math investigation (bounces)

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.

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 62 5 7 33 6 8 104 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?

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

RowFigure Number of

Bounces

2 3 1

2 4 2

2 5 3

2 6 4

2 7 5

2 8 6

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 Row Figure Number of

Bounces

3 2 1

4 2 2

5 2 3

6 2 4

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.

C. On what corner of the pool table or rectangle will the ball drops off?


Number of Dots in a

RowFigure

Corner that the Ball

Drops Off

3 5 Adjacent Corner

4 6 Opposite Corner

5 8 Adjacent Corner

10 9 Adjacent Corner

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−32

=b



Conjecture G

If the number of dots in a column is three (3) and the number of dots in a row is even, the


3 r− (2r+1 )=b



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


5 r+55

=b



Conjecture K

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

10 r+1010

+5=b; except Z, Z+3, Z+3+3, Z+3+3+3,…; where Z is 4



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

2 c−1=r



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

RowFigure

Corner that the Ball

Drops Off

2 4 Opposite Corner

2 6 Opposite Corner

2 8 Opposite 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 Number of Figure Corner that

Dots in a Column

Dots in a Row

the Ball Drops Off

3 5 Adjacent Corner

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

FigureNumber of

Bounces

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

RowFigure Number of

Bounces

3 5 1

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.

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.

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 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)+1010

+5=9.

If r = 5, then 10(5)+10

10+5=11.

If r = 6, then 10(6)+1010

+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.



6. Reorganising

Here are some of the data obtained from the investigation.

Number of Dots

in a Column

Number of

Dots in a Row

FigureNumber

of Bounces

Corner that the

Ball Drops

Off

2 3 1 Adjacent Corner

2 4 2 Opposite Corner










4 10 2Opposite

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.

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.

math investigation (bounces)

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