The aim if this IA is to investigate Lacsap’s Fractions and to come up with a general statement for finding the terms.
When I noticed that Lacsap was Pascal spelt backwards I decided to look for a connection with Pascal’s triangle.
Pascal’s triangle is used to show the numbers of ‘n’ choose ‘r’(nCr). The row number represents the value of ‘and the column number represents the ‘r’ value. Eg. Row 3, colomn 2 = 3C2 = 2.
I noticed that all the numerators of the fractions in Lascap’s fraction (3,6,10,15) are also found in Pascal’s triangle. So I tried to see if I would get the denominator of the fractions by using the row as ‘n’ and the colomn (or element) as ‘r’. This did not work out because Lascap’s triangle does not have a row with only one element like Pascal’s does. To solve this I just added 1 to each row number. This gives me the formula[pic].
Now that we have found an equation to solve to numerator of the fractions, we now have to try and work out how to solve the denominator. As we already know the numerator, we can presume that there is some sort of relationship between this and the denominator.
I noticed that the difference between the numerator and denominator increased by one with each row I went down (1,2,3,4)
Because the denominators in one row are different from each other (row 3 = 7, 6, 7), it can be presumed that the unknown ‘r’ which stands for the elements/column will be part of the general statement.
Because the first and last element in each row is 1 (or 1/1) and no general statement is needed to find these, they can be ignored and erased from the triangle for now.
Row number (n) Difference between numerator and denominator in the 1st element (see   above)  1 /  2 1  3 2  4...
...In Lacsap’sFractions, 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’sFractions, 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...
...Lacsap’sFractions
Laurie Scott
SLMath Internal Assessment
Mr. Winningham
9/5/12
Instructions: In this task you will consider a set of numbers that are presented in a symmetrical pattern.
Pascal’s Triangle
n=0 1 
1 0 
2 3 
3 6 
4 10 
5 15 
6 21 
Table 1: Relationship between Row Number and Numerator of Figure 2
[pic]
Figure 3: Graph of the relationship between Row Number and Numerator of Figure 2
In order to find the sixth and seventh rows, a pattern for determining the denominator must be found:
First it is helpful to determine a relationship between the numerator and denominator of the first term in each row:
Row Number ( n ) Difference of Numerator and 
 Denominator (1st term) 
1 0 
2 1 
3 2 
4 3 
5 4 
Table 2: Relationship between Row Number and the difference of the...
...
MathSL PortfolioLacsap’sFractions 
Type 1: Investigation Portfolio
Greenwood High (An International School) 




Table of Contents:
Introduction……………………………………………………………………………………………………..……..…...Page 2
Patterns in Numerator………………………………………………………………………………….………………Page 2 and Page 3
Plotting Graph of Row Number and Numerator……………………………………………………………Page 4 to Page 7
Finding Denominator………………………………………………….………………………………………..………Page 8 to Page 9
Finding Further Rows……………………………………………………………………..…………………………… Page 10
General Statement……………………………………………………………………………………………………….Page 10
Scope and Limitations…………………………………………………………………………………………………..Page 15
Conclusion…………………………………………………………………………………………………….………………Page15
Pascal’s Triangle, a graphical representation by the French mathematician, Blaise Pascal, is used to show the relationship of numbers in the binomial theorem. It is shown in Figure 1 below:
1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Fig. 1Pascal’s Triangle
This portfolio is on “Lacsap’sFractions”, and finding a pattern in the numerators and denominators of the fractions, as well as creating a general statement for En(r)where r is the element in the nth row; I shall start with r=0.
Row 1 (n=1)
Row 1 (n=1)
As you can tell, Lacsap is just Pascal written backwards! To investigate further on Lacsap’sfractions,...
...INTRODUCTION
Lacsap’s triangleThe set of numbers in concern are basically an inverse of the Pascal’s triangle. These terms themselves are fractions which follow different series themselves. There is a specific function that can accurately predict the fractional numbers accurately. Using the graph plots we can calculate this function and predict the numbers accurately. The whole process for finding the adequate function would involve the use of different smaller function and thus create a general function.
The numerator follows a different equation and the denominator a different one.
1 1
1 1
1 1
1 1
1 1
Therefore if we only consider the numerators in the triangle
1 1
1 1
1 1
1 1
1 1
One can observe that the numerator values increase in a specific order.
So, Let r = number of rows; N= numerator.
Since the first row does not contain a fraction we keep its value as 1; we start from the second row onwards since it has a fraction.
Difference between two r values –
Nr3 – Nr2 = 63= 3
Nr4 – Nr3 = 106=4
Nr5 – Nr4 =1510=5
As you can observe that the differences highlighted in yellow, increase by 1 as we the r increases.
Using this knowledge we could also find...
...Mathematics
SLLACSAP’SFraction‐
Portfolio
Type
I
LACSAP’SFractions  MathSL Type I
Name: Yao Cia Hua
Date: March 22nd, 2012
Teacher: Mr. Mark Bethune
School: Sinarmas World Academy
1
Yao
Cia
Hua
Mathematics
SLLACSAP’SFraction‐
Portfolio
Type
I
Lacsap triangle is a reversed Pascal triangle. This task focuses mainly on finding the
relationship between the number of row n and the numerator N and also the
relationship between the element of a row r and the denominator D . Through this,
a general statement base for En (r) on N and D are suppose to be stated and
explained. For this task various technologies, such as Geogebra, MathType and
calculator, are needed in order to produce a more organized piece of work and
clearer graphs and diagrams.
Finding The Numerator In The Sixth Row:
In order to find the sixth row of this of this triangle, a pattern must firstly be found.
As it is seen from the diagram on the above, the pattern shows that, by adding 1 to
each of the difference between the 2 previous numerators, this will equate the
numerator of the next row.
For example: (using the first 2 numerators)
Difference: 3 – 1 = 2
The following numerator: 2 + 1...
...Lacsap’sFractions
IB Math 20 Portfolio
By: Lorenzo Ravani
Lacsap’sFractions Lacsap is backward for Pascal. If we use Pascal’s triangle we can identify patterns in Lacsap’sfractions. The goal of this portfolio is to ﬁnd an equation that describes the pattern presented in Lacsap’sfraction. This equation must determine the numerator and the denominator for every row possible.
Numerator
Elements of the Pascal’s triangle form multiple horizontal rows (n) and diagonal rows (r). The elements of the ﬁrst diagonal row (r = 1) are a linear function of the row number n. For every other row, each element is a parabolic function of n. Where r represents the element number and n represents the row number. The row numbers that represents the same sets of numbers as the numerators in Lacsap’s triangle, are the second row (r = 2) and the seventh row (r = 7). These rows are respectively the third element in the triangle, and equal to each other because the triangle is symmetrical. In this portfolio we will formulate an equation for only these two rows to ﬁnd Lacsap’s pattern. The equation for the numerator of the second and seventh row can be represented by the equation: (1/2)n * (n+1) = Nn (r) When n represents the row number. And Nn(r) represents the numerator Therefore the numerator of the sixth row is Nn(r) = (1/2)n *...
...Jonghyun Choe
March 25 2011
Math IB SL
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...
...MathSL I.A:
Lacsap'sFractions
Introduction
In my internal assessment, type 1, I was given Lacsap'sFractions task. To do my calculations I used a TI84 graphic calculator. To type the I.A I used Apple's Pages, Microsoft Excel 2011 and Microsoft Word 2011.
Lacsap'sFractions
To find the numerator of the sixth row I looked at the difference between each of the numerators.
1 1
1 32 1
1 64 64 1
1 107 106 107 1

1 1511 159 159 1511 1
Row  Numerator  Difference between numerators nnn1 
1  1  1 
2  3  2 
3  6  3 
4  10  4 
5  15  5 
6  21  6 
7  28  7 
From this table one can notice that the difference between each numerator on each row is always d+1, where d represents the difference between the two previous rows. The equation representing this is: un=un1+(d+1), where un represents the numerator you are looking for, n is the row number and un1 is the numerator from the previous row.
Therefore to find the numerator of the 6th row I did:
u6=15+5+1
u6=21
After that, I plotted a graph doing numerator vs. row:
The relationship between the numerator and the row is best described by the equation of the line:
y=0.5x2+0.5x
The equation is quadratic and can be used to determine any numerator at any row. In the equation, x stands for...