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MATME/PF/M09/N09/M10/N10
20 pages For final assessment in 2009 and 2010
MATHEMATICS
Standard Level
The portfolio - tasks
For use in 2009 and 2010
© International Baccalaureate Organization 2008
– 2 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
CONTENTS
Introduction
Type I tasks
Infinite Surds
Logarithm Bases
Matrix Binomials
Shady Areas
Parallels and Parallelograms
Type II tasks
Body Mass Index
Fishing Rods
Crows dropping nuts
Logan’s Logo
Criteria
Developing your own tasks
Old tasks
– 3 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Introduction
What is the purpose of this document
This document contains new tasks for the portfolio in mathematics SL. These tasks have been produced
by the IB, for teachers to use in 2009 and 2010. It should be noted that any tasks previously produced and
published by the IB will no longer be valid for assessment after November 2008. These include all the
tasks in any teacher support material (TSM). To assist teachers to identify these tasks, a list is included at
the end of this document.
What happens if teachers use these old tasks?
The inclusion of these old tasks in the portfolio will make the portfolio non-compliant, and such
portfolios will therefore attract a 10-mark penalty. Teachers may continue to use the old tasks as practice
tasks, but they should not be included in the portfolio for final assessment.
Why are these changes being made?
An interim version of the TSM for the current course was first published in 2004, with the full TSM
published in 2005. There were concerns that these documents were available for sale, potentially giving
students access to the student work and its accompanying assessment. Teachers also expressed concerns
that model answers soon became easily available on the internet and felt that this made it difficult to
ensure students’ work was their own. There were also frequent requests for more tasks to be published by
the IB, as many teachers are apprehensive about producing their own tasks.
What other documents should I use?
All teachers should have copies of the mathematics SL subject guide (second edition, September 2006),
including the teaching notes appendix, and the TSM (September 2005). Further information, including
additional notes on applying the criteria, are available on the Online Curriculum Centre (OCC).
Important news items are also available on the OCC, as are the diploma programme coordinator notes,
which contain updated information on a variety of issues.
Can I use these tasks before May 2009?
These tasks should only be submitted for final assessment from May 2009 to November 2010. Students
should not include them in portfolios before May 2009. If they are included, they will be subject to a
10-mark penalty.
– 4 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Type I – mathematical investigation
While many teachers incorporate a problem-solving approach into their classroom practice, students also
should be given the opportunity formally to carry out investigative work. The mathematical investigation
is intended to highlight that:
• the idea of investigation is fundamental to the study of mathematics
• investigation work often leads to an appreciation of how mathematics can be applied to solve
problems in a broad range of fields
• the discovery aspect of investigation work deepens understanding and provides intrinsic motivation
• during the process of investigation, students acquire mathematical knowledge, problem-solving
techniques, a knowledge of fundamental concepts and an increase in self-confidence.
All investigations develop from an initial problem, the starting point. The problem must be clearly stated
and contain no ambiguity. In addition, the problem should:
• provide a challenge and the opportunity for creativity
• contain multi-solution paths, that is, contain the potential for students to choose different courses of
action from a range of options.
Essential skills to be assessed
• Producing a strategy
• Generating data
• Recognizing patterns or structures
• Searching for further cases
• Forming a general statement
• Testing a general statement
• Justifying a general statement
• Appropriate use of technology
– 5 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
INFINITE SURDS SL TYPE I
The following expression is an example of an infinite surd.
1 1 1 1 1 ...
Consider this surd as a sequence of terms na where:
1 1 1a
2 1 1 1a
3 1 1 1 1a etc.
Find a formula for 1na in terms of na .
Calculate the decimal values of the first ten terms of the sequence. Using technology, plot the relation
between n and na . Describe what you notice. What does this suggest about the value of 1n na a as n
gets very large? Use your results to find the exact value for this infinite surd.
Consider another infinite surd ...2222 where the first term is 2 2 .
Repeat the entire process above to find the exact value for this surd.
Now consider the general infinite surd ...kkkk where the first term is k k .
Find an expression for the exact value of this general infinite surd in terms of k.
The value of an infinite surd is not always an integer.
Find some values of k that make the expression an integer. Find the general statement that represents all
the values of k for which the expression is an integer.
Test the validity of your general statement using other values of k.
Discuss the scope and/or limitations of your general statement.
Explain how you arrived at your general statement.
– 6 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
LOGARITHM BASES SL TYPE I
Consider the following sequences. Write down the next two terms of each sequence.
8log 2 , 8log 4 , 8log 8 , 8log16 , 8log 32 , …
81log 3 , 81log 9 , 81log 27 , 81log 81 , …
25log 5 , 25log 25 , 25log125 , 25log 625 , …
:
:
: k
m mlog , k
mm2log , 3log k
mm , 4log k
mm , …
Find an expression for the nth
term of each sequence. Write your expressions in the form q
p,
where p, q . Justify your answers using technology.
Now calculate the following, giving your answers in the form q
p, where p, q .
4 8 32log 64, log 64, log 64
7 49 343log 49, log 49, log 49
1 1 1
5 125 625
log 125, log 125, log 125
8 2 16log 512, log 512, log 512
Describe how to obtain the third answer in each row from the first two answers. Create two more
examples that fit the pattern above.
Let cxalog and dxblog . Find the general statement that expresses xablog , in terms of c and d.
Test the validity of your general statement using other values of a, b, and x.
Discuss the scope and/or limitations of a, b, and x.
Explain how you arrived at your general statement.
– 7 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
MATRIX BINOMIALS SL TYPE I
Let 1 1
1 1X and
1 1
1 1Y . Calculate 2 3 4 2 3 4, , ; , ,X X X Y Y Y
By considering integer powers of X and Y , find expressions for , , ( )n n nX Y X Y .
Let aA = X and bB = Y , where a and b are constants.
Use different values of a and b to calculate 2 3 4 2 3 4, , ; , ,A A A B B B
By considering integer powers of A and B, find expressions for , , ( )n n nA B A B .
Now consider a b a b
a b a bM .
Show that M = A + B, and that 2 2 2M A B .
Hence, find the general statement that expresses nM in terms of aX and bY .
Test the validity of your general statement by using different values of a, b, and n.
Discuss the scope and/or limitations of your general statement.
Use an algebraic method to explain how you arrived at your general statement.
– 8 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
SHADY AREAS SL TYPE I
In this investigation you will attempt to find a rule to approximate the area under a curve (i.e. between the
curve and the x-axis) using trapeziums (trapezoids).
Consider the function 2( ) 3g x x .
The diagram below shows the graph of g. The area under this curve from 0x to 1x is approximated
by the sum of the area of two trapeziums. Find this approximation.
Increase the number of trapeziums to five and find a second approximation for the area.
With the help of technology, create diagrams showing an increasing number of trapeziums. For each
diagram, find the approximation for the area. What do you notice?
(This task continues on the following page)
– 9 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Use the diagram below to find a general expression for the area under the curve of g, from 0x to 1x ,
using n trapeziums.
Use your results to develop the general statement that will estimate the area under any curve
( )y f x from x a to x b using n trapeziums. Show clearly how you developed your statement.
Consider the areas under the following three curves, from 1 to 3x x . 2
3
12
xy
23
9
9
xy
x
3 2
3 4 23 40 18y x x x
Use your general statement, with eight trapeziums, to find approximations for these areas.
Find
2
33
1d
2
xx ,
3
31
9d
9
xx
x,
33 2
1(4 23 40 18)dx x x x , and compare these answers with
your approximations. Comment on the accuracy of your approximations.
Use other functions to explore the scope and limitations of your general statement. Does it always work?
Discuss how the shape of a graph influences your approximation.
– 10 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
PARALLELS AND PARALLELOGRAMS SL TYPE I
This task will consider the number of parallelograms formed by intersecting parallel lines.
Figure 1 below shows a pair of horizontal parallel lines and a pair of parallel transversals. One parallelogram
(A1) is formed.
A third parallel transversal is added to the diagram as shown in Figure 2. Three parallelograms are
formed: A1, A2, and 1 2A A .
We can go on drawing additional transversals and forming new parallelograms.
Show that six parallelograms are formed when a fourth transversal is added to Figure 2. List all these
parallelograms, using set notation.
Repeat the process with 5, 6 and 7 transversals. Show your results in a table. Use technology to find a
relation between the number of transversals and the number of parallelograms. Develop a general
statement, and test its validity.
Next consider the number of parallelograms formed by three horizontal parallel lines intersected by
parallel transversals. Develop and test another general statement for this case.
Now extend your results to m horizontal parallel lines intersected by n parallel transversals.
Display the results in a spreadsheet and use this to find the general statement for the overall pattern.
Test the validity of your statement.
Discuss its scope and/or limitations.
Explain how you arrived at this generalization.
– 11 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Type II – mathematical modelling
Problem solving usually elicits a process-oriented approach, whereas mathematical modelling requires an
experimental approach. By considering different alternatives, students can use modelling to arrive at a
specific conclusion, from which the problem can be solved. To focus on the actual process of modelling,
the assessment should concentrate on the appropriateness of the model selected in relation to the given
situation, and on a critical interpretation of the results of the model in the real-world situation chosen.
Mathematical modelling involves the following skills.
• Translating the real-world problem into mathematics
• Constructing a model
• Solving the problem
• Interpreting the solution in the real-world situation (that is, by the modification or amplification of
the problem)
• Recognizing that different models may be used to solve the same problem
• Comparing different models
• Identifying ranges of validity of the models
• Identifying the possible limits of technology
• Manipulating data
Essential skills to be assessed
• Identifying the problem variables
• Constructing relationships between these variables
• Manipulating data relevant to the problem
• Estimating the values of parameters within the model that cannot be measured or calculated from
the data
• Evaluating the usefulness of the model
• Communicating the entire process
• Appropriate use of technology
– 12 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
BODY MASS INDEX SL TYPE II
Body mass index (BMI) is a measure of one’s body fat. It is calculated by taking one’s weight (kg) and
dividing by the square of one’s height (m).
The table below gives the median BMI for females of different ages in the US in the year 2000.
Age (yrs) BMI
2 16.40
3 15.70
4 15.30
5 15.20
6 15.21
7 15.40
8 15.80
9 16.30
10 16.80
11 17.50
12 18.18
13 18.70
14 19.36
15 19.88
16 20.40
17 20.85
18 21.22
19 21.60
20 21.65
(Source: http://www.cdc.gov)
Using technology, plot the data points on a graph. Define all variables used and state any
parameters clearly.
What type of function models the behaviour of the graph? Explain why you chose this function.
Create an equation (a model) that fits the graph.
On a new set of axes, draw your model function and the original graph. Comment on any differences.
Refine your model if necessary.
Use technology to find another function that models the data. On a new set of axes, draw your model
function and the function you found using technology. Comment on any differences.
Use your model to estimate the BMI of a 30-year-old woman in the US. Discuss the reasonableness of
your answer.
Use the Internet to find BMI data for females from another country. Does your model also fit this data?
If not, what changes would you need to make? Discuss any limitations to your model.
– 13 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
FISHING RODS SL TYPE II
A fishing rod requires guides for the line so that it does not tangle and so that the line casts easily and
efficiently. In this task, you will develop a mathematical model for the placement of line guides on a
fishing rod.
The diagram shows a fishing rod with eight guides, plus a guide at the tip of the rod.
Leo has a fishing rod with overall length 230 cm. The table shown below gives the distances for each of
the line guides from the tip of his fishing rod.
Guide number (from tip) 1 2 3 4 5 6 7 8
Distance from tip (cm) 10 23 38 55 74 96 120 149
Define suitable variables, discuss parameters/constraints.
Using technology, plot the data points on a graph.
Using matrix methods or otherwise, find a quadratic function and a cubic function which model this
situation. Explain the process you used. On a new set of axes, draw these model functions and the
original data points. Comment on any differences.
Find a polynomial function which passes through every data point. Explain your choice of function, and
discuss its reasonableness. On a new set of axes, draw this model function and the original data points.
Comment on any differences.
Using technology, find one other function that fits the data. On a new set of axes, draw this model
function and the original data points. Comment on any differences.
Which of your functions found above best models this situation? Explain your choice.
Use your quadratic model to decide where you could place a ninth guide. Discuss the implications of
adding a ninth guide to the rod.
Mark has a fishing rod with overall length 300 cm. The table shown below gives the distances for each of
the line guides from the tip of the Mark’s fishing rod.
Guide number (from tip) 1 2 3 4 5 6 7 8
Distance from tip (cm) 10 22 34 48 64 81 102 124
How well does your quadratic model fit this new data? What changes, if any, would need to be made for
that model to fit this data? Discuss any limitations to your model.
– 14 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
CROWS DROPPING NUTS SL TYPE II
Crows love nuts but their beaks are not strong enough to break some nuts open. To crack open the shells,
they will repeatedly drop the nut on a hard surface until it opens.
The following table shows the average number of drops it takes to break open a large nut from
varying heights.
Large Nuts
Height of drop (m) 1.7 2.0 2.9 4.1 5.6 6.3 7.0 8.0 10.0 13.9
Number of drops 42.0 21.0 10.3 6.8 5.1 4.8 4.4 4.1 3.7 3.2
Using technology, plot the data points on a graph. Define all variables used and state any parameters clearly.
What type of function models the behaviour of the graph? Explain why you chose this function.
Create an equation (a model) that fits the graph.
On a new set of axes, draw your model and the original graph. Comment on any differences. Refine your
model if necessary.
Use technology to find another function that models the data. On a new set of axes, draw your model
function and the function you found using technology. Comment on any differences.
The following tables show the average number of drops it takes to break open a medium nut, and a small
nut, from varying heights.
Medium Nuts
Height of drop (m) 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10.0 15.0
Number of drops - - 27.1 18.3 12.2 11.1 7.4 7.6 5.8 3.6
Small Nuts
Height of drop (m) 1.5 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10.0 15.0
Number of drops - - - 57.0 19.0 14.7 12.3 9.7 13.3 9.5
How well does your first model apply to nuts of different sizes. What changes, if any, need to be made
to your model to fit the data for medium and small nuts? Discuss any limitations to your models.
– 15 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
LOGAN’S LOGO SL TYPE II
Note to teachers: The size of the square is not critical until it is measured. Variations may result when
copies of the task are made. Students should measure the diagram as it is presented. It will be very helpful
to moderators if you include a copy of the task with any work selected for the sample.
Logan has designed the logo below.
The diagram shows a square which is divided into three regions by two curves. The logo is the shaded region between the two curves. Logan wishes to develop mathematical functions that model these curves.
Using an appropriate set of axes, identify and record a number of data points on the curves which will allow you to develop model functions for them. Define all variables used and state any parameters clearly.
Using technology, plot these two sets of data points on a graph. What type of functions model the behaviour of the data? Explain why you chose these functions.
Find functions that represent the upper and lower curves forming the logo. Discuss any limitations.
Logan wishes to print T-shirts with the logo on the back. She must double the dimensions of the logo for this purpose. Describe how your functions must be modified.
Logan also wishes to print business cards. A standard business card is 9 cm by 5 cm. How must your functions be modified so that the logo extends from one end of the card to the other? Use technology to show the results.
What fraction of the area of the card does the logo occupy? Why might this be an important aspect of a business card?
– 16 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Overview of assessment criteria for type I tasks
Crit
eri
on
F:
Qu
ali
ty o
f w
ork
Th
e st
ud
ent
has
sho
wn
a p
oor
qual
ity
of
wo
rk.
Th
e st
ud
ent
has
sho
wn
a
sati
sfa
cto
ry
qual
ity
of
wo
rk.
Th
e st
ud
ent
has
sho
wn
an
ou
tsta
nd
ing
qual
ity
of
wo
rk.
Crit
eri
on
E:
Use
of
tech
nolo
gy
Th
e st
ud
ent
use
s a
calc
ula
tor
or
com
pute
r fo
r o
nly
ro
uti
ne
calc
ula
tion
s.
Th
e st
ud
ent
att
emp
ts t
o u
se
a ca
lcula
tor
or
com
pu
ter
in a
man
ner
th
at c
ou
ld e
nh
ance
the
dev
elop
men
t o
f th
e ta
sk.
Th
e st
ud
ent
mak
es l
imit
ed
use
of
a ca
lcula
tor
or
com
pu
ter
in a
man
ner
that
enh
ance
s th
e d
evel
op
men
t o
f
the
task
.
Th
e st
ud
ent
mak
es f
ull
and
reso
urce
ful
use
of
a
calc
ula
tor
or
com
pu
ter
in a
man
ner
th
at s
ign
ific
an
tly
enh
ance
s th
e d
evel
op
men
t o
f
the
task
.
Crit
eri
on
D:
Resu
lts
—
gen
erali
za
tio
n
Th
e st
ud
ent
do
es
no
t
pro
du
ce a
ny
gen
eral
stat
emen
t co
nsi
sten
t w
ith
the
pat
tern
s an
d/o
r st
ruct
ure
s
gen
erat
ed.
Th
e st
ud
ent
att
emp
ts t
o
pro
duce
a g
ener
al s
tate
men
t
that
is
con
sist
ent
wit
h t
he
pat
tern
s an
d/o
r st
ruct
ure
s
gen
erat
ed.
Th
e st
ud
ent
corre
ctl
y
pro
duce
s a
gen
eral
sta
tem
ent
that
is
con
sist
ent
wit
h t
he
pat
tern
s an
d/o
r st
ruct
ure
s
gen
erat
ed.
Th
e st
ud
ent
ex
press
es
the
co
rrec
t gen
eral
sta
tem
ent
in
ap
pro
pria
te m
ath
em
ati
ca
l
term
inolo
gy
.
Th
e st
ud
ent
corre
ctl
y s
tate
s
the
scop
e o
r li
mit
atio
ns
of
the
gen
eral
sta
tem
ent.
Th
e st
ud
ent
giv
es a
co
rrect,
info
rmal
ju
stif
icat
ion o
f th
e
gen
eral
sta
tem
ent.
Crit
eri
on
C:
Ma
them
ati
cal
pro
cess
— s
earc
hin
g f
or
pa
ttern
s
Th
e st
ud
ent
do
es
no
t
att
em
pt
to u
se a
mat
hem
atic
al s
trat
egy.
Th
e st
ud
ent
use
s a
mat
hem
atic
al s
trat
egy t
o
pro
duce
dat
a.
Th
e st
ud
ent
org
an
izes
the
dat
a g
ener
ated
.
Th
e st
ud
ent
att
emp
ts t
o
an
aly
se d
ata
to e
nab
le t
he
form
ula
tio
n o
f a
gen
eral
stat
emen
t.
Th
e st
ud
ent
succe
ssfu
lly
an
aly
ses
the
correc
t d
ata
to
enab
le t
he
form
ula
tio
n o
f a
gen
eral
sta
tem
ent.
Th
e st
ud
ent
test
s th
e val
idit
y
of
the
gen
eral
sta
tem
ent
by
con
sid
erin
g f
urt
her
exam
ple
s.
Crit
eri
on
B:
Co
mm
un
ica
tio
n
Th
e st
ud
ent
neit
her
pro
vid
es
exp
lanat
ion
s n
or
use
s ap
pro
pri
ate
form
s o
f re
pre
sen
tati
on
(fo
r ex
amp
le,
sym
bols
, ta
ble
s, g
raph
s an
d/o
r
dia
gra
ms)
.
Th
e st
ud
ent
att
emp
ts t
o p
rov
ide
exp
lanat
ion
s o
r u
ses
som
e ap
pro
pri
ate
form
s o
f re
pre
sen
tati
on
(fo
r ex
amp
le,
sym
bols
, ta
ble
s, g
raph
s an
d/o
r
dia
gra
ms)
.
Th
e st
ud
ent
pro
vid
es a
deq
ua
te
exp
lanat
ion
s o
r ar
gu
men
ts, an
d
com
mun
icat
es t
hem
usi
ng
app
rop
riat
e
form
s o
f re
pre
sen
tati
on
(fo
r ex
amp
le,
sym
bols
, ta
ble
s, g
raph
s an
d/o
r
dia
gra
ms)
.
Th
e st
ud
ent
pro
vid
es c
om
ple
te,
co
her
en
t ex
pla
nat
ion
s o
r ar
gu
men
ts,
and
co
mm
un
icat
es t
hem
cle
arly
usi
ng
app
rop
riat
e fo
rms
of
rep
rese
nta
tion
(fo
r ex
amp
le,
sym
bols
, ta
ble
s, g
raph
s
and
/or
dia
gra
ms)
.
Crit
eri
on
A:
Use
of
no
tati
on
an
d
term
inolo
gy
Th
e st
ud
ent
do
es
no
t u
se a
pp
rop
riat
e
nota
tio
n a
nd
term
ino
logy
.
Th
e st
ud
ent
use
s
som
e a
pp
rop
riat
e
nota
tio
n a
nd
/or
term
ino
logy
.
Th
e st
ud
ent
use
s
app
rop
riat
e nota
tio
n
and
ter
min
olo
gy
in
a co
nsi
sten
t
man
ner
and
do
es s
o
thro
ugh
ou
t th
e
wo
rk.
0
1
2
3
4
5
– 17 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Overview of assessment criteria for type II tasks
Crit
eri
on
F:
Qu
ali
ty o
f w
ork
Th
e st
ud
ent
has
sho
wn
a p
oor
qual
ity
of
wo
rk.
Th
e st
ud
ent
has
sho
wn
a
sati
sfa
cto
ry
qual
ity
of
wo
rk.
Th
e st
ud
ent
has
sho
wn
an
ou
tsta
nd
ing
qual
ity
of
wo
rk.
Crit
eri
on
E:
Use
of
tech
nolo
gy
Th
e st
ud
ent
use
s a
calc
ula
tor
or
com
pu
ter
for
on
ly r
ou
tin
e c
alcu
lati
on
s.
Th
e st
ud
ent
att
emp
ts t
o
use
a c
alcu
lato
r o
r
com
pu
ter
in a
man
ner
that
cou
ld e
nh
ance
the
dev
elop
men
t o
f th
e ta
sk.
Th
e st
ud
ent
mak
es
lim
ited
use
of
a ca
lcu
lato
r
or
com
pute
r in
a m
anner
that
enhan
ces
the
dev
elop
men
t o
f th
e ta
sk.
Th
e st
ud
ent
mak
es f
ull
and
reso
urce
ful
use
of
a
calc
ula
tor
or
com
pu
ter
in
a m
ann
er t
hat
sig
nif
ica
ntl
y e
nh
ance
s th
e
dev
elop
men
t o
f th
e ta
sk.
Crit
eri
on
D:
Resu
lts
—
inte
rp
reta
tio
n
Th
e st
ud
ent
ha
s n
ot
arr
ived
at
any
resu
lts.
Th
e st
ud
ent
has
arr
ived
at
som
e
resu
lts.
Th
e st
ud
ent
ha
s n
ot
inte
rp
rete
d t
he
reas
on
able
nes
s o
f th
e re
sult
s o
f th
e
mo
del
in
the
con
tex
t o
f th
e t
ask
.
Th
e st
ud
ent
has
att
em
pte
d t
o
inte
rpre
t th
e re
aso
nab
len
ess
of
the
resu
lts
of
the
mo
del
in
the
con
tex
t o
f
the t
ask
, to
the
app
rop
riat
e deg
ree
of
accu
racy
.
Th
e st
ud
ent
has
co
rrec
tly
in
terp
rete
d
the
reas
on
able
nes
s o
f th
e re
sult
s o
f
the
mo
del
in
the
con
text
of
the
task
,
to t
he
app
rop
riat
e d
egre
e o
f ac
cura
cy.
Th
e st
ud
ent
has
co
rrec
tly
and
crit
ica
lly i
nte
rpre
ted
the
reas
on
able
nes
s o
f th
e re
sult
s o
f th
e
mo
del
in
the
con
text
of
the
task
,
inclu
din
g p
oss
ible
lim
itat
ion
s an
d
mo
dif
icat
ion
s o
f th
ese
resu
lts,
to
th
e
app
rop
riat
e deg
ree
of
accu
racy
.
Crit
eri
on
C:
Ma
them
ati
cal
pro
cess
— d
evel
op
ing a
mo
del
Th
e st
ud
ent
do
es
no
t d
efi
ne
var
iable
s, p
aram
eter
s o
r co
nst
rain
ts
of
the
task
.
Th
e st
ud
ent
def
ines
so
me
var
iable
s, p
aram
eter
s o
r co
nst
rain
ts
of
the
task
.
Th
e st
ud
ent
def
ines
var
iab
les,
par
amet
ers
an
d c
onst
rain
ts o
f th
e
task
an
d a
ttem
pts
to c
reat
e a
mo
del
.
Th
e st
ud
ent
corre
ctl
y a
na
lyse
s
var
iable
s, p
aram
eter
s an
d
con
stra
ints
of
the
task
to e
nab
le t
he
form
ula
tio
n o
f a
mat
hem
atic
al
mo
del
th
at i
s re
leva
nt
to t
he
task
and
con
sist
ent
wit
h t
he
lev
el o
f th
e
cou
rse.
Th
e st
ud
ent
co
nsi
der
s h
ow
wel
l
the
mo
del
fit
s th
e dat
a.
Th
e st
ud
ent
ap
pli
es
the
mod
el t
o
oth
er s
itu
atio
ns.
Crit
eri
on
B:
Co
mm
un
ica
tio
n
Th
e st
ud
ent
neit
her
pro
vid
es
exp
lanat
ion
s n
or
use
s
app
rop
riat
e fo
rms
of
rep
rese
nta
tio
n (
for
exam
ple
,
sym
bols
, ta
ble
s, g
raph
s an
d/o
r
dia
gra
ms)
.
Th
e st
ud
ent
att
emp
ts t
o p
rov
ide
exp
lanat
ion
s o
r u
ses
som
e
app
rop
riat
e fo
rms
of
rep
rese
nta
tio
n (
for
exam
ple
,
sym
bols
, ta
ble
s, g
raph
s and
/or
dia
gra
ms)
.
Th
e st
ud
ent
pro
vid
es a
deq
ua
te
exp
lanat
ion
s o
r ar
gu
men
ts, an
d
com
mun
icat
es t
hem
usi
ng
app
rop
riat
e fo
rms
of
rep
rese
nta
tio
n (
for
exam
ple
,
sym
bols
, ta
ble
s, g
raph
s an
d/o
r
dia
gra
ms)
.
Th
e st
ud
ent
pro
vid
es c
om
ple
te,
co
her
en
t ex
pla
nat
ion
s o
r
argu
men
ts,
and
co
mm
un
icat
es
them
cle
arly
usi
ng
app
rop
riat
e
form
s o
f re
pre
sen
tati
on
(fo
r
exam
ple
, sy
mbo
ls, ta
ble
s, g
raph
s
and
/or
dia
gra
ms)
.
Crit
eri
on
A:
Use
of
no
tati
on
an
d
term
inolo
gy
Th
e st
ud
ent
do
es n
ot
use
ap
pro
pri
ate
nota
tio
n a
nd
term
ino
logy
.
Th
e st
ud
ent
use
s
som
e a
pp
rop
riat
e
nota
tio
n a
nd
/or
term
ino
logy
.
Th
e st
ud
ent
use
s
app
rop
riat
e nota
tio
n
and
ter
min
olo
gy
in
a
co
nsi
sten
t m
ann
er
and
does
so
thro
ugh
ou
t th
e w
ork
.
0
1
2
3
4
5
– 18 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Tasks developed by teachers
Introduction
As stated in the Mathematics SL guide (2006), portfolio tasks must be integrated into the course of study.
This course of study should be devised before the start of the course and suitable tasks identified that can
be incorporated into it to support the learning process. Students need to submit two pieces of work, but it
is a good idea for them to be allowed to complete more than two and choose the best ones.
When setting tasks, the background of the students and the purpose of each task should be considered, as
well as the types of technology available to students. The tasks should be:
presented to students at appropriate times, periodically over the two-year course
meaningful and relevant to the topic being studied at the time of the task
considered as part of normal classwork and homework, not as something extra.
It may be helpful to provide students with a timetable of tasks at an early stage to assist them in managing
their time. The following section deals with the cycle of development from possible starting points to the
writing of a task.
Starting points
The process of developing a task can start from a number of different points.
A task written by someone else
It will be necessary to work the task first to check suitability. Amendments will almost certainly be
needed for the task to be incorporated into a particular course of study. This includes the tasks in
this document.
A syllabus topic to be covered
Some syllabus topics are suited to particular types of task. For example, sequences and series invite
investigative work using a graphic display calculator (GDC), and exponential functions can be applied in
a modelling task.
Outside sources
A report in a newspaper or journal can often provide the starting point for a modelling task or an
investigation. Such a report provides an ideal opportunity to apply mathematics to real-life contexts.
These reports may not appear at appropriate times in the course, so starting points of this kind usually
require long-term planning.
Interesting points that arise in class discussion
Sometimes an interesting mathematics problem is exchanged among colleagues or arises from class
discussion. If it is relevant to the syllabus it could be developed into a portfolio task.
– 19 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Questions before starting
The following questions need to be considered before starting to develop a portfolio task.
What is the purpose of the task?
The purpose of each task should be clearly understood in terms of whether it is being used to introduce a
topic, reinforce mathematical meaning or take the place of a revision exercise.
What type should it be (type I or type II)?
It is important to make a decision about the type of task at an early stage and to make sure the task
addresses the particular requirements of that type.
What part of the syllabus does this assess?
Portfolio tasks must relate directly to the syllabus. Choosing topics outside the syllabus, or extending
work on topics beyond the intended level of study, will create extra work for the student and the teacher.
What knowledge and skills are involved?
Teachers should consider the prior knowledge and skills that are required in order for students to
complete the task successfully. Teachers should also consider the mathematical knowledge and skills
they wish the students to obtain, develop and review as they work through the task.
What follow-up work will be needed?
The extent of the follow-up work required will vary with the nature of the task and should be planned in
advance.
The cycle of development
In developing a portfolio task it will be necessary to work through a number of stages.
Stage 1
Draft the task, or select a task that has been written by someone else. The assessment criteria should be
consulted at this point.
Stage 2
Work the task yourself in full, as if you were a student.
Stage 3
Refer to the assessment criteria. Will the task provide an opportunity for students to gain the highest
achievement levels?
Stage 4
Consider whether the task has achieved its aims. Is it of an appropriate length? Is it at an appropriate
level? What will the students learn?
Stage 5
What flaws in the task have been exposed? How could the task be improved?
Stage 6
Redraft the task so that it will be ready to use with your students.
Stage 7
Present the task to your students, and then repeat stages 3 to 6.
– 20 – MATME/PF/M09/N09/M10/N10
For final assessment in 2009 and 2010
Titles of tasks taken from old teacher support materials.
These are the titles of tasks which appear in TSMs published for the old course. They should not be
included in portfolios after the November 2008 examination session. In the second edition of the TSM,
some tasks were not published in all three languages, so the titles for all three languages are included
here for reference.
TSM (Mathematical methods SL, first edition, November 1998)
Title Título Titre
Investigating the quadratic
function
Investigación de la función
cuadrática
Exploration de fonctions du
second degré
Investigating the graphs of
sine functions
Investigación de las gráficas de
las funciones seno
Recherche sur les graphes de
fonctions sinus
Transforming data Transformación de datos Modification de données
An investigation into the
Newton-Raphson method
Una investigación sobre el
método de Newton-Raphson
Une étude de la méthode de
Newton-Raphson
Transformation matrices Transformación de
matrices
Les matrices de
transformation
Webs and staircases:
investigating fixed-point
iteration
Telarañas y escaleras:
investigación de la iteración
con un punto fijo
Toiles d’araignée et escaliers: une
étude des itérations vers un point
fixe
Radio transmitters Radiotransmisores Émetteurs radios
The decibel scale La escala de decibeles L’échelle décibel
Equations of lines in vector
form
Ecuaciones de rectas en
forma vectorial
Équations vectorielles de
droites
Modelling a can of drink Elección de un modelo para
una lata de bebida
Modélisation d’une canette
Population growth Crecimiento de la
población
Croissance d’une
population
Investing money Inversión de dinero Investir de l’argent
The water in a lake El agua de un lago L’eau du lac
Speed limits Límites de velocidad Limite de vitesse
Crossing a river Bote que cruza un río Traverse rune rivière
TSM (Mathematical methods SL, second edition, November 2000)
Title Título Titre
Investigating logarithms Investigación de logaritmos Investigation sur les logarithmes
Absolute value (modulus)
graphs
Gráficas de valor absoluto
(modulo)
Courbes et valeurs absolues
Areas under curves
Resolución de problemas
cerrados extensos
Fish Pond Una piscina para peces L’étang à poisons
Geometría