Regions and Lines

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Aim
To investigate the relationships that exists when lines are drawn on a plane. Strategies
To develop some conjectures, rules and patterns by investigating the relationships that form between the number of lines, intersections points, and bounded and unbounded regions. I plan to use rules to further define the relationship.

The following relationships will be investigated:
# Lines# Lines# Lines# Lines
# Lines# Intersects# Lines# Lines# Lines# Lines-

# unbounded regions
min # bounded shapes
max # bounded shapes
# possibilities of intersections
max. # Intersections
# bounded regions
min # bounded triangles
max # bounded triangles
# bounded regions (not perpendicular lines)
# bounded quads

General Rules:
1.
When considering minimum regions formed, lines cannot be drawn parallel. After the first 2 lines have been drawn, it is possible for region(s) to form using the 3rd line. When investigating minimum regions, at LEAST 1 region must be created with every line.

2.
Two consecutive lines cannot be drawn parallel
3.
If a line crosses through an intersection already formed by other lines, it does not count as an extra point of intersection
4.
n: number of lines
y: Number of bounded or unbounded regions
x: number of intersects

When x number of lines are drawn, how many unbounded regions are formed?
*RULES: 1. The first 2 lines drawn must be perpendicular
2. The line drawn must go from one side all the way across to the other (it cannot only cross halfway)

1 line

2 lines

3 lines

# Lines

# Unbounded
Regions
2
4
6
8
10

1
2
3
4
5

4 lines

5 lines

# Bounded
Regions
0
0
1
2
3

Conjecture A
The data above suggests that there is a relationship between the number of lines drawn and the number of unbounded regions formed. When 1 line is drawn, there are 2 unbounded regions. When 2 lines are drawn, 4 unbounded regions form. The number of unbounded regions formed is always double the number of lines.

y=2n

(where y is the number of unbounded regions)

Testing Conjecture A
How many unbounded regions will form when 10 lines are drawn? 8

Prediction
y=2n
y=2 x 10
y=20

7

6

9

4

3

5

2

10

1

11

20
19
12

13

16
14

15

17

18

Justifying Conjecture A
The number of unbounded regions in double the number of lines because every time another line is drawn, it is splitting the diagram creating 2 more regions. As another 2 regions are created with every line added, 2 more regions are added to the whole diagram.

When investigating Conjecture A, the data produced showed that another relationship exists between the number of lines and the number of bounded regions.
# Lines
1
2
3
4
5
6
7
8

# Bounded
Regions MIN
0
0
1
2
3
4
5
6

# Bounded
Regions MAX
0
0
1
3
6
10
15
21

Conjecture B
What is the relationship between the number lines and the minimum and maximum number of bounded shapes formed?

3 lines

4 lines

5 lines

6 lines
7 lines

8 lines
From the table of results, it can be observed that the number of lines subtract 2 equals to the minimum number of bounded regions.
The relationship between the number of lines and minimum regions is: y= n – 2 (where y is the minimum number of regions formed)

The maximum number of bounded regions can be observed to be increasing the first time by 1 region, then 2, then 3, etc. Because of this pattern, a relationship must exist between the number of lines and the maximum regions. As it does not have a common difference (it does not increase by the same amount each time), a table can be constructed to see if there is a second common difference (in which case, the relationship will be quadratic). y=ax2+bx+c

n

0123456

y

1 0 0 1 3 6 10 15 21
-1

0
1

1
1

2 34
5
11 1 1

7

8
6

1

a = ½ the 2nd common difference
(1 x ½ )= ½
b= (a+b)= the common diff. between 0 and 1
(½ + b)= -1
b = - 3/ 2
c= when n...
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