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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.
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