Ib Maths Sl Portfolio Lacsap Fraction

Topics: Fraction, Elementary arithmetic, Number Pages: 14 (2368 words) Published: August 22, 2013
IB Maths SL TYPE I

Lacsap’s Fractions Portfolio

Lacsap’s fraction
Introduction:
Lacsap’s fraction is a symmetrical triangle that has the following pattern in the first five rows

The shape is similar to Pascal triangle. It has the same quantity of symmetrical triangle as Pascal triangle. And Lacsap is the inverse alphabet order of Pascal. These make me think about Pascal triangle and I made an assumption that elements in Lacsap triangle may have the same relationship as in Pascal triangle. However, the elements in Lacsap’s fraction triangle may or may not have the same relationship as Pascal triangle so I ignore my hypothesis about Pascal triangle and decide to find the relationship by not referring to Pascal triangle. I believe that every element in Lacsap’s triangle must be in a sequence and the task for this portfolio is to find the relationship between each element.

This portfolio will be divided into five parts; finding numerator relationship, finding denominator relationship, test the accuracy of the statement, finding additional rows and limitations of the statement.

The notations in this portfolio are
* X = the element place
* N = row number
* En(x) = The xthelement on the nthrow

Numerator Relationship:
First thing I notice about the Lacsap is that the numerators are the same in the same row. The numerators are the same in each row

My first statement about Lacsap’s fraction is the numerators are the same in each row.

Considering the second diagonal row may give me some clue about how to find the numerator. If I can find the numerator for the second diagonal, the numerator for that row is revealed because the numerators are the same in the same horizontal row. Second diagonal row

Row number| Numerator| 1st difference| 2nd difference|
1| 1| -| -|
2| 3| 2| -|
3| 6| 3| 1|
4| 10| 4| 1|
5| 15| 5| 1|

I can see that the numerator’s 1st difference is 2, 3, 4 and 5 for the first five rows and the 2nd difference is 1. The numerators are triangular number. Please see the next page for the explanation of triangular number.

Triangular numbers are a sequence of number that follows the following pattern: Term| Triangle| Number|
1| | 1|
2| | 3|
3| | 6|
4| | 10|
And so on….

The table below shows how the numerators in each row are derived. Row number| Numerator| Numerator|
1| 1| 1|
2| 1+2| 3|
3| 1+2+3| 6|
4| 1+2+3+4| 10|
5| 1+2+3+4+5| 15|

Plot the numerators in the graph to find the relationship between row number and the numerator. n= row number| f(n)= Numerator|
1| 1|
2| 3|
3| 6|
4| 10|
5| 15|

I used Casio FX9860 Calculator to plot the graph and find the equation. Given that f(n) = row number and n= numerator.
Plot
Data input

Graph
Equation

The equation of the graph isfn=0.5n2+0.5n
Therefore, I can make a statement about finding the numerator that: numerator=0.5n2+0.5n
numerator=nn+12
Where: n=row number

Conclusion for finding the numerator:
* All the numerators are the same in the same row.
* numerator=nn+12 when n = row number

Denominator Relationship:
Turning the first and last element of each row (they are 1 for every row) to fraction may give me some clue of how to find the denominator. The numerator and denominator of the fraction will be the same to make the fraction equal to 1. From the generalisation I have made before “the numerators are the same in each row” the numerator of the first and last element will equal to the numerator of the other elements in that row. This is a new Lacsap triangle.

The denominators in each row seem to have a quadratic relationship. I think this is because first and last denominators of each row are the highest and the middle denominator is the lowest. The denominators are symmetrical which means everything on the left half is equal to the right half.

I used Logger Pro to find the relationship of...
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