• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7

# LACSAP's Functions

Extracts from this document...

Introduction

LACSAP’S FRACTIONS

Tanya Zhandria Waqanika
Prince Andrew High School
IB Math SL Portfolio #1
Mr Brown

Task 1 – Find the Numerator
Task 2 – Plot relation between the row number and Numerator, write general statement about it
Task 3 – Find the 6
th and 7th rows
Task 4 – Find the general statement for E
n(r)
Task 6 – Discuss the scope/limitations of the General Statement

Task 1 - Finding the Numerator

Since ‘LACSAP’ is just ‘Pascal’ reversed, I decided to do some research about Pascal’s triangle. Through my research, I was able to come across the equation nCr, or n!/r!(n –r)! where n represents the row number and r represents the ‘element’ number, or the diagonal row number.

e.g. 3C2 = 3!/2!(3 – 2)! = 1 x 2 x 3/1 x 2 x 1 = 3

This equation is used to find any number within Pascal’s triangle. With 1 at the top of both pyramids representing n = 0, r = 0, I decided to compare Pascal’s triangle to LACSAP’s fractions.

Pascal’s triangle

 1 1 1 1 1 1 1 1 1 1

LACSAP’S Fractions.

Since I’m trying to find the numerator, I was looking for any similarities between the two triangles.

Middle

th an 7th Numerators

6th Numerator
(6+1)C2
(7)
C2
6!/2!(6 – 2)!
6!/2!4! =
21

7th Numerator

(7+1)C2
(8)
C2
8!/2!(8– 2)!
8!/2!6! =
28

These numbers can also be found going down r = 2 of Pascal’s triangle.

Finding the Denominator

Whilst it took a bit of trial of error, I found that the difference between the Numerator and the Denominator were following a pattern similar to Pascal’s Triangle, eg. In this grid, I have replaced the numbers with the differences between the Numerator and Denominator to clearly see the pattern.

 0 0 1 1 1 1 1 1 1 1 1

Comparing the new triangle to Pascal’s Triangle, it almost appears that the new triangle is a mirror image of Pascal’s triangle. Using this new information, I decided to plot the row numbers of LACSAP’s triangle against r = 1 of the new triangle as it’s the only diagonal row that appears consistent with Pascal’s triangle.

 n (Row number) Difference of Numerator and Denominator 1 0 2 1 3 2 4 3 5 4

According to the table, the difference of the Numerator and Denominator is (n -1) as the values for the differences, though the same, are only the same once they’ve been bumped down one.

Conclusion

n(r), can be written as:

Numerator/Denominator = (n+1)C2/( (n+1)C2) - r(n - r)

Row 8
(8+1)C2/( (8+1)C2) - r(8 - r)

 Element Number (n+1)C2/( (n+1)C2) - r(n - r) 0 (8+1)C2/( (8+1)C2) - 0(8 - 0) = 1 1 (8+1)C2/( (8+1)C2) - 1(8 - 1) = 36/29 2 (8+1)C2/( (8+1)C2) - 2(8 - 2) = 36/24 3 (8+1)C2/( (8+1)C2) - 3(8 - 3) = 36/21 4 (8+1)C2/( (8+1)C2) - 4(8 - 4) = 36/20

After this, the rest of the row is reflected, thus it looks like
1, 36/29, 36/24, 36/21, 36/20, 36/21, 36/24, 36/29, 1

Row 9
(9+1)C2/( (9+1)C2) - r(9 - r)

 Element Number (n+1)C2/( (n+1)C2) - r(n - r) 0 (9+1)C2/( (9+1)C2) - 0(9 - 0) = 1 1 (9+1)C2/( (9+1)C2) - 1(9 - 1) = 45/37 2 (9+1)C2/( (9+1)C2) - 2(9 - 2) = 45/31 3 (9+1)C2/( (9+1)C2) - 3(9 - 3) = 45/27 4 (9+1)C2/( (9+1)C2) - 4(9 - 4) = 45/25

1, 55/47, 55/41, 55/37, 55/35, 55/35, 55/37, 55/41, 55/47, 1

Row 10
(10+1)C2/( (10+1)C2) - r(10 - r)

 Element Number (n+1)C2/( (n+1)C2) - r(n - r) 0 (10+1)C2/( (10+1)C2) - 0(10 - 0) = 1 1 (10+1)C2/( (10+1)C2) - 1(10 - 1) = 55/46 2 (10+1)C2/( (10+1)C2) - 2(10 - 2) = 55/39 3 (10+1)C2/( (10+1)C2) - 3(10 - 3) = 55/34 4 (10+1)C2/( (10+1)C2) - 4(10 - 4) = 55/31 5 (10+1)C2/( (10+1)C2) - 5(10 - 5) = 55/20

1, 55/46, 55/39, 55/34, 55/31, 55/20, 55/31, 55/34, 55/39, 55/46, 1

Task 6 - Discuss the scope/limitations of the General Statement

Some limitations of General Statement are that:

• The numerator must be greater than 0
• In the equation (n+1)C2,  n +1 must be greater than 2/r in order for the General Statement to work

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related International Baccalaureate Maths essays

1. ## Extended Essay- Math

ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ï¿½^ï¿½ï¿½ï¿½ï¿½Cï¿½ï¿½?ï¿½ï¿½3/4ï¿½ï¿½ï¿½\ ï¿½ ï¿½uï¿½ï¿½!?ï¿½ï¿½noï¿½/? ï¿½_jï¿½ï¿½ï¿½@P@P@P@P@P@P@P@P@P@P@/ï¿½wï¿½ï¿½ï¿½ï¿½Åï¿½pï¿½ï¿½ï¿½ï¿½(r)P ï¿½Eï¿½ ï¿½:ï¿½ï¿½ï¿½ï¿½ï¿½l?ï¿½7ï¿½-ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ ï¿½ï¿½ï¿½ ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½ ( ï¿½?-ï¿½ï¿½;ï¿½Pï¿½ï¿½ï¿½8oï¿½ï¿½tï¿½(ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Oï¿½6Ûï¿½ï¿½ï¿½ï¿½_ï¿½Wï¿½...ptP@P@P@P@P@P@P@P@P@P@P@ï¿½ï¿½ï¿½(pï¿½gï¿½ï¿½7ï¿½ï¿½ï¿½:k"ï¿½Qtï¿½ï¿½ï¿½ï¿½d'ï¿½ï¿½ï¿½ï¿½eï¿½ï¿½"ï¿½Bï¿½?ï¿½:(1/2ï¿½ï¿½ï¿½ï¿½]]ï¿½4ï¿½ko=ï¿½"úï¿½JÈWï¿½ ï¿½...2I`ï¿½]ï¿½rï¿½ï¿½ßï¿½ï¿½QOÙ·ï¿½oï¿½tï¿½ï¿½zï¿½ï¿½1/4ï¿½kR|6ï¿½ ï¿½ï¿½&'iï¿½jV>3ï¿½ï¿½(r)k~ï¿½3/4-ï¿½-~"ï¿½Í¡xï¿½Gï¿½6ï¿½4=n Wï¿½+ï¿½ï¿½ï¿½ï¿½51/2{Mï¿½ï¿½ï¿½ï¿½;- ï¿½ï¿½1/4Jï¿½ï¿½hï¿½-1/4 yï¿½(tm)ï¿½3/4ï¿½tï¿½ï¿½ï¿½ï¿½Cd\$ï¿½|0ï¿½-+ï¿½Xd3/4"3/4ï¿½ï¿½ ï¿½!ï¿½ï¿½%ï¿½ï¿½5Y1/4Gï¿½Eï¿½[]I8Pï¿½ï¿½ï¿½ï¿½ï¿½>ï¿½ï¿½ï¿½xï¿½ï¿½j_4ï¿½ï¿½_|ï¿½ï¿½kï¿½(r)ï¿½º"ï¿½ï¿½ï¿½^oï¿½wï¿½1/2ï¿½'ï¿½ï¿½-ï¿½[ï¿½]ï¿½uï¿½oï¿½Ì*ï¿½(c)"i"@ï¿½ï¿½\$ï¿½szï¿½ï¿½&kï¿½ ï¿½w/ï¿½ï¿½wHï¿½1/4á-ï¿½+Z ï¿½ï¿½1/4Wï¿½%"Mï¿½^B&"Rï¿½ï¿½Tï¿½ï¿½ï¿½(tm)bï¿½-Zï¿½(c)ï¿½ï¿½Iï¿½ï¿½oï¿½ï¿½'ï¿½oï¿½ï¿½ï¿½kï¿½/ï¿½ï¿½;ï¿½> ï¿½ï¿½ x.ï¿½Oï¿½5ï¿½cï¿½>ï¿½eÕµ[ï¿½0-'iVpï¿½ï¿½ ...ï¿½ï¿½*(c)ï¿½Z}ï¿½ï¿½ kï¿½"uï¿½ï¿½sï¿½1ï¿½1/4?fï¿½ x

2. ## Population trends. The aim of this investigation is to find out more about different ...

This being true means that and should be estimates. If is equal to 2774 and is equal 4 then the equation results in 554.8, this was done by trial and error. These constants can be tested to see if they are useful when time has a value that isn't zero.

1. ## Maths Project. Statistical Analysis of GCSE results at my secondary school summer 2010 ...

12 We 9 40 f 40 11 We 10 34 m 34 10 We 9 58 f 58 9 Wh 10 46 m 46 8 Wh 9 46 f 46 7 Wi 11 52 f 52 6 Wi 10 40 m 40 5 Wi 10 28 m 28 4 Wo

2. ## Stellar Numbers. In this task geometric shapes which lead to special numbers ...

181 253 6n2 0 6 24 54 96 150 216 Difference between Sequence and 0.5n2 5 11 17 23 29 35 41 Second difference 6 6 6 6 6 6 This second difference tells me the value for 'b' which is equal to 6.

1. ## MATH IB SL INT ASS1 - Pascal's Triangle

r Y5(r) Y6(r) Y7(r) Y8(r) Y8(9) 0 15 21 28 36 45 1 11 16 22 29 37 2 9 13 18 24 31 3 9 12 16 21 27 4 11 13 16 20 25 5 15 16 18 21 25 6 - 21 22 24 27 7

2. ## Lascap's Fractions. I was able to derive a general statement for both the ...

The numerator formula is divided by the denominator formula. The numerator formula can also be written as . Now that we know that we can substitute it in the denominator formula for y. Hence, the equation below represents the general statement of . To prove that the general statement works correctly, the following example is provided, that clearly proves the general statement right.

1. ## LACSAP FRACTIONS - I will begin my investigation by continuing the pattern and finding ...

In order to find a general statemnet for the denominator I will be considering the relationship between the numerator and denominator. I began to notice a pattern formulating between the r value and the difference between the numerator and denominator. The table below shows the relationship between the row number(n)

2. ## Lacsap's fractions - IB portfolio

Knowing that 1, 3, 6, 10, 15 are terms of a sequence called ’triangle (or triangular) numbers’ (which can be calculated by the formula: n×(n2+n)/2) it can be figured out that another way to calculate kn is using the formula for the calculation of the triangle numbers (by changing the original formula a little).

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to