Concepts: similarity; ratios; length and area; least common multiples; greatest common factors; analyzing, testing, and extending patterns; expressing patterns algebraically

Beginning

I notice a path that travels at 45° angles to the sides of a rectangle.
I notice that the path ends when it hits a corner.
I notice that the path reminds me of the path of a ball on a pool table.
I notice that the "ball" bounces 5 times, always at right angle from where in hit.
I notice that there is always a gap of 4 units between each bounce, start, or end point—even around the corners of the rectangle.

Exploring

I wonder how long the path is. (It may be easiest to measure it in "diagonal" units.)
I wonder what the paths would look like on rectangles of other dimensions.
I wonder if there are patterns to the path length, bounces, and gap length for tables of different dimensions.
I wonder if there is a quick way to predict which corner the path will end at.
I wonder if a path could ever continue forever without hitting a corner.

Creating

Students may

Create and explore their own paths on rectangles of different dimensions.
Think in reverse: Create tables with paths having a chosen path length and number of hits.
Change the rules by allowing the ball to travel at different angles to the sides.
Imagine paths inside a three-dimensional box.

Reflecting and Extending

I notice that when two rectangular "tables" are mathematically similar, their paths are also similar, with corresponding scale factors.
I notice that I can calculate a scale factor for a table and its path by using the greatest common factor (GCF) of the dimensions of the table.
I notice that I can describe the length of a path using the least common multiple of the dimensions of table.
I notice that the gap between two bounces (including the start and end points) is two times the GCF mentioned above.
I notice that the number of bounces for a table whose dimensions have a GCF of 1 is always equal to the sum of the two dimensions.
I wonder what causes all of these patterns.
I notice that the ball never lands in the pocket where it started.
I notice that I can determine where it lands by paying attention to whether the dimensions are even or odd.
I wonder if there is a formula for the number of points where a path intersects itself.
I wonder if I can create tables whose paths satisfy conditions that I choose.
I wonder what would happen on tables whose dimensions are not whole numbers.

Notes

The image is an example of a "game" popularly known as Paper Pool. You can find a more structured activity built around this image in in Exploration 8: Paper Pool from my book, Advanced Common Core Math Explorations: Factors and Multiples.

By thinking carefully about the relationship between path length and gap length (possibly via the number of hits), students can derive a formula for the relationship between the GCF and LCM of two numbers.