Regions and Lines
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
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