Cone Exploration and Optimization : : : : Copyright © 2014 National Math + Science Initiative, Dallas, Texas

conceptual understanding. and reinforcemathematicalconceptstoenhance darkened pointsofthestartointroduce,explore, for usingthoserepresentationsindicatedbythe lesson providesmultiplestrategiesandmodels order toincreasestudentunderstanding. to connectvariousapproachesasituationin NMSI emphasizesusingmultiplerepresentations MODALITY and VolumesAreas MODULE/CONNECTION TO AP* on polynomials Algebra 2,Math3,Pre-Calculus,or4inaunit LEVEL Board wasnotinvolvedintheproduction ofthisproduct. of theCollegeEntranceExaminationBoard. TheCollege *Advanced Placementand AP are registered trademarks G I H INITIATIVE MATH +SCIENCE NATIONAL J N Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at at online us Visit reserved. rights All Texas. Dallas, Initiative, +Science Math National ©2014 Copyright G K P F A L V G –Graphical N –Numerical A– Analytical – VerbalV P –Physical E M

D N Q C

The O B P A I function thatmodelsarelationship,whilereinforcing This lessonfocusesonunderstandingandcreatinga construction. domain issueswhileexploringthelimitsofcone of theconewithgreatestvolume.Studentsconsider conclude thelessonbydeterminingdimensions amount ofoverlapincreasesordecreases.Students in whichthecone’s dimensions changeasthe LESSON THIS ABOUT and Optimization ExplorationCone Students will OBJECTIVES practices. a varietyofmathematicalconcepts,skills,and by developingcoherenceandconnectionsamong enhances studentunderstandingofthesestandards table featureontheirgraphingcalculator. The lesson volumes. Studentswillgraphfunctionsandusethe graphing anon-linearfunction,andcalculating the skillsofdeterminingareasonabledomain, ● ● overlapping theedges. They explorethemanner by cuttingalongtheradiusofacircleand n thislesson,studentsmanipulateapapercone determine theconeofmaximumvolume. of acone. explore theeffect ofchangingthedimensions www.nms.org Mathematics . i T E A C H E R P A G E S

T E A C H E R P A G E S ii G-GMD.3 A-CED.2 F-IF.5 Reinforced/Applied Standards F-IF.4 Targeted Standards standard isconnectedtomodeling. of aspecificstandardindicatesthatthehighschool to thespecificskill. standards ratherthanprovidingtheinitialintroduction requires thatstudentsrecallandapplyeachofthese State StandardsforMathematicalContent. The lesson This lessonaddressesthefollowingCommonCore MATHEMATICAL CONTENT COMMON CORE STATE STANDARDS FOR Mathematics—Cone Exploration and Optimization : : : : Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at at online us Visit reserved. rights All Texas. Dallas, Initiative, +Science Math National ©2014 Copyright Create equations in two or more variables Create equationsintwoormorevariables Use volume formulas for cylinders, Use volumeformulasfor cylinders, See question7 problems. pyramids, cones,andspheres tosolve See questions7,9 axes withlabelsandscales. quantities; graphequationsoncoordinate to representrelationshipsbetween See questions5-6,8 domain forthefunction. positive integerswouldbeanappropriate assemble nenginesinafactory, thenthe the numberofperson-hoursittakesto For example,ifthefunctionh(n)gives quantitative relationshipitdescribes. graph and,whereapplicable,tothe See questions5-15 and periodicity. minimums; symmetries;endbehavior; or negative;relative maximumsand is increasing, decreasing, positive, intercepts; intervalswhere thefunction of therelationship. key featuresgivenaverbaldescription the quantities,andsketchgraphsshowing features ofgraphsandtablesinterms between twoquantities,interpretkey Relate the domain of a function to its Relate thedomainofafunctiontoits For afunctionthatmodelsrelationship The starsymbol( ★

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Key features include: ★ ★

) at the end ) attheend ★

MP.1 Standards forMathematicalPractice. teachers toaddressthefollowingCommonCoreState connections acrossgradelevels. This lessonallows understanding andtoassisttheminmakingimportant proficiencies tohelpstudentsdevelopknowledgeand NMSI incorporatestheseimportantprocessesand that arecriticalformathematicsinstruction. practices basedonprocessesandproficiencies These standardsdescribeavarietyofinstructional MATHEMATICAL PRACTICE COMMON CORE STATE STANDARDS FOR G-SRT.8 MP.5 MP.2 : : :

Make sense of problems and persevere in Make senseofproblemsandperseverein determine themaximumvolume ofthecone. Students usegraphingcalculatorsto dimensions. overlap theedgestocreate conesofvarying Students cutalongaradiusofcircle and Use appropriatetoolsstrategically. volume functionintermsofasinglevariable. and manipulatethesymbolstocomposea manipulation ofaconcrete modelofacircle the dimensionsofaconebasedon Students create equationsinvolving Reason abstractlyandquantitatively. maximum volume. determine thedimensionsofconewith an equationtodeterminethevolume,and of overlapincreases ordecreases, write the cone’s dimensionschangeastheamount dimensions, explore themannerinwhich overlap theedgestocreate conesofvarying Students cutalongaradiusofcircle and solving them. : Use trigonometric ratios and the Use trigonometricratiosandthe See question7 triangles inappliedproblems. Pythagorean Theorem tosolveright

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our website: The followingadditionalassessmentsarelocatedon embedded inthislesson: The followingtypesofformativeassessmentsare ASSESSMENTS included inthislesson: The followingskillslaythefoundationforconcepts FOUNDATIONAL SKILLS ● ● ● ● ● ● ● ● Choice Questions Areas and Volumes –Pre-CalculusMultiple Response Questions Areas and Volumes –Pre-CalculusFree Choice Questions Areas and Volumes – Algebra 2Multiple Response Questions Areas and Volumes – Algebra 2Free Students applyknowledgetoanewsituation. Students engageinindependentpractice. using agraphingcalculator Calculate themaximumvalueofafunction area, volume) slant height,circumference,surface Identify featuresofacone(radius,height, Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at at online us Visit reserved. rights All Texas. Dallas, Initiative, +Science Math National ©2014 Copyright Mathematics—Cone Exploration and Optimization MATERIALS ANDRESOURCES ● ● ● ● ● ● Tape Rulers Protractors Skewers Scissors pages Student Activity www.nms.org . iii

T E A C H E R P A G E S

T E A C H E R P A G E S T iv NMSI TI-Nspire SkillBuilders. Equations andFindingPointsofInterest Nspire™ technology. See You maywishtosupportthisactivitywith TI- figures. provided forthislessonconsiderthedegenerate maximum volumeeasiertojustify. The answers extension makestheidentificationofanabsolute zero oraradiusofzero.Forcalculusstudents,this to permitdegenerateconeswitheitheraheightof In calculus,thedefinitionofaconeisextended correct conclusion extremes ofthedomainhelpsstudentsdraw Observing theverysmallvolumeofconeat of theconealwaysincreasesordecreases. sometimes mistakenlyconcludethatthevolume and surfaceareaallchangemonotonically, students provide guidanceasneeded.Sincetheradius,height, their observations.Listentodiscussionsand TEACHING SUGGESTIONS Mathematics—Cone Exploration and Optimization circles toformvariousconesanddiscuss work ingroupsastheymanipulatetheir his lessonismostpowerfulwhenstudents Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at at online us Visit reserved. rights All Texas. Dallas, Initiative, +Science Math National ©2014 Copyright . Graphing Functionsand inthe 7 5 1-4 include thefollowing: Suggested modificationsforadditionalscaffolding fixed at10cm.Leadthestudenttorecognize the legslabeled triangular cross formula forthevolumeofacone, Ask thestudenttobeginwithfamiliar domains forradiusandheight. the studentvisualizelimitednatureof Refer tothesketchesinquestions1–4help before attemptingtodrawconclusions. student tolabelthedimensionsofcone look likeatvariouspointsofoverlap. Ask the Provide sketchesofwhattheconewould volume formula. be solvedfor a relationshipbetween that thePythagorean Theorem provides . Provideasketchoftheright h andsubstitutedintothe sectionofthecone,with r and h r andthehypotenuse and www.nms.org h thatcan . Mathematics—Cone Exploration and Optimization

Objectives Solve and use literal equations in real life and mathematical applications. AP Calculus Skills/ AP Pre-Calculus Skills/Objectives Solve literal equations area, (perimeter, and volume). Objectives Algebra 2 Skills/ Solve literal equations area, (perimeter, and volume). Objectives Geometry Skills/ Solve literal equations area, (perimeter, and volume). S E G A Objectives P Algebra 1 Skills/

Solve literal equations area, (perimeter, and volume). R E H C Objectives A 7th Grade Skills/ Solve literal equations area, (perimeter, and volume). E T Objectives 6th Grade Skills/ Solve literal equations area, (perimeter, and volume). Objectives 5th Grade Skills/ Isolate the variable for length or width in the formulas for area and perimeter of a rectangle. Objectives 4th Grade Skills/ Isolate the variable for length or width in the formula for area of a rectangle. Objectives 3rd Grade Skills/ Write the formula Write for the area of a rectangle using a variable for the missing length or width. NMSI CONTENT PROGRESSION CHART PROGRESSION NMSI CONTENT goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how In the spirit of NMSI’s specific skills build and develop from third grade through pre-calculus in an accelerated program that enables students to take college-level Algebra 1 are compacted into three courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and courses. Grade 6 includes all of the content and some from 7, 7 contains remainder Algebra 1 Algebra 1 includes the remainder of content from Grade 8 and all content and some of the from Grade 8, content. This portion The complete Content Progression Chart for this module is provided on our website and at the beginning of training manual. of the chart illustrates how skills included in this particular lesson develop as students advance through accelerated course sequence.

T E A C H E R P A G E S vi Mathematics—Cone Exploration and Optimization Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at at online us Visit reserved. rights All Texas. Dallas, Initiative, +Science Math National ©2014 Copyright www.nms.org . 8. 7. 6. 5. 4. 3. 2. 1. Answers b. b. d. b. d. b. radius andthevolumeto takeonvaluesof0or eliminate anynegativevalues for the volumeisgreaterthan orequalto0.Since the volumeofacone. This functionisalwaysnon-negativesothereareno restrictionstoassurethat From astrictlyalgebraicperspective,thedomain ofthisfunctionisdeterminedbytheradicand. angle withthebasesothat Accept anyanswers. change. The circumferenceofthebaseconedecreasesfrom . 0cm a. a. Theradiusincreasesfrom0cmto a. a. c. c. c. INITIATIVE MATH +SCIENCE NATIONAL Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at at online us Visit reserved. rights All Texas. Dallas, Initiative, +Science Math National ©2014 Copyright If adegenerate coneisconsidered,thelateralsurfaceareaatitsgreatestwhenheightof The areaofthebaseconedecreasesfrom The heightincreasesfrom0cmto The radiusdecreasesfrom 0 cm increase andthendecrease. As theradiusofconechangesbetweenitsmaximum cone willbe0. cone isequalto If adegenerateconeisconsidered,thelateralsurfaceareaatitssmallestwhenheightof the conewillbe0. cone is0. The lateralsurfaceareaincreases. The areaofthebaseconeincreasesfrom The heightdecreasesfrom The lateralsurfaceareadecreasesfrom The slantheightoftheconeisradiusoriginalcircle(length ≤ ≤

r h

≤ so ≤ The radiusoftheconeissamelengthasoriginalcircle. 10cmifdegenerateconepermittedor0 10cmifdegenerateconepermittedor0 Cone ExplorationCone Optimization and Q , theradiusoforiginalcircle. A common(butincorrect)conjectureisthattheslant heightformsa45degree

and Q Q

cmto0cm. cmto0cm. . r Another common(butincorrect)conjectureisthat . The situationaldomainistheneither

Q 0 Q cm. cm. c m 2 to . Inthecontextofsituation,thisfunctionrepresents r 10 0 representstheradiusof the cone,domainmust <<

0 r to c m The radiusoftheconeis0.

0 2 . to

c ifwewanttohold m < < to 2

. r h

< cmto0cm.

< 0 anditsminimumof0,thevolumemust 10cmifnotpermitted c 10cmifnotpermitted m 2 . 10 0 ≤≤ The slantheightdoesnot r r Q and ).

The volumeofthe www.nms.org V ifweallowthe to positivevalues.

The volumeof Mathematics . . vii

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T E A C H E R P A G E S viii 16. 15. 14. 13. 12. 11. 10. 9. Mathematics—Cone Exploration andOptimization Students comparetheirmeasurementstothedimensionsinquestion13. See graph: 1.530 100 150 200 250 300 350 400 450 50 V Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at at online us Visit reserved. rights All Texas. Dallas, Initiative, +Science Math National ©2014 Copyright = 1 360 cm, θ 2 cm; ( 2 3 ⋅ 10 4 ⋅ π ) 5 = cm.Comparisonswithstudentconjectureswillvary. 6.01 6 7 ° 8 9 10 cm. r www.nms.org . NATIONAL MATH + SCIENCE INITIATIVE Mathematics

Cone Exploration and Optimization

For the following activities, cut out the circle and cut along Q.

E D F

I Q A

N L M

Consider that the radius of the circle is length Q. Overlap the sides of the cut in the circle, moving point A counterclockwise to various points on the circumference of the circle to form a cone. 1. What happens to the circumference of the base of the cone as point A moves over the circle through points B, C, D, and continuing around the circle until it reaches its original position? What happens to the slant height of the cone? What is the relationship between the radius of the original circle and the slant height of the cone?

2. When the circumference of the cone decreases, what happens to a. the radius of the base of the cone?

b. the height of the cone?

c. the area of the base?

d. the lateral surface area of the cone?

3. When the circumference is its smallest length, point A then moves clockwise around the circle back to its starting point. As the circumference of the cone increases, what happens to a. the radius of the base of the cone?

b. the height of the cone?

c. the area of the base?

d. the lateral surface area of the cone?

4. a. At what height is the cone’s lateral surface area the greatest? What is the radius of the cone when the lateral surface area is the greatest? What is the volume of the cone when the lateral surface area is the greatest?

b. At what height is the cone’s lateral surface area the least? What is the radius of the cone when the lateral surface area is the least? What is the volume of the cone when the lateral surface area is the least?

c. What happens to the volume of the cone between the instances when the radius of the cone is Q and the radius of the cone is 0?

For questions 5 – 12, assume that the radius of the original circle is 10 cm. 5. a. Determine the domain for the height of the cone formed.

b. Determine the domain for the radius of the cone formed.

6. Make a conjecture as to what the cone’s radius should be to maximize the volume of the cone.

7. Determine the equation for the volume of the cone as a function of r, the radius of the base of the cone.

8. What is the domain of the function V 450 determined in question 7? Explain the domain algebraically, based on the equation. 400 350

300

250 9. Graph the volume of the cone with respect to the radius of the cone. 200 150

100

50 r 0 1 2 3 4 5 6 7 8 9 10

10. Verify that the volume of the cone is 0 when r is 10 cm.

11. What is the volume of the cone when r is 5 cm?

12. Using a graphing calculator, determine the maximum volume of the cone created from a circle with a radius of 10 cm.

13. Determine the radius and the height of the cone with maximum volume. How do these results compare to your conjecture in question 6?

14. What is the circumference of the base of the cone with maximum volume? What is the length of the arc that should be cut from the circumference of the original circle in order to form the cone with the maximum volume?

15. What is the measure of the arc, in degrees, that is cut from the circumference of the circle?

16. Based on the answers to questions 14 and 15, remove the appropriate sector from the circle and tape the cut edges together to form the cone. Measure the radius and the height of the cone and compare their lengths to the calculated dimensions.