01 GEOMETRY.doc

7/27/2019 01 GEOMETRY.doc

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Geometry


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Geometry

Table of Contents

Unit 1: Geometric Patterns and Reasoning.....................................................................1

Unit 2: Reasoning and Proof...........................................................................................15

Unit 3: Parallel and Perpendicular Relationsips........................................................2!

Unit ": Triangles and #uadrilaterals.............................................................................3$

Unit 5: %imilarity and Trigonometry.............................................................................53

Unit $: &rea' Polyedra' %urface &rea' and (olume...................................................$!

Unit !: Circles and %peres.............................................................................................!)

Unit ): Transformations..................................................................................................*1

Most of the math symbols in this document were made withMath Type software.Specific fonts must be installed on the users computer for the symbols to be read.It is best to use thepdf format of a document if a printed copy is needed.

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Louisiana Comprehensive Curriculum' Re+ised 2,,)

Course -ntroduction

The ouisiana !epartment of "ducation issued the Comprehensive Curriculumin #$$%. Thecurriculum has been re&ised based on teacher feedbac', an e(ternal re&iew by a team of content

e(perts from outside the state, and input from course writers. )s in the first edition, theLouisiana Comprehensive Curriculum, re&ised #$$* is ali+ned with state content standards, asdefined by rade-e&el "(pectations "s, and or+ani0ed into coherent, time-bound unitswith sample acti&ities and classroom assessments to +uide teachin+ and learnin+. The order ofthe units ensures that all "s to be tested are addressed prior to the administration of i")1assessments.

istrict -mplementation Guidelines

ocal districts are responsible for implementation and monitorin+ of theLouisianaComprehensive Curriculumand ha&e been dele+ated the responsibility to decide if

units are to be tau+ht in the order presented

substitutions of e2ui&alent acti&ities are allowed "S can be ade2uately addressed usin+ fewer acti&ities than presented permitted chan+es are to be made at the district, school, or teacher le&el

!istricts ha&e been re2uested to inform teachers of decisions made.

-mplementation of &cti+ities in te Classroom

Incorporation of activities into lesson plans is critical to the successful implementation of the

Louisiana Comprehensive Curriculum. esson plans should be desi+ned to introduce students toone or more of the acti&ities, to pro&ide bac'+round information and follow-up, and to preparestudents for success in masterin+ the rade-e&el "(pectations associated with the acti&ities.esson plans should address indi&idual needs of students and should include processes for re-

teachin+ concepts or s'ills for students who need additional instruction. )ppropriateaccommodations must be made for students with disabilities.

/e0 eatures

Content Area Literacy Strategiesare an inte+ral part of appro(imately one-third of the acti&ities.Strate+y names are italici0ed. The lin' &iew literacy strate+y descriptions opens a documentcontainin+ detailed descriptions and e(amples of the literacy strate+ies. This document can alsobe accessed directly athttp://www.louisianaschools.net/lde/uploads/33$%4.doc.

)Materials List is pro&ided for each acti&ity andBlacline Masters !BLMs" are pro&ided toassist in the deli&ery of acti&ities or to assess student learnin+. ) separate 5lac'line Master

document is pro&ided for each course.

TheAccess #uide to the Comprehensive Curriculumis an online database ofsuggested strategies' accommodations' assisti+e tecnology' and assessment

options tat may pro+ide greater access to te curriculum acti+ities. TheAccess #uidewill be piloted durin+ the #$$*-#$$6 school year in rades 7 and *,with other +rades to be added o&er time. 8lic' on theAccess #uideicon found on the first pa+eof each unit or by +oin+ directly to the url http://mconn.doe.state.la.us/access+uide/default.asp(.
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Louisiana Comprehensive Curriculum$ %evised &''(

Geometry

Unit 1: Geometric Patterns and Reasoning

Time rame: )ppro(imately three wee's

Unit escription

This unit introduces the use of inducti&e reasonin+ to e(tend a pattern, and then find therule for +eneratin+ the nth term in a se2uence. )dditionally, countin+ techni2ues andmathematical modelin+, includin+ line of best fit, will be used to find solutions to real-life problems.

%tudent Understandings

Students apply inducti&e reasonin+ to identify terms of a se2uence by +eneratin+ a rulefor the nth term. Students reco+ni0e linear &ersus non-linear sets of data and can 9ustifytheir reasonin+. Students can apply countin+ techni2ues to sol&e real-life problems.

Guiding #uestions

3. 8an students +i&e e(amples of correct and incorrect usa+e of inducti&ereasonin+

#. 8an students use countin+ techni2ues with patterns to determine the number ofdia+onals and the sums of an+les in poly+ons

;. 8an students state the characteristics of a linear set of data7. 8an students determine the formula for findin+ the nth term in a linear data

set%. 8an students sol&e a real-life se2uence problem based on countin+

Unit 1 Gradee+el 4pectations 6G4s7

G4 8 G4 Tet and 9encmars

&lgebra

%. Write the e2uation of a line of best fit for a set of #-&ariable real-life datapresented in table or scatter plot form, with or without technolo+y )-#-


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#$. Show or 9ustify the correlation match between a linear or non-linear data setand a +raph !-#-


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Solving or*e.tprompts. If teachers want students to reflect critically on the topic 9ustlearned, they would use Special )owers$ Alternative ,iewpoints$ orWhat If-prompts.

In this particular acti&ity, usin+ the*e.tcate+ory, +i&e students the followin+ prompt:

i&en the pattern BBBBB, -4, 3#, BBBBB, 7*, ...answer the followin+ e(ercises:a. Cill in the missin+ numbers.b. !etermine the ne(t two numbers in this se2uence.c. !escribe how you determined what numbers completed the se2uence. 5e

sure to e(plain your reasonin+.d. )re there any other numbers that would complete this se2uence "(plain

your reasonin+.

Students will ha&e to thin' critically to determine which numbers ma'e the se2uencewor'. Some will create a linear pattern while others will create a non-linear pattern.nit 3eometric 1atterns and ?easonin+ ;
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&cti+ity 3: Recognisin+ ?ules to enerate a Se2uence 5M

Teacher note/ Information for activities 1 and 5 can +e found in most Alge+ra I and:or

Alge+ra & te.t+oos0 While this sill should have +een mastered in Alge+ra 4$ the reviewis used to help students distinguish the difference +etween inductive and deductive

reasoning !#L9 4;"0

>sin+ the inear or @on-linear 5Ms, ha&e students complete a modifiedopinionnaire&iew literacy strate+y descriptions before discussin+ the definition of linear.


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!iscuss the differences between the data sets to determine what ma'es a data set linear ornot linear.

The s'ills listed in the followin+ acti&ity will re&iew concepts that were to be mastered in)l+ebra I. nit 3eometric 1atterns and ?easonin+ %


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7, *, 3#, 34, #$ G Cind the 3$$thterm. Solution/Formula 5n6 4''thterm 5'' 7, 6, 37, 36, #7, G Cind the 4=thterm. Solution/Formula n246 3;thterm 115 Students should also be re2uired to de&elop the formulas without the use of

technolo+y.Students should also be re2uired to +enerate the nthterm for picture patterns. >se the

eneratin+ the nthTerm for 1icture 1atterns 5M for e(amples.

&cti+ity 5: igurate /umbers 6G4s: 5' 2,' 22' 2$' 2!7

Materials ist: pencil, paper, +raphin+ calculator, S2uare Ci+urate @umbers 5M,?ectan+ular Ci+urate @umbers 5M, Trian+ular Ci+urate @umbers 5M

In this acti&ity, students will +enerate the formulas for findin+ the nthterm in s2uare,rectan+ular, or trian+ular number patterns. "ach of these is a non-linear number pattern.Ci+urate numbers are numbers that can be represented by a re+ular +eometrical

arran+ement of e2ually spaced points. They may be in the shape of any re+ular poly+on,or other +eometric arran+ements. "ach set of fi+urate numbers represents a distinct non-linear pattern. This acti&ity concentrates on +eometrical fi+ures students are familiar withwhich aid in findin+ the al+ebraic rule for findin+ the nth term.

Teacher note/ More information can +e found through a search on ahoo or #oogle0

S2uare @umbers

>se the S2uare Ci+urate @umbers 5M to present the followin+ dia+ram.

Cirst, ha&e students translate the picture pattern into a number pattern by countin+ thenumber of dots in each fi+ure. The number pattern is 3, 7, 6, 34, #%G. )s' the students ifthe pattern is a linear one. They should tell you that the data cannot be linear since thedifference between &alues is not constant. Some students may reco+ni0e immediately that

the numbers are perfect s2uares, but many will not unless the teacher pro&ides leadin+2uestions for class discussion. If needed, as' students why the picture pattern is called as2uare number pattern. ead students to reco+ni0e that the dots form s2uares, and that thenumber of dots in each s2uare is the same as the area of the s2uare. It may be necessaryto as' them what is meant by the termperfect s?uare0 The students will understand thatthe numbers in the number se2uence are the s2uares of the countin+ numbers

, , , ...& & & &4 & 1 5 . The formula for +eneratin+ the nth term is &n . nit 3eometric 1atterns and ?easonin+ 4


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that it is important to 'now the characteristics of linear data sets common differencebetween each two terms in order to 2uic'ly identify those that are non-linear.


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Trian+ular @umbers

>se the Trian+ular Ci+urate @umbers 5M to present the followin+ dia+ram.

e that

they can add & to the first value to get the second value$ add 1 to the second value to get

the third value$ etc0 They will want to say that the rule is to add the ne.t whole num+er to

theprevious one0 They need to understand that this indicates that the pattern is notlinear$ since the difference +etween values is not the same0 Lead them to understand that

this pattern cannot +e the formula or rule for generating the nthterm$ since the patternthey see is +ased upon nowing a previous term.

The formula for +eneratin+ the nthterm is. /n n 4

&

+. The calculator will show the

re+ression e2uation as $.%n#L $.%n. @otice that this is half of n!n E 4"which was the rulefor the rectan+ular numbers. Show students that the trian+ular number pattern could alsobe drawn as

"ach of the patterns abo&e is one-half of each of the rectan+les below.

eometry>nit 3eometric 1atterns and ?easonin+ *


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Therefore, if the area of the rectan+le is ( )3n n + , the area of the trian+le would be half as

much.

1ro&ide students with a &ariety of number patterns, some linear and some non-linear,with which to practice their s'ills. )cti&ity 4 +i&es an e(ample of some +eometricsituations in which these s'ills must be applied.

)t this point, ha&e students refer to the inear or @on-linear 5M completed in )cti&ity;. The teacher should ha&e the students decide whether their first instincts were correct.The teacher should lead a discussion about which patterns are linear, and how thestudents 'now they are linear usin+ the terminolo+y and strate+ies presented in )cti&ites;, 7, and %.

&cti+ity $: &pplying Patterns and Counting to Geometric Concepts 6G4s: 5' 2,'

22' 2$' 2!7

Materials ist: pencil, paper, +raphin+ calculator


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5)8 58)8)5 85)

Nne way to thin' about this is that for any &erte(, there are two different possible names.So three &ertices times two names each is si( possibilities.

)nother way to thin' about this is that there are ; positions to fill when namin+ thetrian+le. There are ; &ertices from which to choose for the first position, but only #remain as choices for the second position. Nnce the second position is filled, there is onlyone &erte( remainin+ with which to fill the last position. ?e&iew with students that ;O is ;( # ( 3 P 4 which is the same as the number of possible names. This is a concept tau+ht in)l+ebra I. i&e a few more e(amples in which the total number of possibilities can bedetermined.

?elate the idea of determinin+ how many choices one has to name a trian+le to 1roblem):Gow many ways can 1 +oos +e arranged on a shelf if they are chosen from aselection of ( different +oos-

There would be * ways to fill the first position, = ways to fill the second position, and 4ways to fill the third position. * ( = ( 4 P ;;4. In situations in which order is importante.+., )58 is different than )85, the number of possibilities is called a permutation.

Cor situations in which order is @NT important i.e., )58 and )85 would be consideredduplicates since they are the same three letters, the number of possibilities is called acombination. To 'now the number of combinations of ; boo's that can be put on theshelf, ta'e into account how many arran+ements would be considered to be the same foreach set of ; boo's. This is ;O or 4, so di&idin+ ;;4 by 4 is %4. There would be %4different combinations to put on the shelf. In other words, one could display a differentcombination of ; boo's for %4 ways before he/she would ha&e to repeat a set.

nit 3eometric 1atterns and ?easonin+ 33


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Introduce the class to circular permutations to answer such 2uestions as, D


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!. i&en = distinct points in a plane, how many line se+ments will be drawn ife&ery pair of points is connected Solution/ &4

". Suppose there are * points in a plane such that no three points are collinear.nit 3eometric 1atterns and ?easonin+ 3;
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indicate, orally or in writin+, the process for +eneratin+ the ne(t item. Thestudents will also state the rule for +eneratin+ the nthterm in each se2uence.

)cti&ity ; : The student will use a +raphin+ calculator to plot table entries for a+i&en non-linear se2uence in order to determine the re+ression e2uation for

the data set.

)cti&ity = : The student will participate in a simulation e(ercise to determine atournament schedule for his/her district, re+ional, or state hi+h school baseballteam, bas'etball team, etc.

eometry>nit 3eometric 1atterns and ?easonin+ 37


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Geometry

Unit 2: Reasoning and Proof

Time rame: )ppro(imately four wee's

Unit escription

This unit introduces the de&elopment of ar+uments for +eometric situations. 8on9ecturesand con&incin+ ar+uments are first based on e(perimental data, then are de&eloped frominducti&e reasonin+, and, finally, are presented usin+ deducti&e proofs in two-column,flow patterns, para+raphs, and indirect formats.


Students understand the basic role proof plays in mathematics. Students learn todistin+uish proofs from con&incin+ ar+uments. They understand that proof may be+enerated by first pro&idin+ numerical ar+uments such as measurements, and then byreplacin+ the measurements with &ariables.

Guiding #uestions

3. 8an students de&elop inducti&e ar+uments for con9ectures and offer reasonssupportin+ their &alidity

#. 8an students de&elop short al+orithmic-based proofs that +enerali0e numericalar+uments

;. 8an students de&elop more +eneral ar+uments based on definitions and basica(ioms and postulates



Geometry

3$. Corm and test con9ectures concernin+ +eometric relationships includin+ lines,

an+les, and poly+ons i.e., trian+les, 2uadrilaterals, and n-+ons, with andwithout technolo+y -3-


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ata &nalysis' Probability' and iscrete ;at

#;. !raw and 9ustify conclusions based on the use of lo+ic e.+., conditionalstatements, con&erse, in&erse, contrapositi&e !-*-


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Students should be encoura+ed to use the cards to study and to refer to them as theyencounter &arious symbols throu+hout the rest of the course. Students can buy rin+s to'eep the cards to+ether, or they can punch holes in the cards in order to 'eep them intheir binders. )nother option for or+ani0ation is to ha&e students 'eep the cards in a0ippered ba+ which could also be 'ept in their binders. ?emember, students should use

these cards to help them study and when they are doin+ assi+nments, so it is important tochec' that the students are creatin+ their cards correctly.

nit #?easonin+ and 1roof 3=

ad?acent angles

efinition:two an+lesthat share a common&erte( and a common side,but no common interior

points

4ample:

/oneample:

3

#

are ad9acent an+les

3 #

Relationsip:

)ll linear pairs are ad9acentan+les.


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aily =armups =it Tecnology

>sin+ a computer drawin+ pro+ram such as The #eometer8s SetchpadK,ha&e studentsre&iew basic terminolo+y by constructin+ &arious fi+ures such as an+les, se+ments,se+ment bisectors, an+le bisectors, etc. i&e students characteristics of the fi+ure they are

to draw, such as the measure of the an+le or se+ment.


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&cti+ity 2: Comparing Reasoning: -nducti+e +s. educti+e 6G4: 1!7

Materials ist: e(amples of inducti&e and deducti&e reasonin+

?e&iew the definition of inducti&e lo+ic and pro&ide the students with the followin+

scenario.

Qudys 1roblem/Solution

DMy dad is in the @a&y and he says that food is +reat on submarines,E offeredQudy. DMy mom,E added 5obbie, Dwor's for the airlines and she says that airlinefood is notoriously bad.E DMy mom is an astronaut trainee,E added reer, Dandshe says that their food is the worst ima+inable.E DUou 'now,E concluded Qudy, DIbet no life e(ists beyond earthOE 5obbie and reer both loo'ed at her, pu00led.DWhatE DSure,E e(plained Qudy, D)t e(treme altitudes, food must taste so badthat no creature could stand to eatK therefore, no life e(ists out there.E

What do you thin' of Qudys inducti&e reasonin+ What possible other con9ectures couldbe made What are the problems with her conclusion

Students should reco+ni0e that Qudy has identified a patternAeach of Qudys friendsparents wor's at a different altitude, and as the altitude increases below sea le&el, toairplane altitudes, to outer space the food +ets worse. >sin+ this pattern, Qudy ma'es acon9ecture which she does not stateAas the altitude increases the food +ets worse. Sheuses that con9ecture to ma'e the statement that no life e(ists beyond earth. Theconclusion is in&alid because it does not come directly from the pattern Qudy hasobser&ed. Students should be able to identify the process of inducti&e reasonin+ asidentifyin+ a pattern to ma'e a con9ecture, but they should reali0e Qudys con9ecture isin&alid.

)fter this discussion, define deducti&e reasonin+ and offer the followin+ e(ample for thestudents to analy0e.

)le(s rades

)le(s math teacher always tells him that homewor' is practice at home. She alsotells him that the more he practices his math, the better his +rades will be. )le(did all of his homewor' this wee'. When he +ets to class before the test, he tellshis teacher, DIm +oin+ to do well on the test today.E

What do you thin' of )le(s deducti&e reasonin+ )re there any problems with hisconclusion

The teacher should then lead a discussion about the differences between inducti&e anddeducti&e reasonin+. Students should be +i&en other e(amples of reasonin+ and be as'edto determine if the reasonin+ used was inducti&e or deducti&e.

eometry>nit #?easonin+ and 1roof 36


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&cti+ity 3: istinguising 9et0een -nducti+e and educti+e Reasoning 6G4: 1!7

Materials ist: Internet access or presentation e2uipment for classroom useK pencilK paper

nit #?easonin+ and 1roof #$
http://www.sparknotes.com/math/geometry3/inductiveanddeductivereasoning/http://www.sparknotes.com/math/geometry3/inductiveanddeductivereasoning/


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the conditional statement is also true, and li'ewise, a false hypothesis does not mean thatthe conditional is false. Show students the followin+ truth table and ha&e them determineif &arious conditionals are true or false based on a +i&en set of conditions.

p ? p ?

T T TT C C

C T T

C C C

!isplay the followin+ conditional statement for all students to see: DIf two an+les ha&ethe same measure, then they are con+ruent.E ead a brief discussion about thetruthfulness of the statement. @e(t, display the con&erse: DIf two an+les are con+ruent,then they ha&e the same measure.E ead a discussion about how these two statements arerelated and about the truthfulness. @ow, display the in&erse of the conditional statement:DIf two an+les do not ha&e the same measure, then they are not con+ruent.E ead another

discussion about how the conditional and the in&erse statements are related and about thetruthfulness. Cinally, display the contrapositi&e of the conditional: DIf two an+les are notcon+ruent, then they do not ha&e the same measure.E ead a discussion about how thecontrapositi&e is related to the con&erse and about its truthfulness.

!efine the term logically e?uivalent. >sin+ the conditional DIf two an+les form a linearpair, then they are supplementary,E ha&e students write the con&erse, in&erse, andcontrapositi&e of each statement and then determine the truthfulness of each statement.>sin+ these statements and the definition of lo+ically e2ui&alent, ha&e the studentsdetermine which statements are lo+ically e2ui&alent. Students should see that aconditional and its contrapositi&e are lo+ically e2ui&alent as are the con&erse and in&erse

of a conditional. Show the students the truth table below and +i&e students more practicewritin+ the con&erse, in&erse, and contrapositi&e and determinin+ the truthfulness of eachstatement.

p ? 8onditionalp ?

8on&erse? p

In&ersep ?: :

8ontrapositi&e? p: :

T T T T T T

T C C T T C

C T T C C T

C C T T T T

)fter students ha&e demonstrated an understandin+ of the relationships abo&e, thendisplay other conditional statements.


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the con&erse of his or her conditional statements to the class and e(plain why thecon&erse is true or false.

&cti+ity $: a0s of %yllogism and etacment 6G4: 237

Materials ist: pencil, paper

!isplay statements similar to the followin+: D)ll do+s are mammals. 5uster is a do+.EThese statements illustrate the law of detachment. )s' students to determine a lo+icalconclusion from these statements. nit #?easonin+ and 1roof ##


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Nnce +roups ha&e had the opportunity to complete their proofs, ha&e them present theproofs to the class. >sin+ a modified 2uestioning the ?uthor !tA"&iew literacy strate+ydescriptions techni2ue, ha&e students as' 2uestions about the proofs to clarify their ownunderstandin+. The +oal of tAis to help students construct meanin+ from te(t. Insteadof students as'in+ the 2uestions durin+ readin+, in this acti&ity students will be as'in+

2uestions after they ha&e re&iewed the proofs of other +roups. Some possible 2uestionsmi+ht be:

!oes the flow of the proof ma'e sense lo+ically Is the correct reason +i&en for the statement presented )re the statement and reason necessary to complete the proof Is there a step missin+ that would help the reasonin+ sound more lo+ical

Teachers should direct students to thin' about their classmates presentations and de&elop2uestions that may hi+hli+ht incorrect lo+ic or missin+ information. Teachers should setrules that create an en&ironment conduci&e to this process. Students should as' andanswer most of the 2uestionsK howe&er, where necessary the teacher should offer his/her

own 2uestions/e(planations to a&oid misconceptions and incorrect answers.

Nnce students ha&e ad9usted to or+ani0in+ the proofs, introduce proofs with unnecessaryinformation. ?e2uire that students use only information that is rele&ant to the proof andor+ani0e the information into a lo+ical order. 1ro&ide students with the opportunity topro+ress from basic al+ebraic proofs to basic +eometric proofs based on al+ebraicconcepts definition of con+ruence, an+le and se+ment addition postulates, properties ofe2uality. 5oth of the strate+ies employed earlier in this acti&ity process guides and tAcan also be used here to promote hi+her order thin'in+ and understandin+.

&cti+ity ): Proofs 6G4: 1*7


The al+ebraic proofs were two column proofs, but some students find flow proofs orpara+raph proofs easier to follow. "mphasis should be placed on pro&idin+ a con&incin+,easy-to-follow ar+ument with reasons rather than on usin+ one particular format. Thecontent for these proofs should focus on basic +eometric concepts se+ment and an+leaddition, con+ruent se+ments, and an+les. While these proofs may be similar to those in)cti&ity =, these proofs are different because students ha&e to come up with thestatements and reasons as opposed to 9ust arran+in+ them in the correct order. Students

must determine the ar+uments and reasons with their classmates. Cacilitate studentswor' with proofs by ha&in+ the students create a modified mathstory chain&iewliteracy strate+y descriptions. )story chaintypically has a +roup of students create astory based on content that has already been presented. "ach member of the +roup adds aline to the story until the story is completed. In this acti&ity, thestory chainhas beenmodified by ha&in+ students complete a proof based on the al+ebraic and +eometricconcepts learned in this unit.


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In each +roup, each member should ta'e a turn writin+ one statement andreason for the proof. The first member will write the first statement andreason. The second member will read the first persons statement and reasonand decide if it is lo+ical. Then he/she will add his/her own statement andreason. This process will continue until the entire proof is written. "ach time a

member recei&es the proof, he/she should read the entire proof to be surehe/she a+rees with the lo+ic and flow of the proof. If any person in the +rouphas a concern with any of the pre&ious information, he/she should help his/herclassmate correct the statement then add his/her new information. roupsshould be allowed to use a two-column proof, a para+raph proof, or a flowproof.

oo' for correct proofs. When most +roups ha&e completed their proofs,encoura+e them to discuss their ideas with other +roups. )t this point, studentsshould 2uestion each other if they feel as thou+h there are errors in any of thewor'.

8hoose three different +roups to write a particular correct proof on the

board. )s a class, discuss &ariations and similarities of the three proofs, andtal' about e(tra steps that could be added or omitted.

&cti+ity *: un 0it &ngles 6G4s: 11' 1*' 237


?e&iew the relationships amon+ an+les formed by the intersection of two parallel linesand a trans&ersal that were learned in +rade *. 1ro&ide students with a +raphic similar to!ia+ram 3 in which lines aand +are parallel. Cirst, pro&ide a number that represents the

measure of an+le 3. se acti&ities that re2uire students to pro&ide proofs or con&incin+ ar+uments foranswers throu+hout the year.

eometry>nit #?easonin+ and 1roof #7


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16151413

1211109

8765

4321

87

65

43

21

Diagram 2Diagram 1

b

a

t

%ample &ssessments

General &ssessments

The student will answer prompts that include the followin+ concepts in his/hermath learning logs &iew literacy strate+y descriptions:

o 8omparin+ inducti&e and deducti&e reasonin+.

o !escribin+ a situation in which he/she had se&eral e(periences that led

him/her to ma'e a true con9ecture. The student will describe a situation inwhich he/she had se&eral e(periences that led to a false con9ecture.

o

?espondin+ to an ad&ertisement, such as the followin+: DThose whochoose Tint-and-Trim


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)cti&ity # : The student will find instances in real-life in which lo+ical conclusionsha&e been made. nit #?easonin+ and 1roof #4


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Geometry

Unit 3: Parallel and Perpendicular Relationsips

Time rame: )ppro(imately three wee's

Unit escription

This unit demonstrates the basic role played by "uclids fifth postulate in +eometry."uclids fifth postulate is stated in most te(tboo's usin+ the wordin+ found in 1layfairs)(iom: Through a given point$ only one line can +e drawn parallel to a given line0 Thisa(iom and se&eral others are considered by some mathematicians to be e2ui&alent to"uclids fifth postulate. The focus is on basic an+le measurement relationships forparallel and perpendicular lines, e2uations of lines that are parallel and perpendicular inthe coordinate plane, and pro&in+ that two or more lines are parallel usin+ &arious

methods includin+ distance between two lines.


Students should 'now the basic an+le measurement relationships and slope relationshipsbetween parallel and perpendicular lines in the plane. Students can write and identifye2uations of lines that represent parallel and perpendicular lines. They can reco+ni0e theconditions that must e(ist for two or more lines to be parallel. Three-dimensional fi+urescan be connected to their #-dimensional counterparts when possible.

Guiding #uestions

3. 8an students relate parallelism to "uclids fifth postulate and its ramificationsfor "uclidean eometry

#. 8an students use parallelism to find and de&elop the basic an+lemeasurements related to trian+les and to trans&ersals intersectin+ parallellines

;. 8an students lin' perpendicularity to an+le measurements and to itsrelationship with parallelism in the plane and ;-dimensional space

7. 8an students sol&e problems +i&en the e2uations of lines that areperpendicular or parallel to a +i&en line in the coordinate plane and discuss theslope relationships +o&ernin+ these situations

%. 8an students sol&e problems that deal with distance on the number line or inthe coordinate plane

eometry>nit ;1arallel and 1erpendicular ?elationships #=


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/umber and /umber Relations

3. Simplify and determine the &alue of radical e(pressions @-#-


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DmatchE when folded o&er one another. 5ecause a line measures 3*$ de+rees, themeasures of the two an+les are 6$ de+rees each.

Students will then determine the slope of the DcreaseE line and compare it to that of theori+inal line. )ll data from the class should be recorded in a chart. The chart should

include a column for the slope of the ori+inal line and a column for the slope of theDcreaseE line. >sin+ the class data, student pairs should ma'e a con9ecture about the slopeof perpendicular lines.

>sin+ the +raph with the +i&en parallel lines, ha&e students fold the +raph so each linelies on itself to create a DcreaseE line which passes throu+h the pair of parallel lines. Thenha&e students discuss how the slopes of the parallel lines are related, and whether or notthe DcreaseE is perpendicular to one or both of the +i&en lines. This should lead to adiscussion about the theorem that states Dif a line is perpendicular to one of two parallellines, then it is perpendicular to the other.E

&cti+ity 2: Parallel and Perpendicular ines 6G4: $7

Materials ist: pencil, paper, +raph paper, computer drawin+ pro+ram optional

1ro&ide students with se&eral e2uations of pairs of lines that are parallel, and se&erale2uations of pairs of lines that are perpendicular, but dont +i&e the students therelationships.


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To end the acti&ity, ha&e students wor'in+ alone write e2uations of any two lines that areparallel and any two lines that are perpendicular. !o not pro&ide them with anyinformation such as slopes ory-intercepts.

&cti+ity 3: Pro+ing ines are Parallel 6G4s: 1,' 11' 1*7

Materials ist: pencil, paper, dia+rams for discussion, learnin+ lo+

Teacher note/ The names of the special angle pairs formed +y two lines and a

transversal$ and the relationships of these special angle pairs are found in the #rade (

#L9s so they are not discussed in detail here0 A review may +e necessary depending on

the students in the class0

i&e students dia+rams of parallel lines and trans&ersals, and dia+rams of lines that arenot parallel with trans&ersals. ead a discussion to determine what characteristics of

parallel lines will +uarantee that two lines are parallel e.+., two lines are parallel ifcorrespondin+ an+les are con+ruent, alternate interior an+les are con+ruent, alternatee(terior an+les are con+ruent, consecuti&e interior an+les are supplementary, parallellines are e&erywhere e2uidistant.

sin+ these postulates as truth, studentscan pro&e the other theorems and con&erses. )llow students to initially use an+lemeasures to write proofs for specific sets of lines to pro&e these theorems, but alsore2uire them to use +eneral proofs that pro&e lines parallel throu+h +eneralities. Theseproofs can ta'e any form informal, para+raph, two-column, flow. 1ro&ide opportunitiesfor students to pro&e the other theorems which are based on the postulate forcorrespondin+ an+les. nit 7.

eometry>nit ;1arallel and 1erpendicular ?elationships ;$


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1ro&ide dia+rams of two lines that are perpendicular to one line.


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discussion should re&eal that the distance is always the shortest line se+ment between twopoints or the shortest distance between a point and a line.


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8orrespondin+ an+les are con+ruent when lines are parallel. Same-side interior an+les are supplementary when lines are parallel. The measure of the e(terior an+le of a trian+le is e2ual to the sum of the

measures of the two remote interior an+les of the trian+le.

Two lines parallel to a third line are parallel to each other.

Npposite sides of a parallelo+ram are parallel and con+ruent. Npposite an+les of a parallelo+ram are con+ruent. )d9acent an+les of a parallelo+ram are supplementary. The se+ment 9oinin+ the midpoints of two sides of a trian+le is parallel to the

third side and has a len+th e2ual to half the len+th of the third side.

The ratio of the perimeters of two similar trian+les is the same as the scalefactor of the similar trian+les.

The sum of the e(terior an+le measures of any con&e( poly+on is ;4$V. The sum of the an+le measures of a 2uadrilateral is ;4$V. The sum of the an+le measures of a he(a+on is =#$V.

Teachers should insure that students understand the relationships amon+ the an+lesformed by the parallel lines and trans&ersals as discussed in >nit # )cti&ity =. nit 7. While students may ha&e been introduced to much ofthis information prior to this course, this acti&ity connects their 'nowled+e of parallellines and the an+le relationships formed by those lines, to their prior 'nowled+e oftrian+les and 2uadrilaterals.

&cti+ity !: Parallel ine acts 6G4s: 1,' 11' 1*7

Materials ist: pencil, paper, 1arallel ine Cacts 5M, ruler

Teacher *ote/ The focus of this activity is to apply the properties of parallel lines

learned in earlier activities0 This is a good application of proof using the parallel line

properties0 It also introduces information relative to angle relationships in triangles$connecting this unit to nit 50

i&e students copies of the 1arallel ine Cacts 5M and ha&e them complete the

followin+ steps.

nit ;1arallel and 1erpendicular ?elationships ;;


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>se the same dia+ram to write a proof toshow that the measure of an e(terioran+le in a trian+le has the same measure as the sum of the two remote interioran+les. This the "(terior )n+le Sum theorem.

nit ;1arallel and 1erpendicular ?elationships ;7


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find the slope of the +i&en line and write the e2uation of the +i&en line. write the e2uation for a line which passes throu+h the +i&en point and is

parallel to the +i&en line.

write the e2uation of the line which is perpendicular to the +i&en line andpasses throu+h the +i&en point.

) Whats My ine ?ubric 5M is pro&ided for this assessment.

)cti&ity 4 : The teacher will pro&ide the student with nets or dia+rams formed byintersectin+ lines parallel and nonparallel and a minimal number of an+lemeasures for the dia+ram. The student will calculate the missin+ an+le measures

usin+ either the formula 3*$ #S n= or the an+les created by trans&ersals thatintersect parallel lines.

eometry>nit ;1arallel and 1erpendicular ?elationships ;%


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Geometry

Unit ": Triangles and #uadrilaterals

Time rame:)ppro(imately fi&e wee's

Unit escription

This unit introduces the &arious postulates and theorems that outline the study ofcon+ruence and similarity. The focus is on similarity and con+ruence treated as similaritywith a ratio of 3 to 3. It also includes the definitions of special se+ments in trian+les,classic theorems that de&elop the total concept of a trian+le, and relationships betweentrian+les and 2uadrilaterals that underpin measurement relationships. The properties ofthe special 2uadrilaterals parallelo+rams, trape0oids, and 'ites are also de&eloped anddiscussed.


Students should 'now definin+ properties and basic relationships for all forms oftrian+les and 2uadrilaterals. They should also be able to discuss and apply the con+ruencepostulates and theorems and compare and contrast them with their similarity counterparts.Students should be able to apply basic classical theorems, such as the Isosceles Trian+letheorem, Trian+le Ine2uality theorem, and others.

Guiding #uestions

3. 8an students illustrate the basic properties and relationships tied tocon+ruence and similarity

#. 8an students de&elop and pro&e con9ectures related to con+ruence andsimilarity

;. 8an students draw and use fi+ures to 9ustify ar+uments and con9ectures aboutcon+ruence and similarity

7. 8an students state and apply classic theorems about trian+les, based oncon+ruence and similarity patterns

%. 8an students construct the special se+ments of a trian+le and apply theirproperties

4. 8an students determine the appropriate name of a 2uadrilateral +i&en specificproperties of the fi+ure

=. 8an students apply properties of 2uadrilaterals to find missin+ an+le and sidemeasures

eometry>nit 7Trian+les and uadrilaterals ;4


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Unit " Gradee+el 4pectations 6G4s7





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&cti+ity 3: Corresponding Parts 6CPCTC7 6G4s: 1,' 1)7


*ote:While this activity does not as students to find the measures of segments orangles$ it does re?uire them to determine corresponding parts of congruent triangles0

This is a sill necessary when determining corresponding sides to write proportions for

similarity0

The focus of this lesson is to ma'e students aware of correct ways to name con+ruenttrian+les to preser&e correspondin+ parts.


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&cti+ity 5: &re Tey Congruent@ 6G4s: 1,7

Materials ist: pencil, paper, ruler, protractor, compass, +eometry drawin+ softwareoptional, computers optional

1ro&ide students with the measures of two sides and a non-included an+le for a trian+le,or the measures of three an+les and no sides.


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To end the acti&ity, ha&e students employ techni2ues used in class to pro&e two trian+lesfrom a dia+ram are con+ruent. This indi&idual wor' will show that students ha&emastered the s'ill.

&cti+ity !: &ltitudes' &ngle 9isectors' ;edians' and Perpendicular 9isectors of a

Triangle 6G4: 1,7

Materials ist: pencil, paper, patty paper, +eometry software, computers


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the acute trian+le, the perpendicular bisectors will intersect inside the trian+le. Cor theri+ht trian+le, the perpendicular bisectors will intersect on the hypotenuse. Cor the obtusetrian+le, the perpendicular bisectors will intersect outside the trian+le.

1ro&ide students with the definition of altitudein a trian+le.


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nit 7Trian+les and uadrilaterals 7;


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C0 ( ) ;4 & && 7 7 3 1$ $ m D 2 $ y D 2 .E

eometry>nit 7Trian+les and uadrilaterals 77


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&cti+ity *: ;ore on &ngle 9isectors' ;edians' and Perpendicular 9isectors of a

Triangle 6G4s: 1,' 1*7

Materials ist: pencil, paper, +eometry software, computers

sin+ an automatic drawer, such as that found in #eometer8s SetchpadQ,

draw scalene trian+leABCand measure the len+ths of AB and AC.

#. 8onstruct m$the an+le bisector of BAC .;. 8onstruct the midpointand the perpendicular bisector of BC .

7. !raw the median from pointAto BC .

%. Mo&e pointAuntil the an+le bisector, perpendicular bisector, and the median

coincide. ?ecord the len+ths of AB and AC.

4. !ra+ point ) to find two other positions for point ) in which an+le bisector,perpendicular bisector, and the median coincide. )+ain, record the len+ths of

AB and AC0

)s' students to ma'e a con9ecture about ABC when the an+le bisector of BAC , themedian fromAto BC , and the perpendicular bisector of BC coincide. nit 7Trian+les and uadrilaterals 7%

!

8

5)


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Solution/ Since B is a perpendicular +isector$ BA and BC are rightangles +ecause perpendicular lines form 5 right angles0 That maesAB and CBV V right triangles0 By the definition of perpendicular +isector$

is the midpoint of AC0 By the definition of midpoint$ AI CI 0 Also$ B B +ecause congruence of segments is refle.ive0 Therefore$ since two pairs of legs of

AB and CBV V are congruent$ AB CBV V +y LL0

&cti+ity 11: -neAualities for %ides and &ngles in a Triangle 6G4s: 1' 1,7

Materials ist: pencil, paper, )n+le and Side ?elationships 5M, patty paper, ruler,protractor


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"ach student should measure N@" of the sides of the trian+le, record themeasurement in the correct place on the )n+le and Side ?elationships 5M, thenpass the trian+le to the ne(t +roup member. This step should be repeated with theremainin+ two sides of the trian+les.

)fter the third side has been measured and recorded in all three trian+les, each

student in the +roup should write down all of the measurements from the +roup.Then each student should ta'e one trian+le and chec' to be sure the side measuresare sensible chec' units, be sure the stated measure Dloo'sE close.


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5. 9FV has &ertices ( ) ( ) ( )#,3 , %, ; , and 7, ; 9 F . ist the an+les inorder from +reatest to least.

Solution/ , ,F 9

"nd this acti&ity with S)AW* writing &iew literacy strate+y descriptions from the What

if- cate+ory. i&e the students the prompt DWhat relationships between sides and an+lesoccur if the trian+le is isosceles !o these relationships still fit with the obser&ations youfound completin+ this acti&ity "(plain.E Since students ha&e already studied isoscelestrian+les in )cti&ity 3, they should reali0e that if the measures of the base an+les are+reater than the &erte( an+le, then the le+s of the isosceles trian+le are lon+er than thebase and &ice &ersa. This does fit with the con9ectures they will ma'e throu+h thisdisco&ery acti&ity. They should also understand that when listin+ the sides of an isoscelestrian+le in order by its len+ths, two of the sides will need to be desi+nated as e2ual, e.+.,)8 P 58 X )8. Students may write this as an entry in their math learning logs&iewliteracy strate+y descriptions or they may turn it in as a separate assi+nment.

&cti+ity 13: Te Triangle -neAuality 6G4: 1,7

Materials ist: straws, rulers, timer, pencil, paper

Students should wor' in +roups of two or three. i&e each +roup a set of straws whichha&e been cut into different len+ths. Cirst, ha&e students measure the len+th of eachstraw. Instruct students to ma'e as many different trian+les with the se+ments as possiblewithin a certain time.


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)llow the students to ponder the statement for a moment and as' them to thin' of some2uestions they mi+ht ha&e, related to the statement. )fter a minute or two, ha&e studentspair up and +enerate two or three 2uestions they would li'e to ha&e answered that relateto the statement. When all of the pairs ha&e de&eloped their 2uestions, ha&e one memberfrom each pair share their 2uestions with the class. )s the 2uestions are read aloud, write

them on the board or o&erhead. Students should also copy these in their noteboo's. When2uestions are repeated or are &ery similar to others which ha&e already been posed, those2uestions should be starred or hi+hli+hted in some way. Nnce all of the students2uestions ha&e been shared, loo' o&er the list and determine if the teacher needs to addhis/her own 2uestions. The list should include the followin+ 2uestions:

What is the definition of similar poly+ons 8an isosceles trian+les be similar to scalene or e2uilateral trian+les 8an acute trian+les be similar to obtuse or ri+ht trian+les )re con+ruent trian+les similar What must be true about two trian+les in order for them to be similar


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class. Students should be told that they are re2uired to be able to support their statements,con9ectures, and answers with e&idence from their in&esti+ation with the process +uide.sin+ thegraphic organi>er +ie0 literacy strategy descriptions pro&idedin the uadrilateral Camily 5M, fill in the names of the 2uadrilaterals so that each ofthe followin+ is used e(actly once:

1)?)"N?)M YIT" S>)?" >)!?I)T"?) T?)1"ZNI! ?"8T)@" ISNS8""S T?)1"ZNI! ?S

eometry>nit 7Trian+les and uadrilaterals %3


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4planation: Collowin+ the arrows: The properties of each fi+ure are also properties ofthe fi+ure that follows it. ?e&ersin+ the arrows: "&ery fi+ure is also the one that precedesit.

er+i&en in the uadrilateral Camily 5Ms,

and then lead a class discussion to summari0e how different 2uadrilaterals are related toone another. Students should be able to identify a s2uare as bein+ a rectan+le, rhombus,parallelo+ram, and 2uadrilateral and be able to 9ustify their reasonin+.

&cti+ity 1): ;edian of a Trape


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eometry>nit 7Trian+les and uadrilaterals %7


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Geometry

Unit 5: %imilarity and Trigonometry

Time rame:)ppro(imately four wee's

Unit escription

This unit addresses the measurement side of the similarity relationship which is e(tendedto the 1ytha+orean theorem, its con&erse, and their applications. The three basictri+onometric relationships are defined and applied to ri+ht trian+le situations.


Students apply their 'nowled+e of similar trian+les to findin+ the missin+ measures ofsides of similar trian+les, and to usin+ the 1ytha+orean theorem to find the len+th ofmissin+ sides in a ri+ht trian+le. The con&erse of the 1ytha+orean theorem is used todetermine whether a +i&en trian+le is a ri+ht, acute, or obtuse trian+le. Students can usesine, cosine, and tangentto find len+ths of sides or measures of an+les in ri+ht trian+lesand the relationship to similarity.

Guiding #uestions

3. 8an students use proportions to find the len+ths of missin+ sides of similartrian+les

#. 8an students use similar trian+les and other properties to pro&e and apply the1ytha+orean theorem and its con&erse

;. 8an students relate tri+onometric ratio use to 'nowled+e of similar trian+les7. 8an students usesine$ cosine, and tangentto find the measures of missin+

sides or an+le measures in a ri+ht trian+le

eometry>nit %Similarity and Tri+onometry %%


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the 5M and the new drawin+.


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trian+lesE @e(t, ha&e students use pattern bloc's to create a trian+le similarto the ori+inal trian+le so the ratio of side len+ths is ;:3. )s': DWhat is theratio of the areas of these two similar trian+lesE

The s'etches below are not included in the 5M but are pro&ided here to

illustrate what the students should be creatin+ at their des's as they arewor'in+ throu+h the 5M.

Sample setches/


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description of similar ;-! shapes ratio of ed+es ratio of &olumes

. . .

. . .

. . .

. . .

)t the completion of this acti&ity, ha&e students answer the followin+ prompt in theirmath learning logs &iew literacy strate+y descriptions:

>sin+ what you ha&e learned about the relationships between the ratio ofthe sides and the ratio of the areas of similar fi+ures, determine therelationship between the ratio of the sides and the ratio of the perimetersof those same similar fi+ures. 5e sure to e(plain your reasonin+ and

pro&ide e(amples/calculations to aid your e(planation.

) learning logis a noteboo' students 'eep in order to record ideas, 2uestions, reactions,and new understandin+s. Students should use their math learning logother times in classin addition to those listed throu+hout the curriculum. This will pro&ide opportunities todemonstrate understandin+.

&cti+ity 3: 4ploring %imilarity Using %cale ra0ings 6G4s: 2' "' 1,7

Materials ist: Internet access or printed copy of instructions, bo(es full or emptyAenou+h to ha&e one for each +roup, rulers, pencil, +raph paper, scissors, tape, calculator

isit the websitehttp://ericir.syr.edu/irtual/essons/Mathematics/eometry/"N$$$;.htmlto access theinformation for this acti&ity which allows students to use their 'nowled+e of similarfi+ures. There should be no need for students to access the website since the material canbe printed for the class. This website pro&ides instructions for students to create scalemodels applyin+ their 'nowled+e of similar fi+ures. When the students are creatin+ theirscale models, they will ha&e to decide on a different scale so that the model is not thesame si0e as the bo( they were +i&en. They will calculate the scale factor in the last stepof the acti&ity. 5efore the students calculate the surface area and &olume of the ori+inaland scale model, ha&e students predict what they thin' the surface area and &olumeshould be based on the measurements of the two fi+ures. This will assist students indeterminin+ if their solutions are reasonable and allows them to apply the informationlearned in )cti&ity #. Then ha&e the students calculate the surface area and &olume of thebo(es, and the scale factors for the len+th, surface area, and &olume. Since students aremeasurin+ the items themsel&es, help them to understand why their results may not bee(actly what theory says they should be.

eometry>nit %Similarity and Tri+onometry %6
http://www.louisianaschools.net/lde/uploads/11056.dochttp://ericir.syr.edu/Virtual/Lessons/Mathematics/Geometry/GEO0003.htmlhttp://www.louisianaschools.net/lde/uploads/11056.dochttp://ericir.syr.edu/Virtual/Lessons/Mathematics/Geometry/GEO0003.html


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&cti+ity ": %potligt on %imilarity 6G4s: 2' "' 1*' 237

Materials ist: pencil, paper, Spotli+ht on Similarity 5M, o&erhead pro9ector

>se the Spotli+ht on Similarity 5M to ma'e a transparency for the o&erhead or a copycan be made for each student to help students in&esti+ate the followin+ problem:

) spotli+ht at point 1 throws out a beam of li+ht. The li+ht shines on a screen that can be mo&ed closer to or farther from the

li+ht. The screen at position ) is a hori0ontal distance ) from the li+ht and atposition 5 is a hori0ontaldistance 5.

The len+ths aand +indicate the len+ths of the li+ht patch on the screen.

Show that the ratio of the len+th +to the len+th adepends only on thedistances ) and 5, and not on the measure ofan+ley of the beam to theperpendicular, nor on the measure of an+le. of the beam itself.

>se similar trian+le relationships,a c + d

A B

+ += . This means

a c + d

A A B B+ = + . and

c d

A B=

because the trian+les are similar. nit %Similarity and Tri+onometry 4$


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is independent of the an+les.andyand is thescale factorrelatin+ the distances of thetwo screens and the si0es of the ima+es on the two screens.

Nnce students de&elop an understandin+ of the term scale factor, +i&e students themeasurements of certain fi+ures and a scale factor.


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&cti+ity !: ;idsegment Teorem for Triangles 6G4s: "' 1,' 1)' 1*7

Materials ist: pencil, tracin+ or patty paper, scissors, ruler, protractors

Separate the class into +roups of four. i&e each +roup a sheet of tracin+ paper or patty

paper and ha&e each draw a trian+le of any type and cut it out. nfold the trian+le and ma'e any obser&ations that seem to be true about thetrian+le and the midse+ment. Students should use rulers and protractors to &erifythe obser&ations they ma'e concernin+ the midse+ment of the trian+le.

7. Cold and unfold the remainin+ two midse+ments of the trian+le.%.


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pro&ide De(pertE answers to 2uestions from their peers about similar fi+ures. )s' the+roups to +enerate ; [ % 2uestions about similar fi+ures they thin' they mi+ht be as'ed orthat they would li'e to as' other e(perts. 1ro&ide no&elty items li'e ties, +raduation caps,lab coats, clipboards, etc., to don when the students are the Math Masters.

)fter +i&in+ the students time to re&iew material and create their 2uestions, call a +roupto the front of the room and as' its members to face the class standin+ shoulder toshoulder. The Math Masters in&ite 2uestions from the other +roups. With the first2uestion, model how the Math Masters should answer their peers 2uestions. Studentsshould huddle to+ether after each 2uestion to discuss and decide upon the answer, thenha&e the spo'esperson +i&e the answer.

!irect the students to thin' carefully about the answer they recei&e and to challen+e orcorrect the Math Masters if the answers are not correct or need additional information.)fter % minutes or so, ha&e a new +roup ta'e its place as Math Masters and continue theprocess.

Some 2uestions that mi+ht be as'ed are:

What is the definition of similar fi+ures What information must be pro&ided to pro&e that two trian+les are similar )re all trian+les similar )re all 2uadrilaterals similar )re s2uares similar to rectan+les


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"mploy a directed reading2thining activity !%TA"&iew literacy strate+y descriptionsto aid students in readin+ and processin+ the information presented in the website. The%TAapproach in&ites students to ma'e predictions and chec' their predictions throu+hthe readin+. It re2uires students to pause as they read the information to as'/answer

2uestions. Ta'e the students throu+h the followin+ steps: Introduce +acground nowledge05e+in the lesson with a discussion abouttri+onometry. "licit information students may already 'now about tri+onometry.Many students may ha&e limited prior 'nowled+e of tri+onometry, and that iso'ay. !iscuss the title of the acti&ity. ?ecord students ideas on the board or chartpaper.

Mae predictions0)s' 2uestions that in&ite predictions, such as: DWhat do youe(pect to learn from this acti&ity 5ased on what we ha&e learned already, whatinformation do you thin' the author will includeE


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ha&e a short le+ of 3 unit. ) ;$V-4$V-6$V trian+le whose short le+ is 3 is called the unittrian+le. sin+ the 1ytha+orean theorem, students will calculate the len+th ofthe hypotenuse in simplified radical form.

?epeat the acti&ity se&eral times, but use different measures for the sides of thee2uilateral trian+le. Start with e2uilateral trian+les whose sides are 7 units, and then 4units and also use the isosceles ri+ht trian+le. !o this se&eral times until students see apattern in the numbers. The +oal is to ha&e students write these as formulas:

short leg D 3# hypotenuseand long leg D ; short legin ;$V-4$V-6$V trian+les. Cor

7%V-7%V-6$V trian+les, the relationship is hypotenuse D # (leg. )dditionally, show

students how proportions are an alternati&e way of calculatin+ the same &alues.

To help students become familiar with the definition ofsineand cosine, ha&e themcalculate the ratios usin+ the side len+ths of special ri+ht trian+les. nit %Similarity and Tri+onometry 44


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calculate the hei+ht of the chosen ob9ect usin+ tri+ functions. Instructions forma'in+ a clinometer can be found in most +eometry te(tboo's and onnumerous websites, such as:http://web793.lane.edu/partners/eweb/ttr/mc'en0ie/resources/ideaban'/clin.html.

eometry>nit %Similarity and Tri+onometry 4*
http://web4j1.lane.edu/partners/eweb/ttr/mckenzie/resources/ideabank/clin.htmlhttp://web4j1.lane.edu/partners/eweb/ttr/mckenzie/resources/ideabank/clin.htmlhttp://web4j1.lane.edu/partners/eweb/ttr/mckenzie/resources/ideabank/clin.htmlhttp://web4j1.lane.edu/partners/eweb/ttr/mckenzie/resources/ideabank/clin.html


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Geometry

Unit $: &rea' Polyedra' %urface &rea' and (olume

Time rame:)ppro(imately fi&e wee's

Unit escription

This unit pro&ides an e(amination of properties of measurement in +eometry. Whilestudents are familiar with the area, surface area, and &olume formulas from pre&iouswor', this unit pro&ides 9ustifications and e(tensions of students pre&ious wor'.Si+nificant emphasis is +i&en to ;-dimensional fi+ures and their decomposition forsurface area and &olume considerations.


Students understand that measurement is a choice of unit, an application of that unit tothe ob9ect to be measured, a countin+ of the units, and a reportin+ of the measurement.Students should ha&e a solid understandin+ of poly+ons and polyhedra, the meanin+ ofre+ular, the meanin+ of parallel and perpendicular in ;-dimensional space, and the reason

pyramids and cones ha&e a factor of3; in their formulas.

Guiding #uestions

3. 8an students find the perimeters and areas of trian+les, standard2uadrilaterals, and re+ular poly+ons, as well as irre+ular fi+ures for whichsufficient information is pro&ided

#. 8an students pro&ide ar+uments for the &alidity of the standard planar areaformulas

;. 8an students define and pro&ide 9ustifications for poly+onal and polyhedralrelationships in&ol&in+ parallel bases and perpendicular altitudes and theo&erall +eneral , Bh= formula, whereBis the area of the base

7. 8an students use the surface area and &olume formulas for rectan+ular solids,prisms, pyramids, and cones

%. 8an students find distances in ;-dimensional space for rectan+ular solids

usin+ +enerali0ations of the 1ytha+orean theorem4. 8an students use area models to substantiate the calculations for

conditional/+eometric probability ar+uments

eometry>nit 4)rea, 1olyhedra, Surface )rea, and olume 46


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Unit $ Gradee+el 4pectations 6G4s7


;easurement

=. Cind the &olume and surface area of pyramids, spheres, and cones M-;-


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)fter in&esti+atin+ the parallelo+ram, ha&e students e(amine the formulas for other planefi+ures [ trian+les, trape0oids, and rhombi.


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>sin+ these cate+ories, create thou+ht-pro&o'in+ prompts related to the topic. In thiscase, students should be +i&en a prompt which will cause them to predict the relationshipbetween two cylinders with different bases and hei+hts. This is modified because it doesnot as' students to predict what will happen ne(t, but ha&in+ them ma'e the predictiondoes fall under the*e.tcate+ory.

To present the writin+ topic ta'e two sheets of paper and create two baseless cylindersli'e the ones shown below.

1resent the cylinders to the class and ha&e them write a few sentences to answer thefollowin+ topic from the*e.tcate+ory post the topic on the board or the o&erhead:

*e.t/ 5ased on your prior 'nowled+e of &olume, predict whether you belie&e the&olumes of the two cylinders are e2ual or whether the shorter or taller cylinderhas the +reater &olume. "(plain why you predicted as you did.

!o not ha&e the students calculate the &olume at this stepAtheir predictions should bebased solely on their prior 'nowled+e and their obser&ations.

)fter +i&in+ the students time to write their responses, ha&e students share theirpredictions.


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What If- i&en a cylinder which has a base with a diameter of 3$ inches and ahei+ht of 3# inches, what would happen to the hei+ht of the cylinder if the&olume was to remain the same and the base was increased by %$] Whatwould happen to the base of the cylinder if the &olume was to remain thesame and the hei+ht was increased by =%]

Nnce students complete their writin+s, ha&e students share their ideas with a partner, thena +roup, and then the entire class as they did after the*e.tprompt. These discussionsshould include their calculations of &olume and the new measures for the hei+ht and base.

&cti+ity ": Cube Coloring Problem 6G4s: *' 1,7

Materials ist: a lar+e 2uantity of unit cubes su+ar cubes can be used, +raph paper,colored pencils or mar'ers, pencils, learnin+ lo+

Teacher *ote/This activity provides a review of surface area and volume of prismsmastered in grade (0

In this acti&ity, students will in&esti+ate what happens when different si0ed cubes areconstructed from unit cubes, the surface areas are painted, and the lar+e cubes are ta'enapart.

nit 4)rea, 1olyhedra, Surface )rea, and olume =7


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Students will note that the three painted faces are always the cornersAei+ht on a cube.The cubes with two faces painted occur on the ed+es between the corner and increase bytwel&e each time. The cubes with one painted face occur as s2uares on the si( faces of theori+inal cube. The cubes with no painted faces are the cubes within the cube. )s an entryin the students math learning logs&iew literacy strate+y descriptions, ha&e the students

e(plain the patterns they ha&e obser&ed. )lso, ha&e the students e(plain how they belie&ethe surface area and &olume formulas for a cube were de&eloped based on their findin+sin this acti&ity. ) learning logis typically a noteboo' a student 'eeps in order to recordideas, 2uestions, and new understandin+s. Students should 'eep this as a separatenoteboo' or as a separate section in their binders if binders are used. Students shoulduse their math learning logsother times in class in addition to those listed throu+hout thecurriculum to pro&ide opportunities to assess understandin+.

To end the acti&ity, ha&e students complete a S)AW* writing &iew literacy strate+ydescriptions from the What If- cate+ory. 1ost the followin+ prompt by writin+ it on theboard or on the o&erhead: D


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measurements identify the radius and hei+ht to +enerate the +eneral formula for findin+the &olume.


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&cti+ity !:%urface &rea 6G4: !7

Materials ist: models of pyramids from )cti&ity 4, pencil, paper

>sin+ their models from )cti&ity 4, ha&e students determine the total surface area of their

constructed pyramids and describe the process of findin+ the surface area of thepyramids. Students should find the surface area of the pyramids with s2uare bases andother re+ular poly+on bases as well.

&cti+ity ): (olumes of Pyramids and Cones 6G4: !7

Materials ist: &olume model 'its, rice or other filler, pencil, paper

8ompare the &olumes of a pyramid and prism with the same base and hei+ht as ademonstration usin+ a &olume model 'it. If enou+h model 'its are a&ailable, ha&e

students wor' in +roups of # or ; when completin+ the acti&ity. Students could also ma'etheir own models usin+ old manila file folders and then complete the acti&ity.

Cill the pyramid from the 'it with rice or unpopped popcorn. )s' students to estimatehow many times the pyramid must be filled in order for the prism to be filled. !o thesame thin+ with a cone and cylinder. !e&elop the concept that the &olume of a conepyramid is one-third the &olume of a cylinder prism if the two solids ha&e thecon+ruent hei+hts and bases. )s an e(tension, as' students to estimate the relationshipbetween the cone and the sphere which are also a part of the 'it. Since the cone must befilled twice before the sphere is filled, the sphere is twice as lar+e as the cones &olume or#; the &olume of the cylinder. 1ro&ide real-life applications in which students must find

the &olumes of cones, pyramids, prisms, and spheres. Students will ha&e to be +i&en theactual formula for the &olume of a sphere in order to accurately find the &olume ofspheres.

&cti+ity *: ;ore 0it (olume and %urface &rea 6G4: !7


nit 4)rea, 1olyhedra, Surface )rea, and olume ==


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&cti+ity 1,: (olume of -rregular Bb?ects 6G4: !7

Materials ist: cylindrical container, water, irre+ular ob9ects li'e an e++-shapedpaperwei+ht, tube of toothpaste with the bo(, pencil, paper, calculator

Teacher *ote/ Although #L9 ; refers to only the volume of pyramids$ cones$ andspheres$ this activity gives students another opportunity to determine the volume of an

o+Hect for which no formula e.ists0

1ro&ide students with a cylindrical ob9ect whose &olume can be calculated and withmar'in+s to measure a predetermined amount of water a bea'er from a science classwould do well. )s' them to place water in the bea'er but not to fill it to the top. !iscusswith the students the &olume of water in the bea'er. )s' them to place an irre+ularob9ect, li'e an e++-shaped paperwei+ht, into the water, bein+ careful not to spill anywater. @ote the displacement of the water and determine the &olume of the paperwei+ht.

@e(t, repeat the acti&ity with a tube of toothpaste. se this acti&ity for a discussion aboutpac'a+in+ efficiency.

&cti+ity 11: Geometric Probability 6G4 217

Materials ist: pencil, paper, calculator

Students should be +i&en problems that re2uire them to find the area of a &ariety ofshapes. This should include basic fi+ures as well as fi+ures within other fi+ures, orcombinations of fi+ures. )s' students to find the probability of randomly selectin+ apoint in a shaded re+ion of the +i&en fi+ure.

"(ample:Mar' created a +ame consistin+ of ;# s2uares on a rectan+ular +ame board. Theboard measures 3-foot by #-feet. 34 of the s2uares are ;-inches by ;-inches whilethe other 34 s2uares are #-inches by #-inches.


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"(ample:) radio station will play the son+ of the day once durin+ each hour. The 3$3stcaller will win ^3$$. If you turn on the radio at #:;% p.m., what is the probabilitythat you ha&e missed the start of the son+ durin+ the #:$$ p.m. to ;:$$ p.m. hour

Solution/ ;% =4$ 3# %*]= = 0

%ample &ssessments

1erformance and other types of assessments can be used to ascertain studentachie&ement. "(amples include:

General &ssessments

The student will complete learnin+ lo+ entries for this unit. Su++ested topics

include:o "(plain why pyramids and cones ha&e 3; as a factor in their formulas for

&olume.o Show how to find the &olume and surface area of a solid that combines a

cylinder with a cone, or a prism with a pyramid. 5e specific.

1ro&ide the student with three-dimensional models. The student will s'etchdia+rams and ta'e appropriate measurements from actual ob9ects needed tocalculate &olume and surface area. The student will label s'etches withmeasurements and then show the process used to calculate &olume and surfacearea. Since this tas' is time consumin+, the student will be +i&en no more thanthree ob9ects, some type of prism, either a cone or pyramid, and a cylinder.

&cti+ity%pecific &ssessments

)cti&ity 3 : The student will determine the total li&in+ area and total area of a+i&en floor plan. This floor plan should ha&e odd-shaped rooms that wouldre2uire usin+ most of the formulas discussed in this acti&ity.

)cti&ity 4 : The student will build a pyramid with a minimum surface area andminimum &olume. The student will show the measurements of the bases,hei+ht of the faces, and the hei+ht of the pyramid based on the +i&enminimum surface area and &olume.

)cti&ity 3$ : The student will desi+n a container to hold a specific &olume of aspecified product. )ssi+n each student a different &olume and specific shapeor assi+n the &olume and allow the student to choose the shape. The studentwill desi+n a label for the container and determine the area of that label. The

eometry>nit 4)rea, 1olyhedra, Surface )rea, and olume =6


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Geometry

Unit !: Circles and %peres

Time rame: )ppro(imately fi&e wee's

Unit escription

This unit focuses on 9ustifications for circular measurement relationships in two and threedimensions, as well as the relationships dealin+ with measures of arcs, chords, secants,and tan+ents related to a circle. It also pro&ides a re&iew of formulas for determinin+ thecircumference and area of circles.


Students can find the surface area and &olume of spheres. Students can apply therelationship of the measures of minor and ma9or arcs to the measures of central an+lesand inscribed an+les, and to the circumference in &arious situations. They can alsoe(plain the rele&ance of tan+ents in real-life situations and the power of a pointrelationship for intersectin+ chords.

Guiding #uestions

3. 8an students pro&ide an ar+ument for the &alue of and the way in which it

can be appro(imated by poly+ons#. 8an students pro&ide con&incin+ ar+uments for the surface area and &olumeformulas for spheres

;. 8an students apply the circumference, surface area, and &olume formulas forcircles, cylinders, cones, and spheres

7. 8an students apply +eometric probability concepts usin+ circular area modelsand usin+ area of a sector

%. 8an students find the measures of inscribed and central an+les in circles, aswell as measures of sectors, chords, and tan+ents to a circle from an e(ternalpoint

4. 8an students use the power of a point theorem intersectin+ chords and

intersectin+ secants to determine measures of intersectin+ chords in a circle

eometry>nit =8ircles and Spheres *3


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Unit ! Gradee+el 4pectations 6G4s7


;easurement

=. Cind &olume and surface area of pyramids, spheres, and cones M-;-


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well as re&ise definitions and e(amples from their initial self-assessment. The +oal is toreplace all the chec' mar's and minus si+ns with a plus si+n. 5ecause studentscontinually re&isit their &ocabulary charts to re&ise their entries, they ha&e multipleopportunities to practice and e(tend their understandin+ of 'ey terms related to circlesand spheres.

&cti+ity 2: eri+ation of te &rea of a Circle ormula 6G4s: 13' 1!7

Materials ist: pencil, paper, paper circles, scissors, automatic drawin+ pro+ram

ead students in an e(ercise to show how the formula for the area of a circle can bede&eloped. sin+ this process allows students to re&iew the circumferenceformula and area formula of a parallelo+ram.

)nother way to show the deri&ation is to increase the number of sides of a re+ularpoly+on inscribed in a circle. >sin+ an automatic drawin+ pro+ram such as The#eometer8s Setchpad, ha&e students draw a circle and inscribe an e2uilateral trian+le.


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)s an entry in the students math learning logs&iew literacy strate+y descriptions, ha&ethe students e(plain the process they should employ to find the probability of a darthittin+ a certain re+ion of the board. The prompt could be:

8reate a dart board that includes at least one circle and one other poly+on. Shadeat least one re+ion of the dart board. 8alculate the probability of a dart hittin+ theshaded re+ions. "(plain the process you used to find the probability for yourdart board.

)learning log is typically a noteboo' students 'eep in order to record their ideas,2uestions, and new understandin+s. Students should 'eep this as a separate noteboo' oras a separate section in their binders. Students should use their math learning logsothertimes in class, in addition to those listed throu+hout the curriculum, to pro&ideopportunities to assess understandin+.

&cti+ity ": Central &ngles and &rcs 6G4: 137

Materials ist: Sample @otes 5M, Split-1a+e @otes Model 5M, paper, pencil, circledia+rams, strin+ or tailors tape,

In this acti&ity, students will use asplit2page notetaing&iew literacy strate+ydescriptions format to ta'e notes on central an+les and arcs. Split2page notetaingissimply a different way for students to or+ani0e their notes to help them use their notesmore effecti&ely for study. Model the approach by placin+ on the board or o&erheadsample split-pa+e notes from the topic of circles. "(plain the &alue of ta'in+ notes in thisformat by sayin+ it or+ani0es information and ideas from &arious sources. It helpsseparate bi+ ideas from supportin+ details, and it promotes acti&e readin+ and listenin+.

@e(t, as' students tousesplit2page notetaingwhile listenin+ to a brief presentation oncentral an+les and arcs. This presentation should define the terms central an+les, arcs,ma9or arcs, minor arcs, and semicircles. Tell students to draw a line from the top tobottom of their pa+es about one-third from the left side of the pa+e. Nn the left side theywill write the terms or concepts and on the ri+ht side they will write the definition ore(planation of the term. They can also draw e(amples on the ri+ht side of the pa+e. )fterthe presentation, ha&e students compare notes with a partner, then answer 2uestions andpro&ide clarification usin+ the Split-1a+e @otes Model 5M as a +uide. Show studentshow they can prompt recall by bendin+ the sheet of notes so that the information in theri+ht or left column is co&ered. Then proceed with the remainder of the acti&ityremindin+ students to add information to the concepts they ha&e recorded already li'eformulas and other e(amples.

1ro&ide pairs of students with a dia+ram containin+ a circle with a +i&en radius len+thand with central an+les labeled 3, #, and ;. The measures of the central an+les should bein the ratio of #:;:7. nit =8ircles and Spheres *7
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measures of the respecti&e arcs. ?epeat this acti&ity for other circles and other ratios. Inaddition, ha&e students identify ma9or and minor arcs +i&en a central an+le. Studentsshould reali0e that the measure of the ma9or arc is e2ual to ;4$ [ the measure of theminor arc. To help students &isuali0e ma9or and minor arcs, as' them to determine thetype of arc associated with &arious times of day displayed on an analo+ cloc'.

?e&iew with students the fact that the sum of the central an+les of a circle is ;4$V. Ifneeded, ha&e students use strin+ or a tailors tape measure to find the circumference ofthe circle and to determine the len+th of one of its arcs.


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&cti+ity $: Grap -t 6G4s: 13' 227

Materials ist: results from +roup sur&eys, protractors, calculators, pencil, paper, colorpencils or colors, teacher-created data sets and matchin+ circle +raphs, Internet access,

ma+a0ines, newspapers The last three items are optional and for teacher use


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sectors. nit =8ircles and Spheres *=


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&cti+ity 1,: -nscribed angles 6G4s: 13' 1!' 1*7

Materials ist: paper, pencil, ruler, protractor

se the 5M to discussthe three theorems listed below:

3. If a tan+ent and a secant intersect at a point on a circle, then the measure of eachan+le formed is half of the measure of its intercepted arc.

#. If two secants intersect in the interior of a circle, then the measure of each an+le ishalf the sum of the measures of the arcs intercepted by the an+le and its &ertical

an+le.;. If a tan+ent and a secant, two tan+ents, or two secants intersect in the e(terior of a

circle, then the measure of the an+le formed is half the difference of the measuresof the intercepted arcs.

Throu+hout the discussion, ha&e students write down the relationships formed by eachpair of intersectin+ lines.

eometry>nit =8ircles and Spheres **


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&cti+ity 13: %urface &rea of a %pere 6G4: !7

Materials ist: Surface )rea of a Sphere 5M, small spheres, wrappin+ paper, pencil,paper, scissors, tape


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1ro&ide practical applications problems which re2uire the use of the surface area of asphere for students to wor'.

&cti+ity 1": %urface &rea and (olume of %peres 6G4s: !' 1,7

Materials ist: Internet access for research, paper, pencil, scissors, tape, centimeter orinch cubes, se&eral types of balls

Spherical balls are used in many sports e.+., +olf, soccer, baseball, bas'etball.


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Geometry

Unit ): Transformations

Time rame:)ppro(imately two wee's

Unit escription

This unit pro&ides a deeper mathematical understandin+ and 9ustifications fortransformations that students ha&e seen in pre&ious +rades. The focus is pro&idin+9ustifications for the con+ruence and similarity relationships associated with translations,reflections, rotations, and dilations centered at the ori+in.


Students determine what transformations ha&e been performed on a fi+ure and candetermine a composition of transformations that can be performed to mimic othertransformations li'e rotations. They are also able to find new coordinates fortransformations without actually performin+ the indicated transformation.

Guiding #uestions

3. 8an students find transformations and mappin+s that relate one con+ruentfi+ure in the plane to another

#. 8an students pro&ide an ar+ument for the preser&ation of measures of fi+uresunder reflections, translations, and rotations

;. 8an students find the dilation enlar+ement or reduction, centered at theori+in, of a specified fi+ure in the plane and relate it to

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