Math SL PortfolioLacsap’s Fractions
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
121
1331
14641
15101051
Fig. 1Pascal’s Triangle
This portfolio is on “Lacsap’s Fractions”, 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’s fractions, I will now take the given set of numbers involved in Lacsap’s triangle, which are presented in a symmetrical fashion, similar to Pascal’s triangle. It is shown in Figure 2. 11
Row 3 (n=3)
Row 3 (n=3)
Row 2 (n=2)
Row 2 (n=2)
1321
164641
Row 5 (n=5)
Row 5 (n=5)
Row 4 (n=4)
Row 4 (n=4)
11071061071
1151115915915111
Fig. 2Lacsap’s Triangle
The focus of this portfolio is to find 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, starting with r=0. For example, E52=15; The 2nd element in the 5th row is 15.
When observing the triangle, there are many patterns that could be found among the numbers. However, I will only investigate on the following patterns: For convenience, I will be taking the numerator of a row as Sn and the row number as n. I see that the row number is related to the numerator of each row. I noticed that the sum of the previous row’s numerator and the row number of that row gives the numerator of that row. This pattern is shown in the table 1 below to calculate the numerators of the different rows.
Row (n) Numerator of previous row (Sn1) Numerator (Sn) n=1  S1=1
n=2 S1=1 S2=3
n=3 S2=3 S3=6
n=4 S3=6 S4=10
n=5 S4=10 S5=15
n=6 S5=15 S6=21
n=7 S6=21 S7=28
n=8 S7=28 S8=36
Table 1Relation Between Numerator and Row Number and Previous Row Number’s Numerator Therefore, I will say that the general formula for finding the numerators is: Sn=n+Sn1
where Sn is the new numerator, Sn1is the previous row’s numerator, and n is the row number. I shall prove this statement by substituting values of Sn, n, and Sn1. If I take n=4, I know that S3=6. I will substitute these values in the above statement to find S4. S4=4+S3
S4=4+6
∴S4=10
The general statement works! Hence, to find the numerator of the 6th row, I can use the formula mentioned above:
I know that when n=5, S5=15. Hence we can use the method that we had found earlier to find the numerator value of Row 6. If I take n=6 and S5=15 (since s=15 when n=5), I substitute these values in the above equation to obtain the numerator when n=6. S6=6+S5
S6=6+15
∴S6=21
I will confirm our findings about the relation between the row number and numerator by plotting the results on Table 2 below, followed by a graph.
Table 2Relation Between Numerator and Row Number
Row Number (n)(Xaxis) Numerator (S)(yaxis)
1 1
2 3
3 6
4 10
5 15
6 21
7 28
8 36
Fig. 3 Graph of Relation of...