IB MATHEMATICS PORTFOLIO IA TASKS GOLD MEDAL HEIGHTS SL TYPE 2/II TUTORS HELP SAMPLES EXAMPLES SOLUTION GUIDANCE by rsingh33

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Mathematics/Math portfolio (SL/HL Type 1/I Type 2/II) IA
(Internal Assessment) Circles/ Lacsap’s Fractions/ Gold
Medal Heights/ Fish Production/ Shadow Functions/
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MATHEMATICS
Standard Level
The portfolio – IA tasks
For use in 2012 and 2013
CIRCLES SL TYPE I
Aim: The aim of this task is to investigate positions of points in
intersecting circles.
 The following diagram shows a circle with centre O and radius
                    r, and any point P. 1 C
The circle has centre P and radius OP. Let A be one of the points
 of intersection of and . Circle has centre A, and radius r. The
   point is the intersection of with (OP). This is shown in the
                          diagram below.
Let . Use an analytic approach to find , when , and . Describe
what you notice and write a general statement to represent this.
Let . Find , when , and . Describe what you notice and write a
general statement to represent this. Comment whether or not this
statement is consistent with your earlier statement.
Use technology to investigate other values of r and OP. Find the

Test the validity of your general statement by using different
values of OP and r.
Discuss the scope and/or limitations of the general statement.
       Explain how you arrived at the general statement.


LACSAP’S FRACTIONS SL TYPE I
Aim: In this task you will consider a set of numbers that are
presented in a symmetrical pattern.
        Consider the five rows of numbers shown below.
Describe how to find the numerator of the sixth row.
Using technology, plot the relation between the row number, n,
and the numerator in each row. Describe what you notice from
your plot and write a general statement to represent this.
Find the sixth and seventh rows. Describe any patterns you used.
Let be the element in the nth row, starting with . Example: .
Find the general statement for.
Test the validity of the general statement by finding additional
rows.
Discuss the scope and/or limitations of the general statement.
Explain how you arrived at your general statement.
FISH PRODUCTION SL TYPE II
Aim: This task considers commercial fishing in a particular
country in two different environments – the sea and fish farms
(aquaculture). The data is taken from the UN Statistics Division
Common Database.
The following table gives the total mass of fish caught in the
sea, in thousands of tonnes (1 tonne = 1000 kilograms).
Define suitable variables and discuss any parameters/constraints.
Using technology, plot the data points from the table on a graph.
Comment on any apparent trends in your graph and suggest
suitable models.
Analytically develop a model that fits the data points. (You may
find it useful to consider a combination of functions.)
On a new set of axes, draw your model function and the original
data points. Comment on any differences. Revise your model if
necessary.
The table below gives the total mass of fish, in thousands of
tonnes, from fish farms.
Plot the data points from this table on a graph, and discuss
whether your analytical model for the original data fits the new
data.
Use technology to find a suitable model for the new data. On a
new set of axes, draw both models.
Discuss how trends in the first model could be explained by
trends in the second model.
By considering both models, discuss possible future trends in
both types of fishing.
GOLD MEDAL HEIGHTS SL TYPE II
Aim: The aim of this task is to consider the winning height for
the men’s high jump in the Olympic Games.
The table below gives the height (in centimeters) achieved by
the gold medalists at various Olympic Games.
Note: The Olympic Games were not held in 1940 and 1944.
Using technology, plot the data points on a graph. Define all
variables used and state any parameters clearly. Discuss any
possible constraints of the task.
What type of function models the behaviour of the graph?
Explain why you chose this function. Analytically create an
equation to model the data in the above table.
On a new set of axes, draw your model function and the original
graph. Comment on any differences. Discuss the limitations of
your model. Refine your model if necessary.
Use technology to find another function that models the data. On
a new set of axes, draw both your model functions. Comment on
any differences.
Had the Games been held in 1940 and 1944, estimate what the
winning heights would have been and justify your answers.
Use your model to predict the winning height in 1984 and in
2016. Comment on your answers.
The following table gives the winning heights for all the other
Olympic Games since 1896.
How well does your model fit the additional data?
Discuss the overall trend from 1896 to 2008, with specific
references to significant fluctuations.
What modifications, if any, need to be made to your model to fit
the new data?

MATHEMATICS
Higher Level
The portfolio – IA tasks
For use in 2012 and 2013
SHADOW FUNCTIONS HL TYPE I
To the student: The work that you produce to address the
questions in this task should be a report
that can stand on its own. It is best to avoid copying the
qu stions in th task to adopt a “qu stion
and answ r” format.
While real zeros of polynomial functions may be easily read off
the graph of the
polynomial, the same is not true for complex zeros. In this task,
you will investigate the
method of shadow functions and their generators, which helps
identify the real and
imaginary components of complex zeros from key points along
the x-axis.
Part A (Quadratic Polynomials)
Consider the quadratic function 2 2
Write down the coordinates of the vertex.
Show that 1 y has zeros, where.
The “shadow function” to 1 y is another quadratic 2 y which
shares the same vertex as.
However, 2 y has opposite concavity to that of 1 y and its zeros
are in the form.
Use various values for a and b to generate pairs of functions 1 y
and.
Hence, or otherwise, express 2 y in terms of 1 y and m y , where
m y is call d th “shadow
g n rating function”.
_ On a labelled diagram, illustrate how the zeros of 2 y may be
helpful in the determination
of the real and imaginary components of the complex zeros of 1
y.
Part B (Cubic Polynomials)
Now consider the cubic function 1 _ _ _ _ y _ (x _ 2) x _ (3_
2i) x _ (3_ 2i) .
The shadow function in this case is another cubic function 2 y
which shares two points with 1 y ,
has opposite concavity and its zeros are _2 and 3_ 2 as depicted
in the figure below. The shadow
generating function in this case passes through the two points of
intersection.
Write down an expression for 2 y and hence find the points of
intersection between 1 y and 2 y .
_ Hence, or otherwise, determine the equation of the shadow
generating function m y in this
case.
_ Once again, express 2 y in terms of 1 y and m y . Use
technology to investigate similar
cubic functions.
_ Write a general statement about the functions 1 y , 2 y and m y
and prove your statement.
_ On a labelled diagram, illustrate how the zeros of 2 y may be
helpful in the determination
of the real and imaginary components of the complex zeros of 1
y.
_ How can the findings with quadratics and cubics be applied to
quartics?
PATTERNS FROM COMPLEX NUMBERS HL TYPE I
To the student: The work that you produce to address the
questions in this task should be a report
that can stand on its own. It is best to avoid copying the
qu stions in th task to adopt a “qu stion
and answ r” format.
Part A
Us d Moi r ’s th or m to obtain solutions to th quation z3
_1 _ 0 .
_ Use graphing software to plot these roots on an Argand
diagram as well as a unit circle
with centre origin.
_ Choose a root and draw line segments from this root to the
other two roots.
_ Measure these line segments and comment on your results.
_ Repeat the above for the equations z4 _1 _ 0 and z5 _1 _ 0 .
Comment on your results and
try to formulate a conjecture.
_ Factorize zn _1 for n _ 3, 4 and 5.
_ Use graphing software to test your conjecture for some more
values of n �� __ and make
modifications to your conjecture if necessary.
_ Prove your conjecture.
Part B
Us d Moi r ’s th or m to obtain solutions to zn _ i for n _ 3,
4 and 5.
_ Use graphing software to represent each of these solutions on
an Argand diagram.
_ Generalize and prove your results for zn _ a _ bi , where a _
bi _1.
_ What happens when a _ bi _1?
THE DICE GAME HL Type II
To the student: The work that you produce to address the
questions in this task should be a report
that can stand on its own. It is best to avoid copying the
qu stions in th task to adopt a “qu stion
and answ r” format.
The aim of this task is to create a dice game in a casino and
model this using
probability. It is important to examine how best to run the game
from both the
perspective of a player and the casino. In doing so analyse the
game to consider the
optimal payments by the player and payouts by the casino.
_ Consider a game with two players, Ann and Bob. Ann has a
red die and Bob a white die.
They roll their dice and note the number on the upper face. Ann
wins if her score is higher
than Bob’s (not that Bob wins if th scor s ar th sam ). If
both players roll their dice
once each what is the probability that Ann will win the game?
_ Now consider the same game where Ann can roll her die a
second time and will note the
higher score of the two rolls but Bob rolls only once. In this case
what is the probability
that Ann will win?
_ Investigate the game when both players can roll their dice
twice, and also when both
players can roll their dice more than twice, but not necessarily
the same number of times.
_ Consider the game in a casino where the player has a red die
and the bank has a white die.
Find a model for a game so that the casino makes a reasonable
profit in the case where the
player rolls the red die once and the bank rolls the white die
once. (When creating your
model you will need to consider how much a player must pay to
play a game and how
much the bank will pay out if the player wins. Do this from the
perspective of both the
player and the casino and consider carefully the criteria for
whether the game can be
considered worthwhile for both the player and the casino.)
_ Now consider other models for the game including cases such
as where the player or the
bank rolls their dice multiple times, or where multiple players
are involved in the game.
FILLING UP THE PETROL TANK HL Type II
To the student: The work that you produce to address the
questions in this task should be a report
that can stand on its own. It is best to avoid copying the
qu stions in th task to adopt a “qu stion
and answ r” format.
In this task, you will develop a mathematical model that helps
motorists decide which of
the following two options is more economical.
Option 1: to buy petrol from a station on their normal route at a
relatively higher price.
Option 2: to drive an extra distance out of their normal route to
buy cheaper petrol.
Consider a situation in which two motorists, Arwa and Bao,
share the same driving route but own
diff r nt siz d hicl s. Arwa fills up h r hicl ’s tank at a
station along her normal route for
US$ 1 p per litre. On the other hand, Bao drives an extra d
kilometres out of his normal route to fill
up his hicl ’s tank for US$ p2 per litre where p2 _ p1 .
_ Choos suitabl param t rs for Arwa’s and Bao’s hicl s.
Justify your choices.
_ Suppose 1 p _ US$1.00 per litre, 2 p _ US$0.98 per litre and d
_10 km. Which motorist is getting the better deal? Show all
calculations and justify any assumptions you make.
_ On a spreadsheet, choose several sets of values for 1 p , 2 p
and d to investigate further.
_ Define a set of variables that would be relevant in the above
situation.
“Eff cti litr s” is a m thod of comparing th cost of p trol
bought under the two options
described above. Effective litres, for a given vehicle, are those
litres used when the vehicle travels
its normal route.
_ Use your variables and parameters to write algebraic
expressions for 1 E and 2 E which
represent the cost per effective litre under options 1 and 2
respectively.
_ Write a model that helps motorists decide on the more
economical option for their
vehicles.
In the remainder of this task, you need to consider the two
vehicles you have chosen for Arwa
and Bao.
_ Use your model to find the farthest distance that Bao should
drive to obtain a 2 % price
saving.
_ Investigate the relationship between d and 2 p when 2 E is
kept constant (e.g. US$0.90,
US$1.00, … etc.). Use technology to draw a family of curves for
Arwa’s hicl . R p at
for Bao’s hicl .
_ For 2 E _ US$0.90 , provide Arwa with information on three
different stations that yield
this same cost per effective litre. Discuss how such information
may be useful to Arwa.
_ For 2 E _ US$1.00 and 2 p _ US$0.80 , compare the
maximum distance that each motorist
should drive and still save money.
Arwa is a busy person and wonders whether the saving in
money would be worth the time she
would lose in extra driving.
_ Modify your model to account for the time taken to drive to
and from an off-route station.
Clearly justify any assumptions you make.


IB DP Math SL type 2 IA portfolio task Fish production, Gold
Medal heights
IB DP Math HL type 2 IA portfolio task The dice game, Filling
up the petrol tank.
IB DP Math HL type 1 IA portfolio task shadow function,
Patterns from complex number.
IB DP Math SL typ 1 IA portfolio task Lacsap’s fraction,
Circles
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