Ib Math Portfolio Lacsap's Fractions
Exploration of Lacsap’s Fractions
The following will be an investigation of Lacsap’s Fractions, 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 changed the 1s on the outside of the triangle to their fraction forms, thus making all the numerators the same. Row Number (n) | Numerator in Each Row (N) | 1 | 1 | 2 | 3 | 3 | 6 | 4 | 10 | 5 | 15 |
From this table, I can see that the numerator is the result of the current row number added to the numerator of the previous row. That is, if you’re looking for the numerator of row 4, take that number and add the numerator from row 3, which is 6. 4+6=10, and 10 is the numerator of row 4, as shown in the table above. Knowing this, I predict the numerator of
The following will be an investigation of Lacsap’s Fractions, 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 changed the 1s on the outside of the triangle to their fraction forms, thus making all the numerators the same. Row Number (n) | Numerator in Each Row (N) | 1 | 1 | 2 | 3 | 3 | 6 | 4 | 10 | 5 | 15 |
From this table, I can see that the numerator is the result of the current row number added to the numerator of the previous row. That is, if you’re looking for the numerator of row 4, take that number and add the numerator from row 3, which is 6. 4+6=10, and 10 is the numerator of row 4, as shown in the table above. Knowing this, I predict the numerator of