In my internal assessment, type 1, I was given Lacsap's Fractions 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's Fractions
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 the row and y represents the numerator. To prove the equation right, I chose three random points:
A (2,3), B (3,6), C (4,10) and plugged it into the equation y=ax2+bx+c
3=a22+b2+c
6=a(32)+b3+c
10=a42+b4+c
3=4a+2b+c
6=9a+3b+c
10=16a+4b+c
3610=4219311641abc 36100.510.53.562.5683=abc
abc=0.50.50
Therefore the line equation for the numerator is 12n2+12n=u

The numerator (u) happens to always be bigger than the denominator (v) no matter what row (n) it is present on, therefore one can conclude that the n, v and r are all...
...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 * (n+1) Nn(r) =...
...Investigation
Lacsap’sFractions
The focus of this investigation is surrounding Lascap’s Fractions. They are a group of numbers set up in a certain pattern. A similar mathematical example to Lacsap’sFractions is Pascal’s Triangle. Pascal’s Triangle represents the coefficients of the binomial expansion of quadratic equations. It is arranged in such a way that the number underneath the two numbers above it, is the sum.
Ex. 1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
In the example of Pascal’s Triangle below, the highlighted numbers represent this pattern. The two numbers above the third add up to equal the third. (e.g. 2+1=3)
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Although Pascal’s Triangle is set up similarly to Lacsap’sFractions, the patterns are different and the numbers are set up in fractions as opposed to whole numbers. The first five rows of Lacsap’sFractions are set up like this:
1
1 3/2 1
1 6/4 6/4 1
1 10/7 10/6 10/7 1
1 15/11 15/9 15/9 15/11 1
In the next part of the investigation, we will explore the patterns...
...
Math SL 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, I will now take the given...
...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...
...Exploration of Lacsap’sFractions
The following will be an investigation of Lacsap’sFractions, that is, a set of numbers that are presented in a symmetrical pattern. It is an interesting point that ‘Lacsap’ is ‘Pascal’ backwards, which hints that the triangle below will be similar to “Pascal’s Triangle”.
1 1
1 1
1 1
1 1
1 1
There are many patterns evident in this triangle, for instance I can see that there is a vertical axis of symmetry down the middle of the triangle. Each row starts and ends with the number 1. Each row has one more variable than the number of rows, i.e. row 1 has 2 variables. The numerators in the middle stay the same and the diagonals form sequences.
In order to decipher the pattern in the numerators and denominators, I had to look at the triangle a different way. Knowing that the numerators of the row don’t change, it occurred to me that the number 1s on the outside of the triangle could be expressed as fractions.
This proves that all the numerators of the row are the same.
To further investigate the numerators, I will examine the relationship between the
row number and the numerator, which is shown in the table below. These are the numerators after having...
...Lacsap’sFractions
The aim if this IA is to investigate Lacsap’sFractions 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].
(Row number +1)C2 Numerator 
(2+1)C2 = 3 
(3+1) C2 = 6 
(4+1)C2 =10 
(5+1)C2 =15 
Now that we have found an equation to solve to...
...Lacsap’sFractions
Laurie Scott
SL Math 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 Numerator and Denominator of the...
...In elementary math there are several concepts about fractions. One concept students in fourth grade will need to master is learning how to tell if fractions are equivalent with unlike denominators. There are a few prerequisite skills that are necessary in order for the students to understand this concept. The first thing students need to know is what fractions are. Fractions are a way of counting parts of a whole. Secondly, the students need to know how to identify parts of a fraction. The top number in a fraction is the numerator. The numerator is the number of parts in a whole (Eather). The bottom number in a fraction is the denominator. The denominator is the number of parts the whole is divided into (Eather). Lastly, the student will need to have a basic knowledge of their multiplication and division facts. This will help the students in deciding whether or not the fraction is indeed equivalent or not.
The first step in teaching students about equivalent fractions is to have a whole class conversation using manipulatives or visual aides. I would start the lesson with an overhead projection or use of a mimeo board in order to show the students what equivalent fractions look like. I would start with two circles on the board, one divided into two pieces and one divided into four. You can show the students by coloring in one of the...