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

Type I Internal Assessment (Lascap's Triangle)

Extracts from this document...

Introduction


Mathematics Coursework

Introduction

In this coursework, we have to look for a pattern in the numbers given to us in a triangular format as seen in the image below.

The image is a Lacsap’s Fraction. Lacsap’s fractions look a lot like Pascal’s Triangle and the work “lacsap” is “pascal” backwards which points further to their relationship.

image10.png

Objectives

Our objectives are as follows

  1. Find the numerator in the sixth row.
  2. Find and plot the relation between the row number and the numerator in each row, write a general formula
  3. Find the 6th and 7th rows

Using the patterns in task 3, find the general statement forimage00.png

. Where image00.png

  1.  is the (r + 1) th element in the nth row, starting with r = 0.
  2. Find additional rows and test if the general statement agrees and is correct
  3. Discuss the limitations of the general statement

Finding the Numerator of the sixth row

By looking at the Lacsap’s fraction and comparing it to the Pascal’s triangle, I observed that the numerators of the Lacsap’s fraction are the same as the third element number of the Pascal’s Triangle as seen in the image below. By this theory, the sixth numerator should be 21.image11.png

...read more.

Middle

. I tried to find a relationship again using row 2 and found image03.png

  for row 2.

Therefore, image04.png

 and so image05.png

This can be rewritten as, image06.png

I used a graphic calculator to validate the general formula using a GDC and the QuadReg function.image24.gifimage25.gif

Rewritten,                      or

Validating the formula manually

We can validate the formula by trying it on the numbers that we already know.

image26.gif

image27.gif

image28.gif

Finding the 6th and 7th rows

We can use the general formula to find the numerators of the 6th and 7th row.

6th row:

image29.gif

7th row:

image12.gif

Finding the denominator

After taking a hard look at the Lacsap’s Fractions, brain storming through all the possibilities and then through some trial and error, I found that the denominator too is related to the Pascal’s triangle. I have made an image which shows a Pascal’s triangle. In the image, I have highlighted the numbers which represent the difference in the numerator and the denominator of the Lacsap’s fractions.

image13.png

We also observe that the difference in the numerator and the denominator if we go diagonally for the first element number is 1, 2,3,4,5 respectively.

...read more.

Conclusion

Conclusion

I used mathematics and a few observations to find the different formulas required to find the final general formula. I have also used modern technology to validate the formula of my numerator. This was done using my GDC – QuadReg function to validate the formula before moving on to finding the denominator. Using some observation and trial and error, I was able to find a relationship between the numerator and the denominator which in turn allowed me to find a general formula for the denominator. Finally, after finding these two formulas, I combined them in a simple way to get the general formula and used this formula to find additional rows to prove the formula works.

...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. Maths Internal Assessment -triangular and stellar numbers

    the harder it will be to create that 'n- stellar number' and thus harder to prove. With logical reasoning, it is obvious that no decimal numbers can fill the 'p' value when creating the diagram for it. There is no such thing as 'half-a-dot' (p+0.5)

  2. Mathematics Higher Level Internal Assessment Investigating the Sin Curve

    If we look at the first equation, then the first thing to do in that equation would be to split the fractions so that the equation now looks like: . Once that has been done we can now factorize the brackets' section by , and therefore the equation would look like: .

  1. Stellar Numbers. After establishing the general formula for the triangular numbers, stellar (star) shapes ...

    the general statement that generates the sequence p-stellar numbers for any values of p at stage Sn.) It can thereby be claimed that when p=7 Sn = 7n2 - 7n + 1, even though this has not been tested and drawn, but this seems to be a logical formula when studying previous results and applying mathematical reasoning.

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

    + 1 = 46 Therefore the general statement for this shape is: n2 + n + 1 This time, the rule generated in step 7 does apply to a shape with 3 vertices. This evidently shows the effect of having a constant central dot.

  1. Lacsap's Fractions : Internal Assessment

    By using the formula of 0.5n2 + 0.5n achieved from the method above, I entered it into the GDC. When the formula was entered into the GDC, the graph above was shown. This proves that it is a quadratic equation.

  2. Mathematics internal assessment type II- Fish production

    Plotting of points (Fish caught in the sea) The data points were plotted using GeoGebra 4 for Mac OSX, and the x values have been substituted for a better fit of the graph. The points have been plotted into the scatter point graph below: Trends found in the graph Upon first glance of the graph, we can notice a number of trends and characteristics of the graph.

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

    Let us now take a look at the expanded data points by themselves. Graph 9: Raw data plots The overall trend from the graph (previous page) appears to demonstrate a positive increase throughout the passing of time; although there are parts where the values decrease; in generality, the values keep increasing.

  2. Parabola investigation. The property that was investigated was the relationship between the parabola and ...

    sequence in the difference if we omit the first four values of a. This can be proved if we take the value of D for the graph of y = 1.26x 2 ? 6x + 11. The graph is shown below.

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