Thought for the dayCreative Math Prompts lead to discussions in which you (the teacher) do not always know the answers. Be honest about this. Enjoy teaming with your students to learn new things.

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.