Today we looked at a problem involving rebounding balls on a billiard table.
The rules of our problem:
- A “table” can have different dimensions (measurements of height and width).
- A “ball,” modeled by a straight line, is launched at 45 degrees from the bottom left corner of a table.
In preparation for working on the problem, we talked about conjectures, examples, and counter-examples, ending with our own working definitions for today.
An interesting discussion arose over the difference between these two conjectures:
All prime numbers are odd. (Counter-example: 2)
vs.
All odd numbers are prime. (Counter-example: 9, 27, 33, etc.)As a full group, we looked at some examples:
Looking above, we can see:
A table with a height of 6 and a width of 4 will result in the ball landing in the bottom right pocket after 3 rebounds.
A table with a height of 8 and a width of 4 will result in the ball landing in the top left pocket after 1 rebound.
What else do you see?
We looked at the examples we had generated and did a round of notice/wonder:
- What do you notice?
- For some of them, the ball passes through every single square on the table.
- I notice that for three of the rectangles, the ball lands in a pocket after 1 rebound.
- I don’t see any where the ball passes through the same square in two different directions.
- When the height and the width are half of one another, there is only one rebound.
- When one was even and one was odd, there a lot of rebounds.
- If the bigger number can be divided by the smaller number, there is only one rebound.
- What do you wonder?
- I wonder what it would look like if you drew a path from the bottom left and a path from the bottom right on the same table in two different colors.
- Why are there so many rebounds when there is an odd number?
- Why is there only one rebound when the width is half of the length?
- What is special about 4×8 and 6×4? (Fewer rebounds)
- I wonder how the dimensions of the table affect the numbers of squares the ball passes through and the number of squares it does not pass through.
- I wonder if we can predict what corner the ball will eventually land in. (And whether we are sure it will always make it to a corner.)
- Can you predict the number of rebounds, with different sized tables?
- How can reduce the number of rebounds in any dimension table?
Work in breakout rooms
This group focused on trying to reduce the number of bounces, finding that common factors reduce the number of bounces.
This group decided to change the rule of starting at the bottom right corner. Instead of changing the dimensions of the table, they used one 4 x 5 table and experimented launching the ball from different positions.
Resources
- Our billiards work in Jamboard
- Original problem from Math is Love
- Simulator
- Donald in MathMagicLand (video)
