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

Lacsap's Fraction Math Portfolio

Extracts from this document...

Introduction

Lacsap’s Fraction

Thomas Ullrich

Internal Assessment Type 1

Math SL

Introduction:

In this IA Type 1 I’ll have to find patterns in a set of numbers, which are presented in a symmetrical pattern.

Step 1: Finding the numerator of the 6th row

At first, I had a look at the first 4 numerator, which are 3, 6, 10, 15. The first step between 6 and 3 is 3 and increases by 1 in each following row. Therefore we can say that the numerator in the 6th row is 21 (and the 7th is 28.)

Figure 1 Relation between the numerator and n

As you can see the numerator develops not linearly, but in some way exponentially.

image00.png

Figure 2 Lacsap's Triangle

Step 2:

The next step is about finding the complete 6th and 7th row.  For the numerator, I found this specific pattern that describes it. As we see in our first graph, the numerator in row number 6 is 21 and in row number 7 it is 28.

Now it is about finding a pattern for the denominator.image01.png

To show the specific pattern for the denominator I marked the sequences which are responsible for the characteristical features of the Pascal’s triangle.

The sequences are:

2  (+2)  4 (+3)  7 (+4)  11

4  (+2)  6 (+3)  9

7  (+2)  9

Figure 4 Denominator pattern

As we can see in Figure 3, the denominator increases by 1 starting with a step of 2. Following this rule, the denominators are:

6th

...read more.

Middle

To find b, we put a in equation IV:

5=9x0.5+b

5=4.5+b

b=0.5

To find our last unknown variable c, we put a and b in I:

10=16x0.5+4x0.5+c

10=10+c

c=0

Now after we found all three unknown variables, we put them in our original, general quadratic equation.

Numerator=0.5n²+0.5n

Numerator=image09.png

To check, if our expression is correct, we check put in a row number and check if the associated numerator is correct:

Validation test 1:

n=3    

Numerator=(3²+3)/2

  • Numerator=6

Validation test 2:

n=6

Numerator=(6²+6)/2

  • Numerator=21

Validation test 3:

n=7

Numerator=(7²+7)/2

  • Numerator=28

Step 4:

The next step is , to find a general expression for the denominator as a function of row number n and element number r.

image01.png

First of all I’m going to show how the denominator for r=1 (the red circled numbers).

Figure 6 is showing the relation between the denominator and n

To find the general expression for the denominator, we have to use the general, quadratic approach again:

Denominator=an²+bn+c

We need three equations to find all three variables. Therefore we put in the numbers from row 3, 4 and 5 and their associated denominators. Remember that we for now look at the red circled numbers

Quadratic equation for r=1

I:4=9a+3b+c

II:7=16a+4b+c

III:11=25a+5b+c

...read more.

Conclusion

So in line n, r=[0;n]

Step 7:

I will shortly explain, how I arrived at my general statement.

At first, I had a look at the given set of number, where I found patterns between the numerators and the row numbers and also for the denominator and the row numbers.

With that information I came up with the approach using the general quadratic equation. For the numerator it wasn’t too difficult because I only had to solve 3 different equations in order to get the three unknown variables for Numerator (n) =an²+bn+c.

For the denominator, it was a bit trickier. The first step was similar. Using the denominators values to solve Denominator=an²+bn+c. After solving those three equations, I noticed that the first part of the equation looked similar to the general numerator expression. So I tried to make it the same by adding +0,5n-0,5n in order to get 0.5n²+0,5n in each equation. After that was done, I saw that the last part of the equation was always r²-rn.

After finding the formula for the numerator and denominator, finding the general statement for En(r) wasn’t much of a big deal. All I had to do was dividing the numerator by the denominator.

...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

    _a�M���""+\��hY�0�2 o~�-v�...?�;�� �-l��t��=�t�@tr^��0�L�F#Z�S���1/2<�k��;"�zqy�2�P�L×´0 n����Sw�Ml�u� '���"�:�b S5S���Q(r)��"�\���R#(c)(r)pz��0-ha-U ��)��3��z-� Ö3gI?G���8���C3,P��4"tN-ghF�Ð�γ80"��"�B�L%L�S$l'�.���]� �� �@1/28=qq'AC�+OC G-�� -�...K��(tm)}"�e����xHYPl�-b�Ê�>�}"nt�/�8 =D 8�C1/2>=�����-��Dfk�-"�1eOFp ��"���B�|U�f�9�&`Fk �4��z.�L~$�Q[["�N�&*0A��\S+VP�� .i]O�<(tm)��mA�Up4��)���U�!p 4#����-_άi��SU[��)�scWQ�EpY�=&�%y@6^�*�"=B��{v��E�ÎG"ï±/(6p�X�B� � \�rT-�,��!�;]�J"Q���U�P\H�h Î�][\��UkÕH"`< �4V�1/2�"O:>h.��S)# �� P -Ç"EhÞ_0=" ��wMl�1/4���Z�dqLÓ¯_?������̡1,�sr��zu�U��i�Q�ض�"Q\` �*X��HjA ( ��rT4!:(r)��<=�Þ� �4 ��p�Kì¤ï¿½ï¿½ï¿½ï¿½k>�1/4��KR�pM�Tvg���R"&q��Õ�'.@+++��0���DYV�)"<��$Λ ï¿½zq�ä¨ï¿½z ���G�΢ W�P� T��1�j�� ~�8<Zh\ �ȱ g�1/4}:��...�Cq'T`#z�Y���K�t�,x�- / �Ϭi .t"�Q�I��� �-'l�����Q�� -L["~�%8�r� o��A ���"^�"`�kÔ¨Q,���Å���[Z��q;)iR�A� '�Y���s�W1/2Xv��G��.�(�u:(c)�0G"�-[Æs�w�����g��qg"_Q';'U...v����|fJ?��8 -[�w�yG@Ö��)b��...���Ì=��#�<bT��I"�DG��p-�� ��%�&�bÅ�(X��C="*(tm)�O?��o.Ñp�I_j9�"���\�Ù�%` ��d�qq���Ø����1/4lÙ²Y�f�2��A�@�...^8��"�;�1/43f��1/2[�f�G}� �(tm)\S�r��C�_z��"wOg�-���"�3/4z��á§z*��d�Ĵj'e���/�

  2. Math Studies I.A

    During the interpretation of results the weak correlation was partially explained by factors such as war, food shortage, economic crisis, epidemic of diseases, and so on. I addition, it could well be that there is a non linear correlation instead.

  1. Math IA - Logan's Logo

    Therefore, to find the value of c, I must first determine the center line of my curve - the middle line of the total height. In fact, we already determined this when we found variable a. (Recall that we divided the total height of the curve by 2 to find the amplitude).

  2. Math Studies - IA

    square regression line, the greater the European win in the Majors is the smaller their victory in the Ryder Cup is. Hence there is a negative correlation between the two variables. In other words, the better the US performs in the majors (going toward zero on the x-axis, since any value below on represents a US win)

  1. Math Portfolio Type II Gold Medal heights

    the y value when x=1, as the expression of log(1)=0, so by default, meaning k equals zero, the y value at x=1 is 0, the effect of the k can be expressed by stating (1,k). As whenever the x-value is 1 the y value equals k.This all seems very abstract

  2. IB Math IA- evaluating definite integrals

    the graph is concave up 0.8 < b < 2.4 & 4.8 < b < 5 the graph is concave down Using this information, we can develop a formula without an integral for I(b). Trials: f(x) = 3cos(2x) (red) f(x) = cos(x) (green) f(x) = 3sin(2x) (blue) (original function) f(x)

  1. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    197 203 198 204 212 216 218 224 223 225 236 Looking at these value tables and the graph above (Graph 3); we see that while the function approximately interests with four data points; at the 24th, 28th, 36th, and the 48th year-elapsed point; with approximately the same value.

  2. High Jump Gold Medals Portfolio Type 2 Math

    Table 4 shows all data collected 1896-2008. The ?Year? row shows the number of years since 1896. Table 4 Year 0 8 12 16 24 32 36 40 44 48 52 56 60 64 68 72 76 80 84 Height 190 180 191 193 193 194 197 203 198 204

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