Post on 24-Mar-2018
transcript
Given name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Family name: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
School: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MathematicsQueensland Comparable Assessment Tasks (QCATs) 2012 9Behind the scenesStudent booklet
2
© The State of Queensland (QSA) 2012 Please read our copyright notice <www.qsa.qld.edu.au/copyright.html>.Queensland Studies Authority PO Box 307 Spring Hill Qld 4004Phone: (07) 3864 0299 Fax: (07) 3221 2533 Email: office@qsa.qld.edu.au Website: www.qsa.qld.edu.auImages p. 1 Dancers: <www.123rf.com/photo_7887714_hip-hop-men-performing-and-act-over-an-urban-background.html>; p. 2 Dance montage: <www.123rf.com/photo_10448221_three-dancers>, <www.123rf.com/photo_8343118_portrait-of-an-attractive-young-bboy-dancer-showing-moves-against-grey-background.html>; p. 3 Director: PhotoDisc_OS29005; Stage manager: PhotoDisc_OS35077; p. 4, pp. 11–12 Silhouettes of dancers: <www.sxc.hu/photo/1206163>, <www.sxc.hu/photo/1206163>. All other images © QSA.
| QCATs 2012 Student booklet Year 9 Mathematics
Setting the sceneBehind the scenes of a successful dance competition are a director and stage manager.
In this assessment, you will:Follow the instructions in the director’s notes to:
• calculate areas of projected logos for painting on the backdrop• choose suitable colours for dance competition logos• make decisions about stage lighting• apply your understandings to accommodate the director’s last-minute changes.
The following formulas may be useful:
Show your working• Your teacher is looking for mathematical thinking and reasoning, not just correct answers.• When using a calculator, show enough working so that your teacher can see the method you used.• Ensure all answers are rounded to an appropriate number of decimal places.• If you cannot complete a question, show what you have been able to do.• Credit will be given if an incorrect answer is used correctly in a later question.
Pythagoras’ theorem
Distance formula
Director’s notesDirector’s notes
. . . . . . . . . .
I make decisions about
, , …venue colours lighting
DirectorStage
managerMy job is to put the director’sideas into practice. I use mathsto make the ideas work.
•
•
c2 a2 b2+= a
b
c
d x2 x1– 2 y2 y1– 2+=
Distance (d) from A to B y
x
A (x1, y1)
B (x2, y2)d
| 3Queensland Studies Authority
4
Projecting and painting logos on the backdrop
1. Calculate the total area of Logo A.
Diagram 1: Logo A
backdrop
3 m
3 m
3 m
3 m
projector
1 m
1 m
Logo A
Logo BDirector’s note #1
Use both Logo A and Logo B for the
dance competition.
Project them onto the backdrop so
they can be painted onto the surface.
Calculate the area of each logo so
you know how much paint you need.
•
•
•
Director’s note #1
Use both Logo A and Logo B for the
dance competition.
Project them onto the backdrop so
they can be painted onto the surface.
Calculate the area of each logo so
you know how much paint you need.
•
•
•
Show all working.
Total area = . . . . . . . . . . . . . . . . . . . . . . . . .
| QCATs 2012 Student booklet Year 9 Mathematics
2. Calculate the total area of Logo B, correct to 1 decimal place.
Key:
= 1.0 m
= 2.0 m
= 3.0 m
Diagram 2: Logo B
This curve is part of a circle.
Show all working.
Total area = . . . . . . . . . . . . . . . . . . . . . . . . .
| 5Queensland Studies Authority
6
Choosing colours for the dance competition logos
3. Plot and label the colours in Table 1 on the cyan/magenta colour graph below.
Director’s note #2
Use cyan and magenta tints
to mix colours for the logos.•
Director’s note #2
Use cyan and magenta tints
to mix colours for the logos.•
Table 1: Colour chart
Colour
Name blue turquoise purple cobalt slate blue periwinkle
(c, m) (100, 100) (100, 15) (39, 81) (60, 60) (49, 56) (33, 33)
When and tints are mixedcyan magenta
into white paint, the code for the resulting colour is apair of numbers , where is the quantity of
and is the quantity of tint added toeach litre of paint.
(c, m) ccyan m magenta
(c, m) is used in place of (x, y)
0 20 40 60 80 1000
20
40
60
80
100
Units of cyan (c) per litre c
Units
ofm
agenta
(m)
per
litre
m
white
slate blue
Graph 1: Colour graph
| QCATs 2012 Student booklet Year 9 Mathematics
4. Find the colour difference between slate blue and periwinkle.
Use Pythagoras’ theorem or the distance formula (see page 3).
5. List all colours from Table 1 that could match director’s note #3.
Justify your decisions. (You may not need to calculate the colour difference for all colours.)
Colour difference is the distance between two colours when plotted on a colour graph. Colours used for logos should have a high contrast (a large colour difference).
Director’s note #3
Use two colours:
slate blue
another colour from Table 1 with a colour
difference of more than 25 units from slate blue.
•
•
Director’s note #3
Use two colours:
slate blue
another colour from Table 1 with a colour
difference of more than 25 units from slate blue.
•
•
Show all working.
Distance = . . . . . . . . . . . . . . . . . . . . . . . . .
Show all working.
| 7Queensland Studies Authority
8
Lighting the dance floor
6. Find the radius (r) of the circle of light that each light shines on the floor from 3.9 m above.
Remember: tan ���opposite
adjacent
Each light shines acircle of light on thefloor as shown.
Director’s note #4
As a special effect I want the
rectangular dance floor lit from
above.
Use the 10 lights from the
storeroom.
•
•
Director’s note #4
As a special effect I want the
rectangular dance floor lit from
above.
Use the 10 lights from the
storeroom.
•
•
Diagram 3: Lighting the dance floor
floor r
3.9 m
20°
light
Diagram 4: The largest square lit by one light
length of sideof square
s
square lit by onecircle of light
r radius ofcircle of light
on floor
r
The rectangular dance floor can be divided into squares, each lit by one circle of light.
Show all working.
Radius = . . . . . . . . . . . . . . . . . . . . . . . . .
| QCATs 2012 Student booklet Year 9 Mathematics
7. (a) Use the radius found in Question 6 and Pythagoras’ theorem to calculate the length of the side (s) of the largest square that can be lit by one circle of light (see Diagram 4).
(b) Find the length (l) and width (w) of the largest rectangular dance floor that can be completely lit by 10 lights at 3.9 m high (see Diagram 5).
Dancefloor
Diagram 5: Rectangular dance floor seen from above
length ( )l
width ( )w
Show all working.
Length of side of square = . . . . . . . . . . . . . . . . . . . . . . . . .
Show all working.
Length (l) = . . . . . . . . . . . . . . . . . . . . . . . . . Width (w) = . . . . . . . . . . . . . . . . . . . . . . . . .
| 9Queensland Studies Authority
10
The director�s last minute changes
8. What will be the new height (h) of Logo B, if the projector is moved 1 metre further away from the backdrop (see Diagram 6)?
Logo B was enlarged from itsoriginal height of 5 m bymoving the projector furtheraway from the backdrop.
Director’s note #5
Sorry, a last minute change.
The logos on the backdrop
need to be larger.
Director’s note #5
Sorry, a last minute change.
The logos on the backdrop
need to be larger.
Diagram 6: Making Logo B larger
projector
original position of projectorprojector moved back 1 metre
h
1 m
5 m
4 m
Show all working.
New height = . . . . . . . . . . . . . . . . . . . . . . . . .
| QCATs 2012 Student booklet Year 9 Mathematics
9. If Logo A is enlarged so that a = 4.5,
(a) find the scale factor
(b) find the lengths of the sides marked b in Diagram 7.
a
Logo A(from Q1)
3 m
1 m
3 m
3 m
3 m
1 m
Making Logo A larger
The director is not sure how large to make Logo A.
Diagram 7: Making Logo A larger a
b
a
b
a
The original and enlarged logos are similar figures.
EnlargedLogo A
Show all working.
Scale factor = . . . . . . . . . . . . . . . . . . . . . . . . .
Show all working.
Length of sides marked b = . . . . . . . . . . . . . . . . . . . . . . . . .
| 11Queensland Studies Authority
12
10. (a) Expand the expression (x + 1)2
The total area of Enlarged Logo A = 2(3x + 3)2 – (x + 1)2
(b) Expand and simplify the expression for the total area of Enlarged Logo A.
(c) Find the total area of Enlarged Logo A, by substituting the director’s value for x into the equation in Question 10 (b). See director’s note #6.
If the side of the small square is increased from 1 m to (x + 1) m, the area of the small square = (x + 1)2 square metres (see Diagram 8).
3 + 3x
3 + 3x
x + 1
3 + 3x
x + 1
3 + 3xEnlargedLogo A
Dimensionsare in metres.
(x + 1)2 =Diagram 8: Enlarged Logo A
The total area of Enlarged Logo A = 2(3x + 3)2 – (x + 1)2
=
Total area = . . . . . . . . . . . . . . . . . . . . . . . . .
Director’s note #6
I’ve decided how large
Logo A should be.
Make = 0.2 metresx
Director’s note #6
I’ve decided how large
Logo A should be.
Make = 0.2 metresx
| QCATs 2012 Student booklet Year 9 Mathematics
11. (a) Clearly show that sides a of the dance floor are 9 metres long (to the nearest metre).
Diagram 9: Triangular dance floor seen from above
Director's note #7
Oops, sorry – another last
minute change.
• We have a new venue and the
dance floor is in the corner of
the hall.
• We will need to change the
arrangement of the lights.
Director's note #7
Oops, sorry – another last
minute change.
• We have a new venue and the
dance floor is in the corner of
the hall.
• We will need to change the
arrangement of the lights.
Dance floor
(Area 40 m )�2
a
a
Show all working.
| 13Queensland Studies Authority
14
Director's note #8
We can adjust the height of the lights.
The higher the lights are, the dimmer the
light on the floor.
Make sure the radius (r) of each circle of
light isor the
light will be too dim.
6 lights should
be enough.
•
•
•
•
no more than 1.8 metres
Director's note #8
We can adjust the height of the lights.
The higher the lights are, the dimmer the
light on the floor.
Make sure the radius (r) of each circle of
light isor the
light will be too dim.
6 lights should
be enough.
•
•
•
•
no more than 1.8 metres
s
r
r
I agree with . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , that . . . . . . . . . . lights are needed.
Show all working.
(b) Who do you agree with?
Justify your answer, including:
• the dimensions of the largest square lit by each light without being too dim
• a diagram of the triangular dance floor, showing the arrangement of squares lit by the lights. See Diagram 5 (page 9).
The director thinks 6 lights will be enough to completely light the triangular dance floor without being too dim.
The stage manager thinks 10 lights are needed.
| QCATs 2012 Student booklet Year 9 Mathematics
(c) At what height (h) should the lights in Question 11 (b) be placed to light the triangular dance floor as brightly as possible?
Justify your answer, including:
• the dimensions of the square lit by each light
• diagram/s to aid your explanations.
To make the stage brighter, the stage manager can adjust the height of the lights.
r
20°h
Show all working.
Height = . . . . . . . . . . . . . . . . . . . . . . . . .
| 15Queensland Studies Authority
Gui
de to
mak
ing
judg
men
ts �
Yea
r 9 M
athe
mat
ics
Stu
dent
nam
e . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
Focu
s: T
o ap
ply
and
just
ify s
trate
gies
to s
olve
pro
blem
s, w
ith fl
uent
use
of m
athe
mat
ical
pro
cedu
res.
Feed
back
: . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
. . .
.
Und
erst
andi
ngSk
ills
Und
erst
andi
ngR
easo
ning
Prob
lem
sol
ving
Flue
ncy
Sel
ects
and
app
lies
mat
hem
atic
al c
once
pts
and
inte
rpre
ts in
form
atio
n to
:
•fin
d di
stan
ces
on a
Car
tesi
an p
lane
(Q
uest
ion
4)•
calc
ulat
e un
know
n di
men
sion
s of
righ
t tri
angl
es a
nd s
imila
r sha
pes.
(Que
stio
ns 6
, 7a,
8, 9
, 11a
)
App
lies
prob
lem
sol
ving
stra
tegi
es to
:
•ca
lcul
ate
the
area
s of
com
posi
te s
hape
s (Q
uest
ions
1, 2
)
•ch
oose
col
ours
that
mee
t the
dire
ctor
's n
eeds
(Que
stio
n 5)
•fin
d th
e di
men
sion
s of
a re
ctan
gle
that
can
be
lit b
y 10
ligh
ts (Q
uest
ion
7b)
•de
term
ine
the
optim
um n
umbe
r, ar
rang
emen
t and
hei
ght o
f lig
hts
for t
he tr
iang
ular
dan
ce fl
oor
and
just
ifies
dec
isio
ns (Q
uest
ion
11b,
c).
Acc
urat
ely
plot
s po
ints
on
a C
arte
sian
pla
ne
(Que
stio
n 3)
Exp
ands
, sim
plifi
es, m
anip
ulat
es a
nd
subs
titut
es in
to a
lgeb
raic
exp
ress
ions
(Q
uest
ions
6, 1
0)M
akes
use
of m
athe
mat
ical
pro
cedu
res,
ro
undi
ng a
nd u
nits
. (Q
uest
ions
2, 4
, 5, 6
,7, 1
0, 1
1)
A B C D E
C
orre
ctly
exp
ands
and
sim
plifi
es th
e ex
pres
sion
for t
he a
rea
of e
xpan
ded
Logo
A
in Q
10b.
M
akes
cle
ar, a
ccur
ate
and
effic
ient
use
of
mat
hem
atic
al p
roce
dure
s.
Con
sist
ently
use
s ac
cura
te a
nd a
ppro
pria
te
roun
ding
and
uni
ts.
A
ccur
atel
y lo
cate
s al
l poi
nts
on a
col
our
grap
h an
d co
rrec
tly e
xpan
ds a
sim
ple
bino
mia
l exp
ress
ion
in Q
10a.
R
ecal
ls a
nd s
ubst
itute
s in
to re
leva
nt
form
ulas
for t
he c
alcu
latio
n of
mos
t are
as.
P
lots
som
e po
ints
on
a co
lour
gra
ph, r
ecal
ls
som
e fo
rmul
as a
nd m
akes
use
of s
impl
e m
athe
mat
ical
pro
cedu
res.
C
orre
ctly
sub
stitu
tes
into
and
man
ipul
ates
al
gebr
aic
expr
essi
ons
in Q
6, 1
0c.
Mak
es s
yste
mat
ic u
se o
f mat
hem
atic
al
proc
edur
es a
nd re
gula
r use
of a
ppro
pria
te
roun
ding
and
uni
ts.
P
artia
lly ju
stifi
es c
hoic
e of
col
ours
.
P
rovi
des
a w
ell-r
easo
ned
just
ifica
tion
of
the
stra
tegy
use
d to
det
erm
ine
the
num
ber o
f lig
hts
for t
he tr
iang
ular
floo
r an
d th
eir o
ptim
um h
eigh
t.
Ju
stifi
es c
hoic
e of
col
ours
that
mat
ch
dire
ctor
's n
ote
#3. J
ustif
ies
the
num
ber
of li
ghts
requ
ired,
usi
ng a
dia
gram
of t
he
trian
gula
r flo
or.
C
orre
ctly
cal
cula
tes
colo
ur d
iffer
ence
, th
e ra
dius
of t
he c
ircle
of l
ight
and
the
dim
ensi
ons
of th
e sq
uare
lit b
y th
at
circ
le. (
Q4,
6, 7
a)Fi
nds
the
new
hei
ght o
f Log
o B
and
cl
early
con
firm
s th
e di
men
sion
s of
the
trian
gula
r flo
or.
U
ses
scal
e fa
ctor
to fi
nd th
e ne
w
dim
ensi
ons
of L
ogo
A.
Mak
es s
igni
fican
t pro
gres
s in
mos
t of t
he
follo
win
g, le
adin
g to
som
e so
lutio
ns:
calc
ulat
ing
colo
ur d
iffer
ence
, fin
ding
the
radi
us o
f a c
ircle
of l
ight
and
cal
cula
ting
the
dim
ensi
ons
of a
squ
are
lit b
y th
at
circ
le. (
Q4,
6, 7
a)
M
akes
som
e pr
ogre
ss in
som
e of
the
follo
win
g: c
alcu
latin
g co
lour
diff
eren
ce,
findi
ng th
e ra
dius
of a
circ
le o
f lig
ht o
r ca
lcul
atin
g th
e di
men
sion
s of
a s
quar
e lit
by
that
circ
le. (
Q4,
6, 7
a)
S
ucce
ssfu
lly a
pplie
s a
stra
tegy
to
dete
rmin
e th
e re
quire
d nu
mbe
r of l
ight
s fo
r the
tria
ngul
ar d
ance
floo
r and
thei
r op
timum
hei
ght.
C
alcu
late
s th
e ar
ea o
f Log
o B
. U
ses
an a
ppro
pria
te-s
ized
squ
are
to
dete
rmin
e th
e nu
mbe
r of l
ight
s re
quire
d to
ligh
t the
tria
ngul
ar d
ance
floo
r.
C
alcu
late
s th
e ar
ea o
f Log
o A
, mak
es
sign
ifica
nt p
rogr
ess
in c
alcu
latin
g th
e ar
ea o
f Log
o B
and
list
s so
me
colo
urs
that
mat
ch d
irect
ors
note
#3.
Fi
nds
the
dim
ensi
ons
of th
e re
ctan
gula
r flo
or a
nd m
akes
som
e pr
ogre
ss in
de
term
inin
g th
e nu
mbe
r of l
ight
s re
quire
d to
ligh
t the
tria
ngul
ar fl
oor.
M
akes
som
e pr
ogre
ss in
cal
cula
ting
the
area
of a
com
posi
te s
hape
and
find
ing
the
dim
ensi
ons
of th
e re
ctan
gula
r dan
ce
floor
.