-->

Thursday 16 June 2011

GCSE Lines investigation

If you order your research paper from our custom writing service you will receive a perfectly written assignment on GCSE Lines investigation. What we need from you is to provide us with your detailed paper instructions for our experienced writers to follow all of your specific writing requirements. Specify your order details, state the exact number of pages required and our custom writing professionals will deliver the best quality GCSE Lines investigation paper right on time.

Out staff of freelance writers includes over 120 experts proficient in GCSE Lines investigation, therefore you can rest assured that your assignment will be handled by only top rated specialists. Order your GCSE Lines investigation paper at affordable prices with livepaperhelp.com!



The starting point I have been given for this investigation is three lines which intersect each other, as shown below.


R I


L


The lines, which we will refer to as L, are of infinitive length, though I will only draw them long enough so that we can see the diagrams clearly. The points that the lines cross each other (intersections) I will refer to as I. The areas that are bounded by the lines, I will call regions and refer to them as R. To help me find a relationship between I, L and R, I will have to set a number of boundaries for my lines, if I did not do this then I would have an endless investigation. In this investigation we wish to investigate the maximum number of intersections (I) and regions (R) that a certain number of lines (L) can make. To do this I will start the investigations with simple diagrams and then build them up to complicated sketches, I will then put the results of the findings of the sketches in tables. If I find it necessary I will also draw graphs to show my results clearer, when I have done this, using the tables I will look for a relationship between the no. of lines, intersections and regions.


Write your GCSE Lines investigation research paper


I am now going to set boundaries for the lines so as my investigation will have a limit and so I will be able to find a pattern between the lines, intersections and regions.


1) The lines must be straight.


e.g.


1 4 5 6


If the lines weren’t straight, and they were curved, as above, I could have 1 line with I=5 and R=7.


If this rule wasn’t used, then I couldn’t find any relationship because 1 line could intersect itself numerous times.


) Lines must be of a standard length.


e.g.


As you can see, it would be very difficult to find a pattern if the lines were not of similar length. To cancel this out I will make all the diagrams of similar length, so as I don’t have diagrams as above.


) Lines must not be parallel. I will now do a mini investigation to see what happens if the lines are parallel.


1 L=1 1 L=


R= R=


I=0 I=0


1 4 L=


R=4


I=0


I will now put these results into a table, so I can understand them easier


L I R


1 0


0


0 4


From looking at the table I can now say, that for parallel lines, there is no intersections and the no. of regions equals the no. of lines plus 1.


I can now say that for all parallel lines I=0 and R=L+1.


I can now predict that for 5 lines, there will be no intersections and 6 regions. I will now sketch this out to see if I am correct.


L=5


1 4 5 6 R=6


I=0


I can now see from this diagram that my prediction is correct, L=5 R=6 and I=0.


In this mini investigation I found that R=L+1 and I=0.


As I=0, I now know parallel lines don’t intersect. In the main investigation I need the lines to intersect, so I will make the boundary that no lines can be parallel.


4) I will now investigate, to see what happens, if all the lines intersect at 1 point. I will do this by drawing diagrams.


I=0 1 I=1


1 R= R=4


L=1 L=


4


I=1


1 R=6


L=


6 4


5


I can now say that I will have the constraint that only two lines can cross at one intersection. I have done this because I need to find to maximum no. of intersections.


5) All the lines must cross each other.


e.g.


As you can see, all the lines don’t intersect each other. I want to maximise the no. of intersections so I will say that all the lines must cross each other.


I am now ready to start the main part of the investigation, I will do this by using all of the boundaries that I have found which are


1) All the lines must be straight


) Lines must be of a standard length


) Lines must not be parallel


4) Only lines can intersect at 1 point


5) All lines must cross each other


Using the boundaries I have found, I will start this investigation with simple diagrams and then work to harder diagrams.


I=0 I=1


L=1 L=


R= R=4


I=


L=


R=7


From these diagrams we can see that when I use the boundaries, we can make a clear diagram and I can see now that R=I+L+1. I will now put these results into a table, using the formula so I can see the results more clearly.


I L +1 = R


0 1 =


1 4 = 4


7 = 7


From this table I can see that the formula is correct. I can now predict that if I have 4 lines and 6 intersections, I will have 11 regions.


I=6


L=4


R=11


I can now see that my prediction is correct. I will now try to find a relationship between L & I. First I will put my results into a table and find the 1st and nd differences and work from there.


L I 1st Diff nd Diff


1 0 1 1


1 1





4 6


From this I can now say that because the nd difference is constant, the relationship must be quadratic.


The general equation for a quadratic equation is


ax +bx +c


I know from research that the x coefficient is half of the nd difference. I can now say, knowing this that


a= ½ x L= ½


The equation now becomes I= ½ L + bL + c


I will now put the results of I, ½ L & L into another table, and I will see from that if I can find a relationship between L & I.


L I ½ L Error in ½ L & I


1 0 ½ ½


1 1


4 ½ 1 ½


4 6 8


From this table, I can see that to get from ½ L to I each time we need to subtract ½ L.


We can now say that I= ½ L � ½ L


Predict


I can now say using this equation, that if I have 5L, I will equal;


I= ½ 5 � ½ 5


= 1.5 � .5 =10


I will now draw a diagram to show that this equation is correct


I can now say that my prediction is correct, because I=10 and I have used 5L. This diagram shows that when we have 5L, I will equal 10 as long as we use the boundaries set.





An alternate method can also be used to find a similar equation, which is by putting the results on the table for L & I, onto a graph and then work from there. I will now plot the results onto a graph and study my result.


From the graph I can see that, when I join the points together, it becomes a curved line, & from my mathematical knowledge I know that a curved line means that the relationship between L & I must be quadratic, and the general quadratic equation is


ax + bx + c





In this case the equation now becomes I= aL + bL + c


Instead of using the second difference, I will fill the results of the table into simultaneous equations to work out what a, b & c are.


1) I=1, L= I=aL + bL + c


1=4a + b + c


In this equation I have used results from my table and subbed them into this equation, I will now do the same for more equations, and then I will subtract them from each other to find out the results.


) I=, L= =a + b + c


) I=6, L=4 6=16a + 4b + c


Using these equations I will now be able to find out the values for a, b, & c.


If I now subtract 1) from ) then I will be able to cancel out the c term.


) =a + b + c


- 1) 1=4a + b + c I will call the new


4) =5a + 1b + c equation 4)


I will now do the same as before, for & .


) 6=16a + 4b + c


- ) = a + b + c I will call the new equation,


5) =7a + 1b + c 5)


I will now take 4) away from 5), so I can find out what a is,


5) =7a + b


- 4) =5a + b


1=a


a= ½


I will now fill a into equation 4), so as I can find out what b is,


= 5a + b


=.5 + b


-.5=b


b= -0.5


Using what I have found I can fill a & b into equation 4) to find c


=5a + b + c


=.5 + (-0.5) + c


= +c


c= 0


I can now sub a, b, & c into the general quadratic equation.


I= aL + bL + c


I= ½ L + (-0.5) b + 0





I now will use the example L=5 to test out this equation


I= ( ½ 5) + (-0.5 5) + 0


I=10


I now know that this equation is correct because as I found out in the previous experiment, when L=5, I will equal 10, and there is a diagram to prove that this is correct.


I will now try to find a relationship between L & R, using the methods that I previously used. I will use the results of the diagrams that I drew at the start of this investigation and put them onto a table and try o find a relationship.


L R 1st Diff nd Diff


1 1


4 1


7 4


4 11


From this I can say that the equation is quadratic, because the nd diff is constant.


The general quadratic equation is ax + bx + c


From my research, I can say that the x coefficient is ½ of the second difference, knowing this I can say that a= ½ L= ½


The equation now becomes R= ½ L + bL + c


I will now put the results of R, ½ L & L into another table, and I will be able to see from it, if there is a relationship between L & I.


L R ½ L Error in ½ L & R


1 ½ 1 ½


4


7 4 ½ ½


4 11 8


From the table, I can see that to get from ½ L to R, each time we need to add ½ L and I


We can now say that R= ½ L + ½ L + 1


Predict


I can now say, using this equation, that if I have 5L, R will equal


R= ½ L + ½ L + 1


R= ½ 5 + ½5 + 1


R=1.5 + .5 + 1


R=16


I will now draw a diagram to see if my prediction is correct..


I can now say that my prediction is correct because from the diagram we can see that when we have 5L, R will equal 16 which it does.


I now will use an alternate method to solve this equation


First I need to plot these results between R & I onto a graph.


L R


1


4


7


4 11


I can see from the graph that when the points are joined together, it produces a curved line, & from my research I know that a curved line on a graph, means that the relationship must be quadratic.


The general quadratic equation is ax + bx + c, which in our case becomes R= aL + bL + c


As I did before, I will fill values from my table into the equations, to find the values for a, b, & c.


1) L=, R=4 R= aL + bL + c


4=4a + b + c


In this equation I have used results from my table and subbed them into this equation, I will now do the same for more equations, and then I will subtract them from each other to find out the results.


) L=, R=7 7=a + b + c


) L=4, R=11 11=16a + 4b +c


If we subtract 1 from we will then be able to cancel out the c term.


) 7=a + b + c I will call the new


1) 4=4a + b + c equation 4, I will now do


4) =5a + b the same for &


) 11=16a + 4b + c


- ) 7=a + b + c I will now take 4 away


5) 4=7a + b from 5, so as to find out a.


5) 4=7a + b I will now fill my result for a into


4) =5a + b equation 4, so as to find b


1=a


a= ½


4) =5a + b I will now fill my results for a &


=.5 + b b into equation 1, so as to find c


-.5 = b


b= ½


1) 4=4a + b + c


4= + 1 + c


4=+c


c=1


Now I will sub a, b, & c into the general quadratic equation so it becomes R= ½ L + ½ L + 1


Predict


If the number of lines were 5, then the number of regions would be R= (½ 5) + (½ 5) + 1


R= 1.5 + .5 + 1


R=16


I will now draw a diagram to prove that this equation works


I can now say that my equation & prediction were correct.


At the start of this investigation we where given lines and told to find a relationship between I, L & R, I did this by setting boundaries so I could find a relationship between them. First I drew out some diagrams using the boundaries set, and then I entered the results into a table. From there, using graphs and tables, I found equations that linked L, R, & I together. To further this investigation I would like to try and find a relationship between R & I, to see if it is quadratic or not.





Please note that this sample paper on GCSE Lines investigation is for your review only. In order to eliminate any of the plagiarism issues, it is highly recommended that you do not use it for you own writing purposes. In case you experience difficulties with writing a well structured and accurately composed paper on GCSE Lines investigation, we are here to assist you. Your persuasive essay on GCSE Lines investigation will be written from scratch, so you do not have to worry about its originality.

Order your authentic assignment from livepaperhelp.com and you will be amazed at how easy it is to complete a quality custom paper within the shortest time possible!



No comments:

Post a Comment

Note: only a member of this blog may post a comment.