Let's look at a sample student
data table for touches. It appears that for the
first 5 examples the rule "add length and width
and you get the number of touches." So why the
surprise in example 6?
W
|
L
|
Touches
|
5
|
7
|
12
|
8
|
3
|
11
|
5
|
4
|
9
|
10
|
13
|
23
|
5
|
9
|
14
|
6
|
9
|
5 (surprise!)
|
Why did it come up 5 and
not 15? Let's look at the path:
6 by
9 table
Here are two other
paths that also have 5 "touches".
4 by 6 table
2 by 3 table
Note that for the
smallest of the three tables the "adding the
length and width rule" once again works. The
tables have the same path because they are
geometrically similar which means that their
sides are in proportion to each other. Another
way of thinking about this is that the ratios of
the dimensions are all equal. So if the
dimensions are relatively prime, the path will
traverse every square and the number of touches
will be the sum. Otherwise, find the the
smallest rectangle that will have the same path.
The sum of its dimensions will give you the sum.
Part B -
finding which corner the ball ends up in also
has to do with reduced form rectangles. Similar
paths will always end in the same corner. so the
investigation should be done with paths of
relatively prime dimensions.
W
|
L
|
Touches
|
Corner?
|
5
|
7
|
12
|
Top Right (UR) |
8
|
3
|
11
|
Top Left (TL) |
5
|
4
|
9
|
Bottom Right (BR) |
10
|
13
|
23
|
Top Left (TL) |
5
|
9
|
14
|
Top Right (UR) |
6
|
9
|
5
|
Top Left (TL)* |
7
|
10
|
17
|
Bottom Right (BR) |
Note that the two odd,
relatively prime dimensions both end up in the
upper right or opposite corner. If the width is
even and the length is odd and they are
relatively prime, the width "dominates" the path
and it ends up in the Top Left corner. If the
opposite is true the, then the length dominates
and the ball ends up in the Bottom Right corner.
Does this always work?
If the dimensions are
even, which corner will welcome the ball? (It
could wind up in any of the three corners. Two
even numbers are NOT relatively prime.)
Why can't the ball
return to the Bottom Left corner?
Does switching length
and width numbers change the final destination?
(For example, does the ball end up in the same
corner for a 7 by 5 table as a 5 by 7 table?
(NO!))
*Use the "reduced"
2 by 3 table.
Other Resources
Student's Page
|