IB SL Maths Syllabus

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



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



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= + .


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.

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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




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.

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Syllabus content



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



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.


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= = − ° = .

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Syllabus content



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




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→

.


Syllabus contentSy

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s con

tent

Mat

hem

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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.


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.


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.


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.


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.


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



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.


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


Syllabus content



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




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.


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

=∫

.


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.


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.


• 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.


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.


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.


• 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.


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.


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.


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


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.


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.


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.



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.



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.



1 There is evidence of limited or superficial reflection.

2 There is evidence of meaningful reflection.

3 There is substantial evidence of critical reflection.


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.



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.

Documents

IB SL Maths Syllabus