• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Math IA type I. Here is Lacsaps Fractions (the symmetrical pattern given) from n=1 until n=5, again with red numbers representing n:

Extracts from this document...

Introduction

Lacsap’s Fractions

A Math Internal Assessment

By Kelsey Kennelly

Lacsap’s Fractions…Ha

Clever, IB. You took the word “Pascal” and spelled it backwards to make “Lacsap”. I thought this was simply a Math Assessment but I guess it’s also a word scramble.

Anyways, with that being said, here is Pascal’s Triangle starting at n=0 until n=9, with red numbers representing n:

  1. 1
  2. 1        1
  3. 1        2        1
  4. 1        33        1
  5. 1        4        6        4        1
  6. 1        5        1010        5        1
  7. 1        6        15        20        15        6        1
  8. 1        7        21        35        35        21        7        1
  9. 1        8        28        46        70        46        28        8        1
  10. 1        9        36        84        126        126        84        36        9        1

*Column         0        1        2        3        4        5        6        7        8        9

*Columns rise diagonally from left to right. Column numbers will be represented by “c” followed by a subscript number.

Here is “Lacsap’s” Fractions (the symmetrical pattern given)

...read more.

Middle

The numbers of these numerators can be found in the above example of Pascal’s Triangle (column 2 or c2; or c7 but we’ll focus on c2). The number 1 can be found when n=2 in the second column (c2) of the triangle, which is the numerator of row 1 in Lacsap’s Fractions. Furthermore, when n=3 in the second column (c2) of Pascal’s Triangle, the number 3 is found, which is the numerator in Lacsap’s Fractions when n=2. This pattern continues, which brings me to the idea that the numerator is found by adding 1 to n in Lacsap’s fractions and finding that number in c2 of Pascal’s Triangle. For example, when n=3 in Lacsap’s Fractions, the numerator is 6. When you add 1 to 3, the result is 4; find n=4 in Pascal’s Triangle and find column 2.

...read more.

Conclusion

a =

a=   =

I used the quadratic equation to solve for row 2 and row 4. I used substitution to find the missing variables a and b to arrive at a general statement of

y = x2 + x

where y is the numerator and x is the row number (n)

I have already discovered that the numerator when n=6 is 21.

y = (6)2 + (6)

 y = (36)+ (6)

y = 18 + 3

y = 21

Now I will test the statement when n=7.

y = (7)2 + (7)

y = (49)+ (7)

y= 24.5 + 3.5

y = 28

This is correct because when n=7, 7+1=8. Find n=8 in Pascal’s Triangle in c2 and you will see the number 28. Now that I have discovered the general statement for the numerator, it is time to work on the denominator.

1        1

1                1

1                        1

1                                1

1                                        1

Column        1234        5

There is a pattern between the differences of numerator and denominator for each row. In row 1, for example,  is displayed and there is a difference of zero for numerator and denominator.  This continues in the symmetrical pattern.

Row

Column

Numerator – Denominator

1

1

0

2

1

...read more.

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

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    �0�5�8�����z#b��0�3/4�-�Ji�ã°ï¿½"8J(tm)��� �r�A¸G[ñ5H�9"3�%[m�:(r)�å�4J'TF�cC���Â�>%�1-�'�*�� .1/2K�7<g���Â�0o~Ð���<��á¤ï¿½1/4�9t���| <HN� �B���mZH�� 3_Ah-k<e!�&-�#� +?=�0h=��An�"�i�1/4�>Mzau�?Ö¥r�'#'u'%�+ %è§ï¿½8���x��z�;�e&�hNHW×£- k�s�Q[�\�9����~��'{1/2���"�X'1/4Q�v�...-�1���Z"�V�MX-X�N8 -�;K:ԧ�>cIO'jD�����Ӧ'I'_�5(r)X�&��H;@6�1/4U-rdV"'�b��8e��VÆ~%�1/4��\`OxÒP_(G}L V ���%��,�KR�H'59��SV�K�<w>�(tm)~O����>bM$��`�źd*~'P"@�����(r)"�x���V3/4]�@o�;(r)��w�"{3 ""=�!�W�_���⦦lJz�����71ÿB� -�fÛ­g�JEV ���� �d` gRM/-jZ`^��Dc߸L ��"M�i��&"�5o�z�nqi|����K�J�-\�T"TV�%�B<��z/Ì(r)�"��=-� N71/2��pv}�E�pg˶�p(c)OB�Hd���L��xP���Kf[_-JM��q�1�O1/4��Oѱ�é§ï¿½ï¿½ï¿½c�A�)�-}�t3/4���5����6��t��o w��|+]*(tm)I,�-"�j��/��N��i""n���< �� O-@�x-]"�e�5o�}"��� t�"B�$ ϯ� b(c) $�C��ezU..._������fÒ�-r$Jg|�l�Je�PWJn1/4f6�𭨪"'�� .~q��"8��I(aE'G`C��\���(tm)��S��0��ccMG�����X~c��$��>9?�...o&�9p �� �-����>W���\�?*�8% ���KÄ°g3y-� ���0��>)�V���...��(r)Ss�'1/4L�'Y�v���� ](c)L.1/4s�XÆ��(tm)W3/4��\V�2h(c)�*%�>�È�9�(tm)W���:�| ��61/2��!+�R�h^�>J�$Y����-b�g�-�����B=0ZP� C=�,K�D�z�,�k�w�"� s�=1/2���� ��#N ��@:[}]�"�,�����"�`�C �"� ��o �'DC���!�C0 �1/48س�f�-��"~"V�'�LS��^(c)��-�C*�3/4��B�Ͽ��E�R�x������^�B�M,�]#6of'"��|x�Æ2c9�P}�;*�Y s����~��B>�k8T��?�Ð��0�C" (tm)n� ��� ��<bk"Ts8ï¿½ï¿½Þ ï¿½YF�-0�O���GxV�...y���-|1 <� 9�� +�[Nd �

  2. Math Studies I.A

    ,??-????.=?????,,???.,???.-??. ,??-????. is known as the covariance of X and Y. ,??-?? .= ,-?,??-2.-,,(???)-2.-??.. ,??-?? .is called the standard deviation of X. ,??-?? .= ,-?,??-2.-,(?,??-2.)-??.. ,??-?? .is the standard deviation of Y. r=,?????- ,,???.,???.-??.-,-?,??-2.-,,(???)-2.-??..,-?,??-2.-,(?,??-2.)-??... So, r = ,130013258-,,1707100.(7914.39)-115.-,-51470030000-,,(1707100)-2.-115..,-559261.8111-,,(7914.39)-2.-115... =,12529300.01-161645.4262 �120.7778847.

  1. Math IA type 2. In this task I will be investigating Probabilities and investigating ...

    The possibilities for losing for both the players Total number of possibilities 4 _ _ _ - = 1 = 1 2 2+8+20+40 = 70 5 _ _ _ _ - 4 4 8 6 _ _ _ _ _ - 10 10 20 7 _ _ _ _ _

  2. Math IA Type 1 In this task I will investigate the patterns in the ...

    and as long as the lines intersect the parabola at 2 points each, the value of where m and n are the slopes of the intersecting lines. Now I will look to prove my new conjecture of The equation of any parabola is .

  1. Math Studies - IA

    This can be summarized as this: US win < 1 < EU win Various statistical measures can then be used to find if there is a relationship between the relative size of victory in the majors and either a loss or win in the Ryder Cup.

  2. How many pieces? In this study, the maximum number of parts obtained by n ...

    Plug Y, (X + S), respectively: P = (1/6)n 3 - (1/2)n 2 + (1/3)n + (1/2)n 2 + (1/2)n + 1 Simplify P = (1/6)n 3 + (5/6)n + 1 > Four Dimensional Object For a finite four-dimensional object, a recursive formula can be determined by looking at the

  1. Math IA- Type 1 The Segments of a Polygon

    One can see that the denominators are actually squares of integers such as 1, 2 and 3. The following table helps to analyze the situation better. Ratios of sides of triangles = 1:n Denominator values of the area ratio of equilateral triangles 1:2 12 1:3 22 1:4 32 Looking over

  2. MATH Lacsap's Fractions IA

    Therefore, the numerator of the sixth row will be 15 + 6 equaling to 21 (as stated before). This reinforces the correlation between Lacsap?s fractions and Pascal?s triangle. Also, there is a common 2nd difference of 1 in the numbers, which means that the numerators can be plugged into a quadratic formula.

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