Date post: | 22-Dec-2015 |
Category: |
Documents |
Upload: | frankyp1234 |
View: | 800 times |
Download: | 26 times |
Mathematics SL guideFirst examinations 2014
Diploma Programme
10 Mathematics SL guide
Syllabus outline
Syllabus
Syllabus componentTeaching 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
Algebra
9
Topic 2
Functions and equations
24
Topic 3
Circular functions and trigonometry
16
Topic 4
Vectors
16
Topic 5
Statistics and probability
35
Topic 6
Calculus
40
Mathematical exploration
Internal assessment in mathematics SL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics.
10
Total teaching hours 150
Mathem
atics SL guide17
Syllabus
Syllabus contentSyllabus content
Mathematics SL guide 1
Topic 1—Algebra 9 hours The aim of this topic is to introduce students to some basic algebraic concepts and applications.
Content Further guidance Links
1.1 Arithmetic sequences and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series.
Sigma notation.
Technology may be used to generate and display sequences in several ways.
Link to 2.6, exponential functions.
Int: The chess legend (Sissa ibn Dahir).
Int: Aryabhatta is sometimes considered the “father of algebra”. Compare with al-Khawarizmi.
TOK: How did Gauss add up integers from 1 to 100? Discuss the idea of mathematical intuition as the basis for formal proof.
TOK: Debate over the validity of the notion of “infinity”: finitists such as L. Kronecker consider that “a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps”.
TOK: What is Zeno’s dichotomy paradox? How far can mathematical facts be from intuition?
Applications. Examples include compound interest and population growth.
Mathem
atics SL guide18 Syllabus content
Syllabus content
Mathematics SL guide 2
Content Further guidance Links
1.2 Elementary treatment of exponents and logarithms. Examples:
3416 8= ; 16
3 log 84= ;
log32 5log 2= ; 43 12(2 ) 2− −= .
Appl: Chemistry 18.1 (Calculation of pH ).
TOK: Are logarithms an invention or discovery? (This topic is an opportunity for teachers to generate reflection on “the nature of mathematics”.) Laws of exponents; laws of logarithms.
Change of base. Examples: 4
ln 7log
ln 47 = ,
255
5
loglog
log125 312525 2
= = .
Link to 2.6, logarithmic functions.
1.3 The binomial theorem:
expansion of ( ) ,na b n+ ∈` .
Counting principles may be used in the development of the theorem.
Aim 8: Pascal’s triangle. Attributing the origin of a mathematical discovery to the wrong mathematician.
Int: The so-called “Pascal’s triangle” was known in China much earlier than Pascal.
Calculation of binomial coefficients using
Pascal’s triangle andnr
.
nr
should be found using both the formula
and technology.
Example: finding 6r
from inputting
6n ry C X= and then reading coefficients from the table.
Link to 5.8, binomial distribution. Not required: formal treatment of permutations and formula for n rP .
Mathem
atics SL guide19
Syllabus contentSyllabus content
Mathematics SL guide 3
Topic 2—Functions and equations 24 hours The aims of this topic are to explore the notion of a function as a unifying theme in mathematics, and to apply functional methods to a variety of mathematical situations. It is expected that extensive use will be made of technology in both the development and the application of this topic, rather than elaborate analytical techniques. On examination papers, questions may be set requiring the graphing of functions that do not explicitly appear on the syllabus, and students may need to choose the appropriate viewing window. For those functions explicitly mentioned, questions may also be set on composition of these functions with the linear function y ax b= + .
Content Further guidance Links
2.1 Concept of function : ( )f x f x6 .
Domain, range; image (value).
Example: for 2x x−6 , domain is 2x ≤ , range is 0y ≥ .
A graph is helpful in visualizing the range.
Int: The development of functions, Rene Descartes (France), Gottfried Wilhelm Leibniz (Germany) and Leonhard Euler (Switzerland).
Composite functions. ( )( ) ( ( ))f g x f g x=D . TOK: Is zero the same as “nothing”?
TOK: Is mathematics a formal language? Identity function. Inverse function 1f − . 1 1( )( ) ( )( )f f x f f x x− −= =D D .
On examination papers, students will only be asked to find the inverse of a one-to-one function.
Not required: domain restriction.
2.2 The graph of a function; its equation ( )y f x= . Appl: Chemistry 11.3.1 (sketching and interpreting graphs); geographic skills.
TOK: How accurate is a visual representation of a mathematical concept? (Limits of graphs in delivering information about functions and phenomena in general, relevance of modes of representation.)
Function graphing skills.
Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry, and consideration of domain and range.
Note the difference in the command terms “draw” and “sketch”.
Use of technology to graph a variety of functions, including ones not specifically mentioned.
An analytic approach is also expected for simple functions, including all those listed under topic 2.
The graph of 1 ( )y f x−= as the reflection in the line y x= of the graph of ( )y f x= .
Link to 6.3, local maximum and minimum points.
Mathem
atics SL guide20 Syllabus content
Syllabus content
Mathematics SL guide 4
Content Further guidance Links
2.3 Transformations of graphs. Technology should be used to investigate these transformations.
Appl: Economics 1.1 (shifting of supply and demand curves).
Translations: ( )y f x b= + ; ( )y f x a= − .
Reflections (in both axes): ( )y f x= − ; ( )y f x= − .
Vertical stretch with scale factor p: ( )y pf x= .
Stretch in the x-direction with scale factor 1q
:
( )y f qx= .
Translation by the vector 32−
denotes
horizontal shift of 3 units to the right, and vertical shift of 2 down.
Composite transformations. Example: 2y x= used to obtain 23 2y x= + by a stretch of scale factor 3 in the y-direction
followed by a translation of 02
.
2.4 The quadratic function 2x ax bx c+ +6 : its graph, y-intercept (0, )c . Axis of symmetry.
The form ( )( )x a x p x q− −6 , x-intercepts ( , 0)p and ( , 0)q .
The form 2( )x a x h k− +6 , vertex ( , )h k .
Candidates are expected to be able to change from one form to another.
Links to 2.3, transformations; 2.7, quadratic equations.
Appl: Chemistry 17.2 (equilibrium law).
Appl: Physics 2.1 (kinematics).
Appl: Physics 4.2 (simple harmonic motion).
Appl: Physics 9.1 (HL only) (projectile motion).
Mathem
atics SL guide21
Syllabus contentSyllabus content
Mathematics SL guide 5
Content Further guidance Links
2.5 The reciprocal function 1x
x6 , 0x ≠ : its
graph and self-inverse nature.
The rational function ax bx
cx d+
+6 and its
graph.
Examples: 4 2( ) , 3 2 3
h x xx
= ≠−
;
7 5, 2 5 2xy xx+
= ≠−
.
Vertical and horizontal asymptotes. Diagrams should include all asymptotes and intercepts.
2.6 Exponential functions and their graphs: xx a6 , 0a > , exx6 .
Int: The Babylonian method of multiplication: 2 2 2( )2
a b a bab + − −= . Sulba Sutras in ancient
India and the Bakhshali Manuscript contained an algebraic formula for solving quadratic equations.
Logarithmic functions and their graphs: logax x6 , 0x > , lnx x6 , 0x > .
Relationships between these functions: lnex x aa = ; log x
a a x= ; loga xa x= , 0x > .
Links to 1.1, geometric sequences; 1.2, laws of exponents and logarithms; 2.1, inverse functions; 2.2, graphs of inverses; and 6.1, limits.
Mathem
atics SL guide22 Syllabus content
Syllabus content
Mathematics SL guide 6
Content Further guidance Links
2.7 Solving equations, both graphically and
analytically.
Use of technology to solve a variety of
equations, including those where there is no
appropriate analytic approach.
Solutions may be referred to as roots of
equations or zeros of functions.
Links to 2.2, function graphing skills; and 2.3–
2.6, equations involving specific functions.
Examples: 45 6 0e sin ,
x x xx + − == .
Solving 2
0ax bx c+ + = , 0a ≠ .
The quadratic formula.
The discriminant 2
4b ac∆ = − and the nature
of the roots, that is, two distinct real roots, two
equal real roots, no real roots.
Example: Find k given that the equation 2
3 2 0kx x k+ + = has two equal real roots.
Solving exponential equations.
Examples: 12 10x− = ,
119
3
xx+= .
Link to 1.2, exponents and logarithms.
2.8 Applications of graphing skills and solving
equations that relate to real-life situations.
Link to 1.1, geometric series. Appl: Compound interest, growth and decay;
projectile motion; braking distance; electrical
circuits.
Appl: Physics 7.2.7–7.2.9, 13.2.5, 13.2.6,
13.2.8 (radioactive decay and half-life)
Mathem
atics SL guide23
Syllabus contentSyllabus content
Mathematics SL guide 7
Topic 3—Circular functions and trigonometry 16 hours The aims of this topic are to explore the circular functions and to solve problems using trigonometry. On examination papers, radian measure should be assumed unless otherwise indicated.
Content Further guidance Links
3.1 The circle: radian measure of angles; length of an arc; area of a sector.
Radian measure may be expressed as exact multiples of π , or decimals.
Int: Seki Takakazu calculating π to ten decimal places.
Int: Hipparchus, Menelaus and Ptolemy.
Int: Why are there 360 degrees in a complete turn? Links to Babylonian mathematics.
TOK: Which is a better measure of angle: radian or degree? What are the “best” criteria by which to decide?
TOK: Euclid’s axioms as the building blocks of Euclidean geometry. Link to non-Euclidean geometry.
3.2 Definition of cosθ and sinθ in terms of the unit circle.
Aim 8: Who really invented “Pythagoras’ theorem”?
Int: The first work to refer explicitly to the sine as a function of an angle is the Aryabhatiya of Aryabhata (ca. 510).
TOK: Trigonometry was developed by successive civilizations and cultures. How is mathematical knowledge considered from a sociocultural perspective?
Definition of tanθ as sincos
θθ
. The equation of a straight line through the origin is tany x θ= .
Exact values of trigonometric ratios of π π π π0, , , ,6 4 3 2
and their multiples.
Examples:
π 3 3π 1 3sin , cos , tan 2103 2 4 32= = − ° = .
Mathem
atics SL guide24 Syllabus content
Syllabus content
Mathematics SL guide 8
Content Further guidance Links
3.3 The Pythagorean identity 2 2cos sin 1θ θ+ = .
Double angle identities for sine and cosine.
Simple geometrical diagrams and/or technology may be used to illustrate the double angle formulae (and other trigonometric identities).
Relationship between trigonometric ratios. Examples:
Given sinθ , finding possible values of tanθ without finding θ .
Given 3cos4
x = , and x is acute, find sin 2x
without finding x.
3.4 The circular functions sin x , cos x and tan x : their domains and ranges; amplitude, their periodic nature; and their graphs.
Appl: Physics 4.2 (simple harmonic motion).
Composite functions of the form ( )( ) sin ( )f x a b x c d= + + .
Examples:
( ) tan4
f x x π= − , ( )( ) 2cos 3( 4) 1f x x= − + .
Transformations. Example: siny x= used to obtain 3sin 2y x= by a stretch of scale factor 3 in the y-direction
and a stretch of scale factor 12
in the
x-direction.
Link to 2.3, transformation of graphs.
Applications. Examples include height of tide, motion of a Ferris wheel.
Mathem
atics SL guide25
Syllabus contentSyllabus content
Mathematics SL guide 9
Content Further guidance Links
3.5 Solving trigonometric equations in a finite interval, both graphically and analytically.
Examples: 2sin 1x = , 0 2πx≤ ≤ ,
2sin 2 3cosx x= , o o0 180x≤ ≤ ,
( )2 tan 3( 4) 1x − = , π 3πx− ≤ ≤ .
Equations leading to quadratic equations in sin , cos or tanx x x .
Not required: the general solution of trigonometric equations.
Examples: 22sin 5cos 1 0x x+ + = for 0 4x≤ < π ,
2sin cos2x x= , π πx− ≤ ≤ .
3.6 Solution of triangles. Pythagoras’ theorem is a special case of the cosine rule.
Aim 8: Attributing the origin of a mathematical discovery to the wrong mathematician.
Int: Cosine rule: Al-Kashi and Pythagoras. The cosine rule.
The sine rule, including the ambiguous case.
Area of a triangle, 1sin
2ab C .
Link with 4.2, scalar product, noting that: 2 2 2 2= − ⇒ = + − ⋅c a b c a b a b .
Applications. Examples include navigation, problems in two and three dimensions, including angles of elevation and depression.
TOK: Non-Euclidean geometry: angle sum on a globe greater than 180°.
Mathematics SL guide26
Syllabus contentSy
llabu
s con
tent
Mat
hem
atic
s SL
guid
e 10
Topi
c 4—
Vect
ors
16 h
ours
The
aim
of t
his
topi
c is
to p
rovi
de a
n el
emen
tary
intro
duct
ion
to v
ecto
rs, i
nclu
ding
bot
h al
gebr
aic
and
geom
etric
app
roac
hes.
The
use
of d
ynam
ic g
eom
etry
so
ftwar
e is
ext
rem
ely
help
ful t
o vi
sual
ize
situ
atio
ns in
thre
e di
men
sion
s.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.1
Vec
tors
as d
ispl
acem
ents
in th
e pl
ane
and
in
thre
e di
men
sion
s. Li
nk to
thre
e-di
men
sion
al g
eom
etry
, x, y
and
z-ax
es.
Appl:
Phys
ics 1
.3.2
(vec
tor s
ums a
nd
diff
eren
ces)
Phy
sics
2.2
.2, 2
.2.3
(vec
tor
resu
ltant
s).
TOK: H
ow d
o w
e re
late
a th
eory
to th
e au
thor
? W
ho d
evel
oped
vec
tor a
naly
sis:
JW
Gib
bs o
r O H
eavi
side
?
Com
pone
nts o
f a v
ecto
r; co
lum
n
repr
esen
tatio
n;
1 21
23
3v vv
vv
v=
=+
+v
ij
k.
Com
pone
nts a
re w
ith re
spec
t to
the
unit
vect
ors i
, j a
nd k
(sta
ndar
d ba
sis)
.
Alg
ebra
ic a
nd g
eom
etric
app
roac
hes t
o th
e fo
llow
ing:
A
pplic
atio
ns to
sim
ple
geom
etric
figu
res a
re
esse
ntia
l.
• th
e su
m a
nd d
iffer
ence
of t
wo
vect
ors;
the
zero
vec
tor,
the
vect
or −v;
Th
e di
ffer
ence
of v
and w
is
()
−=
+−
vw
vw
. Vec
tor s
ums a
nd d
iffer
ence
s ca
n be
repr
esen
ted
by th
e di
agon
als o
f a
para
llelo
gram
.
• m
ultip
licat
ion
by a
scal
ar,
kv; p
aral
lel
vect
ors;
Mul
tiplic
atio
n by
a sc
alar
can
be
illus
trate
d by
en
larg
emen
t.
• m
agni
tude
of a
vec
tor, v
;
• un
it ve
ctor
s; b
ase
vect
ors;
i, j
and k;
• po
sitio
n ve
ctor
s O
A→
=a
;
• A
BO
BO
A→
→→
=−
=−ba
. D
ista
nce
betw
een
poin
ts A
and
B is
the
mag
nitu
de o
f A
B→
.
Mathematics SL guide 27
Syllabus contentSy
llabu
s con
tent
Mat
hem
atic
s SL
guid
e 11
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
4.2
The
scal
ar p
rodu
ct o
f tw
o ve
ctor
s. Th
e sc
alar
pro
duct
is a
lso
know
n as
the
“dot
pr
oduc
t”.
Link
to 3
.6, c
osin
e ru
le.
Perp
endi
cula
r vec
tors
; par
alle
l vec
tors
. Fo
r non
-zer
o ve
ctor
s, 0
⋅=
vw
is e
quiv
alen
t to
the
vect
ors b
eing
per
pend
icul
ar.
For p
aral
lel v
ecto
rs,
k=
wv
, ⋅
=vw
vw
.
The
angl
e be
twee
n tw
o ve
ctor
s.
4.3
Vec
tor e
quat
ion
of a
line
in tw
o an
d th
ree
dim
ensi
ons:
t
=+
ra
b.
Rel
evan
ce o
f a
(pos
ition
) and
b (d
irect
ion)
.
Inte
rpre
tatio
n of
t a
s tim
e an
d b
as v
eloc
ity,
with
b re
pres
entin
g sp
eed.
Aim
8: V
ecto
r the
ory
is u
sed
for t
rack
ing
disp
lace
men
t of o
bjec
ts, i
nclu
ding
for p
eace
ful
and
harm
ful p
urpo
ses.
TOK
: Are
alg
ebra
and
geo
met
ry tw
o se
para
te
dom
ains
of k
now
ledg
e? (V
ecto
r alg
ebra
is a
go
od o
ppor
tuni
ty to
dis
cuss
how
geo
met
rical
pr
oper
ties a
re d
escr
ibed
and
gen
eral
ized
by
alge
brai
c m
etho
ds.)
The
angl
e be
twee
n tw
o lin
es.
4.4
Dis
tingu
ishi
ng b
etw
een
coin
cide
nt a
nd p
aral
lel
lines
.
Find
ing
the
poin
t of i
nter
sect
ion
of tw
o lin
es.
Det
erm
inin
g w
heth
er tw
o lin
es in
ters
ect.
Mathematics SL guide28
Syllabus contentSy
llabu
s con
tent
Mat
hem
atic
s SL
guid
e 12
Top
ic 5
—St
atist
ics a
nd p
roba
bilit
y 35
hou
rs Th
e ai
m o
f thi
s to
pic
is to
intro
duce
bas
ic c
once
pts.
It is
exp
ecte
d th
at m
ost o
f the
cal
cula
tions
requ
ired
will
be
done
usi
ng te
chno
logy
, but
exp
lana
tions
of
calc
ulat
ions
by
hand
may
enh
ance
und
erst
andi
ng. T
he e
mph
asis
is o
n un
ders
tand
ing
and
inte
rpre
ting
the
resu
lts o
btai
ned,
in c
onte
xt. S
tatis
tical
tabl
es w
ill n
o lo
nger
be
allo
wed
in e
xam
inat
ions
. Whi
le m
any
of th
e ca
lcul
atio
ns re
quire
d in
exa
min
atio
ns a
re e
stim
ates
, it i
s lik
ely
that
the
com
man
d te
rms “
writ
e do
wn”
, “f
ind”
and
“ca
lcul
ate”
will
be
used
.
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.1
Con
cept
s of p
opul
atio
n, sa
mpl
e, ra
ndom
sa
mpl
e, d
iscr
ete
and
cont
inuo
us d
ata.
Pres
enta
tion
of d
ata:
freq
uenc
y di
strib
utio
ns
(tabl
es);
frequ
ency
hist
ogra
ms w
ith e
qual
clas
s in
terv
als;
Con
tinuo
us a
nd d
iscr
ete
data
. A
ppl:
Psyc
holo
gy: d
escr
iptiv
e st
atis
tics,
rand
om sa
mpl
e (v
ario
us p
lace
s in
the
guid
e).
Aim
8: M
isle
adin
g st
atis
tics.
Int:
The
St P
eter
sbur
g pa
rado
x, C
heby
chev
, Pa
vlov
sky.
bo
x-an
d-w
hisk
er p
lots
; out
liers
. O
utlie
r is d
efin
ed a
s mor
e th
an 1
.5IQ
R×
from
th
e ne
ares
t qua
rtile
.
Tech
nolo
gy m
ay b
e us
ed to
pro
duce
hi
stog
ram
s and
box
-and
-whi
sker
plo
ts.
Gro
uped
dat
a: u
se o
f mid
-inte
rval
val
ues f
or
calc
ulat
ions
; int
erva
l wid
th; u
pper
and
low
er
inte
rval
bou
ndar
ies;
mod
al c
lass
.
Not
req
uire
d:
freq
uenc
y de
nsity
his
togr
ams.
Mathematics SL guide 29
Syllabus contentSy
llabu
s con
tent
Mat
hem
atic
s SL
guid
e 13
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.2
Stat
istic
al m
easu
res a
nd th
eir i
nter
pret
atio
ns.
Cen
tral t
ende
ncy:
mea
n, m
edia
n, m
ode.
Qua
rtile
s, pe
rcen
tiles
.
On
exam
inat
ion
pape
rs, d
ata
will
be
treat
ed a
s th
e po
pula
tion.
Cal
cula
tion
of m
ean
usin
g fo
rmul
a an
d te
chno
logy
. Stu
dent
s sho
uld
use
mid
-inte
rval
va
lues
to e
stim
ate
the
mea
n of
gro
uped
dat
a.
App
l: Ps
ycho
logy
: des
crip
tive
stat
istic
s (v
ario
us p
lace
s in
the
guid
e).
App
l: St
atis
tical
cal
cula
tions
to sh
ow p
atte
rns
and
chan
ges;
geo
grap
hic
skill
s; st
atis
tical
gr
aphs
.
App
l: B
iolo
gy 1
.1.2
(cal
cula
ting
mea
n an
d st
anda
rd d
evia
tion
); B
iolo
gy 1
.1.4
(com
parin
g m
eans
and
spre
ads b
etw
een
two
or m
ore
sam
ples
).
Int:
Dis
cuss
ion
of th
e di
ffer
ent f
orm
ulae
for
varia
nce.
TOK
: Do
diff
eren
t mea
sure
s of c
entra
l te
nden
cy e
xpre
ss d
iffer
ent p
rope
rties
of t
he
data
? A
re th
ese
mea
sure
s inv
ente
d or
di
scov
ered
? C
ould
mat
hem
atic
s mak
e al
tern
ativ
e, e
qual
ly tr
ue, f
orm
ulae
? W
hat d
oes
this
tell
us a
bout
mat
hem
atic
al tr
uths
?
TOK
: How
eas
y is
it to
lie
with
stat
istic
s?
Dis
pers
ion:
rang
e, in
terq
uarti
le ra
nge,
va
rianc
e, st
anda
rd d
evia
tion.
Effe
ct o
f con
stan
t cha
nges
to th
e or
igin
al d
ata.
Cal
cula
tion
of st
anda
rd d
evia
tion/
varia
nce
usin
g on
ly te
chno
logy
.
Link
to 2
.3, t
rans
form
atio
ns.
Examples
:
If 5
is su
btra
cted
from
all
the
data
item
s, th
en
the
mea
n is
dec
reas
ed b
y 5,
but
the
stan
dard
de
viat
ion
is u
ncha
nged
.
If al
l the
dat
a ite
ms a
re d
oubl
ed, t
he m
edia
n is
do
uble
d, b
ut th
e va
rianc
e is
incr
ease
d by
a
fact
or o
f 4.
App
licat
ions
.
5.3
Cum
ulat
ive
freq
uenc
y; c
umul
ativ
e fr
eque
ncy
grap
hs; u
se to
find
med
ian,
qua
rtile
s, pe
rcen
tiles
.
Val
ues o
f the
med
ian
and
quar
tiles
pro
duce
d by
tech
nolo
gy m
ay b
e di
ffer
ent f
rom
thos
e ob
tain
ed fr
om a
cum
ulat
ive
freq
uenc
y gr
aph.
Mathematics SL guide30
Syllabus contentSy
llabu
s con
tent
Mat
hem
atic
s SL
guid
e 14
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.4
Line
ar c
orre
latio
n of
biv
aria
te d
ata.
In
depe
nden
t var
iabl
e x,
dep
ende
nt v
aria
ble
y.
App
l: C
hem
istry
11.
3.3
(cur
ves o
f bes
t fit)
.
App
l: G
eogr
aphy
(geo
grap
hic
skill
s).
Mea
sure
s of c
orre
latio
n; g
eogr
aphi
c sk
ills.
App
l: B
iolo
gy 1
.1.6
(cor
rela
tion
does
not
im
ply
caus
atio
n).
TOK
: Can
we
pred
ict t
he v
alue
of x
from
y,
usin
g th
is e
quat
ion?
TOK
: Can
all
data
be
mod
elle
d by
a (k
now
n)
mat
hem
atic
al fu
nctio
n? C
onsi
der t
he re
liabi
lity
and
valid
ity o
f mat
hem
atic
al m
odel
s in
desc
ribin
g re
al-li
fe p
heno
men
a.
Pear
son’
s pro
duct
–mom
ent c
orre
latio
n co
effic
ient
r.
Tech
nolo
gy sh
ould
be
used
to c
alcu
late
r.
How
ever
, han
d ca
lcul
atio
ns o
f r m
ay e
nhan
ce
unde
rsta
ndin
g.
Posi
tive,
zer
o, n
egat
ive;
stro
ng, w
eak,
no
corr
elat
ion.
Scat
ter d
iagr
ams;
line
s of b
est f
it.
The
line
of b
est f
it pa
sses
thro
ugh
the
mea
n po
int.
Equa
tion
of th
e re
gres
sion
line
of y
on
x.
Use
of t
he e
quat
ion
for p
redi
ctio
n pu
rpos
es.
Mat
hem
atic
al a
nd c
onte
xtua
l int
erpr
etat
ion.
Not
req
uire
d:
the
coef
ficie
nt o
f det
erm
inat
ion
R2 .
Tech
nolo
gy sh
ould
be
used
find
the
equa
tion.
Inte
rpol
atio
n, e
xtra
pola
tion.
5.5
Con
cept
s of t
rial,
outc
ome,
equ
ally
like
ly
outc
omes
, sam
ple
spac
e (U
) and
eve
nt.
The
sam
ple
spac
e ca
n be
repr
esen
ted
diag
ram
mat
ical
ly in
man
y w
ays.
TOK
: To
wha
t ext
ent d
oes m
athe
mat
ics o
ffer
m
odel
s of r
eal l
ife?
Is th
ere
alw
ays a
func
tion
to m
odel
dat
a be
havi
our?
Th
e pr
obab
ility
of a
n ev
ent A
is
()
P()
()
nA
An
U=
.
The
com
plem
enta
ry e
vent
s A a
nd A′
(not
A).
Use
of V
enn
diag
ram
s, tre
e di
agra
ms a
nd
tabl
es o
f out
com
es.
Expe
rimen
ts u
sing
coi
ns, d
ice,
car
ds a
nd so
on,
ca
n en
hanc
e un
ders
tand
ing
of th
e di
stin
ctio
n be
twee
n (e
xper
imen
tal)
rela
tive
freq
uenc
y an
d (th
eore
tical
) pro
babi
lity.
Sim
ulat
ions
may
be
used
to e
nhan
ce th
is to
pic.
Link
s to
5.1,
freq
uenc
y; 5
.3, c
umul
ativ
e fr
eque
ncy.
Mathematics SL guide 31
Syllabus contentSy
llabu
s con
tent
Mat
hem
atic
s SL
guid
e 15
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.6
Com
bine
d ev
ents
, P(
)A
B∪
.
Mut
ually
exc
lusi
ve e
vent
s: P
()
0A
B∩
=.
Con
ditio
nal p
roba
bilit
y; th
e de
finiti
on
()
P()
P|
P()
AB
AB
B∩=
.
Inde
pend
ent e
vent
s; th
e de
finiti
on
()
()
P|
P()
P|
AB
AAB′
==
.
Prob
abili
ties w
ith a
nd w
ithou
t rep
lace
men
t.
The
non-
excl
usiv
ity o
f “or
”.
Prob
lem
s are
ofte
n be
st so
lved
with
the
aid
of a
V
enn
diag
ram
or t
ree
diag
ram
, with
out e
xplic
it us
e of
form
ulae
.
Aim
8: T
he g
ambl
ing
issu
e: u
se o
f pro
babi
lity
in c
asin
os. C
ould
or s
houl
d m
athe
mat
ics h
elp
incr
ease
inco
mes
in g
ambl
ing?
TOK
: Is m
athe
mat
ics u
sefu
l to
mea
sure
risk
s?
TOK
: Can
gam
blin
g be
con
side
red
as a
n ap
plic
atio
n of
mat
hem
atic
s? (T
his i
s a g
ood
oppo
rtuni
ty to
gen
erat
e a
deba
te o
n th
e na
ture
, ro
le a
nd e
thic
s of m
athe
mat
ics r
egar
ding
its
appl
icat
ions
.)
5.7
Con
cept
of d
iscr
ete
rand
om v
aria
bles
and
thei
r pr
obab
ility
dis
tribu
tions
. Si
mpl
e ex
ampl
es o
nly,
such
as:
1
P()
(4)
18X
xx
==
+ fo
r {
}1,
2,3
x∈;
56
7P(
),
,18
1818
Xx
==
.
Expe
cted
val
ue (m
ean)
, E(
)X
for d
iscr
ete
data
.
App
licat
ions
.
E()
0X
= in
dica
tes a
fair
gam
e w
here
X
repr
esen
ts th
e ga
in o
f one
of t
he p
laye
rs.
Exam
ples
incl
ude
gam
es o
f cha
nce.
Mathematics SL guide32
Syllabus contentSy
llabu
s con
tent
Mat
hem
atic
s SL
guid
e 16
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
5.8
Bin
omia
l dis
tribu
tion.
Mea
n an
d va
rianc
e of
the
bino
mia
l di
strib
utio
n.
Not
req
uire
d:
form
al p
roof
of m
ean
and
varia
nce.
Link
to 1
.3, b
inom
ial t
heor
em.
Con
ditio
ns u
nder
whi
ch ra
ndom
var
iabl
es h
ave
this
dis
tribu
tion.
Tech
nolo
gy is
usu
ally
the
best
way
of
calc
ulat
ing
bino
mia
l pro
babi
litie
s.
5.9
Nor
mal
dis
tribu
tions
and
cur
ves.
Stan
dard
izat
ion
of n
orm
al v
aria
bles
(z-v
alue
s,
z-sc
ores
).
Prop
ertie
s of t
he n
orm
al d
istri
butio
n.
Prob
abili
ties a
nd v
alue
s of t
he v
aria
ble
mus
t be
foun
d us
ing
tech
nolo
gy.
Link
to 2
.3, t
rans
form
atio
ns.
The
stan
dard
ized
val
ue (z)
giv
es th
e nu
mbe
r of
stan
dard
dev
iatio
ns fr
om th
e m
ean.
App
l: B
iolo
gy 1
.1.3
(lin
ks to
nor
mal
di
strib
utio
n).
App
l: Ps
ycho
logy
: des
crip
tive
stat
istic
s (v
ario
us p
lace
s in
the
guid
e).
Mathem
atics SL guide33
Syllabus contentSyllabus content
Mathematics SL guide 17
Topic 6—Calculus 40 hours The aim of this topic is to introduce students to the basic concepts and techniques of differential and integral calculus and their applications.
Content Further guidance Links
6.1 Informal ideas of limit and convergence. Example: 0.3, 0.33, 0.333, ... converges to 1
3.
Technology should be used to explore ideas of limits, numerically and graphically.
Appl: Economics 1.5 (marginal cost, marginal revenue, marginal profit).
Appl: Chemistry 11.3.4 (interpreting the gradient of a curve).
Aim 8: The debate over whether Newton or Leibnitz discovered certain calculus concepts.
TOK: What value does the knowledge of limits have? Is infinitesimal behaviour applicable to real life?
TOK: Opportunities for discussing hypothesis formation and testing, and then the formal proof can be tackled by comparing certain cases, through an investigative approach.
Limit notation. Example:
2 3lim
1x
xx→∞
+
−
Links to 1.1, infinite geometric series; 2.5–2.7, rational and exponential functions, and asymptotes.
Definition of derivative from first principles as
0
( ) ( )( ) limh
f x h f xf xh→
+ −′ = .
Use of this definition for derivatives of simple polynomial functions only.
Technology could be used to illustrate other derivatives.
Link to 1.3, binomial theorem.
Use of both forms of notation, ddyx
and ( )f x′ ,
for the first derivative.
Derivative interpreted as gradient function and as rate of change.
Identifying intervals on which functions are increasing or decreasing.
Tangents and normals, and their equations.
Not required: analytic methods of calculating limits.
Use of both analytic approaches and technology.
Technology can be used to explore graphs and their derivatives.
Mathem
atics SL guide34 Syllabus content
Syllabus content
Mathematics SL guide 18
Content Further guidance Links
6.2 Derivative of ( )nx n∈_ , sin x , cos x , tan x , ex and ln x .
Differentiation of a sum and a real multiple of these functions.
The chain rule for composite functions.
The product and quotient rules.
Link to 2.1, composition of functions.
Technology may be used to investigate the chain rule.
The second derivative. Use of both forms of notation, 2
2
ddyx
and ( )f x′′ .
Extension to higher derivatives. dd
n
n
yx
and ( ) ( )nf x .
Mathem
atics SL guide35
Syllabus contentSyllabus content
Mathematics SL guide 19
Content Further guidance Links
6.3 Local maximum and minimum points.
Testing for maximum or minimum.
Using change of sign of the first derivative and using sign of the second derivative.
Use of the terms “concave-up” for ( ) 0f x′′ > , and “concave-down” for ( ) 0f x′′ < .
Appl: profit, area, volume.
Points of inflexion with zero and non-zero gradients.
At a point of inflexion , ( ) 0f x′′ = and changes sign (concavity change).
( ) 0f x′′ = is not a sufficient condition for a point of inflexion: for example,
4y x= at (0,0) .
Graphical behaviour of functions, including the relationship between the graphs of f , f ′ and f ′′ .
Optimization.
Both “global” (for large x ) and “local” behaviour.
Technology can display the graph of a derivative without explicitly finding an expression for the derivative.
Use of the first or second derivative test to justify maximum and/or minimum values.
Applications.
Not required: points of inflexion where ( )f x′′ is not defined: for example, 1 3y x= at (0,0) .
Examples include profit, area, volume.
Link to 2.2, graphing functions.
Mathematics SL guide36
Syllabus contentSy
llabu
s co
nten
t
Mat
hem
atic
s SL
gui
de
20
Co
nten
t Fu
rthe
r gui
danc
e Li
nks
6.4
Inde
fini
te in
tegr
atio
n as
ant
i-di
ffer
entia
tion
.
Inde
fini
te in
tegr
al o
f (
)n xn∈_
, si
nx
, co
sx,
1 x a
nd ex.
1d
lnx
xC
x=
+∫
, 0
x>
.
The
com
posi
tes
of a
ny o
f th
ese
wit
h th
e lin
ear
func
tion
axb
+.
Example:
1(
)co
s(2
3)(
)si
n(2
3)2
fx
xfx
xC
′=
+⇒
=+
+.
Inte
grat
ion
by in
spec
tion
, or
subs
titut
ion
of th
e
form
(
())
'()d
fgxgxx
∫.
Examples
:
()4
22
21
d,
sin
d,
dsi
n
cos
xx
xx
xx
xx x
+∫
∫∫
.
6.5
Ant
i-di
ffer
entia
tion
wit
h a
boun
dary
con
diti
on
to d
eter
min
e th
e co
nsta
nt te
rm.
Example:
if
2d
3dy
xx
x=
+ a
nd
10y=
whe
n 0
x=
, the
n
32
110
2yx
x=
++
.
Int:
Suc
cess
ful c
alcu
latio
n of
the
volu
me
of
the
pyra
mid
al f
rust
um b
y an
cien
t Egy
ptia
ns
(Egy
ptia
n M
osco
w p
apyr
us).
Use
of
infi
nite
sim
als
by G
reek
geo
met
ers.
Def
inite
inte
gral
s, b
oth
anal
ytic
ally
and
usi
ng
tech
nolo
gy.
()d
()
()
b agxxgb
ga
′=
−∫
.
The
val
ue o
f so
me
defi
nite
inte
gral
s ca
n on
ly
be f
ound
usi
ng te
chno
logy
.
Acc
urat
e ca
lcul
atio
n of
the
volu
me
of a
cy
lind
er b
y C
hine
se m
athe
mat
icia
n L
iu H
ui
Are
as u
nder
cur
ves
(bet
wee
n th
e cu
rve
and
the
x-ax
is).
Are
as b
etw
een
curv
es.
Vol
umes
of
revo
lutio
n ab
out t
he x
-axi
s.
Stud
ents
are
exp
ecte
d to
fir
st w
rite
a c
orre
ct
expr
essi
on b
efor
e ca
lcul
atin
g th
e ar
ea.
Tec
hnol
ogy
may
be
used
to e
nhan
ce
unde
rsta
ndin
g of
are
a an
d vo
lum
e.
Int:
Ibn
Al H
ayth
am: f
irst
mat
hem
atic
ian
to
calc
ulat
e th
e in
tegr
al o
f a
func
tion,
in o
rder
to
find
the
volu
me
of a
par
abol
oid.
6.6
Kin
emat
ic p
robl
ems
invo
lvin
g di
spla
cem
ent s
, ve
loci
ty v
and
acc
eler
atio
n a.
d ds
vt
=;
2
2
dd
dd
vs
at
t=
=.
App
l: Ph
ysic
s 2.
1 (k
inem
atic
s).
Tot
al d
ista
nce
trav
elle
d.
Tot
al d
ista
nce
trav
elle
d 2 1
dt tvt
=∫
.
Mathematics SL guide 39
Assessment
Assessment outline
First examinations 2014
Assessment component Weighting
External assessment (3 hours)Paper 1 (1 hour 30 minutes)No calculator allowed. (90 marks)
Section ACompulsory short-response questions based on the whole syllabus.
Section BCompulsory extended-response questions based on the whole syllabus.
80%40%
Paper 2 (1 hour 30 minutes)Graphic display calculator required. (90 marks)
Section ACompulsory short-response questions based on the whole syllabus.
Section BCompulsory extended-response questions based on the whole syllabus.
40%
Internal assessmentThis component is internally assessed by the teacher and externally moderated by the IB at the end of the course.
Mathematical explorationInternal assessment in mathematics SL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. (20 marks)
20%
40 Mathematics SL guide
Assessment
External assessment
GeneralMarkschemes are used to assess students in both papers. The markschemes are specific to each examination.
External assessment detailsPaper 1 and paper 2These papers are externally set and externally marked. Together, they contribute 80% of the final mark for the course. These papers are designed to allow students to demonstrate what they know and what they can do.
CalculatorsPaper 1Students are not permitted access to any calculator. Questions will mainly involve analytic approaches to solutions, rather than requiring the use of a GDC. The paper is not intended to require complicated calculations, with the potential for careless errors. However, questions will include some arithmetical manipulations when they are essential to the development of the question.
Paper 2Students must have access to a GDC at all times. However, not all questions will necessarily require the use of the GDC. Regulations covering the types of GDC allowed are provided in the Handbook of procedures for the Diploma Programme.
Mathematics SL formula bookletEach student must have access to a clean copy of the formula booklet during the examination. It is the responsibility of the school to download a copy from IBIS or the OCC and to ensure that there are sufficient copies available for all students.
Awarding of marksMarks may be awarded for method, accuracy, answers and reasoning, including interpretation.
In paper 1 and paper 2, full marks are not necessarily awarded for a correct answer with no working. Answers must be supported by working and/or explanations (in the form of, for example, diagrams, graphs or calculations). Where an answer is incorrect, some marks may be given for correct method, provided this is shown by written working. All students should therefore be advised to show their working.
Mathematics SL guide 41
External assessment
Paper 1Duration: 1 hour 30 minutesWeighting: 40%• This paper consists of section A, short-response questions, and section B, extended-response questions.
• Students are not permitted access to any calculator on this paper.
Syllabus coverage• Knowledge of all topics is required for this paper. However, not all topics are necessarily assessed in
every examination session.
Mark allocation• This paper is worth 90 marks, representing 40% of the final mark.
• Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily each be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.
Section AThis section consists of compulsory short-response questions based on the whole syllabus. It is worth approximately 45 marks.
The intention of this section is to test students’ knowledge and understanding across the breadth of the syllabus. However, it should not be assumed that the separate topics are given equal emphasis.
Question type• A small number of steps is needed to solve each question.
• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.
Section BThis section consists of a small number of compulsory extended-response questions based on the whole syllabus. It is worth approximately 45 marks. Individual questions may require knowledge of more than one topic.
The intention of this section is to test students’ knowledge and understanding of the syllabus in depth. The range of syllabus topics tested in this section may be narrower than that tested in section A.
Question type• Questions require extended responses involving sustained reasoning.
• Individual questions will develop a single theme.
• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.
• Normally, each question ref lects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problem-solving.
Paper 2Duration: 1 hour 30 minutesWeighting: 40%This paper consists of section A, short-response questions, and section B, extended-response questions. A GDC is required for this paper, but not every question will necessarily require its use.
Mathematics SL guide42
External assessment
Syllabus coverage• Knowledge of all topics is required for this paper. However, not all topics are necessarily assessed in
every examination session.
Mark allocation• This paper is worth 90 marks, representing 40% of the final mark.
• Questions of varying levels of difficulty and length are set. Therefore, individual questions may not necessarily each be worth the same number of marks. The exact number of marks allocated to each question is indicated at the start of the question.
Section AThis section consists of compulsory short-response questions based on the whole syllabus. It is worth approximately 45 marks.
The intention of this section is to test students’ knowledge and understanding across the breadth of the syllabus. However, it should not be assumed that the separate topics are given equal emphasis.
Question type• A small number of steps is needed to solve each question.
• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.
Section BThis section consists of a small number of compulsory extended-response questions based on the whole syllabus. It is worth approximately 45 marks. Individual questions may require knowledge of more than one topic.
The intention of this section is to test students’ knowledge and understanding of the syllabus in depth. The range of syllabus topics tested in this section may be narrower than that tested in section A.
Question type• Questions require extended responses involving sustained reasoning.
• Individual questions will develop a single theme.
• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of these.
• Normally, each question ref lects an incline of difficulty, from relatively easy tasks at the start of a question to relatively difficult tasks at the end of a question. The emphasis is on problem-solving.
Mathematics SL guide 43
Assessment
Internal assessment
Purpose of internal assessmentInternal assessment is an integral part of the course and is compulsory for all students. It enables students to demonstrate the application of their skills and knowledge, and to pursue their personal interests, without the time limitations and other constraints that are associated with written examinations. The internal assessment should, as far as possible, be woven into normal classroom teaching and not be a separate activity conducted after a course has been taught.
Internal assessment in mathematics SL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. It is marked according to five assessment criteria.
Guidance and authenticityThe exploration submitted for internal assessment must be the student’s own work. However, it is not the intention that students should decide upon a title or topic and be left to work on the exploration without any further support from the teacher. The teacher should play an important role during both the planning stage and the period when the student is working on the exploration. It is the responsibility of the teacher to ensure that students are familiar with:
• the requirements of the type of work to be internally assessed
• the IB academic honesty policy available on the OCC
• the assessment criteria—students must understand that the work submitted for assessment must address these criteria effectively.
Teachers and students must discuss the exploration. Students should be encouraged to initiate discussions with the teacher to obtain advice and information, and students must not be penalized for seeking guidance. However, if a student could not have completed the exploration without substantial support from the teacher, this should be recorded on the appropriate form from the Handbook of procedures for the Diploma Programme.
It is the responsibility of teachers to ensure that all students understand the basic meaning and significance of concepts that relate to academic honesty, especially authenticity and intellectual property. Teachers must ensure that all student work for assessment is prepared according to the requirements and must explain clearly to students that the exploration must be entirely their own.
As part of the learning process, teachers can give advice to students on a first draft of the exploration. This advice should be in terms of the way the work could be improved, but this first draft must not be heavily annotated or edited by the teacher. The next version handed to the teacher after the first draft must be the final one.
All work submitted to the IB for moderation or assessment must be authenticated by a teacher, and must not include any known instances of suspected or confirmed malpractice. Each student must sign the coversheet for internal assessment to confirm that the work is his or her authentic work and constitutes the final version of that work. Once a student has officially submitted the final version of the work to a teacher (or the coordinator) for internal assessment, together with the signed coversheet, it cannot be retracted.
Mathematics SL guide44
Internal assessment
Authenticity may be checked by discussion with the student on the content of the work, and scrutiny of one or more of the following:
• the student’s initial proposal
• the first draft of the written work
• the references cited
• the style of writing compared with work known to be that of the student.
The requirement for teachers and students to sign the coversheet for internal assessment applies to the work of all students, not just the sample work that will be submitted to an examiner for the purpose of moderation. If the teacher and student sign a coversheet, but there is a comment to the effect that the work may not be authentic, the student will not be eligible for a mark in that component and no grade will be awarded. For further details refer to the IB publication Academic honesty and the relevant articles in the General regulations: Diploma Programme.
The same piece of work cannot be submitted to meet the requirements of both the internal assessment and the extended essay.
Group workGroup work should not be used for explorations. Each exploration is an individual piece of work.
It should be made clear to students that all work connected with the exploration, including the writing of the exploration, should be their own. It is therefore helpful if teachers try to encourage in students a sense of responsibility for their own learning so that they accept a degree of ownership and take pride in their own work.
Time allocationInternal assessment is an integral part of the mathematics SL course, contributing 20% to the final assessment in the course. This weighting should be reflected in the time that is allocated to teaching the knowledge, skills and understanding required to undertake the work as well as the total time allocated to carry out the work.
It is expected that a total of approximately 10 teaching hours should be allocated to the work. This should include:
• time for the teacher to explain to students the requirements of the exploration
• class time for students to work on the exploration
• time for consultation between the teacher and each student
• time to review and monitor progress, and to check authenticity.
Using assessment criteria for internal assessmentFor internal assessment, a number of assessment criteria have been identified. Each assessment criterion has level descriptors describing specific levels of achievement together with an appropriate range of marks. The level descriptors concentrate on positive achievement, although for the lower levels failure to achieve may be included in the description.
Mathematics SL guide 45
Internal assessment
Teachers must judge the internally assessed work against the criteria using the level descriptors.
• The aim is to find, for each criterion, the descriptor that conveys most accurately the level attained by the student.
• When assessing a student’s work, teachers should read the level descriptors for each criterion, starting with level 0, until they reach a descriptor that describes a level of achievement that has not been reached. The level of achievement gained by the student is therefore the preceding one, and it is this that should be recorded.
• Only whole numbers should be recorded; partial marks, that is fractions and decimals, are not acceptable.
• Teachers should not think in terms of a pass or fail boundary, but should concentrate on identifying the appropriate descriptor for each assessment criterion.
• The highest level descriptors do not imply faultless performance but should be achievable by a student. Teachers should not hesitate to use the extremes if they are appropriate descriptions of the work being assessed.
• A student who attains a high level of achievement in relation to one criterion will not necessarily attain high levels of achievement in relation to the other criteria. Similarly, a student who attains a low level of achievement for one criterion will not necessarily attain low achievement levels for the other criteria. Teachers should not assume that the overall assessment of the students will produce any particular distribution of marks.
• It is expected that the assessment criteria be made available to students.
Internal assessment detailsMathematical explorationDuration: 10 teaching hoursWeighting: 20%
IntroductionThe internally assessed component in this course is a mathematical exploration. This is a short report written by the student based on a topic chosen by him or her, and it should focus on the mathematics of that particular area. The emphasis is on mathematical communication (including formulae, diagrams, graphs and so on), with accompanying commentary, good mathematical writing and thoughtful reflection. A student should develop his or her own focus, with the teacher providing feedback via, for example, discussion and interview. This will allow the students to develop area(s) of interest to them without a time constraint as in an examination, and allow all students to experience a feeling of success.
The final report should be approximately 6 to 12 pages long. It can be either word processed or handwritten. Students should be able to explain all stages of their work in such a way that demonstrates clear understanding. While there is no requirement that students present their work in class, it should be written in such a way that their peers would be able to follow it fairly easily. The report should include a detailed bibliography, and sources need to be referenced in line with the IB academic honesty policy. Direct quotes must be acknowledged.
The purpose of the explorationThe aims of the mathematics SL course are carried through into the objectives that are formally assessed as part of the course, through either written examination papers, or the exploration, or both. In addition to testing the objectives of the course, the exploration is intended to provide students with opportunities to increase their understanding of mathematical concepts and processes, and to develop a wider appreciation of mathematics. These are noted in the aims of the course, in particular, aims 6–9 (applications, technology, moral, social
Mathematics SL guide46
Internal assessment
and ethical implications, and the international dimension). It is intended that, by doing the exploration, students benefit from the mathematical activities undertaken and find them both stimulating and rewarding. It will enable students to acquire the attributes of the IB learner profile.
The specific purposes of the exploration are to:
• develop students’ personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics
• provide opportunities for students to complete a piece of mathematical work over an extended period of time
• enable students to experience the satisfaction of applying mathematical processes independently
• provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics
• encourage students, where appropriate, to discover, use and appreciate the power of technology as a mathematical tool
• enable students to develop the qualities of patience and persistence, and to reflect on the significance of their work
• provide opportunities for students to show, with confidence, how they have developed mathematically.
Management of the explorationWork for the exploration should be incorporated into the course so that students are given the opportunity to learn the skills needed. Time in class can therefore be used for general discussion of areas of study, as well as familiarizing students with the criteria.
Further details on the development of the exploration are included in the teacher support material.
Requirements and recommendationsStudents can choose from a wide variety of activities, for example, modelling, investigations and applications of mathematics. To assist teachers and students in the choice of a topic, a list of stimuli is available in the teacher support material. However, students are not restricted to this list.
The exploration should not normally exceed 12 pages, including diagrams and graphs, but excluding the bibliography. However, it is the quality of the mathematical writing that is important, not the length.
The teacher is expected to give appropriate guidance at all stages of the exploration by, for example, directing students into more productive routes of inquiry, making suggestions for suitable sources of information, and providing advice on the content and clarity of the exploration in the writing-up stage.
Teachers are responsible for indicating to students the existence of errors but should not explicitly correct these errors. It must be emphasized that students are expected to consult the teacher throughout the process.
All students should be familiar with the requirements of the exploration and the criteria by which it is assessed. Students need to start planning their explorations as early as possible in the course. Deadlines should be firmly established. There should be a date for submission of the exploration topic and a brief outline description, a date for the submission of the first draft and, of course, a date for completion.
In developing their explorations, students should aim to make use of mathematics learned as part of the course. The mathematics used should be commensurate with the level of the course, that is, it should be similar to that suggested by the syllabus. It is not expected that students produce work that is outside the mathematics SL syllabus—however, this is not penalized.
Mathematics SL guide 47
Internal assessment
Internal assessment criteriaThe exploration is internally assessed by the teacher and externally moderated by the IB using assessment criteria that relate to the objectives for mathematics SL.
Each exploration is assessed against the following five criteria. The final mark for each exploration is the sum of the scores for each criterion. The maximum possible final mark is 20.
Students will not receive a grade for mathematics SL if they have not submitted an exploration.
Criterion A Communication
Criterion B Mathematical presentation
Criterion C Personal engagement
Criterion D Reflection
Criterion E Use of mathematics
Criterion A: CommunicationThis criterion assesses the organization and coherence of the exploration. A well-organized exploration includes an introduction, has a rationale (which includes explaining why this topic was chosen), describes the aim of the exploration and has a conclusion. A coherent exploration is logically developed and easy to follow.
Graphs, tables and diagrams should accompany the work in the appropriate place and not be attached as appendices to the document.
Achievement level Descriptor
0 The exploration does not reach the standard described by the descriptors below.
1 The exploration has some coherence.
2 The exploration has some coherence and shows some organization.
3 The exploration is coherent and well organized.
4 The exploration is coherent, well organized, concise and complete.
Criterion B: Mathematical presentationThis criterion assesses to what extent the student is able to:
• use appropriate mathematical language (notation, symbols, terminology)
• define key terms, where required
• use multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs and models, where appropriate.
Students are expected to use mathematical language when communicating mathematical ideas, reasoning and findings.
Mathematics SL guide48
Internal assessment
Students are encouraged to choose and use appropriate ICT tools such as graphic display calculators, screenshots, graphing, spreadsheets, databases, drawing and word-processing software, as appropriate, to enhance mathematical communication.
Achievement level Descriptor
0 The exploration does not reach the standard described by the descriptors below.
1 There is some appropriate mathematical presentation.
2 The mathematical presentation is mostly appropriate.
3 The mathematical presentation is appropriate throughout.
Criterion C: Personal engagementThis criterion assesses the extent to which the student engages with the exploration and makes it their own. Personal engagement may be recognized in different attributes and skills. These include thinking independently and/or creatively, addressing personal interest and presenting mathematical ideas in their own way.
Achievement level Descriptor
0 The exploration does not reach the standard described by the descriptors below.
1 There is evidence of limited or superficial personal engagement.
2 There is evidence of some personal engagement.
3 There is evidence of significant personal engagement.
4 There is abundant evidence of outstanding personal engagement.
Criterion D: ReflectionThis criterion assesses how the student reviews, analyses and evaluates the exploration. Although reflection may be seen in the conclusion to the exploration, it may also be found throughout the exploration.
Achievement level Descriptor
0 The exploration does not reach the standard described by the descriptors below.
1 There is evidence of limited or superficial reflection.
2 There is evidence of meaningful reflection.
3 There is substantial evidence of critical reflection.
Mathematics SL guide 49
Internal assessment
Criterion E: Use of mathematicsThis criterion assesses to what extent students use mathematics in the exploration.
Students are expected to produce work that is commensurate with the level of the course. The mathematics explored should either be part of the syllabus, or at a similar level or beyond. It should not be completely based on mathematics listed in the prior learning. If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion.
The mathematics can be regarded as correct even if there are occasional minor errors as long as they do not detract from the flow of the mathematics or lead to an unreasonable outcome.
Achievement level Descriptor
0 The exploration does not reach the standard described by the descriptors below.
1 Some relevant mathematics is used.
2 Some relevant mathematics is used. Limited understanding is demonstrated.
3 Relevant mathematics commensurate with the level of the course is used. Limited understanding is demonstrated.
4 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is partially correct. Some knowledge and understanding are demonstrated.
5 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is mostly correct. Good knowledge and understanding are demonstrated.
6 Relevant mathematics commensurate with the level of the course is used. The mathematics explored is correct. Thorough knowledge and understanding are demonstrated.