Project Based Learning
Project 4
Space Station Design
IntroductiontoC ollegeAlgebra LESSONPLAN Project4
SpaceStationD esign
MATRICES
LESSOND ESCRIPTION
Students w ill deepen their understanding of matrices b y d esigning a real–world space station
usingreal–worlds pacestationm odulesa ndlaunchvehicles.
VOCABULARY
● B330Module:A BigelowA erospaceinflatablespaces tationmodule.
● B330|DeltaI VS tack:AB330a ttachedtoitsL Vthatiso ntheL aunchP ad.
● CrewCapsule:As pacecraftthatisusedtoferryc rewtoandfroma s paces tation.
● CrewSize:T henumbero fpeopleonb oarda s paces tation.
● DockingNode(DN):Modulethatallowss pacestationmodulestobeattachedtogether.
● Dragon2|F alcon9S tack:AcrewcapsuleattachedtoitsLVthatiso ntheL aunchP ad.
● LaunchVehicle(LV):Avehiclethatisusedtoliftpayloadandp assengerstoLEO.
● SpaceXDragon2:A c rewcapsulethatc anlift6 astronautstoL EOandback.
● SpaceXF alcon9 :AL Vthatcanlift2 2,800k gtoLEO.
● InternationalS paceStation(ISS.):Thespaces tationthatiscurrentlyorbitingtheearth.
● LaunchPad:W herearockettakesofffrom.
● LowE arthO rbit(LEO):A bodycirclingtheEartha ta m inimumorbitala ltitude.
● PB/DN:C ombinationofaPBattachedtoaD Nandw eighs17,000k g.
● PB/DN|F alcon9 Stack:APB/DNa ttachedtoitsLVthatisontheLaunchP ad.
● PressurizedVolume:T hev olumeofs ea-levelpressurea irthatisina B igelowmodule.
● PropulsionBus(PB):Theu nitusedtoreboostthes pacestationd uetoo rbitaldecay.
● SpaceStation:A placewhereastronautscangathertoe xplorethew ondersofs pace.
● UnitedL aunchAllianceDeltaI VH eavy:aLVthatc anlift28,790kgtoL EO.
ENDURINGUNDERSTANDINGS
● Thep arametersofaspaces tationcanb esolvedusingM atrices.
● Differentmodulesc anbeattachedinLEOtocreatealargerspaces tation.
● Spaces tationsandrealisticanimationingamingusethee xacts amem athematics.
● Thea mounto fpayloadinLEOisd ependento nthec apabilitieso ftheELV.
ESSENTIALQ UESTIONS
● Whoa rethem anyspaces tationpioneers?
● Whatisaspaces tation?
● WherecanaB igelows paces tationbespottedinthen ightsky?
● Whenwillthefirstc ommercialB igelowspaces tationbeflown?
● Whyd opeoplew anttoliveo naspacestationino rbit?
● Howm anypeoplecanliveonaspacestation?
Matrices SPACESTATIOND ESIGN Page1 o f 1 2
IntroductiontoCollegeAlgebra LESSONPLAN Project4
MATHEMATICS
● Algebra
● Matrices
SCIENCE
● Physics
● Aerospace
● Astronautics
NOTESTOT HET EACHER
The real–world e xamples used in this lesson plan include the Bigelow A erospace inflatable s pace
station m odule a nd o ther s pace station c omponents s uch as the D N and PB. T he SpaceX F alcon 9
andF alconHeavyELVsw illb eu sedtoplacethes paces tationintoL EO.
COMMONC ORESTANDARDS
● CCSS.MATH.CONTENT.HSN.VM.C.6
○ Usematricestorepresenta ndmanipulatedata
● CCSS.MATH.CONTENT.HSN.VM.C.8
○ Add,s ubtract,a ndmultiplym atriceso fa ppropriatedimensions.
● CCSS.MATH.CONTENT.HSA.SSE.A.1.A
○ Interpretpartsofane xpression,s uchasterms,factors,andcoefficients.
● CCSS.MATH.CONTENT.HSF.BF.A.1
○ Writeafunctionthatd escribesarelationshipb etweentwoquantities.
LESSOND URATION
Approximately3weeks,orabout15c lass–hours.
ACTIVITIES
● Solvem atricesu singpencilandp aper.
● Create a nd use a spreadsheet app to d esign a s pace station including the ELVs n ecessary
top lacethes pacestationintoLEOusingm atrices.
MATERIALSNEEDED
● StudentFormulaSheet
○ Formulas,C onversions,andC onstants
● FormativeA ssessment
○ FormativeA ssessmentAnswerKey
● SummativeA ssessment
○ SummativeA ssessmentAnswerKey
Matrices SPACESTATIOND ESIGN Page2of 12
IntroductiontoCollegeAlgebra LESSONP LAN Project4
ANALYSIS
Students w ill b e required to d esign and construct (on p aper, s o to s peak) a real-world space
station based o n the Bigelow A erospace design. T he B igelow space s tation m odules w ill b e
launched a board either the S paceX Falcon 9 or the F alcon H eavy, d epending on which m odule
willfly.
The Bigelow s pace station m odules a re u nique in that w hen p laced in L EO, it inflates to increase
thep ressurizedv olumeoftheinterior.
Based on the type a nd number o f modules that w ill make up the space station design, s tudents
will construct their o wn u nique s pace s tation, calculating the volume, w eight, c rew s ize, crew
volume,a ndthetotalc osto ftheirc reation.
PROCEDURE
MATRICES
FromWikipedia:
A m atrix (plural matrices) is a rectangular a rray o r table o f n umbers,
symbols, or e xpressions, a rranged in rows a nd columns, w hich is used
torepresentamathematicalo bjectorapropertyofs uchanobject.
Matrices u sually use letters to d esignate the matrix itself. The dimensions o f a m atrix u ses the
row–column format. A m atrix w ith 2 rows a nd 3 columns is labeled as a “2 x 3 ” m atrix
(pronounced “ 2 b y 3”). D ifferent e lements o f the matrix a re identified u sing s ubscripts. F or
example,ifa m atrixislabeledasA,the2 ndrowa nd1 stc olumnislabeledA 2,1.
Fore xample,let
A1,1 = 7 A1,2 = 2
A2,1 =4 A2,2 = 8 A1,3 =9
A2,3 = 1
Matrices
SPACES TATIONDESIGN Page3o f 1 2
IntroductiontoC ollegeA lgebra LESSONPLAN Project4
MATRIXO PERATIONS
Matricescanbea ddeda ndm ultiplied.Butc ertainrulesm usta pply,o fc ourse.
Thislessonp lanw illconcentrateonmultiplyingtwom atricestogether.
When m ultiplying two matrices, the n umber o f columns in the first matrix h as to b e equal to the
number of rows in the s econd matrix. T he result of the m ultiplication will y ield a matrix that is
thesamed imensionsa sthesecondm atrix.
Ingeneral,
IfA= [n,m]a ndB =[m,o ],thenA x B =[n,m]x [m,o ]=[m,o ]
For this p roject, we w ill m ultiply a scalar matrix b y another matrix. S uppose we wish to m ultiply
scalarm atrix A =1x3b ymatrixB =3x4.Inthelanguageofm atrices,thattranslatesto
Ax B= [1,3]x [3,4]= [3,4]
To multiply the matrices, multiply the row the the first matrix by e ach column o f the second
matrix.T hus,
Forexample,wem ultiplythetwomatricesb elowb yu singthetechniquedescribeda bove.
Matrices SPACESTATIONDESIGN Page4 o f 1 2
IntroductiontoC ollegeAlgebra LESSONP LAN Project4
GUIDEDP RACTICE
Multiplythes calar1x3matrixbythe3 x4m atrix:
AnswerKey
PROCEDURE
SPACES TATIONDESIGN
Students w ill first learn the b asics of s pace s tation design using pencil, p aper, and s cientific
calculators. W e w ill b e u sing the p roducts from B igelow A erospace, which makes inflatable
habitat modules that o nce placed into o rbit, w ell, inflate. This a llows for a g reater volume o f
space inside for the crew. It h as s olar p anels for electrical p ower a nd radiators to d ispose of w aste
heat. The m odule h as a m ass of 2 5,000 k g, h as a p rice tag of $125,000,000 USD, and can house
ac rewofsix.
Matrices SPACESTATIOND ESIGN Page5of 12
IntroductiontoC ollegeAlgebra LESSONPLAN Project4
Attached to the inflatable modules is a Propulsion Bus /Docking Node combination. The P B/DN
is u sed to maneuver the s pace station to avoid o rbital s pace d ebris o r to boost the station to a
higherorbitala ltitude.TheP B/DNhasam asso f17,000kga ndisp ricedat$75,000,000U SD.
The SpaceX D ragon 2 c rew capsule w ill ferry 6 astronauts to the s pace station a nd b ack. O nce
docked,theDragon2 isputintos leepmode,andtheastronautss pendsixmonthsinspace.
Matrices SPACESTATIOND ESIGN Page6 o f 12
IntroductiontoCollegeA lgebra LESSONPLAN Project4
In the image a bove, three B igelow B330 m odules a re docked with one PB/DN, with three
SpaceXD ragon2sd eliveringacrewo fastronautstothes pacestation.
Below are d ifferent variations p ossible with the B330–PB/DN combinations. The s pace s tation
can begin life a s a s ingle o r double B 330 and expand to a triple. In a ll the c onfigurations, o nly
onePB/DNmoduleisn eededtoa ttachtheB330s.
It also suggests that the space s tation c an b e further expanded a s needed. The image below
represents a s pace station that starts out a s a s ingle, then becomes a triple, and then triples again.
Atotalo fnineB 330sandthreeP B/DNsm akeu pthisremarkables pacestationc onfiguration.
To launch these e xcellent habitat modules into s pace, w e obviously n eed a launch vehicle.
Shopping a round for what’s a vailable to u se to launch our c ity in s pace, we find two Launch
Vehicles (LVs) that are capable of lifting the required mass to LEO, n amely the U nited Launch
Alliance(ULA)D eltaIVHeavyandtheSpaceXF alcon9 .
ULA c harges $ 350,000,000 USD to lift 28,790 k g into Low Earth Orbit (LEO). One B330
weighs 25,000 kg, so the Delta IV Heavy LV c an c arry o ne unit at a time. W e will call this the
“B330|D eltaIVH LV”Stack.
Matrices SPACESTATIOND ESIGN Page7of 1 2
IntroductiontoC ollegeA lgebra LESSONPLAN Project4
SpaceX c harges $ 50,000,000 USD to u se the Falcon 9 LV to lift 2 2,800 k g to LEO. E ach PB/DN
has a mass of 1 7,000 k g so the LV can c arry o ne unit at a time, called the “PB/DN | F alcon 9”
Stack.Eitherone,two,o rthreeB330sc anb ed ockedtoaPB/DNatthesametime
SpaceX a lso c harges $50,000,000 U SD to u se the Falcon 9 LV to c arry the Dragon 2 Crew
Capsule to LEO a nd the B igelow space station for h alf a y ear. SpaceX a lso c harges $50,000,000
USDforeachs eat,fora totalo f$ 300,000,000USDforthes ixseats.
To k eep a c rew aboard the s pace station y ear–round would b e a grand total of $600,000,000 U SD
per annum for the astronauts a nd $100,000,000 USD p er y ear for two L Vs. We w ill c all this the
“Dragon 2 | Falcon 9” S tack. Kindly note that the n umber o f “ Dragon 2 | Falcon 9” Stacks w ill
equaltothenumberof“ B330|D eltaIVHLV”Stacks.
Tos ummarize:
● B330|DeltaIVHLVS tack
○ B330
■ Cost:$ 125,000,000USD
■ Mass:2 5,000k g
■ PressurizedVolume:3 30m 3
■ Capacity:6 A stronauts
○ DeltaIVHLV
■ Cost:$350,000,000USD
● PB/DN|Falcon9 S tack
○ PB/DN
■ Cost:$75,000,000U SD
■ Mass:17,000kg
○ Falcon9LV
■ Cost:$ 50,000,000USD
● Dragon2 |F alcon9Stack
○ Dragon2
■ CostperS eat:$50,000,000U SD
■ Numbero fSeats:1 2peryear
■ TotalC ostofP assengers
● $600,000,000U SD
○ Falcon9
■ Cost:$100,000,000USDp ery ear
All of this information c an b e further s ummarized in a table, which is really n othing more than a
thinlydisguisedmatrix.
Matrices SPACESTATIONDESIGN Page8 o f 12
IntroductiontoCollegeA lgebra LESSONP LAN Project4
The Bigelow s pace s tation depicted earlier in this manuscript c onsists o f three B 330s and one
PB/DN. The n umber o f “ Dragon 2 | F alcon 9” Stacks needed a re equal to the n umber of B330s in
LEO. We will therefore n eed three o f these c rew c apsule s tacks to k eep the space station
continuouslyoccupied.Thisinformationcanalsobetranslatedintoa 1 x3m atrix.
Biglow3S paceStation= [Numbero fB 330s Numbero fPB/DNs CrewRotation]
KindlyNote:theC rewR otationwilla lwaysbeequaltotheNumberofB 330s.
Summingthec olumnsyieldsthetotalc ostforthefirsty earo fassemblya ndo perations.
The last row of the resultant matrix represents the y early cost of continuously o ccupying the
station.T hetotalyearlyc ostisthes umoftheD ragon2 |F alcon9S tack.
Biglow3S paceS tationTotal
ComponentCost LVCost Mass Volume CrewS ize
$2,250M $1,400M 92,000kg 990m3 18
FirstY earCost:$ 3,650,000,000USD
Afterthat:$ 2,100,000,000USD/year
Matrices SPACESTATIOND ESIGN Page9 of 1 2
IntroductiontoCollegeA lgebra LESSONPLAN Project4
GUIDEDPRACTICE
Determine the p arameters of a B igelow 7 S pace Station, w hich is c omposed o f s even B330s and
threePB/DNs(seeimageb elow).
Biglow7 SpaceStationT otal Mass Volume
ComponentCost LVCost CrewSize
Volume
Mass 2,310m 3
CrewSize
FirstY earCost:$__________U SD 42
Afterthat:$___________USD/year
ANSWERKEY Page1 0of 1 2
Biglow7S paceStationT otal
ComponentC ost LVC ost
$5,300M $3,300M 226,000k g
FirstY earCost:$ 8,600,000,000U SD
Afterthat:$ 4,900,000,000U SD/year
CODING
Constants
B330=$125,000,000
DeltaIV=$350,000,000
PB/DN=$75,000,000
Dragon2=$600,000,000
Falcon9=$50,000,000
B330Mass=25,000
Matrices SPACES TATIOND ESIGN
IntroductiontoCollegeA lgebra LESSONP LAN Project4
PB/DNMass=17,000
B330Volume=330
B330CrewSize=6
Input
NumberOfB330s
NumberOfPB/DNs
Output
B330Cost=NumberOfB330s*B330
DeltaIVCost=NumberOfB330s*DeltaIV
B330TotalMass=NumberOfB330s*B330Mass
PB/DNCost=NumberOfPB/DNs*PB/DN
Falcon9Cost=NumberOfPB/DNs*Falcon9
PB/DNTotalMass=NumberOfPB/DNs*PB/DNMass
Dragon2Cost=NumberOfB330s*Dragon2
Dragon2/Falcon9Cost=NumberOfB330s*Falcon9
CrewRotation=NumberOfB330s*B330CrewSize
TotalComponentCost=B330Cost+PB/DNCos+Dragon2Cos
TotalLaunchVehicleCost=DeltaIVCost+Falcon9Cos+Dragon2/Falcon9Cost
TotalMass=B330TotalMass+PB/DNTotalMass
TotalVolume=NumberOfB330s*B330Volume
TotalCrew=NumberOfB330s*B330CrewSiz
TotalFirstYearCost=TotalComponentCos+TotalLaunchVehicleCost
YearlyCost=Dragon2Cost+Dragon2/Falcon9Cost
ARTWORK
Create a p iece o f a rt d epicting the Bigelow space s tation in L ow E arth Orbit. The p iece s hould
include some o f the m athematics involved in this p roject. The a rtwork c an b e m ade in any
medium that the student desires, including drawing, v ideo, a nimation, d ance, a dvertisement,
infographic,etc.Alla rtworkwillb ecollectedandp resentedtotherestoftheclass.
WRITING
Describe in d etail everything that you have learned a bout the B igelow B330 s pace station
module a s it is related to the s tudy of m atrices. You m ust use all eighteen v ocabulary terms in
Matrices SPACESTATIONDESIGN
Page11of 12
IntroductiontoC ollegeAlgebra LESSONPLAN Project4
your writing. The a ssignment c an b e c ompleted in any format the s tudent d esires, including
technical p aper, short story, s creenplay, poetry, e tc. A ll w ritings will b e c ollected and presented
totherestofthec lass.
DISCUSSIONTOPICS
● WhoisRobertBigelow?
● Whatdoy outhinkitw ouldfeelliketoliveina s paces tationfors ixmonths?
● WhereonE arthis3 6.212o Latitudeand– 115.169o L ongitude?
● Whend idtheBigelowinflatablespacestationcomponentfirstflyintospace?
ADDITIONALRESOURCES
● BigelowAerospace
○ https://www.bigelowaerospace.com/
● UnitedLaunchAlliance
○ https://www.ulalaunch.com/
● SpaceX
○ https://www.spacex.com
● Youtube
○ https://www.youtube.com/results?search_query=bigelow+aerospace
● GraphicDesign
○ https://www.canva.com
○ https://www.lunapic.com
HANDOUTS
● Worksheet
○ Formulas,C onstants,a ndConversions
● FormativeA ssessment
○ Matrices
○ SpaceS tationD esign
● SummativeA ssessment
ENDO FLESSONP LAN
Matrices SPACES TATIOND ESIGN
Page1 2of 1 2
Project 4
Space Station Design
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