Ib Math Sl Type1 Ia 2012
SL TYPE 1-LACSAP’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 1-1 illustrates Lacsap’s Fraction.
Fig.1-1
* 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 2-1
According to the table 2-1 and the foundational patterns present on the first page, it is sure there must be a mathematical relationship of row number and numerator number in between. To search the deeper relationship, the plot diagram is the meaningful and practical choice due to the shared feature with function diagram. I decide to let numerator as the y-axis and nth row as the x-axis.
Numerator
Row number
Fig.2-1
By looking through the diagram (Fig.2-1), the shape of the curve seem to be similar with the quadratic
* 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 1-1 illustrates Lacsap’s Fraction.
Fig.1-1
* 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 2-1
According to the table 2-1 and the foundational patterns present on the first page, it is sure there must be a mathematical relationship of row number and numerator number in between. To search the deeper relationship, the plot diagram is the meaningful and practical choice due to the shared feature with function diagram. I decide to let numerator as the y-axis and nth row as the x-axis.
Numerator
Row number
Fig.2-1
By looking through the diagram (Fig.2-1), the shape of the curve seem to be similar with the quadratic