Math 2
Unit 6 ‐ Similarity & Right Triangle Trigonometry
Thinking Through the Lesson Protocol (TTLP) Template
Concepts /
Key Question
Task /
Activity
Page
Describing the essential features of a dilation
6.1 Photocopy Faux
What features of a dilation are important and how can I identify these? Pas
5
Examining proportional relationships in triangles that are known to be similar to
6.2 Triangle Dilations each other based on dilations
How can I describe the relationships between the sides of two triangles when one is a dilation of the other?
10
Comparing definitions of similarity based on dilations and relationships between
6.3 Similar Triangles corresponding sides and angles and Other Figures
What statements can I prove about similar polygons?
What other criteria can I use to determine if triangles are similar?
17
Examining proportional relationships of segments when two transversals intersect 6.4 Cut By a sets of parallel lines
Transversal
What proportionality statements can I make when a triangle is intersected by a line parallel to one of its sides?
23
Applying theorems about lines, angles and proportional relationships based on
6.5 Measured similar triangles
Reasoning
How can I use theorems about shapes, angles and parallel lines to find missing values in a diagram?
How can I prove that an observed pattern always holds?
28
Applying understanding of similar and congruent triangles to find midpoint or any
6.6 Yard Work in point on a line segment that partitions the segment in a given ratio
Segments
How can I find the point that partitions a segment by a particular ratio?
33
Using similar triangles to prove the Pythagorean theorem and theorems about
6.7 Pythagoras by geometric means in right triangles
Proportions
How can I use similar triangles to prove the Pythagorean Theorem and theorems about triangle altitudes?
40
Developing an understanding of right triangle trigonometric relationships based on
6.8 Are Relationships similar triangles?
Predictable?
What side ratios are consistent between similar triangles?
45
Finding relationships between sine and cosine ratios for right triangles, including
6.9 Relationships with the Pythagorean identity
Meaning
How are the sine and cosine ratios for right triangles related?
50
Solving for unknown values in right triangles using trigonometric ratios
6.10 Finding the Value
How can trig functions and their inverses help us find missing parts of a triangle? of a Relationship
54
Practice setting up and solving right triangles to model real world contexts
6.11 Solving Right
How can I use right triangle trigonometry to find lengths, angles and distances Triangles Using between objects? Trigonometric
Relationships
58
1
Secondary Two Mathematics:
An Integrated Approach
Module 6
Similarity and Right Triangle
Trigonometry
By
The Mathematics Vision Project:
Scott Hendrickson, Joleigh Honey,
Barbara Kuehl, Travis Lemon, Janet Sutorius www.mathematicsvisionproject.org In partnership with the
Utah State Office of Education
© 2013 Mathematics Vision Project | MVP
In partnership with the Utah State Office of Education
Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.
2
Module'6'–'Similarity'&'Right'Triangle'Trigonometry'
Classroom(Task:((6.1(Photocopy(Faux(Pas(–(A(Develop(Understanding(Task(
Describing*the*essential*features*of*a*dilation**(G.SRT.1)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.1(
(
Classroom(Task:((6.2(Triangle(Dilations(–(A(Solidify(Understanding(Task(
Examining*proportionality*relationships*in*triangles*that*are*know*to*be*similar*to*each*other*based*on* dilations**(G.SRT.2,(G.SRT.4)( Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.2(
(
Classroom(Task:((6.3(Similar(Triangles(and(Other(Figures(–(A(Solidify(Understanding(Task(
Comparing*definitions*of*similarity*based*on*dilations*and*relationships*between*corresponding*sides*and*
angles***(G.SRT.2,(G.SRT.3)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.3(
(
Classroom(Task:((6.4(Cut(by(a(Transversal(–(A(Solidify(Understanding(Task(
Examining*proportional*relationships*of*segments*when*two*transversals*intersect*sets*of*parallel*lines(
(G.SRT.4)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.4(
(
Classroom(Task:((6.5(Measured(Reasoning(–(A(Practice(Understanding(Task(
Applying*theorems*about*lines,*angles*and*proportional*relationships*when*parallel*lines*are*crossed*by*
multiple*transversals*(G.CO.9,(G.CO.10,(G.SRT.4,(G.SRT.5)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.5(
(
Classroom(Task:((6.6(Yard(Work(in(Segments(–(A(Solidify(Understanding(Task(
Applying*understanding*of*similar*and*congruent*triangles*to*find*midpoint*or*any*point*on*a*line* segment*that*partitions*the*segment*in*a*given*ratio.((G.GPE.6)( Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.6(
(
Classroom(Task:((6.7(Pythagoras(by(Proportions(–(A(Practice(Understanding(Task(
Using*similar*triangles*to*prove*the*Pythagorean*theorem*and*theorems*about*geometric*means*in*right*
triangles*(G.SRT.4,(G.SRT.5)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.7(
(
Classroom(Task:((6.8(Are(Relationships(Predictable?(–(A(Develop(Understanding(Task(
Developing*and*understanding*of*right*triangle*trigonometric*relationships*based*on*similar*triangles(
(G.SRT.6,(G.SRT.8)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.8(
(
Classroom(Task:((6.9(Relationships(with(Meaning(–(A(Solidify(Understanding(Task(
Finding*relationships*between*the*sine*and*cosine*ratios*for*right*triangles,*including*the*Pythagorean*
identity*((G.SRT.6,(G.SRT.7,(F.TF.8)(
©"2013"Mathematics"Vision"Project"|"M
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Licensed(under(the(Creative(Commons(Attribution4NonCommercial4ShareAlike(3.0(Unported(license."
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Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.9(
(
Classroom(Task:((6.10(Finding(the(Value(of(a(Relationship(–(A(Solidify(Understanding(Task(
Solving*for*unknown*values*in*right*triangles*using*trigonometric*ratios((G.SRT.7,(G.SRT.8)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.10(
(
Classroom(Task:((6.11(Solving(Right(Triangles(Using(Trigonometric(Relationships(–(A(Practice(
Understanding(Task(
Practicing*setting*up*and*solving*right*triangles*to*model*real*world*contexts*(G.SRT.6,(G.SRT.7,(F.TF.8)(
Ready,(Set,(Go(Homework:((Similarity(&(Right(Triangle(Trigonometry(6.11(
(
(
(
(
©"2013"Mathematics"Vision"Project"|"M
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Licensed(under(the(Creative(Commons(Attribution4NonCommercial4ShareAlike(3.0(Unported(license."
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4
A"Develop"Understanding"Task"
Mr."and"Mrs."Donahue"are"making"a"scrapbook"for"Mr."Donahue’s"grandfather’s"
75th"birthday"party,"and"they"want"to"enlarge"a"sketch"of"their"grandfather"which"was"drawn"when"
he"was"in"college."""They"have"purchased"some"very"expensive"scrapbook"paper,"and"they"would"like" this"image"to"be"centered"on"the"page.""Because"they"are"unfamiliar"with"the"process"of"enlarging"an" image,"they"have"come"to"Burnell"for"help."
“We"would"like"to"make"a"copy"of"this"image"that"is"twice"as"big,"and"centered"in"the"middle"of"this"
very"expensive"scrapbook"paper,”"Mrs."Donahue"says.""“Can"you"help"us"with"that?”"
“Certainly,”"says"Burnell.""“Glad"to"be"of"service.”"
Burnell"taped"the"original"image"in"the"middle"of"a"white"piece"of"paper,"placed"it"on"the"glass"of"the" photocopy"machine,"inserted"the"expensive"scrapbook"paper"into"the"paper"tray,"and"set"the" enlargement"feature"at"200%."
In"a"moment,"this"image"was"produced."
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“You’ve"ruined"our"expensive"paper,”"cried"Mrs."Donahue."“Much"of"the"image"is"off"the"paper"
instead"of"being"centered.”"
“And"this"image"is"more"than"twice"as"big,”"Mr."Donahue"complained.""“One"fourth"of"grandpa’s"
picture"is"taking"up"as"much"space"as"the"original.”"
"
©"2013"Mathematics"Vision"Project"|"M
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"
Burnell"has"a"new"job"at"a"copy"center"helping"people"use"the"photocopy" machines.""Burnell"thinks"he"knows"everything"about"making"photocopies,"and" so"he"didn’t"complete"his"assignment"to"read"the"training"manual."
2013(www.flickr.com/photos/anorwood(
6.1"Photocopy"Faux"Pas"
In"the"diagram"below,"both"the"original"image—which"Burnell"taped"in"the"middle"of"a"sheet"of" paper—and"the"copy"of"the"image"have"been"reproduced"in"the"same"figure."" 1. Explain"how"the"photocopy"machine"produced"the"partial"copy"of"the"original"image."
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2. Using"a"“rubber"band"stretcher”"finish"the"rest"of"the"enlarged"sketch."
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3. "Where"should"Burnell"have"placed"the"original"image"if"he"wanted"the"final"image"to"be" centered"on"the"paper?" "
4. Mr."Donahue"complained"that"the"copy"was"four"times"bigger"than"the"original.""What"do" you"think?""Did"Burnell"double"the"image"or"quadruple"it?""What"evidence"would"you"use"to" support"your"claim?"
"
5. Transforming"a"figure"by"shrinking"or"enlarging"it"in"this"way"is"a"called"a"dilation.""Based"on" your"thinking"about"how"the"photocopy"was"produced,"list"all"of"the"things"you"need"to"pay" attention"to"when"dilating"an"image."
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6
Similarity)&)Right)Triangle)Trigonometry) 6.1)
Name:!
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Ready,'Set,'Go!'
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2013!www.flickr.com/photos/anorwood!
Ready'
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Topic:!Scale!factors!for!similar!shapes.!
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Give'the'factor'by'which'each'pre6image'was'multiplied'to'create'the'image.'Use'the'scale'
factor'to'fill'in'any'missing'lengths.'
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1.!
2.!
Pre4image!
Pre4image!
Image!
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3.!!
4.!!
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5.!
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6.!
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Licensed!under!the!Creative!Commons!Attribution4NonCommercial4ShareAlike!3.0!Unported!license"
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7
Similarity)&)Right)Triangle)Trigonometry) 6.1)
!
Set'
Topic:!Dilations!in!real!world!contexts!
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For'each'real6world'context'or'circumstance'determine'the'center'of'the'dilation'and'the'
tool'being'used'to'do'the'dilation.'
'
7.!!
8.!!
Fran!walks!backward!to!a!distance!that!will!
The!theatre!technician!plays!with!the!zoom!in!
allow!her!family!to!all!show!up!in!the!photo!she! and!out!buttons!in!effort!to!fill!the!entire!movie! is!about!to!take.! screen!with!the!image.!
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9.!!
10.!!
Melanie!estimates!the!height!of!the!waterfall!by! A!digital!animator!creates!artistic!works!on!her! holding!out!her!thumb!and!using!it!to!see!how! computer.!She!is!currently!doing!an!animation! many!thumbs!tall!to!the!top!of!the!waterfall! that!has!several!telephone!poles!along!a!street! from!where!she!is!standing.!She!then!uses!her! that!goes!off!into!the!distance.! thumb!to!see!that!a!person!at!the!base!of!the! ! waterfall!is!half!a!thumb!tall.! !
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11.!!
12.!!
Ms.!Sunshine!is!having!her!class!do!a!project!
A!copy!machine!is!set!at!300%!for!making!a!
with!a!rubber4band!tracing!device!that!uses! photo!copy.! three!rubber!bands.!
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8
Similarity)&)Right)Triangle)Trigonometry) 6.1)
!
Go'
Topic:!Rates!of!change!related!to!linear,!exponential!and!quadratic!functions!
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Determine'whether'the'given'representation'is'representative'of'a'linear,'exponential'or'
quadratic'function,'classify'as'such'and'justify'your'reasoning.'
13.!
14.!
!
X!
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7!
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12!
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19!
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5!
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Type!of!function:!
Justification:!
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Type!of!function:!
Justification:!
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15.!!!! = 3! ! + 3!!
Type!of!function:!
Justification:!
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16.!!!!! = !7! − 10!
Type!of!function:!
Justification:!
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17.!
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18.!
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!Type!of! function:! !
Justification:!
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7!
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25!
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5!
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Type!of!function:!
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Justification:!
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©"2013"MATHEMATICS"VISION"PROJECT"|"M
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Licensed!under!the!Creative!Commons!Attribution4NonCommercial4ShareAlike!3.0!Unported!license"
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9
A"Solidify"Understanding"Task"
1. Given"ΔABC,"use"point"M"as"the"center"of"a"dilation" to"locate"the"vertices"of"a"triangle"that"has"side" lengths"that"are"three"times"longer"than"the"sides" of"ΔABC." "
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2. Now"use"point"N"as"the"center"of"a"dilation"to" locate"the"vertices"of"a"triangle"that"has"side"lengths"that"are"oneKhalf"the"length"of"the" sides"of"ΔABC."
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M"
A
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B"
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©"2013"Mathematics"Vision"Project"|"M
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In"partnership"with"the"Utah"State"Office"of"Education"""
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Licensed(under(the(Creative(Commons(Attribution4NonCommercial4ShareAlike(3.0(Unported(license."
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2013(www.flickr.com/photos/tshinya.omachi(
6.2"Triangle"Dilations"
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3. Label"the"vertices"in"the"two"triangles"you"created"in"the"diagram"above.""Based"on"this" diagram,"write"several"proportionality"statements"you"believe"are"true.""First"write"your" proportionality"statements"using"the"names"of"the"sides"of"the"triangles"in"your"ratios.""Then" verify"that"the"proportions"are"true"by"replacing"the"side"names"with"their"measurements," measured"to"the"nearest"millimeter."
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My$list$of$proportions:""(try"to"find"at"least"10"proportionality"statements"you"believe"are" true)" ©"2013"Mathematics"Vision"Project"|"M
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4. Based"on"your"work"above,"under"what"conditions"are"the"corresponding"line"segments"in" an"image"and"its"preKimage"parallel"after"a"dilation?"That"is,"which"word"best"completes"this" statement?"
"
After"a"dilation,"corresponding"line"segments"in"an"image"and"its"pre 