This document summarizes an investigation into the path and number of bounces of a ball on a rectangular pool table represented by a dot grid. The investigation explored relationships between the number of dots in columns and rows and the resulting number of bounces and corner where the ball drops off. Several conjectures were proposed and tested, with some found to be true and others false. Key findings included formulas for determining the number of bounces based on the column and row sizes.

1. 1. Getting started (preliminary skirmishing)
Notes:
a) Since it was stated on the problem that the ball will only drop off the table or stop
when it reaches a corner, the speed of the ball must be constant.
b) Since the investigation involves a pool table which is a rectangle, the number of dots
is also considered. Dot papers having the same number of dots in a column and in a
row are discarded.
c) Although this investigation is dealing with the use of a pool table, there are only four
(4) holes or four (4) exits considered because it was stated in the problem that the ball
will only drop off if it reaches a corner. A rectangle has only four (4) corners.
d) The dots are always equally distant. A column or a row having only one (1) dot is
also discarded because no rectangle will be produced.
e) The direction of the starting point of the ball must always be 45° from its sides.
If we will change (increasing or decreasing) the number of dots in a column or in a row,
the number of bounces will also change. The change in the number of bounces might be
Investigation 22
BOUNCES
Imagine a rectangle on a dot paper. Suppose it is a pool table.
Investigate the path of a ball which starts at one corner of the table, is pushed to
an edge, bounces off that edge to another, and so on, as shown in the diagram. When
the ball finally reaches a corner it drops off the table.

2. increasing or decreasing depending on the patterns that the investigator might discover.
Consider the following illustrations.
Illustration 1 Illustration 2
Illustration 3 Illustration 4
The observation was noted on a table.
Illustration Number of Dots in a
Column
Number of Dots in a
Row
Number of Bounces
1 4 6 6
2 5 7 3
3 6 8 10
4 7 9 5
How many bounces can be made considering the number of dots in a column?
How many bounces can be made considering the number of dots in a row?
On what corner of the pool table will the ball drops off?

3. 2. Taking a break (gestating)
What do you observe on the number of bounces if the numbers of dots in a column and in
a row are both odd?
What do you observe on the number of bounces if the numbers of dots in a column and in
a row are both even?
What do you observe on the number of bounces if the number of dots in a column is odd
and the number of dots in a row is even?
What do you observe on the number of bounces if the number of dots in a column is even
and the number of dots in a row is odd?
3. Exploring systematically
A. How many bounces can be made considering the number of dots in a column?
In here, I consider the number of dots in a column to be constant. Thus, the
number of dots in a row is only changing. Let us consider the lowest possible number of
dots in starting this investigation. (Note: One is always not included. Equal number of
dots in a row and in a column is also always excluded.)
Number of
Dots in a
Column
Number of
Dots in a
Row
Figure
Number of
Bounces
2 3 1
2 4 2
2 5 3
2 6 4
2 7 5
2 8 6

4. 2 9 7
B. How many bounces can be made considering the number of dots in a row?
In here, I consider the number of dots in a row to be constant. Thus, the number of
dots in a column is only changing. Let us consider the lowest possible number of dots in
starting this investigation. (Note: One is always not included. Equal number of dots in a
row and in a column is also always excluded.)
Number of Dots in
a Column
Number of Dots in
a Row
Figure
Number of
Bounces
3 2 1
4 2 2
5 2 3
6 2 4

5. 7 2 5
8 2 6
9 2 7
I had also done this with larger number of dots in a column and in a row. With
this observation, I had concluded that the number of bounces is the same if the number of
dots in a column and in a row is interchange. It seems that the figure was only rotated 90°
to the right.

6. C. On what corner of the pool table or rectangle will the ball drops off?
Number of
Dots in a
Column
Number of
Dots in a
Row
Figure
Corner that
the Ball
Drops Off
3 5
Adjacent
Corner
4 6
Opposite
Corner
5 8
Adjacent
Corner
10 9
Adjacent
Corner

7. 4. Making conjectures
Conjecture A
If the numbers of dots in a column and in a row are both even, the ball drops off on the
corner opposite to the corner where the ball started rolling.
Conjecture B
If the numbers of dots in a column and in a row are both odd, the ball drops off on one of
the corners adjacent to the corner where the ball started rolling.
Conjecture C
If the numbers of dots in a column and in a row are both even, the number of bounces is
always even.
Conjecture D
If either the number of dots in a column or in a row is odd or the numbers of dots in a
column and in a row are both odd, the number of bounces is always odd.
Conjecture E
If the number of dots in a column is two (2), we can get the number of bounces by using
the formula
𝑟 − 2 = 𝑏
where 𝑟 = number of dots in a row
b = number of bounces
Conjecture F
If the number of dots in a column is three (3) and the number of dots in a row is odd, the
number of bounces can be determined using the formula
𝑟 − 3
2
= 𝑏

8. where 𝑟 = number of dots in a row
b = number of bounces
Conjecture G
If the number of dots in a column is three (3) and the number of dots in a row is even, the
number of bounces can be determined using the formula
3𝑟 − (2𝑟 + 1) = 𝑏
where 𝑟 = number of dots in a row
b = number of bounces
Conjecture H
If the number of dots in a column is four (4) and the number of dots in a row is any
positive integer, the number of bounces is equal to the number of dots in a row except Z,
Z+3, Z+3+3, Z+3+3+3…; where Z is 7.
Conjecture I
If the number of dots in a column is five (5) and the number of dots in a row is odd, the
first digit of its product or area is the number of bounces if and only if the interval of the
numbers of dots in a column and in a row is divisible by two (2) but not by four (4). If the
product involves three (3) digits, the first two (2) digits is the number of bounces.
Conjecture J
If the number of dots in a column is five (5) and the number of dots in a row is even, the
number of bounces can be determined using the formula
5𝑟 + 5
5
= 𝑏
where 𝑟 = number of dots in a row
b = number of bounces
Conjecture K

9. If the number of dots in a column is ten (10) and the number of dots in a row is any
positive integer, we can get the number of bounces by using the formula
10𝑟+10
10
+ 5 = 𝑏; except Z, Z+3, Z+3+3, Z+3+3+3,…; where Z is 4
where 𝑟 = number of dots in a row
b = number of bounces
Conjecture L
If the number of dots in a column is equal to one less twice the number of dots in a row,
then the number of bounce is always one (1). We can use the formula
2𝑐 − 1 = 𝑟
where 𝑟 = number of dots in a row
b = number of bounces
Conjecture M
If the sum of the numbers of dots in a column and in a row is fifteen (15), then the number of
bounces is always eleven (11).
5. Testing Conjectures
Conjecture A
Considering the numbers of dots in a column and in a row as both even, the ball always
drops off on the corner opposite to the corner where the ball started rolling.
Number of
Dots in a
Column
Number of
Dots in a
Row
Figure
Corner that
the Ball
Drops Off
2 4
Opposite
Corner
2 6
Opposite
Corner
2 8 Opposite

10. Corner
6 4
Opposite
Corner
10 4
Opposite
Corner
Conjecture B
Considering the numbers of dots in a column and in a row as both odd, the ball does not
always drops off on one of the corners adjacent to the corner where the ball started rolling.
Although most of the figures made agree with the conjecture, I had found a counter example.
For instance, if the number of dots in a column is eleven (11) and the number
Number of
Dots in a
Column
Number of
Dots in a
Row
Figure
Corner that
the Ball
Drops Off
3 5
Adjacent
Corner

11. 5 7
Adjacent
Corner
7 9
Adjacent
Corner
11 3
Opposite
Corner
False Conjecture!
Conjecture C
Number of
Dots in a
Column
Number of
Dots in a
Row
Figure
Number of
Bounces

12. 2 4 2
2 6 4
2 8 6
4 6 6
4 10 2
Conjecture D
Considering that the numbers of dots in a column or in a row are both odd just even either of
them. I came up with the following observation. I had found a counter example which is 11 and 3.
Look at last row on the next page. This justifies that conjecture D is not true.
Number of
Dots in a
Column
Number of
Dots in a
Row
Figure
Number of
Bounces
3 5 1

13. 5 7 3
7 9 5
11 3 4
False Conjecture!
Conjecture E
If r = 3, then 3 – 2 = 1.
If r = 4, then 4 – 2 = 2.
If r = 5, then 5 – 2 = 3.

14. If r = 6, then 6 – 2 = 4.
If r = 7, then 7 – 2 = 5.
Conjecture F
If r = 5, then
5−3
2
= 1.
If r = 7, then
7−3
2
= 2.
If r = 9, then
9−3
2
= 3.
If r = 11, then
11−3
2
= 4.
If r = 13, then
13−3
2
= 5.
Conjecture G
If r = 2, 3(2) – (2(2) + 1) = 1.
If r = 4, 3(4) – (2(4) + 1) = 3.
If r = 6, 3(6) – (2(6) + 1) = 5.
If r = 8, 3(8) – (2(8) + 1) = 7.
If r = 10, 3(10) – (2(10) + 1) = 9.
Conjecture H
If r = 2, then b = 2.
If r = 3, then b = 3.
If r = 5, then b = 5.

15. If r = 6, then b = 6.
If r = 8, then b = 8.
Conjecture I
If r = 3, then 5(3) = 15. 1 is the first digit. Thus, 1 is the number of bounces.
If r = 7, then 5(7) = 35. 3 is the first digit. Thus, 3 is the number of bounces.
If r = 11, then 5(11) = 55. 5 is the first digit. Thus, 5 is the number of bounces.
If r = 15, then 5(15) = 75. 75 is the first digit. Thus, 7 is the number of bounces.
If r = 19, then 5(19) = 95. 3 is the first digit. Thus, 9 is the number of bounces.
If the product involves three (3) digits, the first two (2) digits is the number of bounces.
If r = 35, then 5(35) = 175. 17 is the first two digits. Thus, 17 is the number of bounces.
Conjecture J
If r = 2, then
5(2)+5
5
= 3.
If r = 4, then
5(4)+5
5
= 5.
If r = 6, then
5(6)+5
5
= 7.
If r = 8, then
5(8)+5
5
= 9.
If r = 10, then
5(10)+5
5
= 11.
Conjecture K
If r = 2, then
10(2)+10
10
+ 5 = 8.
If r = 3, then
10(3)+10
10
+ 5 = 9.
If r = 5, then
10(5)+10
10
+ 5 = 11.

16. If r = 6, then
10(6)+10
10
+ 5 = 12.
If r = 8, then
10(8)+10
10
+ 5 = 14.
Conjecture L
If c = 4 and r = 7, then 2(4) – 1 = 7.
If c = 5 and r = 9, then 2(5) – 1 = 9.
If c = 6 and r = 11, then 2(6) – 1 = 11.
Since the examples above justifies the conditions of conjecture L, the number of their
bounce is always one (1).
Conjecture M
If r = 11 and c = 4, then 11 + 4 = 15. Thus, the number of bounces is 11.
If r = 10 and c = 5, then 5 + 10 = 15. Thus, the number of bounces is 11.
If r = 9 and c = 6, then 6 + 9 = 15. Thus, the number of bounces is 11.
6. Reorganising
Here are some of the data obtained from the investigation.
Number
of Dots
in a
Column
Number
of Dots
in a
Row
Figure
Number
of
Bounces
Corner
that the
Ball
Drops
Off
2 3 1
Adjacent
Corner
2 4 2
Opposite
Corner
2 5 3
Adjacent
Corner
2 6 4
Opposite
Corner

17. 2 7 5
Adjacent
Corner
2 8 6
Opposite
Corner
2 9 7
Adjacent
Corner
3 2
1 Adjacent
Corner
3 5 1
Adjacent
Corner
4 2 2
Opposite
Corner
4 6 6
Opposite
Corner
4 10 2
Opposite
Corner

18. 5 2 3
Adjacent
Corner
5 8 9
Adjacent
Corner
6 2 4
Opposite
Corner
7 2 5
Adjacent
Corner

19. 8 2 6
Opposite
Corner
9 2 7
Adjacent
Corner
10 9 15
Adjacent
Corner

20. 11 3 4
Opposite
Corner
7. Elaborating
Early in the investigation, several questions were recorded for possible consideration.
These provide elaboration for the investigation.
The questions involved the numbers of dots in a column and in a row and the path of the
ball.
The investigation involves dots on a rectangle. The situation could be varied by
considering the number of dots on a rectangle.
8. Summarising
In this investigation, some aspects were examined:
a. the account of the aspect involving the numbers of dots in a column or/and in a row;
b. the figures drawn for the cases considered;
c. the table showing the data obtained from the investigation;
d. the patterns observed
e. the presentation of conjectures from A to M;
f. the testing of conjectures from A to M, and;
g. the elaboration of this investigation.

21. Extension:
Investigate the number of regions inside the rectangle.
Investigate the number of remaining dots not covered by the ball.
Investigate the number of square inside the rectangle.