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
Examples of noticing and wondering
I notice a path inside a rectangle that travels at 45° angles to the sides.
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.
I notice that the path forms three squares in the middle of the table.
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 Something New!
Students may create and explore their own paths on rectangles of different dimensions. They may even change the rules by allowing the ball to travel at different angles to the sides or by imagining something bouncing around in a three-dimensional box or some other type of shape altogether!
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 drawing and analyzing paths on tables of different sizes, you can discover countless patterns. A few possibilities include:
When two rectangular "tables" are similar, their paths are also similar, with corresponding scale factors.
You can calculate a scale factor for a table and its path by using the greatest common factor (GCF) of the dimensions of the table.
You can describe the length of a path using the least common multiple of the dimensions of table.
The gap between two bounces (including the start and end points) is always an even number. In fact, it is two times the GCF mentioned above.
The number of bounces for a table whose dimensions have a GCF of 1 is always equal to the sum of the two dimensions.
The ball never lands in the pocket where it started. You can determine where it lands in by paying attention to whether the dimensions are even or odd.