Grade6:
Represen-ngRa-osinVariousFormats
NCTMInterac-veIns-tute,2016
Name
Title/Posi-on
Affilia-on
EmailAddress
Introduc-ons….
•  Introduceyourselftoothersatyourtable.
•  Discusssuccessandchallengesyouencounter
whenteachingthetopicofra-ostostudents
inyourclassroom.
2
CommonCoreProgressions
GRADE6
GRADE7
GRADE8
Understandra-oconcepts
andusera-oreasoningto
solveproblems.
Analyzepropor-onal
rela-onshipsanduse
themtosolvereal-world
andmathema-cal
problems.
Understandthe
connec-onsbetween
propor-onalrela-onships,
lines,andlinearequa-ons.
• 
• 
• 
• 
Conceptofra;o
•  Computeunitrates
Usera;olanguage
•  Representpropor;onal
Conceptofunitrate
rela;onshipsbetween
Usera;oandrate
quan;;es
reasoningtosolvereal- •  Usepropor;onal
worldandmathema;cal
rela;onshipstosolve
problems(tables,
mul;stepra;oand
diagrams,double
percentproblems
numberlines,
equa;ons)
•  Graphpropor;onal
rela;onships,
interpre;ngtheunit
rateastheslopeofthe
graph
•  Usesimilartrianglesto
explainwhytheslopeis
thesamebetweenany
twodis;nctpointsona
non-ver;calline
BigIdeasforRa-os
Ara-oisanorderedpairofnumbersor
measurementsthatexpressesacomparison
betweenthenumbersormeasures.
–  Reasoningwithra;osinvolvesaFendingtoand
coordina;ngtwoquan;;es.
Formingara;oinvolvesisola;ngoneaFributefromother
aFributes.
–  Ara;oisamul;plica;vecomparisonoftwo
quan;;esoritisajoiningoftwoquan;;esina
composedunit.
Ra-os
•  Ra-osareexpressedinfrac-onnota-on.
Ra;osandfrac;onsdonothaveiden;calmeaning.
Ra;osandfrac;onscanbeconceivedasoverlappingsets.
•  Frac-onsarera-os,butnotallra-osarefrac-ons.
Somera;osmake“part-part”comparisonsorrelatemorethan
twoparts.
•  Ratesarera-os,butnotallra-osarerates.
Ra-osarebuildingblocksfor
propor-onsandpropor-onalreasoning.
TypesofRa-os
•  Part-to-WholeRa-os
Comparetwomeasuresofsametype
Nonfic;onbookstoallbooksinlibrary,percentages,probabili;es
•  Part-to-PartRa-os
Comparetwomeasuresofsametype
Fic;onbookstononfic;onbooksinlibrary,oddsofanevent
•  RatesasRa-os
Comparisonofmeasuresoftwodifferentthings/quan;;es
Prices,;meanddistance,milespergallon,inchesperfoot
•  SpecialRa-os
Goldenra;o,π,slopeofline,geometricsimilarity,trigonometric
func;onsfromrighttriangles
Ra-os:
StudentThinking
Twoweeksago,twofloweringplantswere
measuredat8inchesand12inches.Todaythey
are11inchesand15inchestall,respec;vely.
Whichfloweringplantgrewmore–
the8-inchor12-inchflower?
Defendtwodifferent“answers”tothisproblem.
Addi-veVersusMul-plica-veReasoning
Mul-pleRepresenta-ons
Duringthissession,wearegoingtousevarious
representa;onstohelpstudentsdevelop
conceptualunderstandingofra;os.
•  Unitra;o
•  Ra;otable
•  Doublenumberline
•  Tapediagram
8
ComparingRa-os
Unitrate
Dis;nguishequivalency
between2ormorera;os
ComparingRa-os
Two camps of Scouts are having pizza parties.
The Bear Camp ordered enough so that every 3
campers will have 2 pizzas.
The leader of the Raccoons ordered enough so
that there would be 3 pizzas for every 5 campers.
Which campers had more pizza to eat:
the Bear campers or the Raccoon campers?
UnitRate
“PizzasperCamper”Approach
Bear Campers
Each of the 3 campers will
get ​𝟏/𝟐 pizza and ​𝟏/𝟔 pizza.
​𝟐/𝟑 pizza per camper
Raccoon Campers
Each of the 5 campers will
get ​𝟏/𝟐 pizza and ​𝟏/𝟏𝟎 pizza.
​𝟑/𝟓 pizza per camper
UnitRate
“CampersperPizza”Approach
Bear Campers
1 ​𝟏/𝟐 campers per pizza
Raccoon Campers
1 ​𝟐/𝟑 campers per pizza
UnitRate
Compareequivalentnumberofpizzas
tonumberofcampers
Bears
6 pizzas for 9 campers
Raccoons
6 pizzas for 10 campers
UnitRate
Compareequivalentnumberofcampers
tonumberofpizzas
Bears
15 campers for 10 pizzas
Raccoons
15 campers for 9 pizzas
ComparingRa-os
Usingmul-plica-vecomparisonsisa
powerfulpropor-onalreasoning
strategywhichisanimportantelement
inalgebra. Mul-plica-veComparisons
Raccoon Camp
Bear Camp
# of Campers
# of Pizzas
3
2
1
# of Campers
# of Pizzas
5
3
1
1
1
ComparingRa-os
Ra-otable
Rela;onshipoftwovariablequan;;es
Ra-oTable
Apersonwhoweighs160poundsonEarthwill
weigh416poundsontheplanetJupiter.
HowmuchwillapersonweighonJupiterwho
weighs120poundsonearth?
Ra-oTable
X3
÷2
÷2
Earth
weight
160
80
Jupiter
weight
416
208
40
120
104
312
Add
Earth
weight
Jupiter
weight
160
80
416
208
40
120
104
312
Ra-oTable
•  Atyourtable,useara;otabletosolveyour
assignedproblem.
–  Findavarietyofwaystousethera;otablewith
theproblem.
–  Whatsuccess/challengesmightstudents
encounterusingthera;otable?
•  Shareyourproblemwithlargegroup.
20
Ra-oTable
Cheeseis$4.25perpound.
A
Howmuchwill12.13
B
poundscost?
C
D
E
F
G
H
Lbs
Cost
1
4.25
10
42.50
2
8.50
0.1
0.425
12.1 51.125
0.01 0.0425
0.03 0.1275
12.13 51.5525
Notes
Given
A x 10
Ax2
A ÷ 10
B + C +D
D ÷ 10
Fx3
E+G
CompareRa-os
Useatapediagramtosolvetheproblem:
Theschoolparkinglotholds161vehicles.
WhenCarlalookedatthefilledparkinglotat
school,sheno;cedtherewere2minivansfor5
othertypesofvehicles.
Howmanyofthevehiclesarenotminivans?
22
TapeDiagram
161vehiclesinra;oof2:5
161 vehicles
161 ÷7 = 23
Not minivans
Vehicles that are not minivans
23 X 5 = 115
TapeDiagram
161vehiclesinra;oof5:7
161 total vehicles
161 ÷7 = 23
Vehicles that are not minivans
23 X 5 = 115
SolvePercentProblemsUsing
DoubleNumberLine
0
30
Explore percent problems
(1) 45% of 70 = _______
(2) 30% of ______ = 75
(3)
_____% of 75 = 30
60
SolvePercentProblemsUsing
DoubleNumberLine
0
7
14
21 28 35 42 49 56 63 70
31.5
(1) 45% of 70 = _______
SolvePercentProblemsUsing
DoubleNumberLine
30%
0
25 50 75
(2)
125
250
30% of ______
= 75
?
250
SolvePercentProblemsUsing
DoubleNumberLine
40%
0
7.5 15 22.5 30
40
(3) _____%
of 75 = 30
75
DoubleNumberLineforPercents
20%
0
6
12
18
24
30
36
42
48
54
?
60
20% of Ms. Thompson’s show dogs are
Labradors. She has 12 Labradors.
How many show dogs does she have?
DoubleNumberLineforPercents
30%
0
9
18
27
80%
36
45
54
63
?
72
81
Jan has completed 27 items which is 30% of the
test. She needs to complete 80% to move on.
•  How many items does she need to complete
to move on?
•  How many items are there on the test?
?
90
Mul-pleRepresenta-ons
Acarmagazineiswri;ngastoryaboutfour
differentcars,repor;ngthenumberofmiles
drivenfordifferentamountsofgas.
•  WithyourExpertgroup,describethegas
mileageforyourassignedcarusingmul;ple
representa;ons(words,table,equa;on,and
graph).
31
Mul-pleRepresenta-ons
•  EachHomegroupwillhaveonememberfrom
theExpertgrouptodiscusstheirassignedcar.
•  Usethevariousrepresenta;onstodecide:
–  Theorderingofthecarsfromgreatestnumberof
milespergallontoleastnumberofmilesper
gallon.
–  ThecarKrystallikelyboughtifshedrove924miles
andused28gallonsofgas.
32
Summary
Thereareseveralwaysthesamecollec;onof
equivalentra;oscanberepresented.These
include:
• unitra;o,
• ra;otables,
• tapediagrams,and
• doublenumberlines.
33
Reflec-on
(Principles to Actions: Ensuring Mathematical Success for All [NCTM 2014], p. 29)
ExitTicket
Describehowthevarious
representa;onsmightcontribute
tothelearningofra;osby
students.
•  Unitra;o
•  Ra;otable
•  Doublenumberlinediagram
•  Tapediagram
35
Disclaimer
The National Council of Teachers of Mathematics is a public voice
of mathematics education, providing vision, leadership, and
professional development to support teachers in ensuring
equitable mathematics learning of the highest quality for all
students. NCTM’s Institutes, an official professional development
offering of the National Council of Teachers of Mathematics,
supports the improvement of pre-K-6 mathematics education by
serving as a resource for teachers so as to provide more and
better mathematics for all students. It is a forum for the
exchange of mathematics ideas, activities, and pedagogical
strategies, and for sharing and interpreting research. The
Institutes presented by the Council present a variety of
viewpoints. The views expressed or implied in the Institutes,
unless otherwise noted, should not be interpreted as official
positions of the Council.
36
37