session 5: teaching through mathematical processes · mathematical processes for gains mathematics...
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Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 1
Session 5: Teaching through Mathematical Processes
120 min
Math Learning GoalsDevelop strategies to assess understanding of• the Mathematical Processes.Practise assessing through observation.•Connect the Mathematical Processes to the Achievement Chart.•
Rationale
MaterialsBLM 5.1–5.8•Math Process S5 •pptsticky notes in 3 •coloursmanipulatives•calculators•graphing •technology
Individual/Whole Group Retention Activity Create three visual timelines ranging from zero to 10 years – one for each learning method: procedural, conceptual, and using the Mathematical Processes. Participants place a sticky note on the timeline based on how long they believe a student retains knowledge acquired through each method.Individual/Pair Anticipation ActivityShowthevideoclip.Reflectonitsrelevancetoretentionandlife-longlearning.Discuss. Individually participants identify the Mathematical Process they feel best assesses the criteria given. They share their choice with a partner justifying their rationale. Display the answer for each process and allow for questions before proceeding to the remaining processes.Distribute BLM 5.1 which contains the complete rubric.
Use a different coloured sticky note for each of procedural knowledge, conceptual understanding, and the Mathematical Processes.
Five Minute Universitywww.cs.cmu.edu/~pattis/videos/5minuteU.wvx
Minds On…
Whole Group Solve Problems Designate three sections of the room as Agree, Disagree, and Don’t Know.Pose the problem: The height of a can of tennis balls is greater than its circumference. Do you agree? disagree? or don’t know?After some ‘think time,’ participants to go to the section of the room that matches their hypothesis and use a variety of manipulatives and strategies to investigate the problem to prove/disprove their belief. Once they reach a conclusion they have an option to change sides. Participants discuss the Mathematical Processes used in the solutions. Ask: Which processes and criteria from the Generic Rubric for Mathematical Processes could be used to assess the solution for this problem?
Whole Group Make ConnectionsMake connections between problem solving and the Mathematical Processes.Participants choose a problem they would like to solve and form working groups based on their selected problem.Introduce problems (BLM 5.3–5.8) and ask which processes are best suited for assessment with each problem. Answers: BLM 5.2a –5.2f; others may be appropriate.
Small Groups Problem Solving/AssessmentIntroduce and explain the Fishbowl Strategy.
Process Expectations/Observation/Mental Note: Observe participants’ ability to use the rubrics to assess the process.
Differentiate content based on participant choice to make activity personally relevant.
Note: Hidden slides have possible solutions to the problems. Deck problem has a solution that can be done using GSP.
Fishbowl is a collaborative learning strategy in which one group forms an inner circle and completes a task, e.g., solve a problem, discuss. The second group forms an outer circle, quietly observes their peers, and prepares to discuss and question what they observed.
Action!
Whole Group Discussion Debrief the Fishbowl Strategy task as a group.Make connections between assessment and the Mathematical Processes.
Consolidate Debrief
ReflectionHome Activity or Further Classroom ConsolidationInyourJournal,reflectontheroletheGenericRubricfortheMathematicalProcesses will have in your teaching/assessment practices.
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
2
5.1:
Gen
eric
Rub
ric fo
r Mat
hem
atic
al P
roce
sses
Thin
king
Prob
lem
Sol
ving
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s ap
prop
riate
to
the
task
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk w
ith s
igni
fican
t pr
ompt
ing
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk w
ith m
inim
al
prom
ptin
g
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk in
depe
nden
tly
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk in
depe
nden
tly w
ith a
br
oade
r vie
w o
f the
task
U
ses
criti
cal t
hink
ing
skill
s to
sol
ve a
pro
blem
Use
s m
inim
al lo
gic
and
prec
isio
n in
mat
hem
atic
al
reas
onin
g to
sol
ve
prob
lem
s
Use
s lo
gic
to s
olve
pr
oble
ms
but l
acks
pr
ecis
ion
in m
athe
mat
ical
re
ason
ing
Sol
ves
prob
lem
s lo
gica
lly
and
with
pre
cisi
on in
m
athe
mat
ical
reas
onin
g
Dem
onst
rate
s a
soph
istic
ated
leve
l of
mat
hem
atic
al re
ason
ing
and
prec
isio
n in
sol
ving
pr
oble
ms
Rea
soni
ng a
nd P
rovi
ng
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Form
ulat
es a
nd d
efen
ds a
hy
poth
esis
or c
onje
ctur
eFo
rms
a hy
poth
esis
or
conj
ectu
re th
at c
onne
cts
few
asp
ects
of t
he p
robl
em
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s so
me
of th
e pe
rtine
nt
aspe
cts
of th
e pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s pe
rtine
nt a
spec
ts o
f the
pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s as
pect
s of
the
prob
lem
w
ith a
bro
ader
vie
w o
f the
pr
oble
mM
akes
infe
renc
es, d
raw
s co
nclu
sion
s an
d gi
ves
just
ifica
tions
Mak
es li
mite
d co
nnec
tions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es s
ome
conn
ectio
ns
to th
e pr
oble
m-s
olvi
ng
proc
ess
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
s
Mak
es d
irect
con
nect
ions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es d
irect
and
insi
ghtfu
l co
nnec
tions
to th
e pr
oble
m-s
olvi
ng p
roce
ss
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
sIn
terp
rets
mat
hem
atic
al
lang
uage
, cha
rts, a
nd
grap
hs
Mis
inte
rpre
ts a
crit
ical
el
emen
t of t
he in
form
atio
n,
but m
akes
som
e re
ason
able
sta
tem
ents
Mis
inte
rpre
ts p
art o
f th
e in
form
atio
n, b
ut
mak
es s
ome
reas
onab
le
stat
emen
ts
Inte
rpre
ts th
e in
form
atio
n co
rrec
tly a
nd m
akes
re
ason
able
sta
tem
ents
Inte
rpre
ts th
e in
form
atio
n co
rrec
tly, a
nd m
akes
in
sigh
tful s
tate
men
ts
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
3
Refl
ectin
gC
riter
iaB
elow
Lev
el 1
Sp
ecifi
c Fe
edba
ckLe
vel 1
Leve
l 2Le
vel 3
Leve
l 4
Use
s m
etac
ogni
tive
skill
s to
det
erm
ine
whi
ch
mat
hem
atic
al p
roce
sses
to
revi
sit i
n or
der t
o re
ach
the
goal
App
lies
met
acog
nitiv
e sk
ills
with
sig
nific
ant
prom
ptin
g in
det
erm
inin
g w
hich
mat
hem
atic
al
proc
ess
to re
visi
t in
orde
r to
reac
h th
e go
al
App
lies
met
acog
nitiv
e sk
ills
with
min
imal
pr
ompt
ing
in d
eter
min
ing
whi
ch m
athe
mat
ical
pr
oces
s to
revi
sit i
n or
der
to re
ach
the
goal
App
lies
met
acog
nitiv
e sk
ills
inde
pend
ently
in
det
erm
inin
g w
hich
m
athe
mat
ical
pro
cess
to
revi
sit i
n or
der t
o re
ach
the
goal
App
lies
met
acog
nitiv
e sk
ills
inde
pend
ently
in
det
erm
inin
g w
hich
m
athe
mat
ical
pro
cess
to
revi
sit i
n or
der t
o re
ach
the
goal
with
a b
road
er v
iew
of
the
goal
Refl
ects
on
the
reas
onab
lene
ss o
f an
swer
s
Mak
es m
inim
al
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es s
ome
conn
ectio
ns
betw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
and
prov
ides
in
sigh
tful c
omm
ents
App
licat
ion
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Sele
ctin
g To
ols
and
Com
puta
tiona
l Str
ateg
ies
Sel
ects
and
use
s to
ols
and
stra
tegi
es to
sol
ve a
pr
oble
m
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
ajor
er
rors
, om
issi
ons,
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
inor
er
rors
, om
issi
ons
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es a
ccur
atel
y, a
nd
in a
logi
cal s
eque
nce
Sel
ects
and
app
lies
appr
opria
te a
nd e
ffici
ent
tool
s an
d st
rate
gies
, ac
cura
tely
to c
reat
e m
athe
mat
ical
ly e
lega
nt
solu
tions
Con
nect
ing
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Mak
es c
onne
ctio
ns a
mon
g m
athe
mat
ical
con
cept
s an
d pr
oced
ures
Mak
es w
eak
conn
ectio
ns
amon
g m
athe
mat
ical
co
ncep
ts a
nd p
roce
dure
s
Mak
es s
impl
e co
nnec
tions
am
ong
mat
hem
atic
al
conc
epts
and
pro
cedu
res
Mak
es a
ppro
pria
te
conn
ectio
ns a
mon
g m
athe
mat
ical
con
cept
s an
d pr
oced
ures
Mak
es s
trong
con
nect
ions
am
ong
mat
hem
atic
al
conc
epts
and
pro
cedu
res
Rel
ates
mat
hem
atic
al
idea
s to
situ
atio
ns d
raw
n fro
m o
ther
con
text
s
Tran
sfer
s id
eas
to o
ther
co
ntex
ts a
nd m
akes
lim
ited
conn
ectio
ns
Tran
sfer
s id
eas
to o
ther
co
ntex
ts a
nd m
akes
si
mpl
e co
nnec
tions
Tran
sfer
s id
eas
to o
ther
co
ntex
ts a
nd m
akes
ap
prop
riate
con
nect
ions
Tran
sfer
s id
eas
to o
ther
co
ntex
ts a
nd m
akes
un
ique
, orig
inal
or
insi
ghtfu
l con
nect
ions
5.1:
Gen
eric
Rub
ric fo
r Mat
hem
atic
al P
roce
sses
(con
tinue
d)
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
4
Com
mun
icat
ion
Rep
rese
ntin
gC
riter
iaB
elow
Lev
el 1
Sp
ecifi
c Fe
edba
ckLe
vel 1
Leve
l 2Le
vel 3
Leve
l 4
Cre
ates
a m
odel
to
repr
esen
t the
pro
blem
(e
.g.,
num
eric
al, a
lgeb
raic
, gr
aphi
cal,
phys
ical
, or
scal
e m
odel
, by
hand
or
usin
g te
chno
logy
)
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith li
mite
d ef
fect
iven
ess;
re
pres
entin
g lit
tle o
f the
ra
nge
of th
e da
ta
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith s
ome
effe
ctiv
enes
s;
repr
esen
ting
som
e of
the
rang
e of
the
data
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith c
onsi
dera
ble
effe
ctiv
enes
s; re
pres
entin
g m
ost o
f the
rang
e of
the
data
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith a
hig
h de
gree
of
effe
ctiv
enes
s; re
pres
entin
g th
e fu
ll ra
nge
of th
e da
ta
Mak
es c
onne
ctio
ns
betw
een
num
eric
, gr
aphi
cal a
nd a
lgeb
raic
re
pres
enta
tions
Mak
es li
mite
d co
nnec
tions
be
twee
n nu
mer
ic,
grap
hica
l and
alg
ebra
ic
repr
esen
tatio
ns
Mak
es s
ome
conn
ectio
ns
betw
een
num
eric
, gr
aphi
cal a
nd a
lgeb
raic
re
pres
enta
tions
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
num
eric
, gra
phic
al a
nd
alge
brai
c re
pres
enta
tions
Mak
es s
trong
and
in
sigh
tful c
onne
ctio
ns
betw
een
num
eric
, gr
aphi
cal a
nd a
lgeb
raic
re
pres
enta
tions
Tran
slat
es fr
om o
ne
repr
esen
tatio
n to
ano
ther
as
app
ropr
iate
to th
e pr
oble
m
Tran
slat
es re
pres
enta
tion
with
maj
or e
rror
s w
hen
solv
ing
a pr
oble
m
Tran
slat
es re
pres
enta
tions
w
ith s
ome
erro
rs w
hen
solv
ing
a pr
oble
m
Tran
slat
es re
pres
enta
tions
ap
prop
riate
ly w
hen
solv
ing
a pr
oble
m
Tran
slat
es re
pres
enta
tions
ap
prop
riate
ly a
nd w
ith
insi
ght w
hen
solv
ing
a pr
oble
mC
omm
unic
atin
gC
riter
iaB
elow
Lev
el 1
Sp
ecifi
c Fe
edba
ckLe
vel 1
Leve
l 2Le
vel 3
Leve
l 4
Use
s cl
ear l
angu
age
to m
ake
pres
enta
tions
, an
d to
exp
lain
and
just
ify
solu
tions
whe
n re
porti
ng
for v
ario
us p
urpo
ses
and
diffe
rent
aud
ienc
es
Use
s un
clea
r lan
guag
e to
mak
e pr
esen
tatio
ns,
and
to e
xpla
in a
nd ju
stify
so
lutio
ns w
hen
repo
rting
fo
r var
ious
pur
pose
s an
d di
ffere
nt a
udie
nces
Use
s la
ngua
ge th
at is
so
mew
hat u
ncle
ar to
m
ake
pres
enta
tions
, and
to
exp
lain
and
just
ify
solu
tions
whe
n re
porti
ng
for v
ario
us p
urpo
ses
and
diffe
rent
aud
ienc
es
Use
s cl
ear l
angu
age
to m
ake
pres
enta
tions
, an
d to
exp
lain
and
just
ify
solu
tions
whe
n re
porti
ng
for v
ario
us p
urpo
ses
and
diffe
rent
aud
ienc
es
Use
s cl
ear a
nd p
reci
se
lang
uage
to m
ake
pres
enta
tions
, and
to
exp
lain
and
just
ify
solu
tions
whe
n re
porti
ng
for v
ario
us p
urpo
ses
and
diffe
rent
aud
ienc
esU
ses
mat
hem
atic
al
sym
bols
, lab
els,
uni
ts a
nd
conv
entio
ns c
orre
ctly
Som
etim
es u
ses
mat
hem
atic
al s
ymbo
ls,
labe
ls a
nd c
onve
ntio
ns
corr
ectly
Usu
ally
use
s m
athe
mat
ical
sy
mbo
ls, l
abel
s an
d co
nven
tions
cor
rect
ly
Con
sist
ently
use
s m
athe
mat
ical
sym
bols
, la
bels
and
con
vent
ions
co
rrec
tly
Con
sist
ently
use
s m
athe
mat
ical
sym
bols
, la
bels
and
con
vent
ions
, pr
esen
ting
nove
l or
insi
ghtfu
l opp
ortu
nitie
s fo
r th
eir u
seU
ses
mat
hem
atic
al
voca
bula
ry a
ppro
pria
tely
Use
s co
mm
on la
ngua
ge
in p
lace
of m
athe
mat
ical
vo
cabu
lary
or u
ses
key
mat
hem
atic
al te
rms
with
m
ajor
err
ors
Use
s m
athe
mat
ical
vo
cabu
lary
with
min
imal
er
rors
or u
ses
som
e co
mm
on la
ngua
ge in
pl
ace
of v
ocab
ular
y
Use
s m
athe
mat
ical
vo
cabu
lary
app
ropr
iate
lyC
onsi
sten
tly u
ses
mat
hem
atic
al v
ocab
ular
y ap
prop
riate
ly, p
rese
ntin
g no
vel o
r ins
ight
ful
oppo
rtuni
ties
for i
ts u
se
5.1:
Gen
eric
Rub
ric fo
r Mat
hem
atic
al P
roce
sses
(con
tinue
d)
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
5
5.2
(a) V
olum
e of
3-D
Sha
pes
Thin
king
Rea
soni
ng a
nd P
rovi
ngC
riter
iaB
elow
Lev
el 1
Sp
ecifi
c Fe
edba
ckLe
vel 1
Leve
l 2Le
vel 3
Leve
l 4
Form
ulat
es a
nd d
efen
ds a
hy
poth
esis
or c
onje
ctur
eFo
rms
a hy
poth
esis
or
conj
ectu
re th
at c
onne
cts
few
asp
ects
of t
he p
robl
em
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s so
me
of th
e pe
rtine
nt
aspe
cts
of th
e pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s pe
rtine
nt a
spec
ts o
f the
pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s as
pect
s of
the
prob
lem
w
ith a
bro
ader
vie
w o
f the
pr
oble
mM
akes
infe
renc
es, d
raw
s co
nclu
sion
s an
d gi
ves
just
ifica
tions
Mak
es li
mite
d co
nnec
tions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es s
ome
conn
ectio
ns
to th
e pr
oble
m-s
olvi
ng
proc
ess
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
s
Mak
es d
irect
con
nect
ions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es d
irect
and
insi
ghtfu
l co
nnec
tions
to th
e pr
oble
m-s
olvi
ng p
roce
ss
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
sR
eflec
ting
Use
s m
etac
ogni
tive
skill
s to
det
erm
ine
whi
ch
mat
hem
atic
al p
roce
sses
to
revi
sit i
n or
der t
o re
ach
the
goal
App
lies
met
acog
nitiv
e sk
ills
with
sig
nific
ant
prom
ptin
g in
det
erm
inin
g w
hich
mat
hem
atic
al
proc
ess
to re
visi
t in
orde
r to
reac
h th
e go
al
App
lies
met
acog
nitiv
e sk
ills
with
min
imal
pr
ompt
ing
in d
eter
min
ing
whi
ch m
athe
mat
ical
pr
oces
s to
revi
sit i
n or
der
to re
ach
the
goal
App
lies
met
acog
nitiv
e sk
ills
inde
pend
ently
in
det
erm
inin
g w
hich
m
athe
mat
ical
pro
cess
to
revi
sit i
n or
der t
o re
ach
the
goal
App
lies
met
acog
nitiv
e sk
ills
inde
pend
ently
in
det
erm
inin
g w
hich
m
athe
mat
ical
pro
cess
to
revi
sit i
n or
der t
o re
ach
the
goal
with
a b
road
er v
iew
of
the
goal
App
licat
ion
Sele
ctin
g To
ols
and
Com
puta
tiona
l Str
ateg
ies
Sel
ects
and
use
s to
ols
and
stra
tegi
es to
sol
ve a
pr
oble
m
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
ajor
er
rors
, om
issi
ons,
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
inor
er
rors
, om
issi
ons
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es a
ccur
atel
y, a
nd
in a
logi
cal s
eque
nce
Sel
ects
and
app
lies
appr
opria
te a
nd e
ffici
ent
tool
s an
d st
rate
gies
, ac
cura
tely
to c
reat
e m
athe
mat
ical
ly e
lega
nt
solu
tions
Con
nect
ing
Mak
es c
onne
ctio
ns a
mon
g m
athe
mat
ical
con
cept
s an
d pr
oced
ures
Mak
es w
eak
conn
ectio
ns
amon
g m
athe
mat
ical
co
ncep
ts a
nd p
roce
dure
s
Mak
es s
impl
e co
nnec
tions
am
ong
mat
hem
atic
al
conc
epts
and
pro
cedu
res
Mak
es a
ppro
pria
te
conn
ectio
ns a
mon
g m
athe
mat
ical
con
cept
s an
d pr
oced
ures
Mak
es s
trong
con
nect
ions
am
ong
mat
hem
atic
al
conc
epts
and
pro
cedu
res
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
6
5.2
(b) P
aint
ed C
ube
Prob
lem
Thin
king
Prob
lem
Sol
ving
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s ap
prop
riate
to
the
task
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk w
ith s
igni
fican
t pr
ompt
ing
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk w
ith m
inim
al
prom
ptin
g
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk in
depe
nden
tly
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk in
depe
nden
tly w
ith a
br
oade
r vie
w o
f the
task
Use
s cr
itica
l thi
nkin
g sk
ills
to s
olve
a p
robl
emU
ses
min
imal
logi
c an
d pr
ecis
ion
in m
athe
mat
ical
re
ason
ing
to s
olve
pr
oble
ms
Use
s lo
gic
to s
olve
pr
oble
ms
but l
acks
pr
ecis
ion
in m
athe
mat
ical
re
ason
ing
Sol
ves
prob
lem
s lo
gica
lly
and
with
pre
cisi
on in
m
athe
mat
ical
reas
onin
g
Dem
onst
rate
s a
soph
istic
ated
leve
l of
mat
hem
atic
al re
ason
ing
and
prec
isio
n in
sol
ving
pr
oble
ms
Rea
soni
ng a
nd P
rovi
ngFo
rmul
ates
and
def
ends
a
hypo
thes
is o
r con
ject
ure
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s fe
w a
spec
ts o
f the
pro
blem
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s so
me
of th
e pe
rtine
nt
aspe
cts
of th
e pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s pe
rtine
nt a
spec
ts o
f the
pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s as
pect
s of
the
prob
lem
w
ith a
bro
ader
vie
w o
f the
pr
oble
mM
akes
infe
renc
es, d
raw
s co
nclu
sion
s an
d gi
ves
just
ifica
tions
Mak
es li
mite
d co
nnec
tions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es s
ome
conn
ectio
ns
to th
e pr
oble
m-s
olvi
ng
proc
ess
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
s
Mak
es d
irect
con
nect
ions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es d
irect
and
insi
ghtfu
l co
nnec
tions
to th
e pr
oble
m-s
olvi
ng p
roce
ss
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
s
App
licat
ion
Sele
ctin
g To
ols
and
Com
puta
tiona
l Str
ateg
ies
Sel
ects
and
use
s to
ols
and
stra
tegi
es to
sol
ve a
pr
oble
m
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
ajor
er
rors
, om
issi
ons,
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
inor
er
rors
, om
issi
ons
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es a
ccur
atel
y, a
nd
in a
logi
cal s
eque
nce
Sel
ects
and
app
lies
appr
opria
te a
nd e
ffici
ent
tool
s an
d st
rate
gies
, ac
cura
tely
to c
reat
e m
athe
mat
ical
ly e
lega
nt
solu
tions
Com
mun
icat
ion
Rep
rese
ntin
gTr
ansl
ates
from
one
re
pres
enta
tion
to a
noth
er
as a
ppro
pria
te to
the
prob
lem
Tran
slat
es re
pres
enta
tion
with
maj
or e
rror
s w
hen
solv
ing
a pr
oble
m
Tran
slat
es re
pres
enta
tions
w
ith s
ome
erro
rs w
hen
solv
ing
a pr
oble
m
Tran
slat
es
repr
esen
tatio
ns
appr
opria
tely
whe
n so
lvin
g a
prob
lem
Tran
slat
es re
pres
enta
tions
ap
prop
riate
ly a
nd w
ith
insi
ght w
hen
solv
ing
a pr
oble
m
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
7
5.2
(c) T
empe
ratu
re P
robl
em
Thin
king
Prob
lem
Sol
ving
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s ap
prop
riate
to
the
task
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk w
ith s
igni
fican
t pr
ompt
ing
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk w
ith m
inim
al
prom
ptin
g
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk in
depe
nden
tly
Sel
ects
, seq
uenc
es a
nd
appl
ies
mat
hem
atic
al
proc
esse
s to
the
assi
gned
ta
sk in
depe
nden
tly w
ith a
br
oade
r vie
w o
f the
task
Rea
soni
ng a
nd P
rovi
ngFo
rmul
ates
and
def
ends
a
hypo
thes
is o
r con
ject
ure
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s fe
w a
spec
ts o
f the
pro
blem
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s so
me
of th
e pe
rtine
nt
aspe
cts
of th
e pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s pe
rtine
nt a
spec
ts o
f the
pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s as
pect
s of
the
prob
lem
w
ith a
bro
ader
vie
w o
f the
pr
oble
mM
akes
infe
renc
es, d
raw
s co
nclu
sion
s an
d gi
ves
just
ifica
tions
Mak
es li
mite
d co
nnec
tions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es s
ome
conn
ectio
ns
to th
e pr
oble
m-s
olvi
ng
proc
ess
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
s
Mak
es d
irect
con
nect
ions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es d
irect
and
insi
ghtfu
l co
nnec
tions
to th
e pr
oble
m-s
olvi
ng p
roce
ss
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
sSe
lect
ing
Tool
s an
d C
ompu
tatio
nal S
trat
egie
sS
elec
ts a
nd u
ses
tool
s an
d st
rate
gies
to s
olve
a
prob
lem
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
ajor
er
rors
, om
issi
ons,
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
inor
er
rors
, om
issi
ons
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es a
ccur
atel
y, a
nd
in a
logi
cal s
eque
nce
Sel
ects
and
app
lies
appr
opria
te a
nd e
ffici
ent
tool
s an
d st
rate
gies
, ac
cura
tely
to c
reat
e m
athe
mat
ical
ly e
lega
nt
solu
tions
Com
mun
icat
ion
Rep
rese
ntin
gC
reat
es a
mod
el to
re
pres
ent t
he p
robl
em(e
.g.,
num
eric
al, a
lgeb
raic
, gr
aphi
cal,
phys
ical
, or
scal
e m
odel
, by
hand
or
usin
g te
chno
logy
)
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith li
mite
d ef
fect
iven
ess;
re
pres
entin
g lit
tle o
f the
ra
nge
of th
e da
ta
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith s
ome
effe
ctiv
enes
s;
repr
esen
ting
som
e of
the
rang
e of
the
data
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith c
onsi
dera
ble
effe
ctiv
enes
s;
repr
esen
ting
mos
t of t
he
rang
e of
the
data
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith a
hig
h de
gree
of
effe
ctiv
enes
s; re
pres
entin
g th
e fu
ll ra
nge
of th
e da
ta
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
8
5.2
(d) D
art B
oard
Pro
blem
Thin
king
Prob
lem
Sol
ving
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Use
s cr
itica
l thi
nkin
g sk
ills
to s
olve
a p
robl
emU
ses
min
imal
logi
c an
d pr
ecis
ion
in m
athe
mat
ical
re
ason
ing
to s
olve
pr
oble
ms
Use
s lo
gic
to s
olve
pr
oble
ms
but l
acks
pr
ecis
ion
in m
athe
mat
ical
re
ason
ing
Sol
ves
prob
lem
s lo
gica
lly
and
with
pre
cisi
on in
m
athe
mat
ical
reas
onin
g
Dem
onst
rate
s a
soph
istic
ated
leve
l of
mat
hem
atic
al re
ason
ing
and
prec
isio
n in
sol
ving
pr
oble
ms
Rea
soni
ng a
nd P
rovi
ngFo
rmul
ates
and
def
ends
a
hypo
thes
is o
r con
ject
ure
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s fe
w a
spec
ts o
f the
pro
blem
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s so
me
of th
e pe
rtine
nt
aspe
cts
of th
e pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s pe
rtine
nt a
spec
ts o
f the
pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s as
pect
s of
the
prob
lem
w
ith a
bro
ader
vie
w o
f the
pr
oble
mM
akes
infe
renc
es, d
raw
s co
nclu
sion
s an
d gi
ves
just
ifica
tions
Mak
es li
mite
d co
nnec
tions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es s
ome
conn
ectio
ns
to th
e pr
oble
m-s
olvi
ng
proc
ess
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
s
Mak
es d
irect
con
nect
ions
to
the
prob
lem
-sol
ving
pr
oces
s an
d m
odel
s pr
esen
ted
whe
n ju
stify
ing
answ
ers
Mak
es d
irect
and
insi
ghtfu
l co
nnec
tions
to th
e pr
oble
m-s
olvi
ng p
roce
ss
and
mod
els
pres
ente
d w
hen
just
ifyin
g an
swer
s
Refl
ectin
gR
eflec
ts o
n th
e re
ason
able
ness
of
answ
ers
Mak
es m
inim
al
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es s
ome
conn
ectio
ns
betw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
and
prov
ides
in
sigh
tful c
omm
ents
App
licat
ion
Sele
ctin
g To
ols
and
Com
puta
tiona
l Str
ateg
ies
Sel
ects
and
use
s to
ols
and
stra
tegi
es to
sol
ve a
pr
oble
m
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
ajor
er
rors
, om
issi
ons,
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
inor
er
rors
, om
issi
ons
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es a
ccur
atel
y, a
nd
in a
logi
cal s
eque
nce
Sel
ects
and
app
lies
appr
opria
te a
nd e
ffici
ent
tool
s an
d st
rate
gies
, ac
cura
tely
to c
reat
e m
athe
mat
ical
ly e
lega
nt
solu
tions
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
9
5.2
(e) D
eck
Rai
ling
Prob
lem
Thin
king
Prob
lem
Sol
ving
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Use
s cr
itica
l thi
nkin
g sk
ills
to s
olve
a p
robl
emU
ses
min
imal
logi
c an
d pr
ecis
ion
in m
athe
mat
ical
re
ason
ing
to s
olve
pr
oble
ms
Use
s lo
gic
to s
olve
pr
oble
ms
but l
acks
pr
ecis
ion
in m
athe
mat
ical
re
ason
ing
Sol
ves
prob
lem
s lo
gica
lly
and
with
pre
cisi
on in
m
athe
mat
ical
reas
onin
g
Dem
onst
rate
s a
soph
istic
ated
leve
l of
mat
hem
atic
al re
ason
ing
and
prec
isio
n in
sol
ving
pr
oble
ms
Rea
soni
ng a
nd P
rovi
ngFo
rmul
ates
and
def
ends
a
hypo
thes
is o
r con
ject
ure
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s fe
w a
spec
ts o
f the
pro
blem
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s so
me
of th
e pe
rtine
nt
aspe
cts
of th
e pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s pe
rtine
nt a
spec
ts o
f the
pr
oble
m
Form
s a
hypo
thes
is o
r co
njec
ture
that
con
nect
s as
pect
s of
the
prob
lem
w
ith a
bro
ader
vie
w o
f the
pr
oble
mR
eflec
ting
Refl
ects
on
the
reas
onab
lene
ss o
f an
swer
s
Mak
es m
inim
al
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es s
ome
conn
ectio
ns
betw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
a pr
ior e
stim
ate
and
the
solu
tion
and
prov
ides
in
sigh
tful c
omm
ents
App
licat
ion
Sele
ctin
g To
ols
and
Com
puta
tiona
l Str
ateg
ies
Sel
ects
and
use
s to
ols
and
stra
tegi
es to
sol
ve a
pr
oble
m
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
ajor
er
rors
, om
issi
ons,
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es, w
ith m
inor
er
rors
, om
issi
ons
or m
is-
sequ
enci
ng
Sel
ects
and
app
lies
appr
opria
te to
ols
and
stra
tegi
es a
ccur
atel
y, a
nd
in a
logi
cal s
eque
nce
Sel
ects
and
app
lies
appr
opria
te a
nd e
ffici
ent
tool
s an
d st
rate
gies
, ac
cura
tely
to c
reat
e m
athe
mat
ical
ly e
lega
nt
solu
tions
Con
nect
ing
Mak
es c
onne
ctio
ns a
mon
g m
athe
mat
ical
con
cept
s an
d pr
oced
ures
Mak
es w
eak
conn
ectio
ns
amon
g m
athe
mat
ical
co
ncep
ts a
nd p
roce
dure
s
Mak
es s
impl
e co
nnec
tions
am
ong
mat
hem
atic
al
conc
epts
and
pro
cedu
res
Mak
es a
ppro
pria
te
conn
ectio
ns a
mon
g m
athe
mat
ical
con
cept
s an
d pr
oced
ures
Mak
es s
trong
con
nect
ions
am
ong
mat
hem
atic
al
conc
epts
and
pro
cedu
res
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
10
5.2
(e) D
eck
Rai
ling
Prob
lem
(con
tinue
d)
Com
mun
icat
ion
Rep
rese
ntin
gC
riter
iaB
elow
Lev
el 1
Sp
ecifi
c Fe
edba
ckLe
vel 1
Leve
l 2Le
vel 3
Leve
l 4
Cre
ates
a m
odel
to
repr
esen
t the
pro
blem
(e.g
., nu
mer
ical
, al
gebr
aic,
gra
phic
al,
phys
ical
, or s
cale
m
odel
, by
hand
or
usin
g te
chno
logy
)
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith li
mite
d ef
fect
iven
ess;
re
pres
entin
g lit
tle o
f the
ra
nge
of th
e da
ta
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith s
ome
effe
ctiv
enes
s;
repr
esen
ting
som
e of
the
rang
e of
the
data
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
with
co
nsid
erab
le e
ffect
iven
ess;
re
pres
entin
g m
ost o
f the
ra
nge
of th
e da
ta
Cre
ates
a m
odel
that
re
pres
ents
the
prob
lem
w
ith a
hig
h de
gree
of
effe
ctiv
enes
s; re
pres
entin
g th
e fu
ll ra
nge
of th
e da
ta
Mak
es c
onne
ctio
ns
betw
een
num
eric
, gr
aphi
cal a
nd
alge
brai
c re
pres
enta
tions
Mak
es li
mite
d co
nnec
tions
be
twee
n nu
mer
ic,
grap
hica
l and
alg
ebra
ic
repr
esen
tatio
ns
Mak
es s
ome
conn
ectio
ns
betw
een
num
eric
, gr
aphi
cal a
nd a
lgeb
raic
re
pres
enta
tions
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
num
eric
, gra
phic
al a
nd
alge
brai
c re
pres
enta
tions
Mak
es s
trong
and
insi
ghtfu
l co
nnec
tions
bet
wee
n nu
mer
ic, g
raph
ical
and
al
gebr
aic
repr
esen
tatio
ns
Tran
slat
es fr
om o
ne
repr
esen
tatio
n to
an
othe
r as
appr
opria
te
to th
e pr
oble
m
Tran
slat
es re
pres
enta
tion
with
maj
or e
rror
s w
hen
solv
ing
a pr
oble
m
Tran
slat
es
repr
esen
tatio
ns w
ith
som
e er
rors
whe
n so
lvin
g a
prob
lem
Tran
slat
es re
pres
enta
tions
ap
prop
riate
ly w
hen
solv
ing
a pr
oble
m
Tran
slat
es re
pres
enta
tions
ap
prop
riate
ly a
nd w
ith
insi
ght w
hen
solv
ing
a pr
oble
m
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
11
5.2
(f) F
unct
ion
Car
d G
ame
App
licat
ion
Con
nect
ing
Crit
eria
Bel
ow L
evel
1
Spec
ific
Feed
back
Leve
l 1Le
vel 2
Leve
l 3Le
vel 4
Mak
es c
onne
ctio
ns
betw
een
info
rmat
ion
in th
e ch
art a
nd th
e gr
aph.
Mak
es w
eak
conn
ectio
ns
betw
een
info
rmat
ion
in th
e ch
art a
nd th
e gr
aph
Mak
es s
impl
e co
nnec
tions
be
twee
n in
form
atio
n in
the
char
t and
the
grap
h
Mak
es a
ppro
pria
te
conn
ectio
ns b
etw
een
info
rmat
ion
in th
e ch
art
and
the
grap
h
Mak
es s
trong
con
nect
ions
be
twee
n in
form
atio
n in
the
char
t and
the
grap
h
Gat
hers
dat
a th
at c
an b
e us
ed to
sol
ve th
e pr
oble
m
[e.g
., se
lect
crit
ical
x-
valu
es a
nd in
terv
als
for
the
char
t].
Gat
hers
dat
a th
at is
co
nnec
ted
to th
e pr
oble
m,
yet i
napp
ropr
iate
for t
he
inqu
iry
Gat
hers
dat
a th
at is
ap
prop
riate
and
con
nect
ed
to th
e pr
oble
m, y
et m
issi
ng
man
y si
gnifi
cant
cas
es
Gat
hers
dat
a th
at is
ap
prop
riate
and
con
nect
ed
to th
e pr
oble
m, i
nclu
ding
m
ost s
igni
fican
t cas
es
Gat
hers
dat
a th
at is
ap
prop
riate
and
con
nect
ed
to th
e pr
oble
m, i
nclu
ding
al
l sig
nific
ant c
ases
, in
clud
ing
extre
me
case
s
Thin
king
Rea
soni
ng a
nd P
rovi
ngIn
terp
rets
gra
phs.
Mis
inte
rpre
ts a
maj
or p
art
of th
e gi
ven
grap
hica
l in
form
atio
n, b
ut c
arrie
s on
to
mak
e so
me
othe
rwis
e re
ason
able
sta
tem
ents
Mis
inte
rpre
ts p
art o
f th
e gi
ven
grap
hica
l in
form
atio
n, b
ut c
arrie
s on
to
mak
e so
me
othe
rwis
e re
ason
able
sta
tem
ents
Cor
rect
ly in
terp
rets
th
e gi
ven
grap
hica
l in
form
atio
n, a
nd m
akes
re
ason
able
sta
tem
ents
Cor
rect
ly in
terp
rets
th
e gi
ven
grap
hica
l in
form
atio
n, a
nd m
akes
su
btle
or i
nsig
htfu
l st
atem
ents
Mak
es in
fere
nces
in th
e ch
art a
bout
the
requ
ired
grap
h.
Mak
es in
fere
nces
that
ha
ve a
lim
ited
conn
ectio
n to
the
prop
ertie
s of
the
give
n gr
aphs
Mak
es in
fere
nces
that
ha
ve s
ome
conn
ectio
n to
th
e pr
oper
ties
of th
e gi
ven
grap
hs
Mak
es in
fere
nces
that
ha
ve a
dire
ct c
onne
ctio
n to
the
prop
ertie
s of
the
give
n gr
aphs
Mak
es in
fere
nces
that
ha
ve a
dire
ct c
onne
ctio
n to
th
e pr
oper
ties
of th
e gi
ven
grap
hs, w
ith e
vide
nce
of
refle
ctio
n
Com
mun
icat
ion
Rep
rese
ntin
gC
reat
es a
gra
ph to
re
pres
ent t
he d
ata
in th
e ch
art.
Cre
ates
a g
raph
that
re
pres
ents
littl
e of
the
rang
e of
dat
a
Cre
ates
a g
raph
that
re
pres
ents
som
e of
the
rang
e of
dat
a
Cre
ates
a g
raph
that
re
pres
ents
mos
t of t
he
rang
e of
dat
a
Cre
ates
a g
raph
that
re
pres
ents
the
full
rang
e of
dat
a
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 12
5.3 Volume of 3-D Solids
Investigate the relationship between prisms and pyramids with congruent bases and equal heights.1.
Investigate the relationship among a cylinder, a cone and a sphere each with the same radius and 2. the heights of the cylinder and cone are twice their radius.
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 13
5.4 The Painted Cube with Solutions
A 3 a) × 3 × 3 cube made up of small cubes is dipped into a bucket of red paint and removed. (i) How many small cubes will have 3 faces painted?
(ii) How many small cubes will have 2 faces painted?
(iii) How many small cubes will have 1 face painted?
(iv) How many small cubes will have 0 faces painted?
b) Answer the questions in Part (a) when a 10 × 10 × 10 cube is dipped into the bucket of red paint.
c) Look for patterns and answer each of the questions in Part (a) when an n × n × n cube is dipped into the bucket of red paint? Explain your thinking.
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 14
5.4 The Painted Cube with Solutions (continued)
Complete the table.
a) 3 × 3 × 3 cube made up of small cubes is dipped into a bucket of red paint. How many cubes will have 3 faces painted? 2 faces? 1 face? 0 faces?
b) What about a 4 × 4 × 4 cube?
c) What about a n × n × n cube?
Cube Number
3 faces painted
2 faces painted
1 face painted
0 faces painted
3 8 12 6 14 8 24 24 85 8 36 54 276 8 48 96 647 8 60 150 1258 8 72 216 2169 8 84 294 343
10 8 96 384 512. . . . .
n 8 12 2n -( ) 6 2 2n -( ) n -( )2 3
A. Analysis Using Finite DifferencesThe solutions can be arrived at through a modelling approach, with or without the aid of technology (graphing calculator, spreadsheet) or through a spatial/logical approach, using the physical models of the cubes.
Two Faces Painted
The first differences, for the 2 Faces Painted relationship indicate a linear relationship. The rate of change is 12. For any row in the table, the number of cubes with two faces painted is 12 times the cube number, less two.
The algebraic statement of the relationship is thereforeN n2 12 2= -( )
Cube Number 2 Faces Painted
2 03 124 245 366 487 608 729 84
10 96
First Differences
1212121212121212
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 15
5.4 The Painted Cube with Solutions (continued)
One Face Painted
Zero Faces Painted
The second differences, for the 1 Face Painted relationship indicate a quadratic relationship. The number of cubes with one face painted is zero when n is 2. This indicates a quadratic relationship of the form a n -( )2 2 . The second difference equals 12, and also equals 2a .
The algebraic statement of the relationship is therefore:N n1
26 2= -( ) .
Cube Number
1 Faces Painted
2 03 64 245 546 967 1508 2169 294
10 384
First Differences
618304254667890
Second Differences
12121212121212
The third differences, for the 0-Faces-Painted relationship indicate a cubic relationship. The number of cubes with one face painted is zero when n is 2. This indicates a cubic relationship of the form a n -( )2 3 . The second difference equals 6, and also equals 6a.
The algebraic statement of the relationship is therefore:N n0
32= -( ) .
Cube Number
0 Faces Painted
2 03 14 85 276 1647 1258 2169 343
10 512
First Differences
17
19376191
127169
Second Differences
6121824303642
Third Differences
666666
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 16
5.4 The Painted Cube with Solutions (continued)
B. Graphical Perspective
Here is a picture of a Fathom document showing the plots of Two-Faces-Painted, One-face-Painted, and Zero-Faces-Painted. The Two-Faces-Painted plot has a least-squares line fitted to it, and the One-Face-Painted plot has a slider, in which the value of a has been adjusted to 6.
C. Geometric Approach
This strategy is easier to understand when working with linking cubes.Make larger cubes of at least 4 × 4 × 4 to more easily see the solutions:
There will always be 8 “corner cubes,” with 3 faces painted.
There are 12 edges. For each edge, the “corner cubes” will not have 2 faces painted – all the rest will have 2 faces painted. That totals 12(n - 2) for an n × n × n cube.
1 face painted 2 faces painted 3 faces painted
There are 6 “inner squares” with 1 face painted. The dimensions of these squares are (n - 2) × (n - 2). That totals 6(n - 2)2 for an n × n × n cube.
The 2 × 2 × 2 cube that “sits inside” the 4 × 4 × 4 cube represents the 0 faces painted cubes. For an n × n × n cube, the inner cube has dimensions (n - 2) × (n - 2), or (n - 2)3.
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 17
5.4 The Painted Cube with Solutions (continued)
Sample Follow-up Questions
1) For an n × n × n cube, the number of faces with 2 faces painted is 12(n - 2). Refer to the cube to explain where this formula comes from.
2) If a large cube had 512 cubes with one face painted after being dipped into the paint, how many small cubes is it made of?
3) If a large cube had 190 104 cubes with two faces painted after being dipped into the paint, how many small cubes is it made of?
4) A large cube had 17 576 cubes with zero faces painted after being dipped into the paint. Select, without calculating, the number of small cubes the large cube is made of. Explain how you arrived at your answer.
a) 30 b) 56 c) 28 d) 18
5) For what value of n does the number of cubes with 2 faces painted equal the number of cubes with 1 face painted? Verify using the algebraic expressions.
6) For what value of n does the number of cubes with 1 face painted equal the number of cubes with zero faces painted? Verify using the algebraic expressions.
7) Show that the two expressions for the number of small cubes in an n × n cube are equivalent:
n n n n3 3 22 6 2 12 2 8= -( ) + -( ) + -( )+
Extensionsa) A 3 × 4 rectangular prism that is made up of small cubes is dipped into a bucket of red paint. How
many cubes will have 3 faces painted? 2 faces? 1 face? 0 faces?
b) What about a 4 × 5 rectangular prism?
c) What about an (n - 1) × n rectangular prism?
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 18
5.5 Temperature ProblemThe inhabitants off Xenor use two scales for measuring temperature. On the A scale, water freezes at 0° and boils at 80°, whereas on the B scale, water freezes at -20° and boils at 120°. What is the equivalent on the A scale of a temperature of 15° on the B scale?
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 19
5.6 Dart Board ProblemYou have been asked to assign numerical values of 2, 5, and 8 to the three coloured regions on the dart board. Regions with smaller areas are assigned higher scores.
The dart board is designed with a square inside a circle and a square outside the same circle.
Match the three scores with the three coloured regions on the board. Use any tools or strategies available.
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 20
5.7: Deck Problem with SolutionsYou have been hired to build a deck attached to the second floor of a cottage using exactly 30 m of deck railing (Note: the entire outside edge will have railing.).Determine the dimensions of the deck that follow the specifications in the diagram and maximize the area of the deck.
Sample Solution 1: Numeric solution using a table of values
x values (in metres)
y values(in metres)
1 14= - -( )= -
=
14 14 1196 16927
2 2
2m
2 13= - -( )= -
=
13 13 2169 12148
2 2
m2
3 12= - -( )= -
=
12 12 3144 8163
2 2
m2
4 11= - -( )= -
=
11 11 4121 4972
2 2
m2
5 10= - -( )= -
=
10 10 5100 2575
2 2
m2
6 9= - -( )= -
=
9 9 681 972
2 2
m2
7 8= - -( )= -
=
8 8 764 163
2 2
m2
8 7Even though you are able to obtain numerical values for the algebraic representation of area, these values are inadmissible since x can not exceed y.
9 610 511 412 313 214 1
From the table of values the maximum area of the deck is 75 m2.The longer side is 10 m and the shorter side is 5 m.
DECK
COTTAGE
Area of the deck total area cottage section= -
=( )( )- -( ) -y y y x y xx( )Area =( )( )- -( ) -( )
= - -( )
y y y x y x
y y x2 2
DECK
y
y
x
x
COTTAGE
y - x
y - x
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 21
5.7: Deck Problem with Solutions (continued)Sample Solution 2: Algebraic solution using the distributive property and common factoring
Area of the deck total area cottage section= -
=( )( )- -( ) -y y y x y xx( )Since the maximum amount of railing available is 30 m, then when adding up the sides of the deck we obtain:
x x y yx y
x y
+ + - =+ =+ =
302 2 30
15By re-arranging this equation, we have y x= -15 that we can substitute into the formula for the total area of the deck.
Area of the deck =( )( )- -( ) -( )= -( ) -( )- - -( )
y y y x y x
x x x x15 15 15 15-- -( )
= - - + - -( )= - + - - - +
x x
x x x x
x x x x x
225 15 15 15 2
225 30 225 30 30 4
2 2
2 2(( )= - + - - +( )=- + + -
= -
= -
225 30 225 60 4
30 60 430 33 10
2 2
2 2
2
x x x x
x x x xx x
x xx( )
Since this is a quadratic, the vertices are at 3 0x = (that is x = 0) and 10 0- =x (that is x = 0 ).The vertex would be half way between the vertices, so at x = 5 .When x = 5 and x y+ =15 , then y =10 .
The maximum area is obtained when x = 5 m and y =10 m.
Area of the deck
Substitute the values
=( )( )- -( ) -( )=
y y y x y x
10 mm m m m m m
m mm
2
2
( )( )- -( ) -( )= -
=
10 10 5 10 5
100 2575
2
The maximum area of the deck is 75 m2.The longer side is 10 m and the shorter side is 5 m.
common factor
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 22
5.7: Deck Problem with Solutions (continued)Sample Solution 3: Algebraic solution, using a difference of squares
Area of the deck total area cottage section= -
=( )( )- -( ) -y y y x y xx( )
Since the maximum amount of railing available is 30 m, by adding up the sides of the deck we obtain:
x x y yx y
x y
+ + +( )=+ =+ =
302 2 30
15
By re-arranging this equation, we have y x= -15 that we can substitute into the formula for the total area of the deck.
Area of the deck =( )( )- -( ) -( )
= - -( )= - -( )( ) + -
y y y x y x
y y x
y y x y y
2 2
xx
y y x y y x
x y x
( )( )= - +( ) + -( )= ( ) -( )2
By substituting y x= -15 into the area of the deck:Area of the deck =( ) -( )-( )
= ( ) - -( )= - -
=
x x x
x x x
x x x
2 15
30 2
30 230
2 2
xx xx x
-
= -( )3
3 10
2
Since we have a quadratic, the vertices are at 3 0x = (that is x = 0 ) and 10 0- =x (that is x =10 ).The vertex would be half way between the vertices, at x = 5.When x = 5 , we know that x y+ =15, then y =10 .
The maximum area is obtained when x = 5 m and y =10 m.Area of the deck
Substitute the values
=( )( )- -( ) -( )=
y y y x y x
10(( )( )- -( ) -( )= -
=
10 10 5 10 5100 2575 m2
The maximum area of the deck is 75 m2.The longer side is 10 m and the shorter side is 5 m.
using a difference of squares
common factor
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 23
5.7: Deck Problem with Solutions (continued)Sample Solution 4: Graphical solution, developing an equation and interpreting a graph generated using technology
The amount of railing is 30 m. When adding up the outside edges of the deck we obtain:
x x y yx y x y y x
+ + + =+ = + = = -
302 2 30 15 15 or or
This can be substituted into the formula for the total area of the deck.Area of the deck total area cottage section= -
( )= -( ) -A x x15 152--( )2 2x
Re-label the deck and cottage dimensions and obtain the equation to represent the area of the deck.Technology is used to graph the equation and the maximum is identified.
DECK
15 - x
COTTAGE
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 24
5.7: Deck Problem with Solutions (continued)Sample Solution 5: Algebraic and graphical solution, completing the square and graphing
A x x x x A x x x
A
= -( )+ -( ) = + -( )=-
15 15 2 2 15 2
3
2 or
Simplified xx x2 30+
Complete the square to find the vertex of the corresponding graph: A x x( )=- -( ) +3 5 752
Vertex (5, 75)
Parabola opens down, thus the vertex is a maximum.
The maximum of 75 occurs when x = 5.
The dimensions of the deck of maximum area are 5 m and 10 m.The maximum area is 75 m2.
DECK
COTTAGE
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 25
5.7: Deck Problem with Solutions (continued)Sample Solution 6: Algebraic solution, using rate of change
2 2 301515
x yx y
y x
+ =+ =
= -
A xy x y x
xy x
= + -( )= -2 2
A x x x x
x x
( )= -( )-= -
2 15
30 3
2
2
¢( )= -
¢( )= ¢( )
A x x
A x A x
30 6
0
2
Explore critcal points.or does not[ exist]
therefore the point
30 6 05
12
5 0
- ==
¢¢( )=-
¢¢( )<
xx
A x x
A wwhere is a maximumx
A
=
( )= ( )- ( )=
5
5 30 5 3 575
2
Maximum point: (5, 75)
The maximum area of the deck is 75 m2 and occurs when x = 5.The longer side of the deck is 10 m and the shorter side is 5 m.
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 26
5.8a Function Card Game (from MHF4U BLM 6.11.4)
Your team must find a pair of matching cards. To make a matching pair, you find one card that has the graphs of two functions that correspond with a card that shows these functions combined by an operation (addition, subtraction, multiplication, or division).
When you find a matching pair, state how the functions were combined. Discuss why you think it is a match.
Check the answer (BLM 5.8c) and reflect on the result, if you had an error.
Continue until all the cards are collected.
Some of the features to observe in finding a match are:intercepts of combined and original graphs• intersections of original graphs• asymptotes• general motions, e.g., periodic, cubic, exponential• large and small values• odd and even functions• nature of the function between 0 and 1, 0 and –1• domain and range•
Examples
The initial graph of sin x( ) and 2x,can be combined to produce the graphs shown below it. Determine what operations are used to combine them and explain the reasoning. Check answers after you have determined how the functions were combined.
sin x x( ) and 2
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 27
5.8a Function Card Game (from MHF4U BLM 6.11.4) (continued)
Answers
sin x x( )+2 2x xsin( )( )Periodic suggests sine or cosine•Dramatic change for positive • x-values, not existing for negative x-values, suggests exponentialy• -intercept of 1 can be obtained by adding the y-intercepts of each of the original graphs, only addition will produce this
Periodic suggests sine or cosine•Dramatic change for positive • x-values, not existing for negative x-values, suggests exponentialx• -intercepts correspond to the x-intercepts of the sine function therefore multiplication or division.Division by exponential would result •in small y-values in first and fourth quadrant, division by sinusoidal would result in asymptotes, therefore must be multiplication.
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 28
5.8b Function Card Masters for Game (from MHF4U BLM 6.11.5)
x x and 3 x x3 -
x x and 2 4- xx2 4-
2x x and cos( ) 2x x+ ( )cos
cos x x( )-2 2x x- ( )cos
2
3
4
S
J
P
L
G
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 29
5.8b Function Card Masters for Game (from MHF4U BLM 6.11.5) (continued)
sin logx x( ) ( ) and log sinx x( )+ ( )
log sinx x( )- ( ) sin logx x( )- ( )
x x and sin( ) x xsin( )( )
x x2 and sin( ) x x2 sin( )( )
5
6
T
M H
F
R7
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 30
5.8b Function Card Masters for Game (from MHF4U BLM 6.11.5) (continued)
x2 and cos x( ) x x2 cos( )( )
2 2x x and xx
2
2
22
x
x2 2x x and
2x x and cos( ) 2x x and cos( )
8 N
K9
Q 9
4 4
Mathematical Processes for GAINS Mathematics – Professional Learning, 2008 – Session 5 31
5.8b Function Card Masters for Game (from MHF4U BLM 6.11.5) (continued)
sin logx x( ) ( ) and sin logx x( ) ( ) and
5 5
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
32
5.8c
Fun
ctio
n C
ard
Gam
e A
nsw
ers
(from
MH
F4U
BLM
6.1
1.6)
Indi
vidu
al G
raph
sC
ombi
ned
Gra
phs
Key
feat
ures
to h
elp
with
iden
tifica
tion.
Thes
e ar
e no
t int
ende
d to
be
a su
ffici
ent,
nece
ssar
y or
incl
usiv
e lis
t of f
eatu
res;
they
are
a li
st o
f “o
bser
vatio
ns” t
o as
sist
with
mat
chin
g.
diffe
renc
e of
odd
func
tions
is a
n od
d fu
nctio
n•
cann
ot b
e m
ultip
licat
ion
sinc
e od
d m
ultip
lied
by
•od
d is
eve
nge
nera
l mot
ion
is c
ubic
, res
ult i
s cu
bic
•(0
,0) i
s a
poin
t on
both
orig
inal
s an
d co
mbi
natio
n•
cann
ot b
e di
visi
on s
ince
no
asym
ptot
e oc
curs
•x-
•in
terc
epts
occ
ur w
here
the
grap
hs in
terc
ept,
impl
ying
sub
tract
ion
23
.x
x a
nd
J.x
x3-
asym
ptot
es a
t 2 a
nd –
2 su
gges
ts d
ivis
ion
by
•
x24
-di
visi
on re
sults
in
•y-
valu
es o
f 1 o
n th
e co
mbi
ned
grap
h fo
r val
ues
of x
whe
re th
e or
igin
al g
raph
s in
ters
ect
whe
n•
01
<<
y th
e y-
valu
es o
f the
com
bine
d
grap
h be
com
es la
rge,
and
whe
n -<
<1
0y
the
y-
valu
es o
f the
com
bine
d gr
aph
beco
mes
sm
all
(0,0
) is
a po
int o
n th
e co
mbi
ned
grap
h gi
ving
•
info
rmat
ion
abou
t the
num
erat
orod
d fu
nctio
n di
vide
d by
an
even
func
tion
is a
n •
odd
func
tion
3.
and
x
x24
-
P.
xx2
4-
()
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
33
5.8c
Fun
ctio
n C
ard
Gam
e A
nsw
ers
(from
MH
F4U
BLM
6.1
1.6)
(con
tinue
d)
Indi
vidu
al G
raph
sC
ombi
ned
Gra
phs
Key
feat
ures
to h
elp
with
iden
tifica
tion.
Thes
e ar
e no
t int
ende
d to
be
a su
ffici
ent,
nece
ssar
y or
incl
usiv
e lis
t of f
eatu
res;
they
are
a li
st o
f “o
bser
vatio
ns” t
o as
sist
with
mat
chin
g.
perio
dic
sugg
ests
sin
e or
cos
ine
•dr
amat
ic c
hang
e fo
r pos
itive
•
x-va
lues
, not
ex
istin
g fo
r neg
ativ
e x-
valu
es, s
ugge
sts
expo
nent
ial
•y
•-in
terc
ept o
f 2 c
an b
e ob
tain
ed b
y ad
ding
the
y-
inte
rcep
t of 1
of e
ach
of th
e or
igin
al g
raph
s,
only
add
ition
will
pro
duce
this
resu
lt
4.
and
2x
xco
s ()
L. 2
xx
+()
cos
perio
dic
sugg
ests
sin
e or
cos
ine
•dr
amat
ic c
hang
e fo
r pos
itive
•
x-va
lues
, not
ex
istin
g fo
r neg
ativ
e x-
valu
es, s
ugge
sts
expo
nent
ial
y-•
inte
rcep
t of 0
can
be
obta
ined
by
subt
ract
ing
the
y-in
terc
ept o
f 1 o
f eac
h of
the
orig
inal
gra
phs
x•
-inte
rcep
ts a
re w
here
the
orig
inal
gra
phs
inte
rsec
t, im
plyi
ng s
ubtra
ctio
nas
•
x-in
crea
ses,
the
com
bina
tion
decr
ease
s qu
ickl
y, s
ugge
stin
g su
btra
ctio
n of
an
expo
nent
ial
4.
and
2x
xco
s ()
S.
cos
xx
()-
2
perio
dic
sugg
ests
sin
e or
cos
ine
•dr
amat
ic c
hang
e fo
r pos
itive
• x
-val
ues,
not
ex
istin
g fo
r neg
ativ
e x-
valu
es, s
ugge
sts
expo
nent
ial
y•
-inte
rcep
t of 0
can
be
obta
ined
by
subt
ract
ing
the
y-in
terc
ept o
f 1 o
f eac
h of
the
orig
inal
gra
phs
x•
-inte
rcep
ts a
re w
here
the
orig
inal
gra
phs
inte
rsec
t, im
plyi
ng s
ubtra
ctio
nas
•
x-in
crea
ses,
the
com
bina
tion
incr
ease
s qu
ickl
y, s
ugge
stin
g su
btra
ctio
n of
the
perio
dic
from
the
expo
nent
ial
4.
and
2x
xco
s ()
G.
2xx
-()
cos
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
34
5.8c
Fun
ctio
n C
ard
Gam
e A
nsw
ers
(from
MH
F4U
BLM
6.1
1.6)
(con
tinue
d)
Indi
vidu
al G
raph
sC
ombi
ned
Gra
phs
Key
feat
ures
to h
elp
with
iden
tifica
tion.
Thes
e ar
e no
t int
ende
d to
be
a su
ffici
ent,
nece
ssar
y or
incl
usiv
e lis
t of f
eatu
res;
they
are
a li
st o
f “o
bser
vatio
ns” t
o as
sist
with
mat
chin
g.
perio
dic
sugg
ests
sin
e or
cos
ine
func
tion
•do
mai
n >
0 su
gges
ts lo
g fu
nctio
n•
decr
easi
ng s
in
•x
grap
h su
gges
ts s
omet
hing
is
bein
g “ta
ken
away
,” th
us s
ubtra
ctio
nx
•-in
terc
epts
are
whe
re th
e or
igin
al g
raph
s in
ters
ect i
mpl
ying
sub
tract
ion
whe
n lo
g is
ver
y sm
all (
or la
rge
nega
tive)
, the
•
com
bine
d gr
aph
beco
mes
ver
y la
rge,
impl
ying
su
btra
ctio
n th
e lo
g va
lues
5. s
in a
nd
xx
()
()
log
H.
sin
cos
xx
()-
()
perio
dic
sugg
ests
sin
e or
cos
ine
func
tion
•do
mai
n >
0 s
ugge
sts
log
func
tion
•x
•-in
terc
epts
are
whe
re th
e or
igin
al g
raph
s in
ters
ect i
mpl
ying
sub
tract
ion
whe
n lo
g is
ver
y sm
all,
the
com
bine
d gr
aph
•re
mai
ns s
mal
l, im
plyi
ng s
ubtra
ctio
n fro
m th
e lo
g
5.si
nlo
g
and
x
x()
()
M.
log
sin
xx
()-
()
perio
dic
sugg
ests
sin
e or
cos
ine
func
tion
•do
mai
n >
0 su
gges
ts lo
g fu
nctio
n•
whe
n lo
g is
ver
y sm
all,
the
com
bine
d gr
aph
•re
mai
ns s
mal
l, im
plyi
ng lo
g is
not
bei
ng
subt
ract
edth
e si
ne c
urve
is in
crea
sing
, im
plyi
ng s
omet
hing
•
is b
eing
add
ed to
the
sine
.
5.si
nlo
g
and
x
x()
()
T. l
ogsi
nx
x()+
()
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
35
5.8c
Fun
ctio
n C
ard
Gam
e A
nsw
ers
(from
MH
F4U
BLM
6.1
1.6)
(con
tinue
d)
Indi
vidu
al G
raph
sC
ombi
ned
Gra
phs
Key
feat
ures
to h
elp
with
iden
tifica
tion.
Thes
e ar
e no
t int
ende
d to
be
a su
ffici
ent,
nece
ssar
y or
incl
usiv
e lis
t of f
eatu
res;
they
are
a li
st o
f “o
bser
vatio
ns” t
o as
sist
with
mat
chin
g.
perio
dic
sugg
ests
sin
e or
cos
ine
func
tion
•x
•-in
terc
epts
exi
st w
here
ver t
here
is a
n x-
inte
rcep
t in
eith
er o
f the
orig
inal
func
tions
, sug
gest
ing
mul
tiplic
atio
nod
d fu
nctio
n m
ultip
lied
by a
n od
d fu
nctio
n,
•re
sults
in a
n ev
en fu
nctio
n
6.
and
x
xsi
n ()
F. x
xsi
n ()
()
perio
dic
sugg
ests
sin
e or
cos
ine
func
tion
•x
•-in
terc
epts
exi
st w
here
ver t
here
is a
n x-
inte
rcep
t in
eith
er o
f the
orig
inal
func
tions
, sug
gest
ing
mul
tiplic
atio
nev
en fu
nctio
n m
ultip
lied
by a
n od
d fu
nctio
n,
•re
sults
in a
n od
d fu
nctio
n
7.
and
x
x2
sin (
)R
. x
x2
sin (
)(
)
perio
dic
sugg
ests
sin
e or
cos
ine
func
tion
•x
•-in
terc
epts
exi
st w
here
ver t
here
is a
n x-
inte
rcep
t in
eith
er o
f the
orig
inal
func
tions
, sug
gest
ing
mul
tiplic
atio
nev
en fu
nctio
n m
ultip
lied
by a
n ev
en fu
nctio
n,
•re
sults
in a
n ev
en fu
nctio
n
8.
and
x
x2
cos (
)N
. x
x2
cos (
)(
)
Mat
hem
atic
al P
roce
sses
for G
AIN
S M
athe
mat
ics
– P
rofe
ssio
nal L
earn
ing,
200
8 –
Ses
sion
5
36
5.8c
Fun
ctio
n C
ard
Gam
e A
nsw
ers
(from
MH
F4U
BLM
6.1
1.6)
(con
tinue
d)
Indi
vidu
al G
raph
sC
ombi
ned
Gra
phs
Key
feat
ures
to h
elp
with
iden
tifica
tion.
Thes
e ar
e no
t int
ende
d to
be
a su
ffici
ent,
nece
ssar
y or
incl
usiv
e lis
t of f
eatu
res;
they
are
a li
st o
f “o
bser
vatio
ns” t
o as
sist
with
mat
chin
g.
x•
-inte
rcep
t occ
urs
at x
-inte
rcep
t ofx
2su
gges
ting
mul
tiplic
atio
n or
div
isio
nw
here
•
2xis
sm
all t
he c
ombi
ned
grap
h is
larg
e an
d vi
se v
ersa
, sug
gest
ing
divi
sion
by
2x
whe
re th
e gr
aphs
inte
rsec
t at (
2,4)
div
isio
n •
prod
uces
the
poin
t (2,
1)
9.
and
2
2x
xK.
x x2 2
asym
ptot
e at
•
y-ax
is s
ugge
sts
divi
sion
by
a fu
nctio
n go
ing
thro
ugh
the
orig
inco
mbi
ned
func
tion
is s
mal
l as
•x
gets
sm
all,
and
is la
rge
as x
get
s la
rge,
sug
gest
exp
onen
tial
9.
and
2
2x
xQ
. 2 2x x