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10 Mathematical studies SL guide
Syllabus outline
Syllabus
Syllabus component
Teaching hours
SL
All topics are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with the topics listed as prior learning.
Topic 1
Number and algebra
20
Topic 2
Descriptive statistics
12
Topic 3
Logic, sets and probability
20
Topic 4
Statistical applications
17
Topic 5
Geometry and trigonometry
18
Topic 6
Mathematical models
20
Topic 7
Introduction to differential calculus
18
Project
The project is an individual piece of work involving the collection of information or the generation of measurements, and the analysis and evaluation of the information or measurements.
25
Total teaching hours 150
It is essential that teachers are allowed the prescribed minimum number of teaching hours necessary to meet the requirements of the mathematical studies SL course. At SL the minimum prescribed number of hours is 150 hours.
Mathematical studies SL guide 11
Syllabus
Approaches to the teaching and learning of mathematical studies SL
In this course the students will have the opportunity to understand and appreciate both the practical use of mathematics and its aesthetic aspects. They will be encouraged to build on knowledge from prior learning in mathematics and other subjects, as well as their own experience. It is important that students develop mathematical intuition and understand how they can apply mathematics in life.
Teaching needs to be flexible and to allow for different styles of learning. There is a diverse range of students in a mathematical studies SL classroom, and visual, auditory and kinaesthetic approaches to teaching may give new insights. The use of technology, particularly the graphic display calculator (GDC) and computer packages, can be very useful in allowing students to explore ideas in a rich context. It is left to the individual teacher to decide the order in which the separate topics are presented, but teaching and learning activities should weave the parts of the syllabus together and focus on their interrelationships. For example, the connection between geometric sequences and exponential functions can be illustrated by the consideration of compound interest.
Teachers may wish to introduce some topics using hand calculations to give an initial insight into the principles. However, once understanding has been gained, it is envisaged that the use of the GDC will support further workandsimplifycalculation(forexample,theχ2 statistic).
Teachers may take advantage of students’ mathematical intuition by approaching the teaching of probability in a way that does not solely rely on formulae.
The mathematical studies SL project is meant to be not only an assessment tool, but also a sophisticated learning opportunity. It is an independent but well-guided piece of research, using mathematical methods to draw conclusions and answer questions from the individual student’s interests. Project work should be incorporated into the course so that students are given the opportunity to learn the skills needed for the completion of a successful project. It is envisaged that the project will not be undertaken before students have experienced a range of techniques to make it meaningful. The scheme of work should be designed with this in mind.
Teachers should encourage students to find links and applications to their other IB subjects and the core of the hexagon. Everyday problems and questions should be drawn into the lessons to motivate students and keep the material relevant; suggestions are provided in the “Links” column of the syllabus.
For further information on “Approaches to teaching a DP course” please refer to the publication The Diploma Programme: From principles into practice (April 2009). To support teachers, a variety of resources can be found on the OCC and details of workshops for professional development are available on the public website.
Format of the syllabus• Content: this column lists, under each topic, the sub-topics to be covered.
• Further guidance: this column contains more detailed information on specific sub-topics listed in the content column. This clarifies the content for examinations.
Mathematical studies SL guide12
Approaches to the teaching and learning of mathematical studies SL
• Links: this column provides useful links to the aims of the mathematical studies SL course, with suggestions for discussion, real-life examples and project ideas. These suggestions are only a guide for introducing and illustrating the sub-topic and are not exhaustive. Links are labelled as follows.
Appl real-life examples and links to other DP subjects
Aim 8 moral, social and ethical implications of the sub-topic
Int international-mindedness
TOK suggestions for discussion
Note that any syllabus references to other subject guides given in the “Links” column are correct for the current (2012) published versions of the guides.
Course of studyThe content of all seven topics in the syllabus must be taught, although not necessarily in the order in which they appear in this guide. Teachers are expected to construct a course of study that addresses the needs of their students and includes, where necessary, the topics noted in prior learning.
Integration of project workWork leading to the completion of the project must be integrated into the course of study. Details of how to do this are given in the section on internal assessment and in the teacher support material.
Time allocationThe recommended teaching time for standard level courses is 150 hours. For mathematical studies SL, it is expected that 25 hours will be spent on work for the project. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 125 hours allowed for the teaching of the syllabus might be allocated. However, the exact time spent on each topic depends on a number of factors, including the background knowledge and level of preparedness of each student. Teachers should therefore adjust these timings to correspond to the needs of their students.
Time has been allocated in each section of the syllabus to allow for the teaching of topics requiring the use of a GDC.
Use of calculatorsStudents are expected to have access to a GDC at all times during the course. The minimum requirements are reviewed as technology advances, and updated information will be provided to schools. It is expected that teachers and schools monitor calculator use with reference to the calculator policy. Regulations covering the types of calculators allowed in examinations are provided in the Handbook of procedures for the Diploma Programme. Further information and advice is provided in the Mathematical studies SL: Graphic display calculators teacher support material (May 2005) and on the OCC.
Mathematical studies SL guide
Approaches to the teaching and learning of mathematical studies SL
13
Mathematical studies SL formula bookletEach student is required to have access to a clean copy of this booklet during the examination. It is recommended that teachers ensure students are familiar with the contents of this document from the beginning of the course. It is the responsibility of the school to download a copy from IBIS or the OCC, check that there are no printing errors, and ensure that there are sufficient copies available for all students.
Teacher support materialsA variety of teacher support materials will accompany this guide. These materials will include guidance for teachers on the introduction, planning and marking of projects, and specimen examination papers and markschemes.
Command terms and notation listTeachers and students need to be familiar with the IB notation and the command terms, as these will be used without explanation in the examination papers. The “Glossary of command terms” and “Notation list” appear as appendices in this guide.
14 Mathematical studies SL guide
Syllabus
Prior learning topics
As noted in the previous section on prior learning, it is expected that all students have extensive previous mathematical experiences, but these will vary. It is expected that mathematical studies SL students will be familiar with the following topics before they take the examinations, because questions assume knowledge of them. Teachers must therefore ensure that any topics listed here that are unknown to their students at the start of the course are included at an early stage. They should also take into account the existing mathematical knowledge of their students to design an appropriate course of study for mathematical studies SL.
Students must be familiar with SI (Système International) units of length, mass and time, and their derived units.
The reference given in the left-hand column is to the topic in the syllabus content; for example, 1.0 refers to the prior learning for Topic 1—Number and algebra.
Learning how to use the graphic display calculator (GDC) effectively will be an integral part of the course, not a separate topic. Time has been allowed in each topic of the syllabus to do this.
Content Further guidance
1.0 Basic use of the four operations of arithmetic, using integers, decimals and fractions, including order of operations.
Prime numbers, factors and multiples.
Simple applications of ratio, percentage and proportion.
Examples: 2(3 4 7) 62+ × = ; 2 3 4 7 34× + × = .
Basic manipulation of simple algebraic expressions, including factorization and expansion.
Examples: ( );ab ac a b c+ = + 2( 1)( 2) 3 2x x x x+ + = + + .
Rearranging formulae. Example: 1 22
AA bh hb
= ⇒ = .
Evaluating expressions by substitution.
Example: If 3x = − then 2 22 3 ( 3) 2( 3) 3 18x x− + = − − − + = .
Solving linear equations in one variable. Examples: 3( 6) 4( 1) 0x x+ − − = ; 6 4 7
5x+ = .
Solving systems of linear equations in two variables. Example: 3 4 13x y+ = , 1 2 1
3x y− = − .
Evaluating exponential expressions with integer values. Examples: ,ba b∈ ; 4 12
16− = ; 4( 2) 16− = .
Use of inequalities , , ,< ≤ > ≥ . Intervals on the real number line.
Example: 2 5,x x< ≤ ∈ .
Solving linear inequalities. Example: 2 5 7x x+ < − .
Familiarity with commonly accepted world currencies.
Examples: Swiss franc (CHF); United States dollar (USD); British pound sterling (GBP); euro (EUR); Japanese yen (JPY); Australian dollar (AUD).
Mathematical studies SL guide 15
Prior learning topics
Content Further guidance
2.0 The collection of data and its representation in bar charts, pie charts and pictograms.
5.0 Basic geometric concepts: point, line, plane, angle.
Simple two-dimensional shapes and their properties, including perimeters and areas of circles, triangles, quadrilaterals and compound shapes.
SI units for length and area.
Pythagoras’ theorem.
Coordinates in two dimensions.
Midpoints, distance between points.
16 Mathematical studies SL guide
Sylla
bus
cont
ent
Sylla
bus
Topi
c 1—
Num
ber a
nd a
lgeb
ra
20 h
ours
Th
e ai
ms o
f thi
s top
ic a
re to
intro
duce
som
e ba
sic
elem
ents
and
con
cept
s of m
athe
mat
ics,
and
to li
nk th
ese
to fi
nanc
ial a
nd o
ther
app
licat
ions
.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.1
Nat
ural
num
bers
,
; int
eger
s,
; rat
iona
l nu
mbe
rs,
; and
real
num
bers
,
.
Not
req
uire
d:
proo
f of i
rrat
iona
lity,
for e
xam
ple,
of
2.
Link
with
dom
ain
and
rang
e 6.
1.
Int:
His
toric
al d
evel
opm
ent o
f num
ber s
yste
m.
Aw
aren
ess t
hat o
ur m
oder
n nu
mer
als a
re
deve
lope
d fr
om th
e A
rabi
c no
tatio
n.
TOK
: Do
mat
hem
atic
al sy
mbo
ls h
ave
sens
e in
th
e sa
me
way
that
wor
ds h
ave
sens
e? Is
zer
o di
ffer
ent?
Are
thes
e nu
mbe
rs c
reat
ed o
r di
scov
ered
? D
o th
ese
num
bers
exi
st?
1.2
App
roxi
mat
ion:
dec
imal
pla
ces,
sign
ifica
nt
figur
es.
Perc
enta
ge e
rror
s.
Stud
ents
shou
ld b
e aw
are
of th
e er
rors
that
can
re
sult
from
pre
mat
ure
roun
ding
. A
ppl:
Cur
renc
y ap
prox
imat
ions
to n
eare
st
who
le n
umbe
r, eg
pes
o, y
en. C
urre
ncy
appr
oxim
atio
ns to
nea
rest
cen
t/pen
ny, e
g eu
ro,
dolla
r, po
und.
App
l: Ph
ysic
s 1.1
(ran
ge o
f mag
nitu
des)
.
App
l: M
eteo
rolo
gy, a
ltern
ativ
e ro
undi
ng
met
hods
.
App
l: B
iolo
gy 2
.1.5
(mic
rosc
opic
m
easu
rem
ent).
TOK
: App
reci
atio
n of
the
diff
eren
ces o
f sca
le
in n
umbe
r, an
d of
the
way
num
bers
are
use
d th
at a
re w
ell b
eyon
d ou
r eve
ryda
y ex
perie
nce.
Estim
atio
n.
Stud
ents
shou
ld b
e ab
le to
reco
gniz
e w
heth
er
the
resu
lts o
f cal
cula
tions
are
reas
onab
le,
incl
udin
g re
ason
able
val
ues o
f, fo
r exa
mpl
e,
leng
ths,
angl
es a
nd a
reas
.
For e
xam
ple,
leng
ths c
anno
t be
nega
tive.
Mathematical studies SL guide 17
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.3
Expr
essi
ng n
umbe
rs in
the
form
10
ka×
, whe
re
110
a≤
< a
nd k
is a
n in
tege
r. St
uden
ts sh
ould
be
able
to u
se sc
ient
ific
mod
e on
the
GD
C.
App
l: V
ery
larg
e an
d ve
ry sm
all n
umbe
rs, e
g as
trono
mic
al d
ista
nces
, sub
-ato
mic
par
ticle
s;
Phys
ics 1
.1; g
loba
l fin
anci
al fi
gure
s.
App
l: C
hem
istry
1.1
(Avo
gadr
o’s n
umbe
r).
App
l: Ph
ysic
s 1.2
(sci
entif
ic n
otat
ion)
.
App
l: C
hem
istry
and
bio
logy
(sci
entif
ic
nota
tion)
.
App
l: Ea
rth sc
ienc
e (e
arth
quak
e m
easu
rem
ent
scal
e).
Ope
ratio
ns w
ith n
umbe
rs in
this
form
. C
alcu
lato
r not
atio
n is
not
acc
epta
ble.
For e
xam
ple,
5.2
E3 is
not
acc
epta
ble.
1.4
SI (S
ystè
me I
nter
natio
nal)
and
othe
r bas
ic u
nits
of m
easu
rem
ent:
for e
xam
ple,
kilo
gram
(kg)
, m
etre
(m),
seco
nd (s
), lit
re (l
), m
etre
per
seco
nd
(m s–1
), C
elsi
us sc
ale.
Stud
ents
shou
ld b
e ab
le to
con
vert
betw
een
diff
eren
t uni
ts.
Link
with
the
form
of t
he n
otat
ion
in 1
.3, f
or
exam
ple,
6
5km
510
mm
=×
.
App
l: Sp
eed,
acc
eler
atio
n, fo
rce;
Phy
sics
2.1
, Ph
ysic
s 2.2
; con
cent
ratio
n of
a so
lutio
n;
Che
mis
try 1
.5.
Int:
SI n
otat
ion.
TOK
: Doe
s the
use
of S
I not
atio
n he
lp u
s to
thin
k of
mat
hem
atic
s as a
“un
iver
sal
lang
uage
”?
TOK
: Wha
t is m
easu
rabl
e? H
ow c
an o
ne
mea
sure
mat
hem
atic
al a
bilit
y?
1.5
Cur
renc
y co
nver
sion
s. St
uden
ts sh
ould
be
able
to p
erfo
rm c
urre
ncy
trans
actio
ns in
volv
ing
com
mis
sion
. A
ppl:
Econ
omic
s 3.2
(exc
hang
e ra
tes)
.
Aim
8: T
he e
thic
al im
plic
atio
ns o
f tra
ding
in
curr
ency
and
its e
ffec
t on
diff
eren
t nat
iona
l co
mm
uniti
es.
Int:
The
eff
ect o
f flu
ctua
tions
in c
urre
ncy
rate
s on
inte
rnat
iona
l tra
de.
Mathematical studies SL guide18
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.6
Use
of a
GD
C to
solv
e
• pa
irs o
f lin
ear e
quat
ions
in tw
o va
riabl
es
In e
xam
inat
ions
, no
spec
ific
met
hod
of
solu
tion
will
be
requ
ired.
TO
K: E
quat
ions
with
no
solu
tions
. Aw
aren
ess
that
whe
n m
athe
mat
icia
ns ta
lk a
bout
“i
mag
inar
y” o
r “re
al”
solu
tions
they
are
usi
ng
prec
ise
tech
nica
l ter
ms t
hat d
o no
t hav
e th
e sa
me
mea
ning
as t
he e
very
day
term
s. •
quad
ratic
equ
atio
ns.
Stan
dard
term
inol
ogy,
such
as z
eros
or r
oots
, sh
ould
be
taug
ht.
Link
with
qua
drat
ic m
odel
s in
6.3.
1.7
Arit
hmet
ic se
quen
ces a
nd se
ries,
and
thei
r ap
plic
atio
ns.
TO
K: I
nfor
mal
and
form
al re
ason
ing
in
mat
hem
atic
s. H
ow d
oes m
athe
mat
ical
pro
of
diff
er fr
om g
ood
reas
onin
g in
eve
ryda
y lif
e? Is
m
athe
mat
ical
reas
onin
g di
ffer
ent f
rom
sc
ient
ific
reas
onin
g?
TOK
: Bea
uty
and
eleg
ance
in m
athe
mat
ics.
Fibo
nacc
i num
bers
and
con
nect
ions
with
the
Gol
den
ratio
.
Use
of t
he fo
rmul
ae fo
r the
nth
term
and
the
sum
of t
he fi
rst n
term
s of t
he se
quen
ce.
Stud
ents
may
use
a G
DC
for c
alcu
latio
ns, b
ut
they
will
be
expe
cted
to id
entif
y th
e fir
st te
rm
and
the
com
mon
diff
eren
ce.
1.8
Geo
met
ric se
quen
ces a
nd se
ries.
Use
of t
he fo
rmul
ae fo
r the
nth
term
and
the
sum
of t
he fi
rst n
term
s of t
he se
quen
ce.
Not
req
uire
d:
form
al p
roof
s of f
orm
ulae
.
Stud
ents
may
use
a G
DC
for c
alcu
latio
ns, b
ut
they
will
be
expe
cted
to id
entif
y th
e fir
st te
rm
and
the
com
mon
ratio
.
Not
req
uire
d:
use
of lo
garit
hms t
o fin
d n,
giv
en th
e su
m o
f th
e fir
st n
term
s; su
ms t
o in
finity
.
Mathematical studies SL guide 19
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.6
Use
of a
GD
C to
solv
e
• pa
irs o
f lin
ear e
quat
ions
in tw
o va
riabl
es
In e
xam
inat
ions
, no
spec
ific
met
hod
of
solu
tion
will
be
requ
ired.
TO
K: E
quat
ions
with
no
solu
tions
. Aw
aren
ess
that
whe
n m
athe
mat
icia
ns ta
lk a
bout
“i
mag
inar
y” o
r “re
al”
solu
tions
they
are
usi
ng
prec
ise
tech
nica
l ter
ms t
hat d
o no
t hav
e th
e sa
me
mea
ning
as t
he e
very
day
term
s. •
quad
ratic
equ
atio
ns.
Stan
dard
term
inol
ogy,
such
as z
eros
or r
oots
, sh
ould
be
taug
ht.
Link
with
qua
drat
ic m
odel
s in
6.3.
1.7
Arit
hmet
ic se
quen
ces a
nd se
ries,
and
thei
r ap
plic
atio
ns.
TO
K: I
nfor
mal
and
form
al re
ason
ing
in
mat
hem
atic
s. H
ow d
oes m
athe
mat
ical
pro
of
diff
er fr
om g
ood
reas
onin
g in
eve
ryda
y lif
e? Is
m
athe
mat
ical
reas
onin
g di
ffer
ent f
rom
sc
ient
ific
reas
onin
g?
TOK
: Bea
uty
and
eleg
ance
in m
athe
mat
ics.
Fibo
nacc
i num
bers
and
con
nect
ions
with
the
Gol
den
ratio
.
Use
of t
he fo
rmul
ae fo
r the
nth
term
and
the
sum
of t
he fi
rst n
term
s of t
he se
quen
ce.
Stud
ents
may
use
a G
DC
for c
alcu
latio
ns, b
ut
they
will
be
expe
cted
to id
entif
y th
e fir
st te
rm
and
the
com
mon
diff
eren
ce.
1.8
Geo
met
ric se
quen
ces a
nd se
ries.
Use
of t
he fo
rmul
ae fo
r the
nth
term
and
the
sum
of t
he fi
rst n
term
s of t
he se
quen
ce.
Not
req
uire
d:
form
al p
roof
s of f
orm
ulae
.
Stud
ents
may
use
a G
DC
for c
alcu
latio
ns, b
ut
they
will
be
expe
cted
to id
entif
y th
e fir
st te
rm
and
the
com
mon
ratio
.
Not
req
uire
d:
use
of lo
garit
hms t
o fin
d n,
giv
en th
e su
m o
f th
e fir
st n
term
s; su
ms t
o in
finity
.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
1.9
Fina
ncia
l app
licat
ions
of g
eom
etric
sequ
ence
s an
d se
ries:
• co
mpo
und
inte
rest
• an
nual
dep
reci
atio
n.
Not
req
uire
d:
use
of lo
garit
hms.
Use
of t
he G
DC
is e
xpec
ted,
incl
udin
g bu
ilt-in
fina
ncia
l pac
kage
s.
The
conc
ept o
f sim
ple
inte
rest
may
be
used
as
an in
trodu
ctio
n to
com
poun
d in
tere
st b
ut w
ill
not b
e ex
amin
ed.
In e
xam
inat
ions
, que
stio
ns th
at a
sk st
uden
ts to
de
rive
the
form
ula
will
not
be
set.
Com
poun
d in
tere
st c
an b
e ca
lcul
ated
yea
rly,
half-
year
ly, q
uarte
rly o
r mon
thly
.
Link
with
exp
onen
tial m
odel
s 6.4
.
App
l: Ec
onom
ics 3
.2 (e
xcha
nge
rate
s).
Aim
8: E
thic
al p
erce
ptio
ns o
f bor
row
ing
and
lend
ing
mon
ey.
Int:
Do
all s
ocie
ties v
iew
inve
stm
ent a
nd
inte
rest
in th
e sa
me
way
?
Mathematical studies SL guide20
Syllabus content To
pic
2—D
escr
iptiv
e st
atis
tics
12 h
ours
Th
e ai
m o
f thi
s top
ic is
to d
evel
op te
chni
ques
to d
escr
ibe
and
inte
rpre
t set
s of d
ata,
in p
repa
ratio
n fo
r fur
ther
stat
istic
al a
pplic
atio
ns.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
2.1
Cla
ssifi
catio
n of
dat
a as
disc
rete
or c
ontin
uous
. St
uden
ts sh
ould
und
erst
and
the
conc
ept o
f po
pula
tion
and
of re
pres
enta
tive
and
rand
om
sam
plin
g. S
ampl
ing
will
not
be
exam
ined
but
ca
n be
use
d in
inte
rnal
ass
essm
ent.
App
l: Ps
ycho
logy
3 (r
esea
rch
met
hodo
logy
).
App
l: B
iolo
gy 1
(sta
tistic
al a
naly
sis)
.
TOK
: Val
idity
of d
ata
and
intro
duct
ion
of
bias
.
2.2
Sim
ple
disc
rete
dat
a: fr
eque
ncy
tabl
es.
2.3
Gro
uped
dis
cret
e or
con
tinuo
us d
ata:
freq
uenc
y ta
bles
; mid
-inte
rval
val
ues;
upp
er a
nd lo
wer
bo
unda
ries.
Freq
uenc
y hi
stog
ram
s.
In e
xam
inat
ions
, fre
quen
cy h
isto
gram
s will
ha
ve e
qual
cla
ss in
terv
als.
App
l: G
eogr
aphy
(geo
grap
hica
l ana
lyse
s).
2.4
Cum
ulat
ive
freq
uenc
y ta
bles
for g
roup
ed
disc
rete
dat
a an
d fo
r gro
uped
con
tinuo
us d
ata;
cu
mul
ativ
e fr
eque
ncy
curv
es, m
edia
n an
d qu
artil
es.
Box
-and
-whi
sker
dia
gram
.
Not
req
uire
d:
treat
men
t of o
utlie
rs.
Use
of G
DC
to p
rodu
ce h
isto
gram
s and
box
-an
d-w
hisk
er d
iagr
ams.
2.5
Mea
sure
s of c
entra
l ten
denc
y.
For s
impl
e di
scre
te d
ata:
mea
n; m
edia
n; m
ode.
For g
roup
ed d
iscr
ete
and
cont
inuo
us d
ata:
es
timat
e of
a m
ean;
mod
al c
lass
.
Stud
ents
shou
ld u
se m
id-in
terv
al v
alue
s to
estim
ate
the
mea
n of
gro
uped
dat
a.
In e
xam
inat
ions
, que
stio
ns u
sing
∑ n
otat
ion
will
not
be
set.
Aim
8: T
he e
thic
al im
plic
atio
ns o
f usi
ng
stat
istic
s to
mis
lead
.
Mathematical studies SL guide 21
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
2.6
Mea
sure
s of d
ispe
rsio
n: ra
nge,
inte
rqua
rtile
ra
nge,
stan
dard
dev
iatio
n.
Stud
ents
shou
ld u
se m
id-in
terv
al v
alue
s to
estim
ate
the
stan
dard
dev
iatio
n of
gro
uped
da
ta.
In e
xam
inat
ions
:
• st
uden
ts a
re e
xpec
ted
to u
se a
GD
C to
ca
lcul
ate
stan
dard
dev
iatio
ns
• th
e da
ta se
t will
be
treat
ed a
s the
po
pula
tion.
Stud
ents
shou
ld b
e aw
are
that
the
IB n
otat
ion
may
diff
er fr
om th
e no
tatio
n on
thei
r GD
C.
Use
of c
ompu
ter s
prea
dshe
et so
ftwar
e is
enco
urag
ed in
the
treat
men
t of t
his t
opic
.
Int:
The
ben
efits
of s
harin
g an
d an
alys
ing
data
fr
om d
iffer
ent c
ount
ries.
TOK
: Is s
tand
ard
devi
atio
n a
mat
hem
atic
al
disc
over
y or
a c
reat
ion
of th
e hu
man
min
d?
Mathematical studies SL guide22
Syllabus content To
pic
3—Lo
gic,
set
s an
d pr
obab
ility
20
hou
rs
The
aim
s of t
his t
opic
are
to in
trodu
ce th
e pr
inci
ples
of l
ogic
, to
use
set t
heor
y to
intro
duce
pro
babi
lity,
and
to d
eter
min
e th
e lik
elih
ood
of ra
ndom
eve
nts u
sing
a
varie
ty o
f tec
hniq
ues.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
3.1
Bas
ic c
once
pts o
f sym
bolic
logi
c: d
efin
ition
of
a pr
opos
ition
; sym
bolic
not
atio
n of
pr
opos
ition
s.
3.2
Com
poun
d st
atem
ents
: im
plic
atio
n, ⇒
; eq
uiva
lenc
e, ⇔
; neg
atio
n, ¬
; con
junc
tion,
∧
; disj
unct
ion,
∨; e
xclu
sive
disj
unct
ion,
∨.
Tran
slat
ion
betw
een
verb
al st
atem
ents
and
sy
mbo
lic fo
rm.
3.3
Trut
h ta
bles
: con
cept
s of l
ogic
al c
ontra
dict
ion
and
taut
olog
y.
A m
axim
um o
f thr
ee p
ropo
sitio
ns w
ill b
e us
ed
in tr
uth
tabl
es.
Trut
h ta
bles
can
be
used
to il
lust
rate
the
asso
ciat
ive
and
dist
ribut
ive
prop
ertie
s of
conn
ectiv
es, a
nd fo
r var
iatio
ns o
f im
plic
atio
n an
d eq
uiva
lenc
e st
atem
ents
, for
exa
mpl
e,
qp
¬⇒
¬.
3.4
Con
vers
e, in
vers
e, c
ontra
posi
tive.
Logi
cal e
quiv
alen
ce.
A
ppl:
Use
of a
rgum
ents
in d
evel
opin
g a
logi
cal e
ssay
stru
ctur
e.
App
l: C
ompu
ter p
rogr
amm
ing;
dig
ital c
ircui
ts;
Phys
ics H
L 14
.1; P
hysi
cs S
L C
1.
TOK
: Ind
uctiv
e an
d de
duct
ive
logi
c, fa
llaci
es.
Test
ing
the
valid
ity o
f sim
ple
argu
men
ts
thro
ugh
the
use
of tr
uth
tabl
es.
The
topi
c m
ay b
e ex
tend
ed to
incl
ude
syllo
gism
s. In
exa
min
atio
ns th
ese
will
not
be
test
ed.
Mathematical studies SL guide 23
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
3.5
Bas
ic c
once
pts o
f set
theo
ry: e
lem
ents
xA
∈,
subs
ets A
B⊂
; int
erse
ctio
n A
B∩
; uni
on
AB
∪; c
ompl
emen
t A′
.
Ven
n di
agra
ms a
nd si
mpl
e ap
plic
atio
ns.
Not
req
uire
d:
know
ledg
e of
de
Mor
gan’
s law
s.
In e
xam
inat
ions
, the
uni
vers
al se
t U w
ill
incl
ude
no m
ore
than
thre
e su
bset
s.
The
empt
y se
t is d
enot
ed b
y ∅
.
3.6
Sam
ple
spac
e; e
vent
A; c
ompl
emen
tary
eve
nt,
A′.
Prob
abili
ty o
f an
even
t.
Prob
abili
ty o
f a c
ompl
emen
tary
eve
nt.
Expe
cted
val
ue.
Prob
abili
ty m
ay b
e in
trodu
ced
and
taug
ht in
a
prac
tical
way
usi
ng c
oins
, dic
e, p
layi
ng c
ards
an
d ot
her e
xam
ples
to d
emon
strat
e ra
ndom
be
havi
our.
In e
xam
inat
ions
, que
stio
ns in
volv
ing
play
ing
card
s will
not
be
set.
App
l: A
ctua
rial s
tudi
es, p
roba
bilit
y of
life
sp
ans a
nd th
eir e
ffec
t on
insu
ranc
e.
App
l: G
over
nmen
t pla
nnin
g ba
sed
on
proj
ecte
d fig
ures
.
TOK
: The
oret
ical
and
exp
erim
enta
l pr
obab
ility
.
3.7
Prob
abili
ty o
f com
bine
d ev
ents
, mut
ually
ex
clus
ive
even
ts, i
ndep
ende
nt e
vent
s. St
uden
ts sh
ould
be
enco
urag
ed to
use
the
mos
t ap
prop
riate
met
hod
in so
lvin
g in
divi
dual
qu
estio
ns.
App
l: B
iolo
gy 4
.3 (t
heor
etic
al g
enet
ics)
; B
iolo
gy 4
.3.2
(Pun
nett
squa
res)
.
App
l: Ph
ysic
s HL1
3.1
(det
erm
inin
g th
e po
sitio
n of
an
elec
tron)
; Phy
sics
SL
B1.
Aim
8: T
he e
thic
s of g
ambl
ing.
TOK
: The
per
cept
ion
of ri
sk, i
n bu
sine
ss, i
n m
edic
ine
and
safe
ty in
trav
el.
Use
of t
ree
diag
ram
s, V
enn
diag
ram
s, sa
mpl
e sp
ace
diag
ram
s and
tabl
es o
f out
com
es.
Prob
abili
ty u
sing
“w
ith re
plac
emen
t” a
nd
“with
out r
epla
cem
ent”
.
Con
ditio
nal p
roba
bilit
y.
Prob
abili
ty q
uest
ions
will
be
plac
ed in
con
text
an
d w
ill m
ake
use
of d
iagr
amm
atic
re
pres
enta
tions
.
In e
xam
inat
ions
, que
stio
ns re
quiri
ng th
e ex
clus
ive
use
of th
e fo
rmul
a in
sect
ion
3.7
of
the
form
ula
book
let w
ill n
ot b
e se
t.
Mathematical studies SL guide24
Syllabus content To
pic
4—St
atis
tical
app
licat
ions
17
hou
rs
The
aim
s of t
his t
opic
are
to d
evel
op te
chni
ques
in in
fere
ntia
l sta
tistic
s in
orde
r to
anal
yse
sets
of d
ata,
dra
w c
oncl
usio
ns a
nd in
terp
ret t
hese
.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.1
The
norm
al d
istri
butio
n.
The
conc
ept o
f a ra
ndom
var
iabl
e; o
f the
pa
ram
eter
s µ
and
σ; o
f the
bel
l sha
pe; t
he
sym
met
ry a
bout
xµ
=.
Stud
ents
shou
ld b
e aw
are
that
app
roxi
mat
ely
68%
of t
he d
ata
lies b
etw
een µ
σ±
, 95%
lies
be
twee
n 2
µσ
± a
nd 9
9% li
es b
etw
een
3µ
σ±
.
App
l: Ex
ampl
es o
f mea
sure
men
ts, r
angi
ng
from
psy
chol
ogic
al to
phy
sica
l phe
nom
ena,
th
at c
an b
e ap
prox
imat
ed, t
o va
ryin
g de
gree
s, by
the
norm
al d
istri
butio
n.
App
l: B
iolo
gy 1
(sta
tistic
al a
naly
sis)
.
App
l: Ph
ysic
s 3.2
(kin
etic
mol
ecul
ar th
eory
). D
iagr
amm
atic
repr
esen
tatio
n.
Use
of s
ketc
hes o
f nor
mal
cur
ves a
nd sh
adin
g w
hen
usin
g th
e G
DC
is e
xpec
ted.
Nor
mal
pro
babi
lity
calc
ulat
ions
. St
uden
ts w
ill b
e ex
pect
ed to
use
the
GD
C
whe
n ca
lcul
atin
g pr
obab
ilitie
s and
inve
rse
norm
al.
Expe
cted
val
ue.
Inve
rse
norm
al c
alcu
latio
ns.
In e
xam
inat
ions
, inv
erse
nor
mal
que
stio
ns w
ill
not i
nvol
ve fi
ndin
g th
e m
ean
or st
anda
rd
devi
atio
n.
Not
req
uire
d:
Tran
sfor
mat
ion
of a
ny n
orm
al v
aria
ble
to th
e st
anda
rdiz
ed n
orm
al.
Tran
sfor
mat
ion
of a
ny n
orm
al v
aria
ble
to th
e st
anda
rdiz
ed n
orm
al v
aria
ble,
z, m
ay b
e ap
prop
riate
in in
tern
al a
sses
smen
t.
In e
xam
inat
ions
, que
stio
ns re
quiri
ng th
e us
e of
z s
core
s will
not
be
set.
Mathematical studies SL guide 25
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.2
Biv
aria
te d
ata:
the
conc
ept o
f cor
rela
tion.
St
uden
ts sh
ould
be
able
to m
ake
the
dist
inct
ion
betw
een
corr
elat
ion
and
caus
atio
n.
App
l: B
iolo
gy; P
hysi
cs; C
hem
istry
; Soc
ial
scie
nces
.
TOK
: Doe
s cor
rela
tion
impl
y ca
usat
ion?
Sc
atte
r dia
gram
s; li
ne o
f bes
t fit,
by
eye,
pa
ssin
g th
roug
h th
e m
ean
poin
t.
Pear
son’
s pro
duct
–mom
ent c
orre
latio
n co
effic
ient
, r.
Han
d ca
lcul
atio
ns o
f r m
ay e
nhan
ce
unde
rsta
ndin
g.
In e
xam
inat
ions
, stu
dent
s will
be
expe
cted
to
use
a G
DC
to c
alcu
late
r.
Inte
rpre
tatio
n of
pos
itive
, zer
o an
d ne
gativ
e,
stro
ng o
r wea
k co
rrel
atio
ns.
4.3
The
regr
essi
on li
ne fo
r y o
n x.
H
and
calc
ulat
ions
of t
he re
gres
sion
line
may
en
hanc
e un
ders
tand
ing.
In e
xam
inat
ions
, stu
dent
s will
be
expe
cted
to
use
a G
DC
to fi
nd th
e re
gres
sion
line
.
App
l: C
hem
istry
11.
3 (g
raph
ical
tech
niqu
es).
TOK
: Can
we
relia
bly
use
the
equa
tion
of th
e re
gres
sion
line
to m
ake
pred
ictio
ns?
Use
of t
he re
gres
sion
line
for p
redi
ctio
n pu
rpos
es.
Stud
ents
shou
ld b
e aw
are
of th
e da
nger
s of
extra
pola
tion.
Mathematical studies SL guide26
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.4
The
2χ
test
for i
ndep
ende
nce:
form
ulat
ion
of
null
and
alte
rnat
ive
hypo
thes
es; s
igni
fican
ce
leve
ls; c
ontin
genc
y ta
bles
; exp
ecte
d fr
eque
ncie
s; d
egre
es o
f fre
edom
; p-v
alue
s.
In e
xam
inat
ions
:
• th
e m
axim
um n
umbe
r of r
ows o
r col
umns
in
a c
ontin
genc
y ta
ble
will
be
4
• th
e de
gree
s of f
reed
om w
ill a
lway
s be
grea
ter t
han
one
• th
e 2
χ c
ritic
al v
alue
will
alw
ays b
e gi
ven
• on
ly q
uesti
ons o
n up
per t
ail t
ests
with
co
mm
only
use
d si
gnifi
canc
e le
vels
(1%
, 5%
, 10
%) w
ill b
e se
t.
Cal
cula
tion
of e
xpec
ted
freq
uenc
ies b
y ha
nd is
re
quire
d.
Han
d ca
lcul
atio
ns o
f 2
χm
ay e
nhan
ce
unde
rsta
ndin
g.
In e
xam
inat
ions
stud
ents
will
be
expe
cted
to
use
the
GD
C to
cal
cula
te th
e 2
χ st
atis
tic.
If us
ing
2χ
test
s in
inte
rnal
ass
essm
ent,
stud
ents
shou
ld b
e aw
are
of th
e lim
itatio
ns o
f th
e te
st fo
r sm
all e
xpec
ted
freq
uenc
ies;
ex
pect
ed fr
eque
ncie
s mus
t be
grea
ter t
han
5.
If th
e de
gree
of f
reed
om is
1, t
hen
Yat
es’s
co
ntin
uity
cor
rect
ion
shou
ld b
e ap
plie
d.
App
l: B
iolo
gy (i
nter
nal a
sses
smen
t);
Psyc
holo
gy; G
eogr
aphy
.
TOK
: Sci
entif
ic m
etho
d.
Mathematical studies SL guide 27
Syllabus content To
pic
5—G
eom
etry
and
trig
onom
etry
18
hou
rs
The
aim
s of
this
topi
c ar
e to
dev
elop
the
abili
ty to
dra
w c
lear
dia
gram
s in
two
dim
ensi
ons,
and
to a
pply
app
ropr
iate
geo
met
ric a
nd tr
igon
omet
ric te
chni
ques
to
prob
lem
-sol
ving
in tw
o an
d th
ree
dim
ensi
ons.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.1
Equa
tion
of a
line
in tw
o di
men
sion
s: th
e fo
rms
ym
xc
=+
and
0
axby
d+
+=
. Li
nk w
ith li
near
func
tions
in 6
.2.
App
l: G
radi
ents
of m
ount
ain
road
s, eg
C
anad
ian
Hig
hway
. Gra
dien
ts o
f acc
ess r
amps
.
App
l: Ec
onom
ics 1
.2 (e
lasti
city
).
TOK
: Des
carte
s sho
wed
that
geo
met
ric
prob
lem
s can
be
solv
ed a
lgeb
raic
ally
and
vic
e ve
rsa.
Wha
t doe
s thi
s tel
l us a
bout
m
athe
mat
ical
repr
esen
tatio
n an
d m
athe
mat
ical
kn
owle
dge?
Gra
dien
t; in
terc
epts
.
Poin
ts o
f int
erse
ctio
n of
line
s. Li
nk w
ith so
lutio
ns o
f pai
rs o
f lin
ear e
quat
ions
in
1.6
.
Line
s with
gra
dien
ts,
1m a
nd
2m
.
Para
llel l
ines
1
2m
m=
.
Perp
endi
cula
r lin
es,
12
1m
m×
=−
.
5.2
Use
of s
ine,
cos
ine
and
tang
ent r
atio
s to
find
the
side
s and
ang
les o
f rig
ht-a
ngle
d tri
angl
es.
Ang
les o
f ele
vatio
n an
d de
pres
sion
.
Prob
lem
s may
inco
rpor
ate
Pyth
agor
as’
theo
rem
.
In e
xam
inat
ions
, que
stio
ns w
ill o
nly
be se
t in
degr
ees.
App
l: Tr
iang
ulat
ion,
map
-mak
ing,
find
ing
prac
tical
mea
sure
men
ts u
sing
trig
onom
etry
.
Int:
Dia
gram
s of P
ytha
gora
s’ th
eore
m o
ccur
in
early
Chi
nese
and
Indi
an m
anus
crip
ts. T
he
earli
est r
efer
ence
s to
trigo
nom
etry
are
in In
dian
m
athe
mat
ics.
Mathematical studies SL guide28
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.3
Use
of t
he si
ne ru
le:
sin
sin
sin
ab
cA
BC
==
. In
all
area
s of t
his t
opic
, stu
dent
s sho
uld
be
enco
urag
ed to
sket
ch w
ell-l
abel
led
diag
ram
s to
supp
ort t
heir
solu
tions
.
The
ambi
guou
s cas
e co
uld
be ta
ught
, but
will
no
t be
exam
ined
.
In e
xam
inat
ions
, que
stio
ns w
ill o
nly
be se
t in
degr
ees.
App
l: V
ecto
rs; P
hysi
cs 1
.3; b
earin
gs.
Use
of t
he c
osin
e ru
le2
22
2co
sa
bc
bcA
=+
−;
22
2
cos
2b
ca
Abc
+−
=.
Use
of a
rea
of a
tria
ngle
= 1
sin
2ab
C.
Con
stru
ctio
n of
labe
lled
diag
ram
s fro
m v
erba
l st
atem
ents
.
TO
K: U
se th
e fa
ct th
at th
e co
sine
rule
is o
ne
poss
ible
gen
eral
izat
ion
of P
ytha
gora
s’ th
eore
m
to e
xplo
re th
e co
ncep
t of “
gene
ralit
y”.
Mathematical studies SL guide 29
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.4
Geo
met
ry o
f thr
ee-d
imen
sion
al so
lids:
cub
oid;
rig
ht p
rism
; rig
ht p
yram
id; r
ight
con
e; c
ylin
der;
sphe
re; h
emis
pher
e; a
nd c
ombi
natio
ns o
f the
se
solid
s.
The
dist
ance
bet
wee
n tw
o po
ints
; eg
betw
een
two
verti
ces o
r ver
tices
with
mid
poin
ts o
r m
idpo
ints
with
mid
poin
ts.
The
size
of a
n an
gle
betw
een
two
lines
or
betw
een
a lin
e an
d a
plan
e.
Not
req
uire
d:
angl
e be
twee
n tw
o pl
anes
.
In e
xam
inat
ions
, onl
y rig
ht-a
ngle
d tri
gono
met
ry q
uest
ions
will
be
set i
n re
fere
nce
to th
ree-
dim
ensi
onal
shap
es.
TOK
: Wha
t is a
n ax
iom
atic
syst
em?
Do
the
angl
es in
a tr
iang
le a
lway
s add
to 1
80°?
Non
-Euc
lidea
n ge
omet
ry, s
uch
as R
iem
ann’
s. Fl
ight
map
s of a
irlin
es.
App
l: A
rchi
tect
ure
and
desi
gn.
5.5
Vol
ume
and
surf
ace
area
s of t
he th
ree-
dim
ensi
onal
solid
s def
ined
in 5
.4.
Mathematical studies SL guide30
Syllabus content To
pic
6—M
athe
mat
ical
mod
els
20 h
ours
Th
e ai
m o
f thi
s top
ic is
to d
evel
op u
nder
stan
ding
of s
ome
mat
hem
atic
al fu
nctio
ns th
at c
an b
e us
ed to
mod
el p
ract
ical
situ
atio
ns. E
xten
sive
use
of a
GD
C is
to b
e en
cour
aged
in th
is to
pic.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.1
Con
cept
of a
func
tion,
dom
ain,
rang
e an
d gr
aph.
Func
tion
nota
tion,
eg
(),
(
), (
)f
xv
tC
n.
Con
cept
of a
func
tion
as a
mat
hem
atic
al
mod
el.
In e
xam
inat
ions
:
• th
e do
mai
n is
the
set o
f all
real
num
bers
un
less
oth
erw
ise
stat
ed
• m
appi
ng n
otat
ion
:fx
y
will
not
be
used
.
TO
K: W
hy c
an w
e us
e m
athe
mat
ics t
o de
scrib
e th
e w
orld
and
mak
e pr
edic
tions
? Is
it
beca
use
we
disc
over
the
mat
hem
atic
al b
asis
of
the
wor
ld o
r bec
ause
we
impo
se o
ur o
wn
mat
hem
atic
al st
ruct
ures
ont
o th
e w
orld
?
The
rela
tions
hip
betw
een
real
-wor
ld p
robl
ems
and
mat
hem
atic
al m
odel
s.
6.2
Line
ar m
odel
s.
Line
ar fu
nctio
ns a
nd th
eir g
raph
s, (
)f
xm
xc
=+
.
Link
with
equ
atio
n of
a li
ne in
5.1
. A
ppl:
Con
vers
ion
grap
hs, e
g te
mpe
ratu
re o
r cu
rren
cy c
onve
rsio
n; P
hysi
cs 3
.1; E
cono
mic
s 3.
2.
6.3
Qua
drat
ic m
odel
s.
Qua
drat
ic fu
nctio
ns a
nd th
eir g
raph
s (p
arab
olas
): 2
()
fx
axbx
c=
++
;0
≠a
Link
with
the
quad
ratic
equ
atio
ns in
1.6
. Fu
nctio
ns w
ith z
ero,
one
or t
wo
real
root
s are
in
clud
ed.
App
l: C
ost f
unct
ions
; pro
ject
ile m
otio
n;
Phys
ics 9
.1; a
rea
func
tions
.
Prop
ertie
s of a
par
abol
a: sy
mm
etry
; ver
tex;
in
terc
epts
on
the
x-ax
is a
nd y
-axi
s.
Equa
tion
of th
e ax
is o
f sym
met
ry,
2bx
a=−
.
The
form
of t
he e
quat
ion
of th
e ax
is of
sy
mm
etry
may
initi
ally
be
foun
d by
in
vest
igat
ion.
Prop
ertie
s sho
uld
be il
lust
rate
d w
ith a
GD
C o
r gr
aphi
cal s
oftw
are.
Mathematical studies SL guide 31
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.4
Expo
nent
ial m
odel
s.
Expo
nent
ial f
unct
ions
and
thei
r gra
phs:
(
);
,1,
0+
=+
∈≠
≠
xf
xka
ca
ak
.
()
;,
1,0
−+
=+
∈≠
≠
xf
xka
ca
ak
.
Con
cept
and
equ
atio
n of
a h
oriz
onta
l as
ympt
ote.
In e
xam
inat
ions
, stu
dent
s will
be
expe
cted
to
use
grap
hica
l met
hods
, inc
ludi
ng G
DC
s, to
so
lve
prob
lem
s.
App
l: B
iolo
gy 5
.3 (p
opul
atio
ns).
App
l: B
iolo
gy 5
.3.2
(pop
ulat
ion
grow
th);
Phys
ics 1
3.2
(rad
ioac
tive
deca
y); P
hysi
cs I2
(X
-ray
atte
nuat
ion)
; coo
ling
of a
liqu
id; s
prea
d of
a v
irus;
dep
reci
atio
n.
6.5
Mod
els u
sing
func
tions
of t
he fo
rm
()
...;
,=
++
∈
mn
fx
axbx
mn
. In
exa
min
atio
ns, s
tude
nts w
ill b
e ex
pect
ed to
us
e gr
aphi
cal m
etho
ds, i
nclu
ding
GD
Cs,
to
solv
e pr
oble
ms.
Func
tions
of t
his t
ype
and
thei
r gra
phs.
The
y-ax
is a
s a v
ertic
al a
sym
ptot
e.
Exam
ples
: 4
()
35
3f
xx
x=
−+
, 2
4(
)3
gx
xx
=−
.
6.6
Dra
win
g ac
cura
te g
raph
s.
Cre
atin
g a
sket
ch fr
om in
form
atio
n gi
ven.
Tran
sfer
ring
a gr
aph
from
GD
C to
pap
er.
Rea
ding
, int
erpr
etin
g an
d m
akin
g pr
edic
tions
us
ing
grap
hs.
Stud
ents
shou
ld b
e aw
are
of th
e di
ffer
ence
be
twee
n th
e co
mm
and
term
s “dr
aw”
and
“ske
tch”
.
All
grap
hs sh
ould
be
labe
lled
and
have
som
e in
dica
tion
of sc
ale.
TO
K: D
oes a
gra
ph w
ithou
t lab
els o
r in
dica
tion
of sc
ale
have
mea
ning
?
Incl
uded
all
the
func
tions
abo
ve a
nd a
dditi
ons
and
subt
ract
ions
. Ex
ampl
es:
32
()
5f
xx
x=
+−
, (
)3
xg
xx
−=
+.
Mathematical studies SL guide32
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.7
Use
of a
GD
C to
solv
e eq
uatio
ns in
volv
ing
com
bina
tions
of t
he fu
nctio
ns a
bove
. Ex
ampl
es:
32
23
1x
xx
+=
+−
, 53x
x=
.
Oth
er fu
nctio
ns c
an b
e us
ed fo
r mod
ellin
g in
in
tern
al a
sses
smen
t but
will
not
be
set o
n ex
amin
atio
n pa
pers
.
Mathematical studies SL guide 33
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.7
Use
of a
GD
C to
solv
e eq
uatio
ns in
volv
ing
com
bina
tions
of t
he fu
nctio
ns a
bove
. Ex
ampl
es:
32
23
1x
xx
+=
+−
, 53x
x=
.
Oth
er fu
nctio
ns c
an b
e us
ed fo
r mod
ellin
g in
in
tern
al a
sses
smen
t but
will
not
be
set o
n ex
amin
atio
n pa
pers
.
Topi
c 7—
Intr
oduc
tion
to d
iffer
entia
l cal
culu
s 18
hou
rs
The
aim
of t
his t
opic
is to
intro
duce
the
conc
ept o
f the
der
ivat
ive
of a
func
tion
and
to a
pply
it to
opt
imiz
atio
n an
d ot
her p
robl
ems.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
7.1
Con
cept
of t
he d
eriv
ativ
e as
a ra
te o
f cha
nge.
Tang
ent t
o a
curv
e.
Not
req
uire
d:
form
al tr
eatm
ent o
f lim
its.
Teac
hers
are
enc
oura
ged
to in
trodu
ce
diff
eren
tiatio
n th
roug
h a
grap
hica
l app
roac
h,
rath
er th
an a
form
al tr
eatm
ent.
Emph
asis
is p
lace
d on
inte
rpre
tatio
n of
the
conc
ept i
n di
ffer
ent c
onte
xts.
In e
xam
inat
ions
, que
stio
ns o
n di
ffer
entia
tion
from
firs
t prin
cipl
es w
ill n
ot b
e se
t.
App
l: R
ates
of c
hang
e in
eco
nom
ics,
kine
mat
ics a
nd m
edic
ine.
Aim
8: P
lagi
aris
m a
nd a
ckno
wle
dgm
ent o
f so
urce
s, eg
the
conf
lict b
etw
een
New
ton
and
Leib
nitz
, who
app
roac
hed
the
deve
lopm
ent o
f ca
lcul
us fr
om d
iffer
ent d
irect
ions
TOK
: Is i
ntui
tion
a va
lid w
ay o
f kno
win
g in
m
aths
?
How
is it
pos
sibl
e to
reac
h th
e sa
me
conc
lusi
on
from
diff
eren
t res
earc
h pa
ths?
7.2
The
prin
cipl
e th
at
1(
)(
)n
nf
xax
fx
anx
−′
=⇒
=.
The
deriv
ativ
e of
func
tions
of t
he fo
rm
1(
)...
,−
=+
+n
nf
xax
bxw
here
all
expo
nent
s are
in
tege
rs.
Stud
ents
shou
ld b
e fa
mili
ar w
ith th
e al
tern
ativ
e
nota
tion
for d
eriv
ativ
es d dy x
or
d dV r.
In e
xam
inat
ions
, kno
wle
dge
of th
e se
cond
de
rivat
ive
will
not
be
assu
med
.
Mathematical studies SL guide34
Syllabus content
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
7.3
Gra
dien
ts o
f cur
ves f
or g
iven
val
ues o
f x.
Val
ues o
f x w
here
(
)f
x′
is g
iven
.
The
use
of te
chno
logy
to fi
nd th
e gr
adie
nt a
t a
poin
t is a
lso
enco
urag
ed.
Equa
tion
of th
e ta
ngen
t at a
giv
en p
oint
. Th
e us
e of
tech
nolo
gy to
dra
w ta
ngen
t and
no
rmal
line
s is a
lso
enco
urag
ed.
Equa
tion
of th
e lin
e pe
rpen
dicu
lar t
o th
e ta
ngen
t at a
giv
en p
oint
(nor
mal
). Li
nks w
ith p
erpe
ndic
ular
line
s in
5.1.
7.4
Incr
easi
ng a
nd d
ecre
asin
g fu
nctio
ns.
Gra
phic
al in
terp
reta
tion
of
()
0f
x′
>,
()
0f
x′
=
and
()
0f
x′
<.
7.5
Val
ues o
f x w
here
the
grad
ient
of a
cur
ve is
ze
ro.
Solu
tion
of
()
0f
x′
=.
Stat
iona
ry p
oint
s.
The
use
of te
chno
logy
to d
ispl
ay
()
fx
and
(
)f
x′
, and
find
the
solu
tions
of
()
0f
x′
= is
al
so e
ncou
rage
d.
Loca
l max
imum
and
min
imum
poi
nts.
Aw
aren
ess t
hat a
loca
l max
imum
/min
imum
w
ill n
ot n
eces
saril
y be
the
grea
test
/leas
t val
ue
of th
e fu
nctio
n in
the
give
n do
mai
n.
Aw
aren
ess o
f poi
nts o
f inf
lexi
on w
ith z
ero
grad
ient
is to
be
enco
urag
ed, b
ut w
ill n
ot b
e ex
amin
ed.
7.6
Opt
imiz
atio
n pr
oble
ms.
Exam
ples
: Max
imiz
ing
prof
it, m
inim
izin
g co
st,
max
imiz
ing
volu
me
for g
iven
surf
ace
area
.
In e
xam
inat
ions
, que
stio
ns o
n ki
nem
atic
s will
no
t be
set.
App
l: Ef
ficie
nt u
se o
f mat
eria
l in
pack
agin
g.
App
l: Ph
ysic
s 2.1
(kin
emat
ics)
.