# Lacsap's fractions

Topics: Elementary arithmetic, Number, Fraction Pages: 17 (2839 words) Published: September 28, 2014

Yao
Cia
Hua

Mathematics
SL

LACSAP’S
Fraction-­‐
Portfolio
Type
I

LACSAP’S Fractions - Math SL Type I
Name: Yao Cia Hua
Date: March 22nd, 2012
Teacher: Mr. Mark Bethune

1

Yao
Cia
Hua

Mathematics
SL

LACSAP’S
Fraction-­‐
Portfolio
Type
I

Lacsap triangle is a reversed Pascal triangle. This task focuses mainly on finding the relationship between the number of row n and the numerator N and also the relationship between the element of a row r and the denominator D . Through this, a general statement base for En (r) on N and D are suppose to be stated and explained. For this task various technologies, such as Geogebra, MathType and calculator, are needed in order to produce a more organized piece of work and clearer graphs and diagrams.

Finding The Numerator In The Sixth Row:
In order to find the sixth row of this of this triangle, a pattern must firstly be found.

As it is seen from the diagram on the above, the pattern shows that, by adding 1 to each of the difference between the 2 previous numerators, this will equate the numerator of the next row.
For example: (using the first 2 numerators)
Difference: 3 – 1 = 2
The following numerator: 2 + 1 = 3

2

Yao
Cia
Hua

Mathematics
SL

LACSAP’S
Fraction-­‐
Portfolio
Type
I

There is also another way of looking at these patterns. If the difference between the 2 previous numerators is already known, the following numerator can be found by summing the difference and the value of the current numerator, finally adding 1 . For example: (using the first 2 numerators)

Difference: 3 - 1 = 2
Current numerator: 3
The following numerator: ( 2 + 3 ) + 1 = 6
So, in order to find the numerator of the sixth row, either pattern 1 or 2 can be applied. By using pattern 1 , the value 6 can simply be added to the current numerator value, which is 15 . This will then give an answer of 21 . By using the second pattern, the value 5 , which is the difference between 15 and 10 , can be added to 15 then add 1 to find the value. Either way will result the same value.

The Relationship & Finding A General Statement For The Numerator (N):

3

Yao
Cia
Hua

Mathematics
SL

LACSAP’S
Fraction-­‐
Portfolio
Type
I

When only looking at these points above, the relationship between the number of row, n , and the value of the numerator, N , without drawing any curve, it seems to be exponential. However, when the graph is continued as shown above, it can be identified that all the co-ordinate points lay of the same parabola. A parabola has a general function of y = Ax 2 + Bx + C . As there are 3 variables in this function, A , B and C , 3 functions are required.

Process Of Finding A General Statement For Numerator ( N ):

(1;1),(2; 3),(3;6)
1 = (A × (1 )) + (B × 1) + C
1= A+ B+C
2

3 = (A × (2 2 )) + (B × 2) + C
3 = 4A + 2B + C
6 = (A × (32 )) + (B × 3) + C
6 = 9A + 3B + C

sub.A =

1 = A + B + C.........1
3 = 4A + 2B + C........2
6 = 9A + 3B + C.........3
2 − 1 : 2 = 3A + B........4
3 − 2 : 3 = 5A + B...........5
5 − 4 :1 = 2A
1
A=
2

1
2

1
+ B + C.........1'
2
1
3 = (4 × ) + 2B + C.......2'
2
1
= B + C.........1''
2
3 = 2 + 2B + C..........2''
1 = 2B + C..........2'''
1
2'''− 1'' : = B
2
1
B=
2
1=

1
1
&B=
2
2
1 1
1 = + + C..........1'''
2 2
1 = 1+ C
C=0
sub.A =

1
1
Sub. A = , B = ,C = 0 to y = Ax 2 + Bx + C
2
2
1
1
y = x2 + x
2
2
1
y = x(x + 1)
2

4

Yao
Cia...