Lesson: Pool Paths
Student
 Teacher
 Extensions
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
 10
8
3
 9
5
4
 7
10
13
 21
 5
 9
 12
 6
 9
 3 (surprise!)

Why did it come up 3 and not 13? Let's look at the path:
 

6 by 9 table

Here are two other paths that also have 3 "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 and subtracting 2" 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 minus 2 will give you the number of touches.

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
 10
 Top Right (UR)
8
3
 9
 Top Left (TL)
5
4
 7
 Bottom Right (BR)
10
13
 21
 Top Left (TL)
 5
 9
 12
 Top Right (UR)
 6
 9
 3
 Top Left (TL)*
 7
 10
15
 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, then the length dominates and the ball ends up in the Bottom Right corner. Does this always work? Will the ball ever return to the corner from which it starts?

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

  • See my blog entry "An Online Math Activity: Pool Paths"
  • See NCTM's Illumination activity Paper Pool (You have to be an NCTM member to use this.) The applet in the Illuminations activity is no longer available. An alternative is to use the Phet sim which is a nice simulation for this activity. However, you will have to count bounces.
  • Kate Nowak's blog entry: "Good Problems: Follow that Diagonal". How does this activity compare to the Pool Paths activity?