRVLWHVLGHVRIDWUDQVYHUVDO — 283 — 9. If two angles are both exteriorDQJOHVWKH\PLJKWEH DVXSSOHPHQWDU\DQJOHV EREWXVHDQJOHV FHLWKHURIWKHDERYH 10. Which two things are true about alternate exterior angles" DWKH\DUHRQRSSRVLWHVLGHVRIDWUDQVYHUVDODQGRQWKHRXWVLGHRIWKHOLQHV that are being crossed EWKH\DUHRQWKHVDPHVLGHRIDWUDQVYHUVDODQGRQWKHRXWVLGHRIWKHOLQHV that are being crossed FWKH\DUHDGMDFHQWDQGRQWKHRXWVLGHRIWKHOLQHVWKDWDUHEHLQJFURVVHG 11. Alternate interior angles cannot also be DULJKWDQJOHV EFRQJUXHQWDQJOHV FDGMDFHQWDQJOHV 12. Which must be true about consecutive interior angles" DWKH\DUHDOZD\VDGMDFHQWWRHDFKRWKHU EWKH\DUHDOZD\VFRPSOHPHQWDU\ FWKH\DUHDOZD\VRQWKHVDPHVLGHRIDWUDQVYHUVDO Corresponding angles might be found on DDGHVLJQRQDTXLOW EDGRRUIUDPH FDVWRSVLJQ :KLFKRIWKHIROORZLQJZRXOGDOZD\VEHcongruent angles" DWZRFRUQHUVRIDURRP EWZRULJKWDQJOHV FWZRFRUUHVSRQGLQJDQJOHV $QVZHUV 1. B 2. $ B $ & $ 7. & $ 9. & 10. $ 11. & 12. & — $ B 284 — Complete the Sentence 1. $SRLQWDSHQWDJRQDVSKHUHDQGDSDLURIDOWHUQDWHLQWHULRUDQJOHVDUHDOOH[DPSOHVRI __________________________________. 2. The door of Debbie’s classroom was ______________ from the two ends of the hallway. :KHQDWUDQVYHUVDOLQWHUVHFWVWZROLQHVWKHDQJOHVLQWKHXSSHUULJKWFRUQHURIHDFKLQWHUsection would be ______________________________. _________________________were represented when two bars on a window intersected at a right angle. 7KHZKLWHOLQHVRQWKHURDGLQWHUVHFWHGERWKVLGHVRIDFURVVZDONDWDQDQJOHDQGIRUPHGD ________________________________. __________________________ are in between two lines. 7. $QJOHVWKDWDUHRQRSSRVLWHVLGHVRIDWUDQVYHUVDODUHBBBBBBBBBBBBBBBBBBBBXQOHVVWKH\ DUHH[WHULRUDQJOHV ,IDQJOHVGRQRWLQFOXGHDQ\UHJLRQRIVSDFHEHWZHHQWZROLQHVWKH\DUHBBBBBBBBBBBBB 9. ³2XWVLGH´DQJOHVWKDWDUHRQRSSRVLWHVLGHVRIDWUDQVYHUVDODUHBBBBBBBBBBBBBBBBBBB 10. 3DLUVRIDFXWHDQJOHVFRPSOHPHQWDU\DQJOHVLQWHULRUDQJOHVH[WHULRUDQJOHVRUYHUWLFDO angles all might be ________________________________. 11. $VWKHERDWWRVVHGDURXQGRQWKHZDYHVDSDLURIBBBBBBBBBBBBBBBBBBBBBZHUHUHSUHVHQWHGE\LWVPDVWDQGWKHÀDJSROHRQVKRUH 12. ,IDSDLURIDQJOHVLVEHWZHHQWZROLQHVDQGRQWKHVDPHVLGHRIDWUDQVYHUVDOWKH\DUH ___________________ angles. 7KHFURVVRQ0W5REHUWVKDVWZRSLHFHVDQGRQHSLHFHLVWKHBBBBBBBBBBBBBBBBBBBRI the other. 7KHWHOHSKRQHZLUHUDQDORQJVLGHE\VLGHDOZD\VWKHVDPHGLVWDQFHDSDUWMXVWOLNHDSDLU of ____________________________. $QVZHUV 1. *HRPHWULF¿JXUHV 2. Equidistant &RUUHVSRQGLQJDQJOHV Perpendicular lines Transversal Interior angles 7. $OWHUQDWHLQWHULRUDQJOHV ([WHULRUDQJOHV 9. $OWHUQDWHH[WHULRUDQJOHV 10. &RQJUXHQWDQJOHV 11. Skew lines 12. &RQVHFXWLYHLQWHULRUDQJOHV Perpendicular bisector Parallel lines — 285 — Creative Writing Write about the picture using words from this unit. Label or number things on the picture or XVHGLDJUDPVWRKHOSH[SODLQWKHZRUGVLIQHHGHG — 286 — Place-Based Practice Activity )LQGPDSVRUDHULDOSKRWRVRI\RXUWRZQRU¿QGDQGSULQWSDJHVIURP*RRJOH 0DSV&KRRVHDUHDVZLWKORWVRIURDGLQWHUVHFWLRQVRURWKHULQWHUVHFWLQJOLQHV Other options might be to obtain highway blueprints from the local road departPHQWRUWR¿QGPDSVRIDQDUHDWKDW\RXDUHSODQQLQJWRYLVLWRQDFODVVWULS 6WXG\WKHPDSVDQGRUSKRWRVWR¿QGH[DPSOHVRIHTXLGLVWDQWSRLQWVSDUDOOHO OLQHVSHUSHQGLFXODUOLQHVDQGELVHFWRUVDQGWUDQVYHUVDOVDQGDOORIWKHLUDVVRFLated angles. 6WXGHQWVFDQGRWKLVDFWLYLW\LQVPDOOJURXSV7KHQWKH\FDQVKDUHWKHLU¿QGLQJV ZLWKDQRWKHUJURXS7KHRWKHUJURXSRIVWXGHQWVFDQGHFLGHLIWKHH[DPSOHVDUH FRUUHFWDQGFDQWU\WRLGHQWLI\PRUHH[DPSOHV — 287 — — 288 — Unit Assessment — 289 9 — — 290 — Geometry: Unit 4 Name: ___________________ Date: ___________________ Match the key vocabulary words on the left with their definition on the right. Place the letter of the definition in front of the word that it matches. a. lines that are on the same plane (coplanar) but that do not intersect 1) _____ skew lines 2) _____ transversals b. lines that do not intersect and that are not parallel and cannot be coplanar. 3) _____ perpendicular lines 4) _____ perpendicular bisector c. lines that intersect at a right (90o) angle 5) _____ parallel lines d. is a line that is perpendicular to a segment, and that passes through its midpoint e. lines that cuts across another set of lines Fill in the Blank: Fill in the blank for each statement with the word that best fits. Choose the words from the Word Bank. Some words will not be used. Word Bank equidistant geometric figure parallel perpendicular skew line transversal 6) A line is 7) 8) A to another lines if the lines meet at 90 degrees. means equally distant, or at the same distance. is a line that cuts across another set of lines. 9) Boards on a dock would be an example of 10) A lines. is any set of points on a plane or in space — 291 — Multiple Choice: Read each statement carefully and circle the best answer. 11) An angle in the interior inside of a geometric figure is a/an _______________ angle. a) exterior b) interior c) alternate interior d) alternate exterior 12) Two exterior angles on opposite sides of a transversal that lie on different parallel lines are __________________ angles. a) exterior b) interior c) alternate interior d) alternate exterior 13) The four angles that do not include a region of the space between two lines intersected by a transversal are the ______________ angles. a) exterior b) interior c) alternate interior d) alternate exterior 14) Two interior angles which lie on different parallel lines and on opposite sides of a transversal are _____________________ angles. a) exterior b) interior c) alternate interior d) alternate exterior — 292 — 15) In the illustration below of the old chapel building in Yakutat, put an X on the congruent angles that are found on this building. 16) In the space below, illustrate or define consecutive interior angles. 17) In the space below, Illustrate OR write a definition for corresponding angles. — 293 — — 294 — Geometry: Unit 4 Name: ___________________ Date: ___________________ Match the key vocabulary words on the left with their definition on the right. Place the letter of the definition in front of the word that it matches. a. lines that are on the same plane (coplanar) but that do not intersect 1) b skew lines 2) e transversals 3) c perpendicular lines 4) d perpendicular bisector 5) a parallel lines b. lines that do not intersect and that are not parallel and cannot be coplanar. c. lines that intersect at a right (90o) angle d. is a line that is perpendicular to a segment, and that passes through its midpoint e. lines that cuts across another set of lines Fill in the Blank: Fill in the blank for each statement with the word that best fits. Choose the words from the Word Bank. Some words will not be used. Word Bank equidistant geometric figure parallel perpendicular skew line transversal 6) A line is perpendicular to another lines if the lines meet at 90 degrees. 7) equidistant means equally distant, or at the same distance. 8) A transversal is a line that cuts across another set of lines. 9) Boards on a dock would be an example of parallel lines. 10) A geometric figure is any set of points on a plane or in space — 295 — Multiple Choice: Read each statement carefully and circle the best answer. 11) An angle in the interior inside of a geometric figure is a/an _______________ angle. a) exterior b) interior c) alternate interior d) alternate exterior 12) Two exterior angles on opposite sides of a transversal that lie on different parallel lines are __________________ angles. a) exterior b) interior c) alternate interior d) alternate exterior 13) The four angles that do not include a region of the space between two lines intersected by a transversal are the ______________ angles. a) exterior b) interior c) alternate interior d) alternate exterior 14) Two interior angles which lie on different parallel lines and on opposite sides of a transversal are _____________________ angles. a) exterior b) interior c) alternate interior d) alternate exterior 15) In the illustration below of the old chapel building in Yakutat, put an X on the congruent angles that are found on this building. Arrows point to congruent angles on building. Illustration of old chapel building with x's on congruent triangles. — 296 — 16) In the space below, illustrate or define consecutive interior angles. Illustration ORConsecutive interior angles are two interior angles lying on the same side of the transversal cutting across two lines.. 17) In the space below, Illustrate OR write a definition for corresponding angles. OR When two lines are crossed by another line. the angles in matching corners or positions are called corresponding angles. — 297 — — 298 — UNIT 5 Triangles Sealaska Heritage Institute Grade Level Expectations for Unit 5 Unit 5—Triangles Alaska State Mathematics Standard A $VWXGHQWVKRXOGXQGHUVWDQGPDWKHPDWLFDOIDFWVFRQFHSWVSULQFLSOHVDQGWKHRULHV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG $FRQVWUXFWGUDZPHDVXUHWUDQVIRUPFRPSDUHYLVXDOL]HFODVVLI\DQGDQDO\]H WKHUHODWLRQVKLSVDPRQJJHRPHWULF¿JXUHVDQG Alaska State Mathematics Standard C $VWXGHQWVKRXOGXQGHUVWDQGDQGEHDEOHWRIRUPDQGXVHDSSURSULDWHPHWKRGVWRGH¿QH DQGH[SODLQPDWKHPDWLFDOUHODWLRQVKLSV $VWXGHQWZKRPHHWVWKHFRQWHQWVWDQGDUGVKRXOG &H[SUHVVDQGUHSUHVHQWPDWKHPDWLFDOLGHDVXVLQJRUDODQGZULWWHQSUHVHQWDWLRQV SK\VLFDOPDWHULDOVSLFWXUHVJUDSKVFKDUWVDQGDOJHEUDLFH[SUHVVLRQV &UHODWHPDWKHPDWLFDOWHUPVWRHYHU\GD\ODQJXDJH GLEs The student demonstrates an understanding of geometric relationships by >@*LGHQWLI\LQJDQDO\]LQJFRPSDULQJRUXVLQJSURSHUWLHVRISODQH¿JXUHV VXSSOHPHQWDU\FRPSOHPHQWDU\RUYHUWLFDODQJOHV VXPRILQWHULRURUH[WHULRUDQJOHVRIDSRO\JRQ The student communicates his or her mathematical thinking by >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\WUDQVODWLQJDPRQJWKHVHDOWHUQDWLYHUHSUHVHQWDWLRQVRUXVLQJDSSURSULDWHYRFDEXODU\ V\PEROVRUWHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQGVROXWLRQV >@36UHSUHVHQWLQJPDWKHPDWLFDOSUREOHPVQXPHULFDOO\JUDSKLFDOO\DQGRUV\PEROLFDOO\FRPPXQLFDWLQJPDWKLGHDVLQZULWLQJRUXVLQJDSSURSULDWHYRFDEXODU\V\PEROVRU WHFKQRORJ\WRH[SODLQMXVWLI\DQGGHIHQGVWUDWHJLHVDQGVROXWLRQV Vocabulary & Definitions Introduction of Math Vocabulary polygon $SRO\JRQLVDSODQHVKDSHZLWKWKUHHRUPRUHVWUDLJKWVLGHV 7ULDQJOHVUHFWDQJOHVWUDSH]RLGVSHQWDJRQVDQGRFWDJRQVDUHD IHZH[DPSOHVRISRO\JRQV$VNVWXGHQWV³:KLFKRIWKHVHDUHSRO\JRQV"´7KHDQVZHULV±DOORIWKHP ,QQRUWKHUQDUHDVZLWKSHUPDIURVWSHUPDQHQWO\IUR]HQJURXQG a pattern of polygons often develops on the ground surface. <RXKDYHWRLPDJLQHWKDWDOOWKHVLGHVDUHVWUDLJKW triangle $WULDQJOHLVDQ\SRO\JRQZLWKWKUHHVLGHV 7KHWULDQJOHRQWKLVEDVNHWEDOOKRRSLVDQH[DPSOHRIDWULDQJOH WKDW\RXFDQVHHDURXQG\RXLQWKHVFKRROWKHFRPPXQLW\RULQ nature. interior The interior is the set of points enclosed by a geometULF¿JXUH,WGRHVQRWLQFOXGHWKHYHUWLFHVRUWKHVLGHVRU HGJHVRIWKH¿JXUH The miners in the picture are in the interior of a mine. — 305 — Introduction of Math Vocabulary acute triangle $QDFXWHWULDQJOHLVDWULDQJOHLQZKLFKDOORIWKHLQWHULRUDQJOHVDUHDFXWHRUOHVVWKDQo. ,QWKHSLFWXUHWKHERRNVRQWKHVKHOIIRUPDQDFXWHWULDQJOH obtuse triangle $QREWXVHWULDQJOHLVDWULDQJOHWKDWKDVDQREWXVH angle as one of its interior angles. The turnovers in the picture are obtuse triangles. right triangle $ULJKWWULDQJOHLVDWULDQJOHZKLFKKDVDULJKW LQWHULRUDQJOH ,QWKLVSLFWXUHWKHNLWWHQKDVIRXQG a sheltered spot inside a right triangle. — 306 — Introduction of Math Vocabulary scalene $VFDOHQHWULDQJOHLVDWULDQJOHIRUZKLFKDOOWKUHHVLGHVKDYHGLIIHUent lengths. The triangle shown on this railing is a scalene triangle. isosceles $QLVRVFHOHVWULDQJOHLVDWULDQJOHZLWKWZRVLGHVWKDWDUHWKHVDPH length. This coat hangar is shaped like an isosceles triangle. equilateral $QHTXLODWHUDOWULDQJOHLVDWULDQJOHIRUZKLFKDOOWKUHH VLGHVDUHWKHVDPHOHQJWKFRQJUXHQW These candles are shaped like equilateral triangles. — 307 — Introduction of Math Vocabulary equiangular Equiangular means that all of the angles are equal. $PXVLFDOWULDQJOHLVDQH[DPSOHRIDQHTXLDQgular triangle. altitude 7KHDOWLWXGHRIDWULDQJOHLVGLVWDQFHEHWZHHQDYHUWH[RIDWULDQJOHDQGWKHRSSRVLWHVLGH,QWKLVGLDJUDPVHJPHQW³D´LV the altitude. $OOWKUHHDOWLWXGHVRIDWULDQJOHDUHVKRZQLQWKLVSLFWXUH base The base of a triangle is the side of the triangle that is SHUSHQGLFXODUWRWKHDOWLWXGH,QWKHGLDJUDPWKHFP is the base. ,QWKLVSLFWXUHRQHEDVHRIWKHWULangle is on the man’s head! — 308 — Introduction of Math Vocabulary base angle The base angle of a triangle is either of two angles that have the base for a side. ,QWKLVLVRVFHOHVWULDQJOHDQJOHV³5´DQG³7´DUHWKHEDVHDQJOHV vertex angle 7KHYHUWH[DQJOHRIDWULDQJOHLVWKHDQJOHRSSRVLWHWKHEDVH ,QWKLVWULDQJOHDQJOH³%´LVWKHYHUWH[DQJOH leg The leg of a triangle is one of its sides. In a right trianJOHWKHWZRVLGHVRSSRVLWHWKHDFXWHDQJOHVDUHFDOOHG WKHOHJVDQGLQDQLVRVFHOHVWULDQJOHWKHVLGHVRSSRVLWH the congruent angles are called the legs. Here’s a triangle in which two of the legs really are legs! hypotenuse $K\SRWHQXVHLVWKHVLGHRIDULJKWWULDQJOHRSSRVLWHWKHULJKW angle. In this picture the bird is sitting on a hypotenuse — 309 — Introduction of Math Vocabulary exterior angle (of a triangle) $QH[WHULRUDQJOHLVDQDQJOHIRUPHGRXWVLGHDWULDQJOH ZKHQRQHVLGHLVH[WHQGHG,QWKLVGLDJUDPDQJOH:LVDQ H[WHULRUDQJOH ,QWKHSLFWXUHWKHoDQJOHLVDQH[DPSOHRIDQH[WHULRU angle of a triangle formed by the spokes of a bicycle wheel. remote interior angle ³5HPRWH´PHDQVIDUDZD\5HPRWHLQWHULRU angles of a triangle are the two angles that DUHIDUWKHVWIURPDQH[WHULRUDQJOH$UHPRWH LQWHULRUDQJOHLVQRWDGMDFHQWWRWKHH[WHULRU angle. — 310 — Introduction of Math Vocabulary median The median of a triangle is a line segment drawn from RQHYHUWH[WRWKHPLGSRLQWRIWKHRSSRVLWHVLGH This picture shows the three medians of a triangle. angle bisector ,QDWULDQJOHDQDQJOHELVHFWRULVDOLQHWKDWGLYLGHVDQLQWHrior angle in half. 7KHYHUWLFDOSRVWRQWKLVURRIWUXVVLVDQH[DPSOHRIDQDQJOHELVHFWRU — 311 — — 312 — Language and Skills Development Using the Math Vocabulary Terms Language & Skills Development LISTENING Use the activity pages from the Student Support Materials. Three Sentences 3URYLGHHDFKVWXGHQWZLWKWKUHHEODQNÀDVKFDUGV(DFKVWXGHQWVKRXOGWKHQZULWH WKHQXPEHUVWRRQKLVKHUFDUGVRQHQXPEHUSHUFDUG6D\WKUHHVHQWHQFHV only one of which contains a vocabulary word. The students should listen careIXOO\WRWKHWKUHHVHQWHQFHVWKDW\RXVD\$IWHUVD\LQJWKHWKUHHVHQWHQFHVHDFK student should then show his/her number card that represents the number of WKH VHQWHQFH ZKLFK FRQWDLQHG WKH YRFDEXODU\ ZRUG 5HSHDW ZLWK RWKHU VHWV RI sentences. SPEAKING Numbered Boxes %HIRUH WKH DFWLYLW\ EHJLQV SUHSDUH D SDJH WKDW FRQWDLQV RU PRUH ER[HV 1XPEHUHDFKRIWKHER[HV3URYLGHHDFKVWXGHQWZLWKDFRS\RIWKHQXPEHUHG ER[HV(DFKVWXGHQWVKRXOGWKHQVKDGHLQKDOIRIWKHER[HVZLWKDSHQFLODQ\ WHQER[HV:KHQWKHVWXGHQWVDUHUHDG\PRXQWWKHYRFDEXODU\LOOXVWUDWLRQVRQ WKHFKDONERDUGDQGVD\WKHQXPEHURIDER[EHWZHHQDQGWRRQHRIWKH VWXGHQWV7KHVWXGHQWVKRXOGORRNRQKLVKHUIRUPWRVHHLIWKDWER[QXPEHULV VKDGHGLQ ,I WKDW ER[ LV VKDGHG LQ WKH VWXGHQW PD\ ³SDVV´ WR DQRWKHU SOD\HU +RZHYHU LI WKH ER[ LV QRW VKDGHGLQ KHVKH VKRXOG VD\ D FRPSOHWH VHQWHQFH DERXWDYRFDEXODU\LOOXVWUDWLRQ\RXSRLQWWR7KHVWXGHQWVPD\H[FKDQJHSDJHV SHULRGLFDOO\GXULQJWKLVDFWLYLW\5HSHDWXQWLOPDQ\VWXGHQWVKDYHUHVSRQGHGLQ this way. READING Circle of Words %HIRUHWKHDFWLYLW\EHJLQVSUHSDUHDSDJHWKDWFRQWDLQVWKHVLJKWZRUGV3URYLGH each student with a copy of the page. The students should cut the sight words IURPWKHLUSDJHV:KHQDVWXGHQWKDVFXWRXWWKHVLJKWZRUGVKHVKHVKRXOGOD\ WKHPRQKLVKHUGHVNLQDFLUFOH7KHQHDFKVWXGHQWVKRXOGSODFHDSHQRUSHQFLO in the center of the circle of sight word cards. Each student should spin the pen/ SHQFLO6D\DVLJKWZRUG$Q\VWXGHQWRUVWXGHQWVZKRVHSHQVSHQFLOVDUHSRLQWLQJ WRWKHVLJKWZRUG\RXVDLGVKRXOGFDOO³%,1*2´7KHVWXGHQWRUVWXGHQWVVKRXOG WKHQUHPRYHWKRVHVLJKWZRUGVIURPWKHLUGHVNV&RQWLQXHLQWKLVZD\XQWLODVWXdent or students have no sight words left on their desks. Use the activity pages from the Student Support Materials. Every Second Letter :ULWHDVLJKWZRUGRQWKHFKDONERDUGRPLWWLQJHYHU\VHFRQGOHWWHU3URYLGHWKH students with writing paper and pens. The students should look at the incomplete word on the chalkboard and then write the sight word for it on their papers. 5HSHDWXVLQJRWKHUVLJKWZRUGV WRITING Use the activity pages from the Student Support Materials. 7KLVDFWLYLW\PD\DOVREHGRQHLQWHDPIRUP,QWKLVFDVHKDYHWKHLQFRPSOHWH ZRUGVSUHSDUHGRQVHSDUDWHÀDVKFDUGV0RXQWRQHRIWKHFDUGVRQWKHFKDONERDUG :KHQ \RX VD\ ³*R´ WKH ¿UVW SOD\HU IURP HDFK WHDP PXVW UXVK WR WKH chalkboard and write the sight word for it - adding all of the missing letters. 5HSHDWXQWLODOOSOD\HUVKDYHSDUWLFLSDWHG — 315 — Student Support Materials — 319 — — 321 — — 323 — — 325 — — 327 — — 329 — — 331 — — 333 — — 335 — — 337 — — 339 — — 341 — — 343 — — 345 — — 347 — — 349 — — 351 — — 353 — — 355 — — 357 — True-False Sentences (Listening and/or Reading Comprehension) 1. $SRO\JRQDOZD\VKDVVWUDLJKWVLGHV 2. The three sides of a triangle are always the same length. 6LGHVDQGYHUWLFHVDUHSDUWRIWKHLQWHULRURIDJHRPHWULF¿JXUH ,QDQDFXWHWULDQJOHDOOWKUHHDQJOHVPXVWEHDFXWH ,QDQREWXVHWULDQJOHDOOWKUHHDQJOHVDUHJUHDWHUWKDQo. ,QDULJKWWULDQJOHWZRVLGHVDUHSHUSHQGLFXODU 7. ,QDVFDOHQHWULDQJOHWKHEDVHLVDOZD\VWKHVDPHOHQJWKDVWKHDOWLWXGH $ULJKWWULDQJOHFDQQRWDOVREHDQLVRVFHOHVWULDQJOH 9. Equiangular triangles have three angles that are the same size. 10. ,IWKHWKUHHVLGHVRIDWULDQJOHDUHFRQJUXHQWLWLVDQHTXLODWHUDOWULDQJOH 11. The base of a triangle forms a right angle with the altitude. 12. The longest side of a triangle is also known as its altitude. $EDVHDQJOHRIDWULDQJOHKDVWKHEDVHRIWKHWULDQJOHDVRQHRILWVVLGHV 7KHDQJOHRIDWULDQJOHRSSRVLWHLWVEDVHLVFDOOHGWKHYHUWH[DQJOH Most triangles have two legs. Every triangle has a hypotenuse. 17. $QH[WHULRUDQJOHRIDWULDQJOHLVFRPSOHPHQWDU\WRLWVDGMDFHQWLQWHULRUDQJOH )RUHYHU\H[WHULRUDQJOHWKHUHDUHWZRUHPRWHLQWHULRUDQJOHV 19. $PHGLDQLVDOZD\VSHUSHQGLFXODUWRWKHVLGHWKDWLWELVHFWV 20. ,QDWULDQJOHWKHDQJOHELVHFWRUDOVRELVHFWVWKHVLGHRSSRVLWHWKDWDQJOH $QVZHUV7))7)7))777)77)))7)) 1. $Q\¿JXUHZLWKWKUHHRUPRUHVWUDLJKWOLQHVLVDSRO\JRQ 2. $WULDQJOHKDVWKUHHVLGHVDQGWKUHHDQJOHV 7KHLQWHULRURIDVKDSHRU¿JXUHLQFOXGHVDOORIWKHSRLQWVHQFORVHGE\LW You might have a right angle in an acute triangle. $QREWXVHWULDQJOHKDVRQHREWXVHDQJOH $OORIWKHDQJOHVLQDULJKWWULDQJOHDUHULJKWDQJOHV 7. $OOWKUHHVLGHVRIDVFDOHQHWULDQJOHDUHGLIIHUHQWOHQJWKV Isosceles triangles have two legs that are the same length. 9. $QREWXVHWULDQJOHFRXOGDOVREHDQHTXLDQJXODUWULDQJOH 10. $OOLVRVFHOHVWULDQJOHVDUHHTXLODWHUDO 11. The base of a triangle is always the side toward the bottom of the page. 12. 7KHDOWLWXGHRIDWULDQJOHLVWKHVKRUWHVWVHJPHQWWKDWFDQEHGUDZQIURPDYHUWH[WRWKHRSSRVLWHVLGH The base angle of a triangle is opposite its base. $WULDQJOHFDQKDYHRQO\RQHYHUWH[DQJOH The side of a triangle is also the leg of a triangle. The side opposite the right angle is the hypotenuse of a right triangle. 17. $QH[WHULRUDQJOHRIDWULDQJOHLVIRUPHGE\H[WHQGLQJRQHVLGHRIWKHWULDQJOH $UHPRWHLQWHULRUDQJOHLVDGMDFHQWWRDQRWKHUUHPRWHLQWHULRUDQJOH 19. ,QDWULDQJOHDPHGLDQELVHFWVRQHRIWKHVLGHV 20. The angle bisector of a triangle divides one of the interior angles in half. $QVZHUV)77)7)77)))7))777)77 — 359 — Match the Halves 1. Squares and rectangles are types of $$QDFXWHWULDQJOH $Q\SRO\JRQZLWKWKUHHVLGHV B. Equiangular. 7KHLQWHULRURIDWULDQJOHGRHVQRWLQFOXGH &$EDVHDQJOH ,IQRQHRIWKHDQJOHVRIDWULDQJOHPHDVXUH more than 90o it is $WULDQJOHZLWKDZLGHDQJOHJUHDWHUWKDQo is '$OHJRIWKHWULDQJOH ,IWZRVLGHVRIDWULDQJOHDUHSHUSHQGLFXODULWLV )+DVWREHDWULDQJOH 7. In a scalene triangle *$QH[WHULRUDQJOHLVIRUPHG ,IWZRVLGHVRIDWULDQJOHDUHHTXDOLQOHQJWKLW is 9. If all three angles of a triangle have the same PHDVXUHLWLV 10. In an equilateral triangle H. None of the sides are the same length. J. Polygons. 11. The base of a triangle K. Into two equal parts. 12. The altitude of a triangle is the shortest segment from $QDQJOHIRUZKLFKWKHEDVHRIDWULDQJOH forms one side is $YHUWH[DQJOHLV /$PHGLDQ $Q\VLGHRIDVFDOHQHWULDQJOHLV O. The sides and the vertices. $ULJKWWULDQJOHLVWKHRQO\W\SHRIWULDQJOHWKDW 3$QREWXVHWULDQJOH :KHQDVLGHRIDWULDQJOHLVH[WHQGHG 41RWDGMDFHQWWRWKHH[WHULRUDQJOH $UHPRWHH[WHULRUDQJOHLV 5$QLVRVFHOHVWULDQJOH $VLGHRIDWULDQJOHLVELVHFWHGE\ 6$YHUWH[WRWKHRSSRVLWHVLGH $QDQJOHELVHFWRUGLYLGHVDQDQJOH 7$ULJKWWULDQJOH E. Has a hypotenuse. I. Opposite the base of a triangle. 0$OORIWKHVLGHVDUHWKHVDPHOHQJWK N. Is perpendicular to its altitude. $QVZHUV-)2$37+5%016&,'(*4/ 20K — 360 — 'H¿QLWLRQV polygon - a plane shape with three or more straight sides. triangle - any polygon with three sides. interior - WKHVHWRISRLQWVHQFORVHGE\DJHRPHWULF¿JXUH acute (triangle) - DWULDQJOHLQZKLFKDOORIWKHLQWHULRUDQJOHVDUHDFXWHRUOHVVWKDQo. obtuse (triangle) - a triangle that has an obtuse angle as one of its interior angles. right (triangle) - DWULDQJOHZKLFKKDVDULJKWLQWHULRUDQJOH scalene (triangle) - a triangle for which all three sides have different lengths. isosceles (triangle) - a triangle with two sides that are the same length. equilateral (triangle) - DWULDQJOHIRUZKLFKDOOWKUHHVLGHVDUHWKHVDPHOHQJWKFRQJUXHQW equiangular (triangle) - a triangle for which all three angles are congruent. altitude - WKHGLVWDQFHEHWZHHQDYHUWH[RIDWULDQJOHDQGWKHRSSRVLWHVLGH base - the side of the triangle that is perpendicular to the altitude. base angle - either of two angles of a triangle that have the base for a side. vertex angle - the angle opposite the base of a triangle. leg - one of the sides of a triangle. hypotenuse - the side of a right triangle opposite the right angle. exterior angle (of a triangle) - DQDQJOHIRUPHGRXWVLGHDWULDQJOHZKHQRQHVLGHLVH[WHQGHG remote interior angle (of a triangle) - DQDQJOHRIDWULDQJOHWKDWLVQRWDGMDFHQWWRDQH[WHULRU angle. median (of a triangle) - DOLQHVHJPHQWGUDZQIURPRQHYHUWH[WRWKHPLGSRLQWRIWKHRSSRVLWH side. angle bisector - a line that divides an interior angle of a triangle in half. — 361 — Which Belongs 1. ,QDULJKWWULDQJOHWKHEDVHK\SRWHQXVHDOWLWXGHLVRSSRVLWHWKHYHUWH[ 2. 7KHFDSLWDOOHWWHU³$´LVDQLVRVFHOHVDVFDOHQHDULJKWWULDQJOH $QLQWHULRUH[WHULRUREWXVHDQJOHLQFOXGHVSRLQWVHQFORVHGE\DJHRPHWULF¿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o angle to the base of a triangle. 12. $QDFXWHVFDOHQHREWXVHWULDQJOHFDQQRWKDYHDo angle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riangle Obtuse 9HUWH[DQJOH 17. Equilateral Base angle 19. Leg 20. Median 1. Base 2. Isosceles Interior 5LJKW Base $QJOHELVHFWRU 7. Scalene Equiangular 9. Polygon 10. ([WHULRU — 362 — Multiple Choice $QHQFORVHG¿JXUHPDGHXSRIWKUHHVHJPHQWVLVD DVFDOHQHWULDQJOH EWULDQJOH FSRO\JRQ 2. The longest side of a right triangle is its DK\SRWHQXVH EDOWLWXGH FEDVH 7KHVLGHRIDWULDQJOHWKDWLVRSSRVLWHWKHYHUWH[DQJOHLVWKH DK\SRWHQXVH EEDVH FOHJ $VHJPHQWWKDWGLYLGHVDQDQJOHLQKDOILV DDPHGLDQ EDQDOWLWXGH FDQDQJOHELVHFWRU $ULJKWWULDQJOHFDQQRWDOVREH DHTXLDQJXODU EVFDOHQH FLVRVFHOHV :KLFKRIWKHVHDQJOHVLVRSSRVLWHWKHEDVHRIDWULDQJOH DYHUWH[DQJOH EEDVHDQJOH FH[WHULRUDQJOH $OORIWKHSRLQWVHQFORVHGE\DJHRPHWULF¿JXUHPDNHXSLWV DH[WHULRU EDOWLWXGH FLQWHULRU <RXPLJKW¿QGDo angle in an DDFXWHWULDQJOH EULJKWWULDQJOH FREWXVHWULDQJOH $VHJPHQWWKDWGLYLGHVRQHVLGHRIDWULDQJOHLQWRWZRHTXDOSDUWVLVFDOOHG DDQDQJOHELVHFWRU EDPHGLDQ FDQLVRVFHOHV $WULDQJOHLQZKLFKDOORIWKHDQJOHVDUHVPDOOHUWKDQULJKWDQJOHVLVDQ DDFXWHWULDQJOH EREWXVHWULDQJOH FLVRVFHOHVWULDQJOH — 363 — $VLGHRIDQLVRVFHOHVWULDQJOHWKDWLVQRWWKHEDVHLVFDOOHGD DDOWLWXGH EYHUWH[VLGH FOHJ ,IDWULDQJOHKDVWZRVLGHVWKDWDUHSHUSHQGLFXODULWLV DDQLVRVFHOHVWULDQJOH EDULJKWWULDQJOH FDVFDOHQHWULDQJOH :KHQDWOHDVWWZRVLGHVRIDWULDQJOHDUHFRQJUXHQWWKHWULDQJOHLV DLVRVFHOHV EHTXLODWHUDO FHTXLDQJXODU ,IDOOWKUHHVLGHVRIDWULDQJOHDUHWKHVDPHLWLV DHTXLDQJXODU ELVRVFHOHV FHTXLODWHUDO 7KHKHLJKWRIDWULDQJOHIURPLWVEDVHLVFDOOHGLWV DDOWLWXGH EPHGLDQ FK\SRWHQXVH $QHQFORVHG¿JXUHZLWKVWUDLJKWFRSODQDUVLGHVLVFDOOHGD DSRO\JRQ EH[WHULRUDQJOH FPHGLDQ $QDQJOHRIDWULDQJOHWKDWLVQRWDEDVHDQJOHZRXOGEHWKH DUHPRWHLQWHULRUDQJOH EPHGLDQDQJOH FYHUWH[DQJOH $QDQJOHWKDWIRUPHGE\WKHH[WHQVLRQRIRQHRIDWULDQJOH¶VVLGHVLVLWV DYHUWH[DQJOH EH[WHULRUDQJOH FEDVHDQJOH ,IDQDQJOHGRHVQRWIRUPDOLQHDUSDLUZLWKWKHH[WHULRUDQJOHRIDWULDQJOHLWZRXOGEHD DYHUWH[DQJOH EUHPRWHLQWHULRUDQJOH FEDVHDQJOH ,IDWULDQJOHLVQRWDQLVRVFHOHVWULDQJOHLWLV DVFDOHQH EULJKW FHTXLODWHUDO $QVZHUV EDEFDDFFEDFEDFDDFEED — 364 — Complete the Sentence 1. 7KHVHWRISRLQWVHQFORVHGE\DJHRPHWULF¿JXUHLVLWVBBBBBBBBBBBBB 2. $WULDQJOHLVDBBBBBBBBBBBBBBBBBBBBBLIWZRRILWVVLGHVDUHSHUSHQGLFXODU 6HJPHQW$'ZDVWKHBBBBBBBBBBBBBBBBBBBBRIWULDQJOH$%&EHFDXVHLWGLYLGHGDQJOH$ into two equal parts. $Q\VLGHRIDWULDQJOHLVDBBBBBBBBBBBBDOWKRXJKWKDWWHUPLVQRWXVXDOO\XVHGIRUD hypotenuse or a base. Three sides of unequal lengths would make up a ______________triangle. )URPWKHIURQWRIWKHKRXVHWKHURRIZDVVKDSHGOLNHDQBBBBBBBBBBBBBBBBBBBWULDQJOH because the two sides were of equal lengths. 7. $FRDWKDQJHUKDVPRUHRUOHVVWKHVKDSHRIDQBBBBBBBBBBBBBWULDQJOH ,QDWULDQJOHDBBBBBBBBBBBBBBBBBBBBBBLVQRWDGMDFHQWWRLWVH[WHULRUDQJOH 9. $OORIWKHDQJOHVRIWKDWWULDQJOHKDYHWKHVDPHPHDVXUHVRLWLVDQBBBBBBBBBBBBBBWULangle. 10. (DFKVLGHRIDQBBBBBBBBBBBBBBBBBBBBBBBBWULDQJOHKDGDOHQJWKRIFHQWLPHWHUV 11. $WULDQJOH¶VWKUHHDQJOHVPHDVXUHooDQGoVRLWLVDQBBBBBBBBBBWULDQJOH 12. $BBBBBBBBBBBBBBBBBBBBBRIDWULDQJOHELVHFWVWKHEDVHRIWKHWULDQJOH :KHQ\RX¿QGWKHDOWLWXGHRIDWULDQJOH\RXPHDVXUHSHUSHQGLFXODUWRWKH _______________________. 7KHVKRUWHVWGLVWDQFHIURPWKHEDVHRIDWULDQJOHWRWKHRSSRVLWHYHUWH[LVWKH _______________. $QDQJOHRIDWULDQJOHPXVWEHWKHBBBBBBBBBBBBBBBBBLILWLVQRWDEDVHDQJOH $BBBBBBBBBBBBBBBBBBBBBLVDW\SHRISRO\JRQZLWKH[DFWO\WKUHHVLGHV 17. ,QDULJKWWULDQJOHWKHBBBBBBBBBBBBBBBBBBBLVQRWSHUSHQGLFXODUWRDQ\RWKHUVLGH ,IDVLGHRIDQWULDQJOHLVH[WHQGHGDQBBBBBBBBBBBBBBBBBBBBBBLVFUHDWHG 19. $BBBBBBBBBBBBBBBBBBBBBBBBBBBPLJKWKDYHHOHYHQVLGHV 20. ,IWKHEDVHRIDWULDQJOHIRUPVRQHVLGHRIDQDQJOHWKDWDQJOHLVFDOOHGDBBBBBBBBBBBBB $QVZHUV 1. interior 2. right triangle angle bisector leg scalene isosceles 7. obtuse remote interior angle 9. equiangular 10. equilateral 11. acute 12. median base altitude YHUWH[DQJOH triangle 17. hypotenuse H[WHULRUDQJOH 19. polygon 20. base angle — 365 — Creative Writing 7KLVSLFWXUHZDVWDNHQLQWKH¶VZKHQWKH¿UVWEULGJHIURP-XQHDXWR'RXJODV,VODQG was under construction. — 366 — Place-Based Practice Activity 'LYLGHVWXGHQWVLQWRVPDOOJURXSVDQGDVVLJQHDFKJURXSRQHW\SHRIWUL angle: Scalene Isosceles Equilateral $FXWH Obtuse 5LJKW (DFKJURXSVKRXOG¿QGDJRRGH[DPSOHRIWKHLUWULDQJOHVRPHZKHUHLQWKH VFKRRORUFRPPXQLW\DQGVNHWFKRUSKRWRJUDSKLW7KHQWKH\VKRXOGODEHODV many of the following as possible on their triangle: Base $OWLWXGH Median Base angle 9HUWH[DQJOH ([WHULRUDQJOH 5HPRWHLQWHULRUDQJOH $QJOHELVHFWRU Leg (DFKJURXSVKRXOGVKDUHWKHLUZRUNZLWKWKHFODVVE\PDNLQJDSRVWHURUDEULHI class presentation. $VNVWXGHQWVLQVPDOOJURXSVWRFKRRVHRQHRIWKHVHWRSLFV $UW &RQVWUXFWLRQ Navigation $VNHDFKJURXSWRGRUHVHDUFKWR¿QGRXWKRZWULDQJOHVDUHLPSRUWDQWRUUHOHYDQW WRWKHWRSLFWKH\FKRVH$VWKH\UHVHDUFKWKH\VKRXOGLGHQWLI\WKHW\SHVDQG SDUWVRIWULDQJOHVWKDWWKH\¿QGRXWDERXW — 367 — — 368 — Unit Assessment — 369 9 — — 370 — Geometry: Unit 5-Triangles Quiz Name: ___________________ Date: ___________________ Match the key vocabulary word on the left with correct shape on the right. Put the letter of the shape, in front of the word it matches. a. illustration of scalene triangle 1) _____ polygon 2) _____ acute triangle 3) _____ right triangle b. 4) _____ scalene triangle 5) _____ isosceles triangle illustration of acute triangle 6) _____ equilateral triangle 7) _____ equiangular triangle c. illustration of right triangle d. illustration of equiangular triangle e. illustration of polygon f. illustration of equilateral triangle g. illustration of isosceles triangle — 371 — Illustrations: In the next three text items, follow the instructions for each item. 8) Look at the shape below and label it in the space provided. Matt...insert an illustration of a triangle ____________________________ label the shape 9) In the illustration below, put an X on the base of the triangle. illustration of triangle 10) in the space below, draw a picture showing an angle bisector. 11) In the illustration below, put an X on the vertex angle. illustration of triangle — 372 — Fill in the blank: Complete each sentence with the word that fits best. Choose the word from the Word Bank. Word Bank angle bisector base base exterior hypotenuse leg median remote interior vertex 12) The 13) A /an 14) A/an extended. 15) of a triangle is the angle opposite the base. I s the side of a right triangle opposite the right angle. angle of a triangle is an angle formed outside a triangle when one line is angles of a triangle are the two angles that are farthest from an exterior angle. 16) The angle of a triangle is either of two angles that have the base for a side. 17) The of a triangle is a line segment drawn from one vertex to the midpoint of the opposite side. 18) The angle of a triangle is either of two angles that have the base for the side. True/False: Read each item carefully and decide if the statement is true or false. Circle the correct answer. 19) The exterior is the set of points enclosed by a geometric figure. a) True b) False 20) Triangles, rectangles, trapezoids, pentagons, and octagons are all examples of polygons. a) True b) False — 373 — — 374 — Geometry: Unit 5-Triangles Quiz Name: ___________________ Date: ___________________ Match the key vocabulary word on the left with correct shape on the right. Put the letter of the shape, in front of the word it matches. 1) e polygon 2) b acute triangle 3) c right triangle 4) a scalene triangle 5) g isosceles triangle 6) f equilateral triangle 7) d equiangular triangle a. illustration of scalene triangle b. llustration of acute triangle C illustration of right triangle d. illustration of equiangular triangle e. Illustration of polygon f. illustration of equilateral triangle g. llustration of isosceles triangle — 375 — Illustrations: In the next three text items, follow the instructions for each item. 8) Look at the shape below and label it in the space below it. Matt...insert an illustration of a triangle ___________________________________ Triangle 9) In the illustration below, put an X on the base of the triangle. illustration of triangle illustration of triangle with an X on the base. 10) in the space below, draw a picture showing an angle bisector. Show illustration of angle bisector. 11) In the illustration below, put an X on the vertex angle. illustration of triangle illustration of triangle with an X on the vertex — 376 — Fill in the blank: Complete each sentence with the word that fits best. Choose the word from the Word Bank. Word Bank angle bisector base base exterior hypotenuse leg median remote interior vertex 12) The leg of a triangle is the angle opposite the base. 13) A /an hypotenuse is the side of a right triangle opposite the right angle. 14) A/an exterior angle of a triangle is an angle formed outside a triangle when one line is extended. 15) remote interior angles of a triangle are the two angles that are farthest from an exterior angle. 16) The base angle of a triangle is either of two angles that have the base for a side. 17) The median of a triangle is a line segment drawn from one vertex to the midpoint of the opposite side. 18) The base angle of a triangle is either of two angles that have the base for the side. True/False: Read each item carefully and decide if the statement is true or false. Circle the correct answer. 19) The exterior is the set of points enclosed by a geometric figure. a) True b) False 20) Triangles, rectangles, trapezoids, pentagons, and octagons are all examples of polygons. a) True b) False — 377 — — 378 —
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