Year 9 Mathematics QCAT 2012 student booklet | Behind the ... · PDF file4 | QCATs 2012 Student booklet Year 9 Mathematics Projecting and painting logos on the backdrop 1. Calculate

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: [email protected] 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 = . . . . . . . . . . . . . . . . . . . . . . . . .


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 = . . . . . . . . . . . . . . . . . . . . . . . . .


6

Choosing colours for the dance competition logos

3. Plot and label the colours in Table 1 on the cyan/magenta colour graph below.


Use cyan and magenta tints

to mix colours for the logos.•


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


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).


Use two colours:

slate blue

another colour from Table 1 with a colour

difference of more than 25 units from slate blue.

•

•


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.


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.


As a special effect I want the

rectangular dance floor lit from

above.

Use the 10 lights from the

storeroom.

•

•


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 = . . . . . . . . . . . . . . . . . . . . . . . . .


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) = . . . . . . . . . . . . . . . . . . . . . . . . .


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.


Sorry, a last minute change.

The logos on the backdrop

need to be larger.


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 = . . . . . . . . . . . . . . . . . . . . . . . . .


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 = . . . . . . . . . . . . . . . . . . . . . . . . .


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 = . . . . . . . . . . . . . . . . . . . . . . . . .


I’ve decided how large

Logo A should be.

Make = 0.2 metresx


I’ve decided how large

Logo A should be.

Make = 0.2 metresx


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.


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.


(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 = . . . . . . . . . . . . . . . . . . . . . . . . .


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

.

Documents

Year 9 Mathematics QCAT 2012 student booklet | Behind the ... · PDF file4 | QCATs 2012 Student booklet Year 9 Mathematics Projecting and painting logos on the backdrop 1. Calculate