Math SL Portfolio – Tips and Reminders Checklist
Notation and Terminology
Check for the following:
•I did not use calculator notation. (I didn’t include things like ‘x^2’ for or Sn for Sn) •I used appropriate mathematical vocabulary.
Communication
Check for the following:
•The reader will not need to refer to the list of questions in order to understand my work. •My responses are not numbered.
•I have an introduction, conclusion, title page, and table of contents. •All graphs are labeled – Each graph has a title, labeled axes, and appropriate scale. •My graphs and tables are within the body of my work. They are not separate or in an appendix. •I have explained why I made the choices I did when going through the task. •I did not include key stroke sequences, e.g. “I pressed the 2nd key, then TRACE…” •My tables do not straddle pages.
•My tables are labeled well, including my variable definitions in each column.
Use of Technology
Check for the following:
•I used technology to illustrate my points and ideas. I didn’t just “stick in” a graph. •Each graph or table (or other piece of tech.) is accompanied by explanations and my ideas. •I did not include too many graphs on the same axes – my graphs are easy to read.
Mathematical Process (Type 1)
Check for the following:
•I explicitly defined variables and parameters the first time I used them, even if they were already defined in the portfolio handout.
•I stated any constraints/limits on the values of the variables (e.g., “time cannot be negative”). •I did not state conjectures/ideas as facts.
•I used proper mathematical strategies to produce my data. •I organized my data in a table, chart, or other useful representation. •I tested my general statement by using known data.
•After testing my general statement with known data, I considered further examples in order to further test and apply my general statement.
...Jonghyun Choe
March 25 2011
MathIBSL
Internal Assessment – LASCAP’S Fraction
The goal of this task is to consider a set of fractions which are presented in a symmetrical, recurring sequence, and to find a general statement for the pattern.
The presented pattern is:
Row 1
1 1
Row 2
1 32 1
Row 3
1 64 64 1
Row 4
1 107 106 107 1
Row 5
1 1511 159 159 1511 1
Step 1: This pattern is known as Lascap’s Fractions. En(r) will be used to represent the values involved in the pattern. r represents the element number, starting at r=0, and n represents the row number starting at n=1. So for instance, E52=159, the second element on the fifth row. Additionally, N will represent the value of the numerator and D value of the denominator.
To begin with, it is clear that in order to obtain a general statement for the pattern, two different statements will be needed to combine to form one final statement. This means that there will be two different...
...Alma Guadalupe Luna
MathIA (SL TYPE1)
Circles
Circles
Introduction
The objective of this task is to explore the relationship between the positions of points within circles that intersect.
The first figure illustrates circle C1 with radius r, centre O, and any point P. r is the distance between the centre O and any point (such as A) of circle C1.
Figure 1
The second diagram shows circle C2 with radius OP and centre P, as well as circle C3 with radius r and centre A. An intersection between C1 and C2 is marked by point A. The intersection of C3 with OP is marked by point P’.
Figure 2
Through this investigation I will examine how the r values correlate with the values of OP in determining the length of OP’ when r is held as a constant variable and the value of OP is the variable that is subject to change. I will then venture on to study the inverse, the relationship when the r values becomes the variable that is changed and the OP value is held constant.
r as a Constant
If we let the value of r be equal to 1, we can use that information to find the length of OP’ when OP=2, 3, and 4. The first thing one can deduce is that by using the points A, O, P’, and P two isosceles triangles can be formed; ∆AOP and ∆AOP’. To rationalize this assertion through an analytic approach it should first be understood...
...Taipei European SchoolMath Portfolio

VINCENT CHEN 
Gold Medal Heights
Aim: To consider the winning height for the men’s high jump in the Olympic games
Years  1932  1936  1948  1952  1956  1960  1964  1968  1972  1976  1980 
Height (cm)  197  203  198  204  212  216  218  224  223  225  236 
Height (cm)
Height (cm)
As shown from the table above, showing the height achieved by the gold medalists at various Olympic games, the Olympic games were not held in 1940 and 1944 due to World War II.
Year (1=1932, 2 = 1936 and so on)
Year (1=1932, 2 = 1936 and so on)
Using autograph, the graph above is a scatter graph showing the high jump results from the table.
The plot suggests that the high jump heights start off with a steep positive slope then coming to a decreasing negative slope, however without the 1940 and 1944 high jump competitions, it may not be certain. Then finally, it starts increasing again with a fairly steep positive slope.
However, it would not be realistic if the function has an infinitely increasing range, such as quadratic, exponential and linear because of the limitations that humans have due to natural forces like gravity. Therefore, narrowing down the options that may fit this graph to natural logarithm and logistics
Since the statistics given starts from year 1896, in order to make sure that calculations can be as simplified as possible, I have decided to rearrange the table with the...
...In Lacsap’s Fractions, En(r) refers to the (r+1)th term in the nth row. The numerator and denominator are found separately, therefore to find the general statement, two different equations, one for the numerator and one for the denominator, must be found. Let M=numerator and let D=denominator so that En(r) = M/D.
To find the numerator for any number of Lacsap’s Fractions, an equation must be made that uses the row number to find the numerator. Because the numerator changes depending on the row, the two variables (row number and numerator) must be compared. To find this equation, the relationship between the row number and numerator must be found, put it graph form, and the equation for the graph will be the equation needed.
Row Number, n  Numerator, N 
1  1 
2  3 
3  6 
4  10 
5  15 
Numerator
Numerator
Row Number
Row Number
The equation for the numerator can be derived by using quadratic regression on a graphing calculator. The equation is; y = .5x2 + .5x. This translates into; M=.5n2+.5n, where n=row number, and M=numerator. This means that any numerator from a certain row number can be found by using this equation. For example, to find the numerator of the sixth row, “6” needs to be substituted in for n.
M= .5n2 + .5n
M= .5(6)2 + .5(6)
M= .5(36) + .5(6)
M= 18 + 3
M= 21
The Numerator for row six is 21
They method to find the equation for the denominator is similar, but slightly more difficult...
...Lacsap’s Fractions
IBMathSL
Internal Assessment Paper 1
Lacsap’s Fractions
Lacsap is Pascal spelled backward. Therefore, Pascal’s Triangle can be used practically especially with this diagram.
(Diagram 1)
This diagram is of Pascal’s Triangle and shows the relationship of the row number, n, and the diagonal columns, r. This is evident in Lacsap’s Fractions as well, and can be used to help understand some of the following questions.
Solutions
Describe how to find the numerator of the sixth row.
There are multiple methods for finding the numerator of each consecutive row; one way is with the use of a formula, and another by using a diagonal method of counting illustrated by a diagram.
The following image can be used to demonstrate both techniques to finding the numerator:
(Diagram 2)
This formula uses “n” as the row number and the outcome is the numerator of the requested sixth row.
n2 + n
2
As indicated, inputting the number 6, as the requested sixth row, for n gives the solution of 21.
X = n2 + n
2
X = (6)2 + (6)
2
X = 36 + 6
2
X = 42
2
X = 21
Therefore, as shown, the numerator of the sixth row is 21, and this can be checked for validity by entering each number, 1 through 5, into the formula and making sure that the answer corresponds with the numerator in the above diagram.
Where n = 5: Where n = 4:
X = n2 + n X = n2 + n...
...SL TYPE 1LACSAP’S FRACTIONS
* INTRODUCTION
This investigation is going to do research patterns relates to the Lacsap’s Fractions. For its external structure, Lacsap’s Fraction is analogous to Pascal’s Triangle. Lacsap’s Fraction presents the way of generating and organizing the binomial coefficients. Within this investigation, the work is planning to be divided into two parts. In the first part, the content will relate to the pattern of numerators. In the second part, I am going to do the research on the patterns of denominator and the general statement for. Admittedly, the technology of computing will be involved into this investigation (E.g. Geogebro and GSP5chs). The following figure 11 illustrates Lacsap’s Fraction.
Fig.11
* PART A  CALCULATIONS and ANALYSIS
Firstly, I am going to research the numerator patterns. By observing the numerators of these fractions, it is illustrated that the first row of numerator is 1, second row of numerator is 3, third row of numerator is 6, fourth row of numerator is 10 and fifth row of numerator is 15. Let’s present it into the mathematical way: (= numerator of the row)
Continued
I realize that , , , Thus it is easily to find the numerator of the sixth row which is getting from. In order to do further investigation, it is essential to make a data table.
 numerator 
1  1 
2  3 
3  6 
4  10 
5  15 
6  21 
…  … 
n  ? 
Table 21
According to the...
...OPA Circle Style
The three circles; O, P and A intersect to create an interesting investigation regarding circles. Since this is a Calculus course, the investigation does have to deal with Derivatives. The most important and the focus of this portfolio is the line segment, OP’. Using the given diagram above, this investigation consists of finding the general equation for discovering OP’.
The values that are given are that r, is the radius of C1 and C2. OP and AP are the radii of C3. This information allows for the information to be manipulated too create two isosceles triangles. The first triangle and the one that is given, ∆OPA is an isosceles triangle therefore it can be concluded, thanks to the Isosceles Triangle Theorem that angle O and A are congruent to each other in this triangle. ∆OPA is not the only triangle that can be created, ∆OP’A is the second triangle created with a radius from C2. Therefore ∆OP’A is also an isosceles triangle. Now in both the triangles stated above, they share a common angle, O. With the help of this information we can analyze the preliminary relationship between OP and OP’.
We can find OP’, we have the tools. The two tools that we will be using mainly are the sine law and cosine law (respectively);
Sin Aa = Sin Bb = Sin Cc a2= b2+ c22bc cosA
and the two triangles that are going to be used;
For the first calculations, r=1 and OP =2. By finding the ∠O in one triangle, I have found the ∠O in both triangles,...
...
AP/IB Biology Lab Assessment
The effect of various fruit and vegetable cell membranes on their water potential
Independent variable: Type of fruit or vegetable. (Produce used: Russet potatoes, Pascal celery, Gala apple, Navel orange, and Imperator carrot). The fruit or vegetable will be placed in six 56.7 gram cups, ranging with sucrose molarities of 0 (distilled water), 0.2, 0.4, 0.6, 0.8, 1.0, with 5 trials, leading to 30 cups for each produce variable.
Dependent variable: The water potential of the produce, found by placing the produce in different molarities of sucrose and finding the isotonic state of the produce with a plotted line graph.
Controlled variables: The controlled variables include:
The type of produce used: Each variable produce is of the same type. Only Russet potatoes, Pascal celery, Gala apple, Naval orange, and an Imperator carrot were used.
The produce to solution ratio. Each sample of produce for each trial was completely submerged in 24 ml (measured with a graduated syringe) of the solution: either distilled water or sucrose of specific molarity. If the produce was not completely submerged, the measuring of the mass difference and the water potential may not be accurate.
The produce, while submerged, received no light. The 56.7 gram cups were covered with aluminum foil, to prevent the sucrose or water from evaporating.
All samples of produce were weighed for initial mass and final mass on a digital gram scale, to...
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