Math Lacsap's Fraction Portfolio
| Math SL Portfolio-Lacsap’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
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1 Fig. 1-Pascal’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.
1 1
Row 3 (n=3)
Row 3 (n=3)
Row 2 (n=2)
Row 2 (n=2)
1 32 1
1 64 64 1
Row 5 (n=5)
Row 5 (n=5)
Row 4 (n=4)
Row 4 (n=4)
1 107 106 107 1
1 1511 159 159 1511 1 Fig. 2-Lacsap’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
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. 1-Pascal’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.
1 1
Row 3 (n=3)
Row 3 (n=3)
Row 2 (n=2)
Row 2 (n=2)
1 32 1
1 64 64 1
Row 5 (n=5)
Row 5 (n=5)
Row 4 (n=4)
Row 4 (n=4)
1 107 106 107 1
1 1511 159 159 1511 1 Fig. 2-Lacsap’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