Geometry, Teacher's Edition

399 Pages·2012·4.42 MB·English

Geometry Teacher’s Edition
Jen Kershaw, (JenK)
Melissa Kramer, (MelissaK)
Teresa Zwack, (TeresaZ)
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AUTHORS
Jen Kershaw,(JenK)
To access a customizable version of this book, as well as other
Melissa Kramer,(MelissaK)
interactivecontent,visitwww.ck12.org
Teresa Zwack, (TeresaZ)
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Contents
1 GeometryTE-TeachingTips 1
1.1 BasicsofGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 ReasoningandProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 ParallelandPerpendicularLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 CongruentTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 RelationshipsWithinTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.6 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.7 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8 RightTriangleTrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.9 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.10 PerimeterandArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
1.11 SurfaceAreaandVolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
1.12 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2 GeometryTE-CommonErrors 67
2.1 BasicsofGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.2 ReasoningandProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.3 ParallelandPerpendicularLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.4 CongruentTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.5 RelationshipsWithinTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.6 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.7 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.8 RightTriangleTrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
2.9 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.10 PerimeterandArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
2.11 SurfaceAreaandVolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
2.12 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3 GeometryTE-Enrichment 140
3.1 BasicsofGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.2 ReasoningandProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
3.3 ParallelandPerpendicularLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
3.4 CongruentTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.5 RelationshipsWithinTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
3.6 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
3.7 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
3.8 RightTriangleTrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
3.9 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
3.10 PerimeterandArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
3.11 SurfaceAreaandVolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
3.12 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
Contents
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4 GeometryTE-DifferentiatedInstruction 223
4.1 BasicsofGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
4.2 ReasoningandProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
4.3 ParallelandPerpendicularLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
4.4 CongruentTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
4.5 RelationshipsWithinTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
4.6 Quadrilateral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
4.7 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
4.8 RightTriangleTrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
4.9 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
4.10 PerimeterandArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
4.11 SurfaceAreaandVolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
4.12 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
5 GeometryTE-ProblemSolving 310
5.1 BasicsofGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
5.2 ReasoningandProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
5.3 ParallelandPerpendicularLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
5.4 CongruentTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
5.5 RelationshipswithinTriangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
5.6 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
5.7 Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
5.8 RightTriangleTrigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
5.9 Circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
5.10 PerimeterandArea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
5.11 SurfaceAreaandVolume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
5.12 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Contents
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C 1
HAPTER
Geometry TE - Teaching
Tips
Chapter Outline
1.1 BASICS OF GEOMETRY
1.2 REASONING AND PROOF
1.3 PARALLEL AND PERPENDICULAR LINES
1.4 CONGRUENT TRIANGLES
1.5 RELATIONSHIPS WITHIN TRIANGLES
1.6 QUADRILATERALS
1.7 SIMILARITY
1.8 RIGHT TRIANGLE TRIGONOMETRY
1.9 CIRCLES
1.10 PERIMETER AND AREA
1.11 SURFACE AREA AND VOLUME
1.12 TRANSFORMATIONS
Chapter1. GeometryTE-TeachingTips
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1.1 Basics of Geometry
Points, Lines, and Planes
Pacing: Thislessonshouldtakeapproximatelythreeclassperiods.
Goal: Thislessonintroducesstudentstothebasicprinciplesofgeometry. Studentswillbecomefamiliarwiththree
primary undefined geometric terms and how these terms are used to define other geometric vocabulary. Finally,
studentsareintroducedtotheconceptofdimensions.
Study Skills Tips! Start your students off on the correct foot – vocabulary is a necessity in geometry success!
Devote five minutes of each class period to creating flash cards of the major terminology of this text. Use personal
whiteboards to perform quick vocabulary checks. Or, better yet, visit Discovery School’s puzzle maker and make
yourownwordsearchesandcrosswords(http://puzzlemaker.discoveryeducation.com/)!
Language Arts Connection! To give an example of why some words are undefined, use the concept of circularity.
Students use a dictionary, either electronic or paper (yes, they are still printed!) to complete this activity. Ask
students to look up the word point in their reference. Find a key word in that definition. Students should continue
thisprocessuntilthewordpoint isfound. Repeatthisprocessforlineandplane. Therationalebehindthisactivity
is for students to see there is no one way to define these geometric terms, thus allowing them to be undefined but
recognizable.
Real World Connection! Have students identify real-life examples of points, lines, planes in the classroom, as well
as sets of collinear and coplanar. For example, points could be chairs, lines could be the intersection of the ceiling
andwall,andthefloorisagreatmodelofaplane. Ifyourchairsarefour-legged,thisisafantasticexampleofwhy
3pointsdetermineaplane,notfour. Fourleggedchairstendtowobble,while3−leggedstoolsremainstable.
Tohelpstudentsunderstanddimension,usethefollowingtable:
TABLE 1.1:
Zero-dimensional 1-dimensional(length) 2-dimensional(lengthand 3-dimensional (length,
width) width,andheight)
Have studentswrite abstractexamples of eachdimension (point, line, plane, prism, etc) inthe first row. Then have
studentsbrainstormreal-lifeexamplesofeachdimension. Completethetablebygatheringtheresponsesofvarious
students.
Segments and Distances
Pacing: Thislessonshouldtakeoneclassperiod
Goal: Students should be familiar with using rulers to measure distances. This lesson incorporates geometric
postulatesandpropertiestomeasurement,suchastheSegmentAdditionProperty.
RealWorldConnection! Toreviewtheconceptofmeasurement,useamapofyourcommunity. Labelseveralthings
on your map important to students – high school, grocery store, movie theatre, etc. Have students practice finding
1.1. BasicsofGeometry
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thedistancesbetweenlandmarks“asthecrowflies.”
Extension! Discusswithyourstudentstherationaleofusingdifferentunitsofdistance–inch,foot,centimeter,mile,
etc. Why are things measured in inches as opposed to fractional feet? This is also a great time to introduce the
differencebetweenthemetricsystemandtheU.S.measurementsystem. Havestudentsperformresearchregarding
whytheUnitedStatescontinuestouseitssystemwhilethemajorityofothercountriesusethemetricsystem. Provide
prosandconstousingeachtypeofsystem.
Funtip! Havestudentsdevisetheirownmeasurementdevice. Studentscanusetheirinventiontomeasureaschool
hallway,parkinglot,orfootballfield. Engageinawhole-classdiscussionregardingtheresults.
Refresher! Studentsmayneedarefresherregardingmultiplyingunits. Havethestudentswriteoutthecompleteunit,
asonpage18,andshowstudentshowunitscanbecross-canceled.
Look out! While the Segment Addition Property seems simple, students begin to struggle once proofs come into
play. Remind students that the Segment Addition Property allows an individual to combine smaller measurements
ofalinesegmentintoitswhole.
Rays and Angles
Pacing: Thislessonshouldtakeoneclassperiod
Goal: Thislessonintroducesstudentstoraysandanglesandhowtouseaprotractortomeasureangles. Severalreal
worldmodelsareusedtoillustratetheconceptsofangles.
Real World Connection! Have students Think-Pair-Share their answers to the opening question, “Can you think of
otherreal-lifeexamplesofrays?”Chooseseveralgroupstosharewiththeclass.
Notation Tip! Beginning geometry students may get confused regarding the ray notation. Draw rays in different
directions so students become comfortable with the concept that ray notation always points to the right, regardless
ofthedrawnray’sorientation.
Teaching Strategy! Using a classroom sized protractor will allow students to check to make sure their calculations
arethesameasyours. Betteryet, useanoverheadprojectorordigitalimagertodemonstratetheproperwaytouse
aprotractor.
TeachingStrategy! Agoodhabitforstudentsistonameanangleusingofallthreeletters. Thisbecomesimportant
whenlabelingverticesoftrianglesandlabelingsimilarandcongruentfiguresusingthesimilaritystatement. Further-
more,stresstostudentstheuseofdoubleandtriplearcstodenoteanglesofdifferentmeasurements. Studentscanget
caughtupinthemassamountsofnotationandforgetthisimportantconcept,especiallyduringtrianglecongruency.
StresstheparallelismbetweentheSegmentAdditionPropertyandAngleAdditionProperty. Studentswilldiscover
thatmanygeometricaltheoremsandpropertiesarequitesimilar,withperhapsonewordschanged. Yet,themeaning
remainsthesame.
ArtsandCraftsTime! Havestudentstakeapieceofpaperandfolditatanyangleoftheirchoosingfromthecorner
of the paper. Open the fold and refold the paper at a different angle, forming two “rays” and three angles. Show
howtheangleadditionpropertycanbeusedbyaskingstudentstomeasuretheircreatedanglesandfindingthesum
–theyshouldequal90degrees!
Physical Models! The angle formed at a person’s elbow is a useful physical model of angles. Have the students
puttheirarmstraightout, illustratingastraightangle. Thenhavethestudentgraduallyturntheirarmup(ordown)
gradually to demonstrate how the degree changes. Use several students as examples to show that the length of the
forearmandbicepdonotchangetheanglemeasurement.
Chapter1. GeometryTE-TeachingTips
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Segments and Angles
Pacing: Thislessonshouldtakeonetooneandone-halfclassperiods
Goal: Thelessonintroducesstudentstotheconceptofcongruencyandbisectors. Studentswillusealgebratowrite
equivalencestatementsandsolveforunknownvariables.
Have Fun! Have students do a call-back, similar to what cheerleaders do. You call out “AB” and students would
retort, “The distance between!” Continue this for several examples so students begin to see the difference between
thedistancenotationandsegmentnotation.
This is a great lesson for students to create a “dictionary” of all the notation and definitions learned thus far. In
additiontotheflashcardsstudentsaremaking,thedictionaryprovidesaninvaluablereferencebeforeassessments.
WhenteachingtheMidpointPostulate,reiteratetostudentsthatthisreallyisthearithmeticaverageoftheendpoints,
incorporatingalgebraandstatisticsintothelesson.
Visualization! Studentshavenotlearnedaboutaperpendicularbisector. HavestudentscompleteExample3without
usingtheirtextsasguides. Havestudentsshowtheirbisectors. Hopefullyyourclasswillconstructmultiplebisectors,
not simply those that are perpendicular. This helps students visualize that there are an infinite amount of bisectors,
butonlyonethatisperpendicular.
FunTip! Tovisualizetheanglecongruencetheoremandprovideameansofassessingtheabilitytouseaprotractor,
give students entering your class an angle measure on a slip of paper (the measurements should repeat). Have the
students construct the angle as a warm up. Then have the students find their “matching” partner and check their
partner’sangleusingaprotractor.
Physical Models! Once students have reviewed Example 5, have them copy the angle onto a sheet of notebook
paper or patty paper and measure the degree of the bisector. Students will construct a fold at that particular angle
measurementtoseetheanglebisectorray.
RealLifeApplication! Anothermethodofillustratinganglebisectorsistoshowacompassrose,asshownbelow.
http://commons.wikimedia.org/wiki/File:Compass_rose_browns_00.svg
StudentscanseehowdirectionssuchasSWS,NNW,etcbisectthetraditionalfour-cornerdirections.
Angle Pairs
Pacing: Thislessonshouldtakeonetooneandone-halfclassperiods
1.1. BasicsofGeometry
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Goal: Anglepairsareimperativetogeometry. Thislessonintroducesstudentstocommonanglepairs.
Inquiry Learning! Students should be encouraged to learn through self-discovery whenever possible. To illustrate
the concept of the Linear Pair Postulate, offer several examples of linear pairs. Have students measure each angle
andfindthesumofthelinearpair. Studentsshoulddiscoveranylinearpairofanglesissupplementary.
To further illustrate the idea of vertical angles, extend the adjacent ray of the previous linear pairs to a line. Have
studentsrepeattheprocessofmeasuringtheangles, notatingthelinearpairs. Studentswillcometotheconclusion
thattheanglesoppositeinthe[U+0080][U+009C]X[U+0080][U+009D]areequal.
Students tend to get confused with the term vertical, as in vertical angles. Vertical angles are named because the
anglesshareavertex,notnecessarilybecausetheyareinaverticalmanner.
InterdisciplinaryConnection! NASAhasdevelopedmanylessonplansthatinfusescience, technology, andmathe-
matics. Thefollowinglinkwilltakeyoutoalessonplanincorporatingtheseasonsandverticalangles. http://sunea
rthday.nasa.gov/2005/educators/AOTK_lessons.pdf
Classifying Triangles
Pacing: Thislessonshouldtakeoneclassperiod
Goal: Students have previously experienced triangle terminology: scalene, equilateral, isosceles. This lesson
incorporatesthesetermswithotherdefiningcharacteristics.
InClassActivity: Givepairsofstudentsthreerawpiecesofspaghetti(youcanalsousenon-bendystraws). Instruct
onepartnertorecreatethebelowtablewhilethesecondmakestwobreaksinthespaghetti. Itisokayifsomebreaks
away!
Thestudentsaretomeasurethethreepiecesformedbythetwobreaksandattempttoconstructatriangleusingthese
segments. Students will reach the conclusion that the sum of two segments must always be larger than the third
if a triangle is to be formed. The Triangle Inequality Theorem can be found in the lesson entitled Inequalities in
Triangles
TABLE 1.2:
Segment1Length(incm) Segment2Length(incm) Segment3Length(incm) Can a triangle be formed
(Yes/No)
Showing students the difference between line segments and curves, introduce cooked spaghetti. The flexibility of
the spaghetti demonstrates to students that segments must be straight in order to provide rigidity and follow the
definitionsofpolygons.
Students can express the concepts presented in this lesson using a Venn diagram or a hierarchy. If students are not
familiarwithahierarchy,remindstudentsahierarchyisanorderingofrelatedobjectsfromthemostgeneraltothe
mostspecific. Anexampleisshownbelow.
Chapter1. GeometryTE-TeachingTips