Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics...

25
10 Mathematical studies SL guide Syllabus outline Syllabus Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with the topics listed as prior learning. Topic 1 Number and algebra 20 Topic 2 Descriptive statistics 12 Topic 3 Logic, sets and probability 20 Topic 4 Statistical applications 17 Topic 5 Geometry and trigonometry 18 Topic 6 Mathematical models 20 Topic 7 Introduction to differential calculus 18 Project The project is an individual piece of work involving the collection of information or the generation of measurements, and the analysis and evaluation of the information or measurements. 25 Total teaching hours 150 It is essential that teachers are allowed the prescribed minimum number of teaching hours necessary to meet the requirements of the mathematical studies SL course. At SL the minimum prescribed number of hours is 150 hours.

Transcript of Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics...

Page 1: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

10 Mathematical studies SL guide

Syllabus outline

Syllabus

Syllabus component

Teaching hours

SL

All topics are compulsory. Students must study all the sub-topics in each of the topics in the syllabus as listed in this guide. Students are also required to be familiar with the topics listed as prior learning.

Topic 1

Number and algebra

20

Topic 2

Descriptive statistics

12

Topic 3

Logic, sets and probability

20

Topic 4

Statistical applications

17

Topic 5

Geometry and trigonometry

18

Topic 6

Mathematical models

20

Topic 7

Introduction to differential calculus

18

Project

The project is an individual piece of work involving the collection of information or the generation of measurements, and the analysis and evaluation of the information or measurements.

25

Total teaching hours 150

It is essential that teachers are allowed the prescribed minimum number of teaching hours necessary to meet the requirements of the mathematical studies SL course. At SL the minimum prescribed number of hours is 150 hours.

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Mathematical studies SL guide 11

Syllabus

Approaches to the teaching and learning of mathematical studies SL

In this course the students will have the opportunity to understand and appreciate both the practical use of mathematics and its aesthetic aspects. They will be encouraged to build on knowledge from prior learning in mathematics and other subjects, as well as their own experience. It is important that students develop mathematical intuition and understand how they can apply mathematics in life.

Teaching needs to be flexible and to allow for different styles of learning. There is a diverse range of students in a mathematical studies SL classroom, and visual, auditory and kinaesthetic approaches to teaching may give new insights. The use of technology, particularly the graphic display calculator (GDC) and computer packages, can be very useful in allowing students to explore ideas in a rich context. It is left to the individual teacher to decide the order in which the separate topics are presented, but teaching and learning activities should weave the parts of the syllabus together and focus on their interrelationships. For example, the connection between geometric sequences and exponential functions can be illustrated by the consideration of compound interest.

Teachers may wish to introduce some topics using hand calculations to give an initial insight into the principles. However, once understanding has been gained, it is envisaged that the use of the GDC will support further workandsimplifycalculation(forexample,theχ2 statistic).

Teachers may take advantage of students’ mathematical intuition by approaching the teaching of probability in a way that does not solely rely on formulae.

The mathematical studies SL project is meant to be not only an assessment tool, but also a sophisticated learning opportunity. It is an independent but well-guided piece of research, using mathematical methods to draw conclusions and answer questions from the individual student’s interests. Project work should be incorporated into the course so that students are given the opportunity to learn the skills needed for the completion of a successful project. It is envisaged that the project will not be undertaken before students have experienced a range of techniques to make it meaningful. The scheme of work should be designed with this in mind.

Teachers should encourage students to find links and applications to their other IB subjects and the core of the hexagon. Everyday problems and questions should be drawn into the lessons to motivate students and keep the material relevant; suggestions are provided in the “Links” column of the syllabus.

For further information on “Approaches to teaching a DP course” please refer to the publication The Diploma Programme: From principles into practice (April 2009). To support teachers, a variety of resources can be found on the OCC and details of workshops for professional development are available on the public website.

Format of the syllabus• Content: this column lists, under each topic, the sub-topics to be covered.

• Further guidance: this column contains more detailed information on specific sub-topics listed in the content column. This clarifies the content for examinations.

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Mathematical studies SL guide12

Approaches to the teaching and learning of mathematical studies SL

• Links: this column provides useful links to the aims of the mathematical studies SL course, with suggestions for discussion, real-life examples and project ideas. These suggestions are only a guide for introducing and illustrating the sub-topic and are not exhaustive. Links are labelled as follows.

Appl real-life examples and links to other DP subjects

Aim 8 moral, social and ethical implications of the sub-topic

Int international-mindedness

TOK suggestions for discussion

Note that any syllabus references to other subject guides given in the “Links” column are correct for the current (2012) published versions of the guides.

Course of studyThe content of all seven topics in the syllabus must be taught, although not necessarily in the order in which they appear in this guide. Teachers are expected to construct a course of study that addresses the needs of their students and includes, where necessary, the topics noted in prior learning.

Integration of project workWork leading to the completion of the project must be integrated into the course of study. Details of how to do this are given in the section on internal assessment and in the teacher support material.

Time allocationThe recommended teaching time for standard level courses is 150 hours. For mathematical studies SL, it is expected that 25 hours will be spent on work for the project. The time allocations given in this guide are approximate, and are intended to suggest how the remaining 125 hours allowed for the teaching of the syllabus might be allocated. However, the exact time spent on each topic depends on a number of factors, including the background knowledge and level of preparedness of each student. Teachers should therefore adjust these timings to correspond to the needs of their students.

Time has been allocated in each section of the syllabus to allow for the teaching of topics requiring the use of a GDC.

Use of calculatorsStudents are expected to have access to a GDC at all times during the course. The minimum requirements are reviewed as technology advances, and updated information will be provided to schools. It is expected that teachers and schools monitor calculator use with reference to the calculator policy. Regulations covering the types of calculators allowed in examinations are provided in the Handbook of procedures for the Diploma Programme. Further information and advice is provided in the Mathematical studies SL: Graphic display calculators teacher support material (May 2005) and on the OCC.

Page 4: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide

Approaches to the teaching and learning of mathematical studies SL

13

Mathematical studies SL formula bookletEach student is required to have access to a clean copy of this booklet during the examination. It is recommended that teachers ensure students are familiar with the contents of this document from the beginning of the course. It is the responsibility of the school to download a copy from IBIS or the OCC, check that there are no printing errors, and ensure that there are sufficient copies available for all students.

Teacher support materialsA variety of teacher support materials will accompany this guide. These materials will include guidance for teachers on the introduction, planning and marking of projects, and specimen examination papers and markschemes.

Command terms and notation listTeachers and students need to be familiar with the IB notation and the command terms, as these will be used without explanation in the examination papers. The “Glossary of command terms” and “Notation list” appear as appendices in this guide.

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14 Mathematical studies SL guide

Syllabus

Prior learning topics

As noted in the previous section on prior learning, it is expected that all students have extensive previous mathematical experiences, but these will vary. It is expected that mathematical studies SL students will be familiar with the following topics before they take the examinations, because questions assume knowledge of them. Teachers must therefore ensure that any topics listed here that are unknown to their students at the start of the course are included at an early stage. They should also take into account the existing mathematical knowledge of their students to design an appropriate course of study for mathematical studies SL.

Students must be familiar with SI (Système International) units of length, mass and time, and their derived units.

The reference given in the left-hand column is to the topic in the syllabus content; for example, 1.0 refers to the prior learning for Topic 1—Number and algebra.

Learning how to use the graphic display calculator (GDC) effectively will be an integral part of the course, not a separate topic. Time has been allowed in each topic of the syllabus to do this.

Content Further guidance

1.0 Basic use of the four operations of arithmetic, using integers, decimals and fractions, including order of operations.

Prime numbers, factors and multiples.

Simple applications of ratio, percentage and proportion.

Examples: 2(3 4 7) 62+ × = ; 2 3 4 7 34× + × = .

Basic manipulation of simple algebraic expressions, including factorization and expansion.

Examples: ( );ab ac a b c+ = + 2( 1)( 2) 3 2x x x x+ + = + + .

Rearranging formulae. Example: 1 22

AA bh hb

= ⇒ = .

Evaluating expressions by substitution.

Example: If 3x = − then 2 22 3 ( 3) 2( 3) 3 18x x− + = − − − + = .

Solving linear equations in one variable. Examples: 3( 6) 4( 1) 0x x+ − − = ; 6 4 7

5x+ = .

Solving systems of linear equations in two variables. Example: 3 4 13x y+ = , 1 2 1

3x y− = − .

Evaluating exponential expressions with integer values. Examples: ,ba b∈ ; 4 12

16− = ; 4( 2) 16− = .

Use of inequalities , , ,< ≤ > ≥ . Intervals on the real number line.

Example: 2 5,x x< ≤ ∈ .

Solving linear inequalities. Example: 2 5 7x x+ < − .

Familiarity with commonly accepted world currencies.

Examples: Swiss franc (CHF); United States dollar (USD); British pound sterling (GBP); euro (EUR); Japanese yen (JPY); Australian dollar (AUD).

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Mathematical studies SL guide 15

Prior learning topics

Content Further guidance

2.0 The collection of data and its representation in bar charts, pie charts and pictograms.

5.0 Basic geometric concepts: point, line, plane, angle.

Simple two-dimensional shapes and their properties, including perimeters and areas of circles, triangles, quadrilaterals and compound shapes.

SI units for length and area.

Pythagoras’ theorem.

Coordinates in two dimensions.

Midpoints, distance between points.

Page 7: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

16 Mathematical studies SL guide

Sylla

bus

cont

ent

Sylla

bus

Topi

c 1—

Num

ber a

nd a

lgeb

ra

20 h

ours

Th

e ai

ms o

f thi

s top

ic a

re to

intro

duce

som

e ba

sic

elem

ents

and

con

cept

s of m

athe

mat

ics,

and

to li

nk th

ese

to fi

nanc

ial a

nd o

ther

app

licat

ions

.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.1

Nat

ural

num

bers

,

; int

eger

s,

; rat

iona

l nu

mbe

rs,

; and

real

num

bers

,

.

Not

req

uire

d:

proo

f of i

rrat

iona

lity,

for e

xam

ple,

of

2.

Link

with

dom

ain

and

rang

e 6.

1.

Int:

His

toric

al d

evel

opm

ent o

f num

ber s

yste

m.

Aw

aren

ess t

hat o

ur m

oder

n nu

mer

als a

re

deve

lope

d fr

om th

e A

rabi

c no

tatio

n.

TOK

: Do

mat

hem

atic

al sy

mbo

ls h

ave

sens

e in

th

e sa

me

way

that

wor

ds h

ave

sens

e? Is

zer

o di

ffer

ent?

Are

thes

e nu

mbe

rs c

reat

ed o

r di

scov

ered

? D

o th

ese

num

bers

exi

st?

1.2

App

roxi

mat

ion:

dec

imal

pla

ces,

sign

ifica

nt

figur

es.

Perc

enta

ge e

rror

s.

Stud

ents

shou

ld b

e aw

are

of th

e er

rors

that

can

re

sult

from

pre

mat

ure

roun

ding

. A

ppl:

Cur

renc

y ap

prox

imat

ions

to n

eare

st

who

le n

umbe

r, eg

pes

o, y

en. C

urre

ncy

appr

oxim

atio

ns to

nea

rest

cen

t/pen

ny, e

g eu

ro,

dolla

r, po

und.

App

l: Ph

ysic

s 1.1

(ran

ge o

f mag

nitu

des)

.

App

l: M

eteo

rolo

gy, a

ltern

ativ

e ro

undi

ng

met

hods

.

App

l: B

iolo

gy 2

.1.5

(mic

rosc

opic

m

easu

rem

ent).

TOK

: App

reci

atio

n of

the

diff

eren

ces o

f sca

le

in n

umbe

r, an

d of

the

way

num

bers

are

use

d th

at a

re w

ell b

eyon

d ou

r eve

ryda

y ex

perie

nce.

Estim

atio

n.

Stud

ents

shou

ld b

e ab

le to

reco

gniz

e w

heth

er

the

resu

lts o

f cal

cula

tions

are

reas

onab

le,

incl

udin

g re

ason

able

val

ues o

f, fo

r exa

mpl

e,

leng

ths,

angl

es a

nd a

reas

.

For e

xam

ple,

leng

ths c

anno

t be

nega

tive.

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Mathematical studies SL guide 17

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.3

Expr

essi

ng n

umbe

rs in

the

form

10

ka×

, whe

re

110

a≤

< a

nd k

is a

n in

tege

r. St

uden

ts sh

ould

be

able

to u

se sc

ient

ific

mod

e on

the

GD

C.

App

l: V

ery

larg

e an

d ve

ry sm

all n

umbe

rs, e

g as

trono

mic

al d

ista

nces

, sub

-ato

mic

par

ticle

s;

Phys

ics 1

.1; g

loba

l fin

anci

al fi

gure

s.

App

l: C

hem

istry

1.1

(Avo

gadr

o’s n

umbe

r).

App

l: Ph

ysic

s 1.2

(sci

entif

ic n

otat

ion)

.

App

l: C

hem

istry

and

bio

logy

(sci

entif

ic

nota

tion)

.

App

l: Ea

rth sc

ienc

e (e

arth

quak

e m

easu

rem

ent

scal

e).

Ope

ratio

ns w

ith n

umbe

rs in

this

form

. C

alcu

lato

r not

atio

n is

not

acc

epta

ble.

For e

xam

ple,

5.2

E3 is

not

acc

epta

ble.

1.4

SI (S

ystè

me I

nter

natio

nal)

and

othe

r bas

ic u

nits

of m

easu

rem

ent:

for e

xam

ple,

kilo

gram

(kg)

, m

etre

(m),

seco

nd (s

), lit

re (l

), m

etre

per

seco

nd

(m s–1

), C

elsi

us sc

ale.

Stud

ents

shou

ld b

e ab

le to

con

vert

betw

een

diff

eren

t uni

ts.

Link

with

the

form

of t

he n

otat

ion

in 1

.3, f

or

exam

ple,

6

5km

510

mm

.

App

l: Sp

eed,

acc

eler

atio

n, fo

rce;

Phy

sics

2.1

, Ph

ysic

s 2.2

; con

cent

ratio

n of

a so

lutio

n;

Che

mis

try 1

.5.

Int:

SI n

otat

ion.

TOK

: Doe

s the

use

of S

I not

atio

n he

lp u

s to

thin

k of

mat

hem

atic

s as a

“un

iver

sal

lang

uage

”?

TOK

: Wha

t is m

easu

rabl

e? H

ow c

an o

ne

mea

sure

mat

hem

atic

al a

bilit

y?

1.5

Cur

renc

y co

nver

sion

s. St

uden

ts sh

ould

be

able

to p

erfo

rm c

urre

ncy

trans

actio

ns in

volv

ing

com

mis

sion

. A

ppl:

Econ

omic

s 3.2

(exc

hang

e ra

tes)

.

Aim

8: T

he e

thic

al im

plic

atio

ns o

f tra

ding

in

curr

ency

and

its e

ffec

t on

diff

eren

t nat

iona

l co

mm

uniti

es.

Int:

The

eff

ect o

f flu

ctua

tions

in c

urre

ncy

rate

s on

inte

rnat

iona

l tra

de.

Page 9: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide18

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.6

Use

of a

GD

C to

solv

e

• pa

irs o

f lin

ear e

quat

ions

in tw

o va

riabl

es

In e

xam

inat

ions

, no

spec

ific

met

hod

of

solu

tion

will

be

requ

ired.

TO

K: E

quat

ions

with

no

solu

tions

. Aw

aren

ess

that

whe

n m

athe

mat

icia

ns ta

lk a

bout

“i

mag

inar

y” o

r “re

al”

solu

tions

they

are

usi

ng

prec

ise

tech

nica

l ter

ms t

hat d

o no

t hav

e th

e sa

me

mea

ning

as t

he e

very

day

term

s. •

quad

ratic

equ

atio

ns.

Stan

dard

term

inol

ogy,

such

as z

eros

or r

oots

, sh

ould

be

taug

ht.

Link

with

qua

drat

ic m

odel

s in

6.3.

1.7

Arit

hmet

ic se

quen

ces a

nd se

ries,

and

thei

r ap

plic

atio

ns.

TO

K: I

nfor

mal

and

form

al re

ason

ing

in

mat

hem

atic

s. H

ow d

oes m

athe

mat

ical

pro

of

diff

er fr

om g

ood

reas

onin

g in

eve

ryda

y lif

e? Is

m

athe

mat

ical

reas

onin

g di

ffer

ent f

rom

sc

ient

ific

reas

onin

g?

TOK

: Bea

uty

and

eleg

ance

in m

athe

mat

ics.

Fibo

nacc

i num

bers

and

con

nect

ions

with

the

Gol

den

ratio

.

Use

of t

he fo

rmul

ae fo

r the

nth

term

and

the

sum

of t

he fi

rst n

term

s of t

he se

quen

ce.

Stud

ents

may

use

a G

DC

for c

alcu

latio

ns, b

ut

they

will

be

expe

cted

to id

entif

y th

e fir

st te

rm

and

the

com

mon

diff

eren

ce.

1.8

Geo

met

ric se

quen

ces a

nd se

ries.

Use

of t

he fo

rmul

ae fo

r the

nth

term

and

the

sum

of t

he fi

rst n

term

s of t

he se

quen

ce.

Not

req

uire

d:

form

al p

roof

s of f

orm

ulae

.

Stud

ents

may

use

a G

DC

for c

alcu

latio

ns, b

ut

they

will

be

expe

cted

to id

entif

y th

e fir

st te

rm

and

the

com

mon

ratio

.

Not

req

uire

d:

use

of lo

garit

hms t

o fin

d n,

giv

en th

e su

m o

f th

e fir

st n

term

s; su

ms t

o in

finity

.

Page 10: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide 19

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.6

Use

of a

GD

C to

solv

e

• pa

irs o

f lin

ear e

quat

ions

in tw

o va

riabl

es

In e

xam

inat

ions

, no

spec

ific

met

hod

of

solu

tion

will

be

requ

ired.

TO

K: E

quat

ions

with

no

solu

tions

. Aw

aren

ess

that

whe

n m

athe

mat

icia

ns ta

lk a

bout

“i

mag

inar

y” o

r “re

al”

solu

tions

they

are

usi

ng

prec

ise

tech

nica

l ter

ms t

hat d

o no

t hav

e th

e sa

me

mea

ning

as t

he e

very

day

term

s. •

quad

ratic

equ

atio

ns.

Stan

dard

term

inol

ogy,

such

as z

eros

or r

oots

, sh

ould

be

taug

ht.

Link

with

qua

drat

ic m

odel

s in

6.3.

1.7

Arit

hmet

ic se

quen

ces a

nd se

ries,

and

thei

r ap

plic

atio

ns.

TO

K: I

nfor

mal

and

form

al re

ason

ing

in

mat

hem

atic

s. H

ow d

oes m

athe

mat

ical

pro

of

diff

er fr

om g

ood

reas

onin

g in

eve

ryda

y lif

e? Is

m

athe

mat

ical

reas

onin

g di

ffer

ent f

rom

sc

ient

ific

reas

onin

g?

TOK

: Bea

uty

and

eleg

ance

in m

athe

mat

ics.

Fibo

nacc

i num

bers

and

con

nect

ions

with

the

Gol

den

ratio

.

Use

of t

he fo

rmul

ae fo

r the

nth

term

and

the

sum

of t

he fi

rst n

term

s of t

he se

quen

ce.

Stud

ents

may

use

a G

DC

for c

alcu

latio

ns, b

ut

they

will

be

expe

cted

to id

entif

y th

e fir

st te

rm

and

the

com

mon

diff

eren

ce.

1.8

Geo

met

ric se

quen

ces a

nd se

ries.

Use

of t

he fo

rmul

ae fo

r the

nth

term

and

the

sum

of t

he fi

rst n

term

s of t

he se

quen

ce.

Not

req

uire

d:

form

al p

roof

s of f

orm

ulae

.

Stud

ents

may

use

a G

DC

for c

alcu

latio

ns, b

ut

they

will

be

expe

cted

to id

entif

y th

e fir

st te

rm

and

the

com

mon

ratio

.

Not

req

uire

d:

use

of lo

garit

hms t

o fin

d n,

giv

en th

e su

m o

f th

e fir

st n

term

s; su

ms t

o in

finity

.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

1.9

Fina

ncia

l app

licat

ions

of g

eom

etric

sequ

ence

s an

d se

ries:

• co

mpo

und

inte

rest

• an

nual

dep

reci

atio

n.

Not

req

uire

d:

use

of lo

garit

hms.

Use

of t

he G

DC

is e

xpec

ted,

incl

udin

g bu

ilt-in

fina

ncia

l pac

kage

s.

The

conc

ept o

f sim

ple

inte

rest

may

be

used

as

an in

trodu

ctio

n to

com

poun

d in

tere

st b

ut w

ill

not b

e ex

amin

ed.

In e

xam

inat

ions

, que

stio

ns th

at a

sk st

uden

ts to

de

rive

the

form

ula

will

not

be

set.

Com

poun

d in

tere

st c

an b

e ca

lcul

ated

yea

rly,

half-

year

ly, q

uarte

rly o

r mon

thly

.

Link

with

exp

onen

tial m

odel

s 6.4

.

App

l: Ec

onom

ics 3

.2 (e

xcha

nge

rate

s).

Aim

8: E

thic

al p

erce

ptio

ns o

f bor

row

ing

and

lend

ing

mon

ey.

Int:

Do

all s

ocie

ties v

iew

inve

stm

ent a

nd

inte

rest

in th

e sa

me

way

?

Page 11: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide20

Syllabus content To

pic

2—D

escr

iptiv

e st

atis

tics

12 h

ours

Th

e ai

m o

f thi

s top

ic is

to d

evel

op te

chni

ques

to d

escr

ibe

and

inte

rpre

t set

s of d

ata,

in p

repa

ratio

n fo

r fur

ther

stat

istic

al a

pplic

atio

ns.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

2.1

Cla

ssifi

catio

n of

dat

a as

disc

rete

or c

ontin

uous

. St

uden

ts sh

ould

und

erst

and

the

conc

ept o

f po

pula

tion

and

of re

pres

enta

tive

and

rand

om

sam

plin

g. S

ampl

ing

will

not

be

exam

ined

but

ca

n be

use

d in

inte

rnal

ass

essm

ent.

App

l: Ps

ycho

logy

3 (r

esea

rch

met

hodo

logy

).

App

l: B

iolo

gy 1

(sta

tistic

al a

naly

sis)

.

TOK

: Val

idity

of d

ata

and

intro

duct

ion

of

bias

.

2.2

Sim

ple

disc

rete

dat

a: fr

eque

ncy

tabl

es.

2.3

Gro

uped

dis

cret

e or

con

tinuo

us d

ata:

freq

uenc

y ta

bles

; mid

-inte

rval

val

ues;

upp

er a

nd lo

wer

bo

unda

ries.

Freq

uenc

y hi

stog

ram

s.

In e

xam

inat

ions

, fre

quen

cy h

isto

gram

s will

ha

ve e

qual

cla

ss in

terv

als.

App

l: G

eogr

aphy

(geo

grap

hica

l ana

lyse

s).

2.4

Cum

ulat

ive

freq

uenc

y ta

bles

for g

roup

ed

disc

rete

dat

a an

d fo

r gro

uped

con

tinuo

us d

ata;

cu

mul

ativ

e fr

eque

ncy

curv

es, m

edia

n an

d qu

artil

es.

Box

-and

-whi

sker

dia

gram

.

Not

req

uire

d:

treat

men

t of o

utlie

rs.

Use

of G

DC

to p

rodu

ce h

isto

gram

s and

box

-an

d-w

hisk

er d

iagr

ams.

2.5

Mea

sure

s of c

entra

l ten

denc

y.

For s

impl

e di

scre

te d

ata:

mea

n; m

edia

n; m

ode.

For g

roup

ed d

iscr

ete

and

cont

inuo

us d

ata:

es

timat

e of

a m

ean;

mod

al c

lass

.

Stud

ents

shou

ld u

se m

id-in

terv

al v

alue

s to

estim

ate

the

mea

n of

gro

uped

dat

a.

In e

xam

inat

ions

, que

stio

ns u

sing

∑ n

otat

ion

will

not

be

set.

Aim

8: T

he e

thic

al im

plic

atio

ns o

f usi

ng

stat

istic

s to

mis

lead

.

Page 12: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide 21

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

2.6

Mea

sure

s of d

ispe

rsio

n: ra

nge,

inte

rqua

rtile

ra

nge,

stan

dard

dev

iatio

n.

Stud

ents

shou

ld u

se m

id-in

terv

al v

alue

s to

estim

ate

the

stan

dard

dev

iatio

n of

gro

uped

da

ta.

In e

xam

inat

ions

:

• st

uden

ts a

re e

xpec

ted

to u

se a

GD

C to

ca

lcul

ate

stan

dard

dev

iatio

ns

• th

e da

ta se

t will

be

treat

ed a

s the

po

pula

tion.

Stud

ents

shou

ld b

e aw

are

that

the

IB n

otat

ion

may

diff

er fr

om th

e no

tatio

n on

thei

r GD

C.

Use

of c

ompu

ter s

prea

dshe

et so

ftwar

e is

enco

urag

ed in

the

treat

men

t of t

his t

opic

.

Int:

The

ben

efits

of s

harin

g an

d an

alys

ing

data

fr

om d

iffer

ent c

ount

ries.

TOK

: Is s

tand

ard

devi

atio

n a

mat

hem

atic

al

disc

over

y or

a c

reat

ion

of th

e hu

man

min

d?

Page 13: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide22

Syllabus content To

pic

3—Lo

gic,

set

s an

d pr

obab

ility

20

hou

rs

The

aim

s of t

his t

opic

are

to in

trodu

ce th

e pr

inci

ples

of l

ogic

, to

use

set t

heor

y to

intro

duce

pro

babi

lity,

and

to d

eter

min

e th

e lik

elih

ood

of ra

ndom

eve

nts u

sing

a

varie

ty o

f tec

hniq

ues.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

3.1

Bas

ic c

once

pts o

f sym

bolic

logi

c: d

efin

ition

of

a pr

opos

ition

; sym

bolic

not

atio

n of

pr

opos

ition

s.

3.2

Com

poun

d st

atem

ents

: im

plic

atio

n, ⇒

; eq

uiva

lenc

e, ⇔

; neg

atio

n, ¬

; con

junc

tion,

; disj

unct

ion,

∨; e

xclu

sive

disj

unct

ion,

∨.

Tran

slat

ion

betw

een

verb

al st

atem

ents

and

sy

mbo

lic fo

rm.

3.3

Trut

h ta

bles

: con

cept

s of l

ogic

al c

ontra

dict

ion

and

taut

olog

y.

A m

axim

um o

f thr

ee p

ropo

sitio

ns w

ill b

e us

ed

in tr

uth

tabl

es.

Trut

h ta

bles

can

be

used

to il

lust

rate

the

asso

ciat

ive

and

dist

ribut

ive

prop

ertie

s of

conn

ectiv

es, a

nd fo

r var

iatio

ns o

f im

plic

atio

n an

d eq

uiva

lenc

e st

atem

ents

, for

exa

mpl

e,

qp

¬⇒

¬.

3.4

Con

vers

e, in

vers

e, c

ontra

posi

tive.

Logi

cal e

quiv

alen

ce.

A

ppl:

Use

of a

rgum

ents

in d

evel

opin

g a

logi

cal e

ssay

stru

ctur

e.

App

l: C

ompu

ter p

rogr

amm

ing;

dig

ital c

ircui

ts;

Phys

ics H

L 14

.1; P

hysi

cs S

L C

1.

TOK

: Ind

uctiv

e an

d de

duct

ive

logi

c, fa

llaci

es.

Test

ing

the

valid

ity o

f sim

ple

argu

men

ts

thro

ugh

the

use

of tr

uth

tabl

es.

The

topi

c m

ay b

e ex

tend

ed to

incl

ude

syllo

gism

s. In

exa

min

atio

ns th

ese

will

not

be

test

ed.

Page 14: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide 23

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

3.5

Bas

ic c

once

pts o

f set

theo

ry: e

lem

ents

xA

∈,

subs

ets A

B⊂

; int

erse

ctio

n A

B∩

; uni

on

AB

∪; c

ompl

emen

t A′

.

Ven

n di

agra

ms a

nd si

mpl

e ap

plic

atio

ns.

Not

req

uire

d:

know

ledg

e of

de

Mor

gan’

s law

s.

In e

xam

inat

ions

, the

uni

vers

al se

t U w

ill

incl

ude

no m

ore

than

thre

e su

bset

s.

The

empt

y se

t is d

enot

ed b

y ∅

.

3.6

Sam

ple

spac

e; e

vent

A; c

ompl

emen

tary

eve

nt,

A′.

Prob

abili

ty o

f an

even

t.

Prob

abili

ty o

f a c

ompl

emen

tary

eve

nt.

Expe

cted

val

ue.

Prob

abili

ty m

ay b

e in

trodu

ced

and

taug

ht in

a

prac

tical

way

usi

ng c

oins

, dic

e, p

layi

ng c

ards

an

d ot

her e

xam

ples

to d

emon

strat

e ra

ndom

be

havi

our.

In e

xam

inat

ions

, que

stio

ns in

volv

ing

play

ing

card

s will

not

be

set.

App

l: A

ctua

rial s

tudi

es, p

roba

bilit

y of

life

sp

ans a

nd th

eir e

ffec

t on

insu

ranc

e.

App

l: G

over

nmen

t pla

nnin

g ba

sed

on

proj

ecte

d fig

ures

.

TOK

: The

oret

ical

and

exp

erim

enta

l pr

obab

ility

.

3.7

Prob

abili

ty o

f com

bine

d ev

ents

, mut

ually

ex

clus

ive

even

ts, i

ndep

ende

nt e

vent

s. St

uden

ts sh

ould

be

enco

urag

ed to

use

the

mos

t ap

prop

riate

met

hod

in so

lvin

g in

divi

dual

qu

estio

ns.

App

l: B

iolo

gy 4

.3 (t

heor

etic

al g

enet

ics)

; B

iolo

gy 4

.3.2

(Pun

nett

squa

res)

.

App

l: Ph

ysic

s HL1

3.1

(det

erm

inin

g th

e po

sitio

n of

an

elec

tron)

; Phy

sics

SL

B1.

Aim

8: T

he e

thic

s of g

ambl

ing.

TOK

: The

per

cept

ion

of ri

sk, i

n bu

sine

ss, i

n m

edic

ine

and

safe

ty in

trav

el.

Use

of t

ree

diag

ram

s, V

enn

diag

ram

s, sa

mpl

e sp

ace

diag

ram

s and

tabl

es o

f out

com

es.

Prob

abili

ty u

sing

“w

ith re

plac

emen

t” a

nd

“with

out r

epla

cem

ent”

.

Con

ditio

nal p

roba

bilit

y.

Prob

abili

ty q

uest

ions

will

be

plac

ed in

con

text

an

d w

ill m

ake

use

of d

iagr

amm

atic

re

pres

enta

tions

.

In e

xam

inat

ions

, que

stio

ns re

quiri

ng th

e ex

clus

ive

use

of th

e fo

rmul

a in

sect

ion

3.7

of

the

form

ula

book

let w

ill n

ot b

e se

t.

Page 15: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide24

Syllabus content To

pic

4—St

atis

tical

app

licat

ions

17

hou

rs

The

aim

s of t

his t

opic

are

to d

evel

op te

chni

ques

in in

fere

ntia

l sta

tistic

s in

orde

r to

anal

yse

sets

of d

ata,

dra

w c

oncl

usio

ns a

nd in

terp

ret t

hese

.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.1

The

norm

al d

istri

butio

n.

The

conc

ept o

f a ra

ndom

var

iabl

e; o

f the

pa

ram

eter

s µ

and

σ; o

f the

bel

l sha

pe; t

he

sym

met

ry a

bout

=.

Stud

ents

shou

ld b

e aw

are

that

app

roxi

mat

ely

68%

of t

he d

ata

lies b

etw

een µ

σ±

, 95%

lies

be

twee

n 2

µσ

± a

nd 9

9% li

es b

etw

een

σ±

.

App

l: Ex

ampl

es o

f mea

sure

men

ts, r

angi

ng

from

psy

chol

ogic

al to

phy

sica

l phe

nom

ena,

th

at c

an b

e ap

prox

imat

ed, t

o va

ryin

g de

gree

s, by

the

norm

al d

istri

butio

n.

App

l: B

iolo

gy 1

(sta

tistic

al a

naly

sis)

.

App

l: Ph

ysic

s 3.2

(kin

etic

mol

ecul

ar th

eory

). D

iagr

amm

atic

repr

esen

tatio

n.

Use

of s

ketc

hes o

f nor

mal

cur

ves a

nd sh

adin

g w

hen

usin

g th

e G

DC

is e

xpec

ted.

Nor

mal

pro

babi

lity

calc

ulat

ions

. St

uden

ts w

ill b

e ex

pect

ed to

use

the

GD

C

whe

n ca

lcul

atin

g pr

obab

ilitie

s and

inve

rse

norm

al.

Expe

cted

val

ue.

Inve

rse

norm

al c

alcu

latio

ns.

In e

xam

inat

ions

, inv

erse

nor

mal

que

stio

ns w

ill

not i

nvol

ve fi

ndin

g th

e m

ean

or st

anda

rd

devi

atio

n.

Not

req

uire

d:

Tran

sfor

mat

ion

of a

ny n

orm

al v

aria

ble

to th

e st

anda

rdiz

ed n

orm

al.

Tran

sfor

mat

ion

of a

ny n

orm

al v

aria

ble

to th

e st

anda

rdiz

ed n

orm

al v

aria

ble,

z, m

ay b

e ap

prop

riate

in in

tern

al a

sses

smen

t.

In e

xam

inat

ions

, que

stio

ns re

quiri

ng th

e us

e of

z s

core

s will

not

be

set.

Page 16: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide 25

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.2

Biv

aria

te d

ata:

the

conc

ept o

f cor

rela

tion.

St

uden

ts sh

ould

be

able

to m

ake

the

dist

inct

ion

betw

een

corr

elat

ion

and

caus

atio

n.

App

l: B

iolo

gy; P

hysi

cs; C

hem

istry

; Soc

ial

scie

nces

.

TOK

: Doe

s cor

rela

tion

impl

y ca

usat

ion?

Sc

atte

r dia

gram

s; li

ne o

f bes

t fit,

by

eye,

pa

ssin

g th

roug

h th

e m

ean

poin

t.

Pear

son’

s pro

duct

–mom

ent c

orre

latio

n co

effic

ient

, r.

Han

d ca

lcul

atio

ns o

f r m

ay e

nhan

ce

unde

rsta

ndin

g.

In e

xam

inat

ions

, stu

dent

s will

be

expe

cted

to

use

a G

DC

to c

alcu

late

r.

Inte

rpre

tatio

n of

pos

itive

, zer

o an

d ne

gativ

e,

stro

ng o

r wea

k co

rrel

atio

ns.

4.3

The

regr

essi

on li

ne fo

r y o

n x.

H

and

calc

ulat

ions

of t

he re

gres

sion

line

may

en

hanc

e un

ders

tand

ing.

In e

xam

inat

ions

, stu

dent

s will

be

expe

cted

to

use

a G

DC

to fi

nd th

e re

gres

sion

line

.

App

l: C

hem

istry

11.

3 (g

raph

ical

tech

niqu

es).

TOK

: Can

we

relia

bly

use

the

equa

tion

of th

e re

gres

sion

line

to m

ake

pred

ictio

ns?

Use

of t

he re

gres

sion

line

for p

redi

ctio

n pu

rpos

es.

Stud

ents

shou

ld b

e aw

are

of th

e da

nger

s of

extra

pola

tion.

Page 17: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide26

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

4.4

The

test

for i

ndep

ende

nce:

form

ulat

ion

of

null

and

alte

rnat

ive

hypo

thes

es; s

igni

fican

ce

leve

ls; c

ontin

genc

y ta

bles

; exp

ecte

d fr

eque

ncie

s; d

egre

es o

f fre

edom

; p-v

alue

s.

In e

xam

inat

ions

:

• th

e m

axim

um n

umbe

r of r

ows o

r col

umns

in

a c

ontin

genc

y ta

ble

will

be

4

• th

e de

gree

s of f

reed

om w

ill a

lway

s be

grea

ter t

han

one

• th

e 2

χ c

ritic

al v

alue

will

alw

ays b

e gi

ven

• on

ly q

uesti

ons o

n up

per t

ail t

ests

with

co

mm

only

use

d si

gnifi

canc

e le

vels

(1%

, 5%

, 10

%) w

ill b

e se

t.

Cal

cula

tion

of e

xpec

ted

freq

uenc

ies b

y ha

nd is

re

quire

d.

Han

d ca

lcul

atio

ns o

f 2

χm

ay e

nhan

ce

unde

rsta

ndin

g.

In e

xam

inat

ions

stud

ents

will

be

expe

cted

to

use

the

GD

C to

cal

cula

te th

e 2

χ st

atis

tic.

If us

ing

test

s in

inte

rnal

ass

essm

ent,

stud

ents

shou

ld b

e aw

are

of th

e lim

itatio

ns o

f th

e te

st fo

r sm

all e

xpec

ted

freq

uenc

ies;

ex

pect

ed fr

eque

ncie

s mus

t be

grea

ter t

han

5.

If th

e de

gree

of f

reed

om is

1, t

hen

Yat

es’s

co

ntin

uity

cor

rect

ion

shou

ld b

e ap

plie

d.

App

l: B

iolo

gy (i

nter

nal a

sses

smen

t);

Psyc

holo

gy; G

eogr

aphy

.

TOK

: Sci

entif

ic m

etho

d.

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Mathematical studies SL guide 27

Syllabus content To

pic

5—G

eom

etry

and

trig

onom

etry

18

hou

rs

The

aim

s of

this

topi

c ar

e to

dev

elop

the

abili

ty to

dra

w c

lear

dia

gram

s in

two

dim

ensi

ons,

and

to a

pply

app

ropr

iate

geo

met

ric a

nd tr

igon

omet

ric te

chni

ques

to

prob

lem

-sol

ving

in tw

o an

d th

ree

dim

ensi

ons.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

5.1

Equa

tion

of a

line

in tw

o di

men

sion

s: th

e fo

rms

ym

xc

=+

and

0

axby

d+

+=

. Li

nk w

ith li

near

func

tions

in 6

.2.

App

l: G

radi

ents

of m

ount

ain

road

s, eg

C

anad

ian

Hig

hway

. Gra

dien

ts o

f acc

ess r

amps

.

App

l: Ec

onom

ics 1

.2 (e

lasti

city

).

TOK

: Des

carte

s sho

wed

that

geo

met

ric

prob

lem

s can

be

solv

ed a

lgeb

raic

ally

and

vic

e ve

rsa.

Wha

t doe

s thi

s tel

l us a

bout

m

athe

mat

ical

repr

esen

tatio

n an

d m

athe

mat

ical

kn

owle

dge?

Gra

dien

t; in

terc

epts

.

Poin

ts o

f int

erse

ctio

n of

line

s. Li

nk w

ith so

lutio

ns o

f pai

rs o

f lin

ear e

quat

ions

in

1.6

.

Line

s with

gra

dien

ts,

1m a

nd

2m

.

Para

llel l

ines

1

2m

m=

.

Perp

endi

cula

r lin

es,

12

1m

=−

.

5.2

Use

of s

ine,

cos

ine

and

tang

ent r

atio

s to

find

the

side

s and

ang

les o

f rig

ht-a

ngle

d tri

angl

es.

Ang

les o

f ele

vatio

n an

d de

pres

sion

.

Prob

lem

s may

inco

rpor

ate

Pyth

agor

as’

theo

rem

.

In e

xam

inat

ions

, que

stio

ns w

ill o

nly

be se

t in

degr

ees.

App

l: Tr

iang

ulat

ion,

map

-mak

ing,

find

ing

prac

tical

mea

sure

men

ts u

sing

trig

onom

etry

.

Int:

Dia

gram

s of P

ytha

gora

s’ th

eore

m o

ccur

in

early

Chi

nese

and

Indi

an m

anus

crip

ts. T

he

earli

est r

efer

ence

s to

trigo

nom

etry

are

in In

dian

m

athe

mat

ics.

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Mathematical studies SL guide28

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

5.3

Use

of t

he si

ne ru

le:

sin

sin

sin

ab

cA

BC

==

. In

all

area

s of t

his t

opic

, stu

dent

s sho

uld

be

enco

urag

ed to

sket

ch w

ell-l

abel

led

diag

ram

s to

supp

ort t

heir

solu

tions

.

The

ambi

guou

s cas

e co

uld

be ta

ught

, but

will

no

t be

exam

ined

.

In e

xam

inat

ions

, que

stio

ns w

ill o

nly

be se

t in

degr

ees.

App

l: V

ecto

rs; P

hysi

cs 1

.3; b

earin

gs.

Use

of t

he c

osin

e ru

le2

22

2co

sa

bc

bcA

=+

−;

22

2

cos

2b

ca

Abc

+−

=.

Use

of a

rea

of a

tria

ngle

= 1

sin

2ab

C.

Con

stru

ctio

n of

labe

lled

diag

ram

s fro

m v

erba

l st

atem

ents

.

TO

K: U

se th

e fa

ct th

at th

e co

sine

rule

is o

ne

poss

ible

gen

eral

izat

ion

of P

ytha

gora

s’ th

eore

m

to e

xplo

re th

e co

ncep

t of “

gene

ralit

y”.

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Mathematical studies SL guide 29

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

5.4

Geo

met

ry o

f thr

ee-d

imen

sion

al so

lids:

cub

oid;

rig

ht p

rism

; rig

ht p

yram

id; r

ight

con

e; c

ylin

der;

sphe

re; h

emis

pher

e; a

nd c

ombi

natio

ns o

f the

se

solid

s.

The

dist

ance

bet

wee

n tw

o po

ints

; eg

betw

een

two

verti

ces o

r ver

tices

with

mid

poin

ts o

r m

idpo

ints

with

mid

poin

ts.

The

size

of a

n an

gle

betw

een

two

lines

or

betw

een

a lin

e an

d a

plan

e.

Not

req

uire

d:

angl

e be

twee

n tw

o pl

anes

.

In e

xam

inat

ions

, onl

y rig

ht-a

ngle

d tri

gono

met

ry q

uest

ions

will

be

set i

n re

fere

nce

to th

ree-

dim

ensi

onal

shap

es.

TOK

: Wha

t is a

n ax

iom

atic

syst

em?

Do

the

angl

es in

a tr

iang

le a

lway

s add

to 1

80°?

Non

-Euc

lidea

n ge

omet

ry, s

uch

as R

iem

ann’

s. Fl

ight

map

s of a

irlin

es.

App

l: A

rchi

tect

ure

and

desi

gn.

5.5

Vol

ume

and

surf

ace

area

s of t

he th

ree-

dim

ensi

onal

solid

s def

ined

in 5

.4.

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Mathematical studies SL guide30

Syllabus content To

pic

6—M

athe

mat

ical

mod

els

20 h

ours

Th

e ai

m o

f thi

s top

ic is

to d

evel

op u

nder

stan

ding

of s

ome

mat

hem

atic

al fu

nctio

ns th

at c

an b

e us

ed to

mod

el p

ract

ical

situ

atio

ns. E

xten

sive

use

of a

GD

C is

to b

e en

cour

aged

in th

is to

pic.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

6.1

Con

cept

of a

func

tion,

dom

ain,

rang

e an

d gr

aph.

Func

tion

nota

tion,

eg

(),

(

), (

)f

xv

tC

n.

Con

cept

of a

func

tion

as a

mat

hem

atic

al

mod

el.

In e

xam

inat

ions

:

• th

e do

mai

n is

the

set o

f all

real

num

bers

un

less

oth

erw

ise

stat

ed

• m

appi

ng n

otat

ion

:fx

y

will

not

be

used

.

TO

K: W

hy c

an w

e us

e m

athe

mat

ics t

o de

scrib

e th

e w

orld

and

mak

e pr

edic

tions

? Is

it

beca

use

we

disc

over

the

mat

hem

atic

al b

asis

of

the

wor

ld o

r bec

ause

we

impo

se o

ur o

wn

mat

hem

atic

al st

ruct

ures

ont

o th

e w

orld

?

The

rela

tions

hip

betw

een

real

-wor

ld p

robl

ems

and

mat

hem

atic

al m

odel

s.

6.2

Line

ar m

odel

s.

Line

ar fu

nctio

ns a

nd th

eir g

raph

s, (

)f

xm

xc

=+

.

Link

with

equ

atio

n of

a li

ne in

5.1

. A

ppl:

Con

vers

ion

grap

hs, e

g te

mpe

ratu

re o

r cu

rren

cy c

onve

rsio

n; P

hysi

cs 3

.1; E

cono

mic

s 3.

2.

6.3

Qua

drat

ic m

odel

s.

Qua

drat

ic fu

nctio

ns a

nd th

eir g

raph

s (p

arab

olas

): 2

()

fx

axbx

c=

++

;0

≠a

Link

with

the

quad

ratic

equ

atio

ns in

1.6

. Fu

nctio

ns w

ith z

ero,

one

or t

wo

real

root

s are

in

clud

ed.

App

l: C

ost f

unct

ions

; pro

ject

ile m

otio

n;

Phys

ics 9

.1; a

rea

func

tions

.

Prop

ertie

s of a

par

abol

a: sy

mm

etry

; ver

tex;

in

terc

epts

on

the

x-ax

is a

nd y

-axi

s.

Equa

tion

of th

e ax

is o

f sym

met

ry,

2bx

a=−

.

The

form

of t

he e

quat

ion

of th

e ax

is of

sy

mm

etry

may

initi

ally

be

foun

d by

in

vest

igat

ion.

Prop

ertie

s sho

uld

be il

lust

rate

d w

ith a

GD

C o

r gr

aphi

cal s

oftw

are.

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Mathematical studies SL guide 31

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

6.4

Expo

nent

ial m

odel

s.

Expo

nent

ial f

unct

ions

and

thei

r gra

phs:

(

);

,1,

0+

=+

∈≠

xf

xka

ca

ak

.

()

;,

1,0

−+

=+

∈≠

xf

xka

ca

ak

.

Con

cept

and

equ

atio

n of

a h

oriz

onta

l as

ympt

ote.

In e

xam

inat

ions

, stu

dent

s will

be

expe

cted

to

use

grap

hica

l met

hods

, inc

ludi

ng G

DC

s, to

so

lve

prob

lem

s.

App

l: B

iolo

gy 5

.3 (p

opul

atio

ns).

App

l: B

iolo

gy 5

.3.2

(pop

ulat

ion

grow

th);

Phys

ics 1

3.2

(rad

ioac

tive

deca

y); P

hysi

cs I2

(X

-ray

atte

nuat

ion)

; coo

ling

of a

liqu

id; s

prea

d of

a v

irus;

dep

reci

atio

n.

6.5

Mod

els u

sing

func

tions

of t

he fo

rm

()

...;

,=

++

mn

fx

axbx

mn

. In

exa

min

atio

ns, s

tude

nts w

ill b

e ex

pect

ed to

us

e gr

aphi

cal m

etho

ds, i

nclu

ding

GD

Cs,

to

solv

e pr

oble

ms.

Func

tions

of t

his t

ype

and

thei

r gra

phs.

The

y-ax

is a

s a v

ertic

al a

sym

ptot

e.

Exam

ples

: 4

()

35

3f

xx

x=

−+

, 2

4(

)3

gx

xx

=−

.

6.6

Dra

win

g ac

cura

te g

raph

s.

Cre

atin

g a

sket

ch fr

om in

form

atio

n gi

ven.

Tran

sfer

ring

a gr

aph

from

GD

C to

pap

er.

Rea

ding

, int

erpr

etin

g an

d m

akin

g pr

edic

tions

us

ing

grap

hs.

Stud

ents

shou

ld b

e aw

are

of th

e di

ffer

ence

be

twee

n th

e co

mm

and

term

s “dr

aw”

and

“ske

tch”

.

All

grap

hs sh

ould

be

labe

lled

and

have

som

e in

dica

tion

of sc

ale.

TO

K: D

oes a

gra

ph w

ithou

t lab

els o

r in

dica

tion

of sc

ale

have

mea

ning

?

Incl

uded

all

the

func

tions

abo

ve a

nd a

dditi

ons

and

subt

ract

ions

. Ex

ampl

es:

32

()

5f

xx

x=

+−

, (

)3

xg

xx

−=

+.

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Mathematical studies SL guide32

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Co

nten

t Fu

rthe

r gui

danc

e Li

nks

6.7

Use

of a

GD

C to

solv

e eq

uatio

ns in

volv

ing

com

bina

tions

of t

he fu

nctio

ns a

bove

. Ex

ampl

es:

32

23

1x

xx

+=

+−

, 53x

x=

.

Oth

er fu

nctio

ns c

an b

e us

ed fo

r mod

ellin

g in

in

tern

al a

sses

smen

t but

will

not

be

set o

n ex

amin

atio

n pa

pers

.

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Mathematical studies SL guide 33

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

6.7

Use

of a

GD

C to

solv

e eq

uatio

ns in

volv

ing

com

bina

tions

of t

he fu

nctio

ns a

bove

. Ex

ampl

es:

32

23

1x

xx

+=

+−

, 53x

x=

.

Oth

er fu

nctio

ns c

an b

e us

ed fo

r mod

ellin

g in

in

tern

al a

sses

smen

t but

will

not

be

set o

n ex

amin

atio

n pa

pers

.

Topi

c 7—

Intr

oduc

tion

to d

iffer

entia

l cal

culu

s 18

hou

rs

The

aim

of t

his t

opic

is to

intro

duce

the

conc

ept o

f the

der

ivat

ive

of a

func

tion

and

to a

pply

it to

opt

imiz

atio

n an

d ot

her p

robl

ems.

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

7.1

Con

cept

of t

he d

eriv

ativ

e as

a ra

te o

f cha

nge.

Tang

ent t

o a

curv

e.

Not

req

uire

d:

form

al tr

eatm

ent o

f lim

its.

Teac

hers

are

enc

oura

ged

to in

trodu

ce

diff

eren

tiatio

n th

roug

h a

grap

hica

l app

roac

h,

rath

er th

an a

form

al tr

eatm

ent.

Emph

asis

is p

lace

d on

inte

rpre

tatio

n of

the

conc

ept i

n di

ffer

ent c

onte

xts.

In e

xam

inat

ions

, que

stio

ns o

n di

ffer

entia

tion

from

firs

t prin

cipl

es w

ill n

ot b

e se

t.

App

l: R

ates

of c

hang

e in

eco

nom

ics,

kine

mat

ics a

nd m

edic

ine.

Aim

8: P

lagi

aris

m a

nd a

ckno

wle

dgm

ent o

f so

urce

s, eg

the

conf

lict b

etw

een

New

ton

and

Leib

nitz

, who

app

roac

hed

the

deve

lopm

ent o

f ca

lcul

us fr

om d

iffer

ent d

irect

ions

TOK

: Is i

ntui

tion

a va

lid w

ay o

f kno

win

g in

m

aths

?

How

is it

pos

sibl

e to

reac

h th

e sa

me

conc

lusi

on

from

diff

eren

t res

earc

h pa

ths?

7.2

The

prin

cipl

e th

at

1(

)(

)n

nf

xax

fx

anx

−′

=⇒

=.

The

deriv

ativ

e of

func

tions

of t

he fo

rm

1(

)...

,−

=+

+n

nf

xax

bxw

here

all

expo

nent

s are

in

tege

rs.

Stud

ents

shou

ld b

e fa

mili

ar w

ith th

e al

tern

ativ

e

nota

tion

for d

eriv

ativ

es d dy x

or

d dV r.

In e

xam

inat

ions

, kno

wle

dge

of th

e se

cond

de

rivat

ive

will

not

be

assu

med

.

Page 25: Teaching Syllabus component SL - sjsd.k12.mo.us · Syllabus component Teaching hours SL All topics are compulsory. Students must study all the sub-topics in each of the topics in

Mathematical studies SL guide34

Syllabus content

Co

nten

t Fu

rthe

r gui

danc

e Li

nks

7.3

Gra

dien

ts o

f cur

ves f

or g

iven

val

ues o

f x.

Val

ues o

f x w

here

(

)f

x′

is g

iven

.

The

use

of te

chno

logy

to fi

nd th

e gr

adie

nt a

t a

poin

t is a

lso

enco

urag

ed.

Equa

tion

of th

e ta

ngen

t at a

giv

en p

oint

. Th

e us

e of

tech

nolo

gy to

dra

w ta

ngen

t and

no

rmal

line

s is a

lso

enco

urag

ed.

Equa

tion

of th

e lin

e pe

rpen

dicu

lar t

o th

e ta

ngen

t at a

giv

en p

oint

(nor

mal

). Li

nks w

ith p

erpe

ndic

ular

line

s in

5.1.

7.4

Incr

easi

ng a

nd d

ecre

asin

g fu

nctio

ns.

Gra

phic

al in

terp

reta

tion

of

()

0f

x′

>,

()

0f

x′

=

and

()

0f

x′

<.

7.5

Val

ues o

f x w

here

the

grad

ient

of a

cur

ve is

ze

ro.

Solu

tion

of

()

0f

x′

=.

Stat

iona

ry p

oint

s.

The

use

of te

chno

logy

to d

ispl

ay

()

fx

and

(

)f

x′

, and

find

the

solu

tions

of

()

0f

x′

= is

al

so e

ncou

rage

d.

Loca

l max

imum

and

min

imum

poi

nts.

Aw

aren

ess t

hat a

loca

l max

imum

/min

imum

w

ill n

ot n

eces

saril

y be

the

grea

test

/leas

t val

ue

of th

e fu

nctio

n in

the

give

n do

mai

n.

Aw

aren

ess o

f poi

nts o

f inf

lexi

on w

ith z

ero

grad

ient

is to

be

enco

urag

ed, b

ut w

ill n

ot b

e ex

amin

ed.

7.6

Opt

imiz

atio

n pr

oble

ms.

Exam

ples

: Max

imiz

ing

prof

it, m

inim

izin

g co

st,

max

imiz

ing

volu

me

for g

iven

surf

ace

area

.

In e

xam

inat

ions

, que

stio

ns o

n ki

nem

atic

s will

no

t be

set.

App

l: Ef

ficie

nt u

se o

f mat

eria

l in

pack

agin

g.

App

l: Ph

ysic

s 2.1

(kin

emat

ics)

.