**Thought for the day**: *Creative 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.